Unnamed: 0
int64 0
7.24k
| id
int64 1
7.28k
| raw_text
stringlengths 9
124k
| vw_text
stringlengths 12
15k
|
|---|---|---|---|
1,100 | 2,003 | Incremental A
S. Koenig and M. Likhachev
Georgia Institute of Technology
College of Computing
Atlanta, GA 30312-0280
skoenig, mlikhach @cc.gatech.edu
Abstract
Incremental search techniques find optimal solutions to series of similar
search tasks much faster than is possible by solving each search task
from scratch. While researchers have developed incremental versions of
uninformed search methods, we develop an incremental version of A*.
The first search of Lifelong Planning A* is the same as that of A* but
all subsequent searches are much faster because it reuses those parts of
the previous search tree that are identical to the new search tree. We then
present experimental results that demonstrate the advantages of Lifelong
Planning A* for simple route planning tasks.
1 Overview
Artificial intelligence has investigated knowledge-based search techniques that allow one
to solve search tasks in large domains. Most of the research on these methods has studied
how to solve one-shot search problems. However, search is often a repetitive process,
where one needs to solve a series of similar search tasks, for example, because the actual
situation turns out to be slightly different from the one initially assumed or because the
situation changes over time. An example for route planning tasks are changing traffic
conditions. Thus, one needs to replan for the new situation, for example if one always
wants to display the least time-consuming route from the airport to the conference center
on a web page. In these situations, most search methods replan from scratch, that is, solve
the search problems independently. Incremental search techniques share with case-based
planning, plan adaptation, repair-based planning, and learning search-control knowledge
the property that they find solutions to series of similar search tasks much faster than is
possible by solving each search task from scratch. Incremental search techniques, however,
differ from the other techniques in that the quality of their solutions is guaranteed to be as
good as the quality of the solutions obtained by replanning from scratch.
Although incremental search methods are not widely known in artificial intelligence and
control, different researchers have developed incremental search versions of uninformed
search methods in the algorithms literature. An overview can be found in [FMSN00].
We, on the other hand, develop an incremental version of A*, thus combining ideas from
the algorithms literature and the artificial intelligence literature. We call the algorithm
Lifelong Planning A* (LPA*), in analogy to ?lifelong learning? [Thr98], because it reuses
We thank Anthony Stentz for his support. The Intelligent Decision-Making Group is partly supported by NSF awards under contracts IIS9984827, IIS-0098807, and ITR/AP-0113881. The views and conclusions contained in this document are those of the authors and should not be
interpreted as representing the official policies, either expressed or implied, of the sponsoring organizations and agencies or the U.S. government.
information from previous searches. LPA* uses heuristics to focus the search and always
finds a shortest path for the current edge costs. The first search of LPA* is the same as that
of A* but all subsequent searches are much faster. LPA* produces at least the search tree
that A* builds. However, it achieves a substantial speedup over A* because it reuses those
parts of the previous search tree that are identical to the new search tree.
2 The Route Planning Task
Lifelong Planning A* (LPA*) solves the following search task: It applies to finite graph
search problems on known graphs whose edge costs can increase or decrease over time.
denotes the finite set of vertices of the graph.
denotes the set of successors
of
vertex . Similarly,
denotes the set of predecessors of vertex .
!"#
%$'& denotes the cost of moving from vertex to vertex "()*+,
. LPA*
always determines a shortest path from a given start vertex ,-/.10324.56 to a given goal
vertex 879 0;: , knowing both the topology of the graph and the current edge costs. We
use <>=,
to denote the start distance of vertex ?@ , that is, the length of a shortest path
from -/.102A. to .
To motivate and test LPA*, we use a special case of these search tasks that is easy to
visualize. We apply LPA* to navigation problems in known eight-connected gridworlds
with cells whose traversability can change over time. They are either traversable (with cost
one) or untraversable. LPA* always determines a shortest path between two given cells of
the gridworld, knowing both the topology of the gridworld and which cells are currently
blocked. This is a special case of the graph search problems on eight-connected grids whose
edge costs are either one or infinity. As an approximation of the distance between two cells,
we use the maximum of the absolute differences of their x and y coordinates. This results
in consistent heuristics that are for eight-connected grids what Manhattan distances are for
four-connected grids.
3 Reusing Information from Previous Searches
The graph search problems can be solved with traditional graph-search methods, such as
breadth-first search, if they update the shortest path every time some edge costs change.
They typically do not take advantage of information from previous searches. The following
example, however, shows that this can be advantageous.
5CED F
Consider the gridworlds of size B
shown in Figure 1. The original gridworld is
shown on top and the changed gridworld is shown at the bottom. The traversability of only
a few cells has changed. In particular, three blocked cells became traversable (namely, B3,
C5, and D2) and three traversable cells became blocked (namely, A1, A4, D3). Thus, two
percent of the cells changed their status but the obstacle density remained the same. The
figure shows the shortest paths in both cases, breaking ties towards the north. Note that we
assume that one can squeeze through diagonal obstacles. (This is just an artifact of how we
generated the underlying graphs from the mazes.) The shortest path changed since one cell
on the original shortest path became blocked.
Once the start distances of all cells are known, one can easily trace back a shortest path
from the start cell to the goal cell by always greedily decreasing the start distance, starting
at the goal cell. This is similar to how A* traces the shortest path back from 79 03: to
-/.10324. using the search tree it has constructed. Thus, we only need to determine the start
distances. The start distances are shown in each traversable cell of the original and changed
gridworlds. Those cells whose start distances in the changed gridworld have changed from
the corresponding ones in the original gridworld are shaded gray.
There are two different ways of decreasing the search effort for determining the start distances for the changed gridworld. First, some start distances have not changed and thus
need not get recomputed. This is what DynamicSWSF-FP [RR96] does. (DynamicSWSF-
Original Eight-Connected Gridworld
1
7
6
4
A
3
3
4
B
C
D
6
7
8
7
7
2
7
6
3
7
6
5
4
3
2
5
5
4
3
2
2
2
3
4
5
6
4
3
2
1
1
1 sstart 1
9
3
3
2
3
6
6
7
8
5
5
4
5
6
7
8
6
7
7
11
7
6
7
8
8
4
4
7
8
9
10
9
8
7
6
5
5
10
9
8
9
8
7
4
11
10
9
8
5
11
10
11
10
6
5
6
8
8
8
9
8
9
9
11
7
8
7
7
7
8
9
8
8
8
9
10
9
9
9
9
10
10
10
10
10
10
10
11
12
12
13
12
11
11
11
11
11
11
16
15
14
13
16
14
14
14
15
15
15
15
12 sgoal 14
12 13
14
12
12
14
12 13 14
13 14
14
14
15
15
15
15
15
15
15
15
12
16
16
16
16
16
16
16
16
16
Changed Eight-Connected Gridworld
7
6
7
5
5
5
4
6
7
8
7
6
7
6
5
4
3
2
5
3
3
4
7
2
2
3
4
5
6
4
3
2
1
1
1 sstart 1
9
3
3
2
3
6
6
7
8
6
7
7
8
5
5
6
4
5
6
7
8
11
7
6
7
8
4
4
7
8
9
10
9
8
7
6
5
5
10
9
8
9
8
7
11
10
9
8
11
10
14
13
6
12
7
5
6
8
8
8
9
8
9
9
7
7
7
8
9
10
11
8
8
8
9
10
9
9
9
10
11
10
10
10
10
10
11
11
19
18
17
16
12
16
15
14
12
11
11
11
11
19
17
17
17
18
18
18
18
12 sgoal 14
12 13
14
12
12
14
12 13 14
13 14
14
14
15
15
15
15
15
15
15
15
15
19
19
18
17
16
16
16
16
16
Figure 1: Simple Gridworld
FP, as originally stated, searches from the goal vertex to the start vertex and thus maintains
estimates of the goal distances rather than the start distances. It is a simple matter of
restating it to search from the start vertex to the goal vertex. Furthermore, DynamicSWSFFP, as originally stated, recomputes all goal distances that have changed. To avoid biasing our experimental results in favor of LPA*, we changed the termination condition of
DynamicSWSF-FP so that it stops immediately after it is sure that it has found a shortest
path.) Second, heuristic knowledge, in form of approximations of the goal distances, can
be used to focus the search and determine that some start distances need not get computed
at all. This is what A* [Pea85] does. We demonstrate that the two ways of decreasing the
search effort are orthogonal by developing LPA* that combines both of them and thus is
able to replan faster than either DynamicSWSF-FP or A*.
Figure 2 shows in gray those cells whose start distances each of the four algorithms recomputes. (To be precise: it shows in gray the cells that each of the four algorithms expands.)
During the search in the original gridworld, DynamicSWSF-FP computes the same start
distances as breadth-first search during the first search and LPA* computes the same start
distances as A*. During the search in the changed gridworld, however, both incremental
search (DynamicSWSF-FP) and heuristic search (A*) individually decrease the number of
start distances that need to get recomputed compared to breadth-first search, and together
(LPA*) decrease the number even more.
4 Lifelong Planning A*
Lifelong Planning A* (LPA*) is an incremental version of A* that uses heuristics
to
control its search. As for A*, the heuristics approximate the goal distances of the vertices
. They need to be consistent, that is, satisfy 79 03:
and
$E, ; "
"
for
all vertices 5 and " 5+
with
79 0;: .
LPA* maintains an estimate <
of the start distance <+=
of each vertex . These values
directly correspond to the g-values of an A* search. They are carried forward from search to
search. LPA* also maintains a second kind of estimate of the start distances. The rhs-values
are one-step lookahead values based on the g-values and thus potentially better informed
complete search
Original Eight-Connected Gridworld
uninformed search
heuristic search
breadth-first search
A*
sstart
sgoal
sstart
incremental search
DynamicSWSF-FP (with early termination)
sstart
sgoal
sgoal
Lifelong Planning A*
sstart
sgoal
complete search
Changed Eight-Connected Gridworld
sstart
uninformed search
heuristic search
breadth-first search
A*
sgoal
sstart
incremental search
DynamicSWSF-FP (with early termination)
sstart
sgoal
sgoal
Lifelong Planning A*
sstart
sgoal
Figure 2: Performance of Search Methods in the Simple Gridworld
than the g-values. They always satisfy the following relationship:
"!$#%'&)( * ,+ .-%/021435/6 ,
if
otherwise.
(1)
A vertex is called locally consistent iff its g-value equals its rhs-value. This is similar to
satisfying the Bellman equation for undiscounted deterministic sequential decision problems. Thus, this concept is important because the g-values of all vertices equal their start
distances iff all vertices are locally consistent. However, LPA* does not make every vertex
locally consistent. Instead, it uses the heuristics
to focus the search and update only
the g-values that are relevant for computing a shortest path from ,-/.10324. to 739 0;: .
LPA* maintains a priority queue 7 that always contains exactly the locally inconsistent
vertices. These are the vertices whose g-values LPA* potentially needs to update to make
them locally consistent. The keys of the vertices in the priority queue correspond to the
f-values used by A*, and LPA* always expands the vertex in the priority queue with the
smallest key, similar to A* that always expands the vertex in the priority queue with the
smallest f-value. By expanding a vertex, we mean executing 10-16 (numbers in brackets
refer to line numbers in Figure 3). The key 8
of vertex is a vector with two components:
* ,+
* * * + ,+ * +.+* ,+ * * ,+ " * ,++
! * ,+ * ,+
* "!$#&%'(# + )
* !$#&%'*# * !#%+'*# + ) +
*-5, +
*,/ . !$#&%'(# +
*, +
! 1032 '"4$56-798 * * +;: * ,5+.+
*,! +
*-,5+
* *-5, +< . *-5, +.+
*,=
*-5, +.+
* +
* "A"B %
C +
* "A"B %C +< . * *A"B %
C ++
, * * *-5, +ED * + * +?@ > *-5, +.+
*-5, + ! F ,3*-:$5, : + *-,5+
* ,+
*-5, + ! F,3:$: *-,5+3G
,
* ,+
*+
*+
The pseudocode uses the following functions
to manage the priority
*+
* + queue: U.TopKey returns the smallest priority of all vertices in priority queue
. (If
is empty,
.) U.Pop deletes the vertex with the smallest
priority in priority
queue
and returns the
* then
+ U.TopKey returns
* ,+
vertex. U.Insert
inserts vertex into priority queue with priority . Finally, U.Remove
removes vertex from priority queue .
procedure CalculateKey
01 return
procedure Initialize
02
;
03 for all
04
05 U.Insert
;
;
;
;
procedure UpdateVertex
06 if
07 if
U.Remove
;
08 if
U.Insert
;
CalculateKey
;
procedure ComputeShortestPath
09 while U.TopKey
CalculateKey
OR
10
U.Pop ;
11
if
12
;
13
for all
UpdateVertex
;
14
else
15
;
16
for all
UpdateVertex
;
procedure Main
*+
17 Initialize ;
18 forever
* +
19
ComputeShortestPath ;
20
Wait for changes in edge
* costs;
+
21
for all directed edges
* with
+ changed edge costs
22
Update the edge
cost
;
* +
23
UpdateVertex
;
H
-,F : H -,F H
Figure 3: Lifelong Planning A*.
I
KJ IEL
M I=N $O"6
and 8?W
X
RUT&V <
A
4
where 8QP
SRUT&V <
A
4
(2)
1 . Keys are
compared according to a lexicographic ordering. For example, a key 8(
is smaller than
or equal to a key 8 "/
, denoted by 8(
Z$ Y 8 "
, iff either 8 P
8 P"
or ( 8 P
8 P"
used
and 8[W
$ 8 W "
). 8QP,
corresponds directly to the f-values \
<+=
by A* because both the g-values and rhs-values of LPA* correspond to the g-values of
A* and the h-values of LPA* correspond to the h-values of A*. 8 W
corresponds to the
g-values of A*. LPA* expands vertices in the order of increasing k P -values and vertices
with equal k P -values in order of increasing k W -values. This is similar to A* that expands
vertices in the order of increasing f-values (since the heuristics are consistent) and vertices
with equal f-values that are on the same branch of the search tree in order of increasing
g-values (since it grows the search tree).
A locally inconsistent vertex is called overconsistent iff <
^]
. When LPA*
expands a locally overconsistent vertex 12-13 , then +
<+=,
because vertex
has the smallest key among all locally inconsistent vertices.
<+=,
implies that
8
`_ \
baA< =
*c and thus LPA* expands overconsistent vertices in the same order
as A*. During the expansion of vertex , LPA* sets the g-value of vertex to its rhsvalue and thus its start distance 12 , which is the desired value and also makes the vertex
locally consistent. Its g-value then no longer changes until
LPA* terminates. A locally
. When LPA* expands
inconsistent vertex is called underconsistent iff <
a locally underconsistent vertex 15-16 , then it simply sets the g-value of the vertex to
infinity 15 . This makes the vertex either locally consistent or locally overconsistent. If
the expanded vertex was locally overconsistent, then the change of its g-value can affect
the local consistency of its successors 13 . Similarly, if the expanded vertex was locally
underconsistent, then it and its successors can be affected 16 . LPA* therefore updates
rhs-values of these vertices, checks their local consistency, and adds them to or removes
them from the priority queue accordingly.
LPA* expands vertices until 739 0;: is locally consistent and the key of the vertex to expand
next is no smaller than the key of 79 0;: . This is similar to A* that expands vertices until it
expands 879 03: at which point in time the g-value of 739 0;: equals its start distance and the
f-value of the vertex to expand next is no smaller than the f-value of 79 0;: . It turns out
that LPA* expands a vertex at most twice, namely at most once when it is underconsistent
and at most once when it is overconsistent. Thus, ComputeShortestPath
returns after a
number of vertex expansions that is at most twice the number of vertices.
If < 879 0;:
& after the search, then there is no finite-cost path from - .1032A. to 879 0;: .
Otherwise, one can trace back a shortest path from -/.10324. to 79 0;: by always moving from
the current vertex , starting at 79 03: , to any predecessor " that minimizes < "#
" ;
until -/.10324. is reached (ties can be broken arbitrarily), similar to what A* can do if it does
not use backpointers.
The resulting version of LPA* is shown in Figure 3. The main function Main() first calls
Initialize() to initialize the search problem 17 . Initialize() sets the initial g-values of
all vertices to infinity and sets their rhs-values according to Equation 1 03-04 . Thus,
initially -/.10324. is the only locally inconsistent vertex and is inserted into the otherwise
empty priority queue with a key calculated according to Equation 2 05 . This initialization
guarantees that the first call to ComputeShortestPath() performs exactly an A* search, that
is, expands exactly the same vertices as A* in exactly the same order, provided that A*
breaks ties between vertices with the same f-values suitably. Notice that, in an actual
implementation, Initialize() only needs to initialize a vertex when it encounters it during
the search and thus does not need to initialize all vertices up front. This is important because
the number of vertices can be large and only a few of them might be reached during the
search. LPA* then waits for changes in edge costs 20 . If some edge costs have changed,
it calls UpdateVertex() 23 to update the rhs-values and keys of the vertices potentially
affected by the changed edge costs as well as their membership in the priority queue if they
become locally consistent or inconsistent, and finally recalculates a shortest path 19 .
5 Optimizations of Lifelong Planning A*
There are several simple ways of optimizing LPA* without changing its overall operation.
The resulting version of LPA* is shown in Figure 4. First, a vertex sometimes gets removed
from the priority queue and then immediately reinserted with a different key. For example,
a vertex can get removed on line 07 and then be reentered on line 08 . In this case,
it is often more efficient to leave the vertex in the priority queue, update its key, and only
change its position in the priority queue. Second, when UpdateVertex
on line 13 computes the rhs-value for a successor of an overconsistent vertex it is unnecessary to take the
minimum over all of its respective predecessors. It is sufficient to compute the rhs-value
as the minimum of its old rhs-value and the sum of the new g-value of the overconsistent
vertex and the cost of moving from the overconsistent vertex to the successor. The reason
is that only the g-value of the overconsistent vertex has changed. Since it decreased, it can
only decrease the rhs-values of the successor. Third, when UpdateVertex
on line 16
computes the rhs-value for a successor of an underconsistent vertex, the only g-value that
has changed is the g-value of the underconsistent vertex. Since it increased, the rhs-value
of the successor can only get affected if its old rhs-value was based on the old g-value of
the underconsistent vertex. This can be used to decide whether the successor needs to get
updated and its rhs-value needs to get recomputed 21? . Fourth, the second and third optimization concerned the computations of the rhs-values of the successors after the g-value
of a vertex has changed. Similar optimizations can be made for the computation of the
rhs-value of a vertex after the cost of one of its incoming edges has changed.
6 Analytical and Experimental Results
We can prove the correctness of ComputeShortestPath().
* +
1
The pseudocode uses the following
functions to manage the priority queue: U.Top returns a vertex with the smallest priority
*+
* + of all vertices in
. (If
is empty, then U.TopKey
returns
.)
priority queue
* + . U.TopKey returns the smallest priority of all vertices in priority
* queue
+
U.Insert
inserts vertex into priority queue
with priority . U.Update * ,+ changes the priority
of
vertex
in
priority
queue
to . (It
does nothing if the current priority of vertex already equals .) Finally, U.Remove
removes vertex from priority queue .
$
* ,+
* + * +.+ * ,+ * * ,+ * ,+.+
! * + * ,+
* !$#%+'*# + )
* !$#%+'*# * !#%+'*# + ) +
*, *+< *- . ,5+ .
*-5, *-5, + + *,+ , ! , ! + + *-,F *-,=
*,+.+
*-,5+.+
* *-5, +
*, +
, ! +
*-,+
* +
*
* + @>
* A"B %
C +
* A"B %
C + . * A"B %
C +.+
, * *-5, + D * + *-5, +.+
*-5, +
*, +
*-,5+
!Z =,9:(: *-5, +
* . !$#%+'*# + * ,+
* * ,+ 1 *, +/: *-,F" ,+.+
* ,+
B *-,5C5 + *-, +
=,9:(: *-5, +G
,
!Z
* * ,+ B C5 : *-,F" ,+
, + * * + :* ,+.+
* . !#%+'*# + * ,+
!
0
3
2 '*4(56 !$8
* ,+
procedure CalculateKey
* * ,+
01? return
procedure Initialize
02?
;
03? for all
04?
05? U.Insert
procedure UpdateVertex
06? if (
07? else if
08? else if
;
;
;
;
AND
AND
AND
procedure ComputeShortestPath
09? while U.TopKey
CalculateKey
10?
U.Top ;
11?
if
12?
;
13?
U.Remove
;
14?
for all
15?
if
16?
UpdateVertex
;
17?
else
18?
;
19?
;
20?
for all
21?
if
22?
if
23?
UpdateVertex
;
U.Update
CalculateKey
;
U.Insert
CalculateKey
U.Remove
;
;
OR
;
OR
;
*+
procedure Main* +
24? Initialize ;
25? forever
* +
26?
ComputeShortestPath ;
27?
Wait for changes in edge
* costs;
+
28?
for all directed
with changed edge costs
* edges+
29?
; *
+
30?
Update the edge
;
* cost
+
31?
if ( *
+ ) * +
* * +
32?
if * * +
* 5+
+
33?
else if*
+ * +
34?
if
* +
35?
UpdateVertex
;
: B C5 : -,F H -,F : H -,= 1H
:*B -C5H D. : !$-,=#&% 1'*H # -H
-H -, ;:"B C5 -H *-,5+ :*-,= 1H+.+
-H . !$#&%'*# -H ! 1032 '"4$5
6 8 * * + ;:* 1H+.+
-H
;
;
Figure 4: Lifelong Planning A* (optimized version)
Theorem 1 ComputeShortestPath() terminates and one can then trace back a shortest path from
' to ' by always moving from the current vertex , starting at ' , to any predecessor /
that minimizes -% / 21435 / 6 until is reached (ties can be broken arbitrarily).
(The proofs can be found in [LK01].) We now compare breadth-first search, A*,
DynamicSWSF-FP, and the optimized version of LPA* experimentally. (We use
DynamicSWSF-FP with the same optimizations that we developed for LPA*, to avoid biasing our experimental results in favor of LPA*.) The priority queues of all four algorithms
were implemented as binary heaps. Since all algorithms determine the same paths (if they
break ties suitably), we need to compare their total search time until a shortest path has
been found. Since the actual runtimes are implementation-dependent, we instead use three
measures that all correspond to common operations performed by the algorithms and thus
heavily influence their runtimes: the total number of vertex expansions (that is, updates
of the g-values, similar to backup operations of dynamic programming for sequential decision problems), the total number of vertex accesses
(for example, to read or change
their values), and the total number of heap percolates (exchanges of a parent and child
in the heap). Note that we count two vertex expansions, not just one vertex expansion, if
LPA* expands the same vertex twice, to avoid biasing our experimental results in favor of
LPA*.
All of our experiments use fifty eight-connected gridworlds that have size
C
and an
obstacle density of 40 percent. The start cell is at coordinates (34, 20) and the goal cell
is at coordinates (5, 20), where the upper leftmost cell is at coordinates (0, 0). For each
gridworld, the initial obstacle configuration was generated randomly. Then, it was changed
500 times in a row, each time by making eight randomly chosen blocked cells traversable
and eight randomly chosen traversable cells blocked. Thus, each time one percent of the
cells changed their status but the obstacle density remained the same. After each of the
500 changes, the algorithms recomputed a shortest path from the start cell to the goal cell.
For each of the four algorithms and each of the three performance measures, the following
table reports the mean of the performance measure for the 500 changes: both its average
over the fifty mazes and its 95-percent confidence interval over the fifty mazes (assuming a
normal distribution with unknown variance). The table confirms the observations made in
Section 3: LPA* outperforms the other three search methods according to all three performance measures.
complete search
ve =
va =
hp =
incremental search
ve =
va =
hp =
uninformed search
breadth-first search
1331.7
4.4
26207.2
84.0
5985.3
19.7
DynamicSWSF-FP
173.0
4.9
5697.4
167.0
956.2
26.6
heuristic search
A*
ve =
284.0
5.9
va = 6177.3
129.3
hp = 1697.3
39.9
Lifelong Planning A*
ve =
25.6
2.0
va = 1235.9
75.0
hp =
240.1
16.9
We have also applied LPA* successfully to more complex planning tasks, including the
kind of route planning tasks that Focussed Dynamic A* [Ste95] applies to. The results will
be reported separately.
References
[FMSN00] D. Frigioni, A. Marchetti-Spaccamela, and U. Nanni. Fully dynamic algorithms for maintaining shortest paths trees. Journal of Algorithms, 34(2):251?
281, 2000.
[LK01]
M. Likhachev and S. Koenig. Lifelong Planning A* and Dynamic A* Lite: The
proofs. Technical report, College of Computing, Georgia Institute of Technology, Atlanta (Georgia), 2001.
[Pea85]
J. Pearl. Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, 1985.
[RR96]
G. Ramalingam and T. Reps. An incremental algorithm for a generalization of
the shortest-path problem. Journal of Algorithms, 21:267?305, 1996.
[Ste95]
A. Stentz. The focussed D* algorithm for real-time replanning. In Proceedings
of the International Joint Conference on Artificial Intelligence, pages 1652?
1659, 1995.
[Thr98]
Sebastian Thrun. Lifelong learning algorithms. In S. Thrun and L. Pratt, editors, Learning To Learn. Kluwer Academic Publishers, 1998.
| 2003 |@word version:9 advantageous:1 suitably:2 termination:3 d2:1 confirms:1 shot:1 initial:2 configuration:1 series:3 contains:1 document:1 outperforms:1 current:5 subsequent:2 remove:8 update:11 intelligence:4 accordingly:1 constructed:1 predecessor:4 become:1 prove:1 combine:1 planning:20 bellman:1 decreasing:3 actual:3 increasing:4 provided:1 underlying:1 what:4 kind:2 interpreted:1 minimizes:2 developed:3 informed:1 guarantee:1 every:2 expands:14 tie:5 exactly:4 control:3 reuses:3 local:2 topkey:6 path:21 ap:1 might:1 twice:3 initialization:1 studied:1 shaded:1 directed:2 procedure:10 confidence:1 wait:3 get:8 ga:1 influence:1 deterministic:1 center:1 starting:3 independently:1 immediately:2 his:1 coordinate:4 updated:1 heavily:1 programming:1 us:5 satisfying:1 bottom:1 inserted:1 solved:1 connected:9 ordering:1 decrease:4 removed:2 traversable:6 substantial:1 agency:1 broken:2 dynamic:4 motivate:1 solving:3 easily:1 joint:1 recomputes:2 artificial:4 whose:6 heuristic:12 widely:1 solve:4 otherwise:3 favor:3 traversability:2 advantage:2 analytical:1 adaptation:1 relevant:1 combining:1 iff:5 lookahead:1 squeeze:1 parent:1 empty:3 undiscounted:1 produce:1 incremental:15 executing:1 leave:1 develop:2 uninformed:5 solves:1 implemented:1 implies:1 differ:1 successor:10 exchange:1 government:1 generalization:1 insert:8 normal:1 visualize:1 achieves:1 early:2 smallest:7 heap:3 currently:1 replanning:2 individually:1 correctness:1 successfully:1 lexicographic:1 always:11 rather:1 avoid:3 gatech:1 focus:3 check:1 greedily:1 dependent:1 membership:1 typically:1 initially:2 expand:2 overall:1 among:1 denoted:1 plan:1 special:2 airport:1 initialize:10 equal:7 once:3 runtimes:2 identical:2 report:2 intelligent:2 few:2 randomly:3 ve:4 lite:1 recalculates:1 atlanta:2 organization:1 navigation:1 bracket:1 edge:17 respective:1 orthogonal:1 tree:9 old:3 desired:1 increased:1 obstacle:5 cost:19 vertex:91 front:1 reported:1 density:3 international:1 contract:1 together:1 manage:2 priority:30 return:9 reusing:1 north:1 matter:1 satisfy:2 performed:1 view:1 break:2 traffic:1 reached:3 start:25 maintains:4 became:3 variance:1 skoenig:1 correspond:5 cc:1 researcher:2 sebastian:1 ed:1 proof:2 stop:1 knowledge:3 back:4 wesley:1 originally:2 spaccamela:1 furthermore:1 just:2 until:6 koenig:2 hand:1 web:1 gridworlds:4 quality:2 gray:3 artifact:1 grows:1 b3:1 concept:1 read:1 during:6 replan:3 sponsoring:1 leftmost:1 ramalingam:1 complete:3 demonstrate:2 performs:1 percent:4 common:1 pseudocode:2 qp:2 overview:2 kluwer:1 refer:1 blocked:6 grid:3 consistency:2 similarly:2 hp:4 moving:4 access:1 longer:1 add:1 optimizing:1 route:5 binary:1 arbitrarily:2 rep:1 ced:1 minimum:2 determine:3 shortest:19 ii:1 branch:1 technical:1 faster:5 academic:1 award:1 a1:1 va:4 baa:1 repetitive:1 sometimes:1 cell:25 want:1 separately:1 decreased:1 interval:1 else:5 publisher:1 fifty:3 sure:1 inconsistent:6 call:4 easy:1 concerned:1 pratt:1 affect:1 topology:2 idea:1 itr:1 knowing:2 whether:1 effort:2 likhachev:2 queue:22 sgoal:10 locally:18 nsf:1 notice:1 sstart:10 affected:3 group:1 recomputed:4 four:5 key:13 deletes:1 d3:1 changing:2 breadth:7 graph:8 underconsistent:7 sum:1 fourth:1 decide:1 decision:3 guaranteed:1 display:1 lpa:42 infinity:3 stentz:2 expanded:2 speedup:1 developing:1 according:4 smaller:3 slightly:1 terminates:2 making:2 repair:1 equation:3 turn:2 count:1 addison:1 operation:3 apply:1 eight:10 encounter:1 original:7 top:3 denotes:4 a4:1 maintaining:1 build:1 implied:1 already:1 strategy:1 traditional:1 diagonal:1 distance:25 thank:1 thrun:2 restating:1 reason:1 assuming:1 length:1 relationship:1 potentially:3 trace:4 stated:2 marchetti:1 implementation:2 policy:1 unknown:1 upper:1 observation:1 finite:3 situation:4 precise:1 gridworld:16 namely:3 optimized:2 pop:2 pearl:1 rut:1 able:1 fp:11 biasing:3 overconsistent:10 including:1 representing:1 technology:2 carried:1 kj:1 literature:3 determining:1 manhattan:1 fully:1 analogy:1 sufficient:1 consistent:11 editor:1 share:1 row:1 changed:24 supported:1 allow:1 institute:2 lifelong:15 focussed:2 absolute:1 calculated:1 maze:3 computes:4 author:1 c5:1 forward:1 made:2 approximate:1 forever:2 status:2 incoming:1 assumed:1 unnecessary:1 consuming:1 percolate:1 search:84 table:2 learn:1 expanding:1 expansion:5 investigated:1 complex:1 anthony:1 domain:1 official:1 main:4 rh:16 backup:1 nothing:1 child:1 georgia:3 position:1 breaking:1 third:2 theorem:1 remained:2 sequential:2 iel:1 simply:1 expressed:1 contained:1 applies:2 dynamicswsf:11 corresponds:2 determines:2 goal:11 towards:1 change:13 experimentally:1 called:3 total:4 partly:1 experimental:5 college:2 support:1 calculatekey:7 scratch:4 |
1,101 | 2,004 | Scaling laws and local minima in Hebbian ICA
Magnus Rattray and Gleb Basalyga
Department of Computer Science, University of Manchester,
Manchester M13 9PL, UK.
magnus@cs.man.ac.uk, basalygg@cs.man.ac.uk
Abstract
We study the dynamics of a Hebbian ICA algorithm extracting a single non-Gaussian component from a high-dimensional Gaussian background. For both on-line and batch learning we find that a surprisingly
large number of examples are required to avoid trapping in a sub-optimal
state close to the initial conditions. To extract a skewed signal at least
examples are required for -dimensional data and
examples are required to extract a symmetrical signal with non-zero kurtosis.
1 Introduction
Independent component analysis (ICA) is a statistical modelling technique which has attracted a significant amount of research interest in recent years (for a review, see Hyv?arinen,
1999). The goal of ICA is to find a representation of data in terms of a combination of statistically independent variables. A number of neural learning algorithms have been applied
to this problem, as detailed in the aforementioned review.
Theoretical studies of ICA algorithms have mainly focussed on asymptotic stability and
efficiency, using the established results of stochastic approximation theory. However, in
practice the transient stages of learning will often be more significant in determining the
success of an algorithm. In this paper a Hebbian ICA algorithm is analysed in both on-line
and batch mode, highlighting the critical importance of the transient dynamics. We find that
a surprisingly large number of training examples are required in order to avoid trapping in
a sub-optimal state close to the initial conditions. To detect a skewed signal at least
examples are required for -dimensional data, while
examples are required for a
symmetric signal with non-zero kurtosis. In addition, for on-line learning we show that
the maximal initial learning rate which allows successful learning is unusually low, being
for a skewed signal and
for a symmetric signal.
In order to obtain a tractable model, we consider the limit of high-dimensional data and
study an idealised data set in which a single non-Gaussian source is mixed into a large
number of Gaussian sources. Recently, one of us considered a more general model in
which an arbitrary, but relatively small, number of non-Gaussian sources were mixed into
a high-dimensional Gaussian background (Rattray, 2002). In that work a solution to the
dynamics of the on-line algorithm was obtained in closed form for
learning iterations
and a simple solution to the asymptotic dynamics under the optimal learning rate decay was
obtained. However, it was noted there that modelling the dynamics on an
timescale
is not always appropriate, because the algorithm typically requires much longer in order to
escape from a class of metastable states close to the initial conditions. In order to elucidate
this effect in greater detail we focus here on the simplest case of a single non-Gaussian
source and we will limit our analysis to the dynamics close to the initial conditions.
In recent years a number of on-line learning algorithms, including back-propagation and
Sanger?s PCA algorithm, have been studied using techniques from statistical mechanics
(see, for example, Biehl (1994); Biehl and Schwarze (1995); Saad and Solla (1995) and
contributions in Saad (1998)). These analyses exploited the ?self-averaging? property of
certain macroscopic variables in order to obtain ordinary differential equations describing
the deterministic evolution of these quantities over time in the large limit. In the present
case the appropriate macroscopic quantity does not self-average and fluctuations have to
be considered even in the limit. In this case it is more natural to model the on-line learning
dynamics as a diffusion process (see, for example Gardiner, 1985).
2 Data Model
In order to apply the Hebbian ICA algorithm we must first sphere the data, ie. linearly
transform the data so that it has zero mean and an identity covariance matrix. This can be
achieved by standard transformations in a batch setting or for on-line learning an adaptive
sphering algorithm, such as the one introduced by Cardoso and Laheld (1996), could be
used. To simplify the analysis it is assumed here that the data has already been sphered.
Without loss of generality it can also be assumed that the sources each have unit variance.
Each data point is generated from a noiseless linear mixture of sources which are decomposed into a single non-Gaussian source and
uncorrelated Gaussian components,
. We will also decompose the mixing matrix into a column vector
and a
rectangular matrix
associated with the non-Gaussian and Gaussian
components respectively,
! #"$% '&
*
)(
(1)
is presented to the
We will consider both the on-line case, in which a new IID example
algorithm at each time and then discarded, and also the batch case, in which a finite set
of examples are available to the algorithm. To conform with the model assumptions the
mixing matrix must be unitary, which leads to the following constraints,
+ ,- /. 10 ,2 0 "3 0 4
0 + %0
1 0 10 % 7
&
50 . %0 ,6%0
(2)
(3)
3 On-line learning
B<:'8%0CC 8 such that the projection 9;:<8=013>@?A . Defining
9 8 0 #"D
(4)
BE#"GFIH JKJ 8LJMJ B where F NO P
The goal of ICA is to find a vector
the overlap
we obtain,
8
RB >S? 8
where we have made use of the constraint in eqn. (2). This assumes zero correlation between and which is true for on-line learning but is only strictly true for the first iteration
of batch learning (see section 4). In the algorithm described below we impose a normalisation constraint on such that
. In this case we see that the goal is to find such
that
.
JMJ 8LJKJQ
8
A simple Hebbian (or anti-Hebbian) learning rule was studied by Hyv?arinen and Oja
(1998), who showed it to have a remarkably simple stability condition. We will consider
the deflationary form in which a single source is learned at one time. The algorithm is
closely related to Projection Pursuit algorithms, which seek interesting projections in highdimensional data. A typical criteria for an interesting projection is to find one which is
maximally non-Gaussian in some sense. Maximising some such measure (simple examples would be skewness or kurtosis) leads to the following simple algorithm (see Hyv?arinen
and Oja, 1998, for details). The change in at time is given by,
8
*
& (5)
( followed
by normalisation such that JKJ 8LJKJ
Here, is the learning rate and 9 is some non-linear function which
we will take to be
at least three times differentiable. An even non-linearity, eg. 9 9 , is appropriate
for
detecting asymmetric signals
while a more common choice is an odd function, eg.
9 9
or 9
9 E, which
used to detect symmetric non-Gaussian
P hascanto bebe chosen
signals. In the latter case
in order to ensure stability of the
correct solution, as described by
Hyv?
a
rinen
and
Oja
(1998),
either adaptively or using a?
in the case of an even non-linearity.
Remarkably, the same
priori knowledge. We set
non-linearity can be used to separate both sub and super-Gaussian signals, in contrast to
8
9 (
maximum likelihood methods for which this is typically not the case.
We can write the above algorithm as,
(6)
8 ( H "8% ( 9"( 9 ( " 9/ ( ,( 9 ( JMJ ( JMJ &
and (two different scalings will be considered below) we can
For large
expand out to get a weight decay normalisation,
8 ( 8 ( " 9 ( ! ( 9 ( 8 (#" $ 9 ( 8 ( &
(7)
Taking the dot-product with gives the following update increment for the overlap B ,
% B< & 9 ( B ( 9 ( " 9 ( B (
(8)
where we used %the constraint in eqn. (3) to set ,0
. Below we calculate the mean
and variance of B for two different scalings of the learning rate. Because the conditional
in eqn. 4) these expressions
distribution for 9 given only depends on B (setting JKJ 8LJKJ
will depend only on B and statistics of the non-Gaussian source distribution.
3.1 Dynamics close to the initial conditions
, 8
B<
'
If the entries in and are initially of similar order then one would expect
.
This is the typical case if we consider a random and uncorrelated choice for and the initial
entries in . Larger initial values of could only be obtained with some prior knowledge
of the mixing matrix which we will not assume. We will set
in the following
discussion, where is assumed to be an
quantity. The discussion below is therefore
restricted to describing the dynamics close to the initial conditions. For an account of the
transient dynamics far from the initial conditions and the asymptotic dynamics close to an
optimal solution, see Rattray (2002).
8
3.1.1
B
(
(;: B*)
9 even, +
- , N
9 9
If the signal is asymmetrical then an even non-linearity can be used, for example
is a common choice. In this case the appropriate (ie. maximal) scaling for the learning rate
is
and we set
where is an
scaled learning rate parameter. In
/.10
.
even,
%
odd,
%
(: 8=01 )
Figure 1: Close to the initial conditions (where
) the learning
dynamics is equivalent to diffusion in a polynomial potential. For asymmetrical source
distributions we can use an even non-linearity in which case the potential is cubic, as shown
on the left. For symmetrical source distributions with non-zero kurtosis we should use
an odd non-linearity in which case the potential is quartic, as shown on the right.
The
.
with a potential barrier
dynamics is initially confined in a metastable state near
(E N
%
( at each iteration are given
+ % F . ( " +
F .( "
E ( .
(9)
+
%
(10)
Var ( .
F .
this case we find that the mean and variance of the change in
by (to leading order in
),
+
+% ( .
F ON
where
is the third cumulant of the source distribution (third central moment), which
measures skewness, and brackets denote averages over
. We also find that
E
for integer !
. In this case the system can be described by a
Fokker-Planck equation for large (see, for example, Gardiner, 1985) with a characteristic
timescale of
. The system is locally equivalent to a diffusion in the following cubic
potential,
"
#
(11)
F . ( +
F .(
with a diffusion coefficient $ %
F . which is independent% ofmust
( . The shape of this
be overcome to
potential is shown on the left of fig. 1. A potential barrier of
(
escape a metastable state close to the initial conditions.
3.1.2
9 odd, + " , N
9 9
9
9
5 . 0
If the signal is symmetrical, or only weakly asymmetrical, it will be necessary to use an
odd non-linearity, for example
or
are popular choices. In
this case a lower learning rate is required in order to achieve successful separation. The
appropriate scaling for the learning rate is
and we set
where again is
an
scaled learning rate parameter. In this case we find that the mean and variance of
the change in at each iteration are given by,
(
.
+ % ( . F . ( " # + " F .(
"
(12)
+
%
F .
Var ( .
(13)
where + " is the fourth cumulant
source distribution (measuring kurtosis) and brackets
NO Pof .theAgain
denote averages over F
the system can be described by a Fokker-Planck
E
equation for large
but in this case the timescale for learning is
, an order of
slower than in the asymmetrical case. The system is locally equivalent to diffusion in the
following quartic potential,
" F . ( " J + " F J .( " (14)
with a diffusion coefficient $ F . . We have assumed < Sign + " which is
a necessary condition for successful learning. In the case of a cubic non-linearity this is
(
also the condition for stability of the optimal fixed point, although in general these two
conditions may not be equivalent (Rattray, 2002). The
of this potential is shown
shape
on the right of fig. 1 and again a potential barrier of
must be overcome to escape a
metastable state close to the initial conditions.
%
B< N
For large . the dynamics of ( corresponds to an Ornstein-Uhlenbeck process with a Gaussian stationary distribution of fixed unit variance. Thus, if one chooses too large . initially
N (recall, B (0 ) ). As . is reduced
the dynamics will become localised close to B
3.1.3 Escape times from a metastable state at
.
the potential barrier confining the dynamics is reduced. The timescale for escape for large
(mean first passage time) is mainly determined by the effective size of the barrier (see,
for example, Gardiner, 1985),
%
where
*
escape
%
$
(15)
* is a unit of time in
N , + .5: .
for even 9 , +
,
N . + .%:& . (16)
for odd 9 , + " ,
is the potential barrier, $ is the diffusion coefficient and
the diffusion process. For the two cases considered above we obtain,
even
escape
odd
escape
F .
+
F
F .
J + " F J
The constants of proportionality depend on the shape of the potential and not on . As the
learning rate parameter is reduced so the timescale for escape is also reduced. However, the
choice of optimal learning rate is non-trivial and cannot be determined by considering only
the leading order terms in as above, because although small will result in a quicker
, this in turn will lead to a very slow
escape from the unstable fixed point near
learning transient after escape. Notice that escape time is shortest when the cumulants
or " are large, suggesting that deflationary ICA algorithms will tend to find these signals
first.
B
.
B N
+
+
From the above discussion one can draw two important conclusions. Firstly, the initial
learning rate must be less than
initially in order to avoid trapping close to the
initial conditions. Secondly, the number of iterations required to escape the initial transient
will be greater than
, resulting in an extremely slow initial stage of learning for large
. The most extreme case is for symmetric source distributions with non-zero kurtosis, in
which case
learning iterations are required.
In fig. 2 we show results of learning with an asymmetric source (top) and uniform source
(bottom) for different scaled learning rates. As the learning rate is increased (left to right)
we observe that the dynamics becomes increasingly stochastic, with the potential barrier
becoming increasingly significant (potential maxima are shown as dashed lines). For the
largest value of learning rate ( ) the algorithm becomes trapped close to the initial
conditions for the whole simulation time. From the time axis we observe that the learning
timescale is
for the asymmetrical signal and
for the symmetric signal, as
predicted by our theory.
.$
?=0.1
1
?=1
?=5
1
1
R
R
0.5
0.5
0.5
0
0
0
2
?(x)=x , ? ? 0
R
3
0
5
10
t/N2
15
1
R ?(x)=x3, ? ? 0
4
0.5
0
5
10
t/N2
15
0
1
R
0.5
1
R
0.5
0
0
0
?0.5
?0.5
?0.5
?1
0
5
t/N3
+
&
10
?1
0
NQN
5
t/N3
10
5
?1
0
10
t/N2
15
5
t/N3
10
Figure 2: 100-dimensional data (
) is produced from a mixture containing a single
non-Gaussian source. In the top row we show results for a binary, asymmetrical source with
skewness
. In the bottom row we show results for a uniformly
and
distributed source and
. Each row shows learning with the same initial conditions
and data but with different scaled learning rates (left to right
and ) where
(top) or
(bottom). Dashed lines are maxima of the potentials in fig. 1.
. :
.
:
.R N & Q
4 Batch learning
The batch version of eqn. (5) for sufficiently small learning rates can be written,
% 8 9 ( ( 9 ( 8 ( "
(
(17)
where is the number of training examples. Here we argue that such an update requires
at least the same order of examples as in the on-line case, in order to be successful. Less
data will result in a low signal-to-noise ratio initially and the possibility of trapping in a
sub-optimal fixed point close to the initial conditions.
As in the on-line case we can write the update in terms of
B
B
,
% B 9 ( ( 9 ( B ( " &
(
N
(18)
We make an assumption that successful learning is unlikely unless the initial increment in
is in the desired direction. For example, with an asymmetric signal and quadratic nonlinearity we require
initially, while for a symmetric signal and odd non-linearity
we require
% . We have carried out simulations of batch learning which confirm
that a relatively low percentage of runs in which the intial increment was incorrect result
in successful learning compared to typical performance. As in the on-line case we observe
that runs either succeed, in which case
, or fail badly with remaining
.
%
%B B +
N B
'
B
>
?
B
' initially and we can therefore expand the right-hand side of
As before, B
%
%
eqn. (18) in orders of B for large . B
( B at the first iteration) is a sum over raninit
%B
domly sampled terms, and the central limit theorem states that for large the distribution
init
from which
is sampled will be Gaussian, with mean and variance given by (to leading order in ),
B
% B +
F B " # + " F B
(19)
%
&
Var B
(20)
F
Notice that the +
term disappears in the case of an asymmetrical non-linearity, which
is why we have left both % terms in eqn. (19). The algorithm will be likely to fail when
is of the same order (or greater) than the mean. Since
the standard deviation of B
B ' initially, we see that this is true for R
in the case of an even nonE
init
init
init
R
linearity and asymmetric signal, or for
in the case of an odd non-linearity and
a signal with non-zero kurtosis. We expect these results to be necessary but not necessarily
sufficient for successful learning, since we have only shown that this order of examples is
the minimum required to avoid a low signal-to-noise ratio in the first learning iteration. A
complete treatment of the batch learning problem would require much more sophisticated
formulations such as the mean-field theory of Wong et al. (2000).
5 Conclusions and future work
In both the batch and on-line Hebbian ICA algorithm we find that a surprisingly large number of examples are required to avoid a sub-optimal fixed point close to the initial conditions. We expect simialr scaling laws to apply in the case of small numbers of non-Gaussian
sources. Analysis of the square demixing problem appears to be much more challenging
as in this case there may be no simple macroscopic description of the system for large .
It is therefore unclear at present whether ICA algorithms based on Maximum-likelihood
and Information-theoretic principles (see, for example, Bell and Sejnowski, 1995; Amari
et al., 1996; Cardoso and Laheld, 1996), which estimate a square demixing matrix, exhibit
similar classes of fixed point to those studied here.
Acknowledgements: This work was supported by an EPSRC award (ref. GR/M48123). We would
like to thank Jon Shapiro for useful comments on a preliminary version of this paper.
References
S-I Amari, A Cichocki, and H H Yang. In D S Touretzky, M C Mozer, and M E Hasselmo, editors, Neural Information Processing Systems 8, pages 757?763. MIT Press,
Cambridge MA, 1996.
A J Bell and T J Sejnowski. Neural Computation, 7:1129?1159, 1995.
M Biehl. Europhys. Lett., 25:391?396, 1994.
M Biehl and H Schwarze. J. Phys. A, 28:643?656, 1995.
J-F Cardoso and B. Laheld. IEEE Trans. on Signal Processing, 44:3017?3030, 1996.
C. W. Gardiner. Handbook of Stochastic Methods. Springer-Verlag, New York, 1985.
A Hyv?arinen. Neural Computing Surveys, 2:94?128, 1999.
A Hyv?arinen and E Oja. Signal Processing, 64:301?313, 1998.
M Rattray. Neural Computation, 14, 2002 (in press).
D Saad, editor. On-line Learning in Neural Networks. Cambridge University Press, 1998.
D Saad and S A Solla. Phys. Rev. Lett., 74:4337?4340, 1995.
K Y M Wong, S Li, and P Luo. In S A Solla, T K Leen, and K-R M?uller, editors, Neural
Information Processing Systems 12. MIT Press, Cambridge MA, 2000.
| 2004 |@word version:2 polynomial:1 proportionality:1 hyv:6 gfih:1 seek:1 simulation:2 covariance:1 moment:1 initial:22 luo:1 analysed:1 attracted:1 must:4 written:1 shape:3 update:3 stationary:1 trapping:4 deflationary:2 detecting:1 jkj:3 firstly:1 differential:1 become:1 incorrect:1 ica:11 mechanic:1 decomposed:1 considering:1 becomes:2 pof:1 linearity:12 skewness:3 transformation:1 unusually:1 scaled:4 uk:3 unit:3 planck:2 before:1 local:1 limit:5 fluctuation:1 becoming:1 studied:3 challenging:1 statistically:1 practice:1 x3:1 laheld:3 bell:2 projection:4 get:1 cannot:1 close:15 wong:2 equivalent:4 deterministic:1 rectangular:1 survey:1 rule:1 stability:4 increment:3 elucidate:1 rinen:1 asymmetric:4 bottom:3 epsrc:1 quicker:1 calculate:1 solla:3 mozer:1 dynamic:17 depend:2 weakly:1 efficiency:1 effective:1 sejnowski:2 europhys:1 larger:1 biehl:4 amari:2 statistic:1 gleb:1 timescale:6 transform:1 differentiable:1 kurtosis:7 maximal:2 product:1 mixing:3 achieve:1 description:1 manchester:2 ac:2 odd:9 c:2 predicted:1 direction:1 closely:1 correct:1 stochastic:3 transient:5 require:3 arinen:5 decompose:1 preliminary:1 secondly:1 strictly:1 pl:1 sufficiently:1 considered:4 magnus:2 domly:1 largest:1 hasselmo:1 uller:1 mit:2 gaussian:19 always:1 super:1 avoid:5 focus:1 modelling:2 likelihood:2 mainly:2 contrast:1 detect:2 sense:1 typically:2 unlikely:1 initially:8 expand:2 aforementioned:1 priori:1 field:1 jon:1 future:1 simplify:1 escape:13 oja:4 interest:1 normalisation:3 possibility:1 mixture:2 bracket:2 extreme:1 necessary:3 unless:1 desired:1 theoretical:1 increased:1 column:1 cumulants:1 measuring:1 ordinary:1 deviation:1 entry:2 intial:1 uniform:1 successful:7 gr:1 too:1 chooses:1 adaptively:1 ie:2 again:2 central:2 containing:1 leading:3 li:1 account:1 potential:16 suggesting:1 coefficient:3 depends:1 ornstein:1 closed:1 contribution:1 square:2 variance:6 who:1 characteristic:1 produced:1 iid:1 none:1 phys:2 touretzky:1 associated:1 sampled:2 treatment:1 popular:1 recall:1 knowledge:2 sophisticated:1 back:1 appears:1 maximally:1 formulation:1 leen:1 generality:1 stage:2 correlation:1 hand:1 eqn:6 propagation:1 mode:1 schwarze:2 effect:1 true:3 asymmetrical:7 idealised:1 evolution:1 symmetric:6 eg:2 skewed:3 self:2 noted:1 criterion:1 complete:1 theoretic:1 passage:1 recently:1 common:2 significant:3 cambridge:3 nonlinearity:1 dot:1 longer:1 recent:2 showed:1 quartic:2 certain:1 verlag:1 binary:1 success:1 exploited:1 minimum:2 greater:3 impose:1 shortest:1 signal:22 dashed:2 hebbian:7 sphere:1 award:1 noiseless:1 iteration:8 uhlenbeck:1 achieved:1 confined:1 background:2 remarkably:2 addition:1 source:20 macroscopic:3 saad:4 comment:1 tend:1 integer:1 extracting:1 unitary:1 near:2 yang:1 whether:1 expression:1 pca:1 york:1 useful:1 detailed:1 cardoso:3 amount:1 locally:2 simplest:1 reduced:4 shapiro:1 percentage:1 notice:2 sign:1 trapped:1 rb:1 rattray:5 conform:1 write:2 diffusion:8 year:2 sum:1 run:2 fourth:1 separation:1 draw:1 scaling:6 followed:1 quadratic:1 badly:1 gardiner:4 constraint:4 n3:3 extremely:1 relatively:2 sphering:1 department:1 metastable:5 combination:1 increasingly:2 rev:1 restricted:1 equation:3 describing:2 turn:1 fail:2 tractable:1 available:1 pursuit:1 apply:2 observe:3 appropriate:5 batch:10 slower:1 assumes:1 top:3 ensure:1 remaining:1 sanger:1 already:1 quantity:3 unclear:1 exhibit:1 separate:1 thank:1 argue:1 unstable:1 trivial:1 maximising:1 ratio:2 basalyga:1 localised:1 confining:1 discarded:1 finite:1 anti:1 defining:1 arbitrary:1 introduced:1 required:11 learned:1 established:1 trans:1 below:4 including:1 critical:1 overlap:2 natural:1 disappears:1 axis:1 carried:1 extract:2 cichocki:1 review:2 prior:1 acknowledgement:1 determining:1 asymptotic:3 law:2 loss:1 expect:3 mixed:2 interesting:2 var:3 sufficient:1 principle:1 editor:3 uncorrelated:2 row:3 surprisingly:3 supported:1 side:1 focussed:1 taking:1 barrier:7 distributed:1 overcome:2 lett:2 made:1 adaptive:1 far:1 confirm:1 handbook:1 symmetrical:3 assumed:4 why:1 init:4 m13:1 necessarily:1 linearly:1 whole:1 noise:2 n2:3 ref:1 fig:4 cubic:3 slow:2 sub:5 third:2 theorem:1 decay:2 demixing:2 importance:1 likely:1 jmj:3 highlighting:1 springer:1 fokker:2 corresponds:1 ma:2 succeed:1 conditional:1 goal:3 identity:1 man:2 change:3 typical:3 determined:2 uniformly:1 averaging:1 highdimensional:1 latter:1 sphered:1 cumulant:2 |
1,102 | 2,005 | Audio-Visual Sound Separation Via
Hidden Markov Models
John Hershey
Department of Cognitive Science
University of California San Diego
Michael Casey
Mitsubishi Electric Research Labs
Cambridge, Massachussets
jhershey@cogsci.ucsd.edu
casey@merl.com
Abstract
It is well known that under noisy conditions we can hear speech
much more clearly when we read the speaker's lips. This suggests the utility of audio-visual information for the task of speech
enhancement. We propose a method to exploit audio-visual cues
to enable speech separation under non-stationary noise and with
a single microphone. We revise and extend HMM-based speech
enhancement techniques, in which signal and noise models are factori ally combined, to incorporate visual lip information and employ novel signal HMMs in which the dynamics of narrow-band
and wide band components are factorial. We avoid the combinatorial explosion in the factorial model by using a simple approximate inference technique to quickly estimate the clean signals in
a mixture. We present a preliminary evaluation of this approach
using a small-vocabulary audio-visual database, showing promising
improvements in machine intelligibility for speech enhanced using
audio and visual information.
1
Introduction
We often take for granted the ease with which we can carryon a conversation in the
proverbial cocktail party scenario: guests chatter, glasses clink, music plays in the
background: the room is filled with ambient sound. The vibrations from different
sources and their reverberations coalesce translucently yielding a single time series at
each ear, in which sounds largely overlap even in the frequency domain. Remarkably
the human auditory system delivers high-quality impressions of sounds in conditions
that perplex our best computational systems. A variety of strategies appear to be at
work in this, including binaural spatial analysis, and inference using prior knowledge
of likely signals and their contexts. In speech perception, vision often plays a crucial
role, because we can follow in the lips and face the very mechanisms that modulate
the sound, even when the sound is obscured by acoustic noise.
It has been demonstrated that the addition of visual cues can enhance speech recog-
nition as much as removing 15 dB of noise [1]. Vision provides speech cues that are
complementary to audio cues such as components of consonants and vowels that
are likely to be obscured by acoustic noise [2]. Visual information is demonstra-
bly beneficial to HMM-based automatic speech recognition (ASR) systems, which
typically suffer tremendously under moderate acoustical noise [3].
We introduce a method of audio-visual speech enhancement using factorial hidden
Markov models (fHMMs). We focus on speech enhancement rather than speech
recognition for two reasons: first, speech conveys useful paralinguistic information,
such as prosody, emotion, and speaker identity, and second, speech contains useful
cues for separation from noise, such as pitch. In automatic speech recognition (ASR)
systems, these cues are typically discarded in an effort to reduce irrelevant variance
among speakers and utterances within a phonetic class.
Whereas the benefit of vision to speech recognition is well known, we may well
wonder if visual input offers similar benefits to speech enhancement. In [4] a nonparametric density estimator was used to adapt audio and video transforms to
maximize the mutual information between the face of a target speaker and an audio
mixture containing both the target voice and a distracter voice. These transforms
were then used to construct a stationary filter for separating the target voice from
the mixture without any prior knowledge or training. In [5] a multi-layer perceptron
is trained to map noisy estimates of formants to clean ones, employing lip parameters
(width, height and area of the lip opening) extracted from video as additional input.
The re-estimated formant contours were used to filter the speech to enhance the
signal. In both cases video information improved signal separation. Neither system,
however, made use of the dynamics of speech.
In speech recognition, HMMs are commonly used because of the advantages of
modeling signal dynamics. This suggests the following strategy: train an audiovisual HMM on clean speech, infer the likelihoods of its state sequences, and use
the inferred state probabilities of the signal and noise to estimate a sequence of filters
to clean the data. In cases where background noise also has regularity, such as the
combination of two voices, another HMM can be used to model the background
noise. Ephraim [6] first proposed an approach to factorially combining two HMMs
in such an enhancement system. In [7] an efficient variational learning rule for the
factorial HMM is formulated, and in [8, 9] fHMM speech enhancement was recently
revived using some clever tricks to allow more complex models.
The fHMM approach is amenable to audio-visual speech enhancement in many
different forms. In the simplest formulation, which we pursue here, the signal observation model includes visual features. These visual inputs constrain the signal
HMM and produce more accurate filters. Below we present a prototype architecture
for such a system along with preliminary results. 1
1.1
Factorial Speech Models
One of the challenges of using speech HMMs for enhancement is to model speech
in sufficient detail. Typically, speech models, following the practice in ASR, ignore
narrow-band, spectral details (corresponding to upper cepstral components) which
carry pitch information, because they tend to vary across speakers and utterances
for the same word or phoneme. Instead such systems focus on the smooth, or wideband, spectral characteristics (corresponding to lower cepstral components) such as
are produced by the articulation of the mouth. Such wide-band spectral patterns
loosely represent formant patterns, a well-known cue for vowel discrimination. In
cases where the pitch or other narrow-band properties, of the background signals
differ from the foreground speech, and have predictable dynamics, such as with
lWe defer a detailed mathematical development to subsequent publications. Contact
jhershey@cogsci.ucsd.edu for further information
two simultaneous speech signals, these components may be helpful in separating
the two signals. Figure 1 illustrates the analysis of two words into wide-band and
narrow-band components.
"one"
" two"
Full band:
Narrow band:
Wide band:
Figure 1: full-band, narrow-band, and wide-band log spectrograms of two words.
The wide-band log spectrograms (bottom) are derived by low-pass filtering the
log spectra (across the frequency domain), and the narrow-band log spectrograms
(middle) derived by high pass filtering the log spectra The full log spectrogram
(top) is the sum of the two.
However, the wide-band and narrow-band variations in speech are only loosely coupled. For instance, a given formant is likely to be uttered with many different
pitches and a given pitch may be used to utter any formant. Thus a model of the
full spectrum of speech would have to have enough states to represent every combination of pitches and formants. Such a model requires a large amount of training
data and imposes serious computational burdens. For instance in [8] a model with
8000 states is employed. When combined with a similarly complex noise model, the
composite model has 64 million states. This is expensive in terms of computation
as well as the number of data points required for inference.
To parsimoniously model the complexity of speech, we employ a factorial HMM for
a single speech signal, in which wide and narrow-band components are represented
in sub-models with independent dynamics. We therefore train the two submodels
independently using Gaussian observation probability density functions (p.d.f.) on
the wide-band or narrow-band log spectra, with diagonal covariances for the sake of
simplicity. Figure 2(a) depicts the graphical model for a single wide or narrow-band
component.
Narrow-Band Slate
Di screte States
Wide- Band Stale
Continuou s
Observation s
Combined
Observmions
(a) simple HMM
(b) factorial speech HMM
Figure 2: single HMMs are trained separately on wide-band and narrow-band speech
signals (a) and then combined factorially in (b) by adding the means and variances
of their observation distributions
To combine the sub-models, we have to specify the observation p.d.f. for a combination of a wide and a narrow-band state, over the log-spectrum of speech prior to
liftering. Because the observation densities of each component are Gaussian, and
the log-spectra of the wide and narrow-band components add in the log spectrum,
the composite state has a Gaussian observation p.d.f., whose mean and variance is
the sum of the component observation means and variances. Although the states
of the two sub-models are marginally independent they are typically conditionally
dependent given the observation sequence. In other words we assume that the state
dependencies between the sub-models for a given speech signal can be explained
entirely via the observations. Figure 2(b) depicts the combination of the wide and
narrow-band models, where the observation p.d.f. 's are a function of two state variables.
When combining the signal and noise models (or two different speech models) in
contrast, the signals add in the frequency domain, and hence in the log spectral
domain they longer simply add. In the spectral domain the amplitudes of the two
signals have log-normal distributions, and the relative phases are unknown. There
is no closed form distribution for the sum of two random variables with log-normal
amplitudes and a uniformly distributed phase difference. Disregarding phase differences we apply a well-known approximation to the sum of two lognormal random
variables, in which we match the mean and variance of a lognormal random variable
to the sum of the means and variances of the two component lognormal random
variables [10]. Phase uncertainty can also be incorporated into an approximation;
however in practice the costs appear to outweigh the benefits.2 Figure 3(a) depicts
the combination of two factorial speech models, where the observation p.d.f.s are a
function of two state variables.
-'-- .. .
Video Observations
~
Audio
Observations
(a) dual factorial HMM
6
(1f
0
(b) speech fHMM with video
Figure 3: combining two speech fHMMs (a) and adding video observations to a
speech fHMM (b).
Using the log-normal observation distribution of the composite model we can estimate the likelihood of the speech and noise states for each frame using the well
known forward-backward recursion. For each frame of the test data we can compute
the expected value of the amplitude of each model in each frequency bin. Taking
2The uncertainty of the phase differences can be incorporated by modeling the sum as
a mixture of lognormals that uniformly samples phase differences. Each mixture element
is approximated by taking as its mean the length of the sum of the mean amplitudes when
added in the complex plane according a particular phase difference, and as its variance
the sum of the two variances. This estimation is facilitated by the assumption of diagonal
covariances in the log spectral domain.
the expected value of the signal in the numerator and the expected value of the
signal plus noise in the denominator yields a Wiener filter which is applied to the
original noisy signal enhancing the desired component. When we have two speech
signals one person's noise is another's signal and we can separate both by the same
method.
2
Incorporating vision
We incorporate vision after training the audio models in order to test the improvement yielded by visual input while holding the audio model constant. A video observation distribution is added to each state in the model by obtaining the probability
of each state in each frame of the audio training data using the forward-backward
procedure, then estimating the parameters of the video observation distributions
for each state, in the manner of the Baum-Welch observation re-estimation formula.
This procedure is iterated until it converges. In this way we construct a system in
which the visual observations are modular. Figure 3(b) depicts the structure ofthe
resulting speech model.
Such a method in which audio and visual features are integrated early in processing
is only one of several approaches. We envision other late integration approaches
in which audio and visual dynamics are more loosely coupled. What method of
audio-visual integration may be best for this task is an open question.
3
Efficient inference
In the models described above, in which we factorially combine two speech models ,
each of which is itself factorial , the complexity of inference in the composite model,
using the forward-backward recursion, can easily become unmanageable. If K is
the number of states in each subcomponent, then K4 is the number of states in
the composite HMM. In our experiments K is on the order of 40 states, so there
are 2,560,000 states in the composite model. Naively each composite state must be
searched when computing the probabilities of state sequences necessary for inference. Interesting approximation schemes for similar models are developed in [8, 9].
We develop an approximation as follows.
Rather than computing the forward-backward procedure on the composite HMM,
we compute it sequentially on each sub-HMM to derive the probability of each state
in each frame. Of course, in order to evaluate the observation probabilities of the
current sub-HMMs for a given frame, we need to consider the state probabilities of
the other three sub-HMMs, because their means and variances are combined in the
observation model. These state probabilities and their associated observation probabilities comprise a mixture model for a given frame. The composite mixture model
still has K4 states, so to defray this complexity during forward-backward analysis
of the current sub-HMM, for each frame we approximate the observation mixtures
of each of the other three sub-HMMs with a single Gaussian, whose mean and variance matches that of the mixture. Thus we only have to consider the K states of
the current model, and use the summarized means and variances of the other three
HMMs as auxiliary inputs to the observation model. We initialize the state probabilities in each frame with the equilibrium distribution for each sub-HMM. In our
experiments, after a handful of iterations, the composite state probabilities tend to
converge. This method is closely related to a structured variational approximation
for factorial HMMs [7] and can be also be seen as an approximate belief propagation
or sum-product algorithm [11].
4
Data
We used a small-vocabulary audio-visual speech database developed by Fu Jie
Huang at Carnegie Mellon University 3 [12]. These data consist of audio and video
recordings of 10 subjects (7 males and 3 females) saying 78 isolated words commonly used for numbers and time, such aS,"one" "Monday", "February", "night",
etc. The sequence of 78 words is repeated in 10 different takes. Half of these takes
were used for training, and one of the remaining takes was used as the test set.
The data set included outer lip parameters extracted from video using an automatic
lip tracker, including height of the upper and lower lips relative to the corners the
width from corner to corner. We interpolated these lip parameters to match the
audio frame rate, and calculate time derivatives.
Audio consisted of 16-bit, 44.1 kHz recordings which we resample to 8000 kHz.
The audio was framed at 60 frames per second, with an overlap of 50% , yielding
264 samples per frame. 4 The frames were analyzed into cepstra: the wide-band log
spectrum is derived from the lower 20 cepstral components and the wide-band log
spectrum from the upper cepstra.
5
Results
Speaker dependent wide and narrow-band HMMs having 40 states each were trained
on data from two subjects (" Anne" and" Chris") selected from the training set. A
PCA basis was used to reduce the log spectrograms to a more manageable size of
30 dimensions during training. This resulted in some non-zero covariances near
the diagonal in the learned observation covariance matrices, which we discarded.
An entropic prior and parameter extinction were used to sparsify the transition
matrices during training [13].
The narrow-band model learned states that represented different pitches and had
transition probabilities that were non-zero mainly between neighboring pitches. The
narrow-band model's video observation probability distributions were largely overlapping, reflecting the fact that video tells us little about pitch. The wide-band
model learned states that represented different formant structures. The video observation distributions for several states in the wide-band model were clearly separated, reflecting the information that video provides about the formant structure.
Subjectively the enhanced signals sound well separated from each other for the
most part. Figure 4(a) (bottom) shows the estimated spectrograms for a mixture
of two different words spoken by the same speaker - an extremely difficult task.
To quantify these results we evaluate the system using speech recognizer, on the
slightly easier task of separating the speech of the two different speakers, whose
voices were in different but overlapping pitch ranges.
A test set was generated by mixing together 39 randomly chosen pairs of words, one
from each subject, such that no word was used twice. Each word pair was mixed
at five different signal to noise ratios (SNRs), with the SNR provided to the system
at test time. 5 The total number of test mixtures for each subject was thus 195.
3 see
http://amp.ece.cmu.edu/projects/ Audio VisualSpeechProcessing/
4Sine windows were used in analysis and synthesis such that their product forms windows that sum to unity when overlapped 50%. The windowed frames were analyzed using
a 264-point fast Fourier transform (FFT) . The phases of the resulting spectra were discarded.
5Estimation of the SNR is necessary in practice; however this subject has been treated
The separated test sounds were estimated by the system under two conditions: with
and without the use of video information.
We evaluated the estimates on the test set using a speech recognition system developed by Bhiksha Raj, using the eMU Sphinx ASR engine. 6 Existing speech
HMMs trained on 60 hours of broadcast news data were used for recognition. 7 The
models were adapted in an unsupervised manner to clean speech from each speaker,
by learning a single affine transformation of all the state means, using a maximum
likelihood linear regression procedure [14]. The recognizer adapted to each speaker
was tested with the enhanced speech produced by the speech model for that speaker,
as well as with no enhancement.
Results are shown in figure 4(b). Recognition was greatly facilitated by the enhancement, with additional gains resulting from the use of video. It is somewhat
surprising that the gains for video occur mostly in areas of higher SNR, whereas in
human speech perception they occur under lower SNR. Little subjective difference
was noted with the use of video in the case of two speakers. However in other
experiments, when both voices came from the same speaker, the video was crucial
in disambiguating which signal came from which voice.
"one"
"two"
Originals
Mixture
Separated
SNR dB
(a) signal separation spectrograms
(b) automatic speech recognition
Figure 4: spectrograms of separated speech signals for a mixture two words spoken
by the same speaker (a), and speech recognition performance for 39 mixtures of two
words spoken by different speakers (b)
6
Discussion
We have presented promising techniques for audio-visual speech enhancement. We
introduced a factorial HMM to track both formant and pitch information, as well
as video, in a unified probabilistic model, and demonstrated its effectiveness in
speech enhancement. We are not aware of any other HMM-based audio-visual
elsewhere [6] and is beyond the scope of this paper.
6 see http://www.speech.cs.cmu.edu/sphinxj.
7These models represented every combination of three phones (triphones) using 6000
states tied across trip hone models, with a 16-element Gaussian mixture observation model
for each state. The data were processed at 8 kHz in 25ms windows overlapped by 15ms,
with a frame rate of 100 frames per second, and analyzed into 31 Mel frequency components
from which 13 cepstral coefficients were derived. These coefficients with the mean vector
removed, and supplemented with their time differences, comprised the observed features
speech enhancement systems in the literature. The results are tentative given the
small sample of voices used ; however they suggest that further study with a larger
sample of voices is warranted. It would be useful to compare the performance of
a factorial speech model to that of each factor in isolation, as well as to a fullspectrum model. Measures of quality and intelligibility by human listeners in terms
of speech and emotion recognition , as well as speaker identity, will also be helpful
in further demonstrating the utility of these techniques. We look forward to further
development of these techniques in future research.
Acknowledgments
We wish to thank Mitsubishi Electric Research Labs for hosting this research. Special thanks to Bhiksha Raj for devising and producing the evaluation using speech
recognition , and to Matt Brand for his entropic HMM toolkit.
References
[1] W. H. Sumby and I. Pollack. Visual contribution to speech intelligibility in noise.
Journal of th e Acoustical Society of America, 26:212- 215, 1954.
[2] Jordi Robert-Ribes, Jean-Luc Schwartz, Tahar Lallouache, and Pierre Escudier. Complementarity and synergy in bimodal speech. Journ el of the Acoustical Society of
America, 103(6):3677- 3689, 1998.
[3] Stepmane Dupont and Juergen Luettin. Audio-visual speech modeling for continuous
speech recognition. IEEE transactions on Multimedia, 2(3):141- 151, 2000.
[4] John W. Fisher, Trevor Darrell , William T. Freeman, and Paul Viola. Learning joint
statistical models for audio-visual fusion and segregation. In Advances in Neural
Information Processing Systems 13. 200l.
[5] Laurent Girin , Jean-Luc Schwartz, and Gang Feng. Audio-visual enhancement of
speech in noise. Journ el of the Acoustical Society of America, 109(6):3007- 3019,
200l.
[6] Yariv Ephraim. Statistical-model based speech enhancement systems. Proceedings of
th e IEEE, 80(10):1526- 1554, 1992.
[7] Z. Ghahramani and M. Jordan. Factorial hidden markov models. In David S. Touretzky, Michael C. Mozer , and M.E. Hasselmo, editors, Advances in Neural Information
Processing Systems 8, 1996.
[8] Sam T. Roweis. One microphone source separation. In Advances in Neural Information Processing Systems 13. 200l.
[9] Hagai Attias, John C. Platt , Alex Acero, and Li Deng. Speech denoising and dereverb eration using probabilistic models. In Advances in Neural Information Processing
Systems 13. 200l.
[10] M. J. F . Gales. Mod el-Bas ed Techniques for Noise Robust Speech R ecognition. PhD
thesis, Cambridge University, 1996.
[11] F . R. Kschischang, B. Frey, and H .-A. Loeliger. Factor graphs and the sum-product
algorithm. IEEE Trans. Inform. Theory, 47(2):498- 519, 200l.
[12] F. J. Huang and T. Chen. Real-time lip-synch face animation driven by human voice.
In IEEE Wo rkshop on Multimedia Signal Processing, Los Angeles, California, Dec
1998.
[13] Matt Brand. Structure learning in conditional probability models via an entropic
prior and parameter extinction. Neural Computation, 11(5):1155- 1182, 1999.
[14] C. J. Leggetter and P. C. Woodland. Maximum likelihood linear regression for speaker
adaptation of the parameters of continuous density hidden markov models. Computer
Speech and Language, 9: 171- 185, 1995.
| 2005 |@word middle:1 manageable:1 extinction:2 open:1 mitsubishi:2 covariance:4 carry:1 series:1 contains:1 loeliger:1 envision:1 amp:1 subjective:1 existing:1 current:3 com:1 anne:1 surprising:1 must:1 john:3 subsequent:1 subcomponent:1 dupont:1 discrimination:1 stationary:2 cue:7 half:1 selected:1 devising:1 plane:1 provides:2 monday:1 five:1 height:2 mathematical:1 along:1 windowed:1 become:1 combine:2 manner:2 introduce:1 expected:3 multi:1 formants:2 audiovisual:1 freeman:1 little:2 window:3 provided:1 estimating:1 project:1 what:1 pursue:1 developed:3 spoken:3 unified:1 transformation:1 every:2 schwartz:2 platt:1 appear:2 producing:1 frey:1 laurent:1 plus:1 twice:1 suggests:2 hmms:12 ease:1 wideband:1 range:1 acknowledgment:1 yariv:1 practice:3 procedure:4 area:2 composite:10 word:12 distracter:1 suggest:1 clever:1 acero:1 context:1 www:1 outweigh:1 map:1 demonstrated:2 baum:1 uttered:1 independently:1 welch:1 simplicity:1 estimator:1 rule:1 his:1 variation:1 diego:1 enhanced:3 play:2 target:3 trick:1 element:2 overlapped:2 recognition:13 expensive:1 approximated:1 complementarity:1 database:2 bottom:2 role:1 recog:1 observed:1 calculate:1 news:1 removed:1 ephraim:2 mozer:1 predictable:1 complexity:3 dynamic:6 trained:4 basis:1 binaural:1 easily:1 joint:1 slate:1 represented:4 listener:1 america:3 train:2 separated:5 snrs:1 fast:1 cogsci:2 tell:1 whose:3 modular:1 larger:1 jean:2 formant:7 transform:1 noisy:3 itself:1 sequence:5 advantage:1 propose:1 product:3 adaptation:1 neighboring:1 combining:3 mixing:1 roweis:1 los:1 enhancement:16 regularity:1 darrell:1 produce:1 converges:1 derive:1 develop:1 auxiliary:1 c:1 quantify:1 differ:1 closely:1 filter:5 human:4 enable:1 bin:1 preliminary:2 hagai:1 tracker:1 normal:3 equilibrium:1 scope:1 vary:1 early:1 entropic:3 resample:1 recognizer:2 estimation:3 combinatorial:1 vibration:1 hasselmo:1 clearly:2 gaussian:5 rather:2 avoid:1 sparsify:1 publication:1 derived:4 focus:2 casey:2 improvement:2 likelihood:4 mainly:1 greatly:1 contrast:1 tremendously:1 glass:1 helpful:2 inference:6 dependent:2 el:3 typically:4 integrated:1 hidden:4 journ:2 among:1 dual:1 development:2 spatial:1 integration:2 initialize:1 mutual:1 special:1 emotion:2 construct:2 asr:4 comprise:1 having:1 aware:1 look:1 unsupervised:1 foreground:1 future:1 serious:1 employ:2 opening:1 randomly:1 resulted:1 parsimoniously:1 phase:8 vowel:2 william:1 evaluation:2 male:1 mixture:15 analyzed:3 yielding:2 amenable:1 accurate:1 ambient:1 fu:1 explosion:1 necessary:2 filled:1 loosely:3 re:2 desired:1 obscured:2 isolated:1 pollack:1 merl:1 lwe:1 modeling:3 instance:2 juergen:1 cost:1 snr:5 comprised:1 wonder:1 dependency:1 combined:5 person:1 density:4 thanks:1 probabilistic:2 michael:2 enhance:2 quickly:1 together:1 synthesis:1 thesis:1 ear:1 containing:1 huang:2 broadcast:1 gale:1 cognitive:1 corner:3 sphinx:1 derivative:1 li:1 summarized:1 includes:1 coefficient:2 sine:1 lab:2 closed:1 defer:1 contribution:1 wiener:1 variance:11 largely:2 phoneme:1 characteristic:1 yield:1 ofthe:1 fhmm:4 iterated:1 produced:2 marginally:1 simultaneous:1 inform:1 touretzky:1 trevor:1 ed:1 frequency:5 conveys:1 associated:1 di:1 jordi:1 gain:2 auditory:1 revise:1 conversation:1 knowledge:2 amplitude:4 reflecting:2 higher:1 follow:1 hershey:1 fhmms:2 improved:1 specify:1 formulation:1 evaluated:1 until:1 ally:1 night:1 overlapping:2 propagation:1 bly:1 quality:2 escudier:1 stale:1 bhiksha:2 matt:2 consisted:1 hence:1 read:1 conditionally:1 numerator:1 width:2 during:3 speaker:17 noted:1 mel:1 m:2 impression:1 delivers:1 variational:2 novel:1 recently:1 khz:3 million:1 extend:1 mellon:1 cambridge:2 framed:1 automatic:4 similarly:1 language:1 had:1 toolkit:1 longer:1 etc:1 add:3 subjectively:1 triphones:1 female:1 raj:2 moderate:1 irrelevant:1 phone:1 scenario:1 phonetic:1 driven:1 came:2 nition:1 seen:1 additional:2 somewhat:1 spectrogram:8 employed:1 deng:1 converge:1 maximize:1 signal:30 full:4 sound:8 infer:1 smooth:1 match:3 adapt:1 offer:1 pitch:11 regression:2 denominator:1 vision:5 enhancing:1 cmu:2 iteration:1 represent:2 bimodal:1 luettin:1 dec:1 background:4 remarkably:1 addition:1 whereas:2 separately:1 source:2 crucial:2 recording:2 tend:2 subject:5 db:2 mod:1 effectiveness:1 jordan:1 near:1 enough:1 fft:1 variety:1 isolation:1 architecture:1 reduce:2 prototype:1 attias:1 angeles:1 pca:1 utility:2 granted:1 effort:1 wo:1 suffer:1 speech:74 cocktail:1 jie:1 useful:3 woodland:1 detailed:1 factorial:14 hosting:1 transforms:2 nonparametric:1 revived:1 amount:1 band:35 processed:1 simplest:1 http:2 estimated:3 per:3 track:1 carnegie:1 demonstrating:1 k4:2 neither:1 clean:5 backward:5 graph:1 sum:11 facilitated:2 uncertainty:2 saying:1 submodels:1 separation:6 bit:1 entirely:1 layer:1 yielded:1 adapted:2 occur:2 gang:1 handful:1 constrain:1 alex:1 cepstra:2 sake:1 interpolated:1 fourier:1 extremely:1 department:1 structured:1 according:1 combination:6 beneficial:1 across:3 slightly:1 sam:1 unity:1 explained:1 segregation:1 mechanism:1 apply:1 intelligibility:3 spectral:6 pierre:1 voice:10 original:2 top:1 remaining:1 graphical:1 music:1 exploit:1 ghahramani:1 february:1 society:3 contact:1 feng:1 added:2 question:1 strategy:2 diagonal:3 ecognition:1 separate:1 thank:1 separating:3 hmm:18 outer:1 chris:1 acoustical:4 reason:1 length:1 ratio:1 guest:1 difficult:1 mostly:1 robert:1 holding:1 reverberation:1 ba:1 unknown:1 upper:3 observation:29 markov:4 discarded:3 hone:1 viola:1 incorporated:2 frame:15 ucsd:2 inferred:1 introduced:1 david:1 pair:2 required:1 trip:1 coalesce:1 tentative:1 california:2 acoustic:2 learned:3 narrow:19 engine:1 emu:1 hour:1 trans:1 beyond:1 below:1 perception:2 pattern:2 articulation:1 hear:1 challenge:1 including:2 video:20 belief:1 mouth:1 overlap:2 eration:1 treated:1 recursion:2 clink:1 scheme:1 jhershey:2 utterance:2 coupled:2 prior:5 literature:1 relative:2 mixed:1 interesting:1 filtering:2 utter:1 affine:1 sufficient:1 imposes:1 editor:1 course:1 elsewhere:1 allow:1 perceptron:1 wide:20 face:3 cepstral:4 lognormal:3 factorially:3 taking:2 unmanageable:1 benefit:3 distributed:1 dimension:1 vocabulary:2 transition:2 sumby:1 contour:1 forward:6 commonly:2 made:1 san:1 party:1 employing:1 transaction:1 approximate:3 ignore:1 synergy:1 sequentially:1 consonant:1 spectrum:10 continuous:2 lip:10 promising:2 robust:1 kschischang:1 obtaining:1 warranted:1 complex:3 electric:2 domain:6 synch:1 noise:19 massachussets:1 paul:1 animation:1 repeated:1 complementary:1 depicts:4 sub:10 wish:1 tied:1 late:1 removing:1 formula:1 showing:1 supplemented:1 disregarding:1 fusion:1 burden:1 incorporating:1 naively:1 consist:1 adding:2 phd:1 illustrates:1 chen:1 easier:1 simply:1 likely:3 visual:25 extracted:2 conditional:1 modulate:1 identity:2 formulated:1 room:1 disambiguating:1 luc:2 fisher:1 included:1 uniformly:2 denoising:1 microphone:2 total:1 multimedia:2 pas:2 ece:1 brand:2 searched:1 prosody:1 incorporate:2 evaluate:2 audio:28 tested:1 |
1,103 | 2,006 | .
Information-geometric decomposition In
spike analysis
Hiroyuki Nakahara; Shun-ichi Amari
Lab. for Mathematical Neuroscience, RIKEN Brain Science Institute
2-1 Hirosawa, Wako, Saitama, 351-0198 Japan
{him, amari} @brain.riken.go.jp
Abstract
We present an information-geometric measure to systematically
investigate neuronal firing patterns, taking account not only of
the second-order but also of higher-order interactions. We begin
with the case of two neurons for illustration and show how to test
whether or not any pairwise correlation in one period is significantly
different from that in the other period. In order to test such a hypothesis of different firing rates, the correlation term needs to be
singled out 'orthogonally' to the firing rates, where the null hypothesis might not be of independent firing. This method is also shown
to directly associate neural firing with behavior via their mutual
information, which is decomposed into two types of information,
conveyed by mean firing rate and coincident firing, respectively.
Then, we show that these results, using the 'orthogonal' decomposition, are naturally extended to the case of three neurons and n
neurons in general.
1
Introduction
Based on the theory of hierarchical structure and related invariant decomposition
of interactions by information geometry [3], the present paper briefly summarizes
methods useful for systematically analyzing a population of neural firing [9].
Many researches have shown that the mean firing rate of a single neuron may carry
significant information on sensory and motion signals. Information conveyed by
populational firing, however, may not be only an accumulation of mean firing rates.
Other statistical structure, e.g., coincident firing [13, 14], may also carry behavioral
information. One obvious step to investigate this issue is to single out a contribution
by coincident firing between two neurons, i.e., the pairwise correlation [2, 6].
In general, however, it is not sufficient to test a pairwise correlation of neural firing,
because there can be triplewise and higher correlations. For example, three variables
(neurons) are not independent in general even when they are pairwise independent.
We need to establish a systematic method of analysis, including these higher-order
? also affiliated with Dept. of Knowledge Sci., Japan Advanced Inst. of Sci. & Tech.
correlations [1, 5,7, 13] . We propose one approach, the information-geometric measure that uses the dual orthogonality of the natural and expectation parameters in
exponential family distributions [4]. We represent a neural firing pattern by a binary
random vector x. The probability distribution of firing patterns can be expanded
by a log linear model, where the set {p( x)} of all the probability distributions forms
a (2n - I)-dimensional manifold 8 n . Each p(x) is given by 2n probabilities
pi1???in=Prob{X1=i1,???,Xn=in}, ik=O,I, subjectto
L
Pi1???in=1
il ,"',i n
and expansion in log p( x) is given by
logp(x) =
L BiXi + L BijXiXj + L
i<j
BijkXiXjXk???
+ B1... nX1 ... Xn
- 'Ij;,
i<j<k
where indices of Bijk, etc. satisfy i < j < k, etc. We can have a general theory of
this n neuron case [3, 9], however , to be concrete given the limited space, we mainly
discuss two and three neuron cases in the present paper. Our method shares some
features with previous studies (e.g. [7]) in use of the log linear model. Yet, we make
explicit use of the dual orthogonality so that the method becomes more transparent
and more systematic.
In the present paper, we are interested in addressing two issues: (1) to analyze correlated firing of neurons and (2) to connect such a technique with behavioral events.
In (1), previous studies often assumed independent firing as the null hypothesis.
However, for example, when we compare firing patterns in two periods, as control
and 'test' periods, there may exist a weak correlation in the control period. Hence,
benefiting from the 'orthogonal' coordinates, we develop a method applicable to the
null hypothesis of non-independent firing, irrespective of firing rates. It is equally
important to relate such a method with investigation of behavioral significance as
(2). We show that we can do so, using orthogonal decomposition of the mutual
information (MI) between firing and behavior [11, 12].
In the following , we discuss first the case of two neurons and then the case of three
neurons , demonstrating our method with artificial simulated data. The validity of
our method has been shown also with experimental data[9, 10] but not shown here
due to the limited space.
2
Information-geometric measure: case of two neurons
?
We denote two neurons by Xl and X 2 (Xi = 1, indicates if neuron i has a spike or
not in a short time bin). Its joint probability p(x), x = (X1,X2), is given by Pij =
Prob{x1 = i;X2 = j} > 0, i,j = 0, 1. Among four probabilities, {POO ,P01,P10,Pl1},
only three are free. The set of all such distributions of x forms a three-dimensional
manifold 8 2. Any three of Pij can be used as a coordinate system of 8 2.
There are many different coordinate systems of 8 2 . The coordinates of the expectation parameters, called 17-coordinates, 'TI = (171,172,1712), is given by
17i = Prob {Xi = I} = E [Xi], i = 1,2, 173 = 1712 = E [X1 X2] = P12,
where E denotes the expectation and 17i and 1712 correspond to the mean firing rates
and the mean coincident firing, respectively.
As other coordinate systems, we can also use the triplet, (171,172, Cov [Xl, X 2]) , where
Cov [Xl , X 2] is the covariance,and/or the triplet (171,172, p), where p is the correlation
coefficient (COR), p = J '112 -,/11 '12
, often called N-JPSTH [2].
'/11 (l -
'7d'72 (1 - '72)
Which quantity would be convenient to represent the pairwise correlational component? It is desirable to define the degree of the correlation independently from
the marginals (171,172), To this end, we use the 'orthogonal' coordinates (171 , 172 , B),
originating from information geometry of 8 2 , so that the coordinate curve of B is
always orthogonal to those of 171 and 172.
The orthogonality of two directions in 8 2 (8 n in general) is defined by the Riemannian metric due to the Fisher information matrix [8, 4]. Denoting any coordinates
in 8 n by ~ = (6, ... , ~n)' the Fisher information matrix G is given by
(1)
where l
9ij(~)
(x;~) =
logp (x; ~). The orthogonality between
~i
and
= O. In case of 8 2 , we desire to have E [tel (X ;171 , 172, B)
~j
is defined by
8~il(x;171'172,B)]
=
o
(i = 1, 2). When B is orthogonal to (171, 172), we say that B represents pure
correlations independently of marginals. Such B is given by the following theorem.
Theorem 1.
The coordinate
B = log PuPoo
P01PlO
is orthogonal to the marginals 171 and 172 .
(2)
We have another interpretation of B. Let's expand p(x) by logp(x) = L;=l BiXi +
B12X1X2 - 'IjJ. Simple calculation lets us get the coefficients, B1 = log Pia,
B2 =
paa
log EQl,
'
I
jJ
=
-logpoo,
and
B
=
B12
(as
Eq
2).
The
triplet
()
=
(B1'
B2,
B
)
forms
12
paa
another coordinate system, called the natural parameters, or B-coordinates. We
remark that B12 is 0 when and only when Xl and X 2 are independent.
The triplet
C== (171,172,B12 )
forms an 'orthogonal' coordinate system of 8 2 , called the mixed coordinates [4].
We use the Kullback-Leibler divergence (KL) to measure the discrepancy between
two probabilities p(x) and q(x) , defined by D[p:q] = LxP(x)log~t~}. In the
following , we denote any coordinates of p by
etc (the same for q). Using the
orthogonality between 17- and B-coordinates, we have the decomposition in the KL.
e
Theorem 2.
D [q : p] = D [q : r**] + D [r** : p] ,
+ D [r* : q],
(3)
are given by Cr = (17f, 17~, Bj) and Cr = (17f, 17g, B~), respectively.
D [p : q] = D [p : r*]
where r* and r**
>
?
The squared distance ds 2 between two nearby distributions p(x , ~) and p(x,~, +d~)
is given by the quadratic form of d~,
L
ds 2 =
9ij(~)d~id~j,
i,jE(1,2,3)
which is approximately twice the KL, i.e. , ds 2 ~ 2D [P(x , ~) : p(x,~
Now suppose
~
is the mixed coordinates
is of the form gfj
=[
gll g~2
gl2
gf2
o
0
0
0
g~3
C.
+ ~)].
Then, the Fisher information matrix
1and we have ds 2
= dsi
g~3(dB3) 2, ds~ = Li,j E(1,2) 9fjd17id17j, corresponding to Eq. 3.
+ ds~,
where dsi =
This decomposition comes from the choice of the orthogonal coordinates and gives
us the merits of simple procedure in statistical inference. First, let us estimate
the parameter TI = (1}1,1}2) and B from N observed data Xl, ... , XN. The maximum likelihood estimator (mle) ( , which is asymptotically unbiased and efficient,
is easily obtained by 1)' . = l..#{x? = I} and 8 = log fh?(1-=-fh-.ib+~12) using
?
N
(1]1-1]12)(1]2-1]12) ,
?
fj12 = tt#{XIX2 = I}. The covariance of estimation error, f::J.TI and f::J.B, is given
asymptotically by Cov [
~~
] = ttGZ1. Since the cross terms of G or G- 1 vanish
for the orthogonal coordinates, we have Cov [f::J.TI, f::J.B] = 0, implying that the estimation error f::J.TI of marginals and that of interaction are mutually independent.
Such a property does not hold for other non-orthogonal parameterization such as
the COR p, the covariance etc. Second, in practice, we often like to compare many
spike distributions, q(x(t)) (i.e, (q(t)) for (t = 1", T), with a distribution in the
control period p( x) , or (P. Because the orthogonality between TI and B allows us to
treat them independently, these comparisons become very simple.
These properties bring a simple procedure of testing hypothesis concerning the null
hypothesis
against
(4)
Ho : B = Bo
where Bo is not necessarily zero, whereas Bo = 0 corresponds to the null hypothesis
of independent firing , which is often used in literature in different setting. Let the
log likelihood of the models Ho and HI be, respectively,
lo
= maxlogp(Xl ' ... , XN ; TI , Bo)
TI
and h
= maxlogp(Xl' ... , XN; TI, B).
TI,e
The likelihood ratio test uses the test statistics A = 2log ~. By the mle with respect
to TI and which can be performed independently, we have
e,
(5)
lo = logp(x ,r"B o),
where r, are the same in both models. A similar situation holds in the case of testing
TI = Tlo against TI =I Tlo for unknown B.
Under the hypothesis H o, A is approximated for a large N as
A= 2
t
log
i=l
P(Xi;~' B~)
';::;j
N gi3 (8 - BO)2 '" X2(1).
(6)
p(Xi; TI, B)
Thus, we can easily submit our data to a hypothetical testing of significant coincident firing against null hypothesis of any correlated firing, independently from the
mean firing rate modulation 1 .
We now turn to relate the above approach with another important issue, which is
to relate such a coincident firing with behavior. Let us denote by Y a variable of
discrete behavioral choices. The MI between X = (X1 ,X2 ) and Y is written by
J(X, Y) =
Ep(x ,y)
p(x , y)]
[ log p(x)p(y)
=
Ep(Y)
[D [P(Xly) : p(X)]].
Using the mixed coordinates for p(Xly) and p(X) , we have D [P(Xly) : p(X)]
D [?(Xly) : ?(X)] = D [?(Xly) : ('J + D [(I : ?(X)J, where (' = ('(X,y)
((1 (Xly), (2 (X Iy) , (3 (X)) = (1}1 (Xly), 1}2(Xly), B3(X)).
1 A more proper formulation in this hypothetical testing can be derived, resulting in
using p value from X2 (2) distribution , but we omit it here due to the limited space [9]
Theorem 3.
+ h(X, Y) ,
J(X, Y) = It (X , Y)
where It (X , Y), h(X, Y) are given by
It (X, Y) =
Ep(Y)
[D [?(Xly) : ('(X,y)]] ,h(X, Y) =
(7)
Ep(Y)
[D [('(X,y) : ?(X)]] .
Obviously, the similar result holds with respect to p(YIX). By this theorem, J is the
sum of the two terms: It is by modulation of the correlation components of X, while
h is by modulation of the marginals of X. This observation helps us investigate the
behavioral significance by modulating either coincident firing or mean firing rates.
0.1
0 . 1 ,-----~-~--~-~-____,
~ A (al /
'J,
~ 0.05~ .?.??.? __ I~)? ? _ ~, .? .? .?.? .?.? .? ~C)? ? .? .? .? .? .?
u;
o
a
00
- ........................ _12 .
..
-
100
300
500
100
300
lime Ims)
500
,-------~--~-~-____,
B
\ll
. , .. , .. . , . .. , .. ". , .. ' .. , .. " . . .. ' .. , . ... . . , .. . . , .. , _ .# ' ...... .. , .. . . ' .. -" .. . , ?? ,
!-
~0.05
W :
112
~
_________
'J ,
~
.. __ '_12 '- - - .-- - --- -"" -_ ..
?0:------=:-:
10:::0 ~~~~3=00=---~---:-:
500
00
100
"300
lime Ims)
500
Figure 1: Demonstration of information-geometric measure in two neuron case,
using simulated neural data, where two behavioral choices (sl, s2) are assumed.
A,B. (1]1 , 1]2 , 1]12) with respect to sl, s2. C,D . COR,B, computed by using ",
L-iP(Si)",(Si) with P(Si) = 1/2 (i = 1, 2). E. p-values. F. MI.
Fig 1 succinctly demonstrates results in this section. Figs 1 A, B are supposed to
show mean firing rates of two neurons and mean coincident firing for two different
stimuli (sl, s2). The period (a) is assumed as the control period, i.e. , where no
stimuli is shown yet, whereas the stimulus is shown in the periods (b,c). Fig 1 C,
D gives COR, B. They look to change similarly over periods, which is reasonable
because both COR and B represent the same correlational component, but indeed
change slightly differently over periods (e.g., the relative magnitudes between the
periods (a) and (c) are different for COR and B) , which is also reasonable because
both represent the correlational component as in different coordinate systems. Using
B in Fig 1 D, Fig 1 E shows p-values derived from X2 (1) (i.e., P > 0.95 in Fig 1 E is
'a significance with P < 0.05') for two different null hypotheses , one of the averaged
firing in the control period (by solid line) and the other of independent firing (by
dashed line) , which is of popular use in literature.
In general, it becomes complicated to test the former hypothesis , using COR. This
is because the COR, as the coordinate component, is not orthogonal to the mean
firing rates so that estimation errors among the COR and mean firing rates are
entangled and that the proper metric among them is rather difficult to compute.
Once using B, this testing becomes simple due to orthogonality between B and mean
firing rates.
Notably, we would draw completely different conclusions on significant coincident
firing given each null hypothesis in Fig 1 E. This difference may be striking when we
are to understand the brain function with these kinds of data. Fig 1 F shows the MI
between firing and behavior, where behavioral event is with respect to stimuli, and
its decomposition. There is no behavioral information conveyed by the modulation
of coincident firing in the period (b) (i.e., h = 0 in the period (b)). The increase
in the total MI (i.e., I) in the period (c), compared with the period (b), is due not
to the MI in mean firing (h) but to the MI correlation (h). Thus, with a great
ease, we can directly inspect a function of neural correlation component in relation
to behavior.
3
Three neuron case
With more than two neurons, we need to look not only into a pairwise interaction
but also into higher-order interactions. Our results in the two neuron case are
naturally extended to n neuron case and here, we focus on three neuron case for
illustration.
For three neurons X = (X 1,X2,X3), we let p(x), x = (X1,X2,X3), be their joint
probability distribution and put Pijk = Prob {Xl = i, X2 = j, X3 = k}, i, j, k = 0,1.
The set of all such distributions forms a 7-dimensional manifold 8 3 due to "L.Pijk =
1. The 1]-coordinates 'fI = ('fI1; 'fI2; 'fI3) = (1]1,1]2,1]3; 1]12,1]23,1]13; 1]123) is defined by
(i, j = 1,2, 3; i i- j), 1]123 = E [X1X2X3].
To single out the purely triplewise correlation, we utilize the dual orthogonality of 8- and 1]-coordinates. By using expansion of log p( x) = "L. 8iXi +
"L.8ijXiXj + 8123X1X2X3 - 'ljJ, we obtain 8-coordinates, () = (()1;()2;()3) =
(8 1,82,83; 812 ,823 ,8 13 ; 8123 ). It's easy to get the expression of these coefficients
(e.g. ,123
8
= log P110PIOIP0l1POOO
P111 PIOO POIOP001). Information geometry gives the following theorem.
1]i
= E [Xi]
(i
= 1,2,3),
1Jij
= E [XiXj]
Theorem 4.
8123 represents the pure triplewise interaction in the sense that
it is orthogonal to any changes in the single and pairwise marginals, i.e., 'fIl and 'fI2.
We use the following two mixed coordinates to utilize the dual orthogonality,
(I = ('fIl; ()2; ()3), (2 = ('fIl; 'fI2; ()3).
Here (2 is useful to single out the triple wise interaction (()3 = 8123 ), while (I is to
single out the pairwise and triplewise interactions together (()2; ()3). Note that 8123 is
not orthogonal to {8 ij }. In other words , except the case of no triple wise interaction
(8 123 = 0), 8ij do not directly represent the pairwise correlation of two random
variables Xi, X j . The case of independent firing is given by 1]ij = 1]i1]j, 1]123 = 1]11]21]3
or equivalently by ()2 = 0, ()3 = o.
The decomposition in the KL is now given as follows.
Theorem 5.
D [p : q]
= D [p : p] + D [p : q] = D [p : fi] + D [p : q] = D [p : p] + D [p : fi] + D [p : q] .
where, using the mixed coordinates, we have
(g =
(8)
('fIi; 'fI~; ()?), (f = ('fIi; ()~; ()?).
A hypothetical testing is formulated similarly to the two neuron case. We can examine a significance of the triplewise interaction by A2 = 2ND [p : p] ~ N g~7 (~) (8f238i23)2 ~ X2(1). For a significance of triplewise and pairwise interactions together,
we have Al = 2ND [p : fi] ~ N "L.J,j=4 gfj(f)((f - (f)((f - (f) ~ X2(4).
For the decomposition of the MI between firing X and behavior Y, we have
Theorem 6.
J(X, Y) = h (X, Y)
+ h(X, Y)
= h(X, Y)
+ J4(X, Y)
(9)
where
h(X, Y) =
Ep(Y)
[D [( I(X ly ) : ( I(X,y)] ] , h(X, Y) =
Ep(Y)
[D [(I (X,y) : ( I(X) ]] ,
h(X, Y) =
Ep(Y)
[D [(2(X ly ) : ( 2(X,y)] ] ,
Ep(Y)
[D [( 2(X,y) : ( 2(X) ]],
14 (X,
Y) =
By t he first equality, I is decomposed into two parts: II is conveyed by the pairwise
and triplewise interactions of firing, and h by the mean firing rate modulation.
By the second equality, I is decomposed differently: h, conveyed by t he triplewise
interaction, and 14 , by the other terms.
00
(e)
(a)
100
r:- ~ : -
8~~I----'
[:
0.04
'"0.02
(d)
300
500
700
-0. 10
300
500
700
0
100
300
500
700
..
~ ~~j ~.~~
it~,-~
I?
?
-2
0
100
1- _____ _
N205 E
x
- - -
i: -. -. --, --) ----
." ,t: t' .....~I/"'\:""''! ~I
~....,:..;..".u,,'rV! '/ J.~'"
~
t.
I
oo'-'--"1""
oo:"-'-'-'-'-"'-'----::3~
00c--~---'c5-:-:00-~---=1l700
time (ms)
95
100
300
500
700
N~05011-F- - - - - - - - - - - .':,f?- - - - ? - - - - - - - - .- - - -
95
......
0
100
300
time (ms)
500
700
Figure 2: Demonstration in three neuron case. A '11 = ('111> '112, '113) ~ ('T/i , 'T/ij,'T/ijk)
from top to bottom, since we treated a homogeneous case in this simulation for
simplicity. B. COR. C. (}12,(}13 , (}23' D (}12 3 . E p-value,...., X2 (1). F p-value,...., X2(4).
We emphasize that all the above decompositions come from the choice of the 'ort hogonal' coordinates. Fig 2 highlights some of the results in this section. Fig 2 A
shows t he mean firing rates (see legend). The period (a) is assumed as t he control
period. Fig 2 B indicates t hat COR changes only in the periods (c,d), while Fig 2
C indicates that (}123 changes only in t he period (d). Taken together, we observe
that t he triplewise correlation (}123 can be modulated independently from COR. Fig
2 E indicates the p-value from X2(1) against the null hypothesis of the activity in
t he control period. The triplewise coincident firing becomes significant only in the
period (d). Fig 2 F indicates the p-value from X2(4) . The coincident firing, taking
t he triplewise and pairwise interaction together, becomes significant in both periods
(c,d). We cannot observe these differences in modulation of pairwise and triplewise
interactions over periods (c, d), when we inspect only COR.
Remark: For a general n neuron case, we can use the k-cut mixed coordinates,
(k = ('111 ' ... , '11 k; 0 k+l, .. . , On) = ('I1k- ; 0k+)' Using the orthogonality between 'I1kand 0 k+, the similar results hold. To meet the computational complexity involved
in this general case, some practical difficulties should be resolved in practice [9] .
4
Discussions
We presented the information-geometric measures to analyze spike firing patterns,
using two and three neuron cases for illustration. The choice of 'orthogonal' coordinates provides us with a simple, transparent and systematic procedure to test
significant firing patterns and to directly relate such a pattern with behavior. We
hope that this method simplifies and strengthens experimental data analysis.
Acknowledgments
HN thanks M. Tatsuno, K. Siu and K. Kobayashi for their assistance. HN is supported by Grants-in-Aid 13210154 from the Ministry of Edu. Japan.
References
[1] M. Abeles, H. Bergman, E. Margalit, and E. Vaadia. Spatiotemporal firing
patterns in the frontal cortex of behaving monkeys. J Neurophysiol, 70(4):162938.,1993.
[2] A. M. H. J. Aertsen, G. 1. Gerstein, M. K. Habib, and G. Palm. Dynamics of
neuronal firing correlation: Modulation of "effective connectivity". Journal of
Neurophysiology, 61(5):900- 917, May 1989.
[3] S. Amari. Information geometry on hierarchical decomposition of stochastic
interactions. IEEE Transaction on Information Theory, pages 1701- 1711,2001.
[4] S. Amari and H. Nagaoka. Methods of Information Geometry. AMS and Oxford
University Press, 2000.
[5] S. Griin. Unitary joint-events in multiple-neuron spiking activity: detection,
significance, and interpretation. Verlag Harri Deutsch, Reihe Physik, Band 60 .
Thun, Frankfurt/Main, 1996.
[6] H. Ito and S. Tsuji. Model dependence in quantification of spike interdependence by joint peri-stimulus time histogram. Neural Computation, 12:195- 217,
2000.
[7] L. Martignon, G. Deco, K. Laskey, M. Diamond, W. A. Freiwald, and E. Vaadia. Neural coding: Higher-order temporal patterns in the neurostatistics of
cell assemblies. Neural Computation, 12(11):2621- 2653, 2000.
[8] H. Nagaoka and S. Amari. Differential geometry of smooth families of probability distributions. Technical report , University of Tokyo, 1982.
[9] H. Nakahara and S. Amari. Information geometric measure for neural spikes.
in prepration.
[10] H. Nakahara, S. Amari, M. Tatsuno, S. Kang, K. Kobayashi , K. Anderson,
E. Miller, and T. Poggio. Information geometric measures for spike firing.
Society for Neuroscience Abstracts, 27:821.46 (page.2178), 2001.
[11] M. W . Oram, N. G. Hatsopoulos, B. J. Richmond, and J . P. Donoghue. Excess
synchrony in motor cortical neurons provides redundant direction information
with that from coarse temporal measures. J Neurophysiol., 86(4):1700- 1716,
2001.
[12] S. Panzeri and S. R. Schultz. A unified approach to the study of temporal,
correlational, and rate coding. Neural Computation, 13(6):1311-49., 2001a.
[13] A. Riehle, S. Griin, M. Diesmann, and A. Aertsen. Spike synchronization and
rate modulation differentially involved in motor cortical function. Science,
278:1950- 1953, 12 Dec 1997.
[14] E. Vaadia, I. Haalman, M. Abeles, H. Bergman, Y. Prut, H. Slovin, and
A. Aertsen. Dynamics of neuronal interactions in monkey cortex in relation to
behavioural events. Nature, 373:515- 518, 9 Feb 1995.
| 2006 |@word neurophysiology:1 briefly:1 nd:2 physik:1 simulation:1 decomposition:11 covariance:3 solid:1 carry:2 tlo:2 denoting:1 wako:1 si:3 yet:2 written:1 motor:2 implying:1 parameterization:1 short:1 coarse:1 provides:2 bixi:2 mathematical:1 become:1 differential:1 ik:1 behavioral:8 interdependence:1 pairwise:13 notably:1 indeed:1 behavior:7 examine:1 brain:3 decomposed:3 bijk:1 becomes:5 begin:1 null:9 kind:1 monkey:2 unified:1 temporal:3 hypothetical:3 ti:14 demonstrates:1 control:7 ly:2 omit:1 grant:1 kobayashi:2 treat:1 analyzing:1 id:1 oxford:1 meet:1 firing:54 modulation:8 approximately:1 might:1 twice:1 b12:3 gll:1 limited:3 ease:1 averaged:1 practical:1 acknowledgment:1 testing:6 practice:2 x3:3 procedure:3 significantly:1 convenient:1 word:1 get:2 cannot:1 put:1 accumulation:1 poo:1 go:1 independently:6 simplicity:1 pure:2 freiwald:1 estimator:1 population:1 coordinate:30 suppose:1 homogeneous:1 us:2 ixi:1 hypothesis:13 bergman:2 associate:1 approximated:1 strengthens:1 cut:1 observed:1 ep:8 bottom:1 db3:1 hatsopoulos:1 complexity:1 dynamic:2 purely:1 completely:1 neurophysiol:2 easily:2 joint:4 resolved:1 differently:2 harri:1 riken:2 effective:1 artificial:1 griin:2 fi3:1 say:1 amari:7 cov:4 statistic:1 nagaoka:2 singled:1 ip:1 obviously:1 vaadia:3 propose:1 interaction:17 jij:1 benefiting:1 supposed:1 differentially:1 help:1 oo:2 develop:1 ij:7 eq:2 come:2 xly:9 deutsch:1 direction:2 tokyo:1 stochastic:1 shun:1 bin:1 transparent:2 investigation:1 hold:4 fil:3 great:1 panzeri:1 bj:1 a2:1 fh:2 estimation:3 applicable:1 him:1 modulating:1 hope:1 always:1 yix:1 rather:1 cr:2 derived:2 focus:1 indicates:5 mainly:1 likelihood:3 tech:1 richmond:1 sense:1 am:1 inst:1 inference:1 paa:2 margalit:1 relation:2 originating:1 expand:1 i1:2 interested:1 issue:3 dual:4 among:3 mutual:2 once:1 represents:2 look:2 discrepancy:1 report:1 stimulus:5 divergence:1 geometry:6 detection:1 investigate:3 poggio:1 orthogonal:15 logp:4 pia:1 addressing:1 saitama:1 siu:1 connect:1 spatiotemporal:1 abele:2 thanks:1 peri:1 systematic:3 together:4 iy:1 concrete:1 hirosawa:1 connectivity:1 squared:1 deco:1 hn:2 li:1 japan:3 account:1 b2:2 coding:2 coefficient:3 satisfy:1 performed:1 lab:1 analyze:2 complicated:1 synchrony:1 contribution:1 il:2 prut:1 miller:1 correspond:1 weak:1 against:4 martignon:1 involved:2 obvious:1 naturally:2 mi:8 riemannian:1 oram:1 popular:1 knowledge:1 hiroyuki:1 higher:5 formulation:1 anderson:1 correlation:17 d:6 laskey:1 b3:1 validity:1 unbiased:1 former:1 hence:1 equality:2 leibler:1 riehle:1 haalman:1 ll:1 assistance:1 m:2 tt:1 motion:1 p12:1 bring:1 wise:2 fi:5 spiking:1 jp:1 interpretation:2 he:8 marginals:6 ims:2 significant:6 frankfurt:1 similarly:2 j4:1 cortex:2 behaving:1 etc:4 ort:1 feb:1 fii:2 pl1:1 verlag:1 binary:1 p10:1 ministry:1 period:25 redundant:1 signal:1 dashed:1 ii:1 multiple:1 desirable:1 rv:1 smooth:1 technical:1 calculation:1 cross:1 concerning:1 equally:1 mle:2 expectation:3 metric:2 histogram:1 represent:5 cell:1 dec:1 whereas:2 entangled:1 p01:1 legend:1 unitary:1 easy:1 simplifies:1 donoghue:1 whether:1 expression:1 jj:1 remark:2 useful:2 band:1 sl:3 exist:1 neuroscience:2 discrete:1 ichi:1 four:1 demonstrating:1 utilize:2 asymptotically:2 sum:1 prob:4 striking:1 family:2 reasonable:2 draw:1 gerstein:1 summarizes:1 lime:2 hi:1 quadratic:1 pijk:2 activity:2 orthogonality:10 i1k:1 x2:16 nearby:1 diesmann:1 pi1:2 expanded:1 palm:1 neurostatistics:1 slightly:1 xixj:1 invariant:1 taken:1 behavioural:1 mutually:1 discus:2 turn:1 merit:1 gfj:2 cor:13 end:1 fi1:1 observe:2 hierarchical:2 subjectto:1 ho:2 hat:1 denotes:1 top:1 assembly:1 establish:1 society:1 quantity:1 spike:8 dependence:1 aertsen:3 distance:1 sci:2 simulated:2 manifold:3 index:1 illustration:3 ratio:1 demonstration:2 equivalently:1 difficult:1 relate:4 affiliated:1 proper:2 unknown:1 diamond:1 inspect:2 neuron:28 observation:1 coincident:12 situation:1 extended:2 kl:4 kang:1 fi2:3 pattern:9 including:1 event:4 natural:2 treated:1 difficulty:1 quantification:1 advanced:1 orthogonally:1 irrespective:1 ljj:1 geometric:8 literature:2 relative:1 synchronization:1 dsi:2 highlight:1 mixed:6 triple:2 degree:1 conveyed:5 slovin:1 sufficient:1 pij:2 systematically:2 share:1 lo:2 succinctly:1 populational:1 supported:1 free:1 understand:1 institute:1 taking:2 curve:1 xn:5 cortical:2 sensory:1 c5:1 schultz:1 transaction:1 excess:1 emphasize:1 kullback:1 b1:3 assumed:4 xi:7 triplet:4 nature:1 tel:1 expansion:2 necessarily:1 submit:1 significance:6 main:1 s2:3 x1:6 neuronal:3 fig:14 je:1 aid:1 explicit:1 exponential:1 xl:8 ib:1 vanish:1 ito:1 theorem:9 hogonal:1 jpsth:1 magnitude:1 ijj:1 desire:1 bo:5 corresponds:1 lxp:1 xix2:1 formulated:1 nakahara:3 fisher:3 habib:1 change:5 except:1 correlational:4 called:4 total:1 experimental:2 ijk:1 nx1:1 modulated:1 frontal:1 dept:1 correlated:2 |
1,104 | 2,007 | Quantizing Density Estimators
Peter Meinicke
Neuroinformatics Group
University of Bielefeld
Bielefeld, Germany
pmeinick@techfak.uni-bielefeld.de
Helge Ritter
Neuroinformatics Group
University of Bielefeld
Bielefeld, Germany
helge@techfak.uni-bielefeld.de
Abstract
We suggest a nonparametric framework for unsupervised learning of
projection models in terms of density estimation on quantized sample
spaces. The objective is not to optimally reconstruct the data but instead the quantizer is chosen to optimally reconstruct the density of the
data. For the resulting quantizing density estimator (QDE) we present a
general method for parameter estimation and model selection. We show
how projection sets which correspond to traditional unsupervised methods like vector quantization or PCA appear in the new framework. For a
principal component quantizer we present results on synthetic and realworld data, which show that the QDE can improve the generalization of
the kernel density estimator although its estimate is based on significantly
lower-dimensional projection indices of the data.
1 Introduction
Unsupervised learning is essentially concerned with finding alternative representations for
unlabeled data. These alternative representations usually reflect some important properties
of the underlying distribution and usually they try to exploit some redundancy in the data.
In that way many unsupervised methods aim at a complexity-reduced representation of the
data, like the most common approaches, namely vector quantization (VQ) and principal
component analysis (PCA). Both approaches can be viewed as specific kinds of quantization, which is a basic mechanism of complexity reduction.
The objective of our approach to unsupervised learning is to achieve a suitable quantization
of the data space which allows for an optimal reconstruction of the underlying density from
a finite sample. In that way we consider unsupervised learning as density estimation on a
quantized sample space and the resulting estimator will be referred to as quantizing density
estimator (QDE). The construction of a QDE first requires to specify a suitable class of
parametrized quantization functions and then to select from this set a certain function with
good generalization properties. While the first point is common to unsupervised learning,
the latter point is addressed in a density estimation framework where we tackle the model
selection problem in a data-driven and nonparametric way.
It is often overlooked that modern Bayesian approaches to unsupervised learning and model
selection are almost always based on some strong assumptions about the data distribution.
Unfortunately these assumptions usually cannot be inferred from human knowledge about
the data domain and therefore the model building process is usually driven by computational considerations. Although our approach can be interpreted in terms of a generative
model of the data, in contrast to most other generative models (see [10] for an overview),
the present approach is nonparametric, since no specific assumptions about the functional
form of the data distribution have to be made. In that way our approach compares well with
other quantization methods, like principal curves and surfaces [4, 13, 6], which only have
to make rather general assumptions about the underlying distribution. The QDE approach
can utilize these methods as specific quantization techniques and shows a practical way
how to further automatize the construction of unsupervised learning machines.
2 Quantization by Density Estimation
We will now explain how the QDE may be derived from a generalization of the kernel
density estimator (KDE), one of the most popular methods for nonparametric density estimation [12, 11]. If we construct a kernel density estimator on the basis of a quantized
sample, we have the following estimator
(1)
$#% #&
"! a sample from the target distribution,
where
denotes the kernel
('*)+-is,/
.*0213)+
function and
with
parameter
vector
is
a
given
quantization or
4 to a parametrized subset . 0 Sof the sample space
projection function4which
maps a point
576 98 :
8T:UV#W
;<8 :=>7?A@ B>CEDGF POQ6*R
HJILKNM
M
(2)
Thereby the projection
index
associates a data point with its nearest neighbour in the
. 0
Y to
which is parametrized according
projection set
. 0
]
X'AQ76 R
;ZR\[P]^1_)a` !
(3)
where
is the set of allR possible projection indices which are realizations of the deterministic latent variable . For a fixed non-zero kernel bandwidth the parameters of the
quantization function may be determined by nonparametric maximum likelihood (ML) estimation, as will be introduced in the next section.
For an intuitive motivation of the QDE, one may ask from
perspec a data
compression
! for the realization
tive whether it is necessary to store all the sample data
of the kernel density estimator or if it is possible to first reduce the data by some suitable quantization method and then construct the estimator from the more parsimonious
complexity-reduced data set. Clearly, we would prefer a quantizer which does not decrease
the performance of the estimator on unseen and unquantized data.
To get an idea of how to select a suitable quantization function let us consider an example
from a 1D data space. In one dimension a natural
. 0 projection
set
can! be specified by a set
of b quantization levels on the real line, i.e.
dc
cfe . For a fixed kernel
bandwidth, we can now perform maximum likelihood estimation of the level coordinates.
In that way we obtain a maximum likelihood estimator of the form
agh5
3j
'mlno?A@ BCEDpFZqnj gdrfO
e
qsj j
=i
gY
c
(4)
k
c ! counting the number of data points which are
with
l
i
quantized
to level . In this case, it remains the question how to choose the number of
quantization levels.
From a different starting point the authors in [3] proposed the same functional form of a
nonparametric
ML density estimator with respect to Gaussian kernels of equal width cen
tered
on variable positions. As with the traditional Gaussian KDE (fixed kernel centers
on data points), for consistency of the estimator the bandwidth has to be decreased as the
sample size increases.
In [3] the authors reported that for a fixed non-zero bandwidth, ML
estimation of the kernel centers always resulted in a smaller number of actually distinct
centers,
i.e. several kernels coincided to maximize the
likelihood. Therefore the resulting
estimator had the form of (4) where b corresponds to the number of distinct centers with
counting the number of kernels coinciding at c . The optimum number of effective
i
quantization
levels for a given bandwidth therefore arises as an automatic byproduct of ML
estimation.
Finally one has to choose an appropriate kernel width which implicitly determines the complexity of the quantizer. The bandwidth selection problem has been tackled in the domain
of kernel density estimation for some time and many approaches have been proposed (see
e.g. [5] for an overview), among which the cross-validation methods are most common. In
the next section we will adopt the method of likelihood cross-validation to find a practical
answer to the bandwidth selection problem.
3 General Learning Scheme
By applying the method of sieves
as proposed in [3], for a fixed non-zero bandwidth we can
estimate
the parameters of the quantization function via maximization of the log-likelihood
BZ9a
w.r.t. to . For consistency of the resulting density estimator the band
width has to be decreased as the sample size increases, since asymptotically the estimator
must converge to a mixture of delta functions centered on the data points. Thus, for decreasing bandwidth, the quantization function of the QDE must converge to the identity
function, i.e. the QDE must converge to the kernel density estimator.
For a fixed bandwidth, maximization of the likelihood can be achieved by applying the EMalgorithm [2] which provides a convenient optimization scheme, especially for Gaussian
kernels. The EM-scheme requires to iterate the following two steps
Yr
;=q
E-Step:
(5)
q
Y
?A@ B5C 0 ?
B ; r
r
M-Step:
(6)
r
s
for a sequence
with suitable initial parameter vector
and sufficient
Y
r
l
convergence at . Thereby i
denotes the posterior probability
that
; r data point has
been ?generated? by mixture component k with density
. For further insight
one may realize that the M-Step requires to solve a constrained
optimization problem by
r
searching for
C 0 ?
B 6*R
"! #
R R
r
Y
R 7?A@ B CEDpF X
O 6 R
subject to
HJILK M
M
r
(7)
(8)
In general this optimization problem can only be solved by iterative techniques. Therefore
it may be convenient not to maximize but only to increase the log-likelihood at the M-Step
which then corresponds to an application of the generalized EM-algorithm. Without (8)
unconstrained maximization according to (7) yields another class of interesting learning
schemes which for reasons of space will not be considered in this paper.
For Gaussian kernels and an Euclidean metric for the projection, in the limiting case of a
vanishing bandwidth, EM-optimization of the QDE parameters corresponds to minimization of the following error or risk
Y
CEDpF OX6 R
J
H
ILK M
M
Minimization of such error functions corresponds to traditional approaches to unsupervised
learning of projection models which can be viewed as a special case of QDE-based learning.
3.1 Bandwidth Selection
It is easy to see that the kernel bandwidth cannot be determined by ML-estimation since
maximization of the likelihood would drive the bandwidth towards zero. For selection of
the kernel bandwidth, we therefore apply the method of likelihood cross-validation (see
e.g. [12]), which can be realized by a slight extension of the above EM-scheme. With the
4
leave-one-out QDE
;Yr
O r
(9)
BZ
the idea is to maximize
with respect to the kernel bandwidth. For a
Gaussian kernel with bandwidth an appropriate EM scheme requires the following MStep update rule
Y
r
r OX r
M
r
M
(10)
The posterior probabilities are easily derived from a leave-one-out version of (5). In an
overall optimization scheme one may now alter the estimation of and or alternatively
one may estimate both by likelihood cross-validation.
4 Projection Sets in Multidimensions
By the specification of a certain class of quantization functions we can incorporate domain
knowledge into the density estimation process, in order to improve generalization. Thereby
the idea is to reduce the variance of the density estimator by reducing the variation of the
quantized training set. The price is an increase of the bias which requires a careful selection
of the set of admissible quantization functions. Then the QDE offers the chance to find a
better bias/variance trade-off then with the ?non-quantizing? KDE.
We will now show how to utilize existing methods for unsupervised learning within the
current density estimation framework. Because many unsupervised methods can be stated
in terms of finding optimal projection sets, it is straightforward to apply the corresponding
types of quantization functions within the current framework. Thus in the following we
shall consider specific parametrizations of the general projection set (3) which correspond
to traditional unsupervised learning methods.
4.1 Vector Quantization
Vector quantization (VQ) is a standard technique among unsupervised methods and it is
easily incorporated into the current density estimation framework by straightforward gen-
eralization of the one-dimensional quantizer in section 2 to the multi-dimensional case.
Again with a fixed kernel bandwidth ML estimation yields a certain number of b
distinct (?effective?) quantization levels, similar to maximum entropy clustering [9, 1].
The projection set of a vector quantizer can be parametrized according to a general basis
function representation [7] 4
6
> ;N[
!
$#&
with
-dimensional vector of basis functions
rL
s5 rq
for component k .
(11)
containing discrete delta functions, i.e.
The QDE on the basis of a vector quantizer can be expected to generalize well if some
cluster structure is present within the data. In multi-dimensional spaces the data are often
concentrated in certain regions which allows for a sparse representation by some reference
vectors well-positioned in those regions. An alternative approach has been proposed in
[14] where the application of the support vector formalism to density estimation results in
a sparse representation of the data distribution.
4.2 Principal Component Analysis
projection set yields candidate functions of the form
A linear affine parametrization of 4
the
6 R
5
Rs RX[N)a`
(12)
with
. The PCA approach reflects our knowledge that in most high-dimensional
data spaces, the data are concentrated around some manifold of lower dimensionality.
To exploit this structure PCA divides the sample space into two subspaces which are quantized in different ways: within the ?inner? subspace spanned by the directions of the projection manifold we have no quantization at all; within the orthogonal ?outer? subspace the
data are quantized to a single level.
With a Gaussian kernel with fixed bandwidth the constrained optimization problem at the
M-Step takes a convenient form which facilitates further analysis of the learning algorithm.
From (7) and (8) it follows that one has to maximize the following objective function
sN>
const.
O
r
O r O L5O
M
r
M
(13)
matrix has orthogonal
where
columns which span the subspace of the projection
manifold. From the consideration
one finds
Q
of the corresponding stationarity conditions
that the sample mean
is an estimator of the shift vector .
Maximization of (13) with respect to
tr
then requires to maximize the following trace
!
O" O #a$ S
with symmetric matrices
!
r
Thus (14) is maximized
if
values, i.e. with ( (
dimensionality
r # r
tr
&%'
r
r # ;" r #r
r
(14)
(15)
%
contains all eigenvectors of % , associated with positive eigenwe have the optimal subspace
(*) being the eigenvalues of
j
+(
'
(
,
sal
- ! j
(16)
which complements a recent result about parametric dimensionality estimation with respect
to a -factor model with isotropic
noise [8]. For the QDE, the two limiting cases
Gaussian
of zero and infinite bandwidth,
areO ofparticular
O interest. With the positive definite sample
covariance matrix
one can show
DpC
O
%
%
DpC
%
(17)
Thus for sufficiently large bandwidth becomes negative definite, which implies a zero
subspace dimensionality estimator
, i.e. all data are quantized to the sample mean.
%
For sufficiently small bandwidth
becomes positive definite implying
, i.e. no
quantization takes place.
4.3 Independent Component Analysis
The PCA method provides a rather coarse quantization scheme since it only decides between one-level and no quantization for each subspace dimension. A natural refinement
would therefore be to allow for a certain number of effective quantization levels for each
component. Such an approach may be viewed as a nonparametric variant of independent
component analysis (ICA). The idea is to quantize each coordinate axis separately, which
yields a multi-dimensional quantization grid according to
6 R
Y
>
+
$#&
#s R\[
! +
(18)
[ )+ [ )
with
,
as in (11). Thereby
the components of contain the
l and
quantization levels of the -th coordinate axis with direction . Further, it makes sense to
. There are strong similarities with a
normalize the direction vectors according to M M
parametric ICA model which has been suggested in [10], where source densities have been
mixtures of delta functions and additive noise has been isotropic Gaussian.
Other unsupervised learning methods which correspond to different projection sets, like
principal curves or multilayer perceptrons (see [7] for an overview) can as well be incorporated into the QDE framework and will be considered elsewhere.
5 Experiments
In the following experiments we investigated the PCA based QDE with Gaussian kernel
and compared the generalization performance with that of the ?non-quantizing? KDE.
All parameters, including the bandwidth of the KDE, were estimated by likelihood crossvalidation. In the first experiment we sampled
h 100 points from a stretched and rotated
rectangle. In this case the QDE extracted
uniform distribution with support on a
a one-dimensional ?unquantized? subspace. Generalization performance was measured by
the average log-likelihood on an independent
1000-point test set. With an automatically se
lected 1D subspace (compression
ratio
) the PCA-QDE could improve the performance
of the KDE from
to
. Thus, the PCA-QDE could successfully exploit the elongated structure of the distribution. The estimated
density functions are depicted in figure 1,
grid. From the images one can
where grey-values are proportional to
on a
see, that the QDE better captures the global structure of the distribution while the KDE is
more sensitive to local variations in the data.
In a second experiment we trained PCA-QDEs with
-dimensional real-world data
(
images) which had been derived from the MNIST database of handwritten digits
(http://www.research.att.com/
yann/ocr/mnist/). For each digit class a
-point training set and a
-point test set were used to compare the PCA-QDE with
Figure 1: Left: stretched uniform distribution in 2D with white points indicating 100 data
points used for estimation; middle: Estimated density using the PCA-QDE; right: kernel
density estimate.
the KDE, with results shown in table 1. Again the PCA-QDE improved the generalization performance of the KDE although the QDE decided to remove about 40 ?redundant?
dimensions per digit class.
Table 1: Results on -dimensional digit data for different digit classes ?0?...?9? (first row);
second row: difference between average log-likelihoods of (PCA-)QDE and KDE on test
set; third row: optimal subspace dimensionality of QDE
Digit:
:
:
0
1.87
22
1
0.66
29
2
1.02
26
3
1.38
24
4
1.58
24
5
1.54
25
6
1.44
24
7
0.64
27
8
1.53
21
9
1.33
25
6 Conclusion
The QDE offers a nonparametric approach to unsupervised learning of quantization functions which can be viewed as a generalization of the kernel density estimator. While the
KDE is directly constructed from the given data set the QDE first creates a quantized representation of the data. Unlike traditional quantization methods which minimize the associated reconstruction error of the data points, the QDE adjusts the quantizer to optimize
an estimate of the data density. This feature allows for a convenient model selection procedure, since the complexity of the quantizer can be controlled by the kernel bandwidth,
which in turn can be selected in a data-driven way. For a practical realization we outlined EM-schemes for parameter estimation and bandwidth selection. As an illustration,
we discussed examples with different projection sets which correspond to VQ, PCA and
ICA methods. We presented experiments which demonstrate that the bias imposed by the
quantization can lead to an improved generalization as compared to the ?non-quantizing?
KDE. This suggests that QDEs offer a promising approach to unsupervised learning that
allows to control bias without the usually rather strong distributional assumptions of the
Bayesian approach.
Acknowledgement
This work was funded by the Deutsche Forschungsgemeinschaft within the project SFB
360.
References
[1] J. M. Buhmann and N. Tishby. Empirical risk approximation: A statistical learning
theory of data clustering. In C. M. Bishop, editor, Neural Networks and Machine
Learning, pages 57?68. Springer, Berlin Heidelberg New York, 1998.
[2] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete
data via the EM algorithm. Journal of the Royal Statistical Society Series B, 39:1?38,
1977.
[3] Stuart Geman and Chii-Ruey Hwang. Nonparametric maximum likelihood estimation
by the method of sieves. The Annals of Statistics, 10(2):401?414, 1982.
[4] T. Hastie and W. Stuetzle. Principal curves. Journal of the American Statistical
Association, 84:502?516, 1989.
[5] M. C. Jones, J. S. Marron, and S. J. Sheather. A brief survey of bandwidth selection
for density estimation. Journal of the American Statistical Association, 91(433):401?
407, 1996.
[6] B. K?egl, A. Krzyzak, T. Linder, and K. Zeger. Learning and design of principal
curves. IEEE Transaction on Pattern Analysis and Machine Intelligence, 22(3):281?
297, 2000.
[7] Peter Meinicke. Unsupervised Learning in a Generalized Regression Framework. PhD thesis, Universitaet Bielefeld, 2000. http://archiv.ub.unibielefeld.de/disshabi/2000/0033/.
[8] Peter Meinicke and Helge Ritter. Resolution-based complexity control for Gaussian
mixture models. Neural Computation, 13(2):453?475, 2001.
[9] K. Rose, E. Gurewitz, and G. C. Fox. Statistical mechanics and phase transitions in
clustering. Physical Review Letters, 65(8):945?948, 1990.
[10] Sam Roweis and Zoubin Ghahramani. A unifying review of linear Gaussian models.
Neural Computation, 11(2):305?345, 1999.
[11] D. W. Scott. Multivariate Density Estimation. Wiley, 1992.
[12] B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman and
Hall, London and New York, 1986.
[13] Alex J. Smola, Robert C. Williamson, Sebastian Mika, and Bernhard Sch?olkopf. Regularized principal manifolds. In Proc. 4th European Conference on Computational
Learning Theory, volume 1572, pages 214?229. Springer-Verlag, 1999.
[14] Vladimir N. Vapnik and Sayan Mukherjee. Support vector method for multivariate
density estimation. In S. A. Solla, T. K. Leen, and K.-R. M?uller, editors, Advances in
Neural Information Processing Systems, volume 12, pages 659?665. The MIT Press,
2000.
| 2007 |@word middle:1 version:1 compression:2 meinicke:3 grey:1 covariance:1 thereby:4 tr:2 reduction:1 initial:1 contains:1 att:1 series:1 existing:1 current:3 com:1 must:3 realize:1 additive:1 zeger:1 mstep:1 remove:1 update:1 implying:1 generative:2 selected:1 yr:2 intelligence:1 mln:1 isotropic:2 parametrization:1 vanishing:1 quantizer:9 quantized:9 provides:2 coarse:1 constructed:1 ica:3 expected:1 mechanic:1 multi:3 unquantized:2 decreasing:1 automatically:1 becomes:2 project:1 underlying:3 deutsche:1 kind:1 interpreted:1 finding:2 tackle:1 control:2 appear:1 positive:3 local:1 mika:1 suggests:1 decided:1 practical:3 definite:3 silverman:1 digit:6 stuetzle:1 procedure:1 empirical:1 significantly:1 projection:19 convenient:4 suggest:1 zoubin:1 get:1 cannot:2 unlabeled:1 selection:11 risk:2 applying:2 optimize:1 www:1 map:1 imposed:1 center:4 elongated:1 deterministic:1 straightforward:2 starting:1 survey:1 resolution:1 estimator:22 insight:1 rule:1 adjusts:1 spanned:1 searching:1 coordinate:3 variation:2 limiting:2 techfak:2 construction:2 target:1 annals:1 hjilk:1 associate:1 areo:1 mukherjee:1 distributional:1 database:1 geman:1 solved:1 capture:1 region:2 solla:1 decrease:1 trade:1 rose:1 dempster:1 complexity:6 trained:1 creates:1 basis:4 easily:2 distinct:3 effective:3 london:1 neuroinformatics:2 solve:1 reconstruct:2 statistic:2 unseen:1 laird:1 sequence:1 eigenvalue:1 quantizing:6 reconstruction:2 realization:3 gen:1 parametrizations:1 achieve:1 roweis:1 intuitive:1 normalize:1 olkopf:1 crossvalidation:1 convergence:1 cluster:1 optimum:1 pmeinick:1 leave:2 rotated:1 measured:1 nearest:1 strong:3 implies:1 direction:3 centered:1 human:1 generalization:9 extension:1 around:1 considered:2 sufficiently:2 hall:1 cen:1 adopt:1 lected:1 estimation:26 proc:1 sensitive:1 successfully:1 reflects:1 minimization:2 uller:1 mit:1 clearly:1 gaussian:11 always:2 aim:1 rather:3 derived:3 likelihood:16 contrast:1 sense:1 germany:2 overall:1 among:2 constrained:2 special:1 equal:1 construct:2 chapman:1 stuart:1 jones:1 unsupervised:18 alter:1 modern:1 neighbour:1 resulted:1 phase:1 multidimensions:1 stationarity:1 interest:1 mixture:4 byproduct:1 necessary:1 orthogonal:2 fox:1 incomplete:1 euclidean:1 divide:1 eralization:1 formalism:1 column:1 maximization:5 subset:1 uniform:2 tishby:1 optimally:2 reported:1 answer:1 marron:1 synthetic:1 density:34 ritter:2 off:1 again:2 reflect:1 thesis:1 containing:1 choose:2 american:2 de:3 gy:1 try:1 minimize:1 variance:2 maximized:1 correspond:4 yield:4 generalize:1 chii:1 bayesian:2 handwritten:1 rx:1 drive:1 explain:1 sebastian:1 associated:2 tered:1 sampled:1 popular:1 ask:1 knowledge:3 dimensionality:5 positioned:1 actually:1 specify:1 coinciding:1 improved:2 leen:1 ox:1 smola:1 hwang:1 building:1 contain:1 sieve:2 symmetric:1 white:1 width:3 generalized:2 demonstrate:1 gh:1 image:2 consideration:2 common:3 functional:2 rl:1 overview:3 physical:1 volume:2 discussed:1 association:2 slight:1 stretched:2 automatic:1 unconstrained:1 outlined:1 grid:2 consistency:2 had:2 funded:1 specification:1 similarity:1 surface:1 posterior:2 multivariate:2 recent:1 driven:3 store:1 certain:5 verlag:1 converge:3 maximize:5 redundant:1 cross:4 offer:3 controlled:1 variant:1 basic:1 regression:1 multilayer:1 essentially:1 metric:1 bz:2 kernel:27 sof:1 achieved:1 separately:1 addressed:1 decreased:2 source:1 sch:1 unlike:1 subject:1 facilitates:1 sheather:1 counting:2 forschungsgemeinschaft:1 easy:1 concerned:1 iterate:1 hastie:1 bandwidth:26 reduce:2 idea:4 inner:1 shift:1 whether:1 pca:14 sfb:1 krzyzak:1 peter:3 york:2 se:1 eigenvectors:1 nonparametric:9 band:1 concentrated:2 reduced:2 http:2 delta:3 estimated:3 per:1 discrete:1 shall:1 group:2 redundancy:1 utilize:2 rectangle:1 asymptotically:1 realworld:1 letter:1 place:1 bielefeld:7 almost:1 yann:1 parsimonious:1 prefer:1 tackled:1 ilk:1 alex:1 archiv:1 span:1 ruey:1 according:5 smaller:1 em:7 sam:1 helge:3 vq:3 remains:1 turn:1 mechanism:1 apply:2 ocr:1 appropriate:2 alternative:3 denotes:2 clustering:3 unifying:1 const:1 exploit:3 ghahramani:1 especially:1 society:1 objective:3 question:1 realized:1 parametric:2 traditional:5 subspace:10 berlin:1 parametrized:4 outer:1 manifold:4 reason:1 index:3 illustration:1 ratio:1 vladimir:1 unfortunately:1 robert:1 kde:12 trace:1 stated:1 negative:1 design:1 perform:1 finite:1 incorporated:2 dc:1 overlooked:1 inferred:1 introduced:1 tive:1 namely:1 complement:1 specified:1 suggested:1 usually:5 pattern:1 scott:1 including:1 royal:1 suitable:5 natural:2 regularized:1 buhmann:1 zr:1 dpc:2 scheme:9 improve:3 brief:1 axis:2 gurewitz:1 review:2 acknowledgement:1 interesting:1 proportional:1 validation:4 affine:1 sufficient:1 rubin:1 editor:2 row:3 elsewhere:1 cfe:1 bias:4 allow:1 sparse:2 curve:4 dimension:3 world:1 transition:1 author:2 made:1 refinement:1 transaction:1 uni:2 implicitly:1 bernhard:1 ml:6 global:1 decides:1 universitaet:1 alternatively:1 latent:1 iterative:1 table:2 promising:1 heidelberg:1 quantize:1 williamson:1 investigated:1 european:1 domain:3 motivation:1 noise:2 referred:1 wiley:1 position:1 candidate:1 third:1 coincided:1 admissible:1 sayan:1 specific:4 bishop:1 quantization:33 mnist:2 vapnik:1 phd:1 egl:1 entropy:1 depicted:1 springer:2 corresponds:4 determines:1 chance:1 extracted:1 viewed:4 identity:1 careful:1 towards:1 price:1 determined:2 infinite:1 reducing:1 principal:8 perceptrons:1 indicating:1 select:2 linder:1 support:3 latter:1 arises:1 ub:1 incorporate:1 |
1,105 | 2,008 | Model Based Population Tracking and
Automatic Detection of Distribution Changes
Igor V. Cadez ?
Dept. of Information and Computer Science,
University of California,
Irvine, CA 92612
icadez@ics.uci.edu
P. S. Bradley
digiMine, Inc.
10500 NE 8th Street,
Bellevue, WA 98004-4332
paulb@digimine.com
Abstract
Probabilistic mixture models are used for a broad range of data analysis tasks such as clustering, classification, predictive modeling, etc.
Due to their inherent probabilistic nature, mixture models can easily be
combined with other probabilistic or non-probabilistic techniques thus
forming more complex data analysis systems. In the case of online data
(where there is a stream of data available) models can be constantly updated to reflect the most current distribution of the incoming data. However, in many business applications the models themselves represent a
parsimonious summary of the data and therefore it is not desirable to
change models frequently, much less with every new data point. In such
a framework it becomes crucial to track the applicability of the mixture
model and detect the point in time when the model fails to adequately
represent the data. In this paper we formulate the problem of change
detection and propose a principled solution. Empirical results over both
synthetic and real-life data sets are presented.
1 Introduction and Notation
Consider a data set D = {x1 , x2 , . . . , xn } consisting of n independent, identically distributed (iid) data points. In context of this paper the data points could be vectors, sequences, etc. Further, consider a probabilistic mixture model that maps each data set to a
real number, the probability of observing the data set:
P (D|?) =
n
Y
i=1
P (xi |?) =
n X
K
Y
?k P (xi |?k ),
(1)
i=1 k=1
where the model is parameterized by ? = {?1 , . . . , ?K , ?1 , . . . , ?K }. Each P (.|?k ) represents a mixture component, while ?i represents mixture weights. It is often more convenient
?
Work was done while author was at digiMine, Inc., Bellevue, WA.
to operate with the log of the probability and define the log-likelihood function as:
l(?|D) = log P (D|?) =
n
X
log P (xi |?) =
i=1
n
X
LogPi
i=1
which is additive over data points rather than multiplicative. The LogPi terms we introduce
in the notation represent each data point?s contribution to the overall log-likelihood and
therefore describe how well a data point fits under the model. For example, Figure 3 shows
a distribution of LogP scores using a mixture of conditionally independent (CI) models.
Maximizing probability1 of the data with respect to the parameters ? can be accomplished
by the Expectation-Maximization (EM) algorithm [6] in linear time in both data complexity
(e.g., number of dimensions) and data set size (e.g., number of data points). Although EM
guarantees only local optimality, it is a preferred method for finding good solutions in
linear time. We consider an arbitrary but fixed parametric form of the model, therefore
we sometimes refer to a specific set of parameters ? as the model. Note that since the
logarithm is a monotonic function, the optimal set of parameters is the same whether we
use likelihood or log-likelihood.
Consider an online data source where there are data sets Dt available at certain time intervals t (not necessarily equal time periods or number of data points). For example, there
could be a data set generated on a daily basis, or it could represent a constant stream of
data from a monitoring device. In addition, we assume that we have an initial model ?0
that was built (optimized, fitted) on some in-sample data D0 = {D1 , D2 , . . . , Dt0 }. We
would like to be able to detect a change in the underlying distribution of data points within
data sets Dt that would be sufficient to require building of a new model ?1 . The criterion
for building a new model is loosely defined as ?the model does not adequately fit the data
anymore?.
2 Model Based Population Similarity
In this section we formulate the problem of model-based population similarity and tracking.
In case of mixture models we start with the following observations:
? The mixture model defines the probability density function (PDF) that is used to
score each data point (LogP scores), leading to the score for the overall population
(log-likelihood or sum of LogP scores).
? The optimal mixture model puts more PDF mass over dense regions in the data
space. Different components allow the mixture model to distribute its PDF over
disconnected dense regions in the data space. More PDF mass in a portion of the
data space implies higher LogP scores for the data points lying in that region of
the space.
? If model is to generalize well (e.g., there is no significant overfitting) it cannot put
significant PDF mass over regions of data space that are populated by data points
solely due to the details of a specific data sample used to build the model.
? Dense regions in the data space discovered by a non-overfitting model are the
intrinsic property of the true data-generating distribution even if the functional
form of the model is not well matched with the true data generating distribution. In
the latter case, the model might not be able to discover all dense regions or might
not model the correct shape of the regions, but the regions that are discovered (if
any) are intrinsic to the data.
1
This approach is called maximum-likelihood estimation. If we included parameter priors we
could equally well apply results in this paper to the maximum a posteriori estimation.
? If there is confidence that the model is not overfitting and that it generalizes well
(e.g., cross-validation was used to determine the optimal number of mixture components), the new data from the same distribution as the in-sample data should be
dense in the same regions that are predicted by the model.
Given these observations, we seek to define a measure of data-distribution similarity based
on how well the dense regions of the data space are preserved when new data is introduced.
In model based clustering, dense regions are equivalent to higher LogP scores, hence we
cast the problem of determining data distribution similarity into one of determining LogP
distribution similarity (relative to the model). For example, Figure 3 (left) shows a histogram of one such distribution. It is important to note several properties of Figure 3: 1)
there are several distinct peaks from which distribution tails off toward smaller LogP values, therefore simple summary scores fail to efficiently summarize the LogP distribution.
For example, log-likelihood is proportional to the mean of LogP distribution in Figure 3,
and the mean is not a very useful statistic when describing such a multimodal distribution
(also confirmed experimentally); 2) the histogram itself is not a truly non-parametric representation of the underlying distribution, given that the results are dependent on bin width.
In passing we also note that the shape of the histogram in Figure 3 is a consequence of the
CI model we use: different peaks come from different discrete attributes, while the tails
come from continuous Gaussians. It is a simple exercise to show that LogP scores for
a 1-dimensional data set generated by a single Gaussian have an exponential distribution
with a sharp cutoff on the right and tail toward the left.
To define the similarity of the data distributions based on LogP scores in a purely nonparametric way we have at our disposal the powerful formalism of Kolmogorov-Smirnov
(KS) statistics [7]. KS statistics make use of empirical cumulative distribution functions
(CDF) to estimate distance between two empirical 1-dimensional distributions, in our case
distributions of LogP scores. In principle, we could compare the LogP distribution of the
new data set Dt to that of the training set D0 and obtain the probability that the two came
from the same distribution. In practice, however, this approach is not feasible since we do
not assume that the estimated model and the true data generating process share the same
functional form (see Section 3). Consequently, we need to consider the specific KS score
in relation to the natural variability of the true data generating distribution. In the situation
with streaming data, the model is estimated over the in-sample data D0 . Then the individual
in-sample data sets D1 , D2 , . . . , Dt0 are used to estimate the natural variability of the KS
statistics. This variability needs to be quantified due to the fact that the model may not
truly match the data distribution. When the natural variance of the KS statistics over the
in-sample data has been determined, the LogP scores for a new dataset Dt , t > t0 are
computed. Using principled heuristics, one can then determine whether or not the LogP
signature for Dt is significantly different than the LogP signatures for the in-sample data.
To clarify various steps, we provide an algorithmic description of the change detection
process.
Algorithm 1 (Quantifying Natural Variance of KS Statistics):
Given an ?in-sample? dataset D0 = {D1 , D2 , . . . , Dt0 }, proceed as follows:
1. Estimate the parameters ?0 of the mixture model P (D|?) over D0 (see equation (1)).
2. Compute
ni
X
(2)
LogP (Di ) =
log P (x?i |?0 ), x?i ? Di , ni = |Di |, i = 1, . . . , t0 .
?i=1
3. For 1 ? i, j ? t0 , compute LKS (i, j) = log [PKS (Di , Dj )]. See [7] for details on
PKS computation.
4. For 1 ? i ? t0 , compute the KS measure MKS (i) as
Pt 0
j=1 LKS (i, j)
.
MKS (i) =
t0
(3)
5. Compute ?M = M ean[MKS (i)] and ?M = ST D[MKS (i)] to quantify the natural
variability of MKS over the ?in-sample? data.
Algorithm 2 (Evaluating New Data):
Given a new dataset Dt , t > t0 , ?M and ?M proceed as follows:
1.
2.
3.
4.
Compute LogP (Dt ) as in (2).
For 1 ? i ? t0 , compute LKS (i, t).
Compute MKS (t) as in (3).
Apply decision criteria using MKS (t), ?M , ?M to determine whether or not ?0 is
a good fit for the new data. For example, if
|MKS (t) ? ?M |
> 3,
?M
then ?0 is not a good fit any more.
(4)
Note, however, that the 3-? interval be interpreted as a confidence interval only in the
limit when number of data sets goes to infinity. In applications presented in this paper we
certainly do not have that condition satisfied and we consider this approach as an ?educated
heuristic? (gaining firm statistical grounds in the limit).
2.1 Space and Time Complexity of the Methodology
The proposed methodology was motivated by a business application with large data sets,
hence it must have time complexity that is close to linear in order to scale well. In order
to assess the time complexity, we use the following notation: nt = |Dt | is the number of
data points in the data set Dt ; t0 is the index ofP
the last in-sample data set, but is also the
t0
number of in-sample data sets; n0 = |D0 | = t=1
nt is the total number of in-sample
data points (in all the in-sample data sets); n = n0 /t0 is the average number of data points
in the in-sample data sets. For simplicity of argument, we assume that all the data sets are
approximately of the same size, that is nt ? n.
The analysis presented here does not take into account the time and space complexity
needed to estimated the parameters ? of the mixture model (1). In the first phase of the
methodology, we must score each of the in-sample data points under the model (to obtain
the LogP distributions) which has time complexity of O(n0 ). Calculation of KS statistics
for two data sets is done in one pass over the LogP distributions, but it requires that the
LogP scores be sorted, hence it has time complexity of 2n + 2O(n log n) = O(n log n).
Since we must calculate all the pairwise KS measures, this step has time complexity of
t0 (t0 ? 1)/2 O(n log n) = O(t20 n log n). In-sample mean and variance of the KS measure
are obtained in time which is linear in t0 hence the asymptotic time complexity does not
change. In order to evaluate out-of-sample data sets we must keep LogP distributions for
each of the in-sample data sets as well as several scalars (e.g., mean and variance of the
in-sample KS measure) which requires O(n0 ) memory.
To score an out-of-sample data set Dt , t > t0 , we must first obtain the LogP distribution
of Dt which has time complexity of O(n) and then calculate the KS measure relative to
each of the in-sample data sets which has time complexity O(n log n) per in-sample data
set, or t0 O(n log n) = O(t0 n log n) for the full in-sample period. The LogP distribution
for Dt can be discarded once the pairwise KS measures are obtained.
3000
3000
2500
2500
2000
2000
Count
3500
Count
3500
1500
1500
1000
1000
500
500
0
?5.5
?5
?4.5
?4
?3.5
?3
0
?2.5
?5.5
?5
?4.5
LogP
?4
?3.5
?3
?2.5
?4
?3.5
?3
?2.5
LogP
3000
3000
2500
2500
2000
2000
Count
3500
Count
3500
1500
1500
1000
1000
500
500
0
?5.5
?5
?4.5
?4
?3.5
?3
LogP
?2.5
0
?5.5
?5
?4.5
LogP
Figure 1: Histograms of LogP scores for two data sets generated from the first model
(top row) and two data sets generated from the second model (bottom row). Each data
set contains 50,000 data points. All histograms are obtained from the model fitted on the
in-sample period.
Overall, the proposed methodology requires O(n0 ) memory, O(t20 n log n) time for preprocessing and O(t0 n log n) time for out-of-sample evaluation. Further, since t0 is typically a
small constant (e.g., t0 = 7 or t0 = 30), the out-of-sample evaluation practically has time
complexity of O(n log n).
3 Experimental Setup
Experiments presented consist of two parts: experiments on synthetic data and experiments
on the aggregations over real web-log data.
3.1 Experiments on Synthetic Data
Synthetic data is a valuable tool when determining both applicability and limitations of the
proposed approach. Synthetic data was generated by sampling from a a two component CI
model (the true model is not used in evaluations). The data consist of a two-state discrete
dimension and a continuous dimension. First 100 data sets where generated by sampling
from a mixture model with parameters: [?1 , ?2 ] = [0.6, 0.4] as weights, ?1 = [0.8, 0.2]
and ?2 = [0.4, 0.6] as discrete state probabilities, [?1 , ?12 ] = [10, 5] and [?2 , ?22 ] = [0, 7]
as mean and variance (Gaussian) for the continuous variable. Then the discrete dimension
probability of the second cluster was changed from ?2 = [0.4, 0.6] to ?0 2 = [0.5, 0.5]
keeping the remaining parameters fixed and an additional 100 data sets were generated by
sampling from this altered model. This is a fairly small change in the distribution and the
underlying LogP scores appear to be very similar as can be seen in Figure 1. The figure
shows LogP distributions for the first two data sets generated from the first model (top row)
and the first two data sets generated from the second model (bottom row). Plots within each
0
0
?1
?5
<log(KS probability)>
<log(KS probability)>
?2
?3
?4
?10
?5
(b)
(a)
?6
0
20
40
60
80
100
Data set Dt
120
140
160
180
?15
200
0
0
20
40
60
80
100
Data set Dt
120
140
160
180
200
40
60
80
100
Data set Dt
120
140
160
180
200
0
?5
?2
?10
?15
<log(KS probability)>
<log(KS probability)>
?4
?6
?8
?20
?25
?30
?35
?10
?40
?12
?45
(c)
?14
0
20
40
60
80
100
Data set Dt
120
140
160
180
200
?50
(d)
0
20
Figure 2: Average log(KS probability) over the in-sample period for four experiments on
synthetic data, varying the number of data points per data set: a) 1,000; b) 5,000; c) 10,000;
d) 50,000. The dotted vertical line separates in-sample and out-of-sample periods. Note
that y-axes have different scales in order to show full variability of the data.
row should be more similar than plots from different rows, but this is difficult to discern by
visual inspection.
Algorithms 1 and 2 were evaluated by using the first 10 data sets to estimate a two component model. Then pairwise KS measures were calculated between all possible data set
pairs relative to the estimated model. Figure 2 shows average KS measures over in-sample
data sets (first 10) for four experiments varying the number of data points in each experiment. Note that the vertical axes are different in each of the plots to better show the range
of values. As the number of data points in the data set increases, the change that occurs
at t = 101 becomes more apparent. At 50,000 data points (bottom right plot of Figure 2)
the change in the distribution becomes easily detectable. Since this number of data points
is typically considered to be small compared to the number of data points in our real life
applications we expect to be able to detect such slight distribution changes.
3.2 Experiments on Real Life Data
Figure 3 shows a distribution for a typical day from a content web-site. There are almost
50,000 data points in the data set with over 100 dimensions each. The LogP score distribution is similar to that of synthetic data in Figure 1 which is a consequence of the CI model
used. Note, however, that in this data set the true generating distribution is not known
and is unlikely to be purely a CI model. Therefore, the average log KS measure over insample data has much lower values (see Figure 3 right, and plots in Figure 2). Another
way to phrase this observation is to note that since the true generating data distribution is
most likely not CI, the observed similarity of LogP distributions (the KS measure) is much
lower since there are two factors of dissimilarity: 1) different data sets; 2) inability of the
CI model to capture all the aspects of the true data distribution. Nonetheless, the first 31
?100
5000
?200
4500
4000
?300
<log(KS probability)>
3500
Count
3000
2500
2000
?400
?500
?600
1500
1000
?700
500
0
?100
?800
?80
?60
?40
?20
LogP
0
20
40
60
0
10
20
30
40
50
Data set D
60
70
80
90
100
t
Figure 3: Left: distribution of 42655 LogP scores from mixture of conditional independence models. The data is a single-day of click-stream data from a commercial web site.
Right: Average log(KS probability) over the 31 day in-sample period for a content website showing a glitch on day 27 and a permanent change on day 43, both detected by the
proposed methodology.
data sets (one month of data) that were used to build the initial model ?0 can be used to
define the natural variability of the KS measures against which additional data sets can be
compared. The result is that in Figure 3 we clearly see a problem with the distribution on
day 27 (a glitch in the data) and a permanent change in the distribution on day 43. Both
of the detected changes correspond to real changes in the data, as verified by the commercial website operators. Automatic description of changes in the distribution and criteria for
automatic rebuilding of the model are beyond scope of this paper.
4 Related Work
Automatic detection of various types of data changes appear in the literature in several
different flavors. For example, novelty detection ([4], [8]) is the task of determining unusual
or novel data points relative to some model. This is closely related to the outlier detection
problem ([1], [5]) where the goal is not only to find unusual data points, but the ones that
appear not to have been generated by the data generating distribution. A related problem
has been addressed by [2] in the context of time series modeling where outliers and trends
can contaminate the model estimation. More recently mixture models have been applied
more directly to outlier detection [3].
The method proposed in this paper addesses a different problem. We are not interested in
new and unusual data points; on the contrary, the method is quite robust with respect to
outliers. An outlier or two do not necessarily mean that the underlying data distribution has
changed. Also, some of the distribution changes we are interested in detecting might be
considered uninteresting and/or not-novel; for example, a slight shift of the population as
a whole is something that we certainly detect as a change but it is rarely considered novel
unless the shift is drastic.
There is also a set of online learning algorithms that update model parameters as the new
data becomes available (for variants and additional references, e.g. [6]). In that framework there is no such concept as a data distribution change since the models are constantly
updated to reflect the most current distribution. For example, instead of detecting a slight
shift of the population as a whole, online learning algorithms update the model to reflect
the shift.
5 Conclusions
In this paper we introduced a model-based method for automatic distribution change detection in an online data environment. Given the LogP distribution data signature we further
showed how to compare different data sets relative to the model using KS statistics and how
to obtain a single measure of similarity between the new data and the model. Finally, we
discussed heuristics for change detection that become principled in the limit as the number
of possible data sets increases.
Experimental results over synthetic and real online data indicate that the proposed methodology is able to alert the analyst to slight distributional changes. This methodology may be
used as the basis of a system to automatically re-estimate parameters of a mixture model on
an ? as-needed? basis ? when the model fails to adequately represent the data after a certain
point in time.
References
[1] V. Barnett and T. Lewis. Outliers in statistical data. Wiley, 1984.
[2] A. G. Bruce, J. T. Conor, and R. D. Martin. Prediction with robustness towards outliers, trends,
and level shifts. In Proceedings of the Third International Conference on Neural Networks in
Financial Engineering, pages 564?577, 1996.
[3] I. V. Cadez, P. Smyth, and H. Mannila. Probabilistic modeling of transaction data with applications to profiling, visualization, and prediction. In F. Provost and R. Srikant, editors, Proceedings
of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 37?46. ACM, 2001.
[4] C. Campbell and K. P. Bennett. A linear programming approach to novelty detection. In T. K.
Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems
13, pages 395?401. MIT Press, 2001.
[5] T. Fawcett and F. J. Provost. Activity monitoring: Noticing interesting changes in behavior. In
Proceedings of the Fifth ACM SIGKDD International Conference on Knowledge Discovery and
Data Mining, pages 53?62, 1999.
[6] R. Neal and G. Hinton. A view of the em algorithm that justifies incremental, sparse and other
variants. In M. I. Jordan, editor, Learning in Graphical Models, pages 355?368. Kluwer Academic Publishers, 1998.
[7] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C:
The Art of Scientific Computing, Second Edition. Cambridge University Press, Cambridge, UK,
1992.
[8] B. Sch?olkopf, R. C. Williamson, A. J. Smola, J. Shawe-Taylor, and J. C. Platt. Support vector
method for novelty detection. In S. A. Solla, T. K. Leen, and K.-R. Mller, editors, Advances in
Neural Information Processing Systems 12, pages 582?588. MIT Press, 2000.
| 2008 |@word smirnov:1 d2:3 seek:1 bellevue:2 initial:2 contains:1 score:20 series:1 cadez:2 bradley:1 current:2 com:1 nt:3 must:5 numerical:1 additive:1 shape:2 plot:5 update:2 n0:5 device:1 website:2 inspection:1 detecting:2 insample:1 alert:1 become:1 introduce:1 pairwise:3 behavior:1 themselves:1 frequently:1 automatically:1 becomes:4 discover:1 notation:3 underlying:4 matched:1 mass:3 interpreted:1 finding:1 guarantee:1 contaminate:1 every:1 uk:1 platt:1 appear:3 educated:1 engineering:1 local:1 limit:3 consequence:2 solely:1 approximately:1 might:3 k:26 quantified:1 range:2 practice:1 mannila:1 empirical:3 significantly:1 convenient:1 confidence:2 cannot:1 close:1 operator:1 put:2 context:2 equivalent:1 map:1 logpi:2 maximizing:1 go:1 formulate:2 simplicity:1 financial:1 population:6 updated:2 pt:1 commercial:2 smyth:1 programming:1 probability1:1 trend:2 distributional:1 bottom:3 observed:1 capture:1 calculate:2 region:11 solla:1 valuable:1 principled:3 environment:1 complexity:12 signature:3 predictive:1 purely:2 basis:3 easily:2 multimodal:1 various:2 kolmogorov:1 distinct:1 describe:1 detected:2 rebuilding:1 firm:1 dt0:3 heuristic:3 apparent:1 quite:1 statistic:8 itself:1 online:6 sequence:1 propose:1 mks:8 uci:1 description:2 olkopf:1 recipe:1 cluster:1 generating:7 incremental:1 predicted:1 implies:1 come:2 quantify:1 indicate:1 closely:1 correct:1 attribute:1 bin:1 require:1 clarify:1 lying:1 practically:1 considered:3 ic:1 ground:1 algorithmic:1 scope:1 mller:1 estimation:3 tool:1 mit:2 clearly:1 gaussian:2 rather:1 varying:2 ax:2 likelihood:7 sigkdd:2 detect:4 posteriori:1 dependent:1 streaming:1 vetterling:1 typically:2 unlikely:1 relation:1 interested:2 overall:3 classification:1 art:1 fairly:1 equal:1 once:1 sampling:3 barnett:1 represents:2 broad:1 igor:1 inherent:1 individual:1 phase:1 consisting:1 detection:11 mining:2 evaluation:3 certainly:2 mixture:18 truly:2 daily:1 unless:1 loosely:1 logarithm:1 taylor:1 re:1 fitted:2 formalism:1 modeling:3 conor:1 logp:36 maximization:1 phrase:1 applicability:2 uninteresting:1 seventh:1 synthetic:8 combined:1 st:1 density:1 peak:2 international:3 probabilistic:6 off:1 reflect:3 satisfied:1 leading:1 account:1 distribute:1 inc:2 permanent:2 stream:3 multiplicative:1 view:1 observing:1 portion:1 start:1 aggregation:1 bruce:1 contribution:1 ass:1 ni:2 variance:5 efficiently:1 correspond:1 generalize:1 iid:1 monitoring:2 confirmed:1 against:1 nonetheless:1 di:4 irvine:1 dataset:3 knowledge:2 campbell:1 disposal:1 higher:2 dt:16 day:7 methodology:7 leen:2 done:2 evaluated:1 smola:1 web:3 defines:1 scientific:1 building:2 dietterich:1 concept:1 true:8 adequately:3 hence:4 neal:1 conditionally:1 width:1 criterion:3 pdf:5 novel:3 recently:1 functional:2 tail:3 slight:4 discussed:1 kluwer:1 refer:1 significant:2 cambridge:2 automatic:5 populated:1 shawe:1 dj:1 similarity:8 etc:2 something:1 showed:1 certain:2 came:1 life:3 accomplished:1 seen:1 additional:3 determine:3 novelty:3 period:6 full:2 desirable:1 d0:6 match:1 academic:1 calculation:1 cross:1 profiling:1 dept:1 ofp:1 equally:1 prediction:2 variant:2 expectation:1 histogram:5 represent:5 sometimes:1 fawcett:1 preserved:1 addition:1 interval:3 addressed:1 source:1 crucial:1 publisher:1 sch:1 operate:1 contrary:1 jordan:1 identically:1 independence:1 fit:4 click:1 shift:5 t0:20 whether:3 motivated:1 passing:1 proceed:2 useful:1 nonparametric:1 pks:2 srikant:1 dotted:1 estimated:4 track:1 per:2 discrete:4 four:2 cutoff:1 verified:1 sum:1 parameterized:1 powerful:1 noticing:1 discern:1 almost:1 parsimonious:1 decision:1 activity:1 infinity:1 x2:1 aspect:1 argument:1 optimality:1 martin:1 disconnected:1 smaller:1 em:3 outlier:7 equation:1 visualization:1 describing:1 count:5 fail:1 detectable:1 needed:2 drastic:1 unusual:3 generalizes:1 available:3 gaussians:1 apply:2 anymore:1 robustness:1 top:2 clustering:2 remaining:1 graphical:1 build:2 occurs:1 parametric:2 distance:1 separate:1 street:1 toward:2 analyst:1 index:1 glitch:2 setup:1 difficult:1 vertical:2 observation:3 discarded:1 situation:1 hinton:1 variability:6 discovered:2 provost:2 arbitrary:1 sharp:1 introduced:2 cast:1 pair:1 optimized:1 california:1 able:4 beyond:1 summarize:1 built:1 gaining:1 t20:2 memory:2 business:2 natural:6 altered:1 ne:1 tresp:1 prior:1 literature:1 discovery:2 determining:4 relative:5 asymptotic:1 expect:1 interesting:1 limitation:1 proportional:1 validation:1 sufficient:1 principle:1 editor:4 share:1 row:6 summary:2 changed:2 last:1 keeping:1 allow:1 fifth:1 sparse:1 distributed:1 dimension:5 xn:1 evaluating:1 cumulative:1 calculated:1 author:1 preprocessing:1 transaction:1 preferred:1 keep:1 overfitting:3 incoming:1 xi:3 continuous:3 nature:1 lks:3 robust:1 ca:1 ean:1 williamson:1 complex:1 necessarily:2 dense:7 whole:2 edition:1 x1:1 site:2 wiley:1 fails:2 exponential:1 exercise:1 third:1 specific:3 showing:1 intrinsic:2 consist:2 ci:7 dissimilarity:1 justifies:1 flavor:1 flannery:1 likely:1 forming:1 visual:1 tracking:2 scalar:1 monotonic:1 constantly:2 lewis:1 cdf:1 acm:3 teukolsky:1 conditional:1 sorted:1 month:1 goal:1 consequently:1 quantifying:1 towards:1 bennett:1 feasible:1 change:23 experimentally:1 included:1 determined:1 typical:1 content:2 called:1 total:1 pas:1 experimental:2 rarely:1 support:1 latter:1 inability:1 evaluate:1 d1:3 |
1,106 | 2,009 | MIME: Mutual Information Minimization
and Entropy Maximization for Bayesian
Belief Propagation
Anand Rangarajan
Dept. of Computer and Information Science and Engineering
University of Florida
Gainesville, FL 32611-6120, US
anand@cise.ufl.edu
Alan L. Yuille
Smith-Kettlewell Eye Research Institute
2318 Fillmore St.
San Francisco, CA 94115, US
yuille@ski.org
Abstract
Bayesian belief propagation in graphical models has been recently
shown to have very close ties to inference methods based in statistical physics. After Yedidia et al. demonstrated that belief propagation fixed points correspond to extrema of the so-called Bethe
free energy, Yuille derived a double loop algorithm that is guaranteed to converge to a local minimum of the Bethe free energy.
Yuille?s algorithm is based on a certain decomposition of the Bethe
free energy and he mentions that other decompositions are possible and may even be fruitful. In the present work, we begin with
the Bethe free energy and show that it has a principled interpretation as pairwise mutual information minimization and marginal
entropy maximization (MIME). Next, we construct a family of free
energy functions from a spectrum of decompositions of the original
Bethe free energy. For each free energy in this family, we develop
a new algorithm that is guaranteed to converge to a local minimum. Preliminary computer simulations are in agreement with
this theoretical development.
1
Introduction
In graphical models, Bayesian belief propagation (BBP) algorithms often (but not
always) yield reasonable estimates of the marginal probabilities at each node [6].
Recently, Yedidia et al. [7] demonstrated an intriguing connection between BBP
and certain inference methods based in statistical physics. Essentially, they demonstrated that traditional BBP algorithms can be shown to arise from approximations
of the extrema of the Bethe and Kikuchi free energies. Next, Yuille [8] derived new
double-loop algorithms which are guaranteed to minimize the Bethe and Kikuchi
energy functions while continuing to have close ties to the original BBP algorithms.
Yuille?s approach relies on a certain decomposition of the Bethe and Kikuchi free
energies. In the present work, we begin with a new principle?pairwise mutual information minimization and marginal entropy maximization (MIME)?and derive
a new energy function which is shown to be equivalent to the Bethe free energy.
After demonstrating this connection, we derive a family of free energies closely related to the MIME principle which also shown to be equivalent, when constraint
satisfaction is exact, to the Bethe free energy. For each member in this family of
energy functions , we derive a new algorithm that is guaranteed to converge to a
local minimum. Moreover, the resulting form of the algorithm is very simple despite
the somewhat unwieldy nature of the algebraic development. Preliminary comparisons of the new algorithm with BBP were carried out on spin glass-like problems
and indicate that the new algorithm is convergent when BBP is not. However, the
effectiveness of the new algorithms remains to be seen.
2
Bethe free energy and the MIME principle
In this section, we show that the Bethe free energy can be interpreted as pairwise
mutual information minimization and marginal entropy maximization.
The Bethe free energy for Bayesian belief propagation is written as
FBethe ({pij , pi , ?ij , ?ij }) =
P
P
P
P
pij (xi ,xj )
pi (xi )
ij:i>j
xi ,xj pij (xi , xj ) log ?ij (xi ,xj ) ?
i (ni ? 1)
xi pi (xi ) log ?i (xi )
P
P
P
+ ij:i>j xj ?ij (xj )[ xi pij (xi , xj ) ? pj (xj )]
P
P
P
+ ij:i>j xi ?ji (xi )[ xj pij (xi , xj ) ? pi (xi )]
P
P
+ ij:i>j ?ij ( xi ,xj pij (xi , xj ) ? 1)
(1)
def
where ?ij (xi , xj ) = ?ij (xi , xj )?i (xi )?j (xj ) and ni is the number of neighbors of
node i. Link functions ?ij > 0 are available relational data between nodes i and j.
The
P singleton function ?i is also available at each node i. The double summation
ij:i>j is carried out only over the nodes that are connected. The Lagrange parameters {?ij , ?ij } are needed in the Bethe free energy (1) to satisfy the following
constraints relating the joint probabilities {pij } with the marginals {pi }:
X
X
X
pij (xi , xj ) = pj (xj ),
pij (xi , xj ) = pi (xi ), and
pij (xi , xj ) = 1. (2)
xi
xi ,xj
xj
The pairwise mutual information is defined as
X
pij (xi , xj )
M Iij =
pij (xi , xj ) log
p
i (xi )pj (xj )
x ,x
i
(3)
j
The mutual information is minimized when the joint probability pij (xi , xj ) =
pi (xi )pj (xj ) or equivalently when nodes i and j are independent. When nodes i and
j are connected via a non-separable link ?ij (xi , xj ) they will not be independent.
We now state the MIME principle.
Statement of the MIME principle: Maximize the marginal entropy and minimize the pairwise mutual information using the available marginal and pairwise
link function expectations while satisfying the joint probability constraints.
The pairwise MIME principle leads to the following free energy:
FMIME ({pij , pi , ?ij , ?ij }) =
P P
P
P
pij (xi ,xj )
i
xi pi (xi ) log pi (xi )
ij:i>j
xi ,xj pij (xi , xj ) log pi (xi )pj (xj ) +
P
P
P P
? ij:i>j xi ,xj pij (xi , xj ) log ?ij (xi , xj ) ? i xi pi (xi ) log ?i (xi )
P
P
P
+ ij:i>j xj ?ij (xj )[ xi pij (xi , xj ) ? pj (xj )]
P
P
P
+ ij:i>j xi ?ji (xi )[ xj pij (xi , xj ) ? pi (xi )]
P
P
+ ij:i>j ?ij ( xi ,xj pij (xi , xj ) ? 1).
(4)
In the above free energy, we minimize the pairwise mutual information and maximize
the marginal entropies. The singleton and pairwise link functions are additional
information which do not allow the system to reach its ?natural? equilibrium?a
uniform i.i.d. distribution on the nodes. The Lagrange parameters enforce the
constraints between the pairwise and marginal probabilities. These constraints are
the same as in the Bethe free energy (1). Note that the Lagrange parameter terms
vanish if the constraints in (2) are exactly satisfied. This is an important point
when considering equivalences between different energy functions.
Lemma 1 Provided the constraints in (2) are exactly satisfied, the MIME free
energy in (4) is equivalent to the Bethe free energy in (1).
Proof: Using the fact that constraint satisfaction is exact and using the identity
pij (xi , xj ) = pji (xj , xi ), we may write
X X
X X
pij (xi , xj ) log pi (xi )
pij (xi , xj ) log pi (xi )pj (xj ) = ?
?
ij:i>j xi ,xj
ij:i6=j xi ,xj
=?
X
ni
X X
ij:i>j xi ,xj
pij (xi , xj ) log ?i (xi )?j (xj ) =
X
i
pi (xi ) log pi (xi ),
xi
i
and
X
ni
X
pi (xi ) log ?i (xi ).
(5)
xi
We have shown that a marginal entropy term emerges from the mutual information
term in (4) when constraint satisfaction is exact. Collecting the marginal entropy
terms together and rearranging the MIME free energy in (4), we get the Bethe free
energy in (1).
3
A family of decompositions of the Bethe free energy
Recall that the Bethe free energy and the energy function resulting from application
of the MIME principle were shown to be equivalent. However, the MIME energy
function is merely one particular decomposition of the Bethe free energy. As Yuille
mentions [8], many decompositions are possible. The main motivation for considering alternative decompositions is for algorithmic reasons. We believe that certain
decompositions may be more effective than others. This belief is based on our previous experience with closely related deterministic annealing algorithms [3, 2]. In
this section, we derive a family of free energies that are equivalent to the Bethe
free energy provided constraint satisfaction is exact. The family of free energies is
inspired by and closely related to the MIME free energy in (4).
Lemma 2 The following family of energy functions indexed by the free parameters
? > 0 and {?i } is equivalent to the original Bethe free energy (1) provided the
constraints in (2) are exactly satisfied and the parameters q and r are set to {q i =
(1 ? ?)ni } and {ri = 1 ? ni ?i } respectively.
Fequiv ({pij , pi , ?ij , ?ij }) =
P
P
P
ij:i>j
xi ,xj pij (xi , xj ) log [
xj
P P
P
pij (xiP
,xj )
pij (xi ,xj )]? [
pij (xi ,xj )]?
xi
P
+ i xi pi (xi ) log pi (xi ) ? i qi xi pi (xi ) log pi (xi )
P
P
?
? ij:i>j xi ,xj pij (xi , xj ) log ?ij (xi , xj )?i?i (xi )?j j (xj )
P P
? i ri xi pi (xi ) log ?i (xi )
P
P
P
+ ij:i>j xj ?ij (xj )[ xi pij (xi , xj ) ? pj (xj )]
P
P
P
+ ij:i>j xi ?ji (xi )[ xj pij (xi , xj ) ? pi (xi )]
P
P
+ ij:i>j ?ij ( xi ,xj pij (xi , xj ) ? 1).
(6)
In (6), the first term is no longer the pairwise mutual information as in (4). And
unlike (4), pi (xi ) no longer appears in the pairwise mutual information-like term.
P
P
Proof: We selectively substitute
xj pij (xi , xj ) =
xi pij (xi , xj ) = pj (xj ) and
pi (xi ) to show the equivalence. First
X X
X
X
X X
pij (xi , xj )]? [
pij (xi , xj ) log[
ij:i>j xi ,xj
xj
X X
pij (xi , xj )]? = ?
?
xj
i
?
pij (xi , xj ) log ?i i (xi )?j j (xj ) =
ij:i>j xi ,xj
pi (xi ) log pi (xi ),
ni
xi
X
n i ?i
i
X
pi (xi ) log ?i (xi ).
(7)
xj
Substituting the identities in (7) into (6), we see that the free energies are algebraically equivalent.
4
A family of algorithms for belief propagation
We now derive descent algorithms for the family of energy functions in (6). All
the algorithms are guaranteed to converge to a local minimum of (6) under mild
assumptions regarding the number of fixed points. For each member in the family
of energy functions, there is a corresponding descent algorithm. Since the form of
the free energy in (6) is complex and precludes easy minimization, we use algebraic
(Legendre) transformations [1] to simplify the optimization.
X
X
pij (xi , xj ) =
pij (xi , xj ) log
?
xj
xj
min?ji (xi ) ?
?
X
P
xj
pij (xi , xj ) log
xi
pij (xi , xj ) log ?ji (xi ) + ?ji (xi ) ?
X
pij (xi , xj ) =
P
xj
pij (xi , xj )
xi
min?ij (xj ) ?
P
xi
pij (xi , xj ) log ?ij (xj ) + ?ij (xj ) ?
P
xi
pij (xi , xj )
?pi (xi ) log pi (xi ) = min ?pi (xi ) log ?i (xi ) + ?i (xi ) ? pi (xi ).
?i (xi )
(8)
We now apply the above algebraic transforms. The new free energy is (after some
algebraic manipulations)
X X
pij (xi , xj )
Fequiv ({pij , pi , ?ij , ?i , ?ij , ?ij }) =
pij (xi , xj ) log
ij:i>j xi ,xj
? (x )? ? (x )
?ji
i
j
ij
+?
XX
ij:i6=j
?
X X
?ij (xj ) +
xi
XX
i
pi (xi ) log
xi
?
?
pij (xi , xj ) log ?ij (xi , xj )?i i (xi )?j j (xj ) ?
XX
ij:i>j
?ij (xj )[
xj
+
q
?i i (xi )
X
XX
pij (xi , xj ) ? pj (xj )] +
xi
ij:i>j
X X
ri
xi
qi
?i (xi )
xi
pi (xi ) log ?i (xi )
xi
?ji (xi )[
+
X X
i
i
ij:i>j xi ,xj
+
pi (xi )
X
pij (xi , xj ) ? pi (xi )]
xj
X
ij:i>j
?ij (
X
pij (xi , xj ) ? 1).
(9)
xi ,xj
We continue to keep the parameters {qi } and {ri } in (9). However, from Lemma 2,
we know that the equivalence of (9) to the Bethe free energy is predicated upon
appropriate setting of these parameters. In the rest of the paper, we continue to
use q and r for the sake of notational simplicity.
Despite the introduction of new variables via Legendre transforms, the optimization problem in (9) is still a minimization problem over all the variables. The
algebraically transformed energy function in (9) is separately convex w.r.t. {p ij , pi }
and w.r.t. {?ij , ?i } provided ? ? [0, 1]. Since the overall energy function is not
convex w.r.t. all the variables, we pursue an alternating algorithm strategy similar
to the double loop algorithm in Yuille [8]. The basic idea is to separately minimize
w.r.t. the variables {?ij , ?i } and the variables {pij , pi }. The linear constraints in
(2) are enforced when minimizing w.r.t the latter and do not affect the convergence
properties of the algorithm since the energy function w.r.t. {pij , pi } is convex .
We evaluate the fixpoints of {?ij , ?i }. Note that (9) is convex w.r.t. {?ij , ?i }.
X
X
pij (xi , xj ), and ?i (xi ) = pi (xi ).
(10)
pij (xi , xj ), ?ji (xi ) =
?ij (xj ) =
xj
xi
The fixpoints of {pij , pi } are evaluated next. Note that (9) is convex w.r.t. {pij , pi }.
?
?
?
= ?ji
(xi )?ij
(xj )?ij (xi , xj )?i?i (xi )?j j (xj )e??ij (xj )??ji (xi )??ij ?1
P
pi (xi ) = ?qi i (xi )?iri (xi )e k ?ki (xi )?1 .
(11)
pij (xi , xj )
The constraint satisfaction equations from (2) can be rewritten as
X
pij (xi , xj ) = pi (xi ) ?
xj
e
2?ji (xi )
=
P
xj
?
?
?
?
?ji
(xi )?ij
(xj )?ij (xi ,xj )?i i (xi )?j j (xj )e??ij (xj )??ij ?1
q
r
?i i (xi )?i i (xi )e
P
k6=j
?ki (xi )?1
(12)
Similar relations can be obtained for the other constraints in (2). Consider a Lagrange parameter update sequence where the Lagrange parameter currently being
updated is tagged as ?new? with the rest designated as ?old.? We can then rewrite
the Lagrange parameter updates using ?old? and ?new? values. Please note that
each Lagrange parameter update corresponds to one of the constraints in (2). It
can be shown that the iterative update of the Lagrange parameters is guaranteed
to converge to the unique solution of (2) [8]. While rewriting (12), we multiply the
old
left and right sides with e?2?ji (xi ) .
new
e2?ji
P
(xi )?2?old
ji (xi )
xj
=
?
??old (xj )??old (xi )?? old ?1
?
?
?
ji
ij
?ji
(xi )?ij
(xj )?ij (xi ,xj )?i i (xi )?j j (xj )e ij
q
r
?i i (xi )?i i (xi )e
P
k
?old (xi )?1
ki
.
(13)
Using (11), we relate each Lagrange parameter update with an update of p ij (xi , xj )
and pi (xi ). We again invoke the ?old? and ?new? designations, this time on the
probabilities. From (11), (12) and (13), we write the joint probability update
s
pnew
old
pold (xi )
ij (xi , xj )
??new
(x
)+?
(x
)
(14)
= e ji i ji i = P i old
old
pij (xi , xj )
xj pij (xi , xj )
and for the marginal probability update
new
old
pnew
(xi )
i
= e?ji (xi )??ji (xi ) =
pold
(x
)
i
i
sP
xj
pold
ij (xi , xj )
pold
i (xi )
(15)
.
From (14) and (15), the update equations for the probabilities are
s
s
X
pold (xi )
new
old
pij (xi , xj ) = pij (xi , xj ) P i old
, pnew
(x
)
=
pold
pold
i
i
i (xi )
ij (xi , xj )
xj pij (xi , xj )
x
j
(16)
With the probability updates in place, we may write down new algorithms minimizing the family of Bethe equivalent free energies using only probability updates.
The update equations (16) can be seen to satisfy the first constraint in (2). Similar
update equations can be derived for the other constraints in (2). For each Lagrange
parameter update, an equivalent, simultaneous probability (joint and marginal)
update can be derived similar to (16). The overall family of algorithms can be summarized as shown in the pseudocode. Despite the unwieldy algebraic development
preceding it, the algorithm is very simple and straightforward.
Set free parameters ? ? [0, 1] and {?i }.
Initialize {pij , pi }. Set {qi = (1 ? ?)ni } and {ri = 1 ? ni ?i }.
Begin A: Outer Loop
P
?ij (xj ) ? xi pij (xi , xj )
P
?ji (xi ) ? xj pij (xi , xj )
?i (xi ) ? pi (xi )
?
?
?
pij (xi , xj ) ? ?ji
(xi )?ij
(xj )?ij (xi , xj )?i?i (xi )?j j (xj )
pi (xi ) ? ?qi i (xi )?iri (xi )
P
P
Begin B: Inner Loop: Do B until N1 ij:i>j [( xj pij (xi , xj ) ?
P
pi (xi ))2 + ( xi pij (xi , xj ) ? pj (xj ))2 ] < cthr
Simultaneously update pij (xi , xj ) and pi (xi ) below.
r
pij (xi , xj ) ? pij (xi , xj ) P pi (xi )
x
pij (xi ,xj )
j
q
P
pi (xi ) ? pi (xi ) xj pij (xi , xj )
Simultaneously update pij (xi , xj ) and pj (xj ) below.
r
p (x )
pij (xi , xj ) ? pij (xi , xj ) P j j
x
pij (xi ,xj )
i
q
P
pj (xj ) ? pj (xj ) xi pij (xi , xj )
Normalize pij (xi , xj ).
p (x ,x )
pij (xi , xj ) ? P ij i j
xi ,xj
pij (xi ,xj )
End B
End A
In the above family of algorithms, the MIME algorithm corresponds to free parameter settings ? = 1 and ?i = 0 which in turn lead to parameter settings qi = 0
and ri = 1. The Yuille [8] double loop algorithm corresponds to the free parameter
settings ? = 0 and ?i = 0 which in turn leads to parameter settings qi = ni and
ri = 1. A crucial point is that the energy function for every valid parameter setting
is equivalent to the Bethe free energy provided constraint satisfaction is exact. The
inner loop constraint satisfaction threshold parameter cthr setting is very important
in this regard. We are obviously not restricted to the MIME parameter settings.
At this early stage of exploration of the inter-relationships between Bayesian belief
propagation and inference methods based in statistical physics [7], it is premature
to speculate regarding the ?best? parameter settings for ? and {?i }. Most likely,
the effectiveness of the algorithms will vary depending on the problem setting which
enters into the formulation via the link functions {?ij } and the singleton functions
{?i }.
5
Results
We implemented the family of algorithms in C++ and conducted tests on locally
connected 50 node graphs and binary state variables. The ?i (xi ) and ?ij (xi , xj ) are
of the form e?hi and e?hij where hi and hij are drawn from uniform distributions
(in the interval [?1, 1]). Provided the constraint satisfaction theshold parameter
cthr was set low enough, the algorithm (for ? = 1 and other parameter settings
as described in Figure 1) exhibited monotonic convergence. Figure 2 shows the
number of inner loop iterations corresponding to different settings of the constraint
satisfaction threshold parameter. We also implemented the BBP algorithm and
empirically observed that it often did not converge for these graphs. These results
are quite preliminary and far more validation experiments are required. However,
they provide a proof of concept for our approach.
6
Conclusion
We began with the MIME principle and showed the equivalence of the MIMEbased free energy to the Bethe free energy assuming constraint satisfaction to be
exact. Then, we derived new decompositions of the Bethe free energy inspired
by the MIME principle, and driven by our belief that certain decompositions may
be more effective than others. We then derived a convergent algorithm for each
member in the family of MIME-based decompositions. It remains to be seen if the
MIME-based algorithms are efficient for a reasonable class of problems. While the
MIME-based algorithms derived here use closed-form solutions in the constraint
satisfaction inner loop, it may turn out that the inner loop is better handled using
preconditioned gradient-based descent algorithms. And it is important to explore
the inter-relationships between the convergent MIME-based descent algorithms and
other recent related approaches with interesting convergence properties [4, 5].
References
[1] E. Mjolsness and C. Garrett. Algebraic transformations of objective functions. Neural
Networks, 3:651?669, 1990.
[2] A. Rangarajan. Self annealing and self annihilation: unifying deterministic annealing
and relaxation labeling. Pattern Recognition, 33:635?649, 2000.
[3] A. Rangarajan, S. Gold, and E. Mjolsness. A novel optimizing network architecture
with applications. Neural Computation, 8(5):1041?1060, 1996.
[4] Y. W. Teh and M. Welling. Passing and bouncing messages for generalized inference.
Technical Report GCNU 2001-01, Gatsby Computational Neuroscience Unit, University College, London, 2001.
[5] M. Wainwright, T. Jaakola, and A. Willsky. Tree-based reparameterization framework
for approximate estimation of stochastic processes on graphs with cycles. Technical
Report LIDS P-2510, MIT, Cambridge, MA, 2001.
[6] Y. Weiss. Correctness of local probability propagation in graphical models with loops.
Neural Computation, 12:1?41, 2000.
[7] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Bethe free energy, Kikuchi approximations and belief propagation algorithms. In Advances in Neural Information Processing
Systems 13, Cambridge, MA, 2001. MIT Press.
[8] A. L. Yuille. A double loop algorithm to minimize the Bethe and Kikuchi free energies.
Neural Computation, 2001. (submitted).
?0.5
0
500
1000
MIME energy
?0.4
MIME energy
?0.4
MIME energy
?0.4
?0.5
1500
0
500
1000
?0.5
1500
0
500
1000
iteration
iteration
iteration
(a)
(b)
(c)
1500
Figure 1: MIME energy versus outer loop iteration: 50 node, local topology,
? = 1. Constraint satisfaction threshold parameter cthr was set to (a) 10?8 (b) 10?4
(c) 10?2
7
20
2
18
1.9
6
1.8
12
10
8
5
total # of inner loop iterations
14
total # of inner loop iterations
total # of inner loop iterations
16
4
3
6
1.7
1.6
1.5
1.4
1.3
4
1.2
2
2
0
1.1
0
500
1000
outer loop iteration index
(a)
1500
1
0
500
1000
outer loop iteration index
(b)
1500
1
0
500
1000
1500
outer loop iteration index
(c)
Figure 2: Inner loop iterations versus outer loop: 50 node, local topology,
? = 1. Constraint satisfaction threshold parameter cthr was set to (a) 10?8 (b)
10?4 (c) 10?2
| 2009 |@word mild:1 fixpoints:2 simulation:1 gainesville:1 decomposition:12 pold:7 mention:2 intriguing:1 written:1 update:17 smith:1 node:11 org:1 kettlewell:1 pairwise:12 inter:2 inspired:2 freeman:1 considering:2 begin:4 provided:6 moreover:1 xx:4 interpreted:1 pursue:1 extremum:2 transformation:2 every:1 collecting:1 tie:2 exactly:3 unit:1 engineering:1 local:7 despite:3 equivalence:4 jaakola:1 unique:1 get:1 close:2 fruitful:1 equivalent:10 demonstrated:3 deterministic:2 straightforward:1 iri:2 convex:5 simplicity:1 reparameterization:1 updated:1 exact:6 agreement:1 satisfying:1 recognition:1 observed:1 enters:1 connected:3 cycle:1 mjolsness:2 principled:1 fbethe:1 rewrite:1 yuille:10 upon:1 joint:5 effective:2 london:1 labeling:1 quite:1 precludes:1 obviously:1 sequence:1 loop:21 gold:1 normalize:1 convergence:3 double:6 rangarajan:3 kikuchi:5 derive:5 develop:1 depending:1 ij:87 implemented:2 indicate:1 closely:3 stochastic:1 exploration:1 preliminary:3 summation:1 annihilation:1 equilibrium:1 algorithmic:1 substituting:1 vary:1 early:1 estimation:1 currently:1 correctness:1 minimization:6 mit:2 always:1 derived:7 notational:1 glass:1 inference:4 xip:1 relation:1 transformed:1 overall:2 k6:1 development:3 initialize:1 mutual:11 marginal:12 construct:1 minimized:1 others:2 report:2 simplify:1 simultaneously:2 n1:1 message:1 multiply:1 experience:1 indexed:1 tree:1 continuing:1 old:15 theoretical:1 maximization:4 uniform:2 conducted:1 st:1 physic:3 invoke:1 together:1 again:1 satisfied:3 singleton:3 speculate:1 summarized:1 satisfy:2 mime:25 closed:1 minimize:5 spin:1 ni:10 correspond:1 yield:1 bayesian:5 submitted:1 simultaneous:1 reach:1 energy:59 e2:1 gcnu:1 proof:3 recall:1 emerges:1 garrett:1 appears:1 wei:2 formulation:1 evaluated:1 predicated:1 stage:1 until:1 propagation:9 believe:1 concept:1 tagged:1 alternating:1 self:2 please:1 generalized:1 novel:1 recently:2 began:1 pseudocode:1 ji:24 empirically:1 he:1 interpretation:1 relating:1 marginals:1 cambridge:2 i6:2 longer:2 showed:1 recent:1 optimizing:1 driven:1 manipulation:1 certain:5 binary:1 continue:2 seen:3 minimum:4 additional:1 somewhat:1 preceding:1 converge:6 maximize:2 algebraically:2 alan:1 technical:2 dept:1 qi:8 basic:1 essentially:1 expectation:1 iteration:12 separately:2 annealing:3 interval:1 crucial:1 rest:2 unlike:1 exhibited:1 ufl:1 member:3 anand:2 effectiveness:2 easy:1 enough:1 xj:179 affect:1 architecture:1 topology:2 inner:9 regarding:2 idea:1 handled:1 algebraic:6 passing:1 transforms:2 locally:1 neuroscience:1 theshold:1 write:3 pnew:3 demonstrating:1 threshold:4 drawn:1 pj:14 rewriting:1 graph:3 relaxation:1 merely:1 enforced:1 bouncing:1 place:1 family:16 reasonable:2 fl:1 def:1 ki:3 guaranteed:6 hi:2 convergent:3 constraint:25 ri:7 sake:1 min:3 separable:1 designated:1 legendre:2 lid:1 restricted:1 equation:4 remains:2 turn:3 needed:1 know:1 end:2 available:3 rewritten:1 yedidia:3 apply:1 enforce:1 appropriate:1 alternative:1 pji:1 florida:1 substitute:1 original:3 graphical:3 unifying:1 objective:1 strategy:1 traditional:1 gradient:1 link:5 outer:6 reason:1 preconditioned:1 willsky:1 assuming:1 index:3 relationship:2 minimizing:2 equivalently:1 statement:1 relate:1 hij:2 ski:1 teh:1 descent:4 relational:1 required:1 connection:2 below:2 pattern:1 belief:10 wainwright:1 satisfaction:13 natural:1 eye:1 carried:2 designation:1 interesting:1 versus:2 validation:1 pij:86 principle:9 pi:56 free:46 side:1 allow:1 institute:1 neighbor:1 regard:1 valid:1 san:1 premature:1 far:1 welling:1 approximate:1 keep:1 francisco:1 xi:248 spectrum:1 iterative:1 bethe:29 nature:1 ca:1 rearranging:1 complex:1 sp:1 did:1 main:1 motivation:1 arise:1 fillmore:1 gatsby:1 iij:1 vanish:1 unwieldy:2 down:1 entropy:8 likely:1 explore:1 lagrange:10 monotonic:1 corresponds:3 relies:1 ma:2 identity:2 cise:1 bbp:7 lemma:3 called:1 total:3 selectively:1 college:1 latter:1 evaluate:1 |
1,107 | 201 | 364
Jain and Waibel
Incremental Parsing by Modular Recurrent
Connectionist Networks
Ajay N. Jain Alex H. Waibel
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
ABSTRACT
We present a novel, modular, recurrent connectionist network architecture which learns to robustly perform incremental parsing of complex
sentences. From sequential input, one word at a time, our networks
learn to do semantic role assignment, noun phrase attachment, and
clause structure recognition for sentences with passive constructions and
center embedded clauses. The networks make syntactic and semantic
predictions at every point in time, and previous predictions are revised
as expectations are affirmed or violated with the arrival of new information. Our networks induce their own "grammar rules" for dynamically
transforming an input sequence of words into a syntactic/semantic interpretation. These networks generalize and display tolerance to input
which has been corrupted in ways common in spoken language.
1 INTRODUCTION
Previously, we have reported on experiments using connectionist models for a small parsing task using a new network formalism which extends back-propagation to better fit the
needs of sequential symbolic domains such as parsing (Jain, 1989). We showed that connectionist networks could learn the complex dynamic behavior needed in parsing. The
task included passive sentences which require dynamic incorporation of previously unseen right context information into partially built syntactic/semantic interpretations. The
trained parsing network exhibited predictive behavior and was able to modify or confirm
Incremental Parsing by Modular Recurrent Connectionist Networks
IInterclause Units I
I
Clause RolfS Units
Clause M
Phra-s-e1-'1 .. . Phn-,-eJ-"
Clause 1
Phra-s-e1""""11 .. . Phn-s-e-'1
r-I
r-I
I
I
II
r-I
Clause Structure Units
r-I
II
IPhrase Level Gating U n l t s l t - - - - - - - 1
Word Level
, Word Units
I
Figure 1: High-level Parsing Architecture.
hypotheses as sentences were sequentially processed. It was also able to generalize well
and tolerate iII-formed input
In this paper, we describe work on extending our parsing architecture to grammatically
complex sentences. 1 The paper is organized as follows. First, we briefly outline the
network formalism and the general architecture. Second, the parsing task is defined and
the procedure for constructing and training the parser is presented. Then the dynamic
behavior of the parser is illustrated, and the performance is characterized.
2
NETWORK ARCHITECTURE
We have developed an extension to back-propagation networks which is specifically
designed to perform tasks in sequential domains requiring symbol manipulation (Jain,
1989). It is substantially different from other connectionist approaches to sequential
problems (e.g. Elman, 1988; Jordan, 1986; Waibel et al., 1989). There are four major
features of this formalism. One, units retain partial activation between updates. They
can respond to repetitive weak stimuli as well as singular sharp stimuli. Two, units
are responsive to both static activation values of other units and their dynamic changes.
Three, well-behaved symbol buffers can be constructed using groups of units whose
connections are gated by other units. Four. the formalism supports recurrent networks.
The networks are able to learn complex time-varying behavior using a gradient descent
procedure via error back-propagation.
Figure 1 shows a high-level diagram of the general parsing architecture. It is organized
into five hierarchical levels: Word, Phrase, Clause Structure, Clause Roles, and Inter1 Another
presentation of this work appears in Jain and Waibel (1990).
365
366
Jain and Waibel
clause. The description will proceed bottom up. A word is presented to the network by
stimulating its associated word unit for a short time. This produces a pattern of activation
across the feature units which represents the meaning of the word. The connections from
the word units to the feature units which encode semantic and syntactic information about
words are compiled into the network and are fixed. 2 The Phrase level uses the sequence
of word representations from the Word level to build contiguous phrases. Connections
from the Word level to the Phrase level are modulated by gating units which learn the
required conditional assignment behavior. The Clause Structure level maps phrases into
the constituent clauses of the input sentence. The Clause Roles level describes the roles
and relationships of the phrases in each clause of the sentence. The final level, Interclause, represents the interrelationships among the clauses. The following section defines
a parsing task and gives a detailed description of the construction and training of a parsing
network which performs the task.
3 INCREMENTAL PARSING
In parsing spoken language, it is desirable to process input one word at a time as words
are produced by the speaker and to incrementally build an output representation. This
allows tight bi-directional coupling of the parser to the underlying speech recognition
system. In such a system, the parser processes information as soon as it is produced and
provides predictive information to the recognition system based on a rich representation
of the current context As mentioned earlier, our previous work applying connectionist architectures to a parsing task was promising. The experiment described below extends our
previous work to grammatically complex sentences requiring a significant scale increase.
3.1
Parsing Task
The domain for the experiment was sentences with up to three clauses including nontrivial center-embedding and passive constructions.3 Here are some example sentences:
? Fido dug up a bone near the tree in the garden.
? I know the man who John says Mary gave the book.
? The dog who ate the snake was given a bone.
Given sequential input, one word at a time, the task is to incrementally build a representation of the input sentence which includes the following infonnation: phrase structure,
clause structure, semantic role assignment, and interclause relationships. Figure 2 shows
a representation of the desired parse of the last sentence in the list above.
2Connectionist networks have been used for lexical acquisition successfully (Miikkulainen and Dyer, 1989).
However, in building large systems, it makes sense from an efficiency perspective to precompile as much lexical
information as possible into a network. This is a pragmatic design choice in building large systems.
3The training set contained over 200 sentences. These are a subset of the sentences which form the example
set of a parser based on a left associative grammar (Hausser, 1988). These sentences are grammatically
interesting, but they do not reflect the statistical structure of common speech.
Incremental Parsing by Modular Recurrent Connectionist Networks
[Clause 1:
[Clause 2:
[The dog RECIP] [was given ACTION] [a bone PATIENT]]
[who AGENT] [ate ACTION] [the snake PATIENT]
(RELATIVE to Clause 1, Phrase 1))
Figure 2: Representation of an Example Sentence.
3.2
Constructing the Parser
The architecture for the network follows that given in Figure 1. The following paragraphs
describe the detailed network structure bottom up. The constraints on the numbers of
objects and labels are fixed for a particular network. but the architecture itself is scalable.
Wherever possible in the network construction. modularity and architectural constraints
have been exploited to minimize training time and maximize generalization. A network
was constructed from three separate recurrent subnetworks trained to perform a portion
of the parsing task on the training sentences. The performance of the full network will
be discussed in detail in the next section.
The Phrase level contains three types of units: phrase block units. gating units. and hidden
units. There are 10 phrase blocks. each being able to capture up to 4 words forming
a phrase. The phrase blocks contain sets of units (called slots) whose target activation
patterns correspond to word feature patterns of words in phrases. Each slot has an
associated gating unit which learns to conditionally assign an activation pattern from the
feature units of the Word level to the slot. The gating units have input connections from
the hidden units. The hidden units have input connections from the feature units. gating
units, and phrase block units. The direct recurrence between the gating and hidden units
allows the gating units to learn to inhibit and compete with one another. The indirect
recurrence arising from the connections between the phrase blocks and the hidden units
provides the context of the current input word. The target activation values for each
gating unit are dynamically calculated during training; each gating unit must learn to
become active at the proper time in order to perform the phrasal parsing. Each phrase
block with its associated gating and hidden units has its weights slaved to the other phrase
blocks in the Phrase level. Thus. if a particular phrase construction is only present in one
position in the training set. all of the phrase blocks still learn to parse the construction.
The Clause Roles level also has shared weights among separate clause modules. This
level is trained by simulating the sequential building and mapping of clauses to sets of
units containing the phrase blocks for each clause (see Figure 1). There are two types
of units in this level: labeling units and hidden units. The labeling units learn to label
the phrases of the clauses with semantic roles and attach phrases to other (within-clause)
phrases. For each clause. there is a set of units which assigns role labels (agent. patient.
recipient. action) to phrases. There is also a set of units indicating phrasal modification.
The hidden units are recurrently connected to the labeling units to provide context and
competition as with the Phrase level; they also have input connections from the phrase
blocks of a single clause. During training. the targets for the labeling units are set at
the beginning of the input presentation and remain static. In order to minimize global
error across the training set. the units must learn to become active or inactive as soon as
367
368
Jain and Waibel
possible in the input. This forces the network to learn to be predictive.
The Clause Structure and Interclause levels are trained simultaneously as a single module.
There are three types of units at this level: mapping, labeling, and hidden units. The
mapping units assign phrase blocks to clauses. The labeling units indicate relative clause
and a subordinate clause relationships. The mapping and labeling units are recurrently
connected to the hidden units which also have input connections from the phrase blocks
of the Phrase level. The behavior of the Phrase level is simulated during training of this
module. This module utilizes no weight sharing techniques. As with the Clause Roles
level, the targets for the labeling and mapping units are set at the beginning of input
presentation, thus inducing the same type of predictive behavior.
4 PARSING PERFORMANCE
The separately trained submodules described above were assembled into a single network
which performs the full parsing task. No additional training was needed to fine-tune the
full parsing network despite significant differences between actual subnetwork performance and the simulated subnetwork performance used during training. The network
successfully modeled the large diverse training set. This section discusses three aspects
of the parsing network's performance: dynamic behavior of the integrated network, generalization, and tolerance to noisy input.
4.1
Dynamic Behavior
The dynamic behavior of the network will be illustrated on the example sentence from
Figure 2: "The dog who ate the snake was given a bone." This sentence was not in
the training set. Due to space limitations, actual plots of network behavior will only be
presented for a small portion of the network.
Initially, all of the units in the network are at their resting values. The units of the phrase
blocks all have low activation. The word unit corresponding to "the" is stimulated,
causing its word feature representation to become active across the feature units of the
Word level. The gating unit associated with the slot 1 of phrase block 1 becomes active,
and the feature representation of "the" is assigned to the slot; the gate closes as the next
word is presented. The remaining words of the sentence are processed similarly, resulting
in the final Phrase level representation shown in Figure 2. While this is occurring, the
higher levels of the network are processing the evolving Phrase level representation.
The behavior of some of the mapping units of the Clause Structure Level is shown
in Figure 3. Early in the presentation of the first word, the Clause Structure level
hypothesizes that the first 4 phrase blocks will belong to the first clause-reflecting
the dominance of single clause sentences in the training set. After "the" is assigned
to the first phrase block, this hypothesis is revised. The network then believes that
there is an embedded clause of 3 (possibly 4) phrases following the first phrase. This
predictive behavior emerged spontaneously from the training procedure (a large majority
of sentences in the training set beginning with a determiner had embedded clauses after
the first phrase). The next two words ("dog who") confirm the network's expectation. The
word "ate" allows the network to firmly decide on an embedded clause of 3 phrases within
Incremental Parsing by Modular Recurrent Connectionist Networks
ClllUse_1
The
dog
who
ate
the
snake was
gl\leO
1 1 1 1 1 1 1 1 1 1~1 1 1 1 1 1 ~1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ClllUse_2
The
dog
who
ate
the
snake was
gi\leO
-... -._-_............... -.... _............... -......................................
p
h
r
a
s
~
111111111111
I111111
11111111111111,,"11111111111..
..11111111111111111111111111111111111111
11111111 .. 11 .... 111,'11111111 ....................
.,1111111111111111111111111111111111111111111111111111111111111111111111111111111111
p
h
r
a
s
~
~
~ .Illllllllll1l1llll11ll1l1l1lh
.4
I ...................
P
i. . . ,ll m
111111111111111111111111111111111111
11111111111""'""'""'1111
~I I I I I I I I I I I I I I
j . . . . Idl l l l l l l l l l l l l l~~I I I I I I I I I I I I I I I I I I I I
11111111111111111111111111111111111111111111111111 111111111111111111111111111111111
.1111111111111111111111 II ........ .. 111"' ... 11 ...... . ......... . .. ......... """ ....
........
Figure 3: Example of Clause Structure Dynamic Behavior.
the main clause. This is the correct clausal structure of the sentence and is confirmed by
the remainder of the input. The Interclause level indicates the appropriate relative clause
relationship during the initial hypothesis of the embedded clause.
The Clause Roles level processes the individual clauses as they get mapped through
the Clause Structure level. The labeling units for clause 1 initially hypothesize an
Agent/Action/Patient role structure with some competition from a Rec/Act/Pat role structure (the Agent and Patient units' activation traces for clause I, phrase 1 are shown in
Figure 4). This prediction occurs because active constructs outnumbered passive ones
during training. The final decision about role structure is postponed until just after the
embedded clause is presented. The verb phrase "was given" immediately causes the
Rec/Act/Pat role structure to dominate. Also, the network indicates that a fourth phrase
(e.g. "by Mary'') is expected to be the Agent. As with the first clause, an AgjAct/Pat
role structure is predicted for clause 2; this time the prediction is borne out
4.2
Generalization
One type of generalization is automatic. A detail of the word representation scheme was
omitted from the previous discussion. The feature patterns have two parts: a syntactic/semantic part and an identification part. The representations of "John" and "Peter"
differ only in their ID parts. Units in the network which learn do not have any input
connections from the ID portions of the word units. Thus, when the network learns to
369
370
Jain and Waibel
CUUiELPlfw.)El-[TIE_DXl
The
dog
who
ate
the
snake was
given a
bone
I
i ". . . . . ',", 1 1I1I1I1 I1Iml l ~ 1 ~1I1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1'1111111 .....
""oIllm
111 . . . . . . . . . . . . .
m
Figure 4: Example of Clause Roles Dynamic Behavior.
parse "John gave the bone to the dog:' it will know how to parse "Peter promised the
mitt to the boy:' This type of generalization is extremely useful, both for addition of
new words to the network and for processing many sentences not explicitly trained on.
The network also generalizes to correctly process truly novel sentences-sentences which
are distinct (ignoring ID features) from those in the training set. The weight sharing techniques at the Phrase and Clause Structure levels have an impact here. While being difficult
to measure generalization quantitatively, some statements can be made about the types
of novel sentences which can be correctly processed relative to the training sentences.
Substitution of single words resulting in a meaningful sentence is tolerated almost without exception. Substitution of entire phrases by different phrases causes some errors in
structural parsing on sentences which have few similar training exemplars. However, the
network does quite well on sentences which can be formed from composition between
familiar sentences (e.g. interchanging clauses).
4.3 Tolerance to Noise
Several types of noise tolerance are interesting to analyze: ungrammaticality, word deletions (especially poorly articulated short function words), variance in word speed, interword silences, interjections, word/phrase repetitions, etc. The effects of noise were
simulated by testing the parsing network on training sentences which had been corrupted
in the ways listed above. Note that the parser was trained only on well-formed sentences.
Sentences in which verbs were made ungrammatical were processed without difficulty
(e.g. "We am happy."). Sentences in which verb phrases were badly corrupted produced
reasonable interpretations. For example, the sentence "Peter was gave a bone to Fido:'
received an AgJAct/Pat/Rec role structure as if "was gave" was supposed to be either
"gave" or "has given". Interpretation of corrupted verb phrases was context dependent.
Single clause sentences in which determiners were randomly deleted to simulate speech
recognition errors were processed correctly 8S percent of the time. Multiple clause
sentences degraded in a similar manner produced more parsing errors. There were fewer
examples of multi-clause sentence types, and this hurt performance. Deletion of function
words such as prepositions beginning prepositional phrases produced few errors, but
deletions of critical function words such as "to" in infinitive constructions introducing
subordinate clauses caused serious problems.
Incremental Parsing by Modular Recurrent Connectionist Networks
The network was somewhat sensitive to variations in word presentation speed (it was
trained on a constant speed), but tolerated inter-word silences. Interjections of "ahhn and
partial phrase repetitions were also tested. The network did not perform as well on these
sentences as other networks trained for less complex parsing tasks. One possibility is
that the weight sharing is preventing the formation of strong attractors for the training
sentences. There appears to be a tradeoff between generalization and noise tolerance.
5 CONCLUSION
We have presented a novel connectionist network architecture and its application to a
non-trivial parsing task. A hierarchical, modular, recurrent connectionist network was
constructed which successfully learned to parse grammatically complex sentences. The
parser exhibited predictive behavior and was able to dynamically revise hypotheses.
Techniques for maximizing generalization were also discussed. Network performance
on novel sentences was impressive. Results of testing the parser's sensitivity to several
types of noise were somewhat mixed, but the parser performed well on ungrammatical
sentences and sentences with non-critical function word deletions.
Acknowledgments
This research was funded by grants from ATR Interpreting Telephony Research Laboratories and the National Science Foundation under grant number EET-87 16324. We thank
Dave Touretzky for helpful comments and discussions.
References
J. L. Elman. (1988) Finding Structure in Time. Tech. Rep. 8801, Center for Research
in Language, University of California, San Diego.
R. Hausser. (1988) Computation of Language. Springer-Verlag.
A. N. Jain. (1989) A Connectionist Architecturefor Sequential Symbolic Domains. Tech.
Rep. CMU-CS-89-187, School of Computer Science, Carnegie Mellon University.
A. N. Jain and A. H. Waibel. (1990) Robust connectionist parsing of spoken language.
In Proceedings of the 1990 IEEE International Conference on Acoustics. Speech.
and Signal Processing.
M. I. Jordan. (1986) Serial Order: A Parallel Distributed Processing Approach. Tech.
Rep. 8604, Institute for Cognitive Science, University of California, San Diego.
R. Miikkulainen and M. O. Dyer. (1989) Encoding input/output representations in
connectionist cognitive systems. In D. Touretzky. G. Hinton. and T. Sejnowski
(eds.) , Proceedings of the 1988 Connectionist Models Summer School, pp. 347356. Morgan Kaufmann Publishers.
A. Waibel, T. Hanazawa, O. Hinton, K. Shikano, and K. Lang. (1989) Phoneme recognition using time-delay neural networks. IEEE Transactions on Acoustics. Speech.
and Signal Processing 37(3):328-339.
371
| 201 |@word briefly:1 idl:1 initial:1 substitution:2 contains:1 current:2 activation:8 lang:1 must:2 parsing:32 john:3 hypothesize:1 designed:1 plot:1 update:1 fewer:1 beginning:4 short:2 provides:2 five:1 constructed:3 direct:1 become:3 paragraph:1 manner:1 inter:1 expected:1 behavior:16 elman:2 multi:1 actual:2 affirmed:1 becomes:1 underlying:1 substantially:1 developed:1 spoken:3 finding:1 every:1 act:2 unit:61 grant:2 modify:1 despite:1 encoding:1 id:3 dynamically:3 bi:1 acknowledgment:1 spontaneously:1 testing:2 block:16 procedure:3 evolving:1 word:42 induce:1 symbolic:2 get:1 close:1 context:5 applying:1 map:1 center:3 lexical:2 maximizing:1 assigns:1 immediately:1 rule:1 dominate:1 embedding:1 variation:1 hurt:1 phrasal:2 construction:7 target:4 parser:10 diego:2 us:1 hypothesis:4 pa:1 recognition:5 rec:3 bottom:2 role:17 module:4 capture:1 connected:2 inhibit:1 mentioned:1 transforming:1 recip:1 dynamic:9 trained:9 tight:1 predictive:6 efficiency:1 indirect:1 leo:2 articulated:1 jain:10 distinct:1 describe:2 sejnowski:1 labeling:9 formation:1 whose:2 modular:7 emerged:1 quite:1 say:1 grammar:2 gi:1 unseen:1 syntactic:5 itself:1 noisy:1 final:3 associative:1 hanazawa:1 sequence:2 remainder:1 causing:1 fido:2 poorly:1 supposed:1 description:2 inducing:1 competition:2 constituent:1 extending:1 produce:1 incremental:7 object:1 coupling:1 recurrent:9 exemplar:1 interclause:4 school:3 received:1 strong:1 predicted:1 c:1 indicate:1 differ:1 correct:1 subordinate:2 require:1 assign:2 generalization:8 extension:1 mapping:6 major:1 early:1 omitted:1 determiner:2 label:3 infonnation:1 sensitive:1 repetition:2 successfully:3 ej:1 varying:1 encode:1 indicates:2 tech:3 sense:1 am:1 helpful:1 dependent:1 el:1 snake:6 integrated:1 entire:1 initially:2 hidden:10 i1:1 among:2 noun:1 construct:1 represents:2 interchanging:1 connectionist:17 stimulus:2 quantitatively:1 serious:1 few:2 randomly:1 simultaneously:1 national:1 individual:1 familiar:1 attractor:1 possibility:1 truly:1 partial:2 tree:1 desired:1 formalism:4 earlier:1 contiguous:1 assignment:3 phrase:54 introducing:1 subset:1 delay:1 reported:1 corrupted:4 tolerated:2 international:1 sensitivity:1 retain:1 reflect:1 containing:1 possibly:1 borne:1 interjection:2 book:1 cognitive:2 includes:1 hypothesizes:1 explicitly:1 caused:1 performed:1 bone:7 analyze:1 portion:3 parallel:1 dug:1 minimize:2 formed:3 degraded:1 variance:1 who:8 kaufmann:1 phoneme:1 correspond:1 directional:1 generalize:2 weak:1 identification:1 produced:5 confirmed:1 dave:1 touretzky:2 sharing:3 ed:1 acquisition:1 ungrammaticality:1 pp:1 associated:4 static:2 revise:1 organized:2 back:3 reflecting:1 appears:2 tolerate:1 higher:1 just:1 until:1 parse:5 propagation:3 incrementally:2 defines:1 behaved:1 mary:2 building:3 effect:1 requiring:2 contain:1 assigned:2 laboratory:1 semantic:8 illustrated:2 conditionally:1 ll:1 during:6 recurrence:2 speaker:1 outline:1 performs:2 interrelationship:1 interpreting:1 passive:4 percent:1 meaning:1 novel:5 common:2 clause:58 discussed:2 interpretation:4 belong:1 resting:1 mellon:2 significant:2 composition:1 automatic:1 similarly:1 language:5 had:2 funded:1 impressive:1 compiled:1 etc:1 own:1 showed:1 perspective:1 manipulation:1 buffer:1 verlag:1 rep:3 postponed:1 exploited:1 infinitive:1 morgan:1 additional:1 somewhat:2 maximize:1 signal:2 ii:3 full:3 desirable:1 multiple:1 characterized:1 serial:1 e1:2 impact:1 prediction:4 scalable:1 ajay:1 expectation:2 patient:5 cmu:1 repetitive:1 addition:1 separately:1 fine:1 diagram:1 singular:1 publisher:1 exhibited:2 comment:1 grammatically:4 jordan:2 structural:1 near:1 iii:1 submodules:1 fit:1 gave:5 architecture:10 tradeoff:1 inactive:1 peter:3 speech:5 proceed:1 cause:2 action:4 useful:1 detailed:2 listed:1 tune:1 processed:5 arising:1 correctly:3 clausal:1 diverse:1 carnegie:2 group:1 dominance:1 four:2 promised:1 deleted:1 compete:1 fourth:1 respond:1 extends:2 almost:1 reasonable:1 decide:1 architectural:1 utilizes:1 decision:1 summer:1 display:1 badly:1 nontrivial:1 incorporation:1 constraint:2 alex:1 aspect:1 speed:3 simulate:1 extremely:1 waibel:9 across:3 describes:1 ate:7 remain:1 prepositional:1 wherever:1 modification:1 previously:2 discus:1 needed:2 know:2 dyer:2 subnetworks:1 generalizes:1 hierarchical:2 appropriate:1 simulating:1 robustly:1 responsive:1 gate:1 recipient:1 remaining:1 build:3 especially:1 occurs:1 subnetwork:2 gradient:1 separate:2 mapped:1 simulated:3 atr:1 majority:1 thank:1 trivial:1 modeled:1 relationship:4 happy:1 difficult:1 statement:1 boy:1 trace:1 design:1 proper:1 perform:5 gated:1 revised:2 descent:1 pat:4 hinton:2 sharp:1 verb:4 dog:8 required:1 sentence:46 connection:9 california:2 hausser:2 learned:1 deletion:4 acoustic:2 assembled:1 able:5 below:1 pattern:5 rolf:1 built:1 including:1 garden:1 belief:1 critical:2 difficulty:1 force:1 attach:1 scheme:1 firmly:1 attachment:1 relative:4 embedded:6 mixed:1 interesting:2 limitation:1 telephony:1 foundation:1 agent:5 i111111:1 preposition:1 gl:1 last:1 soon:2 silence:2 institute:1 tolerance:5 distributed:1 calculated:1 phn:2 rich:1 preventing:1 made:2 san:2 miikkulainen:2 transaction:1 eet:1 confirm:2 global:1 sequentially:1 active:5 pittsburgh:1 shikano:1 modularity:1 stimulated:1 promising:1 learn:11 robust:1 ignoring:1 ungrammatical:2 complex:7 constructing:2 domain:4 did:1 main:1 noise:5 arrival:1 position:1 learns:3 gating:12 symbol:2 list:1 recurrently:2 sequential:7 occurring:1 forming:1 contained:1 partially:1 springer:1 slaved:1 stimulating:1 conditional:1 slot:5 presentation:5 shared:1 man:1 change:1 included:1 specifically:1 called:1 meaningful:1 indicating:1 pragmatic:1 exception:1 support:1 modulated:1 violated:1 tested:1 |
1,108 | 2,010 | Information Geometrical Framework for
Analyzing Belief Propagation Decoder
Shiro Ikeda
Kyushu Inst. of Tech., & PRESTO, JST
Wakamatsu, Kitakyushu, Fukuoka, 808-0196 Japan
shiro@brain.kyutech.ac.jp
Toshiyuki Tanaka
Tokyo Metropolitan Univ.
Hachioji, Tokyo, 192-0397 Japan
tanaka@eei.metro-u.ac.jp
Shun-ichi Amari
RIKEN BSI
Wako, Saitama, 351-0198 Japan
amari@brain.riken.go.jp
Abstract
The mystery of belief propagation (BP) decoder, especially of the turbo
decoding, is studied from information geometrical viewpoint. The loopy
belief network (BN) of turbo codes makes it difficult to obtain the true
?belief? by BP, and the characteristics of the algorithm and its equilibrium are not clearly understood. Our study gives an intuitive understanding of the mechanism, and a new framework for the analysis. Based on
the framework, we reveal basic properties of the turbo decoding.
1 Introduction
Since the proposal of turbo codes[2], they have been attracting a lot of interests because
of their high performance of error correction. Although the thorough experimental results
strongly support the potential of this iterative decoding method, the mathematical background is not sufficiently understood. McEliece et al.[5] have shown its relation to the
Pearl?s BP, but the BN for the turbo decoding is loopy, and the BP solution gives only an
approximation.
The problem of the turbo decoding is a specific example of a general problem of marginalizing an exponential family distribution. The distribution includes higher order correlations,
and its direct marginalization is intractable. But the partial model with a part of the correlations, can be marginalized with BP algorithm exactly, since it does not have any loop. By
collecting and exchanging the BP results of the partial models, the true ?belief? is approximated. This structure is common among various iterative methods, such as Gallager codes,
Beth?e approximation in statistical physics[4], and BP for loopy BN.
We investigate the problem from information geometrical viewpoint[1]. It gives a new
framework for analyzing these iterative methods, and shows an intuitive understanding of
them. Also it reveals a lot of basic properties, such as characteristics of the equilibrium, the
condition of stability, the cost function related to the decoder, and the decoding error. In this
paper, we focus on the turbo decoding, because its structure is simple, but the framework
is general, and the main results can be generalized.
2 Information Geometrical Framework
2.1 Marginalization, MPM Decoding, and Belief
which is defined as follows
!"$#%&!'(" )'*
+'," -./$
(1)
#
"
/
1
2
"
3
is the linear function of 0
, and each
is the higher order correlations
where,
of 0 1 2 . The problem of turbo codes and similar iterative methods are to marginalize
1=<> 1 this
&
6
;
9
7
8
:
distribution. Let 4 denote the operator of marginalization as, 4 5
. The
marginalization is equivalent to take the expectation of as
?@6 ;798 A BC & D?E&F
/F HG
In the case of MPM (maximization of the posterior marginals) decoding, 1JI 0LKM $' M 2
and the sign of each FN1 is the decoding result. In the belief network, 1 I 0PO M 2 and FN1 is
the belief. In these iterative methods, the marginalization of eq.(1) is not tractable, but the
marginalization of the following distribution is tractable.
Q &SR/T* % U" # )'," Q &>'VT.
KW Q XT/ ZY M
\[] ^T I`_ G (2)
Each Q &SRT includes only one of the 0 " 3 &$2 in eq.(1), and additional parameter T is
used to adjust linear part of . The iterative methods are exchanging information through
T for each aQ , and finally approximate 4 5 & .
Let us consider a distribution of
2.2 The Case of Turbo Decoding
bcNd9e%fghNi\f
jNk\d
|
|
l |
m noXprqsXtvu w x
|
|
m
noXprqsXtNy {w z
| l+} } ~?L???X?? | l+? } ? } ? |
w x w\z
w x w{z
bcNd9e%f??k\i\f+jNk\d
l+? } ?
w$x || ?
|?
? sXoXpr? qsX?tLu z ??
x sXoXprqsXt
y l+? } ?
wz
Figure 1: Turbo codes
? ,??v\
/?v/?
?? ?&: ? ? 1 +
1 ? ? ?) ?%U?N ? ? ? I
&S ? ? ?
0LM
?V2
S ? ?? ? ? ?? ? ? 1 ?v? 9?N ? ? ? ? I 0vKM $' M 2
? ? M+?
?
? ? ? + ?? ? .
The ultimate goal of the turbo decoding is the MPM decoding of based on ?? S
Since the channel is memoryless, the following relation holds
S ? ?? ? ? ?? ? ? * % ?? ?
?'??@? ? ? ? ?'??@?? ? ??SK ??' ?N? r???>
??? O ? M M?K??{?
??? ?> ???&?> 6
?798 ? ? ? ??'(?N?? G
?
In the case of turbo codes, is the information bits, from which the turbo encoder generates
two sets of parity bits,
, and
,
(Fig.1). Each parity bit is expressed as the form
, where the product is
taken over a subset of
. The codeword
is then transmitted over a
noisy channel, which we assume BSC (binary symmetric channel) with flipping probability
. The receiver observes
,
.
0LKM {' M 2
By assuming the uniform prior on , the posterior distribution is given as follows
B S ? ? ? + ?? ? ? % ?&? ?
?'?? ? '?? ?
? ?
? ? ? ? (3)
+S ? ?? ? + ?? ? ?
? U" # &!'("
+!'(" ? / G
? , " Q (?? ? ? Q ? Q Y M ? .
Here is the normalizing factor, and "$#v&(?? S
^
[
? . When ? is large, marginalization of
to eq.(1), where
??rS ? ? ? ? ??? ? is equivalent
Equation(3)
is intractable since it needs summation over ?
terms. Turbo codes utilize two decoders which solve the MPM decoding of aQ SRT?Y M ? in eq.(2). The
? ? ? Q ? and the prior of which has the form of
distribution is derived from ?
SRT !XT K ??XT G
&SR/T is a factorizable distribution. The marginalization of ?
? ? ? Q ? is feasible since
T
its BN is loop free. The parameter serves as the window of exchanging the information
between the two decoders. The MPM decoding is approximated by updating T iteratively
&?? ? ? ? ? ? ? ?
in ?turbo? like way.
2.3 Information Geometrical View of MPM Decoding
Let us consider the family of all the probability distributions over . We denote it by ,
which is defined as
& &S? O I 0LKM $' M 2 A B M G
We consider an ? ?flat submanifold # in . This is the submanifold of # SR
defined
as
G
# # &SR
? ?" # !' ` KW # )? E
I _
(4)
? , every distribution of # can be rewritten as follows
Since " # &?*??
'
KW # ???]'?
KW # / G
# &SR
? U" # !
It shows that every distribution of # is decomposable, or factorizable. From the information geometry[1], we have the following theorem of ?projection.
Theorem 1. Let be an ? ?flat submanifold in , and let & I . The point in that
minimizes the KL-divergence from & to , is denoted by,
4?5 &? ?
%B $' & ? $R &-,U
and is called the ?projection of
&
!#"
to
)(+*
. The ?projection is unique.
#
It is easy to show that the marginalization corresponds to the ?projection to [7]. Since
MPM decoding and marginalization is equivalent, MPM decoding is also equivalent to the
?projection to .
#
2.4 Information Geometry of Turbo Decoding
5 &
Let ./
denote the parameters in
?5 ?
of the ?projected distribution,
#& 576
',
4 1 2 3
(8*
.0 #
?
The turbo decoding process is written as follows,
$R &SR G
T O for O , and M .
by
2. Project ? &SR/T onto # as ./?5 ? &SR/T , and calculate T ?
T ? .0P5 ? &SRT K T G
by
3. Project &SR/T ? onto # as . ?5 &SRT ? , and calculate T
T ./?5
&SR/T ? K T ? G
.0P5 ? &SR/T , go to step 2.
4. If .0P5
&SRT ?
the estimated parameter
, the projection of
?? S ? turbo
? onto # approximates
?
T
V
'
T ? , where the estimated distribution is
The
?? ? ?? decoding
, as
# &SR
?&" # &!'?T P '?T ?
KVW # XT 'VT ? / G
(5)
T
eq.(5) is
An intuitive understanding of the turbo decoding is as follows. In step 2, ?
T , and T ? is estimated byinprojecting
replaced with " ? & . The distribution becomes ? SR
it onto # . In step 3, XT
in eq.(5) is replaced with "
& , and T is estimated by ?
T ? .
projection of +SR
We now define the submanifold corresponding to each decoder,
Q Q SRT "$#v&!'(" Q !'?T P KVW Q XT ? T
I`_
Y? M ? G
T is the coordinate system of Q . Q is also an ? ?flat submanifold. ?? and
" ? in general.
Q # hold because " Q & includes cross terms of and "
N&J
1. Let
The information geometrical view of the turbo decoding is schematically shown in Fig.2.
3 The Properties of Belief Propagation Decoder
3.1 Equilibrium
SRT ? SRT ? #LSR
M GG 4 5 +T &SR/'VT ? T H G 4 5 ? SRT ? # &SR G
?
?
as
Let us define a manifold
& I ?A B &? A B # &SR
G
?
From its definition, for any & I , the expectation of is the same, and its ?
projection to # coincides with # SR
. This is an ?flat submanifold[1], and we call
an equimarginal submanifold. Since eq.(6) holds, #LSR
$ SRT ? $ ? SRT I
is satisfied.
an ? ?flat version of the submanifold as
, which connects # SR
,
, and ? &SR/
T in log-linear manner
Let
&SRus/ T define
?
# &SR +&SR T ? ? SR T A ? Q M G
Q < #
&?? ? ? ? ? ? ?? ? is included in the . It can be proved by taking
Since
# eq.(7)
holds,
KM , ? M .
When the the turbo decoding converges, equilibrium solution defines three important dis ,
, and
. They satisfy the following two conditions:
tributions,
(6)
(7)
#% $'&)( +* #,
"!
"!
# - -/& . 2''3 457698 : ;=<>7?
- -/. ! !
0
#
1&! 1!
- -/. $ # ( +* ,
@
&!
"&!
Figure 2: Turbo decoding
A
R UWvurwW x+wy pqxzwy t _
KLCFEHGJI
BNQ R t UXWZY=s p ] _
BPO
RpqUWrY9s t ] _
BDCFEHGJI
BNM
RTSVUXWZY\[^]`_
a+b=cVdfehg ejik'l
cJm=n
k'o
Figure 3:
and
# &SR +SRT ?
&SR/T
?
?
&?? ? ? ? ? ? ? ?
? ? ? + ?? ? , &SR
is the true marginalization of ?? S ? ? ? + ?? ? .
If
includes ?? ?
? ? ? ? ? ? . This fact means that
necessarily include ?? ?
??rS ? ? ? ? ??? ? and does
are not necessarily equimarginal, which is the origin of the
However,
# &SR
not
Theorem 2. When the turbo decoding procedure converges, the convergent probability
,
, and
belong to equimarginal submanifold
distributions
, while its ?flat version
includes these three distributions and also the posterior distribution
(Fig.3).
decoding error.
3.2 Condition of Stability
W # in eq.(4) and W Q in eq.(2)
?#L %6 E798 A B #v&SR
L{ 4 W #L $ Z? Q XT%6
798 A B Q &SRTL{}| W Q XT^Y? M ? G
The expectation parameters are defined as follows with
? # ?>+T ? ? ? T G
We give a sufficiently small perturbation ~ to T and apply one turbo decoding step. The
?projection from ? &SRT ' ~ to # gives,
? # ?
' *? ? XT ' ~
?
? #v ? ? ? XT ~ G
Here, ? #L is the Fisher information matrix of #v&SR
, and ? Q T is that of Q &SRT ,
Y? M ? . Note that ? # is a diagonal matrix. The Fisher information matrix is defined
as follows
? # L
{ 44"? W # L
{ 4 ? # {?
? Q XT??
{ |\| ? W Q XTL
{ | ? Q XT ZY M ? G
T ? in step 2 will be,
T ? T ? ?
' ?? # $? ? ? XT K?? ? ? ~ G
Here, ? is an identity matrix of size ? . Following the same line for step 3, we derive the
Equation (6) is rewritten as follows with these parameters,
theorem which coincides with the result of Richardson[6].
Theorem 3. Let
When ? 1{? ? M
1 be the eigenvalues
of the matrix defined as
? ? #% {? ? T ? K ? ?Z? ? #% $ ? ? ? T K?? ?? G
holds for all , the equilibrium point is stable.
3.3 Cost Function and Characteristics of Equilibrium
XT /T ? W L# K W T ? !' W ? XT / G
*T '?T ?
T G
G G /T - is the critical point of .
Theorem 4. The equilibrium state
T * ? # K ? ? XT? , { | *? # K ?>?XT ? . For the
Proof. Direct calculation gives { |
?
L
#
?
T
?
equilibrium,
? ? holds, and the proof is completed.
T Q K T Q is small,
When X
T T ? #% ? {}| G
T ?
K T ? K ? #% ?
{|
We give the cost function which plays an important role in turbo decoding.
Here,
. This function is identical to the ?free energy? defined in [4].
{{ || || {{ || || ? # ? K # ?? ? # ? K # ? ? G
T '?T ? T K T ?
{ {
M ? # K ??H' ? ? X?? K ? ? G
X?? K ? ? K ??'?? ?
{
{
{
{
This shows how the algorithm works, but it does not give the characteristics of the equilibrium point. The Hessian of is
And by transforming the variables as,
44
4
4
Most probably, 44
is positive definite but
saddle at equilibrium.
and
, we have
is always negative, and
is generally
& SR
! as
SR ! ?" # !' ` ]' KW !/ ] ?
W !H ? ? A B ?&"$#v&!'
' / a&%6
(798 &" &$ \" ? &/ G
,
), and Q &SR/T
? ? ? ? ? ? (
This distribution includes #v&SR
( ), &?? S
( ,;T , Q ), where ? M M / , @ M O / , and ? O M / . The expectation
parameter ?? ! is defined as,
?? ! L{ 4 W ! A B SR
! G
Let us consider , where every distribution SR
! I has the same expectation parameter, that is, ?S !H ?S
holds. Here, we define, ?S
H ?S
+! a . From
the Taylor expansion, we have,
FL1 ! FL1 !' A {
?PFN1/ 9 ? ' A { Q FN1\ " Q ' M A { Q { $ FL1 " Q $
Q
?
? Q# $
'A { Q { ? F 1 " Q ? ' M A { 3 { % F 1 9 3 % ' ! &'(&)P!' &1 *&+) G (8)
? #Q
? 3 #%
3.4 Perturbation Analysis
For the following discussion, we define a distribution
0
L F 1 r 2 !?EF 1 0 Y+ N2 ? F 1 J 1 a? 6
798 K
0 1? 2 1
&
SR
?
!
a
Q
Q
11
1 K 11 A
1 Q Q K A { Q K A 3 3 3 Q { 3 { $ K A ?r? ? $ { ? F 1 " Q $
?
3
? Q # $
Q
(9)
1 1 1=1
L
{
F
1
1
Q
M?? , and
where,
.
&
SR
$
+SR
I holds, T ? and
a
Let
since
T ? K
K , T and
V
. Also
when we put
? , ? K T ? holds. From eq.(9), we have the
following result,
1 #
1=1 1 Q 1 1 { Q A 3 3 3 Q {%3 { Q A ?r? ? Q {N? FN1\ G
Q
(10)
K K
K ? K 3
K ?
Next, let , and we consider &SR !
% I
, where is the parameter which
? ? ? ? ? ? is not necessarily included in
satisfies this equation. Since SR!! &?? S
, is generally not equal to . From eq.(9),
1 1#
1 1 A 1 Q 1 1 A { Q A 3 3 3 Q { 3 { Q A ?r? ? Q {N? FN1 G
K
K ?
K Q K1 ? 1 Q K 3
From the condition T 'VT ? and eq.(10), we have the following approximation,
11
1
A { Q K A 3 3 3 Q { 3 { $ K A ?r? ? $ {
? FL1 G
K
?
? Q< $ 3
. After adding some
The indexes are for , are for , and +
{
definitions, that is,
, and
, where
is the Fisher
information matrix of
which is a diagonal matrix, we substitute with function of up to its 2nd order, and neglect the higher orders of . And we have,
?
?
?S
#
?
?? )
%
?? ! ' M A { Q K A 3 3 3 Q { 3 { $ K A ?r? ? $ { ? ?S G
?
3
? Q< $
This result gives the approximation accuracy of the BP decoding. Let the true belief be
, and we evaluate the difference between and
on . The result is
summarized in the following theorem.
Theorem 5. The true expectation of , which is
, is approximated as,
?
??
Where
is the solution of the turbo decoding.
Equation (11) is related to the ?embedded?curvature of
extended to general case where [3, 8].
[
(11)
(Fig.3). The result can be
4 Discussion
We have shown a new framework for understanding and analyzing the belief propagation
decoder.
Since the BN of turbo codes is loopy, we don?t have enough theoretical results for BP
algorithm, while a lot of experiments show that it works surprisingly well in such cases.
The mystery of the BP decoders is summarized in 2 points, the approximation accuracy
and the convergence property.
Our results elucidate the mathematical background of the BP decoding algorithm. The
information geometrical structure of the equilibrium is summarized in Theorem 2. It shows
?
the ?flat submanifold
plays an important role. Furthermore, Theorem 5 shows that
the relation between
and the ?flat submanifold
causes the decoding error,
and the principal component of the error is the curvature of
. Since the curvature
strongly depends on the codeword, we can control it by the encoder design. This shows a
room for improvement of the ?near optimum error correcting code?[2].
For the convergent property, we have shown the energy function, which is known as Beth?e
free energy[4, 9]. Unfortunately, the fixed point of the turbo decoding algorithm is generally a saddle of the function, which makes further analysis difficult. We have only shown a
local stability condition, and the global property is one of our future works.
This paper gives a first step to the information geometrical understanding of the belief
propagation decoder. The main results are for the turbo decoding, but the mechanism is
common with wider class, and the framework is valid for them. We believe further study
in this direction will lead us to better understanding and improvements of these methods.
Acknowledgments
We thank Chiranjib Bhattacharyya who gave us the opportunity to face this problem. We
are also grateful to Yoshiyuki Kabashima and Motohiko Isaka for useful discussions.
References
[1] S. Amari and H. Nagaoka. (2000) Methods of Information Geometry, volume 191 of
Translations of Mathematical Monographs. American Mathematical Society.
[2] C. Berrou and A. Glavieux. (1996) Near optimum error correcting coding and decoding: Turbo-codes. IEEE Transactions on Communications, 44(10):1261?1271.
[3] S. Ikeda, T. Tanaka, and S. Amari. (2001) Information geometry of turbo codes and
low-density parity-check codes. submitted to IEEE transaction on Information Theory.
[4] Y. Kabashima and D. Saad. (2001) The TAP approach to intensive and extensive connectivity systems. In M. Opper and D. Saad, editors, Advanced Mean Field Methods ?
Theory and Practice, chapter 6, pages 65?84. The MIT Press.
[5] R. J. McEliece, D. J. C. MacKay, and J.-F. Cheng. (1998) Turbo decoding as an instance of Pearl?s ?belief propagation? algorithm. IEEE Journal on Selected Areas in
Communications, 16(2):140?152.
[6] T. J. Richardson. (2000) The geometry of turbo-decoding dynamics. IEEE Transactions on Information Theory, 46(1):9?23.
[7] T. Tanaka. (2001) Information geometry of mean-field approximation. In M. Opper and
D. Saad, editors, Advanced Mean Field Methods ? Theory and Practice, chapter 17,
pages 259?273. The MIT Press.
[8] T. Tanaka, S. Ikeda, and S. Amari. (2002) Information-geometrical significance of
sparsity in Gallager codes. in T. G. Dietterich et al. (eds.), Advances in Neural Information Processing Systems, vol. 14 (this volumn), The MIT Press.
[9] J. S. Yedidia, W. T. Freeman, and Y. Weiss. (2001) Bethe free energy, Kikuchi approximations, and belief propagation algorithms. Technical Report TR2001?16, Mitsubishi
Electric Research Laboratories.
| 2010 |@word version:2 nd:1 r:2 mitsubishi:1 bn:5 equimarginal:3 bc:1 bhattacharyya:1 wako:1 written:1 ikeda:3 selected:1 mpm:8 mathematical:4 direct:2 manner:1 brain:2 freeman:1 window:1 tlu:1 becomes:1 project:2 minimizes:1 thorough:1 every:3 collecting:1 exactly:1 control:1 positive:1 understood:2 local:1 analyzing:3 studied:1 unique:1 acknowledgment:1 practice:2 definite:1 sxt:1 procedure:1 area:1 projection:9 onto:4 marginalize:1 operator:1 put:1 equivalent:4 cjm:1 go:2 decomposable:1 correcting:2 stability:3 coordinate:1 elucidate:1 play:2 origin:1 approximated:3 updating:1 tributions:1 role:2 p5:3 calculate:2 observes:1 monograph:1 transforming:1 tr2001:1 dynamic:1 grateful:1 po:1 various:1 chapter:2 riken:2 univ:1 solve:1 amari:5 encoder:2 richardson:2 nagaoka:1 noisy:1 eigenvalue:1 kvw:2 product:1 loop:2 intuitive:3 convergence:1 optimum:2 converges:2 kikuchi:1 wider:1 derive:1 ac:2 eq:13 direction:1 tokyo:2 jst:1 shun:1 summation:1 correction:1 hold:9 sufficiently:2 equilibrium:11 lm:1 metropolitan:1 bsc:1 clearly:1 beth:2 always:1 mit:3 derived:1 focus:1 vk:1 improvement:2 check:1 tech:1 inst:1 relation:3 among:1 denoted:1 mackay:1 equal:1 field:3 identical:1 bnm:1 future:1 report:1 hni:1 fn1:5 divergence:1 replaced:2 geometry:6 connects:1 interest:1 investigate:1 adjust:1 hg:1 partial:2 taylor:1 theoretical:1 instance:1 metro:1 exchanging:3 maximization:1 loopy:4 cost:3 subset:1 saitama:1 uniform:1 submanifold:11 density:1 physic:1 decoding:38 connectivity:1 satisfied:1 american:1 japan:3 potential:1 summarized:3 coding:1 includes:6 satisfy:1 eei:1 depends:1 view:2 lot:3 accuracy:2 characteristic:4 who:1 toshiyuki:1 zy:2 kabashima:2 submitted:1 ed:1 definition:2 energy:4 proof:2 proved:1 higher:3 wei:1 strongly:2 furthermore:1 correlation:3 mceliece:2 propagation:7 defines:1 reveal:1 believe:1 dietterich:1 true:5 bsi:1 memoryless:1 symmetric:1 iteratively:1 laboratory:1 coincides:2 generalized:1 gg:1 geometrical:9 ef:1 common:2 ji:1 jp:3 volume:1 belong:1 approximates:1 marginals:1 shiro:2 aq:2 stable:1 attracting:1 curvature:3 posterior:3 codeword:2 binary:1 vt:4 transmitted:1 additional:1 berrou:1 fl1:4 eji:1 technical:1 calculation:1 cross:1 basic:2 expectation:6 yoshiyuki:1 proposal:1 background:2 schematically:1 saad:3 sr:58 probably:1 call:1 near:2 easy:1 enough:1 marginalization:11 gave:1 fukuoka:1 intensive:1 ultimate:1 hessian:1 cause:1 generally:3 useful:1 sign:1 estimated:4 vol:1 ichi:1 utilize:1 mystery:2 family:2 bit:3 convergent:2 cheng:1 turbo:33 bp:11 flat:8 bpo:1 generates:1 kyushu:1 taken:1 equation:4 chiranjib:1 mechanism:2 wvu:1 tractable:2 serf:1 presto:1 rewritten:2 yedidia:1 apply:1 v2:1 substitute:1 include:1 completed:1 opportunity:1 marginalized:1 neglect:1 k1:1 especially:1 society:1 flipping:1 diagonal:2 thank:1 decoder:10 manifold:1 assuming:1 code:13 index:1 difficult:2 unfortunately:1 negative:1 design:1 extended:1 communication:2 perturbation:2 kl:1 extensive:1 tap:1 pearl:2 tanaka:5 wy:1 sparsity:1 wz:1 belief:14 critical:1 advanced:2 lk:3 prior:2 understanding:6 marginalizing:1 embedded:1 viewpoint:2 editor:2 translation:1 surprisingly:1 parity:3 free:4 dis:1 hachioji:1 taking:1 face:1 opper:2 valid:1 kyutech:1 projected:1 transaction:3 approximate:1 global:1 reveals:1 receiver:1 don:1 iterative:6 sk:1 channel:3 bethe:1 expansion:1 necessarily:3 electric:1 factorizable:2 significance:1 main:2 n2:1 fig:4 exponential:1 theorem:10 specific:1 xt:18 normalizing:1 intractable:2 adding:1 jnk:2 sx:1 saddle:2 gallager:2 expressed:1 corresponds:1 satisfies:1 glavieux:1 goal:1 identity:1 room:1 fisher:3 feasible:1 included:2 principal:1 called:1 experimental:1 support:1 evaluate:1 |
1,109 | 2,011 | Orientational and geometric
determinants
place and head-
Neil Burgess & Tom Hartley
Institute of Cognitive Neuroscience & Department of Anatomy, UCL
17 Queen Square, London WCIN 3AR, UK
n. burgess@ucl.ac.uk. t.hartley@ucl.ac.uk
Abstract
We present a model of the firing of place and head-direction cells in
rat hippocampus. The model can predict the response of individual
cells and populations to parametric manipulations of both geometric (e.g. O'Keefe & Burgess, 1996) and orientational (Fenton et
aI., 2000a) cues, extending a previous geometric model (Hartley
et al., 2000). It provides a functional description of how these
cells' spatial responses are derived from the rat's environment and
makes easily testable quantitative predictions. Consideration of the
phenomenon of remapping (Muller & Kubie, 1987; Bostock et aI.,
1991) indicates that the model may also be consistent with nonparametric changes in firing, and provides constraints for its future
development.
1
Introduction
'Place cells' recorded in the hippocampus of freely moving rats encode the rat's
current location (O'Keefe & Dostrovsky, 1971; Wilson & McNaughton, 1993). In
open environments a place cell will fire whenever the rat enters a specific portion
of the environment (the 'place field'), independent of the rat's orientation (Muller
et aI., 1994). This location-specific firing appears to be present on the rat's first
visit to an environment (e.g. Hill, 1978), and does not depend on the presence
of local cues such as odors on the floor or walls. The complementary pattern of
firing has also been found in related brain areas: 'head-direction cells' that fire
whenever the rat faces in a particular direction independent of its location (Taube
et aI., 1990). Experiments involving consistent rotation of cues at or beyond the
edge of the environment (referred to as 'distal' cues) produce rotation of the entire
place (O'Keefe & Speakman, 1987; Muller et aI., 1987) or head-direction (Taube
et aI., 1990) cell representation. Rotating cues within the environment does not
produce this effect (Cressant et aI., 1997). Here we suggest a predicitive model of
the mechanisms underlying these spatial responses.
2
Geometric influences given consistent orientation
Given a stable directional reference (e.g. stable distal cues across trials), fields are
determined by inputs tuned to detect extended obstacles or boundaries at particular
bearings. That is, they respond whenever a boundary or obstacle occurs at a given
distance along a given allocentric direction, independent of the rat's orientation.
These inputs are referred to below as putative 'boundary vector cells' (BVCs). The
functional form of these inputs has been estimated by recording from the same
place cell in several environments of differing geometry within the same set of distal
orientation cu~s (O'Keefe & Burgess, 1996; Hartley et al., 2000). That is, for a
BVC i tuned to a boundary at distance di and bearing <Pi relative to the rat, the
response to a houndary segment at distance r and bearing 9, subtending an angle
cfJ at the rat, is given by:
Cli == gi(r, fJ)CfJ,
gi (r,
fJ)
ex:
exp[-(r - d i )2/2a;ad(di )]
V21r0";ad( d
i)
exp[-(fJ - <Pi)2 /2a~ng]
X -----r===========---
(1)
- /21ru
V
any
2
where the angular width aang is a constant but the radial width Urad == uo(1+di //3)
so that the width of tuning to distance increases with the distance of peak response
diD Constants 0"0 and /3 determine width at zero distance and its rate of increase
with distance. The firing rate of BVC i, when the rat is at a location z, is found
by integrating eli over (1 (this is done numerically as the distance r to the nearest
boundary in direction fJ is a function of z, fJ and the geometry of the environment).
A place cell's firing rate F(Z) is then simply the thresholded linear sum of the firing
rates of the n Bves connected to it, Le.
where e(z) is the Heaviside function (S(z) == z if x > 0; Sex) == 0 otherwise). All
simulations have /3 == 183cm, Uo == 12.2cm, Urad == 0.2rad, while the threshold T
can vary between simulations (e.g. between Figs. 1 and 2) but not between cells,
and A is an arbitrary constant as absolute firing rates are not shown.
Thus, in this model, a place cell's response is simply determined by the parameters
d i and ifJi chosen for the set of BVes connected to it. Assuming a random selection
of BVCs for each place cell, and a single value for T, the model provides a good
fit to the characteristics of populations of place fields across different environments,
such as the distribution of firing rates and field shapes and sizes. Inputs can also
be chosen so as to fit a given place field so that its behavior in a new environment
of different shape can be predicted. See Hartley et al. (2000) and Fig. 1.
Like other models relying on the bearing to a landmark (Redish & Touretsky, 1996;
McNaughton et al., 1996), the basic geometrical model assumes an accurate directionalreference, but does not state how this depends on the sensory input. Note
that, as such, this model already captures effects of consistent rotation of orientation cues around an environment as a reorientation of the directional reference
frame that in turn affects the directions along which BVCs are tuned to respond.
Indeed, the effect of consistent rotation of orientation cues about a environment of
fixed geometry is identical to the rotation of the environment within a fixed directional reference frame, and can be modelled in this way (see e.g. the square and
diamond in Figs. 1b,c).
3
Model of geometric and orientation influences
Models of head direction (Skaggs et al., 1995; Zhang, 1996) indicate how orientation
might be derived. Internal inputs (e.g. vestibular or proprioceptive) maintain a
consistent representation of heading within a ring of head-direction cells arranged
to form a continuous attractor . Correlational learning of associations from visual
inputs to head direction cells then allows the representation of head direction to
be maintained in synch with the external world. These models account for the
preferential influence of large cues at a stable bearing (i.e. at or beyond the edge
of the environment), and effects of instability caused by continual movement of
cues or disorientation of the rat. They also allow orientation to be maintained
in the face of cue removal, unless all cues are removed in which case orientation
is wholly reliant on internal inputs and will drift over time. In this paper we
take a step towards providing a quantitative model for the combined influences
of orientation cues and boundaries on the firing of place and head direction cells.
Such a model should be able to predict the behaviour of these cells under arbitrary
environmental manipulations, bearing in mind that some (extended) objects may
be both orientation cues and boundaries.
We focus on a series of experiments regarding inconsistent rotations of two extended
cue cards (one white, one black) around the perimeter of a cylinder in the absence
of any other orientation cues (Fenton et aI., 2000a). Each of these cards controls the
orientation of the set of place fields when rotated together or alone (after removal of
the other cue). When both are rotated inconsistently, place fields are displaced in a
non-uniform manner, with the displacement of a field being a function of its location
within the environment. These findings cannot be explained by a simple rotation
of the reference frame. Fig. 2A shows how place fields are displaced following
counter rotation of the two cue cards. Since the cue cards are orientation cues and
also walls of the environment, explaining these data within the current framework
requires two separate considerations: i) how the movement of the cards affects the
BVC's directional reference frame, and ii) how the movement of the cards, acting
as boundaries, directly affects the BVCs.
We make the following assumptions:
1. The influence of a distal visual cue on the directional reference system is
proportional to its proximity to the rat.
2. In the continued presence of color (or contrast) variation along a boundary to which a BVC responds, the BVC will become modulated by color:
responding preferentially to, say, a white section of wall rather than the
adjacent grey wall. In the absence of such variation it will revert to its
unmodulated response.
We note that assumption 1) is consistent with most implementations of the head
direction model discussed above, in that the influence of an extended distal cue will
increase with the angle subtended by it at the rat. We also note that assumption
2) implies the presence of synaptic learning (something not required by the rest of
the model), albeit outside of the hippocampus.
To avoid having to simulate enough random selections of BVCs to produce place
fields at all locations within the environment and with all combinations of distance,
bearing and color preferences, the model must be further simplified. To model the
effect of cue manipulation on a place field in a location from which there are two
cue cards at distances D i and bearings qli, we simulate a place cell for that location
which receives inputs from two BVCs tuned to the distances D i and bearings qli,
and to the most common color of boundary segments to which it respondes (across
all positions of the rat). That is, di .= D i and 4>i == q>i in equation 1. For each
location in the environment, we compute the shape of the place field formed by the
thresholded sum of these BVCs, before and after the cue card manipulation. This
simplification is broadly representative of the qualitative effect of the manipulation
on the locations of place fields!.
How does this model campare to the Fenton et aI. data? First we note that (due
to assumption 1) each cue card can control the overall orientation of the place and
head-direction representations. Similarly removing a cue card will have little effect,
save for a slight rotation and/or transverse spreading of the Bve that responds to it
(as it is no longer constrained by the color boundary, see assumption 2). When the
cues are rotated inconsistently, the firing fields of the BVCs move relative to each
other. The net effect of this on place fields and their centroids (Fig. 2B) compares
well with the data (Fig. 2A) and is composed of two separate effects. First, the
rotation of the cues produces a non-uniform distortion of the head direction system.
The extent of rotation depends on the location of the animal relative to the cues
as the closer a cue the more it affects the directional reference at that location
(assumption 1) see Fig. 2C (ii). This distortion of the directional reference frame
affects the orientations to which the boundary vector cells are tuned, and thus
affects the location of place fields in an approximately rotational manner see Fig.
2C (iii).. Second, the movement of the cue cards directly affects the firing fields
of the BVCs due to their color preferrence. This 'translational' effect is shown
in Fig. 2C (iv). Note that neither translational nor rotational effects alone are
sufficient to explain the observed data. Fenton et aI. (2000b) also make a distinction
between translation and rotation in producing a phenomenological model of their
data. However, as such, their model does not provide a mechanistic account at the
level of cells, is specific to the cue-card manipulation they made and so does not
make any prediction for head-direction cells or place cells in other experiments.
4
Non-parametric changes: 'remapping'
Our model considers the pattern of firing of place cells when the rat is put into an
environment of different shape, or when two very familiar landmarks are moved or
removed. In these situations changes to patterns of firing tend to be parametric,
and the model aims to capture the parametric relationships between firing pattern
and environmental manipulation. However we note that, after several days or weeks
of experience, the place cell representations of two environments of different shape
gradually diverge (Lever et aI., 2002), such that the final representations can be
said to have 'remapped' (MUller and Kubie, 1987). After 'remapping' a given cell
might fire in only one of the environments, or might fire in both but in unrelated
locations. Additionally, changing the color of the cue card in a grey cylinder from
white to black can cause more rapid remapping such that the effect on the first
day is probably best described as a slight rotation, with remapping occurring by
the second day (Bostock et aI., 1991). Note that simply removing the cue card just
causes the overall orientation of the place field representation to drift.
Could the current model be extended to begin to understand these apparently nonparametric changes? The effects of replacing the cue card with a novel one are
consistent with assumption 2 and the extra-hippocampal learning it implies: BVCs
initially respond to the new color as they would upon removal of the cue card, with
1 Simulations of place fields with a larger number of BVCs indicate similar field movements, but of reduced magnitude in locations far from the cue cards. However the good
match between the simple model and the data (Figs. 2A,B) suggests that the cue cards
do provide the majority of BVC input. This might be due to learned salience over the
extensive training period, and to the learning process implied by assumption 2. Against
this, place fields formed by more the two BVC inputs (e.g. the four BVCs in Fig. Ic)
generally give a better match to field shape, especially in locations far from the two cue
cards.
the slight rotation or spreading of the firing field noted above. Over time in the
presence of the new color, the color modulation of BVCs sharpens such that those
previously responding to white or grey no longer respond to black, while new BVCs
that do respond to black begin to fire. Thus the original place fields (particularly
those nearest the card and so most reliant on BVCs from that direction) will tend
to fall below threshold, unless receiving a connection from a newly active blacksensitive BVC, in which case the field will change location. Equally, some previously
silent place cells will become active due to input from a newly-active black-sensitive
BVC. By contrast, the slow shape-dependent remapping would appear to require
some additional mechanism. This may be related to the evidence of shorter-term
learning of associations between place cells (M~hta et aI., 1997) or the NMDAdependent stability of place fields (Kentros et aI., 1998) or postulated processes of
learned orthogonalisation of hippocampal representations (Marr, 1971; McClelland
et aI., 1995; Treves & Rolls, 1992; Fuhs & Touretzky, 2000; Kali & Dayan, 2000).
5
Conclusion
The model we have presented is consistent with a large body of detailed data on
the effects of parametric environmental manipulations on place and head-direction
cells. More importantly, it is a predictive model at the level of individual cells. Fig.
2C (ii) shows the prediction resulting from assumption 1) regarding the effect of
the inconsistent cue card manipulation on head-direction cells. We note that there
is an alternative to this location-dependent warping of head direction responses:
a direction-dependent warping such that responses to north directions are tilted
northwestwards while responses to south directions are tilted southwestwards. This
would correspond to the alternative assumption that the influence of a distal visual
cue on a head direction cell is proportional to the similarity of the average direction
of the cue from the rat and the preferred direction of the cell. We chose to simulate
the former (assumption 1) as this is consistent with current head-direction models in
keeping the angular separation of preferred directions constant (but rotating all of
them together as a function of the proximity of the rat to one or other cue card). The
alternative assumption breaks this constancy, but would produce roughly equivalent
results for place cell firing. Thus, on the basis of the Fenton et al. experiment on
place cells we must predict one or other of the two effects on head-direction, or some
combination of both. Beyond this, the model can predict the effect of essentially
arbitrary parametric movements of cues and boundaries on place and head-direction
'cells over the short term. It also appears to be at least consistent with the nonparametric 'remapping' changes induced by color changes. Whether or not it can
also predict the statistics of remapping over longer timescales in response to purely
geometric changes is a question for future work.
Acknowledgements: We thank John O'Keefe, Colin Lever and Bob Muller for
many useful discussions.
6
References
Bostock E, Muller RU, Kubie JL (1991) Experience-dependent modifications of hippocampal place cell firing Hippocampus 1, 193-206.
Cressant A, Muller RU, Poucet B (1997) Failure of centrally placed objects to control the
firing fields of hippocampal place cells. J. Neurosci. 17, 2531-2542.
Fenton AA, Csizmadia G, & Muller RU (2000a). Conjoint control of hippocampal place
cell firing by two visual stimuli. I. The effects of moving the stimuli on firing field positions.
J. Gen. Physiol, 116, 191-209.
Fenton AA, Csizmadia G, &. Muller RU (2000b). Conjoint control of hippocampal place
cell firing by two visual stimuli. Ii. A vector-field theory that predicts modifications of the
representation of the environment. J. Gen. Physiol, 116, 211-221.
Fuhs MC, Touretzky DS (2000) Synaptic learning models of map separation in the hippocampus. Neurocomputing, 32:379-384.
Hartley T, Burgess N, Lever C, Cacucci F, O'Keefe J (2000) Modeling place fields in terms
of the cortical inputs to the hippocampus. Hippocampus, 10, 369-379.
Hill AJ (1978) First occurrence of hippocampal spatial firing in a new environment. Exp.
Neural 62, 282-297.
Kali S, Dayan P (2000) The Involvement of Recurrent Connections in Area CA3 in Establishing the Properties of Place Fields: A Model. J. Neurosci. 20, 7463-7477.
Kentros C, Hargreaves E, Hawkins RD, Kandel ER, Shapiro M, Muller RU (1998) Abolition of long-term stability of new hippocampal place cell maps by NMDA receptor blockade. Science, 280, 2121-2126.
McNaughton BL, Knierim JJ, Wilson MA (1994) 'Vector encoding and the vestibular
foundations of spatial cognition: a neurophysiological and computational hypothesis', In
The Cognitive NeuroJJciences, (ed. Gazzaniga, M.) 585-596 (MIT Press, Boston, 1994).
Lever CL, Wills T, Cacucci F, Burgess N, O'Keefe J (2002) Long-term plasticity in the
hippocampal place cell representation of environmental geometry. Nature, in press.
Marr D (1971) Simple memory: a theory for archicortex. Phil. Trans. Roy. Soc. Lond B
262, 23-81.
McClelland JL, McNaughton BL, O'Reilly RC (1995) Why there are complementary
learning-systems in the hippocampus and neocortex - insights from the successes and
failures of connectionist models of learning and memory. Psychological Review 102, 419457.
Mehta MR, Barnes CA, McNaughton BL (1997) Experience-dependent, asymmetric expansion of hippocampal place fields. Proc. Nat. Acad. Sci. 94, 8918-8921.
Muller RU, Bostock E, Taube JS, Kubie JL (1994) On the directional firing properties of
hippocampal place cells. J. Neurosci. 14 7235-7251.
Muller RU, Kubie JL (1987) The effects of changes in the environment on the spatial firing
of hippocampal complex-spike cells. J. Neurosci 7, 1951-1968.
Muller RD, Kubie JL, Ranck JB (1987) Spatial firing patterns of hippocampal complexspike cells in a fixed environment. J. Neurosci., 7, 1935-1950.
O'Keefe J, Burgess N (1996) Geometric Determinants of the Place Fields of Hippocampal
Neurones. Nature 381, 425-428.
O'Keefe J, Dostrovsky J (1971) The hippocampus as a spatial map: preliminary evidence
from unit activity in the freely moving rat. Brain Res 34, 171-175.
O'Keefe J, Speakman A (1987) Single unit activity in the rat hippocampus during a spatial
memory task. Exp. Brain Res 68, 1-27.
Redish AD, Touretzky DS (1996) Modeling interactions of the rat's place and head direction systems Advances in Neural Information Processing Systems, 8. D Touretzky, MC
Mozer, ME Hasselmo (eds) pp. 61-67. MIT Press, Cambridge MA.
Skaggs WE, Knierim JJ, Kudrimoti HS, McNaughton BL (1995) 'A model of the neural
basis of the rat's sense of direction' Advances in Neural Information Processing Systems,
7. G Tesauro, D Touretzky &. TK Le'en (eds) pp. 51-58. MIT Press, Cambridge MA.
Taube JS, Muller RD, Ranck JB (1990) Head-direction cells recorded from the postsubiculum in freely moving rats. 1. Description .and quantitative analysis. J. Neurosci 10, 420-435.
Treves A, Rolls ET (1992) Computational constraints suggest the need for two distinct
input systems to the hippocampal CA3 network. Hippocampus 2, 189-200.
Wilson MA, McNaughton BL (1993) Dynamics of the hippocampal ensemble code for
space. Science 261, 1055-1058.
Zhang K (1996) Representation of spatial orientation by the intrinsic dynamics of the
head-direction cell ensemble: a theory. J .Neurosci., 16, 2112-2126.
b)
e)
Figure 1: Model of the geometrical influence on place fields (adapted from Hartley et
aI., 2000), assuming a stable directional reference frame. Place fields are composed
from thresholded linear sums of the firing rates of 'boundary vector cells' (BVCs).
a) Above: Each BVC has a Gaussian tuned response to the presence of a boundary
at a given distance and bearing from the rat (independent of its orientation). Below:
The sharpness of tuning of a BVC decreases as the distance to which it is tuned
increases. The only free parameters of a BVC are the distance and direction of
peak response. b) Place fields recorded from the same cell in four environments of
different shape or orientation relative to distal cues. c) Simulation of the place fields
in b) by the best fitting set of 4 BVCs constrained to be in orthogonal directions
(BVCs shown on the left, simulated fields on the right). The simulated cell can now
be used to predict firing in novel situations. Real and predicted data from three
novel environments are shown in d) and e) respectIvely, showing good qualitative
agreement.
A
B
=:==-~"
k""
,
i)
iii)
ii)
c
<It'"
I
\
-
-
:-
=:==-"~. . .,
--:q:::::'"
,=:.~
. .,
~
/'
t
f
:
i)
'''t'--
ii)
t
.......
.
......~
iii)
:
~
.
:- ....-?3:;:/??
iv)
Figure 2: Changes to place fields in a cylinder following inconsistent rotation of two
cue cards. A) Experimental data shown in a birds-eye view of the cyclinder including
the black and white cue-cards (adapted from Fenton et aI., 2000a). i) A place field
with the cue cards in the 'standard' condition (used throughout training). ii) The
place field after inconsistent rotation of each cue card by 12.5? further apart ('apart'
condition)~ iii) The movement of the centroid of place field from the standard
condition (tail of arrow) to the apart condition (head of arrow). B) Simulation of
21 place fields in the cyclinder in standard and apart conditions. Cue card locations
are indicated by a black line (initial card positions are indicated by a dotted line
to illustrate changes from one condition to another). i) and ii) show the place field
nearest in location to that shown in A) in standard and apart conditions. iii) shows
the movement of the centroids of simulated place fields between standard and apart
conditions. C) i) Simulation of the movement of place field centroids between the
standard and 'together' conditions (cue cards rotated 12.5? closer together). ii)
The distortion of the preferred direction of a head direction cell. Arrows show the
preferred direction in the 'apart' condition, the preferred direction was 'up' in the
standard condition. iii) the movement of place field centroids between the standard
and apart condition due solely to the directional distortion shown in ii). iv) the
movement of place field centroids due solely to the movement of the cue cards acting
as distinct cues (without any directional distortion shown in ii). The net effect of
fields iii) and iv) is that shown in B iii).
| 2011 |@word h:1 trial:1 determinant:2 cu:1 sharpens:1 hippocampus:11 sex:1 open:1 mehta:1 grey:3 simulation:6 initial:1 bvc:12 series:1 tuned:7 ranck:2 current:4 must:2 john:1 tilted:2 physiol:2 plasticity:1 shape:8 alone:2 cue:54 short:1 cacucci:2 provides:3 location:20 preference:1 zhang:2 rc:1 along:3 become:2 qualitative:2 fitting:1 manner:2 indeed:1 rapid:1 roughly:1 behavior:1 nor:1 brain:3 relying:1 little:1 begin:2 underlying:1 unrelated:1 remapping:8 cm:2 differing:1 finding:1 quantitative:3 continual:1 uk:3 control:5 unit:2 uo:2 unmodulated:1 producing:1 appear:1 before:1 local:1 acad:1 receptor:1 encoding:1 establishing:1 firing:28 modulation:1 approximately:1 solely:2 might:4 black:7 chose:1 bird:1 suggests:1 speakman:2 cfj:2 kubie:6 displacement:1 wholly:1 area:2 reilly:1 radial:1 integrating:1 suggest:2 cannot:1 selection:2 put:1 influence:8 instability:1 equivalent:1 map:3 phil:1 sharpness:1 insight:1 continued:1 importantly:1 marr:2 population:2 stability:2 variation:2 mcnaughton:7 hypothesis:1 agreement:1 roy:1 particularly:1 asymmetric:1 predicts:1 observed:1 constancy:1 fuhs:2 enters:1 capture:2 connected:2 movement:12 removed:2 counter:1 decrease:1 mozer:1 environment:27 dynamic:2 depend:1 segment:2 predictive:1 purely:1 upon:1 basis:2 easily:1 revert:1 distinct:2 london:1 outside:1 larger:1 say:1 distortion:5 otherwise:1 statistic:1 neil:1 gi:2 final:1 orthogonalisation:1 net:2 ucl:3 interaction:1 gen:2 description:2 moved:1 cli:1 qli:2 wcin:1 extending:1 produce:5 ring:1 rotated:4 object:2 bve:1 tk:1 recurrent:1 ac:2 illustrate:1 nearest:3 soc:1 predicted:2 indicate:2 implies:2 direction:40 anatomy:1 hartley:7 require:1 behaviour:1 wall:4 preliminary:1 proximity:2 around:2 hawkins:1 ic:1 exp:4 cognition:1 predict:6 week:1 vary:1 proc:1 spreading:2 sensitive:1 hasselmo:1 kudrimoti:1 mit:3 gaussian:1 aim:1 rather:1 avoid:1 wilson:3 encode:1 derived:2 focus:1 indicates:1 contrast:2 centroid:6 detect:1 sense:1 dependent:5 dayan:2 entire:1 initially:1 overall:2 translational:2 orientation:21 kali:2 development:1 animal:1 spatial:9 constrained:2 field:49 having:1 ng:1 identical:1 future:2 connectionist:1 stimulus:3 jb:2 composed:2 neurocomputing:1 individual:2 familiar:1 geometry:4 fire:5 attractor:1 maintain:1 cylinder:3 perimeter:1 accurate:1 edge:2 closer:2 preferential:1 experience:3 shorter:1 orthogonal:1 unless:2 iv:4 rotating:2 re:2 psychological:1 dostrovsky:2 modeling:2 obstacle:2 ar:1 queen:1 subtending:1 ca3:2 uniform:2 combined:1 peak:2 receiving:1 diverge:1 together:4 lever:4 recorded:3 cognitive:2 external:1 fenton:8 account:2 redish:2 north:1 postulated:1 caused:1 ad:3 depends:2 break:1 view:1 apparently:1 portion:1 poucet:1 square:2 formed:2 roll:2 characteristic:1 ensemble:2 correspond:1 directional:11 modelled:1 mc:2 bob:1 explain:1 touretzky:5 whenever:3 synaptic:2 ed:3 against:1 failure:2 pp:2 di:4 newly:2 color:11 orientational:2 nmda:1 appears:2 day:3 tom:1 response:13 arranged:1 done:1 angular:2 just:1 subtended:1 d:2 receives:1 replacing:1 aj:1 indicated:2 effect:20 former:1 proprioceptive:1 white:5 distal:7 adjacent:1 blockade:1 during:1 width:4 maintained:2 noted:1 rat:25 hippocampal:16 hill:2 fj:5 geometrical:2 consideration:2 novel:3 archicortex:1 common:1 rotation:16 functional:2 jl:5 association:2 discussed:1 slight:3 tail:1 numerically:1 cambridge:2 ai:17 tuning:2 rd:3 postsubiculum:1 similarly:1 phenomenological:1 moving:4 stable:4 longer:3 similarity:1 something:1 j:2 involvement:1 apart:8 tesauro:1 manipulation:9 success:1 muller:14 additional:1 floor:1 mr:1 freely:3 determine:1 taube:4 period:1 colin:1 ii:11 match:2 long:2 equally:1 visit:1 prediction:3 involving:1 basic:1 essentially:1 cell:50 extra:1 rest:1 probably:1 south:1 recording:1 tend:2 induced:1 inconsistent:4 presence:5 iii:8 enough:1 affect:7 skaggs:2 burgess:7 fit:2 silent:1 regarding:2 whether:1 neurones:1 cause:2 jj:2 generally:1 useful:1 detailed:1 nonparametric:3 neocortex:1 mcclelland:2 reduced:1 shapiro:1 dotted:1 neuroscience:1 estimated:1 broadly:1 four:2 threshold:2 changing:1 neither:1 thresholded:3 sum:3 hta:1 angle:2 eli:1 respond:5 place:64 throughout:1 separation:2 putative:1 simplification:1 centrally:1 barnes:1 activity:2 adapted:2 constraint:2 simulate:3 lond:1 department:1 combination:2 across:3 modification:2 explained:1 gradually:1 equation:1 previously:2 turn:1 mechanism:2 mind:1 mechanistic:1 occurrence:1 save:1 alternative:3 odor:1 allocentric:1 original:1 inconsistently:2 assumes:1 responding:2 testable:1 especially:1 bl:5 implied:1 move:1 warping:2 already:1 question:1 occurs:1 spike:1 parametric:6 responds:2 said:1 distance:14 separate:2 card:30 thank:1 sci:1 landmark:2 majority:1 simulated:3 me:1 extent:1 considers:1 assuming:2 ru:8 touretsky:1 code:1 relationship:1 providing:1 rotational:2 preferentially:1 implementation:1 diamond:1 displaced:2 situation:2 extended:5 head:25 frame:6 arbitrary:3 bostock:4 knierim:2 drift:2 treves:2 transverse:1 required:1 extensive:1 connection:2 rad:1 distinction:1 learned:2 vestibular:2 trans:1 beyond:3 able:1 gazzaniga:1 below:3 pattern:5 remapped:1 including:1 memory:3 eye:1 review:1 geometric:7 acknowledgement:1 removal:3 relative:4 proportional:2 conjoint:2 foundation:1 sufficient:1 consistent:11 pi:2 translation:1 placed:1 keeping:1 free:1 heading:1 salience:1 allow:1 understand:1 institute:1 explaining:1 fall:1 face:2 absolute:1 boundary:15 cortical:1 world:1 sensory:1 made:1 simplified:1 reorientation:1 far:2 preferred:5 active:3 continuous:1 why:1 additionally:1 nature:2 ca:1 expansion:1 bearing:10 cl:1 complex:1 synch:1 did:1 timescales:1 neurosci:7 arrow:3 complementary:2 body:1 fig:12 referred:2 representative:1 en:1 slow:1 position:3 kandel:1 removing:2 specific:3 showing:1 er:1 evidence:2 intrinsic:1 albeit:1 keefe:10 magnitude:1 nat:1 occurring:1 boston:1 simply:3 reliant:2 visual:5 neurophysiological:1 aa:2 environmental:4 ma:4 towards:1 absence:2 change:11 determined:2 acting:2 correlational:1 experimental:1 internal:2 modulated:1 heaviside:1 phenomenon:1 ex:1 |
1,110 | 2,012 | Optimising Synchronisation Times for
Mobile Devices
Neil D. Lawrence
Department of Computer Science,
Regent Court, 211 Portobello Road,
Sheffield, Sl 4DP, U.K.
neil~dcs.shef . ac.uk
Antony 1. T. Rowstron Christopher M . Bishop Michael J. Taylor
Microsoft Research
7 J. J. Thomson Avenue
Cambridge, CB3 OFB, U.K.
{antr,cmbishop,mitaylor}~microsoft.com
Abstract
With the increasing number of users of mobile computing devices
(e.g. personal digital assistants) and the advent of third generation
mobile phones, wireless communications are becoming increasingly
important. Many applications rely on the device maintaining a
replica of a data-structure which is stored on a server, for example news databases, calendars and e-mail. ill this paper we explore
the question of the optimal strategy for synchronising such replicas.
We utilise probabilistic models to represent how the data-structures
evolve and to model user behaviour. We then formulate objective
functions which can be minimised with respect to the synchronisation timings. We demonstrate, using two real world data-sets, that
a user can obtain more up-to-date information using our approach.
1
Introduction
As the available bandwidth for wireless devices increases, new challenges are presented in the utilisation of such bandwidth. Given that always up connections are
generally considered infeasible an important area of research in mobile devices is
the development of intelligent strategies for communicating between mobile devices
and servers. ill this paper we consider the scenario where we are interested in maintaining, on a personal digital assistant (PDA) with wireless access, an up-to-date
replica of some, perhaps disparate, data-structures which are evolving in time. The
objective is to make sure our replica is not 'stale'. We will consider a limited number
of connections or synchronisations. Each synchronisation involves a reconciliation
between the replica on the mobile device and the data-structures of interest on the
server. Later in the paper we shall examine two typical examples of such an application,an internet news database and a user's e-mail messages. Currently the typical
strategy! for performing such reconciliations is to synchronise every M minutes,
lSee, for example, AvantGo http://vvv. avantgo. com.
where M is a constant, we will call this strategy the uniformly-spaced strategy. We
will make the timings of the synchronisations adaptable by developing a cost function that can be optimised with respect to the timings, thereby improving system
performance.
2
Cost Function
We wish to minimise the staleness of the replica, where we define staleness as the
time between an update of a portion of the data-structure on the server and the time
of the synchronisation of that update with the PDA. For simplicity we shall assume
that each time the PDA synchronises all the outstanding updates are transferred.
Thus, after synchronisation the replica on the mobile device is consistent with the
master copy on the server. Therefore, if skis the time of the kth synchronisation
in a day, and updates to the data-structure occur at times Uj then the average
staleness of the updates transferred during synchronisation Sk will be
(1)
As well as staleness, we may be interested in optimising other criteria. For example,
mobile phone companies may seek to equalise demand across the network by introducing time varying costs for the synchronisations, c(t). Additionally one could
argue that there is little point in keeping the replica fresh during periods when the
user is unlikely to check his PDA, for example when he or she is sleeping. We might
therefore want to minimise the time between the user's examination of the PDA
and the last synchronisation. If the user looks at the PDA at times ai then we can
express this as
(2)
Given the timings Uj and ai, the call cost schedule c(t) and K synchronisations, the
total cost function may now be written
K
C=
L (-aFk + fJSk + C(Sk)) '
(3)
k=l
where a and fJ are constants with units of ~~~:y which express the relative importance of the separate parts of the cost function. Unfortunately, of course, whilst
we are likely to have knowledge of the call cost schedule, c(t), we won't know the
true timings {Uj} and {ai} and the cost function will be a priori incomputable. If,
though, we have historic data2 relating to these times, we can seek to make progress
by modelling these timings probabilistically. Then, rather than minimising the actual cost function, we can look to minimise the expectation of the cost function
under these probabilistic models.
3
Expected Cost
There are several different possibilities for our modelling strategy. A sensible assumption is that there is independence between different parts of the data-structure
(i.e. e-mail and business news can be modelled separately), however, there may
be dependencies between update times which occur within the same part. The
2When modelling user ru:cess times, if historic data is not available, models could also
be constructed by querying the user about their likely ru:tivities.
periodicity of the data may be something we can take advantage of, but any nonstationarity in the data may cause problems. There are various model classes we
could consider; for this work however, we restrict ourselves to stationary models,
and ones in which updates arrive independently and in a periodic fashion . .
Let us take T to be the largest period of oscillation in the data arrivals, for a
particular portion of a data-structure. We model this portion with a probability
distribution, Pu(t). Naturally more than one update may occur in that interval,
therefore our probability distribution really specifies a distribution over time given
one that one update (or user access) has occurred. To fully specify the model we
also are required to store the expected number of updates, Ju , (or accesses, J a ) that
occur in that interval.
The expected value of Sk may now be written,
(4)
where Op(:v) is an expectation under the distribution p(x), Au(t) = JuPu(t) can be
viewed as the rate at which updates are occurring and So = SK - T.
We can model the user access times, ai, in a similar manner, which leads us to the
expected value of the freshness, (Fk)Pa(t) = kk +l Aa(t)(t - sk)dt, where Aa(t) =
JaPa(t) The overall expected cost, which we will utilise as our objective function,
may therefore be written
J:
K
(C) =
L
(Sk)p" - (Fk)Pa
+ C(Sk)) .
(5)
k=l
3.1
Probabilistic Models.
We now have an objective function which is a function of the variables we wish to
optimise, the synchronisation times, but whilst we have mentioned some characteristics of the models Pu(t) and Pa(t) we have not yet fully specified their form.
We have decreed that the models should be periodic and that they may consider
each datum to occur independently. In effect we are modelling data which is mapped
to a circle. Various options are available for handling such models; for this work,
we constrain our investigations to kernel density estimates (KDE).
In order to maintain periodicity, we must select a basis function for our KDE which
represents a distribution on a circle, one simple way of achieving this aim is to wrap
a distribution that is defined along a line to the circle (Mardia, 1972). A traditional
density which represents a distribution on the line, p(t), may be wrapped around
a circle of circumference T to give us a distribution defined on the circle, p( 0),
where 0 = t mod T. This means a basis function with its centre at T - 8, that will
typically have probability mass when u > T, wraps around to maintain a continuous
density at T. The wrapped Gaussian distribution 3 that we make use of takes the
form
(6)
The final kernel density estimate thus consists of mapping the data points tn
-t
On
3In practice we must approximate the wrapped distribution by restricting the number
of terms in the sum.
"-'
~60
.?
~40
"-'
.S
1)520
~
OJ
H
Co>
OJ
""0
Thu Fri Sat Sun
Thu Fri Sat Sun
~
-20
Figure 1: Left: part of the KDE developed for the business category together with a
piecewise constant approximation. Middle: the same portion of the KDE for the FA Carling Premiership data. Right: percent decrease in staleness vs number of synchronisations
per day for e-mail data.
and obtaining a distribution
1 N
p(()) = N
L W N(()I()n, (}"2),
(7)
n=l
where N is the number of data-points and the width parameters, (}", can be set
through cross validation. Models of this type may be made use of for both Pu(t)
and Pa(t) .
3.2
Incorporating Prior Knowledge.
The underlying component frequencies of the data will clearly be more complex than
simply a weekly or daily basis. Ideally we should be looking to incorporate as much
of our prior knowledge about these component frequencies as possible. IT we were
modelling financial market's news, for example, we would expect weekdays to have
similar characteristics to each other, but differing characteristics from the weekend.
For this work, we considered four different scenarios of this type. For the first
scenario, we took T = 1 day and placed no other constraints on the model. For the
second we considered the longest period to be one week, T = 1 week, and placed no
further constraints on the model. For the remaining two though we also considered
T to be one week, but we implemented further assumptions about the nature of the
data. Firstly we split the data into weekdays and weekends. We then modelled these
two categories separately, making sure that we maintained a continuous function
for the whole week by wrapping basis functions between weekdays and weekends.
Secondly we split the datainto weekdays, Saturdays and Sundays, modelling each
category separately and again wrapping basis functions across the days.
3.3
Model Selection.
To select the basis function widths, and to determine which periodicity assumption
best matched the data, we ?utilised ten fold cross validation. For each different
periodicity we used cross validation to first select the basis function width. We
then compared the average likelihood across the ten validation sets, selecting the
periodicity with the highest associated value.
4
Optimising the Synchronisation Times
Given that our user model, Pa(t), and our data model, Pu(t) will be a KDE based
on wrapped Gaussians, we should be in a position to compute the required integrals
'"
C/J
Q)
Q)
t{360
>=1
t{360
>=1
<il
.....,40
X
X
C/J
+
)I(
.S
]40
C/J
.S
S520
S520
~
~
Q)
Q)
!-<
!-<
U
~
-20
X
~
C/J
C/J
Q)
!I
~
.....,
4
X
"0
~
j
~
C/J
Q)
2
2
X
X
-20
Figure 2: Results from the news database tests. Left:
February /March based models tested on April. Middle:
March/ April testing on May. Right: April/May testing on
June. The results are in the form of box plots. The lower
line of the box represents the 25th percentile of the data, the
upper line the 75th percentile and the central line the median. The 'whiskers' represent the maximum extent of the
data up to 1.5 x (75th percentile - 25th percentile). Data
which lies outside the whiskers is marked with crosses.
.S
X
12
Q)
C/J
~
~20
u
+
+
40
>=1
.$20
U
Q)
"0
60
x
!i I~!
24
X
X
xx
Q)
"0
X
X
X
~40
X
X
-60
+
xx
X
xXx
in (5) and evaluate our objective function and derivatives thereof.
First though, we must give some attention to the target application for the algorithm. A known disadvantage of the standard kernel density estimate is the high
storage requirements of the end model. The model requires that N floating point
numbers must be stored, where N is the quantity of training data. Secondly, integrating across the cost function results in an objective function which is dependent
on a large number of evaluations of the cumulative Gaussian distribution. Given
that we envisage that such optimisations could be occurring within a PDA or mobile
phone, it would seem prudent to seek a simpler approach to the required minimisation.
An alternative approach that we explored is to approximate the given distributions
with a functional form which is more amenable to the integration. For example, a
piecewise constant approximation to the KDE simplifies the integral considerably.
It leads to a piecewise constant approximation for Aa(t) and Au(t). Integration
over which simply leads to a piecewise linear function which may be computed in a
straightforward manner. Gradients may also be computed. We chose to reduce the
optimisation to a series of one-dimensional line minimisations. This can be achieved
in the following manner. First, note that the objective function, as a function of a
particular synchronisation time Sk, may be written:
(8)
In other words, each synchronisation is only dependent on that of its neighbours. We
may therefore perform the optimisation by visiting each synchronisation time, Sk, in
a random order and optimising its position between its neighbours, which involves a
one dimensional line minimisation of (8). This process, which is guaranteed to find
a (local) minimum in our objective function, may be repeated until convergence.
5
Results
In this section we mainly explore the effectiveness of modelling the data-structures
of interest. We will briefly touch upon the utility of modelling the cost evolution
and user accesses in Section 5.2 but we leave a more detailed exploration of this
area to later works.
5.1
Modelling Data Structures
To determine the effectiveness of our approach, we utilised two different sources of
data: a news web-site and e-mail on a mail server.
The news database data-set was collected from the BBC
News web site4 . This site maintains a database of articles which are categorised according to subject, for example, UK News, Business News, Motorsport etc .. We
had six months of data from February to July 2000 for
24 categories of the database.
We modelled the data by decomposing it into the different categories and modelling each separately. This
allowed us to explore the periodicity of each category independently. This is a sensible approach given that the
nature of the data varies considerably across the categories. .Two extreme examples are Business news and
FA Carling Premiership news 5 , Figure 1. Business news
predominantly arrives during the week whereas FA Carling Premiership news arrives typically just after soccer
games finish on a Saturday. Business news was best
modelled on a Weekday/Weekend basis, and FA Carling Premiership news was best modelled on a Weekday /Saturday /Sunday basis. To evaluate the feasibility
of our approach, we selected three consecutive months
of data. The inference step consisted of constructing our
models on data from the first two months. To restrict
our investigations to the nature of the data evolution
only, user access frequency was taken to be uniform and
cost of connection was considered to be constant. For
the decision step we considered 1 to 24 synchronisations
a day. The synchronisation times were optimised for
each category separately, they were initialised with a
uniformly-spaced strategy, optimisation of the timings
then proceeded as described in Section 4. The staleness associated with these timings was then computed
for the third month. This value was compared with the
staleness resulting from the uniformly-spaced strategy
containing the same number of synchronisations 6 . The
percentage decrease in staleness is shown in figures 2
and 3 in the form of box-plots.
60
x
x
x
40
x
x
20
12
-20 X X
00
g}40
?
'"@
t260
.S
Xx
x
~XX ~
x
~,(?<:
~x
~
x
\xx
++
.,?
+ )(
+
x
x x
x
x
+
+
+ +
+
-120
+
+
-140
+
+
+
-160
+
+ + +
-180
+
-200
Figure 3:
May/June
based models tested on
July.
+ signifies the
FA Carling Premiership
Stream.
Generally an improvement in performance is observed,
however, we note that in Figure 3 the performance for several categories is extremely
4http://news.bbc.co.uk.
5The FA Carling Premiership is England's premier division soccer.
6The uniformly-spaced strategy's staleness varies with the timing of the first of the K
synchronisations. This figure was therefore an average of the staleness from all possible
starting points taken at five minute intervals.
poor. In particular the FA Carling Premiership stream in Figure 3. The poor
performance is caused by the soccer season ending in May. As a result relatively
few articles are written in July, most of them concerning player transfer speculation,
and the timing of those articles is very different from those in May. In other words
the data evolves in a non-stationary manner which we have not modelled. The other
poor performers are also sports related categories exhibiting non-stationarities.
The e-mail data-set was collected by examining the logs of e-mail arrival times for
9 researchers from Microsoft's Cambridge research lab. This data was collected for
January and February 2001. We utilised the January data to build the probabilistic models and the February data to evaluate the average reduction in staleness.
Figure 1 shows the results obtained.
In practice, a user is more likely to be interested in a combination of different categories of data. Perhaps several different streams of news and his e-mail. Therefore,
to recreate a more realistic situation where a user has a combination of interests,
we also collected e-mail arrivals for three users from February, March and April
2000. We randomly generated user profiles by sampling, without replacement, five
categories from the available twenty-seven, rejecting samples where more than one
e-mail stream was selected. We then modelled the users' interests by constructing
an unweighted mixture of the five categories and proceeded to optimise the synchronisation times based on this model. This was performed one hundred times. The
average staleness for the different numbers of synchronisations per day is shown in
Figure 4.
Note that the performance for the combined categories is worse than it is for each
individually. This is to be expected as the entropy of the combined model will always
be greater than that of its constituents, we therefore have less information about
arrival times, and as a result there are less gains to be made over the uniformlyspaced strategy7.
5.2
Affect of Cost and User Model
In the previous sections we focussed on modelling the evolution of the databases.
Here we now briefly turn our attention to the other portions of the system, user
behaviour and connection cost. For this preliminary study, it proved difficult to
obtain high quality data representing user access times. We therefore artificially
generated a model which represents a user who accesses there device frequently
at breakfast, lunchtime and during the evening, and rarely at night. Figure 4
simply shows the user model, Pa(t), along with the result of optimising the cost
function for uniform data arrivals and fixed cost under this user model. Note how
synchronisation times are suggested just before high periods of user activity are
about to occur. Also in Figure 4 is the effect of a varying cost, c(t), under uniform
Pa(t) and Pa(t).
Currently most mobile internet access providers appear to be charging a flat fee for
call costs (typically in the U.K. about 15 cents per minute). However, when demand
on their systems rise they may wish to incorporate a varying cost to flatten peak
demands. This cost could be an actual cost for the user, or alternatively a 'shadow
price' specified by service provider for controlling demand (Kelly, 2000). We give a
simple example of such a call cost in Figure 4. For this we considered user access
and data update rates to be constant. Note how the times move away from periods
of high cost.
7The uniformly-spaced strategy can be shown to be optimal when the entropy of the
underlying distribution is maximised (a uniform distribution across the interval).
"-'
gj 60
>=1
Cl)
7Lo
0.3
0.25
.....,
0.2
"-'
1200
~ 900
~20
U
0.15
'...."
Cl)
:::::::: 600
'"
0.1
U
0.05
0
00:00
U
Cl)
300
"'0
~
08:00
X
.S
00:00
08:00
16:00
00:00
X
-20
Figure 4: Left: change in synchronisation times for variable user access rates. x shows
the initialisation points, + the end points. Middle: change in synchronisation times for a
variable cost. Right: performance improvements for the combination of news and e-mail.
6
Discussion
The optimisation strategy we suggest could be sensitive to local minima, we did
not try a range of different initialisations to explore this phenomena. However, by
initialising with the uniformly-spaced strategy we ensured that we increased the
objective function relative to the standard strategy. The month of July showed how
a non-stationarity in the data structure can dramatically affect our performance.
We are currently exploring on-line Bayesian models which we hope will track such
non-stationarities.
The system we have explored in this work assumed that the data replicated on the
mobile device was only modified on the server. A more general problem is that of
mutable replicas where the data may be modified on the server or the client. Typical
applications of such technology include mobile databases, where sales personnel
modify portions of the database whilst on the road, and a calendar application on
a PDA, where the user adds appointments on the PDA.
Finally there are many other applications of this type of technology beyond mobile
devices. Web crawlers need to estimate when pages are modified to maintain a
representative cache (eho and Garcia-Molina, 2000). Proxy servers could also
be made to intelligent maintain their caches of web-pages up-to-date (Willis and
Mikhailov, 1999; Wolman et al., 1999) .
References
Cho, J. and H. Garcia-Molina (2000). Synchronizing a database to improve freshness.
In Proceedings 2000 ACM International Conference on Management of Data (SIG-
MOD).
Kelly, F. P. (2000). Models for a self-managed internet. Philosophical Transactions of
the Royal Society A358, 2335-2348.
Mardia, K. V. (1972). Statistics of Directional Data. London: Academic Press.
Rowstron, A. 1. T., N. D. Lawrence, and C. M. Bishop (2001). Probabilistic modelling of
replica divergence. In Proceedings of the 8th Workshop on Hot Topics in Operating
Systems HOTOS (VIII).
Willis, C. E. and M. Mikhailov (1999). Towards a better understanding of web resources
and server responses for improved caching. In Proceedings of the 8th International
World Wide Web Conference, pp. 153-165.
Wolman, A., G. M. Voelker, N. Sharma, N. Cardwell, A. Karlin, and H. M. Levy
(1999). On the scale and performance of co-operative web proxy caching. In 17th
ACM Symposium Operating System Principles (SOSP'99), pp. 16-3l.
Yu, H. and A. Vahdat (2000). Design and evaluation of a continuous consistency model
for replicated services. In 4th Symposium on Operating System Design and Implementation (OSDI).
| 2012 |@word proceeded:2 briefly:2 middle:3 seek:3 weekday:6 thereby:1 reduction:1 series:1 selecting:1 initialisation:2 com:2 yet:1 written:5 must:4 realistic:1 plot:2 update:12 v:1 stationary:2 selected:2 device:11 maximised:1 data2:1 firstly:1 simpler:1 five:3 along:2 constructed:1 saturday:3 symposium:2 consists:1 regent:1 manner:4 market:1 expected:6 examine:1 frequently:1 company:1 little:1 actual:2 cache:2 increasing:1 xx:5 underlying:2 matched:1 mass:1 advent:1 developed:1 whilst:3 differing:1 cmbishop:1 synchronisation:27 every:1 stationarities:2 weekly:1 ensured:1 uk:3 sale:1 unit:1 appear:1 before:1 service:2 timing:10 local:2 modify:1 vahdat:1 optimised:2 becoming:1 might:1 chose:1 au:2 co:3 limited:1 range:1 testing:2 practice:2 area:2 evolving:1 word:2 road:2 integrating:1 flatten:1 suggest:1 selection:1 storage:1 circumference:1 straightforward:1 attention:2 starting:1 independently:3 formulate:1 simplicity:1 communicating:1 his:2 financial:1 lsee:1 target:1 controlling:1 user:30 sig:1 pa:8 database:10 observed:1 news:19 sun:2 decrease:2 highest:1 mentioned:1 pda:9 ideally:1 rowstron:2 personal:2 upon:1 bbc:2 division:1 basis:9 lunchtime:1 various:2 weekend:4 london:1 outside:1 voelker:1 calendar:2 statistic:1 neil:2 envisage:1 final:1 advantage:1 took:1 karlin:1 date:3 constituent:1 convergence:1 requirement:1 leave:1 ac:1 op:1 progress:1 implemented:1 involves:2 shadow:1 exhibiting:1 exploration:1 behaviour:2 really:1 investigation:2 preliminary:1 secondly:2 exploring:1 around:2 considered:7 lawrence:2 mapping:1 week:5 consecutive:1 assistant:2 currently:3 sensitive:1 individually:1 largest:1 hope:1 clearly:1 always:2 gaussian:2 aim:1 sunday:2 rather:1 modified:3 caching:2 season:1 mobile:13 varying:3 probabilistically:1 minimisation:3 june:2 she:1 longest:1 modelling:12 check:1 likelihood:1 mainly:1 improvement:2 afk:1 inference:1 osdi:1 dependent:2 unlikely:1 typically:3 interested:3 overall:1 ill:2 prudent:1 priori:1 development:1 integration:2 sampling:1 optimising:5 represents:4 synchronizing:1 look:2 yu:1 intelligent:2 piecewise:4 few:1 randomly:1 neighbour:2 divergence:1 floating:1 ourselves:1 replacement:1 microsoft:3 maintain:4 stationarity:1 interest:4 message:1 possibility:1 evaluation:2 mixture:1 extreme:1 arrives:2 amenable:1 integral:2 daily:1 taylor:1 circle:5 increased:1 disadvantage:1 signifies:1 cost:27 introducing:1 uniform:4 hundred:1 examining:1 stored:2 dependency:1 varies:2 antony:1 periodic:2 considerably:2 combined:2 cho:1 ju:1 density:5 peak:1 international:2 probabilistic:5 minimised:1 michael:1 together:1 again:1 central:1 management:1 containing:1 worse:1 derivative:1 caused:1 stream:4 later:2 performed:1 utilised:3 lab:1 try:1 thu:2 portion:6 option:1 maintains:1 il:1 characteristic:3 who:1 spaced:6 directional:1 modelled:7 bayesian:1 rejecting:1 provider:2 researcher:1 premier:1 nonstationarity:1 frequency:3 initialised:1 pp:2 thereof:1 naturally:1 associated:2 gain:1 proved:1 knowledge:3 schedule:2 adaptable:1 dt:1 day:6 xxx:1 breakfast:1 specify:1 response:1 april:4 improved:1 though:3 box:3 just:2 until:1 web:7 christopher:1 touch:1 night:1 quality:1 perhaps:2 stale:1 effect:2 consisted:1 true:1 managed:1 evolution:3 staleness:12 utilisation:1 wrapped:4 during:4 width:3 game:1 self:1 maintained:1 portobello:1 percentile:4 soccer:3 won:1 criterion:1 thomson:1 demonstrate:1 willis:2 tn:1 fj:1 percent:1 predominantly:1 functional:1 he:1 occurred:1 relating:1 cambridge:2 ai:4 fk:2 consistency:1 centre:1 had:1 access:11 operating:3 gj:1 etc:1 pu:4 add:1 something:1 showed:1 phone:3 scenario:3 store:1 server:10 freshness:2 molina:2 minimum:2 greater:1 performer:1 fri:2 determine:2 period:5 sharma:1 july:4 reconciliation:2 england:1 academic:1 minimising:1 cross:4 concerning:1 feasibility:1 sheffield:1 optimisation:5 expectation:2 represent:2 kernel:3 achieved:1 sleeping:1 whereas:1 want:1 separately:5 shef:1 interval:4 appointment:1 operative:1 median:1 source:1 sure:2 subject:1 vvv:1 mod:2 seem:1 effectiveness:2 call:5 split:2 independence:1 finish:1 affect:2 bandwidth:2 restrict:2 reduce:1 simplifies:1 avenue:1 court:1 minimise:3 recreate:1 six:1 utility:1 cause:1 dramatically:1 generally:2 detailed:1 ten:2 category:14 http:2 specifies:1 sl:1 percentage:1 per:3 track:1 shall:2 express:2 four:1 achieving:1 cb3:1 ce:1 replica:10 sum:1 master:1 arrive:1 oscillation:1 decision:1 initialising:1 fee:1 internet:3 guaranteed:1 datum:1 fold:1 activity:1 occur:6 constraint:2 constrain:1 flat:1 extremely:1 performing:1 relatively:1 transferred:2 department:1 developing:1 according:1 march:3 poor:3 combination:3 across:6 increasingly:1 evolves:1 making:1 taken:2 resource:1 turn:1 know:1 end:2 available:4 gaussians:1 decomposing:1 away:1 alternative:1 cent:1 remaining:1 include:1 maintaining:2 eho:1 uj:3 build:1 february:5 society:1 objective:9 move:1 question:1 quantity:1 wrapping:2 strategy:13 fa:7 traditional:1 visiting:1 gradient:1 dp:1 kth:1 wrap:2 separate:1 mapped:1 sensible:2 seven:1 mail:12 argue:1 extent:1 collected:4 topic:1 fresh:1 viii:1 ru:2 kk:1 difficult:1 unfortunately:1 kde:6 disparate:1 rise:1 design:2 implementation:1 ski:1 twenty:1 perform:1 upper:1 january:2 situation:1 communication:1 looking:1 dc:1 ofb:1 required:3 specified:2 speculation:1 connection:4 philosophical:1 beyond:1 suggested:1 challenge:1 optimise:2 oj:2 royal:1 charging:1 hot:1 business:6 rely:1 examination:1 client:1 representing:1 improve:1 technology:2 prior:2 kelly:2 understanding:1 evolve:1 relative:2 fully:2 historic:2 expect:1 whisker:2 generation:1 querying:1 digital:2 validation:4 consistent:1 proxy:2 article:3 principle:1 lo:1 course:1 periodicity:6 placed:2 wireless:3 copy:1 keeping:1 infeasible:1 last:1 wide:1 focussed:1 world:2 cumulative:1 ending:1 unweighted:1 made:3 replicated:2 wolman:2 transaction:1 approximate:2 categorised:1 sat:2 assumed:1 alternatively:1 continuous:3 evening:1 sk:9 additionally:1 nature:3 transfer:1 obtaining:1 improving:1 complex:1 artificially:1 constructing:2 cl:3 did:1 whole:1 arrival:5 profile:1 repeated:1 allowed:1 site:2 representative:1 fashion:1 position:2 wish:3 lie:1 mardia:2 levy:1 third:2 synchronise:1 minute:3 bishop:2 explored:2 incorporating:1 workshop:1 restricting:1 importance:1 occurring:2 demand:4 entropy:2 garcia:2 simply:3 explore:4 likely:3 personnel:1 sport:1 aa:3 utilise:2 acm:2 viewed:1 marked:1 month:5 towards:1 price:1 change:2 typical:3 uniformly:6 total:1 player:1 rarely:1 select:3 crawler:1 outstanding:1 incorporate:2 evaluate:3 tested:2 phenomenon:1 handling:1 |
1,111 | 2,013 | Grouping with Bias
Stella X. Yu
Robotics Institute
Carnegie Mellon University
Center for the Neural Basis of Cognition
Pittsburgh, PA 15213-3890
Jianbo Shi
Robotics Institute
Carnegie Mellon University
5000 Forbes Ave
Pittsburgh, PA 15213-3890
stella. yu@es. emu. edu
jshi@es.emu.edu
Abstract
With the optimization of pattern discrimination as a goal, graph
partitioning approaches often lack the capability to integrate prior
knowledge to guide grouping. In this paper, we consider priors
from unitary generative models, partially labeled data and spatial
attention. These priors are modelled as constraints in the solution
space. By imposing uniformity condition on the constraints, we
restrict the feasible space to one of smooth solutions. A subspace
projection method is developed to solve this constrained eigenproblema We demonstrate that simple priors can greatly improve image
segmentation results.
1 " Introduction
Grouping is often thought of as the process of finding intrinsic clusters or group
structures within a data set. In image segmentation, it means finding objects or
object segments by clustering pixels and segregating them from background. It
is often considered a bottom-up process. Although never explicitly stated, higher
level of knowledge, such as familiar object shapes, is to be used only in a separate
post-processing step.
The need for the integration of prior knowledge arises in a number of applications. In
computer vision, we would like image segmentation to correspond directly to object
segmentation. In data clustering, if users provide a few examples of clusters, we
would like a system to adapt the grouping process to achieve the desired properties.
In this case, there is an intimate connection to learning classification with partially
labeled data.
We show in this paper that it is possible to integrate both bottom-up and top-down
information in a single grouping process. In the proposed method, the bottom-up
grouping process is modelled as a graph partitioning [1, 4, 12, 11, 14, 15] problem, and the top-down knowledge is encoded as constraints on the solution space.
Though we consider normalized cuts criteria in particular, similar derivation can be
developed for other graph partitioning criteria as well. We show that it leads to a
constrained eigenvalue problem, where the global optimal solution can be obtained
by eigendecomposition. Our model is expanded in detail in Section 2. Results and
conclusions are given in Section 3.
2
Model
In graph theoretic methods for grouping, a relational graph GA == (V, E, W) is first
constructed based on pairwise similarity between two elements. Associated with
the graph edge between vertex i and j is weight Wij , characterizing their likelihood
of belonging in the same group.
For image segmentation, pixels are taken as graph nodes, and pairwise pixel similarity can be evaluated based on a number of low level grouping cues. Fig. Ic shows
one possible defini~ion, where the weight b.etween two pixels is inversely proportional
to the magnitude of the strongest intervening edge [9].
a)Image.
d)NCuts.
e)Segmentation.
Figure 1: Segmentation by graph partitioning. a)200 x 129 image with a few pixels
marked( +). b)Edge map extracted using quadrature filters.c)Local affinity fields of
marked pixels superimposed together. For every marked pixel, we compute its affinity
with its neighbours within a radius of 10. The value is determined by a Gaussian function
of the maximum magnitude of edges crossing the straight line connecting the two pixels
[9]. When there is a strong edge separating the two, the affinity is low. Darker intensities
mean larger values. d)Solution by graph partitioning. It is the second eigenvector from
normalized cuts [15] on the affinity matrix. It assigns a value to each pixel. Pixels of
similar values belong to the same group. e)Segmentation by thresholding the eigenvector
with o. This gives a bipartitioning of the image which corresponds to the best cuts that
have maximum within-region coherence and between-region distinction.
After an image is transcribed into a graph, image segmentation becomes a vertex
partitioning problem. Consider segmenting an image into foreground and background. This corresponds to vertex bipartitioning (VI, V2 ) on graph G, where
V = VI U V2 and VI n V2 = 0. A good segmentation seeks a partitioning such
that nodes within partitions are tightly connected and nodes across partitions are
loosely connected. A number of criteria have been proposed to achieve this goal.
For normalized cuts [15], the solution is given by some eigenvector of weight matrix
W (Fig. Id). Thresholding on it leads to a discrete segmentation (Fig. Ie). W.hile
we will focus on normalized cuts criteria [15], most of the following discussions apply
to other criteria as well.
2.1
Biased grouping as constrained optimization
Knowledge other than the image itself can greatly change the segmentation we might
obtain based on such low level cues. Rather than seeing boundaries between black
and white regions, we see objects. The sources of priors we consider in this paper
are: unitary generative models (Fig. 2a), which could arise from sensor models
in MRF [5], partial grouping (Fig. 2b), which could arise from human computer
interaction [8], and spatial attention (Fig. 2c). All of these provide additional, often
long-range, binding information for grouping.
We model such prior knowledge in the form of constraints on a valid grouping
configuration. In particular, we see that all such prior knowledge defines a partial
a)Bright foreground.
b)Partial grouping.
c)Spatial attention.
Figure 2: Examples of priors considered in this paper. a)Local constraints from unitary
generative models. In this case, pixels of light (dark) intensities are likely to be the foreground(background). This prior knowledge is helpful not only for identifying the tiger as
the foreground, but also for perceiving the river as one piece. How can we incorporate
these unitary constraints into a- graph that handles only pairwise relationships between
pixels? b )Global configuration constraints from partial grouping a priori. In this case,
we have manually selected two sets of pixels to be grouped together in foreground (+)
and background (JJ.) respectively. They are distributed across the image and often have
distinct local features. How can we force them to be in the same group and further bring
similar pixels along and push dissimilar pixels apart? c)Global constraints from spatial
attention. We move our eyes to the place of most interest and then devote our limited
visual processing to it. The complicated scene structures in the periphery can thus be
ignored while sparing the parts associated with the object at fovea. How can we use this
information to facilitate figural popout in segmentation?
grouping solution, indicating which set of pixels should belong to one partition.
Let Hz, 1 == 1"" ,n, denote a partial grouping. H t have pixels known to be in
Vt , t == 1,2. These sets are derived as follows.
Unitary generative models: H l and H 2 contains a set of pixels that satisfy the
unitary generative models for foreground and background respectively. For example,
in Fig. 2a, H l (H2 ) contains pixels of brightest(darkest) intensities.
Partial grouping: Each Hz, 1 == 1, ... ,n, contains a set of pixels that users specify to
belong together. The relationships between Hz, 1 > 2 and Vt , t == 1,2 are indefinite.
Spatial attention: H l == 0 and H 2 contains pixels randomly selected outside the
visual fovea, since we want to maintain maximum discrimination at the -fovea but
merging pixels far away from the fovea to be one group.
To formulate these constraints induced on the graph partitioning, we introduce
binary group indicators X == [Xl, X 2 ]. Let N == IVI be the number of nodes in the
graph. For t == 1,2, X t is an N x 1 vector where Xt(k) == 1 if vertex k E Vt and 0
otherwise. The constrained grouping problem can be formally written as:
min
s.t.
?(Xl ,X2 )
Xt(i) == Xt(j), i, j E HE, 1 == 1"" ,n, t == 1,2,
Xt(i) =1= Xt(j), i E H l , j E H 2 , t == 1,2,
where ?(X1 ,X2 ) is some graph partitioning cost function, such as minimum cuts
[6], average cuts [7], or normalized cuts [15]. The first set of constraints can be
re-written in matrix form: U T X == 0 , where, e.g. for some column k, Uik == 1,
Ujk == ~1. We search for the optimal solution only in the feasible set determined
by all the constraints.
2.2
Conditions on grouping constraints
The above formulation can be implemented by the maximum-flow algorithm for
minimum cuts criteria [6, 13, 3], where two special nodes called source and sink are
introduced,.with infinite weights set up between nodes in HI (H2 ) and source(sink).
In the context of learning from labeled and unlabeled data, the biased mincuts
are linked to minimizing leave-one-out cross validation [2]. In the normalize cuts
formulation, this leads to a constrained eigenvalue problem, as soon to be seen.
However, simply forcing a few nodes to be in the same group can produce some
undesirable graph partitioning results, illustrated in Fig. 3. Without bias, the data
points are naturally first organized into top and bottom groups, and then subdivided
into left and right halves (Fig. 3a). When we assign points from top and bottom
clusters to be together, we do not just want one of the groups to lose its labeled
point to the other group (Fig. 3b), but rather we desire the biased grouping process
to explore their neighbouring connections and change the organization to left and
right division accordingly.
Larger Cut
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
0
a
a
a
a
a
a
a
a
Desired Cut
a a a
a a a
0
a
0
a a a
a a a
a a a
a a
a a
a a a a a a
a a a a a a
0
0
0
a a
a a
a a
0
0
0
0
I
0
a a a a a a
a a a a a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
.A.
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Min
Cut
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
0
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
0
a
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a II
a a
a
0
0
a
a a
a a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
Perturbed
Min
Cut
b)With bias.
a)No bias.
Figure 3: Undesired grouping caused by simple grouping constraints. a)Data points
are distributed in four groups, with a larger spatial gap between top and bottom
groups than that between left and right groups. Defining weights based on proximity, we find the top-bottom grouping as the optimal bisection. b)Introduce two
pairs of filled nodes to be together. Each pair has one point from the top and the
other from the bottom group. The desired partitioning should now be the left-right
division. However, perturbation on the unconstrained optimal cut can lead to a
partitioning that satisfies the constraints while producing the smallest cut cost.
The desire of propagating partial grouping information on the constrained nodes
is, however, not reflected in the constrained partitioning criterion itself. Often,
a slightly perturbed version of the optimal unbiased cut becomes the legitimate
optimum. One reason for such a solution being undesirable is that some of the
"perturbed" nodes-are isolated from their close neighbours.
To fix this problem, we introduce the notion of uniformity of a graph partitioning.
Intuitively, if two labeled nodes, i and j, have similar connections to their neighbours, we desire a cut to treat them fairly so that if i gets grouped with i's friends,
j also gets grouped with j's friends (Fig. 3b). This uniformity condition is one way
to propagate prior grouping information from labeled nodes to their neighbours.
For normalized cuts criteria, we define the normalized cuts of a single node to be
?.X)-
NC u t s ( ~,-
EXt(k)=I=Xt(i),YtWik
D..
n
.
This value is high for a node isolated from its close neighbours in partitioning X.
We may not know in advance what this value is for the optimal partitioning, but
we desire this value to be the same for any pair of nodes preassigned together:
NCuts(i;X) == NCuts(j;X), \li,j E Hz, l == 1,??? ,no
While this condition does not force NCuts(i; X) to be small for each labeled node,
it is unlikely for all of them to have a large value while producing the minimum
NCuts for the global bisection. Similar measures can be defined for other criteria.
In Fig. 4, we show that the uniformity condition on the bias helps preserving
the smoothness of solutions at every labeled point. Such smoothing is necessary
especially when partially labeled data are scarce.
0.5
0.5
: -.... .... -....
~
_o5A~,S/
-0.5
-1 [l...--V_-_l...--_-----'---_----'l
o
a)Point set data.
-1[?.....
300
b)Simple bias.
0
I
100
300
c) Conditioned bias.
0.5
0.5
o
o
0
-0.2
-0.4
-0.5
-1
-1
100
d)NCuts
wlo bias.
e)NCuts
100
wi bias b).
f)NCuts
wi bias c).
Figure 4: Condition constraints with uniformity. a)Data consist of three strips, with
100 points each, numbered from left toright. Two points from the side strips are randomly chosen to be pre-assigned together. b)Simple constraint U T X == 0 forces any
feasible solution to have equal valuation on the two marked points. c)Conditioned
constraint UTpX == o. Note that now we cannot tell which points are biased. We
compute W using Gaussian function of distance with u == 3. d) Segmentation without bias gives three separate groups. e)Segmentation with simple bias not only fails
to glue the two side strips into one, but also has two marked points isolated from
their neighbours. f)Segmentaion with conditioned bias brings two side strips into
one group. See the definition of P below.
2.3
COlllpntation: subspace projection
To develop a computational solution for the c9nstrained optimization problem, we
introduce some notations. Let the degree matrix D be a diagonal matrix, D ii ==
Ek Wik, \Ii. Let P == D-IW be the normalized weight matrix. It is a transition
probability matrix for nonnegative weight matrix W [10]. Let a == xI~~l be
the degree ratio of VI, where 1 is the vector of ones. We define a new variable
x == (1 - a)XI - aX2 ? We can show that for normalized cuts, the biased grouping
with the uniformity condition is translated into:
.
mIn E(X)
==
xT(D-W)x
T
TD
' s.t. U Px ==
x
x
Note, we have dropped the constraint Xt(i)
=1=
o.
Xt(j), i E HI, j E H 2 , t == 1,2.
Using Lagrange multipliers, we find that the optimal solution x* satisfies:
QPx* == AX*,
E(X*) == 1 - A,
where Q is a projector onto the feasible solution space:
Q == I - D-1V(VTD-1V)-lVT , V
==
pTU.
Here we assume that the conditioned constraint V is of full rank, thus V T D- 1V is
invertible. Since 1 is still the trivial 'Solution corresponding to the largest eigenvalue
of 1, the second leading right eigenvector of the matrix QP is the solution we seek.
To summarize, given weight matrix W, partial grouping in matrix form UT x
we do the following to find the optimal bipartitioning:
== 0,
Compute degree matrix D, D ii == E j Wij , Vi.
Compute normalized matrix P == D-1W.
Compute conditioned constraint V == pTU.
Compute projected weight matrix W == QP==p-n-1V(VTn-1V)-lVTp.
Compute the second largest eigenvector x*: Wx* == AX*.
Threshold x* to get a discrete segmentation.
Step
Step
Step
Step
Step
Step
1:
2:
3:
4:
5:
6:
3
Results and conclusions
We apply our method to the images in Fig. 2. For all the examples, we compute
pixel affinity W as in Fig. 1. All the segmentation results are obtained by thresholding the eigenvectors using their mean values. The results without bias, with
simple bias U T x == 0 and conditioned bias U T Px == 0 are compared in Fig. 5, 6, 7.
e)Simple bias.
b) Prior.
c)NCuts' on W.
f)Seg. on e)
g)Conditioned bias.
d)Seg.
wlo bias.
h)Seg. w/ bias.
Figure 5: Segmentation with bias from unitary generative models. a)Edge map of the
100 x 150 image. N = 15000. b)We randomly sample 17 brightest pixels for HI (+),48
darkest pixels for H2 (~), producing 63 constraints in total. c) and d) show the solution
without bias. It picks up the optimal bisection based on intensity distribution. e) and
f) show the solution with simple bias. The labeled nodes have an uneven influence on
grouping. g) and h) show the solution with conditioned bias. It successfully breaks the
image into tiger and river as our general impression of this image. The computation for
the three cases takes 11, 9 and 91ms respectively.
Prior knowledge is particularly useful in supplying long-range grouping information
which often lacks in data grouping based on low level cues. With our model, the
partial grouping prior can be integrated into the bottom-up grouping framework by
seeking the optimal solution in a restricted domain. We show that the uniformity
constraint is effective in eliminating spurious solutions resulting from simple perturbation on the optimal unbiased solutions. Segmentation from the discretization
of the continuous eigenvectors also becomes trivial.
e)Simple bias.
f)Seg. on e)
g)Conditioned bias.
h)Seg. w / bias.
Figure 6: Segmentation with bias from hand-labeled partial grouping. a)Edge map of
the 80 x 82 image. N = 6560. b)Hand-labeled partial grouping includes 21 pixels for HI
(+), 31 pixels for H 2 (A), producing 50 constraints in total. c) and d) show the solution
without bias. It favors a few largest nearby pieces of similar intensity. e) and f) show the
solution with simple bias. Labeled pixels in cluttered contexts are poor at binding their
segments together. g) and h) show the solution with conditioned bias. It successfully pops
out the pumpkin made of many small intensity patches. The computation for the three
cases takes 5, 5 and 71ms respectively.
f)4th eig. b)
g)6th eig. b)
h)4th eig. d)
i)6 th eig. d)
j)8 th eig. d)
Figure 7: Segmentation with bias from spatial attention. N = 25800. a)We randomly
choose 86 pixels far away from the fovea (Fig. 2c) for H 2 (A), producing 85 constraints.
b) and c) show the solution with simple bias. It is similar to the solution without bias
(Fig. 1). d) and e) show the solution with conditioned bias. It ignores the variation
in the background scene, which includes not only large pieces of constant intensity, but
also many small segments of various intensities. The foreground .successfully clips out the
human figure. f) and g) are two subsequent eigenvectors with simple bias. h), i) and
j.) are those with conditioned bias. There is still a lot of structural organization in the
former, but almost none in the latter. This greatly simplifies the task we face when seeking
a segmentation from the continuous eigensolution. The computation for the three cases
takes 16, 25 and 220ms respectively.
All these benefits come at a computational cost that is 10 times that for the original
unbiased grouping problem. We note that we can also impose both UT x == 0 and
U T Px == 0, or even U T pBX == 0, S > 1. Little improvement is observed in our
examples.' Since projected weight matrix W becomes denser, the computation slows
down. We hope that this problem can be alleviated by using multi-scale techniques.
It remains open for future research.
Acknowledgelllents
This research is supported by (DARPA HumanID) ONR
NSF IRI-9817496.
NOOOI4-00-1-091~ and
References
[1] A. Amir? and M. Lindenbaum. Quantitative analysis of grouping process. In
European Conference on Computer Vision, pages 371-84, 1996.
[2] A. Blum and S. Chawla. Learning from labeled and unlabeled data using graph
mincuts, 2001.
[3] Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization
via graph cuts. In International Conference on Computer Vision, 1999.
[4] Y. Gdalyahu, D. Weinshall, and M. Werman. A randpmized algorithm for
pairwise clustering. ill Neural Information Processing Systems, pages 424-30,
1998.
[5] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the
Bayesian restoration of images. IEEE Transactions on Pattern Analysis and
Machine Intelligence, 6(6):721-41, 1984.
[6] H. Ishikawa and D. Geiger. Segmentation by grouping junctions. In IEEE
Conference on Computer Vision and Pattern Recognition, 1998.
[7] I. H. Jermyn and H. Ishikawa. Globally optimal regions and boundaries. In
International Conference on Computer Vision, 1999.
[8] M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active contour models.
International Journal of Computer Vision, pages 321-331, 1988.
[9] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for
image segmentation. International Journal of Computer Vision, 2001.
[10] M. Meila and J. Shi. Learning segmentation with random walk. ill Neural
Information Processing Systems, 2001.
[11] P. Perona and W. Freeman. A factorization approach to grouping. In European
Conference on Computer Vision, pages 655-70, 1998.
[12] J. Puzicha, T. Hofmann, and J. Buhmann. Unsupervised texture segmentation in a deterministic annealing framework. IEEE Transactions on Pattern
Analysis and Machine Intelligence, 20(8):803-18, 1998.
[13] S. Roy and I. J. Cox. A maximum-flow formulation of then-camera stereo
correspondence problem. In International Conference on Computer Vision,
1998.
[14] E. Sharon, A. Brandt, and R. Basri. Fast multiscale image segmentation. In
IEEE Conference on Computer Vision and Pattern Recognition, pages 70-7,
2000.
[15] J. Shi and J. Malik. Normalized cuts and image segmentation. In IEEE Conference on Computer Vision and Pattern Recognition, pages 731-7, June 1997.
| 2013 |@word cox:1 version:1 eliminating:1 glue:1 open:1 seek:2 propagate:1 pick:1 configuration:2 contains:4 pbx:1 segmentaion:1 discretization:1 ka:1 written:2 subsequent:1 partition:3 wx:1 hofmann:1 shape:1 discrimination:2 v_:1 generative:6 cue:3 selected:2 half:1 amir:1 accordingly:1 intelligence:2 supplying:1 node:17 brandt:1 along:1 constructed:1 introduce:4 pairwise:4 multi:1 freeman:1 globally:1 td:1 little:1 becomes:4 notation:1 what:1 weinshall:1 eigenvector:5 developed:2 finding:2 quantitative:1 every:2 jianbo:1 partitioning:16 producing:5 segmenting:1 dropped:1 local:3 treat:1 preassigned:1 ext:1 id:1 vtn:1 might:1 black:1 limited:1 factorization:1 range:2 gdalyahu:1 camera:1 thought:1 projection:2 alleviated:1 pre:1 acknowledgelllents:1 seeing:1 numbered:1 get:3 cannot:1 ga:1 unlabeled:2 onto:1 undesirable:2 close:2 context:2 influence:1 lindenbaum:1 map:3 projector:1 center:1 shi:4 deterministic:1 attention:6 iri:1 cluttered:1 bipartitioning:3 formulate:1 identifying:1 assigns:1 legitimate:1 handle:1 notion:1 variation:1 user:2 neighbouring:1 pa:2 element:1 crossing:1 recognition:3 particularly:1 roy:1 cut:23 geman:2 labeled:14 bottom:9 observed:1 sparing:1 seg:5 region:4 connected:2 defini:1 uniformity:7 segment:3 division:2 basis:1 sink:2 translated:1 darpa:1 various:1 derivation:1 distinct:1 fast:2 effective:1 tell:1 outside:1 encoded:1 larger:3 solve:1 denser:1 otherwise:1 favor:1 itself:2 eigenvalue:3 interaction:1 figural:1 achieve:2 intervening:1 normalize:1 cluster:3 optimum:1 produce:1 leave:1 object:6 help:1 friend:2 develop:1 propagating:1 strong:1 implemented:1 come:1 radius:1 filter:1 stochastic:1 human:2 subdivided:1 assign:1 fix:1 proximity:1 considered:2 ic:1 brightest:2 cognition:1 werman:1 smallest:1 lose:1 iw:1 grouped:3 largest:3 successfully:3 hope:1 minimization:1 sensor:1 gaussian:2 rather:2 derived:1 focus:1 ax:2 june:1 improvement:1 rank:1 likelihood:1 superimposed:1 greatly:3 ave:1 helpful:1 leung:1 unlikely:1 integrated:1 snake:1 spurious:1 perona:1 wij:2 pixel:29 classification:1 ill:2 priori:1 spatial:7 constrained:7 integration:1 special:1 fairly:1 field:1 smoothing:1 never:1 equal:1 manually:1 ishikawa:2 yu:2 unsupervised:1 foreground:7 future:1 few:4 randomly:4 neighbour:6 tightly:1 familiar:1 maintain:1 organization:2 interest:1 light:1 edge:7 partial:11 necessary:1 filled:1 loosely:1 walk:1 desired:3 re:1 isolated:3 column:1 restoration:1 etween:1 cost:3 vertex:4 veksler:1 perturbed:3 eigensolution:1 international:5 river:2 ie:1 invertible:1 together:8 connecting:1 choose:1 transcribed:1 ek:1 leading:1 li:1 includes:2 satisfy:1 explicitly:1 caused:1 vi:5 ax2:1 piece:3 break:1 lot:1 linked:1 capability:1 complicated:1 forbes:1 bright:1 correspond:1 modelled:2 bayesian:1 bisection:3 none:1 straight:1 strongest:1 strip:4 definition:1 energy:1 ptu:2 naturally:1 associated:2 wlo:2 noooi4:1 knowledge:9 ut:2 segmentation:28 organized:1 higher:1 reflected:1 specify:1 formulation:3 evaluated:1 though:1 just:1 hand:2 multiscale:1 eig:5 lack:2 defines:1 brings:1 facilitate:1 normalized:11 unbiased:3 ncuts:9 multiplier:1 former:1 assigned:1 illustrated:1 white:1 undesired:1 criterion:9 m:3 impression:1 theoretic:1 demonstrate:1 bring:1 image:21 boykov:1 jshi:1 vtd:1 qp:2 belong:3 he:1 mellon:2 imposing:1 gibbs:1 smoothness:1 unconstrained:1 meila:1 similarity:2 apart:1 forcing:1 periphery:1 binary:1 onr:1 vt:3 preserving:1 seen:1 additional:1 minimum:3 impose:1 ii:4 full:1 witkin:1 smooth:1 adapt:1 cross:1 long:2 post:1 mrf:1 vision:11 popout:1 robotics:2 ion:1 background:6 want:2 annealing:1 source:3 biased:5 ivi:1 hz:4 induced:1 flow:2 unitary:7 structural:1 ujk:1 restrict:1 simplifies:1 hile:1 stereo:1 jj:1 ignored:1 useful:1 eigenvectors:3 dark:1 clip:1 zabih:1 nsf:1 carnegie:2 discrete:2 group:16 indefinite:1 segregating:1 four:1 threshold:1 blum:1 sharon:1 graph:19 relaxation:1 place:1 almost:1 patch:1 geiger:1 coherence:1 toright:1 hi:4 correspondence:1 nonnegative:1 constraint:24 scene:2 x2:2 nearby:1 min:4 expanded:1 px:3 poor:1 belonging:1 across:2 slightly:1 wi:2 pumpkin:1 intuitively:1 restricted:1 taken:1 remains:1 know:1 junction:1 apply:2 v2:3 away:2 chawla:1 darkest:2 original:1 top:7 clustering:3 especially:1 seeking:2 move:1 malik:2 diagonal:1 devote:1 affinity:5 subspace:2 fovea:5 separate:2 distance:1 separating:1 valuation:1 trivial:2 reason:1 relationship:2 ratio:1 minimizing:1 nc:1 stated:1 slows:1 defining:1 relational:1 perturbation:2 intensity:8 introduced:1 pair:3 connection:3 distinction:1 emu:2 pop:1 below:1 pattern:6 summarize:1 force:3 indicator:1 buhmann:1 scarce:1 wik:1 improve:1 inversely:1 eye:1 stella:2 prior:14 proportional:1 validation:1 eigendecomposition:1 integrate:2 h2:3 degree:3 thresholding:3 supported:1 soon:1 bias:37 guide:1 side:3 terzopoulos:1 institute:2 characterizing:1 face:1 distributed:2 benefit:1 boundary:2 valid:1 transition:1 contour:2 ignores:1 made:1 projected:2 far:2 transaction:2 approximate:1 basri:1 global:4 active:1 pittsburgh:2 belongie:1 xi:2 search:1 continuous:2 european:2 domain:1 arise:2 quadrature:1 x1:1 fig:17 uik:1 darker:1 fails:1 xl:2 intimate:1 down:3 xt:9 grouping:38 intrinsic:1 consist:1 merging:1 texture:2 magnitude:2 conditioned:12 push:1 gap:1 simply:1 likely:1 explore:1 visual:2 lagrange:1 desire:4 partially:3 binding:2 corresponds:2 satisfies:2 extracted:1 goal:2 marked:5 feasible:4 change:2 tiger:2 determined:2 perceiving:1 infinite:1 called:1 mincuts:2 total:2 e:2 indicating:1 formally:1 uneven:1 puzicha:1 latter:1 arises:1 dissimilar:1 incorporate:1 |
1,112 | 2,014 | A New Discriminative Kernel From Probabilistic Models
K. Tsuda,*tM. Kawanabe,* G. Ratsch,?*S . Sonnenburg,* and K.-R. Muller*+
t AIST CBRC, 2-41-6, Aomi, Koto-ku , Tokyo, 135-0064, Japan
*Fraunhofer FIRST, Kekulestr. 7, 12489 Berlin , Germany
? Australian National U ni versi ty,
Research School for Information Sciences and Engineering,
Canberra, ACT 0200 , Australia
+University of Pot sdam, Am Neuen Palais 10, 14469 Pot sdam, Germany
ko ji. tsuda@aist.go.jp, nabe @first.fraunhofer.de ,
Gunnar.Raetsch@anu.edu.au , {sonne, klaus }@fir st.fraunhofer.d e
Abstract
Recently, Jaakkola and Haussler proposed a method for constructing kernel functions from probabilistic models. Their so called
"Fisher kernel" has been combined with discriminative classifiers
such as SVM and applied successfully in e.g. DNA and protein
analysis. Whereas the Fisher kernel (FK) is calculated from the
marginal log-likelihood, we propose the TOP kernel derived from
Tangent vectors Of Posterior log-odds . Furthermore we develop
a theoretical framework on feature extractors from probabilistic
models and use it for analyzing FK and TOP. In experiments our
new discriminative TOP kernel compares favorably to the Fisher
kernel.
1 Introduction
In classification tasks , learning enables us to predict the output y E {-1 , + 1} of
some unknown system given the input a! E X based on the training examples
{a!i ' y;}i=l' The purpose of a feature extractor f : X --+ ]RD is to convert the
representation of data without losing the information needed for classification [3] .
When X is a vector space like ]Rd , a variety of feature extractors have been proposed
(e.g. Chapter 10 in [3]) . However, they are typically not applicable when X is a
set of sequences of symbols and does not h ave the structure of a vector space as in
DNA or protein analysis [2].
Recently, the Fisher kernel (FK) [6] was proposed to compute features from a probabilistic model p( a!, Y 18). At first, the parameter estimate 9 is obtained from training
examples . Then , the tangent vector of the log m arginal likelihood log p( ~ 1
9) is used
as a feature vector. The Fisher kernel refers to the inner product in this feature
space, but the method is effectively a feature extractor (also since the features are
computed explicitly). The Fisher kernel was combined with a discriminative classifier such as SVM and achieved excellent classification result s in several fields, for
example in DNA and protein analysis [6 , 5]. Empirically, it is reported that the
FK-SVM system often outperforms the classification performance of the plug-in es-
timate. 1 Note that the Fisher kernel is only one possible member in the family of
feature extractors f iJ (re ) : X --+ ]RD that can be derived from probabilistic models .
We call this family "model-dependent feature extractors" . Exploring this family is
a very import ant and interesting subject.
Since model-dependent feature extractors have been newly developed, the performance measures for them are not yet established. We therefore first propose two
performance measures . Then, we define a new kernel (or equivalently a feature
extractor) derived from t he Tangent vector Of Posterior log-odds - that we denote
as TOP kernel. vVe will analyze the performance of the TOP kernel and the Fisher
kernel in terms of our performance measures. Then the TOP kernel is compared
favorably to the Fisher kernel in a protein classification experiment.
2 Performance Measures
To b egin with, let us describ e the notations. Let re E X b e the input 'point' and
y E { -1 , +1 } be the class label. X may be a finite set or an infini te set like ]Rd.
Let us assume that we know the parametric model of the joint probability p( re, y19)
where 9 E]RP is the parameter vector. Assume that the model p(re,yI9) is regular
[7] and contains t he true distribution. Then the true parameter 9 * is uniquely
determined. Let iJ be a consistent est imator [1] of 9 *, which is obtained by n training
examples drawn i.i.d. from p(re , YI9*). Let oed = of 108i , Vof = (OeJ, ... ,Oep !) T ,
and V~f denote a p X P matrix whose (i,j) element is 0 2 f 1(08i 08 j ) .
As the Fisher kernel is commonly used in combination with linear classifiers such
as SVMs, one reasonable performance measure is the classification error of a linear
classifier wTfiJ (re) + b (w E]RD and b E]R) in the feature space. Usually wand b
are learned, so the optimal feature extractor is different wit h regard to each learning
algorithm. To cancel out this ambiguity and to make a theoretical analysis possible,
we assume the optimal learning algorithm is used. When wand b are optimally
chosen, the classification error is
R(fiJ) =
min
wES ,bE~
E""y<I>[-y(w T fiJ(re ) + b)],
(2 .1 )
where S = {w l llwi l = 1,w E ]RD }, <I> [a ] is the step function which is 1 when
a > 0 and otherwise 0, and E""y denotes the expectation with respect to the true
distribution p( re, y 19*). R(f iJ) is at least as large as the Bayes error L * [3] and
R(f iJ) = L * only if the linear classifier implements the same decision rule as the
Bayes optimal rule.
As a related measure , we consider the estimation error of the posterior probability
by a logistic regressor F(w T fiJ(re ) + b), with e.g. F(t) = 1/ (1 + exp( -t)):
D(fiJ) =
min
wE~D ,bE~
E",IF(w T fo(re ) + b) - P(y = +1Ire,9*)I.
(2 .2)
The relationship between D(fiJ ) and R(fiJ) is illustrated as follows: Let L be t he
classification error rate of a posterior probability estimator P(y
+ lire). With
regard t o L, the following inequality is known[l]:
L - L* :s; 2E.,IP(y =
+l lre ) - P(y = +1 Ire , 9*)I.
(2 .3)
When P(y
+llre):= F(w T fiJ(re) + b) , this inequality leads to the following
relationship between the two measures
(2.4)
R(fiJ) - L* :s; 2D(fiJ)?
1 In classification by plug-in estimate, re is classified by t hresholding the posterior probability fj = sign(P(y = +llre, 0) - 0.5) [1].
---------------------------
Since D(fo ) is an upper bound on R(fo), it is useful to derive a new kernel to
minimize D(f 0) ' as will be done in Sec. 4.
3 The Fisher kernel
The Fisher kernel (FK) is defined 2 as K (;e , ;e' ) = s(;e ,iJ )TZ-1(iJ)s (;e' ,iJ) , where s
is the Fisher score
s(;e ,iJ ) = (otl1logp(;eliJ) , ... ,Otlp 10gp( ;eliJ ))
T =
\7 e logp(;e ,iJ ),
and Z is the Fisher information matrix: Z(9) = E", [s(;e,9)s(;e,9)TI9]. The theoretical foundation of FK is described in the following theorem [6]: "a kernel classifier
employed the Fisher kernel derived from a model that cont ains the lab el as a la tent
variable is , asymptotically, at least as good a classifier as t he MAP labeling based
on the model" . The theorem says that the Fisher kernel can perform at least as
well as the plug-in estimate, if the parameters of linear classifier are properly det ermined (cf. Appendix A of [6]). With our p erforman ce measure, this t heorem can
be represented more concisely: R(f 0) is bounded by the classificat ion error of t he
plug-in estimate
R(fo) :S; E""y<I> [- y(P(y
=
+ ll;e,iJ ) - 0. 5)] .
(3.1 )
Not e that the classification rule constructed b y the plug-in estimate P( y = + 11;e , iJ)
can also be realized by a linear classifier in feat ure space. Propert y (3.1) is important
since it gu arantees that the Fisher kernel performs better when t he optimal w and b
are available. However, the Fisher kernel is not the only one to satisfy t his inequality.
In the next section, we present a new kernel which satisfies (3.1) and has a more
app ealing theoretical property as well.
4 The TOP Kernel
Definition Now we proceed to propose a new kernel: Our aim is to obtain a feature
extractor that achieves small D(f 0). When a feature extractor!0 (;e) satisfies3
W T
!o(;e ) + b = p -1( p(y
=
+ 11;e , 9 *)) for all;e E X
(4 .1 )
with certain values of w and b, we have D(f 0) = O. However, since the true
parameter 9 * is unknown, all we can do is t o construct ! 0 which approximately
satisfies (4.1). Let us define
v( ;e,9)
=
p-1 (p (y
=
+11;e , 9 )) = 10g( P( y = +11;e,9 )) -log(P (y = -11;e,9) ),
which is called the posterior log-odds of a probabilistic model [1]. By Taylor expansion around the estimate iJ up to t he first order 4 , we can approximate v( ;e,9*)
as
l'
v( ;e,9*) ~ v( ;e,iJ) + L0tliv( ;e ,iJ)(e: -
ad.
(4.2)
i=l
2In practice, some variants of the Fisher kernel are used. For example, if the derivative
of each class distribution , not marginal , is taken, the feature vector of FK is quite simila r
to that of our kernel. However , th ese variants should b e deliberately discrimin at ed from
the Fisher kernel in theoretical disc ussions. Throughout this pap er including ex p erim ents,
we adopt the o rigi nal defi ni t ion of the F isher kern el from [6] .
3Notice t h at p- l (t) = log t - log(l - t )
40 ne can easily derive TOP kern els from higher order Taylor ex pansions . Howeve r, we
will only deal wit h t he first order expansion here, because higher order ex pansio ns would
induce extremely high dimensional feature vectors in practical cases.
Thus , by setting
( 4.3)
and
w:= w * = (1, 8; -
e
l , ???
, 8; - ep)T, b = 0,
(4.4)
equation (4.1) is approximately satisfied. Since a Tangent vector Of the Posterior
log-odds const itutes the main p art of the feature vector, we call the inner product
of t he two feature vectors "TOP kernel" :
(4.5)
It is easy to verify t hat the TOP kernel satisfies (3.1) , b ecause we can construct
the same decision rule as the plug-in estimate by using the first element only (i.e.
w = (1 , 0, . .. ,0), b = 0).
A Theoretical Analysis In this section , we compare the TOP kernel with the plugin es timate in terms of p erformance measures . Later on , we assume that 0 < P (y =
+1Ial,8 ) < 1 to prevent IV( al,8)1 from going to infinity. Also, it is assumed t hat
VeP (y = +1Ial , 8) and V~P ( y = +1Ial,8 ) are bounded. Substituting the plug-in
estimate denoted by the subscript IT into D(fo ), we have
Define 68 =
8-
8*. By Taylor expansion around 8* , we have
where 8 0 = 8* + "(68 (O :S "( :S 1). When the TOP kernel is used,
D(fo) :S E",IF((w* )T fo(al)) - P (y = +1 Ial,8*)I ,
(4.7)
where w * is defined as in (4.4). Since P is Lipschitz-continuous, there is a finit e
positive constant M such that IP(a) - P (b)1 :S Mia - bl. Thus,
D(fo) :S ME",I(w *)T fo (rn ) - P-l (P (y
=
+ 1Irn , 8* )) I?
(4.8)
Since (w* )T f 0 (al ) is the T aylor expansion of p - 1 (P(y = + 11al , 8 *)) up to the first
order (4.2) , the first order t erms of 68 are excluded from the right side of (4.8 ),
thus D(fo ) = 0 (11 68 112 ) . Since both, the plug-in and the TOP kernel, dep end on
the parameter estimate 8, the errors D,,(8) and D(fo) become smaller as 1168 11
decreases. This shows t h at if w and b are optimally chosen , t he rate of convergence
of the TOP kernel is much faster than that of the plug-in estimate.
This result is closely related to large sample p erforman ces : Assuming t hat 8 is a
n 1/ 2 consistent estimator with asymptotic normality (e.g. the maximum likelihood
estimator) , we have 11681 1 = Op(n- l / 2 )[7J, where 01' denotes stochastic order cf. [1].
So we can directly derive the convergen ce order as D,,(8) = Op (n- l / 2 ) and D(f 0) =
Op( n - l ). By using t he rela tion (2.4) , it follows that R,, (8 ) - L* = Op( n - l / 2 ) and
R(f 0) - L * = Op (n- l ).5 Therefore, t he TOP kernel h as a much b etter convergen ce
rate in R(f 0)' which is a strong motiva tion to use the TOP kernel instead of plug-in
estimate.
5Fo r detail ed disc ussion a bout t he conve rgence orders of classificatio n e rror, see C ha pte r
6 of [1]
However, we must notice that this fast rate is possible only when the optimal linear classifier is combined with the TOP kernel. Since non-optimal linear classifiers
typically have the rate Op(n- 1 / 2 ) [1 ], the overall rate is dominated by the slower
rate and turns out to be Op (n - 1 / 2 ) . But this theoretical analysis is still meaningful, because it shows the existence of a very efficient linear boundary in the TOP
feature space. This result encourages practical efforts to improve linear boundaries
by engineering loss functions and regularization terms with e.g. cross validation,
bootstrapping or other model selection criteria [1].
Exponential Family: A Special Case ?When the distribution of two classes belong
to the exponential family, the TOP kernel can achieve an even better result than
shown above . Distributions of the exponential family can be written as q( re , 11) =
exp( 11 T t (;I!) +~( 11)) , where t (;I!) is a vector-valued function called sufficient statistics
and ~ ( 11) is a normalization factor [4]. Let 0' denote the parameter for class prior
probability of the positive model P( y = +1). Then, the probabilistic model IS
described as
where 8 = {O' , 11+1 ' 11 - 1}? The posterior log-odds reads
The TOP feature vector is described as
A
A
AT
ATT
iiJ(;I! ) = (v( ;I! ,8) ,Oav(re ,8 ), V'7 +1 v(re , 8 ) , V'7 _1 v (;I!,8 ) ) .
where V'7 ,v(;I!,iJ ) = s(ts(re) + V'7,~s(f,s)) for s = {+1,-1}. So, when w =
( 1,0, 11+1 - "'+1,11"-1 - "'- 1) T and b is properly set, the true log-odds p - l (P(y =
+11;I!,8*)) can be constructed as a linear function in the feature space (4.1). Thus
DUiJ) = 0 and RUiJ) = L*. Furthermore, since each feature is represented as
a linear function of sufficient statistics t+1 (re) and t - l (re), one can construct an
equivalent feature space as (t + l (re) T, Ll (re) T) T without knowing iJ. This result is
important because all graphical models without hidden states can be represented
as members of the exponential family, for example markov models [4] .
5 Experiments on Protein Data
In order to illustrate that the TOP kernel works well for real-world problems , we
will show t he result s on protein classification. The protein sequence data is obtained
from the Superfamily websit e. 6 This site provides sequence files with different degrees of redundancy filtering ; we used the one with 10% redundancy filtering. Here,
4541 sequences are hierarchically labeled into 7 classes, 558 folds, 845 superfamilies and 1343 families according to the SCOP(1.53) scheme. In our experiment ,
we determine the top category "classes" as the learning target. The numbers of
sequences in the classes are listed as 791, 1277, 1015 , 915,84,76,383 . We only use
the first 4 classes, and 6 two-class problem s are generated from all pairs among t he
4 classes . The 5th and 6th classes are not used because t he number of examples is
too small. Also, the 7th class is not used because this class is quite different from
the others and too easy to classify. In each two-class problem , the examples are
randomly divided into 25 % training set, 25 % validation set and 50% t est set. The
validation set is used for model selection.
6http://stash.mrc-lmb.cam.ac.uk/S UPERFAMILY /
As a probabilistic model for protein sequences, we make use of hidden markov models [2] with fully connected states. 7 The Baum- Welch algorithm (e.g. [2]) is used
for maximum likelihood training. To construct FK and TOP kernels , the derivatives with respect to all paramet ers of the HMMs from both classes are included.
The derivative with respect to the class prior probability is included as well: Let
q( OIl , e) be the probability density function of a HMM. Then, the marginal di stribution is written as p(ocI8) = aq( oc, e+1 ) + (1- a)q( oc, L1) , where a is a parameter
corresponding to the class prior. The feature vector of FK consists of the following:
V'e, logp( oc I8)
00: logp(oc I8)
P (y=s loc , 8)V'e , logq(oc ,es)
1
SE {-l ,+l }
,
1
'
--;;-P (y = +1 1?c , 9) - - -, P(y = -11?c, 9) ,
a
I -a
(5.1 )
(5.2)
while the feature vector of TOP includes V'e ,v( oc ,8)
sV'e , logq( oc ,e s) s =
{+ 1, _ 1}.8 We obtained e+1 and e-1 from the training examples of respective
classes and set a = 0.5. In the definition of the TOP kernel (4.5), we did not
include any normalization of feature vectors. However, in practical situations, it is
effective to normalize feature s for improving classification performance. Here, each
feature of the TOP kernel is normalized to have mean 0 and variance 1. Also in FK,
we normalized the features in the same way instead of using the Fisher information
matrix, because it is difficult to estimate it reliably in a high dimensional parameter
space. Both, the TOP kernel and FK are combined with SVMs with bias terms.
When classifying with HMMs , one observes the difference of the log-likelihoods for
the two classes and discriminat es by thresholding at an appropriate value. Theoretically, this threshold should be determined by the (true) class prior probability.
But, this is typically not available. Furthermore the estimation of the prior probability from training data often leads to poor results [2] . To avoid this problem,
the threshold is determined such that the false positive rate and the false negative
rate are equal in the test set. This threshold is determined in the same way for
FK-SVMs and TOP-SVMs.
The hybrid HMM-TOP-SVM system has several model parameters: the number
of HMM states, the pseudo count value [2] and the regularization parameter C of
the SVM. vVe determine these parameters as follows: First, the number of states
and the pseudo count value are determined such that the error of the HMM on
the validation set (i. e. validation error) is minimized. Based on the chosen HMM
model, the paramet er C is det ermined such that the validation error of TOP-SVM is
minimized. Here, the number of states and the pseudo count value are chosen from
{3 , 5,7,10,15,20,30,40, 60} and {l0 -1 0, 10 - 7 , 10 - 5 , 10 - 4 ,10 - 3 , 1O- 2 }, respectively.
For C, 15 equally spaced points on the log scale are taken from [10- 4 ,10 1]. Note
that the model selection is performed in the same manner for the Fisher kernel as
well.
The error rates over 15 different training/validation/test divisions are shown in Figure 1 and 2. The results of stat istical tests are shown in Table 1 as well. Compared
with the plug-in est imate, the Fisher kernel performed significantly better in several sett ings (i.e. 1-3, 2-3, 3-4). This result partially agrees with observations in
[6]. However, our TOP approach significantly outperforms the Fisher kernel: According to the Wilcoxson signed ranks test, the TOP kernel was significantly better
7Several HMM models have been engineered for protein classification [2]. However, we
do not use such HMMs because the main purpose of experiment is to compare FK and
TOP.
8 0aV (OC, 0) is a constant which does not depend on OIl.
So it is not included in the
feature vector.
1-2
1-3
0.3
0.1 6
~
0.1 4
0.1 2
0.1
0.08
0.25
0.2
~
FK
TOP
2-3
0.18
0.16
0.14
~
P
""I
0. 32
0.3
0. 28 1
0.26
FK
TOP
P
2-4
0.32
FK
TOP
3-4
04
0.3
t8
~
0.12
0.1
P
~$ ~~~
1
0. 24
0.15
P
1-4
0. 36
FK
0.28
0.26
0.24
ffi
0.22
0. 2
TOP
!~ ,~ ~
1
:::1
034
032 ,
03
0 28
1
I
P
FK
TOP
P
FK
TOP
Figure 1: The error rates of SVMs with two feature extractors in t he protein classification experiments. T he labels 'P ','FK' ,'T OP' denote t he p lug-in estimate , the
F isher kernel and t he TOP kernel, respect ively. T he t itle on each subfigure shows
the t wo prot ein classes used for t he experiment.
1-2
F igure 2: Comparison of the error rates
of t he F isher kernel and t he TO P kernel in discrim ination between class 1 and
2. Every point corresponds to one of
15 differen t t raining/validation /test set
splits. Except t wo cases, t he T OP kernel
achieves smaller error rates .
0.14
0.12
0.1
0.08
0?ct~06
0.08
0.1
0.12
0.14
TOP
in all settings . Also, the t-test j udged t h at t he difference is significant except for
1-4 and 2-4. This indicates that the T OP kernel was able to capture discrim inative
information better than t he Fisher kernel.
6
Conclusion
In this study, we presented the new discrim inative TOP kernel derived from probabilistic models. Since the theoret ical framework for su ch kernels has so far not
been established, we proposed two performance measures to analyze them and gave
bounds an d rates to gain a bett er insigh t into model depen dent feat ure extractors
from probabilistic models. Exp erimentally, we showed that the T OP kernel compares favorably to F K in a realistic protein classification experim ent . Note t h at
Sm ith and Gales[8] h ave sh own t h at a similar approach works excellent ly in speech
recogni tion tasks as well. Fu t ure research will focus on constructing sm all sam ple
bounds fo r t he T OP kernel to exten d the validity of t his work. Since other nonlinear
transformat ions F are possible for obtaining different and possibly b etter features,
we will furt hermore consider to learn t he nonlinear transformat ion F from training samples . An interes ting point is that so far T OP kernels perform local linear
approximations, it would be interesting to move in the direction of local or even
Table 1: P-values of statistical test s in the protein classification experiments . Two
kinds of tests, t- test (denoted as T in the table) and Wilcoxson signed ranks test
(i.e. WX) , are used. When the difference is significant (p-value < 0.05), a single
star * is put beside the value. Double stars ** indicate that the difference is very
significant (p-value < 0.01).
I Methods I Test II
P, FK
P, TOP
FK,TOP
T
WX
T
WX
T
WX
I Methods I Test II
P, FK
P, TOP
FK,TOP
T
WX
T
WX
T
WX
1-2
0.95
0.85
0.015*
4.3 X 10- 4 **
0.0093**
8.5 X 10 - 4 **
1-3
0.14
0.041 *
1.7 X 10 - ~**
6.1 X 10- 5 **
2.2 X 10 -4**
6.1 X 10 - 5 **
1-4
0.78
0.24
0.11
0.030*
0.21
0.048*
2-3
0.0032**
0.0040**
3.0 X 10 -1~**
6.1 X 10 - 5 **
2.6 X 10 -M*
6.1 X 10 - 5 **
2-4
0.79
0.80
0.059
0.035*
0.079
0.0034**
3-4
0.12
0.026*
5.3 X 10 -0**
3.1 X 10- 4 **
0.0031 **
1.8 X 10- 4 **
global nonlinear expansions.
Acknowledgment s
vVe thank T. Tanaka, M . Sugiyama, S.-I. Amari, K. Karplus,
R. Karchin, F. Sohler and A. Zien for valuable discussions. Moreover, we gratefully
acknowledge partial support from DFG (JA 379/9-1, MU 987/1-1) and travel grants
from EU (Neurocolt II).
References
[1] L. Devroye, L. Gyorfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer , 1996.
[2] R. Durbin, S. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis:
Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press ,
1998.
[3] K. Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press,
San Diego, 2nd edition, 1990.
[4] D. Geiger and C. Meek. Graphical models and exponential famili es . Technical
Report MSR- TR-98-10, Microsoft Research, 1998.
[5] T.S. Jaakkola, M. Diekhans, and D. Haussler. A discriminative framework for
detecting remote protein homologies. J. Compo Biol. , 7:95-114, 2000.
[6] T.S. Jaakkola and D. Haussler. Exploiting generative models in discriminative
classifiers. In M.S. Kearns, S.A. SoHa, and D.A. Cohn, editors, Advances in
Neural Information Processing Systems 11 , pages 487- 493. MIT Press, 1999.
[7] N. Murata, S. Yoshizawa, and S. Amari. Network information criterion determining the number of hidden units for an artificial neural network model.
IEEE Trans. Neural Networks, 5:865- 872, 1994.
[8] N. Smith and M. Gales. Speech recognition using SVMs. In T.G. Dietterich,
S. Becker, and Z. Ghahramani , editors, Advances in Neural Information Processing Systems 14. MIT Press, 2002. to appear.
| 2014 |@word msr:1 nd:1 tr:1 contains:1 score:1 att:1 loc:1 outperforms:2 erms:1 yet:1 import:1 must:1 written:2 realistic:1 wx:7 enables:1 generative:1 ial:4 ith:1 smith:1 compo:1 ire:2 provides:1 detecting:1 istical:1 constructed:2 become:1 consists:1 manner:1 theoretically:1 hresholding:1 notation:1 bounded:2 moreover:1 kind:1 developed:1 bootstrapping:1 pseudo:3 every:1 act:1 classifier:12 prot:1 yi9:2 uk:1 unit:1 ly:1 grant:1 appear:1 positive:3 engineering:2 local:2 plugin:1 analyzing:1 ure:3 subscript:1 approximately:2 lugosi:1 signed:2 au:1 hmms:3 gyorfi:1 practical:3 acknowledgment:1 practice:1 implement:1 erformance:1 lira:1 significantly:3 induce:1 refers:1 regular:1 kern:2 protein:14 sonne:1 selection:3 put:1 equivalent:1 map:1 baum:1 go:1 welch:1 wit:2 rule:4 haussler:3 estimator:3 his:2 target:1 diego:1 losing:1 element:2 defi:1 recognition:3 labeled:1 logq:2 ep:1 capture:1 connected:1 sonnenburg:1 eu:1 decrease:1 remote:1 observes:1 valuable:1 mu:1 cam:1 oed:1 ings:1 depend:1 division:1 gu:1 finit:1 easily:1 joint:1 chapter:1 represented:3 fast:1 effective:1 artificial:1 labeling:1 klaus:1 whose:1 quite:2 valued:1 say:1 otherwise:1 amari:2 statistic:2 gp:1 ip:2 sequence:7 propose:3 product:2 achieve:1 normalize:1 ent:1 exploiting:1 convergence:1 lmb:1 double:1 derive:3 develop:1 transformat:2 stat:1 illustrate:1 ac:1 ij:16 op:13 school:1 dep:1 strong:1 krogh:1 pot:2 indicate:1 australian:1 direction:1 fij:9 closely:1 tokyo:1 koto:1 stochastic:1 australia:1 engineered:1 ja:1 discriminat:1 biological:1 paramet:2 exploring:1 dent:1 around:2 exp:3 predict:1 heorem:1 substituting:1 achieves:2 adopt:1 purpose:2 estimation:2 travel:1 applicable:1 label:2 agrees:1 successfully:1 mit:2 aim:1 avoid:1 jaakkola:3 derived:5 l0:1 focus:1 properly:2 rank:2 likelihood:5 indicates:1 ave:2 am:1 dependent:2 el:3 typically:3 hidden:3 irn:1 ical:1 going:1 germany:2 overall:1 classification:16 among:1 denoted:2 art:1 special:1 neuen:1 marginal:3 field:1 construct:4 equal:1 sohler:1 cancel:1 minimized:2 others:1 report:1 randomly:1 duij:1 national:1 dfg:1 ents:1 microsoft:1 sh:1 furt:1 ively:1 fu:1 partial:1 respective:1 iv:1 taylor:3 re:20 tsuda:2 karplus:1 theoretical:7 subfigure:1 classify:1 versi:1 logp:3 too:2 optimally:2 reported:1 sv:1 combined:4 st:1 density:1 infini:1 probabilistic:12 regressor:1 aomi:1 ambiguity:1 satisfied:1 lug:1 gale:2 exten:1 fir:1 possibly:1 tz:1 derivative:3 japan:1 de:1 scop:1 star:2 sec:1 includes:1 satisfy:1 explicitly:1 ad:1 later:1 tion:3 performed:2 lab:1 analyze:2 bayes:2 minimize:1 kekulestr:1 ni:2 variance:1 acid:1 murata:1 spaced:1 ant:1 disc:2 mrc:1 app:1 classified:1 mia:1 fo:13 ed:2 definition:2 ty:1 yoshizawa:1 di:1 timate:2 gain:1 newly:1 eddy:1 recogni:1 higher:2 done:1 furthermore:3 su:1 nonlinear:3 cohn:1 logistic:1 convergen:2 oil:2 dietterich:1 validity:1 verify:1 true:6 homology:1 deliberately:1 normalized:2 regularization:2 excluded:1 read:1 aist:2 illustrated:1 deal:1 llwi:1 ll:2 uniquely:1 encourages:1 oc:8 criterion:2 palais:1 performs:1 l1:1 fj:1 recently:2 ji:1 empirically:1 lre:1 jp:1 belong:1 he:29 raetsch:1 significant:3 cambridge:1 rd:6 fk:24 vep:1 depen:1 sugiyama:1 gratefully:1 aq:1 pte:1 posterior:8 own:1 showed:1 certain:1 inequality:3 muller:1 employed:1 determine:2 ii:3 zien:1 technical:1 faster:1 academic:1 plug:11 cross:1 divided:1 equally:1 variant:2 ko:1 expectation:1 kernel:63 normalization:2 achieved:1 ion:4 cbrc:1 whereas:1 ratsch:1 file:1 subject:1 member:2 odds:6 call:2 split:1 easy:2 variety:1 discrim:3 gave:1 nabe:1 inner:2 tm:1 knowing:1 det:2 diekhans:1 ese:1 becker:1 effort:1 wo:2 speech:2 proceed:1 useful:1 se:1 listed:1 svms:6 category:1 dna:3 wes:1 http:1 bett:1 notice:2 sign:1 gunnar:1 redundancy:2 threshold:3 drawn:1 prevent:1 ce:4 nal:1 asymptotically:1 pap:1 convert:1 wand:2 family:8 reasonable:1 throughout:1 geiger:1 decision:2 appendix:1 bound:3 ct:1 meek:1 igure:1 fold:1 ti9:1 durbin:1 infinity:1 propert:1 dominated:1 min:2 extremely:1 fukunaga:1 according:2 combination:1 poor:1 smaller:2 sam:1 tent:1 taken:2 equation:1 imate:1 turn:1 count:3 ffi:1 needed:1 know:1 end:1 available:2 etter:2 experim:1 kawanabe:1 appropriate:1 differen:1 slower:1 rp:1 hat:3 existence:1 top:46 denotes:2 cf:2 include:1 graphical:2 const:1 ting:1 ghahramani:1 bl:1 move:1 realized:1 parametric:1 thank:1 berlin:1 neurocolt:1 hmm:6 me:1 assuming:1 devroye:1 cont:1 relationship:2 equivalently:1 difficult:1 favorably:3 negative:1 reliably:1 unknown:2 perform:2 upper:1 av:1 observation:1 nucleic:1 markov:2 sm:2 finite:1 acknowledge:1 t:1 situation:1 rn:1 pair:1 concisely:1 learned:1 bout:1 established:2 tanaka:1 trans:1 able:1 usually:1 pattern:2 including:1 hybrid:1 normality:1 scheme:1 improve:1 ne:1 rela:1 fraunhofer:3 prior:5 tangent:4 determining:1 asymptotic:1 beside:1 loss:1 fully:1 interesting:2 filtering:2 conve:1 validation:8 foundation:1 degree:1 sufficient:2 consistent:2 thresholding:1 editor:2 i8:2 classifying:1 side:1 bias:1 superfamily:2 regard:2 boundary:2 calculated:1 raining:1 world:1 commonly:1 san:1 ple:1 far:2 approximate:1 feat:2 global:1 inative:2 assumed:1 discriminative:6 mitchison:1 continuous:1 table:3 ku:1 learn:1 obtaining:1 improving:1 expansion:5 excellent:2 constructing:2 t8:1 did:1 main:2 hierarchically:1 edition:1 vve:3 site:1 canberra:1 theoret:1 ein:1 n:1 iij:1 exponential:5 extractor:13 theorem:2 er:4 symbol:1 svm:6 false:2 effectively:1 te:1 anu:1 soha:1 partially:1 rror:1 springer:1 ch:1 corresponds:1 satisfies:3 classificat:1 lipschitz:1 fisher:25 included:3 determined:5 except:2 kearns:1 called:3 e:5 la:1 est:3 meaningful:1 support:1 biol:1 ex:3 |
1,113 | 2,015 | Geometrical Singularities in the
Neuromanifold of Multilayer Perceptrons
Shun-ichi Amari, Hyeyoung Park, and Tomoko Ozeki
Brain Science Institute, RIKEN
Hirosawa 2-1, Wako, Saitama, 351-0198, Japan
{amari, hypark, tomoko} @brain.riken.go.jp
Abstract
Singularities are ubiquitous in the parameter space of hierarchical
models such as multilayer perceptrons. At singularities, the Fisher
information matrix degenerates, and the Cramer-Rao paradigm
does no more hold, implying that the classical model selection theory such as AIC and MDL cannot be applied. It is important to
study the relation between the generalization error and the training
error at singularities. The present paper demonstrates a method
of analyzing these errors both for the maximum likelihood estimator and the Bayesian predictive distribution in terms of Gaussian
random fields, by using simple models.
1
Introduction
A neural network is specified by a number of parameters which are synaptic weights
and biases. Learning takes place by modifying these parameters from observed
input-output examples. Let us denote these parameters by a vector () = (0 1 , .. . , On).
Then, a network is represented by a point in the parameter space S, where () plays
the role of a coordinate system. The parameter space S is called a neuromanifold.
A learning process is represented by a trajectory in the neuromanifold. The dynamical behavior of learning is known to be very slow, because of the plateau
phenomenon. The statistical physical method [1] has made it clear that plateaus
are ubiquitous in a large-scale perceptron. In order to improve the dynamics of
learning, the natural gradient learning method has been introduced by taking the
Riemannian geometrical structure of the neuromanifold into account [2, 3]. Its
adaptive version, where the inverse of the Fisher information matrix is estimated
adaptively, is shown to have excellent behaviors by computer simulations [4, 5].
Because of the symmetry in the architecture of the multilayer perceptrons, the
parameter space of the MLP admits an equivalence relation [6, 7]. The residue class
divided by the equivalence relation gives rise to singularities in the neuromanifold,
and plateaus exist at such singularities [8]. The Fisher information matrix becomes
singular at singularities, so that the neuromanifold is strongly curved like the spacetime including black holes.
In the neighborhood of singularit ies, the Fisher-Cramer-Rao paradigm does not
hold, and the estimator is no more subject to the Gaussian distribution even asymptotically. This is essential in neural learning and model selection. The AlC and MDL
criteria of model selection use the Gaussian paradigm, so that it is not appropriate.
The problem was first pointed out by Hagiwara et al. [9]. Watanabe [10] applied
algebraic geometry to elucidate the behavior of the Bayesian predictive estimator in
MLP, showing sharp difference in regular cases and singular cases. Fukumizu [11]
gives a general analysis of the maximum likelihood estimators in singular statistical
models including the multilayer perceptrons.
The present paper is a first step to elucidate effects of singularities in the neuromanifold of multilayer perceptrons. We use a simple cone model to elucidate how
different the behaviors of the maximum likelihood estimator and the Bayes predictive distribution are from the regular case. To this end, we introduce the Gaussian
random field [11, 12, 13], and analyze the generalization error and training error for
both the mle (maximum likelihood estimator) and the Bayes estimator.
2
Topology of neuromanifold
Let us consider MLP with h hidden units and one output unit,
h
Y=
L
Vi<{J (Wi? x)
+ n.
(1)
i= l
where y is output, x is input and n is Gaussian noise. Let us summarize all the
parameters in a single parameter vector () = (Wl , ???, Wh; Vl , ???, Vh) and write
h
f(x; ()) =
L
Vi<{J (Wi?
x).
(2)
i=l
Then, () is a coordinate system of the neuromanifold. Because of the noise, the
input-output relation is stochastic, given by the conditional probability distribution
p(ylx,()) =
1 {I
-2(y-f(x;())) 2} ,
J2 exp
(3)
where we normalized the scale of noise equal to 1. Each point in the neuromanifold
represents a neural network or its probability distribution.
It is known that the behavior of MLP is invariant under 1) permutations of hidden
units , and 2) sign change of both Wi and Vi at the same time. Two networks
are equivalent when they are mapped by any of the above operations which form
a group. Hence, it is natural to treat the residual space SI ::::J, where ::::J is the
equivalence relation. There are some points which are invariant under a some nontrivial isotropy subgroup, on which singularities occurs.
When Vi = 0, vi<{J (Wi? x) = 0 so that all the points on the sub manifold Vi = 0 are
equivalent whatever Wi is. We do not need this hidden unit. Hence, in M = SI ::::J,
all of these points are reduced to one and the same point. When Wi = Wj hold,
these two units may be merged into one, and when Vi +Vj is the same, the two points
are equivalent even when they differ in Vi - Vj. Hence, the dimension reduction takes
place in the subspace satisfying Wi = Wj. Such singularities occur on the critical
submanifolds of the two types
(4)
3
Simple toy models
Given training data, the parameters of the neural network are estimated or trained
by learning. It is important to elucidate the effects of singularities on learning or
estimation. We use simple toy models to attack this problem. One is a very simple
multilayer percept ron having only one hidden unit. The other is a simple cone
model: Let x be Gaussian random variable x E R d +2 , with mean p, and identity
covariance matrix I ,
(5)
and let 5 = {p,Ip, E R d +2 } be the parameter space. The cone model M is a subset
of 5, embedded as
M : p,
(6)
where c is a constant, IIa 2 11 = 1,
When d = 1, 51 is a circle so that
p,
=
W
W
E 5 d and 5 d is a d-dimensional unit sphere.
is replaced by angle B, and we have
~
VI + c2
See Figure 1. The M is a cone, having
is the singular point.
(~,
1 B) .
( ccos
(7)
csinB
w) as coordinates, where the apex
~
= 0
,,
Figure 1: One-dimensional cone model
The input-output relation of a simple multilayer perceptron is given by
y = v<p(w . x)
+n
(8)
When v = 0, the behavior is the same whatever w is. Let us put w = (3w , where
(3 = Iwl and W E 5 d , and ~ = vlwl, 'l/J( x;(3 , w) = <p{(3(w? x)} /(3. We then have
y = ~'l/J(x;(3,w)
+n
(9)
which shows the cone structure with apex at ~ = O. In this paper, we assume that
(3 is knwon and does not need to be estimateed.
4
Asymptotic statistical inference: generalization error and
training error
Let D = {Xl,???, XT} be T independent observations from the true distribution
Po(x) which is specified by ~ = 0, that is, at the singular point. In the case
of neural networks , the training set D is T input-output pairs (Xt, Yt), from the
conditional probability distributions p(Ylx;~, w) and the true one is at ~ = O. The
discussions go in parallel, so that we show here only the cone model. We study the
characteristics of both the mle and the Bayesian predictive estimator.
Let p(x) be the estimated distribution from data D . In the case of mle, it is given
by p(x; 0) where 0 is the mle given by the maximizer of the log likelihood. For the
Bayes estimator, it is given by the Bayes predictive distribution p(xID).
We evaluate the estimator by the generalization error defined by the KL-divergence
from Po(x) to p(x),
Eg en
= ED [K[po : pll,
K[Po: p] =
Epo
[log
~(~i] .
(10)
Similarly, the training error is defined by using the empirical expectation,
(11)
In order to evaluate the estimator p, one uses E gen , but it is not computable.
Instead, one uses the Etrain which is computable. Hence, it is important to see the
difference between Egen and Etrain- This is used as a principle of model selection.
When the statistical model M is regular, or the true distribution Po (x) is at a regular
point , the mle-based p(x, 0) and the Bayes predictive distribution are asymptotically equivalent, and are Fisher efficient under reasonable regularity conditions,
Eg en
~
d
2T '
Eg en ~ Etrain
d
+ T'
where d is the dimension number of parameter vector
(12)
(j.
All of these good relations do not hold in the singular case. The mle is no more
asymptotically Gaussian, the mle and the Bayes estimators have different asymptotic characteristics, although liT consistency is guaranteed. The relation between
the generalization and training error is different, so that we need a different model
selection criterion to determine the number of hidden units.
5
Gaussian random fields and mle
Here, we introduce the Gaussian random field [11, 12, 13] in the case of the cone
model. The log likelihood of data D is written as
L(D,~,w) =
1 T
-"2l: Ilxt -
~a(w)112.
(13)
t=l
Following Hartigan [13] (see also [11] for details), we first fix wand search for the ~
that maximizes L. This is easy since L is a quadratic function of ( The maximum
t is given by
(14)
=
Y(w)
(15)
a(w) ? X,
By the central limit theorem, Y (w) = a( w) . x is a Gaussian random field defined
on Sd = {w}. By substituting t(w) in (14) the log likelihood function becomes
T
,
I",
1
2
-2 ~ IIXtl1 + 2Y
L(w) =
2
(w).
(16)
t=l
Therefore, the mle
Theorem 1.
w is given by the maximizer of L(w), w= argmaxwy2(w).
In the case of the cone model, the mle satisfies
2~ED h~p y
Egen
Etrain
Corollary 1.
=
-
2~ED
2
h:p y
(w)] ,
2
(w)] .
(17)
(18)
When d is large, the mle satisfies
Egen
::::::
Etrain
::::::
c2 d
2T(1 + c2 ) '
c2 d
2T(1 + c2 )?
(19)
(20)
It should be remarked that the generalization and training errors depend on the
shape parameter c as well as the dimension number d.
6
Bayesian predictive distribution
The Bayes paradigm uses the posterior probability of the parameters based on the
set of observations D. The posterior probability density is written as,
T
p(~,wID)
=
c(D)1f(~,w) rrp(xtl~, w) ,
(21)
t= l
where c(D) is the normalization factor depending only on data D , 1f(~ , w) is a prior
distribution on the parameter space. The Bayesian predictive distribution p(xID) is
obtained by averaging p(xl~, w) with respect to the posterior distribution p(~, wiD),
and can be written as
p(xID) =
Jp(xl~, w)p(~, wID)d~dw.
(22)
The Bayes predictive distribution depends on the prior distribution 1f( ~, w) . As long
as the prior is a smooth function, the first order asymptotic properties are the same
for the mle and Bayes estimators in the regular case. However, at singularities, the
situation is different. Here, we assume a uniform prior for w. For C we assume two
different priors, the uniform prior and the Jeffreys prior.
We show here a sketch of calculations in the case of Jeffreys prior, 7f(~,w) ex 1~ld .
By introducing
Id(u) =
J
~ Iz + uldexp {_~Z2} dz,
(23)
after lengthy calculations, we obtain
(24)
where
XT+! =
~(x + VTx) ,
Pd(x) =
J
Id(Y(w)) exp
{~Y2(W)} dw.
(25)
Here Y(w) has the same form defined in (15), and Pd(x) is the function of the
sufficient statistics x. By using the Edgeworth expansion, we have
p(xID)
(26)
where \7 is the gradient and H2 (x) is the Hermite polynomial. We thus have the
following theorem.
Theorem 2.
Under the Jeffreys prior for ~, the generalization error and the
training error of the predictive distribution are given by
Egen
(27)
Etr ain
(28)
Under the uniform prior, the above results hold by replacing Id(Y) in the definition
of Pd(X) by 1. In addition, From (24), we can easily obtain Egen = (d + 1)/2T for
the Jeffreys prior, and Egen = 1/2T for the uniform prior.
The theorem shows rather surprising results: Under the uniform prior, the generalization error is constant and does not depend on d. This is completely different from
the regular case. However, this striking result is given rise to by the uniform prior
on f The uniform prior puts strong emphasis on the singularity, showing that one
should be very careful for choosing a prior when the model includes singularities. In
the case of J effreys prior, the generalization error increases in proportion to d, which
is the same result as the regular case. In addition, the symmetric duality between
Egen and E train does not hold for both of the uniform prior and the Jeffreys prior.
7
Gaussian random field of MLP
In the case of MLP with one hidden unit , the log likelihood is written as
1 T
2
L(D;~, w)=-22:{Yt-~CPi9(w.Xt)} .
t=l
(29)
Let us define a Gaussian random field depending on D and w,
1
Y(w) =
1m
yT
T
LYt<P,6 (w? Xt) '"" N(O,A(w,w'))
(30)
t= l
where A(w, w') = Ex [<p,6(w . x)<p,6 (w' . x)].
Theorem 3.
For the mle, we have
(31)
Egen
(32)
Etrain
(33)
where A(w) = A(w, w).
In order to analyze the Bayes predictive distribution , we define
Sd(D,w) =
1
J A(w) d+1
Y (W))
{ 1 y2 (w) }
Id ( JA(W)
exp --A() .
A(w)
2
w
(34)
We then have the Edgeworth expansion of the predictive distribution of the form,
p(Ylx, D)
_1_ exp {_ y2 } {I
~
f(L
y27f
2
+ -'!L EW[V'Sd(D, w)<p,6 (w
. x)]
(35)
Ew [Sd( D , w )]
~ EW[V'V'Sd(D,w)A(w)] H ( )}
+ 2T
EW[Sd(D,w)]
2 Y
,
1m
yT
where V' is the gradient with respect to Y(w). We thus have the following theorem.
Theorem 4.
Under the Jeffreys prior for ~ , the generalization error and the
training error of the predictive distribution are given by
Egen
Etrain
(36)
=
Under the uniform prior, the above results hold by redefining
(37)
We can also obtain Egen
the uniform prior.
= (d + 1)/2T for
the Jeffreys prior, and Egen
= 1/2T for
There is a nice correspondence between the cone model and MLP. However, there
is no sufficient statistics in the MLP case, while all the data are summarized in the
sufficient statistics x in the cone model.
8
Conclusions and discussions
We have analyzed the asymptotic behaviors of the MLE and Bayes estimators in
terms of the generalization error and the training error by using simple statistical
models (cone model and simple MLP), when the true parameter is at singularity.
Since the classic paradigm of statistical inference based on the Cramer-Rao theorem
does not hold in such a singular case, we need a new theory. The Gaussian random
field has played a fundamental role. We can compare the estimation accuracy of
the maximum likelihood estimator and the Bayesian predictive distribution from the
results of analysis. Under the proposed framework , the various estimation methods
can be studied and compared to each other.
References
[1] Saad, D. and Solla, S. A. (1995). Physical Review E, 52,4225-4243.
[2] Amari, S. (1998). N eural Computation, 10,251-276.
[3] Amari S. and Nagaoka, H. (2000). Methods of Information Geometry, AMS.
[4] Amari, S., Park, H. , and Fukumizu, F. (2000). Neural Computation, 12, 13991409.
[5] Park, H., Amari, S. and Fukumizu, F. (2000). Neural Networks, 13, 755-764.
[6] Chen, A. M., Lu, H. , and Hecht-Nielsen, R. (1993). Neural Computations, 5,
910-927.
[7] Riigger, S. M. and Ossen, A. (1997). Neural Processing Letters, 5, 63-72.
[8] Fukumizu, K. and Amari, S. (2000) Neural Networks, 13 317-327.
[9] Hagiwara, K. , Hayasaka, K. , Toda, N., Usui, S., and Kuno, K . (2001). Neural
Networks, 14 1419-1430.
[10] Watanabe, S. (2001). Neural Computation, 13, 899-933.
[11] Fukumizu, K. (2001). Research Memorandum, 780, lnst. of Statistical Mathematics.
[12] Dacunha-Castelle, D. and Gassiat, E. (1997). Probability and Statistics, 1,285317.
[13] Hartigan, J. A. (1985). Proceedings of Berkeley Conference in Honor of J.
Neyman and J. Kiefer, 2, 807-810.
| 2015 |@word effect:2 toda:1 normalized:1 version:1 true:4 polynomial:1 proportion:1 differ:1 y2:3 hence:4 classical:1 merged:1 symmetric:1 occurs:1 modifying:1 simulation:1 stochastic:1 eg:3 covariance:1 wid:3 gradient:3 subspace:1 shun:1 xid:4 ja:1 mapped:1 ld:1 reduction:1 criterion:2 fix:1 generalization:11 manifold:1 etr:1 singularity:15 wako:1 hold:8 z2:1 geometrical:2 surprising:1 cramer:3 si:2 exp:4 written:4 substituting:1 physical:2 shape:1 rise:2 jp:2 estimation:3 implying:1 observation:2 ain:1 wl:1 ozeki:1 neuromanifold:10 consistency:1 mathematics:1 iia:1 fukumizu:5 pointed:1 similarly:1 curved:1 situation:1 ron:1 gaussian:13 attack:1 apex:2 rather:1 hermite:1 sharp:1 c2:5 corollary:1 posterior:3 xtl:1 introduced:1 pair:1 specified:2 kl:1 redefining:1 likelihood:9 introduce:2 iwl:1 honor:1 behavior:7 am:1 subgroup:1 inference:2 brain:2 dynamical:1 vl:1 hidden:6 relation:8 determine:1 paradigm:5 summarize:1 becomes:2 lnst:1 including:2 maximizes:1 isotropy:1 critical:1 smooth:1 submanifolds:1 natural:2 etrain:7 calculation:2 usui:1 sphere:1 long:1 residual:1 field:8 equal:1 divided:1 having:2 mle:14 y:1 hecht:1 improve:1 berkeley:1 represents:1 park:3 lit:1 multilayer:7 expectation:1 demonstrates:1 normalization:1 whatever:2 unit:9 vh:1 prior:23 nice:1 review:1 addition:2 divergence:1 residue:1 asymptotic:4 embedded:1 treat:1 sd:6 limit:1 replaced:1 geometry:2 singular:7 permutation:1 saad:1 analyzing:1 id:4 mlp:9 subject:1 black:1 h2:1 emphasis:1 studied:1 sufficient:3 equivalence:3 mdl:2 principle:1 analyzed:1 easy:1 architecture:1 topology:1 edgeworth:2 bias:1 perceptron:2 computable:2 institute:1 taking:1 circle:1 empirical:1 dimension:3 regular:7 rao:3 algebraic:1 made:1 cannot:1 adaptive:1 selection:5 put:2 introducing:1 subset:1 clear:1 equivalent:4 saitama:1 uniform:10 yt:4 dz:1 ylx:3 go:2 reduced:1 exist:1 estimator:15 adaptively:1 density:1 fundamental:1 sign:1 estimated:3 search:1 dw:2 classic:1 write:1 coordinate:3 memorandum:1 iz:1 symmetry:1 elucidate:4 play:1 hirosawa:1 ichi:1 central:1 group:1 expansion:2 us:3 excellent:1 hartigan:2 vj:2 satisfying:1 noise:3 gassiat:1 asymptotically:3 toy:2 japan:1 account:1 cone:12 observed:1 role:2 wand:1 inverse:1 angle:1 summarized:1 letter:1 includes:1 striking:1 place:2 wj:2 reasonable:1 en:3 slow:1 vi:9 solla:1 depends:1 sub:1 watanabe:2 xl:3 analyze:2 pd:3 bayes:11 parallel:1 guaranteed:1 aic:1 spacetime:1 dynamic:1 quadratic:1 correspondence:1 played:1 trained:1 depend:2 nontrivial:1 theorem:9 occur:1 accuracy:1 predictive:14 kiefer:1 characteristic:2 percept:1 xt:5 showing:2 completely:1 hagiwara:2 eural:1 admits:1 po:5 easily:1 essential:1 bayesian:6 represented:2 various:1 alc:1 lu:1 riken:2 train:1 trajectory:1 hole:1 chen:1 plateau:3 neighborhood:1 choosing:1 synaptic:1 ed:3 lengthy:1 definition:1 wi:7 egen:11 amari:7 remarked:1 statistic:4 jeffreys:7 riemannian:1 nagaoka:1 invariant:2 ip:1 satisfies:2 wh:1 neyman:1 conditional:2 ubiquitous:2 identity:1 nielsen:1 careful:1 lyt:1 j2:1 end:1 fisher:5 change:1 gen:1 operation:1 degenerate:1 averaging:1 hierarchical:1 appropriate:1 called:1 strongly:1 pll:1 duality:1 kuno:1 regularity:1 sketch:1 vtx:1 perceptrons:5 replacing:1 ew:4 maximizer:2 depending:2 evaluate:2 ex:2 phenomenon:1 strong:1 |
1,114 | 2,016 | ALGONQUIN - Learning dynamic noise
models from noisy speech for robust
speech recognition
Brendan J. Freyl, Trausti T. Kristjansson l , Li Deng2 , Alex Acero 2
1
Probabilistic and Statistical Inference Group, University of Toronto
http://www.psi.toronto.edu
2 Speech Technology Group , Microsoft Research
Abstract
A challenging, unsolved problem in the speech recognition community is recognizing speech signals that are corrupted by loud,
highly nonstationary noise. One approach to noisy speech recognition is to automatically remove the noise from the cepstrum sequence before feeding it in to a clean speech recognizer. In previous
work published in Eurospeech, we showed how a probability model
trained on clean speech and a separate probability model trained
on noise could be combined for the purpose of estimating the noisefree speech from the noisy speech. We showed how an iterative 2nd
order vector Taylor series approximation could be used for probabilistic inference in this model. In many circumstances, it is not
possible to obtain examples of noise without speech. Noise statistics may change significantly during an utterance, so that speechfree frames are not sufficient for estimating the noise model. In this
paper, we show how the noise model can be learned even when the
data contains speech. In particular, the noise model can be learned
from the test utterance and then used to de noise the test utterance.
The approximate inference technique is used as an approximate E
step in a generalized EM algorithm that learns the parameters of
the noise model from a test utterance. For both Wall Street J ournal data with added noise samples and the Aurora benchmark, we
show that the new noise adaptive technique performs as well as or
significantly better than the non-adaptive algorithm, without the
need for a separate training set of noise examples.
1
Introduction
Two main approaches to robust speech recognition include "recognizer domain approaches" (Varga and Moore 1990; Gales and Young 1996), where the acoustic
recognition model is modified or retrained to recognize noisy, distorted speech, and
"feature domain approaches" (Boll 1979; Deng et al. 2000; Attias et al. 2001; Frey
et al. 2001), where the features of noisy, distorted speech are first denoised and then
fed into a speech recognition system whose acoustic recognition model is trained on
clean speech.
One advantage of the feature domain approach over the recognizer domain approach
is that the speech modeling part of the denoising model can have much lower com-
plexity than the full acoustic recognition model. This can lead to a much faster
overall system, since the denoising process uses probabilistic inference in a much
smaller model. Also, since the complexity of the denoising model is much lower
than the complexity of the recognizer, the denoising model can be adapted to new
environments more easily, or a variety of denoising models can be stored and applied
as needed.
We model the log-spectra of clean speech, noise, and channel impulse response
function using mixtures of Gaussians. (In contrast, Attias et al. (2001) model
autoregressive coefficients.) The relationship between these log-spectra and the
log-spectrum of the noisy speech is nonlinear, leading to a posterior distribution
over the clean speech that is a mixture of non-Gaussian distributions. We show
how a variational technique that makes use of an iterative 2nd order vector Taylor
series approximation can be used to infer the clean speech and compute sufficient
statistics for a generalized EM algorithm that can learn the noise model from noisy
speech.
Our method, called ALGONQUIN, improves on previous work using the vector
Taylor series approximation (Moreno 1996) by modeling the variance of the noise
and channel instead of using point estimates, by modeling the noise and channel as a
mixture mixture model instead of a single component model, by iterating Laplace's
method to track the clean speech instead of applying it once at the model centers,
by accounting for the error in the nonlinear relationship between the log-spectra,
and by learning the noise model from noisy speech.
2
ALGONQUIN's Probability Model
For clarity, we present a version of ALGONQUIN that treats frames of log-spectra
independently. The extension of the version presented here to HMM models of
speech, noise and channel distortion is analogous to the extension of a mixture of
Gaussians to an HMM with Gaussian outputs.
Following (Moreno 1996), we derive an approximate relationship between the log
spectra of the clean speech, noise, channel and noisy speech. Assuming additive
noise and linear channel distortion, the windowed FFT Y(j) for a particular frame
(25 ms duration, spaced at 10 ms intervals) of noisy speech is related to the FFTs
of the channel H(j), clean speech 5(j) and additive noise N(j) by
Y(j) = H(j)5(j)
+ N(j).
(1)
We use a mel-frequency scale, in which case this relationship is only approximate.
However, it is quite accurate if the channel frequency response is roughly constant
across each mel-frequency filter band.
For brevity, we will assume H(j) = 1 in the remainder of this paper. Assuming
there is no channel distortion simplifies the description of the algorithm. To see
how channel distortion can be accounted for in a nonadaptive way, see (Frey et al.
2001). The technique described in this paper for adapting the noise model can be
extended to adapting the channel model.
Assuming H(j) = 1, the energy spectrum is obtained as follows:
IY(j)1 2 = Y(j)*Y(j) = 5(j)* 5(j) + N(j)* N(j) + 2Re(N(j)* 5(j))
= 15(j)1 2 + IN(j)12 + 2Re(N(j)* 5(j)) ,
where "*,, denotes complex conjugate. If the phase of the noise and the speech are
uncorrelated, the last term in the above expression is small and we can approximate
the energy spectrum as follows:
IYUW
~
ISUW + INUW?
(2)
Although we could model these spectra directly, they are constrained to be nonnegative. To make density modeling easier, we model the log-spectrum instead. An
additional benefit to this approach is that channel distortion is an additive effect in
the log-spectrum domain.
Letting y be the vector containing the log-spectrum log IY(:W, and similarly for s
and n , we can rewrite (2) as
exp(y) ~ exp(s) + exp(n) = exp(s) 0 (1 + exp(n - s)) ,
where the expO function operates in an element-wise fashion on its vector argument
and the "0" symbol indicates element-wise product.
Taking the logarithm, we obtain a function gO that is an approximate mapping of
sand n to y (see (Moreno 1996) for more details):
y
~ g([~]) = s + In(l + exp(n - s)).
(4)
"T" indicates matrix transpose and InO and expO operate on the individual elements
of their vector arguments.
Assuming the errors in the above approximation are Gaussian, the observation
likelihood is
(5)
p(yls,n) =N(y;g([~]),W),
where W is the diagonal covariance matrix of the errors. A more precise approximation to the observation likelihood can be obtained by writing W as a function of
s and n , but we assume W is constant for clarity.
Using a prior p(s, n), the goal of de noising is to infer the log-spectrum of the clean
speech s , given the log-spectrum ofthe noisy speech y. The minimum squared error
estimate of sis s = Is sp(sly) , where p(sly) ex InP(yls, n)p(s, n). This inference is
made difficult by the fact that the nonlinearity g([s n]T) in (5) makes the posterior
non-Gaussian even if the prior is Gaussian. In the next section, we show how an iterative variational method that uses a 2nd order vector Taylor series approximation
can be used for approximate inference and learning.
We assume that a priori the speech and noise are independent - p(s , n) = p(s)p(n)
- and we model each using a separate mixture of Gaussians. cS = 1, ... , NS is the
class index for the clean speech and en = 1, ... ,Nn is the class index for the noise.
The mixing proportions and Gaussian components are parameterized as follows:
p(s) = LP(cS)p(slcS), p(C S) =7r~s , p(slc S) =N(s;JL~s ,~~s ),
CS
We assume the covariance matrices
~~s
and
~~n
are diagonal.
Combining (5) and (6), the joint distribution over the noisy speech, clean speech
class, clean speech vector, noise class and noise vector is
p(y , s , cs, n , en) = N(y; g([~]), w)7r~sN(s; JL~s , ~~s )7r~N(n; JL~n , ~~n).
(7)
Under this joint distribution, the posterior p(s, nly) is a mixture of non-Gaussian
distributions. In fact, for a given speech class and noise class, the posterior
p(s, nics, en , y) may have multiple modes. So, exact computation of s is intractable
and we use an approximation.
3
Approximating the Posterior
For the current frame of noisy speech y, ALGONQUIN approximates the posterior
using a simpler, parameterized distribution, q:
p(s ,cS, n,cnly)
~
q(s,cS,n,c n ).
The "variational parameters" of q are adjusted to make this approximation accurate,
and then q is used as a surrogate for the true posterior when computing ? and
learning the noise model (c.f. (Jordan et al. 1998)).
For each cS and en, we approximate p(s, nics, en, y) by a Gaussian,
(9)
where 1J~'en and 1J~'en are the approximate posterior means of the speech and noise
for classes cS and en, and <P ~~en, <P~.r;,n and <P~::'en specify the covariance matrix for
the speech and noise for classes cS and en. Since rows of vectors in (4) do not interact
and since the likelihood covariance matrix q, and the prior covariance matrices ~ ~.
and ~~n are diagonal, the matrices <P~~ en, <P~.r;,n and <P~::'en are diagonal.
The posterior mixing proportions for classes cS and en are q( cS , en) = Pc' en. The
approximate posterior is given by q(s,n,cs,cn ) = q(s , nlcs ,cn)q(c S, en).
The goal of variational inference is to minimize the relative entropy (KullbackLeibler divergence) between q and p:
"''''11 (
K=~~
c'
en
s
n
,cS,cn nI ) ).
q s , n , cS ,cn) In q(s
( ,n
S
P s, c , n , c y
This is a particularly good choice for a cost function, because, since lnp(y) doesn't
depend on the variational parameters, minimizing K is equivalent to maximizing
F
= lnp ()
y -
K
"''''11 (
= ~~
e'
en
s
n
q s , n , cS ,cn) In p(s,cS,n,cn,y)
(
S
n) ,
q s, n, c ,c
which is a lower bound on the log-probability of the data. So, variational inference
can be used as a generalized E step (Neal and Hinton 1998) in an algorithm that
alternatively maximizes a lower bound on lnp(y) with respect to the variational
parameters and the noise model parameters, as described in the next section.
Variational inference begins by optimizing the means and variances in (9) for each
and en. Initially, we set the posterior means and variances to the prior means
and variances. F does not have a simple form in these variational parameters.
So, at each iteration, we make a 2nd order vector Taylor series approximation of
the likelihood, centered at the current variational parameters, and maximize the
resulting approximation to F. The updates are
CS
where g' 0 is a matrix of derivatives whose rows correspond to the noisy speech y
and whose columns correspond to the clean speech and noise [s n].
The inverse posterior covariance matrix is the sum of the inverse prior covariance
matrix and the inverse likelihood covariance matrix, modified by the Jacobian g' 0
for the mapping from s and n to y
The posterior means are moved towards the prior means and toward values that
match the observation y. These two effects are weighted by the inverse prior covariance matrix and the inverse likelihood covariance matrix.
After iterating the above updates (in our experiments, 3 to 5 times) for each eS and
en, the posterior mixing proportions that maximize :F are computed:
where A is a normalizing constant that is computed so that L e.en Pe'en = 1. The
minimum squared error estimate of the clean speech, s, is
We apply this algorithm on a frame-by-frame basis, until all frames in the test
utterance have been denoised.
4
Speed
Since elements of s, nand y that are in different rows do not interact in (4), the
above matrix algebra reduces to efficient scalar algebra. For 256 speech components,
4 noise components and 3 iterations of inference, our unoptimized C code takes 60
ms to denoise each frame. We are confident that this time can be reduced by an
order of magnitude using standard implementation tricks.
5
Adapting the Noise Model Using Noisy Speech
The version of ALGONQUIN described above requires that a mixture model of the
noise be trained on noise samples, before the log-spectrum of the noisy speech can
be denoised. Here, we describe how the iterative inference technique can be used as
the E step in a generalized EM algorithm for learning the noise model from noisy
speech.
For a set of frames
bound
y(1), . .. , yeT)
in a noisy test utterance, we construct a total
:F = L:F(t) :::; Llnp(y(t)).
t
t
The generalized EM algorithm alternates between updating one set of variational
(t)
...
T? h
parameters Pe.en,
11 n(t)
e'en, et c. ?or each f rame t=I, T
... , ,and maximizIng.r
WIt
respect to the noise model parameters 7r~n, J.t~n and ~~n. Since:F:::; Ltlnp(y(t)),
this procedure maximizes a lower bound on the log-probability of the data. The use
of the vector Taylor series approximations leads to an algorithm that maximizes an
approximation to a lower bound on the log-probability of the data.
Restaurant Street Airport Station Average
dB
2.12
2.96
1.82
1.73
2.16
dB
3.87
4.78
2.27
3.24
3.54
dB
9.18
10.73
5.49
6.48
7.97
dB
20.51
13.52
14.97
15.18
18.54
o dB
47.04
45.68
36.00
37.24
41.49
-5dB
78.69
72.34
69.04
67.26
71.83
Average
16.54
17.53
12.11
12.77
14.74
Table 1: Word error rates (in percent) on set B of the Aurora test set, for the
adaptive version of ALGONQUIN with 4 noise componentsset.
20
15
10
5
Setting the derivatives of :F with respect to the noise model parameters to zero, we
obtain the following M step updates:
~nen +---
(~
' " ' '~
" ' Pe.
(t)en (opnn(t)
e' e n
t
+d?lag (( 11e'n en
(t) -#-t n)(
(t) -#-ten
n )T))) / ('"'
en 11 n
e' en
~ '"'
~ P (t)
e. en )
cB
t
cB
The variational parameters can be updated multiple times before updating the
model parameters, or the variational parameters can updated only once before updating the model parameters. The latter approach may converge more quickly in
some situations.
6
Experimental Results
After training a 256-component speech model on clean speech, we used the
adaptive version of ALGONQUIN to denoise noisy test utterances on two
tasks: the publically available Aurora limited vocabulary speech recognition task
(http://www.etsi.org/technicalactiv/dsr.htm); the Wall Street J ournal (WSJ) large
vocabulary speech recognition task, with Microsoft's Whisper speech recognition
system.
We obtained results on all 48 test sets from partitions A and B of the Aurora
database. Each set contains 24,000 sentences that have been corrupted from one of
4 different noise types and one of 6 different signal to noise ratios. Table 1 gives
t he error rates for the adaptive version of ALGONQUIN, with 4 noise components.
These error rates are superior to error rates obtained by our spectral subtraction
technique for (Deng et al. 2000) , and highly competitive with other results on the
Aurora task.
Table 2 compares the performances of the adaptive version of ALGONQUIN and
t he non-adaptive version. For the non-adaptive version, 20 non-speech frames are
used to estimate the noise model. For the adaptive version, the parameters are
init ialized using 20 non-speech frames and then 3 iterations of generalized EM are
used to learn the noise model. The average error rate over all noise types and
SNRs for set B of Aurora drops from 17.65% to 15.19% when t he noise adaptive
algorithm is used to update the noise model. This is a relative gain of 13.94%.
When 4 components are used there is a further gain of 2.5%.
The Wall Street J ournal test set consists of 167 sentences spoken by female speakers. The Microsoft Whisper recognizer with a 5,000 word vocabulary was used to
recognize these sentences. Table 2 shows that the adaptive version of algonquin
.
WER
WER
Reduction
WER
Reduction
20 frames 1 comp
in WER
4 comps
in WER
Aurora, Set A
18.10%
15.91%
12.10%
15.62%
13.70%
Aurora, Set B
14.74%
17.65%
15.19%
13.94%
16.49%
WSJ, XD14, 10dB
30.00%
21.8%
27.33%
21.50%
28.33%
WSJ, XD10, 10dB
21.80%
20.6%
20.6%
5.50'70
5.50 '70
Table 2: Word error rates (WER) and percentage reduction in WER for the Aurora
test data and the Wall Street J ournal test data, without scaling.
performs better than the non-adaptive version, especially on noise type "XD1 4" ,
which consists of the highly-nonstationary sound of a jet engine shutting down. For
noise type "XD1O", which is stationary noise, we observe a gain, but we do not see
any further gain for multiple noise components.
7
Conclusions
A far as variational methods go, ALGONQUIN is a fast technique for denoising logspectrum or cepstrum speech feature vectors. ALGONQUIN improves on previous
work using the vector Taylor series approximation, by using multiple component
speech and noise models, and it uses an iterative variational method to produce
accurate posterior distributions for speech and noise. By employing a generalized
EM method, ALGONQUIN can estimate a noise model from noisy speech data.
Our results show that the noise adaptive ALGONQUIN algorithm can obtain better results than the non-adaptive version. This is especially important for nonstationary noise, where the non-adaptive algorithm relies on an estimate of the
noise based on a subset of the frames , but the adaptive algorithm uses all the
frames of the utterance, even those that contain speech.
A different approach to denoising speech features is to learn time-domain models.
Attias et al. (2001) report results on a non-adaptive time-domain technique. Our
results cannot be directly compared with theirs, since our results are for unscaled
data. Eventually, the two approaches should be thoroughly compared.
References
Attias, H. , Platt , J . C., Acero, A., and Deng, L. 2001. Speech denoising and dereverberation using probabilistic models. In Advances in Neural Information Processing
Systems 13. MIT Press, Cambridge MA.
Boll, S. 1979. Suppression of acoustic noise in speech using spectral subtraction. IEEE
Transactions on Acoustics, Speech and Signal Processing, 27:114- 120.
Deng, L. , Acero, A., Plumpe, M., and Huang, X. D. 2000. Large-vocabulary speech
recognition under adverse acoustic environments. In Proceedings of the International
Conference on Spoken Language Processing, pages 806- 809.
Frey, B. J. , Deng, L. , Acero, A., and Kristjansson, T. 2001. ALGONQUIN: Iterating
Laplace's method to remove multiple types of acoustic distortion for robust speech
recognition. In Proceedings of Eurospeech 2001.
Gales, M. J. F. and Young, S. J . 1996. Robust continuous speech recognition using parallel
model combination. IEEE Speech and Audio Processing, 4(5):352- 359.
Jordan, M. 1. , Ghahramani, Z., J aakkola, T. S., and Saul , L. K. 1998. An introduction
to variational methods for graphical models. In Jordan, M. 1., editor, Learning in
Graphical Models. Kluwer Academic Publishers, Norwell MA.
Moreno, P. 1996. Speech R ecognition in Noisy Environments. Carnegie Mellon University,
Pittsburgh PA. Doctoral dissertation.
Neal, R. M. and Hinton, G. E. 1998. A view of the EM algorithm that justifies incremental,
sparse, and other variants . In Jordan , M. 1. , editor, Learning in Graphical Models ,
pages 355- 368. Kluwer Academic Publishers, Norwell MA .
Varga, A. P. and Moore, R. K. 1990. Hidden Markov model decomposition of speech and
noise. In Proceedings of th e International Conference on Acoustics, Speech and Signal
Processing, pages 845- 848. IEEE Press.
| 2016 |@word version:13 proportion:3 nd:4 kristjansson:2 accounting:1 covariance:10 decomposition:1 reduction:3 series:7 contains:2 current:2 com:1 si:1 yet:1 additive:3 partition:1 remove:2 moreno:4 drop:1 update:4 stationary:1 dissertation:1 toronto:2 org:1 simpler:1 windowed:1 consists:2 roughly:1 automatically:1 begin:1 estimating:2 maximizes:3 spoken:2 platt:1 before:4 frey:3 treat:1 whisper:2 doctoral:1 challenging:1 limited:1 procedure:1 significantly:2 adapting:3 word:3 inp:1 cannot:1 noising:1 acero:4 applying:1 writing:1 www:2 equivalent:1 center:1 maximizing:2 go:2 independently:1 duration:1 wit:1 laplace:2 analogous:1 updated:2 exact:1 us:4 trick:1 element:4 pa:1 recognition:14 particularly:1 updating:3 database:1 trausti:1 environment:3 complexity:2 dynamic:1 trained:4 depend:1 rewrite:1 algebra:2 basis:1 easily:1 joint:2 htm:1 snrs:1 fast:1 describe:1 whose:3 quite:1 lag:1 distortion:6 statistic:2 noisy:21 sequence:1 advantage:1 product:1 remainder:1 combining:1 mixing:3 description:1 moved:1 produce:1 wsj:3 incremental:1 derive:1 c:17 filter:1 centered:1 sand:1 feeding:1 wall:4 adjusted:1 extension:2 exp:6 cb:2 mapping:2 purpose:1 recognizer:5 weighted:1 mit:1 gaussian:8 modified:2 indicates:2 likelihood:6 contrast:1 brendan:1 suppression:1 inference:11 publically:1 nn:1 aurora:9 initially:1 nand:1 hidden:1 unoptimized:1 overall:1 priori:1 constrained:1 airport:1 once:2 noisefree:1 construct:1 report:1 recognize:2 divergence:1 individual:1 phase:1 microsoft:3 highly:3 mixture:8 pc:1 algonquin:16 accurate:3 norwell:2 taylor:7 logarithm:1 re:2 column:1 modeling:4 cost:1 subset:1 recognizing:1 eurospeech:2 kullbackleibler:1 stored:1 corrupted:2 combined:1 confident:1 thoroughly:1 density:1 international:2 probabilistic:4 iy:2 quickly:1 squared:2 containing:1 huang:1 gale:2 derivative:2 leading:1 li:1 de:2 slc:1 coefficient:1 view:1 competitive:1 denoised:3 parallel:1 minimize:1 ni:1 variance:4 spaced:1 ofthe:1 correspond:2 comp:2 published:1 energy:2 frequency:3 psi:1 unsolved:1 gain:4 improves:2 response:2 specify:1 cepstrum:2 sly:2 until:1 nonlinear:2 mode:1 impulse:1 effect:2 contain:1 true:1 moore:2 neal:2 during:1 speaker:1 mel:2 m:3 generalized:7 plexity:1 performs:2 percent:1 variational:16 wise:2 superior:1 jl:3 he:3 approximates:1 kluwer:2 theirs:1 mellon:1 cambridge:1 similarly:1 nonlinearity:1 language:1 posterior:15 showed:2 female:1 optimizing:1 yls:2 nics:2 lnp:3 minimum:2 additional:1 deng:5 subtraction:2 converge:1 maximize:2 signal:4 full:1 multiple:5 sound:1 infer:2 reduces:1 faster:1 academic:2 match:1 jet:1 variant:1 circumstance:1 iteration:3 interval:1 publisher:2 operate:1 db:8 jordan:4 nonstationary:3 fft:1 variety:1 restaurant:1 simplifies:1 cn:6 attias:4 expression:1 speech:72 varga:2 iterating:3 band:1 ten:1 reduced:1 http:2 percentage:1 track:1 carnegie:1 group:2 clarity:2 clean:16 nonadaptive:1 sum:1 inverse:5 parameterized:2 wer:7 distorted:2 scaling:1 bound:5 nonnegative:1 adapted:1 alex:1 expo:2 speed:1 argument:2 freyl:1 alternate:1 combination:1 conjugate:1 smaller:1 across:1 em:7 lp:1 eventually:1 needed:1 letting:1 fed:1 available:1 gaussians:3 nen:1 apply:1 observe:1 spectral:2 denotes:1 include:1 graphical:3 ghahramani:1 especially:2 approximating:1 added:1 diagonal:4 surrogate:1 ecognition:1 separate:3 street:5 hmm:2 toward:1 assuming:4 code:1 index:2 relationship:4 ratio:1 minimizing:1 difficult:1 implementation:1 observation:3 markov:1 benchmark:1 situation:1 extended:1 ino:1 precise:1 hinton:2 frame:14 station:1 retrained:1 community:1 rame:1 boll:2 sentence:3 acoustic:8 engine:1 learned:2 dereverberation:1 dsr:1 nly:1 xd1:1 technology:1 utterance:8 sn:1 prior:7 relative:2 sufficient:2 editor:2 uncorrelated:1 unscaled:1 row:3 accounted:1 last:1 transpose:1 saul:1 taking:1 sparse:1 benefit:1 vocabulary:4 autoregressive:1 doesn:1 made:1 adaptive:17 far:1 employing:1 transaction:1 approximate:10 pittsburgh:1 alternatively:1 spectrum:15 continuous:1 iterative:5 table:5 channel:12 learn:3 robust:4 init:1 interact:2 complex:1 shutting:1 domain:7 sp:1 main:1 noise:63 denoise:2 en:30 fashion:1 n:1 pe:3 jacobian:1 ffts:1 learns:1 young:2 down:1 symbol:1 normalizing:1 intractable:1 magnitude:1 justifies:1 easier:1 entropy:1 scalar:1 loud:1 relies:1 ma:3 goal:2 towards:1 change:1 adverse:1 operates:1 denoising:8 called:1 total:1 e:1 experimental:1 latter:1 brevity:1 audio:1 ex:1 |
1,115 | 2,017 | Stabilizing Value Function
with the
Xin Wang
Department of Computer Science
Oregon State University
Corvallis, OR, 97331
wangxi@cs. orst. edu
Thomas G Dietterich
Department of Computer Science
Oregon State University
Corvallis, OR, 97331
tgd@cs. orst. edu
Abstract
We address the problem of non-convergence of online reinforcement
learning algorithms (e.g., Q learning and SARSA(A)) by adopting an incremental-batch approach that separates the exploration
process from the function fitting process. Our BFBP (Batch Fit
to Best Paths) algorithm alternates between an exploration phase
(during which trajectories are generated to try to find fragments
of the optimal policy) and a function fitting phase (during which
a function approximator is fit to the best known paths from start
states to terminal states). An advantage of this approach is that
batch value-function fitting is a global process, which allows it to
address the tradeoffs in function approximation that cannot be
handled by local, online algorithms. This approach was pioneered
by Boyan and Moore with their GROWSUPPORT and ROUT algorithms. We show how to improve upon their work by applying
a better exploration process and by enriching the function fitting
procedure to incorporate Bellman error and advantage error measures into the objective function. The results show improved performance on several benchmark problems.
1
Introduction
Function approximation is essential for applying value-function-based reinforcement
learning (RL) algorithms to solve large Markov decision problems (MDPs). However, online RL algorithms such as SARSA(A) have been shown experimentally to
have difficulty converging when applied with function approximators. Theoretical
analysis has not been able to prove convergence, even in the case-of linear function
approximators. (See Gordon (2001), however, for a non-divergence result.) The
heart of the problem is that the approximate values of different states (e.g., 81 and
82) are coupled through the parameters of the function approximator. The optimal
policy at state 81 may require increasing a parameter, while the optimal policy at
state 82 may require decreasing it. As a result, algorithms based on local parameter
updates tend to oscillate or even to diverge.
To avoid this problem, a more global approach is called for-an approach that
can consider Sl and S2 simultaneously and find a solution that works well in both
states. One approach is to formulate the reinforcement learning problem as a global
search through a space of parameterized policies as in the policy gradient algorithms
(Williams, 1992; Sutton, McAllester, Singh, & Mansour, 2000; Konda & Tsitsiklis, 2000; Baxter & Bartlett, 2000). This avoids the oscillation problem, but the
resulting algorithms are slow and only converge to local optima.
We pursue an alternative approach that formulates the function approximation
problem as a global supervised learning problem. This approach, pioneered by
Boyan and Moore in their GROWSUPPORT (1995) and ROUT (1996) algorithms,
separates the reinforcement learning problem into two subproblems: the exploration
problem (finding a good partial value function) and the representation problem (representing and generalizing that value function). These algorithms alternate between
two phases. During the exploration phase, a support set of points is constructed
whose optimal values are known within some tolerance. In the function fitting
phase, a function approximator V is fit to the support set.
In this paper, we describe two ways of improving upon GROWSUPPORT and ROUT.
First, we replace the support set with the set of states that lie along the best
paths found during exploration. Second, we employ a combined error function that
includes terms for the supervised error, the Bellman error, and the advantage error
(Baird, 1995) into the function fitting process. The resulting BFBP (Batch Fit to
Best Paths) method gives significantly better performance on resource-constrained
scheduling problems as well as on the mountain car toy benchmark problem.
2
GrowSupport, ROUT, and BFBP
Consider a deterministic, episodic MDP. Let s' == a(s) denote the state s' that
results from performing a in s and r(a, s) denote the one-step reward. Both GROWSUPPORT and ROUT build a support set S == {(Si' V(Si))} of states whose optimal
values V (s) are known with reasonable accuracy. Both algorithms initialize S with
a set of terminal states (with V(s) == 0). In each iteration, a function approximator
V is fit to S to minimize :Ei[V(Si) - V(Si)]2. Then, an exploration process attempts
to identify new points to include in S.
In GROWSUPPORT, a sample of points X is initially drawn from the state space.
In each iteration, after fitting V, GROWSUPPORT computes a new estimate V(s)
for each state sEX according to V(s) == max a r(s, a) + V(a(s)), where V(a(s))
is computed by executing the greedy policy with respect to V starting in a(s). If
V(a(s)) is within c of V(a(s)), for all actions a, then (s, V(s)) is added to S.
ROUT employs a different procedure suitable for stochastic MDPs. Let P(s'ls, a)
be the probability that action a in state s results in state s' and R(s'ls, a) be
the expected one-step reward. During the exploration phase, ROUT generates a
trajectory from the start state to a terminal state and then searches for a state s
along that trajectory such that (i) V(s) is not a good approximation to the backedup value V(s) == maxa :Est P(s'ls, a)[R(s'ls, a) + V(s')], and (ii) for every state s
along a set of rollout trajectories starting at s', V(s) is within c of the backed-up
value maxa :Est P(s'ls, a)[R(s'ls, a) + V(s')]. If such a state is found, then (s, V(s))
is added to S.
Both GROWSUPPORT and ROUT rely on the function approximator to generalize
well at the boundaries of the support set. A new state s can only be added to
S if V has generalized to all of s's successor states. H this occurs consistently,
then eventually the support set will expand to include all of the starting states of
the MDP, at which point a satisfactory policy has been found. However, if this
"boundary generalization" does not occur, then no new points will be added to S,
and both GROWSUPPORT and ROUT. terminate without a solution. Unfortunately,
most regression methods have high bias and variance near the boundaries of their
training data, so failures of boundary generalization are common.
These observations led us to develop the BFBP algorithm. In BFBP, the exploration
process maintains a data structure S that stores the best known path from the start
state to a terminal state and a "tree" of one-step departures from this best path
(Le., states that can be reached by executing an action in some state on the best
path). At each state Si E S, the data structure stores the action
executed in that
state (to reach the next state in the path), the one-step reward ri, and the estimated
value V(Si). S also stores each action a_ that causes a departure from the best path
along with the resulting state S_, reward r_ and estimated value V(s_). We will
denote by B the subset of S that constitutes the best path. The estimated values
V are computed as folloV1S. For states S'i E B, V(Si) is computed 'by summing the
immediate rewards r j for all steps j 2: i along B. For the one-step departure states
s_, V(s_) is computed from an exploration trial in which the greedy policy was
followed starting in state s_.
at
fuitially, S is empty, so a random trajectory is generated from the start state So to a
terminal state, and it becomes the initial best known path. fu subsequent iterations,
is chosen and executed to
a state Si E B is chosen at random, and an action 1=
produce state s' and reward r'. Then the greedy policy (with respect to the current
V) is executed until a terminal state is reached. The rewards along this new path
are summed to produce V(s'). If V(s') +r' > V(Si), then the best path is revised as
follows. The new best action in state Si becomes al with estimated value V(s') +r' .
This improved value is then propagated backwards to update the V estimates for
in state Si becomes an inferior
all ancestor states in B. The old best action
action a_ with result state s_. Finally all descendants of s_ along the old best
path are deleted. This method of investigating one-step departures from the best
path is inspired by Harvey and Ginsberg's (1995) limited discrepancy search (LDS)
algorithm. In each exploration phase, K one-step departure paths are explored.
a' at
at
After the exploration phase, the value function approximation V is recomputed with
the goal of minimizing a combined error function:
J(V)
==
As
L (V(s) - V(S))2 + Ab L (V(s) sES
Aa
L L
[r(s, a*)
+ V(a*(s))])2 +
sEB
([r(s,a-)
+ V(a-(s))] -
[r(s,a*)
+ V(a*(s))]):.
The three terms of this objective function are referred to as the supervised, Bellman,
and advantage terms. Their relative importance is controlled by the coefficients As,
Ab' and Au. The supervised term is the usual squared error between the V(s) values
stored in S and the fitted values V(s). The Bellman term is the squared error
between the fitted value and the backed-up value of the next state on the best path.
And the advantage term penalizes any case where the backed-up value of an inferior
action a_ is larger than the backed-up value of the best action a*. The notation
(u)+ == u if u 2: 0 and 0 otherwise.
TheoreIll 1 Let M be a deterministic MDP such that (aJ there are only a finite
number of starting states, (bJ there are only? a finite set of actions executable in
each state, and (c) all policies reach a terminal state. Then BFBP applied to M
converges.
Proof: The LDS exploration process is monotonic, since the data structure S is
only updated if a new best path is found. The conditions of the theorem imply
that there are only a finite number of possible paths that? can be explored from the
starting states to the terminal states. Hence, the data structure S will eventually
converge. Consequently, the value function V fit to S will also converge. Q.E.D.
The theorem requires that the MDP contain no cycles. There are cycles in our jobshop scheduling problems, but we eliminate them by remembering all states visited
along the current trajectory and barring any action that would return to a previously
visited state. Note also that the theorem applies to MDPs with continuous state
spaces provided the action space and the start states are finite.
Unfortunately, BFBP does not necessarily converge to an optimal policy. This is
because LDS exploration can get stuck in a local optimum such that all one step
departures using the V-greedy policy produce trajectories that do not improve over
the current best path. Hence, although BFBP resembles policy iteration, it does not
have the same optimality guarantees,. because policy iteration evaluates the current
greedy policy in all states in the state space.
Theoretically, we could prove convergence to the optimal policy under modified conditions. If we replace LDS exploration with ?-greedy exploration, then exploration
will converge to the optimal paths with probability 1. When trained on those paths,
if the function approximator fits a sufficiently accurate V, then BFBS will converge
optimally. hI our experiments, however, we have found that ?-greedy gives no improvement over LDS, whereas LDS exploration provides more complete coverage of
one-step departures from the current best path, and these are used in J(V).
3
Experimental Evaluation
We have studied five domains: Grid World and Puddle World (Boyan & Moore,
1995), Mountain Car (Sutton, 1996), and resource-constrained scheduling problems
ART-1 and ART-2 (Zhang & Dietterich, 1995). For the first three domains, following Boyan and Moore, we compare BFBP with GROWSUPPORT. For the final
domain, it is difficult to draw a sample of states X from the state space to initialize
GROWSUPPORT. Hence, we compare against ROUT instead. As mentioned above,
we detected and removed cycles from the scheduling domain (since ROUT requires
this). We retained the cycles in the first three problems. On mountain car, we also
applied SARSA(A) with the CMAC function approximator developed by Sutton
(1996).
We experimented with two function approximators: regression trees (RT) and
locally-weighted linear regression (LWLR). Our regression trees employ linear separating planes at the internal nodes and linear surfaces at the leaf nodes. The trees
are grown top-down in the usual fashion. To determine the splitting plane at a
node, we choose a state Si at random from S, choose one of its inferior children S_,
and construct the plane that is the perpendicular bisector of these two points. The
splitting plane is evaluated by fitting the resulting child nodes to the data (as leaf
nodes) and computing the value of J (V). A number C of parent-child pairs (Si' S - )
are generated and evaluated, and the best one is retained to be the splitting plane.
This process is then repeated recursively until a node contains fewer than M data
points~ The linear surfaces at the leaves are trained by gradient descent to minimize
J(V). The gradient descent terminates after 100 steps or earlier if J becomes very
small. In our experiments, we tried all combinations of the following parameters
and report the best results: (a) 11 learning rates (from 0.00001 to 0.1), (b) M == 1,
Table 1: Comparison of results on three toy domains.
Problem Domain
Grid World
Algorithms
GROWSUPPORT
BFBP
Puddle World
G ROWSUPPORT
Mountain Car
BFBP
SARSA(A)
GROWSUPPORT
BFBP
Optimal Policyfj
Yes
Yes
Yes
Yes
No
No
Yes
Best Policy Length
39
39
39
39
103
93
88
Table 2: Results of ROUT and BFBP on scheduling problem ART-I-TRNOO
I
Performance
I ROUT (RT) I ROUT (LWLR) I BFBP (RT)
I
Best policy explored
I
1.75
I
1.55
I
1.50
I Best final learned policy I
1.8625
I
1.8125
I
1.55
10, 20, 50, 100, 1000, (c) C == 5, 10, 20, 50, 100, and (d) K == 50, 100, 150, 200.
For locally-weighted linear regression, we replicated the methods of B'oyan and
Moore. To compute V(s), a linear regression is performed using all points Si E S
weighted by their distance to S according to the kernel exp -(Ilsi - sII 2 /a 2 ). We
experimented with all combinations of the following parameters and report the best
results: (a) 29 values (from 0.01 to 1000.0) of the tolerance E that controls the
addition of new points to S, and (b) 39 values (from 0.01 to 1000.0) for a.
We execute ROUT and GROWSUPPORT to termination. We execute BFBP for 100
iterations, but it converges much earlier: 36 iterations for the grid world, 3 for
puddle world, 10 for mountain car, and 5 for the job-shop scheduling problems.
Table 1 compares the results of the algorithms on the toy domains with parameters
for each method tuned to give the best results and with As == 1 and Ab == Aa == o.
In all cases, BFBP matches or beats the other methods. In Mountain Car, in
particular, we were pleased that BFBP discovered the optimal policy very quickly.
Table 2 compares the results of ROUT and BFBP on job-shop scheduling problem
TRNOO from problem set ART-1 (again with As == 1 and Ab == Aa == 0). For ROUT,
results with both LWLR and RT are shown. LWLR gives better results for ROUT.
We conjecture that this is because ROUT needs a value function approximator that
is conservative near the boundary of the training data, whereas BFBP does not.
We report both the best policy found during the iterations and the final policy at
convergence. Figure 1 plots the r,esults for ROUT (LWLR) against BFBP (RT) for
eight additional scheduling problems from ART-I. The figure of merit is RDF, which
is a normalized measure of schedule length (small values are preferred). BFBP's
learned policy out-performs ROUT's in every case.
The experiments above all employed only the supervised term in the error function
J. These experiments demonstrate that LDS exploration gives better training sets
than the support set methods of GROWSUPPORT and ROUT. Now we turn to the
question of whether the Bellman and advantage terms can provide improved results.
For the grid world and puddle world tasks, the supervised term already gives optimal
performance, so we focus on the mountain car and job-shop scheduling problems.
Table 3 summarizes the results for BFBP on the mountain car problem. All parameter settings, except for the last, succeed in finding the optimal policy. To get
2.4
best policy explored
+
y=xbest finalleamed policy
x
2.2
G:'
Q
es
<l)
?
Xx
1.8
+ x
x
?
~
0.-
1.6
+
0...
ffP=l
+
1.4
1.2
1
1
1.4
1.2
1.6
1.8
2.2
2.4
ROUT performance (RDF)
Figure 1: Performance of Rout vs. BFBP over 8 job shop scheduling problems
Table 3: Fraction of parameter settings that give optimal performance for BFBP on the
mountain car problem
.As
0.0
1.0
1.0
1.0
.Ab
0.0
0.0
10.0
100.0
.Aa
1.0
0.0
0.0
0.0
#
settings
2/1311
52/1280
163/1295
4/939
As
0.0
1.0
1.0
1.0
Ab
1.0
0.0
0.0
1000.0
Aa
0.0
10.0
100.0
0.0
#
settings
1/1297
184/1291
133/1286
0/1299
a sense of the robustness of the method, we also report the fraction of parameter
settings that gave the optimal policy. The number of parameter settings tested (the
denominator) should be the same for all combinations of A values. Nonetheless,
for reasons unrelated to the parameter settings, some combinations failed to be
executed by our distributed process scheduler. The best settings combine As == 1
with either Ab == 10 or Aa == 10. However, if we employ either the Bellman or the
advantage term alone, the results are poor. Hence, it appears that the supervised
term is very important for good performance, but that the advantage and Bellman
terms can improve performance substantially .and reduce the sensitivity of BFBP
to the settings of the other parameters.
Table 4 shows the performance of BFBP on ART-I-TRNOO. The best performance
(at convergence) is obtained with As == Aa == 1 and Ab == O. As with mountain car,
these experiments show that the supervised term is the most important, but that
it gives even better results when combined with the advantage term.
All of the above experiments compare performance on single problems. We also
tested the ability of BFBP to generalize to similar problems following the formulation of (Zhang & Dietterich, 1995). Figure 2 compares the performance of neural networks and regression trees as function approximators for BFBP. Both were
trained on job shop scheduling problem set ART-2. Twenty of the problems in
ART-2 were used for training, 20 for cross-validation, and 50 for testing. Eleven
different values for As, Ab' Aa and eight different values for the learning rate were
tried, with the best values selected according to the cross-validation set. Figure 2
shows that BFBP is significantly better than the baseline performance (with RDF
Table 4: Performance ofBFBP on ART-1-TENOO for different settings of the .A parameters.
The ('perform;' column gives the best RDF in any iteratIon and the RDF at convergence.
.A 8
0
0
0
1
1
.Ab
.Aa
0
1
10
0
1
1
1
1
1
1
.A 8
0
0
0
1
1
perform .
1.50/1.75
1.50/1.775
1.50/1.775
1.50/1.488
1.525/1.55
.Ab
.Aa
1
1
100
0
1
0
10
1
10
10
perform.
1.50/1.775
1.50/1.825
1.50/1.738
1.463/1.525
1.50/1.588
.A 8
1
0
1
1
1
.Ab
.Aa
0
1
1
0
1
0
100
a
100
100
perform.
1.50/1.55
1.50/1.65
1.50/1.563
1.50/1.588
1.50/1.675
1.8 . - - - - - - , - - - - . - - - - - - - , . - - - . . . , - - - - , - - - - - - ,
BFBP neural net ----*---.
BFBP regression tree --- -G----
1.75
_________________________RP_E
_
1.7
1.65
LL
o
1.6
0:
~
~
1.55
~
1.45
1.4'?._?4<.,_.
,'"'::-..... ,c
._??
_._._.-!n-._,._.
.
...._._.,.._._._.
._,._.
.
._.,
._.._._.
,,_._.
._.,_T._..O_L__ .
.
1.35
1.3
L . . - -_ _- - - l - -_ _- - ' - -_ _- - L
o
10
15
- ' - - -_ _- - - ' -_ _-----'
20
25
30
LOS iteration
Figure 2: BFBP on ART-2 using neural nets and regression trees. "RDF" is a hand-coded
heuristic, "TDL" is Zhang's TD(.A) neural network.
as a search heuristic) and that its performance is comparable to TD(A) with neural networks (Zhang & Dietterich, 1995). Figure 3 shows that for ART-2, using
parent/inferior-child pair splits gives better results than using axis-parallel splits.
4
Conclusions
This paper has shown that the exploration strategies underlying GROWSUPPORT
and ROUT can be improved by simply remembering and training on the best paths
found between start and terminal states. Furthermore, the paper proved that the
BFBP method converges for arbitrary function approximators, which is a result
that has not yet been demonstrated for online methods such as SARSA(A). In
addition, we have shown that the performance of our BFBP algorithm can be further
improved (and made more robust) by incorporating a penalty for violations of the
Bellman equation or a penalty for preferring inferior actions (an advantage error).
Taken together, these results show that incremental-batch value function approximation can be a reliable, convergent method for solving deterministic reinforcement
learning problems. The key to the success of the method is the ability to separate
the exploration process from the function approximation process and to make the
exploration process convergent. This insight should also be applicable to stochastic
episodic MDPs.
1.9 , - - - - - - - . - - - - - , - - - - - - , - - - - . . , - - - - - - - , - - - - - ,
*...
axis-parallel ....
parent/inferior-child .. ??11 ???.
1.3
L -_ _- - l -
o
- ' - - -_ _- - - ' -
- L -_ _---L._ _- - - - I
10
20
15
25
30
LDS iteration
Figure 3: Axis parallel splits versus parent/inferior-child pair splits on ART-2
Acknowledgments
The authors gratefully acknowledge the support of AFOSR under contract F4962098-1-0375, and the NSF under grants IRl-9626584, I1S-0083292, 1TR-5710001197,
and EIA-9818414. We thank Valentina Zubek for her careful reading of the paper.
References
Baird, L. C. (1995). Residual algorithms: Reinforcement learning with function approximation. In ICML-95, 30-37, San Francisco, CA. Morgan Kaufmann.
Baxter, J., & Bartlett, P. L. (2000). Reinforcement learning in POMDP's via direct
gradient ascent. In ICML-2000, 41-48. Morgan Kaufmann, San Francisco, CA.
Boyan, J. A., & Moore, A. W. (1995). Generalization in reinforcement learning: Safely
approximating the value function. In NIPS-7, 369-376. The MIT Press, Cambridge.
Boyan, J. A., & Moore, A. W. (1996). Learning evaluation functions for large acyclic
domains. In ICML-96, 63-70. Morgan Kaufmann, San Francisco, CA.
Gordon, G. J. (2001). Reinforcement learning with function approximation converge to a
region. In NIPS-13, 1040-1046. The MIT Press.
Harvey, W. D., & Ginsberg, L. P. (1995). Limited discrepancy search. In IJCAI-95,
825-830. Morgan Kaufmann.
Konda, V. R., & Tsitsiklis, J. N. (2000). Policy gradient methods for reinforcement learning
with function approximation. In NIPS-12, 1008-1014 Cambridge, MA. MIT Press.
Moll, R., Barto, A. G., Perkins, T. J., & Sutton, R. S. (1999). Learning instanceindependent value functions to enhance local search. In NIPS-ll, 1017-1023.
Sutton, R. S., McAllester, D., Singh, S., & Mansour, Y. (2000). Policy gradient methods
for reinforcement learning with function approximation. In NIPS-12, 1057-1063.
Sutton, R. S. (1996). Generalization in reinforcement learning: Successful examples using
sparse coarse coding. In NIPS-8, 1038-1044. The MIT Press, Cambridge.
Williams, R. J. (1992). Simple statistical gradient-following algorithms for connectionist
reinforcement learning. Machine Learning, 8,229.
-..
Zhang, W., & Dietterich, T. G. (1995). A reinforcement learning approach to job-shop
scheduling. In IJCAI-95, 1114-1120. Morgan Kaufmann, San Francisco, CA.
| 2017 |@word trial:1 sex:1 termination:1 tried:2 tr:1 recursively:1 initial:1 contains:1 fragment:1 tuned:1 current:5 ginsberg:2 si:14 yet:1 subsequent:1 eleven:1 plot:1 update:2 v:1 alone:1 greedy:7 leaf:3 fewer:1 selected:1 plane:5 provides:1 coarse:1 node:6 zhang:5 five:1 rollout:1 along:8 constructed:1 sii:1 direct:1 descendant:1 prove:2 tdl:1 fitting:8 combine:1 theoretically:1 expected:1 terminal:9 bellman:8 inspired:1 decreasing:1 td:2 increasing:1 becomes:4 provided:1 xx:1 notation:1 unrelated:1 underlying:1 zubek:1 mountain:10 pursue:1 substantially:1 maxa:2 developed:1 finding:2 guarantee:1 safely:1 every:2 control:1 grant:1 local:5 sutton:6 path:24 au:1 resembles:1 studied:1 xbest:1 limited:2 enriching:1 perpendicular:1 acknowledgment:1 testing:1 procedure:2 cmac:1 episodic:2 significantly:2 get:2 cannot:1 seb:1 scheduling:12 applying:2 deterministic:3 demonstrated:1 backed:4 williams:2 starting:6 l:6 pomdp:1 formulate:1 stabilizing:1 splitting:3 insight:1 updated:1 pioneered:2 lwlr:5 wang:1 region:1 cycle:4 removed:1 mentioned:1 reward:7 trained:3 singh:2 solving:1 upon:2 grown:1 describe:1 detected:1 whose:2 heuristic:2 larger:1 solve:1 s:1 otherwise:1 ability:2 final:3 online:4 advantage:10 net:2 los:1 convergence:6 empty:1 optimum:2 parent:4 ijcai:2 produce:3 incremental:2 executing:2 converges:3 develop:1 job:6 coverage:1 c:2 stochastic:2 exploration:22 mcallester:2 successor:1 require:2 generalization:4 sarsa:5 sufficiently:1 exp:1 bj:1 applicable:1 visited:2 weighted:3 mit:4 modified:1 r_:1 avoid:1 barto:1 focus:1 improvement:1 consistently:1 baseline:1 sense:1 eliminate:1 initially:1 her:1 ancestor:1 expand:1 constrained:2 summed:1 initialize:2 art:12 construct:1 barring:1 icml:3 constitutes:1 discrepancy:2 report:4 connectionist:1 gordon:2 employ:4 simultaneously:1 divergence:1 phase:8 attempt:1 ab:12 evaluation:2 violation:1 tgd:1 accurate:1 fu:1 partial:1 tree:7 old:2 penalizes:1 theoretical:1 fitted:2 column:1 earlier:2 formulates:1 subset:1 i1s:1 successful:1 optimally:1 stored:1 combined:3 sensitivity:1 preferring:1 contract:1 diverge:1 enhance:1 together:1 quickly:1 squared:2 again:1 choose:2 return:1 toy:3 coding:1 includes:1 coefficient:1 baird:2 oregon:2 performed:1 try:1 reached:2 start:6 maintains:1 parallel:3 minimize:2 accuracy:1 variance:1 kaufmann:5 identify:1 yes:5 generalize:2 lds:8 trajectory:7 reach:2 failure:1 evaluates:1 against:2 nonetheless:1 proof:1 propagated:1 proved:1 car:10 schedule:1 appears:1 supervised:8 improved:5 formulation:1 evaluated:2 execute:2 eia:1 furthermore:1 until:2 hand:1 ei:1 irl:1 aj:1 mdp:4 dietterich:5 contain:1 normalized:1 hence:4 moore:7 satisfactory:1 ffp:1 ll:2 during:6 inferior:7 generalized:1 complete:1 demonstrate:1 performs:1 theoreill:1 common:1 executable:1 rl:2 corvallis:2 cambridge:3 grid:4 gratefully:1 surface:2 store:3 harvey:2 success:1 approximators:5 morgan:5 additional:1 remembering:2 employed:1 converge:7 determine:1 ii:1 match:1 cross:2 coded:1 controlled:1 converging:1 regression:9 denominator:1 iteration:11 kernel:1 adopting:1 whereas:2 addition:2 ascent:1 tend:1 near:2 backwards:1 split:4 baxter:2 fit:7 gave:1 moll:1 jobshop:1 reduce:1 valentina:1 tradeoff:1 whether:1 handled:1 bartlett:2 penalty:2 oscillate:1 cause:1 action:15 locally:2 sl:1 nsf:1 estimated:4 recomputed:1 key:1 drawn:1 deleted:1 fraction:2 rout:25 parameterized:1 reasonable:1 oscillation:1 draw:1 decision:1 summarizes:1 comparable:1 hi:1 followed:1 convergent:2 occur:1 perkins:1 ri:1 generates:1 optimality:1 performing:1 conjecture:1 department:2 according:3 alternate:2 combination:4 poor:1 terminates:1 heart:1 taken:1 resource:2 equation:1 previously:1 turn:1 eventually:2 merit:1 eight:2 batch:5 alternative:1 robustness:1 thomas:1 top:1 include:2 konda:2 build:1 approximating:1 objective:2 added:4 question:1 occurs:1 already:1 strategy:1 rt:5 usual:2 gradient:7 distance:1 separate:3 thank:1 separating:1 s_:8 reason:1 length:2 retained:2 minimizing:1 difficult:1 unfortunately:2 executed:4 esults:1 subproblems:1 policy:29 twenty:1 perform:4 observation:1 revised:1 markov:1 benchmark:2 finite:4 rdf:6 descent:2 acknowledge:1 beat:1 immediate:1 mansour:2 discovered:1 arbitrary:1 pair:3 orst:2 learned:2 nip:6 address:2 able:1 departure:7 reading:1 max:1 reliable:1 suitable:1 difficulty:1 boyan:6 rely:1 residual:1 representing:1 shop:6 improve:3 mdps:4 imply:1 f4962098:1 axis:3 coupled:1 pleased:1 relative:1 afosr:1 acyclic:1 approximator:8 versus:1 validation:2 last:1 tsitsiklis:2 bias:1 sparse:1 tolerance:2 distributed:1 boundary:5 world:8 avoids:1 computes:1 stuck:1 made:1 reinforcement:14 replicated:1 author:1 san:4 approximate:1 preferred:1 global:4 investigating:1 summing:1 a_:3 francisco:4 search:6 continuous:1 table:8 terminate:1 robust:1 ca:4 improving:1 necessarily:1 domain:8 s2:1 child:6 repeated:1 referred:1 fashion:1 slow:1 scheduler:1 lie:1 theorem:3 down:1 explored:4 experimented:2 essential:1 incorporating:1 importance:1 generalizing:1 led:1 simply:1 failed:1 monotonic:1 applies:1 aa:11 ma:1 succeed:1 goal:1 consequently:1 careful:1 replace:2 experimentally:1 except:1 conservative:1 called:1 experimental:1 xin:1 e:1 est:2 bisector:1 puddle:4 internal:1 support:8 incorporate:1 tested:2 |
1,116 | 2,018 | The Fidelity of Local Ordinal Encoding
Javid Sadr, Sayan Mukherjee, Keith Thoresz, Pawan Sinha
Center for Biological and Computational Learning
Department of Brain and Cognitive Sciences, MIT
Cambridge, Massachusetts, 02142 USA
{sadr,sayan,thorek,sinha}@ai.mit.edu
Abstract
A key question in neuroscience is how to encode sensory stimuli
such as images and sounds. Motivated by studies of response properties of neurons in the early cortical areas, we propose an encoding
scheme that dispenses with absolute measures of signal intensity
or contrast and uses, instead, only local ordinal measures. In this
scheme, the structure of a signal is represented by a set of equalities
and inequalities across adjacent regions. In this paper, we focus
on characterizing the fidelity of this representation strategy. We
develop a regularization approach for image reconstruction from
ordinal measures and thereby demonstrate that the ordinal representation scheme can faithfully encode signal structure. We also
present a neurally plausible implementation of this computation
that uses only local update rules. The results highlight the robustness and generalization ability of local ordinal encodings for the
task of pattern classification.
1
Introduction
Biological and artificial recognition systems face the challenge of grouping together
differing proximal stimuli arising from the same underlying object. How well the
system succeeds in overcoming this challenge is critically dependent on the nature
of the internal representations against which the observed inputs are matched. The
representation schemes should be capable of efficiently encoding object concepts
while being tolerant to their appearance variations.
In this paper, we introduce and characterize a biologically plausible representation
scheme for encoding signal structure. The scheme employs a simple vocabulary
of local ordinal relations, of the kind that early sensory neurons are capable of
extracting. Our results so far suggest that this scheme possesses several desirable
characteristics, including tolerance to object appearance variations, computational
simplicity, and low memory requirements. We develop and demonstrate our ideas in
the visual domain, but they are intended to be applicable to other sensory modalities
as well.
The starting point for our proposal lies in studies of the response properties of
neurons in the early sensory cortical areas. These response properties constrain
Figure 1: (a) A schematic contrast response curve for a primary visual cortex
neuron. The response of the neuron saturates at low contrast values. (b) An
idealization of (a). This unit can be thought of as an ordinal comparator, providing
information only about contrast polarity but not its magnitude.
the kinds of measurements that can plausibly be included in our representation
scheme. In the visual domain, many striate cortical neurons have rapidly saturating
contrast response functions [1, 4]. Their tendency to reach ceiling level responses at
low contrast values render these neurons sensitive primarily to local ordinal, rather
than metric, relations. We propose to use an idealization of such units as the basic
vocabulary of our representation scheme (figure 1). In this scheme, objects are
encoded as sets of local ordinal relations across image regions. As discussed below,
this very simple idea seems well suited to handling the photometric appearance
variations that real-world objects exhibit.
Figure 2: The challenge for a representation scheme: to construct stable descriptions
of objects despite radical changes in appearance.
As figure 2 shows, variations in illumination significantly alter the individual brightness of different parts of the face, such as the eyes, cheeks, and forehead. Therefore,
absolute image brightness distributions are unlikely to be adequate for classifying
all of these images as depicting the same underlying object. Even the contrast
magnitudes across different parts of the face change greatly under different lighting
conditions. While the absolute luminance and contrast magnitude information is
highly variable across these images, Thoresz and Sinha [9] have shown that one can
identify some stable ordinal measurements. Figure 3 shows several pairs of average
brightness values over localized patches for each of the three images included in
figure 2. Certain regularities are apparent. For instance, the average brightness
of the left eye is always less than that of the forehead, irrespective of the lighting
conditions. The relative magnitudes of the two brightness values may change, but
the sign of the inequality does not. In other words, the ordinal relationship between
the average brightnesses of the <left-eye, forehead> pair is invariant under lighting
changes. Figure 3 shows several other such pair-wise invariances. It seems, therefore
that local ordinal relations may encode the stable facial attributes across different
illumination conditions. An additional advantage to using ordinal relations is their
natural robustness to sensor noise. Thus, it would seem that local ordinal representations may be well suited for devising compact representations, robust against
Figure 3: The absolute brightnesses and their relative magnitudes change under different lighting conditions but several pair-wise ordinal relationships stay invariant.
large photometric variations, for at least some classes of objects. Notably, for similar reasons, ordinal measures have also been shown to be a powerful tool for simple,
efficient, and robust stereo image matching [3].
In what follows, we address an important open question regarding the expressiveness of the ordinal representation scheme. Given that this scheme ignores absolute
luminance and contrast magnitude information, an obvious question that arises is
whether such a crude representation strategy can encode object/image structure
with any fidelity.
2
Information Content of Local Ordinal Encoding
Figure 4 shows how we define ordinal relations between an image region pa and
its immediate neighbors pb = {pa1 , . . . , pa8 }. In the conventional rectilinear grid,
when all image regions pa are considered, four of the eight relations are redundant;
we encode the remaining four as {1, 0, ?1} based on the difference in luminance
between two neighbors being positive, zero, or negative, respectively. To demonstrate the richness of information encoded by this scheme, we compare the original
image to one produced by a function that reconstructs the image using local ordinal
relationships as constraints. Our reconstruction function has the form
f (x) = w ? ?(x),
(1)
where x = {i, j} is the position of a pixel, f (x) is its intensity, ? is a map from the
input space into a high (possibly infinite) dimensional space, w is a hyperplane in
this high-dimensional space, and u ? v denotes an inner product.
Infinitely many reconstruction functions could satisfy the given ordinal constraints.
To make the problem well-posed we regularize [10] the reconstruction function subject to the ordinal constraints, as done in ordinal regression for ranking document
Department of Brain Sciences, MIT
Cambridge, Massachusetts, USA.
{sadr,sayan,thorek,sinha}@ai.mit.edu
Neighbors? relations to pixel of interest
???????????????????????????I(pa )
<
=
<
<
>
<
<
<
I(pa1 )
I(pa2 )
I(pa3 )
I(pa4 )
I(pa5 )
I(pa6 )
I(pa7 )
I(pa8 )
(1)
Figure 4: Ordinal relationships between
an image region pa and its neighbors.
???????????????????????????retrieval results [5]. Our regularization term is a norm in a Reproducing Kernel
Hilbert Space (RKHS) [2, 11]. Minimizing the norm in a RKHS subject to the
ordinal constraints corresponds to the following convex constrained quadratic optimization problem:
X
1
min ||w||2 + C
?p
(2)
?,w 2
p
subject to
?(?p )w ? (?(xpa ) ? ?(xpb )) ? |?p | ? ?p , ? p and ? ? 0,
(3)
where the function ?(y) = +1 for y ? 0 and ?1 otherwise, p is the index over
all pairwise ordinal relations between all pixels pa and their local neighbors pb (as
depicted in figure 4), ?p are slack variables which are penalized by C (the trade-off
between smoothness and ordinal constraints), and ?p take integer values {?1, 0, 1}
denoting the ordinal relation (less than, equal to, or greater than, respectively)
between pa and pb ; for the case ?p = 0 the inequality in (3) will be a strict equality.
Taking the dual of (2) subject to constraints (3) results in the following convex
quadratic optimization problem which has only box constraints:
X
1 XX
? pq
?p ?q K
(4)
max
|?p | ?p ?
?
2 p q
p
subject to
0 ? ?p ? C
?C ? ?p ? C
?C ? ?p ? 0
if
if
if
?p > 0,
?p = 0,
?p < 0,
(5)
? have
where ?p are the dual Lagrange multipliers, and the elements of the matrix K
the form
? pq
K
= (?(xpa ) ? ?(xpb )) ? (?(xqa ) ? ?(xqb ))
= K(xpa , xqa ) ? K(xpb , xqa ) ? K(xpa , xqb ) + K(xpb , xqb ),
where K(y, x) = ?(y)??(x) using the standard kernel trick [8]. In this paper we use
only Gaussian kernels K(y, x) = exp(?||x?y||2 /2? 2 ). The reconstruction function,
f (x), obtained from optimizing (4) subject to box constraints (5) has the following
form
X
f (x) =
?p (K(x, xpa ) ? K(x, xpb )) .
(6)
p
Note that in general many of the ?p values may be zero ? these terms do not
contribute to the reconstruction, and the corresponding constraints in (3) were not
300
200
100
0
0
128
255
128
255
300
200
100
0
(a)
(b)
(c)
0
(d)
Figure 5: Reconstruction results from the regularization approach. (a) Original
images. (b) Reconstructed images. (c) Absolute difference between original and
reconstruction. (d) Histogram of absolute difference.
required. The remaining ?p with absolute value less than C satisfy the inequality
constraints in (3), whereas those with absolute value at C violate them.
Figure 5 depicts two typical reconstructions performed by this algorithm. The
difference images and error histograms suggests that the reconstructions closely
match the source images.
3
Discussion
Our reconstruction results suggest that the local ordinal representation can faithfully encode image structure. Thus, even though individual ordinal relations are
insensitive to absolute luminance or contrast magnitude, a set of such relations implicitly encodes metric information. In the context of the human visual system, this
result suggests that the rapidly saturating contrast response functions of the early
visual neurons do not significantly hinder their ability to convey accurate image
information to subsequent cortical stages.
An important question that arises here is what are the strengths and limitations of
local ordinal encoding. The first key limitation is that for any choice of neighborhood size over which ordinal relations are extracted, there are classes of images for
which the local ordinal representation will be unable to encode the metric structure. For a neighborhood of size n, an image with regions of different luminance
embedded in a uniform background and mutually separated by a distance greater
than n would constitute such an image. In general, sparse images present a problem for this representation scheme, as might foveal or cortical ?magnification,? for
example. This issue could be addressed by using ordinal relations across multiple
scales, perhaps in an adaptive way that varies with the smoothness or sparseness of
the stimulus.
Second, the regularization approach above seems biologically implausible. Our intent in using this approach for reconstructions was to show via well-understood
theoretical tools the richness of information that local ordinal representations pro-
Figure 6: Reconstruction results from the relaxation approach.
vide. In order to address the neural plausibility requirement, we have developed a
simple relaxation-based approach with purely local update rules of the kind that
can easily be implemented by cortical circuitry. Each unit communicates only with
its immediate neighbors and modifies its value incrementally up or down (starting
from an arbitrary state) depending on the number of ordinal relations in the positive
or negative direction. This computation is performed iteratively until the network
settles to an equilibrium state. The update rule can be formally stated as
X
(?(Rpa ,t ? Rpb ,t ) ? ?(Ipa ? Ipb )),
(7)
Rpa ,t+1 = Rpa ,t + ?
pb
where Rpa ,t is the intensity of the reconstructed pixel pa at step t, Ipa is the intensity of the corresponding pixel in the original image, ? is a positive update
rate, and ? and pb are as described above. Figure 6 shows four examples of image
reconstructions performed using a relaxation-based approach.
A third potential limitation is that the scheme does not appear to constitute a
compact code. If each pixel must be encoded in terms of its relations with all of
its eight neighbors, where each relation takes one of three values, {?1, 0, 1}, then
what has been gained over the original image where each pixel is encoded by 8 bits?
There are three ways to address this question.
1. Eight relations per pixel is highly redundant ? four are sufficient. In fact, as
shown in figure 7, the scheme can also tolerate several missing relations.
Figure 7: Five reconstructions, shown here to demonstrate the robustness of local
ordinal encoding to missing inputs. From left to right: reconstructions based on
100%, 80%, 60%, 40%, and 20% of the full set of immediate neighbor relations.
2. An advantage to using ordinal relations is that they can be extracted and transmitted much more reliably than metric ones. These relations share the same spirit
(a)
(b)
Figure 8: A small collection of ordinal relations (a), though insufficient for high
fidelity reconstruction, is very effective for pattern classification despite significant
appearance variations. (b) Results of using a local ordinal relationship based template to detect face patterns. The program places white dots at the centers of
patches classified as faces. (From Thoresz and Sinha, in preparation.)
as loss functions used in robust statistics [6] and trimmed or Winsorized estimators.
3. The intent of the visual system is often not to encode/reconstruct images with
perfect fidelity, but rather to encode the most stable characteristics that can aid in
classification. In this context, a few ordinal relations may suffice for encoding objects
reliably. Figure 8 shows the results of using less than 20 relations for detecting faces.
Clearly, such a small set would not be sufficient for reconstructions, but it works
well for classification. Its generalization arises because it defines an equivalence
class of patterns.
In summary, the ordinal representation scheme provides a neurally plausible strategy for encoding signal structure. While in this paper we focus on demonstrating
the fidelity of this scheme, we believe that its true strength lies in defining equivalence classes of patterns enabling generalizations over appearance variations in
objects. Several interesting directions remain to be explored. These include the
study of ordinal representations across multiple scales, learning schemes for identifying subsets of ordinal relations consistent across different instances of an object,
and the relationship of this work to multi-dimensional scaling [12] and to the use
of truncated, quantized wavelet coefficients as ?signatures? for fast, multiresolution
image querying [7].
Acknowledgements
We would like to thank Gadi Geiger, Antonio Torralba, Ryan Rifkin, Gonzalo Ramos, and
Tabitha Spagnolo. Javid Sadr is a Howard Hughes Medical Institute Pre-Doctoral Fellow.
References
[1] A. Anzai, M. A. Bearse, R. D. Freeman, and D. Cai. Contrast coding by cells in
the cat?s striate cortex: monocular vs. binocular detection. Visual Neuroscience,
12:77?93, 1995.
[2] N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 686:337?404,
1950.
[3] D. Bhat and S. Nayar. Ordinal measures for image correspondence. In IEEE Conf.
on Computer Vision and Pattern Recognition, pages 351?357, 1996.
[4] G. C. DeAngelis, I. Ohzawa, and R. D. Freeman. Spatiotemporal organization of
simple-cell receptive fields in the cat?s striate cortex. i. general characteristics and
postnatal development. J. Neurophysiology, 69:1091?1117, 1993.
[5] R. Herbrich, T. Graepel, and K. Obermeyer. Support vector learning for ordinal
regression. In Proc. of the Ninth Intl. Conf. on Artificial Neural Networks, pages
97?102, 1999.
[6] P. Huber. Robust Statistics. John Wiley and Sons, New York, 1981.
[7] C. E. Jacobs, A. Finkelstein, and D. H. Salesin. Fast multiresolution image querying.
In Computer Graphics Proc., Annual Conf. Series (SIGGRAPH 95), pages 277?286,
1995.
[8] T. Poggio. On optimal nonlinear associative recall. Biological Cybernetics, 19:201?209,
1975.
[9] K. Thoresz and P. Sinha. Qualitative representations for recognition. Vision Sciences
Society Abstracts, 1:81, 2001.
[10] A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill-posed Problems. W. H. Winston,
Washington, D.C., 1977.
[11] G. Wahba. Spline Models for Observational Data. Series in Applied Mathematics,
Vol 59, SIAM, Philadelphia, 1990.
[12] F. W. Young and C. H. Null. Mds of nominal data: the recovery of metric information
with alscal. Psychometika, 53.3:367?379, 1978.
| 2018 |@word neurophysiology:1 seems:3 norm:2 open:1 jacob:1 brightness:7 thereby:1 foveal:1 series:2 denoting:1 document:1 rkhs:2 must:1 john:1 subsequent:1 update:4 v:1 devising:1 postnatal:1 provides:1 detecting:1 contribute:1 quantized:1 math:1 herbrich:1 five:1 ipb:1 qualitative:1 vide:1 introduce:1 pairwise:1 huber:1 notably:1 multi:1 brain:2 freeman:2 xx:1 underlying:2 matched:1 suffice:1 null:1 what:3 kind:3 developed:1 differing:1 fellow:1 unit:3 medical:1 appear:1 positive:3 understood:1 local:19 despite:2 encoding:10 might:1 doctoral:1 equivalence:2 suggests:2 hughes:1 area:2 thought:1 significantly:2 matching:1 word:1 pre:1 suggest:2 context:2 conventional:1 map:1 center:2 missing:2 modifies:1 starting:2 convex:2 simplicity:1 identifying:1 recovery:1 rule:3 estimator:1 regularize:1 variation:7 nominal:1 us:2 pa:7 element:1 trick:1 recognition:3 magnification:1 mukherjee:1 observed:1 region:6 richness:2 trade:1 hinder:1 signature:1 purely:1 easily:1 siggraph:1 represented:1 cat:2 separated:1 alscal:1 fast:2 effective:1 deangelis:1 artificial:2 neighborhood:2 apparent:1 encoded:4 posed:2 plausible:3 pa2:1 otherwise:1 reconstruct:1 ability:2 statistic:2 associative:1 advantage:2 cai:1 propose:2 reconstruction:18 product:1 finkelstein:1 rapidly:2 rifkin:1 multiresolution:2 description:1 regularity:1 requirement:2 intl:1 perfect:1 object:12 depending:1 develop:2 radical:1 keith:1 soc:1 implemented:1 direction:2 closely:1 attribute:1 human:1 settle:1 observational:1 anzai:1 generalization:3 biological:3 ryan:1 considered:1 exp:1 equilibrium:1 circuitry:1 early:4 torralba:1 proc:2 applicable:1 sensitive:1 xpa:5 faithfully:2 tool:2 mit:4 clearly:1 sensor:1 always:1 gaussian:1 rather:2 encode:9 focus:2 greatly:1 contrast:12 detect:1 dependent:1 unlikely:1 relation:26 rpa:4 pixel:8 issue:1 fidelity:6 classification:4 dual:2 ill:1 development:1 constrained:1 equal:1 construct:1 field:1 washington:1 alter:1 photometric:2 stimulus:3 spline:1 employ:1 primarily:1 few:1 individual:2 intended:1 pawan:1 detection:1 organization:1 interest:1 highly:2 gadi:1 accurate:1 capable:2 poggio:1 facial:1 theoretical:1 sinha:6 instance:2 subset:1 uniform:1 graphic:1 characterize:1 varies:1 spatiotemporal:1 proximal:1 siam:1 stay:1 off:1 together:1 reconstructs:1 possibly:1 cognitive:1 conf:3 potential:1 coding:1 coefficient:1 satisfy:2 ranking:1 performed:3 characteristic:3 efficiently:1 identify:1 salesin:1 critically:1 produced:1 lighting:4 cybernetics:1 classified:1 implausible:1 reach:1 against:2 obvious:1 massachusetts:2 recall:1 hilbert:1 graepel:1 tolerate:1 response:8 amer:1 done:1 box:2 though:2 stage:1 binocular:1 until:1 aronszajn:1 nonlinear:1 incrementally:1 defines:1 perhaps:1 believe:1 usa:2 ohzawa:1 concept:1 multiplier:1 true:1 equality:2 regularization:4 iteratively:1 white:1 adjacent:1 demonstrate:4 pro:1 image:31 wise:2 insensitive:1 discussed:1 forehead:3 measurement:2 significant:1 cambridge:2 ai:2 rectilinear:1 smoothness:2 grid:1 mathematics:1 gonzalo:1 pq:2 dot:1 stable:4 cortex:3 optimizing:1 tikhonov:1 certain:1 inequality:4 transmitted:1 additional:1 greater:2 redundant:2 signal:5 neurally:2 desirable:1 violate:1 sound:1 multiple:2 full:1 match:1 plausibility:1 retrieval:1 schematic:1 basic:1 regression:2 vision:2 metric:5 histogram:2 kernel:4 cell:2 proposal:1 background:1 whereas:1 bhat:1 addressed:1 source:1 modality:1 posse:1 strict:1 subject:6 spirit:1 seem:1 integer:1 extracting:1 wahba:1 inner:1 idea:2 regarding:1 whether:1 motivated:1 trimmed:1 stereo:1 render:1 york:1 constitute:2 adequate:1 antonio:1 ipa:2 sign:1 neuroscience:2 arising:1 per:1 vol:1 key:2 four:4 demonstrating:1 pb:5 luminance:5 relaxation:3 idealization:2 powerful:1 place:1 pa1:2 cheek:1 patch:2 geiger:1 scaling:1 bit:1 correspondence:1 quadratic:2 winston:1 annual:1 strength:2 constraint:10 constrain:1 encodes:1 min:1 department:2 across:8 remain:1 son:1 biologically:2 invariant:2 handling:1 ceiling:1 monocular:1 mutually:1 slack:1 ordinal:46 eight:3 robustness:3 original:5 denotes:1 remaining:2 include:1 plausibly:1 society:1 question:5 strategy:3 primary:1 receptive:1 striate:3 md:1 exhibit:1 distance:1 unable:1 thank:1 reason:1 code:1 index:1 polarity:1 relationship:6 providing:1 minimizing:1 insufficient:1 negative:2 stated:1 intent:2 implementation:1 reliably:2 neuron:8 howard:1 enabling:1 truncated:1 immediate:3 defining:1 saturates:1 reproducing:2 ninth:1 arbitrary:1 intensity:4 expressiveness:1 overcoming:1 pair:4 required:1 trans:1 address:3 below:1 pattern:6 challenge:3 program:1 including:1 memory:1 max:1 natural:1 ramos:1 scheme:20 eye:3 irrespective:1 philadelphia:1 acknowledgement:1 relative:2 embedded:1 loss:1 highlight:1 interesting:1 limitation:3 querying:2 localized:1 sufficient:2 consistent:1 classifying:1 share:1 arsenin:1 penalized:1 summary:1 institute:1 neighbor:8 template:1 characterizing:1 face:6 taking:1 absolute:10 sparse:1 tolerance:1 curve:1 cortical:6 vocabulary:2 world:1 sensory:4 ignores:1 collection:1 adaptive:1 far:1 reconstructed:2 compact:2 implicitly:1 tolerant:1 winsorized:1 nature:1 robust:4 dispenses:1 depicting:1 domain:2 noise:1 convey:1 depicts:1 aid:1 wiley:1 position:1 lie:2 crude:1 communicates:1 third:1 wavelet:1 young:1 sayan:3 down:1 explored:1 grouping:1 gained:1 magnitude:7 illumination:2 sparseness:1 suited:2 depicted:1 appearance:6 infinitely:1 visual:7 lagrange:1 saturating:2 corresponds:1 extracted:2 comparator:1 content:1 change:5 included:2 infinite:1 typical:1 hyperplane:1 invariance:1 tendency:1 succeeds:1 formally:1 internal:1 support:1 arises:3 preparation:1 nayar:1 |
1,117 | 2,019 | Learning Body Pose via Specialized Maps
Romer Rosales
Department of Computer Science
Boston University, Boston, MA 02215
rrosales@cs.bu.edu
Stan Sclaroff
Department of Computer Science
Boston University, Boston, MA 02215
sclaroff@cs.bu.edu
Abstract
A nonlinear supervised learning model, the Specialized Mappings
Architecture (SMA), is described and applied to the estimation of
human body pose from monocular images. The SMA consists of
several specialized forward mapping functions and an inverse mapping function. Each specialized function maps certain domains
of the input space (image features) onto the output space (body
pose parameters). The key algorithmic problems faced are those of
learning the specialized domains and mapping functions in an optimal way, as well as performing inference given inputs and knowledge of the inverse function. Solutions to these problems employ
the EM algorithm and alternating choices of conditional independence assumptions. Performance of the approach is evaluated with
synthetic and real video sequences of human motion.
1
Introduction
In everyday life, humans can easily estimate body part locations (body pose) from
relatively low-resolution images of the projected 3D world (e.g., when viewing a
photograph or a video). However, body pose estimation is a very difficult computer
vision problem. It is believed that humans employ extensive prior knowledge about
human body structure and motion in this task [10]. Assuming this , we consider
how a computer might learn the underlying structure and thereby infer body pose.
In computer vision, this task is usually posed as a tracking problem. Typically,
models comprised of 2D or 3D geometric primitives are designed for tracking a
specific articulated body [13, 5, 2, 15]. At each frame, these models are fitted to the
image to optimize some cost function. Careful manual placement of the model on
the first frame is required, and tracking in subsequent frames tends to be sensitive to
errors in initialization and numerical drift. Generally, these systems cannot recover
from tracking errors in the middle of a sequence. To address these weaknesses,
more complex dynamic models have been proposed [14, 13,9]; these methods learn
a prior over some specific motion (such as walking). This strong prior however,
substantially limits the generality of the motions that can be tracked.
Departing from the aforementioned tracking paradigm, in [8] a Gaussian probability
model was learned for short human motion sequences. In [17] dynamic programming was used to calculate the best global labeling according to the learned joint
probability density function of the position and velocity of body features. Still,
in these approaches, the joint locations, correspondences, or model initialization
must be provided by hand. In [1], the manifold of human body dynamics was modeled via a hidden Markov model and learned via entropic minimization. In all of
these approaches models were learned. Although the approach presented here can
be used to model dynamics, we argue that when general human motion dynamics
are intended to be learned, the amount of training data, model complexity, and
computational resources required are impractical. As a consequence, models with
large priors towards specific motions (e .g., walking) are generated. In this paper we
describe a non-linear supervised learning algorithm, the Specialized Maps Architecture (SMA), for recovering articulated body pose from single monocular images.
This approach avoids the need for initialization and tracking per se, and reduces
the above mentioned disadvantages.
2
Specialized Maps
There at least two key characteristics of the problem we are trying to solve which
make it different from other supervised learning problems. First, we have access to
the inverse map. We are trying to learn unknown probabilistic maps from inputs to
outputs space, but we have access to the map (in general probabilistic) from outputs
to inputs. In our pose estimation problem, it is easy to see how we can artificially,
using computer graphics (CG), produce some visual features (e.g., body silhouettes)
given joint positions 1 . Second, it is one-to-many: one input can be associated with
more than one output. Features obtained from silhouettes (and many other visual
features) are ambiguous. Consider an occluded arm, or the reflective ambiguity
generated by symmetric poses. This last observation precludes the use of standard
algorithms for supervised learning that fit a single mapping function to the data.
Given input and output spaces ~c and ~t, and the inverse function ( : ~t -+ ~c, we
describe a solution for these supervised learning problems. Our approach consists
in generating a series of m functions ?k : ~c -+ ~t. Each of these functions is
specialized to map only certain inputs (for a specialized sub-domain) better than
others. For example, each sub-domain can be a region of the input space. However,
the specialized sub-domain of ?k can be more general than just a connected region
in the input space.
Several other learning models use a similar concept of fitting surfaces to the observed
data by splitting the input space into several regions and approximating simpler
functions in these regions (e.g., [11,7, 6]). However, in these approaches, the inverse
map is not incorporated in the estimation algorithm because it is not considered
in the problem definition and the forward model is usually more complex, making
inference and learning more difficult.
The key algorithmic problems are that of estimating the specialized domains and
functions in an optimal way (taking into account the form of the specialized functions), and using the knowledge of the inverse function to formulate efficient inferIThus, ( is a computer graphics rendering, in general called forward kinematics
ence and learning algorithms. We propose to determine the specialized domains
and functions using an approximate EM algorithm and to perform inference using,
in an alternating fashion, the conditional independence assumptions specified by
the forward and inverse models. Fig. l(a) illustrates a learned forward model.
Figure 1: SMA diagram illustrating (a) an already learned SMA model with m specialized
functions mapping subsets of the training data, each subset is drawn with a different color
(at initializations, coloring is random) and (b) the mean-output inference process in which a
given observation is mapped by all the specialized functions , and then a feedback matching
step, using (, is performed to choose the best of the m estimates.
3
Probabilistic Model
Let the training sets of output-input observations be \)! = {1jI1, ... , 1jIN } , and Y =
{Vl , ... ,VN} respectively. We will use Z i = (1jIi,Vi) to define the given output-input
training pair, and Z = {ZI ' ... , ZN } as our observed training set.
We introduce the unobserved random variable y = (Yl , ... , Yn). In our model any Yi
has domain the discrete set C = {l, ... , M} oflabels for the specialized functions , and
can be thought as the function number used to map data point i; thus M is the number of specialized mapping functions. Our model uses parameters 8 = (8 1 , ... , 8M , A) ,
8k represents the parameters of the mapping function k; A = (AI"", AM), where
Ak represents P(Yi = kI8): the prior probability that mapping function with label
i will be used to map an unknown point. As an example, P(Yi lz i, 8) represents the
probability that function number Yi generated data point number i.
Using Bayes' rule and assuming independence of observations given 8, we have the
log-probability of our data given the modellogp(ZI8), which we want to maximize:
argm;x 2:)og LP(1jIi lvi, Yi = k,8)P(Yi = kI8)p(Vi ),
i
(1)
k
where we used the independence assumption p(vI8) = p(v). This is also equivalent
to maximizing the conditional likelihood of the model.
Because of the log-sum encountered, this problem is intractable in general. However, there exist practical approximate optimization procedures, one of them is
Expectation Maximization (EM) [3,4, 12].
3.1
Learning
The EM algorithm is well known, therefore here we only provide the derivations
specific to SMA's. The E-step consists of finding P(y = klz, 8) = P(y). Note that
the variables Yi are assumed independent (given Z i)' Thus, factorizing P(y):
p(y) =
II P(t)(Yi) = II[(AYiP(1/Jilvi,Yi,B))/(2:AkP(1/Jilvi,Yi = k,B))]
(2)
kEC
However, p( 1/Ji lVi, Yi = k, B) is still undefined. For the implementation described in
this paper we use N(1/Ji; ?k(Vi,B k ), ~k)' where Bk are the parameters of the k-th
specialized function, and ~k the error covariance of the specialized function k . One
way to interpret this choice is to think that the error cost in estimating 1/J once
we know the specialized function to use, is a Gaussian distribution with mean the
output of the specialized function and some covariance which is map dependent.
This also led to tractable further derivations. Other choices were given in [16].
The M-step consists of finding B(t) = argmaxoEj>(t) [logp(Z,y IB)]. In our case we
can show that this is equivalent to finding:
argmJn
2: 2: P(t)(Yi = k)(1/Ji i
?k(Vi, Bk))T~kl(Zi - ?k(Zi,B k ))?
(3)
k
This gives the following update rules for Ak and ~k (where Lagrange multipliers
were used to incorporate the constraint that the sum of the Ak'S is 1.
-n1 2:. P(Yi = klzi' B)
(4)
In keeping the formulation general, we have not defined the form of the specialized
functions ?k. Whether or not we can find a closed form solution for the update of
Bk depends on the form of ?k. For example if ?k is a non-linear function, we may
have to use iterative optimization to find Bit). In case ?k yield a quadratic form,
then a closed form update exists. However, in general we have:
(6)
In our experiments, ?k is a I-hidden layer perceptron. Thus, the M-step is an
approximate, iterative optimization procedure.
4
Inference
Once learning is accomplished, each specialized function maps (with different levels
of accuracy) the input space. We can formally state the inference process as that
of maximum-a-posteriori (MAP) estimation where we are interested in finding the
most likely output h given an input configuration x:
h* = argmaxp(hlx) = argmax ' " p(hly, x)P(y),
h
h
~
(7)
Y
Any further treatment depends on the properties of the probability distributions
involved. If p(hlx, y) = N(h ; ?y(x) , ~y), the MAP estimate involves finding the
maximum in a mixture of Gaussians. However, no closed form solution exists and
moreover, we have not incorporated the potentially useful knowledge of the inverse
function C.
4.1
MAP by Using the Inverse Function (
The access to a forward kinematics function ( (called here the inverse function)
allows to formulate a different inference algorithm. We are again interested in
finding an optimal h* given an input x (e.g. , an optimal body pose given features
taken from an image). This can be formulated as:
(8)
h* = arg maxp(hlx) = argmaxp(xlh) "p(hly, x)P(y) ,
h
~
h
y
simply by Bayes' rule, and marginalizing over all variables except h. Note that we
have made the distribution p(xlh) appear in the solution. This is important because
we can know use our knowledge of ( to define this distribution. This solution is
completely general within our architecture, we did not make any assumptions on
the form of the distributions or algorithms used.
5
Approximate Inference using (
Let us assume that we can approximate Lyp(hly, x)P(y) by a set of samples generated according to p(hly,x)P(y) and a kernel function K(h,hs). Denote the set
of samples HSpl = {hs}s=l...s. An approximate to Lyp(hly,x)P(y) is formally
built by ~ L;=l K(h , h s ), with the normalizing condition
any given h s .
J K(h , hs)dh
= 1 for
We will consider two simple forms of K. If K(h, h s ) = J(h - h s ), we have:
argmaxhP(xlh) L;=l J(h - h s).
h=
After some simple manipulations, this can be reduced to the following equivalent
discrete optimization problem whose goal is to find the most likely sample s*:
(9)
where the last equivalence used the assumption p(xlh) = N(x; ((h), ~d.
A
S
If K(h, h s) = N(h ; hs , ~Spl)' we have: h = argmaxhP(xlh) L S =l N(h ; hs , ~Spl).
This case is hard to use in practice, because contrary to the case above (Eq. 9) , in
general, there is no guarantee that the optimal h is among the samples.
5.1
A Deterministic Approximation based on the Functions Mean
Output
The structure of the inference in SMA, and the choice of probabilities p(hlx, y)
allows us to construct a newer approximation that is considerably less expensive to
compute, and it is deterministic. Intuitively they idea consists of asking each of the
specialized functions ?k what their most likely estimate for h is, given the observed
input x. The opinions of each of these specialized functions are then evaluated
using our distribution p(xlh) similar to the above sampling method.
This can be justified by the observation that the probability of the mean is maximal
in a Gaussian distribution. Thus by considering the means ?k(X), we would be
considering the most likely output of each specialized function. Of course, in many
cases this approximation could be very far from the best solution, for example when
the uncertainty in the function estimate is relatively high relative to the difference
between means.
We use Fig. l(b) to illustrate the mean-output (MO) approximate inference process.
When generating an estimate of body pose, denoted h, given an input x (the gray
point with a dark contour in the lower plane), the SMA generates a series of output
hypotheses tl q, = {h!h obtained using hk = (/Jk(x), with k E C (illustrated by each
of the points pointed to by the arrows).
Given the set tlq" the most accurate hypothesis under the mean-output criteria is
the one that minimizes the function:
k*
(10)
where in the last equation we have assumed p(xlh) is Gaussian.
5.2
Bayesian Inference
Note that in many cases, there may not be any need to simply provide a point
estimate, in terms of a most likely output h. In fact we could instead use the whole
distribution found in the inference process. We can show that using the above
choices for K we can respectively obtain.
1 s
p(hlx) =
S 2: N (x; ((hs ), ~d,
(11)
8= 1
s
p(hlx) = N(h; h8' ~Spz) 2:N(x; ((h) , ~d?
(12)
8=1
6
Experiments
The described architecture was tested using a computer graphics rendering as our
( inverse function. The training data set consisted of approx. 7,000 frames of
human body poses obtained through motion capture. The output consisted of 20
2D marker positions (i. e., 3D markers projected to the image plane using a perspective model) but linearly encoded by 8 real values using Principal Component
Analysis (PCA). The input (visual features) consisted of 7 real-valued Hu moments
computed on synthetically generated silhouettes of the articulated figure. For training/testing we generated 120,000 data points: our 3D poses from motion capture
were projected to 16 views along the view-sphere equator. We took 8,000 for training and the rest for testing. The only free parameter in this test, related to the
given SMA, was the number of specialized functions used; this was set to 15. For
this, several model selection approaches could be used instead. Due to space limitations, in this paper we show results using the mean-output inference algorithm only,
readers are referred to http://cs-people.bu.edu/rrosales/SMABodyInference where
inference using multiple samples is shown.
Fig. 2(left) shows the reconstruction obtained in several single images coming from
three different artificial sequences. The agreement between reconstruction and observation is easy to perceive for all sequences. Note that for self-occluding configurations, reconstruction is harder, but still the estimate is close to ground-truth. No
human intervention nor pose initialization was required. For quantitative results,
Fig. 2(right) shows the average marker error and variance per body orientation in
percentage of body height. Note that the error is bigger for orientations closer
to a and 7r radians. This intuitively agrees with the notion that at those angles
(side-views) , there is less visibility of the body parts. We consider this performance
promising, given the complexity of the task and the simplicity of the approach. By
choosing poses at random from training set, the RMSE was 17% of body height. In
related work, quantitative performance have been usually ignored, in part due to
the lack of ground-truth and standard evaluation data sets.
Penormance regarding cameraviewpoinl (16 101al)
2.9 ,-----~~--,-:.----'.----.--:----.;...-_---,----,
2.75
14
16
Figure 2: Left: Example reconstruction of several test sequences with CG-generated
silhouettes. Each set consists of input images and reconstruction (every 5th frame). Right:
Marker root-mean-square-error and variance per camera viewpoint (every 27r/32 rads.).
Units are percentage of body height. Approx. 110,000 test poses were used.
6.1
Experiments using Real Visual Cues
Fig. 3 shows examples of system performance with real segmented visual data,
obtained from observing a human subject. Reconstruction for several relatively
complex sequences are shown. Note that even though the characteristics of the
segmented body differ from the ones used for training, good performance is still
achieved. Most reconstructions are visually close to what can be thought as the
right pose reconstruction. Body orientation is also generally accurate.
7
Conclusion
In this paper, we have proposed the Specialized Mappings Architecture (SMA) . A
learning algorithm was developed for this architecture using ideas from ML estimation and latent variable models. Inference was based on the possibility of alternatively use different sets of conditional independence assumptions specified by the
forward and inverse models. The incorporation of the inverse function in the model
allows for simpler forward models. For example the inverse function is an architectural alternative to the gating networks of Mixture of Experts [11]. SMA advantages
for body pose estimation include: no iterative methods for inference are used, the
Figure 3: Reconstruction obtained from observing a human subject (every 10th frame).
algorithm for inference runs in constant time and scales only linearly O(M) with
respect to the number of specialized functions M; manual initialization is not required; compared to approaches that learn dynamical models, the requirements for
data are much smaller, and also large priors to specific motions are prevented thus
improving generalization capabilities.
References
[1] M. Brand. Shadow puppetry. In ICCV, 1999.
[2] C. Bregler. Tracking people with twists and exponential maps. In CVPR, 1998.
[3] 1. Csiszar and G. Thsnady. Information geometry and alternating minimization procedures. Statistics and Decisions, 1:205- 237, 1984.
[4] A. Dempster, N. Laird, and D. Rubin. Maximum likelihood estimation from incomplete data. Journal of the Royal Statistical Society (B), 39(1), 1977.
[5] J. Deutscher, A. Blake, and 1. Reid. Articulated body motion capture by annealed
particle filtering. In CVPR, 2000.
[6] J.H. Friedman. Multivatiate adaptive regression splines. The Annals of Statistics,
19,1-141 , 1991.
[7] G. Hinton, B. Sallans, and Z. Ghahramani. A hierarchical community of experts.
Learning in Graphical Models, M. Jordan (editor) , 1998.
[8] N. Howe, M. Leventon, and B. Freeman. Bayesian reconstruction of 3d human motion
from single-camera video. In NIPS-1 2, 2000.
[9] M. Isard and A. Blake. Contour tracking by stochastic propagation of conditional
density. In ECCV, 1996.
[10] G. Johansson. Visual perception of biological motion and a model for its analysis.
P erception and Psychophysics, 14(2): 210-211, 1973.
[11] M. 1. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm.
N eural Computation, 6, 181-214, 1994.
[12] R. Neal and G. Hinton. A view of the em algorithm that justifies incremental , sparse,
and other variants. Learning in Graphical Models, M. Jordan (editor) , 1998.
[13] Dirk Ormoneit , Hedvig Sidenbladh, Michael J . Black, and Trevor Hastie. Learning
and tracking cyclic human motion. In NIPS-1 3, 200l.
[14] Vladimir Pavlovic, James M. Rehg, and John MacCormick. Learning switching linear
models of human motion. In NIPS-13, 200l.
[15] J. M. Regh and T. Kanade. Model-based tracking of self-occluding articulated objects.
In ICC V, 1995.
[16] R. Rosales and S. Sclaroff. Specialized mappings and the estimation of body pose
from a single image. In IEEE Human Motion Workshop , 2000.
[17] Y. Song, Xiaoling Feng, and P. Perona. Towards detection of human motion. In
CVPR, 2000.
| 2019 |@word h:6 illustrating:1 middle:1 johansson:1 hu:1 covariance:2 jacob:1 thereby:1 harder:1 moment:1 configuration:2 series:2 cyclic:1 must:1 john:1 subsequent:1 numerical:1 visibility:1 designed:1 update:3 cue:1 isard:1 plane:2 short:1 argm:1 location:2 simpler:2 height:3 along:1 consists:6 fitting:1 introduce:1 nor:1 freeman:1 considering:2 provided:1 estimating:2 underlying:1 moreover:1 what:2 substantially:1 minimizes:1 developed:1 unobserved:1 finding:6 impractical:1 guarantee:1 quantitative:2 every:3 unit:1 intervention:1 yn:1 appear:1 reid:1 tends:1 limit:1 consequence:1 switching:1 puppetry:1 ak:3 might:1 black:1 initialization:6 equivalence:1 practical:1 camera:2 testing:2 practice:1 procedure:3 thought:2 matching:1 onto:1 cannot:1 selection:1 close:2 h8:1 optimize:1 equivalent:3 map:17 deterministic:2 maximizing:1 annealed:1 primitive:1 resolution:1 formulate:2 simplicity:1 splitting:1 perceive:1 rule:3 rehg:1 notion:1 annals:1 programming:1 us:1 hypothesis:2 agreement:1 velocity:1 expensive:1 jk:1 walking:2 observed:3 capture:3 calculate:1 region:4 connected:1 mentioned:1 dempster:1 complexity:2 occluded:1 dynamic:5 completely:1 easily:1 joint:3 derivation:2 articulated:5 describe:2 artificial:1 labeling:1 choosing:1 whose:1 encoded:1 posed:1 solve:1 valued:1 cvpr:3 precludes:1 maxp:1 statistic:2 think:1 laird:1 sequence:7 advantage:1 took:1 propose:1 reconstruction:10 maximal:1 coming:1 everyday:1 requirement:1 produce:1 generating:2 argmaxp:2 incremental:1 object:1 illustrate:1 pose:19 eq:1 strong:1 recovering:1 c:3 involves:1 shadow:1 rosales:2 differ:1 stochastic:1 human:17 viewing:1 opinion:1 generalization:1 biological:1 bregler:1 considered:1 ground:2 blake:2 visually:1 mapping:11 algorithmic:2 mo:1 sma:11 entropic:1 estimation:9 label:1 sensitive:1 agrees:1 minimization:2 gaussian:4 og:1 likelihood:2 hk:1 cg:2 am:1 posteriori:1 inference:17 dependent:1 vl:1 typically:1 hidden:2 perona:1 interested:2 arg:1 aforementioned:1 among:1 orientation:3 denoted:1 psychophysics:1 once:2 construct:1 sampling:1 represents:3 others:1 spline:1 hedvig:1 pavlovic:1 employ:2 intended:1 argmax:1 geometry:1 n1:1 friedman:1 detection:1 possibility:1 evaluation:1 weakness:1 mixture:3 undefined:1 csiszar:1 accurate:2 closer:1 incomplete:1 fitted:1 asking:1 ence:1 disadvantage:1 leventon:1 zn:1 logp:1 maximization:1 cost:2 subset:2 comprised:1 graphic:3 synthetic:1 considerably:1 density:2 bu:3 probabilistic:3 yl:1 michael:1 again:1 ambiguity:1 choose:1 xlh:7 expert:3 account:1 jii:2 vi:4 depends:2 performed:1 view:4 root:1 closed:3 observing:2 recover:1 bayes:2 capability:1 hly:5 rmse:1 square:1 accuracy:1 variance:2 characteristic:2 yield:1 ji1:1 bayesian:2 manual:2 trevor:1 definition:1 involved:1 james:1 associated:1 radian:1 treatment:1 knowledge:5 color:1 coloring:1 supervised:5 formulation:1 evaluated:2 though:1 generality:1 just:1 hand:1 hlx:6 nonlinear:1 marker:4 lack:1 propagation:1 gray:1 concept:1 multiplier:1 consisted:3 alternating:3 symmetric:1 neal:1 illustrated:1 self:2 ambiguous:1 criterion:1 argmjn:1 trying:2 motion:17 image:10 specialized:30 ji:3 tracked:1 twist:1 interpret:1 ai:1 approx:2 pointed:1 particle:1 access:3 surface:1 perspective:1 manipulation:1 certain:2 life:1 yi:13 accomplished:1 determine:1 paradigm:1 maximize:1 ii:2 multiple:1 infer:1 reduces:1 segmented:2 believed:1 sphere:1 prevented:1 bigger:1 variant:1 regression:1 vision:2 expectation:1 kernel:1 achieved:1 equator:1 justified:1 want:1 diagram:1 howe:1 rest:1 subject:2 contrary:1 jordan:3 reflective:1 synthetically:1 easy:2 rendering:2 independence:5 fit:1 zi:3 architecture:6 hastie:1 idea:2 regarding:1 whether:1 pca:1 song:1 ignored:1 generally:2 useful:1 se:1 amount:1 dark:1 reduced:1 http:1 exist:1 percentage:2 per:3 discrete:2 key:3 drawn:1 sum:2 run:1 inverse:14 akp:1 uncertainty:1 angle:1 reader:1 spl:2 architectural:1 vn:1 sallans:1 decision:1 bit:1 layer:1 correspondence:1 quadratic:1 encountered:1 placement:1 constraint:1 incorporation:1 generates:1 erception:1 performing:1 relatively:3 deutscher:1 department:2 according:2 smaller:1 em:6 newer:1 lp:1 making:1 intuitively:2 iccv:1 taken:1 monocular:2 resource:1 equation:1 kinematics:2 know:2 tractable:1 gaussians:1 hierarchical:2 romer:1 alternative:1 include:1 graphical:2 ghahramani:1 approximating:1 society:1 lyp:2 feng:1 already:1 sidenbladh:1 maccormick:1 mapped:1 manifold:1 argue:1 assuming:2 modeled:1 vladimir:1 difficult:2 potentially:1 implementation:1 unknown:2 perform:1 observation:6 markov:1 jin:1 hinton:2 incorporated:2 dirk:1 frame:6 community:1 drift:1 bk:3 pair:1 required:4 specified:2 extensive:1 kl:1 rad:1 learned:7 nip:3 address:1 usually:3 dynamical:1 perception:1 built:1 royal:1 video:3 ormoneit:1 arm:1 stan:1 faced:1 prior:6 geometric:1 icc:1 marginalizing:1 relative:1 limitation:1 filtering:1 rubin:1 viewpoint:1 editor:2 eccv:1 course:1 last:3 keeping:1 free:1 side:1 perceptron:1 taking:1 sparse:1 departing:1 feedback:1 world:1 avoids:1 contour:2 forward:8 made:1 adaptive:1 projected:3 lz:1 far:1 approximate:7 silhouette:4 ml:1 global:1 assumed:2 alternatively:1 factorizing:1 iterative:3 latent:1 promising:1 learn:4 kanade:1 improving:1 complex:3 artificially:1 domain:8 lvi:2 did:1 linearly:2 arrow:1 whole:1 body:26 eural:1 fig:5 referred:1 tl:1 fashion:1 sub:3 position:3 exponential:1 ib:1 specific:5 gating:1 normalizing:1 intractable:1 exists:2 workshop:1 illustrates:1 justifies:1 sclaroff:3 boston:4 led:1 photograph:1 simply:2 likely:5 visual:6 lagrange:1 tracking:10 truth:2 dh:1 ma:2 conditional:5 goal:1 formulated:1 careful:1 towards:2 hard:1 except:1 principal:1 called:2 brand:1 occluding:2 formally:2 people:2 incorporate:1 tested:1 |
1,118 | 202 | 490
Bell
Learning in higher-order' artificial dendritic trees'
Tony Bell
Artificial Intelligence Laboratory
Vrije Universiteit Brussel
Pleinlaan 2, B-1050 Brussels, BELGIUM
(tony@arti.vub.ac.be)
ABSTRACT
If neurons sum up their inputs in a non-linear way, as some simula-
tions suggest, how is this distributed fine-grained non-linearity exploited during learning? How are all the small sigmoids in synapse,
spine and dendritic tree lined up in the right areas of their respective
input spaces? In this report, I show how an abstract atemporal highly
nested tree structure with a quadratic transfer function associated
with each branchpoint, can self organise using only a single global
reinforcement scalar, to perform binary classification tasks. The procedure works well, solving the 6-multiplexer and a difficult phoneme
classification task as well as back-propagation does, and faster.
Furthermore, it does not calculate an error gradient, but uses a statistical scheme to build moving models of the reinforcement signal.
1. INTRODUCTION
The computational territory between the linearly summing McCulloch-Pitts neuron and
the non-linear differential equations of Hodgkin & Huxley is relatively sparsely populated. Connectionists use variants of the former and computational neuroscientists
struggle with the exploding parameter spaces provided by the latter. However, evidence from biophysical simulations suggests that the voltage transfer properties of
synapses, spines and dendritic membranes involve many detailed non-linear interactions, not just a squashing function at the cell body. Real neurons may indeed be
higher-order nets.
For the computationally-minded, higher order interactions means, first of all, quadratic
terms. This contribution presents a simple learning principle for a binary tree with a
logistic/quadratic transfer function at each node. These functions, though highly
nested, are shown to be capable of changing their shape in concert. The resulting tree
structure receives inputs at its leaves, and outputs an estimate of the probability that
the input pattern is a member of one of two classes at the top.
Learning in Higher-Order' Artificial Dendritic Trees'
A number of other schemes exist for learning in higher-order neural nets. Sigma-Pi
units, higher-order threshold logic units (Giles & Maxwell, 87) and product units (Durbin & Rumelhart, 89) are all examples of units which learn coefficients of non-linear
functions. Product unit networks, like Radial Basis Function nets, consist of a layer of
non-linear transformations, followed by a normal Perceptron-style layer. The scheme
presented here has more in common with the work reviewed in Barron (88) (see also
Tenorio 90) on polynomial networks in that it uses low order polynomials in a tree of
low degree. The differences lie in a global rather than layer-by-Iayer learning scheme,
and a transfer function derived from a gaussian discriminant function.
2. THE ARTIFICIAL DENDRITIC TREE (ADT)
The network architecture in Figure I(a) is that of a binary tree which propagates real
number values from its leaf nodes (or inputs) to its root node which is the output. In
this simple formulation, the tree is construed as a binary classifier. The output node
signals a number between 1 and 0 which represents the probability that the pattern
presented to the tree was a member of the positive class of patterns or the negative
class. Because the input patterns may have extremely high dimension and the tree is,
at least initially, constrained to be binary, the depth of the tree may be significant, at
least more than one might like to back-propagate through. A transfer function is associated with each 'hidden' node of the tree and the output node. This will hereafter be
referred to as a Z{unction, for the simple reason that it takes in two variables X and
Y, and outputs Z. A cascade of Z-functions performs the computation of the tree and
the learning procedure consists of changing these functions. The tree is referred to as
an Artificial Dendritic Tree or ADT with the same degree of licence that one may talk
of Artificial Neural Networks, or ANNs.
(a)
z (x ,y) 1.0
z (x)
(b)
I A
(c)
x
lnput nodes
(d)
X
Y
Figure 1: (a) an Artificial Dendritic Tree, (b) a ID Z-node (c) a 2D Z-node (d)
A ID Z-function constructed from2 gaussians (e) approximating a step function
2.1. THE TRANSFER FUNCTION
The idea behind the Z-function is to allow the two variables arriving at a node to
interact locally in a non-linear way which contributes to the global computation of the
tree. The transfer function is derived from statistical considerations. To simplify, consider the one-dimensional case of a variable X travelling on a wire as in Figure 1(b).
A statistical estimation procedure could observe the distribution of values of X when
the global pattern was positive or negative and derive a decision rule from these. In
Figure I(d), the two density functions f+(x) and f-(x) are plotted. Where they meet,
the local computation must answer that, based on its information, the global pattern is
positively classified with probability 0.5. Assuming that there are equal numbers of
positive and negative patterns (ie: that the a priori probability of positive is 0.5), it is
easy to see that the conditional probability of being in the positive class given our
value for X, is given by equation (1).
491
492
Bell
z (x)
= P [class=+ve Ix] =
[+ex)
[+(x)+[-(x)
(1)
This can be also derived from Bayesian reasoning (Therrien, 89). The fonn of z (x) is
shown with the thick line in Figure l(d) for the given [+(x) and [-(x). If [+(x) and
[-ex) can be usefully approximated by normal (gaussian) curves as plotted above,
then (1) translates into (2):
z ex) =
1.
1+e -mp"
t
,input =
~-(x) - ~+(x) + In[ a:]
a
(2)
This can be obtained by substituting equation (4) overleaf into (1) using the definitions
of a and ~ given. The exact form a and ~ take depends on the number of variables
input. The first striking thing is that the form of (2) is exactly that of the backpropagation logistic function. The second is that input is a polynomial quadratic
expression. For Z-functions with 2 inputs (x ,y) using formulas (4.2) it takes the fonn:
w lX2+W2Y2+w~+w 4X+wsY+w6
(3)
The w' s can be thought of as weights just as in backprop, defining a 6D space of
transfer functions. However optimising the w's directly through gradient descent may
not be the best idea (though this is what Tenorio does), since for any error function E,
law 4 = x law 1 =Y law 3. That is, the axes of the optimisation are not independent of each other. There are, however, two sets of 5 independent parameters which
the w's in (3) are actually composed from if we calculate input from (4.2). These are
Jl:, cr;,
cr; and r+, denoting the means, standard deviations and correlation
coefficient defining the two-dimensional distribution of (x ,y) values which should be
positively classified. The other 5 variables define the negative distribution.
Thus 2 Gaussians (hereafter referred to as the positive and negative models) define a
quadratic transfer function (called the Z{unction) which can be interpreted as expressing conditional probability of positive class membership. The shape of these functions
can be altered by changing the statistical parameters defining the distributions which
undedy them. In Figure l(d), a 1-dimensional Z-function is seen to be sigmoidal
though it need not be monotonic at all. Figure 2(b)-(h) shows a selection of 2D Zfunctions. In general the Z-function divides its N-dimensional input space with a N-1
dimensional hypersurface. In 2D, this will be an ellipse, a parabola, a hyperbola or
some combination of the three. Although the dividing surface is quadratic, the Zfunction is still a logistic or squashing function. The exponent input is actually
equivalent to the log likelihood ratio or In(j+(x)/j-(x?. commonly used in statistics.
In this work, 2-dimensional gaussians are used to generate Z-functions. There are
compelling reasons for this. One dimensional Z-functions are of little use since they
do not reduce information. Z-functions of dimension higher than 1 perform optimal
class-based information reduction by propagating conditional probabilities of class
membership. But 2D Z-functions using 2D gaussians are of particular interest because
they include in their function space all boolean functions of two variables (or at least
analogue versions of these functions). For example the gaussians which would come to
represent the positive and negative exemplar patterns for XOR are drawn as ellipses in
Figure 2(a). They have equal means and variances but the negative exemplar patterns
are correlated while the positive ones are anti-correlated. These models automatically
give rise to the XOR surface in Figure 2(b) if put through equation (2). An interesting
aE
aE
11;,
aE
Learning in Higher-Order' Artificial Dendritic Trees'
observation is that a problem of Nth order (XOR is 2nd order, 3-parity is 3rd order
etc) can be solved by a polynomial of degree N (Figure 2d). Since 2nd degree polynomials like (3) are used in our system, there is one step up in power from 1st degree
systems like the Perceptron. Thus 3-parity is to the Z-function unit what XOR is to the
Perceptron (in this case not quadratically separable).
A GAUSSIAN IS:
f (x)=.le-IJ(%)
a
(4)
in one dimension:
a=(21t) 1120'%
(4.1.1)
~(x )
(4.1.2)
(x -Jl% )2
20'x 2
in two dimensions: a=21tO'x O'y(l-r 2)112
1
~(x ,y)= 2(l-r2)
in n dimensions:
(4.2.1)
[ (X-J,1x)2
0'% 2
+
(y -~ )2 2r (x -J,1x )(y -~ )
0'/
O'x O'y
a=(21t)" /2 IK 11/2
(4.n.l)
~<!)= ~ (!-mlK- 1<!-m)
(4.n.2)
Jl%=E [x]
is the expected value or mean of x
O';:E [x 2]-Jl% 2
is the variance of x
r
E[xy]~%Jly
1 (4.2.2)
is the correlation coefficiem of a bivariate gaussian
m=E [!]
is the mean vector of a multivariate gaussian
K=E [<!-m)<!-m)T] is the covariance matrix of a multivariate? gaussian
with IK 1its determinant
(j)
(i)
(k)
/l\.
M
Figure 2: (a) two anti-(;orrelated gaussians seen from above (b) the resulting Zfunction (c)-(h) Some other 20 Z-functions. (i) 3-parity in a cube cannot be
solved by a 30 Z-function (j) but yields to a cascade of 20 ones (k).
2.2. THE LEARNING PROCEDURE
If gaussians are used to model the distribution of inputs x which give positive and
negative classification errors, rather than just the distribution of positively and negatively classified x, then it is possible to formulate an incremental learning procedure
for training Z-functions. This procedure enables the system to deal with data which is
not gaussianly distributed.
493
494
Bell
2.2.1. Without hidden units: learning a step function.
A simple example illustrates this principle. Consider a network consisting entirely of a
I-dimensional Z-function. as in Figure 1(b). The input is a real number from 0 to 1
and the output is to be a step function, such that 0.5-1.0 is classed positively (output
1.0) and 0.0-0.5 should output 0.0. The 4 parameters of the Z-function (Jl+,Jl-,cr+,crl
are initialised randomly and example patterns are presented to the 'tree'. On each
presentation t, the error 0 in the response is calculated by 0, ~ d, -0" the desired
minus the actual output at time t, and 2 of the parameters are altered. If the error is
positive, the positive model is altered, otherwise the negative model is altered. Changing a model consists of 'sliding' the estimates of the appropriate first and second
moments (E[x] and E [x 2]) according to a 'moving-average' scheme:
E [x], ~ to,x,+(1-to,)E [x ]'-1
(5.1)
(5.2)
where t is a plasticity or learning rate, x, is the value input and E [x ]'-1 was the previous estimate of the mean value of x for the appropriate gaussian. This rule means that
at any moment, the parameters determining the positive and negative models are
weighted averages of recent inputs which have generated errors. The influence which a
particular input has had decays over time. This algorithm was run with ?=0.1. After
100 random numbers had been presented, with error signals from the step-function
changing the models, the models come to well represent the distribution of positive
and negative inputs. At this stage the models and their associated Z-function are those
shown in Figure l(d). But now, most of the error reinforcement will be coming from
a small region around 0.5, which means that since the gaussians are modelling the
errors, they will be drawn towards the centre and become narrower. This has the
effect, Figure l(e), of increasing the gain of the sigmoidal Z-function. In the limit, it
will converge to a perfect step function as the gaussians become infinitesimally
separated delta functions. This initial demonstration shows the automatic gain adjustment property of the Z-function.
2.2.2. With hidden units: the 6-multiplexer.
The first example showed how a 1D Z-function can minimise error by modelling it.
This example shows how a cascade of 2-dimensional Z-functions can co-operate to
solve a 3rd order problem. A 6-multiplexer circuit receives as input 6 bits, 4 of which
are data bits and 2 are address bits. If the address bits are 00, it must output the contents of the first data bit, if 01, the second, 10 the third and 11 the fourth. There are
64 different input patterns. Choosing an tree architecture is a difficult problem in general, but the first step is to choose one which we know can solve the problem. This is
illustrated in Figure 3(a). This is an architecture for which there exists a solution using
binary Boolean functions.
The tree's solution was arrived at as follows: each node was initialised with 10 random values: E[x]' E[y], E[x 2], E[y2] and E[xy] for each of its positive and negative
models. The learning rate t was set to 0.02 and input patterns were generated and
propagated up to the top node, where an error measurement was made. The error was
then broadcast globally to all nodes, each one, in effect, being told to respond more
positively (or negatively) should the same circumstances arise again, and adjusting
their Z-functions in the same way as equations (5). This time, however, 5 parameters
Learning in Higher-Order' Artificial Dendritic Trees'
were adjusted per node per presentation. instead of 2. Again. which model (positive
or negative) is adjusted depends on the sign of the error at the top of the tree.
The tree learns after about 200 random bit patterns are presented (7 seconds on a
Symbolics). After 300 presentations (the state depicted in Figure 3a), the mean
squared error is falling steadily to zero. An adequate back-propagation network takes
6000 presentations to converge on a solution. The solution achieved is a rather messy
combination of half-hearted XORs and NXORs, and ambiguous AND/ORs. The problem was tried with different trees. In general any tree of sufficient richness can solve
the problem though larger trees take longer. Trees for which no nice solutions exist.
ie: those with fewer than 6 well-chosen inputs from the address bits can sometimes
still perform rather well. A tree with straight convergence. only one contact per
address bit, can still quickly approach 80% performance, but further training is destructive. Figure 3(b) shows a tree trained to output 1 if half or more of its 8 inputs
were on.
Al rr===---n
7 ...
(a)
8
"~_--'I
(b)
Figure 3: Solving the 6-multiplexer (a) and the 8-majority predicate (b)
2.2.3. Phoneme classification.
A good question was if such a tree could perform well on a large problem, so a typical back-propagation application was attempted. Space does not permit a full account
here. but the details appear in Bell (89). The data came from 100 speakers speaking
the confusable E-set phonemes (B, D, E and V). This was the same data as that used
by Lang & Hinton (88). Four trees were built out of 192 input units and the trees
trained using a learning schedule of E falling from 0.01 to 0.001 over the course of 30
presentations of each of 668 training patterns. Generalisation to a test set was 88.5%,
0.5% worse than an equivalently simple backprop net A more sophisticated backprop
net, using time-delays and multiresolution training could reach 93% generalisation.
Thirty epochs with the trees took some 16 hours on a Sun 3-260 whereas the backprop
experiments were performed on a Convex supercomputer. The conclusion from these
experiments is that trees some 8 levels deep are capable of almost matching normal
back-propagation on a large classification task in a fraction of the training time.
Attempts to build time-symmetry into the trees have not so far been successful.
3. DISCUSSION
Even within the context of other connectionist leaming procedures, there is something
of an air of mystery about this one. The apparatus of gradient descent, either for individual units or for the whole tree is absent or at least hidden.
495
496
Bell
3.1. HOW DOES IT WORK?
It is necessary to reflect on the effect of modelling errors. Models of errors are an
attempt to push a node's outputs towards the edge of its parent's input square. Where
the model is perfect, it is simple for the node above to model the model by applying a
sigmoid, and so on to the top of the tree, where the error disappears. But the modelling is actually done in a totally distributed and collaborative way. The identification
of 1.0 with positive error (top output too small) means that Z-functions are more
likely to be monotonic towards (1,1) the further they are from the inputs.
Two standard problems are overcome in unusual ways. The first, credit assignment, is
solved because different Z-functions are able to model different errors, giving them
different roles. Although all nodes receive the same feedback, some changes to a
node's model will be swiftly undone when the new errors that result from them begin
to be broadcast. Other nodes can change freely either because they are not yet essential to the computation or because there exist alterations of their models tolerable to
the nodes above. The second problem is stability. In backprop, the way the error
diffuses through the net ensures that the upper weights are slaved to the lower ones
because the lower are changing more slowly. In this system, the upper nodes are
slaved to the lower ones because they are explicitly modelling their activities. Conversely, the lower nodes will never be allowed to change too quickly since the errors
generated by sluggish top nodes will throw them back into the behaviour the top
nodes expect For a low enough learning rate e, the solutions are stable.
Amongst the real problems with this system are the following. First, the credit assignment is not solved for units receiving the same input variables, making many normal
connectionist architectures impossible. Second, the system can only deal with 2
classes. Third, as with other algorithms, choice of architecture is a 'black art'.
3.2. BIOPHYSICS & REAL NEURONS
The name' Artificial Dendritic Tree' is perhaps overdoing it. The tree has no dynamic
properties, activation flows in only one direction, the branchpoints of the tree routinely
implement XOR and the 'cell' as a whole implements phoneme recognition (only a
small step from grandmothers). The title was kept because what drove the work was a
search for a computational explanation of how fine-grained local non-linearities of low
degree could combine in a learning process. Work in computational neuroscience, in
particular with compartmental models (Koch & Poggio 87; RaIl & Segev 88; Segev et
al 89, Shepherd & Brayton 87) have shown that it is likely that many non-linear
effects take place between synapse and soma. Synaptic transfer functions can be sigmoidal, spines with active channels may mutually excite each other (even implement
boolean computations) and inhibitory inputs can 'veto' firing in a highly non-linear
fashion (silent inhibition). The dendritic membrane itself is filled with active ion
channels, whose boosting or quenching properties depend in a complex way on the
intracellular voltage levels or Ca'Jn. concentration (itself dependent on voltage). Thus
we may be able to consider the membrane itself as a distributed processing system,
meaning that the synapses are no longer the privileged sites of learning which they
have tended to be since Hebb. Active channels can serve to implement threshold functions just as well at the dendritic branchpoints as at the soma, where they generate
spikes. There are many different kinds of ion channel (Yamada et aI, 89) with inhomogenous distributions over the dendritic tree. A neuron's DNA may generate a certain 'base set' of channel proteins that span a non-linear function space just as our
Learning in Higher-Order' Arti ficial Dendritic Trees'
parameters span the Z-function space. The properties of a part of dendritic membrane
could be seen as a point in channel space. Viewed this way. the neuron becomes one
large computer. When one considers the Purkinje cell of the cerebellum with 100.000
inputs, as many spines. a massive arborisation full of active channels, many of them
Ca-permeable or Ca-dependent. with spiking and plateau potentials occurring in the
dendritic tree. the notion that the cell may be implementing a 99.999 dimensional
hyperplane starts to recede. here is an extra motivation for considering the cell as a
complex computer. Algorithms such as back-propagation would require feedback circuits to send error. If the cell is the feedback unit, then reinforcement can occur as a
spike at the soma rein vades the dendritic tree. Thus nerves may not spike just for
axonal purposes. but also to penetrate the electrotonic length of the dendrites. This
was thought to be a component of Hebbian learning at the synapses, but it could be
the basis of more if the dendritic membrane computes.
4. Acknowledgements
To Kevin Lang for the speech data and to Rolf Pfeifer and Luc Steels for support.
Further credits in Bell (90). The author is funded by ESPRIT B.R.A. 3234.
5. References
Barron A & Barron R (88) Statistical Learning Networks: a unifying view, in Wegman
E (ed) Proc. 20th Symp. on Compo Science & Statistics [see also this volume]
Bell T (89) Artificial Dendritic Learning. in Almeida L. (ed) Proc. EURASIP
Workshop on Neural Networks. Lecture notes in Computer Science. SpringerVerlag. [also VUB AI-lab Memo 89-20].
Durbin R & Rumelhart D (89) Product Units: A Computationally Powerful and Biologically Plausible Extansion to Backpropagation Nets. Neural Computation J
Giles C.L. & Maxwell T (87) Learning. in variance and generalisation in high-order
neural networks. Applied Optics vol 26. no. 23
Koch C & Poggio T (87) Biophysics of Computational Systems: Neurons, synapses
and membranes. in G. Edelman et al (eds). Synaptic Function. John Wiley.
Lang K & Hinton G (88) The Development of the Time-Delay Neural Network Architecture for Speech Recognition. Tech Report CMU-CS-88-J52
RaIl W & Segev I (88) Excitable Dendritic Spine Clusters: non-linear synaptic processing. in R.Cotterill (ed) Computer Simulation in Brain Science. Camb.U.P.
Segev I. Fleshman J & Burke R. (89) Compartmental Models of Complex Neurons. in
Methods in Neuronal Modelling
Shepherd G & Brayton R (87) Logic operations are properties of computer simulated
interactions between excitable dendritic spines. Neuroscience. vol 21, no. 1 1987
Koch C & Segev I (eds) MIT press 1989
Tenorio M & Lee W (90) Self-Organizing Network for Optimal Supervised Learning,
IEEE Transactions in Neural Networks, 1990 [see also this volume]
Therrien C (89) Decision Estimation and Classification.
Yamada W, Koch C & Adams P (89) Multiple Channels and Calcium Dynamics, in
Methods in Neuronal Modelling Koch C & Segev I (eds) MIT press 1989.
497
| 202 |@word determinant:1 version:1 polynomial:5 nd:2 simulation:2 propagate:1 tried:1 covariance:1 arti:2 fonn:2 minus:1 mlk:1 moment:2 reduction:1 initial:1 hereafter:2 denoting:1 unction:2 lang:3 yet:1 activation:1 must:2 john:1 plasticity:1 shape:2 enables:1 concert:1 intelligence:1 leaf:2 half:2 fewer:1 ficial:1 yamada:2 compo:1 boosting:1 node:24 sigmoidal:3 constructed:1 differential:1 become:2 ik:2 edelman:1 consists:2 lx2:1 combine:1 symp:1 jly:1 expected:1 indeed:1 spine:6 brain:1 globally:1 automatically:1 little:1 actual:1 considering:1 increasing:1 totally:1 provided:1 begin:1 linearity:2 becomes:1 circuit:2 mcculloch:1 what:3 kind:1 interpreted:1 transformation:1 usefully:1 exactly:1 esprit:1 classifier:1 unit:13 appear:1 positive:17 local:2 struggle:1 limit:1 apparatus:1 id:2 permeable:1 meet:1 firing:1 might:1 black:1 suggests:1 conversely:1 co:1 thirty:1 implement:4 backpropagation:2 procedure:7 area:1 bell:8 cascade:3 thought:2 matching:1 undone:1 radial:1 suggest:1 protein:1 cannot:1 selection:1 put:1 context:1 influence:1 applying:1 impossible:1 equivalent:1 send:1 convex:1 formulate:1 penetrate:1 rule:2 stability:1 notion:1 drove:1 massive:1 exact:1 us:2 simula:1 rumelhart:2 approximated:1 recognition:2 parabola:1 sparsely:1 role:1 solved:4 calculate:2 region:1 ensures:1 richness:1 sun:1 messy:1 dynamic:2 trained:2 depend:1 solving:2 serve:1 negatively:2 basis:2 routinely:1 talk:1 separated:1 artificial:11 kevin:1 choosing:1 rein:1 adt:2 whose:1 larger:1 solve:3 plausible:1 compartmental:2 otherwise:1 statistic:2 itself:3 rr:1 biophysical:1 net:7 took:1 interaction:3 product:3 coming:1 lnput:1 organizing:1 multiresolution:1 convergence:1 parent:1 cluster:1 incremental:1 perfect:2 arborisation:1 adam:1 tions:1 derive:1 ac:1 propagating:1 exemplar:2 ij:1 ex:3 dividing:1 throw:1 brayton:2 c:1 come:2 direction:1 thick:1 implementing:1 backprop:5 require:1 behaviour:1 licence:1 dendritic:21 adjusted:2 burke:1 pleinlaan:1 around:1 credit:3 normal:4 koch:5 pitt:1 substituting:1 quenching:1 belgium:1 purpose:1 estimation:2 proc:2 title:1 minded:1 weighted:1 mit:2 gaussian:7 rather:4 cr:3 voltage:3 derived:3 ax:1 modelling:7 likelihood:1 tech:1 dependent:2 membership:2 initially:1 hidden:4 diffuses:1 classification:6 priori:1 exponent:1 development:1 constrained:1 art:1 cube:1 equal:2 never:1 optimising:1 represents:1 report:2 connectionist:2 simplify:1 randomly:1 composed:1 ve:1 individual:1 vrije:1 consisting:1 attempt:2 neuroscientist:1 interest:1 highly:3 cotterill:1 behind:1 edge:1 capable:2 necessary:1 xy:2 poggio:2 respective:1 tree:47 filled:1 divide:1 desired:1 plotted:2 confusable:1 vub:2 giles:2 compelling:1 boolean:3 purkinje:1 assignment:2 deviation:1 delay:2 predicate:1 successful:1 too:2 answer:1 swiftly:1 st:1 density:1 ie:2 therrien:2 told:1 receiving:1 lee:1 quickly:2 again:2 squared:1 reflect:1 choose:1 broadcast:2 slowly:1 worse:1 style:1 multiplexer:4 account:1 potential:1 alteration:1 coefficient:2 mp:1 explicitly:1 depends:2 performed:1 root:1 view:1 lab:1 universiteit:1 start:1 contribution:1 construed:1 square:1 air:1 collaborative:1 xor:5 phoneme:4 variance:3 yield:1 hyperbola:1 territory:1 bayesian:1 identification:1 straight:1 classified:3 anns:1 plateau:1 synapsis:4 reach:1 tended:1 ed:6 synaptic:3 definition:1 initialised:2 steadily:1 destructive:1 associated:3 propagated:1 gain:2 adjusting:1 schedule:1 sophisticated:1 actually:3 back:7 nerve:1 maxwell:2 higher:10 supervised:1 response:1 synapse:2 formulation:1 done:1 though:4 furthermore:1 just:6 stage:1 correlation:2 receives:2 propagation:5 logistic:3 perhaps:1 name:1 effect:4 y2:1 former:1 laboratory:1 illustrated:1 deal:2 cerebellum:1 during:1 self:2 ambiguous:1 speaker:1 arrived:1 performs:1 reasoning:1 meaning:1 consideration:1 common:1 sigmoid:1 spiking:1 camb:1 volume:2 jl:6 significant:1 expressing:1 measurement:1 ai:2 rd:2 automatic:1 populated:1 centre:1 had:2 funded:1 moving:2 stable:1 longer:2 surface:2 inhibition:1 etc:1 base:1 something:1 multivariate:2 recent:1 showed:1 certain:1 binary:6 came:1 exploited:1 seen:3 freely:1 converge:2 signal:3 exploding:1 sliding:1 full:2 multiple:1 hebbian:1 faster:1 ellipsis:1 biophysics:2 privileged:1 variant:1 ae:3 optimisation:1 circumstance:1 cmu:1 represent:2 sometimes:1 achieved:1 cell:6 ion:2 receive:1 whereas:1 fine:2 extra:1 operate:1 shepherd:2 thing:1 member:2 veto:1 flow:1 axonal:1 easy:1 enough:1 architecture:6 silent:1 reduce:1 idea:2 translates:1 minimise:1 absent:1 expression:1 speech:2 speaking:1 adequate:1 deep:1 electrotonic:1 detailed:1 involve:1 brussel:1 locally:1 dna:1 generate:3 exist:3 inhibitory:1 sign:1 delta:1 neuroscience:2 per:3 vol:2 four:1 soma:3 threshold:2 falling:2 drawn:2 changing:6 kept:1 fraction:1 sum:1 run:1 mystery:1 fourth:1 respond:1 hodgkin:1 striking:1 powerful:1 place:1 almost:1 decision:2 bit:8 entirely:1 layer:3 followed:1 quadratic:6 durbin:2 activity:1 occur:1 optic:1 huxley:1 segev:6 extremely:1 span:2 separable:1 infinitesimally:1 relatively:1 according:1 brussels:1 combination:2 membrane:6 making:1 biologically:1 inhomogenous:1 computationally:2 equation:5 mutually:1 lined:1 know:1 unusual:1 travelling:1 gaussians:9 operation:1 permit:1 observe:1 barron:3 appropriate:2 tolerable:1 supercomputer:1 jn:1 top:7 symbolics:1 tony:2 include:1 unifying:1 giving:1 build:2 ellipse:1 approximating:1 contact:1 question:1 spike:3 concentration:1 gradient:3 amongst:1 simulated:1 majority:1 considers:1 discriminant:1 reason:2 assuming:1 w6:1 length:1 ratio:1 demonstration:1 equivalently:1 difficult:2 classed:1 sigma:1 negative:13 rise:1 steel:1 memo:1 calcium:1 xors:1 perform:4 upper:2 neuron:8 wire:1 observation:1 descent:2 anti:2 wegman:1 defining:3 hinton:2 connectionists:1 quadratically:1 hour:1 address:4 able:2 pattern:14 grandmother:1 rolf:1 built:1 explanation:1 analogue:1 power:1 nth:1 scheme:5 altered:4 disappears:1 excitable:2 nice:1 epoch:1 acknowledgement:1 determining:1 law:3 expect:1 lecture:1 interesting:1 degree:6 sufficient:1 propagates:1 principle:2 pi:1 squashing:2 course:1 parity:3 arriving:1 organise:1 allow:1 perceptron:3 distributed:4 curve:1 dimension:5 depth:1 calculated:1 overcome:1 feedback:3 computes:1 author:1 commonly:1 reinforcement:4 made:1 far:1 hypersurface:1 transaction:1 logic:2 global:5 active:4 summing:1 excite:1 iayer:1 search:1 reviewed:1 channel:8 learn:1 transfer:10 ca:3 symmetry:1 contributes:1 dendrite:1 interact:1 complex:3 linearly:1 intracellular:1 whole:2 motivation:1 arise:1 allowed:1 body:1 positively:5 site:1 referred:3 neuronal:2 fashion:1 hebb:1 wiley:1 lie:1 rail:2 third:2 pfeifer:1 ix:1 grained:2 learns:1 formula:1 r2:1 decay:1 evidence:1 bivariate:1 consist:1 exists:1 essential:1 workshop:1 sluggish:1 sigmoids:1 illustrates:1 push:1 occurring:1 depicted:1 likely:2 tenorio:3 adjustment:1 scalar:1 monotonic:2 slaved:2 nested:2 conditional:3 viewed:1 presentation:5 narrower:1 towards:3 leaming:1 luc:1 crl:1 content:1 change:3 eurasip:1 springerverlag:1 typical:1 generalisation:3 hyperplane:1 called:1 attempted:1 support:1 almeida:1 latter:1 correlated:2 |
1,119 | 2,020 | On Discriminative vs. Generative
classifiers: A comparison of logistic
regression and naive Bayes
Andrew Y. Ng
Michael I. Jordan
Computer Science Division
C.S. Div. & Dept. of Stat.
University of California, Berkeley University of California, Berkeley
Berkeley, CA 94720
Berkeley, CA 94720
Abstract
We compare discriminative and generative learning as typified by
logistic regression and naive Bayes. We show, contrary to a widelyheld belief that discriminative classifiers are almost always to be
preferred, that there can often be two distinct regimes of performance as the training set size is increased, one in which each
algorithm does better. This stems from the observation- which
is borne out in repeated experiments- that while discriminative
learning has lower asymptotic error, a generative classifier may also
approach its (higher) asymptotic error much faster.
1
Introduction
Generative classifiers learn a model of the joint probability, p( x, y), of the inputs x
and the label y, and make their predictions by using Bayes rules to calculate p(ylx),
and then picking the most likely label y. Discriminative classifiers model the posterior p(ylx) directly, or learn a direct map from inputs x to the class labels. There
are several compelling reasons for using discriminative rather than generative classifiers, one of which, succinctly articulated by Vapnik [6], is that "one should solve
the [classification] problem directly and never solve a more general problem as an
intermediate step [such as modeling p(xly)]." Indeed, leaving aside computational
issues and matters such as handling missing data, the prevailing consensus seems
to be that discriminative classifiers are almost always to be preferred to generative
ones.
Another piece of prevailing folk wisdom is that the number of examples needed to
fit a model is often roughly linear in the number of free parameters of a model.
This has its theoretical basis in the observation that for "many" models, the VC
dimension is roughly linear or at most some low-order polynomial in the number
of parameters (see, e.g., [1, 3]), and it is known that sample complexity in the
discriminative setting is linear in the VC dimension [6].
In this paper, we study empirically and theoretically the extent to which these
beliefs are true. A parametric family of probabilistic models p(x, y) can be fit either
to optimize the joint likelihood of the inputs and the labels, or fit to optimize the
conditional likelihood p(ylx), or even fit to minimize the 0-1 training error obtained
by thresholding p(ylx) to make predictions. Given a classifier hGen fit according
to the first criterion, and a model h Dis fit according to either the second or the
third criterion (using the same parametric family of models) , we call hGen and
h Dis a Generative-Discriminative pair. For example, if p(xly) is Gaussian and p(y)
is multinomial, then the corresponding Generative-Discriminative pair is Normal
Discriminant Analysis and logistic regression. Similarly, for the case of discrete
inputs it is also well known that the naive Bayes classifier and logistic regression
form a Generative-Discriminative pair [4, 5].
To compare generative and discriminative learning, it seems natural to focus on
such pairs. In this paper, we consider the naive Bayes model (for both discrete and
continuous inputs) and its discriminative analog, logistic regression/linear classification, and show: (a) The generative model does indeed have a higher asymptotic
error (as the number of training examples becomes large) than the discriminative
model, but (b) The generative model may also approach its asymptotic error much
faster than the discriminative model- possibly with a number of training examples
that is only logarithmic, rather than linear, in the number of parameters. This
suggests-and our empirical results strongly support-that, as the number of training examples is increased, there can be two distinct regimes of performance, the
first in which the generative model has already approached its asymptotic error and
is thus doing better, and the second in which the discriminative model approaches
its lower asymptotic error and does better.
2
Preliminaries
We consider a binary classification task, and begin with the case of discrete data.
Let X = {O, l}n be the n-dimensional input space, where we have assumed binary
inputs for simplicity (the generalization offering no difficulties). Let the output
labels be Y = {T, F}, and let there be a joint distribution V over X x Y from which
a training set S = {x(i) , y(i) }~1 of m iid examples is drawn. The generative naive
Bayes classifier uses S to calculate estimates p(xiIY) and p(y) of the probabilities
p(xi IY) and p(y), as follows:
,y=b}+1
(1)
P' (x-, = 11Y = b) = #s{xi=l
#s{y-b}+21
(and similarly for p(y = b),) where #s{-} counts the number of occurrences of an
event in the training set S. Here, setting l = corresponds to taking the empirical
estimates of the probabilities, and l is more traditionally set to a positive value such
as 1, which corresponds to using Laplace smoothing of the probabilities. To classify
a test example x, the naive Bayes classifier hGen : X r-+ Y predicts hGen(x) = T if
and only if the following quantity is positive:
?
IGen(x ) = log
(rr~-d) (x i ly
= T))p(y = T)
(rrni=1 P' (X,_IY -_ F)) P'( Y -_
~
p(xilY = T)
F) = L..,log ' ( _I _ F)
i=1
P X, Y -
p(y = T)
+ log P' (Y -_ F)'
(2)
In the case of continuous inputs , almost everything remains the same, except that
we now assume X = [O,l]n, and let p(xilY = b) be parameterized as a univariate
Gaussian distribution with parameters {ti ly=b and if; (note that the j1's, but not
the if's , depend on y). The parameters are fit via maximum likelihood, so for
example {ti ly=b is the empirical mean of the i-th coordinate of all the examples in
the training set with label y = b. Note that this method is also equivalent to Normal
Discriminant Analysis assuming diagonal covariance matrices. In the sequel, we also
let J.tily=b = E[Xi IY = b] and
= Ey[Var(xi ly)] be the "true" means and variances
(regardless of whether the data are Gaussian or not).
In both the discrete and the continuous cases, it is well known that the discriminative analog of naive Bayes is logistic regression. This model has parameters [,8, OJ,
and posits that p(y = Tlx; ,8, O) = 1/(1 +exp(-,8Tx - 0)). Given a test example x,
a;
the discriminative logistic regression classifier ho is : X
and only if the linear discriminant function
I-t
Y predicts hOis (x)
= T if
lDis(x) = L~=l (3ixi + ()
(3)
is positive. Being a discriminative model, the parameters [(3, ()] can be fit either to
maximize the conditionallikelikood on the training set L~= llogp(y(i) Ix(i); (3, ()), or
to minimize 0-1 training error L~= ll{hois(x(i)) 1- y(i)}, where 1{-} is the indicator
function (I{True} = 1, I{False} = 0) . Insofar as the error metric is 0-1 classification
error, we view the latter alternative as being more truly in the "spirit" of discriminative learning, though the former is also frequently used as a computationally
efficient approximation to the latter. In this paper, we will largely ignore the difference between these two versions of discriminative learning and, with some abuse of
terminology, will loosely use the term "logistic regression" to refer to either, though
our formal analyses will focus on the latter method.
Finally, let 1i be the family of all linear classifiers (maps from X to Y); and given a
classifier h : X I-t y, define its generalization error to be c(h) = Pr(x,y)~v [h(x) 1- y].
3
Analysis of algorithms
When V is such that the two classes are far from linearly separable, neither logistic
regression nor naive Bayes can possibly do well, since both are linear classifiers.
Thus, to obtain non-trivial results, it is most interesting to compare the performance
of these algorithms to their asymptotic errors (cf. the agnostic learning setting).
More precisely, let hGen,oo be the population version of the naive Bayes classifier; i.e.
hGen,oo is the naive Bayes classifier with parameters p(xly) = p(xly),p(y) = p(y).
Similarly, let hOis ,oo be the population version of logistic regression. The following
two propositions are then completely straightforward.
Proposition 1 Let
hGen and h Dis be any generative-discriminative pair of classifiers, and hGen,oo and hois, oo be their asymptotic/population versions. Then l
c(hDis,oo) :S c(hGen,oo).
Proposition 2 Let h Dis be logistic regression in n-dimensions.
Then with high
probability
c(hois ) :S c(hois,oo) + 0 (J ~ log ~)
Thus, for c(hOis ) :S c(hOis,oo) + EO to hold with high probability (here,
fixed constant), it suffices to pick m = O(n).
EO
> 0 is some
Proposition 1 states that aymptotically, the error of the discriminative logistic regression is smaller than that of the generative naive Bayes. This is easily shown
by observing that, since c(hDis) converges to infhE1-l c(h) (where 1i is the class of
all linear classifiers), it must therefore be asymptotically no worse than the linear
classifier picked by naive Bayes. This proposition also provides a basis for what
seems to be the widely held belief that discriminative classifiers are better than
generative ones.
Proposition 2 is another standard result, and is a straightforward application of
Vapnik's uniform convergence bounds to logistic regression, and using the fact that
1i has VC dimension n. The second part of the proposition states that the sample
complexity of discriminative learning- that is, the number of examples needed to
approach the asymptotic error- is at most on the order of n. Note that the worst
case sample complexity is also lower-bounded by order n [6].
lUnder a technical assumption (that is true for most classifiers, including logistic regression) that the family of possible classifiers hOis (in the case of logistic regression, this
is 1l) has finite VC dimension.
The picture for discriminative learning is thus fairly well-understood: The error
converges to that of the best linear classifier, and convergence occurs after on the
order of n examples. How about generative learning, specifically the case of the
naive Bayes classifier? We begin with the following lemma.
?
?
Lemma 3 Let any 101,8 > and any l 2: be fixed. Assume that for some fixed
Po > 0, we have that Po :s: p(y = T) :s: 1 - Po. Let m = 0 ((l/Ei) log(n/8)). Then
with probability at least 1 - 8:
1. In case of discrete inputs, IjJ(XiIY = b) - p(xilY
b) - p(y = b) I :s: 101, for all i = 1, ... ,n and bEY.
= b)1 :s: 101 and IjJ(y =
2. In the case of continuous inputs, IPi ly=b - f-li ly=b I :s: 101, laT
IjJ(y = b) - p(y = b) I :s: 101 for all i = 1, ... ,n and bEY.
-
O"T I :s: 101, and
?
Proof (sketch). Consider the discrete case, and let l = for now. Let 101 :s: po/2.
By the Chernoff bound, with probability at least 1 - 81 = 1- 2exp(-2Eim) , the
fraction of positive examples will be within 101 of p(y = T) , which implies IjJ(y =
b) - p(y = b)1 :s: 101, and we have at least 1 m positive and 1m negative examples,
where I = Po - 101 = 0(1). So by the Chernoff bound again , for specific i, b, the
chance that IjJ(XiIY = b) - p(xilY = b)1 > 101 is at most 82 = 2exp(-2Ehm). Since
there are 2n such probabilities, the overall chance of error, by the Union bound, is
at most 81 + 2n82 . Substituting in 81 and 8/s definitions , we see that to guarantee
81 + 2n82 :s: 8, it suffices that m is as stated. Lastly, smoothing (l > 0) adds at most
a small, O(l/m) perturbation to these probabilities , and using the same argument
as above with (say) 101/2 instead of 101, and arguing that this O(l/m) perturbation
is at most 101/2 (which it is as m is at least order l/Ei) , again gives the result. The
result for the continuous case is proved similarly using a Chernoff-bounds based
argument (and the assumption that Xi E [0,1]).
D
Thus, with a number of samples that is only logarithmic, rather than linear, in n, the
parameters of the generative classifier hGen are uniformly close to their asymptotic
values in hGen ,oo . Is is tempting to conclude therefore that c(hGen), the error of the
generative naive Bayes classifier, also converges to its asymptotic value of c(hGen,oo)
after this many examples, implying only 0 (log n) examples are required to fit a
naive Bayes model. We will shortly establish some simple conditions under which
this intuition is indeed correct. Note that this implies that, even though naive Bayes
converges to a higher asymptotic error of c(hGen,oo) compared to logistic regression's
c: (hDis, oo ), it may also approach it significantly faster-after O(log n), rather than
O(n), training examples.
One way of showing c(hGen) approaches c(hGen,oo) is by showing that the parameters' convergence implies that hGen is very likely to make the same predictions as
hGen,oo . Recall hGen makes its predictions by thresholding the discriminant function lGen defined in (2). Let lGen,oo be the corresponding discriminant function
used by hGen,oo. On every example on which both lGen and lGen ,oo fall on the same
side of zero, hGen and hGen,oo will make the same prediction. Moreover, as long as
lGen,oo (x) is, with fairly high probability, far from zero, then lGen (x), being a small
perturbation of lGen ,oo(x), will also be usually on the same side ofzero as lGen ,oo (x).
Theorem 4 Define G(T) = Pr(x,y)~v[(lGen ,oo(x) E [O,Tn] A y = T) V (lG en,oo(X) E
[-Tn, O]A Y = F)]. Assume that for some fixed Po > 0, we have Po :s: p(y = T) :s:
1 - Po, and that either Po :s: P(Xi = 11Y = b) :s: 1 - Po for all i, b (in the case of
discrete inputs), or O"T 2: Po (in the continuous case). Then with high probability,
c:( hGen )
:s: c:(hGen,oo) + G (0 (J ~ logn))
.
(4)
Proof (sketch). c(hGen) - c(hGen,oo) is upperbounded by the chance that
hGen,oo correctly classifies a randomly chosen example, but hGen misclassifies it.
Lemma 3 ensures that, with high probability, all the parameters of hGen are within
O( j(log n)/m) of those of hGen ,oo . This in turn implies that everyone of the n + 1
terms in the sum in lGen (as in Equation 2) is within O( j(1ogn)/m) of the corresponding term in lGen ,oo , and hence that IlGen(x) -lGen,oo(x)1 :S O(nj(1ogn)/m).
Letting T = O( j(logn)/m), we therefore see that it is possible for hGen,oo to be correct and hGen to be wrong on an example (x , y) only if y = T and lGen,oo(X) E [0, Tn]
(so that it is possible that lGen,oo(X) ::::: 0, lGen (x) :S 0), or if y = F and
lGen,oo(X) E [-Tn, 0]. The probability of this is exactly G(T), which therefore upperbounds c(hGen) - c(hGen,oo ).
D
The key quantity in the Theorem is the G(T) , which must be small when T is
small in order for the bound to be non-trivial. Note G(T) is upper-bounded by
Prx[lGen,oo(x) E [-Tn, Tn]]-the chance that lGen, oo(X) (a random variable whose
distribution is induced by x ""' V) falls near zero. To gain intuition about the scaling
of these random variables, consider the following:
Proposition 5 Suppose that, for at least an 0(1) fraction of the features i (i =
1, ... ,n), it holds true that IP(Xi = 11Y = T) - P(Xi = 11Y = F)I ::::: 'Y for some
fixed'Y > 0 (or IJLi ly=T - JLi ly=FI ::::: 'Y in the case of continuous inputs). Then
E[lGen ,oo(x)ly = T] = O(n), and -E[lGen,oo (x)ly = F] = O(n).
Thus, as long as the class label gives information about an 0(1) fraction of the
features (or less formally, as long as most of the features are "relevant" to the class
label), the expected value of IlGen, oo(X) I will be O(n). The proposition is easily
proved by showing that, conditioned on (say) the event y = T, each of the terms
in the summation in lGen, oo(x) (as in Equation (2), but with fi's replaced by p's)
has non-negative expectation (by non-negativity of KL-divergence), and moreover
an 0(1) fraction of them have expectation bounded away from zero.
Proposition 5 guarantees that IlGen,oo (x)1 has large expectation, though what we
want in order to bound G is actually slightly stronger, namely that the random
variable IlGen,oo (x)1 further be large/far from zero with high probability. There
are several ways of deriving sufficient conditions for ensuring that G is small. One
way of obtaining a loose bound is via the Chebyshev inequality. For the rest of
this discussion, let us for simplicity implicitly condition on the event that a test
example x has label T. The Chebyshev inequality implies that Pr[lGen ,oo(x) :S
E[lGen ,oo(X)] - t] :S Var(lGen,oo(x))/t2 . Now, lGen,oo (X) is the sum of n random
variables (ignoring the term involving the priors p(y)). If (still conditioned on y),
these n random variables are independent (i.e. if the "naive Bayes assumption,"
that the xi's are conditionally independent given y, holds), then its variance is O(n);
even if the n random variables were not completely independent, the variance may
still be not much larger than 0 (n) (and may even be smaller, depending on the
signs of the correlations), and is at most O(n 2). So, if E[lGen,oo (x)ly = T] = an (as
would be guaranteed by Proposition 5) for some a > 0, by setting t = (a - T)n,
Chebyshev's inequality gives Pr[lGen,oo(x) :S Tn] :S O(l/(a - T)2n1/) (T < a), where
1} = 0 in the worst case, and 1} = 1 in the independent case. This thus gives
a bound for G(T), but note that it will frequently be very loose. Indeed, in the
unrealistic case in which the naive Bayes assumption really holds , we can obtain
the much stronger (via the Chernoff bound) G(T):S exp(-O((a - T)2n)) , which is
exponentially small in n. In the continuous case, if lGen,oo (x) has a density that,
within some small interval [-m,mJ, is uniformly bounded by O(l/n), then we also
have G(T) = O(T). In any case, we also have the following Corollary to Theorem 4.
Corollary 6 Let the conditions of Theorem 4 hold, and suppose that G(T) :S Eo/2+
F(T) for some function F(T) (independent of n) that satisfies F(T) -+ 0 as T -+ 0,
and some fixed EO > O. Then for ?(hGen) :S c(hGen,oo) + EO to hold with high
pima (continuous)
adult (continuous)
0.5
0.45
0.4
e
;;;
0.35
0.3
0.250
boston (predict it > median price, continuous)
0.5
0.45, - - - - - - - - - - - - - ,
0.45
"
0.4
0.4
~-2~~~,____
~
0.3
'--,
20
40
I::: \ ...~~ __
gO.35
0.25
0"0
60
optdigits (O's and 1 's, continuous)
10
20
02Q- - -""2"'
0- - -4"'0-----"cJ
60
30
optdigits (2's and 3's, continuous)
ionosphere (continuous)
o.4,---- - - - - - - - - - ,
0 .4", - - - - - - - - - ----.
0.3
0 .3
0.5,---- - - - - - - - - - - - - - ,
~0.2
01 ~
0.1
0.2
50
100
150
200
adult (discrete)
liver disorders (continuous)
0.7,---- - - - - - - - - - - - - - ,
0.5, - - - - - - - - - - - - - - - - - ,
0.6
0.45
~
~
0.4
~ 0.35
0.4
0.3
0.350
20
40
60
0.250
20
promoters (discrete)
60
80
100
100
120
lymphography (discrete)
0.5
200
300
400
breast cancer (discrete)
0.5
~:: ~
~
40
,......?...?..??.?.?..?.?..?.?.?.?.?...?...?. ?.. _.
0.2
~ ~'\~.~:::.:~
~0 . 3
0.4
"'"
0.45
~
.....
0.1
0 .2
..... ".""""",,.
0.3
0.10'------,5~
0---.,.
10~0---.---,J
%L--~
20~-~
40---'-6~0---'---'8~0-~
100
150
lenses (predict hard vs. soft, discrete)
0.250
0.4
0.8, - - - - - - - - - - - - - ,
0.4
0'6 \=~_
0.3
gOA
~
0 .2
.....
100
200
300
voting records (discrete)
sick (discrete)
0.5
0.2
0.4
~ 0.35
?...?
gO.2
~
0.1 \--.
------------ -- --- --- --- 0.10'---~-~10c--~
15--2~0-~25
%~----,5~0---~10~0~--,,!
150
00
20
40
60
80
Figure 1: Results of 15 experiments on datasets from the VCI Machine Learning
repository. Plots are of generalization error vs. m (averaged over 1000 random
train/test splits). Dashed line is logistic regression; solid line is naive Bayes.
probability, it suffices to pick m = O(log n).
Note that the previous discussion implies that the preconditions of the Corollary
do indeed hold in the case that the naive Bayes (and Proposition 5's) assumption
holds , for any constant fa so long as n is large enough that fa ::::: exp( -O(o:2n))
(and similarly for the bounded Var(lGen ,oo (x)) case, with the more restrictive fa :::::
O(I/(o: 2n 17))). This also means that either ofthese (the latter also requiring T) > 0)
is a sufficient condition for the asymptotic sample complexity to be 0 (log n).
4
Experiments
The results of the previous section imply that even though the discriminative logistic regression algorithm has a lower asymptotic error, the generative naive Bayes
classifier may also converge more quickly to its (higher) asymptotic error. Thus, as
the number of training examples m is increased, one would expect generative naive
Bayes to initially do better, but for discriminative logistic regression to eventually
catch up to, and quite likely overtake, the performance of naive Bayes.
To test these predictions, we performed experiments on 15 datasets, 8 with continuous inputs, 7 with discrete inputs, from the VCI Machine Learning repository.2
The results ofthese experiments are shown in Figure 1. We find that the theoretical
predictions are borne out surprisingly well. There are a few cases in which logistic
regression's performance did not catch up to that of naive Bayes, but this is observed
primarily in particularly small datasets in which m presumably cannot grow large
enough for us to observe the expected dominance of logistic regression in the large
m limit.
5
Discussion
Efron [2] also analyzed logistic regression and Normal Discriminant Analysis (for
continuous inputs) , and concluded that the former was only asymptotically very
slightly (1/3- 1/2 times) less statistically efficient. This is in marked contrast to our
results, and one key difference is that, rather than assuming P(xly) is Gaussian with
a diagonal covariance matrix (as we did), Efron considered the case where P(xly) is
modeled as Gaussian with a full convariance matrix. In this setting, the estimated
covariance matrix is singular if we have fewer than linear in n training examples, so
it is no surprise that Normal Discriminant Analysis cannot learn much faster than
logistic regression here. A second important difference is that Efron considered
only the special case in which the P(xly) is truly Gaussian. Such an asymptotic
comparison is not very useful in the general case, since the only possible conclusion, if ?(hDis,oo) < ?(hGen,oo), is that logistic regression is the superior algorithm.
In contrast, as we saw previously, it is in the non-asymptotic case that the most
interesting "two-regime" behavior is observed.
Practical classification algorithms generally involve some form of regularization- in
particular logistic regression can often be improved upon in practice by techniques
2To maximize the consistency with the theoretical discussion, these experiments avoided
discrete/continuous hybrids by considering only the discrete or only the continuous-valued
inputs for a dataset where necessary. Train/test splits were random subject to there being
at least one example of each class in the training set, and continuous-valued inputs were also
rescaled to [0 , 1] if necessary. In the case of linearly separable datasets, logistic regression
makes no distinction between the many possible separating planes. In this setting we used
an MCMC sampler to pick a classifier randomly from them (i.e., so the errors reported
are empirical averages over the separating hyperplanes) . Our implementation of Normal
Discriminant Analysis also used the (standard) trick of adding ? to the diagonal of the
covariance matrix to ensure invertibility, and for naive Bayes we used I = 1.
such as shrinking the parameters via an L1 constraint, imposing a margin constraint
in the separable case, or various forms of averaging. Such regularization techniques
can be viewed as changing the model family, however, and as such they are largely
orthogonal to the analysis in this paper, which is based on examining particularly
clear cases of Generative-Discriminative model pairings. By developing a clearer
understanding of the conditions under which pure generative and discriminative
approaches are most successful, we should be better able to design hybrid classifiers
that enjoy the best properties of either across a wider range of conditions.
Finally, while our discussion has focused on naive Bayes and logistic regression, it is
straightforward to extend the analyses to several other models , including generativediscriminative pairs generated by using a fixed-structure , bounded fan-in Bayesian
network model for P(xly) (of which naive Bayes is a special case).
Acknowledgments
We thank Andrew McCallum for helpful conversations. A. Ng is supported by a
Microsoft Research fellowship. This work was also supported by a grant from Intel
Corporation, NSF grant IIS-9988642, and ONR MURI N00014-00-1-0637.
References
[1] M. Anthony and P. Bartlett. Neural Network Learning: Th eoretical Foundations. Cambridge University Press, 1999.
[2] B. Efron. The efficiency of logistic regression compared to Normal Discriminant Analysis. Journ. of the Amer. Statist. Assoc., 70:892- 898 , 1975.
[3] P. Goldberg and M. Jerrum. Bounding the VC dimension of concept classes parameterized by real numbers. Machine Learning, 18:131-148, 1995.
[4] G.J. McLachlan. Discriminant Analysis and Statistical Pattern Recognition. Wiley,
New York, 1992.
[5] Y. D. Rubinstein and T. Hastie. Discriminative vs. informative learning. In Proceedings
of th e Third International Conference on Knowledge Discovery and Data Mining, pages
49- 53. AAAI Press, 1997.
[6] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998.
| 2020 |@word repository:2 version:4 polynomial:1 seems:3 stronger:2 covariance:4 pick:3 solid:1 offering:1 must:2 john:1 informative:1 j1:1 plot:1 v:4 aside:1 generative:24 implying:1 fewer:1 plane:1 mccallum:1 record:1 provides:1 hyperplanes:1 ipi:1 direct:1 pairing:1 theoretically:1 expected:2 indeed:5 behavior:1 roughly:2 frequently:2 nor:1 tily:1 considering:1 becomes:1 begin:2 classifies:1 bounded:6 moreover:2 agnostic:1 what:2 corporation:1 nj:1 guarantee:2 berkeley:4 every:1 ti:2 voting:1 exactly:1 classifier:30 wrong:1 assoc:1 ly:11 enjoy:1 grant:2 positive:5 understood:1 limit:1 abuse:1 suggests:1 range:1 statistically:1 averaged:1 practical:1 acknowledgment:1 arguing:1 union:1 practice:1 empirical:4 significantly:1 cannot:2 close:1 optimize:2 equivalent:1 map:2 missing:1 straightforward:3 regardless:1 go:2 focused:1 simplicity:2 disorder:1 pure:1 rule:1 deriving:1 population:3 jli:1 coordinate:1 traditionally:1 laplace:1 suppose:2 us:1 ixi:1 goldberg:1 trick:1 recognition:1 particularly:2 predicts:2 muri:1 observed:2 worst:2 calculate:2 precondition:1 ensures:1 rescaled:1 intuition:2 complexity:4 depend:1 upon:1 division:1 efficiency:1 basis:2 completely:2 ogn:2 easily:2 joint:3 po:11 various:1 tx:1 llogp:1 articulated:1 train:2 distinct:2 approached:1 rubinstein:1 whose:1 quite:1 widely:1 solve:2 bey:2 say:2 larger:1 valued:2 jerrum:1 ip:1 rr:1 relevant:1 convergence:3 converges:4 wider:1 oo:54 andrew:2 depending:1 stat:1 liver:1 clearer:1 xly:8 implies:6 posit:1 correct:2 vc:5 everything:1 suffices:3 generalization:3 really:1 preliminary:1 convariance:1 proposition:12 summation:1 hold:8 considered:2 normal:6 exp:5 hois:9 presumably:1 predict:2 substituting:1 label:9 saw:1 mclachlan:1 always:2 gaussian:6 rather:5 corollary:3 focus:2 likelihood:3 contrast:2 helpful:1 initially:1 journ:1 issue:1 classification:5 overall:1 logn:2 misclassifies:1 prevailing:2 smoothing:2 fairly:2 special:2 never:1 ng:2 chernoff:4 t2:1 few:1 primarily:1 randomly:2 divergence:1 replaced:1 n1:1 microsoft:1 tlx:1 mining:1 truly:2 analyzed:1 upperbounded:1 held:1 necessary:2 folk:1 orthogonal:1 loosely:1 theoretical:3 increased:3 classify:1 modeling:1 compelling:1 typified:1 soft:1 uniform:1 examining:1 successful:1 reported:1 density:1 overtake:1 international:1 sequel:1 probabilistic:1 picking:1 michael:1 iy:3 quickly:1 again:2 aaai:1 possibly:2 borne:2 worse:1 li:1 invertibility:1 matter:1 piece:1 performed:1 view:1 picked:1 doing:1 observing:1 bayes:28 minimize:2 variance:3 largely:2 wisdom:1 bayesian:1 iid:1 definition:1 proof:2 gain:1 proved:2 dataset:1 recall:1 conversation:1 efron:4 knowledge:1 cj:1 actually:1 higher:4 improved:1 amer:1 though:5 strongly:1 lastly:1 correlation:1 sketch:2 ei:2 vci:2 logistic:28 requiring:1 true:5 concept:1 former:2 hence:1 regularization:2 conditionally:1 ll:1 criterion:2 tn:7 l1:1 goa:1 fi:2 superior:1 multinomial:1 empirically:1 exponentially:1 analog:2 extend:1 eoretical:1 refer:1 cambridge:1 imposing:1 consistency:1 similarly:5 add:1 sick:1 posterior:1 n00014:1 inequality:3 binary:2 onr:1 ey:1 eo:5 converge:1 maximize:2 tempting:1 dashed:1 ii:1 full:1 stem:1 technical:1 faster:4 long:4 ensuring:1 prediction:7 involving:1 regression:28 breast:1 expectation:3 metric:1 want:1 fellowship:1 interval:1 median:1 leaving:1 grow:1 concluded:1 singular:1 rest:1 induced:1 subject:1 contrary:1 spirit:1 jordan:1 call:1 near:1 intermediate:1 split:2 insofar:1 enough:2 fit:9 hastie:1 chebyshev:3 whether:1 bartlett:1 york:1 useful:1 generally:1 clear:1 involve:1 ylx:4 statist:1 nsf:1 eim:1 sign:1 estimated:1 lgen:29 correctly:1 discrete:17 dominance:1 key:2 terminology:1 drawn:1 changing:1 neither:1 asymptotically:2 fraction:4 sum:2 parameterized:2 almost:3 family:5 scaling:1 bound:10 guaranteed:1 fan:1 precisely:1 constraint:2 argument:2 separable:3 developing:1 according:2 smaller:2 slightly:2 across:1 son:1 ofthese:2 pr:4 computationally:1 equation:2 remains:1 previously:1 turn:1 count:1 loose:2 eventually:1 needed:2 letting:1 observe:1 away:1 occurrence:1 alternative:1 ho:1 shortly:1 cf:1 ensure:1 lat:1 upperbounds:1 restrictive:1 establish:1 already:1 quantity:2 occurs:1 parametric:2 fa:3 diagonal:3 div:1 thank:1 separating:2 extent:1 consensus:1 discriminant:10 reason:1 trivial:2 assuming:2 modeled:1 lg:1 pima:1 negative:2 stated:1 implementation:1 design:1 upper:1 observation:2 datasets:4 finite:1 n82:2 xily:4 perturbation:3 pair:6 required:1 kl:1 namely:1 generativediscriminative:1 california:2 distinction:1 adult:2 able:1 usually:1 pattern:1 regime:3 oj:1 including:2 belief:3 everyone:1 unrealistic:1 event:3 natural:1 difficulty:1 hybrid:2 indicator:1 imply:1 igen:1 picture:1 negativity:1 catch:2 naive:27 prior:1 understanding:1 discovery:1 asymptotic:17 expect:1 interesting:2 var:3 foundation:1 hgen:37 sufficient:2 thresholding:2 cancer:1 succinctly:1 surprisingly:1 supported:2 free:1 dis:4 formal:1 side:2 fall:2 taking:1 dimension:6 avoided:1 far:3 lymphography:1 ignore:1 preferred:2 implicitly:1 assumed:1 conclude:1 discriminative:31 xi:9 continuous:20 learn:3 mj:1 ca:2 ignoring:1 obtaining:1 anthony:1 did:2 promoter:1 linearly:2 bounding:1 prx:1 repeated:1 intel:1 en:1 wiley:2 shrinking:1 third:2 ix:1 theorem:4 specific:1 showing:3 ionosphere:1 vapnik:3 false:1 adding:1 conditioned:2 margin:1 xiiy:3 boston:1 surprise:1 logarithmic:2 likely:3 univariate:1 ijj:5 ldis:1 corresponds:2 chance:4 satisfies:1 conditional:1 optdigits:2 marked:1 viewed:1 price:1 hard:1 specifically:1 except:1 uniformly:2 sampler:1 averaging:1 lemma:3 lens:1 formally:1 support:1 latter:4 dept:1 mcmc:1 handling:1 |
1,120 | 2,021 | Group Redundancy Measures Reveal
Redundancy Reduction in the Auditory
Pathway
Gal Chechik
Amir Globerson
Naftali Tishby
School of Computer Science and Engineering
and The Interdisciplinary Center for Neural Computation
Hebrew University of Jerusalem , Israel
ggal@cs.huji.ac.il
Michael J. Anderson
Eric D. Young
Department of Biomedical Engineering
Johns Hopkins University, Baltimore, MD, USA
Israel N elken
Department of Physiology, Hadassah Medical School
and The Interdisciplinary Center for Neural Computation
Hebrew University of Jerusalem, Israel
Abstract
The way groups of auditory neurons interact to code acoustic information is investigated using an information theoretic approach.
We develop measures of redundancy among groups of neurons, and
apply them to the study of collaborative coding efficiency in two
processing stations in the auditory pathway: the inferior colliculus
(IC) and the primary auditory cortex (AI). Under two schemes for
the coding of the acoustic content, acoustic segments coding and
stimulus identity coding, we show differences both in information
content and group redundancies between IC and AI neurons. These
results provide for the first time a direct evidence for redundancy
reduction along the ascending auditory pathway, as has been hypothesized for theoretical considerations [Barlow 1959,2001]. The
redundancy effects under the single-spikes coding scheme are significant only for groups larger than ten cells, and cannot be revealed
with the redundancy measures that use only pairs of cells. The
results suggest that the auditory system transforms low level representations that contain redundancies due to the statistical structure of natural stimuli, into a representation in which cortical neurons extract rare and independent component of complex acoustic
signals, that are useful for auditory scene analysis.
1
Introduction
How do groups of sensory neurons interact to code information and how do these
interactions change along the ascending sensory pathways? According to the a
common view, sensory systems are composed of a series of processing stations,
representing more and more complex aspects of sensory inputs. The changes in
representations of stimuli along the sensory pathway reflect the information processing performed by the system. Several computational principles that govern
these changes were suggested, such as information maximization and redundancy
reduction [2, 3, 11]. In order to investigate such changes in practice, it is necessary to develop methods to quantify information content and redundancies among
groups of neurons, and trace these measures along the sensory pathway.
Interactions and high order correlations between neurons were mostly investigated
within single brain areas on the level of pairs of cells (but also for larger groups of
cells [9]) showing both synergistic and redundant interactions [8, 10, 21, 6, 7, 13].
The current study develops information theoretic redundancy measures for larger
groups of neurons , focusing on the case of stimulus-conditioned independence. We
then compare these measures in electro-physiological recordings from two auditory
stations: the auditory mid-brain and the primary auditory cortex.
2
Redundancy measures for groups of neurons
To investigate high order correlations and interactions within groups of neurons
we start by defining information measures for groups of cells and then develop
information redundancy measures for such groups. The properties of these measures
are then further discussed for the specific case of stimulus-conditioned independence.
Formally, the level of independence of two variables X and Y is commonly quantified
by their mutual information (MI) [17,5]. This well known quantity, now widely used
in analysis of neural data, is defined by
J(X; Y) = DKL[P(X, Y)IIP(X)P(Y)] =
~p(x, y)log (:~~~~~))
(1)
and measures how close the joint distribution P(X, Y) is to the factorization by the
marginal distributions P(X)P(Y) (DKL is the Kullback Leiber divergence [5]).
For larger groups of cells, an important generalized measure quantifies the information that several variables provide about each other. This multi information
measure [18] is defined by
(2)
Similar to the mutual information case, the multi information measures how close
the joint distribution is to the factorization by the marginals. It thus vanishes when
variables are independent and is otherwise positive.
We now turn to develop measures for group redundancies. Consider first the simple
case of a pair of neurons (Xl, X 2 ) conveying information about the stimulus S. In
this case, the redundancy-synergy index ([4, 7]) is defined by
(3)
Intuitively, RSpairs measures the amount of information on the stimulus S gained
by observing the joint distribution of both Xl and X 2 , as compared with observing
the two cells independently. In the extreme case where Xl = X 2 , the two cells
are completely redundant and provide the same information about the stimulus,
yielding RSpairs = I(Xl' X 2 ; S) - I(X l ; S) - I(X2 ; S) = -I(Xl; S), which is always
non-positive. On the other hand, positive RSpairs values testify for synergistic
interaction between Xl and X 2 ([8, 7, 4]).
For larger groups of neurons, several different measures of redundancy-synergy may
be considered, that encompass different levels of interactions. For example, one can
quantify the residual information obtained from a group of N neurons compared
to all its N - 1 subgroups. As with inclusion-exclusion calculations this measure
takes the form of a telescopic sum: RSNIN-l = I(XN; S) - L{X N -l} I(X N -\ S) +
... + (_l)N-l L{Xd I(Xi ; S), where {Xk} are all the subgroups of size k out of the
N available neurons. Unfortunately, this measure involves 2N information terms,
making its calculation infeasible even for moderate N values 1.
A different RS measure quantifies the information embodied in the joint distribution
of N neurons compared to that provided by N single independent neurons, and is
defined by
N
RSN ll = I(Xl ' ... , X N ; S) -
2..: I(Xi ; S)
(4)
i=l
Interestingly, this synergy-redundancy measure may be rewritten as the difference
between two multi-information terms
N
I(Xl ' ... , X N ; S) -
2..: I(Xi ; S) =
(5)
i= l
N
H(Xl' ... ,XN) - H(Xl' ... , XNIS) -
2..: H(Xi ) -
H(XiIS) =
i=l
I(X l ; ... ; XNIS) - I(X l ; ... ;XN)
where H(X) = - L x P(x)log(P(x)) is the entropy of X 2 . We conclude that the
index RSN ll can be separated into two terms: one that is always non-negative,
and measures the coding synergy, and the second which is always non-positive and
quantifies the redundancy. These two terms correspond to two types of interactions
between neurons: The first type are within-stimulus correlations (sometimes termed
noise correlations) that emerge from functional connections between neurons and
contribute to synergy. The second type are between stimulus correlations (or across
stimulus correlations) that reflect the fact that the cells have similar responses per
stimulus, and contribute to redundancy. Being interested in the latter type of
correlations, we limit the discussion to the redundancy term -I(Xl; ... ; XN)'
Formulating RSN ll as in equation 5 proves highly useful when neural activities
are independent given the stimulus P(XIS) = II~l P(XiIS). In this case, the
first (synergy) term vanishes , thus limiting neural interactions to the redundant
lOur results below suggest that some redundancy effects become significant only for
groups larger than 10-15 cells.
2When comparing redundancy in different processing stations, one must consider the
effects of the baseline information conveyed by single neurons. We thus use the normalized
redundancy (compare with [1 5] p.315 and [4]) defined by !iS Nll = RSNldI(Xl; ... ; X N ; S)
regime. More importantly, under the independence assumption we only have to
estimate the marginal distributions P(XiIS = s) for each stimulus s instead of
the full distribution P(XIS = s). It thus allows to estimate an exponentially
smaller number of parameters, which in our case of small sample sizes, provides
more accurate information estimates. This approximation makes it possible to
investigate redundancy among considerably larger groups of neurons than the 2-3
neuron groups considered previously in the literature.
How reasonable is the conditional-independence approximation ? It is a good approximation whenever neuronal activity is mostly determined by the presented stimulus and to a lesser extent by interactions with nearby neurons. A possible example
is the high input regime of cortical neurons receiving thousands of inputs, where
a single input has only a limited influence on the activity of the target cell. The
experimental evidence in this regard is however mixed (see e.g.[9]). One should note
however, that stimulus-conditioned independence is implicitly assumed in analysis
of non-simultaneously recorded data.
To summarize, the stimulus-conditioned independence assumption limits interactions to the redundant regime, but allows to compare the extent of redundancy
among large groups of cells in different brain areas.
3
Experimental Methods
To investigate redundancy in the auditory pathway, we analyze extracellular recordings from two brain areas of gas-anesthetized cats: 16 cells from the Inferior Colliculus (Ie) - the third processing station of the ascending auditory pathway - and
19 cells from the Primary Auditory Cortex (AI) - the fifth station. Neural activity
was recorded non-simultaneously from a total of 6 different animals responding to
a set of complex natural and modified stimuli. Because cortical auditory neurons
respond differently to simple and complex stimuli [12 , 1], we refrain from using artificial over-simplified acoustic stimuli but instead use a set of stimuli based on bird
vocalizations which contains complex 'real-life' acoustic features. A representative
example is shown in figure 1.
7
6
Q)
"0
."1i.e
E
'"
20
40
60
80
time (milliseconds)
100
20
40
60
80
100
time (milliseconds)
Figure 1: A representative stimulus containing a short bird vocalization recorded in
a natural environment. The set of stimuli consisted of similar natural and modified
recordings. A. Signal in time domain B. Signal in frequency domain.
4
Experimental Results
In practice, in order to estimate the information conveyed by neural activity from
limited data, one must assume a decoding procedure, such as focusing on a simple
statistic of the spike trains that encompasses some of its informative properties. In
this paper we consider two extreme cases: coding short acoustic segments with single spikes and coding the stimulus identity with spike counts in a long window. In
addition, we estimated information and redundancy obtained with two other statistics. First, the latency of the first spike after stimulus onset, and secondly, a statistic
which generalizes the counts statistics for a general renewal process [19]. These calculations yielded higher information content on average, but similar redundancies
as presented below. Their detailed results will be reported elsewhere.
0.15
1.2
~
Inlerior Colliculus (IC)
0.1
c
u'"
~0 . 8
C
:::l
.$
:0
U
iO.6
~
0.05
<ii
c
o
0.4
t5
jg
0
Auditory Cortex (AI)
o~==========
5
10
no of cells
15
-0.05 L--".--'--~---~--~---
o
5
10
no of cells
15
20
Figure 2: A. Information about stimulus frames as a function of number of cells.
Information calculation was repeated for several subgroups of each size, and with
several random seed initializations. The dark curve depicts the expected information
provided by independent neurons (this expected curve is corrected for saturation
effects [16] and is thus sub linear). The curved line depicts average information
from joint distribution of sets of neurons Mean[J(Xl' ... Xk; S)]. All information
estimations were corrected for small-samples bias by shuffling methods [14] . B.
Fractional redundancy (difference of the mutual information from the expected
baseline information divided by the baseline) as a function of number of neurons.
4.1
Coding acoustics with single spikes
The current section focuses on the relation between single spikes and short windows
of the acoustic stimuli shortly preceding them (which we denote as frames). As the
set of possible frames is very large and no frame actually repeats itself, we must
first pre-process the stimuli to reduce frames dimensionality.
To this end, we first transformed the stimuli into the frequency domain (roughly approximating the cochlear transformation) and then extracted overlapping windows
of 50 millisecond length, with 1 millisecond spacing. This set was clustered into
32 representatives, using a metric that groups together acoustic segments with the
same spectro-temporal energy structure. This representation allowed us to estimate
the joint distribution (under the stimulus-conditioned independence assumption) of
cells' activity and stimuli, for groups of cells of different sizes. Figure 2A shows the
mutual information between spikes and stimulus frames as a function of the number
of cells for both AI and Ie neurons. Ie neurons convey high information but largely
deviate from the information expected for independent neurons. On the other hand,
AI neurons provide an order of magnitude less information than Ie cells but their
information sums almost linearly, as expected from independent neurons.
The difference between an information curve and its linear baseline measures the
redundancy RSN II of equation 5. Figure 2B presents the normalized redundancy as
a function of number of cells, showing that Ie cells are significantly more redundant
than AI cells.
0.6 rr==--------,--------;
D
_
0.6 rr==--------,--------;
D
Primary Auditory Cortex A 1
Inferior Colliculus IC
_
0.5 ~'-"'=-c.:..::..::.:..:.:::.'--===::...:..::-----"
Primary Auditory Cortex A 1
Inferior Colliculus IC
0.5 ~'-"'=--"-"-':.:..:.:::.'--===::...:..::'---------"
Q)
Q)
u
c
u
c
504
504
u
u
u
u
o
o
00.3
00.3
:?"'
:?"'
~02
~0.2
0.
0.
o
o
0.1
o
0.1
-0.5
-04
-0.3
-0.2
-0.1
pairwise redundancy -I(X ;Y)/I(X ;Y;S)
o
-8.8triplets fractional
-0.6
-04
redundancy
-0.2
0
-I(X;Y;Z)/I(X;Y;Z;S)
Figure 3: Distribution of pairs (A.) and triplets (B.) normalized redundancies. AI
cells (light bars) are significantly more independent than Ie cells (dark bars). Spike
counts were collected over a window that maximizes mean single cells MI. Number
of bins in counts-histogram was optimized separately for every cell. Information
estimations were corrected for small-samples bias by shuffling methods [14].
4.2
Coding stimuli by spike counts
We now turn to investigate a second coding paradigm, and calculate the information conveyed by AI and Ie spike counts about the identity of the presented
stimulus. To this end, we calculate a histogram of spike counts and estimate the
counts' distribution as obtained from repeated presentations of the stimuli.
The distribution of fractional redundancy in pairs of AI and Ie neurons is presented
in figure 3A, and that of triplets in figure 3B 3 . As in the case of coding with
single spikes, single AI cells convey on average less information about the stimulus.
However, they are also more independent, thus making it possible to gain more
information from groups of neurons. Ie neurons on the other hand, provide more
information when considered separately but are more redundant.
As in the case of coding acoustics with single spikes, single Ie cells provide more
information than AI cells (data not shown) but this time AI cells convey half the
information that Ie cells provide, while they convey ten times less information
than Ie cells about acoustics. This suggests that AI cells poorly code the physical
characteristics of the sound but convey information about its global properties. To
illustrate the high information provided by both sets, we trained a neural network
classifier that predicts the identity of the presented stimulus according to spike
counts of a limited set of neurons. Figure 4 shows that both sets of neurons achieve
considerable prediction accuracy, but Ie neurons obtain average accuracy of more
than 90 percent already with five cells, while the average prediction accuracy using
cortical neurons rises continuously 4.
3Unlike the binary case of single spikes, the limited amount of data prevents a robust
estimation of information from spike counts for more than triplets of cells.
4The probability of accurate prediction is exponentially related to the input-output
mutual information, via the relation Pcorrect = exp( -missing nats) yielding Mlnats =
In(no. of stimuli) + In(Pcorrect). Classification thus provides lower bounds on information
content .
Figure 4. Prediction accuracy of stimulus identity as a function of number of
Ie (upper curve) and AI (lower curve)
cells used by the classifier. Error bars
denote standard deviation across several subgroups of the same size. For
each subgroup, a one-hidden layer neural network was trained separately for
each stimulus using some stimulus presentations as a training set and the rest
for testing. Performance reported is
for the testing set.
5
0.95
>-
"[IS
"al
0.9
:0
<:0.85
o
"u
~
0.8
Q.
0.75
I
II I
I
I?I
II I II
IIIII I I I I I
jI" I
0.7 '-------=---~--~---~--~
5
10
number of cells
15
20
Discussion
We have developed information theoretic measures of redundancy among groups of
neurons and applied them to investigate the collaborative coding efficiency in the
auditory modality. Under two different coding paradigms, we show differences in
both information content and group redundancies between Ie and cortical auditory
neurons. Single Ie neurons carry more information about the presented stimulus,
but are also more redundant. On the other hand, auditory cortical neurons carry
less information but are more independent, thus allowing information to be summed
almost linearly when considering groups of few tens of neurons. The results provide
for the first time direct evidence for redundancy reduction along the ascending
auditory pathway, as has been hypothesized by Barlow [2, 3]. The redundancy
effects under the single-spikes coding paradigm are significant only for groups larger
than ten cells, and cannot be revealed with the standard redundancy measures that
use only pairs of cells.
Our results suggest that transformations leading to redundancy reduction are not
limited to low level sensory processing (aimed to reduce redundancy in input statistics) but are applied even at cortical sensory stations. We suggest that an essential
experimental prerequisite to reveal these effects is the use of complex acoustic stimuli whose processing occurs at high level processing stations.
The above findings are in agreement with the view that along the ascending sensory
pathways, the number of neurons increase, their firing rates decrease , and neurons
become tuned to more complex and independent features. Together, these suggest
that the neural representation is mapped into a representation with higher effective
dimensionality. Interestingly, recent advances in kernel-methods learning [20] have
shown that nonlinear mapping into higher dimension and over-complete representations may be useful for learning of complex classifications. It is therefore possible
that such mappings provide easier readout and more efficient learning in the brain.
Acknowledgements
This work supported in part by a Human Frontier Science Project (HFSP) grant
RG 0133/1998 and by a grant from the Israeli Ministry of Science.
References
[1]
O. Bar-Yosef and I. Nelken. Responses of neurons in cat primary auditory cortex to
bird chirps: Effects of temporal and spectral context. J. Neuroscience, in press, 2001.
[2]
H.B. Barlow. Sensory mechanisms, the reduction of redundancy, and intelligence. In
Mechanisation of thought processes, pages 535- 539. Her Majesty's stationary office,
London , 1959.
[3]
H .B. Barlow. Redundancy reduction revisited. Network: Computation in neural
systems, 12:241-253, 200l.
[4]
N . Brenner, S.P . Strong, R . Koberle, R. de Ruyter van Steveninck, and W . Bialek.
Synergy in a neural code. Neural Computation, 13(7):1531, 2000.
[5]
T.M. Cover and J.A. Thomas. The elements of information theory. Plenum Press,
New York, 1991.
[6]
Y. Dan, J.M. Alonso, W.M. Usrey, and R.C. Reid. Coding of visual information
by precisely correlated spikes in the lateral geniculate nucleus. Nature Neuroscience,
[7]
I. Gat and N. Tishby. Synergy and redundancy among brain cells of behaving monkeys. In M.S. Kearns, S.A. Solla, and D.A.Cohn, editors, Advances in Neural Information Proceedings systems, volume 11, Cambridge, MA, 1999. MIT Press.
[8]
T.J. Gawne and B.J. Richmond. How independent are the messages carried by adjacent inferior temporal cortical neurons? J. Neurosci., 13(7):2758- 2771, 1993.
[9]
P.M. Gochin, M. Colombo, G. A. Dorfman, G.L. Gerstein, and C.G. Gross. Neural
ensemble coding in inferior temporal cortex. J. Neurophysiol., 71:2325- 2337, 1994.
1(6):501- 507, 1998.
[10] M. Meister. Multineural codes in retinal signaling. Proc. Natl. Acad. Sci., 93:609- 614,
1996.
[11] J .P. Nadal, N. Brunei, and N . Parga. Nonlinear feedforward networks with stochastic
outputs: infomax implies redundancy reduction. Network: Computation in neural
systems, 9:207- 217, 1998.
[12] I. Nelken, Y. Rotman, and O. Bar-Yosef. Specialization of the auditory system for
the analysis of natural sounds. In J. Brugge and P.F. Poon, editors, Central Auditory
Processing and Neural Modeling. Plenum, New York, 1997.
[13] S. Nirenberg, S.M. Carcieri, A.L. Jacobs, and P.E. Latham. Retinal ganglion cells act
largely as independent encoders. Nature, 411:698- 701, 200l.
[14] LM . Optican , T.J. Gawne, B.J. Richmond, and P .J . Joseph. Unbiased measures of
transmitted information and channel capacity from multivariate neuronal data. Bioi.
Cyber, 65:305- 310, 1991.
[15] E . T. Rolls and A. Treves . Neural Networks and Brain Function. Oxford Univ . Press,
1998.
[16] I. Samengo. Independent neurons representing a fintie set of stimuli: dependence of
the mutual information on the number of units sampled. Network: Comput. Neural
Syst., 12:21- 31 , 200l.
[17] C.E. Shanon. A mathematical theory of communication. The Bell systems technical
journal, 27:379- 423,623- 656, 1948.
[18] M. Studenty and J. Vejnarova. The multiinformation function as a tool for measuring
stochastic dependence. In M.I. Jordan, editor, Learning in Graphical Models, pages
261-297. Dordrecht: Kluwer, 1998.
[19] C. van Vreeswijk. Information trasmission with renewal neurons. In J.M. Bower,
editor, Computational Neuroscience: Trends in Research. Elsevier Press, 200l.
[20] V.N. Vapnik. The nature of statistical learning theory. Springer-Verlag, Berlin, 1995.
[21] DK. Warland, P. Reinagel, and M. Meister. Decoding visual information from a
population of retinal ganglion cells. J. Neurophysiol., 78:2336- 2350, 1997.
| 2021 |@word r:1 jacob:1 carry:2 reduction:8 series:1 contains:1 tuned:1 interestingly:2 optican:1 current:2 comparing:1 must:3 john:1 informative:1 stationary:1 half:1 intelligence:1 amir:1 xk:2 short:3 provides:2 contribute:2 revisited:1 five:1 mathematical:1 along:6 direct:2 become:2 pathway:10 dan:1 pairwise:1 expected:5 roughly:1 multi:3 brain:7 window:4 considering:1 provided:3 project:1 maximizes:1 israel:3 nadal:1 monkey:1 developed:1 finding:1 gal:1 transformation:2 temporal:4 every:1 act:1 xd:1 classifier:2 unit:1 medical:1 grant:2 dorfman:1 reid:1 positive:4 engineering:2 limit:2 io:1 acad:1 oxford:1 firing:1 bird:3 initialization:1 chirp:1 quantified:1 suggests:1 multiinformation:1 factorization:2 limited:5 steveninck:1 globerson:1 testing:2 practice:2 signaling:1 procedure:1 area:3 bell:1 physiology:1 significantly:2 thought:1 chechik:1 pre:1 carcieri:1 suggest:5 cannot:2 synergistic:2 close:2 ggal:1 influence:1 context:1 telescopic:1 center:2 missing:1 jerusalem:2 independently:1 reinagel:1 importantly:1 population:1 limiting:1 plenum:2 target:1 agreement:1 element:1 trend:1 predicts:1 leiber:1 thousand:1 calculate:2 readout:1 solla:1 decrease:1 gross:1 vanishes:2 govern:1 environment:1 nats:1 trained:2 segment:3 eric:1 efficiency:2 completely:1 mechanisation:1 neurophysiol:2 joint:6 differently:1 cat:2 train:1 separated:1 univ:1 effective:1 london:1 artificial:1 dordrecht:1 whose:1 larger:8 widely:1 otherwise:1 nirenberg:1 statistic:5 itself:1 vocalization:2 nll:1 rr:2 interaction:10 poon:1 poorly:1 achieve:1 illustrate:1 develop:4 ac:1 school:2 strong:1 c:1 involves:1 implies:1 quantify:2 stochastic:2 human:1 bin:1 clustered:1 gochin:1 secondly:1 frontier:1 considered:3 ic:5 exp:1 seed:1 mapping:2 lm:1 estimation:3 proc:1 geniculate:1 tool:1 mit:1 always:3 modified:2 colombo:1 office:1 focus:1 majesty:1 richmond:2 baseline:4 elsevier:1 hidden:1 relation:2 her:1 transformed:1 interested:1 among:6 classification:2 rsn:4 animal:1 renewal:2 summed:1 mutual:6 marginal:2 stimulus:44 develops:1 few:1 composed:1 simultaneously:2 divergence:1 testify:1 message:1 brunei:1 investigate:6 highly:1 extreme:2 yielding:2 light:1 natl:1 accurate:2 necessary:1 theoretical:1 modeling:1 cover:1 measuring:1 maximization:1 deviation:1 rare:1 tishby:2 reported:2 encoders:1 considerably:1 huji:1 ie:16 interdisciplinary:2 rotman:1 receiving:1 decoding:2 infomax:1 michael:1 together:2 hopkins:1 continuously:1 reflect:2 recorded:3 central:1 iip:1 containing:1 leading:1 syst:1 de:1 retinal:3 coding:18 onset:1 performed:1 view:2 observing:2 analyze:1 start:1 collaborative:2 il:1 accuracy:4 roll:1 largely:2 characteristic:1 ensemble:1 correspond:1 conveying:1 parga:1 gawne:2 whenever:1 energy:1 frequency:2 mi:2 gain:1 auditory:24 sampled:1 fractional:3 dimensionality:2 brugge:1 actually:1 focusing:2 higher:3 response:2 anderson:1 biomedical:1 correlation:7 hand:4 iiiii:1 nonlinear:2 overlapping:1 cohn:1 reveal:2 usa:1 effect:7 hypothesized:2 contain:1 normalized:3 barlow:4 consisted:1 unbiased:1 adjacent:1 ll:3 inferior:6 naftali:1 generalized:1 theoretic:3 complete:1 latham:1 percent:1 consideration:1 common:1 functional:1 physical:1 ji:1 exponentially:2 volume:1 discussed:1 kluwer:1 marginals:1 significant:3 cambridge:1 ai:15 shuffling:2 inclusion:1 jg:1 cortex:8 behaving:1 multivariate:1 hadassah:1 exclusion:1 recent:1 moderate:1 termed:1 verlag:1 binary:1 refrain:1 life:1 transmitted:1 ministry:1 preceding:1 paradigm:3 redundant:7 signal:3 ii:6 encompass:1 full:1 sound:2 technical:1 calculation:4 long:1 divided:1 dkl:2 prediction:4 metric:1 histogram:2 sometimes:1 kernel:1 cell:44 addition:1 separately:3 spacing:1 baltimore:1 modality:1 rest:1 unlike:1 recording:3 cyber:1 electro:1 jordan:1 revealed:2 feedforward:1 independence:8 reduce:2 lesser:1 specialization:1 york:2 useful:3 latency:1 detailed:1 aimed:1 transforms:1 amount:2 dark:2 mid:1 ten:4 vejnarova:1 millisecond:4 estimated:1 neuroscience:3 per:1 xii:3 group:28 redundancy:46 sum:2 colliculus:5 respond:1 almost:2 reasonable:1 gerstein:1 bound:1 layer:1 yielded:1 activity:6 precisely:1 x2:1 scene:1 nearby:1 aspect:1 formulating:1 extracellular:1 department:2 according:2 yosef:2 across:2 smaller:1 joseph:1 making:2 intuitively:1 equation:2 previously:1 turn:2 count:10 mechanism:1 vreeswijk:1 ascending:5 end:2 meister:2 available:1 generalizes:1 rewritten:1 prerequisite:1 apply:1 spectral:1 shortly:1 thomas:1 responding:1 graphical:1 warland:1 prof:1 approximating:1 already:1 quantity:1 spike:19 occurs:1 primary:6 dependence:2 md:1 bialek:1 mapped:1 lateral:1 sci:1 capacity:1 berlin:1 alonso:1 cochlear:1 extent:2 collected:1 code:5 length:1 index:2 hebrew:2 lour:1 mostly:2 unfortunately:1 trace:1 negative:1 rise:1 allowing:1 upper:1 neuron:50 curved:1 gas:1 defining:1 communication:1 frame:6 station:8 treves:1 pair:6 connection:1 optimized:1 acoustic:13 subgroup:5 israeli:1 suggested:1 bar:5 below:2 regime:3 summarize:1 encompasses:1 saturation:1 natural:5 residual:1 representing:2 scheme:2 carried:1 extract:1 embodied:1 koberle:1 deviate:1 literature:1 acknowledgement:1 mixed:1 nucleus:1 conveyed:3 principle:1 editor:4 elsewhere:1 repeat:1 supported:1 infeasible:1 bias:2 emerge:1 anesthetized:1 fifth:1 van:2 regard:1 curve:5 dimension:1 xn:4 cortical:8 t5:1 sensory:10 nelken:2 commonly:1 simplified:1 spectro:1 implicitly:1 kullback:1 synergy:8 global:1 conclude:1 assumed:1 xi:6 quantifies:3 triplet:4 nature:3 channel:1 robust:1 ruyter:1 interact:2 investigated:2 complex:8 domain:3 linearly:2 neurosci:1 noise:1 repeated:2 allowed:1 convey:5 usrey:1 neuronal:2 representative:3 depicts:2 sub:1 xl:13 comput:1 bower:1 third:1 young:1 specific:1 showing:2 dk:1 physiological:1 evidence:3 essential:1 vapnik:1 gained:1 gat:1 magnitude:1 conditioned:5 easier:1 entropy:1 rg:1 ganglion:2 visual:2 prevents:1 springer:1 extracted:1 ma:1 conditional:1 bioi:1 identity:5 presentation:2 brenner:1 content:6 change:4 considerable:1 determined:1 corrected:3 kearns:1 total:1 hfsp:1 experimental:4 formally:1 latter:1 correlated:1 |
1,121 | 2,022 | Learning Lateral Interactions for
Feature Binding and Sensory Segmentation
Heiko Wersing
HONDA R&D Europe GmbH
Carl-Legien-Str.30, 63073 Offenbach/Main, Germany
heiko.wersing@hre-ftr.f.rd.honda.co.jp
Abstract
We present a new approach to the supervised learning of lateral interactions for the competitive layer model (CLM) dynamic feature binding
architecture. The method is based on consistency conditions, which were
recently shown to characterize the attractor states of this linear threshold
recurrent network. For a given set of training examples the learning problem is formulated as a convex quadratic optimization problem in the lateral interaction weights. An efficient dimension reduction of the learning
problem can be achieved by using a linear superposition of basis interactions. We show the successful application of the method to a medical
image segmentation problem of fluorescence microscope cell images.
1 Introduction
Feature binding has been proposed to provide elegant solution strategies to the segmentation problem in perception [11, 12, 14]. A lot of feature binding models have thus tried
to reproduce groping mechanisms like the Gestalt laws of visual perception, e.g. connectedness and good continuation, using temporal synchronization [12] or spatial coactivation
[9, 14] for binding. Quite generally in these models, grouping is based on lateral interactions between feature-representing neurons, which characterize the degree of compatibility
between features. Currently in most of the approaches this lateral interaction scheme is chosen heuristically, since the experimental data on the corresponding connection patterns in
the visual cortex is insufficient. Nevertheless, in more complex feature spaces this heuristic
approach becomes infeasible, raising the question for more systematic learning methods
for lateral interactions.
Mozer et al. [4] suggested supervised learning for a dynamic feature binding model of
complex-valued directional units, where the connections to hidden units guiding the grouping dynamics were adapted by recurrent backpropagation learning. The application was
limited to synthetic rectangle patterns. Hofmann et al. [2] considered unsupervised texture
segmentation by a pairwise clustering approach on feature vectors derived from Gabor filter
banks at different frequencies and orientations. In their model the pairwise feature compatibilities are determined by a divergence measure of the local feature distributions which
was shown to achieve good segmentation results for a range of image types. The problem
of segmentation can also be phrased as a labeling problem, where relaxation labeling algorithms have been used as a popular tool in a wide range of computer vision applications.
Pelillo & Refice [7] suggested a supervised learning method for the compatibility coefficients of relaxation labeling algorithms, based on minimizing the distance between a target
labeling vector and the output after iterating a fixed number of relaxation steps. The main
problem are multiple local minima arising in this highly nonlinear optimization problem.
Recent results have shown that linear threshold (LT) networks provide interesting architectures for combining properties of digital selection and analogue context-sensitive amplification [1, 13] with efficient hardware implementation options [1]. Xie et al. [16]
demonstrated how these properties can be used to learn winner-take-all competition between groups of neurons in an LT network with lateral inhibition. The CLM binding model
is implemented by a large-scale topographically organized LT network, and it was shown
that this leads to consistency conditions characterizing its binding states [14]. In this contribution we show how these conditions can be used to formulate a learning approach for
the CLM as a quadratic optimization problem. In Section 2 we briefly introduce the competitive layer binding model. Our learning approach is elaborated in Section 3. In Section
4 we show application results of the approach to a cell segmentation problem and give a
discussion in the final Section 5.
2 The CLM architecture
The CLM [9, 14] consists of a set of layers of feature-selective neurons (see Fig. 1). The
activity of a neuron at position in layer is denoted by , and a column denotes the set
of the neuron activities ,
, sharing a common position . With each column
a particular ?feature? is associated, which is described by a set of parameters like e.g. local
edge elements characterized by position and orientation
. A binding between
two features, represented by columns and
, is expressed by simultaneous activities
! and "#
! that share a common layer $ . All neurons in a column are equally
driven by an external input %& , which represents the significance of the detection of feature
by a preprocessing step. The afferent input %& is fed to the activities & with a connection
weight ' ( . Within each layer the activities are coupled via lateral connections ) "
which characterize the degree of compatibility between features and and which is a
symmetric function of the feature parameters, thus ) " *) "+ . The purpose of the layered
arrangement in the CLM is to enforce an assignment of the input features to the layers
by the dynamics, using the contextual information stored in the lateral interactions. The
unique assignment to a single layer is realized by a columnar Winner-Take-All (WTA)
circuit, which uses mutual symmetric inhibitory interactions with absolute strength ' ,
between neural activities - and ./ that share a common column . Due to the WTA
coupling, for a stable equilibrium state of the CLM only a neuron from one layer can be
active within each column [14]. The number of layers does not predetermine the number
of active groups, since for sufficiently many layers only those are active that carry a salient
group. The combination of afferent inputs and lateral and vertical interactions is combined
into the standard linear threshold additive activity dynamics
0
.1324&6587:9
';<%.=2?>
/
&@/.5A>
"
) " . "BDCE
(1)
where 7F&G!HJIK - . For ' large compared to the lateral weights ) " , the single active
neuron in a column reproduces its afferent input, ML % . As was shown [14], the stable
states of (1) satisfy the consistency conditions
>
"
/
) " "N/ O >
"
) " "P
for all
RQTU
S G$ VW
(2)
which express the assignment of a feature to the layer =
$ V with highest lateral support.
xrL
xr?L
layer L
vertical WTA
interaction
xr2 lateral interaction xr?2
layer 2
xr1
xr?1
layer 1
hr
hr?
input
Figure 1: The competitive layer model architecture (see text for description).
3 Learning of CLM Lateral Interactions
Formulation of the Learning Problem. The overall task of the learning algorithm is
to adapt the lateral interactions, given by the interaction coefficients ) " , such that the
CLM architecture performs appropriate segmentation on the labeled training data and also
generalizes to new test data. We assume that the training data consists of a set of
labeled training patterns ,
, where each pattern
consists of a subset
of
different features with their corresponding labels $ . For
each labeled training pattern a target labeling vector is constructed by choosing
Q,U
for all
S $ V
(3)
/
for the labeled columns, assuming %
. Columns for features which are not contained
. In
for all S
in the training pattern are filled with zeroes according to
the following indices run over all possible features, e.g. all edges of different orientations at different image positions, while V run over the subset of features realized in
a particular pattern, e.g. only one oriented edge at each image position. The assignment
vectors =
form the basis of the learning approach since they represent the
target activity distribution, which we want to obtain after iterating the CLM with appropriately adjusted lateral interactions. In the following the abbreviation $ for $ V is used to
keep the notation readable.
The goal of the learning process is to make the training patterns consistent, which is in
accordance with (2) expressed by the inequalities
>
"
/
) " " / OT>
"!
"
) " "
for all
# @Q! S
$
(4)
These 2A
inequalities define the learning problem that we want to solve in the
following. Let us develop a more compact notation. We can rewrite (4) as
>
& $# % / ) %#
" "
O!
for all
# @QTU
S G$
(5)
#$% "
&
@/
where # ' " )( +* ( / # + " / 2,( # + "P" - . The form of the inequalities can be simplified by
introducing multiindices . and / which correspond to /10 32E , .40 5# RQE and
# &98 & #$ %/ . The index . runs over all 2
! consistency relations de)5670 ) % " , 6 0
"
8 with components
fined
for
the
labeled
columns
of the assignment vectors. The vectors :
&;6 8 are called consistency vectors
and represent the consistency constraints for the lateral
interaction. The index / runs over all entries in the lateral interaction matrix. The vector
)
@)
) ) with components contains the corresponding
matrix entries. The inequalities (4) can then be written in the form
>
& 6 8 )56
6
O
.F
for all
(6)
This illustrates the nature of the learning problem. The problem is to find a weight vector
which leads to a lateral interaction matrix, such that the consistency vectors lie in the opposite half space of the weight state space. Since the conditions (6) determine the attractivity
to achieve
of the training patterns, it is customary to introduce a positive margin
greater robustness. This gives the target inequalities
>
& 6 8 )5665
6
O!
.F
for all
(7)
which we want to solve in for given training data. If the system of inequalities admits a
solution for it is called compatible. If there is no satisfying all constraints, the system
is called incompatible.
% #
Superposition of Basis Interactions. If the number of features
is large, the number
of parameters in the complete interaction matrix ) " may be too large to be robustly estimated from a limited number of training examples. To achieve generalization from the
training data, it is necessary to reduce the number of parameters which have to be adapted
during learning. This is also useful to incorporate a priori knowledge into the interaction.
An example is to choose basis functions which incorporate invariances such as translation
and rotation invariance, or which satisfy the constraint that the interaction is equal in all
layers. A simple but powerful approach is to choose a set of fixed basis interactions
with compatibilities "
J
W , with an interaction ) " obtained by linear
superposition
%#
' #
#
) % " *>
% # "
>
6
(8)
with weight coefficients
W . Now the learning problem of solving the inequalities (7) can be recast in the new free parameters
. After inserting (8) into (7) we
obtain the transformed problem
& 68
6
8
(9)
5 OA for all F
.
6
8
&8
8 in the basis interacwhere 1! 6 6 6 is the component of the consistency vector :
tion . The basis interactions can thus be used to reduce the dimensionality of the learning
>
>
51 >
problem. To avoid any redundancy, the basis interactions should be linearly independent.
Although the functions are here denoted ?basis? functions,
they need neither be orthogonal
.
nor span the whole space of interactions
Quadratic Consistency Optimization. The generic case in any real world application is
that the majority of training vectors contains relevant information, while single spurious
vectors may be present due to noise or other disturbing factors. Consequently, in most
applications the equations (7) or (9) will be incompatible and can only be satisfied approximately. This will be especially the case, if a low-dimensional embedding is used for the
basis function templates as described above. We therefore suggest to adapt the interactions
by minimizing the following convex cost function
QCO
>
8
9 >
6
& 68 ) 6
5 C
(10)
A similar minimization approach was suggested for the imprinting of attractors for the
Brain-State-in-a-Box (BSB) model [8], and a recent study has shown that the approach is
competitive with other methods for designing BSB associative memories [6].
For a fixed positive margin A , the cost function (10) is minimized by making the inner
products
of the weight vector and the consistency vectors negative. The global minimum
is attained if the inner products are all equal to 2 , which can be interpreted
with QCO
such that all consistency inequalities are fulfilled in an equal manner. Although this additional regularizing constraint is hard to justify on theoretical grounds, the later application
shows that it works quite well for the application examples considered.
If we insert the expansion of in the basis of function templates we obtain according to (8)
QCO
>
9 >
8
8
5 C
(11)
which results in a -dimensional convex quadratic minimization problem in the
parameters. The coefficients , which give the components of the training patterns in the basis
@/
interactions, are given by
" "N/ " 2
" " " . The quadratic
optimization problem is then given by minimizing
8
8
! 6 & 6 8 6
!
!
5 > 5
(12)
8
8 8
where ! 8 and ! 8 . If the coefficients are unconstrained,
then
by solving the linear system of equations
the minimum
! of 5 (12) can beforobtained
all .
QCO
>
4 Application to Cell Segmentation
The automatic detection and segmentation of individual cells in fluorescence micrographs
is a key technology for high-throughput analysis of immune cell surface proteins [5]. The
strong shape variability of cells in tissue, however, poses a strong challenge to any automatic recognition approach. Figure 2a shows corresponding fluorescence microscopy
images from a tissue section containing lymphocyte cells (courtesy W. Schubert). In the
bottom row corresponding image patches are displayed, where individual cell regions were
manually labeled to obtain training data for the learning process.
For each of the image patches, a training vector consists of a list of labeled edge features
parameterized by E
, where E is the position in the image and E is a unit local
edge orientation vector computed from the intensity gradient. For a
pixel image
this amounts to a set of labeled edge features. Since the figure-ground separating mechanism as implemented by the CLM [14] is also used for this cell segmentation application,
features which are not labeled as part of a cell obtain the corresponding background label,
given by
. Each training pattern contains one additional free layer, to enable the
learning algorithm to generalize over the number of layers.
The lateral interaction to be adapted is decomposed into the following weighted basis components: i) A constant negative interaction between all features, which facilitates group
separation, ii) a self-coupling interaction in the background layer which determines the attractivity of the background for figure-ground segmentation, and iii) an angular interaction
with limited range, which is in itself decomposed into templates, capturing the interaction for a particular combination of the relative angles between two edges. This angular decomposition is done using a discretization of the space of orientations, turning the
unit-vector representation into an angular orientation variable
. To achieve rotation invariance of the interaction, it is only dependent on the edge orientations relative
6
8
7
8
9
7
6
5
4
4
2
3
5
2
3
a) Manually labelled training patterns
b) Grouping results after learning
Figure 2: a) Original images and manually labeled training patterns from a fluorescence
micrograph. b) Test patterns and resulting CLM segmentation with learned lateral interaction. Grayscale represents different layer activations, where a total of 20 layers plus one
background layer (black) was used.
2
to their mutual position difference vector
. The angles and are dis
cretized by partitioning the interval
into 8 subintervals. For each combination of
the two discretized edge orientations there is an interaction template generated, which is
only responding in this combined orientation interval. Thus the angular templates do not
overlap in the combined space, i.e. if W
for a particular
, then
P
for all S . Since the interaction must be symmetric under feature exchange, this does not result in
different combinations, but
only 36 independent templates. Apart form the discretization, the interaction represents
the most arbitrary angular-dependent interaction within the local neighborhood, which is
symmetric under feature exchange. We use two sets of angular templates for
O
and
O
O respectively, where is the maximal local interaction range. With
the abovementioned two components, the resulting optimization problem is 36+36+2=74dimensional. Figure 3 compares the optimized interaction field to earlier heuristic lateral
interactions for contour grouping. See [15] for a more detailed discussion.
#
#
The performance of the learning approach was investigated by choosing a small number
of the manually labeled patterns as training patterns. For all the training examples we
used, the resulting inequalities (9) were in fact incompatible, rendering a direct solution of
(9) infeasible. After training was completed by minimizing (12), a new image patch was
selected as a test pattern and the CLM grouping was performed with the lateral interaction
learned before, using the dynamical model as described in [14]. The quadratic consistency
optimization was performed as described in the previous section, exploring the free margin
parameter . For a set of two training patterns as shown in Fig. (2)a with a total of 1600
features each, a learning sweep takes about 4 minutes on a standard desktop computer.
Typical segmentation results obtained with the quadratic consistency optimization approach are shown in Figure 2b, where the margin was given by *
. The grouping
results were not very sensitive to in a range of O O
. The grouping results show
a good segmentation performance where most of the salient cells are detected as single
groups. There are some spurious groups where a dark image region forms an additional
group and some smaller cells are rejected into the background layer. Apart from these
minor errors, the optimization has achieved an adequate balancing of the different lateral
interaction components for this segmentation task.
n2
n1
n2
n2
d
n1
p
1
p
2
b) Edge parameters
a) Plotting scheme
c) Standard continuity interaction field
d) Learned interaction field
Figure 3: Comparison between heuristic continuity grouping interaction field and a learned
lateral interaction field for cell segmentation. The interaction depends on the difference
vector and two unit vectors , shown in b), encoding directed orientation. a) ex
plains the interaction visualizations c) and d) by showing a magnification of the plot c) of
the interaction field of a single horizontal edge pointing to the left. The plots are generated by computing the interaction of the central directed edge with directed edges of all
directions (like a cylindrical plot) at a spatial grid. Black edges share excitatory, white
edges share inhibitory interaction with the central edge and length codes for interaction
strength. The cocircular continuity field in c) depends on position and orientation but is not
direction-selective. It supports pairs of edges which are cocircular, i.e. lie tangentially to a
common circle and has been recently used for contour segmentation [3, 14]. The learned
lateral interaction field is shown in d). It is direction-selective and supports pairs of edges
which ?turn right?. The strong local support is balanced by similarly strong long-range
inhibition.
5 Discussion
The presented results show that appropriate lateral interactions can be obtained for the
CLM binding architecture from the quadratic consistency optimization approach. The only
a priori conditions which were used for the template design were the properties of locality,
symmetry, and translation as well as rotation invariance. This supervised learning approach
has clear advantages over the manual tuning of complex feature interactions in complex feature spaces with many parameters. We consider this as an important step towards practical
applicability of the feature binding concept.
The presented quadratic consistency optimization method is based on choosing equal margins for all consistency inequalities. There exist other approaches to large margin classifica-
tion, like support vector machines [10], where more sophisticated methods were suggested
for appropriate margin determination. The application of similar methods to the supervised
learning of CLM interactions provides an interesting field for future work.
Acknowledgments: This work was supported by DFG grant GK-231 and carried out at the
Faculty of Technology, University of Bielefeld. The author thanks Helge Ritter and Tim
Nattkemper for discussions and Walter Schubert for providing the cell image data.
References
[1] R. Hahnloser, R. Sarpeshkar, M. A. Mahowald, R. J. Douglas, and H. S. Seung. Digital selection
and analogue amplification coexist in a cortex-inspired silicon circuit. Nature, 405:947?951,
2000.
[2] T. Hofmann, J. Puzicha, and J. Buhmann. Unsupervised texture segmentation in a deterministic
annealing framework. IEEE Trans. Pattern Analysis and Machine Intelligence, 20(8):803?818,
1998.
[3] Z. Li. A neural model of contour integration in the primary visual cortex. Neural Computation,
10:903?940, 1998.
[4] M. Mozer, R. S. Zemel, M. Behrmann, and C. K. I. Williams. Learning to segment images
using dynamic feature binding. Neural Computation, 4(5):650?665, 1992.
[5] T. W. Nattkemper, H. Ritter, and W. Schubert. A neural classificator enabling high-throughput
topological analysis of lymphocytes in tissue sections. IEEE Trans. Inf. Techn. in Biomed.,
5(2):138?149, 2001.
[6] J. Park, H. Cho, and D. Park. On the design of BSB associative memories using semidefinite
programming. Neural Computation, 11:1985?1994, 1999.
[7] M. Pelillo and M Refice. Learning compatibility coefficients for relaxation labeling processes.
IEEE Trans. Pattern Analysis and Machine Intelligence, 16(9):933?945, 1994.
[8] Renzo Perfetti. A synthesis procedure for Brain-State-in-a-Box neural networks. IEEE Transactions on Neural Networks, 6(5):1071?1080, September 1995.
[9] H. Ritter. A spatial approach to feature linking. In Proc. International Neural Network Conference Paris Vol.2, pages 898?901, 1990.
[10] V. Vapnik. The nature of statistical learning theory. Springer, New York, 1995.
[11] C. von der Malsburg. The what and why of binding: The modeler?s perspective. Neuron,
24:95?104, 1999.
[12] D. Wang and D. Terman. Image segmentation based on oscillatory correlation. Neural Computation, 9(4):805?836, 1997.
[13] H. Wersing, W.-J. Beyn, and H. Ritter. Dynamical stability conditions for recurrent neural networks with unsaturating piecewise linear transfer functions. Neural Computation, 13(8):1811?
1825, 2001.
[14] H. Wersing, J. J. Steil, and H. Ritter. A competitive layer model for feature binding and sensory
segmentation. Neural Computation, 13(2):357?387, 2001.
[15] Heiko Wersing. Spatial Feature Binding and Learning in Competitive Neural Layer Architectures. PhD thesis, University of Bielefeld, 2000. Published by Cuvillier, Goettingen.
[16] X. Xie, R. Hahnloser, and H.S. Seung. Learning winner-take-all competition between groups of
neurons in lateral inhibition networks. In Advances in Neural Information Processing Systems,
volume 13. The MIT Press, 2001.
| 2022 |@word cylindrical:1 faculty:1 briefly:1 heuristically:1 tried:1 decomposition:1 carry:1 reduction:1 contains:3 contextual:1 discretization:2 activation:1 written:1 must:1 additive:1 shape:1 hofmann:2 plot:3 half:1 selected:1 intelligence:2 desktop:1 provides:1 honda:2 constructed:1 direct:1 consists:4 manner:1 introduce:2 pairwise:2 nor:1 brain:2 discretized:1 inspired:1 decomposed:2 str:1 becomes:1 notation:2 circuit:2 what:1 interpreted:1 temporal:1 partitioning:1 unit:5 medical:1 grant:1 positive:2 before:1 local:7 accordance:1 encoding:1 micrographs:1 approximately:1 connectedness:1 black:2 plus:1 co:1 limited:3 range:6 coactivation:1 directed:3 unique:1 practical:1 acknowledgment:1 bsb:3 imprinting:1 backpropagation:1 xr:3 procedure:1 gabor:1 suggest:1 protein:1 selection:2 layered:1 coexist:1 context:1 deterministic:1 demonstrated:1 courtesy:1 williams:1 convex:3 formulate:1 embedding:1 stability:1 target:4 programming:1 carl:1 us:1 designing:1 element:1 satisfying:1 recognition:1 magnification:1 labeled:11 bottom:1 wang:1 region:2 highest:1 balanced:1 mozer:2 seung:2 dynamic:6 rewrite:1 solving:2 segment:1 topographically:1 basis:13 represented:1 sarpeshkar:1 walter:1 detected:1 zemel:1 labeling:6 choosing:3 neighborhood:1 quite:2 heuristic:3 valued:1 solve:2 itself:1 final:1 associative:2 advantage:1 interaction:60 product:2 maximal:1 inserting:1 relevant:1 combining:1 achieve:4 amplification:2 description:1 competition:2 tim:1 coupling:2 recurrent:3 develop:1 pose:1 minor:1 qt:1 pelillo:2 strong:4 implemented:2 direction:3 filter:1 enable:1 exchange:2 generalization:1 goettingen:1 adjusted:1 insert:1 exploring:1 sufficiently:1 considered:2 ground:3 equilibrium:1 pointing:1 purpose:1 proc:1 label:2 currently:1 superposition:3 fluorescence:4 sensitive:2 tool:1 weighted:1 minimization:2 mit:1 heiko:3 avoid:1 derived:1 dependent:2 hidden:1 relation:1 spurious:2 reproduce:1 selective:3 transformed:1 germany:1 biomed:1 compatibility:6 overall:1 schubert:3 orientation:11 pixel:1 denoted:2 priori:2 spatial:4 integration:1 mutual:2 equal:4 field:9 manually:4 represents:3 park:2 unsupervised:2 throughput:2 future:1 minimized:1 terman:1 piecewise:1 oriented:1 divergence:1 individual:2 dfg:1 attractor:2 n1:2 detection:2 highly:1 semidefinite:1 clm:15 edge:18 necessary:1 orthogonal:1 filled:1 circle:1 theoretical:1 column:10 earlier:1 assignment:5 mahowald:1 cost:2 introducing:1 applicability:1 subset:2 entry:2 lymphocyte:2 successful:1 too:1 characterize:3 stored:1 refice:2 synthetic:1 combined:3 cho:1 thanks:1 international:1 systematic:1 ritter:5 synthesis:1 von:1 central:2 satisfied:1 thesis:1 containing:1 choose:2 external:1 li:1 de:1 coefficient:6 satisfy:2 nattkemper:2 afferent:3 depends:2 tion:2 later:1 lot:1 performed:2 competitive:6 option:1 elaborated:1 contribution:1 tangentially:1 correspond:1 directional:1 generalize:1 published:1 tissue:3 simultaneous:1 oscillatory:1 sharing:1 manual:1 frequency:1 associated:1 modeler:1 popular:1 knowledge:1 dce:1 dimensionality:1 segmentation:21 organized:1 sophisticated:1 attained:1 supervised:5 xie:2 classifica:1 formulation:1 done:1 box:2 angular:6 rejected:1 correlation:1 horizontal:1 nonlinear:1 continuity:3 concept:1 symmetric:4 white:1 during:1 self:1 complete:1 performs:1 image:16 recently:2 steil:1 common:4 rotation:3 winner:3 jp:1 volume:1 linking:1 silicon:1 rd:1 unconstrained:1 consistency:16 automatic:2 grid:1 similarly:1 tuning:1 immune:1 stable:2 europe:1 cortex:3 surface:1 inhibition:3 recent:2 perspective:1 inf:1 driven:1 apart:2 inequality:10 der:1 minimum:3 greater:1 additional:3 attractivity:2 determine:1 ii:1 multiple:1 characterized:1 adapt:2 determination:1 fined:1 long:1 equally:1 predetermine:1 ftr:1 vision:1 represent:2 achieved:2 cell:14 microscope:1 microscopy:1 background:5 want:3 interval:2 annealing:1 appropriately:1 ot:1 elegant:1 facilitates:1 iii:1 rendering:1 architecture:7 opposite:1 reduce:2 inner:2 york:1 adequate:1 generally:1 iterating:2 useful:1 detailed:1 clear:1 xr2:1 amount:1 dark:1 hardware:1 continuation:1 exist:1 inhibitory:2 xr1:1 estimated:1 arising:1 fulfilled:1 hji:1 vol:1 express:1 group:8 redundancy:1 salient:2 key:1 threshold:3 nevertheless:1 micrograph:1 neither:1 douglas:1 rectangle:1 relaxation:4 run:4 angle:2 parameterized:1 powerful:1 bielefeld:2 patch:3 separation:1 incompatible:3 capturing:1 layer:26 quadratic:9 topological:1 activity:8 adapted:3 strength:2 constraint:4 phrased:1 span:1 according:2 combination:4 smaller:1 helge:1 wta:3 making:1 equation:2 visualization:1 turn:1 mechanism:2 fed:1 generalizes:1 enforce:1 appropriate:3 generic:1 robustly:1 robustness:1 customary:1 original:1 denotes:1 clustering:1 responding:1 completed:1 readable:1 malsburg:1 rqt:1 especially:1 sweep:1 question:1 arrangement:1 realized:2 strategy:1 primary:1 abovementioned:1 september:1 gradient:1 distance:1 lateral:28 oa:1 majority:1 separating:1 assuming:1 length:1 code:1 index:3 insufficient:1 providing:1 minimizing:4 gk:1 negative:2 implementation:1 design:2 vertical:2 neuron:10 enabling:1 displayed:1 variability:1 arbitrary:1 intensity:1 pair:2 paris:1 connection:4 optimized:1 raising:1 learned:5 trans:3 suggested:4 dynamical:2 perception:2 pattern:20 challenge:1 recast:1 memory:2 analogue:2 overlap:1 turning:1 hr:2 buhmann:1 representing:1 scheme:2 technology:2 carried:1 coupled:1 text:1 relative:2 law:1 synchronization:1 interesting:2 digital:2 degree:2 consistent:1 plotting:1 bank:1 share:4 balancing:1 translation:2 row:1 compatible:1 excitatory:1 supported:1 free:3 infeasible:2 dis:1 cocircular:2 wide:1 template:8 characterizing:1 absolute:1 dimension:1 plain:1 world:1 contour:3 sensory:2 author:1 disturbing:1 preprocessing:1 simplified:1 gestalt:1 transaction:1 compact:1 keep:1 ml:1 reproduces:1 active:4 global:1 grayscale:1 why:1 learn:1 nature:3 transfer:1 subintervals:1 symmetry:1 expansion:1 investigated:1 complex:4 significance:1 main:2 linearly:1 whole:1 noise:1 n2:3 gmbh:1 fig:2 position:8 guiding:1 lie:2 behrmann:1 minute:1 showing:1 list:1 admits:1 grouping:8 vapnik:1 texture:2 phd:1 illustrates:1 margin:7 columnar:1 locality:1 lt:3 visual:3 expressed:2 contained:1 binding:16 springer:1 determines:1 abbreviation:1 hahnloser:2 goal:1 formulated:1 consequently:1 towards:1 labelled:1 hard:1 wersing:5 determined:1 typical:1 justify:1 hre:1 called:3 total:2 techn:1 invariance:4 experimental:1 puzicha:1 support:5 incorporate:2 regularizing:1 ex:1 |
1,122 | 2,023 | Pranking with Ranking
Koby Crammer and Yoram Singer
School of Computer Science & Engineering
The Hebrew University, Jerusalem 91904, Israel
{kobics,singer}@cs.huji.ac.il
Abstract
We discuss the problem of ranking instances. In our framework
each instance is associated with a rank or a rating, which is an
integer from 1 to k. Our goal is to find a rank-prediction rule that
assigns each instance a rank which is as close as possible to the
instance's true rank. We describe a simple and efficient online algorithm, analyze its performance in the mistake bound model, and
prove its correctness. We describe two sets of experiments, with
synthetic data and with the EachMovie dataset for collaborative
filtering. In the experiments we performed, our algorithm outperforms online algorithms for regression and classification applied to
ranking.
1
Introduction
The ranking problem we discuss in this paper shares common properties with both
classification and regression problems. As in classification problems the goal is to
assign one of k possible labels to a new instance. Similar to regression problems,
the set of k labels is structured as there is a total order relation between the labels.
We refer to the labels as ranks and without loss of generality assume that the ranks
constitute the set {I, 2, .. . , k} . Settings in which it is natural to rank or rate instances rather than classify are common in tasks such as information retrieval and
collaborative filtering. We use the latter as our running example. In collaborative
filtering the goal is to predict a user's rating on new items such as books or movies
given the user's past ratings of the similar items. The goal is to determine whether
a movie fan will like a new movie and to what degree, which is expressed as a
rank. An example for possible ratings might be, run-to-see , very-good, good,
only-if-you-must, and do-not-bother. While the different ratings carry meaningful semantics, from a learning-theoretic point of view we model the ratings as a
totally ordered set (whose size is 5 in the example above).
The interest in ordering or ranking of objects is by no means new and is still the
source of ongoing research in many fields such mathematical economics, social science, and computer science. Due to lack of space we clearly cannot cover thoroughly
previous work related to ranking. For a short overview from a learning-theoretic
point of view see [1] and the references therein. One of the main results of [1] underscores a complexity gap between classification learning and ranking learning. To
sidestep the inherent intractability problems of ranking learning several approaches
have been suggested. One possible approach is to cast a ranking problem as a
regression problem. Another approach is to reduce a total order into a set of pref-
Correct interval
#l \
I
Figure 1: An Illustration of the update rule.
erences over pairs [3, 5]. The first case imposes a metric on the set of ranking rules
which might not be realistic, while the second approach is time consuming since it
requires increasing the sample size from n to O(n 2 ).
In this paper we consider an alternative approach that directly maintains a totally
ordered set via projections. Our starting point is similar to that of Herbrich et. al [5]
in the sense that we project each instance into the reals. However, our work then
deviates and operates directly on rankings by associating each ranking with distinct
sub-interval of the reals and adapting the support of each sub-interval while learning. In the next section we describe a simple and efficient online algorithm that
manipulates concurrently the direction onto which we project the instances and
the division into sub-intervals. In Sec. 3 we prove the correctness of the algorithm
and analyze its performance in the mistake bound model. We describe in Sec. 4
experiments that compare the algorithm to online algorithms for classification and
regression applied to ranking which demonstrate the merits of our approach.
2
The PRank Algorithm
This paper focuses on online algorithms for ranking instances. We are given a
sequence (Xl, yl), ... , (xt , yt) , ... of instance-rank pairs. Each instance xt is in IR n
and its corresponding rank yt is an element from finite set y with a total order
relation. We assume without loss of generality that y = {I , 2, ... ,k} with ">"
as the order relation. The total order over the set Y induces a partial order over
the instances in the following natural sense. We say that xt is preferred over X S
if yt > yS. We also say that xt and x S are not comparable if neither yt > yS nor
yt < yS. We denote this case simply as yt = yS. Note that the induced partial order
is of a unique form in which the instances form k equivalence classes which are totally
ordered l . A ranking rule H is a mapping from instances to ranks, H : IR n -+ y.
The family of ranking rules we discuss in this paper employs a vector w E IR n and
a set of k thresholds bl :::; ... :::; bk - l :::; bk = 00. For convenience we denote by
b = (b l , . .. ,bk-d the vector of thresholds excluding bk which is fixed to 00. Given a
new instance x the ranking rule first computes the inner-product between w and x .
The predicted rank is then defined to be the index of the first (smallest) threshold
br for which w . x < br . This type of ranking rules divide the space into parallel
equally-ranked regions: all the instances that satisfy br - l < W? x < br are assigned
the same rank r. Formally, given a ranking rule defined by wand b the predicted
rank of an instance x is, H(x) = minrE{l, ... ,k}{r : w . x - br < O}. Note that the
above minimum is always well defined since we set bk = 00.
The analysis that we use in this paper is based on the mistake bound model for
online learning. The algorithm we describe works in rounds. On round t the learning
algorithm gets an instance xt. Given x t , the algorithm outputs a rank , il = minr {r :
W? x - br < O}. It then receives the correct rank yt and updates its ranking rule by
modifying wand b. We say that our algorithm made a ranking mistake if il f:. yt.
IFor a discussion of this type of partial orders see [6] .
Initialize: Set wI = 0 , b~ , ... ,
Loop: Fort=1 ,2, ... ,T
?
?
?
?
bLl = 0, bl =
00 .
Get a new rank-value xt E IRn.
Predict fl = min r E{I, ... ,k} {r: w t . xt - b~ < o}.
Get a new label yt.
If fl t- yt update w t (otherwise set w t+! = w t , \;fr : b~+! = bn :
1. For r = 1, ... , k - 1
If yt :::; r Then y~ = -1
Else y~ = 1.
2. For r = 1, ... , k - 1
If (w t . xt - b~)y~ :::; 0 Then T; = y~
Else
= o.
3. Update w t+! f- w t + CL r T;)xt.
For r = 1, . .. , k - 1 update:
b~+1 f- b~ -
T;
T;
Output:
H(x) = min r E{1, ... ,k} {r : w T +! . x - b;+!
< O}.
Figure 2: The PRank algorithm.
We wish to make the predicted rank as close as possible to the true rank. Formally,
the goal of the learning algorithm is to minimize the ranking-loss which is defined to
be the number of thresholds between the true rank and the predicted rank. Using
the representation of ranks as integers in {I ... k}, the ranking-loss after T rounds
is equal to the accumulated difference between the predicted and true rank-values,
'?'[=1 W- yt I. The algorithm we describe updates its ranking rule only on rounds
on which it made ranking mistakes. Such algorithms are called conservative.
We now describe the update rule of the algorithm which is motivated by the perceptron algorithm for classification and hence we call it the PRank algorithm (for
Perceptron Ranking). For simplicity, we omit the index of the round when referring to an input instance-rank pair (x, y) and the ranking rule wand h. Since
b1 :::; b2 :::; ... :::; bk - 1 :::; bk then the predicted rank is correct if w . x > br for
r = 1, ... ,y - 1 and w . x < br for y, . .. , k - 1. We represent the above inequalities by expanding the rank y into into k - 1 virtual variables Yl , ... ,Yk-l. We set
Yr = +1 for the case W? x > br and Yr = -1 for w . x < br . Put another way, a
rank value y induces the vector (Yl, ... , Yk-d = (+1, ... , +1 , -1 , ... , -1) where the
maximal index r for which Yr = +1 is y-1. Thus, the prediction of a ranking rule is
correct if Yr(w? x - br ) > 0 for all r. If the algorithm makes a mistake by ranking x
as fj instead of Y then there is at least one threshold, indexed r, for which the value
of W? x is on the wrong side of br , i.e. Yr(w? x - br ) :::; O. To correct the mistake, we
need to "move" the values of W? x and br toward each other. We do so by modifying
only the values of the br's for which Yr (w . x - br ) :::; 0 and replace them with br - Yr.
We also replace the value of w with w + ('? Yr)x where the sum is taken over the
indices r for which there was a prediction error, i.e., Yr (w . x - br ) :::; o.
An illustration of the update rule is given in Fig 1. In the example, we used the
set Y = {I ... 5}. (Note that b5 = 00 is omitted from all the plots in Fig 1.) The
correct rank of the instance is Y = 4, and thus the value of w . x should fall in the
fourth interval, between b3 and b4 . However, in the illustration the value of w . x
fell below b1 and the predicted rank is fj = 1. The threshold values b1 , b2 and b3 are
a source of the error since the value of b1 , b2 , b3 is higher then W? x. To mend the
mistake the algorithm decreases b1 , b2 and b3 by a unit value and replace them with
b1 -1 , b2 -1 and b3 -1. It also modifies w to be w+3x since '?r:Yr(w.x- br):SOYr = 3.
Thus, the inner-product W? x increases by 311x11 2 . This update is illustrated at the
middle plot of Fig. 1. The updat ed prediction rule is sketched on the right hand
side of Fig. 1. Note that after the update, the predicted rank of x is Y = 3 which is
closer to the true rank y = 4. The pseudocode of algorithm is given in Fig 2.
To conclude this section we like to note that PRank can be straightforwardly combined with Mercer kernels [8] and voting techniques [4] often used for improving the
performance of margin classifiers in batch and online settings.
3
Analysis
Before we prove the mistake bound of the algorithm we first show that it maintains a consistent hypothesis in the sense that it preserves the correct order of the
thresholds. Specifically, we show by induction that for any ranking rule that can
be derived by the algorithm along its run, (w 1 , b 1 ) , ... , (w T +1 , b T +1) we have
that b~ :S ... :S bL1 for all t. Since the initialization of the thresholds is such that
b~ :S b~ :S ... :S bL1' then it suffices to show that the claim holds inductively. For
simplicity, we write the updating rule of PRank in an alternative form. Let [7f] be
1 if the predicate 7f holds and 0 otherwise. We now rewrite the value of
(from
Fig. 2) as
= y~[(wt . xt - bny~ :S 0]. Note that the values of b~ are integers for
all r and t since for all r we initialize b; = 0, and b~+l - b~ E {-1 , 0, +1}.
T;
T;
Lemma 1 (Order Preservation) Let w t and b t be the current ranking rule,
where bi :S .. . :S bL1' and let (xt,yt) be an instance-rank pair fed to PRank
on round t. Denote by wt+1 and bt+1 the resulting ranking rule after the update of
PRank, then bi+1 :S ... :S bt~ll?
Proof: In order to show that PRank maintains the order of the thresholds we
use the definition of the algorithm for y~, namely we define y~ = +1 for r < yt and
y~ = -1 for r 2:: yt. We now prove that b~t~ 2:: b~+l for all r by showing that
b~+l - b~
2::
y~+1[(wt . xt - b~+1)Y;+l
:S 0] - y;[(wt . xt - b;)y; :S 0], (1)
which we obtain by substituting the values of bt+1. Since b~+1 :S b~ and b~ ,b~+1 E Z
we get that the value of b~+1 - b~ on the left hand side of Eq. (1) is a non-negative
integer. Recall that y~ = 1 if yt > r and y~ = -1 otherwise, and therefore,
y~+l :S y~. We now analyze two cases. We first consider the case y~+1 :j:. y~ which
implies that y~+l = -1, y~ = +1. In this case, the right hand-side of Eq. (1) is at
most zero, and the claim trivially holds. The other case is when y~+1 = y~. Here
we get that the value of the right hand-side Eq. (1) cannot exceed 1. We therefore
have to consider only the case where b~ = b~+1 and y~+1 = y~. But given these two
conditions we have that y~+1[(wt. xt - b~+1)Y~+1 < 0] and y~[(wt. xt - b~)y~ < 0]
are equal. The right hand side of Eq. (1) is now zero and the inequality holds with
?
equality.
In order to simplify the analysis of the algorithm we introduce the following notation. Given a hyperplane wand a set of k -1 thresholds b we denote by v E ~n+k-1
the vector which is a concatenation of wand b that is v = (w, b). For brevity we refer to the vector vas a ranking rule. Given two vectors v' = (w', b ' ) and v = (w, b)
we have v' . v = w' . w + b' . b and IIvl1 2 = IIwl1 2 + IlbW.
Theorem 2 (Mistake bound) Let (xl, y1), ... , (x T , yT) be an input sequence for
PRank where xt E ~n and yt E {l. .. k}. Denote by R2 = maxt Ilxtl12. Assume
that there is a ranking rule v* = (w* , b*) with
:S ... :S bk- 1 of a unit norm that
classifies the entire sequence correctly with margin "( = minr,t{ (w* . xt - b;)yn > o.
Then, the rank loss of the algorithm '?;=1 Iyt - yt I, is at most (k - 1) (R 2 + 1) / "(2 .
br
Proof: Let us fix an example (xt, yt) which the algorithm received on round t.
By definition the algorithm ranked the example using the ranking rule v t which is
composed of w t and the thresholds b t . Similarly, we denote by vt+l the updated
rule (wt+l , bt+l) after round t.
That
is wt+l = w t + ("
and bt+l
= btr - Ttr
,
ur Tt)xt
r
r
for r = 1, 2, ... , k - 1. Let us denote by n t =
yt 1the difference between the true
rank and the predicted rank. It is straightforward to verify that nt = 2:=r ITn Note
that if there wasn't a ranking mistake on round t then
= for r = 1, ... , k-1, and
thus also nt = 0. To prove the theorem we bound 2:=t nt from above by bounding
IIvtl12 from above and below. First, we derive a lower bound on IIvtl12 by bounding
v* . v H1 . Substituting the values of w H1 and b H1 we get,
W-
T;
?
k-1
v* . vt+l = v* . v t
+ 2:= T;
(w* . xt -
b;)
(2)
r=1
T; from the pseudocode in Fig. 2 we need to analyze two cases. If (w t ?xt - b~)y; :::; ?
then T; = y;. Using the assumption that v* ranks the data correctly with a margin
We further bound the right term by considering two cases. Using the definition of
of at least "( we get that T;(W* . xt - b;) ~ "(. For the other case for which
(w t . xt - b;)y; > we have T; = and thus T;(W* . xt - b;) = 0. Summing
now over r we get,
?
?
k-1
2:= T; (w* . x t - b;) ~ nt"( .
(3)
r= 1
Combining Eq. (2) and Eq. (3) we get v* . vt+l ~ v* . v t + nt"(. Unfolding the
sum, we get that after T rounds the algorithm satisfies, v* . v T+ 1 ~ 2:=t nt"( =
"( 2:=t nt. Plugging this result into Cauchy-Schwartz inequality, (1Iv T+1 11 21Iv* 112 ~
(vT+l . v*) 2) and using the assumption that v* is of a unit norm we get the lower
bound, IIv T+ll1 2 ~ (2:=t nt)2 "(2.
Next, we bound the norm of v from above. As before, assume that an example
(xt, yt) was ranked using the ranking rule v t and denote by vt+l the ranking rule
after the round. We now expand the values ofw t+1 and bt+l in the norm ofv H1 and
get, IIv H1 112 = IIwtl12 + IIb t l1 2 + 2 2:=r T; (w t . xt - b;) + (2:=r T;)21IxtI12 + 2:=r (T;)2.
Since T; E {-1,0,+1} we have that (2:=rT;)2 :::; (nt)2 and 2:=r(T;) 2 = nt and we
therefore get,
IIv H1 112 :::; IIvtl12 + 22:= T; (w t . xt - b~) + (nt)21IxtW + nt .
(4)
r
We further develop the second term using the update rule of the algorithm and get,
2:= T; (w t . xt - b~) = 2:=[(wt . xt - b~)y; :::; 0] ((wt . xt - b~)y~) :::;
r
?.
(5)
r
Plugging Eq. (5) into Eq. (4) and using the bound IIxtl12 :::; R2 we get that
IlvH1112:::; IIvtl12 + (nt)2R2 + nt. Thus, the ranking rule we obtain after T rounds
of the algorithm satisfies the upper bound, IlvT+l W :::; R2 2:=t(nt )2 + 2:=t nt. Combining the lower bound IlvT+l W ~ (2:=t nt)2 "(2 with the upper bound we have that,
(2:=tnt) 2"(2:::; Ilv T+1112:::; R2 2:=t(nt )2 + 2:=t nt . Dividing both sides by "(2 2:=tnt we
finally get,
2:= nt :::; R2 [2:=t(n t )2]
t
f [2:=t ntl + 1 .
(6)
"(
By definition, nt is at most k - 1, which implies that 2:=t(n t )2 :::; 2:=t nt(k - 1) =
(k -1) 2:=t nt. Using this inequality in Eq. (6) we get the desired bound, 2:=;=1 Ig t ytl = 2:=;=1 nt :::; [(k - 1)R2 + 1lh2 :::; [(k - 1)(R2 + 1)lh2 .
?
i"
I
...
~
Figure 3: Comparison of the time-averaged ranking-loss of PRank, WH, and MCP
on synthetic data (left). Comparison of the time-averaged ranking-loss of PRank,
WH, and MCP on the EachMovie dataset using viewers who rated and at least 200
movies (middle) and at least 100 movies (right).
4
Experiments
In this section we describe experiments we performed that compared PRank with
two other online learning algorithms applied to ranking: a multiclass generalization
of the perceptron algorithm [2], denoted MCP, and the Widrow-Hoff [9] algorithm
for online regression learning which we denote by WHo For WH we fixed its learning
rate to a constant value. The hypotheses the three algorithms maintain share
similarities but are different in their complexity: PRank maintains a vector w of
dimension n and a vector of k - 1 modifiable thresholds b, totaling n + k - 1
parameters; MCP maintains k prototypes which are vectors of size n, yielding kn
parameters; WH maintains a single vector w of size n. Therefore, MCP builds the
most complex hypothesis of the three while WH builds the simplest.
Due to the lack of space, we only describe two sets of experiments with two different
datasets. The dataset used in the first experiment is synthetic and was generated in
a similar way to the dataset used by Herbrich et. al. [5]. We first generated random
points x = (Xl, X2) uniformly at random from the unit square [0,1 f. Each point
was assigned a rank y from the set {I, ... , 5} according to the following ranking rule,
y = maxr{r : lO((XI - 0.5)(X2 - 0.5)) + ~ > br } where b = (-00 , -1, -0.1,0.25,1)
and ~ is a normally distributed noise of a zero mean and a standard deviation
of 0.125. We generated 100 sequences of instance-rank pairs each of length 7000.
We fed the sequences to the three algorithms and obtained a prediction for each
instance. We converted the real-valued predictions of WH into ranks by rounding
each prediction to its closest rank value. As in ~5] we used a non-homogeneous
polynomial of degree 2, K(XI' X2) = ((Xl? X2) + 1) as the inner-product operation
between each input instance and the hyperplanes the three algorithms maintain.
At each time step, we computed for each algorithm the accumulated ranking-loss
normalized by the instantaneous sequence length. Formally, the time-averaged loss
Iyt _ytl. We computed these losses for T = 1, ... ,7000.
after T rounds is, (liT)
To increase the statistical significance of the results we repeated the process 100
times, picking a new random instance-rank sequence of length 7,000 each time, and
averaging the instantaneous losses across the 100 runs. The results are depicted
on the left hand side of Fig. 3. The 95% confidence intervals are smaller then the
symbols used in the plot. In this experiment the performance of MPC is constantly
worse than the performance of WH and PRank. WH initially suffers the smallest
instantaneous loss but after about 500 rounds PRank achieves the best performance
and eventually the number of ranking mistakes that PRank suffers is significantly
lower than both WH and MPC.
'L,i
In the second set of experiments we used the EachMovie dataset [7]. This dataset
is used for collaborative filtering tasks and contains ratings of movies provided
by 61 , 265 people. Each person in the dataset viewed a subset of movies from a
collection of 1623 titles. Each viewer rated each movie that she saw using one of
6 possible ratings: 0, 0.2, 0.4, 0.6, 0.8,1. We chose subsets of people who viewed a
significant amount of movies extracting for evaluation people who have rated at
least 100 movies. There were 7,542 such viewers. We chose at random one person
among these viewers and set the person's ratings to be the target rank. We used the
ratings of all the rest of the people who viewed enough movies as features. Thus,
the goal is to learn to predict the "taste" of a random user using the user's past
ratings as a feedback and the ratings of fellow viewers as features. The prediction
rule associates a weight with each fellow viewer an therefore can be seen as learning
correlations between the tastes of different viewers. Next, we subtracted 0.5 from
each rating and therefore the possible ratings are -0.5 , -0.3, -0.1 , 0.1, 0.3, 0.5. This
linear transformation enabled us to assign a value of zero to movies which have not
been rated. We fed these feature-rank pairs one at a time, in an online fashion .
Since we picked viewer who rated at least 100 movies, we were able to perform at
least 100 rounds of online predictions and updates. We repeated this experiment
500 times, choosing each time a random viewer for the target rank. The results are
shown on the right hand-side of Fig. 3. The error bars in the plot indicate 95%
condfidence levels. We repeated the experiment using viewers who have seen at
least 200 movies. (There were 1802 such viewers.) The results of this experiment
are shown in the middle plot of Fig. 3. Along the entire run of the algorithms ,
PRank is significantly better than WH, and consistently better than the multiclass
perceptron algorithm, although the latter employs a bigger hypothesis.
Finally, we have also evaluated the performance of PRank in a batch setting, using
the experimental setup of [5]. In this experiment, we ran PRank over the training
data as an online algorithm and used its last hypothesis to rank unseen test data.
Here as well PRank came out first, outperforming all the algorithms described in [5].
Acknowledgments Thanks to Sanjoy Dagupta and Rob Schapire for numerous
discussions on ranking problems and algorithms. Thanks also to Eleazar Eskin and
Uri Maoz for carefully reading the manuscript.
References
[1] William W. Cohen, Robert E. Schapire, and Yoram Singer. Learning to order things.
Journal of Artificial Int elligence Research, 10:243- 270 , 1999.
[2] K. Crammer and Y. Singer. Ultraconservative online algorithms for multiclass problems. Proc. of the Fourteenth Annual ConI on Computational Learning Theory, 200l.
[3] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for
combining preferences. Machine Learning: Proc. of the Fifteenth Inti. ConI, 1998.
[4] Y. Freund and R. E. Schapire. Large margin classification using the perceptron algorithm. Machine Learning, 37(3): 277-296, 1999.
[5] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal
regression. Advances in Large Margin Classifiers. MIT Press, 2000.
[6] J. Kemeny and J . Snell. Mathematical Models in the Social Sciences. MIT Press, 1962.
[7] Paul McJones. EachMovie collaborative filtering data set. DEC Systems Research
Center, 1997. http://www.research.digital.com/SRC/eachmoviej.
[8] Vladimir N. Vapnik. Statistical Learning Theory. Wiley, 1998.
[9] Bernard Widrow and Marcian E. Hoff. Adaptive switching circuits. 1960 IRE
WESCON Convention Record, 1960. Reprinted in Neurocomputing (MIT Press, 1988).
| 2023 |@word middle:3 polynomial:1 norm:4 bn:1 carry:1 contains:1 outperforms:1 past:2 current:1 com:1 nt:24 must:1 realistic:1 plot:5 update:13 yr:10 item:2 short:1 record:1 eskin:1 ire:1 boosting:1 herbrich:3 hyperplanes:1 preference:1 mathematical:2 along:2 prove:5 introduce:1 nor:1 considering:1 totally:3 increasing:1 project:2 classifies:1 notation:1 provided:1 circuit:1 israel:1 what:1 transformation:1 fellow:2 voting:1 wrong:1 classifier:2 schwartz:1 unit:4 normally:1 omit:1 yn:1 before:2 eleazar:1 engineering:1 mistake:12 switching:1 might:2 chose:2 therein:1 initialization:1 equivalence:1 bi:2 averaged:3 unique:1 acknowledgment:1 adapting:1 significantly:2 projection:1 confidence:1 get:17 cannot:2 close:2 onto:1 convenience:1 put:1 www:1 yt:22 center:1 modifies:1 jerusalem:1 economics:1 starting:1 straightforward:1 simplicity:2 assigns:1 manipulates:1 rule:29 enabled:1 updated:1 target:2 user:4 homogeneous:1 hypothesis:5 associate:1 element:1 iib:1 updating:1 mend:1 region:1 ordering:1 decrease:1 yk:2 ran:1 src:1 complexity:2 inductively:1 iiv:3 rewrite:1 division:1 ofv:1 distinct:1 describe:9 artificial:1 choosing:1 whose:1 pref:1 valued:1 say:3 otherwise:3 unseen:1 btr:1 online:13 sequence:7 product:3 maximal:1 fr:1 loop:1 combining:3 maoz:1 object:1 ilvt:2 derive:1 develop:1 ac:1 widrow:2 school:1 received:1 eq:9 dividing:1 c:1 predicted:9 implies:2 indicate:1 convention:1 direction:1 correct:7 modifying:2 virtual:1 assign:2 suffices:1 fix:1 generalization:1 snell:1 viewer:11 hold:4 mapping:1 predict:3 claim:2 substituting:2 achieves:1 smallest:2 omitted:1 ilv:1 proc:2 label:5 title:1 saw:1 correctness:2 unfolding:1 mit:3 clearly:1 concurrently:1 always:1 rather:1 totaling:1 ytl:2 derived:1 focus:1 she:1 consistently:1 rank:46 underscore:1 sense:3 accumulated:2 bt:6 entire:2 initially:1 relation:3 irn:1 expand:1 semantics:1 sketched:1 x11:1 classification:7 among:1 denoted:1 initialize:2 hoff:2 field:1 equal:2 lit:1 koby:1 simplify:1 inherent:1 employ:2 bl1:3 composed:1 preserve:1 neurocomputing:1 maintain:2 william:1 interest:1 evaluation:1 yielding:1 closer:1 partial:3 indexed:1 iv:2 divide:1 desired:1 instance:25 classify:1 cover:1 minr:2 deviation:1 subset:2 predicate:1 rounding:1 straightforwardly:1 kn:1 synthetic:3 combined:1 thoroughly:1 referring:1 person:3 thanks:2 huji:1 pranking:1 yl:3 picking:1 lh2:2 worse:1 book:1 sidestep:1 converted:1 sec:2 b2:5 int:1 satisfy:1 ranking:46 performed:2 view:2 h1:6 picked:1 analyze:4 maintains:6 parallel:1 collaborative:5 minimize:1 il:3 ir:3 square:1 who:7 suffers:2 ed:1 definition:4 mpc:2 associated:1 proof:2 dataset:7 wh:10 recall:1 graepel:1 carefully:1 manuscript:1 higher:1 evaluated:1 generality:2 erences:1 correlation:1 hand:7 receives:1 lack:2 b3:5 verify:1 true:6 normalized:1 hence:1 assigned:2 equality:1 illustrated:1 round:15 ll:1 ttr:1 theoretic:2 demonstrate:1 tt:1 l1:1 fj:2 instantaneous:3 common:2 pseudocode:2 overview:1 cohen:1 b4:1 refer:2 significant:1 trivially:1 similarly:1 kobics:1 similarity:1 closest:1 inequality:4 outperforming:1 came:1 vt:5 seen:2 minimum:1 determine:1 itn:1 preservation:1 bother:1 eachmovie:4 retrieval:1 equally:1 y:4 bigger:1 plugging:2 va:1 prediction:9 regression:7 metric:1 fifteenth:1 kernel:1 represent:1 dec:1 interval:6 else:2 source:2 rest:1 fell:1 induced:1 thing:1 integer:4 call:1 extracting:1 exceed:1 enough:1 associating:1 reduce:1 inner:3 prototype:1 wasn:1 br:21 multiclass:3 reprinted:1 whether:1 motivated:1 b5:1 constitute:1 amount:1 ifor:1 induces:2 simplest:1 schapire:4 http:1 correctly:2 modifiable:1 write:1 threshold:12 neither:1 sum:2 wand:5 run:4 fourteenth:1 you:1 fourth:1 family:1 comparable:1 bound:15 fl:2 fan:1 annual:1 x2:4 min:2 structured:1 according:1 across:1 smaller:1 ur:1 wi:1 rob:1 inti:1 taken:1 mcp:5 ntl:1 discus:3 eventually:1 singer:5 ordinal:1 merit:1 fed:3 operation:1 subtracted:1 alternative:2 batch:2 running:1 coni:2 yoram:2 build:2 bl:2 move:1 rt:1 obermayer:1 kemeny:1 concatenation:1 cauchy:1 toward:1 induction:1 length:3 index:4 elligence:1 illustration:3 iyt:2 hebrew:1 vladimir:1 setup:1 robert:1 negative:1 ofw:1 perform:1 upper:2 datasets:1 finite:1 prank:20 excluding:1 bll:1 y1:1 tnt:2 rating:14 bk:8 cast:1 pair:6 fort:1 namely:1 able:1 suggested:1 bar:1 below:2 reading:1 natural:2 ranked:3 ll1:1 movie:14 rated:5 numerous:1 deviate:1 taste:2 freund:2 loss:12 filtering:5 iiwl1:1 digital:1 degree:2 consistent:1 imposes:1 mercer:1 intractability:1 share:2 maxt:1 lo:1 last:1 side:9 perceptron:5 fall:1 distributed:1 feedback:1 dimension:1 boundary:1 computes:1 made:2 collection:1 adaptive:1 ig:1 social:2 preferred:1 maxr:1 wescon:1 b1:6 summing:1 conclude:1 consuming:1 xi:2 ultraconservative:1 learn:1 expanding:1 improving:1 cl:1 complex:1 significance:1 main:1 bounding:2 noise:1 paul:1 repeated:3 fig:10 fashion:1 wiley:1 sub:3 wish:1 xl:4 theorem:2 xt:30 showing:1 symbol:1 r2:8 vapnik:1 iyer:1 uri:1 margin:6 gap:1 updat:1 depicted:1 simply:1 expressed:1 ordered:3 satisfies:2 constantly:1 goal:6 viewed:3 replace:3 specifically:1 operates:1 uniformly:1 wt:10 hyperplane:1 averaging:1 lemma:1 conservative:1 total:4 called:1 sanjoy:1 bernard:1 experimental:1 meaningful:1 formally:3 support:1 people:4 latter:2 crammer:2 brevity:1 ongoing:1 |
1,123 | 2,024 | Incorporating Invariances in Nonlinear
Support Vector Machines
Olivier Chapelle
Bernhard Scholkopf
olivier.chapelle@lip6.fr
LIP6, Paris, France
Biowulf Technologies
bernhard.schoelkopf@tuebingen.mpg.de
Max-Planck-Institute, Tiibingen, Germany
Biowulf Technologies
Abstract
The choice of an SVM kernel corresponds to the choice of a representation of the data in a feature space and, to improve performance , it should therefore incorporate prior knowledge such as
known transformation invariances. We propose a technique which
extends earlier work and aims at incorporating invariances in nonlinear kernels. We show on a digit recognition task that the proposed approach is superior to the Virtual Support Vector method,
which previously had been the method of choice.
1
Introduction
In some classification tasks, an a priori knowledge is known about the invariances
related to the task. For instance, in image classification, we know that the label of
a given image should not change after a small translation or rotation.
More generally, we assume we know a local transformation Lt depending on a
parameter t (for instance, a vertical translation of t pixels) such that any point x
should be considered equivalent to LtX, the transformed point. Ideally, the output
of the learned function should be constant when its inputs are transformed by the
desired invariance.
It has been shown [1] that one can not find a non-trivial kernel which is globally
invariant. For this reason, we consider here local invariances and for this purpose
we associate at each training point X i a tangent vector dXi,
dXi =
lim -1 (LtXi t--+o
t
Xi)
81t=o LtXi
= -
8t
In practice dXi can be either computed by finite difference or by differentiation.
Note that generally one can consider more than one invariance transformation.
A common way of introducing invariances in a learning system is to add the perturbed examples LtXi in the training set [7]. Those points are often called virtual
examples. In the SVM framework , when applied only to the SVs, it leads to the
Virtual Support Vector (VSV) method [10]. An alternative to this is to modify
directly the cost function in order to take into account the tangent vectors. This
has been successfully applied to neural networks [13] and linear Support Vector
Machines [11]. The aim of the present work is to extend these methods to the case
of nonlinear SVMs which will be achieved mainly by using the kernel peA trick
[12].
The paper is organized as follows. After introducing the basics of Support Vector
Machines in section 2, we recall the method proposed in [11] to train invariant linear
SVMs (section 3). In section 4, we show how to extend it to the nonlinear case and
finally experimental results are provided in section 5.
2
Support Vector Learning
We introduce some standard notations for SVMs; for a complete description, see
[15]. Let {(Xi, Yi) h<i<n be a set of training examples, Xi E IRd , belonging to classes
labeled by Yi E {-1,1}. In kernel methods, we map these vectors into a feature
space using a kernel function K(Xi' Xj) that defines an inner product in this feature
space. The decision function given by an SVM is the maximal margin hyperplane
in this space,
g(X) = sign(f(x)), where f(x) =
(~a?YiK(Xi'X) + b) .
(1)
The coefficients a? are obtained by maximizing the functional
1
n
W(o:) = Lai
-"2
i=l
n
L aiajYiyjK(Xi,Xj)
i,j=l
under the constraints L:~= 1 aiYi = 0 and
ai
(2)
~ O.
This formulation of the SVM optimization problem is called the hard margin formulation since no training errors are allowed. In the rest of the paper, we will
not consider the so called soft-margin SVM algorithm [4], where training errors are
allowed.
3
Invariances for Linear SVMs
For linear SVMs, one wants to find a hyperplane whose normal vector w is as
orthogonal as possible to the tangent vectors. This can be easily understood from
the equality
f(Xi + dXi) - f(Xi) = w . dXi'
For this purpose, it has been suggested [11] to minimize the functional
n
(1 - ')')w 2
+ ')' L(w, dXi)2
(3)
i=l
subject to the constraints Yi(W . Xi + b) ~ 1. The parameter,), trades off between
normal SVM training (')' = 0) and full enforcement of the orthogonality between
the hyperplane and the invariance directions (')' ---+ 1).
Let us introduce
c, ~
((1-0)[ +0
~dx'dxi) 'i',
the square root of the regularized covariance matrix of the tangent vectors.
(4)
It was shown in [11] that training a linear invariant SVM, i.e. minimizing (3), is
equivalent to a standard SVM training after the following linear transformation of
the input space
-1
X --+ C, x.
This method led to significant improvements in linear SVMs, and to small improvements when used as a linear preprocessing step in nonlinear SVMs. The latter,
however, was a hybrid system with unclear theoretical foundations. In the next
section we show how to deal with the nonlinear case in a principled way.
4
Extension to the nonlinear case
In the nonlinear case, the data are first mapped into a high-dimensional feature
space where a linear decision boundary is computed. To extend directly the previous
analysis to the nonlinear case, one would need to compute the matrix C, in feature
space,
C,
=
(
(1 - '"Y)I + '"Y ~ dlJ> (Xi) dlJ> (Xi) T
and the new kernel function
K(x , y) = C~ llJ>(x) . C~ llJ>(y)
)
1~
(5)
= lJ>(x) T C~ 21J>(y)
(6)
However, due to the high dimension of the feature space, it is impossible to do it
directly. We propose two different ways for overcoming this difficulty.
4.1
Decomposition of the tangent Gram matrix
In order to be able to compute the new kernel (6) , we propose to diagonalize the
matrix C, (eq 5) using a similar approach as the kernel PCA trick [12]. In that
article, they showed how it was possible to diagonalize the feature space covariance
matrix by computing the eigendecomposition of the Gram matrix of those points.
Presently, instead of having a set of training points {1J>(Xi)} , we have a set of tangent
vectors {dlJ> (Xi)} and a tangent covariance matrix (the right term of the sum in (5))
Let us introduce the Gram matrix Kt of the tangent vectors:
Kij = dlJ>(Xi )? dlJ>(xj)
K(Xi +dXi, Xj +dxj) - K(Xi +dXi, Xj) - K(Xi ' Xj +dxj)
+ K(Xi' Xj) (7)
(8)
.
dxiT02K(Xi,Xj)d
~
~
XJ
UXiUXj
This matrix Kt can be computed either by finite differences (equation 7) or with the
analytical derivative expression given by equation (8) . Note that for a linear kernel,
K(x,y) = x T y, and (8) reads Kfj = dxi dXj, which is a standard dot product
between the tangent vectors.
Writing the eigendecomposition of Kt as Kt = U AUT , and using the kernel PCA
tools [12], one can show after some algebra (details in [2]) that the new kernel matrix
reads
K(x,y)
1 Kx y
-I - '"Y
(,)
+
n1(
~ -Ap
U.dx,TOK(Xi' X))
(~
~1
U~
~
'p
~
1
'"Y Ap
+1-
(~U.
~ 'p
~1
'"Y
- -1
-)
1 - '"Y
T
d OK(Xi'
x,
~
U~
y))
4.2
The kernel PCA map
A drawback of the previous approach appears when one wants to deal with multiple
invariances (i.e. more than one tangent vector per training point). Indeed, it
requires to diagonalize the matrix Kt (cf eq 7), whose size is equal to the number of
different tangent vectors. For this reason, we propose an alternative method. The
idea is to use directly the so-called kernel peA map, first introduced in [12] and
extended in [14].
This map is based on the fact that even in a high dimensional feature space 1i, a
training set {Xl , .. . , x n } of size n when mapped to this feature space spans a subspace E C 1i whose dimension is at most n . More precisely, if (VI"'" Vn ) E En is
an orthonormal basis of E with each Vi being a principal axis of {<I>(xd, ... , <I> (x n )} ,
the kernel peA map 'i/J : X -+ ~n is defined coordinatewise as
'i/Jp (x) = <I>(x) . v P '
1:S p:S n.
Each principal direction has a linear expansion on the training points {<I>(Xi)} and
the coefficients of this expansion are obtained using kernel peA [12]. Writing the
eigendecompostion of K as K = U AUT, with U an orthonormal matrix and A a
diagonal one, it turns out that the the kernel peA map reads
'i/J(x) = A-1/2U T k(x),
(9)
where k (x) = (K(x, Xl)"'" K(x, xn)) T .
Note that by definition , for all i and j , <I>(Xi) and <I>(Xj) lie in E and thus K(Xi ' Xj) =
<I>(Xi) . <I>(Xj) = 'i/J(Xi) . 'i/J(Xj). This reflects the fact that if we retain all principal
components, kernel peA is just a basis transform in E, leaving the dot product of
training points invariant.
As a consequence, training a nonlinear SVM on {Xl , ... , xn} is equivalent to training
a linear SVM on {'i/J(xd, . . . ,'i/J(xn )} and thus, thanks to the nonlinear mapping 'i/J,
we can work directly in the linear space E and use exactly the technique described
for invariant linear SVMs (section 3) . However the invariance directions d<I>(Xi) do
not necessarily belong to E. By projecting them onto E, some information might
be lost. The hope is that this approximation will give a similar decision function to
the exact one obtained in section 4.l.
Finally, the proposed algorithm consists in training an invariant linear SVM as
described in section 3 with training set { 'i/J(XI) , ... ,'i/J(xn)} and with invariance
directions {d'i/J(XI) , ... , d'i/J (x n)}, where d'i/J (Xi) = 'i/J (Xi + dXi ) - 'i/J(Xi), which can
be expressed from equation (9) as
4.3
Comparisons with the VSV method
One might wonder what is the difference between enforcing an invariance and just
adding the virtual examples LtXi in the training set. Indeed the two approaches
are related and some equivalence can be shown [6] .
So why not just add virtual examples? This is the idea of the Virtual Support
Vector (VSV) method [10] . The reason is the following: if a training point Xi is
far from the margin, adding the virtual example LtXi will not change the decision
boundary since neither of the points can become a support vector. Hence adding
virtual examples in the SVM framework enforces invariance only around the decision
boundary, which, as an aside, is the main reason why the virtual SV method only
adds virtual examples generated from points that were support vectors in the earlier
iteration.
One might argue that the points which are far from the decision boundary do not
provide any information anyway. On the other hand, there is some merit in not
only keeping the output label invariant under the transformation Lt, but also the
real-valued output. This can be justified by seeing the distance of a given point
to the margin as an indication of its class-conditional probability [8]. It appears
reasonable that an invariance transformation should not affect this probability too
much.
5
Experiments
In our experiments, we compared a standard SVM with several methods taking into
account invariances: standard SVM with virtual examples (cf. the VSV method [10])
[VSV], invariant SVM as described in section 4.1 [ISVM] and invariant hyperplane
in kernel peA coordinates as described in section 4.2 [ IHKPcA ].
The hybrid method described in [11] (see end of section 3) did not perform better
than the VSV method and is not included in our experiments for this reason.
Note that in the following experiments, each tangent vector d<I>(Xi) has been normalized by the average length
Ild<I>(xi)W/n in order to be scale independent.
vI:
5.1
Toy problem
The toy problem we considered is the following: the training data has been generated uniformly from [-1 , 1]2. The true decision boundary is a circle centered at the
origin: f(x) = sign(x2 - 0.7).
The a priori knowledge we want to encode in this toy problem is local invariance
under rotations. Therefore, the output of the decision function on a given training
point Xi and on its image R(Xi,C:) obtained by a small rotation should be as similar
as possible. To each training point, we associate a tangent vector dXi which is
actually orthogonal to Xi.
A training set of 30 points was generated and the experiments were repeated 100
times. A Gaussian kernel K(x,y) = exp (_ II X2~~ 1I 2) was chosen.
The results are summarized in figure 1. Adding virtual examples (VSV method)
is already very useful since it made the test error decrease from 6.25% to 3.87%
(with the best choice of a). But the use of ISVM or IHKPcA yields even better
performance. On this toy problem, the more the invariances are enforced b -+ 1),
the better the performances are (see right side of figure 1), reaching a test error of
1.11%.
When comparing log a = 1.4 and log a = 0 (right side of of figure 1), one notices
that the decrease in the test error does not have the same speed. This is actually
the dual of the phenomenon observed on the left side of this figure: for a same value
of gamma, the test error tends to increase, when a is larger. This analysis suggests
that 'Y needs to be adapted in function of a. This can be done automatically by the
gradient descent technique described in [3].
0.12
-
0. 14
- -
log sigma=-O.8
log sigma=O
10 si ma= 1,4
0 .12
0.1
0.06
0 .08
0.04
0.02
O.02 '----_~
, ------:
-0~
.5--~
0 --0~
.5,------~------'c-"
.5
Log sigma
%'------~-~-~
6 -~
8--,~
0 -~
, 2~
- Log (1-gamma)
Figure 1: Left: test error for different learning algorithms plotted against the width
of a RBF kernel and "( fixed to 0.9. Right: test error of IHKPcA across "( and for
different values of (5. The test errors are averaged over the 100 splits and the error
bars correspond to the standard deviation of the means.
5.2
Handwritten digit recognition
As a real world experiment, we tried to incorporate invariances for a handwritten
digit recognition task. The USPS dataset have been used extensively in the past
for this purpose, especially in the SVM community. It consists of 7291 training and
2007 test examples.
According to [9], the best performance has been obtained for a polynomial kernel
of degree 3, and all the results described in this section were performed using this
kernel. The local transformations we considered are translations (horizontal and
vertical). All the tangent vectors have been computed by a finite difference between
the original digit and its I-pixel translated.
We split the training set into 23 subsets of 317 training examples after a random
permutation of the training and test set. Also we concentrated on a binary classification problem, namely separating digits a to 4 against 5 to 9. The gain in
performance should also be valid for the multiclass case.
Figure 2 compares ISVM, IHKPcA and VSV for different values of "(. From those
figures, it can be seen that the difference between ISVM (the original method) and
IHKPcA (the approximation) is much larger than in the toy example. The difference
to the toy example is probably due to the input dimensionality. In 2 dimensions,
with an RBF kernel, the 30 examples of the toy problem "almost span" the whole
feature space, whereas with 256 dimensions , this is no longer the case.
What is noteworthy in these experiments is that our proposed method is much
better than the standard VSV. As explained in section 4.3, the reason for this
might be that invariance is enforced around all training points and not only around
support vectors. Note that what we call VSV here is a standard SVM with a double
size training set containing the original data points and their translates.
The horizontal invariance yields larger improvements than the vertical one. One
of the reason might be that the digits in the USPS database are already centered
vertically.
-
0 .068
- -
IHKPCA
ISVM
VSV
- -
0 .066
0.066
0 .064
0 .064
0 .062
0.062
0.06
0.06
0 .058
0 .058
0.056
0.056
0 .054
0
0.5
1.5
2
2.5
3.5
-Log (1-gamma)
Vertical translation (to the top)
-
0 .068
0 .054
0
0.5
1.5
2
2.5
IHKPCA
ISVM
VSV
3.5
-Log (1-gamma)
Horizontal translation (to the right)
Figure 2: Comparison of ISVM, IHKPcA and VSV on the USPS dataset. The left
of the plot ("( = 0) corresponds to standard SVM whereas the right part of the plot
h -+ 1) means that a lot of emphasis is put on the enforcement of the constraints.
The test errors are averaged over the 23 splits and the error bars correspond to the
standard deviation of the means.
6
Conclusion
We have extended a method for constructing invariant hyperplanes to the nonlinear
case. We have shown results that are superior to the virtual SV method. The latter
has recently broken the record on the NIST database which is the "gold standard"
of handwritten digit benchmarks [5], therefore it appears promising to also try the
new system on that task. For this propose, a large scale version of this method
needs to be derived. The first idea we tried is to compute the kernel PCA map
using only a subset of the training points. Encouraging results have been obtained
on the lO-class USPS database (with the whole training set), but other methods
are also currently under study.
References
[1] C. J. C. Burges. Geometry and invariance in kernel based methods. In
B. Sch6lkopf, C. J . C. Burges, and A. J . Smola, editors, Advances in Kernel Methods - Support Vector Learning. MIT Press, 1999.
[2] O. Chapelle and B. Sch6lkopf. Incorporating invariances in nonlinear Support
Vector Machines, 2001. Availabe at: www-connex.lip6.frrchapelle.
[3] O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee. Choosing multiple
parameters for support vector machines. Machine Learning, 46:131- 159, 2002.
[4] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273 297,1995.
[5] D. DeCoste and B. Sch6lkopf. Training invariant support vector machines.
Machine Learning, 2001. In press.
[6] Todd K. Leen. From data distributions to regularization in invariant learning.
In Nips, volume 7. The MIT Press, 1995.
[7] P. Niyogi, T. Poggio, and F. Girosi. Incorporating prior information in machine
learning by creating virtual examples. IEEE Proceedings on Intelligent Signal
Processing, 86(11):2196-2209, November 1998.
[8] John Platt. Probabilities for support vector machines. In A. Smola, P. Bartlett,
B. Sch6lkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers. MIT Press, Cambridge, MA, 2000.
[9] B. Sch6lkopf, C. Burges, and V. Vapnik. Extracting support data for a given
task. In U. M. Fayyad and R. Uthurusamy, editors, First International Conference on Knowledge Discovery fj Data Mining. AAAI Press, 1995.
[10] B. Sch6lkopf, C. Burges, and V. Vapnik. Incorporating invariances in support
vector learning machines. In Artificial Neural Networks - ICANN'96, volume
1112, pages 47- 52, Berlin, 1996. Springer Lecture Notes in Computer Science.
[11] B. Sch6lkopf, P. Y. Simard, A. J. Smola, and V. N. Vapnik. Prior knowledge
in support vector kernels. In MIT Press, editor, NIPS, volume 10, 1998.
[12] B. Sch6lkopf, A. Smola, and K.-R. Muller. Nonlinear component analysis as a
kernel eigenvalue problem. Neural Computation, 10:1299- 1310, 1998.
[13] P. Simard, Y. LeCun, J. Denker, and B. Victorri. Transformation invariance
in pattern recognition, tangent distance and tangent propagation. In G. Orr
and K. Muller, editors, Neural Networks: Tricks of the trade. Springer, 1998.
[14] K. Tsuda. Support vector classifier with asymmetric kernel function. In M. Verleysen, editor, Proceedings of ESANN'99, pages 183- 188,1999.
[15] V. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998.
| 2024 |@word version:1 polynomial:1 tried:2 covariance:3 decomposition:1 past:1 comparing:1 si:1 dx:2 john:2 girosi:1 plot:2 aside:1 record:1 hyperplanes:1 become:1 scholkopf:1 consists:2 introduce:3 indeed:2 mpg:1 globally:1 automatically:1 encouraging:1 decoste:1 provided:1 notation:1 what:3 transformation:8 differentiation:1 xd:2 exactly:1 classifier:2 platt:1 planck:1 understood:1 local:4 modify:1 tends:1 vertically:1 consequence:1 todd:1 ap:2 noteworthy:1 might:5 emphasis:1 equivalence:1 suggests:1 averaged:2 lecun:1 enforces:1 practice:1 lost:1 digit:7 seeing:1 onto:1 put:1 impossible:1 writing:2 www:1 equivalent:3 map:7 maximizing:1 orthonormal:2 anyway:1 coordinate:1 exact:1 olivier:2 origin:1 associate:2 trick:3 recognition:4 asymmetric:1 mukherjee:1 labeled:1 database:3 observed:1 schoelkopf:1 trade:2 decrease:2 principled:1 broken:1 ideally:1 algebra:1 basis:2 usps:4 translated:1 easily:1 train:1 isvm:7 artificial:1 choosing:1 whose:3 larger:3 valued:1 niyogi:1 transform:1 indication:1 eigenvalue:1 analytical:1 propose:5 product:3 maximal:1 fr:1 gold:1 description:1 double:1 depending:1 connex:1 eq:2 esann:1 direction:4 drawback:1 pea:7 centered:2 virtual:14 extension:1 around:3 considered:3 normal:2 exp:1 mapping:1 purpose:3 label:2 currently:1 lip6:3 successfully:1 tool:1 reflects:1 hope:1 mit:4 gaussian:1 aim:2 reaching:1 encode:1 derived:1 improvement:3 mainly:1 lj:1 transformed:2 france:1 germany:1 pixel:2 classification:3 dual:1 priori:2 verleysen:1 equal:1 having:1 intelligent:1 gamma:4 geometry:1 n1:1 mining:1 kt:5 poggio:1 orthogonal:2 desired:1 circle:1 plotted:1 tsuda:1 theoretical:1 instance:2 kij:1 earlier:2 soft:1 cost:1 introducing:2 deviation:2 subset:2 wonder:1 too:1 perturbed:1 sv:2 thanks:1 international:1 retain:1 off:1 aaai:1 containing:1 creating:1 derivative:1 simard:2 toy:7 account:2 de:1 orr:1 summarized:1 coefficient:2 vi:3 performed:1 root:1 lot:1 try:1 minimize:1 square:1 yield:2 correspond:2 handwritten:3 definition:1 against:2 dxi:12 gain:1 dataset:2 recall:1 knowledge:5 lim:1 dimensionality:1 organized:1 actually:2 appears:3 ok:1 formulation:2 done:1 leen:1 vsv:13 just:3 smola:4 hand:1 horizontal:3 nonlinear:14 ild:1 propagation:1 defines:1 normalized:1 true:1 equality:1 hence:1 regularization:1 read:3 deal:2 width:1 complete:1 fj:1 aiyi:1 image:3 recently:1 superior:2 rotation:3 common:1 functional:2 jp:1 volume:3 extend:3 belong:1 significant:1 cambridge:1 ai:1 had:1 dot:2 chapelle:4 longer:1 add:3 showed:1 binary:1 yi:3 muller:2 seen:1 aut:2 signal:1 ii:1 full:1 multiple:2 lai:1 basic:1 iteration:1 kernel:30 achieved:1 justified:1 whereas:2 want:3 victorri:1 leaving:1 diagonalize:3 rest:1 probably:1 subject:1 dxj:3 call:1 extracting:1 split:3 xj:13 affect:1 inner:1 idea:3 multiclass:1 translates:1 expression:1 pca:4 bartlett:1 ird:1 svs:1 yik:1 useful:1 generally:2 extensively:1 concentrated:1 svms:8 notice:1 sign:2 per:1 neither:1 sum:1 enforced:2 extends:1 almost:1 reasonable:1 vn:1 decision:8 adapted:1 constraint:3 orthogonality:1 precisely:1 x2:2 bousquet:1 speed:1 span:2 fayyad:1 ltx:1 llj:2 according:1 belonging:1 across:1 son:1 presently:1 projecting:1 invariant:12 explained:1 equation:3 previously:1 turn:1 know:2 enforcement:2 merit:1 end:1 denker:1 uthurusamy:1 alternative:2 original:3 top:1 cf:2 especially:1 already:2 biowulf:2 aiajyiyjk:1 diagonal:1 unclear:1 gradient:1 subspace:1 distance:2 mapped:2 separating:1 berlin:1 argue:1 tuebingen:1 trivial:1 reason:7 enforcing:1 length:1 minimizing:1 sigma:3 perform:1 sch6lkopf:8 vertical:4 benchmark:1 finite:3 nist:1 descent:1 tok:1 november:1 extended:2 community:1 overcoming:1 introduced:1 namely:1 paris:1 learned:1 nip:2 able:1 suggested:1 bar:2 pattern:1 max:1 difficulty:1 hybrid:2 regularized:1 improve:1 technology:2 axis:1 prior:3 discovery:1 tangent:16 lecture:1 permutation:1 foundation:1 eigendecomposition:2 degree:1 article:1 editor:6 translation:5 lo:1 keeping:1 side:3 burges:4 institute:1 taking:1 boundary:5 dimension:4 xn:4 gram:3 world:1 valid:1 made:1 preprocessing:1 far:2 tiibingen:1 bernhard:2 xi:38 why:2 promising:1 schuurmans:1 expansion:2 necessarily:1 constructing:1 did:1 icann:1 main:1 whole:2 coordinatewise:1 allowed:2 repeated:1 en:1 wiley:1 dlj:5 kfj:1 xl:3 lie:1 svm:18 cortes:1 incorporating:5 vapnik:6 adding:4 margin:6 kx:1 lt:2 led:1 expressed:1 springer:2 corresponds:2 ma:2 conditional:1 rbf:2 change:2 hard:1 included:1 uniformly:1 hyperplane:4 principal:3 called:4 invariance:26 experimental:1 support:20 latter:2 incorporate:2 phenomenon:1 |
1,124 | 2,025 | A Model of the Phonological Loop:
Generalization and Binding
Randall C. O'Reilly
Department of Psychology
University of Colorado Boulder
345 UCB
Boulder, CO 80309
Rodolfo Soto
Department of Psychology
University of Colorado Boulder
345 UCB
Boulder, CO 80309
oreilly@psych.colorado.edu
Abstract
We present a neural network model that shows how the prefrontal
cortex, interacting with the basal ganglia, can maintain a sequence
of phonological information in activation-based working memory
(i.e., the phonological loop). The primary function of this phonological loop may be to transiently encode arbitrary bindings of
information necessary for tasks - the combinatorial expressive
power of language enables very flexible binding of essentially arbitrary pieces of information. Our model takes advantage of the
closed-class nature of phonemes, which allows different neural representations of all possible phonemes at each sequential position to
be encoded. To make this work, we suggest that the basal ganglia
provide a region-specific update signal that allocates phonemes to
the appropriate sequential coding slot. To demonstrate that flexible, arbitrary binding of novel sequences can be supported by this
mechanism, we show that the model can generalize to novel sequences after moderate amounts of training.
1
Introduction
Sequential binding is a version of the binding problem requiring that the identity
of an item and its position within a sequence be bound. For example, to encode a
phone number (e.g., 492-0054), one must remember not only the digits, but their
order within the sequence. It has been suggested that the brain may have developed a specialized system for this form of binding in the domain of phonological
sequences, in the form of the phonological loop (Baddeley, 1986; Baddeley, Gathercole, & Papagno, 1998; Burgess & Hitch, 1999). The phonological loop is generally
conceived of as a system that can quickly encode a sequence of phonemes and then
repeat this sequence back repeatedly. Standard estimates place the capacity of
this loop at about 2.5 seconds of "inner speech," and it is widely regarded as depending on the prefrontal cortex (e.g. , Paulesu, Frith, & Frackowiak, 1993). We
have developed a model of the phonological loop based on our existing framework
for understanding how the prefrontal cortex and basal ganglia interact to support
activation-based working memory (Frank, Loughry, & O'Reilly, 2001). This model
performs binding by using different neural substrates for the different sequential
positions of phonemes. This is a viable solution for a small, closed-class set of
items like phonemes. However, through the combinatorial power of language, these
phonological sequences can represent a huge number of distinct combinations of
concepts. Therefore, this basic maintenance mechanism can be leveraged in many
different circumstances to bind information needed for immediate use (e.g., in working memory tasks).
A good example of this form of transient, phonologically-dependent binding comes
from a task studied by Miyake and Soto (in preparation). In this task, participants
saw sequentially-presented colored letters one at a time on a computer display, and
had to respond to targets of a red X or a green Y, but not to any other color-letter
combination (e.g., green X's and red Y's, which were also presented). After an initial
series of trials with this set of targets, the targets were switched to be a green X
and a red Y. Thus, the task clearly requires binding of color and letter information,
and updating of these bindings after the switch condition. Miyake and Soto (in
preparation) found that if they simply had participants repeat the word "the" over
and over during the task (i.e., articulatory suppression), it interfered significantly
with performance. In contrast, performing a similar repeated motor response that
did not involve the phonological system (repeated foot tapping) did not interfere
(but this task did interfere at the same level as articulatory suppression in a control
visual search task, so one cannot argue that the interference was simply a matter of
differential task difficulty). Miyake and Soto (in preparation) interpret this pattern
of results as showing that the phonological loop supports the binding of stimulus
features (e.g., participants repeatedly say to themselves "red X, green y' .. " , which
is supported by debriefing reports), and that the use of this phonological system
for unrelated information during articulatory suppression leads to the observed
performance deficits.
This form of phonological binding can be contrasted with other forms of binding
that can be used in other situations and subserved by other brain areas besides the
prefrontal cortex. O'Reilly, Busby, and Soto (in press) identify two other important
binding mechanisms and their neural substrates in addition to the phonological loop
mechanism:
? Cortical coarse-coded conjunctive binding: This is where each neural unit
codes in a graded fashion for a large number of relatively low-order conjunctions, and many such units are used to represent any given input (e.g.,
Wickelgren, 1969; Mel & Fiser, 2000; O'Reilly & Busby, 2002). This form
of binding takes place within the basic representations in the network that
are shaped by gradual learning processes and provides a long-lasting (nontransient) form of binding. In short, these kinds of distributed representations avoid the binding problem in the first place by ensuring that relevant conjunctions are encoded, instead of representing different features
using entirely separate, localist units (which is what gives rise to binding
problems in the first place). However, this form of binding cannot rapidly
encode novel bindings required for specific tasks - the phonological loop
mechanism can thus complement the basic cortical mechanism by providing
flexible, transient bindings on an ad-hoc basis.
? Hippocampal episodic conjunctive binding: Many theories of hippocampal
function converge on the idea that it binds together individual elements of
an experience into a unitary representation, which can for example be later
recalled from partial cues (see O'Reilly & Rudy, 2001 for a review). These
hippocampal conjunctive representations are higher-order and more spe-
cific than the lower-order coarse-coded cortical conjunctive representations
(i.e., a hippocampal conjunction encodes the combination of many feature
elements, while a cortical conjunction encodes relatively few). Thus, the
hippocampus can be seen as a specialized system for doing long-term binding of specific episodes, complementing the more generalized conjunctive
binding performed by the cortex. Importantly, the hippocampus can also
encode these conjunctions rapidly, and therefore it shares some of the same
functionality as the phonological loop mechanism (i.e., rapidly encoding
arbitrary conjunctions required for tasks). Thus, it is likely that the hippocampus and the prefrontal-mediated working memory system (including
the phonological loop) are partially redundant with each other, and work
together in many tasks (Cohen & O'Reilly, 1996).
2
Prefrontal Cortex and Basal Ganglia in Working Memory
Our model of the phonological loop takes advantage of recent work showing how the
prefrontal cortex and basal ganglia can interact to support activation-based working
memory (Frank et al., 2001). The critical principles behind this work are as follows:
? Prefrontal cortex (PFC) is specialized relative to the posterior cortex for
robust and rapidly updatable maintenance of information in an active state
(i.e., via persistent firing of neurons). Thus, PFC can quickly update to
maintain new information (in this case, the one exposure to a sequence of
phonemes), while being able to also protect maintained information from
interference from ongoing processing (see O'Reilly, Braver, & Cohen, 1999;
Cohen, Braver, & O'Reilly, 1996; Miller & Cohen, 2001 for elaborations and
reviews of relevant data).
? Robust maintenance and rapid updating are in fundamental conflict, and
require a dynamic gating mechanism that can switch between these two
modes of operation (O'Reilly et al., 1999; Cohen et al., 1996).
? The basal ganglia (BG) can provide this dynamic gating mechanism via
modulatory, dis inhibitory connectivity with the PFC. Furthermore, this
BG-based gating mechanism provides selectivity, such that separate regions of the PFC can be independently updated or allowed to perform
robust maintenance. A possible anatomical substrate for these separably
updatable PFC regions are the stripe structures identified by Levitt, Lewis,
Yoshioka, and Lund (1993).
? Active maintenance in the PFC is implemented via a combination of recurrent excitatory connections and intracellular excitatory ionic conductances.
This allows the PFC units to generally reflect the current inputs, except
when these units have their intracellular maintenance currents activated,
which causes them to reflect previously maintained information. See Frank
et al. (2001) for more details on the importance of this mechanism.
3
Phonological Loop Model
The above mechanisms motivated our modeling of the phonological loop as follows (see Figure 1) . First, separate PFC stripes are used to encode each step in
the sequence. Thus, binding of phoneme identity and sequential order occurs in
this model by using distinct neural substrates to represent the sequential information. This is entirely feasible because each stripe can represent all of the possible
phonemes, given that they represent a closed class of items. Second, the storage of a
Figure 1: Phonological loop model. Ten different input symbols are possible at each time
step (one unit out of ten activated in the Input layer) . A sequence is encoded in one pass
by presenting the Input together with the sequential location in the Time input layer for
each step in the sequence. The simulated basal ganglia gating mechanism (implemented
by fiat in script code) uses the time input to trigger intracellular maintenance currents
in the corresponding stripe region of the context (PFC) layer (stripes are shown as the
three separate groups of units within the Context layer; individual context units also
had an excitatory self-connection for maintenance). Thus, the first stripe must learn to
encode the first input, etc. Immediately after encoding, the network is then trained to
produce the correct output in response to the time input, without any Input activation
(the activation state shown is the network correctly recalling the third item in a sequence).
The hidden layer must therefore learn to decode the context representations for this recall
phase. Generalization testing involved presenting untrained sequences.
new sequence involves the basal ganglia gating mechanism triggering updates of the
different PFC stripes in the appropriate order. We assume this can be learned over
experience, and we are currently working on developing powerful learning mechanisms for adapting the basal ganglia gating mechanism in this way. This kind of
gating control would also likely require some kind of temporal/sequential input that
indicates the location within the sequence - such information might come from the
cerebellum (e.g., Ivry, 1996).
In advance of having developed realistic and computationally powerful mechanisms
for both the learning and the temporal/sequential control aspects of the model,
we simply implemented these by fiat in the simulator. For the temporal signal
indicating location within the sequence, we simply activated a different individual
time unit for each point in the sequence (the Time input layer in Figure 1). This
signal was then used by a simulated gating mechanism (implemented in script code
in the simulator) to update the corresponding stripe in prefrontal cortex. Although
the resulting model was therefore simplified, it nevertheless still had a challenging
learning task to perform. Specifically, the stripe context layers had to learn to
encode and maintain the current input value properly, and the Hidden layer had to
be able to decode the context layer information as a function of the time input value.
The model was implemented using the Leabra algorithm with standard parameters
(O'Reilly, 1998; O 'Reilly & Munakata, 2000).
Phonological Loop Generalization
..e
0.3
ali 0.2
I:
.S!
"lii
..
.!::!
iij
Q)
0.1
I:
Q)
Cl
0.0
100
200
300
800
Number of Training Events
Figure 2: Generalization results for the phonological loop model as a function of number
training patterns. Generalization is over 90% correct with training on less than 20% of
the possible input patterns. N = 5.
3 .1
Network Training
The network was trained as follows. Sequences (of length 3 for our initial work)
were presented by sequentially activating an input "phoneme" and a corresponding
sequential location input (in the Time input layer) . We only used 10 different
phonemes, each of which was encoded locally with a different unit in the Input layer.
For example, the network could get Time = 0, Input = 2, then Time = 1, Input =
7, then Time = 2, Input = 3 to encode the sequence 2,7,3. During this encoding
phase, the network was trained to activate the current Input on the Output layer,
and the simulated gating function simply activated the intracellular maintenance
currents for the units in the stripe in the Context (PFC) layer that corresponded
to the Time input (i.e., stripe 0 for Time=O, etc). Then, the network was trained
to recall this sequence, during which t ime no Input activation was present. The
network received the sequence of Time inputs (0,1,2), and was trained to produce
the corresponding Output for that location in the sequence (e.g., 2,7,3). The PFC
context layers just maintained their activation states based on the intracellular ion
currents activated during encoding (and recurrent activation) - once the network
has been trained, the active PFC state represents the entire sequence.
3.2
Generalization Results
A critical test of the model is to determine whether it can perform systematically
with novel sequences - only if it demonstrates this capacity can it serve as a mechanism for rapidly binding arbitrary information (such as the task demands studied
by Miyake & Soto, in preparation). With 10 input phonemes and sequences of
length t hree, there were 1,000 different sequences possible (we allowed phonemes
to repeat). We trained on 100, 200, 300, and 800 of these sequences, and tested
generalization on the remaining sequences. The generalization results are shown
in Figure 2, which clearly shows that the network learned these sequences in a
systematic manner and could transfer its training knowledge to novel sequences.
Interestingly, there appears to be a critical transition between 100 and 200 training
sequences - 100 sequences corresponds to each item within each slot being presented roughly 10 times, which appears to provide sufficient statistical information
regarding the independence of individual slots.
Figure 3: Hidden unit representations (values are weights into a hidden unit from all other
layers). Unit in a) encodes the conjunction of a subset of input/output items at time 2.
(b) encodes a different subset of items at time 2. (c) encodes items over times 2 and 3.
(d) has no selectivity in the input, but does project to the output and likely participates
in recall of items at time step 3.
3.3
Analysis of Representations
To understand how the hidden units encode and retrieve information in the maintained context layer in a systematic fashion that supports the good generalization
observed, we examined the patterns of learned weights. Some representative examples are shown in Figure 3. Here, we see evidence of coarse-coded representations
that encode a subset of items in either one time point in the sequence or a couple
of time points. Also we found units that were more clearly associated with retrieval
and not encoding. These types of representations are consistent with our other
work showing how these kinds of representations can support good generalization
(O'Reilly & Busby, 2002).
4
Discussion
We have presented a model of sequential encoding of phonemes, based on
independently-motivated computational and biological considerations, focused on
the neural substrates of the prefrontal cortex and basal ganglia (Frank et al., 2001).
Viewed in more abstract, functional terms, however , our model is just another in
a long line of computational models of how people might encode sequential order
information. There are two classic models: (a) associative chaining, where the acti-
vation of a given item triggers the activation of the next item via associative links,
and (b) item-position association models where items are associated with their sequential positions and recalled from position cues (e.g., Lee & Estes, 1977). The
basic associative chaining model has been decisively ruled out based on error patterns (Henson, Norris, Page, & Baddeley, 1996), but modified versions of it may
avoid these problems (e.g., Lewandowsky & Murdock, 1989). Probably the most
accomplished current model, Burgess and Hitch (1999), is a version of the itemposition association model with a competitive queuing mechanism where the most
active item is output first and is then suppressed to allow other items to be output.
Compared to these existing models, our model is unique in not requiring fast associational links to encode items within the sequence. For example, the Burgess
and Hitch (1999) model uses rapid weight changes to associate items with a context
representation that functions much like the time input in our model. In contrast,
items are maintained strictly via persistent activation in our model , and the basalganglia based gating mechanism provides a means of encoding items into separate
neural slots that implicitly represent sequential order. Thus, the time inputs act
independently on the basal ganglia, which then operates generically on whatever
phoneme information is presently activated in the auditory input, obviating the
need for specific item-context links.
The clear benefit of not requiring associationallinks is that it makes the model much
more flexible and capable of generalization to novel sequences as we have demonstrated here (see O'Reilly & Munakata, 2000 for extended discussion of this general
issue). Thus, we believe our model is uniquely well suited for explaining the role
of the phonological loop in rapid binding of novel task information. Nevertheless,
the present implementation of the model has numerous shortcomings and simplifications, and does not begin to approach the work of Burgess and Hitch (1999) in
accounting for relevant psychological data. Thus, future work will be focused on
remedying these limitations. One important issue that we plan to address is the
interplay between the present model based on the prefrontal cortex and the binding that the hippocampus can provide - we suspect that the hippocampus will
contribute item-position associations and their associated error patterns and other
phenomena as discussed in Burgess and Hitch (1999).
Acknowledgments
This work was supported by ONR grant N00014-00-1-0246 and NSF grant IBN9873492. Rodolfo Soto died tragically at a relatively young age during the preparation of this manuscript - this work is dedicated to his memory.
5
References
Baddeley, A. , Gathercole, S. , & Papagno, C. (1998). The phonological loop as a language
learning device. Psychological Review, 105, 158.
Baddeley, A. D. (1986). Working memory. New York: Oxford University Press.
Burgess, N. , & Hitch, G. J . (1999). Memory for serial order: A network model of the
phonological loop and its timing. Psychological Review, 106, 551- 581.
Cohen, J. D., Braver, T. S., & O'Reilly, R. C. (1996). A computational approach to prefrontal cortex, cognitive control, and schizophrenia: Recent developments and current
challenges. Philosophical Transactions of the Royal Society (London) B, 351, 1515- 1527.
Cohen, J. D. , & O'Reilly, R. C. (1996). A preliminary theory of the interactions between
prefrontal cortex and hippocampus that contribute to planning and prospective memory.
In M. Brandimonte, G. O. Einstein, & M. A. McDaniel (Eds.) , Prospective memory:
Theory and applications (pp. 267- 296). Mahwah, New Jersey: Erlbaum.
Frank, M. J. , Loughry, B., & O 'Reilly, R . C. (2001). Interactions between the frontal cortex
and basal ganglia in working memory: A computational model. Cognitive, Affective,
and Behavioral Neurosci ence, 1 , 137- 160.
Henson, R. N. A., Norris, D. G ., Page, M. P . A., & Baddeley, A. D . (1996) . Unclaimed
memory: Error patterns rule out chaining models of immediate serial recall. Quarterly
Journal of Experimental Psychology: Human Experim ental Psychology, 49(A) , 80- 115.
Ivry, R. (1996). The representation of temporal information in perception and motor
control. Current Opinion in N eurobiology, 6,851-857.
Lee, C. L. , & Estes, W. K. (1977). Order and position in primary memory for letter
strings. Journal of Verbal Learning and Verbal B ehavior, 16, 395- 418.
Levitt , J . B. , Lewis, D. A. , Yoshioka, T. , & Lund, J. S. (1993). Topography of pyramidal neuron intrinsic connections in macaque monkey prefrontal cortex (areas 9 & 46).
Journal of Comparativ e N eurology, 338, 360- 376.
Lewandowsky, S. , & Murdock, B. B. (1989). Memory for serial order. Psychological R eview,
96, 25- 57.
Mel, B. A., & Fiser, J. (2000). Minimizing binding errors using learned conjunctive features. Neural Computation, 12, 731- 762.
Miller, E. K. , & Cohen, J. D. (2001). An integrative theory of prefrontal cortex function.
Annual Review of Neuroscience , 24, 167- 202.
Miyake, A., & Soto , R. (in preparation). The role of the phonological loop in executive
control.
O'Reilly, R. C. (1998). Six principles for biologically-based computational models of cortical cognition. Trends in Cognitive Sci ences, 2(11), 455- 462.
O 'Reilly, R . C ., Braver, T . S., & Cohen, J . D . (1999) . A biologically based computational
model of working memory. In A. Miyake, & P. Shah (Eds.) , Mod els of working m emory:
M echanisms of active maintenance and executiv e control. (pp. 375- 411) . New York:
Cambridge University Press.
O 'Reilly, R. C. , & Busby, R. S. (2002). Generalizable relational binding from coarsecoded distributed representations. Advances in N eural Information Processing Systems
(NIPS), 2001.
O 'Reilly, R . C. , Busby, R. S., & Soto, R. (in press). Three forms of binding and their neural
substrates: Alternatives to temporal synchrony. In A. Cleeremans (Ed.) , Th e unity of
consciousness: Binding, integration, and dissociation. Oxford: Oxford University Press.
O 'Reilly, R. C ., & Munakata, Y . (2000) . Computational explorations in cognitive n euroscience: Understanding th e mind by simulating th e brain. Cambridge, MA: MIT Press.
O 'Reilly, R. C. , & Rudy, J. W . (2001). Conjunctive representations in learning and memory: Principles of cortical and hippocampal function . Psychological Review, 108, 311345.
Paulesu, E. , Frith, C. D. , & Frackowiak, R. S. J. (1993). The neural correlates of the
verbal component of working memory. Nature, 362,342- 345.
Wickelgren, W. A. (1969). Context-sensitive coding, associative memory, and serial order
in (speech) behavior. Psychological R eview, 76 , 1- 15 .
| 2025 |@word trial:1 version:3 hippocampus:6 ences:1 integrative:1 gradual:1 accounting:1 initial:2 series:1 interestingly:1 existing:2 current:10 emory:1 activation:10 conjunctive:7 must:3 realistic:1 enables:1 motor:2 update:4 cue:2 device:1 item:22 complementing:1 updatable:2 short:1 colored:1 provides:3 coarse:3 murdock:2 location:5 contribute:2 differential:1 viable:1 persistent:2 acti:1 affective:1 behavioral:1 manner:1 rapid:3 behavior:1 themselves:1 planning:1 roughly:1 simulator:2 brain:3 project:1 begin:1 unrelated:1 what:1 kind:4 psych:1 string:1 monkey:1 developed:3 generalizable:1 temporal:5 remember:1 act:1 demonstrates:1 control:7 unit:16 whatever:1 grant:2 bind:2 died:1 timing:1 encoding:7 oxford:3 tapping:1 firing:1 might:2 studied:2 examined:1 challenging:1 co:2 unique:1 acknowledgment:1 testing:1 loughry:2 digit:1 episodic:1 area:2 significantly:1 adapting:1 reilly:22 word:1 suggest:1 get:1 cannot:2 storage:1 context:12 paulesu:2 demonstrated:1 exposure:1 independently:3 focused:2 miyake:6 immediately:1 rule:1 regarded:1 importantly:1 his:1 retrieve:1 classic:1 updated:1 target:3 trigger:2 colorado:3 decode:2 substrate:6 us:2 associate:1 element:2 trend:1 updating:2 stripe:11 observed:2 role:2 cleeremans:1 region:4 episode:1 dynamic:2 trained:7 ali:1 serve:1 basis:1 frackowiak:2 jersey:1 distinct:2 fast:1 shortcoming:1 activate:1 london:1 corresponded:1 encoded:4 widely:1 say:1 associative:4 hoc:1 sequence:37 advantage:2 interplay:1 interaction:2 relevant:3 loop:22 rapidly:5 gathercole:2 produce:2 depending:1 recurrent:2 received:1 implemented:5 involves:1 come:2 foot:1 correct:2 functionality:1 exploration:1 human:1 transient:2 opinion:1 require:2 activating:1 generalization:11 preliminary:1 biological:1 strictly:1 cognition:1 combinatorial:2 currently:1 saw:1 sensitive:1 mit:1 clearly:3 modified:1 avoid:2 conjunction:7 encode:13 properly:1 indicates:1 contrast:2 suppression:3 yoshioka:2 dependent:1 el:1 entire:1 hidden:5 issue:2 flexible:4 development:1 plan:1 integration:1 eurobiology:1 once:1 phonological:27 shaped:1 having:1 represents:1 future:1 report:1 stimulus:1 transiently:1 few:1 ime:1 lewandowsky:2 individual:4 phase:2 maintain:3 recalling:1 conductance:1 decisively:1 huge:1 hitch:6 generically:1 behind:1 activated:6 articulatory:3 capable:1 partial:1 necessary:1 experience:2 allocates:1 ruled:1 psychological:6 modeling:1 ence:1 localist:1 subset:3 erlbaum:1 vation:1 rudy:2 fundamental:1 systematic:2 participates:1 lee:2 together:3 quickly:2 connectivity:1 reflect:2 leveraged:1 prefrontal:15 wickelgren:2 cognitive:4 lii:1 coding:2 matter:1 ad:1 bg:2 piece:1 queuing:1 later:1 performed:1 closed:3 script:2 doing:1 red:4 competitive:1 participant:3 synchrony:1 phoneme:15 miller:2 dissociation:1 identify:1 generalize:1 ionic:1 coarsecoded:1 ed:3 pp:2 involved:1 spe:1 phonologically:1 associated:3 couple:1 auditory:1 recall:4 color:2 knowledge:1 oreilly:1 fiat:2 eview:2 back:1 appears:2 manuscript:1 higher:1 response:2 furthermore:1 just:2 fiser:2 working:12 expressive:1 interfere:2 mode:1 soto:9 believe:1 requiring:3 concept:1 ivry:2 consciousness:1 cerebellum:1 during:6 self:1 uniquely:1 maintained:5 mel:2 chaining:3 generalized:1 busby:5 hippocampal:5 presenting:2 demonstrate:1 performs:1 dedicated:1 consideration:1 novel:7 specialized:3 functional:1 cohen:9 association:3 discussed:1 interpret:1 cambridge:2 language:3 had:6 henson:2 cortex:17 etc:2 posterior:1 recent:2 moderate:1 phone:1 selectivity:2 n00014:1 onr:1 accomplished:1 seen:1 converge:1 determine:1 redundant:1 signal:3 long:3 retrieval:1 elaboration:1 serial:4 schizophrenia:1 coded:3 ensuring:1 basic:4 maintenance:10 essentially:1 circumstance:1 represent:6 ion:1 addition:1 pyramidal:1 probably:1 suspect:1 mod:1 unitary:1 switch:2 independence:1 remedying:1 psychology:4 burgess:6 identified:1 triggering:1 inner:1 idea:1 regarding:1 whether:1 motivated:2 cific:1 six:1 speech:2 york:2 cause:1 repeatedly:2 generally:2 modulatory:1 clear:1 involve:1 amount:1 ten:2 locally:1 mcdaniel:1 nsf:1 inhibitory:1 neuroscience:1 conceived:1 correctly:1 anatomical:1 basal:12 group:1 nevertheless:2 letter:4 powerful:2 respond:1 place:4 entirely:2 bound:1 layer:16 simplification:1 display:1 annual:1 encodes:5 aspect:1 performing:1 relatively:3 rodolfo:2 department:2 developing:1 leabra:1 combination:4 suppressed:1 unity:1 biologically:2 randall:1 lasting:1 presently:1 boulder:4 interference:2 computationally:1 previously:1 mechanism:21 needed:1 mind:1 operation:1 experim:1 quarterly:1 einstein:1 appropriate:2 simulating:1 braver:4 alternative:1 shah:1 hree:1 remaining:1 estes:2 graded:1 society:1 occurs:1 primary:2 deficit:1 separate:5 separably:1 capacity:2 simulated:3 link:3 sci:1 prospective:2 argue:1 besides:1 code:3 length:2 providing:1 tragically:1 minimizing:1 frank:5 rise:1 implementation:1 perform:3 neuron:2 immediate:2 situation:1 extended:1 relational:1 interacting:1 arbitrary:5 complement:1 required:2 connection:3 philosophical:1 conflict:1 recalled:2 learned:4 protect:1 macaque:1 nip:1 address:1 able:2 suggested:1 pattern:7 perception:1 lund:2 challenge:1 green:4 memory:19 including:1 royal:1 power:2 critical:3 event:1 difficulty:1 representing:1 numerous:1 mediated:1 review:6 understanding:2 relative:1 topography:1 limitation:1 age:1 executive:1 switched:1 sufficient:1 consistent:1 principle:3 systematically:1 share:1 excitatory:3 supported:3 repeat:3 dis:1 verbal:3 allow:1 understand:1 explaining:1 distributed:2 benefit:1 cortical:6 transition:1 ental:1 simplified:1 transaction:1 correlate:1 implicitly:1 sequentially:2 active:5 search:1 euroscience:1 subserved:1 nature:2 learn:3 robust:3 transfer:1 frith:2 interact:2 untrained:1 pfc:13 cl:1 domain:1 did:3 intracellular:5 neurosci:1 mahwah:1 repeated:2 allowed:2 obviating:1 eural:1 levitt:2 representative:1 fashion:2 iij:1 position:8 third:1 young:1 specific:4 showing:3 gating:10 symbol:1 evidence:1 intrinsic:1 sequential:14 importance:1 associational:1 interfered:1 demand:1 suited:1 simply:5 likely:3 ganglion:12 visual:1 partially:1 binding:34 norris:2 corresponds:1 lewis:2 ma:1 slot:4 identity:2 viewed:1 feasible:1 change:1 specifically:1 except:1 contrasted:1 operates:1 pas:1 experimental:1 munakata:3 ucb:2 indicating:1 support:5 people:1 frontal:1 preparation:6 ongoing:1 baddeley:6 tested:1 phenomenon:1 |
1,125 | 2,026 | Modeling Temporal Structure in Classical
Conditioning
Aaron C. Courville 1 ,3 and David S. Touretzk y 2,3
1 Robotics Institute, 2Computer Science Department
3Center for the Neural Basis of Cognition
Carnegie Mellon University, Pittsburgh, PA 15213-3891
{ aarone, dst} @es.emu.edu
Abstract
The Temporal Coding Hypothesis of Miller and colleagues [7] suggests that animals integrate related temporal patterns of stimuli
into single memory representations. We formalize this concept
using quasi-Bayes estimation to update the parameters of a constrained hidden Markov model. This approach allows us to account
for some surprising temporal effects in the second order conditioning experiments of Miller et al. [1 , 2, 3], which other models are
unable to explain.
1
Introduction
Animal learning involves more than just predicting reinforcement. The well-known
phenomena of latent learning and sensory preconditioning indicate that animals
learn about stimuli in their environment before any reinforcement is supplied. More
recently, a series of experiments by R. R. Miller and colleagues has demonstrated
that in classical conditioning paradigms, animals appear to learn the temporal structure of the stimuli [8]. We will review three of these experiments. We then present
a model of conditioning based on a constrained hidden Markov model , using quasiBayes estimation to adjust the model parameters online. Simulation results confirm
that the model reproduces the experimental observations, suggesting that this approach is a viable alternative to earlier models of classical conditioning which cannot account for the Miller et al. experiments. Table 1 summarizes the experimental
paradigms and the results.
Expt. 1: Simultaneous Conditioning. Responding to a conditioned stimulus
(CS) is impaired when it is presented simultaneously with the unconditioned stimulus (US) rather than preceding the US. The failure of the simultaneous conditioning
procedure to demonstrate a conditioned response (CR) is a well established result
in the classical conditioning literature [9]. Barnet et al. [1] reported an interesting
Expt. 1
Phase 1
(4)T+ US
Phase 2
(4)C -+ T
Expt.2A
Expt. 2B
(12)T -+ C
(12)T -+ C
(8)T -+ US
(8)T ---+ US
C=> C =>CR
Expt. 3
(96)L -+ US -+ X
(8) B -+ X
X=> -
Test => Result
T=> -
Test => Result
C =>CR
B =>CR
Table 1: Experimental Paradigms. Phases 1 and 2 represent two stages of training trials,
each presented (n) times. The plus sign (+ ) indicates simultaneous presentation of stimuli;
the short arrow (-+) indicates one stimulus immediately following another; and the long
arrow (-----+) indicates a 5 sec gap between stimulus offset and the following stimulus onset.
For Expt. 1, the tone T, click train C, and footshock US were all of 5 sec duration. For
Expt. 2, the tone and click train durations were 5 sec and the footshock US lasted 0.5
sec. For Expt. 3, the light L , buzzer E , and auditory stimulus X (either a tone or white
noise) were all of 30 sec duration, while the footshock US lasted 1 sec. CR indicates a
conditioned response to the test stimulus.
second-order extension of the classic paradigm. While a tone CS presented simultaneously with a footshock results in a minimal CR to the tone, a click train preceding
the tone (in phase 2) does acquire associative strength, as indicated by a CR.
Expt. 2: Sensory Preconditioning. Cole et al. [2] exposed rats to a tone T
immediately followed by a click train C. In a second phase, the tone was paired
with a footshock US that either immediately followed tone offset (variant A), or
occurred 5 sec after tone offset (variant B). They found that when C and US both
immediately follow T , little conditioned response is elicited by the presentation of
C. However, when the US occurs 5 sec after tone offset, so that it occurs later than
C (measured relative to T), then C does come to elicit a CR.
Expt. 3: Backward Conditioning. In another experiment by Cole et al. [3],
rats were presented with a flashing light L followed by a footshock US, followed by
an auditory stimulus X (either a tone or white noise). In phase 2, a buzzer B was
followed by X. Testing revealed that while X did not elicit a CR (in fact, it became
a conditioned inhibitor), X did impart an excitatory association to B.
2
Existing Models of Classical Conditioning
The Rescorla-Wagner model [11] is still the best-known model of classical conditioning, but as a trial-level model, it cannot account for within-trial effects such
as second order conditioning or sensitivity to stimulus timing. Sutton and Barto
developed V-dot theory [14] as a real-time extension of Rescorla-Wagner. Further
refinements led to the Temporal Difference (TD) learning algorithm [14]. These
extensions can produce second order conditioning. And using a memory buffer
representation (what Sutton and Barto call a complete serial compound), TD can
represent the temporal structure of a trial. However, TD cannot account for the empirical data in Experiments 1- 3 because it does not make inferences about temporal
relationships among stimuli; it focuses solely on predicting the US. In Experiment
1, some versions of TD can account for the reduced associative strength of a CS
when its onset occurs simultaneously with the US, but no version of TD can explain
why the second-order stimulus C should acquire greater associative strength than
T. In Experiment 2, no learning occurs in Phase 1 with TD because no prediction
error is generated by pairing T with C. As a result, no CR is elicited by C after
T has been paired with the US in Phase 2. In Experiment 3, TD fails to predict
the results because X is not predictive of the US; thus X acquires no associative
strength to pass on to B in the second phase.
Even models that predict future stimuli have trouble accounting for Miller et al. 's
results. Dayan's "successor representation" [4], the world model of Sutton and
Pinette [15], and the basal ganglia model of Suri and Schultz [13] all attempt to
predict future stimulus vectors. Suri and Schultz's model can even produce one
form of sensory preconditioning. However, none of these models can account for
the responses in any of the three experiments in Table 1, because they do not make
the necessary inferences about relations among stimuli.
Temporal Coding Hypothesis The temporal coding hypothesis (TCH) [7]
posits that temporal contiguity is sufficient to produce an association between stimuli. A CS does not need to predict reward in order to acquire an association with
the US. Furthermore, the association is not a simple scalar quantity. Instead, information about the temporal relationships among stimuli is encoded implicitly and
automatically in the memory representation of the trial. Most importantly, TCH
claims that memory representations of trials with similar stimuli become integrated
in such a way as to preserve the relative temporal information [3].
If we apply the concept of memory integration to Experiment 1, we get the memory
representation, C ---+ T + US. If we interpret a CR as a prediction of imminent
reinforcement, then we arrive at the correct prediction of a strong response to C
and a weak response to T. Integrating the hypothesized memory representations of
the two phases of Experiment 2 results in: A) T ---+ C+US and B) T ---+ C ---+ US. The
stimulus C is only predictive ofthe US in variant B, consistent with the experimental
findings. For Experiment 3, an integrated memory representation of the two phases
produces L+ B ---+ US ---+ X. Stimulus B is predictive of the US while X is not. Thus,
the temporal coding hypothesis is able to account for the results of each of the three
experiments by associating stimuli with a timeline.
3
A Computational Model of Temporal Coding
A straightforward formalization of a timeline is a Markov chain of states. For
this initial version of our model, state transitions within the chain are fixed and
deterministic. Each state represents one instant of time, and at each timestep a
transition is made to the next state in the chain. This restricted representation is
key to capturing the phenomena underlying the empirical results. Multiple timelines (or Markov chains) emanate from a single holding state. The transitions out
of this holding state are the only probabilistic and adaptive transitions in the simplified model. These transition probabilities represent the frequency with which
the timelines are experienced. Figure 1 illustrates the model structure used in all
simulations.
Our goal is to show that our model successfully integrates the timelines of the two
training phases of each experiment. In the context of a collection of Markov chains,
integrating timelines amounts to both phases of training becoming associated with
a single Markov chain. Figure 1 shows the integration of the two phases of Expt. 2B.
Figure 1: A depiction of the state and observation structure of the model. Shown are two
timelines, one headed by state j and the other headed by state k. State i, the holding state,
transitions to states j and k with probabilities aij and aik respectively. Below the timeline
representations are a sequence of observations represented here as the symbols T, C and
US. The T and C stimuli appear for two time steps each to simulate their presentation for
an extended duration in the experiment.
During the second phase of the experiments, the second Markov chain (shown in
Figure 1 starting with state k) offers an alternative to the chain associated with the
first phase of learning. If we successfully integrate the timelines, this second chain
is not used.
As suggested in Figure 1, associated with each state is a stimulus observation.
"Stimulus space" is an n-dimensional continuous space, where n is the number
of distinct stimuli that can be observed (tone, light, shock, etc.) Each state has
an expectation concerning the stimuli that should be observed when that state is
occupied. This expectation is modeled by a probability density function, over this
space, defined by a mixture of two multivariate Gaussians. The probability density
at stimulus observation xt in state i at time tis ,
where Wi is a mixture coefficient for the two Gaussians associated with state i. The
Gaussian means /tiD and /til and variances ufo and ufl are vectors of the same
dimension as the stimulus vector xt. Given knowledge of the state, the stimulus
components are assumed to be mutually independent (covariance terms are zero).
We chose a continuous model of observations over a discrete observation model to
capture stimulus generalization effects. These are not pursued in this paper.
For each state, the first Gaussian pdf is non-adaptive, meaning /tiO is fixed about
a point in stimulus space representing the absence of stimuli. ufo is fixed as well.
For the second Gaussian, /til and Ufl are adaptive. This mixture of one fixed and
one adaptive Gaussian is an approximation to the animal's belief distribution about
stimuli, reflecting the observed tolerance animals have to absent expected stimuli.
Put another way, animals seem to be less surprised by the absence of an expected
stimulus than by the presence of an unexpected stimulus.
We assume that knowledge of the current state st is inaccessible to the learner. This
information must be inferred from the observed stimuli. In the case of a Markov
chain, learning with hidden state is exactly the problem of parameter estimation in
hidden Markov models. That is, we must update the estimates of w, /tl and
for
ur
each state, and aij for each state transition (out of the holding state), in order to
maximize the likelihood of the sequence of observations
The standard algorithm for hidden Markov model parameter estimation is the
Baum-Welch method [10]. Baum-Welch is an off-line learning algorithm that requires all observations used in training to be held in memory. In a model of classical
conditioning, this is an unrealistic assumption about animals' memory capabilities.
We therefore require an online learning scheme for the hidden Markov model, with
only limited memory requirements.
Recursive Bayesian inference is one possible online learning scheme. It offers
the appealing property of combining prior beliefs about the world with current observations through the recursive application of Bayes' theorem, p(Alxt) IX
p(xt lx t - 1 , A)p(AIXt - 1 ). The prior distribution, p(AIX t - 1 ) reflects the belief over
the parameter A before the observation at time t , xt. X t - 1 is the observation history up to time t - l , i.e. X t - 1 = {x t - 1 ,xt - 2 , ... }. The likelihood, p(xtIXt-l,A)
is the probability density over xt as a function of the parameter A.
Unfortunately, the implementation of exact recursive Bayesian inference for a continuous density hidden Markov model (CDHMM) is computationally intractable.
This is a consequence of there being missing data in the form of hidden state.
With hidden state, the posterior distribution over the model parameters, after the
observation, is given by
N
p(Alxt)
IX
LP(xtlst = i, X t - 1 , A)p(st = iIX t - 1 , A)p(AIXt - 1 ),
(2)
i=1
where we have summed over the N hidden states. Computing the recursion for
multiple time steps results in an exponentially growing number of terms contributing
to the exact posterior.
We instead use a recursive quasi-Bayes approximate inference scheme developed
by Huo and Lee [5], who employ a quasi-Bayes approach [12]. The quasi-Bayes
approach exploits the existence of a repeating distribution (natural conjugate) over
the parameters for the complete-data CDHMM. (i.e. where missing data such as the
state sequence is taken to be known). Briefly, we estimate the value of the missing
data. We then use these estimates, together with the observations, to update the
hyperparameters governing the prior distribution over the parameters (using Bayes'
theorem). This results in an approximation to the exact posterior distribution over
CDHMM parameters within the conjugate family of the complete-data CDHMM.
See [5] for a more detailed description of the algorithm.
Estimating the missing data (hidden state) involves estimating transition probabilities between states, ~0 = Pr(sT = i, ST+1 = jlXt , A), and joint state and mixture
component label probabilities ([k = Pr(sT = i, IT = klX t , A). Here zr = k is the
mixture component label indicating which Gaussian, k E {a, I}, is the source of the
stimulus observation at time T. A is the current estimate of all model parameters.
We use an online version of the forward-backward algorithm [6] to estimate ~0 and
([1. The forward pass computes the joint probability over state occupancy (taken to
be both the state value and the mixture component label) at time T and the sequence
of observations up to time T. The backward pass computes the probability of the
observations in a memory buffer from time T to the present time t given the state
occupancy at time T. The forward and backward passes over state/observation
sequences are combined to give an estimate of the state occupancy at time T given
the observations up to the present time t. In the simulations reported here the
memory buffer was 7 time steps long (t - T = 6).
We use the estimates from the forward-backward algorithm together with the observations to update the hyperparameters. For the CDHMM, this prior is taken
to be a product of Dirichlet probability density functions (pdfs) for the transition
probabilities (aij) , beta pdfs for the observation model mixture coefficients (Wi)
and normal-gamma pdfs for the Gaussian parameters (Mil and afl)' The basic hyperparameters are exponentially weighted counts of events, with recency weighting
determined by a forgetting parameter p. For example, "'ij is the number of expected
transitions observed from state i to state j, and is used to update the estimate of
parameter aij. The hyperparameter Vik estimates the number of stimulus observations in state i credited to Gaussian k , and is used to update the mixture parameter
Wi. The remaining hyperparameters 'Ij;, ?, and () serve to define the pdfs over Mil
and afl' The variable d in the equations below indexes over stimulus dimensions.
Si1d is an estimate of the sample variance, and is a constant in the present model.
T _
((T-1)
"'ij -
. I,T
P "'ij
. 1,(T-1)
'l' i1d
=
()T
_ p() ( T- 1)
i1d -
-
P 'I' i1d
i1d
1)
(:T
+ 1 + '>ij
T _
rT
+ '>i1
+
7"
(i1 Sild
2
((T-1)
v ik -
,/,T
P v ik
_ p(,/,(T-1) _
'l'i 1d -
+
0,,( 7"-1 ) ,7"
Po/ i 1d
'il
2(p1/Ji;d 1) H[1)
(xT _
d
-
'l'i1d
1)
rT
+ 1 + '>ik
1)
2
+ 1H[1
2
()
II. T- 1 )2
f"'i 1d
In the last step of our inference procedure, we update our estimate of the model
parameters as the mode of their approximate posterior distribution. While this is
an approximation to proper Bayesian inference on the parameter values, the mode
of the approximate posterior is guaranteed to converge to a mode of the exact
posterior. In the equations below, N is the number of states in the model.
T_
Wi -
4
v[1- 1
vio + viI -2
Results and Discussion
The model contained two timelines (Markov chains). Let i denote the holding
state and j, k the initial states of the two chains. The transition probabilities were
initialized as aij = aik = 0.025 and aii = 0.95. Adaptive Gaussian means Mild were
initialized to small random values around a baseline of 10- 4 for all states. The
exponential forgetting factor was P = 0.9975, and both the sample variances Si1d
and the fixed variances aIOd were set to 0.2.
We trained the model on each of the experimental protocols of Table 1, using the
same numbers of trials reported in the original papers. The model was run continuously through both phases of the experiments with a random intertrial interval.
'+-::::
noCR
CR
'+-::::
4
5
'+
-:g4
noCR
~3
C
OJ
E
OJ
E
OJ
E
~3
.E
~2
OJ
~
()
02
.E
c
C
Oi
~1
&!1
"Qi
0
a:
g;1
0
trr
trr
T
C
Experiment 1
0
noCR
trr
(A)C
(B)C
Experiment 2
0
X
B
Experiment 3
Figure 2: Results from 20 runs of the model simulation with each experimental paradigm.
On the ordinate is the total reinforcement (US) , on a log scale, above the baseline (an
arbitrary perception threshold) expected to occur on the next time step. The error bars
represent two standard deviations away from the mean.
Figure 2 shows t he simulation results from each of the three experiments. If we
assume that the CR varies monotonically with the US prediction, then in each case,
t he model's predicted CR agreed with the observations of Miller et al.
The CR predictions are the result of the model integrating t he two phases of learning
into one t imeline. At the t ime of the presentation of the Phase 2 stimuli, the states
forming the timeline describing the Phase 1 pattern of stimuli were judged more
likely to have produced the Phase 2 stimuli than states in the other t imeline, which
served as a null hypothesis. In another experiment, not shown here , we trained the
model on disjoint stimuli in the two phases. In that situation it correctly chose a
separate t imeline for each phase, rather than merging the two .
We have shown that under the assumption t hat observation probabilities are modeled by a mixture of Gaussians, and a very restrictive state transition structure, a
hidden Markov model can integrate the memory representations of similar temporal
stimulus patterns. "Similarity" is formalized in this framework as likelihood under
the t imeline model. We propose t his model as a mechanism for the integration of
memory representations postulated in the Temporal Coding Hypothesis.
The model can be extended in many ways. The current version assumes t hat event
chains are long enough to represent an entire trial, but short enough that the model
will return to the holding state before the start of the next trial. An obvious
refinement would be a mechanism to dynamically adjust chain lengths based on
experience. We are also exploring a generalization of the model to the semi-Markov
domain, where state occupancy duration is modeled explicitly as a pdf. State transitions would then be tied to changes in observations, rather than following a rigid
progression as is currently the case. Finally, we are experiment ing with mechanisms
that allow new chains to be split off from old ones when the model determines that
current stimuli differ consistently from t he closest matching t imeline.
Fitting stimuli into existing t imelines serves to maximize the likelihood of current
observations in light of past experience. But why should animals learn the temporal
structure of stimuli as t imelines? A collection of timelines may be a reasonable
model of the natural world. If t his is true, t hen learning with such a strong inductive
bias may help t he animal to bring experience of related phenomena to bear in novel
sit uations- a desirable characteristic for an adaptive system in a changing world.
Acknowledgments
Thanks to Nathaniel Daw and Ralph Miller for helpful discussions. This research
was funded by National Science Foundation grants IRI-9720350 and IIS-997S403.
Aaron Courville was funded in part by a Canadian NSERC PGS B fellowship.
References
[1] R. C. Barnet, H. M. Arnold, and R. R. Miller. Simultaneous conditioning demonstrated in second-order conditioning: Evidence for similar associative structure in
forward and simultaneous conditioning. Learning and Motivation, 22:253- 268, 1991.
[2] R. P. Cole, R. C. Barnet, and R. R . Miller. Temporal encoding in trace conditioning.
Animal Learning and Behavior, 23(2) :144- 153, 1995 .
[3] R. P. Cole and R. R. Miller. Conditioned excitation and conditioned inhibition acquired through backward conditioning. Learning and Motivation , 30:129- 156, 1999.
[4] P. Dayan. Improving generalization for temporal difference learning: the successor
representation. Neural Computation, 5:613- 624, 1993.
[5] Q. Huo and C.-H. Lee. On-line adaptive learning of the continuous density hidden
Markov model based on approximate recursive Bayes estimate. IEEE Transactions
on Speech and Audio Processing, 5(2):161- 172, 1997.
[6] V . Krishnamurthy and J . B. Moore. On-line estimation of hidden Markov model
parameters based on the Kullback-Leibler information measure. IEEE Transactions
on Signal Processing, 41(8):2557- 2573, 1993.
[7] L. D. Matzel , F. P. Held, and R. R. Miller. Information and the expression of simultaneous and backward associations: Implications for contiguity theory. Learning and
Motivation, 19:317- 344, 1988.
[8] R. R. Miller and R . C. Barnet. The role of time in elementary associations. Current
Directions in Psychological Sci ence, 2(4):106- 111 , 1993.
[9] 1. P. Pavlov. Conditioned Reflexes. Oxford University Press, 1927.
[10] L. R. Rabiner. A tutorial on hidden Markov models and selected applications
speech recognition. Proceedings of th e IEEE, 77(2) :257- 285, 1989.
III
[11] R. A. Rescorla and A. R. Wagner. A theory of Pavlovian conditioning: Variations in
the effectiveness of reinforcement and nonreinforcement . In A. H. Black and W. F.
Prokasy, editors, Classical Conditioning II. Appleton-Century-Crofts, 1972.
[12] A. F . M. Smith and U. E . Makov . A quasi-Bayes sequential procedure for mixtures.
Journal of th e Royal Statistical Soci ety, 40(1):106- 112, 1978.
[13] R. E. Suri and W. Schultz. Temporal difference model reproduces anticipatory neural
activity. N eural Computation, 13(4):841- 862, 200l.
[14] R. S. Sutton and A. G. Barto. Time-derivative models of Pavlovian reinforcement. In
M. Gabriel and J. Moore, editors, Learning and Computational N euroscience: Foundations of Adaptive N etworks, chapter 12 , pages 497- 537. MIT Press, 1990.
[15] R. S. Sutton and B. Pinette. The learning of world models by connectionist networks.
In L. Erlbaum, editor, Proceedings of the seventh annual conference of the cognitive
science society, pages 54- 64, Irvine, California, August 1985.
| 2026 |@word mild:1 trial:9 version:5 briefly:1 simulation:5 t_:1 accounting:1 covariance:1 initial:2 series:1 past:1 existing:2 current:7 surprising:1 must:2 afl:2 update:7 pursued:1 selected:1 tone:13 huo:2 smith:1 short:2 i1d:5 lx:1 klx:1 become:1 beta:1 viable:1 pairing:1 surprised:1 ik:3 fitting:1 headed:2 g4:1 acquired:1 forgetting:2 expected:4 behavior:1 p1:1 growing:1 td:7 automatically:1 little:1 estimating:2 underlying:1 null:1 what:1 contiguity:2 developed:2 finding:1 temporal:21 ti:1 exactly:1 grant:1 appear:2 before:3 tid:1 timing:1 consequence:1 sutton:5 encoding:1 oxford:1 solely:1 becoming:1 credited:1 black:1 plus:1 chose:2 dynamically:1 suggests:1 pavlov:1 limited:1 acknowledgment:1 testing:1 recursive:5 procedure:3 emanate:1 empirical:2 elicit:2 imminent:1 matching:1 integrating:3 get:1 cannot:3 barnet:4 judged:1 put:1 context:1 recency:1 deterministic:1 demonstrated:2 center:1 baum:2 missing:4 straightforward:1 iri:1 starting:1 duration:5 welch:2 formalized:1 immediately:4 importantly:1 his:2 classic:1 century:1 variation:1 krishnamurthy:1 aik:2 exact:4 soci:1 hypothesis:6 pa:1 recognition:1 observed:5 role:1 capture:1 environment:1 inaccessible:1 reward:1 trained:2 uations:1 xtixt:1 exposed:1 predictive:3 serve:1 learner:1 basis:1 preconditioning:3 po:1 joint:2 aii:1 represented:1 chapter:1 train:4 distinct:1 encoded:1 unconditioned:1 online:4 associative:5 sequence:5 propose:1 rescorla:3 product:1 combining:1 alxt:2 description:1 impaired:1 requirement:1 produce:4 help:1 measured:1 ij:5 strong:2 expt:11 involves:2 indicate:1 c:4 come:1 predicted:1 differ:1 posit:1 direction:1 correct:1 successor:2 require:1 generalization:3 elementary:1 extension:3 exploring:1 around:1 normal:1 cognition:1 predict:4 claim:1 estimation:5 integrates:1 label:3 pinette:2 currently:1 cole:4 footshock:6 successfully:2 reflects:1 weighted:1 impart:1 mit:1 inhibitor:1 gaussian:8 rather:3 occupied:1 cr:15 barto:3 mil:2 focus:1 pdfs:4 consistently:1 indicates:4 likelihood:4 lasted:2 baseline:2 helpful:1 inference:7 dayan:2 rigid:1 integrated:2 entire:1 hidden:15 relation:1 quasi:5 i1:2 ralph:1 among:3 animal:11 touretzk:1 constrained:2 integration:3 summed:1 represents:1 buzzer:2 future:2 cdhmm:5 stimulus:51 connectionist:1 employ:1 simultaneously:3 preserve:1 gamma:1 ime:1 national:1 phase:23 attempt:1 adjust:2 mixture:10 light:4 held:2 chain:15 implication:1 necessary:1 experience:3 old:1 initialized:2 minimal:1 psychological:1 modeling:1 earlier:1 ence:1 deviation:1 seventh:1 erlbaum:1 reported:3 varies:1 combined:1 st:5 density:6 thanks:1 sensitivity:1 probabilistic:1 off:2 lee:2 aix:1 together:2 continuously:1 cognitive:1 derivative:1 til:2 return:1 account:7 suggesting:1 ety:1 makov:1 coding:6 sec:8 coefficient:2 postulated:1 explicitly:1 onset:2 later:1 start:1 bayes:8 elicited:2 capability:1 il:1 oi:1 nathaniel:1 became:1 variance:4 characteristic:1 ufo:2 who:1 miller:12 ofthe:1 rabiner:1 matzel:1 weak:1 bayesian:3 produced:1 none:1 served:1 history:1 explain:2 simultaneous:6 failure:1 colleague:2 frequency:1 intertrial:1 obvious:1 associated:4 irvine:1 auditory:2 knowledge:2 formalize:1 agreed:1 reflecting:1 follow:1 response:6 anticipatory:1 furthermore:1 just:1 stage:1 governing:1 mode:3 indicated:1 effect:3 hypothesized:1 concept:2 true:1 inductive:1 moore:2 leibler:1 white:2 during:1 acquires:1 excitation:1 rat:2 pdf:2 complete:3 demonstrate:1 bring:1 meaning:1 suri:3 novel:1 recently:1 ji:1 conditioning:21 exponentially:2 association:6 occurred:1 he:5 interpret:1 mellon:1 appleton:1 dot:1 funded:2 similarity:1 depiction:1 inhibition:1 etc:1 multivariate:1 posterior:6 closest:1 compound:1 buffer:3 nonreinforcement:1 greater:1 preceding:2 converge:1 paradigm:5 maximize:2 monotonically:1 signal:1 ii:3 semi:1 multiple:2 desirable:1 ing:1 offer:2 long:3 concerning:1 serial:1 paired:2 qi:1 prediction:5 variant:3 basic:1 expectation:2 represent:5 robotics:1 fellowship:1 interval:1 source:1 pass:1 ufl:2 seem:1 effectiveness:1 call:1 presence:1 revealed:1 split:1 enough:2 canadian:1 iii:1 associating:1 click:4 vik:1 absent:1 expression:1 speech:2 gabriel:1 detailed:1 amount:1 repeating:1 reduced:1 supplied:1 tutorial:1 sign:1 disjoint:1 correctly:1 carnegie:1 discrete:1 hyperparameter:1 basal:1 key:1 threshold:1 changing:1 shock:1 backward:7 timestep:1 run:2 dst:1 arrive:1 family:1 reasonable:1 summarizes:1 capturing:1 followed:5 guaranteed:1 courville:2 annual:1 activity:1 strength:4 occur:1 simulate:1 pavlovian:2 department:1 conjugate:2 ur:1 wi:4 appealing:1 lp:1 restricted:1 pr:2 taken:3 computationally:1 equation:2 mutually:1 etworks:1 describing:1 count:1 mechanism:3 serf:1 gaussians:3 apply:1 progression:1 away:1 alternative:2 hat:2 existence:1 original:1 responding:1 dirichlet:1 remaining:1 trouble:1 assumes:1 iix:1 instant:1 exploit:1 restrictive:1 classical:8 society:1 quantity:1 occurs:4 rt:2 unable:1 separate:1 sci:1 length:1 modeled:3 relationship:2 index:1 acquire:3 unfortunately:1 holding:6 trace:1 implementation:1 proper:1 observation:26 markov:18 situation:1 extended:2 arbitrary:1 august:1 inferred:1 ordinate:1 david:1 california:1 emu:1 established:1 timeline:12 daw:1 able:1 suggested:1 bar:1 below:3 pattern:3 perception:1 oj:4 memory:15 royal:1 belief:3 unrealistic:1 event:2 natural:2 predicting:2 zr:1 recursion:1 representing:1 scheme:3 occupancy:4 review:1 literature:1 prior:4 hen:1 contributing:1 relative:2 bear:1 interesting:1 foundation:2 integrate:3 sufficient:1 consistent:1 editor:3 excitatory:1 last:1 aij:5 bias:1 allow:1 trr:3 institute:1 arnold:1 wagner:3 tolerance:1 dimension:2 world:5 transition:13 computes:2 sensory:3 forward:5 made:1 reinforcement:6 refinement:2 adaptive:8 schultz:3 simplified:1 collection:2 transaction:2 prokasy:1 approximate:4 implicitly:1 kullback:1 confirm:1 reproduces:2 vio:1 pittsburgh:1 assumed:1 continuous:4 latent:1 why:2 table:4 euroscience:1 learn:3 improving:1 protocol:1 domain:1 did:2 arrow:2 pgs:1 noise:2 hyperparameters:4 motivation:3 eural:1 tl:1 formalization:1 fails:1 experienced:1 exponential:1 tied:1 weighting:1 ix:2 croft:1 theorem:2 xt:7 symbol:1 offset:4 evidence:1 sit:1 intractable:1 merging:1 sequential:1 flashing:1 conditioned:8 illustrates:1 tio:1 gap:1 vii:1 led:1 likely:1 ganglion:1 forming:1 tch:2 unexpected:1 contained:1 nserc:1 scalar:1 reflex:1 determines:1 goal:1 presentation:4 absence:2 change:1 determined:1 total:1 pas:3 e:1 experimental:6 aaron:2 indicating:1 audio:1 phenomenon:3 |
1,126 | 2,027 | TAP Gibbs Free Energy, Belief Propagation and
Sparsity
Lehel Csat?o and Manfred Opper
Neural Computing Research Group
School of Engineering and Applied Science
Aston University, Birmingham B4 7ET, UK.
[csatol,opperm]@aston.ac.uk
Ole Winther
Center for Biological Sequence Analysis, BioCentrum
Technical University of Denmark, B208, 2800 Lyngby, Denmark.
winther@cbs.dtu.dk
Abstract
The adaptive TAP Gibbs free energy for a general densely connected
probabilistic model with quadratic interactions and arbritary single site
constraints is derived. We show how a specific sequential minimization
of the free energy leads to a generalization of Minka?s expectation propagation. Lastly, we derive a sparse representation version of the sequential
algorithm. The usefulness of the approach is demonstrated on classification and density estimation with Gaussian processes and on an independent component analysis problem.
1 Introduction
There is an increasing interest in methods for approximate inference in probabilistic (graphical) models. Such approximations may usually be grouped in three classes. In the first case
we approximate self-consistency relations for marginal probabilities by a set of nonlinear
equations. Mean field (MF) approximations and their advanced extensions belong to this
group. However, it is not clear in general, how to solve these equations efficiently. This
latter problem is of central concern to the second class, the Message passing algorithms,
like Bayesian online approaches (for references, see e.g. [1]) and belief propagation (BP)
which dynamically update approximations to conditional probabilities. Finally, approximations based on Free Energies allow us to derive marginal moments by minimising entropic
loss measures. This method introduces new possibilities for algorithms and also gives approximations for the log-likelihood of observed data. The variational method is the most
prominent member of this group.
One can gain important insight into an approximation, when it can be derived by different
approaches. Recently, the fixed points of the BP algorithm were identified as the stable minima of the Bethe Free Energy, an insight which led to improved approximation schemes [2].
While BP is good and efficient on sparse tree-like structures, one may look for an approxi-
mation that works well in the opposite limit of densely connected graphs where individual
dependencies are weak but their overall effect cannot be neglected. A interesting candidate is the adaptive TAP (ADATAP) approach introduced in [3] as a set of self-consistency
relations. Recently, a message passing algorithm of Minka (termed expectation propagation) [1] was found to solve the ADATAP equations efficiently for models with Gaussian
Process (GP) priors.
The goal of this paper is three-fold. We will add a further derivation of ADATAP using
an approximate free energy. A sequential algorithm for minimising the free energy generalises Minka?s result. Finally, we discuss how a sparse representation of ADATAP can be
achieved for GP models, thereby extending previous sparse on-line approximations to the
batch case [4].
We will specialize to probabilistic models on undirected graphs with nodes that are of the
type
The
set of
&"
!" $ #%
(1)
?s encodes the dependencies
between
/01
32 the
random variables
(called likelihood in the
, whereas the factorising term
')(+*,*+*,( .
following) usually encodes observed data at sites and also incorporates all local con
straints of the
(the range, discreteness, etc). Hence, depending on these constraints,
maybe discrete or continuous. Eq. (1) is a sufficiently rich and interesting class of models
containing Boltzmann machines, models with Gaussian process priors [3], probabilistic independent component analysis [5] as well as Bayes belief networks and probabilistic neural
networks (when the space of variables is augmented by auxiliary integration variables).
2 ADATAP approach from Gibbs Free Energy
We use the minimization of an approximation to a Gibbs Free Energy
derive the ADATAP approximation.
4
in order to re-
The Gibbs Free Energy provides a method for computing marginal moments of as well
as of 57698
within the same approach. It is defined by a constrained relative entropy
minimization which is, for the present problem defined as
>=@?
I H JLKM/0N A
>: M0O+N A
(2)
8 BEDGF
5G698
AC
(
(
<QP
(
H
M O N A
where the brackets denote expectations with respect to the distribution
and
is
M O N A
H J
TS@UVHW \EA ] X9X9Y[YEZ Z
. Finally, F
698
.
shorthand for a vector with elements
*
respect to its DR
(
SinceH^
at
_
the
total
minimum
of
4
(with
arguments)
the
minimizer
in
(2)
is
=`?
I:
h
5i698 and the desired marginal
just
, we conclude
M/0N M O Ngthat
j
>kElgm!8b=@ad? c ef4
I(g:V
8 adc e 4 < .
moments of are
(
n
Lqsr
which is based on splitting 4
We will search for an approximation to 4
4po
4 ,
where 4to is the
>u Gibbs free energy for a factorising model that is obtained from (1) by
setting all "
attempts [6, 7] were based on a truncation of the power series
r . Previous
expansion of 4
with respect to the &v at second order. While this truncation leads to
the correct TAP equations for the large w limit of the so-called SK-model in statistical
4
;:
(<
physics, its general significance is unclear. In fact, it will not be exact for a simple model
with Gaussian likelihood. To make our approximation exact for such a case,;: we define
r
(generalizing
*
$z;:
an{idea
;: of [8])
for an arbitrary Gaussian likelihood yx , 4 r x ;(: <
z
4
5p4po
. The main reason for this definition is the fact that 4 x
(<
(g<
(<
is independent of the actual Gaussian likelihood x chosen to compute 4 ! This result
depends crucially on the moment constraints in (2). Changes in a Gaussian likelihood
can always be absorbed within the Lagrange-multipliers for the constraints.
use this
\
4po qGr We
r
4 x , which
universal form 4 x to define the ADATAP approximation as 4
by construction is exact for any Gaussian likelihood . Introducing appropriate Lagrange
multipliers and , we get
r 4 x ;:
(<
>=Wk
c
57698
x
S U
x (
n
S U
o
o o o
(
=Wk
5
4 o
"! c !$#
with
( q : R5 <
5 O' O q
O
5 O' o , we have
o o o q :
68
(
R5 6 8
5 O (3)
;y !v * Finally, setting
o&% *
o 5 <
(4)
3 Sequential Algorithm
'
\
;:
The expression of 4
in terms of moments
and Lagrange parameters
\
(
o suggests that we may find local minima(< of 4
and o
by iteratively alternating
(
between updates of moments and Lagrange multipliers. Of special interest is the following
sequential algorithm, which is a generalization of Minka?s EP [1] for Gaussian process
classification to an arbritary model of the type eq. (1).
We choose a site and define the updates by using the saddle points of 4
with respect
to the moments and Lagrange multipliers in the following sequential order (where is a
diagonal matrix with elements ):
)+* , ,
fu .
)78, c - c 9 4 , 4
fu:.
)*, ,
u .
)78! , c c- 9 ! 4, 4
fu:.
5
O /
0 ( 532 4 ' 6
o /
5 ) 5 9 , 4 ' 7<;,
/ 5 - ,! 698 , 7< o ;
=/ 5 o 5 9 4 ' , .
The algorithm proceeds then by choosing a new site. The computation of T
( 5>2 4 ' can
be performed efficiently using the Sherman-Woodbury formula because only one element
is changed in each update.
"/
10 ( 53
7, , 2785; 4 ' 6 v
o / 5 5 9 4 ,
/
) * , 698 o 7 ,
"/
5 o! 5 9 , 4 78;,
(
3.1 Cavity interpretation
,
$? B AX , @&Z
&
&
&
&
O
o 5 O' o
At the fixed point, we may take
as the ADATAP approx
imation to the true marginal distribution of
[3]. The sequential approach may thus be
considered as a belief propagation algorithm for ADATAP.
!
:
Although is usually not Gaussian, we can also derive the moments and from
the
has a
Gaussian distribution
corresponding
to x . This auxiliary Gaussian model < x
O
5 O ' q and
likelihood &x
provides us also with an additional approxi
5 . This is useful when the coupling
mation to the matrix of covariances via
'
matrix must be adapted to a set of observations by maximum likelihood II. We will give
an example of this for independent component analysis below.
8C
D
2
( 32 4
It is important to understand
the role of o and o within the ?cavity?
approach to the
&v , it is easy to show that o
M N and o
TAP equations. Defining
M O N M N O
5
where the brackets denote an expectation with respect to the distribution of
E HF
E F
E
E GF
C 2
c &x + when node is
all remaining variables x
deleted from the graph. This
statistics of corresponds to the empty ?cavity? at site . The
marginal distribution
as computed by ADATAP is equivalent to the approximation
that the cavity distribution is Gaussian.
E
4 Examples
4.1 Models with Gaussian Process Priors
, where the vector
For
this class of models, we assume that the graph is embedded
0 in
is the restriction of
a Gaussian
process
(random field)
with
, to a set of
training inputs via
.
is the posterior distribution corresponding to a local
likelihood model, when
we set
5 ' and
the matrix is obtained
from
a positive
definite covariance kernel as D v
D o ( . The diagonal element D ' 9 is included
in the likelihood term.
4
2
4
Our ADATAP approximation can be extended from the finite
set of inputs to the entire
space
by extending the auxiliary
distribution x with its likelihoods
M 0Gaussian
N
&x
to a Gaussian process with mean
and posterior covariance kernel D
which
(
A calculation similar to [4] leads to the
approximates the posterior process.
representation
D
M 0 N
(
D
o
(
q
D D o (
D o (
c D o (
(5)
(6)
D
Algorithms for the update of ?s and ?s will usually suffer from time consuming matrix
multiplications when w is large. This common problem for GP models can be overcome
by a sparsity approximation which extends previous on-line approaches [4] to the batch
ADATAP approach. The idea is to replace the current version x of the approximate Gaus
sian with a further approximation x for which both the the corresponding as well as
are nonzero only, when the nodes and belong to a smaller subset
of nodes called
?basis vectors? (BV) of size [4]. For fixed BV set, the parameters of x are determined
!
and
by minimizing the relative entropy DGF x
x . This yields
Q
(
with the #"Gw projection matrix
$&'% (' . Here is the kernel matrix between
BVs and and ' the kernel matrix between BVs and all nodes. The new distribution x
a likelihood that contains
can be written in the form (1) with
only BVs
D
4
x
$)% j
&$ % 5 +* &$ % + , O
*
(
(
(7)
Eq. (7) can be used to compute the sparse approximation within the sequential algorithm.
We will only give a brief discussion here. In order to recompute the appropriate ?cavity?
parameters o and o when a new node is chosen by the algorithm, one removes a ?pseudo $&%
variable?
from the likelihood and recomputes
of the remaining
$&
$&% the
statistics
% and the computation
ones. When is in the BV set, then simply
reduces to the previous one. We will demonstrate the significance of this approach for two
examples.
4.2 Independent Component Analysis
We consider
a measured signal -/. which is assumed to be an instantaneous linear mixing
of sources corrupted with additive white Gaussian noise 0 that is,
21i q
-/.
. 03.
(8)
(
1
where
is a (time independent) mixing matrix and the noise vector is assumed to be
without temporal correlations having time independent covariance matrix . We thus have
the following likelihood for parameters and sources at time
4 ; X
4 YZ X
4 YIZ
K1
j
4 ;
- .
(9)
.
*
W ( K 1 (
0i
2 W K 1
and for all times
. - . ( ( . . The aim of independent com( ( the unknown
ponent
analysis
is
to
recover
quantities: the sources , the mixing matrix
1
and the noise covariance Wfrom
the
observed
2 W data using the assumption of statistical
independence
.
. . Following [5], we estimate the mix1 of the sources
an MLII procedure, i.e. by maximizing
ing matrix
and
the
noise
covariance
,
by
W K 1
j
S`U W K 1
0yW
the Likelihood . The corresponding estimates are
4
( (
1
M N (
M N
M
1i
170 N
'
. -. . . . .
and
These
- 5
' - 5
- again the structure of the* model
estimates require averages over the posterior of which has
eq. (1). They can be obtained efficiently using our sequential belief propagation
M N
M/ algorithm
N
in an 1iterative EM fashion, where the E-step amounts
to
estimating
and
.
. . with
1
W
fixed
and
and the M-step consists of updating
and
.
5 Simulations
5.1 Classification with GPs
This problem has been studied before [9, 4] using a sequential, sparse algorithm, based
on a single sweep through the data only. Within the ADATAP approach we are able to
perform multiple sweeps in order to achieve a self-consistent solution.
The outputs are bi* 5
, and the likelihood is based on the probit model W ! K
"bg
$#jl&%0' j
nary !
'G
"b/.0
0
S + U (
.
(
!
and
measures the noise level. The pre' O*) -, 5 O where
4
;
*
"
l/o %, M "bo N
#
!
.1032
0 2
O
40 O q
" b"
!
with
,
dictive distribution for a new test input is
.
.
o to D eqs.
( (5).
which is easily rewritten in terms of the parameters ?s and ?s according
5
5
We
used the USPS dataset1 of
with
6 7
u!u86 gray-scale handwritten digit images of size "
training
patterns
and
test
patterns.
For
the
kernel
we
choose
the
RBF
kernel
" "
:98;
" " O . 0 ;O g
>5
5=0 <; 5 <
where is the dimension6 uEofu!u the inputs (
in
D o (
9 ;
and
are parameters. In the simulations we used
random training
this case), and
examples. We performed simulations for different sizes of the BV set and compared multiple iterations with a single sweep through
the dataset. The results are displayed in Fig. 1.
>
The lines show the average results of runs where the task was to classify the digits into
fours/non-fours. Our results show that, in contrast to the online learning, the fluctuations
caused by the order of presentation are diminished (marked with bars on the figure).
D
5.2 Density estimation with GPs
;
Bayesian non-parametric models for
density estimation can be defined [10] by parametris"K
A @ X X 2 2 ZZ B 2
ing densities ? as ?
and using a Gaussian process prior over the space
@
D"
"
of functions
. Observing w data points C
, we can express the predictive
' (,the
*+*,*+GP
( - prior) as
distribution (again, E denotes the expectation over
;
S
FGIH-J KMLON
P
Q R
=
S
X
FGIH-J TULWV
FGIH Y J TUL\]N
YZ[
S
|
1
a}c
e
Q
fh
p h V
R
p
X
TUilGIHL~V
P
Q ^`_badc
egfhh V
R
X
TUijGIHLkV
TUilGIH Y L \??
YZ[
Available from http://www.kernel-machines.org/data/
YZ[
TUilGIH Y L-monqpsrt
;vuxwy{z*w
\
USPS: 4 <?> non 4
2.4
1 iterations
4 iterations
2.3
2.2
Test error %
2.1
2
1.9
1.8
1.7
1.6
1.5
50
150
250
350
450
550
#BV
Figure 1: Results for classification for different BV sizes (x-axis) and multiple sweeps
through the data.
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Figure
2: The GP estimation (continuous line) of a mixture of Gaussians (dotted line) using
u
BVs.
;
4
In the last expression, we have introduced an expectation
X 2 ZB 2 a new, effective Gaussian
A @ over
obtained by multiplying the old prior and the term
and normalizing by .
We assume that for sufficiently large w the integral over can be"performed
K M O by
"bgLaplace?s
N
C
, where
method, leaving us with an approximate predictor of the form ?
the brackets denote posterior
expectation
for
a
GP
model
with
a
kernel
that
is
a
solution
to
S U
"
"
"
the integral equation D
!
!
5
! . The likelihood of
D
D
D
;
o (
(
"
o
( O
X ( Z
the fields *
at the observation points is
. For any fixed
, we can apply the sparse ADATAP algorithm to this problem.
After convergence of this
inner loop, a new value of must be determined from (following a Laplace argument) -
M O " N
until
" global
convergence is achieved. To give a simplified toy example, we choose
!
a kernel D
which reproduces
itself after convolution. Hence, the dependence is
o ( we work
and normalised at the end. We used a periodic kernel
scaled out and
with
u
for data in
given by
(
D o
D o
"
(
!
5
o
"
5
!
q
?8
1
o
"
5
has constant Fourier coefficients up to a cutoff frequency
4 ;
g
!
o
(
C
o
,,@ ;
"
5
5
!
*
in our simulations).
For the experiment we are using artificial data from a mixture of two Gaussians (dotted
line in Fig. 2). We apply the sparse algorithm with multiple sweeps through the data. The
sparsity also avoids the numerical problems caused by a possible close to singular Gram
matrix. For the experiments, the size of the BV set was not limited a priori, and a similar
criterion as in [4] was chosen in order
to decide whether a data
u point should be included
> uEu
training data, only were retained in the BV set.
in the BV set or not. As a result, for
(continuous line in Fig. 2).
5.3 Independent Component Analysis
We have tested the sequential algorithm on an ICA problem for local feature extraction
in hand written 1 digits, i.e. extracting the different stroke styles [5] . We assumed positive
components of (enforced by Lagrange multipliers) and a positive prior
W
`
.
. 5 .
(10)
As in [5] we used 500 handwritten ?3?s which are assumed to be generated by 25 hidden
images. We compared a traditional parallel update algorithm with the sequential
belief
propagation algorithm. Both algorithms have computational complexity w
. We find
that the sequential algorithm needs only on average 7 sweeps through the sites to reach the
desired accuracy whereas the parallel one fails to reach the desired accuracy in 100 sweeps
using a somewhat larger number of flops. The adaptive TAP method using the sequential
belief propagation approach is also not more computationally expensive than the linear
response method used in [5].
6 Conclusion and Outlook
An obvious future direction for the ADATAP approach is the investigation of other minimization algorithms as an alternative to the EP approach outlined before. Also an extension
of the sparse approximation to other non-GP models will be interesting. A highly important
but difficult problem is the assessment of the accuracy of the approximation.
Acknowledgments
M. Opper is grateful to Lars Kai Hansen for suggesting the non-parametric density model.
O. Winther thanks Pedro H?jen-S?rensen for the use of his Matlab code. The work is
supported by EPSRC grant no. GR/M81601 and by the Danish Research Councils through
Center for Biological Sequence Analysis.
References
[1] T.P. Minka. Expectation propagation for approximate Bayesian inference. PhD thesis, Dep. of
Electrical Eng. and Comp. Sci.; MIT, 2000.
[2] J. S. Yedidia, W. T. Freeman and Y. Weiss, Generalized Belief Propagation, to appear in Advances in Neural Information Processing Systems (NIPS?2000), MIT Press (2001).
[3] M. Opper and O. Winther, Tractable approximations for probabilistic models: The adaptive
TAP approach, Phys. Rev. Lett. 86, 3695 (2001).
[4] L. Csat?o and M. Opper. Sparse Gaussian Processes. Neural Computation accepted (2001).
[5] P.A.d.F.R. H?jen-S?rensen, O. Winther, and L. K. Hansen, Mean Field Approaches to
Independent Component Analysis, Neural Computation accepted (2001). Available from
http://www.cbs.dtu.dk/winther/
[6] T. Plefka, Convergence condition of the TAP equations for the infinite-ranged Ising spin glass
model, J. Phys. A 15, 1971 (1982).
[7] T. Tanaka, Mean-Field Theory of Boltzmann Machine Learning, Phys. Rev. E 58, 2302(1998).
[8] G. Parisi and M. Potters, Mean-Field Equations for Spin Models with Orthogonal Interaction
Matrices, J. Phys. A (Math. Gen.) 28, 5267 (1995).
[9] L. Csat?o, E. Fokou?e, M. Opper, B. Schottky, and O. Winther. Efficient approaches to Gaussian
process classification. In Advances in Neural Information Processing Systems, volume 12,
(2000).
[10] D.M. Schmidt. Continuous probability distributions from finite data. arXiv:physics/9808005
(1998)
| 2027 |@word version:2 simulation:4 crucially:1 covariance:6 eng:1 thereby:1 outlook:1 moment:8 series:1 contains:1 current:1 com:1 must:2 written:2 numerical:1 additive:1 lkv:1 remove:1 update:6 manfred:1 provides:2 recompute:1 node:6 math:1 org:1 specialize:1 shorthand:1 consists:1 ica:1 freeman:1 actual:1 increasing:1 estimating:1 adc:1 pseudo:1 temporal:1 scaled:1 uk:2 grant:1 appear:1 positive:3 before:2 engineering:1 local:4 limit:2 fluctuation:1 studied:1 dynamically:1 suggests:1 limited:1 range:1 bi:1 acknowledgment:1 woodbury:1 definite:1 digit:3 procedure:1 universal:1 projection:1 pre:1 get:1 cannot:1 close:1 restriction:1 equivalent:1 www:2 demonstrated:1 center:2 maximizing:1 splitting:1 insight:2 his:1 laplace:1 construction:1 exact:3 gps:2 element:4 expensive:1 updating:1 ising:1 observed:3 epsrc:1 ep:2 role:1 electrical:1 connected:2 complexity:1 neglected:1 grateful:1 predictive:1 basis:1 usps:2 po:2 easily:1 derivation:1 recomputes:1 effective:1 ole:1 artificial:1 ponent:1 choosing:1 larger:1 solve:2 kai:1 statistic:2 gp:7 itself:1 online:2 sequence:2 parisi:1 interaction:2 loop:1 gen:1 mixing:3 achieve:1 opperm:1 adatap:15 convergence:3 empty:1 extending:2 derive:4 depending:1 ac:2 coupling:1 measured:1 school:1 dep:1 eq:5 auxiliary:3 direction:1 correct:1 lars:1 require:1 generalization:2 tul:1 investigation:1 biological:2 extension:2 sufficiently:2 considered:1 bvs:4 entropic:1 estimation:4 birmingham:1 hansen:2 council:1 grouped:1 minimization:4 mit:2 gaussian:22 mation:2 always:1 aim:1 gaus:1 derived:2 ax:1 likelihood:18 contrast:1 glass:1 inference:2 lehel:1 entire:1 hidden:1 relation:2 overall:1 classification:5 priori:1 constrained:1 integration:1 special:1 marginal:6 field:6 having:1 extraction:1 r5:2 look:1 future:1 densely:2 individual:1 attempt:1 interest:2 message:2 possibility:1 highly:1 introduces:1 mixture:2 bracket:3 fu:3 integral:2 orthogonal:1 tree:1 old:1 re:1 desired:3 classify:1 introducing:1 subset:1 plefka:1 predictor:1 usefulness:1 gr:1 dependency:2 corrupted:1 periodic:1 thanks:1 density:5 winther:7 probabilistic:6 physic:2 again:2 central:1 thesis:1 containing:1 choose:3 m81601:1 dr:1 style:1 toy:1 suggesting:1 wk:2 coefficient:1 dictive:1 caused:2 ad:1 depends:1 performed:3 observing:1 bayes:1 hf:1 recover:1 parallel:2 spin:2 accuracy:3 efficiently:4 yield:1 weak:1 bayesian:3 handwritten:2 multiplying:1 comp:1 stroke:1 reach:2 phys:4 danish:1 definition:1 energy:11 frequency:1 minka:5 obvious:1 con:1 gain:1 dataset:1 ea:1 response:1 improved:1 wei:1 just:1 lastly:1 correlation:1 until:1 hand:1 nonlinear:1 assessment:1 propagation:10 gray:1 effect:1 ranged:1 multiplier:5 true:1 hence:2 alternating:1 iteratively:1 nonzero:1 white:1 gw:1 self:3 criterion:1 generalized:1 prominent:1 demonstrate:1 image:2 variational:1 instantaneous:1 recently:2 common:1 qp:1 b4:1 volume:1 jl:1 belong:2 interpretation:1 approximates:1 gibbs:6 approx:1 consistency:2 outlined:1 sherman:1 stable:1 etc:1 add:1 posterior:5 cbs:2 termed:1 minimum:3 additional:1 somewhat:1 signal:1 ii:1 multiple:4 reduces:1 ing:2 technical:1 generalises:1 calculation:1 minimising:2 expectation:8 arxiv:1 iteration:3 kernel:10 achieved:2 whereas:2 singular:1 source:4 leaving:1 nary:1 undirected:1 member:1 incorporates:1 extracting:1 easy:1 independence:1 identified:1 opposite:1 inner:1 idea:2 whether:1 expression:2 suffer:1 passing:2 matlab:1 useful:1 clear:1 maybe:1 amount:1 gih:3 http:2 rensen:2 dotted:2 csat:3 discrete:1 iz:1 express:1 group:3 four:2 deleted:1 discreteness:1 cutoff:1 schottky:1 graph:4 enforced:1 run:1 extends:1 decide:1 fold:1 quadratic:1 adapted:1 bv:9 constraint:4 bp:3 encodes:2 fourier:1 argument:2 according:1 smaller:1 em:1 rev:2 lyngby:1 equation:8 computationally:1 discus:1 imation:1 tractable:1 end:1 available:2 gaussians:2 rewritten:1 yedidia:1 apply:2 appropriate:2 batch:2 alternative:1 schmidt:1 denotes:1 remaining:2 straints:1 graphical:1 yx:1 k1:1 dgf:1 sweep:7 quantity:1 parametric:2 dependence:1 diagonal:2 traditional:1 unclear:1 sci:1 reason:1 denmark:2 potter:1 code:1 retained:1 minimizing:1 difficult:1 boltzmann:2 unknown:1 perform:1 observation:2 convolution:1 finite:2 t:1 displayed:1 defining:1 extended:1 flop:1 arbitrary:1 introduced:2 tap:8 tanaka:1 nip:1 able:1 bar:1 proceeds:1 usually:4 below:1 pattern:2 sparsity:3 belief:8 power:1 sian:1 advanced:1 scheme:1 aston:2 brief:1 dtu:2 lk:1 axis:1 gf:1 prior:7 multiplication:1 relative:2 embedded:1 loss:1 probit:1 interesting:3 consistent:1 factorising:2 changed:1 supported:1 last:1 free:11 truncation:2 allow:1 understand:1 normalised:1 sparse:10 overcome:1 opper:5 lett:1 gram:1 avoids:1 rich:1 dataset1:1 adaptive:4 mlii:1 simplified:1 approximate:6 cavity:5 approxi:2 global:1 reproduces:1 kmlon:1 conclude:1 assumed:4 consuming:1 continuous:4 search:1 iterative:1 sk:1 bethe:1 expansion:1 significance:2 main:1 noise:5 augmented:1 site:6 fig:3 fashion:1 fails:1 ueu:1 candidate:1 formula:1 specific:1 jen:2 badc:1 dk:2 concern:1 normalizing:1 sequential:14 phd:1 mf:1 entropy:2 generalizing:1 led:1 simply:1 saddle:1 absorbed:1 lagrange:6 bo:1 pedro:1 corresponds:1 minimizer:1 conditional:1 goal:1 presentation:1 marked:1 rbf:1 replace:1 change:1 included:2 determined:2 diminished:1 infinite:1 total:1 called:3 egf:1 accepted:2 latter:1 tested:1 |
1,127 | 2,028 | Learning Discriminative Feature Transforms
to Low Dimensions in Low Dimensions
Kari Torkkola
Motorola Labs, 7700 South River Parkway, MD ML28, Tempe AZ 85284, USA
Kari.Torkkola@motorola.com http://members.home.net/torkkola
Abstract
The marriage of Renyi entropy with Parzen density estimation has been
shown to be a viable tool in learning discriminative feature transforms.
However, it suffers from computational complexity proportional to the
square of the number of samples in the training data. This sets a practical
limit to using large databases. We suggest immediate divorce of the two
methods and remarriage of Renyi entropy with a semi-parametric density
estimation method, such as a Gaussian Mixture Models (GMM). This allows all of the computation to take place in the low dimensional target
space, and it reduces computational complexity proportional to square
of the number of components in the mixtures. Furthermore, a convenient extension to Hidden Markov Models as commonly used in speech
recognition becomes possible.
1 Introduction
Feature selection or feature transforms are important aspects of any pattern recognition system. Optimal feature selection coupled with a particular classifier can be done by actually
training and evaluating the classifier using all combinations of available features. Obviously this wrapper strategy does not allow learning feature transforms, because all possible
transforms cannot be enumerated. Both feature selection and feature transforms can be
learned by evaluating some criterion that reflects the ?importance? of a feature or a number
of features jointly. This is called the filter configuration in feature selection. An optimal criterion for this purpose would naturally reflect the Bayes error rate. Approximations can be
used, for example, based on Bhattacharyya bound or on an interclass divergence criterion.
These are usually accompanied by a parametric estimation, such as Gaussian, of the densities at hand [6, 12]. The classical Linear Discriminant Analysis (LDA) assumes all classes
to be Gaussian with a shared single covariance matrix [5]. Heteroscedastic Discriminant
Analysis (HDA) extends this by allowing each of the classes have their own covariances
[9].
Maximizing a particular criterion, the joint mutual information (MI) between the features
and the class labels [1, 17, 16, 13], can be shown to minimize the lower bound of the
classification error [3, 10, 15]. However, MI according to the popular definition of Shannon
can be computationally expensive. Evaluation of the joint MI of a number of variables is
plausible through histograms, but only for a few variables [17]. As a remedy, Principe et
al showed in [4, 11, 10] that using Renyi?s entropy instead of Shannon?s, combined with
Parzen density estimation, leads to expressions of mutual information with computational
, where is the number of samples in the training set. This method
complexity of
can be formulated to express the mutual information between continuous variables and
discrete class labels in order to learn dimension-reducing feature transforms, both linear
[15] and non-linear [14], for pattern recognition. One must note that regarding finding the
extrema, both definitions of entropy are equivalent (see [7] pages 118,406, and [8] page
325).
This formulation of MI evaluates the effect of each sample to every other sample in the
transformed space through the Parzen density estimation kernel. This effect can also called
as the ?information force?. Thus large/huge databases are hard to use due to the
complexity.
To remedy this problem, and also to alleviate the difficulties in Parzen density estimation
in high-dimensional spaces (
), we present a formulation combining the mutual information criterion based on Renyi entropy with a semi-parametric density estimation method
using Gaussian Mixture Models (GMM). In essence, Parzen density estimation is replaced
by GMMs. In order to evaluate the MI, evaluating mutual interactions between mixture
components of the GMMs suffices, instead of having to evaluate interactions between all
pairs of samples. An approach that maps an output space GMM back to input space and
again to output space through the adaptive feature transform is taken. This allows all of the
computation to take place in the target low dimensional space. Computational complexity
is reduced proportional to the square of the number of components in the mixtures.
This paper is structured as follows. An introduction is given to the maximum mutual information (MMI) formulation for discriminative feature transforms using Renyi entropy
and Parzen density estimation. We discuss different strategies to reduce its computational
complexity, and we present a formulation based on GMMs. Empirical results are presented
using a few well known databases, and we conclude by discussing a connection to Hidden
Markov Models.
2 MMI for Discriminative Feature Transforms
"
)(
+ , . !- 0/
$1 # &% ' + 3 254 ,
* $6
9
7
,
as samples of a continuous-valued random variable
Given a set of training data
,
, and class labels as samples of a discrete-valued random variable ,
, the objective is to find a transformation (or its parameters ) to
such that
that maximizes
, the mutual information
(MI) between transformed data and class labels . The procedure is depicted in Fig. 1.
To this end, we need to express as a function of the data set,
, in a differentiable
form. Once that is done, we can perform gradient ascent on as follows
7 8 9
7 +
7
F7
IH F 7 F +
*;:=<?>@2A*;:CBED F * 25*:GBED F + F *
J >
*
(1)
To derive an expression for MI using a non-parametric density estimation method we apply
Renyi?s quadratic entropy instead of Shannon?s entropy as described in [10, 15] because
of its computational advantages. Estimating the density
of as a sum of spherical
Gaussians each centered at a sample , the expression of Renyi?s quadratic entropy of
is
K +
+
N)O P'QSRUT6K + +
L;M 9 2
2
2
IH
N)O P'Q VR TEWX Y
J
HI I H
N)O P'Q Y Z
J> J>
9
9
IH
+ N + Y ] 7 + N + Z ] 7 _^` +
[
> Z J >\[
+ Y N + Z '] 7
[
(2)
Above, use is made of the fact that the convolution of two Gaussians is a Gaussian. Thus
Renyi?s quadratic entropy can be computed as a sum of local interactions as defined by the
kernel, over all pairs of samples.
In order to use this convenient property, a measure of mutual information making use of
quadratic functions of the densities would be desirable. Between a discrete variable and
a continuous variable such a measure has been derived in [10, 15] as follows:
9
I
I
I
7 8 9 2 " RVT K + + B " RVT6K K + + N " RVT K + K K + +
K Y
(3)
We use
for the number of samples in class , for th sample regardless of its class,
and
for the same sample, but emphasizing that it belongs to class , with index within
the class. Expressing densities as their Parzen estimates with kernel width results in
Z
K
]
HI
I
I
7 + $
2 Y + Y N + ] 7
J> J> J>[
H
I I H I H + Y +
Y
B
N ] 7
J >
[
J
J
>
>
I H I I H + Z + Y
N ] 7
N Z Y
(4)
J> J> J>[
Mutual information 7 +
can now be interpreted as an information potential induced
samples of data in different classes. It is now straightforward to derive partial
F 7 F + by
which can
accordingly be interpreted as an information force that other samples
exert to sample + . The three components of the sum give rise to following three compo
>
nents of the information force: Samples within the same class attract each other, All
samples regardless of class attract each other, and F Samples
F of different classes repel each
other. This force, coupled with the latter factor + * inside the sum in (1), tends to
change the transform in such a way that the samples in transformed space
move
into the
direction of the information force, and thus increase the MI criterion 7 +
. See [15]
for details.
Class labels: c
g(w,x)
Low dimensional
features: y
High-dimensional data: x
Gradient
Mutual Information
I(c,y)
(=Information potential)
?I
?w
Figure 1: Learning feature transforms by maximizing the mutual information between class labels
and transformed features.
Each term in (4) consists of a double sum of Gaussians evaluated using the pairwise distance between the samples. The first component consists of a sum of these interactions
within each class, the second of all interactions regardless of class, and the third of a sum
of the interactions of each class against all other samples. The bulk of computation consists
Gaussians, and forming the sums of those. Information force, the
of evaluating these
gradient of , makes use of the same Gaussians, in addition to pairwise differences of the
samples [15]. For large , complexity of
is a problem. Thus, the rest of the paper
explores possibilities of reducing the computation to make the method applicable to large
databases.
7
'
3 How to Reduce Computation?
In essence, we are trying to learn a transform that minimizes the class density overlap in
the output space while trying to drive each class into a singularity. Since kernel density estimate results in a sum of kernels over samples, a divergence measure between the densities
necessarily requires
operations. The only alternatives to reduce this complexity are
either to reduce , or to form simpler density estimates.
Two straightforward ways to achieve the former are clustering or random sampling. In
this case clustering needs to be performed in the high-dimensional input space, which may
be difficult and computationally expensive itself. A transform is then learned to find a
representation that discriminates the cluster centers or the random samples belonging to
different classes. Details of the densities may be lost, more so with random sampling, but
at least this might bring the problem down to a computable level.
The latter alternative can be accomplished by a GMM, for example. A GMM is learned
in the low-dimensional output space for each class, and now, instead of comparing samples against each other, comparing samples against the components of the GMMs suffices.
However, as the parameters of the transform are being learned iteratively, the
will
change at each iteration, and the GMMs need to be estimated again. There is no guarantee
that the change to the transform and to the
is so small that simple re-estimation based
on previous GMMs would suffice. However, this depends on the optimization method used.
+
+ _
A further step in reducing computation is to compare GMMs of different classes in the
output space against each other, instead of comparing the actual samples. In addition to the
inconvenience of re-estimation, we lack now the notion of ?mapping?. Nothing is being
transformed by from the input space to the output space, such that we could change
the transform in order to increase the MI criterion. Although it would be possible now
to evaluate the effect of each sample
to each mixture
component,
and the effect of each
, due to the double summing, we
component to the MI, that is,
will pursue the mapping strategy outlined in the following section.
4
2
Y
T T
4 Two GMM Mapping Strategies
IO-mapping. If the GMM is available in the high-dimensional input space, those models
can be directly mapped into the output space by the transform. Let us call this case the
IO-mapping.
Writing the density of class as a GMM with
mixture components and
as their mixture weights we get
Z
K
I
K
2 Z Z N Z Z
J> [
(5)
We consider now only linear transforms. The transformed density in the low-dimensional
output space is then simply
I
K + 2 Z Z + N Z Z
J> [
(6)
Now, the mutual information in the output space between class labels and the densities as
transformed GMMs can be expressed as a function of , and it will be possible to evaluate
to insert into (1). A great advantage of this strategy is that once the input space
GMMs have been created (by the EM-algorithm, for example), the actual training data
needs not be touched at all during optimization! This is thus a very viable approach if the
GMMs are already available in the high-dimensional input space (see Section 7), or if it is
not too expensive or impossible to estimate them using the EM-algorithm. However, this
might not be the case.
F7 F
OIO-mapping. An alternative is to construct a GMM model for the training data in the
low-dimensional output space. Since getting there requires a transform, the GMM is constructed after having transformed the data using, for example, a random or an informed
guess as the transform. Density estimated from the samples in the output space for class
is
K
I
K + 2 Z Z +
N Z Z
J> [
(7)
Once the output space GMM is constructed, the same samples are used to construct a GMM
in the input space using the same exact assignments of samples to mixture components
as the output space GMMs have. Running the EM-algorithm in the input space is now
unnecessary since we know which samples belong to which mixture components. Similar
strategy has been used to learn GMMs in high dimensional spaces [2]. Let us now use
the notation of Eq.(5) to denote this density also in the input space. As a result, we have
GMMs in both spaces and a transform mapping between
the two.
The transform
can be
learned as in the IO-mapping, by using the equalities
and
.
This case will be called OIO-mapping. The biggest advantage is now avoiding to operate
in the high-dimensional input space at all, not even the one time in the beginning of the
procedure.
Z 2
Z 2
Z
Z
5 Learning the Transform through Mapped GMMs
We present now the derivation of adaptation equations for a linear transform that apply
to either mapping. The first step is to express the MI as a function of the GMM that
is constructed in the output space. This GMM is a function of the transform matrix ,
through the mapping of the input space GMM to the output space GMM. The second step
is to compute its gradient and to make use of it in the first half of Equation (1).
F7 F
5.1 Information Potential as a Function of GMMs
K +
2 K +
K + 2
GMM in the output space
for each
class
is already expressed in (7). We need the following
equalities:
, where
denotes the class prior, and
.
H
J > K +
H ,
Let us denote the three terms in (3) as
, and
N
. Then we have
H
I
I
_
`
^
H 2 " RVT K + + 2 RUT XW
+
N $
J>
! J > [
HI I
I
2 Z Z N Z B Z
J > J > J >
[
I
Y 2 Y B
Y
N
(
%
C
!
Y " " C
H
I
H 2
)2
J>
C 2
Y Y C 2 Y
(8)
Y 2
[ 2
To compact the notation, we change the indexing, and make the substitutions
,
,
,
,
where
,
and
is
the
total
number
of
mixture
components,
and
. Now we can write ,
, and
in a convenient form.
[
[
H
H
H
H
I I I
J > J > J >
2
IH
Y
IH
J> J>
(9)
5.2 Gradient of the Information Potential
As each Gaussian mixture component is now a function of the corresponding input space
component and the transform matrix , it is straightforward (albeit tedious) to write the
. Since each of the three terms in
gradient
is composed of different sums of
, we need its gradient as
F7 F
Y Y 2 F F Y Y
(10)
[
[
[
where the input space GMM parameters are Y 2 Y N and Y
2 Y B with the
equalities Y
2 Y and
Y 2
Y
.
expresses the convolution of two mixture components in the output space. As we
[ have those components in the high-dimensional input space, the gradient expresses
also
[
F
F
7
C 2 F F
how this convolution in the output space changes, as that maps the mixture components to the output space, is being changed. The mutual information measure is defined
in terms of these convolutions, and maximizing it tends to find a that (crudely stated)
minimizes these convolutions between classes and maximizes them within classes. The
desired gradient of the Gaussian with respect to the transform matrix is as follows:
F
2 N Y
> 7.N Y
Y
Y
> Y B Y
Y
(11)
[
[
F F
The total gradient 7
can now be obtained simply by replacing C in (8) and (9)
[
by the above gradient.
In evaluating 7 , the bulk of computation is inevaluating
the , the
F componentwise
F
. In addition,
convolutions. Computational complexity is now
the 7 requires
F
pairwise sums and differences of the mixture parameters in the input space, but these need
only be computed once.
6 Empirical Results
The first step in evaluating this approach is to compare its performance to the computationally more expensive MMI feature transforms that use Parzen density estimation. To
this end, we repeated the pattern recognition experiments of [15] using exactly the same
LVQ-classifier. These experiments were done using five publicly available databases that
are very different in terms of the amount of data, dimension of data, and the number of
training instances. For details of the data sets, please see [15]. OIO-mapping was used
with 3-5 diagonal Gaussians per class to learn a dimension-reducing linear transform. Gradient ascent was used for optimization1. Results are presented in Tables 1 - 5. The last
column denotes the original dimensionality of the data set.
As a figure of the overall performance, the average over all five databases and all reduced
dimensions, which ranged from one up to the original dimension minus one, was 69.6% for
PCA, 77.8% for the MMI-Parzen combination, and 77.0% for the MMI-GMM combination
(30 tests altogether). For LDA this figure cannot be calculated since some databases had
a small
and LDA can only produce
features. The results are very satisfactory
since the best we could hope for is performance equal to the MMI-Parzen combination.
Thus a very significant reduction in computation caused only a minor drop in performance
with this classifier.
"
" NA
7 Discussion
We have presented a method to learn discriminative feature transforms using Maximum
Mutual Information as the criterion. Formulating MI using Renyi entropy, and Gaussian
1
Example video clips can be viewed at http://members.home.net/torkkola/mmi.
Table 1: Accuracy on the Phoneme test data set using LVQ classifier.
Output dimenson
PCA
LDA
MMI-Parzen
MMI-GMM
1
7.6
5.1
15.5
21.4
2
70.0
66.0
68.5
70.4
3
76.8
74.7
75.2
76.8
4
81.1
80.2
80.2
80.2
6
84.2
82.8
82.6
82.6
9
87.3
86.0
85.3
87.7
20
90.0
-
Table 2: Accuracy on the Landsat test data set using LVQ classifier.
Output dimension
PCA
LDA
MMI-Parzen
MMI-GMM
1
41.2
42.5
65.1
65.0
2
81.5
75.7
82.0
80.4
3
85.8
86.2
86.4
86.1
4
87.8
87.2
86.2
88.3
9
89.4
88.8
87.6
87.4
15
90.3
90.0
89.5
89.1
36
90.4
-
Table 3: Accuracy on the Letter test data set using LVQ classifier.
Output dimension
PCA
LDA
MMI-Parzen
MMI-GMM
1
4.5
13.4
16.4
15.7
2
16.0
38.0
50.3
42.4
3
36.0
53.1
62.8
48.3
4
53.2
68.1
70.9
68.5
6
75.2
80.3
82.4
80.9
8
82.5
86.3
88.6
86.6
16
92.4
-
Table 4: Accuracy on the Pipeline data set using LVQ classifier.
Output dimension
PCA
LDA
MMI-Parzen
MMI-GMM
1
41.5
98.4
99.4
91.3
2
88.0
98.8
99.1
98.8
3
87.8
98.9
99.1
4
89.7
99.2
98.9
5
96.4
98.9
99.1
7
97.2
99.0
98.7
12
99.0
-
Table 5: Accuracy on the Pima data set using LVQ classifier.
Output dimension
PCA
LDA
MMI-Parzen
MMI-GMM
1
64.4
65.8
72.0
73.9
2
73.0
77.5
79.7
3
75.2
78.7
79.4
4
74.1
78.5
77.9
5
75.6
78.3
76.7
6
74.7
78.3
77.5
8
74.7
-
Mixture Models as a semi-parametric density estimation method, allows all of the computation to take place in the low-dimensional transform space. Compared to previous formulation using Parzen density estimation, large databases become now a possibility.
A convenient extension to Hidden Markov Models (HMM) as commonly used in speech
recognition becomes also possible. Given an HMM-based speech recognition system,
the state discrimination can be enhanced by learning a linear transform from some highdimensional collection of features to a convenient dimension. Existing HMMs can be converted to these high-dimensional features using so called single-pass retraining (compute
all probabilities using current features, but do re-estimation using a the high-dimensional
set of features). Now a state-discriminative transform to a lower dimension can be learned
using the method presented in this paper. Another round of single-pass retraining then
converts existing HMMs to new discriminative features.
A further advantage of the method in speech recognition is that the state separation in the
transformed output space is measured in terms of the separability of the data represented
as Gaussian mixtures, not in terms of the data itself (actual samples). This should be
advantageous regarding recognition accuracies since HMMs have the same exact structure.
References
[1] R. Battiti. Using mutual information for selecting features in supervised neural net
learning. Neural Networks, 5(4):537?550, July 1994.
[2] Sanjoy Dasgupta. Experiments with random projection. In Proceedings of the 16th
Conference on Uncertainty in Artificial Intelligence, pages 143?151, Stanford, CA,
June30 - July 3 2000.
[3] R.M. Fano. Transmission of Information: A Statistical theory of Communications.
Wiley, New York, 1961.
[4] J.W. Fisher III and J.C. Principe. A methodology for information theoretic feature
extraction. In Proc. of IEEE World Congress On Computational Intelligence, pages
1712?1716, Anchorage, Alaska, May 4-9 1998.
[5] K. Fukunaga. Introduction to statistical pattern recognition (2nd edition). Academic
Press, New York, 1990.
[6] Xuan Guorong, Chai Peiqi, and Wu Minhui. Bhattacharyya distance feature selection. In Proceedings of the 13th International Conference on Pattern Recognition,
volume 2, pages 195 ? 199. IEEE, 25-29 Aug. 1996.
[7] J.N. Kapur. Measures of information and their applications. Wiley, New Delhi, India,
1994.
[8] J.N. Kapur and H.K. Kesavan. Entropy optimization principles with applications.
Academic Press, San Diego, London, 1992.
[9] Nagendra Kumar and Andreas G. Andreou. Heteroscedastic discriminant analysis
and reduced rank HMMs for improved speech recognition. Speech Communication,
26(4):283?297, 1998.
[10] J.C. Principe, J.W. Fisher III, and D. Xu. Information theoretic learning. In Simon
Haykin, editor, Unsupervised Adaptive Filtering. Wiley, New York, NY, 2000.
[11] J.C. Principe, D. Xu, and J.W. Fisher III. Pose estimation in SAR using an
information-theoretic criterion. In Proc. SPIE98, 1998.
[12] George Saon and Mukund Padmanabhan. Minimum bayes error feature selection for
continuous speech recognition. In Todd K. Leen, Thomas G. Dietterich, and Volker
Tresp, editors, Advances in Neural Information Processing Systems 13, pages 800?
806. MIT Press, 2001.
[13] Janne Sinkkonen and Samuel Kaski. Clustering based on conditional distributions in
an auxiliary space. Neural Computation, 14:217?239, 2002.
[14] Kari Torkkola. Nonlinear feature transforms using maximum mutual information.
In Proceedings of the IJCNN, pages 2756?2761, Washington DC, USA, July 15-19
2001.
[15] Kari Torkkola and William Campbell. Mutual information in learning feature transformations. In Proceedings of the 17th International Conference on Machine Learning, pages 1015?1022, Stanford, CA, USA, June 29 - July 2 2000.
[16] N. Vlassis, Y. Motomura, and B. Krose. Supervised dimension reduction of intrinsically low-dimensional data. Neural Computation, 14(1), January 2002.
[17] H. Yang and J. Moody. Feature selection based on joint mutual information. In Proceedings of International ICSC Symposium on Advances in Intelligent Data Analysis,
Rochester, New York, June 22-25 1999.
| 2028 |@word advantageous:1 retraining:2 nd:1 tedious:1 covariance:2 minus:1 reduction:2 wrapper:1 configuration:1 substitution:1 selecting:1 bhattacharyya:2 existing:2 current:1 com:1 comparing:3 must:1 drop:1 discrimination:1 half:1 intelligence:2 guess:1 accordingly:1 inconvenience:1 beginning:1 haykin:1 compo:1 simpler:1 five:2 anchorage:1 constructed:3 become:1 symposium:1 viable:2 consists:3 inside:1 pairwise:3 spherical:1 motorola:2 actual:3 becomes:2 estimating:1 notation:2 suffice:1 maximizes:2 interpreted:2 minimizes:2 pursue:1 informed:1 finding:1 extremum:1 transformation:2 guarantee:1 sinkkonen:1 every:1 exactly:1 classifier:9 local:1 todd:1 tends:2 limit:1 io:3 congress:1 tempe:1 might:2 exert:1 heteroscedastic:2 hmms:4 practical:1 rvt:3 lost:1 oio:3 procedure:2 empirical:2 convenient:5 projection:1 suggest:1 get:1 divorce:1 cannot:2 selection:7 impossible:1 writing:1 equivalent:1 map:2 center:1 maximizing:3 straightforward:3 regardless:3 notion:1 sar:1 target:2 enhanced:1 diego:1 exact:2 recognition:12 expensive:4 database:8 discriminates:1 complexity:9 joint:3 represented:1 kaski:1 derivation:1 london:1 artificial:1 stanford:2 plausible:1 valued:2 jointly:1 transform:21 itself:2 obviously:1 advantage:4 differentiable:1 net:3 interaction:6 adaptation:1 combining:1 achieve:1 az:1 getting:1 chai:1 double:2 cluster:1 transmission:1 produce:1 xuan:1 derive:2 pose:1 measured:1 minor:1 aug:1 eq:1 auxiliary:1 direction:1 filter:1 centered:1 suffices:2 alleviate:1 singularity:1 enumerated:1 extension:2 insert:1 marriage:1 great:1 mapping:12 nents:1 purpose:1 estimation:17 f7:4 proc:2 applicable:1 label:7 tool:1 reflects:1 hope:1 mit:1 gaussian:9 volker:1 derived:1 june:2 rank:1 attract:2 landsat:1 hidden:3 transformed:9 overall:1 classification:1 mutual:18 equal:1 once:4 construct:2 having:2 extraction:1 sampling:2 washington:1 unsupervised:1 intelligent:1 few:2 composed:1 divergence:2 replaced:1 william:1 huge:1 possibility:2 evaluation:1 mixture:17 partial:1 alaska:1 re:3 desired:1 instance:1 column:1 assignment:1 too:1 combined:1 density:26 explores:1 river:1 international:3 parzen:16 moody:1 na:1 again:2 reflect:1 potential:4 converted:1 accompanied:1 caused:1 depends:1 performed:1 icsc:1 lab:1 bayes:2 simon:1 rochester:1 minimize:1 square:3 publicly:1 accuracy:6 phoneme:1 mmi:17 drive:1 suffers:1 definition:2 evaluates:1 against:4 naturally:1 mi:12 popular:1 intrinsically:1 dimensionality:1 actually:1 back:1 campbell:1 supervised:2 methodology:1 improved:1 formulation:5 done:3 evaluated:1 leen:1 furthermore:1 crudely:1 hand:1 replacing:1 nonlinear:1 lack:1 lda:8 usa:3 effect:4 dietterich:1 ranged:1 remedy:2 former:1 equality:3 iteratively:1 satisfactory:1 round:1 during:1 width:1 please:1 essence:2 samuel:1 criterion:9 trying:2 theoretic:3 bring:1 volume:1 belong:1 expressing:1 significant:1 outlined:1 fano:1 had:1 own:1 showed:1 belongs:1 discussing:1 battiti:1 accomplished:1 minimum:1 george:1 july:4 semi:3 desirable:1 reduces:1 academic:2 motomura:1 hda:1 histogram:1 kernel:5 iteration:1 addition:3 rest:1 operate:1 ascent:2 south:1 induced:1 member:2 gmms:15 call:1 yang:1 iii:3 reduce:4 regarding:2 andreas:1 computable:1 expression:3 pca:6 speech:7 york:4 transforms:14 amount:1 clip:1 reduced:3 http:2 estimated:2 per:1 bulk:2 discrete:3 write:2 dasgupta:1 express:5 gmm:24 sum:11 convert:1 letter:1 uncertainty:1 place:3 extends:1 wu:1 separation:1 home:2 bound:2 hi:3 quadratic:4 ijcnn:1 aspect:1 fukunaga:1 formulating:1 kumar:1 structured:1 according:1 combination:4 saon:1 cbed:1 belonging:1 nagendra:1 em:3 separability:1 making:1 indexing:1 taken:1 pipeline:1 computationally:3 equation:2 discus:1 know:1 end:2 available:4 gaussians:6 operation:1 apply:2 alternative:3 altogether:1 original:2 thomas:1 assumes:1 clustering:3 running:1 denotes:2 xw:1 classical:1 objective:1 move:1 already:2 parametric:5 strategy:6 md:1 diagonal:1 gradient:12 distance:2 mapped:2 hmm:2 discriminant:3 index:1 difficult:1 pima:1 stated:1 rise:1 perform:1 allowing:1 convolution:6 markov:3 padmanabhan:1 kapur:2 january:1 immediate:1 vlassis:1 communication:2 dc:1 interclass:1 pair:2 repel:1 connection:1 componentwise:1 andreou:1 learned:6 delhi:1 rut:1 usually:1 pattern:5 video:1 overlap:1 difficulty:1 force:6 created:1 coupled:2 tresp:1 prior:1 proportional:3 filtering:1 principle:1 editor:2 changed:1 last:1 allow:1 india:1 dimension:14 calculated:1 evaluating:7 world:1 kari:4 commonly:2 adaptive:2 made:1 collection:1 san:1 compact:1 parkway:1 summing:1 conclude:1 unnecessary:1 discriminative:7 continuous:4 optimization1:1 table:6 learn:5 ca:2 necessarily:1 edition:1 nothing:1 repeated:1 xu:2 fig:1 biggest:1 ny:1 vr:1 wiley:3 third:1 renyi:9 touched:1 down:1 emphasizing:1 torkkola:6 mukund:1 ih:5 albeit:1 importance:1 entropy:12 depicted:1 simply:2 forming:1 expressed:2 conditional:1 viewed:1 formulated:1 lvq:6 shared:1 fisher:3 hard:1 change:6 reducing:4 called:4 total:2 pas:2 sanjoy:1 shannon:3 janne:1 principe:4 highdimensional:1 krose:1 latter:2 evaluate:4 avoiding:1 |
1,128 | 2,029 | Hyperbolic Self-Organizing Maps for Semantic
Navigation
J?org Ontrup
Neuroinformatics Group
Faculty of Technology
Bielefeld University
D-33501 Bielefeld, Germany
jontrup@techfak.uni-bielefeld.de
Helge Ritter
Neuroinformatics Group
Faculty of Technology
Bielefeld University
D-33501 Bielefeld, Germany
helge@techfak.uni-bielefeld.de
Abstract
We introduce a new type of Self-Organizing Map (SOM) to navigate
in the Semantic Space of large text collections. We propose a ?hyperbolic SOM? (HSOM) based on a regular tesselation of the hyperbolic
plane, which is a non-euclidean space characterized by constant negative
gaussian curvature. The exponentially increasing size of a neighborhood
around a point in hyperbolic space provides more freedom to map the
complex information space arising from language into spatial relations.
We describe experiments, showing that the HSOM can successfully be
applied to text categorization tasks and yields results comparable to other
state-of-the-art methods.
1 Introduction
For many tasks of exploraty data analysis the Self-Organizing Maps (SOM), as introduced
by Kohonen more than a decade ago, have become a widely used tool [1, 2]. So far, the
overwhelming majority of SOM approaches have taken it for granted to use a flat space
as their data model and, motivated by its convenience for visualization, have favored the
(suitably discretized) euclidean plane as their chief ?canvas? for the generated mappings.
However, even if our thinking is deeply entrenched with euclidean space, an obvious limiting factor is the rather restricted neighborhood that ?fits? around a point on a euclidean 2D
surface. Hyperbolic spaces in contrast offer an interesting loophole. They are characterized
by uniform negative curvature, resulting in a geometry such that the size of a neighborhood
around a point increases exponentially with its radius . This exponential scaling behavior
allows to create novel displays of large hierarchical structures that are particular accessible
to visual inspection [3, 4].
Consequently, we suggest to use hyperbolic spaces also in conjunction with the SOM. The
lattice structure of the resulting hyperbolic SOMs (HSOMs) is based on a tesselation of
the hyperbolic space (in 2D or 3D) and the lattice neighborhood reflects the hyperbolic
distance metric that is responsible for the non-intuitive properties of hyperbolic spaces.
After a brief introduction to the construction of hyperbolic spaces we describe several computer experiments that indicate that the HSOM offers new interesting perspectives in the
field of text-mining.
2 Hyperbolic Spaces
Hyperbolic and spherical spaces are the only non-euclidean geometries that are homogeneous and have isotropic distance metrics [5, 6]. The geometry of H2 is a standard topic in
Riemannian geometry (see, e.g. [7]), and the relationships for the area and the circumference of a circle of radius are given by
(1)
These formulae exhibit the highly remarkable property that both quantities grow exponentially with the radius . It is this property that was observed in [3, 4] to make hyperbolic
spaces extremely useful for accommodating hierarchical structures.
!
we must find suitable
To use this potential for the SOM, we must solve two problems:
discretization lattices on H2 to which we can ?attach? the SOM prototype vectors.
after having constructed the SOM, we must somehow project the (hyperbolic!) lattice into
?flat space? in order to be able to inspect the generated maps.
2.1 Projections of Hyperbolic Spaces
"
To construct an isometric (i.e., distance preserving) embedding of the hyperbolic plane into
a ?flat? space, we may use a Minkowski space [8]. In such a space, the squared distance
between two points
and
is given by
$# %& '& #&( % ( ' (
" )$#+*,# ( .- % * % ( /* ' * ' (
(2)
i.e., it ceases to be positive definite. Still, this is a space with zero curvature and its somewhat peculiar distance measure allows to construct an isometric embedding of the hyperbolic plane H2, given by
#0
1 2 435 687 9%
:$2
:87 9' 365 ; 2
(3)
$
2
7
where are polar coordinates on the H2. Under this embedding,
plane
)= - # - the% hyperbolic
about the ' -axis.
appears as the surface < swept out by rotating the curve '
From this embedding, we can construct two further ones, the so-called Klein model and the
M
Poincar?e model [5, 9] (we will use the latter to
u
visualize HSOMs below). Both achieve a proA
jection of the infinite H2 into the unit disk, how1
B
ever, at the price of distorting distances. The
N
Klein model is obtained by projecting the points
C
onto the plane
along rays passing
of
through
the
origin
(see
Fig.
1). Obviously,
D
O
1
this projects all points of
into the ?flat? unit
disk
of
. (e.g.,
).
S
The Poincar?e Model results if we add two further steps: first a perpendicular projection of
Figure 1: Construction steps underlying
the Klein Model onto the (?northern?) surface
Klein and Poincar?e-models of the space H2
of the unit sphere centered at the origin (e.g.,
), and then a stereographic projection of the ?northern? hemisphere onto the unit
circle about the origin in the ground plane
(point ). It turns out that the resulting projection of H2 has a number of pleasant properties, among them the preservation of
<
GHDF
' >=
?
# - % A@ = B C <
' JI
K
EDF G
angles and the mapping of shortest paths onto circular arcs belonging to circles that intersect the unit disk at right angles. Distances in the original H2 are strongly distorted in its
Poincar?e (and also in the Klein) image (cf. Eq. (5)), however, in a rather useful way: the
mapping exhibits a strong ?fish-eye?-effect. The neighborhood of the H2 origin is mapped
almost faithfully (up to a linear shrinkage factor of 2), while more distant regions become
increasingly ?squeezed?. Since asymptotically the radial distances and the circumference
grow both according to the same exponential law, the squeezing is ?conformal?, i.e., (sufficiently small) shapes painted onto H2 are not deformed, only their size shrinks with increasing distance from the origin. By translating the original H2, the fish-eye-fovea can be
moved to any other part of H2, allowing to selectively zoom-in on interesting portions of a
map painted on H2 while still keeping a coarser view of its surrounding context.
2.2 Tesselations of the Hyperbolic Plane
To complete the set-up for a hyperbolic SOM we still need an equivalent of a regular grid in
the hyperbolic plane. For the hyperbolic plane there exist an infinite number of tesselations
with congruent polygons such that each grid point is surrounded by the same number of
neighbors [9, 10]. Fig. 2 shows two example tesselations (for the minimal value of
and for
), using the Poincar?e model for their visualization. While these tesselations
appear non-uniform, this is only due to the fish-eye effect of the Poincar?e projection. In the
original H2, each tesselation triangle has the same size.
)= I
One way to generate these tesselations algorithmically is by repeated application of a suitable set of generators of their symmetry group to a (suitably sized, cf. below) ?starting
triangle?, for more details cf. [11].
Figure 2: Regular triangle tesselations of the hyperbolic plane, projected into the unit disk using
the Poincar?e mapping. The left tesselation shows the case where the minimal number (
)
of equilateral triangles meet at each vertex, the right figure was constructed with
. In the
Poincar?e projection, only sides passing through the origin appear straight, all other sides appear as
circular arcs, although in the original space all triangles are congruent.
3 Hyperbolic SOM Algorithm
We have now all ingredients required for a ?hyperbolic SOM?. We organize the nodes of
a lattice as described above in ?rings? around an origin node. The numbers of nodes of
such a lattice grows very rapidly (asymptotically exponentially) with the chosen lattice
radius (its number of rings). For instance, a lattice with
contains 1625
nodes. Each lattice node carries a prototype vector
from some -dimensional
feature space (if we wish to make any non-standard assumptions about the metric structure
of this space, we would build this into the distance metric that is used for determining the
best-match node). The SOM is then formed in the usual way, e.g., in on-line mode by
C
C
B C
K
in a radial
#+ *
(4)
:
*
with
. However, since
work on a hyperbolic lattice,
"
weandnow
we
have to determine both the neighborhood
the (squared) node distance "
repeatedly determining the winner node and adjusting all nodes
lattice neighborhood
around according to the familiar rule
according to the natural metric that is inherited by the hyperbolic lattice.
The simplest way to do this is to keep with each node a complex number to identify its
position in the Poincar?e model. The node distance is then given (using the Poincar?e model,
see e.g. [7]) as
" arctanh = * *
(5)
The neighborhood
can be defined as the subset of nodes within a certain graph
distance (which is chosen as a small multiple of the neighborhood radius ) around .
4 Experiments
Some introductory experiments where several examples illustrate the favorable properties
of the HSOM as compared to the ?standard? euclidean SOM can be found in [11, 12]. A
major example of the use of the SOM for text mining is the WEBSOM project [2].
4.1 Text Categorization
In order to apply the HSOM to natural text categorization, i.e. the assignment of natural language documents to a number of predefined categories, we follow the widely used
vector-space-model of Information Retrieval (IR). For each document we construct a fea, where the components are determined by the frequency of which term
ture vector
occurs in that document. Following standard practice [13] we choose a term frequency
inverse document frequency weighting scheme:
"
"
! #"%$'&
(6)
"
where the term frequency ! denotes the number
of times term occurs in ") ( , the
number of documents in the training set and "
the document frequency of , i.e. the
number of documents occurs in.
The HSOM can be utilized for text categorization in the following manner. In a first step,
the training set is used to adapt the weight vectors
according to (4). During the second
step, the training set is mapped onto the HSOM lattice. To this end, for each training
example its best match node is determined such that
"(
*
" ( +*
*
(7)
" ( -,
where ")( denotes the feature vector of document "( , as described above. After all
examples have been presented to the net, each node is labelled with the union . of all
categories that belonged to the documents that were mapped to this node. A new, unknown
text is then classified into the union . of categories which are associated with its winner
node selected in the HSOM.
Text Collection. We used the Reuters-215781 data set since it provides a well known
baseline which is also used by other authors to evaluate their approaches, c.f. [14, 15]. We
/
1
As compiled by David Lewis from the AT&T Research Lab in 1987. The data can be found at
http://www.research.att.com/ lewis/
have used the ?ModApte? split, leading to 9603 training and 3299 test documents. After
preprocessing, our training set contained 5561 distinct terms.
C
C
Performance Evaluation. The classification effectiveness is commonly measured
in terms
of precision
and recall
[16], which can be estimated as
are the numbers of documents correctly classified, and
where and
are the
correctly not classified to category , respectively. Analogous, and
corresponding numbers of falsely classified documents.
For each node and each category a confidence value
is determined. It describes the
number of training documents belonging to class which were mapped to node . When
retrieving documents from a given category , we compare
for each node its associated
against a threshold . Documents from nodes with
become then included
into the retrieval set. For nodes which contain a set of documents
, the order of the
, where
.
retrieval set is ranked by
" ( K
$ " (
K
In this way the number of retrieved documents can be controlled and we obtain the
precision-recall-diagrams as shown in Fig. 3.
In
order to compare the
HSOM?s performance for text categorization, we also evaluated a
-nearest neighbor ( -NN) classifier with our training set. Apart from boosting methods
[16] only support vector machines [14] have shown better performances. The confidence
level of a -NN classifier to assign document to class is
" (
)( $ " ( " (
(8)
! #"%$'&
(
(
documents " for which
$ " ( " is maximum. The assign-
-NN
" (
I"
"(
where
)( is the set of
(
ment factor
is 1,+if*
belongs to category and 0 otherwise. According to [14, 17] we
&
nearest neighbors.
have chosen the
C )
Text Categorization Results. The results of three experiments are shown
-, in Table 1. We
have compared a HSOM with
rings and a tesselation with
neighbors (summing up to 1306 nodes) to a spherical standard euclidean SOM as described in [11] with
approx. 1300 nodes, and
the -NN classifier. Our results indicate that the HSOM does not
perform better than a -NN classifier, but to a certain extent also does not play significantly
worse either. It is noticable that for less dominant categories the HSOM yields superior
results to those of the standard SOM. This is due to the fact, that the nodes in H2 cover
a much broader space and therefore offer more freedom to map smaller portions of the
original dataspace with less distortions as compared to euclidean space.
As the -NN results suggest, other state-of-the-art techniques like support vector machines
will probably lead to better numerical categorization results than the HSOM. However,
since the main purpose of the HSOM is the visualization of relationships between texts
and text categories, we believe that the observed categorization performance of the HSOM
compares sufficiently well with the more specialized (non-visualization) techniques to warrant its efficient use for creating insightful maps of large bodies of document data.
Table 1: Precision-recall breakeven points for the ten most prominent categories.
SOM
HSOM
.
-NN
earn
90.0
90.2
93.8
acq
81.2
81.6
83.7
mny-fx
61.7
68.7
69.3
crude
70.3
78.8
84.7
grain
69.4
76.2
81.9
trade
48.8
56.8
61.9
interest
57.1
66.4
71.0
wheat
61.9
69.3
69.0
ship
54.8
61.8
77.5
corn
50.3
53.6
67.9
1
1
0.9
0.9
0.8
0.8
0.7
0.7
earn
acq
money?fx
0.6
0.6
0.4
0
0.2
0.4
0.6
0.8
0.4
1
0
0.2
.
(a) -NN
0.4
0.6
0.8
1
(b) HSOM
2: 0.69
Figure 3: Precision-recall curves for the three most frequent categories earn, acq and money-fx.
4.2 Text Mining & Semantic Navigation
A major advantage of the HSOM is its remarkable capability to map high-dimensional
similarity relationships to a low-dimensional space which can be more easily handled and
interpreted by the human observer. This feature and the particular ?fish-eye? capability motivates our approach to visualize whole text collections with the HSOM. It can be regarded
as an interface capturing the semantic structure of a text database and provides a way to
guide the users attention. In preliminary experiments we have labelled the nodes with
glyphs corresponding to the categories of the documents mapped to that node. In Fig. 4
two HSOM views of the Reuters data set are shown. Note, that the major amount of data
gets mapped to the outermost region, where the nodes of the HSOM make use of the large
space offered by the hyperbolic geometry. During the unsupervised training process, the
document?s categories were not presented to the HSOM. Nevertheless, several document
clusters can be clearly identified. The two most prominent are the earn and acquisition
region of the map, reflecting the large proportion of these categories in the Reuters-21578
collection. Note, that categories which are semantically similar are located beside each
other, as can be seen in the corn, wheat, grain the interest, money-fx or the crude, ship area
of the map. Additional to the category (glyph type) and the number of training documents
per node (glyph size), the number of test documents mapped to each node is shown as the
height of the symbol above the ground plane. In this way the HSOM can be used as a
novelty detector in chronological document streams. For the Reuters-21578 dataset, a particular node strikes out. It corresponds to the small glyph tagged with the ?ship? label in
Fig. 4. Only a few documents from the training collection are mapped to that node as shown
by it?s relatively small glyph size. The large -value on the other hand indicates that it contains a large number of test documents, and is therefore probably semantically connected
to a significant, novel event only contained in the test collection. The right image of Fig. 4
shows the same map, but the focal view now moved into the direction of the conspicious
?ship? node, resulting in a magnification of the corresponding area. A closer inspection reveals, that the vast majority (35 of 40) of the test documents describe an incident where an
Iranian oil rig was attacked in the gulf. Although no document of the training set describes
this incident (because the text collection is ordered by time and the attack took place ?after?
the split into train and test set), the HSOM generalizes well and maps the semantic content
of these documents to the proper area of the map, located between the regions for crude
and ship.
The next example illustrates that the HSOM can provide more information about an unknown text than just it?s category. For this experiment we have taken movie reviews from
the rec.art.movies.reviews newsgroup. Since all the reviews describe a certain movie, we
retrieved their associated genres from the Internet Movie Database (http://www.imdb.com)
to build a set of category labels for each document. The training set contained 8923 ran-
money?fx
ship
trade
corn wheat
grain
interest
acq
crude
earn
Figure 4: The left figure shows a central view of the Reuters data. We used a HSOM with
rings and a tesselation with
neighbors. Ten different glyphs were used to visualize the ten most
frequent categories. They were manually tagged to indicate the correspondence between category
and symbol type. The glyph sizes and the -values (height above ground plane) reflect the number of
training and test documents mapped to the corresponding node, respectively.
domly selected reviews (without their genre information) from films released before 2000.
We then presented the system with five reviews from the film ?Atlantis?, a Disney cartoon
released in 2001. The HSOM correctly classified all of the five texts as reviews for an animation movie. In Fig. 5 the projection of the five new documents onto the map with the
previously acquired text collection is shown. It can be seen that there exist several clusters
related to the animation genre. By moving the fovea of the HSOM we can now ?zoom?
into that region which contains the five new texts. In the right of Fig. 5 it can be seen
that all of the ?Atlantis? reviews where mapped to a node in immediate vicinity of documents describing other Disney animation movies. This example motivates the approach of
?semantic navigation? to rapidly visualize the linkage between unknown documents and
previously acquired semantic concepts.
Mulan
Beauty and
the beast Anastasia
Pocahontas
Hercules
Aladin
Atlantis
Tarzan
Chicken Run
Dinosaur
South Park
Tarzan
Mulan
The Iron Giant
Antz
A Bug?s Life
The Prince
of Egypt
Figure 5: A HSOM with
and a tesselation with
neighbors was used to map movie
rewies from newsgroup channels. In both figures, glyph size and -value indicate the number of
texts related to the animation genre mapped to the corresponding node. Nodes exceeding a certain
threshold were labelled with the title corresponding to the most frequently occuring movie mapped
to that node. The underlined label in the right figure indicates the position of the node to which five
new documents were mapped to.
5 Conclusion
Efficient navigation in ?Sematic Space? requires to address two challenges: (i) how to create a low dimensional display of semantic relationship of documents, and (ii) how to obtain
these relationships by automated text categorization. Our results show that the HSOM can
provide a good solution to both demands simultaneously and within a single framework.
The HSOM is able to exploit the peculiar geometric properties of hyperbolic space to successfully compress complex semantic relationships between text documents. Additionally,
the use of hyperbolic lattice topology for the arrangement of the HSOM nodes offers new
and attractive features for interactive ?semantic navigation?. Large document databases
can be inspected at a glance while the HSOM provides additional information which was
captured during a previous training step, allowing e.g. to rapidly visualize relationships
between new documents and previously acquired collections.
Future work will address more sophisticated visualization strategies based on the new approach, as well as the exploration of other text representations which might take advantage
of hyperbolic space properties.
References
[1] T. Kohonen. Self-Organizing Maps. Springer Series in Information Sciences. 3rd edition, 2001.
[2] Teuvo Kohonen, Samuel Kaski, Krista Lagus, Jarkko Saloj?arvi, Vesa Paatero, and Antti Saarela.
Organization of a massive document collection. IEEE Transactions on Neural Networks, Special Issue on Neural Networks for Data Mining and Knowledge Discovery, 11(3):574?585, May
2000.
[3] John Lamping and Ramana Rao. Laying out and visualizing large trees using a hyperbolic
space. In Proceedings of UIST?94, pages 13?14, 1994.
[4] T. Munzer. Exploring large graphs in 3D hyperbolic space. IEEE Computer Graphics and
Applications, 18(4):18?23, July/August 1998.
[5] H. S. M. Coxeter. Non Euclidean Geometry. Univ. of Toronto Press, Toronto, 1957.
[6] J.A. Thorpe. Elementary Topics in Differential Geometry. Springer-Verlag, New York, 1979.
[7] Frank Morgan. Riemannian Geometry: A Beginner?s Guide. Jones and Bartlett Publishers,
Boston, London, 1993.
[8] Charles W. Misner, J. A. Wheeler, and Kip S. Thorne. Gravitation. Freeman, 1973.
[9] R. Fricke and F. Klein. Vorlesungen u? ber die Theorie der automorphen Funktionen, volume 1.
Teubner, Leipzig, 1897. Reprinted by Johnson Reprint, New York, 1965.
[10] W. Magnus. Noneuclidean Tesselations and Their Groups. Academic Press, 1974.
[11] Helge Ritter. Self-organizing maps in non-euclidian spaces. In E. Oja and S. Kaski, editors,
Kohonen Maps, pages 97?108. Amer Elsevier, 1999.
[12] J. Ontrup and H. Ritter. Text categorization and semantic browsing with self-organizing maps
on non-euclidean spaces. In Proc. of the PKDD-01, 2001.
[13] G. Salton and C. Buckley. Term-weighting approaches in automatic text retrieval. Information
Processing and Management, 24(5):513?523, 1988.
[14] T. Joachims. Text categorization with support vector machines: learning with many relevant
features. In Proc. of ECML-98, number 1398, pages 137?142, Chemnitz, DE, 1998.
[15] Huma Lodhi, John Shawe-Taylor, Nello Cristianini, and Chris Watkins. Text classification using
string kernels. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances in
Neural Information Processing Systems 13, pages 563?569. MIT Press, 2001.
[16] F. Sebastiani, A. Sperduti, and N. Valdambrini. An improved boosting algorithm and its application to automated text categorization. In Proc. of CIKM-00, pages 78?85, 2000.
[17] Y. Yang. An evaluation of statistical approaches to text categorization. Information Retrieval,
1-2(1):69?90, 1999.
| 2029 |@word deformed:1 faculty:2 proportion:1 suitably:2 disk:4 lodhi:1 euclidian:1 carry:1 contains:3 att:1 series:1 document:42 discretization:1 com:2 must:3 john:2 grain:3 numerical:1 distant:1 shape:1 leipzig:1 selected:2 tesselations:7 plane:13 inspection:2 isotropic:1 provides:4 boosting:2 node:39 toronto:2 attack:1 org:1 five:5 height:2 along:1 constructed:2 become:3 differential:1 retrieving:1 introductory:1 ray:1 manner:1 introduce:1 falsely:1 acquired:3 uist:1 behavior:1 pkdd:1 frequently:1 discretized:1 freeman:1 spherical:2 overwhelming:1 increasing:2 project:3 mulan:2 underlying:1 interpreted:1 string:1 giant:1 chronological:1 interactive:1 classifier:4 unit:6 appear:3 organize:1 atlantis:3 positive:1 before:1 todd:1 painted:2 meet:1 path:1 might:1 perpendicular:1 responsible:1 stereographic:1 practice:1 union:2 definite:1 poincar:10 wheeler:1 area:4 intersect:1 hyperbolic:34 significantly:1 projection:7 confidence:2 radial:2 regular:3 suggest:2 get:1 convenience:1 onto:7 context:1 www:2 equivalent:1 map:20 circumference:2 attention:1 starting:1 rule:1 regarded:1 embedding:4 coordinate:1 fx:5 analogous:1 limiting:1 techfak:2 construction:2 play:1 inspected:1 user:1 massive:1 homogeneous:1 origin:7 magnification:1 utilized:1 located:2 rec:1 coarser:1 database:3 observed:2 breakeven:1 region:5 wheat:3 connected:1 rig:1 trade:2 deeply:1 ran:1 cristianini:1 imdb:1 triangle:5 easily:1 polygon:1 kaski:2 genre:4 equilateral:1 surrounding:1 train:1 univ:1 distinct:1 describe:4 london:1 neighborhood:9 neuroinformatics:2 widely:2 solve:1 film:2 distortion:1 otherwise:1 obviously:1 advantage:2 net:1 took:1 propose:1 ment:1 fea:1 frequent:2 kohonen:4 relevant:1 rapidly:3 organizing:6 achieve:1 intuitive:1 moved:2 bug:1 cluster:2 sematic:1 congruent:2 categorization:13 ring:4 illustrate:1 measured:1 nearest:2 eq:1 strong:1 indicate:4 direction:1 radius:5 centered:1 human:1 exploration:1 translating:1 assign:2 preliminary:1 elementary:1 exploring:1 around:6 sufficiently:2 ground:3 magnus:1 mapping:4 visualize:5 major:3 released:2 purpose:1 domly:1 polar:1 favorable:1 proc:3 label:3 title:1 create:2 successfully:2 tool:1 reflects:1 faithfully:1 mit:1 clearly:1 gaussian:1 rather:2 shrinkage:1 beauty:1 volker:1 broader:1 conjunction:1 joachim:1 indicates:2 contrast:1 baseline:1 elsevier:1 nn:8 relation:1 germany:2 issue:1 among:1 classification:2 favored:1 spatial:1 art:3 special:1 noneuclidean:1 field:1 construct:4 having:1 cartoon:1 manually:1 park:1 jones:1 unsupervised:1 warrant:1 thinking:1 future:1 few:1 thorpe:1 oja:1 simultaneously:1 zoom:2 familiar:1 geometry:8 freedom:2 organization:1 interest:3 mining:4 highly:1 circular:2 evaluation:2 navigation:5 predefined:1 iranian:1 peculiar:2 closer:1 tree:1 euclidean:10 taylor:1 rotating:1 circle:3 prince:1 sperduti:1 minimal:2 instance:1 beginner:1 rao:1 cover:1 assignment:1 lattice:14 vertex:1 subset:1 uniform:2 teuvo:1 johnson:1 graphic:1 accessible:1 ritter:3 earn:5 squared:2 central:1 reflect:1 management:1 choose:1 worse:1 creating:1 leading:1 potential:1 de:3 jection:1 stream:1 vesa:1 teubner:1 view:4 observer:1 lab:1 portion:2 capability:2 inherited:1 acq:4 formed:1 ir:1 yield:2 identify:1 straight:1 ago:1 classified:5 detector:1 against:1 gravitation:1 acquisition:1 frequency:5 obvious:1 associated:3 riemannian:2 salton:1 dataset:1 adjusting:1 recall:4 knowledge:1 iron:1 sophisticated:1 reflecting:1 appears:1 isometric:2 follow:1 improved:1 amer:1 evaluated:1 shrink:1 strongly:1 leen:1 just:1 canvas:1 hand:1 glance:1 somehow:1 mode:1 grows:1 believe:1 glyph:8 effect:2 oil:1 contain:1 concept:1 dietterich:1 tagged:2 vicinity:1 semantic:11 attractive:1 visualizing:1 during:3 self:6 die:1 samuel:1 prominent:2 complete:1 occuring:1 egypt:1 interface:1 image:2 novel:2 charles:1 superior:1 specialized:1 ji:1 winner:2 exponentially:4 volume:1 significant:1 sebastiani:1 approx:1 rd:1 grid:2 focal:1 automatic:1 language:2 shawe:1 moving:1 similarity:1 surface:3 compiled:1 money:4 add:1 dominant:1 curvature:3 perspective:1 retrieved:2 hemisphere:1 apart:1 belongs:1 ship:6 certain:4 verlag:1 underlined:1 life:1 der:1 swept:1 preserving:1 seen:3 additional:2 somewhat:1 captured:1 morgan:1 determine:1 shortest:1 novelty:1 strike:1 july:1 preservation:1 ii:1 multiple:1 match:2 characterized:2 adapt:1 offer:4 sphere:1 retrieval:5 academic:1 controlled:1 metric:5 kernel:1 chicken:1 diagram:1 grow:2 publisher:1 probably:2 south:1 effectiveness:1 yang:1 split:2 ture:1 automated:2 fit:1 identified:1 topology:1 tarzan:2 prototype:2 reprinted:1 motivated:1 distorting:1 handled:1 bartlett:1 granted:1 linkage:1 gulf:1 passing:2 york:2 repeatedly:1 buckley:1 useful:2 pleasant:1 amount:1 ten:3 category:20 simplest:1 funktionen:1 generate:1 http:2 northern:2 exist:2 fish:4 estimated:1 arising:1 algorithmically:1 correctly:3 klein:6 per:1 cikm:1 group:4 threshold:2 nevertheless:1 vast:1 asymptotically:2 graph:2 run:1 angle:2 inverse:1 distorted:1 place:1 bielefeld:6 almost:1 scaling:1 comparable:1 capturing:1 internet:1 display:2 correspondence:1 flat:4 extremely:1 minkowski:1 corn:3 relatively:1 according:5 belonging:2 describes:2 smaller:1 increasingly:1 beast:1 helge:3 projecting:1 restricted:1 thorne:1 taken:2 visualization:5 previously:3 turn:1 describing:1 dataspace:1 conformal:1 end:1 generalizes:1 ramana:1 noticable:1 apply:1 hierarchical:2 original:5 compress:1 denotes:2 thomas:1 cf:3 exploit:1 build:2 arrangement:1 quantity:1 occurs:3 coxeter:1 strategy:1 usual:1 arctanh:1 anastasia:1 exhibit:2 fovea:2 distance:13 mapped:13 ontrup:2 majority:2 accommodating:1 modapte:1 topic:2 chris:1 extent:1 nello:1 laying:1 relationship:7 frank:1 theorie:1 negative:2 motivates:2 proper:1 unknown:3 perform:1 allowing:2 inspect:1 arc:2 attacked:1 ecml:1 immediate:1 ever:1 disney:2 august:1 introduced:1 david:1 required:1 kip:1 huma:1 address:2 able:2 below:2 belonged:1 challenge:1 loophole:1 suitable:2 event:1 natural:3 ranked:1 attach:1 scheme:1 movie:8 technology:2 brief:1 eye:4 axis:1 reprint:1 tresp:1 text:31 review:7 geometric:1 discovery:1 determining:2 law:1 beside:1 squeezed:1 interesting:3 squeezing:1 remarkable:2 ingredient:1 generator:1 h2:15 incident:2 offered:1 edf:1 editor:2 surrounded:1 dinosaur:1 keeping:1 antti:1 side:2 guide:2 ber:1 neighbor:6 outermost:1 curve:2 author:1 collection:10 commonly:1 projected:1 preprocessing:1 far:1 transaction:1 uni:2 keep:1 reveals:1 summing:1 decade:1 chief:1 table:2 additionally:1 channel:1 symmetry:1 complex:3 som:18 main:1 reuters:5 whole:1 animation:4 edition:1 repeated:1 body:1 chemnitz:1 fig:8 precision:4 position:2 wish:1 exceeding:1 exponential:2 crude:4 watkins:1 weighting:2 formula:1 navigate:1 showing:1 insightful:1 krista:1 symbol:2 cease:1 conspicious:1 illustrates:1 demand:1 browsing:1 boston:1 visual:1 contained:3 ordered:1 springer:2 corresponds:1 lewis:2 sized:1 consequently:1 labelled:3 price:1 content:1 included:1 infinite:2 determined:3 semantically:2 called:1 newsgroup:2 selectively:1 support:3 latter:1 evaluate:1 paatero:1 |
1,129 | 203 | 622
Atlas, Cole, Connor, EI-Sharkawi, Marks, Muthusamy and Barnard
Performance Comparisons Between
Backpropagation Networks and Classification Trees
on Three Real-World Applications
Ronald Cole
Dept. of CS&E
Oregon Graduate Institute
Beaverton. Oregon 97006
Les Atlas
Dept. of EE. Fr-10
University of Washington
Seattle. Washington 98195
Jerome Connor, Mohamed EI-Sharkawi, and Robert J. Marks II
University of Washington
Etienne Barnard
Carnegie-Mellon University
Yeshwant Muthusamy
Oregon Graduate Institute
ABSTRACT
Multi-layer perceptrons and trained classification trees are two very
different techniques which have recently become popular. Given
enough data and time, both methods are capable of performing arbitrary non-linear classification. We first consider the important
differences between multi-layer perceptrons and classification trees
and conclude that there is not enough theoretical basis for the clearcut superiority of one technique over the other. For this reason, we
performed a number of empirical tests on three real-world problems
in power system load forecasting, power system security prediction,
and speaker-independent vowel identification. In all cases, even for
piecewise-linear trees, the multi-layer perceptron performed as well
as or better than the trained classification trees.
Performance Comparisons
1 INTRODUCTION
In this paper we compare regression and classification systems. A regression system
can generate an output f for an input X, where both X and f are continuous and,
perhaps, multi-dimensional. A classification system can generate an output class, C,
for an input X, where X is continuous and multi-dimensional and C is a member of a
finite alphabet.
The statistical technique of Classification And Regression Trees (CART) was
developed during the years 1973 (Meisel and Michalpoulos) through 1984 (Breiman el
al). As we show in the next section, CART, like the multi-layer perceptron (MLP) ,
can be trained to solve the exclusive-OR problem. Furthermore, the solution it provides is extremely easy to interpret. Moreover, both CART and MLPs are able to provide arbitrary piecewise linear decision boundaries. Although there have been no
links made between CART and biological neural networks, the possible applications
and paradigms used for MLP and CART are very similar.
The authors of this paper represent diverse interests in problems which have the commonality of being both important and potentially well-suited for trainable classifiers.
The load forecasting problem, which is partially a regression problem, uses past load
trends to predict the critical needs of future power generation. The power security
problem uses the classifier as an interpolator of previously known states of the system.
The vowel recognition problem is representative of the difficulties in automatic
speech recognition due to variability across speakers and phonetic context.
In each problem area, large amounts of real data were used for training and disjoint
data sets were used for testing. We were careful to ensure that the experimental conditions were identical for the MLP and CART. We concentrated only on performance
as measured in error on the test set and did not do any formal studies of training or
testing time. (CART was, in general, quite a bit faster.)
In all cases, even with various sizes of training sets, the multi-layer perceptron performed as well as or better than the trained classification trees. We also believe that
integration of many of CART's well-designed attributes into MLP architectures could
only improve the already promising performance of MLP's.
2 BACKGROUND
2.1
Multi-Layer Perceptrons
The name "artificial neural networks" has in some commumbes become almost
synonymous with MLP's trained by back-propagation. Our power studies made use of
this standard algorithm (Rumelhart el ai, 1986) and our vowel studies made use of a
conjugate gradient version (Barnard and Casasent, 1989) of back-propagation. In all
cases the training data consisted of ordered pairs (X ,f)} for regression, or (X ,C)}
for classification. The input to the network is X and the output is, after training,
hopefully very close to f or C.
When MLP's are used for regression, the output, f, can take on real values between 0
and 1. This normalized scale was used as the prediction value in the power forecasting problem. For MLP classifiers the output is formed by taking the (0,1) range of the
output neurons and either thresholding or finding a peak. For example, in the vowel
623
624
Atlas, Cole, Connor, El-Sharkawi, Marks, Muthusamy and Barnard
study we chose the maximum of the 12 output neurons to indicate the vowel class.
2.2
Classification and Regression Trees (CART)
CART has already proven to be useful in diverse applications such as radar signal
classification, medical diagnosis, and mass spectra classification (Breiman et ai, 1984).
Given a set of training examples {(X ,C)}, a binary tree is constructed by sequentially
partitioning the p -dimensional input space, which may consist of quantitative and/or
qualitative data, into p -dimensional polygons. The trained classification tree divides
the domain of the data into non-overlapping regions, each of which is assigned a class
label C. For regression, the estimated function is piecewise constant over these regions.
The first split of the data space is made to obtain the best global separation of the
classes. The next step in CART is to consider the partitioned training examples as two
completely unrelated sets-those examples on the left of the selected hyper-plane, and
those on the right. CART then proceeds as in the first step, treating each subset of the
training examples independently. A question which had long plagued the use of such
sequential schemes was: when should the splitting stop? CART implements a novel,
and very clever approach; splits continue until every training example is separated
from every other, then a pruning criterion is used to sequentially remove less important splits.
2.3
Relative Expectations of MLP and CART
The non-linearly separable exclusive-OR problem is an example of a problem which
both MLP and CART can solve with zero error. The left side of Figure 1 shows a
trained MLP solution to this problem and the right side shows the very simple trained
CART solution. For the MLP the values along the arrows represent trained multiplicative weights and the values in the circles represent trained scalar offset values. For
the CART figure, y and n represent yes or no answers to the trained thresholds and the
values in the circles represent the output Y. It is interesting that CART did not train
correctly for equal numbers of the four different input cases and that one extra example of one of the input cases was sufficient to break the symmetry and allow CART to
train correctly. (Note the similarity to the well-known requirement of random and
different initial weights for training the MLP).
~
y~
08
Figure 1: The MLP and CART solutions to the exclusive-OR problem.
Performance Comparisons
CART trains on the exclusive-OR very easily since a piecewise-linear partition in the
input space is a perfect solution. In general, the MLP will construct classification regions with smooth boundaries, whereas CART will construct regions with "sharp"
comers (each region being, as described previously, an intersection of half planes).
We would thus expect MLP to have an advantage when classification boundaries tend
to be smooth and CART to have an advantage when they are sharper.
Other important differences between MLP and CART include:
For an MLP the number of hidden units can be selected to avoid overfitting or
underfitting the data. CART fits the complexity by using an automatic pruning technique to adjust the size of the tree. The selection of the number of hidden units or the
tree size was implemented in our experiments by using data from a second training set
(independent of the first).
An MLP becomes a classifier through an ad hoc application of thresholds or peak.picking to the output value(s). Great care has gone into the CART splitting rules
while the usual MLP approach is rather arbitrary.
A trained MLP represents an approximate solution to an optimization problem. The
solution may depend on initial choice of weights and on the optimization technique
used. For complex MLP's many of the units are independently and simultaneously
adjusting their weights to best minimize output error.
MLP is a distributed topology where a single point in the input space can have an
effect across all units or analogously, one weight, acting alone, will have minimal
affect on the outputs. CART is very different in that each split value can be mapped
onto one segment in the input space. The behavior of CART makes it much more
useful for data interpretation. A trained tree may be useful for understanding the
structure of the data. The usefulness of MLP's for data interpretation is much less
clear.
The above points, when taken in combination, do not make a clear case for either
MLP or CART to be superior for the best performance as a trained classifier. We thus
believe that the empirical studies of the next sections, with their consistent performance trends, will indicate which of the comparative aspects are the most significant.
3 LOAD FORECASTING
3.1
The Problem
The ability to predict electric power system loads from an hour to several days in the
future can help a utility operator to efficiently schedule and utilize power generation.
This ability to forecast loads can also provide information which can be used to strategically trade energy with other generating systems. In order for these forecasts to be
useful to an operator, they must be accurate and computationally efficient.
3.2
Methods
Hourly temperature and load data for the Seattle{facoma area were provided for us by
the Puget Sound Power and Light Company. Since weekday forecasting is a more
critical problem for the power industry than weekends, we selected the hourly data for
625
626
Atlas, Cole, Connor, El?Sharkawi, Marks, Muthusamy and Barnard
all Tuesdays through Fridays in the interval of November 1, 1988 through January 31,
1989. These data consisted of 1368 hourly measurements that consisted of the 57
days of data collected.
These data were presented to both the MLP and the CART classifier as a 6dimensional input with a single, real-valued output. The MLP required that all values
be normalized to the range (0,1). These same normalized values were used with the
CART technique. Our training and testing process consisted of training the classifiers
on 53 days of the data and testing on the 4 days left over at the end of January 1989.
Our training set consisted of 1272 hourly measurements and our test set contained 96
hourly readings.
The MLP we used in these experiments had 6 inputs (Plus the trained constant bias
term) 10 units in one hidden layer and one output. This topology was chosen by making use of data outside the training and test sets.
3.3
Results
We used an 11 norm for the calculation of error rates and found that both techniques
worked quite well. The average error rate for the :MLP was 1.39% and CART gave
2.86% error. While this difference (given the number of testing points) is not statistically significant. it is worth noting that the trained MLP offers performance which is
at least as good as the current techniques used by the Puget Sound Power and Light
Company and is currently being verified for application to future load prediction.
4 POWER SYSTEM SECURITY
The assessment of security in a power system is an ongoing problem for the efficient
and reliable generation of electric power. Static security addresses whether. after a
disturbance. such as a line break or other rapid load change. the system will reach a
steady state operating condition that does not violate any operating constraint and
cause a "brown-out" or "black-out."
The most efficient generation of power is achieved when the power system is operating near its insecurity boundary. In fact. the ideal case for efficiency would be full
knowledge of the absolute boundaries of the secure regions. Due to the complexity of
the power systems, this full knowledge is impossible. Load flow algorithms, which
are based on iterative solutions of nonlinearly constrained equations, are conventionally used to slowly and accurately determine points of security or insecurity. In real
systems the trajectories through the regions are not predictable in fine detail. Also
these changes can happen too fast to compute new results from the accurate load flow
equations.
We thus propose to use the sparsely known solutions of the load flow equations as a
training set The test set consists of points of unknown security. The error of the test
set can then be computed by comparing the result of the trained classifier to load flow
equation solutions.
Our technique for converting this problem to a problem for a trainable classifier involves defining a training set ((X ,C?) where X is composed of real power, reactive
power, and apparent power at another bus. This 3-dimensional input vector is paired
with the corresponding security status (C=l for secure and C=O for insecure). Since
Performance Comparisons
the system was small, we were able to generate a large number of data points for
training and testing. In fact, well over 20,000 total data points were available for the
(disjoint) training and test sets.
4.1
Results
We observed that for any choice of training data set size, the error rate for the MLP
was always lower than the rate for the CART classifier. At 10,000 points of training
data, the MLP had an error rate of 0.78% and CART has an error rate of 1.46%.
While both of these results are impressive. the difference was statistically significant
(p>.99).
In order to gain insight into the reasons for differences in importance, we looked at
classifier decisions for 2-dimensional slices of the input space. While the CART
boundary sometimes was a better match, certain pathological difficulties made CART
more error-prone than the MLP. Our other studies also showed that there were worse
interpolation characteristics for CART. especially for sparse data. Apparently, starting
with nonlinear combinations of inputs. which is what the MLP does. is better for the
accurate fit than the stair-steps of CART.
5 SPEAKER-INDEPENDENT VOWEL CLASSIFICATION
Speaker-independent classification of vowels excised from continuous speech is a most
difficult task because of the many sources of variability that influence the physical
realization of a given vowel. These sources of variability include the length of the
speaker's vocal tract, phonetic context in which the vowel occurs, speech rate and
syllable stress.
To make the task even more difficult the classifiers were presented only with information from a single spectral slice. The spectral slice, represented by 64 DFf
coefficients (0-4 kHz), was taken from the center of the vowel, where the effects of
coarticulation with surrounding phonemes are least apparent.
The training and test sets for the experiments consisted of featural descriptions, X,
paired with an associated class, C. for each vowel sample. The 12 monophthongal
vowels of English were used for the classes. as heard in the following words: beat. bit.
bet, bat. roses. the, but, boot, book. bought, cot, bird. The vowels were excised from
the wide variety of phonetic contexts in utterances of the TIMIT database, a standard
acoustic phonetic corpus of continuous speech, displaying a wide range of American
dialectical variation (Fisher et ai, 1986) (Lamel et ai, 1986). The training set consisted of 4104 vowels from 320 speakers. The test set consisted of 1644 vowels (137 occurrences of each vowel) from a different set of 100 speakers.
The MLP consisted of 64 inputs (the DFf coefficients. each nonnalized between zero
and one), a single hidden layer of 40 units, and 12 output units; one for each vowel
category. The networks were trained using backpropagation with conjugate gradient
optimization (Barnard and Casasent, 1989). The procedure for training and testing a
network proceeded as follows: The network was trained on 100 iterations through the
4104 training vectors. The trained network was then evaluated on the training set and
a different set of 1644 test vectors (the test set). The network was then trained for an
additional 100 iterations and again evaluated on the training and test sets. This process was continued until the network had converged; convergence was observed as a
627
628
Atlas, Cole, Connor, EI?Sharkawi, Marks, Muthusamy and Barnard
consistent decrease or leveling off of the classification percentage on the test data over
successive sets of 100 iterations.
The CART system was trained using two separate computer routines. One was the
CART program from California Statistical Software; the other was a routine we
designed ourselves. We produced our own routine to ensure a careful and independent
test of the CART concepts described in (Breiman et ai, 1984).
5.1
Results
In order to better understand the results, we performed listening experiments on a subset of the vowels used in these experiments. The vowels were excised from their sentence context and presented in isolation. Five listeners first received training in the
task by classifying 900 vowel tokens and receiving feedback about the correct answer
on each trial. During testing, each listener classified 600 vowels from the test set (50
from each category) without feedback. The average classification performance on the
test set was 51%, compared to chance performance of 8.3%. Details of this experiment are presented in (Muthusamy et ai, 1990). When using the scaled spectral
coefficients to train both techniques, the MLP correctly classified 47.4% of the test set
while CART employing uni-variate splits performed at only 38.2%.
One reason for the poor performance of CART with un i-variate splits may be that
each coefficient (corresponding to energy in a narrow frequency band) contains little
information when considered independently of the other coefficients. For example, reduced energy in the 1 kHz band may be difficult to detect if the energy in the 1.06
kHz band was increased by an appropriate amount. The CART classifier described
above operates by making a series of inquiries about one frequency band at a time, an
intuitively inappropriate approach.
We achieved our best CART results, 46.4%, on the test set by making use of arbitrary
hyper-planes (linear combinations) instead of univariate splits. This search-based approach gave results which were within 1% of the MLP results.
6 CONCLUSIONS
In all cases the performance of the MLP was, in terms of percent error, better than
CART. However, the difference in performance between the two classifiers was only
significant (at the p >.99 level) for the power security problem.
There are several possible reasons for the sometimes superior performance of the MLP
technique, all of which we are currently investigating. One advantage may stem from
the ability of MLP to easily find correlations between large numbers of variables.
Although it is possible for CART to form arbitrary nonlinear decision boundaries, the
efficiency of the recursive splitting process may be inferior to MLP's nonlinear fit.
Another relative disadvantage of CART may be due to the successive nature of node
growth. For example, if the first split that is made for a problem turns out, given the
successive splits, to be suboptimal, it becomes very inefficient to change the first split
to be more suitable.
We feel that the careful statistics used in CART could also be advantageously applied
to MLP. The superior performance of MLP is not yet indicative of best performance
and it may turn out that careful application of statistics may allow further advance-
Performance Comparisons
ments in the MLP technique. It also may be possible that there would be input
representations that would cause better performance for CART than for MLP.
There have been new developments in trained statistical classifiers since the development of CART. More recent techniques, such as projection pursuit (Friedman and
Stuetzle, 1984), may prove as good as or superior to MLP. This continued interplay
between MLP techniques and advanced statistics is a key part of our ongoing research.
Acknowledgements
The authors wish to thank Professor R.D. Martin and Dr. Alan Lippman of the
University of Washington Department of Statistics and Professors Aggoune, Damborg,
and Hwang of the University of Washington Department of Electrical Engineering for
their helpful discussions. David Cohn and Carlos Rivera assisted with many of the
experiments.
We also would like to thank Milan Casey Brace of Puget Power and Light for providing the load forecasting data.
This work was supported by a National Science Foundation Presidential Young Investigator Award for L. Atlas and also by separate grants from the National Science
Foundation and Washington Technology Center.
References
P. E. Barnard and D. Casasent, "Image Processing for Image Understanding with
Neural Nets," Proc. Int. Joint Con! on Neural Nets, Washington, DC, June 18-22,
1989.
L. Breiman, J.H. Friedman, R.A. Olshen, and CJ. Stone, Classification and Regression Trees, Wadsworth International, Belmont, CA, 1984.
W. Fisher, G. Doddington, and K. Goudie-Marshall, "The DARPA Speech Recognition Research Database: Specification and Status," Proc. of the DARPA Speech
Recognition Workshop, pp. 93-100, February 1986.
J.H. Friedman and W. StuetzIe, "Projection Pursuit Regression," J. Amer. Stat. Assoc. 79, pp. 599-608, 1984.
L. Lamel, R. Kassel, and S. Seneff, "Speech Database Development: Design and
Analysis of the Acoustic-Phonetic Corpus," Proc. of the DARPA Speech Recognition
Workshop, pp. 100-110, February 1986.
W.S. Meisel and D.A. Michalpoulos, "A Partitioning Algorithm with Application in
Pattern Classification and the Optimization of Decision Trees," IEEE Trans. Computers C-22, pp. 93-103. 1973.
Y. Muthusamy. R. Cole, and M. Slaney. "Vowel Information in a Single Spectral
Slice: Cochlcagrams Versus Spectrograms," Proc. ICASSP '90, April 3-6. 1990. (to
appear)
D.E. Rumelhart. G.E. Hinton, and RJ. Williams. "Learning Internal Representations
by Error Propagation," Ch. 2 in Parallel Distributed Processing, D.E. Rumelhart,
J.L. McClelland, and the PDP Research Group, MIT Press, Cambridge. MA, 1986.
629
| 203 |@word proceeded:1 trial:1 version:1 norm:1 weekday:1 rivera:1 initial:2 contains:1 series:1 dff:2 past:1 current:1 comparing:1 yet:1 must:1 belmont:1 ronald:1 happen:1 partition:1 remove:1 atlas:6 designed:2 treating:1 alone:1 half:1 selected:3 indicative:1 plane:3 provides:1 node:1 successive:3 five:1 along:1 constructed:1 become:2 qualitative:1 consists:1 prove:1 underfitting:1 leveling:1 cot:1 rapid:1 behavior:1 multi:8 company:2 little:1 inappropriate:1 becomes:2 provided:1 unrelated:1 moreover:1 mass:1 what:1 developed:1 finding:1 quantitative:1 every:2 growth:1 classifier:15 scaled:1 assoc:1 partitioning:2 unit:7 grant:1 medical:1 superiority:1 appear:1 hourly:5 engineering:1 interpolation:1 black:1 chose:1 plus:1 bird:1 graduate:2 range:3 gone:1 statistically:2 bat:1 testing:8 recursive:1 implement:1 backpropagation:2 lippman:1 procedure:1 stuetzle:1 area:2 excised:3 empirical:2 projection:2 word:1 vocal:1 onto:1 close:1 clever:1 selection:1 operator:2 context:4 impossible:1 influence:1 center:2 williams:1 starting:1 independently:3 splitting:3 rule:1 insight:1 continued:2 variation:1 feel:1 us:2 trend:2 rumelhart:3 recognition:5 sparsely:1 database:3 observed:2 electrical:1 region:7 trade:1 decrease:1 aggoune:1 rose:1 predictable:1 complexity:2 radar:1 trained:23 depend:1 segment:1 efficiency:2 basis:1 completely:1 comer:1 easily:2 joint:1 darpa:3 icassp:1 various:1 polygon:1 represented:1 listener:2 alphabet:1 train:4 separated:1 weekend:1 fast:1 surrounding:1 meisel:2 artificial:1 hyper:2 outside:1 quite:2 apparent:2 solve:2 valued:1 presidential:1 ability:3 statistic:4 hoc:1 advantage:3 interplay:1 net:2 propose:1 fr:1 realization:1 description:1 milan:1 seattle:2 convergence:1 requirement:1 comparative:1 perfect:1 generating:1 tract:1 help:1 stat:1 measured:1 received:1 implemented:1 c:1 involves:1 indicate:2 coarticulation:1 correct:1 attribute:1 biological:1 assisted:1 considered:1 plagued:1 great:1 predict:2 commonality:1 proc:4 label:1 currently:2 cole:6 mit:1 always:1 rather:1 avoid:1 dialectical:1 breiman:4 bet:1 casey:1 june:1 secure:2 detect:1 helpful:1 el:4 synonymous:1 hidden:4 classification:22 development:3 constrained:1 integration:1 wadsworth:1 equal:1 construct:2 washington:7 identical:1 represents:1 future:3 piecewise:4 strategically:1 pathological:1 composed:1 simultaneously:1 national:2 ourselves:1 vowel:22 friedman:3 mlp:47 interest:1 adjust:1 light:3 stair:1 accurate:3 capable:1 tree:15 divide:1 circle:2 theoretical:1 minimal:1 increased:1 industry:1 marshall:1 disadvantage:1 subset:2 usefulness:1 too:1 answer:2 peak:2 international:1 off:1 receiving:1 picking:1 analogously:1 again:1 slowly:1 dr:1 worse:1 slaney:1 book:1 american:1 inefficient:1 friday:1 puget:3 coefficient:5 int:1 oregon:3 ad:1 performed:5 multiplicative:1 break:2 apparently:1 carlos:1 parallel:1 timit:1 mlps:1 formed:1 minimize:1 phoneme:1 characteristic:1 efficiently:1 yes:1 identification:1 accurately:1 produced:1 trajectory:1 worth:1 insecurity:2 converged:1 classified:2 inquiry:1 reach:1 energy:4 frequency:2 mohamed:1 pp:4 associated:1 static:1 con:1 stop:1 gain:1 adjusting:1 popular:1 knowledge:2 cj:1 schedule:1 routine:3 back:2 day:4 april:1 amer:1 evaluated:2 furthermore:1 jerome:1 until:2 correlation:1 ei:3 cohn:1 nonlinear:3 hopefully:1 assessment:1 overlapping:1 propagation:3 perhaps:1 hwang:1 believe:2 name:1 effect:2 consisted:9 normalized:3 brown:1 concept:1 assigned:1 during:2 inferior:1 speaker:7 steady:1 criterion:1 stone:1 stress:1 lamel:2 temperature:1 yeshwant:1 percent:1 image:2 novel:1 recently:1 superior:4 physical:1 khz:3 interpretation:2 interpret:1 mellon:1 significant:4 measurement:2 connor:5 cambridge:1 ai:6 automatic:2 ongoing:2 had:4 specification:1 similarity:1 operating:3 impressive:1 own:1 showed:1 recent:1 phonetic:5 certain:1 binary:1 continue:1 seneff:1 additional:1 care:1 spectrogram:1 converting:1 determine:1 paradigm:1 signal:1 ii:1 violate:1 sound:2 rj:1 full:2 stem:1 smooth:2 alan:1 faster:1 match:1 calculation:1 offer:1 long:1 award:1 casasent:3 paired:2 prediction:3 regression:10 expectation:1 iteration:3 represent:5 sometimes:2 achieved:2 background:1 whereas:1 fine:1 interval:1 source:2 extra:1 brace:1 cart:51 tend:1 member:1 flow:4 bought:1 ee:1 near:1 noting:1 ideal:1 split:10 enough:2 muthusamy:7 easy:1 variety:1 affect:1 fit:3 gave:2 isolation:1 architecture:1 topology:2 variate:2 suboptimal:1 listening:1 whether:1 utility:1 forecasting:6 advantageously:1 speech:8 cause:2 useful:4 heard:1 clear:2 amount:2 band:4 concentrated:1 category:2 mcclelland:1 reduced:1 generate:3 percentage:1 estimated:1 disjoint:2 correctly:3 diverse:2 diagnosis:1 carnegie:1 group:1 key:1 four:1 threshold:2 interpolator:1 verified:1 utilize:1 year:1 almost:1 separation:1 decision:4 bit:2 layer:8 syllable:1 constraint:1 worked:1 software:1 aspect:1 extremely:1 performing:1 separable:1 martin:1 department:2 combination:3 poor:1 conjugate:2 across:2 partitioned:1 making:3 damborg:1 intuitively:1 taken:2 computationally:1 equation:4 previously:2 bus:1 turn:2 end:1 available:1 pursuit:2 spectral:4 appropriate:1 occurrence:1 ensure:2 include:2 beaverton:1 etienne:1 especially:1 february:2 kassel:1 already:2 question:1 looked:1 occurs:1 exclusive:4 usual:1 gradient:2 link:1 mapped:1 separate:2 thank:2 collected:1 reason:4 length:1 providing:1 difficult:3 olshen:1 robert:1 potentially:1 sharper:1 design:1 unknown:1 boot:1 neuron:2 finite:1 november:1 january:2 beat:1 defining:1 hinton:1 variability:3 nonnalized:1 dc:1 pdp:1 arbitrary:5 sharp:1 david:1 pair:1 required:1 nonlinearly:1 sentence:1 security:9 acoustic:2 california:1 narrow:1 hour:1 trans:1 address:1 able:2 proceeds:1 pattern:1 reading:1 program:1 reliable:1 power:22 critical:2 suitable:1 difficulty:2 disturbance:1 advanced:1 scheme:1 improve:1 technology:1 conventionally:1 utterance:1 featural:1 understanding:2 acknowledgement:1 relative:2 expect:1 generation:4 interesting:1 proven:1 versus:1 foundation:2 sufficient:1 consistent:2 thresholding:1 displaying:1 classifying:1 prone:1 token:1 supported:1 english:1 formal:1 side:2 allow:2 perceptron:3 institute:2 bias:1 wide:2 taking:1 understand:1 absolute:1 sparse:1 distributed:2 slice:4 boundary:7 feedback:2 world:2 author:2 made:6 employing:1 pruning:2 approximate:1 uni:1 status:2 global:1 sequentially:2 overfitting:1 investigating:1 corpus:2 conclude:1 spectrum:1 continuous:4 iterative:1 un:1 search:1 goudie:1 promising:1 nature:1 ca:1 symmetry:1 tuesday:1 complex:1 electric:2 domain:1 did:2 linearly:1 arrow:1 representative:1 wish:1 young:1 load:14 offset:1 ments:1 consist:1 workshop:2 sequential:1 importance:1 sharkawi:5 forecast:2 suited:1 intersection:1 univariate:1 ordered:1 contained:1 partially:1 scalar:1 ch:1 chance:1 ma:1 careful:4 barnard:8 fisher:2 professor:2 change:3 operates:1 acting:1 total:1 experimental:1 perceptrons:3 internal:1 mark:5 doddington:1 reactive:1 investigator:1 dept:2 trainable:2 |
1,130 | 2,030 | Classifying Single Trial EEG:
Towards Brain Computer Interfacing
Benjamin Blankertz1?, Gabriel Curio2 and Klaus-Robert M?ller1,3
1 Fraunhofer-FIRST.IDA, Kekul?str. 7, 12489 Berlin, Germany
2 Neurophysics Group, Dept. of Neurology, Klinikum Benjamin Franklin,
Freie Universit?t Berlin, Hindenburgdamm 30, 12203 Berlin, Germany
3 University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany
"!$#$&%
' ( "')*,+ -/.&! 0, 1"%02435
6
*%-/ 3 7 08 #
) 943
Abstract
Driven by the progress in the field of single-trial analysis of EEG, there is
a growing interest in brain computer interfaces (BCIs), i.e., systems that
enable human subjects to control a computer only by means of their brain
signals. In a pseudo-online simulation our BCI detects upcoming finger
movements in a natural keyboard typing condition and predicts their laterality. This can be done on average 100?230 ms before the respective
key is actually pressed, i.e., long before the onset of EMG. Our approach
is appealing for its short response time and high classification accuracy
(>96%) in a binary decision where no human training is involved. We
compare discriminative classifiers like Support Vector Machines (SVMs)
and different variants of Fisher Discriminant that possess favorable regularization properties for dealing with high noise cases (inter-trial variablity).
1 Introduction
The online analysis of single-trial electroencephalogram (EEG) measurements is a challenge for signal processing and machine learning. Once the high inter-trial variability (see
Figure 1) of this complex multivariate signal can be reliably processed, the next logical
step is to make use of the brain activities for real-time control of, e.g., a computer. In this
work we study a pseudo-online evaluation of single-trial EEGs from voluntary self-paced
finger movements and exploit the laterality of the left/right hand signal as one bit of information for later control. Features of our BCI approach are (a) no pre-selection for artifact
trials, (b) state-of-the-art learning machines with inbuilt feature selection mechanisms (i.e.,
sparse Fisher Discriminant Analysis and SVMs) that lead to >96% classification accuracies, (c) non-trained users and (d) short response times. Although our setup was not tuned
for speed, the a posteriori determined information transmission rate is 23 bits/min which
makes our approach competitive to existing ones (e.g., [1, 2, 3, 4, 5, 6, 7]) that will be
discussed in section 2.
? To
whom correspondence should be addressed.
Aims and physiological concept of BCI devices.
Two key issues to start with when
conceiving a BCI are (1) the definition of a behavioral context in which a subject?s brain
signals will be monitored and used eventually as surrogate for a bodily, e.g., manual, input
of computer commands, and (2) the choice of brain signals which are optimally capable to
convey the subject?s intention to the computer.
Concerning the behavioral context, typewriting on a computer keyboard is a highly overlearned motor competence. Accordingly, a natural first choice is a BCI-situation which induces the subject to arrive at a particular decision that is coupled to a predefined (learned)
motor output. This approach is well known as a two alternative forced choice-reaction
task (2AFC) where one out of two stimuli (visual, auditory or somatosensory) has to be
detected, categorised and responded to by issuing one out of two alternative motor commands, e.g., pushing a button with either the left or right hand. A task variant without
explicit sensory input is the voluntary, endogeneous generation of a ?go? command involving the deliberate choice between the two possible motor outputs at a self-paced rate. Here,
we chose this latter approach so as to approximate the natural computer input situation of
self-paced typewriting.
Concerning the selection of brain signals related to such endogeneous motor commands
we focussed here on one variant of slow brain potentials which are specifically related to
the preparation and execution of a motor command, rather than reflecting merely unspecific modulations of vigilance or attention. Using multi-channel EEG-mapping it has been
repeatedly demonstrated that several highly localised brain areas contribute to cerebral motor command processes. Specifically, a negative ?Bereitschaftspotential? (BP) precedes the
voluntary initiation of the movement. A differential scalp potential distribution can be reliably demonstrated in a majority of experimental subjects with larger BP at lateral scalp
positions (C3, C4) positioned over the left or right hemispherical primary motor cortex,
respectively, consistenly correlating with the performing (right or left) hand [8, 9].
Because one potential BCI-application is with paralysed patients, one might consider to
mimic the ?no-motor-output? of these individuals by having healthy experimental subjects
to intend a movement but to withhold its execution (motor imagery). While it is true that
brain potentials comparable to BP are associated with an imagination of hand movements,
which indeed is consistent with the assumption that the primary motor cortex is active with
motor imagery, actual motor performance significantly increased these potentials [10]. We
therefore chose to instruct the experimental subjects to actually perform the typewriting
finger movements, rather than to merely imagine their performance, for two reasons: first,
this will increase the BP signal strength optimising the signal-to-noise ratio in BCI-related
single trial analyses; and second, we propose that it is important for the subject?s task efficiency not to be engaged in an unnatural condition where, in addition to the preparation of a
motor command, a second task, i.e., to ?veto? the very same movement, has to be executed.
In the following section we will briefly review part of the impressive earlier research towards BCI devices (e.g., [1, 2, 3, 4, 5, 6, 7]) before experimental set-up and classification
results are discussed in sections 3 and 4 respectively. Finally a brief conclusion in given.
2 A brief outline of BCI research
Birbaumer et al. investigate slow cortical potentials (SCP) and how they can be selfregulated in a feedback scenario. In their thought translation device [2] patients learn to
produce cortical negativity or positivity at a central scalp location at will, which is fed back
to the user. After some training patients are able to transmit binary decisions in a 4 sec periodicity with accuracy levels up to 85% and therewith control a language support program
or an internet browser.
Pfurtscheller et al. built a BCI system based on event-related (de-)synchronisation
(ERD/ERS, typically of the ? and central ? rhythm) for online classification of movement
imaginations or preparations into 2?4 classes (e.g., left/right index finger, feet, tongue).
Typical preprocessing techniques are adaptive autoregressive parameters, common spatial
patterns (after band pass filtering) and band power in subject specific frequency bands.
Classification is done by Fisher discriminant analysis, multi-layer neural networks or LVQ
variants. In classification of exogeneous movement preparations, rates of 98%, 96% and
75% (for three subjects respectively) are obtained before movement onset 1 in a 3 classes
task and trials of 8 sec [3]. Only selected, artifact free trials (less that 40%) were used. A
tetraplegic patient controls his hand orthosis using the Graz BCI system.
Wolpaw et al. study EEG-based cursor control [4], translating the power in subject specific
frequency bands, or autoregressive parameters, from two spatially filtered scalp locations
over sensorimotor cortex into vertical cursor movement. Users initially gain control by various kinds of motor imagery (the setting favours ?movement? vs. ?no movement? in contrast
to ?left? vs. ?right?), which they report to use less and less as feedback training continues.
In cursor control trials of at least 4 sec duration trained subjects reach accuracies of over
90%. Some subjects acquired also considerable control in a 2-d setup.
3 Acquisition and preprocessing of brain signals
Experimental setup. The subject sat in a normal chair, relaxed arms resting on the table, fingers in the standard typing position at the computer keyboard. The task was to press
with the index and little fingers the corresponding keys in a self-chosen order and timing
(?self-paced key typing?). The experiment consisted of 3 sessions of 6 minutes each, preand postceeded by 60 seconds relaxing phase. All sessions were conducted on the same
day with some minutes break inbetween. Typing of a total of 516 keystrokes was done at
an average speed of 1 key every 2.1 seconds.
Brain activity was measured with 27 Ag/AgCl electrodes at positions of the extended international 10-20 system, 21 mounted over motor and somatosensory cortex, 5 frontal and
one occipital, referenced to nasion (sampled at 1000 Hz, band-pass filtered 0.05?200 Hz).
Besides EEG we recorded an electromyogram (EMG) of the musculus flexor digitorum bilaterally (10?200 Hz) and a horizontal and vertical electrooculogram (EOG). In an event
channel the timing of keystrokes was stored along with the EEG signal. All data were
recorded with a NeuroScan device and converted to Matlab format for further analysis.
The signals were downsampled to 100 Hz by picking every 10th sample. In a moderate
rejection we sorted out only 3 out of 516 trials due to heavy measurement artifacts, while
keeping trials that are contaminated by less serious artifacts or eye blinks. Note that 0.6%
rejection rate is very low in contrast to most other BCI offline studies.
The issue of preprocessing. Preprocessing the data can have a substantial effect on classification in terms of accuracy, effort and suitability of different algorithms. The question
to what degree data should be preprocessed prior to classification is a trade-off between the
danger of loosing information or overfitting and not having enough training samples for
the classifier to generalize from high dimensional, noisy data. We have investigated two
options: unprocessed data and preprocessing that was designed to focus on BP related to
finger movement:
(none) take 200 ms of raw data of all relevant channels;
(<5 Hz) filter the signal low pass at 5 Hz, subsample it at 20 Hz and take 150 ms of all
relevant channels (see Figure 1);
Speaking of classification at a certain time point we strictly mean classification based on
EEG signals until that very time point. The following procedure of calculating features
of a single trial due to (<5 Hz) is easy applicable in an online scenario: Take the last 128
sample points of each channel (to the past relative from the given time point), apply a windowed (w(n) := 1 ? cos(n?/128)) FFT, keep only the coefficients corresponding to the pass
1 Precisely:
before mean EMG onset time, for some trials this is before for others after EMG onset.
20
10
[?V]
F3
F1
FZ
F2
F4
CA5
CA3
CA1
CAZ
CA2
CA4
CA6
C5
C3
C1
CZ
C2
C4
C6
CP5
CP3
CP1
CPZ
CP2
CP4
CP6
O1
average
single trial
feature
0
?10
?20
?260
?500
?400
?300
?200
?100
0
?240
?220
?200
?180
[ms]
Figure 1: Averaged data and two single trials of
right finger movements in channel C3. 3 values (marked by circles) of smoothed signals are
taken as features in each channel.
?160
?140
?120
?
??
Figure 2: Sparse Fisher Discriminant Analysis
selected 68 features (shaded) from 405 input dimensions (27 channels ? 15 samples [150 ms])
of raw EEG data.
band (bins 2?7, as bin 1 just contains DC information) and transform back. Downsampling
to 20 Hz is done by calculating the mean of consecutive 5-tuple of data points. We investigated the alternatives of taking all 27 channels, or only the 21 located over motor and
sensorimotor cortex. The 6 frontal and occipital channels are expected not to give strong
contributions to the classification task. Hence a comparison shows, whether a classifier is
disturbed by low information channels or if it even manages to extract information from
them.
Figure 1 depicts two single trial EEG signals at scalp location C3 for right finger movements. These two single trials are very well-shaped and were selected for resembling the
the grand average over all 241 right finger movements, which is drawn as thick line. Usually the BP of a single trial is much more obscured by non task-related brain activity and
noise. The goal of preprocessing is to reveal task-related components to a degree that they
can be detected by a classifier. Figure 1 shows also the feature vectors due to preprocessing
(<5 Hz) calculated from the depicted raw single trial signals.
4 From response-aligned to online classification
We investigate some linear classification methods. Given a linear classifier (w, b) in separating hyperplane formulation (w> x+b = 0), the estimated label {1, ?1} of an input vector
x ? N is y? = sign(w> x + b). If no a priori knowledge on the probability distribution of the
data is available, a typical objective is to minimize a combination of empirical risk function
and some regularization term that restrains the algorithm from overfitting to the training
set {(xk , yk ) | k = 1, . . . , K}. Taking a soft margin loss function [11] yields the empirical
risk function ?Kk=1 max(0, 1 ? yk (w> xk + b)). In most approaches of this type there is a
hyper-parameter that determines the trade-off between risk and regularization, which has
to be chosen by model selection on the training set2 .
Fisher Discriminant (FD) is a well known classification method, in which a projection
vector is determined to maximize the distance between the projected means of the two
classes while minimizing the variance of the projected data within each class [13]. In the
binary decision case FD is equivalent to a least squares regression to (properly scaled) class
labels.
Regularized Fisher Discriminant (RFD) can be obtained via a mathematical programming approach [14]:
min 1/2 ||w||22 + C/K ||? ||22
w,b,?
yk (w> xk + b) = 1 ? ?k
2 As
subject to
for k = 1, . . . , K
this would be very time consuming in k-fold crossvalidation, we proceed similarly to [12].
filter
<5 Hz
<5 Hz
none
none
ch?s
mc
all
mc
all
FD
3.7?2.6
3.3?2.5
18.1?4.8
29.3?6.1
RFD
3.3?2.2
3.1?2.5
7.0?4.1
7.5?3.8
SFD
3.3?2.2
3.4?2.7
6.4?3.4
7.0?3.9
SVM
3.2?2.5
3.6?2.5
8.5?4.3
9.8?4.4
k-NN
21.6?4.9
23.1?5.8
29.6?5.9
32.2?6.8
Table 3: Test set error (? std) for classification at 120 ms before keystroke; ?mc? refers to the 21
channels over (sensori) motor cortex, ?all? refers to all 27 channels.
where ||?||2 denotes the `2 -norm (||w||22 = w> w) and C is a model parameter. The constraint
yk (w> xk + b) = 1 ? ?k ensures that the class means are projected to the corresponding class
labels, i.e., 1 and ?1. Minimizing the length of w maximizes the margin between the
projected class means relative to the intra class variance. This formalization above gives
the opportunity to consider some interesting variants, e.g.,
Sparse Fisher Discriminant (SFD) uses the `1 -norm (||w||1 = ?|wn |) on the regularizer,
i.e., the goal function is 1/N ||w||1 + C/K ||? ||22 . This choice favours solutions with sparse
vectors w, so that this method also yields some feature selection (in input space). When
applied to our raw EEG signals SFD selects 68 out of 405 input dimensions that allow for a
left vs. right classification with good generalization. The choice coincides in general with
what we would expect from neurophysiology, i.e., high loadings for electrodes close to left
and right hemisphere motor cortices which increase prior to the keystroke, cf. Figure 2. But
here the selection is automatically adapted to subject, electrode placement, etc.
Our implementation of RFD and SFD uses the cplex optimizer [15].
Support Vector Machines (SVMs) are well known for their use with kernels [16, 17].
Here we only consider linear SVMs:
min 1/2 ||w||22 + C/K ||? ||1
w,b,?
yk (w> xk + b)
subject to
1 ? ?k ,
and ?k
0
The choice of regulization keeps a bound on the Vapnik-Chervonenkis dimension small. In
an equivalent formulation the objective is to maximize the margin between the two classes
(while minimizing the soft margin loss function)3.
For comparision we also employed a standard classifier of different type:
k-Nearest-Neighbor (k-NN) maps an input vector to that class to which the majority of the
k nearest training samples belong. Those neighbors are determined by euclidean distance
of the corresponding feature vectors. The value of k chosen by model selection was around
15 for processed and around 25 for unprocessed data.
Classification of response-aligned windows. In the first phase we make full use of the
information that we have regarding the timing of the keystrokes. For each single trial we
calculate a feature vector as described above with respect to a fixed timing relative to the
key trigger (?response-aligned?). Table 3 reports the mean error on test sets in a 10?10fold crossvalidation for classifying in ?left? and ?right? at 120 ms prior to keypress. Figure 4
shows the time course of the classification error. For comparison, the result of EMG-based
classification is also displayed. It is more or less at chance level up to 120 ms before the
keystroke. After that the error rate decreases rapidly. Based upon this observation we
chose t =?120 ms for investigating EEG-based classification. From Table 3 we see that
FD works well with the preprocssed data, but as dimensionality increases the performance
breaks down. k-NN is not successful at all. The reason for the failure is that the variance in
the discriminating directions is much smaller that the variance in other directions. So using
the euclidean metric is not an appropirate similarity measure for this purpose. All regularized discriminative classifiers attain comparable results. For preprocessed data a very low
3 We used the implementation LIBSVM by Chang and Lin, available along with other implementations from
! .
60
12
classification error [%]
?120 ms?
50
10
40
8
30
6
20
4
10
0
?1000
2
EMG
EEG
?800
?600
?400
?200
0 [ms]
0
?200
?100
0 [ms]
Figure 4: Comparison of EEG (<5 Hz, mc, SFD) and EMG based classification with respect to the
endpoint of the classification interval. The right panel gives a details view: -230 to 50 ms.
error rate between 3% and 4% can be reached without a significant difference between the
competing methods. For the classification of raw data the error is roughly twice as high.
The concept of seeking sparse solution vectors allows SFD to cope very well with the high
dimensional raw data. Even though the error is twice as high compared to the the minimum
error, this result is very interesting, because it does not rely on preprocessing. So the SFD
approach may be highly useful for online situations, when no precursory experiments are
available for tuning the preprocessing.
The comparison of EEG- and EMG-based classification in Figure 4 demonstrates the rapid
response capability of our system: 230 ms before the actual keystroke the classification rate
exceeds 90%. To assess this result it has to be recalled that movements were performed
spontaneously. At ?120 ms, when the EMG derived classifier is still close to chance, EEG
based classification becomes already very stable with less than 3.5% errors. Interpreting
the last result in the sense of a 2AFC gives an information transfer rate of 60/2.1B ? 22.9
[bits/min], where B = log2 N + p log2 p+(1? p) log2 (1?p/N?1) is the number of bits per selection from N = 2 choices with success probability p = 1 ? 0.035 (under some uniformity
assumptions).
Classification in sliding windows. The second phase is an important step towards online classification of endogeneous brain signals. We have to refrain from using event timing
information (e.g., of keystrokes) in the test set. Accordingly, classification has to be performed in sliding windows and the classifier does not know in what time relation the given
signals are to the event?maybe there is even no event. Technically classification could be
done as before, as the trained classifiers can be applied to the feature vectors calculated
from some arbitrary window. But in practice this is very likely to lead to unreliable results
since those classifiers are highly specialized to signals that have a certain time relation to
the response. The behavior of the classifier elsewhere is uncertain. The typical way to
make classification more robust to time shifted signals is jittered training. In our case we
used 4 windows for each trial, ending at -240, -160, -80 and 0 ms relative to the response
(i.e., we get four feature vectors from each trial).
Movement detection and pseudo-online classification. Detecting upcoming events is
a crucial point in online analysis of brain signals in an unforced condition. To accomplish
this, we employ a second classifier that distinguishes movement events from the ?rest?.
Figures 5 and 6 display the continuous classifier output w> x + b (henceforth called graded)
for left/right and movement/rest distinction, respectively. For Figure 5, a classifier was
trained as described above and subsequently applied to windows sliding over unseen test
samples yielding ?traces? of graded classifier outputs. After doing this for several train/test
set splits, the borders of the shaded tubes are calculated as 5 and 95 percentile values of
1
0.5
0
?0.5
?1
?1.5
1.5
?1000
?750
?500
?250
0
250
500
750
right
left
[ms]
Figure 5: Graded classifier output for left/right
distinctions.
1
0
?1
?2
2
?1000
?750
?500
?250
0
250
median
90
percentile
10,
5?95
perc.
tube
500
750
[ms]
Figure 6: Graded classifier output for movement
detection in endogenous brain signals.
those traces, thin lines are at 10 and 90 percent, and the thick line indicates the median.
At t =?100 ms the median for right events in Figure 5 is approximately 0.9, i.e., applying
the classifier to right events from the test set yielded in 50% of the cases an output greater
0.9 (and in 50% an output less than 0.9). The corresponding 10-percentile line is at 0.25
which means that the output to 90% of the right events was greater than 0.25. The second
classifier (Figure 6) was trained for class ?movement? on all trials with jitters as described
above and for class ?rest? in multiple windows between the keystrokes. The preprocessing
and classification procedure was the same as for left vs. right.
The classifier in Figure 5 shows a pronounced separation during the movement (preparation
and execution) period. In other regions there is an overlap or even crossover of the classifier
outputs of the different classes. From Figure 5 we observe that the left/right classifier alone
does not distinguish reliably between ?movement? and ?no movement? by the magnitude
of its output, which explains the need for a movement detector. The elevation for the left
class is a little less pronounced (e.g., the median is ?1 at t =0 ms compared to 1.2 for right
events) which is probably due to the fact that the subject is right-handed. The movement
detector in Figure 6 brings up the movement phase while giving (mainly) negative output to
the post movement period. This differentiation is not as decisive as desirable, hence further
research has to be pursued to improve on this. Nevertheless a pseudo-online BCI run on the
recorded data using a combination of the two classifiers gave the very satisfying result of
around 10% error rate. Taking this as a 3 classes choice (left, right, none) this corresponds
to an information transmission rate of 29 bits/min.
5 Concluding discussion
We gave an outline of our BCI system in the experimental context of voluntary self-paced
movements. Our approach has the potential for high bit rates, since (1) it works at a high
trial frequency, and (2) classification errors are very low. So far we have used untrained
individuals, i.e., improvement can come from appropriate training schemes to shape the
brain signals. The two-stage process of first a meta classification whether a movement is
about to take place and then a decision between left/right finger movement is very natural
and an important new feature of the proposed system. Furthermore, we reject only 0.6%
of the trials due to artifacts, so our approach seems ideally suited for the true, highly noisy
feedback BCI scenario. Finally, the use of state-of-the-art learning machines enables us not
only to achieve high decision accuracies, but also, as a by-product of the classification, the
few most prominent features that are found match nicely with physiological intuition: the
most salient information can be gained between 230?100 ms before the movement with a
focus on C3/C4 area, i.e., over motor cortices, cf. Figure 2.
There are clear perspectives for improvement in this BCI approach: our future research
activities will therefore focus on (a) projection techniques like ICA, (b) time-series approaches to capture the (non-linear) dynamics of the multivariate EEG signals, and (c)
construction of specially adapted kernel functions (SVM or kernel FD) in the spirit of, e.g.,
[17] to ultimately obtain a BCI feedback system with an even higher bit rate and accuracy.
Acknowledgements. We thank S. Harmeling, M. Kawanabe, J. Kohlmorgen, J. Laub,
S. Mika, G. R?tsch, R. Vig?rio and A. Ziehe for helpful discussions.
References
[1] J. J. Vidal, ?Toward direct brain-computer communication?, Annu. Rev. Biophys., 2: 157?180,
1973.
[2] N. Birbaumer, N. Ghanayim, T. Hinterberger, I. Iversen, B. Kotchoubey, A. K?bler, J. Perelmouter, E. Taub, and H. Flor, ?A spelling device for the paralysed?, Nature, 398: 297?298,
1999.
[3] B. O. Peters, G. Pfurtscheller, and H. Flyvbjerg, ?Automatic Differentiation of Multichannel
EEG Signals?, IEEE Trans. Biomed. Eng., 48(1): 111?116, 2001.
[4] J. R. Wolpaw, D. J. McFarland, and T. M. Vaughan, ?Brain-Computer Interface Research at the
Wadsworth Center?, IEEE Trans. Rehab. Eng., 8(2): 222?226, 2000.
[5] W. D. Penny, S. J. Roberts, E. A. Curran, and M. J. Stokes, ?EEG-based cummunication: a
pattern recognition approach?, IEEE Trans. Rehab. Eng., 8(2): 214?215, 2000.
[6] J. D. Bayliss and D. H. Ballard, ?Recognizing Evoked Potentials in a Virtual Environment?,
in: S. A. Solla, T. K. Leen, and K.-R. M?ller, eds., Advances in Neural Information Processing
Systems, vol. 12, 3?9, MIT Press, 2000.
[7] S. Makeig, S. Enghoff, T.-P. Jung, and T. J. Sejnowski, ?A Natural Basis for Efficient BrainActuated Control?, IEEE Trans. Rehab. Eng., 8(2): 208?211, 2000.
[8] W. Lang, O. Zilch, C. Koska, G. Lindinger, and L. Deecke, ?Negative cortical DC shifts preceding and accompanying simple and complex sequential movements?, Exp. Brain Res., 74(1):
99?104, 1989.
[9] R. Q. Cui, D. Huter, W. Lang, and L. Deecke, ?Neuroimage of voluntary movement: topography
of the Bereitschaftspotential, a 64-channel DC current source density study?, Neuroimage, 9(1):
124?134, 1999.
[10] R. Beisteiner, P. Hollinger, G. Lindinger, W. Lang, and A. Berthoz, ?Mental representations
of movements. Brain potentials associated with imagination of hand movements?, Electroencephalogr. Clin. Neurophysiol., 96(2): 183?193, 1995.
[11] K. P. Bennett and O. L. Mangasarian, ?Robust Linear Programming Discrimination of two
Linearly Inseparable Sets?, Optimization Methods and Software, 1: 23?34, 1992.
[12] G. R?tsch, T. Onoda, and K.-R. M?ller, ?Soft Margins for AdaBoost?, 42(3): 287?320, 2001.
[13] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, Wiley & Sons, 2nd edn., 2001.
[14] S. Mika, G. R?tsch, and K.-R. M?ller, ?A mathematical programming approach to the Kernel
Fisher algorithm?, in: T. K. Leen, T. G. Dietterich, and V. Tresp, eds., Advances in Neural
Information Processing Systems 13, 591?597, MIT Press, 2001.
[15] ?ILOG Solver, ILOG CPLEX 6.5 Reference Manual?,
, 1999.
[16] V. Vapnik, The nature of statistical learning theory, Springer Verlag, New York, 1995.
[17] K.-R. M?ller, S. Mika, G. R?tsch, K. Tsuda, and B. Sch?lkopf, ?An Introduction to KernelBased Learning Algorithms?, IEEE Transactions on Neural Networks, 12(2): 181?201, 2001.
| 2030 |@word blankertz1:1 trial:27 neurophysiology:1 briefly:1 norm:2 loading:1 seems:1 duda:1 nd:1 simulation:1 eng:4 pressed:1 cp2:1 contains:1 series:1 chervonenkis:1 tuned:1 franklin:1 past:1 existing:1 reaction:1 current:1 ida:1 lang:3 issuing:1 tetraplegic:1 shape:1 enables:1 motor:20 designed:1 v:4 alone:1 pursued:1 selected:3 device:5 cp3:1 discrimination:1 accordingly:2 xk:5 short:2 filtered:2 mental:1 detecting:1 contribute:1 location:3 c6:1 windowed:1 along:2 c2:1 mathematical:2 differential:1 direct:1 agcl:1 laub:1 behavioral:2 acquired:1 inter:2 ica:1 expected:1 behavior:1 indeed:1 rapid:1 growing:1 multi:2 brain:21 roughly:1 detects:1 automatically:1 actual:2 little:2 str:1 window:7 kohlmorgen:1 becomes:1 ller1:1 solver:1 maximizes:1 panel:1 hemispherical:1 what:3 kind:1 ca1:1 ag:1 differentiation:2 freie:1 pseudo:4 synchronisation:1 every:2 universit:1 classifier:24 scaled:1 demonstrates:1 control:10 makeig:1 before:11 timing:5 referenced:1 vig:1 modulation:1 approximately:1 might:1 chose:3 twice:2 mika:3 evoked:1 relaxing:1 shaded:2 co:1 averaged:1 harmeling:1 spontaneously:1 practice:1 wolpaw:2 procedure:2 danger:1 area:2 empirical:2 crossover:1 significantly:1 thought:1 projection:2 attain:1 pre:1 intention:1 refers:2 downsampled:1 reject:1 get:1 close:2 selection:8 context:3 risk:3 applying:1 vaughan:1 disturbed:1 equivalent:2 map:1 demonstrated:2 center:1 resembling:1 go:1 attention:1 occipital:2 duration:1 unforced:1 his:1 transmit:1 imagine:1 trigger:1 construction:1 user:3 programming:3 edn:1 us:2 curran:1 rfd:3 satisfying:1 recognition:1 located:1 continues:1 electromyogram:1 std:1 predicts:1 capture:1 calculate:1 graz:1 ensures:1 region:1 solla:1 movement:38 trade:2 decrease:1 yk:5 substantial:1 benjamin:2 intuition:1 environment:1 ideally:1 tsch:4 dynamic:1 ultimately:1 trained:5 uniformity:1 technically:1 upon:1 efficiency:1 f2:1 basis:1 neurophysiol:1 various:1 finger:11 regularizer:1 train:1 forced:1 sejnowski:1 detected:2 precedes:1 neurophysics:1 klaus:1 hyper:1 larger:1 bci:17 browser:1 bereitschaftspotential:2 unseen:1 transform:1 exogeneous:1 noisy:2 bler:1 online:11 propose:1 product:1 deecke:2 rehab:3 relevant:2 aligned:3 rapidly:1 achieve:1 pronounced:2 crossvalidation:2 inbuilt:1 electrode:3 transmission:2 produce:1 measured:1 nearest:2 progress:1 strong:1 somatosensory:2 come:1 direction:2 foot:1 thick:2 filter:2 f4:1 subsequently:1 human:2 enable:1 translating:1 virtual:1 bin:2 explains:1 f1:1 generalization:1 suitability:1 elevation:1 strictly:1 accompanying:1 around:3 normal:1 exp:1 mapping:1 optimizer:1 consecutive:1 inseparable:1 conceiving:1 purpose:1 favorable:1 applicable:1 label:3 healthy:1 curio2:1 electroencephalogr:1 sfd:7 mit:2 interfacing:1 aim:1 rather:2 command:7 unspecific:1 derived:1 focus:3 properly:1 improvement:2 indicates:1 mainly:1 contrast:2 sense:1 am:1 posteriori:1 rio:1 helpful:1 nn:3 typically:1 initially:1 relation:2 cp6:1 selects:1 germany:3 biomed:1 issue:2 classification:38 priori:1 art:2 spatial:1 neuen:1 wadsworth:1 field:1 once:1 f3:1 nicely:1 having:2 shaped:1 optimising:1 afc:2 thin:1 mimic:1 future:1 report:2 stimulus:1 contaminated:1 serious:1 others:1 employ:1 distinguishes:1 few:1 bodily:1 individual:2 phase:4 cplex:2 detection:2 interest:1 fd:5 highly:5 investigate:2 intra:1 evaluation:1 inbetween:1 yielding:1 predefined:1 paralysed:2 tuple:1 capable:1 respective:1 euclidean:2 circle:1 tsuda:1 re:1 obscured:1 uncertain:1 tongue:1 increased:1 handed:1 earlier:1 soft:3 kekul:1 ca3:1 recognizing:1 successful:1 conducted:1 optimally:1 stored:1 perelmouter:1 emg:9 jittered:1 accomplish:1 density:1 international:1 grand:1 discriminating:1 ghanayim:1 off:2 picking:1 imagery:3 central:2 recorded:3 tube:2 vigilance:1 perc:1 positivity:1 henceforth:1 hinterberger:1 imagination:3 potential:9 converted:1 de:1 sec:3 coefficient:1 onset:4 decisive:1 later:1 break:2 view:1 performed:2 endogenous:1 doing:1 reached:1 competitive:1 hindenburgdamm:1 start:1 option:1 capability:1 contribution:1 ass:1 minimize:1 square:1 accuracy:7 responded:1 variance:4 yield:2 blink:1 sensori:1 generalize:1 lkopf:1 raw:6 manages:1 therewith:1 none:4 mc:4 kotchoubey:1 detector:2 reach:1 manual:2 ed:2 definition:1 failure:1 sensorimotor:2 frequency:3 involved:1 acquisition:1 associated:2 monitored:1 gain:1 auditory:1 sampled:1 logical:1 knowledge:1 dimensionality:1 positioned:1 actually:2 reflecting:1 back:2 higher:1 day:1 adaboost:1 response:8 erd:1 formulation:2 done:5 though:1 leen:2 furthermore:1 just:1 stage:1 bilaterally:1 until:1 hand:6 horizontal:1 brings:1 artifact:5 reveal:1 bcis:1 laterality:2 effect:1 dietterich:1 concept:2 true:2 consisted:1 regularization:3 hence:2 spatially:1 during:1 self:6 rhythm:1 percentile:3 coincides:1 m:21 prominent:1 palais:1 outline:2 electroencephalogram:1 interface:2 interpreting:1 percent:1 cp4:1 mangasarian:1 common:1 specialized:1 overlearned:1 cpz:1 stork:1 birbaumer:2 endpoint:1 cerebral:1 discussed:2 belong:1 resting:1 measurement:2 significant:1 taub:1 tuning:1 automatic:1 session:2 similarly:1 language:1 stable:1 cortex:8 impressive:1 similarity:1 etc:1 multivariate:2 perspective:1 moderate:1 driven:1 hemisphere:1 scenario:3 keyboard:3 certain:2 initiation:1 verlag:1 meta:1 binary:3 success:1 refrain:1 minimum:1 greater:2 relaxed:1 preceding:1 employed:1 maximize:2 period:2 ller:4 signal:27 sliding:3 full:1 multiple:1 desirable:1 enghoff:1 exceeds:1 instruct:1 match:1 long:1 lin:1 concerning:2 post:1 hart:1 variant:5 involving:1 regression:1 patient:4 metric:1 kernel:4 cz:1 c1:1 addition:1 addressed:1 interval:1 median:4 source:1 crucial:1 sch:1 electrooculogram:1 rest:3 specially:1 posse:1 flor:1 probably:1 subject:18 hz:13 veto:1 spirit:1 split:1 enough:1 easy:1 fft:1 wn:1 gave:2 competing:1 regarding:1 shift:1 favour:2 unprocessed:2 whether:2 unnatural:1 effort:1 peter:1 york:1 speaking:1 cp5:1 repeatedly:1 matlab:1 proceed:1 gabriel:1 useful:1 clear:1 maybe:1 band:6 induces:1 svms:4 processed:2 multichannel:1 fz:1 deliberate:1 cp1:1 shifted:1 sign:1 estimated:1 per:1 vol:1 group:1 key:6 four:1 salient:1 nevertheless:1 drawn:1 preprocessed:2 regulization:1 libsvm:1 button:1 merely:2 run:1 jitter:1 flexor:1 ca2:1 arrive:1 place:1 separation:1 decision:6 comparable:2 bit:7 layer:1 internet:1 bound:1 paced:5 distinguish:1 display:1 correspondence:1 fold:2 yielded:1 activity:4 scalp:5 strength:1 adapted:2 placement:1 precisely:1 constraint:1 comparision:1 bp:6 software:1 speed:2 min:5 chair:1 concluding:1 performing:1 format:1 combination:2 cui:1 smaller:1 son:1 appealing:1 rev:1 berthoz:1 restrains:1 taken:1 eventually:1 mechanism:1 know:1 fed:1 available:3 vidal:1 apply:1 observe:1 kawanabe:1 appropriate:1 alternative:3 denotes:1 cf:2 opportunity:1 log2:3 iversen:1 clin:1 pushing:1 calculating:2 exploit:1 giving:1 graded:4 upcoming:2 seeking:1 objective:2 intend:1 question:1 already:1 primary:2 spelling:1 surrogate:1 klinikum:1 distance:2 thank:1 berlin:3 lateral:1 majority:2 separating:1 whom:1 discriminant:7 reason:2 toward:1 besides:1 o1:1 index:2 length:1 kk:1 ratio:1 downsampling:1 minimizing:3 setup:3 executed:1 robert:2 localised:1 negative:3 trace:2 implementation:3 reliably:3 perform:1 vertical:2 observation:1 ilog:2 displayed:1 voluntary:5 situation:3 extended:1 variability:1 communication:1 stokes:1 dc:3 smoothed:1 arbitrary:1 competence:1 c3:5 c4:3 recalled:1 potsdam:2 learned:1 distinction:2 trans:4 able:1 mcfarland:1 usually:1 pattern:3 challenge:1 program:1 built:1 max:1 power:2 event:11 overlap:1 natural:5 typing:4 regularized:2 rely:1 arm:1 scheme:1 improve:1 brief:2 eye:1 fraunhofer:1 negativity:1 coupled:1 extract:1 eog:1 tresp:1 review:1 prior:3 acknowledgement:1 relative:4 loss:2 expect:1 topography:1 generation:1 interesting:2 filtering:1 mounted:1 degree:2 consistent:1 classifying:2 heavy:1 translation:1 periodicity:1 course:1 elsewhere:1 jung:1 last:2 free:1 keeping:1 offline:1 allow:1 neighbor:2 focussed:1 taking:3 sparse:5 penny:1 feedback:4 dimension:3 cortical:3 withhold:1 calculated:3 ending:1 autoregressive:2 sensory:1 c5:1 adaptive:1 preprocessing:10 projected:4 far:1 cope:1 transaction:1 approximate:1 keep:2 dealing:1 categorised:1 unreliable:1 correlating:1 active:1 overfitting:2 sat:1 investigating:1 consuming:1 discriminative:2 neurology:1 continuous:1 table:4 flyvbjerg:1 ballard:1 nature:2 channel:14 learn:1 transfer:1 robust:2 onoda:1 eeg:20 investigated:2 complex:2 untrained:1 linearly:1 border:1 noise:3 subsample:1 convey:1 bayliss:1 depicts:1 slow:2 wiley:1 pfurtscheller:2 formalization:1 neuroimage:2 position:3 explicit:1 minute:2 keystroke:9 down:1 annu:1 specific:2 er:1 physiological:2 svm:2 vapnik:2 sequential:1 gained:1 magnitude:1 execution:3 biophys:1 cursor:3 margin:5 rejection:2 suited:1 depicted:1 likely:1 visual:1 set2:1 chang:1 springer:1 ch:1 corresponds:1 determines:1 chance:2 sorted:1 marked:1 lvq:1 loosing:1 goal:2 towards:3 fisher:8 considerable:1 bennett:1 determined:3 specifically:2 typical:3 hyperplane:1 total:1 called:1 pas:4 engaged:1 experimental:6 caz:1 preand:1 ziehe:1 support:3 scp:1 latter:1 kernelbased:1 frontal:2 preparation:5 dept:1 |
1,131 | 2,031 | Correlation Codes in Neuronal Populations
Maoz Shamir and Haim Sompolinsky
Racah Institute of Physics and Center for Neural Computation,
The Hebrew University
of
Jerusalem,
Jerusalem
91904, Israel
Abstract
Population codes often rely on the tuning of the mean responses to the
stimulus parameters. However, this information can be greatly suppressed by long range correlations. Here we study the efficiency of coding information in the second order statistics of the population responses.
We show that the Fisher Information of this system grows linearly with
the size of the system. We propose a bilinear readout model for extracting information from correlation codes, and evaluate its performance in
discrimination and estimation tasks. It is shown that the main source of
information in this system is the stimulus dependence of the variances of
the single neuron responses.
1 Introduction
Experiments in the last years have shown that in many cortical areas, the fluctuations in
the responses of neurons to external stimuli are significantly correlated [1, 2, 3, 4], raising important questions regarding the computational implications of neuronal correlations.
Recent theoretical studies have addressed the issue of how neuronal correlations affect the
efficiency of population coding [4, 5, 6]. It is often assumed that the information about
stimuli is coded mainly in the mean neuronal responses, e.g., in the tuning of the mean
firing rates, and that by averaging the tuned responses across large populations, an accurate estimate can be obtained despite the significant noise in the single neuron responses.
Indeed, for uncorrelated neurons the Fisher Information of the population is extensive [7];
namely, it increases linearly with the number of neurons in the population. Furthermore, it
has been shown that this extensive information can be extracted by relatively simple linear
readout mechanisms [7, 8]. However, it was recently shown [6] that positive correlations
which vary smoothly with space may drastically suppress the information in the mean responses. In particular, the Fisher Information of the system saturates to a finite value as
the system size grows. This raises questions about the computational utility of neuronal
population codes.
Neuronal population responses can represent information in the higher order statistics of
the responses [3], not only in their means. In this work, we study the accuracy of coding information in the second order statistics. We call such schemes correlation codes.
Specifically, we assume that the neuronal responses obey multivariate Gaussian statistics
governed by a stimulus-dependent correlation matrix. We ask whether the Fisher Information of such a system is extensive even in the presence of strong correlations in the neuronal
noise. Secondly, we inquire how information in the second order statistics can be efficiently
extracted.
2 Fisher Information of a Correlation Code
Our model consists of a system of neurons that code a 2D angle ,
. Their
stochastic
response
is
given
by
a
vector
of
activities
where
is
the
activity
of the
-th neuron in the presence of a stimulus , and is distributed according to a multivariate
Gaussian distribution
"! $#&% )' ( (+* ,.-0/ 21 (3* 04
(1)
Here is the mean activity of the -th
neuron and its dependence on is usually referred
to as the tuning curve of the neuron; / is the correlation matrix; and is a normalization
constant. Here we shall limit ourselves to the case of multiplicative modulation of the
correlations. Specifically we use
5 7
6 8 9
09 6 :<5 ;
=6
(2)
( >6
(
(
(
>
5 ;
76 5 ; ?>
> 6 @
76)A @
=6 !B#%DC
(3)
E F
(
(
K>
(4)
9
8 9 ?>
: G !$#&% CGHBI J LNM F
where and E are the correlation strength
and correlation length respectively; L defines
the
>
tuning width
the angle at which the variance of the -th
of the correlations; and denotes
is
shown
in Fig. 1. It is important to note that the
neuron, 9 M , is maximal. An 5 example
76
variance adds a contribution to
which is larger than the contribution of the smooth part
of the correlations. For reasons that will become clear below, we write,
5
=O 6
where
5 7
6 : G 5P
7O 6 : A P5
Q .@
=6
5
Q
denotes the smooth part of the correlation matrix and
diagonal part, which in the example of Eqs. (2)-(4)
is
5P
Q G ( ,9 M
(5)
the discontinuous
(6)
A useful measure of the accuracy of a population code is the Fisher Information (FI). In
the case of uncorrelated populations it is well known that FI increases linearly with system size [7], indicating that the accuracy of the population coding improves as the system
size is increased. Furthermore, it has been shown that relatively simple, linear schemes
can provide reasonable readout models for extracting the information in uncorrelated populations [8]. In the case of a correlated multivariate Gaussian distribution, FI is given as
, where
R R SUT.V2W A R:XZY0[0[
R SUT.V2W * .\ - / : 1 * ,\
(7)
^
R:XZY0[0[
M
(8)
*
* N] / 1 / 0.\_
where \ and / \ denote derivatives of and / with respect to , respectively. The form
* roles. First they control the
of these terms reveals that in general the correlations play two
: (note the dependence of
efficiency
of
the
information
encoded
in
the
mean
activities
R SUT.V2W on 5 ). Secondly, / provides an additional source of information
bdstimc e , it
R&XZY0[0[ ). When the correlations are independent of the stimulus,
` 9
about
a the
ulus (
gf
E
was shown [6] that positive correlations,
, with long correlation length, ih ,
?=0o
C(?,?)
?=?60o
o
?=60
?=?120o
?180
5 K>
?120
?=120o
?60
?
0
> >
(
60
120
180
[deg]
(
>
E
Figure 1: The stimulus-dependent
correlation matrix, Eqs. (2)-(4), depicted
as a function
of two angles,
, where
and
. Here,
,
and
.
L
i
cause the saturation of FI to a finite limit at large . This implies that in the presence
of such correlations, population averaging cannot overcome the noise even in large networks. This analysis however, [6], did not take into account stimulus-dependent correlations, which is the topic of the present work.
RNXZY,[ [ , Eq. (8), we find it useful to write
R XZY0[0[ R Q A R O
(9)
where
R Q C 5 5
Q
Q , \ M
(10)
F
5
Q ,
( stimulus-dependent
is FI of an uncorrelated population
with
variance which equals
R
:
R
Z
X
0
Y
0
[
[
R
Q . Evaluating
and scales linearly with ; R O
terms for the multiplicative
R R Q these
O
. Furthermore, numerical evaluamodel, Eq. (2), we find that R is positive, so that
tion of this term shows that O saturates at large to a small finite value, so that for large
R:XZY0[ [ R Q
M > C 9 \ ??> > M
(11)
F
9
R XZY0[0[ increases linearly with and is equal, for
as shown in Fig. 2. We thus conclude that
R
large , to the FI of variance coding namely to of an independent population in which
Analyzing the
dependence of
information is encoded in their activity variances.
Since in our system the information is encoded in the second order statistics of the population responses, it is obvious that linear readouts are inadequate. This raises the question of
whether there are relatively simple nonlinear readout models for such systems. In the next
sections we will study bilinear readouts and show that they are useful models for extracting
information from correlation codes.
3 A Bilinear Readout for Discrimination Tasks
M
A @
Ina two-interval
discrimination task the system is given two sets of neuronal activities
generated by two proximal stimuli and
and must infer which stimulus generated which activity. The Maximum-Likelihood (ML) discrimination yields the
?3
1
1
x 10
?2
[deg ]
?2
[deg ]
0.8
0.6
s
0.5
J
Jcorr
0.4
0.2
0
0
200
400
600
800
N
1000
0
0
200
400
600
800
1000
N
R XZY0[0[
Figure 2: (a) Fisher Information,
, of the stimulus-dependent correlations, Eqs. (2)(4), as a function of the number of neurons in the system. In (b) we show the difference
between
the full FI and the contribution of the diagonal term,
- as defined by Eq. (9).
Here
,
and
. Note the different scales in (a) and (b).
E
L
probability of error given by
discriminability equals
\
RO
\ , where
d 1 M
\ @ R
`1
M
and the
(12)
It has been previously shown that in the case of uncorrelated populations with mean coding,
the optimal linear readouts achieves the Maximum-Likelihood discrimination performance
in large N [7].
*
In order to isolate the properties of correlation coding we will assume that no information
is coded in the average firing rates of the neurons, and take hereafter. We suggest a
bilinear readout as a simple generalization of the linear readout to correlation codes. In a
discrimination task the bilinear readout makes a decision according to the sign of
(13)
(
A
where a
=6
76
6 (
M 6 M
A @
76 5 1 7
\ 6
decision refers to
. Maximizing the signal-to-noise ratio of this
rule, the optimal bilinear discriminator (OBD) matrix is given by
(14)
Using the optimal weights to evaluate the discrimination error we obtain that in large the
performance of the OBD saturates the ML performance, Eq. (12). Thus, since FI of this
model increases linearly with the size of the system, the discriminability increases as
.
/
Since the correlation matrix depends on the stimulus, , the OBD matrix, Eq. (14), will
also be stimulus dependent. Thus, although the OBD is locally efficient, it cannot be used
as such as a global efficient readout.
4 A Bilinear Readout for Angle Estimation
4.1 Optimal bilinear readout for estimation
To study the global performance of bilinear readouts we investigate bilinear readouts which
minimize the square error of estimating the angle averaged over the whole
range of . For
convenience we use complex notation for the encoded angle, and write as the estimator
of
`
. Let
76
76
=6
K 26
(15)
where
are stimulus independent complex weights. We define the optimal bilinear estimator (OBE) as the set of weights
that minimizes on average the quadratic estimation
error of an unbiased estimator. This error is given by
G
@ M (
is the Lagrange multiplier of the constraint
(16)
where
. In general,
it is impossible
to find a perfectly unbiased estimator for a continuously varied stimulus,
using a finite number of weights. However, in the case of angle estimation, we can employ
the underlying rotational symmetry to generate such an estimator. For this we use the
this case one can show that the Lagrange
symmetry of the correlation matrix, Eq. (2). In
multipliers have the simple form of
, and the OBE weight matrix is in the form
of
)
76 ?>
(
?> ) ( ?>
`
> 6 !B#%DC
>
A > 6
(17)
F
(
A d . This form of a readout matrix, Eq. (17),
?> G > and
where
K> can
guarantees that the estimator will be unbiased. Using these symmetry properties,
be written in the following form (for even )
M 1 W
( > (
^
b
A
?>
@
> "
(18)
W
HBI J
_
K> . These numerical results (Fig. 3 (a))
Figure 3 (a) presents an exampleK>of the function
also suggest
> that the function is mainly determined by a few harmonics plus a delta
peak at . Below we will use this fact to study simpler forms of bilinear readout.
Further analysis of the OBE performance in the large
totic result
@ : M 1
Q 5 ?>
M
Q
Q 5 ?>
M
Q 0 M
limit yields the following asymp-
M
c >
,
( `
(19)
Figure 3 (b) shows the numerical calculation of the OBE error (open circles) as a function
of . The dashed line is the asymptotic behavior, given by Eq. (19). The dotted line is
the Creamer-Rao bound. From the graph one can see that the estimation efficiency of this
readout grows linearly with the size of the system, , but is lower than the bound.
4.2 Truncated bilinear readout
K>
Motivated by the simple structure of the optimal readout matrix observed in Fig. 3 (a), we
studied a bilinear readout of the form of Eqs. (17) and (18) with
which has a delta
function peak at the origin plus a few harmonics. Restricting the number of harmonics to
relatively
small integers, we evaluated numerically the optimal values of the coefficients
for large systems. Surprisingly we found that for small and large , these coef
ficients
approach a value which is independent of the specifics of the model and equals
, yielding a bilinear weight matrix of the form
W
W
(
B
76
@
=6
(
^ b ( K>
( > 6 _ !B#%DC
>
A > 6
W HBI J
F
(20)
Figure 4 shows the numerical results for the squared average error of this readout for several values of ! and
. The results of Fig. 4 show that for a given the
8
0.1
(b)
(a)
w(?)
J
???2
0.05
?2
[deg ]
0
?2
0
0
0
2
?
100
200
300
400
N
?>
Figure 3: (a) Profile of
, Eq. (17), for the OBE with
. (b) Numerical evaluation
of one over the squared estimation error, for the optimal bilinear readout in the multiplicative modulation model (open circles).
The dashed line is the asymptotic behavior, given by
Eq. (19). Here
, for the optimal bilinear readout in the
multiplicative
modulation model. The dotted line is the FI bound. In these simulations
,
and
were used.
@ & M M
L
OV-
E
inverse square error initially increases linearly with
but saturates in the limit of large
. However, the saturation size
increases rapidly with . The precise form of
depends on the specifics of the correlation model. For the exponentially decaying
. Figure 4 shows that for this range of
correlations assumed in Eq. (2), we find
, and
the deviations of the inverse square error from linearity are small. Thus, in
the regime
,
is given by the asymptotic behavior, Eq. (19),
shown by the dashed line.
OV-
OV-
O V - @ : M
We thus conclude that the OBE (with unlimited ) will generate an inverse square estimawith a coefficient given by Eq. (19), and that
tion error which increases linearly with
this value can be achieved for reasonable values of
by an approximate bilinear weight
matrix, of the form of Eq. (20), with small . The asymptotic result, Eq. (19), is smaller
than the optimal value given by the full FI, Eq. (11), see Fig. 4 (dotted line). In fact, it is
equal to the error of an independent population with a variance which equals
and a
quadratic population vector readout of the form
5
Q
M `
(21)
It is important to note that in the presence of correlations, the quadratic readout of Eq.
(21) is very inefficient, yielding a finite error for large as shown in Fig. 4 (line marked
?quadratic?).
5 Discussion
To understand the reason for the simple form
of the approximately optimal bilinear weight
matrix, Eq. (20), we rewrite Eq. (15) with
of Eq. (20) as
?
`
@ 7
6 (
60
W
(22)
`
W 1
1
26
(23)
0.5
1/ ??2 deg?2
0.4
J
p=3
0.3
p=2
0.2
p=1
0.1
quadratic
0
0
500
1000
1500
2000
N
E L
Figure 4: Inverse square estimation error ofthe
finite- approximation for the OBE, Eq.
(20). Solid curves from the bottom
. The bottom curve is
. The dashed
line is the asymptotic behavior,
given
by
Eq.
(19).
The
FI
bound
is
shown
by
the dotted
,
and
were used.
line. For the simulations
Comparing this form with Eq. (21) it can be seen that our readout is in the form of a bilinear
population vector in which the lowest Fourier modes of the response vector have been
removed. Retaining only the high Fourier modes in the response profile suppresses the
cross-correlations between the different components of the residual responses because
the underlying correlations have smooth spatial dependence, whose power is concentrated
mostly in the low Fourier modes. On the other hand, the information contained in the
variance is not removed because the variance contains a discontinuous spatial component,
. In other words, the variance of a correlation profile which has only high Fourier
modes can still preserve its slowly varying components. Thus, by projecting out the low
Fourier modes of the spatial responses the spatial correlations are suppressed but the information in the response variance is retained.
5
Q
This interpretation of the bilinear readout implies that although all the elements of the
correlation matrix depend on the stimulus, only the stimulus dependence of the diagonal
elements is important. This important conclusion is borne out by our theoretical results
concerning the performance of the system. As Eqs. (11) and (19) show, the asymptotic
performance of both the full FI as well as that of the OBE are equivalent to those of an
.
uncorrelated population with a stimulus dependent variance which equals
5
Q
Although we have presented results here concerning a multiplicative model of correlations,
we have studied other models of stimulus dependent correlations. These studies indicate
that the above conclusions apply to a broad class of populations in which information is
encoded in the second order statistics of the responses. Also, for the sake of clarity we
have assumed here that the mean responses are untuned,
. Our studies have shown
that adding tuned mean inputs does not modify the picture since the smoothly varying
positive correlations greatly suppress the information embedded in the first order statistics.
*
The relatively simple form of the readout Eq. (22) suggests that neuronal hardware may
be able to extract efficiently information embedded in local populations of cells whose
noisy responses are strongly correlated, provided that the variances of their responses are
significantly tuned to the stimulus. This latter condition is not too restrictive, since tuning
of variances of neuronal firing rates to stimulus and motor variables is quite common in the
nervous system.
Acknowledgments
This work was partially supported by grants from the Israel-U.S.A. Binational Science
Foundation and the Israeli Science Foundation. M.S. is supported by a scholarship from
the Clore Foundation.
References
[1] E. Fetz, K. Yoyoma and W. Smith, Cerebral Cortex (Plenum Press, New York,
1991).
[2] D. Lee, N.L. Port, W. Kruse and A.P. Georgopoulos, J. Neurosci. , 1161 (1998).
[3] E.M. Maynard, N.G. Hatsopoulos, C.L. Ojakangas, B.D. Acuna, J.N. Sanes, R.A.
Normann, and J.P. Donoghue, J. Neurosci. 19, 8083 (1999).
[4] E. Zohary, M.N. Shadlen and W.T. Newsome, Nature , 140 (1994).
[5] L.F. Abbott and P. Dayan, Neural Computation , 91 (1999).
[6] H. Sompolinsky, H. Yoon, K. Kang and M. Shamir, Phys. Rev. E,
, 051904 (2001);
H. Yoon and H. Sompolinsky, Advances in Neural Information Processing Systems
11 (pp. 167). Kearns M.J, Solla S.A and Cohn D.A, Eds., (Cambridge, MA: MIT
Press, 1999).
[7] S. Seung and H. Sompolinsky, Proc. Natl. Acad. Sci. USA , 10794 (1993).
[8] E. Salinas and L.F. Abbott, J. Comp. Neurosci. , 89 (1994).
| 2031 |@word open:2 simulation:2 solid:1 ulus:1 contains:1 hereafter:1 tuned:3 comparing:1 must:1 written:1 numerical:5 motor:1 discrimination:7 nervous:1 smith:1 provides:1 simpler:1 become:1 consists:1 indeed:1 behavior:4 zohary:1 provided:1 estimating:1 notation:1 underlying:2 linearity:1 lowest:1 israel:2 sut:3 minimizes:1 suppresses:1 guarantee:1 ro:1 control:1 grant:1 positive:4 local:1 modify:1 limit:4 acad:1 bilinear:21 despite:1 analyzing:1 fluctuation:1 firing:3 modulation:3 approximately:1 plus:2 discriminability:2 studied:2 suggests:1 range:3 averaged:1 acknowledgment:1 area:1 significantly:2 word:1 refers:1 suggest:2 acuna:1 cannot:2 convenience:1 impossible:1 equivalent:1 center:1 maximizing:1 jerusalem:2 rule:1 estimator:6 racah:1 population:24 plenum:1 shamir:2 play:1 origin:1 element:2 totic:1 observed:1 role:1 bottom:2 p5:1 yoon:2 inquire:1 readout:30 sompolinsky:4 solla:1 removed:2 hatsopoulos:1 seung:1 raise:2 ov:3 rewrite:1 depend:1 efficiency:4 ojakangas:1 salina:1 whose:2 encoded:5 larger:1 quite:1 statistic:8 noisy:1 propose:1 maximal:1 rapidly:1 maoz:1 clore:1 eq:29 strong:1 implies:2 indicate:1 discontinuous:2 stochastic:1 generalization:1 secondly:2 obe:8 vary:1 achieves:1 estimation:8 proc:1 mit:1 gaussian:3 varying:2 likelihood:2 mainly:2 greatly:2 dependent:8 dayan:1 initially:1 issue:1 retaining:1 spatial:4 equal:7 broad:1 stimulus:23 employ:1 few:2 preserve:1 ourselves:1 investigate:1 evaluation:1 yielding:2 natl:1 implication:1 accurate:1 asymp:1 circle:2 theoretical:2 hbi:2 increased:1 rao:1 newsome:1 deviation:1 inadequate:1 too:1 proximal:1 peak:2 lee:1 physic:1 continuously:1 squared:2 slowly:1 borne:1 external:1 derivative:1 inefficient:1 account:1 coding:7 coefficient:2 depends:2 multiplicative:5 tion:2 decaying:1 contribution:3 minimize:1 square:5 accuracy:3 variance:14 efficiently:2 yield:2 comp:1 phys:1 coef:1 ed:1 pp:1 obvious:1 ask:1 improves:1 higher:1 response:22 evaluated:1 strongly:1 furthermore:3 correlation:41 hand:1 cohn:1 nonlinear:1 maynard:1 defines:1 v2w:3 mode:5 grows:3 usa:1 unbiased:3 multiplier:2 width:1 harmonic:3 recently:1 fi:12 common:1 binational:1 exponentially:1 cerebral:1 interpretation:1 numerically:1 significant:1 cambridge:1 tuning:5 cortex:1 add:1 multivariate:3 recent:1 seen:1 additional:1 kruse:1 signal:1 dashed:4 full:3 infer:1 smooth:3 calculation:1 cross:1 long:2 concerning:2 coded:2 represent:1 normalization:1 achieved:1 cell:1 addressed:1 interval:1 source:2 isolate:1 call:1 extracting:3 integer:1 presence:4 affect:1 perfectly:1 regarding:1 donoghue:1 whether:2 motivated:1 utility:1 york:1 cause:1 useful:3 clear:1 locally:1 concentrated:1 hardware:1 generate:2 dotted:4 sign:1 delta:2 lnm:1 write:3 shall:1 clarity:1 abbott:2 graph:1 year:1 angle:7 inverse:4 reasonable:2 decision:2 bound:4 haim:1 quadratic:5 activity:7 strength:1 constraint:1 georgopoulos:1 unlimited:1 sake:1 fourier:5 relatively:5 according:2 across:1 smaller:1 suppressed:2 rev:1 projecting:1 previously:1 mechanism:1 apply:1 obey:1 denotes:2 restrictive:1 scholarship:1 question:3 dependence:6 diagonal:3 sci:1 topic:1 reason:2 code:10 length:2 retained:1 ratio:1 rotational:1 hebrew:1 mostly:1 suppress:2 neuron:12 finite:6 truncated:1 saturates:4 precise:1 dc:3 varied:1 namely:2 extensive:3 discriminator:1 raising:1 kang:1 israeli:1 able:1 usually:1 below:2 regime:1 saturation:2 power:1 rely:1 residual:1 scheme:2 picture:1 extract:1 normann:1 gf:1 asymptotic:6 embedded:2 untuned:1 foundation:3 shadlen:1 port:1 uncorrelated:6 obd:4 surprisingly:1 last:1 supported:2 drastically:1 understand:1 institute:1 fetz:1 distributed:1 overcome:1 curve:3 cortical:1 evaluating:1 approximate:1 deg:5 ml:2 global:2 reveals:1 assumed:3 conclude:2 nature:1 symmetry:3 complex:2 did:1 main:1 linearly:9 neurosci:3 whole:1 noise:4 profile:3 neuronal:11 fig:7 referred:1 sanes:1 governed:1 specific:2 ih:1 restricting:1 adding:1 smoothly:2 depicted:1 lagrange:2 contained:1 partially:1 extracted:2 ma:1 marked:1 fisher:7 specifically:2 determined:1 averaging:2 kearns:1 indicating:1 ficients:1 latter:1 evaluate:2 correlated:3 |
1,132 | 2,032 | Stochastic Mixed-Signal VLSI Architecture for
High-Dimensional Kernel Machines
Roman Genov and Gert Cauwenberghs
Department of Electrical and Computer Engineering
Johns Hopkins University, Baltimore, MD 21218
roman,gert @jhu.edu
Abstract
A mixed-signal paradigm is presented for high-resolution parallel innerproduct computation in very high dimensions, suitable for efficient implementation of kernels in image processing. At the core of the externally
digital architecture is a high-density, low-power analog array performing
binary-binary partial matrix-vector multiplication. Full digital resolution
is maintained even with low-resolution analog-to-digital conversion, owing to random statistics in the analog summation of binary products. A
random modulation scheme produces near-Bernoulli statistics even for
highly correlated inputs. The approach is validated with real image data,
and with experimental results from a CID/DRAM analog array prototype
in 0.5 m CMOS.
1 Introduction
Analog computational arrays [1, 2, 3, 4] for neural information processing offer very large
integration density and throughput as needed for real-time tasks in computer vision and
pattern recognition [5]. Despite the success of adaptive algorithms and architectures in reducing the effect of analog component mismatch and noise on system performance [6, 7],
the precision and repeatability of analog VLSI computation under process and environmental variations is inadequate for some applications. Digital implementation [10] offers
absolute precision limited only by wordlength, but at the cost of significantly larger silicon
area and power dissipation compared with dedicated, fine-grain parallel analog implementation, e.g., [2, 4].
The purpose of this paper is twofold: to present an internally analog, externally digital architecture for dedicated VLSI kernel-based array processing that outperforms purely digital
approaches with a factor 100-10,000 in throughput, density and energy efficiency; and to
provide a scheme for digital resolution enhancement that exploits Bernoulli random statistics of binary vectors. Largest gains in system precision are obtained for high input dimensions. The framework allows to operate at full digital resolution with relatively imprecise
analog hardware, and with minimal cost in implementation complexity to randomize the
input data.
The computational core of inner-product based kernel operations in image processing and
pattern recognition is that of vector-matrix multiplication (VMM) in high dimensions:
(1)
with -dimensional
input vector , -dimensional output vector
, and
matrix
to
elements
. In artificial neural networks, the matrix elements
correspond
synapses, between neurons. The elements also represent templates
weights,
or
in a vector quantizer [8], or support vectors in a support vector machine [9]. In
what
follows we concentrate on VMM computation which dominates inner-product based 1
kernel computations for high vector dimensions.
2 The Kerneltron: A Massively Parallel VLSI Computational Array
2.1 Internally Analog, Externally Digital Computation
The approach combines the computational efficiency of analog array processing with the
precision of digital processing and the convenience of a programmable and reconfigurable
digital interface.
The digital representation is embedded in the analog array architecture, with inputs presented in bit-serial fashion, and matrix elements stored
locally
in bit-parallel form:
$#
"
!
#
& (' ! &
%
&
(2)
(3)
#
*
& . ! / &
)
, -
%
+
&
decomposing (1) into:
(4)
with binary-binary VMM partials: #
0#
#
!&
! / &
! ' 2
1
(5)
The key is to compute and accumulate the binary-binary partial products (5) using an analog VMM array, and to combine the quantized results in the digital domain according to (4).
#
Digital-to-analog conversion
at the input interface is inherent in the bit-serial implementation, and row-parallel
analog-to-digital converters (ADCs) are used at the output interface
!
&
/
to quantize
. A 512 128 array prototype using CID/DRAM cells is shown in
Figure 1 (a).
2.2 CID/DRAM Cell and Array
0#
$#
# element
The unit cell in the analog array combines a CID computational
[12,
! 13] with a
DRAM storage element. The cell stores one bit
of
a
matrix
element
, performs
!
' !&
a one-quadrant binary-binary multiplication of
and
in (5), and accumulates
1
Radial basis kernels with 354 -norm can also be formulated in inner product format.
RS(i)m
Vout(i)m
M1
M2
M3
CID
DRAM (i)
w mn
x(j)
n
RS(i)m x(j)n Vout(i)m
0
Vdd/2
Vdd
Write
0
Vdd/2
Vdd
Compute
(a)
0
Vdd/2
Vdd
(b)
Figure 1: (a) Micrograph of the Kerneltron prototype,
containing an array of
CID/DRAM cells, and a row-parallel bank of
flash ADCs. Die size is
in 0.5 m CMOS technology. (b) CID computational
cell with integrated DRAM storage.
Circuit diagram, and charge transfer diagram for active write and compute operations.
#
$#
and indices.# The circuit diagram
and operation
the result across cells with common
of the cell are given in Figure
1
(b).
An
array
of
cells
thus
performs
(unsigned)
binary
!
' !&
! /&
multiplication (5) of matrix
and vector
yielding
, for values of in
parallel across the array, and values of in sequence over time.
The cell contains three MOS transistors connected$# in series as depicted in Figure 1 (b).
$#
Transistors
M1 and M2 comprise a dynamic random-access
memory (DRAM) cell, with
!
switch
M1
controlled
by
Row
Select
signal
.
When
activated,
the binary quantity
!
is written in the form of charge (either or 0) stored under the gate of M2.
Transistors M2 and M3 in turn comprise a charge injection device (CID), which by virtue of
charge conservation moves electric charge between two potential wells in a non-destructive
manner [12, 13, 14].
The charge left under the gate of M2 can only be redistributed between the two CID tran$# is
sistors, M2 and M3. An active charge transfer from M2 to M3 can only occur if there
#
non-zero
charge stored, and if the potential on the gate of M2 drops below that of M3! [12].
This
and
' ! & condition implies a logical AND, i.e., unsigned binary multiplication, of
. The multiply-and-accumulate operation is then completed by capacitively sensing
the amount of charge transferred onto the electrode of M3, the output summing node. To
this end, the voltage on the output line, left floating after being pre-charged to ,
is observed.
When the charge transfer is active, the cell contributes a change in voltage
"! where# # "! is the total capacitance on the output line across cells.
The total response is thus
' ! & proportional to the number of actively transferring cells. After
deactivating the input
, the transferred charge returns to the storage node M2. The
CID computation is non-destructive and intrinsically reversible [12], and DRAM refresh is
only required to counteract junction and subthreshold leakage.
The bottom diagram in Figure 1 (b) depicts the charge transfer timing diagram for write
#
0#
and compute operations in the case when both
!
' !&
and
are of logic level 1.
2.3 System-Level Performance
Measurements on the 512 128-element analog array and other fabricated prototypes show
a dynamic range of 43 dB, and a computational cycle of 10 s with power
consumption of
1
.
50 nW per cell. The size of the CID/DRAM cell is 8 45 with
The overall system resolution is limited by the precision in the quantization of the outputs
from the analog array. Through digital postprocessing, two bits are gained over the resolution of the ADCs used [15], for a total system resolution of 8 bits. Larger resolutions can
be obtained by accounting for the statistics of binary terms in the addition, the subject of
the next section.
3 Resolution Enhancement Through Stochastic Encoding
# be achieved (as if computed
Since the analog inner product (5) is discrete, zero error can
digitally) by matching the quantization levels of the ADC
with
each of the
discrete
! / &
levels in the inner product. Perfect reconstruction of
from the quantized output, for
.
bits, assumes the combined effect of noise and
an overall resolution of
nonlinearity in the analog array and the ADC is within one LSB (least significant bit). For
large arrays, this places stringent requirements on analog precision and ADC resolution,
.
.
The implicit assumption is that all quantization levels are (equally) needed. A straightforward study of the statistics of the inner product, below, reveals that this is poor use of
available resources.
3.1 Bernoulli Statistics
#
0#
In what follows
signed,
' ! & we assume
! rather than unsigned, binary values for inputs and
weights,
and
. This translates to exclusive-OR (XOR), rather
than AND, multiplication on the analog array, an operation that can be easily accomplished
with the CID/DRAM
# architecture by differentially coding input and stored bits using twice
the number $of
columns
and unit cells.
#
#
#
$#
'
!&
the (XOR) product
For input
bits
! ' ! & that are Bernoulli distributed (i.e., fair coin flips),
!
! /&
terms
in (5) are Bernoulli distributed, regardless of
. Their sum
#
thus follows a binomial distribution
! / & ! #" %$'&)( & (
(6)
#+* 1$101 * , which in the Central Limit -,/. approaches a normal
#
1 ,
with &
distribution with zero mean and variance . In other words, for random inputs
10 . in high
dimensions
the
active
range
(or
standard
deviation)
of
the
inner-product
is
, a factor
10 .
smaller than the full range .
In principle, this allows to relax the effective resolution of the ADC. However, any reduction in conversion range will result in a small but non-zero probability of overflow. In
practice, the risk of overflow can be reduced to negligible levels with a few additional bits
in the ADC conversion range. An alternative strategy is to use a variable resolution ADC
which expands the conversion range on rare occurences of overflow. 2
2
Or, with stochastic input encoding, overflow detection could initiate a different random draw.
Inner Product
20
10
0
?10
?20
0.2
0.4
0.6
0.8
0.4
0.6
0.8
Output Voltage (V)
50
40
Count
30
20
10
0
(a)
0.2
Output Voltage (V)
(b)
Figure 2: Experimental results from CID/DRAM analog array. (a) Output voltage on the
sense line computing exclusive-or inner product of 64-dimensional stored and presented
binary vectors. A variable number of active bits is summed at different locations in the
array by shifting the presented bits. (b) Top: Measured output and actual inner product
for 1,024 samples of Bernoulli distributed pairs of stored and presented vectors. Bottom:
Histogram of measured array outputs.
3.2 Experimental Results
While the reduced range of the analog inner product supports lower ADC resolution in
terms of number of quantization levels, it requires low levels of mismatch and noise so that
the discrete levels can be individually resolved, near the center of the distribution. To verify
this, we conducted the following experiment.
Figure 2 shows the measured outputs on one row of 128 CID/DRAM cells, configured differentially to compute signed binary (exclusive-OR) inner products of stored and presented
binary vectors in 64 dimensions. The scope trace in Figure 2 (a) is obtained by storing all
bits, and shifting a sequence of input bits that differ with the stored bits by
bits.
The left and right segment of the scope trace correspond to different selections of active
bit locations along the array that are maximally disjoint, to indicate a worst-case mismatch
scenario. The measured and actual inner products in Figure 2 (b) are obtained by storing and presenting 1,024 pairs of random binary vectors. The histogram shows a clearly
resolved, discrete binomial distribution for the observed analog voltage.
For very large arrays, mismatch and noise may pose a problem in the present implementation with floating sense line. A sense amplifier
with virtual ground on the sense line and
.
feedback capacitor optimized to the
range would provide a simple solution.
10
3.3 Real Image Data
Although most randomly selected patterns do not correlate with any chosen template, patterns from the real world tend to correlate, and certainly those that are of interest to kernel
computation 3 . The key is stochastic encoding of the inputs, as to randomize the bits presented to the analog array.
3
This observation, and the binomial distribution for sums of random bits (6), forms the basis for
the associative recall in a Kanerva memory.
500
500
400
400
Count
600
Count
600
300
300
200
200
100
100
0
?1000
?500
0
Inner Product
500
0
?1000
1000
10
10
9
9
8
8
7
7
6
6
Count
11
Count
11
5
4
3
3
2
2
0
?1000
0
500
1000
0
500
1000
Inner Product
5
4
1
?500
1
?500
0
Inner Product
500
1000
0
?1000
?500
#Inner Product
! /&
Figure 3: Histograms of partial binary inner products
for 256 pairs of randomly
selected 32 32 pixel segments of Lena. Left: with unmodulated 8-bit image data for
both vectors. Right: with 12-bit modulated stochastic encoding of one of
the two vectors.
Top: all bit planes and . Bottom: most significant bit (MSB) plane,
.
Randomizing an informative input while retaining the information is a futile goal, and we
are content with a solution that approaches the ideal performance within observable bounds,
and with reasonable cost in implementation. Given that ?ideal? randomized inputs relax the
ADC resolution by . bits, they necessarily reduce the wordlenght of the output by
the same. To account for the lost bits in the range of the output, it is necessary to increase
the range of the ?ideal? randomized input by the same number of bits.
10
.
One possible stochastic encoding scheme that restores the range is
-fold oversampling
of the input through (digital) delta-sigma modulation. This is a workable solution; however
we propose one that is simpler and less costly to implement.
For each -bit input compo .
in the range
nent , pick a random
integer
, and subtract it to produce
.
a modulated input
with additional bits. It can
be shown that for
inner product for is off at most by
worst-case
deterministic
inputs
the
mean
of
the
.
from the origin. The desired
inner products for are retrieved by digitally adding
and . The random offset can be chosen once, so
the inner products obtained for
$#
its inner product with the templates can be pre-computed upon initializing or programming
the array. The implementation cost is thus limited to component-wise subtraction of !
and , achieved using one full adder cell, one bit register, and ROM storage of the
bits for every column of the array.
10
10
Figure 3 provides a proof of principle, using image data selected at random from Lena.
12-bit stochastic encoding of the 8-bit image, by subtracting a random variable in a range
15 times larger than the image, produces the desired binomial distribution for the partial bit
inner products, even for the most significant bit (MSB) which is most highly correlated.
4 Conclusions
We presented an externally digital, internally analog VLSI array architecture suitable for
real-time kernel-based neural computation and machine learning in very large dimensions,
such as image recognition. Fine-grain massive parallelism and distributed
. memory, in an array of 3-transistor CID/DRAM cells, provides a throughput of
binary MACS (mul
tiply accumulates per second) per Watt of power in a 0.5 m process.
A simple stochastic
encoding scheme relaxes precision requirements in the analog implementation by one bit
for each four-fold increase in vector dimension, while retaining full digital overall system
resolution.
Acknowledgments
This research was supported by ONR N00014-99-1-0612, ONR/DARPA N00014-00-C0315, and NSF MIP-9702346. Chips were fabricated through the MOSIS service.
References
[1] A. Kramer, ?Array-based analog computation,? IEEE Micro, vol. 16 (5), pp. 40-49, 1996.
[2] G. Han, E. Sanchez-Sinencio, ?A general purpose neuro-image processor architecture,? Proc.
of IEEE Int. Symp. on Circuits and Systems (ISCAS?96), vol. 3, pp 495-498, 1996
[3] F. Kub, K. Moon, I. Mack, F. Long, ? Programmable analog vector-matrix multipliers,? IEEE
Journal of Solid-State Circuits, vol. 25 (1), pp. 207-214, 1990.
[4] G. Cauwenberghs and V. Pedroni, ?A Charge-Based CMOS Parallel Analog Vector Quantizer,?
Adv. Neural Information Processing Systems (NIPS*94), Cambridge, MA: MIT Press, vol. 7,
pp. 779-786, 1995.
[5] Papageorgiou, C.P, Oren, M. and Poggio, T., ?A General Framework for Object Detection,? in
Proceedings of International Conference on Computer Vision, 1998.
[6] G. Cauwenberghs and M.A. Bayoumi, Eds., Learning on Silicon: Adaptive VLSI Neural Systems, Norwell MA: Kluwer Academic, 1999.
[7] A. Murray and P.J. Edwards, ?Synaptic Noise During MLP Training Enhances Fault-Tolerance,
Generalization and Learning Trajectory,? in Advances in Neural Information Processing Systems, San Mateo, CA: Morgan Kaufman, vol. 5, pp 491-498, 1993.
[8] A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Norwell, MA:
Kluwer, 1992.
[9] V. Vapnik, The Nature of Statistical Learning Theory, 2nd ed., Springer-Verlag, 1999.
[10] J. Wawrzynek, et al., ?SPERT-II: A Vector Microprocessor System and its Application to Large
Problems in Backpropagation Training,? in Advances in Neural Information Processing Systems, Cambridge, MA: MIT Press, vol. 8, pp 619-625, 1996.
[11] A. Chiang, ?A programmable CCD signal processor,? IEEE Journal of Solid-State Circuits,
vol. 25 (6), pp. 1510-1517, 1990.
[12] C. Neugebauer and A. Yariv, ?A Parallel Analog CCD/CMOS Neural Network IC,? Proc. IEEE
Int. Joint Conference on Neural Networks (IJCNN?91), Seattle, WA, vol. 1, pp 447-451, 1991.
[13] V. Pedroni, A. Agranat, C. Neugebauer, A. Yariv, ?Pattern matching and parallel processing
with CCD technology,? Proc. IEEE Int. Joint Conference on Neural Networks (IJCNN?92),
vol. 3, pp 620-623, 1992.
[14] M. Howes, D. Morgan, Eds., Charge-Coupled Devices and Systems, John Wiley & Sons, 1979.
[15] R. Genov, G. Cauwenberghs ?Charge-Mode Parallel Architecture for Matrix-Vector Multiplication,? IEEE T. Circuits and Systems II, vol. 48 (10), 2001.
| 2032 |@word compression:1 norm:1 nd:1 r:2 accounting:1 pick:1 solid:2 reduction:1 contains:1 series:1 outperforms:1 written:1 refresh:1 john:2 grain:2 informative:1 drop:1 msb:2 selected:3 device:2 plane:2 nent:1 core:2 compo:1 chiang:1 quantizer:2 quantized:2 node:2 location:2 provides:2 deactivating:1 simpler:1 along:1 combine:3 symp:1 manner:1 lena:2 actual:2 circuit:6 what:2 kaufman:1 adc:8 fabricated:2 every:1 expands:1 charge:15 unit:2 internally:3 unmodulated:1 negligible:1 engineering:1 timing:1 service:1 limit:1 despite:1 accumulates:2 encoding:7 modulation:2 signed:2 twice:1 mateo:1 limited:3 range:13 acknowledgment:1 yariv:2 practice:1 lost:1 implement:1 agranat:1 backpropagation:1 area:1 jhu:1 significantly:1 matching:2 imprecise:1 pre:2 radial:1 quadrant:1 word:1 convenience:1 onto:1 selection:1 storage:4 unsigned:3 risk:1 deterministic:1 charged:1 center:1 straightforward:1 regardless:1 resolution:17 occurences:1 m2:9 array:30 gert:2 variation:1 massive:1 programming:1 origin:1 element:8 recognition:3 observed:2 bottom:3 electrical:1 initializing:1 worst:2 connected:1 cycle:1 adv:1 digitally:2 complexity:1 dynamic:2 vdd:6 segment:2 purely:1 upon:1 efficiency:2 basis:2 easily:1 resolved:2 darpa:1 chip:1 joint:2 effective:1 artificial:1 larger:3 relax:2 statistic:6 associative:1 sequence:2 transistor:4 reconstruction:1 redistributed:1 tran:1 product:26 propose:1 subtracting:1 differentially:2 seattle:1 enhancement:2 electrode:1 requirement:2 produce:3 cmos:4 perfect:1 object:1 pose:1 measured:4 edward:1 implies:1 indicate:1 differ:1 concentrate:1 owing:1 stochastic:8 stringent:1 virtual:1 generalization:1 summation:1 ground:1 normal:1 ic:1 nw:1 scope:2 mo:1 purpose:2 proc:3 individually:1 largest:1 mit:2 clearly:1 rather:2 voltage:6 validated:1 neugebauer:2 bernoulli:6 sense:4 integrated:1 transferring:1 vlsi:6 pixel:1 overall:3 retaining:2 restores:1 integration:1 summed:1 comprise:2 once:1 throughput:3 roman:2 vmm:4 inherent:1 few:1 randomly:2 micro:1 floating:2 iscas:1 amplifier:1 detection:2 interest:1 mlp:1 highly:2 multiply:1 workable:1 certainly:1 yielding:1 activated:1 norwell:2 partial:5 necessary:1 poggio:1 wordlength:1 capacitively:1 desired:2 mip:1 minimal:1 column:2 cost:4 mac:1 deviation:1 rare:1 inadequate:1 conducted:1 stored:8 randomizing:1 combined:1 density:3 international:1 randomized:2 off:1 hopkins:1 central:1 containing:1 return:1 actively:1 account:1 potential:2 coding:1 int:3 configured:1 register:1 cauwenberghs:4 parallel:11 xor:2 variance:1 moon:1 correspond:2 subthreshold:1 repeatability:1 vout:2 trajectory:1 processor:2 synapsis:1 ed:3 synaptic:1 energy:1 pp:9 destructive:2 proof:1 gain:1 intrinsically:1 logical:1 recall:1 response:1 maximally:1 implicit:1 adder:1 reversible:1 mode:1 gray:1 effect:2 verify:1 multiplier:1 during:1 maintained:1 die:1 presenting:1 performs:2 dissipation:1 dedicated:2 interface:3 postprocessing:1 image:10 wise:1 common:1 analog:33 m1:3 kluwer:2 accumulate:2 silicon:2 measurement:1 significant:3 cambridge:2 nonlinearity:1 access:1 han:1 retrieved:1 massively:1 store:1 scenario:1 n00014:2 verlag:1 binary:21 success:1 onr:2 fault:1 accomplished:1 morgan:2 additional:2 kub:1 subtraction:1 paradigm:1 signal:5 ii:2 full:5 academic:1 offer:2 long:1 serial:2 equally:1 controlled:1 neuro:1 vision:2 histogram:3 represent:1 kernel:8 oren:1 achieved:2 cell:20 addition:1 fine:2 baltimore:1 diagram:5 lsb:1 howe:1 spert:1 operate:1 subject:1 tend:1 db:1 sanchez:1 capacitor:1 integer:1 near:2 ideal:3 relaxes:1 switch:1 architecture:9 converter:1 inner:23 reduce:1 prototype:4 translates:1 programmable:3 amount:1 locally:1 hardware:1 reduced:2 nsf:1 oversampling:1 delta:1 disjoint:1 per:3 write:3 discrete:4 vol:10 key:2 four:1 micrograph:1 mosis:1 sum:2 counteract:1 place:1 reasonable:1 draw:1 bit:36 bound:1 fold:2 occur:1 ijcnn:2 performing:1 injection:1 relatively:1 format:1 transferred:2 department:1 according:1 watt:1 poor:1 across:3 smaller:1 son:1 wawrzynek:1 mack:1 kanerva:1 resource:1 turn:1 count:5 needed:2 initiate:1 flip:1 gersho:1 end:1 junction:1 operation:6 decomposing:1 available:1 alternative:1 coin:1 gate:3 assumes:1 binomial:4 top:2 completed:1 ccd:3 exploit:1 murray:1 overflow:4 leakage:1 move:1 capacitance:1 quantity:1 randomize:2 strategy:1 exclusive:3 md:1 costly:1 cid:15 enhances:1 consumption:1 rom:1 index:1 trace:2 sigma:1 dram:14 implementation:9 conversion:5 neuron:1 observation:1 pair:3 required:1 optimized:1 nip:1 below:2 pattern:5 mismatch:4 parallelism:1 memory:3 shifting:2 power:4 suitable:2 innerproduct:1 mn:1 scheme:4 technology:2 coupled:1 genov:2 bayoumi:1 multiplication:7 embedded:1 mixed:2 proportional:1 pedroni:2 digital:19 principle:2 bank:1 storing:2 row:4 supported:1 template:3 absolute:1 distributed:4 tolerance:1 feedback:1 dimension:8 world:1 adaptive:2 san:1 correlate:2 observable:1 logic:1 active:6 reveals:1 summing:1 conservation:1 nature:1 transfer:4 ca:1 contributes:1 quantize:1 futile:1 necessarily:1 papageorgiou:1 electric:1 domain:1 microprocessor:1 noise:5 fair:1 mul:1 depicts:1 fashion:1 wiley:1 precision:7 externally:4 reconfigurable:1 sensing:1 offset:1 virtue:1 dominates:1 quantization:5 vapnik:1 adding:1 gained:1 subtract:1 depicted:1 springer:1 environmental:1 ma:4 goal:1 formulated:1 kramer:1 flash:1 twofold:1 content:1 change:1 reducing:1 total:3 experimental:3 m3:6 select:1 support:3 modulated:2 correlated:2 |
1,133 | 2,033 | Grouping and dimensionality reduction by
locally linear embedding
Marzia Polito
Division of Physics, Mathematics and Astronomy
California Institute of Technology
Pasadena, CA, 91125
polito @caltech.edu
Pietro Perona
Division of Engeneering and Applied Mathematics
California Institute of Technology
Pasadena, CA, 91125
perona@caltech.edu
Abstract
Locally Linear Embedding (LLE) is an elegant nonlinear
dimensionality-reduction technique recently introduced by Roweis
and Saul [2]. It fails when the data is divided into separate groups.
We study a variant of LLE that can simultaneously group the data
and calculate local embedding of each group. An estimate for the
upper bound on the intrinsic dimension of the data set is obtained
automatically.
1
Introduction
Consider a collection of N data points Xi E
]RD.
Suppose that , while the dimension
D is large, we have independent information suggesting that the data are distributed
on a manifold of dimension d < < D. In many circumstances it is beneficial to
calculate the coordinates Yi E ]Rd of the data on the lower-dimensional manifold,
both because the shape of the manifold may yield some insight in the process that
produced the data, and because it is cheaper to store and manipulate the data when
it is embedded in fewer dimensions. How can we compute such coordinates?
Principal component analysis (PCA) is a classical technique which works well when
the data lie close to a flat manifold [1]. Elegant methods for dealing with data that
is distributed on curved manifolds have been recently proposed [3, 2]. We study
one of them, Locally Linear Embedding (LLE) [2], by Roweis and Saul. While LLE
is not designed to handle data that are disconnected, i.e. separated into groups,
we show that a simple variation of the method will handle this situation correctly.
Furthermore, both the number of groups and the upper bound on the intrinsic
dimension of the data may be estimated automatically, rather than being given
a-priori.
2
Locally linear embedding
The key insight inspiring LLE is that, while the data may not lie close to a globally linear manifold, it may be approximately locally linear, and in this case each
point may be approximated as a linear combination of its nearest neighbors. The
coefficients of this linear combination carries the vital information for constructing
a lower-dimensional linear embedding.
More explicitly: consider a data set {Xd i=l...,N E ]RD. The local linear structure
can be easily encoded in a sparse N by N matrix W, proceeding as follows.
The first step is to choose a criterion to determine the neighbors of each point.
Roweis and Saul chose an integer number K and pick, for every point, the K points
nearest to it. For each point Xi then, they determine the linear combination of its
neighbors which best approximates the point itself. The coefficients of such linear
combinations are computed by minimizing the quadratic cost function:
f(W) =
L
N
IXi
-
L WijXj 12
(1)
j=1
while enforcing the constraints W ij = 0 if Xj is not a neighbor of Xi , and
L:.f=1 Wij = 1 for every i; these constraints ensure that the approximation of
Xi ~ Xi = L:.f=1 WijXj lies in the affine subspace generated by the K nearest
neighbors of Xi, and that the solution W is translation-invariant . This least square
problem may be solved in closed form [2].
The next step consists of calculating a set {Yih=1, ... ,N of points in ]Rd, reproducing
as faithfully as possible the local linear structure encoded in W. This is done
minimizing a cost function
N
<I>(Y) =
N
L IYi - L Wij Yjl2
(2)
i=1
j =1
To ensure the uniqueness of the solution two constraint are imposed: translation
invariance by placing the center of gravity of the data in the origin, i.e. L:i Yi = 0,
and normalized unit covariance of the
Yi's,
i.e.
tt L:~1 Yi
Q9
Yi
= I.
Roweis and Saul prove that <I>(Y) = tr(yT MY), where M is defined as
M
= (I -
wf (I -
W).
The minimum of the function <I>(Y) for the d-th dimensional representation is then
obtained with the following recipe. Given d, consider the d + 1 eigenvectors associated to the d + 1 smallest eigenvalues of the matrix M. Then discard the very
first one. The rows of the matrix Y whose columns are given by such d eigenvectors
give the desired solution. The first eigenvector is discarded because it is a vector
composed of all ones, with 0 as eigenvalue. As we shall see, this is true when the
data set is 'connected' .
2.1
Disjoint components
In LLE every data point has a set of K neighbors. This allows us to partition of
the whole data set X into K -connected components, corresponding to the intuitive
visual notion of different 'groups' in the data set.
We say that a partition X = UiUi is finer than a partition X = Uj 10 if every Ui
is contained in some 10. The partition in K -connected components is the finest
:
?............................
10
20
30
40
50
60
70
60
90
100
-020"'---------:::---;:;;-----O;---;;;------;';c------:::--~
Figure 1: (Top-left) 2D data Xi distributed along a curve (the index i increases
from left to right for convenience). (Top-right) Coordinates Yi of the same points
calculated by LLE with K = 10 and d = 1. The x axis represents the index
i and the y axis represents Yi. This is a good parametrization which recognizes
the intrinsically I-dimensional structure of the data. (Bottom-left) As above, the
data is now disconnected, i.e. points in different groups do not share neighbors.
(Bottom-right) One-dimensional LLE calculated on the data (different symbols used
for points belonging to the different groups). Notice that the Yi's are not a good
representation of the data any longer since they are constant within each group.
partition of the data set such that if two points have at least one neighbor in
common, or one is a neighbor of the other, then they belong to the same component.
Note that for any two points in the same component, we can find an ordered sequence of points having them as endpoints, such that two consecutive points have
at least one neighbor in common. A set is K -connected if it contains only one
K-connected component .
Consider data that is not K -connected, then LLE does not compute a good
parametrization, as illustrated in Figure 1.
2.2
Choice of d.
How is d chosen? The LLE method [2] is based on the assumption that d is known.
What if we do not know it in advance? If we overestimate d it then LLE behaves
pathologically.
Let us consider a straight line, drawn in 1R3 . Figure 2 shows what happens if d
is chosen equal to 1 and to 2. When the choice is 2 (right) then LLE 'makes up'
information and generates a somewhat arbitrary 2D curve.
As an effect of the covariance constraint, the representation curves the line, the
Figure 2: Coordinates Yi calculated for data Xi distributed along a straight line in
]RD = ]R3 when the dimension d is chosen as d = 1 (Left), and d = 2 (Right). The
index i is indicated along the x axis (Left) and along the 2D curve (Right).
curvature can be very high, and even locally we possibly completely lose the linear structure. The problem is, we chosed the wrong target dimension. The onedimensional LLE works in fact perfectly (see Figure 2, left).
PCA provides a principled way of estimating the intrinsic dimensionality of the
data: it corresponds to the number of large singular values of the covariance matrix
of the data. Is such an estimate possible with LLE as well?
3
Dimensionality detection: the size of the eigenvalues
In the example of Figure 2 the two dimensional representation of the data (d = 2)
is clearly the 'wrong' one, since the data lie in a one-dimensional linear subspace.
In this case the unit covariance constraint in minimizing the function <I>(Y) is not
compatible with the linear structure. How could one have obtained the correct
estimate of d? The answer is that d + 1 should be less or equal to the number of
eigenvalues of M that are close to zero.
Proposition 1. Assume that the data Xi E ]RD is K -connected and that it is
locally fiat, i.e. there exists a corresponding set Yi E ]Rd for some d > 0 such that
Yi = L: j Wij}j (zero-error approximation), the set {Yi} has rank d, and has the
origin as center of gravity: L:~1 Yi =
the matrix M. Then d < z .
o.
Call z the number of zero eigenvalues of
Proof. By construction the N vector composed of all 1 's is a zero-eigenvector of
M. Moreover, since the Yi are such that the addends of <I> have zero error, then the
matrix Y , which by hypothesis has rank d, is in the kernel of I - W and hence in
the kernel of M. Due to the center of gravity constraint, all the columns of Y are
orthogonal to the all 1 's vector. Hence M has at least d + 1 zero eigenvalues. D
Therefore, in order to estimate d, one may count the number z of zero eigenvalues
of M and choose any d < z. Within this range, smaller values of d will yield more
compact representations, while larger values of d will yield more expressive ones,
i.e. ones that are most faithful to the original data.
What happens in non-ideal conditions, i.e. when the data are not exactly locally
fiat , and when one has to contend with numerical noise? The appendix provides an
argument showing that the statement in the proposition is robust with respect to
,,'
,, '
,, '
,, '
,, '
,,'
,,'
,, '
,,'
,, '
,, '
10 ' 0
10"
,,'
2nd e igen value
10"
2nd c igc nvul uc
10 "
10 "
10"
lst eige nva] ue
10"
10"
0
0
.,lst c i'cnv? "'
"
Figure 3: (Left) Eigenvalues for the straight-line data Xi used for Figure 2. (Right)
Eigenvalues for the curve data shown in the top-left panel of Figure 1. In both cases
the two last eigenvalue are orders of magnitude smaller than the other eigenvalues,
indicating a maximal dimension d = 1 for the data.
noise, i.e. numerical errors and small deviations from the ideal locally flat data will
result in small deviations from the ideal zero-value of the first d + 1 eigenvalues,
where d is used here for the 'intrinsic' dimension of the data. This is illustrated in
Figure 3.
In Figure 4 we describe the successful application of the dimensionality detection
method on a data set of synthetically generated grayscale images.
4
LLE and grouping
In the first example (2.1) we pointed out the limits of LLE when applied to multiple
components of data. It appears then that a grouping procedure should always
preceed LLE. The data would be first split into its component groups, each one
of which should be then analyzed with LLE. A deeper analysis of the algorithm
though, suggests that grouping and LLE could actually be performed at the same
time.
Proposition 2. Suppose the data set {Xdi=l ,... ,N E ll~P is partitioned into m Kconnected components. Then there exists an m-dimensional eigenspace of M with
zero eigenvalue which admits a basis {vih=l,... ,m where the Vi have entries that are
either '1' or '0' . More precisely: each Vi corresponds to one of the groups of the
data and takes value V i ,j = 1 for j in the group, V i ,j = 0 for j not in the group.
Proof. Without loss of generality, assume that the indexing of the data X i is such
that the weight matrix W , and consequent ely the matrix M, are block-diagonal
with m blocks, each block corresponding to one of the groups of data. This is
achieved by a permutation of indices, which will not effect any further step of our
algorithm. As a direct consequence of the row normalization of W, each block of
M has exactly one eigenvector composed of all ones, with eigenvalue O. Therefore,
there is an m-dimensional eigenspace with eigenvalue 0, and there exist a basis of
it, each vector of which has value 1 on a certain component, 0 otherwise. D
Therefore one may count the number of connected components by computing the
eigenvectors of M corresponding to eigenvalue 0, and counting the number m of
those vectors Vi whose components take few discrete values (see Figure 6). Each
index i may be assigned to a group by clustering based on the value of Vl, ... , V m .
Figure 4: (Left) A sample from a data set of N=1000, 40 by 40 grayscale images,
each one thought as a point in a 1600 dimensional vector space. In each image,
a slightly blurred line separates a dark from a bright portion. The orientation of
the line and its distance from the center of the image are variable. (Middle) The
non-zero eigenvalues of M. LLE is performed with K=20. The 2nd and 3rd smallest
eigenvalues are of smaller size than the others, giving an upper bound of 2 on the
intrinsic dimension of the data set. (Right) The 2-dimensional LLE representation.
The polar coordinates, after rescaling, are the distance of the dividing line from the
center and its orientation.
".
Figure 5: The data set is analogous to the one used above (N =1000, 40 by 40
grayscale images, LLE performed with K=20). The orientation of the line dividing
the dark from the bright portion is now only allowed to vary in two disjoint intervals.
(Middle) The non-zero eigenvalues of M. (Left and Right) The 3rd and 5th (resp.
4th and 6th) eigenvectors of M are used for the LLE representation of the first (resp.
the second) K-component.
,,'
,, '
,, '
,,'
,, '
10'0
4th. 5th and 6th eigenvalues
10 "
10 "
10 "
1st. 2nd and 3rd eigenvalues
Figure 6: (Left) The last six eigenvectors of M for the broken parabola of Figure 1
shown, top to bottom, in reverse order of magnitude of the corresponding eigenvalue.
The x axis is associated to the index i. (Right) The eigenvalues of the same (log
scale). Notice that the last six are practically zero. The eigenvectors corresponding
to the three last eigenvalues have discrete values indicating that the data is split in
three groups. There are z =6 zero-eigenvalues indicating that the dimension of the
data is d:::; z/m - 1 = 1.
In the Appendix (A) we show that such a process is robust with respect to numerical
noise. It is also robust to small perturbations of the block-diagonal structure of M
(see Figure 7). This makes the use of LLE for grouping purposes convenient. Should
the K-connected components be completely separated, the partition would be easily
obtained via a more efficient graph-search algorithm.
The proof is carried out for ordered indices as in Fig. 3 but it is invariant under
index permutation.
The analysis of Proposition 1 may be extended to the dimension of each of the
m groups according to Proposition 2. Therefore, in the ideal case, we will find z
zero-eigenvalues of M which, together with the number m obtained by counting
the discrete-valued eigenvectors may be used to estimate the maximal d using z ~
m(d + 1). This behavior may be observed experimentally, see Figures 6 and 5.
5
Conclusions
We have examined two difficulties of the Locally Linear Embedding method [2] and
shown that, in a neighborhood of ideal conditions, they may be solved by a careful
exam of eigenvectors of the matrix M that are associated to very small eigenvalues.
More specifically: the number of groups in which the data is partitioned corresponds
to the number of discrete-valued eigenvectors, while the maximal dimension d of
the low-dimensional embedding may be obtained by dividing the number of small
eigenvalues by m and subtracting 1.
Both the groups and the low-dimensional embedding coordinates may be computed
from the components of such eigenvectors.
Our algorithms have mainly been tested on synthetically generated data. Further
investigation on real data sets is necessary in order to validate our theoretical results.
w'
,----~-~-~-~--~______,
w'
w'
w'
w'
w'
w'
3rd and 4th eigenvalues .
10 '?
10 "
10 "
10'"
1st and 2ndeige!\values
o~--=-----::------=-------=------,=--~
Figure 7: (Left) 2D Data Xi distributed along a broken parabola. Nevertheless,
for K=14, the components are not completely K-disconnected (a different symbol is
used for the neighbors of the leftmost point on the rightmost component). (Right)
The set of eigenvalues for M. A set of two almost-zero eigenvalues and a set of two
of small size are visible.
References
[1] C. Bishop, Neural Networks for Pattern Recognition, Oxford Univ. Press,
(1995).
[2] S. T. Roweis, L.K.Saul, Science, 290, p. 2323-2326, (2000).
[3] J. Tenenbaum , V. de Silva, J. Langford, Science, 290, p. 2319-2323, (2000).
A
Appendix
In Proposition 2 of Section 4 we proved that during the LLE procedure we can
automatically detect the number of K -connected components, in case there is no
noise. Similarly, in Proposition 1 of Section 3 we proved that under ideal conditions
(no noise, locally flat data), we can determine an estimate for the intrinsic dimension
of the data. Our next goal is to establish a certain robustness of these results in
the case there is numerical noise, or the components are not completely separated,
or the data is not exactly locally flat .
In general, suppose we have a non degenerate matrix A, and an orthonormal basis
of eigenvectors VI, ... , V m , with eigenvalues AI , ... Am. As a consequence of a small
perturbation of the matrix into A + dA, we will have eigenvectors Vi + dVi with
eigenvalues Ai + dAi' The unitary norm constraint makes sure that dVi is orthogonal to Vi and could be therefore written as dVi = L:k#i O'.ikVk. Using again the
orthonormality, one can derive expressions for the perturbations of Ai and Vi :
dAi
O'.ij (Ai - Aj)
< vi,dAvi >
< Vj,dAVi > .
This shows that if the perturbation dA has order E, then the perturbations dA and
are also of order E. Notice that we are not interested in perturbations O'.ij within
the eigenspace of eigenvalue 0, but rather those orthogonal to it, and therefore
Ai =j:. Aj.
O'.ij
| 2033 |@word middle:2 norm:1 nd:4 covariance:4 pick:1 yih:1 tr:1 carry:1 reduction:2 contains:1 rightmost:1 written:1 finest:1 visible:1 numerical:4 partition:6 shape:1 designed:1 davi:2 fewer:1 parametrization:2 provides:2 along:5 direct:1 consists:1 prove:1 behavior:1 globally:1 automatically:3 estimating:1 moreover:1 panel:1 eigenspace:3 what:3 marzia:1 eigenvector:3 astronomy:1 every:4 xd:1 gravity:3 exactly:3 wrong:2 unit:2 overestimate:1 local:3 limit:1 consequence:2 oxford:1 approximately:1 chose:1 examined:1 suggests:1 range:1 faithful:1 block:5 procedure:2 thought:1 convenient:1 convenience:1 close:3 imposed:1 center:5 yt:1 insight:2 orthonormal:1 embedding:9 handle:2 notion:1 coordinate:6 variation:1 analogous:1 resp:2 target:1 construction:1 suppose:3 ixi:1 hypothesis:1 origin:2 approximated:1 recognition:1 parabola:2 bottom:3 observed:1 solved:2 calculate:2 connected:10 principled:1 broken:2 ui:1 division:2 completely:4 basis:3 easily:2 lst:2 engeneering:1 separated:3 univ:1 describe:1 neighborhood:1 whose:2 encoded:2 larger:1 valued:2 say:1 otherwise:1 itself:1 sequence:1 eigenvalue:34 subtracting:1 maximal:3 degenerate:1 roweis:5 intuitive:1 validate:1 recipe:1 derive:1 exam:1 nearest:3 ij:4 dividing:3 correct:1 investigation:1 proposition:7 cnv:1 practically:1 vary:1 consecutive:1 smallest:2 purpose:1 uniqueness:1 polar:1 lose:1 faithfully:1 clearly:1 always:1 rather:2 rank:2 mainly:1 wf:1 detect:1 am:1 vl:1 pasadena:2 perona:2 wij:3 interested:1 orientation:3 priori:1 nva:1 uc:1 equal:2 having:1 placing:1 represents:2 others:1 few:1 composed:3 simultaneously:1 cheaper:1 detection:2 analyzed:1 necessary:1 orthogonal:3 desired:1 theoretical:1 column:2 cost:2 deviation:2 entry:1 successful:1 xdi:1 answer:1 my:1 st:2 physic:1 together:1 again:1 choose:2 possibly:1 preceed:1 rescaling:1 suggesting:1 de:1 coefficient:2 blurred:1 explicitly:1 ely:1 vi:8 performed:3 closed:1 portion:2 square:1 bright:2 yield:3 produced:1 finer:1 straight:3 addend:1 associated:3 proof:3 proved:2 intrinsically:1 dimensionality:5 fiat:2 actually:1 appears:1 done:1 though:1 generality:1 furthermore:1 langford:1 expressive:1 nonlinear:1 aj:2 indicated:1 effect:2 normalized:1 true:1 vih:1 orthonormality:1 hence:2 assigned:1 illustrated:2 ll:1 during:1 ue:1 chosed:1 criterion:1 leftmost:1 tt:1 silva:1 image:5 recently:2 common:2 behaves:1 endpoint:1 polito:2 belong:1 approximates:1 onedimensional:1 ai:5 rd:11 mathematics:2 similarly:1 pointed:1 iyi:1 longer:1 curvature:1 discard:1 reverse:1 store:1 certain:2 yi:14 caltech:2 minimum:1 dai:2 somewhat:1 wijxj:2 determine:3 multiple:1 divided:1 manipulate:1 variant:1 circumstance:1 kernel:2 normalization:1 achieved:1 interval:1 singular:1 sure:1 elegant:2 integer:1 call:1 unitary:1 counting:2 ideal:6 synthetically:2 vital:1 split:2 xj:1 perfectly:1 six:2 pca:2 expression:1 eigenvectors:12 dark:2 locally:12 tenenbaum:1 inspiring:1 exist:1 notice:3 estimated:1 disjoint:2 correctly:1 discrete:4 shall:1 group:19 key:1 nevertheless:1 drawn:1 graph:1 pietro:1 almost:1 appendix:3 bound:3 q9:1 quadratic:1 constraint:7 precisely:1 flat:4 generates:1 argument:1 according:1 combination:4 disconnected:3 belonging:1 beneficial:1 smaller:3 slightly:1 partitioned:2 happens:2 invariant:2 indexing:1 r3:2 count:2 know:1 robustness:1 original:1 top:4 clustering:1 ensure:2 recognizes:1 igc:1 calculating:1 giving:1 uj:1 establish:1 classical:1 diagonal:2 pathologically:1 subspace:2 distance:2 separate:2 manifold:6 enforcing:1 index:8 minimizing:3 statement:1 contend:1 upper:3 discarded:1 curved:1 situation:1 extended:1 perturbation:6 reproducing:1 arbitrary:1 introduced:1 california:2 pattern:1 difficulty:1 technology:2 igen:1 axis:4 carried:1 embedded:1 loss:1 permutation:2 affine:1 dvi:3 share:1 translation:2 row:2 compatible:1 last:4 lle:25 deeper:1 institute:2 saul:5 neighbor:11 sparse:1 distributed:5 curve:5 dimension:14 calculated:3 collection:1 compact:1 dealing:1 xi:11 grayscale:3 search:1 robust:3 ca:2 constructing:1 da:3 vj:1 whole:1 noise:6 allowed:1 fig:1 fails:1 lie:4 bishop:1 showing:1 symbol:2 admits:1 consequent:1 grouping:5 intrinsic:6 exists:2 magnitude:2 visual:1 contained:1 ordered:2 corresponds:3 goal:1 careful:1 experimentally:1 specifically:1 principal:1 invariance:1 indicating:3 tested:1 |
1,134 | 2,034 | Receptive field structure of flow detectors
for heading perception
Jaap A. B e intema
Dept. Zoology & Neurobiology
Ruhr University Bochum, Germany, 44780
beintema@neurobiologie.ruhr-uni-bochum.de
Albert V. van den Berg
Dept. of Neuro-ethology, Helmholtz Institute,
Utrecht University, The Netherlands
a. v. vandenberg@bio.uu.nl
Markus Lappe
Dept. Zoology & Neurobiology
Ruhr University Bochum, Germany, 44780
lappe@neurobiologie .ruhr-uni-bochum.de
Abstract
Observer translation relative to the world creates image flow that
expands from the observer's direction of translation (heading) from
which the observer can recover heading direction. Yet, the image
flow is often more complex, depending on rotation of the eye, scene
layout and translation velocity. A number of models [1-4] have
been proposed on how the human visual system extracts heading
from flow in a neurophysiologic ally plausible way. These models
represent heading by a set of neurons that respond to large image
flow patterns and receive input from motion sensed at different image locations. We analysed these models to determine the exact
receptive field of these heading detectors. We find most models
predict that, contrary to widespread believe, the contribut ing motion sensors have a preferred motion directed circularly rather than
radially around the detector's preferred heading. Moreover, the results suggest to look for more refined structure within the circular
flow, such as bi-circularity or local motion-opponency.
Introduction
The image flow can be considerably more complicated than merely an expanding
pattern of motion vectors centered on the heading direction (Fig. 1). Flow caused
by eye rotation (Fig. 1b) causes the center of flow to be displaced (compare Fig. 1a
and c). The effect of rotation depends on the ratio ofrotation and translation speed.
A
Translational flow
f~~~~;
,,~
~
t
//
...... ........- 1
?0
?
" II
.J + ?
J
.....
? ?.,. "t
B
<If:-
....
---
...
~
~~~ S::
...
...
....-.... +-.......
~
Combined flow
........
.
-
.,
....
'\
-
4i-
.. .... ..+-..,...
....,..
.
.. * ::
+-
~1
"
.. ...... --... ',,,,
......
\\
.. 1J: ~
...
...............
-+
-+~
C
Rotational flow
+-
.......
... 0
'#
+-
~
I'
~
......
...
1
~.. " ?
",. ~
4
~\ + ~
.1,
Figure 1: Flow during a) observer translation through a 3D-cloud of dots, headed
10? towards the left, during b) observer rotation about the vertical towards the
right, and during c) the combination of both.
Also, since the image motions caused by translation depend on point distance and
the image motions caused by rotation do not, the combined movement results in flow
that is no longer purely expanding for scenes containing depth differences (Fig. lc).
Heading detection can therefore not rely on a simple extrapolation mechanism that
determines the point of intersection of motion vectors.
A number of physiologically-based models [1-4] have been proposed on how the
visual system might arrive at a representation of heading from flow that is insensit ive to parameters other than heading direction. These models assume heading is
encoded by a set of units that each respond best to a specific pattern of flow that
matches their preferred heading. Such units resemble neurons found in monkey
brain area MST. MST cells have large receptive fields (RF), typically covering one
quart or more of the visual field, and receive input from several local motion sensors in brain area MT. The receptive field of MST neurons may thus be defined as
the preferred location, speed and direction of all input local motion sensors. Little
is known yet about the RF structure of MST neurons. We looked for similarities
between current models at the level of the RF structure. First we explain the RF
structure of units in the velocity gain model, because this model makes clear assumptions on the RF structure. Next, we we show the results of reconstructing RF
structure of units in the population model[2] . Finally, we analyse the RF structure
of the template model[3] and motion-opponency model[4].
Velocity gain field model
The velocity gain field model[l] is based on flow templates. A flow template, as
introduced by Perrone and Stone[3] , is a unit that evaluates the evidence that the
flow fits the unit 's preferred flow field by summing the responses of local motion
sensors outputs. Heading is then represented by the preferred heading direction of
the most active template(s) . The velocity gain field model[l] is different from Perrone and Stone's template model[2] in the way it acquires invariance for translation
speed, point distances and eye rotation. Whereas the template model requires a
different template for each possible combination of heading direction and rotation,
t he velocity gain field model obtains rotation invariance using far less templates by
exploiting eye rotation velocity signals.
The general scheme applied in the velocity gain field model is as follows. In a set
of flow templates, each tuned to pure expansion with specific preferred heading,
B
A Circular component
<i!t-
-. l
....
...
t
~
4t"44-
~
....
9'
~
...0
?
....
............
9'9'?
II
,9'
""
+-4-
II
#=
.
~
II
?s- "
?
II
~
~
+
9'Y
9'
?
???
..
Radial component
,\
~
~~
.t.
?
1'f ;J?
"" ..........
r
.J'
t
,-'
Jr
y
y
0
..
"
,//,
J
?
... ... -+
??
\
~.
--.-
.~
-...;
..
~-.
..
Figure 2: The heading-centered circular (a) and radial (b) component of the flow
during combined translation and rotation as in Fig. 2c.
the templates would change their activity during eye rotation. Simply subtracting
the rotation velocity signal for each flow template would not suffice to compensate
because each template is differently affected by rotational flow. However, each
flow template can become approximately rotation-invariant by subtracting a gain
field activity that is a multiplication of the eye velocity t with a derivative template
activity 80/ 8R that is specific for each flow template. The latter reflects the change
in flow template activity 0 given a change in rotational flow 8R. Such derivative
template 80/ 8R can be constructed from the activity difference of two templates
tuned to the same heading, but opposite rotation. Thus, in the velocity gain field
model, templates tuned to heading direction and a component of rotation play an
important role.
To further appreciate the idea behind the RF structure in the velocity gain field
model, note that the retinal flow can be split into a circular and radial component,
centered on the heading point (Fig. 2). Translation at different speeds or through
a different 3D environment will alter the radial component only. The circular component contains a rotational component of flow but does not change with point
distances or translational speed. This observation lead to the assumption implemented in the velocity gain field model that templates should only measure the flow
along circles centered on the point of preferred heading.
An example of the RF structure of a typical unit in the velocity gain field model,
tuned to heading and rightward rotation is shown in Fig. 3. This circular RF structure strongly reduces sensitivity to variations in depth structure or the translational
speed, while the template's tuning to heading direction is preserved, because its preferred structure is centered on its preferred heading direction [1] . Interestingly, the
RF structure of the typical rotation-tuned heading units is bi-circular, because the
direction of circular flow is opponent in the hemifields to either side of an axis (in
this case the horizontal axis) through the heading point. Moreover, the structure
contains a gradient in magnitude along the circle, decreasing towards the horizontal
axis.
F:-
.
~
.
.....
...
.r-
I\.
.....
I\.
1\
?
? ? ?
I\.
It
0
I
"'-
.
- " .?
.... -- ...
"
It
.....
....
.r
Of'
~
?
..
rJ
Figure 3: Bi-circular RF structure of a typical unit in the velocity gain field model,
tuned to leftward heading and simultaneous rightward rotation about the vertical.
Individual vectors show the preferred direction and velocity of the input motion
sensors.
Population model
The population model [2] derives a representation of heading direction that is invariant to the other flow parameters using a totally different approach. This model
does not presume an explicit RF structure. Instead, the connections strengths and
preferred directions of local motion inputs to heading-specific flow units are computed according to an optimizing algorithm[5]. We here present the results obtained
for a restricted version of the model in which eye rotation is assumed to be limited
to pursuit that keeps the eye fixated on a stationary point in the scene during the
observer translation. Specifically, we investigated whether a circular or bi-circular
RF structure as predicted by the velocity gain model emerges in the population
model.
The population model [2 ,6] is an implementation of the subspace algorithm by
Heeger and Jepson [5] into a neural network. The subspace algorithm computes a
residual function R(T j) for a range of possible preferred heading directions. The
residual function is minimized when flow vectors measured at m image locations,
described as one array, are perpendicular to the vectors t hat form columns of a
matrix C ~ (T j). This matrix is computed from the preferred 3-D translation vector
T j and the m image locations. Thus, by finding the matrix that minimizes the
residue, the algorithm has solved the heading, irrespective of the 3D-rotation vector,
unknown depths of points and translation speed.
To implement the subspace algorithm in a neurophysiologically plausible way, the
population model assumes two layers of units. The first MT-like layer contains local
motion sensors that fire linearly with speed and have cosine-like direction tuning.
These sensors connect to units in the second MST-like layer. The activity in a 2nd
layer unit , with specific preferred heading T j, represents the likelihood that the
residual function is zero. The connection strengths are determined by the C ~ (T j)
matrix. As not to have too many motion inputs per 2nd layer unit, the residual
function R(T j) is partitioned into smaller sub residues that take only a few motion
inputs. The likelihood for a specific heading is t hen given by the sum of responses
in a population with same preferred heading.
Given the image locations and the preferred heading, one can reconstruct the RF
structure for 2nd layer units with the same preferred heading. The preferred motion
inputs to a second layer unit are given by vectors t hat make up each column of
C ~ (T j). Hereby, t he vector direction represents the preferred motion direction,
A
B
"-
\
t
I
?
Figure 4: Examples of receptive field structure of a population that encodes heading
100 towards the left (circle) . a-b) Five pairs of MT-like sensors, where the motion
sensors of each pair are at a) the same image location, or b) at image locations
one quarter of a cycle apart. c) Distribution of multiple pairs leading to bi-circular
pattern.
and the vector magnitude represents the strength of the synaptic connection. The
matrix C l..(Tj) is computed from the orthogonal complement of a (2m x m + 3)
matrix C(Tj) [5]. On the assumption that only fixational eye movements occur, the
matrix reduces to (2m x m + 1)[6]. Given only two flow vector inputs (m = 2), the
matrix C l.. (T j) reduces to one column of length m = 4. The orthogonal complement
of this 4 x 3 matrix was solved in Mathematica by first computing the nullspace of
the inverse matrix of C (T j), and then constructing an orthonormal basis for it using
Gram-Schmidt orthogonalisation. We computed the orientation and magnitude of
the two MT-inputs analytically. Instead of giving the mathematics, we here describe
the main results.
Circularity
Independent of the spatial arrangement of the two MT-inputs to a 2nd-layer unit,
their preferred motions turned out to be always directed along a circle centered on
the preferred heading point. Fig. 4 shows examples of the circular RF structures,
for different distributions of motion pairs that code for the same heading direction.
Motion-opponency
For pairs of motion sensors at overlapping locations, the vectors of each pair always
turned out to be opponent and of equal magnitude (Fig. 4a). For pairs of spatially
separated motion sensors, the preferred magnitude and direction of the two motion
inputs depend on their location with respect to the hemispheres divided by the line
through heading and fixation point. We find that preferred motion directions are
opponent if the pair is located within the same hemifield, but uni-directional if the
pair is split across the two hemifields as in Fig. 4b.
Bi-circularity
Interestingly, if pairs of motion sensors are split across hemi fields, with partners at
image locations 90 0 rotated about the heading point, a magnitude gradient appears
in the RF structure (Fig. 4b). Thus, with these pairs a bi-circular RF structure can
be constructed similar to units tuned to rotation about the vertical in the velocity
gain field model (compare with Fig. 3).
Note, that the bi-circular RF structures do differ since the axis along which the
largest magnitude occurs is horizontal for the population model and vertical for the
velocity gain field model. The RF structure of the population model unit resembles
a velocity gain field unit tuned to rotation about the horizontal axis, implying a
Adapted from Perrone and Stone (1994)
A
Effective RF structure
B
Direction and speed
tuned motion sensors
""","" .,"',,",
~
(j
Figure 5: Adapted from Perrone and Stone 1994). a) Each detector sums the
responses of the most active sensor at each location. This most active motion
sensor is selected from a pool of sensors tuned to different depth planes (Ca, Cb,
etc). These vectors are the vector sums of preferred rotation component Rand
translational components Ta, Tb, etc. b) Effective RF structure.
large sensitivity to such rotation. This, however, does not conflict with the expected
performance of the population model. Because in this restricted version rotation
invariance is expected only for rotation that keeps the point of interest in the center
of the image plane (in this case rotation about the vertical because heading is leftward) units are likely to be sensitive to rotation about the horizontal and torsional
axis.
Template model
The template model and the velocity gain field model differ in how invariance for
translation velocities, depth structure and eye rotation is obtained. Here, we investigate whether this difference affects the predicted RF structure. In the template
model of Perrone and Stone [3], a template invariant to translation velocity or depth
structure is obtained by summing the responses of the most active sensor at each
image location. This most active sensor is selected from a collection of motion sensors, each tuned to a different ego-translation speed (or depth plane), but with the
same preferred ego-rotation and heading direction (Fig. 5a). Given a large range of
depth planes, it follows that a different radial component of motion will stimulate
another sensor maximally, but that activity nevertheless remains the same. The
contributing response will change only due to a component of motion along a circle
centered on the heading, such as is the case when heading direction or rotation
is varied. Thus, the contributing response will always be from the motion sensor
oriented along the circle around the template's preferred heading. Effectively, this
leads to a bi-circular RF structure for units tuned to heading and rotation (Fig. 5b).
Motion-opponency model
Royden[4] proposed that the effect of rotation is removed at local motion detection
level before the motion signals are received by flow detectors. This is achieved by
MT-like sensors that compute the difference vector between spatially neighbouring
motion vectors. Such difference vector will always be oriented along lines intersecting at the heading point (Fig. 6). Thus, the resulting input to flow detectors will
be oriented radially. Indeed, Royden's results[4] show that the preferred directions
of the operators with the largest response will be radially, not circularly, oriented.
A
Translational flow
B
Rotational flow
L -________
~
.
~
______
~
Figure 6: Motion parallax, the difference vector between locally neighbouring motion vectors. For translation flow (a) the difference vector will be oriented along
line through the heading point, whereas for rotational flow (b) the difference vector
vanishes (compare vectors within square).
Summary and Discussion
We showed that a circular RF structure, such as proposed by the velocity gain field
model[l] , is also found in the population model[2] and is effectively present in the
template model[3] as well. Only the motion-opponent model [4] prefers radial RF
structures. Furthermore, we find that under certain restrictions, the population
model reveals local motion-opponency and bi-circularity, properties that can be
found in the other models as well.
A circular RF structure turns out to be a prominent property in three models.
This supports the counterintuitive, but computationally sensible idea, that it is not
the radial flow structure, but the structure perpendicular to it, that contributes
to the response of heading-sensitive units in the human brain. Studies on area
MST cells not only report selectivity for expanding motion patterns, but also a
significant proportion of cells that are selective to rotation patterns [7-10]. These
models could explain why cells respond so well to circular motion, in particular to
the high rotation speeds (up to about 80 deg/s) not experienced in daily life.
This model study suggests that selectivity for circular flow has a direct link to
heading detection mechanisms. It also suggests that testing selectivity for expanding
motion might be a bad indicator for determining a cell's preferred heading. This
point has been noted before, as MST seems to be systematically tuned to the focus
of rotation, exactly like model neurons [9].
Little is still known about the receptive field structure of MST cells. So far the
receptive field structure of MST cells has only been roughly probed [10], and the
results neither support a radial nor a circular structure. Also , so far only uni-circular
motion has been tested. Our analyses points out that it would be worthwhile to
look for more refined circular structure such as local motion-opponency. Local
motion opponency has already been found in area MT, where some cells respond
only if different parts of their receptive field are stimulated with different motion
[11]. Another promising structure to look for would be bi-circularity, with gradients
in magnitude of preferred motion along the circles.
Acknowledgments
Supported by the German Science Foundation and the German Federal Ministry of
Education and Research.
References
[1] Beintema, J . A. & van den Berg A. V. (1998) Heading detection using motion templates
and eye velocity gain fields. Vision Research, 38(14):2155-2179.
[2] Lappe M., & Rauschecker J . P. (1993) A neural network for the processing of optic flow
from ego-motion in man and higher mammals. Neural Computation, 5:374-39l.
[3] Perrone J. A. & Stone L. S. (1994) A model for the self-motion estimation within
primate extrastriate visual cortex. Vision Research, 34:2917-2938 .
[4] Royden C. S. (1997) Mathematical analysis of motion-opponent mechanisms used in
the determination of heading and depth. Journal of th e Optical Society of America A,
14(9):2128-2143.
[5] Heeger D . J . & Jepson A. D . (1992) Subspace methods for recovering rigid motion I:
Algorithm and implementation. International Journal of Computational Vision , 7:95-117.
[6] Lappe M. & Rauschecker J.P. (1993) Computation of heading direction from optic flow
in visual cortex. In C.L. Giles, S.J. Hanson and J.D. Cowan (eds.), Advances in Neural
Information Processing Systems 5, pp. 433-440. Morgan Kaufmann.
[7] Tanaka K. & Saito H. (1989) Analysis of the visual field by direction, expansion/contraction, and rotation cells clustered in the dorsal part of the medial superior
temporal area of the macaque monkey Journal of Neurophysiology, 62(3):626-64l.
[8] Duffy C. J. & Wurtz R. H. (1991) Sensitivity of MST neurons to optic flow stimuli.
I. A continuum of response selectivity to large-field stimuli. Journal of Neurophysiology,
65(6) :1329-1345.
[9] Lappe M., Bremmer F ., Pekel M., Thiele A., Hoffmann K.-P. (1996) Optic flow processing in monkey STS: a theoretical and experimental approach . th e Journal of Neuroscience,
16(19):6265-6285.
[10] Duffy C. J. & Wurtz R. H. (1991) Sensitivity of MST neurons to optic flow stimuli.
II. Mechanisms of response selectivity revealed by small-field stimuli. Journal of Neurophysiology, 65(6):1346-1359.
[11] Allman J., Miezin F. & McGuinness E. (1985) Stimulus specific responses from beyond
the classical receptive field: Neurophysiological mechanisms for local-global comparisons
in visual neurons. Ann. R ev. N eurosci., 8:407-430.
| 2034 |@word neurophysiology:3 version:2 proportion:1 seems:1 nd:4 ruhr:4 sensed:1 contraction:1 mammal:1 extrastriate:1 contains:3 tuned:13 interestingly:2 current:1 analysed:1 yet:2 mst:11 medial:1 stationary:1 implying:1 selected:2 plane:4 location:12 five:1 mathematical:1 along:9 constructed:2 direct:1 become:1 fixation:1 headed:1 parallax:1 indeed:1 expected:2 roughly:1 nor:1 brain:3 decreasing:1 little:2 totally:1 moreover:2 suffice:1 minimizes:1 monkey:3 finding:1 temporal:1 expands:1 exactly:1 bio:1 unit:23 before:2 local:11 torsional:1 approximately:1 might:2 resembles:1 suggests:2 limited:1 bi:11 range:2 perpendicular:2 directed:2 acknowledgment:1 testing:1 implement:1 saito:1 area:5 radial:8 suggest:1 bochum:4 operator:1 hemi:1 restriction:1 center:2 layout:1 pure:1 array:1 orthonormal:1 counterintuitive:1 population:13 variation:1 play:1 exact:1 neighbouring:2 velocity:25 helmholtz:1 ego:3 located:1 cloud:1 role:1 solved:2 cycle:1 movement:2 removed:1 thiele:1 environment:1 vanishes:1 depend:2 purely:1 creates:1 basis:1 rightward:2 differently:1 represented:1 america:1 separated:1 describe:1 effective:2 refined:2 encoded:1 plausible:2 ive:1 reconstruct:1 analyse:1 orthogonalisation:1 subtracting:2 turned:2 beintema:2 exploiting:1 rotated:1 depending:1 measured:1 received:1 implemented:1 predicted:2 resemble:1 recovering:1 uu:1 differ:2 direction:27 centered:7 human:2 education:1 clustered:1 around:2 cb:1 predict:1 continuum:1 eurosci:1 estimation:1 sensitive:2 largest:2 reflects:1 federal:1 sensor:22 always:4 rather:1 focus:1 likelihood:2 rigid:1 typically:1 selective:1 germany:2 translational:5 orientation:1 spatial:1 field:32 equal:1 represents:3 look:3 alter:1 minimized:1 report:1 stimulus:5 few:1 oriented:5 individual:1 fire:1 detection:4 interest:1 circular:23 investigate:1 zoology:2 nl:1 circularity:5 behind:1 tj:2 daily:1 bremmer:1 orthogonal:2 circle:7 opponency:7 theoretical:1 column:3 giles:1 too:1 connect:1 considerably:1 combined:3 neurobiologie:2 sensitivity:4 international:1 pool:1 hemifield:1 intersecting:1 containing:1 derivative:2 leading:1 de:2 retinal:1 sts:1 caused:3 depends:1 observer:6 extrapolation:1 jaap:1 recover:1 complicated:1 square:1 kaufmann:1 directional:1 utrecht:1 presume:1 detector:6 explain:2 simultaneous:1 synaptic:1 ed:1 evaluates:1 mathematica:1 pp:1 hereby:1 gain:19 radially:3 emerges:1 appears:1 ta:1 higher:1 response:11 maximally:1 rand:1 strongly:1 furthermore:1 ally:1 horizontal:5 overlapping:1 widespread:1 stimulate:1 believe:1 effect:2 analytically:1 spatially:2 during:6 self:1 covering:1 acquires:1 noted:1 cosine:1 prominent:1 stone:6 motion:55 image:15 superior:1 rotation:37 quarter:1 mt:7 he:2 significant:1 tuning:2 mathematics:1 dot:1 longer:1 similarity:1 cortex:2 etc:2 showed:1 leftward:2 optimizing:1 hemisphere:1 apart:1 selectivity:5 certain:1 life:1 morgan:1 ministry:1 determine:1 signal:3 ii:6 multiple:1 rj:1 reduces:3 ing:1 match:1 determination:1 compensate:1 divided:1 neuro:1 vision:3 wurtz:2 albert:1 represent:1 achieved:1 cell:9 receive:2 whereas:2 preserved:1 residue:2 cowan:1 contrary:1 flow:49 allman:1 revealed:1 split:3 affect:1 fit:1 opposite:1 idea:2 lappe:5 whether:2 cause:1 prefers:1 clear:1 quart:1 netherlands:1 locally:1 fixational:1 neuroscience:1 per:1 probed:1 affected:1 nevertheless:1 neither:1 mcguinness:1 merely:1 sum:3 inverse:1 respond:4 arrive:1 layer:8 activity:7 strength:3 occur:1 adapted:2 optic:5 scene:3 encodes:1 markus:1 speed:11 optical:1 hemifields:2 according:1 combination:2 perrone:6 jr:1 smaller:1 across:2 reconstructing:1 partitioned:1 primate:1 den:2 invariant:3 restricted:2 computationally:1 remains:1 turn:1 german:2 mechanism:5 pursuit:1 opponent:5 worthwhile:1 schmidt:1 hat:2 assumes:1 giving:1 society:1 classical:1 appreciate:1 arrangement:1 already:1 looked:1 occurs:1 receptive:9 hoffmann:1 gradient:3 subspace:4 distance:3 link:1 sensible:1 partner:1 length:1 code:1 ratio:1 rotational:6 implementation:2 unknown:1 vertical:5 neuron:8 displaced:1 observation:1 neurobiology:2 varied:1 introduced:1 complement:2 pair:11 connection:3 rauschecker:2 conflict:1 hanson:1 tanaka:1 macaque:1 beyond:1 perception:1 pattern:6 ev:1 tb:1 rf:27 rely:1 indicator:1 residual:4 scheme:1 eye:11 ethology:1 axis:6 irrespective:1 extract:1 hen:1 multiplication:1 contributing:2 relative:1 determining:1 neurophysiologically:1 foundation:1 systematically:1 translation:16 summary:1 supported:1 heading:56 side:1 institute:1 template:28 van:2 depth:9 world:1 gram:1 computes:1 collection:1 far:3 obtains:1 uni:4 preferred:30 keep:2 deg:1 global:1 active:5 reveals:1 fixated:1 summing:2 assumed:1 physiologically:1 why:1 stimulated:1 promising:1 expanding:4 ca:1 contributes:1 expansion:2 investigated:1 complex:1 constructing:1 jepson:2 main:1 linearly:1 fig:15 lc:1 sub:1 experienced:1 explicit:1 heeger:2 nullspace:1 miezin:1 bad:1 specific:7 evidence:1 derives:1 circularly:2 effectively:2 magnitude:8 duffy:2 intersection:1 simply:1 likely:1 neurophysiological:1 visual:7 determines:1 ann:1 towards:4 man:1 change:5 typical:3 specifically:1 determined:1 invariance:4 experimental:1 berg:2 support:2 latter:1 dorsal:1 dept:3 tested:1 |
1,135 | 2,035 | A Bayesian Network for Real-Time
Musical Accompaniment
Christopher Raphael
Department of Mathematics and Statistics,
University of Massachusetts at Amherst,
Amherst, MA 01003-4515,
raphael~math.umass.edu
Abstract
We describe a computer system that provides a real-time musical accompaniment for a live soloist in a piece of non-improvised
music for soloist and accompaniment. A Bayesian network is developed that represents the joint distribution on the times at which
the solo and accompaniment notes are played, relating the two
parts through a layer of hidden variables. The network is first constructed using the rhythmic information contained in the musical
score. The network is then trained to capture the musical interpretations of the soloist and accompanist in an off-line rehearsal phase.
During live accompaniment the learned distribution of the network
is combined with a real-time analysis of the soloist's acoustic signal, performed with a hidden Markov model, to generate a musically principled accompaniment that respects all available sources
of knowledge. A live demonstration will be provided.
1
Introduction
We discuss our continuing work in developing a computer system that plays the
role of a musical accompanist in a piece of non-improvisatory music for soloist
and accompaniment. The system begins with the musical score to a given piece
of music. Then, using training for the accompaniment part as well as a series of
rehearsals, we learn a performer-specific model for the rhythmic interpretation of the
composition. In performance, the system takes the acoustic signal of the live player
and generates the accompaniment around this signal, in real-time, while respecting
the learned model and the constraints imposed by the score. The accompaniment
played by our system responds both flexibly and expressively to the soloist's musical
interpretation.
Our system is composed of two high level tasks we call "Listen" and "Play." Listen
takes as input the acoustic signal of the soloist and, using a hidden Markov model,
performs a real-time analysis of the signal. The output of Listen is essentially
a running commentary on the acoustic input which identifies note boundaries in
the solo part and communicates these events with variable latency. The HMM
framework is well-suited to the listening task and has several attributes we regard
as indispensable to any workable solution:
1. The HMM allows unsupervised training using the Baum-Welch algorithm.
Thus we can automatically adapt to changes in solo instrument, microphone
placement, ambient noise, room acoustics, and the sound of the accompaniment instrument.
2. Musical accompaniment is inherently a real-time problem. Fast dynamic
programming algorithms provide the computational efficiency necessary to
process the soloist's acoustic signal at a rate consistent with the real-time
demands of our application.
3. Musical signals are occasionally ambiguous locally in time, but become
easier to parse when more context is considered. Our system owes much of
its accuracy to the probabilistic formulation of the HMM. This formulation
allows one to compute the probability that an event is in the past. We delay
the estimation of the precise location of an event until we are reasonably
confident that it is, in fact, past. In this way our system achieves accuracy
while retaining the lowest latency possible in the identification of musical
events.
Our work on the Listen component is documented thoroughly in [1] and we omit a
more detailed discussion here.
The heart of our system, the Play component, develops a Bayesian network consisting of hundreds of Gaussian random variables including both observable quantities,
such as note onset times, and unobservable quantities, such as local tempo. The
network can be trained during a rehearsal phase to model both the soloist's and
accompanist's interpretations of a specific piece of music. This model then forms
the backbone of a principled real-time decision-making engine used in performance.
We focus here on the Play component which is the most challenging part of our
system. A more detailed treatment of various aspects of this work is given in [2- 4].
2
Knowledge Sources
A musical accompaniment requires the synthesis of a number of different knowledge
sources. From a modeling perspective, the fundamental challenge of musical accompaniment is to express these disparate knowledge sources in terms of a common
denominator. We describe here the three knowledge sources we use.
1. We work with non-improvisatory music so naturally the musical score,
which gives the pitches and relative durations of the various notes, as well
as points of synchronization between the soloist and accompaniment, must
figure prominently in our model. The score should not be viewed as a rigid
grid prescribing the precise times at which musical events will occur; rather,
the score gives the basic elastic material which will be stretched in various
ways to to produce the actual performance. The score simply does not
address most interpretive aspects of performance.
2. Since our accompanist must follow the soloist, the output of the Listen
component, which identifies note boundaries in the solo part, constitutes
our second knowledge source. While most musical events, such as changes
between neighboring diatonic pitches, can be detected very shortly after
the change of note, some events, such as rearticulations and octave slurs,
are much less obvious and can only be precisely located with the benefit
of longer term hindsight. With this in mind, we feel that any successful
accompaniment system cannot synchronize in a purely responsive manner.
Rather it must be able to predict the future using the past and base its
synchronization on these predictions, as human musicians do.
3. While the same player's performance of a particular piece will vary from
rendition to rendition, many aspects of musical interpretation are clearly
established with only a few repeated examples. These examples, both of
solo performances and human (MIDI) performances of the accompaniment
part constitute the third knowledge source for our system. The solo data
is used primarily to teach the system how to predict the future evolution
of the solo part. The accompaniment data is used to learn the musicality
necessary to bring the accompaniment to life.
We have developed a probabilistic model, a Bayesian network, that represents all
of these knowledge sources through a jointly Gaussian distribution containing hundreds of random variables. The observable variables in this model are the estimated
soloist note onset times produced by Listen and the directly observable times for
the accompaniment notes. Between these observable variables lies a layer of hidden variables that describe unobservable quantities such as local tempo, change in
tempo, and rhythmic stress.
3
A Model for Rhythmic Interpretation
We begin by describing a model for the sequence of note onset times generated by a
monophonic (single voice) musical instrument playing a known piece of music. For
each of the notes, indexed by n = 0, . . . , N, we define a random vector representing
the time, tn, (in seconds) at which the note begins, and the local "tempo," Sn, (in
secs. per measure) for the note. We model this sequence ofrandom vectors through
a random difference equation:
(1)
n = 0, ... , N - 1, where in is the musical length of the nth note, in measures, and
the {(Tn' CTnY} and (to, so)t are mutually independent Gaussian random vectors.
?
The distributions of the {CT n } will tend concentrate around expressing the notion
that tempo changes are gradual. The means and variances of the {CT n} show where
the soloist is speeding-up (negative mean), slowing-down (positive mean), and tell
us if these tempo changes are nearly deterministic (low variance), or quite variable
(high variance). The {Tn} variables describe stretches (positive mean) or compressions (negative mean) in the music that occur without any actual change in tempo,
as in a tenuto or agogic accent. The addition of the {Tn} variables leads to a more
musically plausible model, since not all variation in note lengths can be explained
through tempo variation. Equally important, however, the {Tn} variables stabilize
the model by not forcing the model to explain, and hence respond to, all note length
variation as tempo variation.
Collectively, the distributions of the (Tn' CTn)t vectors characterize the solo player's
rhythmic interpretation. Both overall tendencies (means) and the repeatability of
these tendencies (covariances) are captured by these distributions.
3.1
Joint Model of Solo and Accompaniment
In modeling the situation of musical accompaniment we begin with the our basic
rhythm model of Eqn. 1, now applied to the composite rhythm. More precisely,
Listen
Update
Composite
Accomp
Figure 1: A graphical description of the dependency structure of our model. The
top layer of the graph corresponds to the solo note onset times detected by Listen.
The 2nd layer of the graph describes the (Tn, 0"n) variables that characterize the
rhythmic interpretation. The 3rd layer of the graph is the time-tempo process
{(Sn, t n )}. The bottom layer is the observed accompaniment event times.
let mo , ... , mivs and mg, ... , m'Na denote the positions, in measures, of the various
solo and accompaniment events. For example, a sequence of quarter notes in 3/ 4
time would lie at measure positions 0, 1/ 3, 2/ 3, etc. We then let mo, ... , mN be
the sorted union of these two sets of positions with duplicate times removed; thus
mo < ml < .. . < mN? We then use the model of Eqn. 1 with In = mn+1 - m n ,
n = 0, . . . , N - 1. A graphical description of this model is given in the middle
two layers of Figure 1. In this figure, the layer labeled "Composite" corresponds
to the time-tempo variables, (tn, sn)t, for the composite rhythm, while the layer
labeled "Update" corresponds to the interpretation variables (Tn, 0"n) t. The directed
arrows of this graph indicate the conditional dependency structure of our model.
Thus, given all variables "upstream" of a variable, x, in the graph, the conditional
distribution of x depends only on the parent variables.
Recall that the Listen component estimates the times at which solo notes begin.
How do these estimates figure into our model? We model the note onset times
estimated by Listen as noisy observations of the true positions {t n }. Thus if m n
is a measure position at which a solo note occurs, then the corresponding estimate
from Listen is modeled as
an = tn + an
2
where an rv N(O, 1I ). Similarly, if m n is the measure position of an accompaniment
event, then we model the observed time at which the event occurs as
bn
= tn + f3n
where f3n rv N(O, ",2). These two collections of observable variables constitute the
top layer of our figure, labeled "Listen," and the bottom layer, labeled "Accomp."
There are, of course, measure positions at which both solo and accompaniment
events should occur. If n indexes such a time then an and bn will both be noisy
observations of the true time tn. The vectors/ variables {(to, so)t, (Tn ' O"n)t, a n , f3n}
are assumed to be mutually independent.
4
Training the Model
Our system learns its rhythmic interpretation by estimating the parameters of the
(Tn,O"n) variables. We begin with a collection of J performances of the accompaniment part played in isolation. We refer to the model learned from this accompaniment data as the "practice room" distribution since it reflects the way the
accompanist plays when the constraint of following the soloist is absent. For each
Listen
Update
Composite
Accomp
Figure 2: Conditioning on the observed accompaniment performance (darkened circles), we use the message passing algorithm to compute the conditional distributions
on the unobservable {Tn' O"n} variables.
such performance, we treat the sequence of times at which accompaniment events
occur as observed variables in our model. These variables are shown with darkened
circles in Figure 2. Given an initial assignment of of means and covariances to the
(Tn , O"n) variables, we use the "message passing" algorithm of Bayesian Networks
[8,9] to compute the conditional distributions (given the observed performance) of
the (Tn,O"n) variables. Several such performances lead to several such estimates,
enabling us to improve our initial estimates by reestimating the (Tn ' O"n) parameters
from these conditional distributions.
More specifically, we estimate the (Tn,O"n) parameters using the EM algorithm, as
follows, as in [7]. We let J-L~, ~~ be our initial mean and covariance matrix for the
vector (Tn, 0"n). The conditional distribution of (Tn, 0"n) given the jth accompaniment performance, and using {J-L~ , ~~} , has a N(m; ,n, S~ ) distribution where the
m;,n and S~ parameters are computed using the message passing algorithm. We
then update our parameter estimates by
1
J
.
} Lmj,n
j=l
~ i+ l
n
The conventional wisdom of musicians is that the accompaniment should follow the
soloist. In past versions of our system we have explicitly modeled the asymmetric
roles of soloist and accompaniment through a rather complicated graph structure
[2- 4] . At present we deal with this asymmetry in a more ad hoc, however, perhaps
more effective, manner , as follows.
Training using the accompaniment performances allows our model to learn some of
the musicality these performances demonstrate. Since the soloist's interpretation
must take precedence, we want to use this accompaniment interpretation only to
the extent that it does not conflict with that of the soloist. We accomplish this
by first beginning with the result of the accompaniment training described above.
We use the practice room distributions , (the distributions on the {(Tn, O"n)} learned
from the accompaniment data) , as the initial distributions , {J-L~ , ~~} . We then run
the EM algorithm as described above now treating the currently available collection
of solo performances as the observed data. During this phase, only those parameters relevant to the soloist's rhythmic interpretation will be modified significantly.
Parameters describing the interpretation of a musical segment in which the soloist
is mostly absent will be largely unaffected by the second training pass.
Listen
Update
Composite
Accomp
Figure 3: At any given point in the performance we will have observed a collection
of solo note times estimated estimated by Listen, and the accompaniment event
times (the darkened circles). We compute the conditional distribution on the next
unplayed accompaniment event, given these observations.
This solo training actually happens over the course of a series of rehearsals. We
first initialize our model to the practice room distribution by training with the
accompaniment data. Then we iterate the process of creating a performance with
our system, (described in the next section), extracting the sequence of solo note
onset times in an off-line estimation process, and then retraining the model using all
currently available solo performances. In our experience, only a few such rehearsals
are necessary to train a system that responds gracefully and anticipates the soloist's
rhythmic nuance where appropriate - generally less than 10.
5
Real Time Accompaniment
The methodological key to our real-time accompaniment algorithm is the computation of (conditional) marginal distributions facilitated by the message-passing machinery of Bayesian networks. At any point during the performance some collection
of solo notes and accompaniment notes will have been observed, as in Fig. 3. Conditioned on this information we can compute the distribution on the next unplayed
accompaniment. The real-time computational requirement is limited by passing
only the messages necessary to compute the marginal distribution on the pending
accompaniment note.
Once the conditional marginal distribution of the pending accompaniment note is
calculated we schedule the note accordingly. Currently we schedule the note to be
played at the conditional mean time, given all observed information, however other
reasonable choices are possible. Note that this conditional distribution depends on
all of the sources of information included in our model: The score information, all
currently observed solo and accompaniment note times, and the rhythmic interpretations demonstrated by both the soloist and accompanist captured during the
training phase.
The initial scheduling of each accompaniment note takes place immediately after
the previous accompaniment note is played. It is possible that a solo note will be
detected before the pending accompaniment is played; in this event the pending
accompaniment event is rescheduled by recomputing the its conditional distribution using the newly available information. The pending accompaniment note is
rescheduled each time an additional solo note is detected until its currently scheduled time arrives, at which time it is finally played. In this way our accompaniment
makes use of all currently available information.
Does our system pass the musical equivalent of the Turing Test? We presume
no more objectivity in answering this question than we would have in judging
the merits of our other children. However, we believe that the level of musicality attained by our system is truly surprising, while the reliability is sufficient for live demonstration. We hope that the interested reader will form
an independent opinion, even if different from ours, and to this end we have
made musical examples demonstrating our progress available on the web page:
http://fafner.math.umass.edu/musicplus_one.
Acknowledgments
This work supported by NSF grants IIS-998789 and IIS-0113496.
References
[1] Raphael C. (1999), "Automatic Segmentation of Acoustic Musical Signals Using Hidden
Markov Models," IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.
21, No.4, pp. 360-370.
[2] Raphael C. (2001), "A Probabilistic Expert System for Automatic Musical Accompaniment," Journal of Computational and Graphical Statistics, vol. 10 no. 3, 487-512.
[3] Raphael C. (2001), "Can the Computer Learn to Play Expressively?" Proceedings of
Eighth International Workshop on Artificial Intelligence and Statistics, 113-120, Morgan
Kauffman.
[4] Raphael C. (2001), "Synthesizing Musical Accompaniments with Bayesian Belief Networks," Journal of New Music Research, vol. 30, no. 1, 59-67.
[5] Spiegelhalter D., Dawid A. P., Lauritzen S., Cowell R. (1993), "Bayesian Analysis in
Expert Systems," Statistical Science, Vol. 8, No.3, pp. 219-283.
[6] Cowell R., Dawid A. P., Lauritzen S., Spiegelhalter D. (1999), "Probabilistic Networks
and Expert Systems," Springer, New York.
[7] Lauritzen S. L. (1995), "The EM Algorithm for Graphical Association Models with
Missing Data," Computational Statistics and Data Analysis, Vol. 19, pp. 191-20l.
[8] Lauritzen S. L. (1992), "Propagation of Probabilities, Means, and Variances in Mixed
Graphical Association Models," Journal of the American Statistical Association, Vol. 87,
No. 420, (Theory and Methods), pp. 1098-1108.
[9] Lauritzen S. L. and F. Jensen (1999), "Stable Local Computation with Conditional
Gaussian Distributions," Technical Report R-99-2014, Department of Mathematic Sciences, Aalborg University.
| 2035 |@word version:1 middle:1 compression:1 nd:1 retraining:1 gradual:1 bn:2 covariance:3 initial:5 series:2 uma:2 score:8 accompaniment:55 ours:1 past:4 surprising:1 must:4 ctn:1 treating:1 update:5 intelligence:2 slowing:1 accordingly:1 beginning:1 provides:1 math:2 location:1 constructed:1 become:1 manner:2 automatically:1 actual:2 provided:1 begin:6 estimating:1 lowest:1 backbone:1 developed:2 hindsight:1 grant:1 omit:1 positive:2 before:1 local:4 treat:1 challenging:1 limited:1 directed:1 acknowledgment:1 union:1 practice:3 significantly:1 composite:6 cannot:1 scheduling:1 context:1 live:5 equivalent:1 conventional:1 imposed:1 deterministic:1 demonstrated:1 musician:2 missing:1 baum:1 flexibly:1 duration:1 welch:1 immediately:1 notion:1 variation:4 feel:1 play:6 programming:1 dawid:2 located:1 asymmetric:1 labeled:4 bottom:2 role:2 observed:10 capture:1 removed:1 principled:2 respecting:1 dynamic:1 trained:2 segment:1 purely:1 efficiency:1 joint:2 various:4 train:1 fast:1 describe:4 effective:1 detected:4 artificial:1 tell:1 objectivity:1 quite:1 plausible:1 statistic:4 jointly:1 noisy:2 hoc:1 sequence:5 mg:1 raphael:6 neighboring:1 relevant:1 description:2 parent:1 asymmetry:1 requirement:1 produce:1 lauritzen:5 progress:1 indicate:1 concentrate:1 attribute:1 human:2 opinion:1 material:1 musically:2 expressively:2 precedence:1 stretch:1 around:2 considered:1 predict:2 mo:3 achieves:1 vary:1 estimation:2 currently:6 reflects:1 hope:1 clearly:1 gaussian:4 modified:1 rather:3 focus:1 methodological:1 rigid:1 mathematic:1 prescribing:1 hidden:5 interested:1 unobservable:3 overall:1 retaining:1 initialize:1 marginal:3 once:1 represents:2 unsupervised:1 constitutes:1 nearly:1 future:2 report:1 develops:1 duplicate:1 few:2 primarily:1 aalborg:1 composed:1 phase:4 consisting:1 message:5 workable:1 truly:1 arrives:1 solo:23 ambient:1 necessary:4 experience:1 machinery:1 owes:1 indexed:1 continuing:1 circle:3 recomputing:1 modeling:2 assignment:1 hundred:2 delay:1 successful:1 characterize:2 dependency:2 accomplish:1 anticipates:1 combined:1 confident:1 thoroughly:1 fundamental:1 amherst:2 international:1 probabilistic:4 off:2 synthesis:1 na:1 containing:1 creating:1 expert:3 american:1 sec:1 stabilize:1 explicitly:1 onset:6 depends:2 piece:6 performed:1 ad:1 complicated:1 accuracy:2 musical:25 variance:4 largely:1 wisdom:1 repeatability:1 bayesian:8 identification:1 produced:1 presume:1 unaffected:1 explain:1 rehearsal:5 pp:4 obvious:1 naturally:1 newly:1 treatment:1 massachusetts:1 recall:1 knowledge:8 listen:15 segmentation:1 schedule:2 actually:1 attained:1 follow:2 formulation:2 until:2 eqn:2 parse:1 christopher:1 web:1 propagation:1 accent:1 scheduled:1 perhaps:1 believe:1 true:2 evolution:1 hence:1 deal:1 during:5 ambiguous:1 rhythm:3 octave:1 stress:1 demonstrate:1 tn:22 performs:1 bring:1 common:1 quarter:1 conditioning:1 association:3 interpretation:15 relating:1 expressing:1 composition:1 refer:1 stretched:1 rd:1 automatic:2 grid:1 mathematics:1 similarly:1 reliability:1 stable:1 longer:1 etc:1 base:1 perspective:1 forcing:1 indispensable:1 occasionally:1 life:1 unplayed:2 captured:2 morgan:1 commentary:1 additional:1 performer:1 signal:8 ii:2 rv:2 sound:1 technical:1 adapt:1 equally:1 prediction:1 pitch:2 basic:2 denominator:1 essentially:1 addition:1 want:1 source:9 tend:1 call:1 extracting:1 iterate:1 isolation:1 listening:1 absent:2 passing:5 york:1 constitute:2 generally:1 latency:2 detailed:2 locally:1 documented:1 generate:1 http:1 nsf:1 judging:1 estimated:4 per:1 vol:6 express:1 diatonic:1 key:1 demonstrating:1 graph:6 run:1 facilitated:1 turing:1 respond:1 place:1 reasonable:1 reader:1 decision:1 layer:11 ct:2 played:7 occur:4 placement:1 constraint:2 precisely:2 generates:1 aspect:3 department:2 developing:1 describes:1 em:3 making:1 happens:1 explained:1 heart:1 equation:1 mutually:2 discus:1 describing:2 mind:1 merit:1 instrument:3 end:1 available:6 appropriate:1 tempo:11 responsive:1 voice:1 shortly:1 top:2 running:1 graphical:5 music:8 lmj:1 question:1 quantity:3 occurs:2 responds:2 darkened:3 hmm:3 gracefully:1 extent:1 nuance:1 length:3 modeled:2 index:1 demonstration:2 mostly:1 teach:1 negative:2 disparate:1 synthesizing:1 observation:3 markov:3 enabling:1 situation:1 precise:2 conflict:1 acoustic:7 engine:1 learned:4 established:1 address:1 able:1 pattern:1 kauffman:1 eighth:1 challenge:1 including:1 belief:1 rendition:2 event:17 synchronize:1 nth:1 representing:1 mn:3 improve:1 spiegelhalter:2 identifies:2 sn:3 speeding:1 relative:1 synchronization:2 mixed:1 reestimating:1 sufficient:1 consistent:1 playing:1 course:2 supported:1 jth:1 rhythmic:10 benefit:1 regard:1 boundary:2 calculated:1 collection:5 made:1 transaction:1 observable:5 midi:1 ml:1 assumed:1 learn:4 reasonably:1 pending:5 inherently:1 elastic:1 upstream:1 monophonic:1 arrow:1 noise:1 repeated:1 child:1 interpretive:1 fig:1 position:7 prominently:1 lie:2 answering:1 communicates:1 third:1 learns:1 down:1 specific:2 jensen:1 workshop:1 conditioned:1 demand:1 easier:1 suited:1 simply:1 contained:1 collectively:1 cowell:2 springer:1 corresponds:3 ma:1 conditional:13 viewed:1 sorted:1 room:4 rescheduled:2 change:7 included:1 specifically:1 microphone:1 f3n:3 pas:2 tendency:2 player:3 |
1,136 | 2,036 | Minimax Probability Machine
Gert R.G. Lanckriet*
Department of EECS
University of California, Berkeley
Berkeley, CA 94720-1770
gert@eecs. berkeley.edu
Laurent EI Ghaoui
Department of EECS
University of California, Berkeley
Berkeley, CA 94720-1770
elghaoui@eecs.berkeley.edu
Chiranjib Bhattacharyya
Department of EECS
University of California, Berkeley
Berkeley, CA 94720-1776
chiru@eecs.berkeley.edu
Michael I. Jordan
Computer Science and Statistics
University of California, Berkeley
Berkeley, CA 94720-1776
jordan@cs.berkeley.edu
Abstract
When constructing a classifier, the probability of correct classification of future data points should be maximized. In the current
paper this desideratum is translated in a very direct way into an
optimization problem, which is solved using methods from convex optimization. We also show how to exploit Mercer kernels in
this setting to obtain nonlinear decision boundaries. A worst-case
bound on the probability of misclassification of future data is obtained explicitly.
1
Introduction
Consider the problem of choosing a linear discriminant by minimizing the probabilities that data vectors fall on the wrong side of the boundary. One way to attempt
to achieve this is via a generative approach in which one makes distributional assumptions about the class-conditional densities and thereby estimates and controls
the relevant probabilities. The need to make distributional assumptions, however,
casts doubt on the generality and validity of such an approach, and in discriminative solutions to classification problems it is common to attempt to dispense with
class-conditional densities entirely.
Rather than avoiding any reference to class-conditional densities, it might be useful
to attempt to control misclassification probabilities in a worst-case setting; that
is, under all possible choices of class-conditional densities. Such a minimax approach could be viewed as providing an alternative justification for discriminative
approaches. In this paper we show how such a minimax programme can be carried
out in the setting of binary classification. Our approach involves exploiting the
following powerful theorem due to Isii [6], as extended in recent work by Bertsimas
? http://robotics.eecs.berkeley.edur gert/
and Sethuraman [2]:
where y is a random vector, where a and b are constants, and where the supremum
is taken over all distributions having mean y and covariance matrix ~y. This
theorem provides us with the ability to bound the probability of misclassifying a
point, without making Gaussian or other specific distributional assumptions. We
will show how to exploit this ability in the design of linear classifiers.
One of the appealing features of this formulation is that one obtains an explicit
upper bound on the probability of misclassification of future data: 1/(1 + rP).
A second appealing feature of this approach is that, as in linear discriminant analysis
[7], it is possible to generalize the basic methodology, utilizing Mercer kernels and
thereby forming nonlinear decision boundaries. We show how to do this in Section
3.
The paper is organized as follows: in Section 2 we present the minimax formulation
for linear classifiers, while in Section 3 we deal with kernelizing the method. We
present empirical results in Section 4.
2
Maximum probabilistic decision hyperplane
In this section we present our minimax formulation for linear decision boundaries.
Let x and y denote random vectors in a binary classification problem, with mean
vectors and covariance matrices given by x '" (x, ~x) and y '" (y, ~y) , respectively,
where ""," means that the random variable has the specified mean and covariance
matrix but that the distribution is otherwise unconstrained. Note that x, x , y , Y E
JRn and ~x, ~y E JRnxn.
We want to determine the hyperplane aT z = b (a, z E JRn and b E JR) that separates
the two classes of points with maximal probability with respect to all distributions
having these means and covariance matrices. This boils down to:
max a
s.t.
a ,a ,b
inf Pr{ aT x 2: b} 2: a
(2)
or,
max a
a,a,b
s.t.
1 - a 2: sup Pr{ aT x
1- a
:s b}
(3)
2: sup Pr{aT y 2: b} .
Consider the second constraint in (3). Recall the result of Bertsimas and Sethuraman [2]:
1
supPr{aTY2:b}=-d2' with
1+
d2 = inf
(Y_Yf~y-1(y_y)
(4)
aTy?b
We can write this as d2 = infcTw>d w Tw, where w = ~y -1 /2 (y_y), c T = aT~y 1/2
and d = b - aTy. To solve this,-first notice that we can assume that aTy :S b (i.e.
y is classified correctly by the decision hyperplane aT z = b): indeed, otherwise we
would find d2 = 0 and thus a = 0 for that particular a and b, which can never be
an optimal value. So, d> o. We then form the Lagrangian:
?(w, >.) = w T w
+ >.(d -
c T w),
(5)
which is to be maximized with respect to A 2: 0 and minimized with respect to w .
At the optimum, 2w = AC and d = c T W , so A =
and w =
This yields:
-!#c
c%c.
(6)
Using (4), the second constraint in (3) becomes 1-0: 2: 1/(I+d2 ) or ~ 2: 0:/(1-0:).
Taking (6) into account, this boils down to:
b-aTY2:,,(o:)/aT~ya
V
where
,,(0:)=) 0:
1-0:
(7)
We can handle the first constraint in (3) in a similar way (just write aT x ::::: b as
_aT x 2: -b and apply the result (7) for the second constraint). The optimization
problem (3) then becomes:
max 0:
-b + aTx 2: ,,(o:)JaT~xa
s.t.
a ,a,b
b - aTy 2:
(8)
"(o:h/aT~ya.
Because "(0:) is a monotone increasing function of 0:, we can write this as:
max"
(9)
s.t.
""a,b
b - aTy 2:
"JaT~ya.
From both constraints in (9), we get
aTy + "JaT~ya::::: b::::: aTx - "JaT~xa,
(10)
which allows us to eliminate b from (9):
max"
I<,a
s.t.
aTy + "JaT~ya::::: aTx - "JaT~xa.
(11)
Because we want to maximize ", it is obvious that the inequalities in (10) will
become equalities at the optimum. The optimal value of b will thus be given by
(12)
where a* and "* are the optimal values of a and " respectively. Rearranging the
constraint in (11), we get:
aT(x - y) 2:"
(JaT~xa+ JaT~ya).
(13)
The above is positively homogeneous in a: if a satisfies (13), sa with s E 114 also
does. Furthermore, (13) implies aT(x - y) 2: O. Thus, we can restrict a to be such
that aT(x - y) = 1. The optimization problem (11) then becomes
max"
I<,a
s.t.
~
2:
JaT~xa + JaT~ya
(14)
a T (x-Y)=I ,
which allows us to eliminate ,,:
m~n JaT~xa + JaT~ya
s.t.
aT(x - y) = 1,
(15)
or, equivalently
(16)
This is a convex optimization problem, more precisely a second order cone program
(SOCP) [8,5]. Furthermore, notice that we can write a = ao +Fu, where U E Il~n-l,
ao = (x - y)/llx - y112, and F E IRnx (n-l) is an orthogonal matrix whose columns
span the subspace of vectors orthogonal to x - y.
Using this we can write (16) as an unconstrained SOCP:
(17)
We can solve this problem in various ways, for example using interior-point methods
for SOCP [8], which yield a worst-case complexity of O(n 3 ). Of course, the first and
second moments of x, y must be estimated from data, using for example plug-in estimates X, y, :Ex, :E y for respectively x, y, ~x, ~y. This brings the total complexity
to O(ln 3 ), where l is the number of data points. This is the same complexity as the
quadratic programs one has to solve in support vector machines.
In our implementations, we took an iterative least-squares approach, which is based
on the following form , equivalent to (17):
(18)
At iteration k , we first minimize with respect to 15 and E by setting 15k = II~x 1/2(ao +
Ek = II~y 1/2(ao + Fu k - 1)112. Then we minimize with respect to U
by solving a least squares problem in u for 15 = 15k and E = Ek, which gives us
Uk. Because in both update steps the objective of this COP will not increase, the
iteration will converge to the global minimum II~xl/2(ao + Fu*)112 + II~yl /2(ao +
Fu*)lb with u* an optimal value of u.
Fu k - d112 and
We then obtain a* as ao + Fu* and b* from (12) with "'* = l/h/ar~xa* +
Jar~ya*).
Classification of a new data point Zn ew is done by evaluating
sign( a;; Zn ew - b*): if this is +1, Zn ew is classified as from class x, otherwise Zn ew is
classified as from class y.
It is interesting to see what happens if we make distributional assumptions; in
particular, let us assume that x "" N(x, ~x) and y "" N(y, ~y). This leads to the
following optimization problem:
max a
o:, a ,b
S.t.
-b + aTx ::::: <I>-l(a)JaT~xa
(19)
where <I>(z) is the cumulative distribution function for a standard normal Gaussian
distribution. This has the same form as (8), but now with ",(a) = <I>-l(a) instead
of ",(a)
=
Vl~a (d. a result by Chernoff [4]). We thus solve the same optimization
problem (a disappears from the optimization problem because ",(a) is monotone
increasing) and find the same decision hyperplane aT z = b. The difference lies in
the value of a associated with "'*: a will be higher in this case, so the hyperplane
will have a higher predicted probability of classifying future data correctly.
Kernelization
3
In this section we describe the "kernelization" of the minimax approach described in
the previous section. We seek to map the problem to a higher dimensional feature
space ]Rf via a mapping cP : ]Rn 1-+ ]Rf, such that a linear discriminant in the feature
space corresponds to a nonlinear discriminant in the original space. To carry out
this programme, we need to try to reformulate the minimax problem in terms of a
kernel function K(Z1' Z2) = cp(Z1)T CP(Z2) satisfying Mercer's condition.
1-+ cp(x) ""' (cp(X) , ~cp(x)) and Y 1-+ cp(y) ""'
(cp(y) , ~cp(y)) where {Xi}~1 and {Yi}~1 are training data points in the classes
Let the data be mapped as x
corresponding to x and Y respectively. The decision hyperplane in ]Rf is then given
by aT cp(Z) = b with a, cp(z) E ]Rf and b E ]R. In ]Rf, we need to solve the following
optimization problem:
mln Jr-aT-~-cp-(-x)-a +
JaT~cp(y)a
aT (cp(X) - cp(y)) = 1,
s.t.
(20)
where, as in (12), the optimal value of b will be given by
b* = a; cp(x) -
"'*Jar~cp(x)a*
+ "'*Jar~cp(y)a*,
= a; cp(y)
(21)
where a* and "'* are the optimal values of a and '" respectively. However, we do
not wish to solve the COP in this form, because we want to avoid using f or cp
explicitly.
If a has a component in ]Rf which is orthogonal to the subspace spanned by CP(Xi),
i = 1,2, ... , N x and CP(Yi), i = 1,2, ... , Ny, then that component won't affect the
objective or the constraint in (20) . This implies that we can write a as
N.
Ny
(22)
a = LaiCP(Xi) + L;)jCP(Yj).
i=1
j=1
Substituting expression (22) for a and estimates
=
2:~1 CP(Xi) , ;p(Y) =
1
Ny
1
N.
.....--..
.....--.. T
Ny 2:i=l cp(Yi), ~cp(x) - N. 2: i=1 (cp(Xi) - cp(X)) (cp(Xi) - cp(x)) and ~cp(y) -
;Pw
A
J
y
N
J.
_
A
.....--..
_
.....--..
2:i~1(CP(Yi)
- cp(y))(cp(Yi) - cp(y))T for the means and the covariance matri-
ces in the objective and the constraint of the optimization problem (20), we see
that both the objective and the constraints can be written in terms of the kernel
function K(Zl' Z2) = CP(Z1)T cp(Z2) . We obtain:
T -
-
(23)
"f (k x - ky) = 1,
T
J.
-
N
N
.
-
[a1 a2 ... aN. ;)1 ;)2 ... ;)Ny l , kx E ]R .+ y WIth [kxl i =
2:f;1 K(xj, Zi), ky E ]RN. +Ny with [kyl i = y 2:f~l K(Yj, Zi), Zi = Xi for
where "f =
J
i = 1,2, ... ,Nx and Zi = Yi - N. for i = N x
as:
+ 1, N x + 2, ... ,Nx + Ny . K is defined
(Kx -IN.~~) = (*x)
(24)
Ky -lNy ky
Ky
where 1m is a column vector with ones of dimension m. Kx and Ky contain
respectively the first N x rows and the last Ny rows of the Gram matrix K (defined
as Kij = cp(zdTcp(zj) = K(Zi,Zj)). We can also write (23) as
K=
-
Kx I
m~n II ~"f12
Ky I
+ II.jlV;"f
12
s.t.
T -
-
"f (kx - ky) = 1,
(25)
which is a second order cone program (SOCP) [5] that has the same form as the
SOCP in (16) and can thus be solved in a similar way. Notice that, in this case,
the optimizing variable is "f E ~Nz +Ny instead of a E ~n. Thus the dimension of
the optimization problem increases, but the solution is more powerful because the
kernelization corresponds to a more complex decision boundary in ~n .
Similarly, the optimal value b* of b in (21) will then become
(26)
"'* are the optimal values of "f and", respectively.
Once "f* is known, we get "'* = 1/ ( J~z "f;K~Kx"f* + J~y "f;K~Ky"f* ) and then
where "f* and
b* from (26). Classification of a new data point Znew is then done by evaluating
sign(a; <p(znew) -b*) = sign ( (L~l+Ny b*]iK(Zi, Znew) ) - b*) (again only in terms
of the kernel function): if this is + 1, Znew is classified as from class x , otherwise
Znew is classified as from class y.
4
Experiments
In this section we report the results of experiments that we carried out to test
our algorithmic approach. The validity of 1 - a as the worst case bound on the
probability of misclassification of future data is checked, and we also assess the
usefulness of the kernel trick in this setting. We compare linear kernels and Gaussian
kernels.
Experimental results on standard benchmark problems are summarized in Table 1.
The Wisconsin breast cancer dataset contained 16 missing examples which were not
used. The breast cancer, pima, diabetes, ionosphere and sonar data were obtained
from the VCI repository. Data for the twonorm problem data were generated as
specified in [3]. Each dataset was randomly partitioned into 90% training and
10% test sets. The kernel parameter (u) for the Gaussian kernel (e-llx-yI12/,,) was
tuned using cross-validation over 20 random partitions. The reported results are
the averages over 50 random partitions for both the linear kernel and the Gaussian
kernel with u chosen as above.
The results are comparable with those in the existing literature [3] and with those
obtained with Support Vector Machines. Also, we notice that a is indeed smaller
Table 1: a and test-set accuracy (TSA) compared to BPB (best performance in [3])
and to the performance of an SVM with linear kernel (SVML) and an SVM with
Gaussian kernel (SVMG)
Dataset
Twonorm
Breast cancer
Ionosphere
Pima diabetes
Sonar
Linear
a
80.2 %
84.4 %
63.3 %
31.2 %
62.4 %
kernel
TSA:
96.0 %
97.2 %
85.4 %
73.8 %
75.1 %
BPB
SVML SVMG
Gaussian kernel
TSA:
a
83.6 % 97.2 % 96.3 % 95.6 % 97.4 %
92.7 % 97.3 % 96.8 % 92.6 % 98.5 %
89.9 % 93.0 % 93.7 % 87.8 % 91.5 %
33.0 % 74.6 % 76.1 % 70.1 % 75.3 %
87.1 % 89.8 %
75.9 % 86.7 %
than the test-set accuracy in all cases. Furthermore, a is smaller for a linear decision
boundary then for the nonlinear decision boundary obtained via the Gaussian kernel. This clearly shows that kernelizing the method leads to more powerful decision
boundaries.
5
Conclusions
The problem of linear discrimination has a long and distinguished history. Many
results on misclassification rates have been obtained by making distributional assumptions (e.g., Anderson and Bahadur [1]) . Our results , on the other hand, make
use of recent work on moment problems and semidefinite optimization to obtain
distribution-free results for linear discriminants. We have also shown how to exploit Mercer kernels to generalize our algorithm to nonlinear classification.
The computational complexity of our method is comparable to the quadratic program that one has to solve for the support vector machine (SVM). While we have
used a simple iterative least-squares approach, we believe that there is much to
gain from exploiting analogies to the SVM and developing specialized, more efficient optimization procedures for our algorithm, in particular tools that break the
data into subsets. The extension towards large scale applications is a current focus of our research, as is the problem of developing a variant of our algorithm for
multiway classification and function regression . Also the statistical consequences of
using plug-in estimates for the mean vectors and covariance matrices needs to be
investigated.
Acknowledgements
We would like to acknowledge support from ONR MURI N00014-00-1-0637, from
NSF grants IIS-9988642 and ECS-9983874 and from the Belgian American Educational Foundation.
References
[1] Anderson, T . W . and Bahadur, R. R . (1962) Classification into two multivariate Normal
distributions with different covariance matrices. Annals of Mathematical Statistics 33(2):
420-431.
[2] Bertsimas, D. and Sethuraman, J. (2000) Moment problems and semidefinite optimization. Handbook of Semidefinite Optimization 469-509, Kluwer Academic Publishers.
[3] Breiman L. (1996) Arcing classifiers. Technical Report 460 , Statistics Department,
University of California, December 1997.
[4] Chernoff H. (1972) The selection of effective attributes for deciding between hypothesis
using linear discriminant functions. In Frontiers of Pattern Recognition, (S. Watanabe,
ed.), 55-60. New York: Academic Press.
[5] Boyd, S. and Vandenberghe, L. (2001) Convex Optimization. Course notes for EE364,
Stanford University. Available at http://www . stanford. edu/ class/ee364.
[6] Isii, K. (1963) On the sharpness of Chebyshev-type inequalities.
Math . 14: 185-197.
Ann. Inst. Stat.
[7] Mika, M. Ratsch, G., Weston, J., SchOikopf, B., and Mii11er, K.-R. (1999) Fisher
discriminant analysis with kernels. In Neural Networks for Signal Processing IX, 41- 48 ,
New York: IEEE Press.
[8] Nesterov , Y. and Nemirovsky, A. (1994) Interior Point Polynomial Methods in Convex
Programming: Theory and Applications. Philadelphia, PA: SIAM.
| 2036 |@word repository:1 pw:1 polynomial:1 d2:5 seek:1 nemirovsky:1 covariance:7 thereby:2 carry:1 moment:3 tuned:1 bhattacharyya:1 existing:1 current:2 z2:4 must:1 written:1 partition:2 update:1 tsa:3 discrimination:1 generative:1 mln:1 provides:1 math:1 mathematical:1 direct:1 become:2 ik:1 indeed:2 increasing:2 becomes:3 what:1 berkeley:13 classifier:4 wrong:1 uk:1 control:2 zl:1 grant:1 consequence:1 laurent:1 might:1 mika:1 nz:1 yj:2 atx:4 procedure:1 empirical:1 boyd:1 get:3 interior:2 selection:1 www:1 equivalent:1 map:1 lagrangian:1 missing:1 educational:1 convex:4 sharpness:1 utilizing:1 spanned:1 vandenberghe:1 handle:1 gert:3 justification:1 annals:1 programming:1 homogeneous:1 hypothesis:1 lanckriet:1 trick:1 diabetes:2 pa:1 satisfying:1 recognition:1 distributional:5 muri:1 solved:2 worst:4 complexity:4 dispense:1 nesterov:1 solving:1 translated:1 various:1 describe:1 effective:1 choosing:1 whose:1 stanford:2 solve:7 otherwise:4 ability:2 statistic:3 cop:2 took:1 maximal:1 relevant:1 achieve:1 ky:9 exploiting:2 optimum:2 ac:1 stat:1 sa:1 c:1 involves:1 implies:2 predicted:1 correct:1 attribute:1 ao:7 extension:1 frontier:1 normal:2 deciding:1 mapping:1 algorithmic:1 substituting:1 a2:1 tool:1 clearly:1 gaussian:8 rather:1 avoid:1 breiman:1 arcing:1 focus:1 inst:1 vl:1 eliminate:2 classification:9 once:1 never:1 having:2 chernoff:2 future:5 minimized:1 report:2 randomly:1 attempt:3 semidefinite:3 fu:6 belgian:1 orthogonal:3 kij:1 column:2 ar:1 zn:4 subset:1 usefulness:1 reported:1 eec:7 density:4 siam:1 twonorm:2 probabilistic:1 yl:1 michael:1 again:1 ek:2 american:1 doubt:1 account:1 socp:5 summarized:1 explicitly:2 try:1 break:1 sup:2 ass:1 il:1 square:3 minimize:2 f12:1 accuracy:2 maximized:2 yield:2 generalize:2 classified:5 history:1 checked:1 ed:1 obvious:1 associated:1 boil:2 gain:1 dataset:3 recall:1 organized:1 higher:3 methodology:1 formulation:3 done:2 generality:1 furthermore:3 just:1 xa:8 anderson:2 hand:1 ei:1 vci:1 nonlinear:5 brings:1 believe:1 validity:2 contain:1 equality:1 deal:1 won:1 cp:37 common:1 specialized:1 discriminants:1 kluwer:1 llx:2 unconstrained:2 similarly:1 multiway:1 multivariate:1 recent:2 optimizing:1 inf:2 n00014:1 inequality:2 binary:2 onr:1 yi:6 kyl:1 aty2:2 minimum:1 determine:1 maximize:1 converge:1 signal:1 ii:7 technical:1 academic:2 plug:2 cross:1 long:1 a1:1 desideratum:1 basic:1 variant:1 breast:3 regression:1 aty:7 iteration:2 kernel:19 robotics:1 want:3 bpb:2 ratsch:1 publisher:1 december:1 jordan:2 affect:1 xj:1 zi:6 restrict:1 y112:1 chebyshev:1 expression:1 schoikopf:1 york:2 useful:1 jrn:2 http:2 bahadur:2 misclassifying:1 zj:2 notice:4 nsf:1 sign:3 svml:2 estimated:1 correctly:2 write:7 ce:1 bertsimas:3 monotone:2 cone:2 znew:5 powerful:3 decision:11 comparable:2 entirely:1 bound:4 matri:1 quadratic:2 constraint:9 precisely:1 span:1 department:4 developing:2 jr:2 smaller:2 partitioned:1 appealing:2 tw:1 making:2 happens:1 pr:3 ghaoui:1 taken:1 ln:1 chiranjib:1 available:1 apply:1 distinguished:1 kernelizing:2 alternative:1 isii:2 rp:1 original:1 exploit:3 objective:4 subspace:2 separate:1 mapped:1 nx:2 discriminant:6 reformulate:1 providing:1 minimizing:1 equivalently:1 pima:2 jat:14 design:1 implementation:1 upper:1 benchmark:1 acknowledge:1 extended:1 rn:2 lb:1 cast:1 specified:2 z1:3 california:5 pattern:1 kxl:1 elghaoui:1 program:4 yi12:1 rf:6 max:7 lny:1 misclassification:5 minimax:7 disappears:1 sethuraman:3 carried:2 philadelphia:1 literature:1 acknowledgement:1 wisconsin:1 interesting:1 analogy:1 validation:1 foundation:1 mercer:4 classifying:1 row:2 cancer:3 course:2 last:1 free:1 side:1 fall:1 taking:1 boundary:8 dimension:2 evaluating:2 cumulative:1 gram:1 programme:2 ec:1 obtains:1 supremum:1 global:1 handbook:1 discriminative:2 xi:7 jrnxn:1 iterative:2 sonar:2 table:2 ca:4 rearranging:1 investigated:1 complex:1 constructing:1 positively:1 ny:10 watanabe:1 explicit:1 wish:1 xl:1 lie:1 ix:1 theorem:2 down:2 specific:1 svm:4 ionosphere:2 kx:6 jar:3 forming:1 contained:1 corresponds:2 satisfies:1 weston:1 chiru:1 conditional:4 viewed:1 ann:1 towards:1 jlv:1 fisher:1 hyperplane:6 total:1 experimental:1 ya:9 ew:4 support:4 kernelization:3 avoiding:1 ex:1 |
1,137 | 2,037 | Escaping the Convex Hull with
Extrapolated Vector Machines.
Patrick Haffner
AT&T Labs-Research, 200 Laurel Ave, Middletown, NJ 07748
haffner@research.att.com
Abstract
Maximum margin classifiers such as Support Vector Machines
(SVMs) critically depends upon the convex hulls of the training
samples of each class, as they implicitly search for the minimum
distance between the convex hulls. We propose Extrapolated Vector Machines (XVMs) which rely on extrapolations outside these
convex hulls. XVMs improve SVM generalization very significantly
on the MNIST [7] OCR data. They share similarities with the
Fisher discriminant: maximize the inter-class margin while minimizing the intra-class disparity.
1
Introduction
Both intuition and theory [9] seem to support that the best linear separation between two classes is the one that maximizes the margin. But is this always true?
In the example shown in Fig.(l), the maximum margin hyperplane is Wo; however ,
most observers would say that the separating hyperplane WI has better chances to
generalize, as it takes into account the expected location of additional training sam-
? ? ? ? ? ? ? ? ? ? ? f\J:- ? ? ? ? ? ? ? ? ? ?
??
. . , --.Q.
~"-
_
~
,
................ ~x~...
---------------
..
??
.'
W
1
~---------------
? K~ ???????????0
'''-,,- / 0- 0 00 00 0
0'\
"OW-0_ o_ _o- o- o::-o
. .. . ....... ............ .
(}
Figure 1: Example of separation where the large margin is undesirable. The convex
hull and the separation that corresponds to the standard SVM use plain lines while
the extrapolated convex hulls and XVMs use dotted lines.
pIes. Traditionally, to take this into account, one would estimate the distribution of
the data. In this paper, we just use a very elementary form of extrapolation ("the
poor man variance") and show that it can be implemented into a new extension to
SVMs that we call Extrapolated Vector Machines (XVMs).
2
Adding Extrapolation to Maximum Margin Constraints
This section states extrapolation as a constrained optimization problem and computes a simpler dual form.
Take two classes C+ and C_ with Y+ = +1 and Y_ = -1 1 as respective targets.
The N training samples {(Xi, Yi); 1 ::::; i ::::; N} are separated with a margin p if there
exists a set of weights W such that Ilwll = 1 and
Vk E {+, -}, Vi E Ck, Yk(w,xi+b) 2: p
(1)
SVMs offer techniques to find the weights W which maximize the margin p. Now,
instead of imposing the margin constraint on each training point, suppose that for
two points in the same class Ck, we require any possible extrapolation within a
range factor 17k 2: 0 to be larger than the margin:
Vi,j E Ck , V)" E [-17k, l+17k], Yk (W.()"Xi + (l-)")Xj) + b) 2: P
(2)
It is sufficient to enforce the constraints at the end of the extrapolation segments,
and
(3)
Keeping the constraint over each pair of points would result in N 2 Lagrange multipliers. But we can reduce it to a double constraint applied to each single point. If
follows from Eq.(3) that:
(4)
(5)
We consider J.Lk = max (Yk(W.Xj)) and Vk = min (Yk(W.Xj)) as optimization varilEC.
lEC.
abIes. By adding Eq.(4) and (5), the margin becomes
2p =
L
((17k+ 1)vk - 17kJ.Lk) =
k
L
(Vk -17dJ.Lk - Vk))
(6)
k
Our problem is to maximize the margin under the double constraint:
Vi E Ck , Vk ::::; Yk(W.Xi) ::::; J.Lk
In other words, the extrapolated margin maximization is equivalent to squeezing
the points belonging to a given class between two hyperplanes. Eq.(6) shows that
p is maximized when Vk is maximized while J.Lk - Vk is minimized.
Maximizing the margin over J.Lk , Vk and
following dual problem:
W
with Lagrangian techniques gives us the
(7)
lIn this paper, it is necessary to index the outputs y with the class k rather than
the more traditional sample index i, as extrapolation constraints require two examples to
belong to the same class. The resulting equations are more concise, but harder to read.
Compared to the standard SVM formulation, we have two sets of support vectors.
Moreover, the Lagrange multipliers that we chose are normalized differently from
the traditional SVM multipliers (note that this is one possible choice of notation,
see Section.6 for an alternative choice). They sum to 1 and allow and interesting
geometric interpretation developed in the next section.
3
Geometric Interpretation and Iterative Algorithm
For each class k, we define the nearest point to the other class convex hull along
the direction of w: Nk = I:iECk f3iXi. Nk is a combination of the internal support vectors that belong to class k with f3i > O. At the minimum of (7), because
they correspond to non zero Lagrange multipliers, they fallon the internal margin
Yk(W,Xi) = Vk; therefore, we obtain Vk = Ykw.Nk?
Similarly, we define the furthest point Fk = I: i ECk ~i Xi' Fk is a combination of the
external support vectors, and we have flk = Ykw.Fk.
The dual problem is equivalent to the distance minimization problem
IILYk
((1Jk+I)Nk _1Jk F k)11
k
min
Nk ,Fk EHk
where
1{k
2
is the convex hull containing the examples of class k.
It is possible to solve this optimization problem using an iterative Extrapolated
Convex Hull Distance Minimization (XCHDM) algorithm. It is an extension of the
Nearest Point [5] or Maximal Margin Percept ron [6] algorithms. An interesting
geometric interpretation is also offered in [3]. All the aforementioned algorithms
search for the points in the convex hulls of each class that are the nearest to each
other (Nt and No on Fig.I) , the maximal margin weight vector w = Nt - N o-'
XCHDM look for nearest points in the extrapolated convex hulls (X+ I and X-I
on Fig.I). The extrapolated nearest points are X k = 1JkNk - 1JkFk' Note that
they can be outside the convex hull because we allow negative contribution from
external support vectors. Here again, the weight vector can be expressed as a
difference between two points w = X+ - X - . When the data is non-separable, the
solution is trivial with w = O. With the double set of Lagrange multipliers, the
description of the XCHDM algorithm is beyond the scope of this paper. XCHDM
with 1Jk = 0 are simple SVMs trained by the same algorithm as in [6].
An interesting way to follow the convergence of the XCHDM algorithm is the following. Define the extrapolated primal margin
1'; =
2p = L
((1Jk+ I )vk - 1Jkflk)
k
and the dual margin
1'; = IIX+ - X-II
Convergence consists in reducing the duality gap 1'~ -1'; down to zero. In the rest
of the paper, we will measure convergence with the duality ratio r = 1'~ .
1'2
To determine the threshold to compute the classifier output class sign(w.x+b) leaves
us with two choices. We can require the separation to happen at the center of the
primal margin, with the primal threshold (subtract Eq.(5) from Eq.(4))
bl =
1
-2" LYk ((1Jk+ I )vk-1JkJ.lk)
k
or at the center of the dual margin, with the dual threshold
b2 =
- ~w. 2:)(T}k+1)Nk -
T}kFk) =
- ~ (IIx+ 112 -lix-in
k
Again, at the minimum, it is easy to verify that b1 = b2 . When we did not let
the XCHDM algorithm converge to the minimum, we found that b1 gave better
generalization results.
Our standard stopping heuristic is numerical: stop when the duality ratio gets over
a fixed value (typically between 0.5 and 0.9).
The only other stopping heuristic we have tried so far is based on the following idea.
Define the set of extrapolated pairs as {(T}k+1)Xi -T}kXj; 1 :S i,j :S N}. Convergence
means that we find extrapolated support pairs that contain every extrapolated pair
on the correct side of the margin. We can relax this constraint and stop when the
extrapolated support pairs contain every vector. This means that 12 must be lower
than the primal true margin along w (measured on the non-extrapolated data)
11 = y+ + Y - . This causes the XCHDM algorithm to stop long before 12 reaches
Ii and is called the hybrid stopping heuristic.
4
Beyond SVMs and discriminant approaches.
Kernel Machines consist of any classifier of the type f(x) = L:i Yi(Xi K(x, Xi). SVMs
offer one solution among many others, with the constraint (Xi > O.
XVMs look for solutions that no longer bear this constraint. While the algorithm
described in Section 2 converges toward a solution where vectors act as support of
margins (internal and external), experiments show that the performance of XVMs
can be significantly improved if we stopped before full convergence. In this case,
the vectors with (Xi =/: 0 do not line up onto any type of margin, and should not be
called support vectors.
The extrapolated margin contains terms which are caused by the extrapolation
and are proportional to the width of each class along the direction of w. We
would observe the same phenomenon if we had trained the classifier using Maximum
Likelihood Estimation (MLE) (replace class width with variance). In both MLE and
XVMs, examples which are the furthest from the decision surface play an important
role. XVMs suggest an explanation why.
Note also that like the Fisher discriminant , XVMs look for the projection that
maximizes the inter-class variance while minimizing the intra-class variances.
5
Experiments on MNIST
The MNIST OCR database contains 60,000 handwritten digits for training and
10,000 for testing (the testing data can be extended to 60,000 but we prefer to
keep unseen test data for final testing and comparisons). This database has been
extensively studied on a large variety of learning approaches [7]. It lead to the
first SVM "success story" [2], and results have been improved since then by using
knowledge about the invariance of the data [4].
The input vector is a list of 28x28 pixels ranging from 0 to 255. Before computing
the kernels , the input vectors are normalized to 1: x = II~II'
Good polynomial kernels are easy to define as Kp(x, y) = (x.y)P. We found these
normalized kernels to outperform the unnormalized kernels Kp(x, y) = (a(x.y)+b)P
that have been traditionally used for the MNIST data significantly. For instance,
the baseline error rate with K4 is below 1.2%, whereas it hovers around 1.5% for
K4 (after choosing optimal values for a and b)2.
We also define normalized Gaussian kernels:
Kp(x, y)
= exp
( - ~ Ilx - y112) = [exp (x.y- 1)JP.
(8)
Eq.(8) shows how they relate to normalized polynomial kernels: when x.y ? 1,
Kp and Kp have the same asymptotic behavior. We observed that on MNIST,
the performance with Kp is very similar to what is obtained with unnormalized
Gaussian kernels Ku(x , y) = exp _(X~Y)2. However, they are easier to analyze and
compare to polynomial kernels.
MNIST contains 1 class per digit, so the total number of classes is M=10. To combine binary classifiers to perform multiclass classifications, the two most common
approaches were considered .
? In the one-vs-others case (lvsR) , we have one classifier per class c, with the
positive examples taken from class c and negative examples form the other
classes. Class c is recognized when the corresponding classifier yields the
largest output .
? In the one-vs-one case (lvs1), each classifier only discriminates one class
from another: we need a total of (MU:;-l) = 45 classifiers.
Despite the effort we spent on optimizing the recombination of the classifiers [8]
1vsR SVMs (Table 1) perform significantly better than 1vs1 SVMs (Table 2). 4
3,
For each trial, the number of errors over the 10,000 test samples (#err) and the
total number of support vectors( #SV) are reported. As we only count SVs which
are shared by different classes once, this predicts the test time. For instance, 12,000
support vectors mean that 20% of the 60,000 vectors are used as support.
Preliminary experiments to choose the value of rJk with the hybrid criterion show
that the results for rJk = 1 are better than rJk = 1.5 in a statistically significant
way, and slightly better than rJk = 0.5. We did not consider configurations where
rJ+ f; rJ -; however, this would make sense for the assymetrical 1vsR classifiers.
XVM gain in performance over SVMs for a given configuration ranges from 15%
(1 vsR in Table 3) to 25% (1 vs1 in Table 2).
2This may partly explain a nagging mystery among researchers working on MNIST:
how did Cortes and Vapnik [2] obtain 1.1% error with a degree 4 polynomial ?
3We compared the Max Wins voting algorithm with the DAGSVM decision tree algorithm and found them to perform equally, and worse than 1vsR SVMs. This is is surprising
in the light of results published on other tasks [8] , and would require further investigations
beyond the scope of this paper.
4Slightly better performance was obtained with a new algorithm that uses the incremental properties of our training procedure (this is be the performance reported in the
tables). In a transductive inference framework , treat the test example as a training example: for each of the M possible labels, retrain the M among (M(":-l) classifiers that use
examples with such label. The best label will be the one that causes the smallest increase
in the multiclass margin p such that it combines the classifier margins pc in the following
manner
~= ,,~
2
~
2
P
c~M Pc
The fact that this margin predicts generalization is "justified" by Theorem 1 in [8].
Kernel
K3
K4
K5
Kg
[(2
[(4
K5
0.40
#err #SV
136
8367
127
8331
125
8834
136
13002
147
9014
125
8668
125
8944
Duality Ratio stop
0.75
#err
#SV
136
11132
117
11807
119
12786
137
18784
128
11663
12222
119
125
12852
# err
132
119
119
141
131
117
125
0.99
#SV
13762
15746
17868
25953
13918
16604
18085
Table 1: SVMs on MNIST with 10 1vsR classifiers
Kernel
K3
K4
K5
SVM/ratio at 0.99
# err
#SV
138
11952
135
13526
191
13526
XVM/Hybrid
# err
#SV
117
17020
110
16066
114
15775
Table 2: SVMjXVM on MNIST with 45 1vs1 classifiers
The 103 errors obtained with K4 and r = 0.5 in Table 3 represent only about 1%
error: t his is t he lowest error ever reported for any learning technique without a
priori knowledge about the fact that t he input data corresponds to a pixel map (the
lowest reproducible error previously reported was 1.2% with SVMs and polynomials
of degree 9 [4], it could be reduced to 0.6% by using invariance properties of the
pixel map). The downside is that XVMs require 4 times as many support vectors
as standards SVMs.
Table 3 compares stopping according to t he duality ratio and t he hybrid criterion.
With t he duality ratio, the best performance is most often reached with r = 0.50 (if
t his happens to be consistent ly true, validation data to decide when to stop could
be spared). The hybrid criterion does not require validation data and yields errors
that, while higher than the best XVM, are lower than SVMs and only require a few
more support vectors. It takes fewer iterations to train than SVMs. One way to
interpret this hybrid stopping criterion is that we stop when interpolation in some
(but not all) directions account for all non-interpolated vectors. This suggest t hat
interpolation is only desirable in a few directions.
XVM gain is stronger in the 1vs 1 case (Table 2). This suggests that extrapolating
on a convex hull that contains several different classes (in the 1vsR case) may be
undesirable.
Kernel
K3
K4
K5
Kg
K2
[(4
0.40
# err #SV
118
46662
112
40274
109
36912
128
35809
114
43909
108
36980
Duality Ratio stop
0. 50
# err
#SV
111
43819
43132
103
44226
106
126
39462
114
46905
111
40329
0.75
# err #SV
116
50216
110
52861
110
49383
131
50233
114
53676
114
51088
Hybrid.
Stop Crit.
# err
#SV
125 20604
18002
107
17322
107
125
19218
119
20152
16895
108
Table 3: XVMs on MNIST wit h 10 1vsR classifiers
6
The Soft Margin Case
MNIST is characterized by the quasi-absence of outliers, so to assume that the
data is fully separable does not impair performance at all. To extend XVMs to
non-separable data, we first considered the traditional approaches of adding slack
variables to allow margin constraints to be violated. The most commonly used approach with SVMs adds linear slack variables to the unitary margin. Its application
to the XVM requires to give up the weight normalization constraint, so that the
usual unitary margin can be used in the constraints [9] .
Compared to standard SVMs, a new issue to tackle is the fact that each constraint
corresponds to a pair of vectors: ideally, we should handle N 2 slack variables ~ij.
To have linear constraints that can be solved with KKT, we need to have the
decomposition ~ij = ('T}k+1)~i+'T}k~; (factors ('T}k+1) and 'T}k are added here to ease
later simplifications).
Similarly to Eq.(3), the constraint on the extrapolation from any pair of points is
Vi,j E Ck, Yk (w. (('T}k+1)xi - 'T}kXj) +b) 2: 1 - ('T}k+1)~i - 'T}k~; with ~i'~; 2: 0
Introducing J.tk = max (Yk(w,xj+b) - ~;) and Vk = min (Yk(W,Xi+b)
.ECk
JECk
+ ~i)'
(9)
we ob-
tain the simpler double constraint
Vi E Ck , Vk -~i ~ Yk(W,Xi+b) ~ J.tk+~; with ~i'~; 2: 0
(10)
It follows from Eq.(9) that J.tk and Vk are tied through (l+'T}k)vk = l+'T}kJ.tk
If we fix J.tk (and thus Vk) instead of treating it as an optimization variable, it would
amount to a standard SVM regression problem with {-I, + I} outputs, the width
of the asymmetric f-insensitive tube being J.tk-Vk = (~~~;)'
This remark makes it possible for the reader to verify the results we reported on
MNIST. Vsing the publicly available SVM software SVMtorch [1] with C = 10 and
f = 0.1 as the width of the f-tube yields a 10-class error rate of 1.15% while the
best performance using SVMtorch in classification mode is 1.3% (in both cases, we
use Gaussian kernels with parameter (J = 1650).
An explicit minimization on J.tk requires to add to the standard SVM regression
problem the following constraint over the Lagrange multipliers (we use the same
notation as in [9]) :
Yi= l
Yi=- l
Yi= l
Yi=- l
Note that we still have the standard regression constraint
I: ai = I: ai
This has not been implemented yet , as we question the pertinence of the ~; slack
variables for XVMs. Experiments with SVMtorch on a variety of tasks where
non-zero slacks are required to achieve optimal performance (Reuters, VCI/Forest,
VCI/Breast cancer) have not shown significant improvement using the regression
mode while we vary the width of the f-tube.
Many experiments on SVMs have reported that removing the outliers often gives
efficient and sparse solutions. The early stopping heuristics that we have presented
for XVMs suggest strategies to avoid learning (or to unlearn) the outliers, and this
is the approach we are currently exploring.
7
Concluding Remarks
This paper shows that large margin classification on extrapolated data is equivalent
to the addition of the minimization of a second external margin to the standard SVM
approach. The associated optimization problem is solved efficiently with convex
hull distance minimization algorithms. A 1 % error rate is obtained on the MNIST
dataset: it is the lowest ever obtained without a-priori knowledge about the data.
We are currently trying to identify what other types of dataset show similar gains
over SVMs, to determine how dependent XVM performance is on the facts that the
data is separable or has invariance properties. We have only explored a few among
the many variations the XVM models and algorithms allow , and a justification
of why and when they generalize would help model selection. Geometry-based
algorithms that handle potential outliers are also under investigation.
Learning Theory bounds that would be a function of both the margin and some
form of variance of the data would be necessary to predict XVM generalization and
allow us to also consider the extrapolation factor 'TJ as an optimization variable.
References
[1] R. Collobert and S. Bengio. Support vector machines for large-scale regression
problems. Technical Report IDIAP-RR-00-17, IDIAP, 2000.
[2] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:1- 25 ,
1995.
[3] D. Crisp and C.J.C. Burges. A geometric interpretation of v-SVM classifiers.
In Advances in Neural Information Processing Systems 12, S. A. Solla, T. K.
Leen, K.-R. Mller, eds, Cambridge, MA, 2000. MIT Press.
[4] D. DeCoste and B. Schoelkopf. Training invariant support vector machines.
Machine Learning, special issue on Support Vector Machines and Methods, 200l.
[5] S.S. Keerthi, S.K. Shevade, C. Bhattacharyya, and K.R.K. Murthy. A fast
iterative nearest point algorithm for support vector machine classifier design.
IEEE transactions on neural networks, 11(1):124 - 136, jan 2000.
[6] A. Kowalczyk. Maximal margin perceptron. In Advances in Large Margin Classifiers, Smola, Bartlett, Schlkopf, and Schuurmans, editors, Cambridge, MA,
2000. MIT Press.
[7] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. proceedings of the IEEE, 86(11), 1998.
[8] J. Platt, N. Christianini, and J. Shawe-Taylor. Large margin dags for multiclass
classification. In Advances in Neural Information Processing Systems 12, S. A.
Solla, T. K. Leen, K.-R. Mller, eds, Cambridge, MA, 2000. MIT Press.
[9] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, New-York, 1998.
| 2037 |@word trial:1 polynomial:5 stronger:1 tried:1 decomposition:1 concise:1 harder:1 configuration:2 contains:4 att:1 disparity:1 document:1 bhattacharyya:1 err:10 com:1 nt:2 surprising:1 yet:1 must:1 john:1 numerical:1 happen:1 reproducible:1 extrapolating:1 treating:1 v:3 leaf:1 fewer:1 jkj:1 ron:1 location:1 hyperplanes:1 simpler:2 along:3 consists:1 combine:2 manner:1 inter:2 expected:1 behavior:1 eck:2 decoste:1 becomes:1 moreover:1 notation:2 maximizes:2 lowest:3 what:2 kg:2 ehk:1 developed:1 nj:1 every:2 act:1 voting:1 tackle:1 classifier:19 k2:1 platt:1 ly:1 before:3 positive:1 treat:1 despite:1 interpolation:2 chose:1 studied:1 suggests:1 ease:1 range:2 statistically:1 lecun:1 testing:3 digit:2 procedure:1 jan:1 significantly:4 projection:1 word:1 suggest:3 get:1 onto:1 undesirable:2 selection:1 crisp:1 equivalent:3 map:2 lagrangian:1 center:2 maximizing:1 convex:14 wit:1 his:2 handle:2 traditionally:2 variation:1 justification:1 target:1 suppose:1 play:1 us:1 recognition:1 jk:5 asymmetric:1 predicts:2 database:2 observed:1 lix:1 role:1 solved:2 schoelkopf:1 solla:2 yk:10 intuition:1 discriminates:1 mu:1 ideally:1 trained:2 segment:1 crit:1 upon:1 kxj:2 differently:1 train:1 separated:1 fast:1 kp:6 outside:2 choosing:1 heuristic:4 larger:1 solve:1 say:1 relax:1 unseen:1 transductive:1 final:1 rr:1 propose:1 maximal:3 achieve:1 description:1 vsing:1 convergence:5 double:4 incremental:1 converges:1 tk:7 spent:1 help:1 measured:1 ij:2 nearest:6 eq:8 implemented:2 idiap:2 direction:4 correct:1 hull:14 require:7 fix:1 generalization:4 preliminary:1 investigation:2 elementary:1 extension:2 exploring:1 c_:1 around:1 considered:2 exp:3 k3:3 scope:2 predict:1 mller:2 vary:1 early:1 smallest:1 estimation:1 label:3 currently:2 largest:1 minimization:5 mit:3 always:1 gaussian:3 ck:6 rather:1 avoid:1 vk:19 improvement:1 likelihood:1 laurel:1 ave:1 spared:1 baseline:1 sense:1 inference:1 dependent:1 stopping:6 typically:1 hovers:1 quasi:1 pixel:3 issue:2 dual:6 aforementioned:1 among:4 classification:4 priori:2 constrained:1 special:1 once:1 look:3 minimized:1 others:2 report:1 few:3 geometry:1 keerthi:1 vsr:7 intra:2 light:1 primal:4 pc:2 tj:1 necessary:2 respective:1 f3ixi:1 tree:1 taylor:1 stopped:1 instance:2 soft:1 downside:1 maximization:1 introducing:1 reported:6 sv:10 again:2 tube:3 containing:1 choose:1 worse:1 external:4 account:3 potential:1 b2:2 caused:1 depends:1 vi:5 collobert:1 later:1 extrapolation:10 lab:1 observer:1 analyze:1 reached:1 contribution:1 publicly:1 variance:5 percept:1 maximized:2 correspond:1 yield:3 efficiently:1 identify:1 generalize:2 handwritten:1 critically:1 schlkopf:1 researcher:1 published:1 murthy:1 explain:1 reach:1 ed:2 associated:1 stop:8 gain:3 dataset:2 knowledge:3 higher:1 follow:1 improved:2 formulation:1 leen:2 just:1 smola:1 shevade:1 working:1 fallon:1 vci:2 mode:2 normalized:5 true:3 multiplier:6 verify:2 contain:2 read:1 width:5 unnormalized:2 criterion:4 trying:1 ranging:1 common:1 unlearn:1 jp:1 insensitive:1 belong:2 interpretation:4 he:5 extend:1 interpret:1 significant:2 cambridge:3 imposing:1 ai:2 dag:1 fk:4 similarly:2 shawe:1 dj:1 had:1 similarity:1 longer:1 surface:1 patrick:1 add:2 optimizing:1 binary:1 success:1 yi:6 minimum:4 additional:1 recognized:1 determine:2 maximize:3 converge:1 ii:4 full:1 desirable:1 ykw:2 rj:2 technical:1 characterized:1 x28:1 offer:2 long:1 y_:1 lin:1 mle:2 equally:1 regression:5 breast:1 iteration:1 kernel:13 represent:1 normalization:1 justified:1 whereas:1 addition:1 rest:1 seem:1 call:1 unitary:2 bengio:2 easy:2 variety:2 xj:4 gave:1 escaping:1 f3i:1 reduce:1 idea:1 haffner:3 y112:1 multiclass:3 bartlett:1 tain:1 effort:1 wo:1 york:1 cause:2 svs:1 remark:2 abies:1 amount:1 extensively:1 svms:19 reduced:1 outperform:1 dotted:1 sign:1 per:2 threshold:3 k4:6 christianini:1 sum:1 mystery:1 reader:1 decide:1 separation:4 decision:2 prefer:1 ob:1 bound:1 simplification:1 lec:1 lyk:1 constraint:19 software:1 interpolated:1 min:3 concluding:1 separable:4 o_:1 according:1 combination:2 poor:1 belonging:1 slightly:2 son:1 sam:1 dagsvm:1 wi:1 happens:1 outlier:4 invariant:1 taken:1 equation:1 previously:1 vs1:3 slack:5 count:1 end:1 available:1 k5:4 observe:1 ocr:2 enforce:1 kowalczyk:1 alternative:1 hat:1 iix:2 recombination:1 bl:1 added:1 question:1 strategy:1 usual:1 traditional:3 ow:1 win:1 gradient:1 distance:4 pertinence:1 separating:1 discriminant:3 trivial:1 toward:1 furthest:2 rjk:4 index:2 ratio:7 minimizing:2 pie:1 relate:1 negative:2 kfk:1 design:1 perform:3 extended:1 ever:2 pair:7 required:1 beyond:3 impair:1 below:1 max:3 explanation:1 rely:1 hybrid:7 improve:1 lk:7 kj:2 geometric:4 asymptotic:1 fully:1 bear:1 interesting:3 proportional:1 squeezing:1 validation:2 degree:2 offered:1 sufficient:1 consistent:1 editor:1 story:1 share:1 cancer:1 extrapolated:16 keeping:1 side:1 allow:5 burges:1 perceptron:1 sparse:1 plain:1 computes:1 commonly:1 far:1 transaction:1 implicitly:1 keep:1 kkt:1 b1:2 xi:14 search:2 iterative:3 why:2 table:11 ku:1 forest:1 schuurmans:1 bottou:1 did:3 reuters:1 fig:3 retrain:1 wiley:1 explicit:1 tied:1 down:1 theorem:1 removing:1 list:1 explored:1 svm:11 cortes:2 exists:1 ilwll:1 mnist:13 consist:1 adding:3 vapnik:3 margin:39 nk:6 gap:1 easier:1 subtract:1 ilx:1 lagrange:5 expressed:1 corresponds:3 chance:1 ma:3 replace:1 fisher:2 man:1 shared:1 absence:1 reducing:1 hyperplane:2 called:2 total:3 duality:7 invariance:3 partly:1 internal:3 support:20 violated:1 svmtorch:3 phenomenon:1 |
1,138 | 2,038 | Spike timing and the coding of naturalistic
sounds in a central auditory area of songbirds
Brian D. Wright,
Kamal Sen,
William Bialek
and Allison J. Doupe
Sloan?Swartz
Center
for
Theoretical
Neurobiology
Departments of Physiology and Psychiatry
University
of California at San Francisco, San Francisco, California 94143?0444
NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540
Department
of Physics, Princeton University, Princeton, New Jersey 08544
bdwright/kamal/ajd @phy.ucsf.edu, wbialek@princeton.edu
Abstract
In nature, animals encounter high dimensional sensory stimuli that have
complex statistical and dynamical structure. Attempts to study the neural coding of these natural signals face challenges both in the selection of
the signal ensemble and in the analysis of the resulting neural responses.
For zebra finches, naturalistic stimuli can be defined as sounds that they
encounter in a colony of conspecific birds. We assembled an ensemble of
these sounds by recording groups of 10-40 zebra finches, and then analyzed the response of single neurons in the songbird central auditory area
(field L) to continuous playback of long segments from this ensemble.
Following methods developed in the fly visual system, we measured the
information that spike trains provide about the acoustic stimulus without any assumptions about which features of the stimulus are relevant.
Preliminary results indicate that large amounts of information are carried
by spike timing, with roughly half of the information accessible only
at time resolutions better than 10 ms; additional information is still being revealed as time resolution is improved to 2 ms. Information can be
decomposed into that carried by the locking of individual spikes to the
stimulus (or modulations of spike rate) vs. that carried by timing in spike
patterns. Initial results show that in field L, temporal patterns give at least
% extra information. Thus, single central auditory neurons can provide an informative representation of naturalistic sounds, in which spike
timing may play a significant role.
1 Introduction
Nearly fifty years ago, Barlow [1] and Attneave [2] suggested that the brain may construct
a neural code that provides an efficient representation for the sensory stimuli that occur
in the natural world. Slightly earlier, MacKay and McCulloch [3] emphasized that neurons that could make use of spike timing?rather than a coarser ?rate code??would have
available a vastly larger capacity to convey information, although they left open the question of whether this capacity is used efficiently. Theories for timing codes and efficient
representation have been discussed extensively, but the evidence for these attractive ideas
remains tenuous. A real attack on these issues requires (at least) that we actually measure
the information content and efficiency of the neural code under stimulus conditions that
approximate the natural ones. In practice, constructing an ensemble of ?natural? stimuli
inevitably involves compromises, and the responses to such complex dynamic signals can
be very difficult to analyze.
At present the clearest evidence on efficiency and timing in the coding of naturalistic stimuli
comes from central invertebrate neurons [4, 5] and from the sensory periphery [6, 7] and
thalamus [8, 9] of vertebrates. The situation for central vertebrate brain areas is much less
clear. Here we use the songbird auditory system as an accessible test case for these ideas.
The set of songbird telencephalic auditory areas known as the field L complex is analogous
to mammalian auditory cortex and contains neurons that are strongly driven by natural
sounds, including the songs of birds of the same species (conspecifics) [10, 11, 12, 13].
We record from the zebra finch field L, using naturalistic stimuli that consist of recordings
from groups of 10-40 conspecific birds. We find that single neurons in field L show robust
and well modulated responses to playback of long segments from this song ensemble, and
that we are able to maintain recordings of sufficient stability to collect the large data sets
that are required for a model independent information theoretic analysis. Here we give a
preliminary account of our experiments.
2 A naturalistic ensemble
Auditory processing of complex sounds is critical for perception and communication in
many species, including humans, but surprisingly little is known about how high level brain
areas accomplish this task. Songbirds provide a useful model for tackling this issue, because each bird within a species produces a complex individualized acoustic signal known
as a song, which reflects some innate information about the species? song as well as information learned from a ?tutor? in early life. In addition to learning their own song, birds use
the acoustic information in songs of others to identify mates and group members, to discriminate neighbors from intruders, and to control their living space [14]. Consistent with
how ethologically critical these functions are, songbirds have a large number of forebrain
auditory areas with strong and increasingly specialized responses to songs [11, 15, 16]. The
combination of a rich set of behaviorally relevant stimuli and a series of high-level auditory areas responsive to those sounds provides an opportunity to reveal general principles
of central neural encoding of complex sensory stimuli. Many prior studies have chosen to
study neural responses to individual songs or altered versions thereof. In order to make the
sounds studied increasingly complex and natural, we have made recordings of the sounds
encountered by birds in our colony of zebra finches. To generate the sound ensemble that
was used in this study we first created long records of the vocalizations of groups of 10-40
zebra finches in a soundproof acoustic chamber with a directional microphone above the
bird cages. The group of birds generated a wide variety of vocalizations including songs
and a variety of different types of calls. Segments of these sounds were then joined to cre
ate the sounds presented in the experiment. One of the segments that was presented (
sec) was repeated in alternation with different segments.
We recorded the neural responses in field L of one of the birds from the group to the
ensemble of natural sounds played back through a speaker, at an intensity approximately
equal to that in the colony recording. This bird was lightly anesthetized with urethane. We
used a single electrode to record the neural response waveforms and sorted single units
offline. Further details concerning experimental techniques can be found in Ref. [13].
A
D
500 ms
B
50 Hz
C
Figure 1: A. Spike raster of 4 seconds of the responses of a single neuron in field L to a
30 second segment of a natural sound ensemble of zebra finch sounds. The stimulus was
repeated 80 times. B. Peri-stimulus time histogram (PSTH) with 1 ms bins. C. Sound
pressure waveform for the natural sound ensemble. D. Blowup of segment shown in the
box in A. The scale bar is 50 ms.
3 Information in spike sequences
The auditory telencephalon of birds consists of a set of areas known as the field L complex,
which receive input from the auditory thalamus and project to increasingly selective auditory areas such as NCM, cHV and NIf [12, 17] and ultimately to the brain areas specialized
for the bird?s own song. Field L neurons respond to simple stimuli such as tone bursts, and
are organized in a roughly tonotopic fashion [18], but also respond robustly to many complex sounds, including songs. Figure 1 shows 4 seconds of the responses of a cell in field L
to repeated presentations of a 30 sec segment from the natural ensemble described above.
Averaging over presentations, we see that spike rates are well modulated. Looking at the
responses on a finer time resolution we see that aspects of the spike train are reproducible
ms time scale. This encourages us to measure the information content of
on at least a
these responses over a range of time scales, down to millisecond resolution.
Our approach to estimating the information content of spike trains follows Ref. [4]. At
some time (defined relative to the repeating stimulus) we open a window of size to look
at the response. Within this window we discretize the spike arrival times with resolution
so that the response becomes a ?word? with
letters. If the time resolution
is
very small, the allowed letters are only 1 and 0, but as
becomes larger one must keep
track of multiple spikes within each bin. Examining the whole experiment, we sample
40
Total Entropy
Noise Entropy
Mutual Info
35
Information Rate (bits/sec)
30
25
20
15
10
5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
1/Nrepeats
Figure 2: Mutual information rate for the spike train is shown as a function of data size for
ms.
ms and
the probability distribution of words,
, and the entropy of this distribution sets the
capacity of the code to convey information about the stimulus:
!"
#%$'&)(+*
(1)
where the notation reminds us that the entropy depends both on the size of the words that
we consider and on the time resolution with which we classify the responses. We can think
of this entropy as measuring the size of the neuron?s vocabulary.
,
Because the whole experiment contributes to defining the vocabulary size, estimating the
distribution
and hence the total entropy is not significantly limited by the problems
of finite sample size. This can be seen in Fig. 2 in the stability of the total entropy with
changing the number of repeats used in the analysis. Here we show the total entropy as a
rate in bits per second by dividing the entropy by the time window .
While the capacity of the code is limited by the total entropy, to convey information particular words in the vocabulary must be associated, more or less reliably, with particular stimulus features. If we look at one time relative to the (long) stimulus, and examine the words
generated on repeated presentations, we sample the conditional distribution
. This
distribution has an entropy that quantifies the noise in the response at time , and averaging
over all times we obtain the average noise entropy,
10 )2 34 5 7689 "/. : ' ;
"/. =<,>?#%$'&)(+*
-/.
(2)
>
where indicates a time average (in general, denotes an average over the variable
). Technically, the above average should be an average over stimuli , however, for
a sufficiently long and rich stimulus, the ensemble average over can be replaced by a
time average. For the noise entropy, the problem of sampling is much more severe, since
each distribution
is estimated from a number of examples given by the number
of repeats. Still, as shown in Fig. 2, we find that the dependence of our estimate on sample
size is simple and regular; specifically, we find
/.
5
4 4 3 5
4 4 3
(3)
This is what we expect for any entropy estimate if the distribution is well sampled, and
if we make stronger assumptions about the sampling process (independence of trials etc.)
we can even estimate the correction coefficient [19]. In systems where much larger
data sets are available this extrapolation procedure has been checked, and the observation
of a good fit to Eq. (3) is a strong indication that larger sample sizes will be consistent
with
; further, this extrapolation can be tested against bounds
on the entropy that are derived from more robust quantities [4]. Most importantly, failure
to observe Eq. (3) means that we are in a regime where sampling is not sufficient to draw
reliable conclusions without more sophisticated arguments, and we exclude these regions
of and
from our discussion.
5
Ideally, to measure the spike train total and noise entropy rates, we want to go to the limit of
infinite word duration. A true entropy is extensive, which here means that it grows linearly
with spike train word duration , so that the entropy rate
is constant. For finite
word duration however, words sampled at neighboring times will have correlations between
them due, in part, to correlations in the stimulus (for birdsong these stimulus autocorrelation time scales can extend up to
ms). Since the word samples are not completely
independent, the raw entropy rate is an overestimate of the true entropy rate. The effect is
larger for smaller word duration and the leading dependence of the raw estimate is
5
*
(4)
where and we have already taken the infinite data size limit. We cannot directly
take the large limit, since for large word lengths we eventually reach a data sampling
limit beyond which we are unable to reliably compute the word distributions. On the other
hand, if there is a range of for which the distributions are sufficiently well sampled,
the behavior in Eq. (4) should be observed and can be used to extrapolate to infinite word
size [4]. We have checked that our data shows this behavior and that it sets in for word
sizes below the limit where the data sampling problem occurs. For example, in the case of
ms, it applies for below the limit of ms (above this we
the noise entropy, for
run into sampling problems). The total entropy estimate is nearly perfectly extensive.
Finally, we combine estimates of total and noise entropies to obtain the information that
words carry about the sensory stimulus,
5 0 2 3 4 5 #%$'&)(
(5)
Figure 2 shows the total and noise entropy rates as well as the mutual information rate for
ms and time resolution
ms. The error bars on the raw
a time window
entropy and information rates were estimated to be approximately ! bits/sec using a
simple bootstrap procedure over the repeated trials. The extrapolation to infinite data size
is shown for the mutual information rate estimate (error bars in the extrapolated values
will be "#! bits/sec) and is consistent with the prediction of Eq. (3). Since the total
entropy is nearly extensive and the noise entropy rate decreases with word duration due to
subextensive corrections as described above, the mutual information rate shown in Fig. 2
grows with word duration. We find that there is an upward change in the mutual information
5
Information Rate (bits/sec)
4.5
Spike Train
4
Independent Events
3.5
3
2.5
2
1.5
0
5
10
15
?? (ms)
20
25
30
35
ms) and single spike events as a
Figure 3: Information rates for the spike train (
function of time resolution
of the spike rasters, corrected for finite data size effects.
ms) of
ms and
rate (computed at
%, in the large limit. For
ms that is in the
simplicity in the following, we shall look at a fixed word duration
well-sampled region for all time resolutions
considered.
The mutual information rate measures the rate at which the spike train removes uncertainty
about the stimulus. However, the mutual information estimate does not depend on identifying either the relevant features of the stimulus or the relevant features of the response,
which is crucial in analyzing the response to such complex stimuli. In this sense, our estimates of information transmission and efficiency are independent of any model for the
code, and provide a benchmark against which such models could be tested.
One way to look at the information results is to fix our time window and ask what
happens as we change our time resolution
. When
, the ?word? describing
the response is nothing but the number of spikes in the window, so we have a rate or
counting code. As we decrease
, we gradually distinguish more and more detail in the
ms
arrangement of spikes in the window. We chose a range of values from
in our analyses to cover previously observed response windows for field L neurons and to
ms) of individual song syllables or notes.
probe the behaviorally relevant time scale (
For
ms, we show the results (extrapolated to infinite data size) in the upper curve of
Fig. 3. The spike train mutual information shows a clear increase as the timing resolution
is improved. In addition, Fig. 3 shows that roughly half of the information is accessible at
time resolutions better than ms and additional information is still being revealed as time
resolution is improved to 2 ms.
4 Information in rate modulation
Knowing the mutual information between the stimulus and the spike train (defined in the
window ), we would like to ask whether this can be accounted for by the information
in single spike events or whether there is some additional information conveyed by the
patterns of spikes. In the latter case, we have precisely what we mean by a temporal or
timing code: there is information beyond that attributable to the probability of single spike
events occurring at time relative to the onset of the stimulus. By event at time , we mean
that the event occurs between time and time
, where
is the resolution at which
we are looking at the spike train. This probability is simply proportional to the firing rate
(or peri-stimulus time histogram (PSTH))
at time normalized
by the mean firing rate
. Specifically if the duration of each repeated trial is
we have
.
spk @
4 4
4 4 *
(6)
4
spk @ .
4
where
denotes the stimulus history ( " ). The probability of a spike
event at ,
a priori of knowing the stimulus history, is flat:
spk @
. Thus, the
mutual information between the stimulus and the single spike events is [20]:
(%$
spk @
6
-< > #%$'&)(+*
(7)
where
is the PSTH binned to resolution
and the stimulus average in the first expression is replaced by a time average in the second (as discussed in the calculation of the
noise entropy in spike train words in the previous section). We find that this information
ms. Supposing that the individual spike events are inis approximately bit for
dependent (i.e. no intrinsic spike train correlations), the information rate in single spike
events is obtained by multiplying the mutual information per spike (Eq. 7) by the mean
Hz). This gives an upper bound to the single spike event
firing rate of the neuron (
contribution to the information rate and is shown in the lower curve of Fig. 3 (error bars
are again " ! bits/sec). Comparing with the spike train information (upper curve), we
ms, there is at least
% of the total information in
see that at a resolution of
the spike train that cannot be attributable to single spike events. Thus there is some pattern
of spikes that is contributing synergistically to the mutual information. The fact discussed,
in the previous section, that the spike train information rate grows subextensively with the
the word duration out to the point where data sampling becomes problematic is further
confirmation of the synergy from spike patterns. Thus we have shown model-independent
evidence for a temporal code in the neural responses.
8
5 Conclusion
Until now, few experiments on neural responses in high level, central vertebrate brain areas
have measured the information that these responses provide about dynamic, naturalistic
sensory signals. As emphasized in earlier work on invertebrate systems, information theoretic approaches have the advantage that they require no assumptions about the features of
the stimulus to which neurons respond. Using this method in the songbird auditory forebrain, we found that patterns of spikes seem to be special events in the neural code of these
neurons, since they carry more information than expected by adding up the contributions
of individual spikes. It remains to be determined what these spike patterns are, what stimulus features they may encode, and what mechanisms may be responsible for reading such
codes at even higher levels of processing.
Acknowledgments
Work at UCSF was supported by grants from the NIH (NS34835) and the Sloan-Swartz
Center for Theoretical Neurobiology. BDW and KS supported by NRSA grants from the
NIDCD. We thank Katrin Schenk and Robert Liu for useful discussions.
References
1. Barlow, H.B. (1961). Possible principles underlying the transformation of sensory messages. In Sensory Communication, W.A. Rosenblith, ed., pp. 217?234 (MIT Press, Cambridge, MA).
2. Attneave, F. (1954). Some informational aspects of visual perception. Psychol. Rev. 61,
183?193.
3. MacKay, D. and McCulloch, W.S. (1952). The limiting information capacity of a neuronal
link. Bull. Math. Biophys. 14, 127?135.
4. Strong, S.P., Koberle, R., de Ruyter van Steveninck, R. and Bialek, W. (1998). Entropy and
information in neural spike trains, Phys. Rev. Lett. 80, 197?200.
5. Lewen, G.D., Bialek, W. and de Ruyter van Steveninck, R.R. (2001). Neural coding of
naturalistic motion stimuli. Network 12, 317?329.
6. Rieke, F., Bodnar, D.A. and Bialek, W. (1995). Naturalistic stimuli increase the rate and
efficiency of information transmission by primary auditory afferents. Proc. R. Soc. Lond.
B 262, 259?265.
7. Berry II, M.J., Warland, D.K. and Meister, M. (1997). The structure and precision of retinal
spike trains. Proc. Nat. Acad. Sci. (USA) 94, 5411?5416.
8. Reinagel, P. and Reid, R.C. (2000). Temporal coding of visual information in the thalamus.
J. Neurosci. 20, 5392?5400.
9. Liu, R.C., Tzonev, S., Rebrik, S. and Miller, K.D. (2001). Variability and information in a
neural code of the cat lateral geniculate nucleus. J. Neurophysiol. 86, 2789?2806.
10. Scheich, H., Langner, G. and Bonke, D. (1979). Responsiveness of units in the auditory
neostriatum of the guinea fowl (Numida meleagris) to species-specific calls and synthetic
stimuli II. Discrimination of Iambus-Like Calls. J. Comp. Physiol. A 132, 257?276.
11. Lewicki, M.S. and Arthur, B.J. (1996). Hierarchical organization of auditory temporal
context sensitivity. J. Neurosci. 16(21), 6987?6998.
12. Janata, P. and Margoliash, D. (1999). Gradual emergence of song selectivity in sensorimotor structures of the male zebra finch song system. J. Neurosci. 19(12), 5108?5118.
13. Theunissen, F.E., Sen, K. and Doupe, A.J. (2000). Spectral temporal receptive fields of
nonlinear auditory neurons obtained using natural sounds. J. Neurosci. 20(6), 2315?2331.
14. Searcy, W.A. and Nowicki, S. (1999). In The Design of Animal Communication, M.D.
Hauser and M. Konishi, eds., pp. 577?595 (MIT Press, Cambridge, MA).
15. Margoliash, D. (1983). Acoustic parameters underlying the responses of song-specific
neurons in the white-crowned sparrow. J. Neurosci. 3(5), 1039?1057.
16. Sen, K., Theunissen, F.E. and Doupe, A.J. (2001). Feature analysis of natural sounds in the
songbird auditory forebrain. J. Neurophysiol. 86, 1445?1458.
17. Stripling, R., Kruse, A.A. and Clayton, D.F. (2001). Development of song responses in the
zebra finch caudomedial neostriatum: role of genomic and electrophysiological activities.
J. Neurobiol. 48, 163?180.
18. Zaretsky, M.D. and Konishi, M. (1976). Tonotopic organization in the avian telencephalon.
Brain Res. 111, 167?171.
19. Treves, A. and Panzeri, S. (1995). The upward bias in measures of information derived
from limited data samples. Neural Comput., 7, 399?407.
20. Brenner, N., Strong, S., Koberle, R. and Bialek, W. (2000). Synergy in a neural code,
Neural Comput. 12, 1531?1552.
| 2038 |@word trial:3 version:1 stronger:1 open:2 gradual:1 pressure:1 carry:2 initial:1 synergistically:1 contains:1 series:1 liu:2 phy:1 comparing:1 tackling:1 must:2 physiol:1 informative:1 remove:1 reproducible:1 v:1 discrimination:1 half:2 ajd:1 tone:1 rebrik:1 record:3 provides:2 math:1 psth:3 attack:1 burst:1 consists:1 combine:1 autocorrelation:1 expected:1 blowup:1 behavior:2 roughly:3 examine:1 brain:6 informational:1 decomposed:1 little:1 window:9 vertebrate:3 chv:1 project:1 estimating:2 becomes:3 notation:1 underlying:2 mcculloch:2 what:6 neurobiol:1 developed:1 transformation:1 temporal:6 control:1 unit:2 grant:2 reid:1 overestimate:1 timing:9 limit:7 acad:1 encoding:1 analyzing:1 firing:3 modulation:2 approximately:3 chose:1 bird:12 studied:1 k:1 collect:1 limited:3 range:3 steveninck:2 acknowledgment:1 responsible:1 practice:1 bootstrap:1 procedure:2 area:11 physiology:1 significantly:1 word:22 regular:1 naturalistic:9 cannot:2 selection:1 context:1 center:2 go:1 duration:9 resolution:17 simplicity:1 identifying:1 reinagel:1 importantly:1 stability:2 rieke:1 konishi:2 analogous:1 limiting:1 margoliash:2 play:1 mammalian:1 coarser:1 theunissen:2 observed:2 role:2 fly:1 region:2 decrease:2 locking:1 ideally:1 dynamic:2 ultimately:1 depend:1 segment:8 compromise:1 technically:1 efficiency:4 completely:1 neurophysiol:2 spk:4 langner:1 jersey:2 cat:1 train:19 wbialek:1 larger:5 tested:2 think:1 emergence:1 vocalization:2 sequence:1 indication:1 advantage:1 sen:3 neighboring:1 relevant:5 electrode:1 transmission:2 produce:1 avian:1 colony:3 measured:2 eq:5 strong:4 dividing:1 soc:1 involves:1 indicate:1 come:1 waveform:2 human:1 bin:2 require:1 fix:1 preliminary:2 brian:1 correction:2 sufficiently:2 considered:1 wright:1 panzeri:1 early:1 proc:2 geniculate:1 reflects:1 mit:2 behaviorally:2 genomic:1 playback:2 rather:1 encode:1 derived:2 indicates:1 psychiatry:1 sense:1 dependent:1 selective:1 upward:2 issue:2 priori:1 development:1 animal:2 special:1 mackay:2 mutual:13 field:12 construct:1 equal:1 sampling:7 look:4 nearly:3 kamal:2 fowl:1 others:1 stimulus:39 few:1 individual:5 replaced:2 conspecific:3 william:1 maintain:1 attempt:1 organization:2 message:1 severe:1 male:1 allison:1 analyzed:1 arthur:1 re:1 theoretical:2 classify:1 earlier:2 cover:1 measuring:1 bull:1 examining:1 hauser:1 accomplish:1 finch:8 synthetic:1 cre:1 peri:2 sensitivity:1 accessible:3 physic:1 vastly:1 central:7 recorded:1 again:1 leading:1 account:1 exclude:1 de:2 retinal:1 coding:5 sec:7 coefficient:1 sloan:2 afferent:1 depends:1 onset:1 extrapolation:3 analyze:1 contribution:2 nidcd:1 efficiently:1 ensemble:12 miller:1 identify:1 directional:1 raw:3 multiplying:1 comp:1 finer:1 ago:1 history:2 reach:1 phys:1 rosenblith:1 checked:2 ed:2 against:2 raster:2 failure:1 sensorimotor:1 pp:2 clearest:1 thereof:1 attneave:2 associated:1 sampled:4 auditory:18 ask:2 electrophysiological:1 organized:1 sophisticated:1 actually:1 back:1 higher:1 response:25 improved:3 box:1 strongly:1 nif:1 correlation:3 hand:1 until:1 nonlinear:1 cage:1 reveal:1 grows:3 innate:1 usa:1 effect:2 normalized:1 true:2 barlow:2 hence:1 nowicki:1 reminds:1 attractive:1 white:1 encourages:1 songbird:8 speaker:1 m:25 theoretic:2 motion:1 nih:1 specialized:2 discussed:3 extend:1 significant:1 cambridge:2 zebra:8 crowned:1 cortex:1 etc:1 own:2 driven:1 periphery:1 selectivity:1 life:1 alternation:1 seen:1 responsiveness:1 additional:3 tzonev:1 swartz:2 kruse:1 living:1 signal:5 ii:2 multiple:1 sound:20 thalamus:3 sparrow:1 calculation:1 long:5 concerning:1 prediction:1 histogram:2 cell:1 receive:1 addition:2 want:1 crucial:1 extra:1 fifty:1 recording:5 hz:2 supposing:1 member:1 seem:1 call:3 counting:1 revealed:2 variety:2 independence:2 fit:1 perfectly:1 idea:2 knowing:2 whether:3 expression:1 birdsong:1 song:16 useful:2 clear:2 amount:1 repeating:1 extensively:1 generate:1 problematic:1 millisecond:1 estimated:2 track:1 per:2 shall:1 group:6 changing:1 year:1 run:1 letter:2 uncertainty:1 respond:3 intruder:1 draw:1 bit:7 bound:2 played:1 distinguish:1 syllable:1 encountered:1 activity:1 binned:1 occur:1 precisely:1 flat:1 katrin:1 invertebrate:2 lightly:1 aspect:2 argument:1 lond:1 department:2 combination:1 tonotopic:2 ate:1 slightly:1 increasingly:3 smaller:1 rev:2 happens:1 gradually:1 taken:1 remains:2 forebrain:3 describing:1 eventually:1 previously:1 mechanism:1 meister:1 available:2 probe:1 observe:1 hierarchical:1 spectral:1 chamber:1 responsive:1 robustly:1 encounter:2 denotes:2 opportunity:1 warland:1 tutor:1 question:1 quantity:1 spike:50 already:1 occurs:2 arrangement:1 dependence:2 primary:1 receptive:1 bialek:5 individualized:1 unable:1 thank:1 capacity:5 link:1 sci:1 lateral:1 code:14 length:1 difficult:1 robert:1 info:1 design:1 reliably:2 discretize:1 ethologically:1 neuron:15 observation:1 upper:3 benchmark:1 mate:1 finite:3 neostriatum:2 inevitably:1 nrsa:1 situation:1 neurobiology:2 communication:3 looking:2 defining:1 variability:1 intensity:1 treves:1 clayton:1 required:1 extensive:3 california:2 acoustic:5 learned:1 assembled:1 able:1 suggested:1 bar:4 dynamical:1 pattern:7 perception:2 beyond:2 below:2 regime:1 reading:1 challenge:1 including:4 reliable:1 critical:2 event:13 natural:12 altered:1 created:1 carried:3 psychol:1 koberle:2 prior:1 lewen:1 berry:1 contributing:1 relative:3 expect:1 proportional:1 nucleus:1 conveyed:1 sufficient:2 consistent:3 principle:2 extrapolated:2 surprisingly:1 repeat:2 accounted:1 supported:2 guinea:1 offline:1 bias:1 institute:1 neighbor:1 wide:1 face:1 anesthetized:1 van:2 curve:3 lett:1 vocabulary:3 world:1 rich:2 sensory:8 made:1 san:2 approximate:1 tenuous:1 keep:1 synergy:2 francisco:2 continuous:1 quantifies:1 nature:1 robust:2 confirmation:1 ruyter:2 contributes:1 complex:10 constructing:1 linearly:1 neurosci:5 whole:2 noise:10 arrival:1 nothing:1 repeated:6 ref:2 convey:3 allowed:1 neuronal:1 fig:6 fashion:1 attributable:2 precision:1 comput:2 down:1 specific:2 emphasized:2 evidence:3 consist:1 intrinsic:1 adding:1 nec:1 nat:1 occurring:1 biophys:1 entropy:29 simply:1 visual:3 ncm:1 joined:1 lewicki:1 applies:1 ma:2 conditional:1 sorted:1 presentation:3 brenner:1 content:3 change:2 specifically:2 infinite:5 corrected:1 determined:1 averaging:2 microphone:1 total:11 specie:5 discriminate:1 experimental:1 doupe:3 latter:1 modulated:2 ucsf:2 princeton:4 bodnar:1 extrapolate:1 |
1,139 | 2,039 | ADynamic HMM for On-line
Segmentation of Sequential Data
Jens Kohlmorgen*
Fraunhofer FIRST.IDA
Kekulestr. 7
12489 Berlin, Germany
Steven Lemm
Fraunhofer FIRST.IDA
Kekulestr. 7
12489 Berlin, Germany
jek@first?fraunhofer.de
lemm @first?fraunhofer.de
Abstract
We propose a novel method for the analysis of sequential data
that exhibits an inherent mode switching. In particular, the data
might be a non-stationary time series from a dynamical system
that switches between multiple operating modes. Unlike other approaches, our method processes the data incrementally and without
any training of internal parameters. We use an HMM with a dynamically changing number of states and an on-line variant of the
Viterbi algorithm that performs an unsupervised segmentation and
classification of the data on-the-fly, i.e. the method is able to process incoming data in real-time. The main idea of the approach is
to track and segment changes of the probability density of the data
in a sliding window on the incoming data stream. The usefulness
of the algorithm is demonstrated by an application to a switching
dynamical system.
1
Introduction
Abrupt changes can occur in many different real-world systems like, for example,
in speech, in climatological or industrial processes, in financial markets, and also
in physiological signals (EEG/MEG). Methods for the analysis of time-varying dynamical systems are therefore an important issue in many application areas. In [12],
we introduced the annealed competition of experts method for time series from nonlinear switching dynamics, related approaches were presented, e.g., in [2, 6, 9, 14].
For a brief review of some of these models see [5], a good introduction is given in
[3].
We here present a different approach in two respects. First, the segmentation does
not depend on the predictability of the system. Instead, we merely estimate the
density distribution of the data and track its changes. This is particularly an improvement for systems where data is hard to predict, like, for example, EEG recordings [7] or financial data. Second, it is an on-line method. An incoming data stream
is processed incrementally while keeping the computational effort limited by a fixed
? http://www.first.fraunhofer.de/..-.jek
upper bound, i.e. the algorithm is able to perpetually segment and classify data
streams with a fixed amount of memory and CPU resources. It is even possible to
continuously monitor measured data in real-time, as long as the sampling rate is
not too high.l The main reason for achieving a high on-line processing speed is the
fact that the method, in contrast to the approaches above, does not involve any
training, i.e. iterative adaptation of parameters. Instead, it optimizes the segmentation on-the-fly by means of dynamic programming [1], which thereby results in an
automatic correction or fine-tuning of previously estimated segmentation bounds.
2
The segmentation algorithm
We consider the problem of continuously segmenting a data stream on-line and
simultaneously labeling the segments. The data stream is supposed to have a sequential or temporal structure as follows: it is supposed to consist of consecutive
blocks of data in such a way that the data points in each block originate from
the same underlying distribution. The segmentation task is to be performed in an
unsupervised fashion, i.e. without any a-priori given labels or segmentation bounds.
2.1
Using pdfs as features for segmentation
Consider Yl, Y2 , Y3, ... , with Yt E Rn, an incoming data stream to be analyzed.
The sequence might have already passed a pre-processing step like filtering or subsampling, as long as this can be done on-the-fly in case of an on-line scenario. As
a first step of further processing, it might then be useful to exploit an idea from
dynamical systems theory and embed the data into a higher-dimensional space,
which aims to reconstruct the state space of the underlying system,
Xt
= (Yt,Yt-n'" ,Yt -(m-l)r )'
(1)
The parameter m is called the embedding dimension and T is called the delay
parameter of the embedding. The dimension of the vectors Xt thus is d = m n. The
idea behind embedding is that the measured data might be a potentially non-linear
projection of the systems state or phase space. In any case, an embedding in a
higher-dimensional space might help to resolve structure in the data, a property
which is exploited, e.g., in scatter plots. After the embedding step one might
perform a sub-sampling of the embedded data in order to reduce the amount of
data for real-time processing. 2 Next, we want to track the density distribution of
the embedded data and therefore estimate the probability density function (pdf) in a
sliding window of length W. We use a standard density estimator with multivariate
Gaussian kernels [4] for this purpose, centered on the data points 3 in the window
~
}W -l
{ Xt-w
w=o,
()
1
~l
1
Pt x = W ~ (27fa 2 )d/2 exp
(x - Xt_w)2)
(
-
2a 2
.
(2)
The kernel width a is a smoothing parameter and its value is important to obtain
a good representation of the underlying distribution. We propose to choose a proportional to the mean distance of each Xt to its first d nearest neighbors, averaged
over a sample set {xt}.
1 In our reported application we can process data at 1000 Hz (450 Hz including display)
on a 1.33 GHz PC in MATLAB/C under Linux, which we expect is sufficient for a large
number of applications.
2In that case, our further notation of time indices would refer to the subsampled data.
3We use if to denote a specific vector-valued point and x to denote a vector-valued
variable.
2.2
Similarity of two pdfs
Once we have sampled enough data points to compute the first pdf according to
eq. (2), we can compute a new pdf with each new incoming data point. In order
to quantify the difference between two such functions, f and g, we use the squared
L2 -Norm, also called integrated squared error (ISE) , d(f, g) = J(f - g)2 dx , which
can be calculated analytically if f and 9 are mixtures of Gaussians as in our case
of pdfs estimated from data windows,
(3)
2.3
The HMM in the off-line case
Before we can discuss the on-line variant, it is necessary to introduce the HMM and
the respective off-line algorithm first. For a given a data sequence, {X'dT=l' we can
obtain the corresponding sequence of pdfs {Pt(X)}tES, S = {W, ... , T}, according
to eq. (2). We now construct a hidden Markov model (HMM) where each of these
pdfs is represented by a state s E S, with S being the set of states in the HMM.
For each state s, we define a continuous observation probability distribution,
-
( (X) I s-~
)
PPt
1
V 21f <;
exp
(
-
d(Ps(X),Pt(x)))
22
<;
'
(4)
for observing a pdf Pt(x) in state s. Next, the initial state distribution {1f s LES
of the HMM is given by the uniform distribution, 1fs = liN, with N = lSI being
the number of states. Thus, each state is a-priori equally probable. The HMM
transition matrix, A = (PijkjES, determines each probability to switch from a
state Si to a state Sj. Our aim is to find a representation of the given sequence of
pdfs in terms of a sequence of a small number of representative pdfs, that we call
prototypes, which moreover exhibits only a small number of prototype changes. We
therefore define A in such a way that transitions to the same state are k times more
likely than transitions to any of the other states,
_ {
Pij -
k+~-l
1
k+N - l
;ifi=J
;ifi-j.J
(5)
This completes the definition of our HMM. Note that this HMM has only two free
parameters, k and <;. The well-known Viterbi algorithm [13] can now be applied
to the above HMM in order to compute the optimal - i.e. the most likely - state
sequence of prototype pdfs that might have generated the given sequence of pdfs.
This state sequence represents the segmentation we are aiming at. We can compute
the most likely state sequence more efficiently if we compute it in terms of costs,
c = -log(p), instead of probabilities p, i.e. instead of computing the maximum of
the likelihood function L , we compute the minimum of the cost function , -log(L),
which yields the optimal state sequence as well. In this way we can replace products
by sums and avoid numerical problems [13]. In addition to that, we can further
simplify the computation for the special case of our particular HMM architecture,
which finally results in the following algorithm:
For each time step, t = w, ... ,T, we compute for all S E S the cost cs(t) of the optimal state sequence from W to t, subject to the constraint that it ends in state S at
time t. We call these constrained optimal sequences c-paths and the unconstrained
optimum 0* -path. The iteration can be formulated as follows, with ds,t being a
short hand for d(ps(x)'pt(x)) and bs,s denoting the Kronecker delta function :
Initialization, Vs E S:
Cs(W)
:=
(6)
ds ,w,
Induction, Vs E S:
cs(t) := ds,t
+ min
sES { cs(t -
1)
+ C (1- bs 's)},
for t = W
+ 1, ... , T,
(7)
Termination:
0* :=
(8)
min { cs(T) } .
sES
The regularization constant C, which is given by C = 2C; 2 10g(k) and thus subsumes
our two free HMM parameters, can be interpreted as transition cost for switching
to a new state in the path. 4 The optimal prototype sequence with minimal costs
0* for the complete series of pdfs, which is determined in the last step, is obtained
by logging and updating the c-paths for all states s during the iteration and finally
choosing the one with minimal costs according to eq. (8).
2.4
The on-line algorithm
In order to turn the above segmentation algorithm into an on-line algorithm, we
must restrict the incremental update in eq. (7), such that it only uses pdfs (and
therewith states) from past data points. We neglect at this stage that memory and
CPU resources are limited.
Suppose that we have already processed data up to T - 1. When a new data point
YT arrives at time T, we can generate a new embedded vector XT (once we have
sampled enough initial data points for the embedding), we have a new pdf pT(X)
(once we have sampled enough embedded vectors Xt for the first pdf window), and
thus we have given a new HMM state. We can also readily compute the distances
between the new pdf and all the previous pdfs, dT,t, t < T, according to eq. (3).
A similarly simple and straightforward update of the costs, the c-paths and the
optimal state sequence is only possible, however, if we neglect to consider potential
c-paths that would have contained the new pdf as a prototype in previous segments.
In that case we can simply reuse the c-paths from T - 1. The on-line update at
time T for these restricted paths, that we henceforth denote with a tilde, can be
performed as follows:
For T = W, we have cw(W) := o*(W) := dw,w = O. For T > W:
1. Compute the cost cT(T - 1) for the new state s
For t =
T - 1, compute
w, ... ,
=T
at time T - 1:
0
ift=W
CT(t) :=dT,t+ { min{cT(t-1) ; o*(t-1)+C}: else
(9)
and update
o*(t) := CT(t), if CT(t) < o*(t).
(10)
Here we use all previous optimal segmentations o*(t), so we don't need to
keep the complete matrix (cs(t))S,tES and repeatedly compute the minimum
4We developed an algorithm that computes an appropriate value for the hyperparameter
C from a sample set {it}. Due to the limited space we will present that algorithm in a
forthcoming publication [8].
over all states. However, we must store and update the history of optimal
segmentations 8* (t).
2. Update from T - 1 to T and compute cs(T) for all states s E S obtained
so far, and also get 8*(T): For s = W, ... , T , compute
cs(T) := ds,T + min {cs(T - 1); 8*(T - 1) + C}
(11)
and finally get the cost of the optimal path
8* (T) := min {cs(T)} .
sES
(12)
As for the off-line case, the above algorithm only shows the update equations for
the costs of the C- and 8* -paths. The associated state sequences must be logged
simultaneously during the computation. Note that this can be done by just storing
the sequence of switching points for each path. Moreover, we do not need to keep
the full matrix (cs(t))s ,tES for the update, the most recent column is sufficient.
So far we have presented the incremental version of the segmentation algorithm.
This algorithm still needs an amount of memory and CPU time that is increasing
with each new data point. In order to limit both resources to a fixed amount, we
must remove old pdfs, i.e. old HMM states, at some point. We propose to do this
by discarding all states with time indices smaller or equal to s each time the path
associated with cs(T) in eq. (11) exhibits a switch back from a more recent state/pdf
to the currently considered state s as a result of the min-operation in eq. (11). In
the above algorithm this can simply be done by setting W := s + 1 in that case,
which also allows us to discard the corresponding old cs(T)- and 8* (t)-paths, for all
s::::: sand t < s. In addition, the "if t = W" initialization clause in eq. (9) must be
ignored after the first such cut and the 8* (W - I)-path must therefore still be kept
to compute the else-part also for t = W now. Moreover, we do not have CT(W -1)
and we therefore assume min {CT(W - 1); 8*(W - 1) + C} = 8*(W - 1) + C (in
eq. (9)).
The explanation for this is as follows: A switch back in eq. (11) indicates that a
new data distribution is established, such that the c-path that ends in a pdf state
s from an old distribution routes its path through one of the more recent states
that represent the new distribution, which means that this has lower costs despite
of the incurred additional transition. Vice versa, a newly obtained pdf is unlikely
to properly represent the previous mode then, which justifies our above assumption
about CT (W -1). The effect of the proposed cut-off strategy is that we discard paths
that end in pdfs from old modes but still allow to find the optimal pdf prototype
within the current segment.
Cut-off conditions occur shortly after mode changes in the data and cause the
removal of HMM states with pdfs from old modes. However, if no mode change
takes place in the incoming data sequence, no states will be discarded. We therefore
still need to set a fixed upper limit", for the number of candidate paths/pdfs that
are simultaneously under consideration if we only have limited resources available.
When this limit is reached because no switches are detected, we must successively
discard the oldest path/pdf stored, which finally might result in choosing a suboptimal prototype for that segment however. Ultimately, a continuous discarding
even enforces a change of prototypes after 2", time steps if no switching is induced
by the data until then. The buffer size", should therefore be as large as possible. In
any case, the buffer overflow condition can be recorded along with the segmentation,
which allows us to identify such artificial switchings.
2.5
The labeling algorithm
A labeling algorithm is required to identify segments that represent the same underlying distribution and thus have similar pdf prototypes. The labeling algorithm
generates labels for the segments and assigns identical labels to segments that are
similar in this respect. To this end, we propose a relatively simple on-line clustering
scheme for the prototypes, since we expect the prototypes obtained from the same
underlying distribution to be already well-separated from the other prototypes as
a result of the segmentation algorithm. We assign a new label to a segment if the
distance of its associated prototype to all preceding prototypes exceeds a certain
threshold
and we assign the existing label of the closest preceding prototype
otherwise. This can be written as
e,
l(R) = { ne.wlabel ,. if min1:'Sr<R {d(Pt(r) (x), Pt(R) (x))} >
1 (mdexmml:'Sr<R {d(Pt(r) (x), Pt(R) (x))} ), else;
e
(13)
with the initialization l(l) = newlabel. Here, r = 1, ... , R, denotes the enumeration
of the segments obtained so far , and t(?) denotes the mapping to the index of the
corresponding pdf prototype. Note that the segmentation algorithm might replace
a number of recent pdf prototypes (and also recent segmentation bounds) during
the on-line processing in order to optimize the segmentation each time new data is
presented. Therefore, a relabeling of all segments that have changed is necessary in
each update step of the labeler.
As for the hyperparameters (J and C, we developed an algorithm that computes a
suitable value for from a sample set {X'd. We refer to our forthcoming publication
[8].
3
e
Application
We illustrate our approach by an application to a time series from switching dynamics based on the Mackey-Glass delay differential equation,
dx(t) = -O.lx(t)
dt
0.2x(t - td) .
td)l?
+ 1 + x( t -
(14)
Eq. (14) describes a high-dimensional chaotic system that was originally introduced
as a model of blood cell regulation [10]. In our example, four stationary operating
modes, A, B, C, and D, are established by using different delays, td = 17, 23, 30,
and 35, respectively. The dynamics operates stationary in one mode for a certain
number of time steps, which is chosen at random between 200 and 300 (referring
to sub-sampled data with a step size 6. = 6) . It then randomly switches to one of
the other modes with uniform probability. This procedure is repeated 15 times, it
thus generates a switching chaotic time series with 15 stationary segments. We then
added a relatively large amount of "measurement" noise to the series: zero-mean
Gaussian noise with a standard deviation of 30% of the standard deviation of the
original series.
The on-line segmentation algorithm was then applied to the noisy data, i.e. processing was performed on-line although the full data set was already available in this
case. The scalar time series was embedded on-the-fly by using m = 6 and T = 1 (on
the sub-sampled data) and we used a pdf window of size W = 50. The algorithm
processed 457 data points per second on a 1.33 GHz PC in MATLAB/C under
Linux, including the display of the ongoing segmentation, where one can observe
the re-adaptation of past segmentation bounds and labels when new data becomes
available.
actual modes
mode D
modeC
mode B
mode A
labels
1
2
3
4
3
561
3
3
6
2
2
bounds
on-line segmentation
xl!)
Figure 1: Segmentation of a switching Mackey-Glass time series with noise (bottom)
that operates in four different modes (top). The on-line segmentation algorithm
(middle) , which receives the data points one by one, but not the mode information,
yields correct segmentation bounds almost everywhere. The on-line labeler, however, assigns more labels (6) than desired (4) , presumably due to the fact that the
segments are very short and noisy.
The final segmentation is shown in Fig. 1. Surprisingly, the bounds of the segments
are almost perfectly recovered from the very noisy data set. The only two exceptions
are the third segment from the right , which is noticeably shorter than the original
mode, and the segment in the middle, which is split in two by the algorithm.
This split actually makes sense if one compares it with the data: there is a visible
change in the signal characteristics at that point (t ~ 1500) even though the delay
parameter was not modified there. This change is recorded by the algorithm since
it operates in an unsupervised way.
The on-line labeling algorithm correctly assigns single labels to modes A, B, and
C, but it assigns three labels (4, 5, and 6) to the segments of mode D, the most
chaotic one. This is probably due to the small sample sizes (of the segments), in
combination with the large amount of noise in the data.
4
Discussion
We presented an on-line method for the unsupervised segmentation and identification of sequential data, in particular from non-stationary switching dynamics. It is
based on an HMM where the number of states varies dynamically as an effect of
the way the incoming data is processed. In contrast to other approaches , it processes the data on-line and potentially even in real-time without training of any
parameters. The method provides and updates a segmentation each time a new
data point arrives. In effect, past segmentation bounds and labels are automatically re-adapted when new incoming data points are processed. The number of
prototypes and labels that identify the segments is not fixed but determined by the
algorithm. We expect useful applications of this method in fields where complex
non-stationary dynamics plays an important role, like, e.g., in physiology (EEG,
MEG), climatology, in industrial applications, or in finance.
References
[1] Bellman, R. E. (1957). Dynamic Programming, Princeton University Press,
Princeton, NJ .
[2] Bengio, Y, Frasconi, P. (1995). An Input Output HMM Architecture. In: Advances in Neural Information Processing Systems 7 (eds. Tesauro, Touretzky,
Leen), Morgan Kaufmann, 427- 434.
[3] Bengio, Y (1999). Markovian Models for Sequential Data. Neural Computing
Surveys, http://www.icsi.berkeley.edu/~jagota/NCS, 2:129-162 .
[4] Bishop, C. M. (1995). Neural Networks for Pattern Recognition , Oxford Univ.
Press, NY.
[5] Husmeier, D. (2000). Learning Non-Stationary Conditional Probability Distributions. Neural Networks 13, 287- 290.
[6] Kehagias , A., Petridis, V. (1997). Time Series Segmentation using Predictive
Modular Neural Networks. Neural Computation 9, 1691- 1710.
[7] Kohlmorgen, J. , Miiller, K.-R., Rittweger, J. , Pawelzik, K. (2000). Identification of Nonstationary Dynamics in Physiological Recordings, Bioi Cybern
83(1),73- 84.
[8] Kohlmorgen, J. , Lemm, S. , to appear.
[9] Liehr, S., Pawelzik, K. , Kohlmorgen, J ., Miiller, K.-R. (1999). Hidden Markov
Mixtures of Experts with an Application to EEG Recordings from Sleep. Theo
Biosci 118, 246- 260.
[10] Mackey, M., Glass, 1. (1977). Oscillation and Chaos in a Physiological Control
System. Science 197, 287.
[11] Packard, N. H., Crutchfield J. P. , Farmer, J . D. , Shaw, R. S. (1980). Geometry
from a Time Series. Phys Rev Letters 45, 712- 716.
[12] Pawelzik, K., Kohlmorgen, J. , Miiller, K.-R. (1996). Annealed Competition of
Experts for a Segmentation and Classification of Switching Dynamics. Neural
Computation 8(2), 340- 356.
[13] Rabiner, L. R. (1989). A Tutorial on Hidden Markov Models and Selected
Applications in Speech Recognition, Proceedings of the IEEE 77(2) , 257- 286.
[14] Ramamurti, V., Ghosh, J. (1999). Structurally Adaptive Modular Networks
for Non-Stationary Environments. IEEE Tr. Neural Networks 10(1), 152- 160.
| 2039 |@word middle:2 version:1 norm:1 termination:1 thereby:1 tr:1 initial:2 series:11 denoting:1 past:3 existing:1 current:1 ida:2 recovered:1 si:1 scatter:1 dx:2 must:7 readily:1 written:1 numerical:1 visible:1 remove:1 plot:1 update:10 v:2 stationary:8 mackey:3 selected:1 oldest:1 short:2 provides:1 lx:1 along:1 differential:1 introduce:1 market:1 bellman:1 td:3 resolve:1 cpu:3 enumeration:1 kohlmorgen:5 window:6 increasing:1 becomes:1 actual:1 automatically:1 underlying:5 notation:1 moreover:3 interpreted:1 developed:2 ghosh:1 nj:1 temporal:1 berkeley:1 y3:1 finance:1 control:1 farmer:1 appear:1 segmenting:1 before:1 limit:3 aiming:1 switching:12 despite:1 oxford:1 path:19 might:9 initialization:3 dynamically:2 limited:4 averaged:1 enforces:1 block:2 chaotic:3 procedure:1 area:1 physiology:1 projection:1 pre:1 get:2 cybern:1 www:2 optimize:1 demonstrated:1 yt:5 annealed:2 straightforward:1 survey:1 abrupt:1 assigns:4 estimator:1 financial:2 dw:1 embedding:6 pt:10 suppose:1 play:1 programming:2 us:1 recognition:2 particularly:1 updating:1 cut:3 bottom:1 steven:1 min1:1 fly:4 role:1 icsi:1 environment:1 dynamic:9 ultimately:1 depend:1 segment:20 predictive:1 logging:1 represented:1 separated:1 univ:1 detected:1 artificial:1 labeling:5 choosing:2 ise:1 modular:2 valued:2 s:3 reconstruct:1 otherwise:1 noisy:3 final:1 sequence:17 propose:4 product:1 adaptation:2 supposed:2 competition:2 p:2 optimum:1 incremental:2 help:1 illustrate:1 measured:2 nearest:1 eq:11 c:13 quantify:1 correct:1 centered:1 noticeably:1 sand:1 assign:2 probable:1 correction:1 considered:1 exp:2 presumably:1 viterbi:2 predict:1 mapping:1 consecutive:1 purpose:1 label:12 currently:1 vice:1 gaussian:2 aim:2 modified:1 avoid:1 varying:1 publication:2 improvement:1 pdfs:16 properly:1 likelihood:1 indicates:1 industrial:2 contrast:2 sense:1 glass:3 integrated:1 unlikely:1 hidden:3 germany:2 issue:1 classification:2 priori:2 smoothing:1 special:1 constrained:1 equal:1 once:3 construct:1 field:1 frasconi:1 sampling:2 labeler:2 identical:1 represents:1 unsupervised:4 simplify:1 inherent:1 ppt:1 randomly:1 simultaneously:3 relabeling:1 subsampled:1 phase:1 geometry:1 analyzed:1 mixture:2 arrives:2 pc:2 behind:1 necessary:2 respective:1 shorter:1 old:6 re:2 desired:1 minimal:2 classify:1 column:1 markovian:1 cost:11 deviation:2 uniform:2 usefulness:1 delay:4 too:1 reported:1 stored:1 varies:1 referring:1 density:5 yl:1 off:5 continuously:2 linux:2 squared:2 recorded:2 successively:1 choose:1 henceforth:1 expert:3 jek:2 potential:1 de:3 subsumes:1 kehagias:1 stream:6 performed:3 observing:1 reached:1 kekulestr:2 kaufmann:1 characteristic:1 efficiently:1 yield:2 identify:3 rabiner:1 identification:2 therewith:1 history:1 phys:1 touretzky:1 ed:1 definition:1 associated:3 sampled:5 newly:1 segmentation:32 actually:1 back:2 higher:2 dt:4 originally:1 leen:1 done:3 though:1 just:1 stage:1 until:1 d:4 hand:1 receives:1 nonlinear:1 incrementally:2 mode:19 effect:3 y2:1 analytically:1 regularization:1 during:3 width:1 climatological:1 pdf:17 complete:2 climatology:1 performs:1 chaos:1 novel:1 consideration:1 clause:1 refer:2 measurement:1 biosci:1 versa:1 automatic:1 tuning:1 unconstrained:1 similarly:1 similarity:1 operating:2 multivariate:1 closest:1 recent:5 optimizes:1 discard:3 scenario:1 store:1 buffer:2 certain:2 route:1 tesauro:1 jagota:1 jens:1 exploited:1 morgan:1 minimum:2 additional:1 preceding:2 husmeier:1 signal:2 sliding:2 multiple:1 full:2 exceeds:1 long:2 lin:1 equally:1 variant:2 iteration:2 kernel:2 represent:3 cell:1 addition:2 want:1 fine:1 completes:1 else:3 unlike:1 sr:2 probably:1 recording:3 hz:2 subject:1 induced:1 call:2 nonstationary:1 split:2 enough:3 bengio:2 switch:6 forthcoming:2 architecture:2 ifi:2 restrict:1 suboptimal:1 reduce:1 idea:3 prototype:18 perfectly:1 crutchfield:1 passed:1 reuse:1 effort:1 f:1 miiller:3 speech:2 cause:1 repeatedly:1 matlab:2 ignored:1 useful:2 involve:1 amount:6 processed:5 http:2 generate:1 lsi:1 tutorial:1 estimated:2 delta:1 track:3 per:1 correctly:1 hyperparameter:1 four:2 threshold:1 monitor:1 achieving:1 blood:1 changing:1 kept:1 merely:1 sum:1 everywhere:1 letter:1 logged:1 place:1 almost:2 oscillation:1 bound:9 ct:8 display:2 sleep:1 adapted:1 occur:2 constraint:1 kronecker:1 lemm:3 generates:2 speed:1 min:7 relatively:2 according:4 combination:1 smaller:1 describes:1 rev:1 b:2 restricted:1 resource:4 equation:2 previously:1 pawelzik:3 discus:1 turn:1 end:4 available:3 gaussians:1 operation:1 observe:1 appropriate:1 shaw:1 shortly:1 original:2 denotes:2 clustering:1 subsampling:1 top:1 neglect:2 exploit:1 overflow:1 already:4 added:1 fa:1 strategy:1 exhibit:3 distance:3 cw:1 berlin:2 hmm:18 originate:1 reason:1 induction:1 meg:2 length:1 index:3 modec:1 nc:1 regulation:1 potentially:2 perform:1 upper:2 observation:1 markov:3 discarded:1 tilde:1 rn:1 introduced:2 required:1 established:2 able:2 dynamical:4 pattern:1 including:2 memory:3 explanation:1 packard:1 suitable:1 scheme:1 brief:1 ne:1 fraunhofer:5 review:1 l2:1 removal:1 embedded:5 expect:3 filtering:1 proportional:1 incurred:1 sufficient:2 pij:1 storing:1 ift:1 changed:1 surprisingly:1 last:1 keeping:1 free:2 theo:1 allow:1 neighbor:1 ghz:2 dimension:2 calculated:1 world:1 transition:5 computes:2 perpetually:1 adaptive:1 far:3 sj:1 keep:2 incoming:8 don:1 continuous:2 iterative:1 rittweger:1 eeg:4 complex:1 main:2 noise:4 hyperparameters:1 repeated:1 fig:1 representative:1 fashion:1 ny:1 predictability:1 sub:3 structurally:1 xl:1 candidate:1 third:1 embed:1 specific:1 xt:7 discarding:2 bishop:1 physiological:3 consist:1 sequential:5 te:3 justifies:1 simply:2 likely:3 contained:1 scalar:1 determines:1 conditional:1 bioi:1 formulated:1 replace:2 change:9 hard:1 determined:2 operates:3 called:3 exception:1 internal:1 ongoing:1 princeton:2 |
1,140 | 204 | 308
Donnett and Smithers
Neuronal Group Selection Theory:
A Grounding in Robotics
Jim Donnett and Tim Smithers
Department of Artificial Intelligence
University of Edinburgh
5 Forrest Hill
Edinburgh EH12QL
SCOTLAND
ABSTRACT
In this paper, we discuss a current attempt at applying the organizational principle Edelman calls Neuronal Group Selection to the
control of a real, two-link robotic manipulator. We begin by motivating the need for an alternative to the position-control paradigm
of classical robotics, and suggest that a possible avenue is to look
at the primitive animal limb 'neurologically ballistic' control mode.
We have been considering a selectionist approach to coordinating
a simple perception-action task.
1
MOTIVATION
The majority of industrial robots in the world are mechanical manipUlators - often
arm-like devices consisting of some number of rigid links with actuators mounted
where the links join that move adjacent links relative to each other, rotationally
or translation ally. At the joints there are typically also sensors measuring the
relative position of adjacent links, and it is in terms of position that manipulators
are generally controlled (a desired motion is specified as a desired position of the end
effector, from which can be derived the necessary positions of the links comprising
the manipulator). Position control dominates largely for historical reasons, rooted
in bang-bang control: manipulators bumped between mechanical stops placed so as
to enforce a desired trajectory for the end effector.
Neuronal Group Selection Theory: A Grounding in Robotics
1.1
SERVOMECHANISMS
Mechanical stops have been superceded by position-controlling servomechanisms,
negative feedback systems in which, for a typical manipulator with revolute joints, a
desired joint angle is compared with a feedback signal from the joint sensor signalling
actual measured angle; the difference controls the motive power output of the joint
actuator proportionally.
Where a manipulator is constructed of a number of links, there might be a servomechanism for each joint. In combination, it is well known that joint motions
can affect each other adversely, requiring careful design and analysis to reduce the
possibility of unpleasant dynamical instabilities. This is especially important when
the manipulator will be required to execute fast movements involving many or all
of the joints. We are interested in such dynamic tasks, and acknowledge some successful servomechanistic solutions (see [Andersson 19881, who describes a ping pong
playing robot), but seek an alternative that is not as computationally expensive.
1.2
ESCAPING POSITION CONTROL
In Nature, fast reaching and striking is a primitive and fundamental mode of control. In fast, time-optimal, neurologically ballistic movements (such as horizontal
rotations of the head where subjects are instructed to turn it as fast as possible,
[Hannaford and Stark 1985]), muscle activity patterns seem to show three phases:
a launching phase (a burst of agonist), a braking phase (an antagonist burst), and a
locking phase (a second agonist burst). Experiments have shown (see [Wadman et
al. 1979]) that at least the first 100 mS of activity is the same even if a movement is
blocked mechanically (without forewarning the subject), suggesting that the launch
is specified from predetermined initial conditions (and is not immediately modified
from proprioceptive information). With the braking and locking phases acting as
a damping device at the end of the motion, the complete motion of the arm is
essentially specified by the initial conditions - a mode radically differing from traditional robot positional control. The overall coordination of movements might even
seem naive and simple when compared with the intricacies of servomechanisms (see
[Braitenberg 1989, N ahvi and Hashemi 19841 who discuss the crane driver's strategy
for shifting loads quickly and time-optimally).
The concept of letting insights (such as these) that can be gained from the biological sciences shape the engineering principles used to create artificial autonomous
systems is finding favour with a growing number of researchers in robotics. As it is
not generally trivial to see how life's devices can be mapped onto machines, there is
a need for some fundamental experimental work to develop and test the basic theoretical and empirical components of this approach, and we have been considering
various robotics problems from this perspective.
Here, we discuss an experimental two-link manipulator that performs a simple manipulation task - hitting a simple object perceived to be within its reach. The
perception of the object specifies the initial conditions that determine an arm mo-
309
310
Donnett and Smithers
tion that reaches it. In relating initial conditions with motor currents, we have been
considering a scheme based on Neuronal Group Selection Theory [Edelman 1987,
Reeke and Edelman 1988], a theory of brain organization. We believe this to be
the first attempt to apply selectionist ideas in a real machine, rather than just in
simulation.
2
NEURONAL GROUP SELECTION THEORY
Edelman proposes Neuronal Group Selection (NGS) [Edelman 1978] as an organizing principle for higher brain function - mainly a biological basis for perception primarily applicable to the mammalian (and specifically, human) nervous system
[Edelman 1981]. The essential idea is that groups of cells, structurally varied as a
result of developmental processes, comprise a population from which are selected
those groups whose function leads to adaptive behaviour of the system. Similar
notions appear in immunology and, of course, evolutionary theory, although the
effects of neuronal group selection are manifest in the lifetime of the organism.
There are two premises on which the principle rests. The first is that the unit of
selection is a cell group of perhaps 50 to 10,000 neurons. Intra-group connections
between cells are assumed to vary (greatly) between groups, but other connections
in the brain (particularly inter-group) are quite specific. The second premise is that
the kinds of nervous systems whose organization the principle addresses are able to
adapt to circumstances not previously encountered by the organism or its species
[Edelman 1978].
2.1
THREE CENTRAL TENETS
There are three important ideas in the NGS theory [Edelman 1987].
? A first selective process (cell division, migration, differentiation, or death)
results in structural diversity providing a primary repertoire of variant cell
groups.
? A second selective process occurs as the organism experiences its environment;
group activity that correlates with adaptive behaviour leads to differential
amplification of intra- and inter-group synaptic strengths (the connectivity
pattern remains unchanged). From the primary repertoire are thus selected
groups whose adaptive functioning means they are more likely to find future
use - these groups form the ,econdary repertoire.
? Secondary repertoires themselves form populations, and the NGS theory additionally requires a notion of reentry, or connections between repertoires,
usually arranged in maps, of which the well-known retinotopic mapping of
the visual system is typical. These connections are critical for they correlate
motor and sensory repertoires, and lend the world the kind of spatiotemporal
continuity we all experience.
Neuronal Group Selection Theory: A Grounding in Robotics
2.2
REQUffiEMENTS OF SELECTIVE SYSTEMS
To be selective, a system must satisfy three requirements IReeke and Edelman 1988].
Given a configuration of input signals (ultimately from the sensory epithelia, but for
'deeper' repertoires mainly coming from other neuronal groups), if a group responds
in a specific way it has matched the input IEdelman 1978]. The first requirement of
a selective system is that it have a sufficiently large repertoire of variant elements to
ensure that an adequate match can be found for a wide range of inputs. Secondly,
enough of the groups in a repertoire must 'see' the diverse input signals effectively
and quickly so that selection can operate on these groups. And finally, there must
be a means for 'amplifying' the contribution, to the repertoire, of groups whose
operation when matching input signals has led to adaptive behaviour.
In determining the necessary number of groups in a repertoire, one must consider
the relationship between repertoire size and the specificity of member groups. On
the one hand, if groups are very specific, repertoires will need to be very large in
order to recognize a wide range of possible inputs. On the other hand, if groups are
not as discriminating, it will be possible to have smaller numbers of them, but in
the limit (a single group with virtually no specificity) different signals will no longer
be distinguishable. A simple way to quantify this might be to assume that each
of N groups has a fixed probability, P, of matching an input configuration; then a
typical measure IEdelman 1978] relating the effectiveness of recognition, r, to the
number of groups is r = 1 - (1 - p)N (see Fig. 1).
r
log N
Figure 1: Recognition as a Function of Repertoire Size
From the shape of the curve in Fig. 1, it is clear that, for such a measure, below
some lower threshold for N, the efficacy of recognition is equally poor. Similarly,
above an upper threshold for N, recognition does not improve substantially as more
groups are added.
3
SELECTIONISM IN OUR EXPERIMENT
Our manipulator is required to touch an object perceived to be within reach. This is
a well-defined but non-trivial problem in motor-sensory coordination. Churchland
proposes a geometrical solution for his two-eyed 'crab' IChurchland 1986]' in which
311
312
Donnett and Smithers
eye angles are mapped to those joint angles (the crab has a two-link arm) that
would bring the end of the arm to the point currently foveated by the eyes. Such
a novel solution, in which computation is implicit and massively parallel, would be
welcome; however, the crab is a simulation, and no heed is paid to the question of
how the appropriate sensory-motor mapping could be generated for a real arm.
Reeke and Edelman discuss an automaton, Darwin III, similar to the crab, but
which by selectional processes develops the ability to manipulate objects presented
to it in its environment [Reeke and Edelman 19881. The Darwin III simulation
does not account for arm dynamics; however, Edelman suggests that the training
paradigm is able to handle dynamic effects as well as the geometry of the problem
[Edelman 19891. We are attempting to implement a mechanical analogue of Darwin
III, somewhat simplified, but which will experience the real dynamics of motion.
S.l
EXPERIMENTAL ARCHITECTURE AND HARDWARE
The mechanical arrangement of our manipulator is shown in Fig. 2. The two links
have agonist/antagonist tendon-drive arrangement, with an actuator per tendon.
There are strain gauges in-line with the tendons. A manipulator 'reach' is specified
by six parameters: burst amplitude and period for each of the three phases, launch,
brake, and lock.
'I.
'I.
'I.
'I.
',tendons
'I.
l
upper-arm
left actuator
,,
,
'I.
" '0
"
Dri
forearm/
left actuator
\
U
~
upper-arm
right actuator
forearm
right actuator
Figure 2: Manipulator Mechanical Configuration
Neuronal Group Selection Theory: A Grounding in Robotics
At the end of the manipulator is an array of eleven pyroelectic-effect infrared detectors arranged in a U-shaped pattern. The relative location of a warm object
presented to the arm is registered by the sensors, and is converted to eleven 8-bit
integers. Since the sensor output is proportional to detected infrared energy flux,
objects at the same temperature will give a more positive reading if they are close
to the sensors than if they are further away. Also, a near object will register on
adjacent sensors, not just on the one oriented towards it. Therefore, for a single,
small object, a histogram of the eleven values will have a peak, and showing two
things (Fig. 3): the sensor 'seeing' the most flux indicates the relative direction
of the object, and the sharpness of the peak is proportional to the distance of the
object.
(object distant
and to the left)
(object near and
straight ahead)
Figure 3: Histograms for Distant Versus Near Objects
Modelled on Darwin III [Reeke and Edelman 1988], the architecture of the selectional perception-action coordinator is as in Fig. 4. The boxes represent repertoires
of appropriately interconnected groups of 'neurons'.
Darwin III responds mainly to contour in a two-dimensional world, analogous to the
recognition of histogram shape in our system. Where Darwin Ill's 'unique response'
network is sensitive to line segment lengths and orientations, ours is sensitive to the
length of subsequences in the array of sensor output values in which values increase
or decrease by the same amount, and the amounts by which they change; similarly,
where Darwin Ill's 'generic response' network is sensitive to presence of or changes
in orientation of lines, ours responds to the presence of the subsequences mentioned
above, and the positions in the array where two subsequences abut.
The recognition repertoires are reciprocally connected, and both connect to the motor repertoire which consists of ballistic-movement 6-tuples. The system considers
'touching perceived object' to be adaptive, so when recognition activity correlates
with a given 6-tuple, amplification ensures that the same response will be favoured
in future.
313
314
Donnett and Smithers
4
WORK TO DATE
As the sensing system is not yet functional, this aspect of the system is currently
simulated in an IBM PC/AT. The rest of the electrical and mechanical hardware
is in place. The major difficulty currently faced is that the selectional system will
become computationally intensive on a serial machine.
WORLD
FEATURE
DETECTOR
FEATURE
CORRELATOR
classification
couple
COMBINATION
RESPONSES
(UNIQU~
~
COMBINATION
RESPONSES
cim"':r"~f,~./(GENERIC)
motor map
MOTOR
ACTIONS
Figure 4: Experimental Architecture
For each possible ballistic 'reach', there must be a representation for the 'reach
6-tuple'. Therefore, the motor repertoire must become large as the dexterity of the
manipulator is increased. Similarly, as the array of sensors is extended (resolution
increased, or field of view widened), the classification repertoires must also grow.
On a serial machine, polling the groups in the repertoires must be done one at
a time, introducing a substantial delay between the registration of object and the
actual touch, precluding the interception by the manipulator of fast moving objects.
We are exploring possibilities for parallelizing the selectional process (and have for
this reason constructed a network of processing elements), with the expectation that
this will lead us closer to fast, dynamic manipulation, at minimal computational
expense.
Neuronal Group Selection Theory: A Grounding in Robotics
References
Russell L. Andersson. A Robot Ping-Pong Player: Experiment in Real- Time Intelligent Control. MIT Press, Cambridge, MA, 1988.
Valentino Braitenberg. "Some types of movement" , in C.G. Langton, ed., Artificial
Life, pp. 555-565, Addison-Wesley, 1989.
Paul M. Churchland. "Some reductive strategies in cognitive neurobiology". Mind,
95:279-309, 1986.
Jim Donnett and Tim Smithers. "Behaviour-based control of a two-link ballistic
arm". Dept. of Artificial Intelligence, University of Edinburgh, Research Paper RP
.158, 1990.
Gerald M. Edelman. "Group selection and phasic reentrant signalling: a theory of
higher brain function", in G.M. Edelman and V.B. Mountcastle, eds., The Mindful
Brain, pp. 51-100, MIT Press, Cambridge, MA, 1978.
Gerald M. Edelman. "Group selection as the basis for higher brain function", in
F.O. Schmitt et al., eds., Organization of the Cerebral Cortex, pp. 535-563, MIT
Press, Cambridge, MA, 1981.
Gerald M. Edelman. Neural Darwinism: The Theory of Neuronal Group Selection.
Basic Books, New York, 1987.
Gerald M. Edelman. Personal correspondence, 1989.
Blake Hannaford and Lawrence Stark. "Roles of the elements of the triphasic control
signal". Experimental Neurology, 90:619-634, 1985.
M.J. Nahvi and M.R. Hashemi. "A synthetic motor control system; possible parallels with transformations in cerebellar cortex", in J .R. Bloedel et al., eds., Cerebellar
Functions, pp. 67-69, Springer-Verlag, 1984.
George N. Reeke Jr. and Gerald M. Edelman. "Real brains and artificial intelligence", in Stephen R. Graubard, ed., The Artificial Intelligence Debate, pp.
143-173, The MIT Press, Cambridge, MA, 1988.
W.J. Wadman, J.J. Denier van der Gon, R.H. Geuse, and C.R. Mol. "Control of
fast goal-directed arm movements". Journal of Human Movement Studies, 5:3-17,
1979.
315
| 204 |@word simulation:3 seek:1 paid:1 reentrant:1 initial:4 configuration:3 efficacy:1 ours:2 precluding:1 current:2 yet:1 must:8 tenet:1 distant:2 predetermined:1 eleven:3 shape:3 motor:9 intelligence:4 selected:2 device:3 nervous:2 signalling:2 scotland:1 location:1 launching:1 burst:4 constructed:2 differential:1 driver:1 become:2 edelman:20 consists:1 epithelium:1 inter:2 themselves:1 growing:1 brain:7 reductive:1 actual:2 correlator:1 considering:3 begin:1 retinotopic:1 matched:1 kind:2 substantially:1 differing:1 finding:1 transformation:1 differentiation:1 interception:1 control:14 unit:1 appear:1 positive:1 engineering:1 limit:1 might:3 suggests:1 range:2 directed:1 unique:1 implement:1 empirical:1 eyed:1 matching:2 specificity:2 seeing:1 suggest:1 onto:1 close:1 selection:15 applying:1 instability:1 crane:1 map:2 brake:1 primitive:2 automaton:1 sharpness:1 resolution:1 immediately:1 insight:1 array:4 valentino:1 his:1 population:2 handle:1 notion:2 autonomous:1 analogous:1 controlling:1 element:3 expensive:1 particularly:1 recognition:7 mammalian:1 infrared:2 gon:1 role:1 electrical:1 ensures:1 connected:1 movement:8 decrease:1 russell:1 mentioned:1 substantial:1 developmental:1 pong:2 locking:2 environment:2 dynamic:5 gerald:5 ultimately:1 personal:1 segment:1 churchland:2 division:1 basis:2 joint:9 various:1 fast:7 artificial:6 detected:1 whose:4 quite:1 ability:1 interconnected:1 coming:1 date:1 organizing:1 amplification:2 requirement:2 object:16 tim:2 develop:1 measured:1 launch:2 quantify:1 direction:1 bumped:1 human:2 premise:2 behaviour:4 repertoire:20 biological:2 secondly:1 exploring:1 sufficiently:1 crab:4 blake:1 lawrence:1 mapping:2 mo:1 major:1 vary:1 perceived:3 applicable:1 ballistic:5 currently:3 coordination:2 amplifying:1 sensitive:3 create:1 gauge:1 mit:4 sensor:9 modified:1 reaching:1 rather:1 derived:1 indicates:1 mainly:3 greatly:1 industrial:1 rigid:1 typically:1 coordinator:1 selective:5 comprising:1 interested:1 polling:1 overall:1 motive:1 ill:2 orientation:2 classification:2 proposes:2 animal:1 field:1 comprise:1 shaped:1 ng:3 look:1 future:2 braitenberg:2 develops:1 intelligent:1 primarily:1 oriented:1 recognize:1 phase:6 consisting:1 geometry:1 attempt:2 organization:3 possibility:2 intra:2 pc:1 tuple:2 closer:1 necessary:2 experience:3 damping:1 desired:4 theoretical:1 darwin:7 minimal:1 effector:2 increased:2 measuring:1 organizational:1 introducing:1 abut:1 delay:1 successful:1 motivating:1 optimally:1 connect:1 spatiotemporal:1 synthetic:1 migration:1 fundamental:2 immunology:1 discriminating:1 peak:2 mechanically:1 quickly:2 connectivity:1 central:1 langton:1 adversely:1 cognitive:1 book:1 stark:2 suggesting:1 account:1 converted:1 diversity:1 satisfy:1 register:1 tion:1 view:1 parallel:2 contribution:1 largely:1 who:2 modelled:1 agonist:3 trajectory:1 researcher:1 drive:1 straight:1 ping:2 donnett:6 reach:6 detector:2 synaptic:1 ed:5 energy:1 pp:5 couple:1 stop:2 manifest:1 amplitude:1 wesley:1 higher:3 response:5 arranged:2 execute:1 box:1 done:1 lifetime:1 just:2 implicit:1 hand:2 ally:1 horizontal:1 touch:2 continuity:1 mode:3 perhaps:1 believe:1 manipulator:16 grounding:5 effect:3 requiring:1 concept:1 functioning:1 death:1 proprioceptive:1 adjacent:3 rooted:1 m:1 antagonist:2 hill:1 complete:1 performs:1 motion:5 bring:1 temperature:1 geometrical:1 dexterity:1 novel:1 rotation:1 reeke:5 heed:1 functional:1 neurologically:2 cerebral:1 organism:3 relating:2 braking:2 blocked:1 cambridge:4 similarly:3 mindful:1 moving:1 robot:4 longer:1 cortex:2 touching:1 perspective:1 manipulation:2 massively:1 verlag:1 graubard:1 life:2 der:1 muscle:1 rotationally:1 george:1 somewhat:1 determine:1 paradigm:2 period:1 signal:6 stephen:1 match:1 adapt:1 serial:2 equally:1 manipulate:1 controlled:1 involving:1 basic:2 variant:2 essentially:1 circumstance:1 smithers:6 expectation:1 histogram:3 represent:1 cerebellar:2 robotics:8 cell:5 grow:1 appropriately:1 rest:2 operate:1 subject:2 virtually:1 dri:1 thing:1 member:1 seem:2 effectiveness:1 call:1 integer:1 structural:1 near:3 presence:2 hashemi:2 iii:5 enough:1 affect:1 architecture:3 escaping:1 reduce:1 idea:3 avenue:1 intensive:1 favour:1 six:1 york:1 action:3 adequate:1 generally:2 proportionally:1 clear:1 selectional:4 amount:2 servomechanism:4 welcome:1 hardware:2 specifies:1 coordinating:1 per:1 diverse:1 group:39 threshold:2 registration:1 angle:4 striking:1 place:1 forrest:1 tendon:4 bit:1 correspondence:1 encountered:1 activity:4 strength:1 ahead:1 aspect:1 attempting:1 department:1 combination:3 poor:1 jr:1 describes:1 smaller:1 computationally:2 previously:1 remains:1 discus:4 turn:1 phasic:1 addison:1 letting:1 mind:1 end:5 operation:1 apply:1 limb:1 actuator:7 away:1 enforce:1 appropriate:1 generic:2 alternative:2 rp:1 ensure:1 lock:1 especially:1 classical:1 unchanged:1 move:1 added:1 question:1 occurs:1 arrangement:2 strategy:2 primary:2 traditional:1 responds:3 evolutionary:1 distance:1 link:11 mapped:2 simulated:1 majority:1 considers:1 trivial:2 reason:2 length:2 relationship:1 providing:1 expense:1 debate:1 negative:1 design:1 upper:3 neuron:2 forearm:2 acknowledge:1 extended:1 neurobiology:1 head:1 jim:2 strain:1 varied:1 parallelizing:1 mechanical:7 specified:4 required:2 connection:4 widened:1 registered:1 address:1 able:2 dynamical:1 perception:4 pattern:3 usually:1 below:1 reading:1 lend:1 reciprocally:1 shifting:1 power:1 critical:1 analogue:1 difficulty:1 warm:1 arm:12 scheme:1 improve:1 eye:2 cim:1 naive:1 denier:1 faced:1 mountcastle:1 determining:1 relative:4 revolute:1 mounted:1 proportional:2 versus:1 schmitt:1 principle:5 playing:1 translation:1 ibm:1 course:1 placed:1 deeper:1 wide:2 edinburgh:3 van:1 feedback:2 curve:1 world:4 contour:1 sensory:4 instructed:1 adaptive:5 simplified:1 historical:1 flux:2 correlate:3 robotic:1 assumed:1 tuples:1 neurology:1 subsequence:3 additionally:1 nature:1 mol:1 motivation:1 paul:1 neuronal:12 fig:5 join:1 favoured:1 structurally:1 position:9 load:1 specific:3 showing:1 sensing:1 dominates:1 essential:1 effectively:1 gained:1 foveated:1 intricacy:1 led:1 distinguishable:1 likely:1 positional:1 visual:1 hitting:1 springer:1 radically:1 ma:4 goal:1 bang:2 careful:1 towards:1 change:2 typical:3 specifically:1 acting:1 andersson:2 specie:1 secondary:1 experimental:5 player:1 unpleasant:1 dept:1 |
1,141 | 2,040 | PAC Generalization Bounds for Co-training
Sanjoy Dasgupta
AT&T Labs?Research
dasgupta@research.att.com
Michael L. Littman
AT&T Labs?Research
mlittman@research.att.com
David McAllester
AT&T Labs?Research
dmac@research.att.com
Abstract
The rule-based bootstrapping introduced by Yarowsky, and its cotraining variant by Blum and Mitchell, have met with considerable empirical success. Earlier work on the theory of co-training has been only
loosely related to empirically useful co-training algorithms. Here we give
a new PAC-style bound on generalization error which justifies both the
use of confidences ? partial rules and partial labeling of the unlabeled
data ? and the use of an agreement-based objective function as suggested by Collins and Singer. Our bounds apply to the multiclass case,
i.e., where instances are to be assigned one of labels for
.
1 Introduction
In this paper, we study bootstrapping algorithms for learning from unlabeled data. The
general idea in bootstrapping is to use some initial labeled data to build a (possibly partial)
predictive labeling procedure; then use the labeling procedure to label more data; then use
the newly labeled data to build a new predictive procedure and so on. This process can
be iterated until a fixed point is reached or some other stopping criterion is met. Here we
give PAC style bounds on generalization error which can be used to formally justify certain
boostrapping algorithms.
One well-known form of bootstrapping is the EM algorithm (Dempster, Laird and Rubin,
1977). This algorithm iteratively updates model parameters by using the current model
to infer (a probability distribution on) labels for the unlabeled data and then adjusting the
model parameters to fit the (distribution on) filled-in labels. When the model defines a joint
probability distribution over observable data and unobservable labels, each iteration of the
EM algorithm can be shown to increase the probability of the observable data given the
model parameters. However, EM is often subject to local minima ? situations in which
the filled-in data and the model parameters fit each other well but the model parameters are
far from their maximum-likelihood values. Furthermore, even if EM does find the globally
optimal maximum likelihood parameters, a model with a large number of parameters will
over-fit the data. No PAC-style guarantee has yet been given for the generalization accuracy
of the maximum likelihood model.
An alternative to EM is rule-based bootstrapping of the form used by Yarowsky (1995),
in which one assigns labels to some fraction of a corpus of unlabeled data and then infers
new labeling rules using these assigned labels as training data. New labels lead to new
rules which in turn lead to new labels, and so on. Unlike EM, rule-based bootstrapping
typically does not attempt to fill in, or assign a distribution over, labels unless there is
compelling evidence for a particular label. One intuitive motivation for this is that by
avoiding training on low-confidence filled-in labels one might avoid the self-justifying local
optima encountered by EM. Here we prove PAC-style generalization guarantees for rulebased bootstrapping.
Our results are based on an independence assumption introduced by Blum and
Mitchell (1998) which is rather strong but is used by many successful applications. Consider, for example, a stochastic context-free grammar. If we generate a parse tree using
such a grammar then the nonterminal symbol labeling a phrase separates the phrase from
its context ? the phrase and the context are statistically independent given the nonterminal symbol. More intuitively, in natural language the distribution of contexts into which a
given phrase can be inserted is determined to some extent by the ?type? of the phrase. The
type includes the syntactic category but might also include semantic subclassifications, for
instance, whether a noun phrase refers to a person, organization, or location. If we think of
each particular occurrence of a phrase as a triple
, where is the phrase itself,
is the ?type? of the phrase, and is the context, then we expect that is conditionally
independent of given . The conditional independence can be made to hold precisely if
we generate such triples using a stochastic context free grammar where is the syntactic
category of the phrase.
Blum and Mitchell introduce co-training as a general term for rule-based bootstrapping in
which each rule must be based entirely on or entirely on . In other words, there are
two distinct hypothesis classes, which consists of functions predicting from , and
which consists of functions predicting from . A co-training algorithm bootstraps
by alternately selecting and . The principal assumption made by Blum
and Mitchell is that is conditionally independent of given . Under such circumstances, they show that, given a weak predictor in , and given an algorithm which can
learn under random misclassification noise, it is possible to learn a good predictor in
. This gives some degree of justification for the co-training restriction on rule-based
bootstrapping. However, it does not provide a bound on generalization error as a function of empirically measurable quantities. Furthermore, there is no apparent relationship
between this PAC-learnability theorem and the iterative co-training algorithm they suggest.
Collins and Singer (1999) suggest a refinement of the co-training algorithm in which one
explicitly optimizes an objective function that measures the degree of agreement between
the predictions based on and those based on . They describe methods for ?boosting?
this objective function but do not provide any formal justification for the objective function
itself. Here we give a PAC-style performance guarantee in terms of this agreement rate.
This guarantee formally justifies the Collins and Singer suggestion.
In this paper, we use partial classification rules, which either output a class label or output
a special symbol indicating no opinion. The error of a partial rule is the probability that
the rule is incorrect given that it has an opinion. We work in the co-training setting where
we have a pair of partial rules and where (sometimes) predicts from and
(sometimes) predicts from . Each of the rules and can be ?composite rules?, such
as decision lists, where each composite rule contains a large set of smaller rules within it.
We give a bound on the generalization error of each of the rules and in terms of the
empirical agreement rate between the two rules. This bound formally justifies both the use
h1
X1
X2
h2
Y
Figure 1: The co-training scenario with rules and .
of agreement in the objective function and the use of partial rules. The bound shows the
potential power of unlabeled data ? low generalization error can be achieved for complex
rules with a sufficient quantity of unlabeled data. The use of partial rules is analogous to
the use of confidence ratings ? a partial rule is just a rule with two levels of confidence.
So the bound can also be viewed as justifying the partial labeling aspect of rule-based
bootstrapping, at least in the case of co-training where an independence assumption holds.
The generalization bound leads naturally to algorithms for optimizing the bound. A simple
greedy procedure for doing this is quite similar to the co-training algorithm suggested by
Collins and Singer.
2 The Main Result
We start with some basic definitions and observations. Let be an i.i.d. sample consisting
of individual samples , , . For any statement
we let
be the subset
. For any two statements and we define the empirical estimate to be
!
. For the co-training bounds proved here we assume data is drawn from
some distribution over triples
with #" %$ , and &" , and
where and are conditionally independent given , that is, ('
and )'
. In the co-training framework we are given an unlabeled
sample +* of pairs
drawn i.i.d. from the underlying distribution, and possibly
some labeled samples +, . We will mainly be interested in making inferences from the
unlabeled data. A partial rule on a set " is a mapping from " to %$ - . We will
be interested in pairs of partial rules and which largely agree on the unlabeled data.
The conditional probability relationships in our scenario are depicted graphically in figure 1. Important intuition is given by the data-processing inequality of information theory
.2 30 . In other words, any mutual information
(Cover and Thomas, 1991): ./ 10
between and must be mediated through . In particular, if and agree to a large
extent, then they must reveal a lot about . And yet finding such a pair requires no
labeled data at all. This simple observation is a major motivation for the proof, but things
are complicated considerably by partial rules and by approximate agreement.
For a given partial rule
with
(' 4
6587
9: <;<='?>A@BDC E1F
H GJIKGML
define a function 9 on %$
by
'ON P
'?;<
We want to be a nearly deterministic function of the actual label ; in other words, we
want 'Q9: ' 4 to be near one. We would also like to carry the same
information as . This is equivalent to saying that 9 should be a permutation of the possible
labels A$ . Here we give a condition using only unlabeled data which guarantees, up
to high confidence, that 9 is a permutation; this is the best we can hope for using unlabeled
data alone. We also bound the error rates D' 4 N :9: ' N ' 4 using only
unlabeled data. In the case of '
, if 9 is a permutation then 9 is either the identity
function or the function reversing the two possible values. We use the unlabeled data to
select and so that 9 is a permutation and has low error rate. We can then use a
smaller amount of labeled data to determine which permutation we have found.
We now introduce a few definitions related to sampling issues. Some measure of the complexity of rules and is needed; rather than VC dimension, we adopt a clean notion of
bit length. We assume that rules are specified in some rule language and write for the
number of bits used to specify the rule . We assume that the rule language is prefix-free
(no proper prefix of the bit string specifying a rule is itself a legal rule specification). A
$ . For given partial rules and
prefix free code satisfies the Kraft inequality
and N A$
we now define the following functions of the sample . The first, as
we will see, is a bound on the sampling error for empirical probabilities conditioned upon
'ON ' 4
. The second is a sampling-adjusted disagreement rate between and .
!#"$
% %$&'% (% )&* ! +
"-% .
0/1 352 467% ,
8
9
10/%:;/ ) '2 46< 9
) 02 /=%:0/> 52 43?<@"
A(
$3C
Note that if the sample size is sufficiently large (relative to and % ) then HB I
is near zero. Also note that if
and
have near perfect agreement when they both are
$3C is near one. We can now state our main result.
not then D I
Theorem 1 With probability at least $FE5C over the choice of the sample , we have that
for all
and, , if D I GC3
5 7 for $ N then (a) 9 is a permutation and (b) for
N
all $
-'?4 N6 ' N -' 4 IH5B I G3C
4 N639: ='ON
-' 4
-'?
D I G3C
The theorem states, in essence, that if the sample size is large, and and largely agree
on the unlabeled data, then ' 4 N6 '?N -' 4 is a good estimate of the error rate
4 N639: ='ON -' 4 .
-'?
The theorem also justifies the use of partial rules. Of course it is possible to convert a
partial rule to a total rule by forcing a random choice when the rule would otherwise return
. Converting a partial rule to a total rule in this way and then applying the above theorem
to the total rule gives a weaker result. An interesting case is when '
, is total and is
a perfect copy of , and -' 4 happens to be $1!
. In this case the empirical error
rate of the corresponding total rule ? the rule that guesses when has no opinion ? will
be statistically indistinguishable from from 1/2. However, in this case theorem 1 can still
establish that the false positive and false negative rate of the partial rule is near zero.
J
3 The Analysis
We start with a general lemma about conditional probability estimation.
Lemma 2 For any i.i.d. sample , and any statements and about individual instances
in the sample, the following holds with probability at least $
over the choice of .
M
MM
1E
A
KELD
P
N
O
MM
!D
M
(1)
MMM A 1E MMM 5
M
'
L
' MM 1E
' D ' D
L
Proof. We have the following where the third step follows by the Chernoff bound.
MM
MM
M 5
>
KE 1 C ! ,
1E
'ON6
'ON
MM
Therefore, with probability at least $
'
BI
GC3 (2)
for any given
. By the union bound and the Kraft inequality, we have that with
N .
probability at least $ E C this must hold simultaneously for all and , and all $
Lemma 3 Pick
any rules and for which equation (2) as well as D I G3C 5 7
'
N
'ON
'4
-' 4
hold for all $
N
. Then 9 is a permutation, and moreover, for any N ,
-' 4
'?N 39: =
'?N
5
$1!
Pick any N %$ . We need to show that there exists some such that
9: ='?N . By the definition of I and condition (2) we know
4 N ' N ' 4
?
' N ' N ' 4
'?
I 3
Since I 3
587 , it follows that '?N6 '?N ' 4 5 $! . Rewriting this
D
Proof.
D
D
E
GC
by conditioning on , we get
G GML
P
'
'ON
-' 4
!
'ON P
'
-' 4
5
$C
$!
"
The summation is a convex combination; therefore there must exist some such that
'QN '
)' 4
5
$! . So for each N there must exist a with 9: %Q
' N,
whereby 9 is a permutation.
Lemma 4 Pick any rules
<9:
='ON
'ON
-' 4
D
$C
and satisfying the conditions of the previous lemma. Then
is at least I 3 .
Proof. By the previous lemma 9 is a permutation, so 9: has the same information
content as . Therefore and are conditionally independent given 9: . For any N ,
D I
GC3
'
'?N
!
'
<9: =
'
<9:
='ON
H $
# I
N
'ON
'ON
4 N6
-'?
-' 4
-' 4
'ON
%
9: ='
1E
-' 4
'4
E
E
'
N
-' 4
!
!
'ON A9: '
N6 9: ='
-' 4
-' 4
-' 4
'ON639: ='?N -' 4
N39: ='?N ' 4
'ON639: ='
' 4
'
39: ='
' 4
-' 4
E
%
where the second step involves conditioning on 9: . Also by the previous lemma, we
have ' & 9: '
'4
5
$! so the second term in the above sum must be
negative, whereby
D I
GC3
<9: =
'ON6
'?N
-' 4
<9: =
'ON6
'?N
-' 4
E
'ON 9: ='ON
4 N6 9: =' N
'?
'4
-' 4
Under the conditions of these lemmas, we can derive the bounds on error rates:
4 N39: ='
-'O
-' 4
N
' 4 N6 ?
' N
9: ='ON6 ' N
4 N6 '
'?
N
D I
'4
'4
'4
IH5BHI
GC3
GC3
4 Bounding Total Error
Assuming that we make a random guess when
written as follows.
='
-' 4
-' 4
'
'4
9:
-
I
GC3
D I
'
$
GC3
4 N
'O
can be
=H 8E $ '
G3C to be the bounds on the
To give a bound on the total error rate we first define I
error rate for label N given in theorem 1.
, the total error rate of
'ON
'4
IH'BHI
GC3
We can now state a bound on the total error rate as a corollary to theorem 1.
E C
GC
Corollary 5 With probability at least $
over the choice of we have the follow and such that for all N we have I !
5 7 and
ing
for
all
pairs
of
rules
I !
8$ ! .
GC
E
E B GC ! C E1F
H E8$ ' IH'B $C !
NPO N O
H ! C
(' 4
B $C3
D
'
GC !
E C
$C
Proof. From our main theorem, we know that with probability at least $
! , for all
N . #' 4 N-M9: ' N #' 4 is bounded by I ! . This implies that with
probability at least $
! ,
KE0C
'4
C 1
E F
A
GC ! IH 8E $ $KE
'4
(3)
GC
E C
E
With probability at least $
! we have that ' 4
' 4 is no larger
than ! . So by the union bound
both
of
these
conditions
hold
simultaneously with
8$ ! we have that the upper
probability at least $
. Since CPE3F% 3 !
bound in (3) is maximized by setting -' 4 equal to (' 4
! .
B
KE0C
$C
E
E B
GC
Corollary 5 can be improved in a variety of ways. One could use a relative Chernoff
bound to tighten the uncertainty in ' 4 in the case where this probability is small.
One could also use the error rate bounds I ! to construct bounds on 9: ='
N6 -' 4 . One could then replace the max over I ! by a convex combination.
Another approach is to use the error rate of a rule that combines and , e.g., the rule
outputs if &' 4 , otherwise outputs if )' 4 , and otherwise guesses a random
value. This combined rule will have a lower error rate and it is possible to give bounds on
the error rate of the combined rule. We will not pursue these refined bounds here. It should
be noted, however, that the algorithm described in section 4 can be used with any bound on
total error rate.
$C
$C
5 A Decision List Algorithm
This section suggests a learning algorithm inspired by both Yarowsky (1995) and Collins
and Singer (1999) but modified to take into account theorem 1 and Corollary 5. Corollary 5, or some more refined bound on total error, provides an objective function that can
be pursued by a learning algorithm ? the objective is to find and so as to minimize
the upper bound on the total error rate. Typically, however, the search space is enormous.
Following Yarowsky, we consider the greedy construction of decision list rules.
and
" we have
be two ?feature sets? such that for
and )" we have
7 $3 . We assume that is to be a
decision list over the features in , i.e., a finite sequence of the form
; ; /L L
and 3I
A$ . A decision list can be viewed as a right-branching
where /I
decision tree. More specifically, if is the list
;
; ; /L L
then 1
is if 1 'Q$ and otherwise equals the value of the list
; ; JL L
on .
We define an empty decision list to have value . For in <7 $ we can define as
follows where is the number of feature-value pairs in .
Let
and
7 $ and for
N
BH
$
N
B
3
KE0B
equals the probability that
a certain
stochastic process
I $ which
. This implies the Kraft inequality
is all that is
'
1
It is possible to show that
$
generates the rule
needed in theorem 1 and corollary 5. We also assume that
features and define similarly.
is a decision list over the
Following Yarowsky we suggest growing the decision lists in a greedy manner adding one
feature value pair at a time. A natural choice of greedy heuristic might be a bound on the
total error rate. However, in many cases the final objective function is not an appropriate
choice for the greedy heuristic in greedy algorithms. A* search, for example, might be
viewed as a greedy heuristic where the heuristic function estimates the number of steps
needed to reach a high-value configuration ? a low value configuration might be one step
away from a high value configuration. The greedy heuristic used in greedy search should
estimate the value of the final configuration. Here we suggest using CPE3F !
as a heuristic estimate of the final total error rate ? in the final configuration we should
GC
GC
have that -' 4 is reasonably large and the important term will be C E1F 3 .
For concreteness, we propose the following algorithm. Many variants of this algorithm also
seem sensible.
1. Initialize and
knowledge.
to ?seed rule? decision lists using domain-specific prior
2. Until ' and '
both rules, do the following.
GC O
are both zero, or all features have been used in
(a) Let B denote if ' 65 ' and otherwise.
7 for some N , then extend B by the pair
which
(b) If I !
I I ! .
most increases C
(c) Otherwise
extend B by a single feature-value pair selected to minimize the
C E1FA A ! .
D
GC
D
$C
3. Prune the rules ? iteratively remove the pair from the end of either or that
greedily minimizes the bound on total error until no such removal reduces the
bound.
6 Future Directions
We have given some theoretical justification for some aspects of co-training algorithms that
have been shown to work well in practice. The co-training assumption we have used in our
theorems are is at best only approximately true in practice. One direction for future research
is to try to relax this assumption somehow. The co-training assumption states that and
are independent given . This is equivalent to the statement that the mutual information
between and given is zero. We could relax this assumption by allowing some
small amount of mutual information between and given and giving bounds on error
rates that involve this quantity of mutual information. Another direction for future work,
of course, is the empirical evaluation of co-training and bootstrapping methods suggested
by our theory.
Acknowledgments
The authors wish to acknowledge Avrim Blum for useful discussions and give special
thanks to Steve Abney for clarifying insights.
Literature cited
Blum, A. & Mitchell, T. (1998) Combining labeled and unlabeled data with co-training. COLT.
Collins, M. & Singer, Y. (1999) Unsupervised models for named entity classification. EMNLP.
Cover, T. & Thomas, J. (1991) Elements of information theory. Wiley.
Dempster, A., Laird, N. & Rubin, D. (1977) Maximum-likelihood from incomplete data via the EM
algorithm. J. Royal Statist. Soc. Ser. B, 39:1-38.
Nigam, K. & Ghani, R. (2000) Analyzing the effectiveness and applicability of co-training. CIKM.
Yarowsky, D. (1995) Unsupervised word sense disambiguation rivaling supervised methods. ACL.
| 2040 |@word pick:3 carry:1 initial:1 configuration:5 contains:1 att:3 selecting:1 prefix:3 current:1 com:3 yet:2 must:7 written:1 remove:1 update:1 alone:1 greedy:9 pursued:1 guess:3 selected:1 provides:1 boosting:1 location:1 incorrect:1 prove:1 consists:2 combine:1 manner:1 introduce:2 growing:1 inspired:1 globally:1 actual:1 underlying:1 moreover:1 bounded:1 string:1 pursue:1 minimizes:1 finding:1 bootstrapping:11 guarantee:5 ser:1 yarowsky:6 positive:1 local:2 analyzing:1 approximately:1 might:5 acl:1 on6:6 specifying:1 suggests:1 co:20 bi:1 statistically:2 acknowledgment:1 union:2 practice:2 bootstrap:1 procedure:4 empirical:6 composite:2 confidence:5 word:4 refers:1 suggest:4 get:1 unlabeled:15 bh:1 context:6 applying:1 restriction:1 measurable:1 deterministic:1 equivalent:2 graphically:1 convex:2 ke:2 assigns:1 rule:60 insight:1 fill:1 notion:1 justification:3 analogous:1 construction:1 hypothesis:1 agreement:7 element:1 satisfying:1 rivaling:1 predicts:2 labeled:6 inserted:1 e8:1 intuition:1 dempster:2 complexity:1 littman:1 predictive:2 upon:1 kraft:3 joint:1 distinct:1 describe:1 labeling:6 refined:2 apparent:1 quite:1 larger:1 heuristic:6 relax:2 otherwise:6 grammar:3 syntactic:2 think:1 itself:3 laird:2 final:4 a9:1 sequence:1 propose:1 combining:1 intuitive:1 empty:1 optimum:1 perfect:2 derive:1 mmm:2 nonterminal:2 strong:1 soc:1 involves:1 implies:2 met:2 direction:3 stochastic:3 vc:1 mcallester:1 opinion:3 assign:1 generalization:9 summation:1 adjusted:1 hold:6 mm:6 sufficiently:1 seed:1 mapping:1 major:1 adopt:1 estimation:1 label:16 hope:1 modified:1 rather:2 avoid:1 corollary:6 likelihood:4 mainly:1 greedily:1 sense:1 inference:1 stopping:1 typically:2 interested:2 unobservable:1 classification:2 issue:1 colt:1 noun:1 special:2 initialize:1 mutual:4 equal:3 construct:1 sampling:3 chernoff:2 unsupervised:2 nearly:1 future:3 few:1 simultaneously:2 individual:2 consisting:1 attempt:1 organization:1 evaluation:1 partial:19 unless:1 filled:3 tree:2 loosely:1 incomplete:1 theoretical:1 instance:3 earlier:1 compelling:1 cover:2 phrase:10 applicability:1 subset:1 predictor:2 successful:1 learnability:1 considerably:1 combined:2 person:1 thanks:1 cited:1 michael:1 possibly:2 emnlp:1 e1f:3 style:5 return:1 account:1 potential:1 m9:1 includes:1 explicitly:1 h1:1 lot:1 lab:3 try:1 doing:1 reached:1 start:2 complicated:1 minimize:2 accuracy:1 largely:2 gc3:9 maximized:1 weak:1 iterated:1 reach:1 definition:3 npo:1 naturally:1 proof:5 newly:1 proved:1 adjusting:1 mitchell:5 knowledge:1 infers:1 steve:1 supervised:1 follow:1 specify:1 improved:1 furthermore:2 just:1 until:3 parse:1 somehow:1 defines:1 reveal:1 true:1 assigned:2 iteratively:2 semantic:1 conditionally:4 indistinguishable:1 self:1 branching:1 essence:1 whereby:2 noted:1 criterion:1 empirically:2 conditioning:2 jl:1 extend:2 similarly:1 language:3 specification:1 mlittman:1 optimizing:1 optimizes:1 forcing:1 scenario:2 certain:2 inequality:4 success:1 minimum:1 converting:1 prune:1 determine:1 infer:1 reduces:1 ing:1 justifying:2 prediction:1 variant:2 basic:1 circumstance:1 iteration:1 sometimes:2 achieved:1 want:2 unlike:1 subject:1 thing:1 seem:1 effectiveness:1 near:5 hb:1 variety:1 independence:3 fit:3 idea:1 multiclass:1 whether:1 useful:2 involve:1 amount:2 statist:1 category:2 generate:2 exist:2 cikm:1 write:1 dasgupta:2 blum:6 enormous:1 drawn:2 clean:1 rewriting:1 concreteness:1 fraction:1 convert:1 sum:1 uncertainty:1 named:1 saying:1 disambiguation:1 decision:10 bit:3 entirely:2 bound:34 q9:1 encountered:1 precisely:1 rulebased:1 x2:1 generates:1 aspect:2 bdc:1 combination:2 smaller:2 em:8 making:1 happens:1 intuitively:1 legal:1 equation:1 agree:3 turn:1 singer:6 needed:3 know:2 end:1 apply:1 away:1 appropriate:1 disagreement:1 occurrence:1 alternative:1 thomas:2 include:1 giving:1 build:2 establish:1 objective:8 quantity:3 separate:1 clarifying:1 entity:1 sensible:1 extent:2 assuming:1 length:1 code:1 relationship:2 bhi:1 statement:4 negative:2 proper:1 allowing:1 upper:2 observation:2 finite:1 acknowledge:1 situation:1 gc:12 rating:1 david:1 introduced:2 pair:10 specified:1 c3:1 alternately:1 suggested:3 max:1 royal:1 power:1 misclassification:1 natural:2 predicting:2 mediated:1 n6:11 prior:1 literature:1 removal:1 relative:2 expect:1 permutation:9 suggestion:1 interesting:1 triple:3 h2:1 degree:2 sufficient:1 rubin:2 course:2 free:4 copy:1 formal:1 weaker:1 dimension:1 qn:1 author:1 made:2 refinement:1 far:1 tighten:1 approximate:1 observable:2 corpus:1 search:3 iterative:1 abney:1 learn:2 reasonably:1 nigam:1 complex:1 domain:1 main:3 motivation:2 noise:1 bounding:1 ghani:1 x1:1 wiley:1 wish:1 cotraining:1 third:1 theorem:12 specific:1 pac:7 symbol:3 list:11 evidence:1 exists:1 ih:3 false:2 adding:1 avrim:1 justifies:4 conditioned:1 depicted:1 gml:1 satisfies:1 conditional:3 viewed:3 identity:1 replace:1 considerable:1 content:1 determined:1 specifically:1 reversing:1 justify:1 principal:1 lemma:8 total:15 sanjoy:1 indicating:1 formally:3 select:1 collins:6 avoiding:1 |
1,142 | 2,041 | Grammatical Bigrams
Mark A. Paskin
Computer Science Division
University of California, Berkeley
Berkeley, CA 94720
paskin@cs.berkeley.edu
Abstract
Unsupervised learning algorithms have been derived for several statistical models of English grammar, but their computational complexity makes applying them to large data sets intractable. This
paper presents a probabilistic model of English grammar that is
much simpler than conventional models, but which admits an efficient EM training algorithm. The model is based upon grammatical bigrams, i.e. , syntactic relationships between pairs of words.
We present the results of experiments that quantify the representational adequacy of the grammatical bigram model, its ability to
generalize from labelled data, and its ability to induce syntactic
structure from large amounts of raw text.
1
Introduction
One of the most significant challenges in learning grammars from raw text is keeping the computational complexity manageable. For example, the EM algorithm
for the unsupervised training of Probabilistic Context-Free Grammars- known as
the Inside-Outside algorithm- has been found in practice to be "computationally
intractable for realistic problems" [1]. Unsupervised learning algorithms have been
designed for other grammar models (e.g. , [2, 3]). However, to the best of our knowledge, no large-scale experiments have been carried out to test the efficacy of these
algorithms; the most likely reason is that their computational complexity, like that
of the Inside-Outside algorithm, is impractical.
One way to improve the complexity of inference and learning in statistical models
is to introduce independence assumptions; however, doing so increases the model's
bias. It is natural to wonder how a simpler grammar model (that can be trained
efficiently from raw text) would compare with conventional models (which make
fewer independence assumptions, but which must be trained from labelled data) .
Such a model would be a useful tool in domains where partial accuracy is valuable
and large amounts of unlabelled data are available (e.g., Information Retrieval,
Information Extraction, etc.) .
In this paper, we present a probabilistic model of syntax that is based upon grammatical bigrams, i.e., syntactic relationships between pairs of words. We show how
this model results from introducing independence assumptions into more conven-
the
quick
brown
fox
jumps
over the
lazy dog
Figure 1: An example parse; arrows are drawn from head words to their dependents. The root word is jumps; brown is a predependent (adjunct) of fox; dog is a
postdependent (complement) of over.
tional models; as a result, grammatical bigram models can be trained efficiently
from raw text using an O(n 3 ) EM algorithm. We present the results of experiments
that quantify the representational adequacy of the grammatical bigram model, its
ability to generalize from labelled data, and its ability to induce syntactic structure
from large amounts of raw text.
2
The Grammatical Bigram Model
We first provide a brief introduction to the Dependency Grammar formalism used
by the grammatical bigram model; then, we present the probability model and
relate it to conventional models; finally, we sketch the EM algorithm for training
the model. Details regarding the parsing and learning algorithms can be found in
a companion technical report [4].
Dependency Grammar Formalism. 1 The primary unit of syntactic structure
in dependency grammars is the dependency relationship, or link- a binary relation
between a pair of words in the sentence. In each link, one word is designated the
head, and the other is its dependent. (Typically, different types of dependency
are distinguished, e.g, subject, complement, adjunct, etc.; in our simple model, no
such distinction is made.) Dependents that precede their head are called pre dependents, and dependents that follow their heads are called postdependents.
A dependency parse consists of a set of links that, when viewed as a directed
graph over word tokens , form an ordered tree. This implies three important properties:
1. Every word except one (the root) is dependent to exactly one head.
2. The links are acyclic; no word is, through a sequence of links, dependent to
itself.
3. When drawn as a graph above the sentence, no two dependency relations
cross-a property known as projectivity or planarity.
The planarity constraint ensures that a head word and its (direct or indirect) dependents form a contiguous subsequence of the sentence; this sequence is the head
word's constituent. See Figure 1 for an example dependency parse.
In order to formalize our dependency grammar model, we will view sentences as sequences of word tokens drawn from some set of word types. Let V = {tl' t2, ... , t M }
be our vocabulary of M word types. A sentence with n words is therefore represented as a sequence S = (Wl, W2 , ... , w n ), where each word token Wi is a variable
that ranges over V. For 1 :S i , j :S n , we use the notation (i,j) E L to express that
Wj is a dependent of Wi in the parse L.
IThe Dependency Grammar formalism described here (which is the same used in [5 , 6])
is impoverished compared to the sophisticated models used in Linguistics; refer to [7] for
a comprehensive treatment of English syntax in a dependency framework.
Because it simplifies the structure of our model , we will make the following three
assumptions about Sand L (without loss of generality): (1) the first word WI of S
is a special symbol ROOT E V; (2) the root of L is WI; and, (3) WI has only one
dependent. These assumptions are merely syntactic sugar: they allow us to treat
all words in the true sentence (i.e., (W2, ... ,W n )) as dependent to one word. (The
true root of the sentence is the sole child of WI.)
Probability Model. A probabilistic dependency grammar is a probability distribution P(S, L) where S = (WI,W2, .. . ,wn ) is a sentence, L is a parse of S, and the
words W2, ... ,Wn are random variables ranging over V. Of course, S and L exist
in high dimensional spaces; therefore, tractable representations of this distribution
make use of independence assumptions.
Conventional probabilistic dependency grammar models make use of what may
be called the head word hypothesis: that a head word is the sole (or primary)
determinant of how its constituent combines with other constituents. The head word
hypothesis constitutes an independence assumption; it implies that the distribution
can be safely factored into a product over constituents:
n
P(S,L) =
II P((Wj: (i,j) E L) is the dependent sequencelwi is the head)
i=1
For example, the probability of a particular sequence can be governed by a fixed
set of probabilistic phrase-structure rules , as in [6]; alternatively, the predependent
and postdependent subsequences can be modeled separately by Markov chains that
are specific to the head word, as in [8].
Consider a much stronger independence assumption: that all the dependents of a
head word are independent of one another and their relative order. This is clearly
an approximation; in general, there will be strong correlations between the dependents of a head word. More importantly, this assumption prevents the model
from representing important argument structure constraints. (For example: many
words require dependents , e.g. prepositions; some verbs can have optional objects,
whereas others require or forbid them.) However, this assumption relieves the parser
of having to maintain internal state for each constituent it constructs, and therefore
reduces the computational complexity of parsing and learning.
We can express this independence assumption in the following way: first , we forego
modeling the length of the sentence, n, since in parsing applications it is always
known; then, we expand P(S, Lin) into P(S I L)P(L I n) and choose P(L I n) as
uniform; finally, we select
II
P(S I L)
P( Wj is a [pre/post]dependent I Wi is the head)
(i ,j)EL
This distribution factors into a product of terms over syntactically related word
pairs; therefore, we call this model the "grammatical bigram" model.
The parameters of the model are
<"(xy
P(predependent is ty I head is t x )
6.
"(~
P(postdependent is ty Ihead is t x )
We can make the parameterization explicit by introducing the indicator variable
wi, whose value is 1 if Wi = tx and a otherwise. Then we can express P(S I L) as
P(S IL)
(i,j)EL x=1 y=1
j<i
(i,j)EL x=1 y=1
i<j
Parsing. Parsing a sentence S consists of computing
L*
L:,
argmaxP(L I S,n) = argmaxP(L, Sin) = argmaxP(S I L)
L
L
L
Yuret has shown that there are exponentially many parses of a sentence with n words
[9], so exhaustive search for L * is intractable. Fortunately, our grammar model falls
into the class of "Bilexical Grammars" , for which efficient parsing algorithms have
been developed. Our parsing algorithm (described in the tech report [4]) is derived
from Eisner's span-based chart-parsing algorithm [5], and can find L* in O(n 3 ) time.
Learning. Suppose we have a labelled data set
where Sk = (Wl,k, W2,k,?? ? , Wnk,k) and Lk is a parse over Sk.
likelihood values for our parameters given the training data are
The maximum
et
where the indicator variable
is equal to 1 if (i,j) E Lk and 0 otherwise. As
one would expect, the maximum-likelihood value of ,;; (resp. ,~ ) is simply the
fraction of tx's predependents (resp. postdependents) that were ty.
In the unsupervised acquisition problem, our data set has no parses; our approach is
to treat the Lk as hidden variables and to employ the EM algorithm to learn (locally)
optimal values of the parameters ,. As we have shown above, the
are sufficient
statistics for our model; the companion tech report [4] gives an adaptation of the
Inside-Outside algorithm which computes their conditional expectation in O(n 3 )
time. This algorithm effectively examines every possible parse of every sentence in
the training set and calculates the expected number of times each pair of words was
related syntactically.
et
3
Evaluation
This section presents three experiments that attempt to quantify the representational adequacy and learnability of grammatical bigram models.
Corpora. Our experiments make use of two corpora; one is labelled with parses,
and the other is not. The labelled corpus was generated automatically from the
phrase-structure trees in the Wall Street Journal portion of the Penn Treebank-III
[10].2 The resultant corpus, which we call C, consists of 49,207 sentences (1,037,374
word tokens). This corpus is split into two pieces: 90% of the sentences comprise
corpus Ctrain (44,286 sentences, 934,659 word tokens), and the remaining 10% comprise Ctest (4,921 sentences, 102,715 word tokens).
The unlabelled corpus consists of the 1987- 1992 Wall Street Journal articles in the
TREC Text Research Collection Volumes 1 and 2. These articles were segmented
on sentence boundaries using the technique of [11], and the sentences were postprocessed to have a format similar to corpus C. The resultant corpus consists of
3,347,516 sentences (66,777,856 word tokens). We will call this corpus U.
2This involved selecting a head word for each constituent, for which the head-word
extraction heuristics described in [6] were employed.
Additionally, punctuation was
removed, all words were down-cased, and all numbers were mapped to a special <#> symbol.
The model's vocabulary is the same for all experiments; it consists of the 10,000 most
frequent word types in corpus U; this vocabulary covers 94.0% of word instances
in corpus U and 93.9% of word instances in corpus L. Words encountered during
testing and training that are outside the vocabulary are mapped to the <unk> type.
Performance metric. The performance metric we report is the link precision of
the grammatical bigram model: the fraction of links hypothesized by the model that
are present in the test corpus Ltest. (In a scenario where the model is not required
to output a complete parse, e.g., a shallow parsing task, we could similarly define
a notion of link recall; but in our current setting, these metrics are identical.) Link
precision is measured without regard for link orientation; this amounts to ignoring
the model's choice of root, since this choice induces a directionality on all of the
edges.
Experiments. We report on the results of three experiments:
I. Retention. This experiment represents a best-case scenario: the model is
trained on corpora Ltrain and Ltest and then tested on Ltest. The model's
link precision in this setting is 80.6%.
II. Generalization. In this experiment, we measure the model's ability to generalize
from labelled data. The model is trained on Ltrain and then tested on Ltest.
The model's link precision in this setting is 61.8%.
III. Induction. In this experiment, we measure the model 's ability to induce grammatical structure from unlabelled data. The model is trained on U and then
tested on Ltest . The model's link precision in this setting is 39.7%.
Analysis. The results of Experiment I give some measure of the grammatical
bigram model's representational adequacy. A model that memorizes every parse
would perform perfectly in this setting, but the grammatical bigram model is only
able to recover four out of every five links. To see why, we can examine an example
parse. Figure 2 shows how the models trained in Experiments I, II, and III parse
the same test sentence. In the top parse, syndrome is incorrectly selected as a
postdependent of the first on token rather than the second. This error can be
attributed directly to the grammatical bigram independence assumption: because
argument structure is not modeled, there is no reason to prefer the correct parse,
in which both on tokens have a single dependent , over the chosen parse, in which
the first has two dependents and the second has none. 3
Experiment II measures the generalization ability of the grammatical bigram model;
in this setting, the model can recover three out of every five links. To see why the
performance drops so drastically, we again turn to an example parse: the middle
parse in Figure 2. Because the forces -+ on link was never observed in the training
data, served has been made the head of both on tokens; ironically, this corrects the
error made in the top parse because the planarity constraint rules out the incorrect
link from the first on token to syndrome. Another error in the middle parse is a
failure to select several as a predependent of forces; this error also arises because
the combination never occurs in the training data. Thus, we can attribute this drop
in performance to sparseness in the training data.
We can compare the grammatical bigram model's parsing performance with the
results reported by Eisner [8]. In that investigation, several different probability models are ascribed to the simple dependency grammar described above and
3 Although the model's parse of acquired immune deficiency syndrome agrees with
the labelled corpus, this particular parse reflects a failure of the head-word extraction
heuristics; acquired and immune should be predependents of deficiency, and deficiency
should be a predependent of syndrome .
1. 843
'. 88'
I.
fi r hr-.
n~
14 . 383
<root> she has also served on several task forces on acquired immune deficiency syndrome
r,m fO ~8~\
fi r hn
' . 803
II.
3.2",-d
r
1r
1. 528
1.358
A
9 . 630
,
1
1 2 . 527
14 . 264
-
<root> she has also served on several task forces on acquired immune deficiency syndrome
1. 990
0.913
III.
tI
k
4 . 124
<root> she has also served
0 .14 9
- 1.709
1--'
on several
f'
(~
13 . 585
~(
task forces on acquired immune deficiency syndrome
Figure 2: The same test sentence, parsed by the models trained in each of the three
experiments. Links are labelled with -log2 IXY I I:~1 IXY, the mutual information
of the linked words; dotted edges are default attachments.
are compared on a task similar to Experiment 11.4 Eisner reports that the bestperforming dependency grammar model (Model D) achieves a (direction-sensitive)
link precision of 90.0%, and the Collins parser [6] achieves a (direction-sensitive) link
precision of 92.6%. The superior performance of these models can be attributed to
two factors: first, they include sophisticated models of argument structure; and second, they both make use of part-of-speech taggers, and can "back-off" to non-lexical
distributions when statistics are not available.
Finally, Experiment III shows that when trained on unlabelled data, the grammatical bigram model is able to recover two out of every five links. This performance
is rather poor, and is only slightly better than chance; a model that chooses parses
uniformly at random achieves 31.3% precision on L\est . To get an intuition for why
this performance is so poor, we can examine the last parse, which was induced from
unlabelled data. Because Wall Street Journal articles often report corporate news,
the frequent co-occurrence of has -+ acquired has led to a parse consistent with
the interpretation that the subject she suffers from AIDS, rather than serving on
a task force to study it. We also see that a flat parse structure has been selected
for acquired immune deficiency syndrome; this is because while this particular
noun phrase occurs in the training data, its constituent nouns do not occur independently with any frequency, and so their relative co-occurrence frequencies cannot
be assessed.
4
Discussion
Future work. As one would expect, our experiments indicate that the parsing
performance of the grammatical bigram model is not as good as that of state-ofthe-art parsers; however, its performance in Experiment II suggests that it may be
useful in domains where partial accuracy is valuable and large amounts of unlabelled
data are available. However, to realize that potential, the model must be improved
so that its performance in Experiment III is closer to that of Experiment II.
To that end, we can see two obvious avenues of improvement. The first involves
increasing the model's capacity for generalization and preventing overfitting. The
4The labelled corpus used in that investigation is also based upon a transformed version of Treebank-III, but the head-word extraction heuristics were slightly different , and
sentences with conjunctions were completely eliminated. However, the setup is sufficiently
similar that we think the comparison we draw is informative.
model presented in this paper is sensitive only to pairwise relationships to words;
however, it could make good use of the fact that words can have similar syntactic
behavior. We are currently investigating whether word clustering techniques can
improve performance in supervised and unsupervised learning. Another way to improve the model is to directly address the primary source of parsing error: the lack of
argument structure modeling. We are also investigating approximation algorithms
that reintroduce argument structure constraints without making the computational
complexity unmanageable.
Related work. A recent proposal by Yuret presents a "lexical attraction" model
with similarities to the grammatical bigram model [9]; however, unlike the present
proposal, that model is trained using a heuristic algorithm. The grammatical bigram model also bears resemblance to several proposals to extend finite-state methods to model long-distance dependencies (e.g., [12, 13]), although these models are
not based upon an underlying theory of syntax.
References
[1] K. Lari and S. J. Young. The estimation of stochastic context-free grammars using
the Inside-Outside algorithm. Computer Speech and Language, 4:35- 56, 1990.
[2] John Lafferty, Daniel Sleator, and Davy Temperley. Grammatical trigrams: A probabilistic model of link grammar. In Proceedings of the AAAI Conference on Probabilistic
Approaches to Natural Language, October 1992.
[3] Yves Schabes. Stochastic lexicalized tree-adjoining grammars. In Proceedings of the
Fourteenth International Conference on Computational Linguistics, pages 426-432,
Nantes, France, 1992.
[4] Mark A. Paskin. Cubic-time parsing and learning algorithms for grammatical bigram
models. Technical Report CSD-01-1148, University of California, Berkeley, 2001.
[5] Jason Eisner. Bilexical grammars and their cubic-time parsing algorithms. In Harry
Bunt and Anton Nijholt, editors, Advances in Probabilistic and Other Parsing Technologies, chapter 1. Kluwer Academic Publishers, October 2000.
[6] Michael Collins. Head-driven Statistical Models for Natural Language Parsing. PhD
thesis, University of Pennsylvania, Philadelphia, Pennsylvania, 1999.
[7] Richard A. Hudson. English Word Grammar. B. Blackwell, Oxford, UK, 1990.
[8] Jason M. Eisner. An empirical comparison of probability models for dependency
grammars. Technical Report ICRS-96-11, CIS Department, University of Pennsylvania, 220 S. 33 rd St. Philadelphia, PA 19104- 6389, 1996.
[9] Deniz Yuret . Discovery of Linguistic R elations Using Lexical Attraction. PhD thesis,
Massachusetts Institute of Technology, May 1998.
[10] M. Marcus, B. Santorini, and M. Marcinkiewicz. Building a large annotated corpus
of english: The penn treebank. Computational Linguistics, 19:313- 330, 1993.
[11] Jeffrey C. Reynar and Adwait Ratnaparkhi. A maximum entropy approach to identifying sentence boundaries. In Proceedings of the Fifth Conference on Appli ed Natural
Language Processing, Washington, D.C. , March 31 - April 3 1997.
[12] S. Della Pietra, V. Della Pietra, J. Gillett, J. Lafferty, H. Printz , and L. Ures. Inference and estimation of a long-range trigram model. In Proceedings of the Second
International Colloquium on Grammatical Inference and Applications, number 862 in
Lecture Notes in Artificial Intelligence, pages 78- 92. Springer-Verlag, 1994.
[13] Ronald Rosenfeld. Adaptive Statistical Language Modeling: A Maximum Entropy
Approach. PhD thesis, Carnegie Mellon University, 1994.
| 2041 |@word determinant:1 middle:2 manageable:1 bigram:21 stronger:1 version:1 efficacy:1 selecting:1 daniel:1 current:1 must:2 parsing:16 john:1 realize:1 ronald:1 realistic:1 deniz:1 informative:1 designed:1 drop:2 intelligence:1 fewer:1 selected:2 parameterization:1 simpler:2 five:3 ironically:1 tagger:1 direct:1 incorrect:1 consists:6 combine:1 inside:4 introduce:1 acquired:7 pairwise:1 ascribed:1 expected:1 behavior:1 examine:2 automatically:1 increasing:1 notation:1 underlying:1 what:1 developed:1 impractical:1 safely:1 berkeley:4 every:7 ti:1 exactly:1 uk:1 unit:1 penn:2 retention:1 hudson:1 treat:2 bilexical:2 adwait:1 oxford:1 planarity:3 suggests:1 co:2 range:2 directed:1 testing:1 practice:1 empirical:1 davy:1 word:48 induce:3 pre:2 get:1 cannot:1 ctest:1 context:2 applying:1 conventional:4 quick:1 lexical:3 independently:1 identifying:1 factored:1 rule:2 yuret:3 examines:1 importantly:1 attraction:2 notion:1 resp:2 suppose:1 parser:3 hypothesis:2 pa:1 observed:1 wj:3 ensures:1 news:1 reintroduce:1 removed:1 valuable:2 projectivity:1 intuition:1 colloquium:1 complexity:6 sugar:1 trained:10 ithe:1 upon:4 division:1 completely:1 indirect:1 represented:1 tx:2 chapter:1 artificial:1 outside:5 exhaustive:1 whose:1 heuristic:4 ratnaparkhi:1 otherwise:2 grammar:23 ability:7 statistic:2 schabes:1 syntactic:7 think:1 itself:1 rosenfeld:1 sequence:5 product:2 adaptation:1 frequent:2 representational:4 constituent:7 argmaxp:3 postprocessed:1 object:1 measured:1 sole:2 strong:1 c:1 involves:1 implies:2 indicate:1 quantify:3 direction:2 correct:1 attribute:1 annotated:1 stochastic:2 appli:1 sand:1 require:2 generalization:3 wall:3 investigation:2 marcinkiewicz:1 sufficiently:1 trigram:2 achieves:3 estimation:2 precede:1 currently:1 sensitive:3 agrees:1 wl:2 tool:1 reflects:1 clearly:1 always:1 rather:3 conjunction:1 linguistic:1 derived:2 she:4 improvement:1 likelihood:2 tech:2 inference:3 tional:1 dependent:18 el:3 typically:1 hidden:1 relation:2 expand:1 transformed:1 nantes:1 france:1 unk:1 orientation:1 noun:2 special:2 art:1 mutual:1 equal:1 construct:1 comprise:2 extraction:4 having:1 never:2 eliminated:1 identical:1 represents:1 washington:1 unsupervised:5 constitutes:1 future:1 report:9 t2:1 others:1 richard:1 employ:1 comprehensive:1 pietra:2 relief:1 jeffrey:1 maintain:1 attempt:1 evaluation:1 punctuation:1 adjoining:1 chain:1 edge:2 closer:1 partial:2 xy:1 fox:2 tree:3 bestperforming:1 instance:2 formalism:3 modeling:3 contiguous:1 cover:1 phrase:3 introducing:2 uniform:1 wonder:1 learnability:1 reported:1 dependency:17 chooses:1 st:1 international:2 forbid:1 probabilistic:9 off:1 corrects:1 michael:1 again:1 aaai:1 thesis:3 choose:1 hn:1 forego:1 potential:1 harry:1 piece:1 root:9 jason:2 view:1 memorizes:1 doing:1 linked:1 portion:1 recover:3 il:1 chart:1 accuracy:2 yves:1 efficiently:2 ofthe:1 generalize:3 anton:1 raw:5 none:1 served:4 fo:1 suffers:1 ed:1 failure:2 ty:3 acquisition:1 frequency:2 involved:1 cased:1 obvious:1 resultant:2 attributed:2 treatment:1 massachusetts:1 recall:1 knowledge:1 formalize:1 impoverished:1 sophisticated:2 back:1 supervised:1 follow:1 improved:1 april:1 generality:1 correlation:1 sketch:1 parse:23 lack:1 resemblance:1 building:1 hypothesized:1 brown:2 true:2 sin:1 during:1 ixy:2 syntax:3 complete:1 syntactically:2 ranging:1 fi:2 superior:1 ltrain:2 exponentially:1 volume:1 extend:1 interpretation:1 kluwer:1 significant:1 refer:1 mellon:1 rd:1 similarly:1 adjunct:2 language:5 immune:6 similarity:1 etc:2 recent:1 driven:1 scenario:2 verlag:1 binary:1 fortunately:1 employed:1 syndrome:8 ii:8 corporate:1 reduces:1 segmented:1 technical:3 unlabelled:6 academic:1 cross:1 long:2 retrieval:1 lin:1 post:1 calculates:1 expectation:1 metric:3 proposal:3 whereas:1 separately:1 ures:1 source:1 publisher:1 printz:1 w2:5 unlike:1 subject:2 induced:1 lafferty:2 adequacy:4 call:3 iii:7 split:1 wn:2 independence:8 pennsylvania:3 perfectly:1 regarding:1 simplifies:1 avenue:1 whether:1 speech:2 useful:2 amount:5 locally:1 induces:1 exist:1 dotted:1 serving:1 carnegie:1 express:3 four:1 drawn:3 graph:2 merely:1 fraction:2 fourteenth:1 draw:1 prefer:1 encountered:1 occur:1 constraint:4 deficiency:7 flat:1 argument:5 span:1 format:1 department:1 designated:1 combination:1 poor:2 march:1 slightly:2 em:5 wi:10 shallow:1 making:1 sleator:1 computationally:1 lari:1 turn:1 tractable:1 end:1 available:3 occurrence:2 distinguished:1 top:2 remaining:1 linguistics:3 include:1 clustering:1 log2:1 parsed:1 eisner:5 wnk:1 occurs:2 primary:3 distance:1 link:22 mapped:2 capacity:1 street:3 reason:2 induction:1 marcus:1 length:1 modeled:2 relationship:4 setup:1 october:2 relate:1 perform:1 markov:1 finite:1 optional:1 incorrectly:1 santorini:1 head:22 trec:1 verb:1 complement:2 pair:5 dog:2 required:1 blackwell:1 sentence:23 california:2 distinction:1 bunt:1 address:1 able:2 challenge:1 natural:4 force:6 indicator:2 hr:1 representing:1 improve:3 technology:2 brief:1 lk:3 attachment:1 carried:1 philadelphia:2 text:6 discovery:1 ltest:5 relative:2 loss:1 par:4 expect:2 bear:1 lecture:1 acyclic:1 sufficient:1 consistent:1 article:3 treebank:3 editor:1 preposition:1 course:1 token:11 last:1 keeping:1 english:5 free:2 drastically:1 bias:1 allow:1 institute:1 fall:1 unmanageable:1 fifth:1 grammatical:24 boundary:2 regard:1 vocabulary:4 default:1 computes:1 preventing:1 made:3 jump:2 collection:1 adaptive:1 overfitting:1 investigating:2 corpus:18 alternatively:1 subsequence:2 search:1 sk:2 why:3 additionally:1 learn:1 ca:1 ignoring:1 domain:2 csd:1 arrow:1 elation:1 child:1 tl:1 cubic:2 aid:1 precision:8 explicit:1 governed:1 young:1 companion:2 down:1 specific:1 paskin:3 symbol:2 admits:1 intractable:3 lexicalized:1 effectively:1 ci:1 phd:3 sparseness:1 entropy:2 led:1 simply:1 likely:1 lazy:1 prevents:1 ordered:1 springer:1 chance:1 conditional:1 viewed:1 labelled:10 directionality:1 except:1 uniformly:1 called:3 est:1 select:2 internal:1 mark:2 arises:1 collins:2 assessed:1 tested:3 della:2 |
1,143 | 2,042 | Boosting and Maximum Likelihood for
Exponential Models
Guy Lebanon
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
John Lafferty
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
lebanon@cs.cmu.edu
lafferty@cs.cmu.edu
Abstract
We derive an equivalence between AdaBoost and the dual of a convex
optimization problem, showing that the only difference between minimizing the exponential loss used by AdaBoost and maximum likelihood
for exponential models is that the latter requires the model to be normalized to form a conditional probability distribution over labels. In addition to establishing a simple and easily understood connection between
the two methods, this framework enables us to derive new regularization
procedures for boosting that directly correspond to penalized maximum
likelihood. Experiments on UCI datasets support our theoretical analysis and give additional insight into the relationship between boosting and
logistic regression.
1 Introduction
Several recent papers in statistics and machine learning have been devoted to the relationship between boosting and more standard statistical procedures such as logistic regression.
In spite of this activity, an easy-to-understand and clean connection between these different techniques has not emerged. Friedman, Hastie and Tibshirani [7] note the similarity
between boosting and stepwise logistic regression procedures, and suggest a least-squares
alternative, but view the loss functions of the two problems as different, leaving the precise
relationship between boosting and maximum likelihood unresolved. Kivinen and Warmuth
[8] note that boosting is a form of ?entropy projection,? and Lafferty [9] suggests the use of
Bregman distances to approximate the exponential loss. Mason et al. [10] consider boosting algorithms as functional gradient descent and Duffy and Helmbold [5] study various
loss functions with respect to the PAC boosting property. More recently, Collins, Schapire
and Singer [2] show how different Bregman distances precisely account for boosting and
logistic regression, and use this framework to give the first convergence proof of AdaBoost.
However, in this work the two methods are viewed as minimizing different loss functions.
Moreover, the optimization problems are formulated in terms of a reference distribution
consisting of the zero vector, rather than the empirical distribution of the data, making the
interpretation of this use of Bregman distances problematic from a statistical point of view.
In this paper we present a very basic connection between boosting and maximum likelihood
for exponential models through a simple convex optimization problem. In this setting, it is
seen that the only difference between AdaBoost and maximum likelihood for exponential
models, in particular logistic regression, is that the latter requires the model to be normalized to form a probability distribution. The two methods minimize the same extended
Kullback-Leibler divergence objective function subject to the same feature constraints. Using information geometry, we show that projecting the exponential loss model onto the
simplex of conditional probability distributions gives precisely the maximum likelihood
exponential model with the specified sufficient statistics. In many cases of practical interest, the resulting models will be identical; in particular, as the number of features increases
to fit the training data the two methods will give the same classifiers. We note that throughout the paper we view boosting as a procedure for minimizing the exponential loss, using
either parallel or sequential update algorithms as in [2], rather than as a forward stepwise
procedure as presented in [7] or [6].
Given the recent interest in these techniques, it is striking that this connection has gone unobserved until now. However in general, there may be many ways of writing the constraints
for a convex optimization problem, and many different settings of the Lagrange multipliers (or Kuhn-Tucker vectors) that represent identical solutions. The key to the connection
we present here lies in the use of a particular non-standard presentation of the constraints.
When viewed in this way, there is no need for special-purpose Bregman distances to give a
unified account of boosting and maximum likelihood, as we only make use of the standard
Kullback-Leibler divergence. But our analysis gives more than a formal framework for
understanding old algorithms; it also leads to new algorithms for regularizing AdaBoost,
which is required when the training data is noisy. In particular, we derive a regularization
procedure for AdaBoost that directly corresponds to penalized maximum likelihood using a
Gaussian prior. Experiments on UCI data support our theoretical analysis, demonstrate the
effectiveness of the new regularization method, and give further insight into the relationship
between boosting and maximum likelihood exponential models.
2 Notation
( '*), +.-0 !/ #"%$&(('*'*-&),+.45-0)6/&/ 1%+ 2) 3
798 :
<; =1>+@??@?6+.
LMN'B)O+.-0/
MPL MNL'*)D'*/ )6/XWZY
MNL 'B)O+.-0/Q) HK A'BHCRI,)DC.KT+ES -%'*CF)U/GCV+EHCJI,)D/K S 'B-%C.+E-0/G?
)[ -TL '*)6/ MNL '*)6/QWY
-\]
MNL '*-&45)6/PW^Y )
)
Let
and be finite sets. We denote by
the set of nonnegative measures on
, and by
the set of conditional probability distributions,
for each
. For
, we will overload
the notation
and
; the latter will be suggestive of a conditional probability
distribution, but in general it need not be normalized. Let
,
,
be given functions, which we will refer to as features. These will correspond to the weak
learners in boosting, and to the sufficient statistics in an exponential model. Suppose that
we have data
with empirical distribution
and marginal
; thus,
We assume, without loss of generality, that
for all . Throughout the paper, we assume (for notational convenience) that the training
data has the following property.
Consistent Data Assumption. For each
with
for which
. This will be denoted
.
, there is a unique
For most data sets of interest, each appears only once, so that the assumption trivially
holds. However, if appears more than once, we require that it is labeled consistently. We
make this assumption mainly to correspond with the conventions used to present boosting
algorithms; it is not essential to what follows.
g<hUikjRlK m nV7 oq8 prm h.sRt9u>v `%w xzyJ{zw r|F}
Given
~FaO+V7c'*),+.-0/.^ #_ `D'*-&8b I,45K)Da/ 8 7 8 '*),a3+.-0/ 3cb
a
, we define the exponential model
where
hood estimation problem is to determine parameters
, for
_`d'B-4e)D/f
, by
. The maximum likeli-
that maximize the conditional log-
%'FaD/ {zw LMN'B)O+.-0/ N_`U'*-&45)6/
%' aU/
'FaD/] HCJI,K Z I ucv `%w x%yR{ w r| %x yR{ w *|B} ?
likelihood
or minimize the log loss
. The objective
function to be minimized in the multi-label boosting algorithm AdaBoost.M2 [2] is the ex
ponential loss given by M2
As has been often
noted, the log loss and the exponential loss are qualitatively different. The exponential loss
grows exponentially with increasing negative ?margin,? while the log loss grows linearly.
3 Correspondence Between AdaBoost and Maximum Likelihood
Since we are working with unnormalized models we make use of the extended conditional
Kullback-Leibler divergence or -divergence, given by
',+E_z/ { MQL '*)D/ ,'B- 45)6/ _ '*'*--4e45)D)D// '*-&45)D/ _ 'B-4e)D/
,' #45)6/
_ ' #45)6/ 78
_
]'*MPL +V7T/N
'*MNL +k7T/ ! [ { MNL 'B)D/ '*-45)D/ 'F7 8 'B)O+.-0/"$#%&(' 7 8 45)*) / Y0+ ;d>?
ML + #,%&(' 7 45-) )X 7c'B)O+V-6L 'B)D/E/
. K
.0/
'1. / /
']43O+V_ '* ML / +V7T/
1' . K@/
' O+V_ /
[2]' MN
L +k7T/
. /
. K
.6K 5
defined on
features
def
(possibly taking on the value ). Note that if
and
then this becomes the more familiar KL divergence for probabilities. Let
and a fixed default distribution
be given. We define
as
all
(1)
, this set is non-empty. Note that under the consistent data assumption, we
Since
have that
. Consider now the following two convex optimization
problems, labeled
and .
minimize
subject to
minimize
subject to
Thus, problem
differs from
only in that the solution is required to be normalized.
As we?ll show, the dual problem
corresponds to AdaBoost, and the dual problem .7/ 5
corresponds to maximum likelihood for exponential models.
{ MQL '*)D/ Z ,'B- 4e)D/07c'B)O+.-0/&
This presentation of the constraints is the key to making the correspondence between AdaBoost and maximum likelihood. Note that the constraint
#%&0' 8) , which is the usual presentation of the constraints for maximum likelihood (as dual
to maximum entropy), doesn?t make sense for unnormalized models, since the two sides
of the equation may not be ?on the same scale.? Note further that attempting to rescale by
dividing by the mass of to get
7
M{ PL 'B)D/ Z,'B-9,4e)D'*-&/07c45)6'B)O/ +E- / :#%&0' 78)
would yield nonlinear constraints.
K )D/
,
B
'
e
4
K '3O+kaU/ { LMN'*)6/ ,'B- 4e)D/ < _ '*-&45)D/ 1,^~FaO+V7c'B)O+.-0/"$#%&0' 7<45)*) ?
a uA v b`%w x%yR{zw @|9D8EFHG x"I*{KJ }
_#` =?>@BA " C ; K9'3O; +VaD/
_`D'*-45)6/X
_ '*-&45)6/
?
L M6K9' aU/N
%
K
B
'
_
U
`
V
+
D
a
/
L M6K9'FaD/
M{ PL '*)6/ _ '*-45)D/ u v `%w xzyJ{zw r|9D8EFHG x"IB{KJ } ?
a N P OQRW,S{ =TOVU ` LMO' aU/ 8,'B-4e)D/ =1 . /
8
We now derive the dual problems formally; the following section gives a precise statement
of the duality result. To derive the dual problem .75 , we calculate the Lagrangian as
;
For
def
, the connecting equation
arg
Thus, the dual function
is given by
is given by
(2)
The dual problem is to determine
simply add additional Lagrange multipliers
. To derive the dual for
for the constraints
.
, we
3.1 Special cases
It is now straightforward to derive various boosting and logistic regression problems as
special cases of the above optimization problems.
_ A'B- 4e)D/ 1
uAv `zw x%yR{ w r| x%yR{ w
*| =TO U `LM K 'FaD/
` C I
- 2 &1%+
u 61%
v `%w x%yR{ B|B} 7 8 'B)O+E- / /K -N7 8 'B)D/
a N OQR =7>@ ` C
1%+ { 1>MNL+@?@'*?)6?T/ +
? '*-&45)6/.7 8 'B)O+.-0/ {zw MNL 'B)O+.-0/E7 8 '*),+.-0/ 8,'B-4e) C /
_u>` v `%'*-&w xzyJ45{z)6w r|/ F} K p K_ >'B-4e)D/ uAv `zw x%yR{%w r|B} {
W {
{ Zc_ >'*- 4 )D/
L '*)O+E-0/ < _`D'B-4e)D/
L M / ' aU/< { MQ
798>'*)O+E-0/
a
Case 1: AdaBoost.M2. Take
lent to computing *N
OVQ! =?><@
problem of AdaBoost.M2.
. Then the dual problem
is equivawhich is the optimization
Case 2: Binary AdaBoost. In addition to the assumptions for the previous case, now assume that
, and take
. Then the dual problem is given by
which is the optimization problem of binary AdaBoost.
Case 3: Maximum Likelihood for Exponential Models. In this case we take the same setup
as for AdaBoost.M2 but add the additional normalization constraints:
If these constraints are satisfied, then the other constraints take the
form
and the connecting
equation becomes
were
is the normalizing term
, which corresponds to setting the Lagrange multiplier
to the appropriate value. In this case, after a simple calculation the dual problem is seen
to be
which corresponds to maximum likelihood for a
conditional exponential model with sufficient statistics
.
_`D'e1 4e)D/& K i jRK l m nEo p sJt ?
Case 4: Logistic Regression. Returning to the case of binary AdaBoost, we see that when
we add normalization constraints as above, the model is equivalent to binary logistic regres
sion, since
We note that it is not necessary to scale the features
by a constant factor here, as in [7]; the correspondence between logistic regression and
boosting is direct.
3.2 Duality
K
K 'F_K %+k7T/ # _ [ &_ '*-&45)6/ _ A'*-&45)6/ u u v `%v `%w xzw x%yJ{zyR{zw r|w @|F} x%yR{%w yR{ |B|B} +Ua][ b
/%'F_K %+k7T/
# _ &_ '*-&45)6/ _ A'*-&45)D/
+Ua][ b A?
/
_ N K
_N
_ N OQR =7>@ #" T { MNL '*)6/d
c_ '*-&45)D/ _ N :OVQ!S=TO U #" O { MPL '*)D/B< _ ' -&45)6/
' Y0+V_z/
'*MNL +E_z/
' MNL +V_ / _ N _ N
_ N OVQ!A " =?><@ 3' O+V_ / OVQ!S" =7>@ '*MNL +E_z/
_ N OVA Q!" =?><@ 3' O+VK_ / OVQ!S" =7>@ '*MNL + /6?
_N
_ N _ N : OVQ! =?>@ & " ' O+V_ N /
Let
and
/ be defined as the following exponential families:
K
K
is unnormalized while is normalized. We now define the boosting solution
and
maximum likelihood solution ml as
boost
Thus
boost
ml
where denotes the closure of the set
. The following theorem corresponds to
Proposition 4 of [3] for the usual KL divergence; in [4] the duality theorem is proved for
a general class of Bregman distances, including the extended KL divergence as a special
as in [2], but rather
case. Note that we do not work with divergences such as
, which is more natural and interpretable from a statistical point-of-view.
Theorem. Suppose that
boost
ml
. Then boost and ml exist, are unique, and satisfy
!
Moreover, ml is computed in terms of boost as ml
#"
!
boost .
K
K
_
PSfrag replacements
PSfrag
replacements
N
boost
_N
_N
boost
ml
ml
Figure 1: Geometric view of the duality theorem. Minimizing the exponential loss finds the member
that intersects with the feasible set of measures satisfying the moment constraints (left). When
of
we impose the additional constraint that each conditional distribution
must be normalized,
we introduce a Lagrange multiplier for each training example , giving a higher-dimensional family
. By the duality theorem, projecting the exponential loss solution onto the intersection of the
feasible set with the simplex gives the maximum likelihood solution.
K
This result has a simple geometric interpretation. The unnormalized exponential family
intersects the feasible set of measures satisfying the constraints (1) at a single point.
The algorithms presented
in [2] determine this point, which is the exponential loss solution
Nboost :OVQ!S=?><@
(see Figure 1, left).
_
'*ML +E_z/
"
_
" ' ML +V_z/
K
On the other hand, maximum conditional likelihood estimation for an exponential
model
with the same features is equivalent to the problem Vml
where
N
OQRS=?>@
is the exponential family with additional Lagrange multipliers, one for each normalization
constraint. The feasible set for this problem is
. Since
, by the Pythagorean
(see Figure 1, right).
equality we have that ml
N
OQR =7>@ A !
3 boost
N
_
"
' O+V _ /
4 Regularization
78
Minimizing the exponential loss or the log loss on real data often fails to produce finite
parameters. Specifically, this happens when for some feature
or
7 8 '*),+.-0/"7 8 '*),+G- L '*)6/./^Y
7 8 '*),+.-0"/ 7 8 '*),+G- L '*)6/./ ^Y
for all
for all
- )
- )
and
and
with
with
MPL 'B)D/PWY
MPL 'B)D/PWY0?
(3)
This is especially harmful since often the features for which (3) holds are the most important for the purpose of discrimination. Of course, even when (3) does not hold, models
trained by maximum likelihood or the exponential loss can overfit the training data. A
standard regularization technique in the case of maximum likelihood employs parameter
priors in a Bayesian framework. See [11] for non-Bayesian alternatives in the context of
boosting.
a
In terms of convex duality, parameter priors for the dual problem correspond to ?potentials? on the constraint values in the primal problem. The case of a Gaussian prior on ,
for example, corresponds to a quadratic potential on the constraint values in the primal
problem.
' O+ /
*' MNL ^+k^7D+ @/ N^ \ b
]' ML +k7D+ @/ [ { MQL '*)D/ '*-&45)6/'F78>'B)O+.-0/"$#%&0' 78 45)*) / V8
. Kkw
' .cKGw /
' ,+E_ /
' @/
[4 '*ML +V7D+ @/
f% b
Y
;
;
' ,+ 9+kaU/&
' ,+VaD/S
' @/
M Kkw 'FaD/ M K 'FaD(
/ K / 5z/'FaD/
5%'FaD/
/
' @/ 8 / 8 8
5%'FaD/X 8 /K 8 a 8/
M6KGw ' aU/
L{ MN'B)D/ _ '*-&45)6/ u v `%w x%yR{%w r| x%yR{%w yR{ |B} 8 a 8/ 8/ ?
We now consider primal problems over 3 where
vector that relaxes the original constraints. Define
and
is a parameter
as
and consider the primal problem
reg
where
(4)
reg given by
minimize
subject to
is a convex function whose minimum is at .
To derive the dual problem, the Lagrangian is calculated as
and the dual function is reg
where
,
we
have
of . For a quadratic penalty
the dual function becomes
reg
is the convex conjugate
and
(5)
A sequential update rule for (5) incurs the small additional cost of solving a nonlinear
equation by Newton-Raphson every iteration. See [1] for a discussion of this technique in
the context of exponential models in statistical language modeling.
5 Experiments
We performed experiments on some of the UC Irvine datasets in order to investigate the
relationship between boosting and maximum likelihood empirically. The weak learner was
the decision stump FindAttrTest as described in [6], and the training set consisted of
a randomly chosen 90% of the data. Table 1 shows experiments with regularized boosting.
Two boosting models are compared. The first model was trained for 10 features generated by FindAttrTest, excluding features satisfying condition (3). Training was carried
out using the parallel update method described in [2]. The second model, / , was trained
using the exponential loss with quadratic regularization. The performance was measured
using the conditional log-likelihood of the (normalized) models over the training and test
set, denoted train and test , as well as using the test error rate test . The table entries were
averaged by 10-fold cross validation.
_zK
_
For the weak learner FindAttrTest, only the Iris dataset produced features that satisfy
(3). On average, 4 out of the 10 features were removed. As the flexibility of the weak
learner is increased, (3) is expected to hold more often. On this dataset regularization
improves both the test set log-likelihood and error rate considerably. In datasets where
shows significant overfitting, regularization improves both the log-likelihood measure
and the error rate. In cases of little overfitting (according to the log-likelihood measure),
regularization only improves the test set log-likelihood at the expense of the training set
log-likelihood, however without affecting test set error.
_K
_
_
differs from ml
Next we performed a set of experiments to test how much boost
N
N , where
the boosting model is normalized (after training) to form a conditional probability distribution. For different experiments, FindAttrTest generated a different number of features
(10?100), and the training set was selected randomly. The top row in Figure 2 shows for
the Sonar dataset
as well as between
N and train boost
N
the relationship between train ml
N and train ml
N
Nboost . As the number of features increases so that the training
train ml
'F_ /
'B_ +V_ /
'F_ /
'F_ /
Data
Promoters
Iris
Sonar
Glass
Ionosphere
Hepatitis
Breast
Pima
train
'F_K/
'B_#K@/
Unregularized
test
-0.29
-0.29
-0.22
-0.82
-0.18
-0.28
-0.12
-0.48
-0.60
-1.16
-0.58
-0.90
-0.36
-0.42
-0.14
-0.53
test
'F_K/
0.28
0.21
0.25
0.36
0.13
0.19
0.04
0.26
train
'B_ / /
-0.32
-0.10
-0.26
-0.84
-0.21
-0.28
-0.12
-0.48
'F_ / /
Regularized
test
-0.50
-0.20
-0.48
-0.90
-0.28
-0.39
-0.14
-0.52
test
'B_ / /
0.26
0.09
0.19
0.36
0.10
0.19
0.04
0.25
Table 1: Comparison of unregularized to regularized boosting. For both the regularized and unregularized cases, the first column shows training log-likelihood, the second column shows test loglikelihood, and the third column shows test error rate. Regularization reduces error rate in some
cases while it consistently improves the test set log-likelihood measure on all datasets. All entries
were averaged using 10-fold cross validation.
'B_ / U Y
data is more closely fit ( train ml 7
), the boosting and maximum likelihood models
become more similar, as measured by the KL divergence. This result does not hold when
the model is unidentifiable and the two models diverge in arbitrary directions.
'B_ /
'F_ /
'B_ /
'F_ /
The bottom row in Figure 2 shows the relationship between the test set log-likelihoods,
, together with the test set error rates test ml
. In
N and test boost
N
N and test boost
N
test ml
these figures the testing set was chosen to be 50% of the total data. In order to indicate the
number of points at each error rate, each circle was shifted by a small random value to avoid
points falling on top of each other. While the plots in the bottom row of Figure 2 indicate
that train ml
, as expected, on the test data the linear trend is reversed,
N
N
train boost
so that test ml
N test boost
N
. Identical experiments on Hepatitis, Glass and Promoters
resulted in similar results and are omitted due to lack of space.
'F_ 'B_ / / W
'B'F_ _ / /
The duality result suggests a possible explanation for the higher performance of boosting
with respect to test . The boosting model is less constrained due to the lack of normalization
constraints, and therefore has a smaller -divergence to the uniform model. This may be
interpreted as a higher extended entropy, or less concentrated conditional model.
%'F_ / D Y
_#`\ +
'B%'_ F_ +V_/ ` / %'%'BB__ / / %'B_`>/r?
ML
However, as ml
, the two models come to agree (up to identifiability). It is easy to
N
show that for any exponential model
By taking
/
N
N "
train ml
ml
N
it is seen that as the difference between Vml
N and boost
N
gets smaller, the
boost
divergence between the two models also gets smaller. The empirical results are consistent
with the theoretical analysis. As the number of features is increased so that the training
data is fit more closely, the model matches the empirical distribution and the normalizing
N
becomes a constant. In this case, normalizing the boosting model " boost
does
term
not violate the constraints, and results in the maximum likelihood model.
_`( _
` 'B)D/
Acknowledgments
We thank Michael Collins, Michael Jordan, Andrew Ng, Fernando Pereira, Rob Schapire,
and Yair Weiss for helpful comments on an early version of this paper. Part of this work was
carried out while the second author was visiting the Department of Statistics, University of
California at Berkeley.
?0.05
boost
?0.1
train
boost
0.04
0.035
0.025
ml
0.045
train
0
0.03
?0.15
0.02
?0.2
0.015
0.01
PSfrag replacements
?0.25
PSfrag replacements
?0.3
?0.3
?0.25
?0.2
?0.15
train
?0.1
?0.05
0.005
0
?0.25
0
?0.2
?0.15
ml
?0.1
train
m
l
?0.05
0
0.4
?10
0.35
0.3
test
test
boost
?5
boost
0.25
?15
0.2
?20
PSfrag replacements
PSfrag replacements
?25
?25
?20
?15
0.15
0.1
?10
test
m
l
?5
0.1
0.15
0.2
0.25
0.3
test
0.35
0.4
ml
Figure 2: Comparison of AdaBoost and maximum likelihood for Sonar dataset. The top row com-
boost
pares train ml
to train boost
(left) and train ml
to train ml
(right). The bottom row
shows the relationship between test ml
and test boost
(left) and test ml
and test boost
(right).
The experimental results for other UCI datasets were very similar.
References
[1] S. Chen and R. Rosenfeld. A survey of smoothing techniques for ME models. IEEE Transactions on Speech and Audio Processing, 8(1), 2000.
[2] M. Collins, R. E. Schapire, and Y. Singer. Logistic regression, AdaBoost and Bregman distances. Machine Learning, to appear.
[3] S. Della Pietra, V. Della Pietra, and J. Lafferty. Inducing features of random fields. IEEE
Transactions on Pattern Analysis and Machine Intelligence, 19(4), 1997.
[4] S. Della Pietra, V. Della Pietra, and J. Lafferty. Duality and auxiliary functions for Bregman
distances. Technical Report CMU-CS-01-109, Carnegie Mellon University, 2001.
[5] N. Duffy and D. Helmbold. Potential boosters? In Neural Information Processing Systems,
2000.
[6] Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm. In International
Conference on Machine Learning, 1996.
[7] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: A statistical view of
boosting. The Annals of Statistics, 28(2), 2000.
[8] J. Kivinen and M. K. Warmuth. Boosting as entropy projection. In Computational Learning
Theory, 1999.
[9] J. Lafferty. Additive models, boosting, and inference for generalized divergences. In Computational Learning Theory, 1999.
[10] L. Mason, J. Baxter, P. Bartlett, and M. Frean. Functional gradient techniques for combining
hypotheses. In A. Smola, P. Bartlett, B. Sch?olkopf, and D. Schuurmans, editors, Advances in
Large Margin Classifiers, 1999.
[11] G. R?atsch, T. Onoda, and K.-R. M?uller. Soft margins for AdaBoost. Machine Learning, 2001.
| 2042 |@word version:1 pw:3 closure:1 incurs:1 moment:1 com:1 must:1 john:1 additive:2 enables:1 plot:1 interpretable:1 update:3 v_:7 discrimination:1 intelligence:1 selected:1 yr:12 warmuth:2 boosting:34 direct:1 become:1 psfrag:6 vad:2 introduce:1 expected:2 multi:1 little:1 increasing:1 becomes:4 ua:3 moreover:2 notation:2 v7t:2 mass:1 qw:1 what:1 interpreted:1 unified:1 unobserved:1 berkeley:1 every:1 returning:1 classifier:2 appear:1 understood:1 establishing:1 au:5 equivalence:1 suggests:2 gone:1 averaged:2 practical:1 unique:2 hood:1 yj:1 testing:1 acknowledgment:1 differs:2 procedure:6 empirical:4 projection:2 spite:1 suggest:1 get:3 onto:2 convenience:1 context:2 writing:1 equivalent:2 lagrangian:2 straightforward:1 convex:7 survey:1 helmbold:2 m2:5 insight:2 rule:1 mq:1 annals:1 suppose:2 hypothesis:1 pa:2 trend:1 satisfying:3 labeled:2 bottom:3 calculate:1 removed:1 trained:3 likeli:1 solving:1 learner:4 easily:1 various:2 intersects:2 train:19 whose:1 emerged:1 loglikelihood:1 statistic:6 rosenfeld:1 noisy:1 unresolved:1 uci:3 combining:1 flexibility:1 inducing:1 olkopf:1 qr:4 convergence:1 empty:1 produce:1 derive:8 andrew:1 n_:1 frean:1 measured:2 rescale:1 school:2 dividing:1 auxiliary:1 c:3 indicate:2 come:1 convention:1 kuhn:1 direction:1 closely:2 require:1 proposition:1 pl:2 c_:1 hold:5 cb:1 prm:1 lm:1 early:1 omitted:1 purpose:2 estimation:2 f7:2 label:2 e45:1 uller:1 gaussian:2 e7:1 rather:3 avoid:1 sion:1 notational:1 consistently:2 likelihood:35 mainly:1 hepatitis:2 hk:1 sense:1 glass:2 helpful:1 inference:1 kkw:2 arg:1 dual:16 denoted:2 constrained:1 special:4 smoothing:1 uc:1 marginal:1 field:1 once:2 ng:1 identical:3 simplex:2 minimized:1 report:1 employ:1 randomly:2 divergence:12 resulted:1 pietra:4 familiar:1 geometry:1 consisting:1 replacement:6 friedman:2 interest:3 investigate:1 primal:4 devoted:1 kt:1 bregman:7 necessary:1 old:1 harmful:1 circle:1 theoretical:3 increased:2 column:3 modeling:1 soft:1 cost:1 entry:2 uniform:1 gcv:1 considerably:1 international:1 e_:1 diverge:1 michael:2 connecting:2 together:1 satisfied:1 possibly:1 guy:1 booster:1 jrk:1 account:2 potential:3 stump:1 satisfy:2 reg:4 performed:2 view:6 parallel:2 identifiability:1 minimize:5 square:1 correspond:4 yield:1 weak:4 bayesian:2 produced:1 tucker:1 proof:1 irvine:1 proved:1 dataset:4 improves:4 appears:2 higher:3 adaboost:19 wei:1 unidentifiable:1 generality:1 smola:1 until:1 overfit:1 working:1 lent:1 hand:1 nonlinear:2 lack:2 logistic:11 grows:2 normalized:8 multiplier:5 consisted:1 regularization:10 equality:1 leibler:3 ll:1 noted:1 unnormalized:4 iris:2 generalized:1 mpl:5 demonstrate:1 neo:1 recently:1 functional:2 empirically:1 exponentially:1 interpretation:2 mellon:3 refer:1 significant:1 trivially:1 language:1 similarity:1 add:3 xwzy:1 recent:2 xzw:1 binary:4 sjt:1 seen:3 minimum:1 additional:6 impose:1 determine:3 maximize:1 fernando:1 violate:1 reduces:1 technical:1 match:1 calculation:1 cross:2 raphson:1 e1:1 uav:2 regression:10 basic:1 breast:1 cmu:3 iteration:1 represent:1 normalization:4 addition:2 affecting:1 leaving:1 sch:1 zw:8 qrw:1 comment:1 subject:4 member:1 lafferty:6 n7:1 effectiveness:1 jordan:1 easy:2 relaxes:1 baxter:1 fit:3 hastie:2 bartlett:2 k9:1 penalty:1 speech:1 v8:1 concentrated:1 schapire:4 exist:1 problematic:1 shifted:1 tibshirani:2 carnegie:3 key:2 falling:1 clean:1 striking:1 throughout:2 family:4 decision:1 def:2 correspondence:3 fold:2 quadratic:3 nonnegative:1 activity:1 precisely:2 constraint:21 attempting:1 department:1 according:1 conjugate:1 smaller:3 y0:3 rob:1 making:2 happens:1 projecting:2 unregularized:3 equation:4 agree:1 singer:2 b_:9 appropriate:1 fad:9 alternative:2 yair:1 original:1 denotes:1 top:3 cf:1 newton:1 giving:1 especially:1 nboost:2 objective:2 usual:2 visiting:1 gradient:2 distance:7 reversed:1 thank:1 me:1 relationship:8 minimizing:5 setup:1 statement:1 pima:1 expense:1 negative:1 ba:1 datasets:5 finite:2 descent:1 extended:4 excluding:1 precise:2 dc:1 arbitrary:1 required:2 specified:1 kl:4 connection:5 california:1 boost:26 pattern:1 including:1 explanation:1 natural:1 regularized:4 kivinen:2 mn:2 carried:2 kj:2 prior:4 understanding:1 geometric:2 freund:1 loss:21 validation:2 sufficient:3 consistent:3 ponential:1 editor:1 lmn:3 row:5 course:1 penalized:2 ovum:1 formal:1 side:1 understand:1 taking:2 default:1 calculated:1 doesn:1 kau:2 forward:1 qualitatively:1 author:1 transaction:2 lebanon:2 approximate:1 kullback:3 ml:36 suggestive:1 overfitting:2 pittsburgh:2 sonar:3 table:3 fao:2 k_:1 zk:1 onoda:1 f_:12 schuurmans:1 da:1 mnl:14 promoter:2 linearly:1 tl:1 fails:1 lmo:1 pereira:1 exponential:29 lie:1 ib:1 third:1 theorem:5 showing:1 pac:1 mason:2 ionosphere:1 a3:1 normalizing:3 essential:1 stepwise:2 sequential:2 duffy:2 margin:3 chen:1 entropy:4 intersection:1 simply:1 lagrange:5 corresponds:7 conditional:12 viewed:2 formulated:1 presentation:3 feasible:4 specifically:1 total:1 duality:8 e:1 experimental:1 atsch:1 formally:1 support:2 latter:3 collins:3 overload:1 pythagorean:1 audio:1 regularizing:1 della:4 ex:1 |
1,144 | 2,043 | Efficient Resources Allocation
for Markov Decision Processes
Remi Munos
CMAP, Ecole Polytechnique, 91128 Palaiseau, France
http://www.cmap.polytechnique.fr/....munos
remi.munos@polytechnique.fr
Abstract
It is desirable that a complex decision-making problem in an uncertain world be adequately modeled by a Markov Decision Process
(MDP) whose structural representation is adaptively designed by a
parsimonious resources allocation process. Resources include time
and cost of exploration, amount of memory and computational time
allowed for the policy or value function representation. Concerned
about making the best use of the available resources, we address
the problem of efficiently estimating where adding extra resources
is highly needed in order to improve the expected performance of
the resulting policy. Possible application in reinforcement learning
(RL) , when real-world exploration is highly costly, concerns the detection of those areas of the state-space that need primarily to be
explored in order to improve the policy. Another application concerns approximation of continuous state-space stochastic control
problems using adaptive discretization techniques for which highly
efficient grid points allocation is mandatory to survive high dimensionality. Maybe surprisingly these two problems can be formulated under a common framework: for a given resource allocation,
which defines a belief state over possible MDPs, find where adding
new resources (thus decreasing the uncertainty of some parameters -transition probabilities or rewards) will most likely increase
the expected performance of the new policy. To do so, we use sampling techniques for estimating the contribution of each parameter's
probability distribution function (Pdf) to the expected loss of using an approximate policy (such as the optimal policy of the most
probable MDP) instead of the true (but unknown) policy.
Introduction
Assume that we model a complex decision-making problem under uncertainty by
a finite MDP. Because of the limited resources used, the parameters of the MDP
(transition probabilities and rewards) are uncertain: we assume that we only know
a belief state over their possible values. IT we select the most probable values of
the parameters, we can build a MDP and solve it to deduce the corresponding
optimal policy. However, because of the uncertainty over the true parameters, this
policy may not be the one that maximizes the expected cumulative rewards of the
true (but partially unknown) decision-making problem. We can nevertheless use
sampling techniques to estimate the expected loss of using this policy. Furthermore,
if we assume independence of the parameters (considered as random variables), we
are able to derive the contribution of the uncertainty over each parameter to this
expected loss. As a consequence, we can predict where adding new resoUrces (thus
decreasing the uncertainty over some parameters) will decrease mostly this loss,
thus improving the MDP model of the decision-making problem so as to optimize
the expected future rewards.
As possible application, in model-free RL we may wish to minimize the amount of
real-world exploration (because each experiment is highly costly). Following [1] we
can maintain a Dirichlet pdf over the transition probabilities of the corresponding
MDP. Then, our algorithm is able to predict in which parts of the state space we
should make new experiments, thus decreasing the uncertainty over some parameters (the posterior distribution being less uncertain than the prior) in order to
optimize the expected payoff.
Another application concerns the approximation of continuous (or large discrete)
state-space control problems using variable resolution grids, that requires an efficient resource allocation process in order to survive the "curse of dimensionality"
in high dimensions. For a given grid, because of the interpolation process, the approximate back-up operator introduces a local interpolation error (see [4]) that may
be considered as a random variable (for example in the random grids of [6]). The
algorithm introduced in this paper allows to estimate where we should add new
grid-points, thus decreasing the uncertainty over the local interpolation error, in
order to increase the expected performance of the new grid representation. The
main tool developed here is the calculation of the partial derivative of useful global
measures (the value function or the loss of using a sub-optimal policy) with respect
to each parameter (probabilities and rewards) of a MDP.
1
Description of the problem
We consider a MDP with a finite state-space X and action-space A. A transition
from a state x, action a to a next state y occurs with probability p(Ylx, a) and the
corresponding (deterministic) reward is r(x, a). We introduce the back-up operator
T a defined, for any function W : X --t JR, as
T a W(x) == (' LP(Ylx, a)W(y)
+ r(x, a)
(1)
y
(with some discount factor 0 < (' < 1). It is a contraction mapping, thus the
dynamic programming (DP) equation V(x) == maxaEA T a V(x) has a unique fixed
point V called the value function. Let. us define the Q-values Q(x, a) == T a V (x).
The optimal policy 1[* is the mapping from any state x to? the action 1[* (x) that
maximizes the Q-values: 1[*(x) == maxaEA Q(x, a).
The parameters of the MDP - the probability and the reward functions - are
not perfectly known: all we know is a pdf over their possible values. This uncertainty comes from the limited amount of allocated resources for estimating those
parameters.
Let us choose a specific policy 1r (for example the optimal policy of the MDP with
the most probable parameters). We can estimate the expected loss of using 1r instead
of the true (but unknown) optimal policy 1[*. Let us write J-t == {Pj} the set of all
parameters (p and r functions) of a MDP. We assume that we know a probability
distribution function pdf(J-Lj) over their possible values. For a MDP MJ.t defined
by its parameters P, we write pJL (y Ix, a), r JL (x, a), V JL, QJL, and 7f1-!' respectively its
transition probabilities, rewards, value function, Q-values, and optimal policy.
1.1
Direct gain optimization
We define the gain ]JL(x; 7f) in the MDP MJL as the expected sum of discounted
rewards obtained starting from state x and using policy 7f:
]JL(x; 1f)
== E[2: rykrJL(Xk' 7f(xk))lxo == x; 7f]
(2)
k
where the expectation is taken for sequences of states Xk --t Xk+l occurring with
probability pP(Xk+llxk, 7fJL(Xk)). By definition, the optimal gain in MJL is VJL(x) ==
]JL (x; 7fJL) which is obtained for the optimal policy 7fIL. Let ~ (x) == ]JL (x; if) be
the approximate gain obtained for some approximate policy .7r in the same MDP
MIL. We define the loss to occur LJL(x) from X when one uses the approximate
policy 7r instead of the optimal one 7fJL in MJL:
LIL(X)
== VIL(x) - ~(x)
(3)
An example of approximate policy 1? would be the optimal policy of the most probable MDP, defined by the most probable parameters fi(ylx, a) and r(x, a).
We also consider the problem of maximizing the global gain from a set of initial
states chosen according to some probability distribution P(x). Accordingly, we
define the global gain of a policy 11"": ]JL(7f) == Ex ]JL(x; 7f)P(x) and the global
loss LIL of using some approximate policy 7r instead of the optimal one nIL
(4)
Thus, knowing the pdf over all parameters J-l we can define the expected global
loss L == EJL[LIL].
Next, we would like to define what is the contribution of each parameter uncertainty
to this loss, so we know where we should add new resources (thus reducing some
parameters uncertainty) in order to decrease the expected global loss. We would
like to estimate, for each parameter J-lj,
(5)
E[8L I Add 8u units of resource for Pj]
1.2
Partial derivative of the loss
ill order to quantify (5) we need to be more explicit about the pdf over JL. First, we
assume the independence of the parameters JLj (considered as random variables).
Suppose that pdf (JLj) == N (0, U j) (normal distribution of mean 0 and standard
deviation Uj). We would like to estimate the variation 8L of the expected loss L
when we make a small change of the uncertainty over Pj (consequence of adding new
resources), for example when changing the standard deviation of 8aj in pdf(J.tj). At
the limit of an infinitesimal variation we obtain the partial derivative
which
3
when computed for all parameters J-lj, provides the respective contributions of each
parameter's uncertainty to the global loss.
Z;.,
Another example is when the pdf(pj) is a uniform distribution of support [-b j , bj ].
Then the partial contribution of JLj'S uncertainty to the global loss can be expressed
More generally, we can define a finite number of characteristic scalar meaas
3
surements of the pdf uncertainty (for example the entropy or the moments) and
gf.
compute the partial derivative of the expected global loss with respect to these coefficients. Finally, knowing the actual resources needed to estimate a parameter J..tj
with some uncertainty defined by pdf (J..tj ), we are able to estimate (5).
1.3
Unbiased estimator
We sample N sets of parameters {J..t i }i=1..N from the pd!(J..t) , which define N-MDPs
Mi. For convenience, we use the superscript i to refer to the i-th MDP sample
and the subscript j for the j-th parameter of a variable. We solve each MDP using
standard DP techniques (see [5]). This expensive computation that can be speed-up
in two ways: first, by using the value function and policy computed for the first
MDP as initial values for the other MDPs; second, since all MDPs have the same
structure, by computing once for all an efficient ordering (using a topological sort,
possibly with loops) of the states that will be used for value iteration.
For each MDP, we compute the global loss L i of using the policy 'if and estimate
the expected global loss: L ~
2:::1 L i . In order to estimate the contribution of
a p-arameter's uncertainty to L, we derive the partial derivative of L with respect
to the characteristic coefficients of pdf (J-tj ). In the case of a reward parameter J..tj
that follows a normal distribution N(O, Uj), we can write J..tj == Uj?j where ?j follows
N(O, 1). The partial derivative of the expected loss L with respect to Uj is
-1
8
8
aL == a E/L~N(o.u)[L/L] = a Ee~N(o.l)[LUe] =
a~
~
~
Ee~N(o.1)[8aLue ~j]
(6)
~
from which we deduce the unbiased estimator
8L '" ~
aUj - N
t
i=l
i
Jt;
(7)
8L
aJ..tj Uj
where ~;; is the partial derivative of the global loss Li of MDP M i with respect to
the parameter J..tj (considered as a variable). For other distributions, we can define
similar results to (6) and deduce analogous estimators (for uniform distributions,
we have the same estimator with bj instead of Uj).
The remainder of the paper is organized as follow. Section 2 introduces useful tools
to derive the partial contribution of each parameter -transition probability and
reward- to the value function in a Markov Chain, Section 3 establishes the partial
contribution of each parameter to the global loss, allowing to calculate the estimator
(7), and Section 4 provides an efficient algorithm. All proofs are given in the full
length paper [2].
2
2.1
Non-local dependencies
Influence of a markov chain
In [3] we introduced the notion of influence of a Markov Chain as a way to measure
value function/rewards correlations between states. Let us consider a set of values
V satisfying a Bellman equation
Vex) == , LP(ylx)V(y) + rex)
(8)
y
We define the discounted cumulative k-chained transition probabilities Pk(ylx):
po(ylx)
Pl(ylx)
(= 1 (if x = y) or 0 (if x
IP(ylx)
Ix =y
=1=
y))
LP1(ylw)Pl(wlx)
w
LP1(ylw)Pk-l(wlx)
w
The influence I(ylx) of a state y on another state x is defined as I(ylx) =
2::%:oPk(ylx). Intuitively, I(ylx) measures the expected discounted number of visits of state y starting from x; it is also the partial derivative of the value function
Vex) with respect to the reward r(y). Indeed Vex) can be expressed, as a linear
combination of the rewards at y weighted by the influence I(ylx)
(9)
Vex) = LI(Ylx)r(y)
y
We can also define the influence of a state y on a function f: I(ylf(?)) =
2::x l(ylx)f(x) and the influence of a function f on another function 9 :
l(f(?)\g(?)) = Y":y I(ylg(?))f(y)? In [3], we showed that the influence satisfies
I(ylx)
=,
LP(ylw)I(wlx)
+ lx=y
(10)
w
which is a fixed-point equation of a contractant operator (in I-norm) thus has
a unique solution -the influence- that can be computed by successive iterations.
Similarly, the influence I(ylf(?)) can be obtained as limit of the iterations
I(ylf(?))
+-, LP(Ylw)I(wlf(?)) + fey)
w
Thus the computation of the influence I(ylf(?)) is cheap (equivalent to solving a
Markov chain).
2.2
Total derivative of V
We wish to express the contribution of all parameters - transition probabilities p
and rewards r - (considered as variables) to the value function V by defining the
total derivative of V as a function of those P?ameters. We recall that the total
f dXI + ... + a8t dx .
derivative of a function f of several variables Xl, ..,' X n is df = 88Xl
n
Xn
We already know that the partial derivative of Vex) with respect to the reward r(z)
is the influence I(zjx) = ~~~1. Now, the dependency with respect to the transition
probabilities has to be expressed more carefully because the probabilities p(wlz) for
a given z are dependent (they sum to one). A way to express that is provided in
the theorem that follows whose proof is in [2].
Theorelll 1 For a given state z, let us alter the probabilities p(wlz), for all w,
with some c5'p(wlz) value, such that 2:: w c5'p(wlz) = o. Then Vex) is altered by
c5'V(x) = I(zlx)[,2:: w V(w)c5'p(wlz)]. We deduce the total derivative of
v:
dV(x)
= L1(zlx)[, L
z
V(w)dp(wlz)
+ dr(z)]
w
under the constraint 2::w dp( wi z) = 0 for all z.
3
Total derivative of the loss
, For a given MDP M with parameters J..L (for notation simplification we do not
write the JL superscript in what follows), we want to estimate the loss of using an
approximate policy 7? instead of the optimal one 1f. First, we define the one-step
loss l(x) at a state x as the difference between the gain obtained by choosing the
best action 7f(x) then using the optimal policy 1f and the gain obtained by choosing
action n(x) then the same optimal policy 7f
l(x) == Q(x,1f(x)) - Q(x,ir(x))
(11)
Now we consider the loss L(x), defined by (3), for an initial state x when we use
the approximate policy n. We can prove that L(x) is the expected discounted
cumulative one-step losses l(Xk) for reachable states Xk:
L(x) == E[L I'k l(Xk)lxo == x;n]
k
with the expectation taken in the same sense as in (2).
3.1
Decomposition of the one-step loss
We use (9) to decompose the Q-values
Q(x, a) == I' LP(wlx, a) L I(ylw)r(y, 1f(Y))
w
+ r(x, a)
y
== r(x,a) + Lq(Ylx,a)r(y,7f(y))
y
using the partial contributions q(ylx,a) == I'Ewp(wlx,a)I(ylw) where I(ylw) is
the influence of y on w in the Markov chain derived from the MDP M by choosing
policy 7f. Similarly, we decompose the one-step loss
l(x) == Q(x,7f(x)) - Q(x, n(x))
== r(x,1f(x)) - r(x,7f(x)) + L [q(ylx,1f(x)) - q(ylx,n(x))] r(y,7f(Y))
y
== r(x, 7f(x)) -r(x, 7?(x)) + Ll(Ylx)r(y, 7f(Y))
y
as a function of the partial contributions l(ylx) == q(ylx,1f(x)) - q(ylx, n(x)) (see
figure 1).
o
q (ylx ,IT )
q (ylx ,11- )
Figure 1: The reward r(y,1r(Y)) at
state y contributes to the one-step
loss l(x) = Q(x, 1r(x)) - Q(x, 1?(x))
with the proportion l(ylx)
q(ylx, 1I"(x)) - q(ylx, 1?(x)).
3.2
Total derivative of the one-step loss and global loss
Similarly to section (2.2), we wish to express the contribution of all parameters transition probabilities p and rewards r - (considered as variables) to the one-step
loss function by defining the total derivative of I as a function of those parameters.
Theorem 2 Let us introduce the (formal) differential back-up operator dT a defined, for any function W : X ~ JR, as
dT a W(x) == ry L W(y)dp(ylx, a)
+ dr(x, a)
y
(similar to the back-up operator (1) but using dp and dr instead of p and r). The
total derivative of the one-step loss is
dl(x).==L 1(zlx)dT 7f (z)V(z)
+ dT7f(x)V(x) -
dT;Cx)V(x)
z
under the constraint
E y dp(ylx, a) == 0 for
all x and a.
Theorem 3 Let us introduce the one-step-loss back-up operator S and its formal
differential version dS defined, for any function W : X ~ JR, as
SW(x)
ry LP(Ylx, 7T"(x))W(y)
+ l(x)
y
dSW(x)
ry L
dp(ylx, 7T"(x))W(y)
+ dl(x)
y
Then, the loss L(x) at x satisfies Bellman's equation L
of the loss L (x) and global loss L are, respectively
dL(x)
== SL. The total derivative
L I(zlx)dSL(z)
Z
dL
L I(zIP(?))dSL(z)
z
from which (after regrouping the contribution to each parameter) we deduce the
partial derivatives of the global loss with respect to the rewards and transition
probabilities
4
A fast algorithm
We use the sampling technique introduced in section 1.3. In order to compute the
estimator (7) we calculate the partial derivatives ~~; based on the result of the
previous section, with the following algorithm~
Given the pdf over the parameters j.L, select a policy 7? (for example the optimal
policy of the most probable MDP). For i == 1..N, solve each MDP M i and deduce
its value function Vi, Q-values Qi, and optimal policy 7ri . Deduce the one-step loss
li(x) from (11). Compute the influence I(xIP(?)) (which depends on the transition probabilities pi of M i ) and the influence I(li(xl?)IP(?)) from which we deduce
i
ar ~(Lix,a ). Then calculate Li(x) by solving Bellman's equation Li = SL and deduce
8P,r~:,a). These partial derivatives enable to compute the unbiased estimator (7).
The complexity of solving a discounted MDP with K states, each one connected to
M next states, is O(KM), as is the complexity of computing the influences. Thus,
the overall complexity of this algorithm is O(NKM).
Conclusion?
Being able to compute the contribution of each parameter -transition probabilities
and rewards- to the value function (theorem 1) and to the loss of the expected
rewards to occur if we use an approximate policy (theorem 3) enables us to use
sampling techniques to estimate what are the parameters whose uncertainty are the
most harmful to the expected gain.. A relev-ant resource allocation process would
consider adding new computational resources to reduce uncertainty over the true
value of those parameters. In the examples given in the introduction, this would be
doing new experiments in model-free RL for defining more precisely the transition
probabilities of some relevant states. In discretization techniques for continuous
control problems, this would be adding new grid points in order to improve the
quality of the interpolation at relevant areas of the state space in order to maximize
the expected gain of the new policy. Initial experiments for variable resolution
discretization using random grids show improved performance compared to [3].
Acknowledgments
I am grateful to Andrew Moore, Drew Bagnell and Auton's Lab members for motivating discussions.
References
[1] Richard Dearden, Nir Friedman, and David Andre. Model based bayesian exploration. Proceeding of Uncertainty in Artificial Intelligence, 1999.
[2] Remi Munos. Decision-making under uncertainty:. Efficiently estimating where
extra ressources are needed. Technical report, Ecole Polytechnique, 2002.
[3] Remi Munos and Andrew Moore. Influence and variance of a markov chain :
Application to adaptive discretizations in optimal control. Proceedings of the
38th IEEE Conference on Decision and Control, 1999.
[4] Remi Munos and Andrew W. Moore. Rates of convergence for variable resolution
schemes in optimal control. International Conference on Machine Learning,
2000.
[5] Martin L. Puterman. Markov Decision Processes, Discrete Stochastic Dynamic
Programming. A Wiley-Interscience Publication, 1994.
[6] John Rust. Using Randomization to Break the Curse of Dimensionality. Computational Economics. 1997.
| 2043 |@word version:1 norm:1 proportion:1 km:1 contraction:1 decomposition:1 moment:1 initial:4 ecole:2 discretization:3 dx:1 john:1 cheap:1 enables:1 lue:1 designed:1 intelligence:1 accordingly:1 xk:9 provides:2 lx:1 successive:1 direct:1 differential:2 prove:1 interscience:1 ressources:1 introduce:3 indeed:1 expected:22 ry:3 bellman:3 discounted:5 decreasing:4 actual:1 curse:2 provided:1 estimating:4 notation:1 maximizes:2 what:3 developed:1 control:6 unit:1 local:3 limit:2 consequence:2 subscript:1 interpolation:4 limited:2 unique:2 acknowledgment:1 area:2 discretizations:1 convenience:1 operator:6 influence:16 www:1 optimize:2 deterministic:1 equivalent:1 maximizing:1 economics:1 starting:2 resolution:3 estimator:7 notion:1 variation:2 analogous:1 suppose:1 programming:2 us:1 jlj:3 expensive:1 satisfying:1 lix:1 calculate:3 connected:1 ordering:1 decrease:2 pd:1 complexity:3 reward:21 dynamic:2 chained:1 grateful:1 solving:3 po:1 fast:1 artificial:1 choosing:3 whose:3 solve:3 superscript:2 ip:2 sequence:1 fr:2 remainder:1 relevant:2 loop:1 description:1 convergence:1 derive:3 andrew:3 come:1 quantify:1 stochastic:2 exploration:4 enable:1 alue:1 f1:1 decompose:2 theorelll:1 randomization:1 probable:6 auj:1 pl:2 fil:1 considered:6 normal:2 mapping:2 predict:2 bj:2 establishes:1 tool:2 weighted:1 nkm:1 mil:1 publication:1 derived:1 sense:1 am:1 dependent:1 xip:1 lj:3 france:1 overall:1 ill:1 once:1 sampling:4 survive:2 a8t:1 future:1 alter:1 report:1 richard:1 primarily:1 maintain:1 friedman:1 detection:1 highly:4 wlf:1 introduces:2 tj:8 chain:6 partial:17 respective:1 harmful:1 uncertain:3 ar:1 opk:1 cost:1 deviation:2 uniform:2 rex:1 motivating:1 dependency:2 adaptively:1 international:1 choose:1 possibly:1 dr:3 derivative:21 li:6 zjx:1 coefficient:2 vi:1 depends:1 break:1 lab:1 doing:1 surements:1 sort:1 contribution:14 minimize:1 ir:1 variance:1 characteristic:2 efficiently:2 ant:1 dsl:2 bayesian:1 vil:1 andre:1 definition:1 infinitesimal:1 pp:1 proof:2 mi:1 dxi:1 gain:10 recall:1 dimensionality:3 organized:1 carefully:1 back:5 dt:4 follow:1 improved:1 furthermore:1 correlation:1 d:1 defines:1 quality:1 aj:2 mdp:26 fjl:3 true:5 unbiased:3 wlz:6 adequately:1 moore:3 puterman:1 ll:1 pdf:13 polytechnique:4 l1:1 fi:1 common:1 rl:3 rust:1 jl:10 ejl:1 refer:1 grid:8 similarly:3 reachable:1 ylf:4 deduce:9 add:3 posterior:1 showed:1 mandatory:1 regrouping:1 zip:1 maximize:1 full:1 desirable:1 technical:1 calculation:1 lxo:2 vjl:1 visit:1 qi:1 expectation:2 df:1 iteration:3 want:1 allocated:1 extra:2 member:1 structural:1 ee:2 concerned:1 independence:2 perfectly:1 reduce:1 knowing:2 action:5 useful:2 generally:1 llxk:1 ylx:33 maybe:1 amount:3 discount:1 http:1 ylw:7 sl:2 qjl:1 discrete:2 write:4 express:3 nevertheless:1 changing:1 pj:4 sum:2 uncertainty:20 parsimonious:1 vex:6 decision:9 simplification:1 topological:1 occur:2 constraint:2 precisely:1 ri:1 speed:1 martin:1 according:1 combination:1 jr:3 fey:1 ewp:1 wi:1 lp:6 making:6 arameter:1 intuitively:1 dv:1 taken:2 resource:17 equation:5 needed:3 know:5 auton:1 available:1 dirichlet:1 include:1 sw:1 build:1 uj:6 already:1 occurs:1 costly:2 bagnell:1 dp:8 length:1 modeled:1 mostly:1 lil:3 policy:36 unknown:3 allowing:1 markov:9 finite:3 payoff:1 defining:3 introduced:3 david:1 zlx:4 address:1 able:4 maxaea:2 memory:1 belief:2 dearden:1 scheme:1 improve:3 altered:1 mdps:4 gf:1 nir:1 prior:1 loss:40 allocation:6 pi:1 surprisingly:1 free:2 formal:2 munos:6 dimension:1 xn:1 world:3 transition:14 cumulative:3 c5:4 reinforcement:1 adaptive:2 palaiseau:1 approximate:10 global:16 continuous:3 ljl:1 mj:1 pjl:1 contributes:1 improving:1 complex:2 pk:2 main:1 allowed:1 wiley:1 sub:1 wish:3 explicit:1 lq:1 xl:3 ix:2 theorem:5 specific:1 jt:1 explored:1 concern:3 dl:4 adding:6 drew:1 cmap:2 occurring:1 entropy:1 cx:1 remi:5 likely:1 expressed:3 partially:1 scalar:1 satisfies:2 formulated:1 change:1 reducing:1 called:1 nil:1 total:9 lp1:2 select:2 support:1 ex:1 |
1,145 | 2,044 | Speech Recognition using SVMs
Nathan Smith
Cambridge University
Engineering Dept
Cambridge, CB2 1PZ, U.K.
ndsl 002@eng.cam.ac.uk
Mark Gales
Cambridge University
Engineering Dept
Cambridge, CB2 1PZ, U.K.
mjfg@eng.cam.ac.uk
Abstract
An important issue in applying SVMs to speech recognition is the
ability to classify variable length sequences. This paper presents
extensions to a standard scheme for handling this variable length
data, the Fisher score. A more useful mapping is introduced based
on the likelihood-ratio. The score-space defined by this mapping
avoids some limitations of the Fisher score. Class-conditional generative models are directly incorporated into the definition of the
score-space. The mapping, and appropriate normalisation schemes,
are evaluated on a speaker-independent isolated letter task where
the new mapping outperforms both the Fisher score and HMMs
trained to maximise likelihood.
1
Introduction
Speech recognition is a complex, dynamic classification task. State-of-the-art systems use Hidden Markov Models (HMMs), either trained to maximise likelihood or
discriminatively, to achieve good levels of performance. One of the reasons for the
popularity of HMMs is that they readily handle the variable length data sequences
which result from variations in word sequence, speaker rate and accent. Support
Vector Machines (SVMs) [1] are a powerful, discriminatively-trained technique that
have been shown to work well on a variety of tasks. However they are typically only
applied to static binary classification tasks. This paper examines the application
of SVMs to speech recognition. There are two major problems to address. First,
how to handle the variable length sequences. Second, how to handle multi-class
decisions. This paper only concentrates on dealing with variable length sequences.
It develops our earlier research in [2] and is detailed more fully in [7]. A similar
approach for protein classification is adopted in [3].
There have been a variety of methods proposed to map variable length sequences
to vectors of fixed dimension. These include vector averaging and selecting a 'representative ' number of observations from each utterance. However, these methods
may discard useful information. This paper adopts an approach similar to that of
[4] which makes use of all the available data. Their scheme uses generative probability models of the data to define a mapping into a fixed dimension space, the
Fisher score-space. When incorporated within an SVM kernel, the kernel is known
as the Fisher kernel. Relevant regularisation issues are discussed in [5]. This paper
examines the suitability of the Fisher kernel for classification in speech recognition
and proposes an alternative, more useful, kernel. In addition some normalisation
issues associated with using this kernel for speech recognition are addressed.
Initially a general framework for defining alternative score-spaces is required. First,
define an observation sequence as 0 = (01 , . . . Ot, ... OT) where Ot E ~D , and a set
of generative probability models of the observation sequences as P = {Pk(OI(h)},
where 9 k is the vector of parameters for the kth member of the set. The observation
sequence 0 can be mapped into a vector of fixed dimension [4],
i{J~ (0)
(1)
f(?) is the score-argument and is a function of the members of the set of generative
models P. i{Jft is the score-mapping and is defined using a score-operator F. i{J~(0)
is the score and occupies the fixed-dimension score-space. Our investigation of
score-spaces falls into three divisions. What are the best generative models, scorearguments and score-operators to use?
2
Score-spaces
As HMMs have proved successful in speech recognition, they are a natural choice
as the generative models for this task. In particular HMMs with state output
distributions formed by Gaussian mixture models. There is also the choice of the
score-argument. For a two-class problem, let Pi(019 i ) represent a generative model,
where i = {g, 1, 2} (g denotes the global2-class generative model, and 1 and 2 denote
the class-conditional generative models for the two competing classes). Previous
schemes have used the log of a single generative model, Inpi (019 i ) representing
either both classes as in the original Fisher score (i = g) [4], or one of the classes
(i = 1 or 2) [6]. This score-space is termed the likelihood score-space, i{J~k(O).
The score-space proposed in this paper uses the log of the ratio of the two classconditional generative models, In(P1(019d / P2(019 2)) where 9 = [9{,9J] T. The
corresponding score-space is called the likelihood-ratio score-space, i{J~(0) . Thus,
i{J~k(O)
(2)
i{J~(0)
(3)
The likelihood-ratio score-space can be shown to avoid some of the limitations of
the likelihood score-space, and may be viewed as a generalisation of the standard
generative model classifier. These issues will be discussed later.
Having proposed forms for the generative models and score-arguments, the scoreoperators must be selected. The original score-operator in [4] was the 1st-order
derivative operator applied to HMMs with discrete output distributions. Consider
a continuous density HMM with N emitting states, j E {I . . . N}. Each state,
j, has an output distribution formed by a mixture of K Gaussian components,
N(J-tjk' ~jd where k E {I ... K}. Each component has parameters of weight Wjk,
mean J-tjk and covariance ~jk. The 1st-order derivatives of the log probability of
the sequence 0 with respect to the model parameters are given below1, where the
derivative operator has been defined to give column vectors,
T
L ')'jk(t)S~,jkl
t= l
lFor fuller details of the derivations see [2).
(4)
V Wjk Inp(OIO)
where
S[t ,jk]
Ijdt) is the posterior probability of component k of state j at time t. Assuming
the HMM is left-to-right with no skips and assuming that a state only appears once
in the HMM (i.e. there is no state-tying), then the 1st-order derivative for the
self-transition probability for state j, ajj, is,
t[/j(t)
t=l ajj
1]
Tajj(l- ajj)
(8)
The 1st-order derivatives for each Gaussian parameter and self-transition probability in the HMM can be spliced together into a 'super-vector' which is the score 2 .
From the definitions above, the score for an utterance is a weighted sum of scores
for individual observations. If the scores for the same utterance spoken at different
speaking rates were calculated, they would lie in different regions of score-space
simply because of differing numbers of observations. To ease the task of the classifier
in score-space, the score-space may be normalised by the number of observations,
called sequence length normalisation. Duration information can be retained in the
derivatives of the transition probabilities. One method of normalisation redefines
score-spaces using generative models trained to maximise a modified log likelihood
function, In( 010). Consider that state j has entry time Tj and duration d j (both in
numbers of observations) and output probability bj(Ot) for observation Ot [7]. So,
1
N
L
In(OIO)
T;+d j- 1
d- ((d j -1) lnajj + Inaj(j+1) +
j=l
L
(Inbj(Ot)))
(9)
t=Tj
J
It is not possible to maximise In(OIO) using the EM algorithm. Hill-climbing techniques could be used. However, in this paper, a simpler normalisation method is
employed. The generative models are trained to maximise the standard likelihood
function. Rather than define the score-space using standard state posteriors Ij(t)
(the posterior probability of state j at time t), it is defined on state posteriors normalised by the total state occupancy over the utterance. The standard component
posteriors 1 j k (t) are replaced in Equations 4 to 6 and 8 by their normalised form
'Yjk(t),
A
.
~k
(t) _
-
Ij(t)
T
(WjkN(Ot; ILjk, ~jk)
K
2:: T=l/j(T) 2:: i = l wjiN(ot; ILji' ~ji)
)
(10)
In effect, each derivative is divided by the sum of state posteriors. This is preferred
to division by the total number of observations T which assumes that when the
utterance length varies, the occupation of every state in the state sequence is scaled
by the same ratio. This is not necessarily the case for speech.
The nature of the score-space affects the discriminative power of classifiers built
in the score-space. For example, the likelihood score-space defined on a two-class
2Due to the sum to unity constraints, one of the weight parameters in each Gaussian
mixture is discarded from the definition of the super-vector, as are the forward transitions
in the left-to-right HMM with no skips.
generative model is susceptible to wrap-around [7] . This occurs when two different
locations in acoustic-space map to a single point in score-subspace. As an example,
consider two classes modelled by two widely-spaced Gaussians. If an observation
is generated at the peak of the first Gaussian, then the derivative relative to the
mean of that Gaussian is zero because S [t ,jk] is zero (see Equation 4). However, the
derivative relative to the mean of the distant second Gaussian is also zero because
of a zero component posterior f jdt). A similar problem occurs with an observation
generated at the peak of the second Gaussian. This ambiguity in mapping two
possible locations in acoustic-space to the zero of the score-subspace of the means
represents a wrapping of the acoustic space onto this subspace. This also occurs
in the subspace of the variances. Thus wrap-around can increase class confusion.
A likelihood-ratio score-space defined on these two Gaussians does not suffer wraparound since the component posteriors for each Gaussian are forced to unity.
So far, only 1st-order derivative score-operators have been considered. It is possible to include the zeroth, 2nd and higher-order derivatives. Of course there is
an interaction between the score-operator and the score-argument. For example,
the zeroth-order derivative for the likelihood score-space is expected to be less useful than its counter-part in the likelihood-ratio score-space because of its greater
sensitivity to acoustic conditions. A principled approach to using derivatives in
score-spaces would be useful. Consider the simple case of true class-conditional
generative models P1(OIOd and P2(OI02) with respective estimates of the same
functional form P1 (0 10d and P2(010 2 ) . Expressing the true models as Taylor series expansions about the parameter estimates 01 and O2 (see [7] for more details,
and [3]) ,
Inpi (OIOi ) + (Oi - Oi ) TV' 9i Inpi (OIOi )
1
A T
T
A
A
(
+"2(Oi - Oi ) [V' 9i V' 9i Inpi (OIOi )](Oi - Oi ) + 0 Oi (?)
will , V'~i' vec(V' 9i V'~) T . . . ]T Inpi (OIOi )
3)
(11)
The output from the operator in square brackets is an infinite number of derivatives
arranged as a column vector. Wi is also a column vector. The expressions for the
two true models can be incorporated into an optimal minimum Bayes error decision
A
rule as follows , where 0
priors,
AT
AT
[0 1 , 02 ]T , W = [w i, WJjT, and b encodes the class
+b
wi[l, V'~1' vec(V' 91V'~1) T ... ]T Inp1 (OIOd-
a
w J [l , V'~,' vec(V' 92 V'~) T ... ]T Inp2(OI02) + b
a
T
T )T
w T[1, V' 9'
vec ( V' 9 V' 9
. . . ]T I n P1(OIOd + b
P2(OI02)
w Tiplr(o) + b
a
Inp1(OIOd -lnp2(OI02)
A
a
(12)
iplr(o) is a score in the likelihood-ratio score-space formed by an infinite number of
derivatives with respect to the parameter estimates O. Therefore, the optimal decision rule can be recovered by constructing a well-trained linear classifier in iplr(o) .
In this case, the standard SVM margin can be interpreted as the log posterior margin. This justifies the use of the likelihood-ratio score-space and encourages the
use of higher-order derivatives. However, most HMMs used in speech recognition
are 1st-order Markov processes but speech is a high-order or infinite-order Markov
process. Therefore, a linear decision boundary in the likelihood-ratio score-space defined on 1st-order Markov model estimates is unlikely to be sufficient for recovering
the optimal decision rule due to model incorrectness. However, powerful non-linear
classifiers may be trained in such a likelihood-ratio score-space to try to compensate
for model incorrectness and approximate the optimal decision rule. SVMs with nonlinear kernels such as polynomials or Gaussian Radial Basis Functions (GRBFs) may
be used. Although gains are expected from incorporating higher-order derivatives
into the score-space, the size of the score-space dramatically increases. Therefore,
practical systems may truncate the likelihood-ratio score-space after the 1st-order
derivatives, and hence use linear approximations to the Taylor series expansions 3 .
However, an example of a 2nd-order derivative is V' J-L jk (V'~;k Inp(OIO)) ,
T
V' J-L;k (V'~;k Inp(OIO))
~
-
L 'Yjk(t)"2';;k1
(13)
t= l
For simplicity the component posterior 'Yj k (t) is assumed independent of J-L j k. Once
the score-space has been defined, an SVM classifier can be built in the score-space.
If standard linear, polynomial or GRBF kernels are used in the score-space, then
the space is assumed to have a Euclidean metric tensor. Therefore, the score-space
should first be whitened (i.e. decorrelated and scaled) before the standard kernels
are applied. Failure to perform such score-space normalisation for a linear kernel
in score-space results in a kernel similar to the Plain kernel [5]. This is expected
to perform poorly when different dimensions of score-space have different dynamic
ranges [2]. Simple scaling has been found to be a reasonable approximation to
full whitening and avoids inverting large matrices in [2] (though for classification
of single observations rather than sequences, on a different database). The Fisher
kernel in [4] uses the Fisher Information matrix to normalise the score-space. This
is only an acceptable normalisation for a likelihood score-space under conditions
that give a zero expectation in score-space. The appropriate SVM kernel to use
between two utterances O i and OJ in the normalised score-space is therefore the
Normalised kernel, kN(Oi, OJ) (where ~sc is the covariance matrix in score-space),
(14)
3
Experimental Results
The ISOLET speaker-independent isolated letter database [8] was used for evaluation. The data was coded at a 10 msec frame rate with a 25.6 msec windowsize. The data was parameterised into 39-dimensional feature vectors including 12
MFCCs and a log energy term with corresponding delta and acceleration parameters. 240 utterances per letter from isolet{ 1,2,3,4} were used for training and
60 utterances per letter from isolet5 for testing. There was no overlap between
the training and test speakers. Two sets of letters were tested, the highly confusible E-set, {B C D E G P T V Z}, and the full 26 letters. The baseline HMM
system was well-trained to maximise likelihood. Each letter was modelled by a
10-emitting state left-to-right continuous density HMM with no skips, and silence
by a single emitting-state HMM with no skips. Each state output distribution had
the same number of Gaussian components with diagonal covariance matrices. The
models were tested using a Viterbi recogniser constrained to a silence-letter-silence
network.
31t is useful to note that a linear decision boundary, with zero bias, constructed in a
single-dimensional likelihood-ratio score-space formed by the zeroth-order derivative operator would, under equal class priors, give the standard minimum Bayes error classifier.
The baseline HMMs were used as generative models for SVM kernels. A modified version of SV Mlight Version 3.02 [9] was used to train 1vI SVM classifiers
on each possible class pairing. The sequence length normalisation in Equation 10,
and simple scaling for score-space normalisation, were used during training and
testing. Linear kernels were used in the normalised score-space, since they gave
better performance than GRBFs of variable width and polynomial kernels of degree 2 (including homogeneous, inhomogeneous, and inhomogeneous with zero-mean
score-space). The linear kernel did not require parameter-tuning and, in initial experiments, was found to be fairly insensitive to variations in the SVM trade-off
parameter C. C was fixed at 100, and biased hyperplanes were permitted. A variety of score-subspaces were examined. The abbreviations rn, v, wand t refer to
the score-subspaces \7 J-L jk Inpi( OIOi), \7 veC (I;jk) Inpi(OIOi), \7Wjk Inpi(OIOi) and
\7 ajj Inpi(OIOi) respectively. 1 refers to the log likelihood Inpi(OIOi) and r to the
log likelihood-ratio In[p2(OI02) /Pl( OIOd]. The binary SVM classification results
(and, as a baseline, the binary HMM results) were combined to obtain a single
classification for each utterance. This was done using a simple majority voting
scheme among the full set of 1v1 binary classifiers (for tied letters, the relevant 1v1
classifiers were inspected and then, if necessary, random selection performed [2]).
Table 1: Error-rates for HMM baselines and SVM score-spaces (E-set)
Num compo
per class
per state
1
2
4
6
HMM
min. Bayes majority
error
voting
11.3
11.3
8.7
8.7
6.7
6.7
7.2
7.2
SVM
lik-ratio
(stat. sign.)
6.9 ~99.8~!
5.0 (98.9%)
5.7 (13.6%)
6.1 (59.5%)
score-space
lik
lik
(I-class) (2-class)
7.6
6.1
6.3
9.3
23.2
8.0
7.8
30.6
Table 1 compares the baseline HMM and SVM classifiers as the complexity of the
generative models was varied. Statistical significance confidence levels are given in
brackets comparing the baseline HMM and SVM classifiers with the same generative models, where 95% was taken as a significant result (confidence levels were
defined by (100 - P), where P was given by McNemar's Test and was the percentage probability that the two classifiers had the same error rates and differences
were simply due to random error; for this, a decision by random selection for tied
letters was assigned to an 'undecided ' class [7]). The baseline HMMs were comparable to reported results on the E-set for different databases [10]. The majority
voting scheme gave the same performance as the minimum Bayes error scheme,
indicating that majority voting was an acceptable multi-class scheme for the E-set
experiments. For the SVMs, each likelihood-ratio score-space was defined using its
competing class-conditional generative models and projected into a rnr score-space.
Each likelihood (I-class) score-space was defined using only the generative model
for the first of its two classes, and projected into a rnl score-space. Each likelihood
(2-class) score-space was defined using a generative model for both of its classes,
and projected into a rnl score-space (the original Fisher score, which is a projection
into its rn score-subspace, was also tested but was found to yield slightly higher error
rates). SVMs built using the likelihood-ratio score-space achieved lower error rates
than HMM systems, as low as 5.0%. The likelihood (I-class) score-space performed
slightly worse than the likelihood-ratio score-space because it contained about half
the information and did not contain the log likelihood-ratio. In both cases, the
optimum number of components in the generative models was 2 per state, possibly
reflecting the gender division within each class. The likelihood (2-class) score-space
performed poorly possibly because of wrap-around. However, there was an excep-
tion for generative models with 1 component per class per state (in total the models
had 2 components per state since they modelled both classes). The 2 components
per state did not generally reflect the gender division in the 2-class data, as first
supposed, but the class division. A possible explanation is that each Gaussian component modelled a class with bi-modal distribution caused by gender differences.
Most of the data modelled did not sit at the peaks of the two Gaussians and was
not mapped to the ambiguous zero in score-subspace. Hence there was still sufficient class discrimination in score-space [7]. This task was too small to fully assess
possible decorrelation in error structure between HMM and SVM classifiers [6] .
Without scaling for score-space normalisation, the error-rate for the likelihood-ratio
score-space defined on models with 2 components per state increased from 5.0% to
11.1%. Some likelihood-ratio mr score-spaces were then augmented with 2nd-order
derivatives ~ J-t jk (~~jk lnp( 018)) . The resulting classifiers showed increases in error
rate. The disappointing performance was probably due to the simplicity of the task,
the independence assumption between component posteriors and component means,
and the effect of noise with so few training scores in such large score-spaces.
It is known that some dimensions of feature-space are noisy and degrade classification performance. For this reason, experiments were performed which selected
subsets of the likelihood-ratio score-space and then built SVM classifiers in those
score-subspaces. First, the score-subspaces were selected by parameter type. Error
rates for the resulting classifiers, otherwise identical to the baseline SVMs, are detailed in Table 2. Again, the generative models were class-conditional HMMs with
2 components per state. The log likelihood-ratio was shown to be a powerful discriminating feature 4 ? Increasing the number of dimensions in score-space allowed
more discriminative classifiers. There was more discrimination, or less noise, in the
derivatives of the component means than the component variances. As expected
in a dynamic task, the derivatives of the transitions were also useful since they
contained some duration information.
Table 2: Error rates for subspaces of the likelihood-ratio score-space (E-set)
score-space
error rate, %
r
v
m
mv
mvt
wmvtr
8.5
7.2
5.2
5.0
4.4
4.1
score-space
dimensionality
1
1560
1560
3120
3140
3161
Next, subsets of the mr and wmvtr score-spaces were selected according to dimensions with highest Fisher-ratios [7] . The lowest error rates for the mr and wmvtr
score-spaces were respectively 3.7% at 200 dimensions and 3.2% at 500 dimensions
(respectively significant at 99.1% and 99.7% confidence levels relative to the best
HMM system with 4 components per state). Generally, adding the most discriminative dimensions lowered error-rate until less discriminative dimensions were added.
For most binary classifiers, the most discriminative dimension was the log likelihoodratio. As expected for the E-set, the most discriminative dimensions were dependent
on initial HMM states. The low-order MFCCs and log energy term were the most
important coefficients. Static, delta and acceleration streams were all useful.
4The error rate at 8.5% differed from that for the HMM baseline at 8.7% because of
the non-zero bias for the SVMs.
The HMM and SVM classifiers were run on the full alphabet. The best HMM classifier, with 4 components per state, gave 3.4% error rate. Computational expense
precluded a full optimisation of the SVM classifier. However, generative models
with 2 components per state and a wmvtr score-space pruned to 500 dimensions by
Fisher-ratios, gave a lower error rate of 2.1% (significant at a 99.0% confidence level
relative to the HMM system). Preliminary experiments evaluating sequence length
normalisation on the full alphabet and E-set are detailed in [7].
4
Conclusions
In this work, SVMs have been successfully applied to the classification of speech
data. The paper has concentrated on the nature of the score-space when handling
variable length speech sequences. The standard likelihood score-space of the Fisher
kernel has been extended to the likelihood-ratio score-space, and normalisation
schemes introduced. The new score-space avoids some of the limitations of the
Fisher score-space, and incorporates the class-conditional generative models directly
into the SVM classifier. The different score-spaces have been compared on a speakerindependent isolated letter task. The likelihood-ratio score-space out-performed the
likelihood score-spaces and HMMs trained to maximise likelihood.
Acknowledgements
N. Smith would like to thank EPSRC; his CASE sponsor, the Speech Group at IBM
U.K. Laboratories; and Thorsten Joachims, University of Dortmund, for BV Mlight.
References
[1] V. Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag, 1995.
[2] N. Smith, M. Gales, and M. Niranjan. Data-dependent kernels in SVM classification
of speech patterns. Tech. Report CUED/F-INFENG/TR.387, Cambridge University
Eng.Dept., April 2001.
[3] K. Tsuda et al. A New Discriminative Kernel from Probabilistic Models. In T.G .
Dietterich, S. Becker and Z. Ghahramani, editors Advances in Neural Information
Processing Systems 14, MIT Press, 2002.
[4] T. Jaakkola and D. Haussler. Exploiting Generative Models in Discriminative Classifiers. In M.S. Kearns, S.A. Solia, and D.A. Cohn, editors, Advances in Neural
Information Processing Systems 11 . MIT Press, 1999.
[5] N. Oliver, B. Scholkopf, and A. Smola. Advances in Large-Margin Classifiers, chapter
Natural Regularization from Generative Models. MIT Press, 2000.
[6] S. Fine, J. Navratil, and R. Gopinath. A hybrid GMM / SVM approach to speaker identification. In Proceedings, volume 1, International Conference on Acoustics, Speech,
and Signal Processing, May 2001. Utah, USA .
[7] N. Smith and M. Gales. Using SVMs to classify variable length speech patterns. Tech.
Report CUED/ F-INFENG/ TR.412, Cambridge University Eng.Dept., June 2001.
[8] M. Fanty and R . Cole. Spoken Letter Recognition. In R.P. Lippmann, J .E. Moody,
and D.S . Touretzky, editors, Neural Information Processing Systems 3, pages 220-226 .
Morgan Kaufmann Publishers, 1991.
[9] T. Joachims. Making Large-Scale SVM Learning Practical. In B. Scholkopf,
C. Burges, and A. Smola, editors, Advances in Kernel Methods - Support Vector
Learning. MIT-Press, 1999.
[10] P.C. Loizou and A.S. Spanias. High-Performance Alphabet Recognition. IEEE Transactions on Speech and Audio Processing, 4(6):430-445, Nov. 1996.
| 2044 |@word version:2 polynomial:3 nd:3 ajj:4 eng:4 covariance:3 tr:2 initial:2 series:2 score:112 selecting:1 outperforms:1 o2:1 recovered:1 comparing:1 must:1 readily:1 jkl:1 distant:1 speakerindependent:1 discrimination:2 generative:30 selected:4 half:1 smith:4 compo:1 num:1 location:2 hyperplanes:1 simpler:1 constructed:1 rnl:2 pairing:1 scholkopf:2 expected:5 p1:4 multi:2 increasing:1 lowest:1 what:1 tying:1 interpreted:1 spoken:2 differing:1 every:1 voting:4 classifier:25 scaled:2 uk:2 before:1 maximise:7 engineering:2 zeroth:3 examined:1 hmms:11 ease:1 oiod:5 range:1 bi:1 practical:2 yj:1 testing:2 oio:5 cb2:2 projection:1 word:1 radial:1 inp:3 refers:1 confidence:4 protein:1 onto:1 selection:2 operator:9 applying:1 map:2 duration:3 simplicity:2 examines:2 rule:4 isolet:2 haussler:1 his:1 handle:3 variation:2 isolet5:1 inspected:1 homogeneous:1 us:3 recognition:10 jk:10 database:3 epsrc:1 loizou:1 region:1 spliced:1 counter:1 trade:1 highest:1 principled:1 complexity:1 cam:2 dynamic:3 trained:9 division:5 basis:1 chapter:1 derivation:1 train:1 undecided:1 forced:1 alphabet:3 sc:1 widely:1 otherwise:1 ability:1 noisy:1 sequence:16 interaction:1 fanty:1 relevant:2 poorly:2 achieve:1 supposed:1 wjk:3 exploiting:1 optimum:1 cued:2 ac:2 stat:1 ij:2 p2:5 recovering:1 skip:4 concentrate:1 inhomogeneous:2 occupies:1 require:1 suitability:1 investigation:1 preliminary:1 extension:1 pl:1 around:3 considered:1 mapping:7 bj:1 viterbi:1 tjk:2 major:1 cole:1 successfully:1 weighted:1 mit:4 gaussian:12 super:2 modified:2 rather:2 incorrectness:2 avoid:1 jaakkola:1 june:1 joachim:2 likelihood:41 tech:2 baseline:9 dependent:2 typically:1 unlikely:1 initially:1 hidden:1 issue:4 classification:10 among:1 lfor:1 proposes:1 art:1 constrained:1 fairly:1 equal:1 once:2 fuller:1 having:1 identical:1 represents:1 report:2 develops:1 jdt:1 few:1 individual:1 replaced:1 normalisation:12 highly:1 evaluation:1 mixture:3 bracket:2 tj:2 oliver:1 necessary:1 respective:1 taylor:2 euclidean:1 tsuda:1 isolated:3 increased:1 classify:2 earlier:1 column:3 entry:1 subset:2 successful:1 too:1 reported:1 kn:1 varies:1 sv:1 combined:1 st:8 density:2 peak:3 sensitivity:1 discriminating:1 international:1 probabilistic:1 off:1 together:1 moody:1 again:1 ambiguity:1 reflect:1 inaj:1 gale:3 possibly:2 worse:1 derivative:23 coefficient:1 caused:1 recogniser:1 vi:1 mv:1 stream:1 later:1 try:1 performed:5 tion:1 bayes:4 ass:1 oi:9 formed:4 square:1 variance:2 kaufmann:1 spaced:1 yield:1 climbing:1 modelled:5 identification:1 dortmund:1 touretzky:1 decorrelated:1 definition:3 failure:1 energy:2 associated:1 static:2 gain:1 proved:1 dimensionality:1 reflecting:1 appears:1 higher:4 permitted:1 modal:1 april:1 arranged:1 evaluated:1 though:1 done:1 parameterised:1 smola:2 until:1 grbf:1 cohn:1 nonlinear:1 accent:1 utah:1 effect:2 dietterich:1 contain:1 true:3 usa:1 hence:2 assigned:1 regularization:1 laboratory:1 during:1 self:2 encourages:1 width:1 ambiguous:1 speaker:5 hill:1 confusion:1 functional:1 ji:1 insensitive:1 volume:1 discussed:2 expressing:1 refer:1 significant:3 cambridge:6 vec:5 tuning:1 mfccs:2 had:3 lowered:1 whitening:1 mlight:2 posterior:11 showed:1 discard:1 termed:1 disappointing:1 verlag:1 binary:5 mcnemar:1 lnp:1 morgan:1 minimum:3 greater:1 mr:3 employed:1 signal:1 lik:3 full:6 compensate:1 jft:1 divided:1 niranjan:1 coded:1 sponsor:1 infeng:2 whitened:1 optimisation:1 metric:1 expectation:1 kernel:23 represent:1 achieved:1 addition:1 fine:1 addressed:1 publisher:1 ot:8 biased:1 probably:1 member:2 incorporates:1 variety:3 affect:1 independence:1 gave:4 competing:2 expression:1 becker:1 rnr:1 suffer:1 speech:17 speaking:1 dramatically:1 useful:8 generally:2 detailed:3 redefines:1 concentrated:1 svms:11 solia:1 percentage:1 sign:1 delta:2 popularity:1 per:14 discrete:1 group:1 gmm:1 v1:2 sum:3 wand:1 run:1 letter:12 powerful:3 reasonable:1 decision:8 acceptable:2 scaling:3 comparable:1 bv:1 constraint:1 encodes:1 nathan:1 argument:4 min:1 pruned:1 tv:1 according:1 truncate:1 slightly:2 em:1 unity:2 wi:2 making:1 classconditional:1 thorsten:1 taken:1 equation:3 adopted:1 available:1 gaussians:3 appropriate:2 mvt:1 alternative:2 jd:1 original:3 denotes:1 assumes:1 include:2 k1:1 ghahramani:1 tensor:1 added:1 occurs:3 wrapping:1 diagonal:1 kth:1 wrap:3 subspace:11 thank:1 mapped:2 hmm:21 normalise:1 majority:4 degrade:1 reason:2 assuming:2 length:12 retained:1 ratio:28 susceptible:1 yjk:2 expense:1 perform:2 observation:13 markov:4 discarded:1 defining:1 extended:1 incorporated:3 frame:1 rn:2 varied:1 wraparound:1 introduced:2 inverting:1 required:1 acoustic:5 address:1 precluded:1 pattern:2 built:4 oj:2 including:2 explanation:1 power:1 overlap:1 decorrelation:1 natural:2 hybrid:1 representing:1 scheme:9 occupancy:1 utterance:9 prior:2 acknowledgement:1 regularisation:1 occupation:1 relative:4 fully:2 discriminatively:2 limitation:3 gopinath:1 degree:1 sufficient:2 editor:4 pi:1 ibm:1 course:1 silence:3 bias:2 normalised:6 burges:1 fall:1 boundary:2 calculated:1 dimension:15 transition:5 avoids:3 plain:1 evaluating:1 adopts:1 forward:1 projected:3 far:1 emitting:3 transaction:1 approximate:1 lippmann:1 nov:1 preferred:1 dealing:1 assumed:2 discriminative:8 continuous:2 table:4 nature:3 expansion:2 complex:1 necessarily:1 constructing:1 did:4 pk:1 significance:1 noise:2 allowed:1 augmented:1 representative:1 differed:1 msec:2 lie:1 tied:2 pz:2 svm:20 sit:1 incorporating:1 vapnik:1 adding:1 justifies:1 margin:3 simply:2 contained:2 springer:1 inpi:10 gender:3 conditional:6 abbreviation:1 viewed:1 acceleration:2 fisher:14 generalisation:1 infinite:3 averaging:1 kearns:1 called:2 total:3 experimental:1 indicating:1 mark:1 support:2 dept:4 audio:1 tested:3 handling:2 |
1,146 | 2,045 | Orientation-Selective aVLSI Spiking Neurons
Shih-Chii Liu, J?org Kramer, Giacomo Indiveri,
Tobias Delbruck,
?
and Rodney Douglas
Institute of Neuroinformatics
University of Zurich and ETH Zurich
Winterthurerstrasse 190
CH-8057 Zurich, Switzerland
Abstract
We describe a programmable multi-chip VLSI neuronal system that
can be used for exploring spike-based information processing models.
The system consists of a silicon retina, a PIC microcontroller, and a
transceiver chip whose integrate-and-fire neurons are connected in a soft
winner-take-all architecture. The circuit on this multi-neuron chip approximates a cortical microcircuit. The neurons can be configured for
different computational properties by the virtual connections of a selected set of pixels on the silicon retina. The virtual wiring between
the different chips is effected by an event-driven communication protocol that uses asynchronous digital pulses, similar to spikes in a neuronal system. We used the multi-chip spike-based system to synthesize orientation-tuned neurons using both a feedforward model and a
feedback model. The performance of our analog hardware spiking
model matched the experimental observations and digital simulations of
continuous-valued neurons. The multi-chip VLSI system has advantages
over computer neuronal models in that it is real-time, and the computational time does not scale with the size of the neuronal network.
1 Introduction
The sheer number of cortical neurons and the vast connectivity within the cortex are difficult to duplicate in either hardware or software. Simulations of a network consisting
of thousands of neurons with a connectivity that is representative of cortical neurons can
take minutes to hours on a fast Pentium, particularly if spiking behavior is simulated. The
simulation time of the network increases as the size of the network increases. We have
taken initial steps in mitigating the simulation time of neuronal networks by developing
a multi-chip VLSI system that can support spike-based cortical processing models. The
connectivity between neurons on different chips and between neurons on the same chip are
reconfigurable. The receptive fields are effected by appropriate mapping of the spikes from
source neurons to target neurons. A significant advantage of these hardware simulation
systems is their real-time property; the simulation time of these systems does not increase
with the size of the network.
In this work, we show how we synthesized orientation-tuned spiking neurons using the
multi-chip system in Figure 1. The virtual connection from a selected set of neurons on
Silicon
retina
Router
(White
matter)
Network of
neurons
Orientationselective
neurons
Figure 1: Block diagram of a neuromorphic multi-chip system in which virtual connections
from a set of neurons on a silicon retina onto another set of neurons on a transceiver chip
are effected by a microcontroller. The retina communicates through the AER protocol to
the PIC when it has an active pixel. The PIC communicates with the multi-neuron chip if
the retina address falls into one of its stored templates. The address from the PIC is decoded
by the multi-neuron transceiver. The address of the active neuron on this array can also be
communicated off-chip to another receiver/transceiver.
the retina to the target neurons on the multi-neuron transceiver chip is achieved with a PIC
microcontroller and an asynchronous event-driven communication protocol. The circuit on
this multi-neuron chip approximates a cortical microcircuit (Douglas and Martin, 1991).
We explored different models that have been proposed for the generation of orientation
tuning in neurons of the V1 cortical area. There have been earlier attempts to use multichip systems for creating orientation-selective neurons (Boahen et al., 1997; Whatley et al.,
1997). In the present work, the receptive fields are created in a manner similar to that described in (Whatley et al., 1997). However we extend their work and quantify the tuning
curves of different models. Visual cortical neurons receive inputs from the lateral geniculate nucleus (LGN) neurons which are not orientation-selective. Models for the emergence
of orientation-selectivity in cortical neurons can be divided into two groups; feedforward
models and feedback models. In a feedforward model, the orientation selectivity of a cortical neuron is conferred by the spatial alignment of the LGN neurons that are presynaptic to
the cortical neuron (Hubel and Wiesel, 1962). In a feedback model, a weak orientation bias
provided by the LGN input is sharpened by the intracortical excitatory and/or inhibitory
feedback (Somers et al., 1995; Ben-Yishai et al., 1995; Douglas et al., 1995). In this work,
we quantify the tuning curves of neurons created using a feedforward model and a feedback
model with global inhibition.
2 System Architecture
The multi-chip system (Figure 1) in this work consists of a 16 16 silicon ON/OFF retina,
a PIC microcontroller, and a transceiver chip with a ring of 16 integrate-and-fire neurons
and a global inhibitory neuron. All three modules communicate using the address event
representation (AER) protocol (Lazzaro et al., 1993; Boahen, 1996). The communication
channel signals consist of the address bits, the REQ signal, and the ACK signal. The PIC
and the multi-neuron chip are both transceivers: They can both receive events and send
events (Liu et al., 2001).
The retina with an on-chip arbiter can only send events. Each pixel is composed of an
adaptive photoreceptor that has a rectifying temporal differentiator (Kramer, 2001) in its
feedback loop as shown in Figure 2. Positive temporal irradiance transients (dark-to-bright
or ON transitions) and negative irradiance transients (bright-to-dark or OFF transitions)
appear at two different outputs of the pixel. The outputs are then coded in the form of
asynchronous binary pulses by two neurons within the pixel. These asynchronous pulses
Arbiter
ON
REQ
ON
ACK
neuron
ON
M1
OFF
REQ
OFF
ACK
neuron
OFF
bias
M3
temporal
differentiator
M2
Figure 2: Pixel of the transient imager. The circuit contains a photodiode with a transistor
in a a source-follower configuration with a high-gain inverting amplifier ( ,
) in a
negative feedback loop. A rectifying temporal differentiator in the feedback loop extracts
transient ON and OFF signals. These signals go to individual neurons that generate the
REQ signals to the arbiter. In this schematic, we only show the REQ and ACK signals to
the X-arbiter. The duration of the ACK signal from the X-arbiter is extended within the
pixel by a global refractory bias. This duration sets the refractory period of the neuron.
are the request signals to the AER communication interface. A global parameter sets the
minimum time (or refractory period) between subsequent pulses from the same output.
Hence, the pixel can respond either with one pulse or multiple pulses to a transient. The
pixels are arranged on a rectangular grid. The position of a pixel is encoded with a 4-bit
column address (X address) and a 4-bit row address (Y address) as shown in Figure 3. An
active neuron makes a request to the on-chip arbiter. If the neuron is selected by the arbiter,
then the X and Y addresses which code the location of this neuron are placed on the output
address bus of the chip. The retina then handshakes with the PIC microcontroller.
The multi-neuron chip has an on-chip address decoder for the incoming events and an onchip arbiter to send events. The X address to the chip codes the identity of the neuron
and the Y address codes the input synapse used to stimulate the neuron. Each neuron
can be stimulated externally through an excitatory synapse or an inhibitory synapse. The
excitatory neurons of this array are mutually connected via hard-wired excitatory synapses.
These excitatory neurons also excite a global inhibitory neuron which in turn inhibits all
the excitatory neurons. The membrane potentials of the neurons can be monitored by an
on-chip scanner and the output spikes of the neurons can be monitored by the chip?s AER
output. The address on the output bus codes the active neuron. In this work, the excitatory
neurons on the multi-neuron chip model the orientation tuning properties of simple cells in
the visual cortex and the global inhibitory neuron models an inhibitory interneuron in the
visual cortex.
The receptive fields of the neurons are created by configuring the connections from a subset
of the source pixels on the retina onto the appropriate target neurons on the multi-neuron
transceiver chip through a PIC 16C74 microcontroller. The subsets of retina pixels are
determined by user-supplied templates. The microcontroller filters each retinal event to
90 deg template
Inhibitory synapse
Excitatory synapse
15
Y
0 deg template
0
15
0
X
Figure 3: Spikes from a selected set of neurons within the two rectangular regions on the
retina are mapped by the PIC onto the corresponding orientation-selective neurons on the
transceiver chip. The light-shaded triangles mark the somas of the excitatory neurons and
the dark-shaded triangle marks the soma of the global inhibitory neuron. Only two neurons,
which are mapped for orthogonal orientations, were used in this experiment.
decide if it lies in one or more of the receptive fields (RFs) of the neurons on the receiver. If
it does, an event is transmitted to the appropriate receiver neuron. The typical transmission
time from a spike from the sender to the receiver is about 15 s. This cycle time can be
reduced by using a faster processor in place of the PIC. The retina and transceiver chips
can handle handshaking cycle times on the order of 100 ns.
3 Neuron Circuit
The circuit of a neuron and an excitatory synapse on the transceiver chip is shown in Figure 4. The synapse circuit (M1?M4) in the left box of the figure was originally described in
(Boahen, 1996). The presynaptic spike drives the transistor M4, which acts like a switch.
The bias voltages and set the the strength and the dynamics of the synapse.
The circuit in the right box of Figure 4 implements a linear threshold integrate and fire
neuron with an adjustable voltage threshold, spike pulse width and refractory period. The
synaptic current
charges up the capacitance of the membrane
. When the membrane potential
exceeds a threshold voltage , the output of the transconductance
amplifier M5?M9 switches to a voltage close to . The output of the two inverters (M10?
M12 and M13?M15), , also switches to . The bias voltage, , limits the current
through the transconductance amplifier and the first inverter. The capacitors "!# and $
implement a capacitive divider that provides positive feedback to
. This feedback
speeds up the circuit?s response and provides hysteresis to ensure that small fluctuations of
around % do not make & switch erratically.
When & is high, (' is discharged through transistors M20 and M21 at a rate that is
dependent on ) . This bias voltage controls the spike?s pulse width. Once
is below
% , the transconductance amplifier switches to ground. The first inverter then switches to
but does not immediately go to zero; it decreases linearly at a rate set by *+!# . In
this way, transistor M21 is kept on, even after ,
has decreased below % . As long
as the gate voltage of M21 is sufficiently high, the neuron is in its refractory period. Once
transistor M21 is turned off, a new spike is generated in a time that is inversely proportional
to the magnitude of - &
. The spike output of the circuit is taken from the output of the first
inverter.
Cfb
Vdd
Ve
M1
Vw
Vspk
Vdd
M2
Cs
Iinj
Vpb
Vdd
M5
Vmem
M4
M6
Vdd
M10
M3
M7
Vthr
M11
M13
M12
M14
Vout
Cm
M8
M9
Vdd
Vrfr
Cr
M15
M16
Vqua
M17
M18
M19
M20
Vpw
Cca
Vtau
M21
Figure 4: Circuit diagram of an excitatory synapse (left box) connected to a linear threshold
integrate-and-fire neuron (right box).
Transistors M16?M19 implement a spike frequency adaptation mechanism (Boahen,
1996). A fixed amount of charge (set by ) is dumped onto the capacitor with
every output spike. The resulting charge on sets the current that is subtracted from the
input current, and the neuron?s output frequency decreases accordingly. The voltages
and are used to set the gain and dynamics of the integrator.
4 System Responses
A rotating drum with a black and white strip was placed in front of the retina. The spike
addresses and spike times generated by the retina and the multi-neuron chip at an image
speed of 7.9 mm/s (or 89 pixels/s) of the rotating stimulus were recorded using a logic
analyzer. The orientations of the stimuli are defined in Figure 3. Each pixel of the retina
responded with only one spike to the transition of an edge of the stimulus because the
refractory period of the pixel was set to 500 s. The spike addresses during the time of
travel of the OFF edge of a 0 deg oriented stimulus through the entire array (Figure 5(a))
indicates that almost all the pixels along a row transmitted their addresses sequentially as
the edge passed by. This sequential ordering can be seen because the stimulus was oriented
slightly different from 0 deg. If the stimulus was perfectly at 0 deg, then there would
be a random ordering of the pixel addresses within each row. The same observation can
be made for the OFF-transient spikes recorded in response to a 90 deg oriented stimulus
(Figure 5(b)).
The receptive fields of two orientation-selective neurons were synthesized by mapping the
OFF transient outputs of a selected set of pixels on the retina as shown in Figure 3. These
two neurons have orthogonal preferred orientations. The local excitatory coupling between
the neurons was disabled. There is no self excitation to each neuron so we explored only
a feedforward model and a feedback model using global inhibition. We varied the size
and aspect ratio of the receptive fields of the neurons by changing the template size used
in the mapping of the retina spikes to the transceiver chip. The template size and aspect
ratio determine the orientation responses of the neurons. The orientation response of these
neurons also depends on the time constant of the neuron. On this multi-neuron chip, we
do not have an explicit transistor that
allows us to control the time constant. Instead, we
generated a leak current through
in Figure 4 by controlling the source voltage of
,
250
Retina address for OFF spikes
Retina address for OFF spikes
250
200
150
100
50
0
200
150
100
50
0
7.88
7.9
7.92
7.94
7.96
7.98
8
8.02
?50
0
50
Time (s)
Time (ms)
(a)
(b)
100
150
Figure 5: The spike addresses from the retina were recorded when a 0 deg (a) and a 90 deg
(b) oriented stimulus moved across the retina. The figure shows the time progression of
the stimulated pixels (OFF spikes are marked with circles) as the 0 deg oriented stimulus
(see Figure 3 for the orientation definition) passed over each row in (a). The address on
the ordinate is defined as 16Y + X. A similar observation is true of (b) for the ordering
of the OFF-transient spikes when each column on the retina was stimulated by the 90 deg
oriented stimulus.
& . By increasing , we decrease the time constant of the neuron. Because the neuron
charges up to threshold through the summation of the incoming EPSPs, it can only spike
if the ISIs of the incoming spikes are small enough. The synaptic weight determines the
number of EPSPs needed to drive the neuron above threshold.
We first investigated the feedforward model by using a template size of 5 7 (3 deg 4.2
deg) for one neuron and 7 5 (4.2 deg 3 deg) for the second neuron. The aspect ratio of
this template was 1.4. (We have repeated the following experiments using smaller template
sizes (3 5 and 1 3) and the experimental results were pretty much the same.) The time
constant of the neuron and synaptic gain and strength were adjusted so that both neurons
responded optimally to the stimulus. The connection from the global inhibitory neuron to
the two excitatory neurons was disabled.
Data was collected from the multi-neuron chip for different orientations of the drum (and
hence of the stimulus). The stimulus was presented approximately 500?1000 times to the
retina. Since the orientation-selective neurons responded with only 1?3 spikes every time
the stimulus moved over the retina, we normalized the total number of spikes collected
in these experiments to the number of stimulus presentations. The results are shown as a
polar plot in Figure 6(a) for the two neurons that are sensitive to orthogonal orientations.
Each neuron was more sensitive to a stimulus at its preferred orientation than the nonpreferred orientations. The neuron responded more to the orthogonal orientation than to the
in-between orientations because there were a small number of retina spikes that arrived with
a small ISI when the orthogonally-oriented stimulus moved across the template space of
the retina (see Figure 3). We used an orientation-selective (OS) index to quantify the orien % !#
& !
tation selectivity of the neuron. This index is defined as
% !#
& !
where R() is the response of the neuron. As an example, R(preferred) for neuron 5, which
is sensitive to vertical orientations, is R(90)+R(270) and R(nonpreferred) is R(0)+R(180).
We next investigated the feedback model. In the presence of global inhibition, the multi-
90
90
60
120
150
120
150
30
180
0
210
330
240
300
60
30
180
0
330
210
300
240
270
270
(a)
(b)
Figure 6: Orientation tuning curves of the two neurons in the (a) absence and (b) presence of global inhibition. The responses of the neurons were measured by the number of
spikes collected per stimulus presentation. The radius of the polar plot is normalized to
the maximum response of both neurons. The data was collected for stimulus orientations
spaced at 30 deg intervals. The neuron that responded preferably to a 90 deg oriented stimulus (solid curve) also had a small response to a stimulus at 0 deg orientation (OS=0.428).
The same observation is true for the other neuron (dashed curve) (OS=0.195). In the presence of global inhibition, each neuron responded less to the non-preferred orientation due
to the suppression from the other neuron (cross-orientation inhibition). The output firing
rates were also lower in this case (approximately half of the firing rates in the absence of
inhibition). The OS indices are 0.546 (solid curve) and 0.497 (dashed curve) respectively.
neuron system acts like a soft winner-take-all circuit. We tuned the coupling strengths
between the excitatory neurons and the inhibitory neuron so that we obtained the optimal
response to the same stimulus presentations as in the feedforward case. The new tuning
curves are plotted in Figure 6(b). The non-preferred response of a neuron was suppressed
by the other neuron through the recurrent inhibition (cross-orientation inhibition).
5 Conclusion
We demonstrated a programmable multi-chip VLSI system that can be used for exploring spike-based processing models. This system has advantages over computer neuronal
models in that it is real-time and the computational time does not scale with the size of
the neuronal network. The spiking neurons can be configured for different computational
properties. Interchip and intrachip connectivity between neurons can be programmed using
the AER protocol. In this work, we created receptive fields for orientation-tuned spiking
neurons by mapping the transient spikes from a silicon retina onto the neurons using a microcontroller. We have not mapped onto all the neurons on the transceiver chip because the
PIC microcontroller we used is not fast enough to create receptive fields for more neurons
without distorting the ISI distribution of the incoming retina spikes.
We evaluated the responses of the orientation-tuned spiking neurons for different receptive
field sizes and aspect ratios and also in the absence and presence of feedback inhibition. In
a feedforward model, the aVLSI spiking neurons show orientation selectivity similar to digital simulations of continuous-valued neurons. Adding inhibition increased the selectivity
of the spiking neurons between orthogonal orientations.
We can extend the multi-chip VLSI system in this work to a more sophisticated system that
supports multiple senders and multiple receivers. Such a system can be used, for example,
to implement multi-scale cortical models. The success of the system in this work opens up
the way for more elaborate spike-based emulations in the future.
6 Acknowledgements
We acknowledge T. Horiuchi for the original design of the transceiver chip and David
Lawrence for the software driver development in this work. This work was supported in
part by the Swiss National Foundation Research SPP grant and the K?obler Foundation.
References
Ben-Yishai, R., Bar-Or, R. L., and Sompolinsky, H. (1995). Theory of orientation tuning
in visual cortex. P. Natl. Acad. Sci. USA, 92(9):3844?3848.
Boahen, K. A. (1996). A retinomorphic vision system. IEEE Micro, 16(5):30?39.
Boahen, K. A., Andreou, A., Hinck, T., Kramer, J., and Whatley, A. (1997).
Computation- and memory-based projective field processors. In Sejnowski, T.,
Koch, C., and Douglas, R., editors, Telluride NSF workshop on neuromorphic engineering, Telluride, CO.
Douglas, R., Koch, C., Mahowald, M., Martin, K., and Suarez, H. (1995). Recurrent
excitation in neocortical circuits. Science, 269(5226):981?985.
Douglas, R. and Martin, K. (1991). A functional microcircuit for cat visual cortex. J.
Physiol., 440:735?769.
Hubel, D. and Wiesel, T. (1962). Receptive fields, binocular interaction and functional
architecture. J. of Physio.(Lond), 160:106?154.
Kramer, J. (2001). An integrated optical transient sensor. Submitted for publication.
Lazzaro, J., Wawrzynek, J., Mahowald, M., Sivilotti, M., and Gillespie, D. (1993). Silicon auditory processors as computer peripherals. IEEE Transactions on Neural Networks, 4(3):523?528.
Liu, S.-C., Kramer, J., Indiveri, G., Delbruck, T., Burg, T., and Douglas, R. (2001).
Orientation-selective aVLSI spiking neurons. Neural Networks, 14(6/7):629?643.
Special Issue on Spiking Neurons in Neuroscience and Technology.
Somers, D., Nelson, S., and Sur, M. (1995). An emergent model of orientation selectivity
in cat visual cortex simple cells. Journal of Neuroscience, 15(8):5448?5465.
Whatley, A., Kramer, J., and Douglas, R. (1997). ON/OFF retina to silicon cortex. In
Sejnowski, T., Koch, C., and Douglas, R., editors, Telluride NSF workshop on neuromorphic engineering, Telluride, CO.
| 2045 |@word wiesel:2 open:1 simulation:7 pulse:8 solid:2 initial:1 liu:3 contains:1 configuration:1 tuned:5 current:5 router:1 follower:1 physiol:1 subsequent:1 plot:2 half:1 selected:5 accordingly:1 provides:2 location:1 org:1 along:1 m7:1 driver:1 transceiver:13 consists:2 manner:1 behavior:1 isi:3 multi:23 integrator:1 m8:1 increasing:1 provided:1 matched:1 circuit:12 sivilotti:1 cm:1 temporal:4 winterthurerstrasse:1 every:2 act:2 charge:4 preferably:1 imager:1 control:2 configuring:1 grant:1 appear:1 positive:2 engineering:2 local:1 limit:1 tation:1 acad:1 fluctuation:1 firing:2 approximately:2 onchip:1 black:1 physio:1 shaded:2 co:2 programmed:1 projective:1 block:1 implement:4 communicated:1 swiss:1 area:1 eth:1 onto:6 close:1 demonstrated:1 send:3 go:2 duration:2 rectangular:2 immediately:1 m2:2 array:3 handle:1 target:3 controlling:1 user:1 us:1 synthesize:1 particularly:1 photodiode:1 vspk:1 m14:1 module:1 suarez:1 thousand:1 interchip:1 region:1 connected:3 cycle:2 sompolinsky:1 ordering:3 decrease:3 boahen:6 leak:1 tobias:1 dynamic:2 vdd:5 m18:1 req:5 triangle:2 chip:40 emergent:1 cat:2 horiuchi:1 fast:2 describe:1 sejnowski:2 neuroinformatics:1 whose:1 encoded:1 valued:2 emergence:1 advantage:3 whatley:4 transistor:7 interaction:1 erratically:1 adaptation:1 turned:1 loop:3 moved:3 m21:5 transmission:1 wired:1 ring:1 ben:2 coupling:2 avlsi:3 recurrent:2 measured:1 dumped:1 epsps:2 c:1 quantify:3 switzerland:1 radius:1 emulation:1 filter:1 transient:10 virtual:4 summation:1 adjusted:1 exploring:2 scanner:1 mm:1 around:1 sufficiently:1 ground:1 koch:3 lawrence:1 mapping:4 iinj:1 inverter:4 polar:2 geniculate:1 travel:1 sensitive:3 create:1 sensor:1 cr:1 voltage:9 publication:1 indiveri:2 indicates:1 pentium:1 suppression:1 dependent:1 entire:1 integrated:1 vlsi:5 selective:8 lgn:3 mitigating:1 pixel:19 issue:1 orientation:40 development:1 retinomorphic:1 spatial:1 special:1 field:11 once:2 future:1 stimulus:22 duplicate:1 micro:1 retina:31 oriented:8 composed:1 ve:1 national:1 individual:1 m4:3 consisting:1 fire:4 attempt:1 amplifier:4 alignment:1 light:1 natl:1 yishai:2 edge:3 orthogonal:5 rotating:2 circle:1 plotted:1 increased:1 column:2 soft:2 earlier:1 delbruck:2 neuromorphic:3 mahowald:2 subset:2 front:1 optimally:1 stored:1 giacomo:1 m10:2 off:16 multichip:1 connectivity:4 sharpened:1 recorded:3 creating:1 potential:2 m9:2 intracortical:1 retinal:1 matter:1 configured:2 hysteresis:1 depends:1 microcontroller:9 effected:3 rodney:1 rectifying:2 bright:2 responded:6 spaced:1 discharged:1 chii:1 weak:1 vout:1 drive:2 processor:3 submitted:1 synapsis:1 synaptic:3 strip:1 definition:1 frequency:2 monitored:2 gain:3 auditory:1 sophisticated:1 irradiance:2 originally:1 response:12 synapse:9 arranged:1 evaluated:1 microcircuit:3 box:4 binocular:1 o:4 stimulate:1 disabled:2 usa:1 divider:1 true:2 normalized:2 hence:2 white:2 wiring:1 during:1 width:2 self:1 excitation:2 m:1 m5:2 ack:5 arrived:1 neocortical:1 interface:1 image:1 functional:2 spiking:11 winner:2 refractory:6 analog:1 extend:2 approximates:2 m1:3 synthesized:2 silicon:8 significant:1 tuning:7 grid:1 analyzer:1 had:1 cortex:7 inhibition:11 driven:2 selectivity:6 binary:1 success:1 transmitted:2 minimum:1 seen:1 determine:1 period:5 dashed:2 signal:9 multiple:3 exceeds:1 faster:1 cross:2 long:1 divided:1 coded:1 schematic:1 vision:1 achieved:1 cell:2 receive:2 decreased:1 interval:1 diagram:2 source:4 capacitor:2 vw:1 presence:4 feedforward:8 enough:2 m6:1 switch:6 architecture:3 perfectly:1 drum:2 distorting:1 passed:2 lazzaro:2 differentiator:3 programmable:2 orientationselective:1 amount:1 dark:3 hardware:3 reduced:1 generate:1 supplied:1 nsf:2 inhibitory:10 neuroscience:2 per:1 group:1 shih:1 sheer:1 soma:2 threshold:6 changing:1 douglas:9 kept:1 v1:1 vast:1 communicate:1 respond:1 somers:2 place:1 almost:1 decide:1 bit:3 cca:1 aer:5 strength:3 software:2 aspect:4 speed:2 lond:1 transconductance:3 optical:1 martin:3 inhibits:1 developing:1 peripheral:1 request:2 membrane:3 across:2 slightly:1 smaller:1 suppressed:1 wawrzynek:1 taken:2 zurich:3 bus:2 mutually:1 turn:1 conferred:1 mechanism:1 needed:1 cfb:1 progression:1 appropriate:3 subtracted:1 gate:1 original:1 capacitive:1 ensure:1 burg:1 c74:1 capacitance:1 spike:36 receptive:10 mapped:3 simulated:1 lateral:1 decoder:1 sci:1 nelson:1 presynaptic:2 collected:4 code:4 sur:1 index:3 ratio:4 difficult:1 negative:2 design:1 adjustable:1 m11:1 m12:2 neuron:126 observation:4 vertical:1 acknowledge:1 extended:1 communication:4 varied:1 pic:12 ordinate:1 david:1 inverting:1 connection:5 andreou:1 hour:1 address:23 bar:1 below:2 spp:1 rf:1 memory:1 gillespie:1 event:10 m15:2 technology:1 inversely:1 orthogonally:1 created:4 extract:1 acknowledgement:1 vmem:1 generation:1 proportional:1 digital:3 foundation:2 integrate:4 nucleus:1 editor:2 row:4 excitatory:14 placed:2 supported:1 asynchronous:4 vthr:1 bias:6 institute:1 fall:1 template:10 feedback:13 curve:8 cortical:11 transition:3 made:1 adaptive:1 transaction:1 preferred:5 logic:1 deg:17 global:12 active:4 hubel:2 incoming:4 sequentially:1 receiver:5 excite:1 arbiter:8 continuous:2 pretty:1 stimulated:3 channel:1 m13:2 investigated:2 protocol:5 linearly:1 repeated:1 neuronal:7 representative:1 m20:2 elaborate:1 n:1 position:1 decoded:1 explicit:1 lie:1 communicates:2 externally:1 minute:1 reconfigurable:1 explored:2 consist:1 workshop:2 sequential:1 adding:1 magnitude:1 interneuron:1 sender:2 visual:6 ch:1 determines:1 identity:1 kramer:6 marked:1 presentation:3 telluride:4 absence:3 hard:1 determined:1 typical:1 total:1 experimental:2 m3:2 photoreceptor:1 nonpreferred:2 support:2 mark:2 handshaking:1 |
1,147 | 2,046 | Multi Dimensional ICA to Separate
Correlated Sources
Roland Vollgraf, Klaus Obermayer
Department of Electrical Engineering and Computer Science
Technical University of Berlin Germany
{ vro, oby} @cs.tu-berlin.de
Abstract
We present a new method for the blind separation of sources, which
do not fulfill the independence assumption. In contrast to standard
methods we consider groups of neighboring samples ("patches")
within the observed mixtures.
First we extract independent features from the observed patches.
It turns out that the average dependencies between these features
in different sources is in general lower than the dependencies between the amplitudes of different sources. We show that it might
be the case that most of the dependencies is carried by only a
small number of features. Is this case - provided these features
can be identified by some heuristic - we project all patches into
the subspace which is orthogonal to the subspace spanned by the
"correlated" features.
Standard ICA is then performed on the elements of the transformed
patches (for which the independence assumption holds) and robustly yields a good estimate of the mixing matrix.
1
Introduction
ICA as a method for blind source separation has been proven very useful in a wide
range of statistical data analysis. A strong criterion, that allows to detect and
separate linearly mixed source signals from the observed mixtures, is the independence of the source signals amplitude distribution. Many contrast functions rely on
this assumption, e.g. in the way, that they estimate the Kullback-Leibler distance
to a (non-Gaussian) factorizing multivariate distribution [1 , 2, 3]. Others consider
higher order moments of the source estimates [4, 5]. Naturally these algorithms
fail when the independence assumption does not hold. In such situations it can be
very useful to consider temporal/spatial statistical properties of the source signals
as well. This has been done in form of suitable linear filtering [6] to achieve a sparse
and independent representation of the signals. In [7] the author suggests to model
the sources as a stochastic process and to do the ICA on the innovations rather
than on the signals them self.
In this work we extend the ICA to multidimensional channels of neighboring realizations. The used data model is explained in detail in the following section. In section
3 it will be shown, that there are optimal features, that carry lower dependencies
between the sources and can be used for source separation. A heuristic is introduced, that allows to discard those features, that carry most of the dependencies.
This leads to the Two-Step algorithm described in section 4. Our method requires
(i) sources which exhibit correlations between neighboring pixels (e.g. continuous
sources like images or sound signals), and (ii) sources from which sparse and almost
independent features can be extracted. In section 5 we show separation results and
benchmarks for linearly mixed passport photographs. The method is fast and provides good separation results even for sources, whose correlation coefficient is as
large as 0.9.
2
Sources and observations
Let us consider a set of N source signals Si(r), i = 1, ... , N of length L, where
r is a discrete sample index. The sample index could be of arbitrary dimension,
but we assume that it belongs to some metric space so that neighborhood relations
can be defined. The sample index might be a scalar for sources which are time
series and a two-dimensional vector for sources which are images 1 . The sources are
linearly combined by an unknown mixing matrix A of full rank to produce a set of
N observations Xi(r),
N
Xi(r) =
l: AijSj(r) ,
(1)
j =l
and we assume that the mixing process is stationary, i.e. that the mixing matrix A is
independent of r. In the following we refer to the vectors S(r) = (Sl (r), ... ,SN(r))T
and X(r) = (X 1 (r), ... , XN(r))T as a source and an observation stack. The goal is
to find an appropriate demixing matrix W which - when applied to the observations
X(r) - recovers good estimates S(r),
S(r) = WX(r) ~ S(r)
(2)
of the original source signals (up to a permutation and scaling of the sources). Since
the mixing matrix A is not known its inverse W has to be detected blindly, i.e. only
properties of the sources which are detectable in the mixtures can be exploited. For
a large class of ICA algorithms one assumes that the sources are non-Gaussian and
independent, i.e. that the random vector S which is sampled by L realizations
S: {S(rd, 1= I, ... ,L}
(3)
has a factorizing and non-Gaussian joint probability distribution 2 . In situations,
however, where the independence assumption does not hold, it can be helpful to
take into account spatial dependencies, which can be very prominent for natural
signals, and have been subject for a number of blind source separation algorithms
[8, 9, 6]. Let us now consider patches si(r),
s(r) =
(4)
1 In the following we will mostly consider images, hence we will refer to the abovementioned neighborhood relations as spatial relations.
2In the following, symbols without sample index will refer to the random variable rather
than to the particular realization.
of M ? L neighboring source samples. si(r) could be a sequence of M adjacent
samples of an audio signal or a rectangular patch of M pixels in an image. Instead
of L realizations of a random N-vector S (cf. eq. (3)) we now obtain a little less
than L realizations of a random N x M matrix s,
s: {s(r)}.
(5)
Because of the stationarity of the mixing process we obtain
x = As
s
and
(6)
= Wx,
where x is an N x M matrix of neighboring observations and where the matrices
A and W operate on every column vector of sand x.
3
Optimal spatial features
Let us now consider a set of sources which are not statistically independent , i.e. for
which
N
p(S) = p(Slk"'" SNk) :j:. IIp(sik)
for all
k = 1 ... M.
(7)
i=1
Our goal is to find in a first step a linear transformation 0 E IRMxM which when applied to every patch - yields transformed sources u = sOT for which the
independence assumption, p(Ulk, ... ,UNk) = rr~1p(Uik) does hold for all k =
1 .. . M, at least approximately. When 0 is applied to the observations x , v = xOT ,
we obtain a modified source separation problem
(8)
where the demixing matrix W can be estimated from the transformed observations
v in a second step using standard ICA. Eq. (7) is tantamount to positive transinformation of the source amplitudes.
(9)
where DKL is the Kullback-Leibler distance. As all elements of the patches are
equally distributed, this quantity is the same for all k. Clearly, the dependencies,
that are carried by single elements of the patches, are also present between whole
patches, i.e. J(S1 , S2,"', SN) > O. However, since neighboring samples are correlated, it holds
M
J(S1 ,S2, "' ,SN ) < LJ(Slk ,S2k"",SNk ) .
k=1
(10)
Only if the sources where spatially white and s would consist of independent column
vectors, this would hold with equality. When 0 is applied to the source patches,
the trans-information between patches is not changed, provided 0 is a non-singular
transformation. Neither information is introduced nor discarded by this transformation and it holds
(11)
For the optimal 0 now the column vectors of u = sOT shall be independent. From
(10) and (11) it follows that
M
I(u1 ,u2, " ',uN)
M
= 2::I(ulk,u2k"",uNk) < 2::I(slk ,s2k"",sNk)
k=1
(12)
k=1
The column vectors of u are in general not equally distributed anymore, however the
average trans-information has decreased to the level of information carried between
the patches. In the experiments we shall see that this can be sufficiently small to
reliably estimate the de-mixing matrix W.
So it remains to estimate a matrix 0 that provides a matrix u with independent
columns. We approach this by estimating 0 so that it provides row vectors of
u that have independent elements, i.e. P(Ui) = IT;!1 P(Uik) for all i. With that
and under the assumption that all sources may come from the same distribution
and that there are no "cross dependencies" in u (i.e. p( Uik) is independent from
p( Ujl) for k :j:. l), the independence is guaranteed also for whole column vectors of
u. Thus, standard leA can be applied to patches of sources which yields 0 as the
de-mixing matrix. For real world applications however , 0 has to be estimated from
the observations xO T = v. It holds the relation v = Au, i.e. A only interchanges
rows. So column vectors of u are independent to each other if, and only if columns
of v are independent 3 . Thus, 0 can be computed from x as well.
According to Eq. (12) the trans-information of the elements of columns of u has
decreased in average, but not necessarily uniformly. One can expect some columns
to have more independent elements than others. Thus, it may be advantageous to
detect these columns rsp. the corresponding rows of 0 and discard them prior to the
second leA step. Each source patch Si can be considered as linear combination of
independent components, that are given by the columns of 0- 1 , where the elements
of Ui are the coefficients. In the result of the leA, the coefficients have normalized
variance. Therefore, those components, that have large Euklidian norm, occur as
features with high entropy in the source patches. At the same time it is clear that,
if there are features , that are responsible for the source dependencies, these features
have to be present with large entropy, otherwise the source dependencies would have
been low. Accordingly we propose a heuristic that discards the rows of 0 with the
smallest Euklidian norm prior to the second leA step. How many rows have to be
discarded and if this type of heuristic is applicable at all , depends of the statistical
nature of the sources. In section 5 we show that for the test data this heuristic is
well applicable and almost all dependencies are contained in one feature.
4
The Two-Step algorithm
The considerations of the previous section give rise to a Two-Step algorithm. In
the first step the transformation 0 has to be estimated. Standard leA [1, 2, 5] is
performed on M -dimensional patches, which are chosen with equal probability from
all of the observed mixtures and at random positions. The positions may overlap
but don't overlap the boundaries of the signals.
The resulting "demixing matrix" 0 is applied to the patches of observations, generating a matrix v(r) = x(r )OT, the columns of which are candidates for the input
for the second leA. A number of M D columns that belong to rows of 0 with small
norm are discarded as they very likely represent features , that carry dependencies
between the sources. M D is chosen as a model parameter or it can be determined
empirically, given the data at hand (for instance by detecting a major jump in the
3We assume non-Gaussian distributions for u and v.
increase of the row norm of n). For the remaining columns it is not obvious which
one represents the most sparse and independent feature. So any of them with equal
probability now serve as input sample for the second ICA , which estimates the
demixing matrix W.
When the number N of sources is large, the first ICA may fail to extract the independent source features, because, according to the central limit theorem, the
distribution of their coefficients in the mixtures may be close to a Gaussian distribution. In such a situation we recommend to apply the abovementioned two steps
repeatedly. The source estimates Wx(r) are used as input for the first ICA to
achieve a better n, which in turn allows to better estimate W.
Figure 1: Results of standard and multidimensional ICA performed on a set of
8 correlated passport images. Top row: source images; Second row: linearly
mixed sources; Third row: separation results using kurtosis optimization (FastICA
Matlab package); Bottom row: separation results using multidimensional ICA
(For explanation see text).
5
Numerical experiments
We applied our method to a linear mixture of 8 passport photographs which are
shown in Fig. 1, top row. The images were mixed (d. Fig. 1, second row) using
a matrix whose elements were chosen randomly from a normal distribution with
mean zero and variance one. The mixing matrix had a condition number of 80.
The correlation coefficients of the source images were between 0.4 and 0.9 so that
standard ICA methods failed to recover the sources: Fig. 1, 3rd row, shows the
results of a kurtosis optimization using the FastICA Matlab package 4 .
Fig. 1, bottom row, shows the result of the Two-Step multidimensional ICA described in section 4. For better comparison images were inverted manually to appear
positive. In the first step n was estimated using FastICA on 105 patches, 6 x 6 pixels in size, which were taken with equal probability from random positions from all
mixtures. The result of the first ICA is displayed in Fig. 2. The top row shows the
row vectors of n sorted by the logarithm of their norm. The second row shows the
features (the corresponding columns of n - 1 ) which are extracted by n . In the dia4http://www.cis.hut.fi/projects/ica/fastica/
V'Lt!1
gram below the stars indicate the logarithm of the row norm, log
0%1' and
the squares indicate the mutual information J(Ulk,U7k) between the k-th features
in sources 1 and 7 5, calculated using a histogram estimator. It is quite prominent that (i) a small norm of a column vector corresponds to a strongly correlated
feature, and (ii) there is only one feature which carries most of the dependencies
between the sources. Thus, the first column of v was discarded. The second ICA
was applied to any of the remaining components, chosen randomly and with equal
probability. A comparison between Figs. 1, top and bottom rows, shows that all
sources were successfully recovered.
Figure 2: Result of an ICA (kurtosis optimization) performed on patches of observations (cf. Fig. 1, 2nd row), 6 x 6 pixels in size. Top row: Row vectors of the
demixing matrix O. Second row: Corresponding column vectors of 0- 1 . Vectors
are sorted by increasing norm of the row vectors; dark and bright pixels indicate
positive and negative values. Bottom diagram: Logarithm of the norm of row
vectors (stars) and mutual information J(Ulk' U7k) (squares) between the coefficients
of the corresponding features in the source images 1 and 7.
In the next experiment we examined the influence of selecting columns of v prior
to the second ICA. In Fig. 3 we show the reconstruction error (cf. appendix A),
that could be achieved with the second ICA when only a single column of v served
as input. From the previous experiment we have seen, that only the first component has considerable dependencies. As expected, only the first column yields poor
reconstruction error. Fig. 4 shows the reconstruction error vs. M D when the M D
smallest norm rows of 0 (rsp. columns of v) are discarded. We see, that for all
values a good reconstruction is achieved (re < 0.6). Even if no row is discarded the
result is only slightly worse than for one or two discarded rows. The dependencies
of the first component are "averaged" by the vast majority of components, that
carry no dependencies, in this case. The conspicuous large variance of the error for
larger numbers M D might be due to convergence instabilities or close to Gaussian
distributed columns of u. In either case it gives rise to discard as few components
as possible. To evaluate the influence of the patch size M, the Two-Step algorithm
was applied to 9 different mixtures of the sources shown in Fig. 1, top row, and
using patch sizes between M = 2 x 2 and M = 6 x 6. Table 1 shows the mean
and standard deviation of the achieved reconstruction error. The mixing matrix A
was randomly chosen from a normal distribution with mean zero and variance one.
FastICA was used for both steps, where 5.105 sample patches were used to extract
the optimal features and 2.5.104 samples were used to estimate W. The smallest
row of 0 was always discarded. The algorithm shows a quite robust performance,
and even for patch sizes of 2 x 2 pixels a fairly good separation result is achieved
5Images no. 1 and 7 were chosen exemplarily as the two most strongly correlated sources.
Jl ? ~. ==1 !C.,,". :..::. :':,. : !?;: =I
1
6
11
16
21
large row norm
26
31
36
0
5
10
small row norm
Figure 3: Every single row of 0 used to
generate input for the second leA. Only
the first (smallest norm) row causes bad
reconstruction error for the second leA
step.
patch size M
J-lr e
(Jre
2x2
3x3
4x4
5x5
6x6
0.4361
0.2322
0.1667
0.1408
0.1270
0.0383
0.0433
0.0263
0.0270
0.0460
Figure 4: M D rows with smallest norm
discarded. All values of M D provide
good reconstruction error in the second
step. Note the slidely worse result for
MD=O!
Table 1: Separation result of the TwoStep algorithm performed on a set of 8
correlated passport images (d. Fig. 1, top
row). The table shows the average reconstruction error J-lr e and its standard deviation (Jr e calculated from 9 different mixtures.
(Note, for comparison, that the reconstruction error of the separation in Fig. 1,
bottom row, was 0.2).
6
Summary and outlook
We extended the source separation model to multidimensional channels (image
patches). There are two linear transformations to be considered, one operating inside the channels (0) and one operating between the different channels (W). The
two transformations are estimated in two adjacent leA steps. There are mainly
two advantages, that can be taken from the first transformation: (i) By arranging
independence among the columns of the transformed patches, the average transinformation between different channels is decreased. (ii) A suitable heuristic can
be applied to discard those columns of the transformed patches, that carry most
of the dependencies between different channels. A heuristic, that identifies the dependence carrying components by a small norm of the corresponding rows of 0
has been introduced. It shows, that for the image data only one component carries
most of the dependencies. Due this fact, the described method works well also when
all components are taken into account . In future work, we are going to establish
a Maximum Likelihood model for both transformations. We expect a performance
gain due to the mutual improvement of the estimates of W and 0 during the iterations. It remains to examine what the model has to be in case some rows of 0
are discarded. In this case the transformations don't preserve the dimensionality of
the observation patches.
A
Reconstruction error
The reconstruction error re is a measure for the success of a source separation.
It compares the estimated de-mixing matrix W with the inverse of the original
mixing matrix A with respect to the indeterminacies: scalings and permutations.
It is always nonnegative and equals zero if, and only if P = W A is a nonsingular
permutation matrix. This is the case when for every row of P exactly one element
is different from zero and the rows of P are orthogonal, i.e. ppT is a diagonal
matrix. The reconstruction error is the sum of measures for both aspects
N
re
N
N
N
N
2LlogL P 7j - Llog LPij
3
i=1
j=1
i=1
j=1
N
N
N
N
i=1
j=1
i=1
j=1
+
Llog L P 7j -log detppT
i=1
L log L P 7j - L log L pi
j -
N
j=1
log det ppT .
(13)
Acknowledgment: This work was funded by the German Science Foundat ion (grant no. DFG SE 931/1-1 and DFG OB 102/3-1 ) and Wellcome Trust
061113/Z/00.
References
[1] Anthony J. Bell and Terrence J . Sejnowski, "An information-maximization
approach to blind separation and blind deconvolution," Neural Computation,
vol. 7, no. 6, pp. 1129-1159, 1995.
[2] S. Amari, A. Cichocki, and H. H. Yang, "A new learning algorithm for blind
signal separation," in Advances in Neural Information Processing Systems,
D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds., 1995, vol. 8.
[3] J . F. Cardoso, "Infomax and maximum likelihood for blind source separation,"
IEEE Signal Processing Lett., 1997.
[4] J ean-Franc;ois Cardoso, Sandip Bose, and Benjamin Friedlander, "On optimal
source separation based on second and fourth order cumulants," in Proc. IEEE
Workshop on SSAP, Co rfou, Greece, 1996.
[5] A. Hyvarinen and E. Oja, "A fast fixed point algorithm for independent component analysis.," Neural Comput., vol. 9, pp. 1483- 1492,1997.
[6] M. Zibulevski and B. A. Pearlmutter, "Blind source separation by sparse
decomposition in a signal dictionary," Neural Computation, vol. 12, no. 3, pp.
863- 882, April 200l.
[7] A. Hyvi:irinen, "Independent component analysis for t ime-dependent stochastic
processes," in Proc. Int. Conf. on Artificial Neural Networks (ICANN'98),
1998, pp. 541-546.
[8] 1. Molgedey and H. G. Schuster, "Separation of a mixture of independent
signals using t ime delayed correlations," Phys. Rev. Lett., vol. 72, pp. 36343637, 1994.
[9] H. Attias and C. E. Schreiner, "Blind source separation and deconvolution:
The dynamic component analysis algorithm," Neural Comput., vol. 10, pp.
1373- 1424, 1998.
[10] Anthony J. Bell and Terrence J. Sejnowski, "The 'independent components' of
natural scenes are edge filters," Vision Res. , vol. 37, pp. 3327- 3338, 1997.
| 2046 |@word advantageous:1 norm:15 nd:1 decomposition:1 outlook:1 carry:7 moment:1 series:1 selecting:1 recovered:1 si:4 numerical:1 wx:3 v:1 stationary:1 accordingly:1 lr:2 detecting:1 provides:3 inside:1 expected:1 ica:20 nor:1 examine:1 multi:1 little:1 increasing:1 provided:2 project:2 estimating:1 what:1 transformation:9 temporal:1 every:4 multidimensional:5 exactly:1 grant:1 appear:1 positive:3 engineering:1 limit:1 approximately:1 might:3 au:1 examined:1 suggests:1 co:1 sandip:1 range:1 statistically:1 averaged:1 acknowledgment:1 responsible:1 x3:1 bell:2 close:2 influence:2 instability:1 www:1 rectangular:1 schreiner:1 estimator:1 spanned:1 arranging:1 element:9 observed:4 bottom:5 electrical:1 mozer:1 benjamin:1 ui:2 dynamic:1 carrying:1 serve:1 molgedey:1 joint:1 fast:2 sejnowski:2 detected:1 oby:1 artificial:1 klaus:1 neighborhood:2 u2k:1 whose:2 heuristic:7 quite:2 larger:1 otherwise:1 amari:1 sequence:1 rr:1 advantage:1 kurtosis:3 propose:1 reconstruction:12 lpij:1 tu:1 neighboring:6 realization:5 mixing:12 achieve:2 convergence:1 produce:1 generating:1 indeterminacy:1 eq:3 strong:1 c:1 ois:1 come:1 indicate:3 filter:1 stochastic:2 euklidian:2 sand:1 hold:8 sufficiently:1 considered:2 hut:1 normal:2 rfou:1 major:1 dictionary:1 smallest:5 foundat:1 proc:2 applicable:2 hasselmo:1 successfully:1 clearly:1 ujl:1 gaussian:6 always:2 modified:1 fulfill:1 rather:2 improvement:1 rank:1 likelihood:2 mainly:1 contrast:2 detect:2 helpful:1 dependent:1 lj:1 relation:4 transformed:5 going:1 germany:1 pixel:6 among:1 unk:2 spatial:4 fairly:1 mutual:3 equal:5 manually:1 x4:1 represents:1 future:1 others:2 recommend:1 few:1 franc:1 ppt:2 randomly:3 oja:1 preserve:1 ime:2 dfg:2 delayed:1 stationarity:1 mixture:10 edge:1 vro:1 orthogonal:2 logarithm:3 re:4 twostep:1 instance:1 column:26 transinformation:2 cumulants:1 maximization:1 deviation:2 fastica:5 dependency:18 combined:1 terrence:2 infomax:1 central:1 iip:1 worse:2 conf:1 account:2 de:4 star:2 coefficient:6 int:1 blind:9 depends:1 performed:5 recover:1 square:2 bright:1 variance:4 yield:4 nonsingular:1 served:1 phys:1 touretzky:1 ed:1 pp:7 obvious:1 naturally:1 recovers:1 sampled:1 gain:1 dimensionality:1 greece:1 amplitude:3 higher:1 x6:1 april:1 done:1 strongly:2 correlation:4 hand:1 trust:1 jre:1 normalized:1 hence:1 equality:1 spatially:1 leibler:2 white:1 adjacent:2 x5:1 during:1 self:1 criterion:1 prominent:2 pearlmutter:1 image:14 consideration:1 fi:1 empirically:1 jl:1 extend:1 belong:1 refer:3 rd:2 had:1 funded:1 operating:2 multivariate:1 belongs:1 discard:5 xot:1 success:1 exploited:1 inverted:1 seen:1 signal:15 ii:3 full:1 sound:1 technical:1 cross:1 roland:1 equally:2 dkl:1 vision:1 metric:1 blindly:1 histogram:1 represent:1 iteration:1 achieved:4 ion:1 lea:9 decreased:3 diagram:1 singular:1 source:61 ot:1 operate:1 subject:1 yang:1 independence:8 identified:1 attias:1 det:1 cause:1 repeatedly:1 matlab:2 useful:2 clear:1 se:1 cardoso:2 slk:3 dark:1 generate:1 sl:1 estimated:6 discrete:1 shall:2 vol:7 group:1 neither:1 vast:1 sum:1 inverse:2 package:2 fourth:1 almost:2 separation:21 patch:29 ob:1 appendix:1 scaling:2 guaranteed:1 nonnegative:1 occur:1 x2:1 scene:1 u1:1 aspect:1 department:1 according:2 combination:1 poor:1 jr:1 slightly:1 conspicuous:1 rev:1 s1:2 explained:1 xo:1 taken:3 wellcome:1 remains:2 turn:2 detectable:1 fail:2 german:1 apply:1 snk:3 appropriate:1 robustly:1 anymore:1 original:2 assumes:1 remaining:2 cf:3 top:7 establish:1 quantity:1 dependence:1 md:1 diagonal:1 abovementioned:2 obermayer:1 exhibit:1 subspace:2 distance:2 separate:2 berlin:2 majority:1 length:1 index:4 innovation:1 mostly:1 negative:1 rise:2 reliably:1 unknown:1 observation:11 discarded:10 benchmark:1 displayed:1 situation:3 extended:1 stack:1 arbitrary:1 bose:1 introduced:3 trans:3 below:1 explanation:1 ulk:4 suitable:2 overlap:2 natural:2 rely:1 s2k:2 identifies:1 carried:3 extract:3 cichocki:1 sn:3 text:1 prior:3 friedlander:1 tantamount:1 exemplarily:1 expect:2 permutation:3 mixed:4 filtering:1 proven:1 sik:1 pi:1 row:42 changed:1 summary:1 wide:1 sparse:4 distributed:3 boundary:1 dimension:1 xn:1 world:1 gram:1 calculated:2 lett:2 author:1 interchange:1 jump:1 hyvarinen:1 passport:4 sot:2 kullback:2 xi:2 don:2 factorizing:2 continuous:1 un:1 table:3 channel:6 nature:1 robust:1 ean:1 necessarily:1 anthony:2 icann:1 linearly:4 whole:2 s2:2 fig:12 uik:3 position:3 comput:2 candidate:1 third:1 theorem:1 bad:1 symbol:1 demixing:5 consist:1 deconvolution:2 workshop:1 ci:1 entropy:2 lt:1 photograph:2 likely:1 failed:1 rsp:2 contained:1 scalar:1 u2:1 irinen:1 corresponds:1 extracted:2 goal:2 sorted:2 considerable:1 determined:1 uniformly:1 llog:2 evaluate:1 audio:1 schuster:1 correlated:7 |
1,148 | 2,047 | The Intelligent Surfer:
Probabilistic Combination of Link and
Content Information in PageRank
Matthew Richardson
Pedro Domingos
Department of Computer Science and Engineering
University of Washington
Box 352350
Seattle, WA 98195-2350, USA
{mattr, pedrod}@cs.washington.edu
Abstract
The PageRank algorithm, used in the Google search engine, greatly
improves the results of Web search by taking into account the link
structure of the Web. PageRank assigns to a page a score proportional to the number of times a random surfer would visit that page,
if it surfed indefinitely from page to page, following all outlinks
from a page with equal probability. We propose to improve PageRank by using a more intelligent surfer, one that is guided by a
probabilistic model of the relevance of a page to a query. Efficient
execution of our algorithm at query time is made possible by precomputing at crawl time (and thus once for all queries) the necessary terms. Experiments on two large subsets of the Web indicate
that our algorithm significantly outperforms PageRank in the (human-rated) quality of the pages returned, while remaining efficient
enough to be used in today?s large search engines.
1
Introduction
Traditional information retrieval techniques can give poor results on the Web, with
its vast scale and highly variable content quality. Recently, however, it was found
that Web search results can be much improved by using the information contained in
the link structure between pages. The two best-known algorithms which do this are
HITS [1] and PageRank [2]. The latter is used in the highly successful Google search
engine [3]. The heuristic underlying both of these approaches is that pages with many
inlinks are more likely to be of high quality than pages with few inlinks, given that
the author of a page will presumably include in it links to pages that s/he believes are
of high quality. Given a query (set of words or other query terms), HITS invokes a
traditional search engine to obtain a set of pages relevant to it, expands this set with
its inlinks and outlinks, and then attempts to find two types of pages, hubs (pages
that point to many pages of high quality) and authorities (pages of high quality). Because this computation is carried out at query time, it is not feasible for today?s
search engines, which need to handle tens of millions of queries per day. In contrast,
PageRank computes a single measure of quality for a page at crawl time. This meas-
ure is then combined with a traditional information retrieval score at query time.
Compared with HITS, this has the advantage of much greater efficiency, but the disadvantage that the PageRank score of a page ignores whether or not the page is relevant to the query at hand.
Traditional information retrieval measures like TFIDF [4] rate a document highly if
the query terms occur frequently in it. PageRank rates a page highly if it is at the center of a large sub-web (i.e., if many pages point to it, many other pages point to
those, etc.). Intuitively, however, the best pages should be those that are at the center
of a large sub-web relevant to the query. If one issues a query containing the word
jaguar, then pages containing the word jaguar that are also pointed to by many other
pages containing jaguar are more likely to be good choices than pages that contain
jaguar but have no inlinks from pages containing it. This paper proposes a search
algorithm that formalizes this intuition while, like PageRank, doing most of its computations at crawl time. The PageRank score of a page can be viewed as the rate at
which a surfer would visit that page, if it surfed the Web indefinitely, blindly jumping from page to page. Our algorithm does something closer to what a human surfer
would do, jumping preferentially to pages containing the query terms.
A problem common to both PageRank and HITS is topic drift. Because they give the
same weight to all edges, the pages with the most inlinks in the network being considered (either at crawl or query time) tend to dominate, whether or not they are the
most relevant to the query. Chakrabarti et al. [5] and Bharat and Henzinger [6] propose heuristic methods for differentially weighting links. Our algorithm can be
viewed as a more principled approach to the same problem. It can also be viewed as
an analog for PageRank of Cohn and Hofmann?s [7] variation of HITS. Rafiei and
Mendelzon'
s [8] algorithm, which biases PageRank towards pages containing a specific word, is a predecessor of our work. Haveliwala [9] proposes applying an optimized version of PageRank to the subset of pages containing the query terms, and
suggests that users do this on their own machines.
We first describe PageRank. We then introduce our query-dependent, contentsensitive version of PageRank, and demonstrate how it can be implemented efficiently. Finally, we present and discuss experimental results.
2
PageRank : The Random Surfer
Imagine a web surfer who jumps from web page to web page, choosing with uniform
probability which link to follow at each step. In order to reduce the effect of deadends or endless cycles the surfer will occasionally jump to a random page with some
small probability ?, or when on a page with no out-links. To reformulate this in
graph terms, consider the web as a directed graph, where nodes represent web pages,
and edges between nodes represent links between web pages. Let W be the set of
nodes, N=|W|, Fi be the set of pages page i links to, and B i be the set pages which
link to page i. For pages which have no outlinks we add a link to all pages in the
graph1. In this way, rank which is lost due to pages with no outlinks is redistributed
uniformly to all pages. If averaged over a sufficient number of steps, the probability
the surfer is on page j at some point in time is given by the formula:
P( j ) =
1
(1 ? ? )
P (i )
+? ?
N
i?B j Fi
(1)
For each page s with no outlinks, we set Fs={all N nodes}, and for all other nodes augment B i with s. (B i ? {s})
The PageRank score for node j is defined as this probability: PR(j)=P(j). Because
equation (1) is recursive, it must be iteratively evaluated until P(j) converges. Typically, the initial distribution for P(j) is uniform. PageRank is equivalent to the primary eigenvector of the transition matrix Z:
?1?
Z = (1 ? ? ) ? ?
+ ? M ,with
? N ? NxN
? 1
?
M ji = ? Fi
?0
?
if there is an edge from i to j
(2)
otherwise
One iteration of equation (1) is equivalent to computing xt+1=Zxt, where xjt=P(j) at
iteration t. After convergence, we have xT+1=xT, or xT=ZxT, which means xT is an
eigenvector of Z. Furthermore, since the columns of Z are normalized, x has an eigenvalue of 1.
3
Directed Surfer Model
We propose a more intelligent surfer, who probabilistically hops from page to page,
depending on the content of the pages and the query terms the surfer is looking for.
The resulting probability distribution over pages is:
Pq ( j ) = (1 ? ? ) Pq? ( j ) + ?
? Pq (i ) Pq (i ? j )
(3)
i?B j
where Pq(i?j) is the probability that the surfer transitions to page j given that he is
on page i and is searching for the query q. Pq?(j) specifies where the surfer chooses to
jump when not following links. Pq(j) is the resulting probability distribution over
pages and corresponds to the query-dependent PageRank score (QD-PageRankq(j) ?
Pq(j)). As with PageRank, QD-PageRank is determined by iterative evaluation of
equation 3 from some initial distribution, and is equivalent to the primary eigenvector of the transition matrix Zq, where Z q ji = (1 ? ? ) Pq? ( j ) + ? ? Pq (i ? j ) . Although
i?B j
Pq(i?j) and Pq?(j) are arbitrary distributions, we will focus on the case where both
probability distributions are derived from Rq(j), a measure of relevance of page j to
query q:
Pq? ( j ) =
Rq ( j )
? Rq ( k )
k?W
Pq (i ? j ) =
Rq ( j )
? Rq (k )
(4)
k?Fi
In other words, when choosing among multiple out-links from a page, the directed
surfer tends to follow those which lead to pages whose content has been deemed
relevant to the query (according to Rq). Similarly to PageRank, when a page?s outlinks all have zero relevance, or has no outlinks, we add links from that page to all
other pages in the network. On such a page, the surfer thus chooses a new page to
jump to according to the distribution Pq? (j).
When given a multiple-term query, Q={q 1,q 2,?}, the surfer selects a q according to
some probability distribution, P(q) and uses that term to guide its behavior (according to equation 3) for a large number of steps1. It then selects another term according
to the distribution to determine its behavior, and so on. The resulting distribution
over visited web pages is QD-PageRankQ and is given by
1
However many steps are needed to reach convergence of equation 3.
QD ? PageRank Q ( j ) ? PQ ( j ) =
? P( q) Pq ( j )
(5)
q?Q
For standard PageRank, the PageRank vector is equivalent to the primary eigenvector
of the matrix Z. The vector of single-term QD-PageRankq is again equivalent to the
primary eigenvector of the matrix Zq. An interesting question that arises is whether
the QD-PageRankQ vector is equivalent to the primary eigenvector of a matrix
ZQ = ? P ( q) Z q (corresponding to the combination performed by equation 5). In
q?Q
fact, this is not the case. Instead, the primary eigenvector of ZQ corresponds to the
QD-PageRank obtained by a random surfer who, at each step, selects a new query
according to the distribution P(q). However, QD-PageRankQ is approximately equal
to the PageRank that results from this single-step surfer, for the following reason.
Let xq be the L2-normalized primary eigenvector for matrix Zq (note element j of xq
is QD-PageRankq(j)), thus satisfying xi=Tixi. Since xq is the primary eigenvector for
Zq, we have [10]: ?q, r ? Q : Z q x q ? Z q x r . Thus, to a first degree of approximation, Z q
? x r ? ?Z q x q . Suppose P(q)=1/|Q|. Consider xQ = ? P( q)x q (see equation
r?Q
q?Q
5). Then
?
??
? 1
?
? 1
1
?
?
ZQ x Q = ? ?
Z ?? x q ?? =
? ?? Z q ? x r ?? ? ? ?Z q x q = ? x q = x Q
? q?Q Q q ?? q?
Q q?Q
n
?
?? ?Q ? Q q?Q ? r?Q ? Q q?Q
and thus xQ is approximately an eigenvector for ZQ. Since xQ is equivalent to QDPageRankQ, and ZQ describes the behavior of the single-step surfer, QD-PageRankQ
is approximately the same PageRank that would be obtained by using the single-step
surfer. The approximation has the least error when the individual random surfers defined by Zq are very similar, or are very dissimilar.
(
)
The choice of relevance function Rq(j) is arbitrary. In the simplest case, Rq(j)=R is
independent of the query term and the document, and QD-PageRank reduces to PageRank. One simple content-dependent function could be Rq(j)=1 if the term q appears
on page j, and 0 otherwise. Much more complex functions could be used, such as the
well-known TFIDF information retrieval metric, a score obtained by latent semantic
indexing, or any heuristic measure using text size, positioning, etc?. It is important
to note that most current text ranking functions could be easily incorporated into the
directed surfer model.
4
Scalability
The difficulty with calculating a query dependent PageRank is that a search engine
cannot perform the computation, which can take hours, at query time, when it is expected to return results in seconds (or less). We surmount this problem by precomputing the individual term rankings QD-PageRankq, and combining them at query
time according to equation 5. We show that the computation and storage requirements for QD-PageRankq for hundreds of thousands of words is only approximately
100-200 times that of a single query independent PageRank.
Let W={q1, q2, ?, qm} be the set of words in our lexicon. That is, we assume all
search queries contain terms in W, or we are willing to use plain PageRank for those
terms not in W. Let d q be the number of documents which contain the term q. Then
S = ? d q is the number of unique document-term pairs.
q?W
4 .1
Disk St o ra g e
For each term q, we must store the results of the computation. We add the minor restriction that a search query will only return documents containing all of the terms 1.
Thus, when merging QD-PageRankq?s, we need only to know the QD-PageRankq for
documents that contain the term. Each QD-PageRankq is a vector of d q values. Thus,
the space required to store all of the PageRanks is S, a factor of S/N times the query
independent PageRank alone (recall N is the number of web pages). Further, note
that the storage space is still considerably less than that required for the search engine?s reverse index, which must store information about all document-term pairs, as
opposed to our need to store information about every unique document term pair.
4 .2
Time Requirement s
If Rq(j)=0 for some document j, the directed surfer will never arrive at that page. In
this case, we know QD-PageRankq(j)=0, and thus when calculating QD-PageRankq,
we need only consider the subset of nodes for which Rq(j)>0. We add the reasonable
constraint that Rq(j)=0 if term q does not appear in document j, which is common for
most information retrieval relevance metrics, such as TFIDF. The computation for
term q then only needs to consider dq documents. Because it is proportional to the
number of documents in the graph, the computation of QD-PageRankq for all q in W
will require O(S) time, a factor of S/N times the computation of the query independent PageRank alone. Furthermore, we have noticed in our experiments that the computation converges in fewer iterations on these smaller sub-graphs, empirically reducing the computational requirements to 0.75*S/N. Additional speedup may be derived from the fact that for most words, the sub-graph will completely fit in memory,
unlike PageRank which (for any large corpus) must repeatedly read the graph structure from disk during computation.
4 .3
Empirica l Sca la bilit y
The fraction S/N is critical to determining the scalability of QD-PageRank. If every
document contained vastly different words, S/N would be proportional to the number
of search terms, m. However, this is not the case. Instead, there are a very few words
that are found in almost every document, and many words which are found in very
few documents2; in both cases the contribution to S is small.
In our database of 1.7 million pages (see section 5), we let W be the set of all unique
words, and removed the 100 most common words3. This results in |W|=2.3 million
words, and the ratio S/N was found to be 165. We expect that this ratio will remain
relatively constant even for much larger sets of web pages. This means QDPageRank requires approximately 165 times the storage space and 124 times the
computation time to allow for arbitrary queries over any of the 2.3 million words
(which is still less storage space than is required by the search engine?s reverse index
alone).
1
Google has this ?feature? as well. See http://www.google.com/technology/whyuse.html.
This is because the distribution of words in text tends to follow an inverse power law
[11]. We also verified experimentally that the same holds true for the distribution of the
number of documents a word is found in.
3
It is common to remove ?stop? words such as the, is, etc., as they do not affect the
search.
2
Table 1: Results on educrawl
Query
QD-PR
PR
Table 2: Results on WebBase
Query
QD-PR
PR
chinese association
computer labs
financial aid
intramural
maternity
president office
sororities
student housing
visitor visa
Average
alcoholism
architecture
bicycling
rock climbing
shakespeare
stamp collecting
vintage car
Thailand tourism
Zen Buddhism
Average
5
10.75
9.50
8.00
16.5
12.5
5.00
13.75
14.13
19.25
12.15
6.50
13.25
12.38
10.25
6.75
11.38
7.38
10.75
12.50
10.13
11.50
8.45
8.45
8.43
11.53
9.13
13.15
16.90
8.63
10.68
11.88
2.93
6.88
5.75
5.03
10.68
8.68
9.75
10.38
7.99
Results
We give results on two data sets: educrawl, and WebBase. Educrawl is a crawl of the
web, restricted to .edu domains. The crawler was seeded with the first 18 results of a
search for ? University? on Google (www.google.com). Links containing ? ?? or ? cgibin? were ignored, and links were only followed if they ended with ? .html? . The
crawl contains 1.76 million pages over 32,000 different domains. WebBase is the first
15 million pages of the Stanford WebBase repository [12], which contains over 120
million pages. For both datasets, HTML tags were removed before processing.
We calculated QD-PageRank as described above, using Rq(j) = the fraction of words
equal to q in page j, and P(q)=1/|Q|. We compare our algorithm to the standard PageRank algorithm. For content ranking, we used the same Rq(j) function as for QDPageRank, but, similarly to TFIDF, weighted the contribution of each search term by
the log of its inverse document frequency. As there is nothing published about merging PageRank and content rank into one list, the approach we follow is to normalize
the two scores and add them. This implicitly assumes that PageRank and content rank
are equally important. This resulted in poor PageRank performance, which we found
was because the distribution of PageRanks is much more skewed than the distribution
of content ranks; normalizing the vectors resulted in PageRank primarily determining
the final ranking. To correct this problem, we scaled each vector to have the same
average value in its top ten terms before adding the two vectors. This drastically improved PageRank.
For educrawl, we requested a single word and two double word search queries from
each of three volunteers, resulting in a total of nine queries. For each query, we randomly mixed the top 10 results from standard PageRank with the top 10 results from
QD-PageRank, and gave them to four volunteers, who were asked to rate each search
result as a 0 (not relevant), 1 (somewhat relevant, not very good), or 2 (good search
result) based on the contents of the page it pointed to. In Table 1, we present the final
rating for each method, per query. This rating was obtained by first summing the ratings for the ten pages from each method for each volunteer, and then averaging the
individual ratings. A similar experiment for WebBase is given in Table 2. For WebBase, we randomly selected the queries from Bharat and Henzinger [6]. The four
volunteers for the WebBase evaluation were independent from the four for the
educrawl evaluation, and none knew how the pages they were asked to rate were obtained.
QD-PageRank performs better than PageRank, accomplishing a relative improvement
in relevance of 20% on educrawl and 34% on WebBase. The results are statistically
significant (p<.03 for educrawl and p<.001 for WebBase using a two-tailed paired ttest, one sample per person per query). Averaging over queries, every volunteer
found QD-PageRank to be an improvement over PageRank, though not all differences were statistically significant.
One item to note is that the results on multiple word queries are not as positive as the
results on single word queries. As discussed in section 3, the combination of single
word QD-PageRanks to calculate the QD-PageRank for a multiple word query is only
an approximation, made for practical reasons. This approximation is worse when the
words are highly dependent. Further, some queries, such as ? financial aid? have a
different intended meaning as a phrase than simply the two words ? financial? and
? aid? . For queries such as these, the words are highly dependent. We could partially
overcome this difficulty by adding the most common phrases to the lexicon, thus
treating them the same as single words.
6
Conclusions
In this paper, we introduced a model that probabilistically combines page content and
link structure in the form of an intelligent random surfer. The model can accommodate essentially any query relevance function in use today, and produces higherquality results than PageRank, while having time and storage requirements that are
within reason for today? s large scale search engines.
Ackno w ledg ment s
We would like to thank Gary Wesley and Taher Haveliwala for their help with WebBase, Frank McSherry for eigen-help, and our experiment volunteers for their time.
This work was partially supported by NSF CAREER and IBM Faculty awards to the
second author.
Ref erences
[1] J. M. Kleinberg (1998). Authoritative sources in a hyperlinked environment. Proceedings
of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms.
[2] L. Page, S. Brin, R. Motwani, and T. Winograd (1998). The PageRank citation ranking:
Bringing order to the web. Technical report, Stanford University, Stanford, CA.
[3] S. Brin and L. Page (1998). The anatomy of a large-scale hypertextual Web search engine.
Proceedings of the Seventh International World Wide Web Conference.
[4] G. Salton and M. J. McGill (1983). Introduction to Modern Information Retrieval.
McGraw-Hill, New York, NY.
[5] S. Chakrabarti, B. Dom, D. Gibson, J. Kleinberg, P. Raghavan, and S. Rajagopalan (1998).
Automatic resource compilation by analyzing hyperlink structure and associated text. Proceedings of the Seventh International World Wide Web Conference.
[6] K. Bharat and M. R. Henzinger (1998). Improved algorithms for topic distillation in a hyperlinked environment. Proceedings of the Twenty-First Annual International ACM SIGIR
Conference on Research and Development in Information Retrieval.
[7] D. Cohn and T. Hofmann (2001). The missing link - a probabilistic model of document
content and hypertext connectivity. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors,
Advances in Neural Information Processing Systems 13. MIT Press, Cambridge, MA.
[8] D. Rafiei and A. Mendelzon (2000). What is this page known for? Computing web page
reputations. Proceedings of the Ninth International World Wide Web Conference.
[9] T. Haveliwala (1999). Efficient computation of PageRank. Technical report, Stanford University, Stanford, CA.
[10] G. H. Golub and C. F. Van Loan (1996). Matrix Computations. Johns Hopkins University
Press, Baltimore, MD, third edition.
[11] G. K. Zipf (1949). Human Behavior and the Principle of Least Effort. Addison-Wesley,
Cambridge, MA.
[12] J. Hirai, S. Raghaven, H. Garcia-Molina, A. Paepcke (1999). WebBase: a repository of
web pages. Proceedings of the Ninth World Wide Web Conference.
| 2047 |@word repository:2 faculty:1 version:2 disk:2 willing:1 q1:1 accommodate:1 initial:2 contains:2 score:8 document:17 outperforms:1 current:1 com:2 must:4 john:1 shakespeare:1 hofmann:2 remove:1 treating:1 alone:3 fewer:1 selected:1 item:1 indefinitely:2 authority:1 node:7 lexicon:2 predecessor:1 symposium:1 chakrabarti:2 combine:1 introduce:1 webbase:11 bharat:3 ra:1 expected:1 behavior:4 frequently:1 underlying:1 what:2 eigenvector:10 q2:1 ended:1 formalizes:1 every:4 collecting:1 expands:1 qm:1 hit:5 scaled:1 appear:1 before:2 positive:1 engineering:1 tends:2 analyzing:1 ure:1 approximately:5 suggests:1 statistically:2 averaged:1 directed:5 unique:3 practical:1 lost:1 recursive:1 sca:1 gibson:1 significantly:1 word:28 cannot:1 graph1:1 storage:5 applying:1 www:2 restriction:1 equivalent:7 center:2 missing:1 sigir:1 assigns:1 dominate:1 financial:3 handle:1 searching:1 variation:1 president:1 mcgill:1 imagine:1 today:4 suppose:1 user:1 us:1 domingo:1 element:1 satisfying:1 database:1 winograd:1 hypertext:1 thousand:1 calculate:1 cycle:1 removed:2 principled:1 intuition:1 rq:14 jaguar:4 environment:2 asked:2 dom:1 efficiency:1 completely:1 easily:1 describe:1 query:49 choosing:2 whose:1 heuristic:3 larger:1 stanford:5 otherwise:2 richardson:1 final:2 housing:1 advantage:1 eigenvalue:1 hyperlinked:2 rock:1 propose:3 redistributed:1 ment:1 relevant:7 combining:1 normalize:1 differentially:1 scalability:2 seattle:1 convergence:2 double:1 requirement:4 motwani:1 produce:1 converges:2 help:2 depending:1 minor:1 implemented:1 c:1 empirica:1 indicate:1 qd:28 guided:1 anatomy:1 correct:1 human:3 raghavan:1 brin:2 require:1 tfidf:4 hold:1 considered:1 presumably:1 surfer:25 matthew:1 intramural:1 visited:1 weighted:1 mit:1 probabilistically:2 office:1 derived:2 focus:1 improvement:2 rank:4 greatly:1 contrast:1 dependent:6 typically:1 selects:3 issue:1 among:1 html:3 augment:1 bicycling:1 outlinks:7 proposes:2 tourism:1 development:1 equal:3 once:1 never:1 having:1 washington:2 hop:1 report:2 intelligent:4 few:3 primarily:1 modern:1 randomly:2 resulted:2 individual:3 intended:1 attempt:1 highly:6 surfed:2 evaluation:3 golub:1 mcsherry:1 compilation:1 endless:1 edge:3 closer:1 necessary:1 jumping:2 column:1 disadvantage:1 phrase:2 subset:3 uniform:2 hundred:1 successful:1 seventh:2 haveliwala:3 considerably:1 combined:1 chooses:2 st:1 person:1 international:4 siam:1 probabilistic:3 hopkins:1 connectivity:1 again:1 vastly:1 containing:9 opposed:1 zen:1 worse:1 return:2 account:1 student:1 ranking:5 performed:1 lab:1 doing:1 ttest:1 contribution:2 accomplishing:1 who:4 efficiently:1 climbing:1 none:1 published:1 reach:1 frequency:1 henzinger:3 salton:1 associated:1 stop:1 steps1:1 recall:1 car:1 improves:1 vintage:1 appears:1 wesley:2 day:1 follow:4 hirai:1 improved:3 leen:1 evaluated:1 box:1 though:1 erences:1 furthermore:2 until:1 hand:1 web:26 cohn:2 google:6 quality:7 usa:1 effect:1 dietterich:1 normalized:2 true:1 contain:4 seeded:1 read:1 iteratively:1 buddhism:1 semantic:1 during:1 skewed:1 ackno:1 hill:1 demonstrate:1 performs:1 meaning:1 recently:1 fi:4 common:5 ji:2 empirically:1 million:7 analog:1 he:2 association:1 discussed:1 significant:2 distillation:1 cambridge:2 zipf:1 automatic:1 similarly:2 pointed:2 pq:17 etc:3 add:5 something:1 own:1 reverse:2 occasionally:1 store:4 molina:1 greater:1 additional:1 somewhat:1 determine:1 multiple:4 reduces:1 technical:2 positioning:1 retrieval:7 equally:1 visit:2 award:1 paired:1 xjt:1 essentially:1 metric:2 volunteer:6 blindly:1 iteration:3 represent:2 baltimore:1 source:1 unlike:1 bringing:1 tend:1 enough:1 affect:1 fit:1 gave:1 architecture:1 reduce:1 whether:3 effort:1 f:1 returned:1 york:1 nine:1 repeatedly:1 ignored:1 thailand:1 ten:3 simplest:1 http:1 specifies:1 nsf:1 per:4 discrete:1 four:3 verified:1 vast:1 graph:6 fraction:2 surmount:1 inverse:2 arrive:1 almost:1 reasonable:1 followed:1 hypertextual:1 annual:2 occur:1 constraint:1 tag:1 kleinberg:2 relatively:1 speedup:1 department:1 according:7 combination:3 poor:2 describes:1 smaller:1 remain:1 visa:1 intuitively:1 restricted:1 pr:5 indexing:1 equation:8 resource:1 discus:1 needed:1 know:2 addison:1 eigen:1 assumes:1 remaining:1 include:1 top:3 calculating:2 invokes:1 chinese:1 noticed:1 question:1 primary:8 md:1 traditional:4 link:18 zxt:2 thank:1 topic:2 reason:3 index:2 reformulate:1 ratio:2 preferentially:1 frank:1 twenty:1 perform:1 datasets:1 looking:1 incorporated:1 inlinks:5 ninth:3 arbitrary:3 drift:1 rating:4 introduced:1 pair:3 required:3 optimized:1 engine:10 hour:1 hyperlink:1 rajagopalan:1 pagerank:56 memory:1 belief:1 power:1 critical:1 difficulty:2 improve:1 pedrod:1 rated:1 technology:1 carried:1 deemed:1 tresp:1 xq:6 text:4 l2:1 determining:2 relative:1 nxn:1 law:1 expect:1 mixed:1 interesting:1 proportional:3 authoritative:1 degree:1 sufficient:1 dq:1 editor:1 principle:1 ibm:1 supported:1 drastically:1 bias:1 guide:1 allow:1 wide:4 taking:1 van:1 overcome:1 plain:1 calculated:1 crawl:6 transition:3 world:4 computes:1 ignores:1 author:2 made:2 jump:4 citation:1 implicitly:1 mcgraw:1 corpus:1 summing:1 knew:1 xi:1 search:22 iterative:1 latent:1 reputation:1 zq:10 tailed:1 table:4 ca:2 career:1 requested:1 complex:1 domain:2 precomputing:2 alcoholism:1 edition:1 nothing:1 ref:1 ny:1 aid:3 sub:4 stamp:1 weighting:1 third:1 formula:1 specific:1 xt:5 hub:1 meas:1 list:1 normalizing:1 merging:2 adding:2 execution:1 garcia:1 simply:1 likely:2 contained:2 partially:2 pedro:1 corresponds:2 gary:1 acm:2 ma:2 viewed:3 towards:1 content:12 feasible:1 experimentally:1 loan:1 determined:1 uniformly:1 reducing:1 averaging:2 total:1 experimental:1 la:1 latter:1 arises:1 dissimilar:1 relevance:7 crawler:1 visitor:1 |
1,149 | 2,048 | Motivated Reinforcement Learning
Peter Dayan
Gatsby Computational Neuroscience Unit
17 Queen Square, London, England, WClN 3AR.
dayan@gatsby.ucl.ac.uk
Abstract
The standard reinforcement learning view of the involvement
of neuromodulatory systems in instrumental conditioning includes a rather straightforward conception of motivation as
prediction of sum future reward. Competition between actions
is based on the motivating characteristics of their consequent
states in this sense. Substantial, careful, experiments reviewed
in Dickinson & Balleine, 12,13 into the neurobiology and psychology of motivation shows that this view is incomplete. In many
cases, animals are faced with the choice not between many different actions at a given state, but rather whether a single response is worth executing at all. Evidence suggests that the
motivational process underlying this choice has different psychological and neural properties from that underlying action
choice. We describe and model these motivational systems,
and consider the way they interact.
1 Introduction
Reinforcement learning (RL 28) bears a tortuous relationship with historical and
contemporary ideas in classical and instrumental conditioning. Although RL
sheds important light in some murky areas, it has paid less attention to research concerning the motivation of stimulus-response (SR) links. RL methods
are mainly concerned with preparatory Pavlovian (eg secondary) conditioning,
and, in instrumental conditioning, the competition between multiple possible
actions given a particular stimulus or state, based on the future rewarding or
punishing consequences of those actions. These have been used to build successful and predictive models of the activity of monkey dopamine cells in conditioning. 22,24 By contrast, SR research starts from the premise that, in many
circumstances, given an unconditioned stimulus (US; such as a food pellet),
there is only one natural set of actions (the habit of approaching and eating
the food), and the main issue is whether this set is worth executing (yes, if hungry, no if sated). This is traditionally conceived as a question of consummatory
motivation. SR research goes on to study how these habits, and also the motivation associated with them, are 'attached' in an appropriately preparatory
sense to conditioned stimuli (CSs) that are predictive of the USs.
The difference between RL's competition between multiple actions and SR's
motivation of a single action might seem trivial, particularly if an extra, nUll,
action is included in the action competition in RL, so the subject can actively
choose to do nothing. However, there is substantial evidence from experi-
ments in which drive states (eg hunger, thirst) are manipulated, that motivation in the SR sense works in a sophisticated, intrinsically goal-sensitive,
way and can exert unexpected effects on instrumental conditioning. By comparison with RL, psychological study of multiple goals within single environments is quite advanced, particularly in experiments in which one goal or set
of goals is effective during learning, and another during performance. Based
on these and other studies, (and earlier theoretical ideas from, amongst others,
Konorski, 18,19 Dickinson, Balleine and their colleagues 13 have suggested that
there are really two separate motivational systems, one associated with Pavlovian motivation, as in SR, and one associated with instrumental action choice.
They further suggest, partly based on related suggestions by Berridge and his
colleagues,? that only the Pavlovian system involves dopamine. Neither the
Pavlovian nor the instrumental system maps cleanly onto the standard view
of RL, and the suggestion about dopamine would clearly Significantly damage
the RL interpretation of the involvement of this neuromodulatory system in
conditioning.
In this paper, we describe some of the key evidence supporting the difference between instrumental and Pavlovian motivation (see also Balkenius 3 and
Spier 25 ), and expand the model of RL in the brain to incorporate SR motivation and concomitant evidence on intrinsic goal sensitivity (as well as intrinsic
habits). Some of the computational properties of this new model turn out to
be rather strange - but this is a direct consequence of equivalently strange
observable behavior.
2
Theoretical and Experimental Background
Figure 1 shows a standard view of the involvement of the dopamine system in
RL. 22 ,24 Dopamine neurons in the ventral tegmental area (VTA) and substantia
nigra pars compacta (SN c) report the temporal difference (TD) error 8 (t). In the
simplest version of the theory, this is calculated as 8 (t) = r (t) + V 1T (x( t + 1? V 1T(x( t) ), where r (t) is the value of the reward at time t, x( t) is an internal
representation of the state at time t, V 1T(x( t? is the expectation of the sum
total future reward expected by the animal based on starting from that state,
following policy IT, and the transition from x( t) to x( t + 1) is occasioned by the
action a selected by the subject. In the actor-critic 6 version of the dopamine
theory, this TD error signal is put to two uses. One is adapting parameters that
underlie the actual predictions V 1T(x(t?. For this, 8(t) > 0 if the prediction
from the state at time t, V 1T(x(t?, is overly pessimistic with respect to the sum
of the actual reward, r (t), and the estimated future reward, V 1T (x( t + 1?, from
the subsequent state. The other use for 8 (t) is criticizing the action a adopted
at time t. For this, 8(t) > 0 implies that the action chosen is worth more than
the average worth of x(t), and that the overall policy IT of the subject can
therefore be improved by choosing it more often. In a Q-Iearning 31 version of
the theory, Q1T(X, a) values are learned using an analogous quantity to 8 (t), for
each pair of states x and actions a, and can directly be used to choose between
the actions to improve the policy.
Even absent an account for intrinsic habits, three key paradigms show the
incompleteness of this view of conditioning: appetitive Pavlovian-instrumental
transfer,15 intrinsic drive preference under speCific deprivation states,8 and
incentive learning, as in the control of chains of instrumental behavior. 5
The SR view of conditioning places its emphasis on motivational control of
a prepotent action. That is, the natural response associated with a stimulus
(presumably as output by an action specification mechanism) is only elicited if
A
x state
B
~muli
x
amygdala
TO prediction
accumbens~--+--'--=--l error
OFC
reward
'"
'"
action
TO predictio
error
8
r
state
stimuli
competition
stRkfu~
a
Figure 1: Actor-critic version of the standard RL model. A) Evaluator: A TD error
signal 8 to learn V 1T (x) to match the sum of future rewards r, putatively via the basolateral nuclei of the amygdala, the orbitofrontal cortex and the nucleus accumbens.
B) Instrumental controller: The TD error 8 is used to choose, and teach the choice
of, appropriate actions a to maximize future reward based on the current state and
stimuli, putatively via the dorsolateral prefrontal cortex and the dorsal striatum.
it is motivationally appropriate, according to the current goals of the animal.
The suggestion is that this is mediated by a separate motivational system. USs
have direct access to this system, and CSs have learned access. A conclusion
used to test this structure for the control of actions is that this motivational
system could be able to energize any action being executed by the animal. Appetitive Pavlovian-instrumental transfer 15 shows exactly this. Animals executing an action for an outcome under instrumental control, will perform more
quickly when a CS predictive of reward is presented, even if the CS predicts
a completely different reward from the instrumental outcome. This effect is
abolished by lesions of the shell of the nucleus accumbens,10 one of the main
targets of DA from the VTA. The standard RL model offers no account of the
speed or force of action (though one could certainly imagine various possible
extensions), and has no natural way to accommodate this finding. *
The second challenge to RL comes from experiments on the effects of changing
speCific and general needs for animals. For instance, Berridge & Schulkin8 first
gave rats sucrose and saline solutions with one of a bitter (quinine) and a sour
(citric) taste. They then artificially induced a strong physiological requirement
for salt, for the first time in the life of the animal. Presented with a choice
between the two flavors (in plain water, ie in extinction), the rats preferred to
drink the flavor associated with the salt. Furthermore, the flavor paired with
the salt was awarded positive hedonic reactions, whereas before pairing (and
if it had been paired with sucrose instead) it was treated as being aversive.
The key feature of this experiment is that this preference is evident without
the opportunity for learning. Whereas the RL system could certainly take the
physiological lack of salt as helping determine part of the state x(t), this could
only exert an effect on behavior through learning, contrary to the evidence.
The final complexity for standard RL comes from incentive learning. One
paradigm involves a sequential chain of two actions (a1 and a2) that rats had
to execute in order to get a reward. 5 The subjects were made hungry, and were
first trained to perform action a2 to get a particular reward (a Noyes pellet),
and then to perform the chain of actions a1 - a2 to get the reward. In a final
test phase, the animals were offered the chance of executing a1 and a2 in extinction, for half of them when they were still hungry; for the other half when
they were sated on their normal diet. Figure 2A shows what happens. Sated
animals perform a1 at the same rate as hungry animals, but perform a2 sig*Note that aversive Pavlovian instrumental transfer, in the form of the suppression
of appetitive instrumental responding, is the conventional method for testing aversive
Pavlovian conditioning. There is an obvious motivational explanation for this as well
as the conventional view of competition between appetitive and protective actions.
100
.~
hungry
80
ieo
s
a
~
40
E20
al
al
al
al
Figure 2: Incentive learning. A) Mean total actions al and a 2 for an animal trained
on the chain schedule al - a2 -Noyes pellets. Hungry and sated rats perform al at
the same rate, but sated animals fail to perform a 2. B) Mean total actions when sated
following prior re-exposure to the Noyes pellets when hungry ('hungry-sated') or when
sated (,sated-sated'). Animals re-exposed when sated are significantly less willing to
perform a 2. Note the change in scale between A and B. Adapted from Balieine et a/. 5
nificantly less frequently. Figure 2B shows the basic incentive learning effect.
Here, before the test, animals were given a limited number of the Noyes pellets (without the availability of the manipulanda) either when hungry or when
sated. Those who experienced them hungry ('hungry-sated') show the same
results as the 'sated' group of figure 2A; whereas those who experienced them
sated (,sated-sated') now declined to perform action al either.
This experiment makes two points about the standard RL model. First, the
action nearest to the reward (a2) is affected by the deprivation state without
additional learning. This is like the effect of specific deprivation states discussed above. Second is that a change in the willingness to execute al happens
after re-exposure to the Noyes pellets whilst sated; this learning is believed to
involve insular cortex (part of gustatory neocortex4). That re-exposure directly
affects the choice of al suggests that the instrumental act is partly determined
by an evaluation of its ultimate consequence, a conclusion that relates to a
long-standing psychological debate about the 'cognitive' evaluation of actions.
Dickinson & Balleine 13 suggest that the execution of a2 is mainly controlled
by Pavlovian contingencies, and that Pavlovian motivation is instantly sensitive to goal devaluation via satiation. At this stage in the experiment, however,
al is controlled by instrumental contingencies. By comparison with Pavlovian motivation, instrumental motivation is powerful (since it can depend on
response-outcome expectancies), but dumb (since, without re-exposure, the
animal works hard doing al when it wouldn't be interested in the food in any
case). Ultimately, after extended training,14 in the birth of a new habit, al
becomes controlled by Pavlovian contingencies too, and so becomes directly
sensitive to devaluation. t
3 New Model
These experiments suggest some major modifications to the standard RL view.
Figure 3 shows a sketch of the new model, whose key principles include
? Pavlovian motivation (figure 3A) is associated with prediction error
8(t) = r(t) + VTT(X(t + 1)) - VTT(x(t)
for long term expected future rewards VTT(X, a), given a policy IT. Adopting this makes the model account for the classical conditioning paradigms
explained by the standard RL model.
t It is not empirically clear whether actions that have become habits are completely
automatic 1 or are subject to Pavlovian motivational influences.
A
B
stimuli
X
!
CS
prior
, bias
S S ~amygdala
,"
:
CS
~
sta~e~ __>~
stimuli
X
sIs
US
8
C
prior
bias
habit
stimuli
X
CS
sIs
shell
~~.~
US
a
I
s~a~:
vrT(X)
Figure 3: Tripartite model. A) Evaluator: USs are evaluated by a hard-wired evaluation
system (HE) which is intrinsically sensitive to devaluation. USs can also be evaluated
via a plastic route, as in figure 1, but which nevertheless has prior biases. CSs undergo
Pavlovian stimulus substitution with the USs they predict, and can also be directly evaluated through the learned route. The two sources of information for VTT(x) compete,
forcing the plastic route to adjust to the hard-wired route. B) Habit system: The SR
mapping suggests an appropriate action based on the state X; the vigor of its execution is controlled by dopaminergic 8, putatively acting via the shell of the accumbens.
C) Instrumental controller: Action choice is based on advantages, which are learned,
putatively via the core of the accumbens. Prefrontal working memory is used to unfold
the consequences of chosen actions.
? ret) is determined by a devaluation-sensitive, 'hard-wired', US evaluator that
provides direct value information about motivationally inappropriate USs.
? 8(t), possibly acting through the shell of the accumbens, provides Pavlovian
motivation for pre-wired and new habits (figure 3B), as in Pavlovian instrumental transfer.
? V 1T (x(t) is determined by two competing sources: one as in the standard
model (involving the basolateral nuclei of the amygdala and the orbitofrontal
cortex (OFC),16,23 and including prior biases (sweet tasting foods are appetitive) expressed in the connections from primary taste cortex to oFC and
the amygdala; the other, which is primary, dependent largely on a stimulus
substitution20 relationship between CSs and USs, that is also devaluationdependent. The latter is important for ultimate Pavlovian control over actions; the former for phenomena such as secondary conditioning, which are
known to be devaluation independentY Figure 4A (dashed) shows the contribution of the hard-wired evaluation route, via stimulus-substitution, on the
prediction of value in classical conditioning. Here, stimulus-substitution was
based on a form of Hebbian learning with a synaptic trace, so the shorter the
CS-US interval, the greater the HE component. This translates into greater immediate sensitivity to devaluation, the main characteristic of the hard-wired
route. The plastic route via the amygdala takes responsibility for the remainder of the prediction; and the sum prediction is always correct (solid line).
? Short-term storage of predictive stimuli in prefrontal working memory is
gated9 by 8(t), so can also be devaluation dependent.
? Instrumental motivation depends on policy-based advantages (3C; Baird 2)
A 1T (x,a)
=
Q 1T(x,a) - V1T(X)
trained by the error signal
8 A (t) = 8(t) -A1T(x,a)
Over the course of policy improvement, the advantage of a sub-optimal action
becomes negative, and of an optimal action tends to O. The latter offers a
natural model of the transition from instrumental action selection to an SR
habit. Note that, in this actor-critic scheme, some aspects of advantages are
not necessary, such as the normalizing updates.
B
A
0.5
1::
bO
0[j
\
\
;s: 0.5
79
8
C
value V7T (xo)
-,-7T--,(:..::
XO,-,-,.:..:.
a ,-) -----,
1 .5 ,----"'-"-''-----''-''''-'---, ~4 ,------:.A
advantages
sum
HE
\ component
'
~~~~
,,
,
,
-0.5 A 7T(X , b) ,
?0~--1-0~~20
0
50
cs-us interval
successor
sum
0.2
iteration
100
o
'HE
,
-reprn ---
1 :-
,
~~~
,
??. plastic
component -0.2
50
iteration
101
0~--50 ---'
1 00
iteration
Figure 4: A) Role of the hard-wired route (dashed line), via stimulus-substitution, in
predicting future reward (r = 1) as a function of the CS-US interval. The solid line shows
that the net prediction is always correct. B) Advantages of useful (a) and worthless (b)
actions at the start state Xo. C) Evolution of the value of Xo over learning. The solid line
shows the mean value; the dashed line the hard-wired component, providing immediate devaluation sensitivity. 0) Construction of A7T (xo, a) via a successor representation
component l l (dashed) and a conventionally learned component (dotted). The former
is sensitive to re-exposure devaluation, as in figure 2B. B-D) Action a produces reward
r = 1 with probability 0.9 after 3 timesteps; curves are averages over 2000 runs.
Figure 4B;C show two aspects of instrumental conditioning. Two actions compete at state xo, one, a, with a small cost and a large future payoff; the other,
b, with no cost and no payoff. Figure 4B shows the development of the advantages of these actions over learning. Action a starts looking worse, because
it has a greater immediate cost; its advantage increases as the worth of a
grows greater than the mean value of xo, and then goes to 0 (the birth of the
habit) as the subject learns to choose it every time. Figure 4C shows the value
component of state Xo. This comes to be responsible for the entire prediction
(as A1T(XO, a) ~ 0). As in figure 4A, there is a hard-wired component to this
value which would result in the immediate decrement of response evident in
figure 2A.
? On-line action choice is dependent on 8A (t) as in learned klinokinesis.21 Incentive learning in chains suggests that the representation underlying the advantage of an action includes information about its future consequences, either through an explicit model,27,29 a successor representation,ll or perhaps
a form of f3-model. 26 One way of arranging this would use a VTE-like 30 mechanism for proposing actions (perhaps using working memory in prefrontal
cortex), in order to test their advantages. Figure 4D shows the consequence of
using a learned successor representation underlying the advantage A1T(XO, a)
shown in figure 4B. The dashed line shows the component of A1T(XO, a) dependent on a learned successor representation, and the prior bias about the
value of the reward, and which is therefore sensitive to re-exposure (when
the value accorded to the reward is decreased); the dotted line shows the remaining component of A 1T(XO, a), learned in the standard way. Re-exposure
sensitivity (ie incentive learning) will exist over roughly iterations 25 - 75 .
? SR models also force consideration of the repertoire of possible actions or responses available at a given state (figure 3B;C). We assume that both corticocortical and cortico-(dorsal) striatal plasticity sculpt this collection, using
8 A (t) directly, and maybe also correlational learning rules.
The details of the model are not experimentally fully determined, although
its general scheme is based quite straightforwardly from the experimental evidence referred to (and many other experiments), and by consistency with the
activity of dopamine cells (recordings of which have so far used only a single
motivational state).
4 Discussion
Experiments pose a critical challenge to our understanding of the psychological and neural implementation of reinforcement learning, 12,13 suggesting the
importance of two different sorts of motivation in controlling behavior. With
both empirical and theoretical bases, we have put these two aspects together
through the medium of advantages. The most critical addition is a hard-wired,
stimulus-substitution sensitive, route for the evaluation of stimuli and states,
which competes with a plastic route through the amygdala and the oFC. This
hard-wired route has the property of intrinsic sensitivity to various sorts of
devaluation, and this leads to motivationally appropriate behavior. The computational basis of the new aspects of the model focus on motivational control
of SR links (via VTT ), to add to motivational control of instrumental actions (via
ATT). We also showed the potential decomposition of the advantages into a
component based on the successor representation and therefore sensitive to
re-exposure as in incentive learning, and a standard, learned, component.
The model is obviously incomplete, and requires testing in richer environments. In particular, we have yet to explore how habits get created from actions as the maximal advantage goes to o.
Acknowledgements
I am very grateful to Christian Balkenius, Bernard Balleine, Tony Dickinson,
Sham Kakade, Emmet Spier and Angela Yu for discussions. Funding was from
the Gatsby Charitable Foundation.
References
[1] Adams, CD (1982) Variations in the sensitivity of instrumental responding
to reinforcer devaluation. QJEP 34B:77-98.
[2] Baird, LC (1993) Advantage Updating. Technical report WL-TR-93-1146,
Wright-Patterson Air Force Base.
[3] Balkenius, C (1995) Natural Intelligence in Artificial Creatures. PhD Thesis,
Department of Cognitive Science, Lund University, Sweden.
[4] Balleine, BW & Dickinson, A (1998) Goal-directed instrumental action:
Contingency and incentive learning and their cortical substrates. Neuropharmacology 37:407-419.
[5] Balleine, BW, Garner, C, Gonzalez, F & Dickinson, A (1995) Motivational
control of heterogeneous instrumental chains. Journal of Experimental
Psychology: Animal Behavior Processes 21:203-217.
[6] Barto, AG, Sutton, RS & Anderson, CW (1983) Neuronlike elements that
can solve difficult learning problems. IEEE SMC 13:834-846.
[7] Berridge, KC (2000) Reward learning: Reinforcement, incentives, and expectations. In DL Medin, editor, The Psychology of Learning and Motivation
40:223-278.
[8] Berridge, KC & Schulkin, J (1989) Palatability shift of a salt-associated incentive during sodium depletion. Quarterly Journal of Experimental Psychology: Comparative & Physiological Psychology 41:121-138.
[9] Braver, TS, Barch, DM & Cohen, JD (1999) Cognition and control in
schizophrenia: A computational model of dopamine and prefrontal function. Biological Psychiatry 46:312-328.
[10] Corbit, LH, Muir, JL & Balleine, BW (200l) The role of the nucleus accumbens in instrumental conditioning: Evidence of a functional dissociation
between accumbens core and shell. Journal of Neuroscience 21:3251-3260.
[11] Dayan, P (1993) Improving generalisation for temporal difference learning:
The successor representation. Neural Computation 5:6l3-624.
[12] Dickinson, A & Balleine, B (1994) Motivational control of goal-directed action. Animal Learning & Behavior 22:1-18.
[l3] Dickinson, A & Balleine, B (200l) The role of learning in motivation. In CR
Gallistel, editor, Learning, Motivation and Emotion, Volume 3 of Steven's
Handbook of Experimental Psychology, Third Edition. New York, NY: Wiley.
[14] Dickinson, A, Balleine, B, Watt, A, Gonzalez, F & Boakes, RA (1995) Motivational control after extended instrumental training. Animal Learning &
Behavior 23:197-206.
[15] Estes, WK (1943). Discriminative conditioning. I. A discriminative property
of conditioned anticipation. JEP 32:150-155.
[16] Holland, PC & Gallagher, M (1999) Amygdala circuitry in attentional and
representational processes. Trends in Cognitive Sciences 3:65-73.
[17] Holland, PC & Rescorla, RA (1975) The effect of two ways of devaluing
the unconditioned stimulus after first- and second-order appetitive conditioning. Journal of Experimental Psychology: Animal Behavior Processes
1:355-363.
[18] Konorski, J (1948) Conditioned Reflexes and Neuron Organization. Cam-
bridge, England: Cambridge University Press.
[19] Konorski, J (1967) Integrative Activity of the Brain: An Interdisciplinary
Approach. Chicago, 11: University of Chicago Press.
[20] Mackintosh, NJ (1974) The Psychology of Animal Learning. New York, NY:
Academic Press.
TJ (1995) Bee foraging in uncertain environments using predictive hebbian learning. Nature
[21] Montague, PR, Dayan, P, Person, C & Sejnowski
377:725-728.
[22] Montague, PR, Dayan, P & Sejnowski,
TJ (1996) A framework for mesencephalic dopamine systems based on predictive Hebbian learning. Journal
of Neuroscience 16: 1936-1947.
[23] Schoenbaum, G, Chiba, AA & Gallagher, M (1999) Neural encoding in or-
[24]
[25]
[26]
[27]
bitofrontal cortex and basolateral amygdala during olfactory discrimination learning. Journal of Neuroscience 19:1876-1884.
Schultz, W, Dayan, P & Montague, PR (1997) A neural substrate of prediction and reward. Science 275:1593-1599.
Spier, E (1997) From Reactive Behaviour to Adaptive Behaviour. PhD Thesis, Balliol College, Oxford.
Sutton, RS (1995) TD models: modeling the world at a mixture of time
scales. In A Prieditis & S Russell, editors, Proceedings of the Twelfth International Conference on Machine Learning. San Francisco, CA: Morgan
Kaufmann, 531-539.
Sutton, RS & Barto, AG (1981) An adaptive network that constructs and
uses an internal model of its world. Cognition and Brain Theory 4:217-
246.
[28] Sutton, RS & Barto, AG (1998) Reinforcement Learning. Cambridge, MA:
MIT Press.
[29] Sutton, RS & Pinette, B (1985) The learning of world models by connectionist networks. Proceedings of the Seventh Annual Conference of the Cognitive Science Society. Irvine, CA: Lawrence Erlbaum, 54-64.
[30] Tolman, EC (1938) The determiners of behavior at a choice point. Psychological Review 45:1-41.
[31] Watkins, CJCH (1989) Learning from Delayed Rewards. PhD Thesis, Uni-
versity of Cambridge, Cambridge, UK.
| 2048 |@word nificantly:1 version:4 instrumental:29 extinction:2 twelfth:1 cleanly:1 willing:1 r:5 integrative:1 decomposition:1 paid:1 tr:1 solid:3 accommodate:1 substitution:5 att:1 reaction:1 current:2 si:2 yet:1 subsequent:1 chicago:2 plasticity:1 christian:1 update:1 discrimination:1 half:2 selected:1 intelligence:1 core:2 short:1 provides:2 putatively:4 preference:2 evaluator:3 direct:3 become:1 gallistel:1 pairing:1 olfactory:1 balleine:10 ra:2 expected:2 vtt:5 roughly:1 nor:1 frequently:1 behavior:10 brain:3 v1t:1 preparatory:2 td:5 food:4 actual:2 versity:1 inappropriate:1 becomes:3 motivational:14 underlying:4 v7t:1 competes:1 medium:1 null:1 what:1 monkey:1 accumbens:8 proposing:1 whilst:1 ret:1 finding:1 ag:3 sour:1 nj:1 cjch:1 temporal:2 every:1 act:1 shed:1 iearning:1 exactly:1 uk:2 control:11 unit:1 underlie:1 positive:1 before:2 tends:1 consequence:6 striatum:1 sutton:5 encoding:1 insular:1 oxford:1 might:1 exert:2 emphasis:1 tasting:1 suggests:4 balkenius:3 limited:1 smc:1 medin:1 directed:2 responsible:1 testing:2 jep:1 habit:12 substantia:1 unfold:1 area:2 empirical:1 significantly:2 adapting:1 pre:1 anticipation:1 suggest:3 get:4 onto:1 selection:1 put:2 storage:1 influence:1 conventional:2 map:1 straightforward:1 attention:1 go:3 starting:1 exposure:8 hunger:1 thirst:1 rule:1 his:1 satiation:1 traditionally:1 variation:1 analogous:1 arranging:1 cs:4 target:1 imagine:1 construction:1 controlling:1 dickinson:9 substrate:2 us:9 sig:1 element:1 trend:1 particularly:2 corticocortical:1 updating:1 predicts:1 role:3 steven:1 russell:1 contemporary:1 substantial:2 environment:3 q1t:1 complexity:1 reward:22 cam:1 ultimately:1 trained:3 depend:1 grateful:1 exposed:1 predictive:6 patterson:1 completely:2 basis:1 montague:3 various:2 describe:2 london:1 effective:1 sejnowski:2 artificial:1 choosing:1 outcome:3 birth:2 sculpt:1 quite:2 whose:1 richer:1 solve:1 unconditioned:2 final:2 obviously:1 advantage:15 motivationally:3 net:1 ucl:1 reinforcer:1 rescorla:1 maximal:1 remainder:1 representational:1 competition:6 requirement:1 wired:11 produce:1 adam:1 executing:4 comparative:1 ac:1 pose:1 nearest:1 strong:1 c:8 involves:2 implies:1 come:3 correct:2 successor:7 premise:1 behaviour:2 really:1 repertoire:1 pessimistic:1 biological:1 extension:1 quinine:1 helping:1 wright:1 normal:1 presumably:1 lawrence:1 mapping:1 predict:1 cognition:2 circuitry:1 major:1 ventral:1 a2:8 determiner:1 pinette:1 sensitive:9 bridge:1 ofc:4 wl:1 pellet:6 mit:1 clearly:1 tegmental:1 always:2 rather:3 cr:1 tolman:1 eating:1 barto:3 basolateral:3 focus:1 improvement:1 mainly:2 contrast:1 psychiatry:1 suppression:1 sense:3 am:1 dayan:6 dependent:4 entire:1 kc:2 expand:1 interested:1 issue:1 overall:1 development:1 animal:20 emotion:1 construct:1 f3:1 vta:2 yu:1 future:10 report:2 stimulus:19 others:1 connectionist:1 sweet:1 sta:1 manipulated:1 delayed:1 phase:1 bw:3 saline:1 neuronlike:1 organization:1 evaluation:5 certainly:2 adjust:1 appetitive:6 mixture:1 light:1 pc:2 tj:2 dumb:1 occasioned:1 chain:6 necessary:1 lh:1 shorter:1 sweden:1 incomplete:2 re:9 theoretical:3 uncertain:1 psychological:5 instance:1 earlier:1 modeling:1 ar:1 devaluation:11 queen:1 cost:3 successful:1 seventh:1 erlbaum:1 too:1 motivating:1 straightforwardly:1 foraging:1 person:1 international:1 sensitivity:6 ie:2 interdisciplinary:1 standing:1 rewarding:1 together:1 quickly:1 thesis:3 choose:4 prefrontal:5 possibly:1 worse:1 berridge:4 cognitive:4 a1t:4 actively:1 account:3 protective:1 suggesting:1 potential:1 wk:1 includes:2 availability:1 baird:2 depends:1 view:8 responsibility:1 doing:1 start:3 sort:2 elicited:1 consummatory:1 contribution:1 square:1 air:1 kaufmann:1 characteristic:2 who:2 largely:1 dissociation:1 yes:1 plastic:5 garner:1 worth:5 drive:2 synaptic:1 colleague:2 obvious:1 dm:1 associated:7 irvine:1 intrinsically:2 murky:1 schedule:1 sophisticated:1 diet:1 response:6 improved:1 execute:2 though:1 evaluated:3 anderson:1 furthermore:1 stage:1 sketch:1 working:3 lack:1 willingness:1 perhaps:2 grows:1 effect:7 former:2 evolution:1 eg:2 ll:1 during:4 rat:4 muir:1 evident:2 vte:1 hungry:11 consideration:1 funding:1 functional:1 rl:18 empirically:1 salt:5 conditioning:17 attached:1 cohen:1 jl:1 discussed:1 interpretation:1 he:4 volume:1 neuropharmacology:1 cambridge:4 neuromodulatory:2 automatic:1 consistency:1 had:2 l3:2 specification:1 actor:3 cortex:7 access:2 base:2 add:1 showed:1 involvement:3 awarded:1 forcing:1 route:11 life:1 morgan:1 additional:1 greater:4 nigra:1 determine:1 paradigm:3 maximize:1 signal:3 dashed:5 relates:1 multiple:3 sham:1 hebbian:3 technical:1 match:1 england:2 academic:1 offer:2 believed:1 long:2 concerning:1 schizophrenia:1 paired:2 a1:4 controlled:4 prediction:11 involving:1 basic:1 controller:2 circumstance:1 expectation:2 heterogeneous:1 dopamine:9 iteration:4 adopting:1 cell:2 background:1 whereas:3 addition:1 interval:3 decreased:1 source:2 appropriately:1 extra:1 sr:12 subject:6 induced:1 undergo:1 recording:1 contrary:1 seem:1 conception:1 concerned:1 affect:1 psychology:8 gave:1 timesteps:1 approaching:1 competing:1 idea:2 prieditis:1 translates:1 absent:1 shift:1 whether:3 motivated:1 ultimate:2 peter:1 york:2 action:54 prepotent:1 useful:1 clear:1 involve:1 maybe:1 simplest:1 exist:1 dotted:2 estimated:1 neuroscience:4 sated:18 conceived:1 overly:1 instantly:1 incentive:10 affected:1 group:1 key:4 nevertheless:1 changing:1 neither:1 sum:7 compete:2 run:1 powerful:1 muli:1 place:1 strange:2 gonzalez:2 incompleteness:1 orbitofrontal:2 dorsolateral:1 drink:1 konorski:3 annual:1 activity:3 adapted:1 aspect:4 speed:1 pavlovian:19 dopaminergic:1 department:1 according:1 watt:1 kakade:1 modification:1 happens:2 sucrose:2 explained:1 pr:3 xo:12 depletion:1 turn:1 mechanism:2 fail:1 adopted:1 available:1 accorded:1 quarterly:1 appropriate:4 braver:1 jd:1 responding:2 remaining:1 include:1 worthless:1 tony:1 angela:1 opportunity:1 vigor:1 estes:1 build:1 classical:3 society:1 question:1 quantity:1 damage:1 primary:2 amongst:1 cw:1 link:2 separate:2 attentional:1 trivial:1 water:1 relationship:2 manipulanda:1 concomitant:1 providing:1 equivalently:1 difficult:1 executed:1 striatal:1 teach:1 debate:1 trace:1 negative:1 aversive:3 implementation:1 policy:6 perform:9 neuron:2 t:1 supporting:1 immediate:4 payoff:2 neurobiology:1 extended:2 looking:1 pair:1 connection:1 learned:10 able:1 suggested:1 corbit:1 lund:1 challenge:2 including:1 memory:3 explanation:1 critical:2 natural:5 force:3 treated:1 predicting:1 advanced:1 sodium:1 scheme:2 improve:1 declined:1 created:1 conventionally:1 mediated:1 sn:1 faced:1 prior:6 understanding:1 taste:2 acknowledgement:1 bee:1 review:1 fully:1 par:1 bear:1 abolished:1 suggestion:3 foundation:1 nucleus:5 contingency:4 offered:1 experi:1 principle:1 editor:3 charitable:1 critic:3 cd:1 course:1 bias:5 cortico:1 hedonic:1 curve:1 calculated:1 amygdala:9 transition:2 plain:1 cortical:1 chiba:1 world:3 made:1 reinforcement:6 expectancy:1 wouldn:1 collection:1 historical:1 far:1 schultz:1 adaptive:2 san:1 ec:1 compacta:1 mesencephalic:1 observable:1 uni:1 preferred:1 handbook:1 francisco:1 discriminative:2 gustatory:1 reviewed:1 learn:1 transfer:4 nature:1 ca:2 improving:1 interact:1 punishing:1 artificially:1 e20:1 da:1 main:3 decrement:1 motivation:20 edition:1 nothing:1 lesion:1 referred:1 creature:1 gatsby:3 ny:2 wiley:1 lc:1 experienced:2 sub:1 explicit:1 watkins:1 third:1 deprivation:3 learns:1 specific:3 physiological:3 consequent:1 ments:1 evidence:7 normalizing:1 intrinsic:5 dl:1 sequential:1 importance:1 barch:1 phd:3 gallagher:2 execution:2 conditioned:3 flavor:3 explore:1 unexpected:1 expressed:1 bo:1 holland:2 reflex:1 aa:1 chance:1 ma:1 shell:5 goal:9 careful:1 change:2 hard:11 included:1 determined:4 experimentally:1 generalisation:1 acting:2 correlational:1 total:3 secondary:2 partly:2 experimental:6 vrt:1 bernard:1 college:1 internal:2 tripartite:1 latter:2 dorsal:2 reactive:1 incorporate:1 phenomenon:1 mackintosh:1 |
1,150 | 2,049 | The Noisy Euclidean Traveling Salesman
Problem and Learning
Mikio L. Braun, Joachim M. Buhmann
braunm@cs.uni-bonn.de, jb@cs.uni-bonn.de
Institute for Computer Science, Dept. III,
University of Bonn
R6merstraBe 164, 53117 Bonn, Germany
Abstract
We consider noisy Euclidean traveling salesman problems in the
plane, which are random combinatorial problems with underlying
structure. Gibbs sampling is used to compute average trajectories,
which estimate the underlying structure common to all instances.
This procedure requires identifying the exact relationship between
permutations and tours. In a learning setting, the average trajectory is used as a model to construct solutions to new instances
sampled from the same source. Experimental results show that the
average trajectory can in fact estimate the underlying structure and
that overfitting effects occur if the trajectory adapts too closely to
a single instance.
1
Introduction
The approach in combinatorial optimization is traditionally single-instance and
worst-case-oriented. An algorithm is tested against the worst possible single instance. In reality, algorithms are often applied to a large number of related instances,
the average-case performance being the measurement of interest. This constitutes
a completely different problem: given a set of similar instances, construct solutions
which are good on average. We call this kind of problem multiple-instances and
average-case-oriented. Since the instances share some information, it might be expected that this problem is simpler than solving all instances separately, even for
NP-hard problems.
We will study the following example of a multiple-instance average-case problem,
which is built from the Euclidean travelings salesman problem (TSP) in the plane.
Consider a salesman who makes weekly trips. At the beginning of each week, the
salesman has a new set of appointments for the week, for which he has to plan
the shortest round-trip. The location of the appointments will not be completely
random, because there are certain areas which have a higher probability of containing an appointment, for example cities or business districts within cities. Instead
of solving the planning problem each week from scratch, a clever salesman will try
to exploit the underlying density and have a rough trip pre-planned, which he will
only adapt from week to week.
An idealizing formulization of this setting is as follows. Fix the number of appointments n E N. Let Xl, ... , Xn E ]R2 and (J E 114. Then, the locations of the
appointments for each week are given as samples from the normally distributed
random vectors (i E {1, ... , n})
(1)
The random vector (Xl, ... ,Xn ) will be called a scenario, sampled appointment
locations (sampled) instance. The task consists in finding the permutation 7r E Sn
which minimizes 7r I-t d7r (n)1f(l) + L~:ll d1f (i)1f(iH) , where dij := IIXi - Xj112' and
Sn being the set of all bijective functions on the set {1, ... , n}. Typical examples
are depicted in figure l(a)- (c).
It turns out that the multiple-instance average-case setting is related to learning
theory, especially to the theory of cost-based unsupervised learning. This relationship becomes clear if one considers the performance measure of interest. The algorithm takes a set of instances It, ... ,In as input and outputs a number of solutions
Sl,???, Sn? It is then measured by the average performance (l/n) L~=l C(Sk, h),
where C(s , I) denotes the cost of solution s on instance I. We now modify the
performance measure as follows. Given a finite number of instances It, ... ,In, the
algorithm has to construct a solution s' on a newly sampled instance I'. The performance is then measured by the expected cost E (C (s' ,I')). This can be interpreted
as a learning task. The instances 11 , ... ,In are then the training data, E(C(s', I'))
is the analogue of the expected risk or cost, and the set of solutions is identified
with the hypothesis class in learning theory.
In this paper, the setting presented in the previous paragraph is studied with the
further restriction that only one training instance is present. From this training instance, an average solution is constructed, represented by a closed curve in the plane.
This average trajectory is supposed to capture the essential structure of the underlying probability density, similar to the centroids in K-means clustering. Then, the
average trajectory is used as a seed for a simple heuristic which constructs solutions
on newly drawn instances. The average trajectories are computed by geometrically
averaging tours which are drawn by a Gibbs sampler at finite temperature. This
will be discussed in detail in sections 2 and 3. It turns out that the temperature
acts as a scale or smoothing parameter. A few comments concerning the selection
of this parameter are given in section 6.
The technical content of our approach is reminiscent of the "elastic net" -approaches
of Durbin and Willshaw (see [2], [5]) , but differs in many points. It is based on
a completely different algorithmic approach using Gibbs sampling and a general
technique for averaging tours. Our algorithm has polynomial complexity per Monte
Carlo step and convergence is guaranteed by the usual bounds for Markov Chain
Monte Carlo simulation and Gibbs sampling. Furthermore, the goal is not to provide
a heuristic for computing the best solution, but to extract the relevant statistics of
the Gibbs distribution at finite temperatures to generate the average trajectory,
which will be used to compute solutions on future instances.
2
The Metropolis algorithm
The Metropolis algorithm is a well-known algorithm which simulates a homogeneous
Markov chain whose distribution converges to the Gibbs distribution. We assume
that the reader is familiar with the concepts, we give here only a brief sketch of the
relevant results and refer to [6], [3] for further details.
Let M be a finite set and f: M -+ lit The Gibbs distribution at temperature T E Il4
is given by (m E M)
9T(m) :=
exp( - f(m)/T~
.
exp( - f(m )/T)
Lm/EM
(2)
The Metropolis algorithm works as follows. We start with any element m E M and
set Xl +- m. For i ~ 2, apply a random local update m':= ?(Xi). Then set
with probability min {I, exp( -(f(Xi) - f(m'))/T)}
else
(3)
This scheme converges to the Gibbs distribution if certain conditions on ? are met.
Furthermore, a L2-law of large numbers holds: For h: M --t ]R, ~ L:~=l h(X k ) --t
L:mEM gT(m)h(m) in L2. For TSP, M = Sn and ? is the Lin-Kernighan two-change
[4], which consists in choosing two indexes i, j at random and reversing the path
between the appointments i and j. Note that the Lin-Kernighan two-change and
its generalizations for neighborhood search are powerful heuristic in itself.
3
Averaging Tours
Our goal is to compute the average trajectory, which should grasp the underlying
structure common to all instances, with respect to the Gibbs measure at non-zero
temperature T . The Metropolis algorithm produces a sequence of permutations
7rl, 7r2, ... with P{ 7rn = .} --t gT(.) for n --t 00. Since permutations cannot be
added, we cannot simply compute the empirical means of 7rn . Instead, we map
permutations to their corresponding trajectories.
Definition 1 (trajectory) The trajectory of 7r E Sn given n points Xl, ... ,X n is a
mapping r( 7r): {I, ... , n} --t ]R2 defined by r( 7r) (i) := X1C(i). The set of all trajectories (for all sets of n points) is denoted by Tn (this is the set of all mappings
T {I , ... , n} --t ]R2 ).
Addition of trajectories and multiplication with scalars can be defined pointwise.
Then it is technically possible to compute L:~=l r(7rk). Unfortunately, this does
not yield the desired results , since the relation between permutations and tours is
not one-to-one. For example, the permutation obtained by starting the tour at a
different city still corresponds to the same tour . We therefore need to define the
addition of trajectories in a way which is independent of the choice of permutation
(and therefore trajectory) to represent the tour. We will study the relationship
between tours and permutations first in some detail, since we feel that the concepts
introduced here might be generally useful for analyzing combinatorial optimization
problems.
t
Definition 2 (tour and length of a tour) Let G = (V, E) be a complete (undirected)
graph with V = {I, ... ,n} and E = (~). A subset tEE is called a tour iff It I = n,
for every v E V, there exist exactly two el, e2 E t such that v E el and v E e2,
and (V, t) is connected. Given a symmetric matrix (d ij ) of distances, the length of
a tour t is defined by C(t) := L:{i,j} Et d ij .
The tour corresponding to a permutation 7r E Sn is given by
n-l
t(7r) :={ {7r(I), 7r(n)}} U
{{7r(i) ,7r(i + I)}}.
U
(4)
i=l
If t(7r) = t for a permutation 7r and a tour t, we say that 7r represents t. We
call two permutations 7r, 7r' equivalent, if they represent the same tour and write
7r ,...., 7r'. Let [7r] denote the equivalence class of 7r as usual. Note that the length of
a permutation is fully determined by its equivalence class. Therefore, ,...., describes
the intrinsic symmetries of the TSP formulated as an optimization problem on Sn ,
denoted by TSP(Sn).
We have to define the addition EB of trajectories such that the sum is independent of
the representation. This means that for two tours h, t2 such that h is represented
by 'lf1, 'If~ and t2 by 'lf2, 'If~ it holds that f('lf1) EB f('lf2) ~ f('lfD EB f('If~). The idea
will be to normalize both summands before addition. We will first study the exact
representation symmetry of TSP(Sn) '
The TSP(Sn) symmetry group Algebraically speaking, Sn is a group with
concatenation of functions as multiplication, so we can characterize the equivalence
classes of ~ by studying the set of operations on a permutation which map to the
same equivalent class. We define a group action of Sn on itself by right translation
('If, 9 E Sn):
" . " : Sn x Sn -+ Sn,
g. 'If:= 'lfg- 1.
(5)
Note that any permutation in Sn can be mapped to another by an appropriate
group action (namely 'If -+ 'If' by ('If,-l'lf) . 'If.), such that the group action of Sn on
itself suffices to study the equivalence classes of ~.
For certain 9 E Sn, it holds that t(g? 'If) = t('If). We want to determine the maximal
set H t of elements which keeps t invariant. It even holds that H t is a subgroup
of Sn: The identity is trivially in H t . Let g, h be t-invariant , then t((gh- 1) . 'If) =
t(g ?(h- 1 . 'If)) = t(h- 1 . 'If) = t(h ?(h- 1 . 'If) = t( 'If). H t will be called the symmetry
group of TSP(Sn) and it follows that ['If] = H t ? 'If :={h ? 'If I hE Hd.
The shift u and reversal (2 are defined by (i E {I, ... , n} )
(.).__ {i + iz < n,n
u z.
1
1
.
=
,
(2(i) :=n
+ 1- i,
(6)
and set H :=((2, u), the group generated by u and (2. It holds that (this result is an
easy consequence of (2(2 = id{l ,... ,n}, (2U = u- 1(2 and un = id{l ,... ,n})
H = {uk IkE {I, ... , n}} U {(2u k IkE {l, ... ,n}}.
(7)
The fundamental result is
Theorem 1 Let t be the mapping which sends permutations to tours as defined in
(4). Then, H t = H , where H t is the set of all t-invariant permutations and H is
defined in (7).
Proof: It is obvious that H ~ H t . Now, let h- 1 E H t . We are going to prove
that t-invariant permutations are completely defined by their values on 1 and 2.
Let hE H t and k:= h(l) . Then, h(2) = u(k) or h(2) = u - 1(k), because otherwise,
h would give rise to a link {{'If(h(1),'If(h(2?}} 1. t('If) . For the same reason, h(3)
must be mapped to u ?2(k). Since h must be bijective, h(3) =I- h(l) , so that the sign
of the exponent must be the same as for h(2). In general, h(i) = u?(i- 1l(k). Now
note that for i,k E {l , ... ,n } , u i(k) = uk(i) and therefore,
h=
{
u k- 1
(2un-k
if h(i) = ui-1 (k)
ifh(i)=u- i+1(k)'
D
Adding trajectories We can now define equivalence for trajectories. First define
a group action of Sn on Tn analogously to (5): the action of h E H t on "( E Tn is
given by h ? "( := "( 0 h- 1 . Furthermore, we say that "( ~ 1} , if H t ? "( = H t ?1}.
Our approach is motivated geometrically. We measure distances between trajectories
as follows. Let d: ]R2 x ]R2 -+ Il4 be a metric. Then define h, 1} E Tn)
dh,1}):=
2::=1 dh(k),1}(k).
(8)
Before adding two trajectories we will first choose equivalent representations "(', 1}'
which minimize d( "(' , 1}'). Because of the results presented so far, searching through
all equivalent trajectories is computationally tractable. Note that for h E H t , it
holds that d( h . ,,(, h . rJ) = db, rJ) as h only reorders the summands. It follows that
it suffices to change the representations only for one argument, since d(h? ,,(, i? rJ) =
db, h - 1 i? rJ)? So the time complexity of one addition reduces to 2n computation of
distances which involve n subtractions each.
The normalizing action is defined by
b, rJ
E Tn)
(9)
n , 1J := argmin d( ,,(, n . rJ)?
n EH t
Assuming that the normalizing action is unique 1 , we can prove
Theorem 2 Let ,,(, rJ be two trajectories, and n , 1J the unique normalizing action as
defined in (9). Then, the operation
"( EB rJ := "(
+ n , 1J . rJ
(10)
is representation invariant.
Proof: Let "(I = g. ,,(, rJl = h? rJ for g, h E H t . We claim that n ,I1J1
The normalizing action is defined by
n,I1J1
= gn' 1Jh-1.
= argmin db /, n l . rJl) = argmin d(g . ,,(, nih? rJ) = argmin db , g-l n lh? rJ),
n l EHt
n l EH t
n l EH t
(11)
by inserting g-l parallelly before both arguments in the last step. Since the normalizing action is unique, it follows that for the n l realizing the minimum it holds
that g-ln l h = n , 1J and therefore n l = n , I1J1 = gn' 1Jh-1. Now, consider the sum
which proves the representation independence.
0
The sum of more than two trajectories can be defined by normalizing everything
with respect to the first summand, so that empirical sums EB~=l f(?ri) are now
well-defined.
t
4
Inferring Solutions on New Instances
We transfer a trajectory to a new set of appointments Xl, .. . ,X n by computing the
relaxed tour using the following finite-horizon adaption technique:
First of all, passing times ti for all appointments are computed. We extend the
domain of a trajectory "( from {I, ... , n} to the interval [1, n + 1) by linear interpolation. Then we define ti such that "((ti) is the earliest point with minimal distance
between appointment Xi and the trajectory. The passing times can be calculated
easily by simple geometric considerations. The permutation which sorts (ti)~l is
the relaxed solution of"( to (Xi) .
In a post-processing step, self-intersections are removed first. Then, segments of
length w are optimized by exhaustive search. Let ?r be the relaxed solution. The
path from ?rei) to ?r(i + w + 2) (index addition is modulo n) is replaced by the
best alternative through the appointments ?r(i + 1), ... , ?r(i + w + 1). Iterate for all
i E {I , . . . , n} until there is no further improvement. Since this procedure has time
complexity w!n, it can only be done efficiently for small w.
lOtherwise, perturb the locations of the appointments by infinitesimal changes.
5
Experiments
For experiments, we used the following set-up: We took the 11.111-norm to determine
the normalizing action. Typical sample-sizes for the Markov chain Monte Carlo
integration were 1000 with 100 steps in between to decouple consecutive samples.
Scenarios were modeled after eq. (1), where the Xi were chosen to form simple
geometric shapes.
Average trajectories for different temperatures are plotted in figures l(a)- (c). As
the temperature decreases, the average trajectory converges to the trajectory of a
single locally optimal tour. The graphs demonstrate that the temperature T acts
as a smoothing parameter.
To estimate the expected risk of an average trajectory, the post-processed relaxed
(PPR) solutions were averaged over 100 new instances (see figure l(d)-(g)) in order
to estimate the expected costs. The costs of the best solutions are good approximations, within 5% of the average minimum as determined by careful simulated
annealing. An interesting effect occurs: the expected costs have their minimum at
non-zero temperature. The corresponding trajectories are plotted in figure l(e),(f).
They recover the structure of the scenario. In other words, average trajectories computed at temperatures which are too low, start to overfit to noise present only in
the instance for which they were computed. So computation of the global optimum
of a noisy combinatorial optimization problem might not be the right strategy, because the solutions might not reflect the underlying structure. Averaging over many
suboptimal solutions provides much better statistics.
6
Selection of the Temperature
The question remains how to select the optimal temperature. This problem is essentially the same as determining the correct model complexity in learning theory,
and therefore no fully satisfying answer is readily available. The problem is nevertheless suited for the application of the heuristic provided by the empirical risk
approximation (ERA) framework [1], which will be briefly sketched here.
The main idea of ERA is to coarse-grain the set of hypotheses M by treating
hypotheses as equivalent which are only slightly different. Hypotheses whose ?1
mutual distance (defined in a similar fashion as (8)) is smaller than the parameter
"( E Il4 are considered statistically equivalent. Selecting a subset of solutions such
that ?l -spheres of radius "( cover M results in the coarse-grained hypothesis set
M,. VC-type large deviation bounds depending on the size of the coarse-grained
hypothesis class can now be derived:
p{ C2 (m"! ) -
min C2 (m)
mEM
> 2c} :::; 21M"! 1 sup exp ( mEM.,
n(c am
+c
(
"()2
c - "(
)
)'
(13)
am depending on the distribution. The bound weighs two competing effects. On
the one hand, increasing "( introduces a systematic bias in the estimation. On the
other hand, decreasing "( increases the cardinality of the hypothesis class. Given a
confidence J > 0, the probability of being worse than c > 0 on a second instance and
"( are linked. So an optimal coarsening "( can be determined. ERA then advocates
to either sample from the ,,(-sphere around the empirical minimizer or average over
these solutions.
Now it is well known, that the Gibbs sampler is concentrated on solutions whose
costs are below a certain threshold. Therefore, the ERA is suited for our approach.
In the relating equation the log cardinality of the approximation set occurs, which
is usually interpreted as micro canonical entropy. This relates back to statistical
physics, the starting point of our whole approach. Now interpreting "( as energy,
we can compute the stop temperature from the optimal T Using the well-known
relation from statistical physics ~ee:t:~:: = T - 1 , we can derive a lower bound on
the optimal temperature depending on variance estimates of the specific scenario
given.
7
Conclusion
In reality, optimization algorithms are often applied to many similar instances. We
pointed out that this can be interpreted as a learning problem. The underlying
structure of similar instances should be extracted and used in order reduce the
computational complexity for computing solutions to related instances.
Starting with the noisy Euclidean TSP, the construction of average tours is studied
in this paper, which involves determining the exact relationship between permutation and tours, and identifying the intrinsic symmetries of the TSP. We hope that
this technique might prove to be useful for other applications in the field of averaging over solutions of combinatorial problems. The average trajectories are able to
capture the underlying structure common to all instances. A heuristic for constructing solutions on new instances is proposed. An empirical study of these procedures
is conducted with results satisfying our expectations.
In terms of learning theory, overfitting effects can be observed. This phenomenon
points at a deep connection between combinatorial optimization problems with noise
and learning theory, which might be bidirectional. On the one hand, we believe that
noisy (in contrast to random) combinatorial optimization problems are dominant
in reality. Robust algorithms could be built by first estimating the undistorted
structure and then using this structure as a guideline for constructing solutions
for single instances. On the other hand , hardness of efficient optimization might be
linked to the inability to extract meaningful structure. These connections, which are
subject of further studies, link statistical complexity to computational complexity.
Acknowledgments
The authors would like to thank Naftali Tishby, Scott Kirkpatrick and Michael
Clausen for their helpful comments and discussions.
References
[1] J. M. Buhmann and M. Held. Model selection in clustering by uniform convergence bounds. Advances in Neural Information Processing Systems, 12:216- 222,
1999.
[2] R. Durbin and D. Willshaw. An analogue approach to the travelling salesman
problem using an elastic net method. Nature, 326:689- 691, 1987.
[3] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchio Optimisation by simulated
annealing. Science, 220:671- 680, 1983.
[4] S. Lin and B. Kernighan. An effective heuristic algorithm for the traveling
salesman problem. Operations Research, 21:498- 516, 1973.
[5] P.D. Simic. Statistical mechanics as the underlying theory of "elastic" and
"neural" optimizations. Network, 1:89-103, 1990.
[6] G. Winkler. Image Analysis, Random fields and Dynamic Monte Carlo Methods,
volume 27 of Application of Mathematics. Springer, Heidelberg, 1995.
-sigma2 = O.03
i
17.7
&:
17.6
"j
17.5
o
o
f 17.4
<"Il
T.,...,:0.15OO:Xl
Lenglt. : 5.179571
0
o
17.3
o
o
temperatureT
o.I H
(d)
e
- si ma = O.025
~
T.,...,: 0.212759
Lenglt. : 6.295844
11.5
o
~
&:
11 .45
"j
11.4
~
~ 11.35
o
CD
o
temperatureT
(f)
n 5O"",11I>1S20_025 1 510
0_7654.2.() _742680 _2 31390 . 057 211.(l.Q1597 0 . 2 1 4 79 0.83 22 4 0 .58 33a1 ~
g
Figure 1: (a) Average trajectories at different temperatures for n = 100 appointments on a circle with a 2 = 0.03. (b) Average trajectories at different temperatures,
for multiple Gaussian sources, n = 50 and a 2 = 0.025. (c) The same for an instance
with structure on two levels. (d) Average tour length of the post-processed relaxed
(PPR) solutions for the circle instance plotted in (a). The PPR width was w = 5.
The average fits to noise in the data if the temperature is too low, leading to overfitting phenomena. Note that the average best solution is :s: 16.5. (e) The average
trajectory with the smallest average length of its PPR solutions in (d). (f) Average
tour length as in (d). The average best solution is :s: 10.80. (g) Lowest temperature
trajectory with small average PPR solution length in (f).
| 2049 |@word briefly:1 polynomial:1 norm:1 simulation:1 selecting:1 si:1 reminiscent:1 readily:1 must:3 grain:1 shape:1 treating:1 update:1 plane:3 beginning:1 realizing:1 coarse:3 provides:1 location:4 district:1 simpler:1 constructed:1 c2:2 consists:2 prove:3 advocate:1 paragraph:1 hardness:1 expected:6 planning:1 mechanic:1 decreasing:1 cardinality:2 increasing:1 becomes:1 provided:1 estimating:1 underlying:10 lowest:1 kind:1 interpreted:3 minimizes:1 argmin:4 finding:1 every:1 lfd:1 act:2 ti:4 braun:1 weekly:1 exactly:1 willshaw:2 uk:2 normally:1 before:3 local:1 modify:1 consequence:1 era:4 analyzing:1 id:2 path:2 interpolation:1 might:7 eb:5 studied:2 equivalence:5 statistically:1 averaged:1 unique:3 acknowledgment:1 differs:1 lf:1 procedure:3 area:1 empirical:5 pre:1 word:1 confidence:1 cannot:2 clever:1 selection:3 risk:3 restriction:1 equivalent:6 map:2 starting:3 identifying:2 hd:1 searching:1 traditionally:1 feel:1 construction:1 modulo:1 exact:3 homogeneous:1 hypothesis:7 element:2 satisfying:2 observed:1 capture:2 worst:2 connected:1 decrease:1 removed:1 complexity:7 ui:1 dynamic:1 solving:2 segment:1 technically:1 completely:4 easily:1 represented:2 d7r:1 effective:1 monte:4 choosing:1 neighborhood:1 exhaustive:1 whose:3 heuristic:6 say:2 otherwise:1 statistic:2 winkler:1 noisy:5 tsp:9 itself:3 sequence:1 net:2 took:1 maximal:1 inserting:1 relevant:2 iff:1 adapts:1 supposed:1 normalize:1 convergence:2 optimum:1 produce:1 converges:3 depending:3 derive:1 oo:1 undistorted:1 measured:2 ij:2 x1c:1 eq:1 c:2 involves:1 met:1 rei:1 radius:1 closely:1 correct:1 vc:1 everything:1 fix:1 generalization:1 suffices:2 hold:7 around:1 considered:1 exp:4 seed:1 algorithmic:1 week:6 mapping:3 lm:1 claim:1 consecutive:1 smallest:1 estimation:1 combinatorial:7 city:3 hope:1 rough:1 gaussian:1 idealizing:1 earliest:1 derived:1 joachim:1 improvement:1 contrast:1 centroid:1 am:2 helpful:1 el:2 relation:2 going:1 germany:1 sketched:1 denoted:2 exponent:1 plan:1 smoothing:2 integration:1 mutual:1 field:2 construct:4 clausen:1 sampling:3 represents:1 lit:1 ike:2 unsupervised:1 constitutes:1 future:1 jb:1 np:1 t2:2 summand:1 few:1 micro:1 oriented:2 familiar:1 replaced:1 interest:2 grasp:1 introduces:1 kirkpatrick:2 held:1 chain:3 lh:1 euclidean:4 desired:1 plotted:3 circle:2 weighs:1 minimal:1 instance:35 gn:2 planned:1 cover:1 cost:8 deviation:1 subset:2 tour:24 uniform:1 dij:1 conducted:1 too:3 tishby:1 characterize:1 answer:1 density:2 fundamental:1 systematic:1 physic:2 michael:1 analogously:1 vecchio:1 reflect:1 containing:1 choose:1 worse:1 leading:1 de:2 try:1 closed:1 linked:2 sup:1 start:2 sort:1 recover:1 minimize:1 il:1 variance:1 who:1 efficiently:1 yield:1 carlo:4 trajectory:38 definition:2 infinitesimal:1 against:1 energy:1 obvious:1 e2:2 proof:2 d1f:1 sampled:4 newly:2 stop:1 back:1 bidirectional:1 higher:1 done:1 furthermore:3 until:1 traveling:4 sketch:1 overfit:1 hand:4 kernighan:3 believe:1 effect:4 concept:2 symmetric:1 round:1 ll:1 self:1 width:1 naftali:1 simic:1 bijective:2 complete:1 demonstrate:1 tn:5 temperature:18 gh:1 interpreting:1 image:1 consideration:1 nih:1 common:3 rl:1 volume:1 discussed:1 he:4 extend:1 relating:1 measurement:1 refer:1 gibbs:10 trivially:1 mathematics:1 pointed:1 gt:2 summands:2 dominant:1 scenario:4 certain:4 tee:1 minimum:3 relaxed:5 subtraction:1 algebraically:1 shortest:1 determine:2 relates:1 multiple:4 rj:12 reduces:1 technical:1 adapt:1 sphere:2 lin:3 concerning:1 post:3 a1:1 essentially:1 metric:1 expectation:1 optimisation:1 represent:2 addition:6 want:1 separately:1 appointment:13 interval:1 else:1 annealing:2 source:2 sends:1 comment:2 subject:1 simulates:1 undirected:1 db:4 coarsening:1 call:2 ee:1 iii:1 easy:1 iterate:1 independence:1 fit:1 sigma2:1 identified:1 suboptimal:1 competing:1 reduce:1 idea:2 shift:1 motivated:1 speaking:1 passing:2 action:11 deep:1 generally:1 useful:2 clear:1 involve:1 iixi:1 locally:1 concentrated:1 processed:2 generate:1 sl:1 exist:1 canonical:1 sign:1 per:1 write:1 iz:1 group:8 nevertheless:1 threshold:1 drawn:2 graph:2 geometrically:2 sum:4 powerful:1 reader:1 bound:5 guaranteed:1 durbin:2 occur:1 ri:1 bonn:4 argument:2 min:2 lf2:2 describes:1 slightly:1 em:1 smaller:1 metropolis:4 invariant:5 computationally:1 ln:1 equation:1 remains:1 turn:2 tractable:1 reversal:1 travelling:1 salesman:8 studying:1 operation:3 available:1 apply:1 appropriate:1 gelatt:1 alternative:1 denotes:1 clustering:2 exploit:1 perturb:1 especially:1 prof:1 added:1 question:1 occurs:2 strategy:1 usual:2 rjl:2 distance:5 link:2 mapped:2 simulated:2 concatenation:1 thank:1 considers:1 reason:1 assuming:1 length:8 index:2 relationship:4 pointwise:1 modeled:1 unfortunately:1 eht:1 rise:1 guideline:1 markov:3 finite:5 rn:2 introduced:1 namely:1 trip:3 optimized:1 connection:2 subgroup:1 able:1 below:1 usually:1 scott:1 built:2 analogue:2 business:1 eh:3 buhmann:2 scheme:1 brief:1 extract:2 ppr:5 sn:22 geometric:2 l2:2 multiplication:2 determining:2 law:1 fully:2 permutation:20 interesting:1 lf1:2 share:1 cd:1 translation:1 last:1 bias:1 jh:2 institute:1 distributed:1 curve:1 calculated:1 xn:2 author:1 far:1 uni:2 keep:1 global:1 overfitting:3 mem:3 reorder:1 xi:5 search:2 un:2 sk:1 reality:3 scratch:1 nature:1 transfer:1 robust:1 elastic:3 symmetry:5 heidelberg:1 constructing:2 domain:1 main:1 whole:1 noise:3 mikio:1 fashion:1 inferring:1 xl:6 parallelly:1 grained:2 rk:1 theorem:2 specific:1 r2:6 normalizing:7 essential:1 intrinsic:2 ih:1 adding:2 horizon:1 suited:2 entropy:1 depicted:1 intersection:1 simply:1 scalar:1 springer:1 corresponds:1 minimizer:1 adaption:1 dh:2 extracted:1 ma:1 goal:2 formulated:1 identity:1 careful:1 content:1 hard:1 change:4 typical:2 determined:3 reversing:1 averaging:5 sampler:2 decouple:1 called:3 experimental:1 meaningful:1 select:1 il4:3 inability:1 dept:1 tested:1 phenomenon:2 |
1,151 | 205 | 516
Grossman
The CHIR Algorithm for Feed Forward
Networks with Binary Weights
Tal Grossman
Department of Electronics
Weizmann Institute of Science
Rehovot 76100 Israel
ABSTRACT
A new learning algorithm, Learning by Choice of Internal Represetations (CHIR), was recently introduced. Whereas many algorithms reduce the learning process to minimizing a cost function
over the weights, our method treats the internal representations as
the fundamental entities to be determined. The algorithm applies
a search procedure in the space of internal representations, and a
cooperative adaptation of the weights (e.g. by using the perceptron
learning rule). Since the introduction of its basic, single output version, the CHIR algorithm was generalized to train any feed forward
network of binary neurons. Here we present the generalised version
of the CHIR algorithm, and further demonstrate its versatility by
describing how it can be modified in order to train networks with
binary (?1) weights. Preliminary tests of this binary version on
the random teacher problem are also reported.
I. INTRODUCTION
Learning by Choice oflnternal Representations (CHIR) was recently introduced
[1,11] as a training method for feed forward networks of binary units.
Internal Representations are defined as the states taken by the hidden units
of a network when patterns (e.g. from the training set) are presented to the input
layer of the network. The CHIR algorithm views the internal representations associated with various inputs as the basic independent variables of the learning process.
Once such representations are formed, the weights can be found by simple and local
learning procedures such as the Percept ron Learning Rule (PLR) [2]. Hence the
problem of learning becomes one of searching for proper internal representations,
The CHIR Algorithm for Feed Forward Networks with Binary Weights
rather than of minimizing a cost function by varying the values of weights, which
is the approach used by back propagation (see, however [3],[4] where "back propagation of desired states" is described). This basic idea, of viewing the internal
representations as the fundamental entities, has been used since by other groups [57]. Some of these works, and the main differences between them and our approach,
are briefly disscussed in [11]. One important difference is that the CHIR algorithm,
as well as another similar algorithm, the MRII [8], try to solve the learning problem
for a fixed architecture, and are not guaranteed to converge. Two other algorithms
[5,6] always find a solution, but at the price of increasing the network size during
learning in a manner that resembles similar algorithms developed earlier [9,10]. Another approach [7] is to use an error minimizing algorithm which treat~ the internal
representations as well as the weights as the relevant variables of the search space.
To be more specific, consider first the single layer perceptron with its Perceptron Learning Rule (PLR) [2]. This simple network consists of N input (source)
units j, and a single target unit i. This unit is a binary linear threshold unit, i.e.
when the source units are set in anyone of Jl = 1, .. M patterns, i.e. Sj =
the
state of unit i, Si = ?1 is determined according to the rule
ef,
Si = sign(L WijSj
+ 0i)
(1)
.
j
Here Wij is the (unidirectional) weight assigned to the connection from unit j to
ij 0i is a local bias. For each of the M input patterns, we require that the target
unit (determined using (1)) will take a preassigned value
Learning takes place
in the course of a training session. Starting from any arbitrary initial guess for the
weights, an input v is presented, resulting in the output taking some value Sf. Now
modify every weight according to the rule
er.
(2)
where TJ > 0 is a step size parameter (ej = 1 is used to modify the bias 0). Another
input pattern is presented, and so on, until all inputs draw the correct output. The
Perceptron convergence theorem states [2] that the PLR will find a solution (if one
exists), in a finite number of steps. Nevetheless, one needs, for each unit, both the
desired input and output states in order to apply the PLR.
Consider now a two layer perceptron, with N input, H hidden and J{ output
units (see Fig.1). The elements of the network are binary linear threshold units i,
whose states Si = ?1 are determined according to (1). In a typical task for such
a network, M specified output patterns, Sf'-,t,1J. = efut,lJ., are required in response
to Jl
1, ... , M input patterns. If a solution is found, it first maps each input onto
an internal representation generated on the hidden layer, which, in turn, produces
the correct output. Now imagine that we are not supplied with the weights that
solve the problem; however the correct internal representations are revealed. That
is, we are given a table with M rows, one for each input. Every row has H bits ef'lJ. I
for i = 1..H, specifying the state of the hidden layer obtained in response to input
=
517
518
Grossman
pattern 1'. One can now view each hidden-layer cell i as the target of the PLR,
with the N inputs viewed as source. Given sufficient time, the PLR will converge
to a set of weights Wii' connecting input unit j to hidden unit i, so that indeed
the input-hidden association that appears in column i of our table will be realized.
In order to obtain the correct output, we apply the PLR in a learning process that
uses the hidden layer as source and each output unit as a target, so as to realize
the correct output. In general, however, one is not supplied with a correct table of
internal representations. Finding such a table is the goal of our approach .
... 0
Figure 1. A typical three layered feed forward network (two layered perceptron) with N input, H hidden and I( output units. The unidirectional weight Wij
connects unit j to unit i. A layer index is implicitely included in each unit's index.
During learning, the CHIR algorithm alternates between two phases: in one it
generates the internal representations, and in the other it uses the updated representations in order to search for weights, using some single layer learning rule. This
general scheme describes a large family of possible algorithms, that use different
ways to change the internal representations. and update the weights.
A simple algorithm based on this general scheme was introduced recently [1,11].
In section II we describe the multiple output version of CHIR [11]. In section III we
present a way to modify the algorithm so it can train networks with binary weights,
and the preliminary results of a few tests done on this new version. In the last
section we shortly discuss our results and describe some future directions.
The CHIR Algorithm for Feed Forward Networks with Binary Weights
II. THE CHIR ALGORITHM
The CHIR algorithm that we describe here implements the basic idea of learning by choice of internal representations by breaking the learning process into four
distinct procedures that are repeated in a cyclic order:
1. SETINREP: Generate a table of internal representations {ef''''} by presenting
each input pattern from the training set and recording the states of the hidden
units, using Eq.(l), with the existing couplings Wij and 0i.
2. LEARN23: The current table of internal representations is used as the training
set, the hidden layer cells are used as source, and each output as the target unit
of the PLR. If weights Wij and 0i that produce the desired outputs are found, the
problem has been solved. Otherwise stop after 123 learning sweeps, and keep the
current weights, to use in CHANGE INREP.
3. CHANGE INREP: Generate a new table of internal representations, which
reduces the error in the output : We present the table sequentially, row by row
(pattern by pattern), to the hidden layer. If for pattern v the wrong output is
obtained, the internal representation h 'lI is changed.
e
This is done simply by choosing (at random) a hidden unit i, and checking
the effect of flipping the sign of
on the total output error, i.e. the number of
wrong bits. If the output error is not increased, the flip is accepted and the table of
internal representations is changed accordingly. Otherwise the flip is rejected and
we try another unit. When we have more than one output unit, it might happen
that an error in one output unit can not be corrected without introducing an error
in another unit. Therefore we allow only for a pre-specified number of attempted
flips, lin, and go on to the next pattern even if the output error was not eliminated
completely. This procedure ends with a "modified, "improved" table which is our
next guess of internal representations. Note that this new table does not necessarily
yield a totally correct output for all the patterns. In such a case, the learning process
will go on even if this new table is perfectly realized by the next stage - LEARN12.
e?'' '
4. LEARN12: Present an input pattern; if the output is wrong, apply the PLR
with the first layer serving as source, treating every hidden layer site separately
as target. If input v does yield the correct output, we insert the current state
of the hidden layer as the internal representation associated with pattern v, and
no learning steps are taken. We sweep in this manner the training set, modifying
weights Wij, (between input and hidden layer), hidden-layer thresholds Oi, and, as
explained above, internal representations. If the network has achieved error-free
performance for the entire training set, learning is completed. Otherwise, after lt2
training sweeps (or if the current internal representation is perfectly realized), abort
the PLR stage, keeping the present values of Wij, Oi, and start SETINREP again.
The idea in trying to learn the current internal representation even if it does not
yield the perfect output is that it can serve as a better input for the next LEARN23
stage. That way, in each learning cycle the algorithm tries to improve the overall
performance of the network.
519
520
Grossman
This algorithm can be further generalized for multi-layered feed forward networks by applying the CHANGE INREP and LEARN12 procedures to each of the
hidden layers, one by one, from the last to the first hidden layer.
There are a few details that need to be added.
a) The "iInpatience" parameters: lt2 and h3, which are rather arbitrary, are
introduced to guarantee that the PLR stage is aborted if no solution is found, but
they have to be large enough to allow the PLR to find a solution (if one exists) with
sufficiently high probability. Similar considerations are valid for the lin parameter,
the number of flip attempts allowed in the CHANGE INREP procedure. If this
number is too small, the updated internal representations may not improve. If it is
too large, the new internal representations might be too different from the previous
ones, and therefore hard to learn.
The optimal values depend, in general, on the problem and the network size.
Our experience indicates, however, that once a "reasonable" range of values is found,
performance is fairly insensitive to the precise choice. In addition, a simple rule of
thumb can always be applied: "Whenever learning is getting hard, increase the
parameters". A detailed study of this issue is reported in [11].
b) The Internal representations updating scheme: The CHANGE INREP
procedure that is presented here (and studied in [11]) is probably the simplest and
"most primitive" way to update the InRep table. The choice of the hidden units to
be flipped is completely blind and relies only on the single bit of information about
the improvement of the total output error. It may even happen that no change in the
internal representaion is made, although such a change is needed. This procedure
can certainly be made more efficient, e.g. by probing the fields induced on all the
hidden units to be flipped and then choosing one (or more) of them by applying a
"minimal disturbance" principle as in [8]. Nevertheless it was shown [11] that even
this simple algorithm works quite well.
c) The weights updating schemes: In our experiments we have used the simple
PLR with a fixed increment (7] 1/2, .6.Wij ?1) for weight learning. It has the
advantage of allowing the use of discrete (or integer) weights. Nevertheless, it is just
a component that can be replaced by other, perhaps more sophisticated methods, in
order to achieve, for example, better stability [12], or to take into account various
constraints on the weights, e.g. binary weights [13]. In the following section we
demonstrate how this can be done.
=
=
III. THE CHIR ALGORITHM FOR BINARY WEIGHTS
In this section we describe how the CHIR algorithm can be used in order to train
feed forward networks with binary weights. According to this strong constraint, all
the weights in the system (including the thresholds) can be either +1 or -1. The
way to do it within the CHIR framework is simple: instead of applying the PLR
(or any other single layer, real weights algorithm) for the updating of the weights,
The CHIR Algorithm for Feed Forward Networks with Binary Weights
we can use a binary perceptron learning rule. Several ways to solve the learning
problem in the binary weight perceptron were suggested recently [13]. The one that
we used in the experiments reported here is a modified version of the directed drift
algorithm introduced by Venkatesh [13]. Like the standard PLR, the directed drift
algorithm works on-line, namely, the patterns are presented one by one, the state of
a unit i is calculated according to (1), and whenever an error occurs the incoming
weights are updated. When there is an error it means that
,<0
~'! hI!
'-'
e.n
Wiie.r '
Namely, the field hi = Ej
(induced by the current pattern
is "wrong".
If so, there must be some weights that pull it to the wrong direction. These are the
weights for which
erWii{r < o.
er
Here
is the desired output of unit i for pattern v. The updating of the weights
is done simply by flipping (i.e. Wii ~ -Wij ) at random k of these weights.
The number of weights to be changed in each learning step, k, can be a prefixed parameter of the algorithm, or, as suggested by Venkatesh, can be decreased
gradually during the learning process in a way similar to a cooling schedule (as in
simulated annealing). What we do is to take k
Ihl/2 + 1, making sure, like in
relaxation algorithms, that just enough weights are flipped in order to obtain the
desired target for the current pattern. This simple and local rule is now "plugged"
into the Learn12 and Learn23 procedures instead of (2), and the initial weights are
chosen to be + 1 or -1 at random.
=
We tested the binary version of CHIR on the "random teacher" problem. In
this problem a "teacher network" is created by choosing a random set of +1/-1
weights for the given architecture. The training set is then created by presenting
M input patterns to the network and recording the resulting output as the desired
output patterns. Ip. what follows we took M
2N (exhaustive learning), and an
N :N :1 architecture.
=
The "time" parameter that we use for measuring performance is the number
of sweeps through the training set of M patterns ("epochs") needed in order to find
the solution. Namely, how many times each pattern was presented to the network.
In the experiments presented here, all possible input patterns were presented sequentially in a fixed order (within the perceptron learning sweeps). Therefore in
each cycle of the algorithm there are 112 + h3 + 1 such sweeps. Note that according
to our definition, a single sweep involves the updating of only one layer of weights
or internal representations. for each network size, N, we created an ensemble of
50 independent runs, with different ranodom teachers and starting with a different
random choice of initial weights.
We calculate, as a performance measure, the following quantities:
a. The median number of sweeps, t m .
b. The "inverse average rate",
T,
as defined by Tesauro and Janssen in [14].
521
522
Grossman
c. The success rate, S, i.e. the fraction of runs in which the algorithm finds a
solution in less than the maximal number of training cycles [max specified.
The results,with the typical parameters, for N=3,4,5,6, are given in Table 1.
Table 1. The Random Teacher problem with N:N:l architecture.
N
lt2
123
lin
[max
tm
T
S
3
4
5
6
20
25
40
70
10
10
15
40
5
7
9
11
20
60
300
900
14
87
430
15000
9
37
60
1100
1.00
1.00
1.00
0.71
As mentioned before, these are only preliminary results. No attempt was made
to to optimize the learning parameters.
IV. DISCUSSION
We presented a generalized version of the CHIR algorithm that is capable
of training networks with multiple outputs and hidden layers. A way to modify
the basic alf$ortihm so it can be applied to networks with binary weights was also
explained and tested. The potential importance of such networks, e.g. in hardware
implementation, makes this modified version particularly interesting.
An appealing feature of the CHIR algorithm is the fact that it does not use
any kind of "global control", that manipulates the internal representations (as is
used for example in [5,6]). The mechanism by which the internal representations are
changed is local in the sense that the change is done for each unit and each pattern
without conveying any information from other units or patterns (representations).
Moreover, the feedback from the "teacher" to the system is only a single bit quantity,
namely, whether the output is getting worse or not (in contrast to BP, for example,
where one informs each and every output unit about its individual error).
Other advantages of our algorithm are the simplicity of the calculations, the
need for only integer, or even binary weights and binary units, and the good performance. It should be mentioned again that the CHIR training sweep involves much
less computations than that of back-propagation. The price is the extra memory of
M H bits that is needed during the learning process in order to store the internal
representations of all M training patterns. This feature is biologically implausible
and may be practically limiting. We are developing a method that does not require
such memory. The learning method that is currently studied for that purpose [15],
is related to the MRII rule, that was recently presented by Widrow and Winter in
[8]. It seems that further research will be needed in order to study the practical
differences and the relative advantages of the CHIR and the MRII algorithms.
The eHIR Algorithm for Feed Forward Networks with Binary Weights
Acknowledgements: I am gratefull to Prof. Eytan Domany for many useful
suggestions and comments. This research was partially supported by a grant from
Minerva.
References
[1] Grossman T., Meir R. and Domany E., Complex Systems 2, 555 (1989). See
also in D. Touretzky (ed.), Advances in Neural Information Processing Systems 1,
(Morgan Kaufmann, San Mateo 1989).
[2] Minsky M. and Papert S. 1988, Perceptrons (MIT, Cambridge);
Rosenblatt F. Principles of neurodynamics (Spartan, New York, 1962).
[3] Plaut D.C., Nowlan S.J., and Hinton G.E., Tech.Report CMU-CS-86-126,
Carnegie-Mellon University (1986).
[4] Le Cun Y., Proc. Cognitiva 85, 593 (1985).
[5] Rujan P. and Marchand M., in the Proc. of the First International Joint Conference Neural Networks - Washington D. C. 1989, Vol.lI, pp. 105. and to appear
in Complex Systems.
[6] Mezard M. and Nadal J.P., J.Phys.A. 22, 2191 (1989).
[7] Krogh A., Thorbergsson G.1. and Hertz J.A., in these Proceedings.
R. Rohwer, to apear in the Proc. of DANIP, GMD Bonn, April 1989, J. Kinderman
and A. Linden eds ;
Saad D. and Merom E., preprint (1989).
[8] Widrow B. and Winter R., Computer 21, No.3, 25 (1988).
[9] See e.g. Cameron S.H., IEEE TEC EC-13,299 (1964) ; Hopcroft J.E. and
Mattson R.L., IEEE, TEC EC-14, 552 (1965).
[10] Honavar V. and' Uhr L. in the Proc. of the 1988 Connectionist Models Summer School, Touretzky D., Hinton G . and Sejnowski T. eds. (Morgan Kaufmann,
San Mateo, 1988).
[11] Grossman T., to be published in Complex Systems (1990).
[12] Krauth W. and Mezard M., J.Phys.A, 20, L745 (1988).
[13] Venkatesh S., preprint (1989) ;
Amaldi E. and Nicolis S., J.Phys.France 50, 2333 (1989).
Kohler H., Diederich S., Kinzel W. and Opper M., preprint (1989).
[14] Tesauro G. and Janssen H., Complex Systems 2, 39 (1988).
[15] Nabutovski D., unpublished.
523
| 205 |@word briefly:1 version:9 seems:1 electronics:1 cyclic:1 initial:3 existing:1 current:7 nowlan:1 si:3 must:1 realize:1 happen:2 treating:1 update:2 guess:2 accordingly:1 plaut:1 ron:1 wijsj:1 consists:1 manner:2 indeed:1 aborted:1 multi:1 increasing:1 becomes:1 totally:1 moreover:1 israel:1 what:2 kind:1 nadal:1 developed:1 finding:1 guarantee:1 every:4 wrong:5 control:1 unit:35 grant:1 appear:1 generalised:1 before:1 local:4 treat:2 modify:4 preassigned:1 might:2 resembles:1 studied:2 mateo:2 specifying:1 kinderman:1 range:1 weizmann:1 directed:2 practical:1 implement:1 procedure:9 pre:1 onto:1 layered:3 applying:3 optimize:1 map:1 go:2 primitive:1 starting:2 simplicity:1 manipulates:1 rule:10 pull:1 stability:1 searching:1 increment:1 limiting:1 updated:3 target:7 imagine:1 us:2 element:1 particularly:1 updating:5 cooling:1 cooperative:1 preprint:3 solved:1 calculate:1 cycle:3 chir:22 mentioned:2 depend:1 serve:1 completely:2 joint:1 hopcroft:1 various:2 train:4 distinct:1 describe:4 sejnowski:1 spartan:1 tec:2 choosing:3 exhaustive:1 whose:1 quite:1 solve:3 otherwise:3 ip:1 advantage:3 took:1 maximal:1 adaptation:1 relevant:1 achieve:1 setinrep:2 getting:2 convergence:1 produce:2 perfect:1 coupling:1 informs:1 widrow:2 ij:1 school:1 h3:2 eq:1 krogh:1 strong:1 c:1 involves:2 direction:2 correct:8 modifying:1 uhr:1 viewing:1 require:2 preliminary:3 insert:1 practically:1 sufficiently:1 alf:1 purpose:1 proc:4 currently:1 mit:1 always:2 modified:4 rather:2 ej:2 varying:1 improvement:1 indicates:1 tech:1 contrast:1 sense:1 am:1 lj:2 entire:1 hidden:22 wij:8 france:1 overall:1 issue:1 fairly:1 field:2 once:2 washington:1 eliminated:1 flipped:3 amaldi:1 future:1 report:1 connectionist:1 few:2 winter:2 individual:1 replaced:1 phase:1 connects:1 minsky:1 versatility:1 attempt:2 certainly:1 tj:1 capable:1 experience:1 iv:1 plugged:1 desired:6 minimal:1 increased:1 column:1 earlier:1 measuring:1 cost:2 introducing:1 too:3 reported:3 teacher:6 fundamental:2 international:1 connecting:1 again:2 worse:1 grossman:7 li:2 account:1 potential:1 inrep:6 blind:1 view:2 try:3 start:1 unidirectional:2 formed:1 oi:2 kaufmann:2 percept:1 ensemble:1 yield:3 conveying:1 thumb:1 published:1 implausible:1 phys:3 touretzky:2 whenever:2 ed:3 diederich:1 definition:1 rohwer:1 pp:1 associated:2 stop:1 ihl:1 schedule:1 sophisticated:1 back:3 appears:1 feed:10 response:2 improved:1 april:1 done:5 learn23:3 rejected:1 stage:4 just:2 until:1 propagation:3 abort:1 perhaps:1 effect:1 hence:1 assigned:1 during:4 generalized:3 trying:1 presenting:2 demonstrate:2 representaion:1 consideration:1 ef:3 recently:5 krauth:1 kinzel:1 insensitive:1 jl:2 association:1 mellon:1 cambridge:1 session:1 tesauro:2 store:1 binary:21 success:1 morgan:2 converge:2 ii:2 multiple:2 reduces:1 calculation:1 lin:3 cameron:1 basic:5 cmu:1 minerva:1 achieved:1 cell:2 whereas:1 addition:1 separately:1 decreased:1 annealing:1 median:1 source:6 extra:1 saad:1 cognitiva:1 probably:1 comment:1 sure:1 recording:2 induced:2 integer:2 revealed:1 iii:2 enough:2 architecture:4 perfectly:2 plr:15 reduce:1 idea:3 tm:1 domany:2 whether:1 york:1 useful:1 detailed:1 hardware:1 simplest:1 gmd:1 generate:2 supplied:2 meir:1 sign:2 serving:1 rosenblatt:1 rehovot:1 discrete:1 carnegie:1 vol:1 group:1 four:1 threshold:4 nevertheless:2 relaxation:1 fraction:1 run:2 inverse:1 place:1 family:1 reasonable:1 draw:1 bit:5 layer:21 hi:2 guaranteed:1 summer:1 marchand:1 constraint:2 bp:1 implicitely:1 tal:1 generates:1 bonn:1 anyone:1 department:1 developing:1 according:6 alternate:1 honavar:1 hertz:1 describes:1 appealing:1 cun:1 making:1 biologically:1 lt2:3 explained:2 gradually:1 taken:2 describing:1 turn:1 discus:1 mechanism:1 needed:4 flip:4 prefixed:1 end:1 wii:2 apply:3 shortly:1 completed:1 prof:1 sweep:9 added:1 realized:3 flipping:2 occurs:1 quantity:2 simulated:1 entity:2 index:2 minimizing:3 implementation:1 proper:1 allowing:1 neuron:1 finite:1 hinton:2 precise:1 arbitrary:2 drift:2 introduced:5 venkatesh:3 namely:4 required:1 specified:3 unpublished:1 connection:1 suggested:2 pattern:27 including:1 max:2 memory:2 disturbance:1 scheme:4 improve:2 created:3 epoch:1 acknowledgement:1 checking:1 relative:1 interesting:1 suggestion:1 sufficient:1 principle:2 row:4 course:1 changed:4 supported:1 last:2 free:1 keeping:1 bias:2 allow:2 perceptron:9 institute:1 taking:1 feedback:1 calculated:1 opper:1 valid:1 forward:10 made:3 san:2 ec:2 sj:1 keep:1 global:1 sequentially:2 incoming:1 search:3 table:15 neurodynamics:1 learn:2 necessarily:1 complex:4 rujan:1 main:1 repeated:1 allowed:1 fig:1 site:1 probing:1 papert:1 mezard:2 sf:2 breaking:1 theorem:1 specific:1 er:2 linden:1 exists:2 janssen:2 importance:1 simply:2 partially:1 applies:1 thorbergsson:1 relies:1 viewed:1 goal:1 price:2 change:9 hard:2 included:1 determined:4 typical:3 corrected:1 total:2 accepted:1 eytan:1 attempted:1 perceptrons:1 internal:32 learn12:4 kohler:1 tested:2 |
1,152 | 2,050 | Linear Time Inference in Hierarchical HMMs
Kevin P. Murphy and Mark A. Paskin
Computer Science Department
University of California
Berkeley, CA 94720-1776
murphyk,paskin @cs.berkeley.edu
Abstract
The hierarchical hidden Markov model (HHMM) is a generalization of
the hidden Markov model (HMM) that models sequences with structure
at many length/time scales [FST98]. Unfortunately, the original infertime, where is
ence algorithm is rather complicated, and takes
the length of the sequence, making it impractical for many domains. In
this paper, we show how HHMMs are a special kind of dynamic Bayesian
network (DBN), and thereby derive a much simpler inference algorithm,
which only takes
time. Furthermore, by drawing the connection
between HHMMs and DBNs, we enable the application of many standard approximation techniques to further speed up inference.
1 Introduction
The Hierarchical HMM [FST98] is an extension of the HMM that is designed to
model domains with hierarchical structure, e.g., natural language, XML, DNA sequences
[HIM 00], handwriting [FST98], plan recognition [BVW00], visual action recogntion
[IB00, ME01, Hoe01], and spatial navigation [TRM01, BVW01]. HHMMs are less expressive than stochastic context free grammars (SCFGs), since they only allows hierarchies
of bounded depth, but they are more efficient and easier to learn. Unfortunately, the original
inference algorithm described in [FST98] is somewhat complicated, and takes
time, where is the length of the sequence, is the depth of the hierarchy, and is the
(maximum) number of states at each level of the hierarchy. In this paper, we show how to
represent an HHMM as a dynamic Bayesian network (DBN), and thereby derive a much
simpler and faster inference algorithm, which takes at most
time; empirically,
we find it takes only
time using the junction tree algorithm. Furthermore,
by drawing the connection between HHMMs and DBNs, we enable the application of approximate inference techniques such as belief propagation, which can perform inference in
time.
!
" By inference, we mean offline smoothing, i.e., conditioning on a fixed-length observation se-
quence. This is needed as a subroutine for EM. Once the model has been learned, it will typically be
used for online inference (filtering).
end
0
2
3
a
end
1
end
4
5
b
6
c
8
x
9
7
end
d
end
y
Figure 1: A 3-level hierarchical automaton representing the regular expression
.
Solid lines represent horizontal transitions, dotted lines represent vertical transitions. Letters below a
production state represent the symbol that is emitted. The unnumbered root node is considered level
0, and could be omitted if we fully interconnected states 0 and 1.
We will describe HHMMs in Section 2, and the original
inference algorithm in
Section 3. The main contribution of the paper is in Section 4, where we show how to
represent an HHMM as a DBN. In Section 5, we discuss how to do efficient inference in
this DBN, and in Section 6, we discuss related work. In the full version of this paper, we
discuss how to do parameter and structure learning using EM.
2 Hierarchical HMMs
HHMMs are like HMMs except the states of the stochastic automaton can emit single
observations or strings of observations. (For simplicity of exposition, we shall assume
all observations are discrete symbols, but HHMMs can easily be generalized to handle
continuous observations, as we discuss in Section 4.1.) Those that emit single symbols
are called ?production states?, and those that emit strings are termed ?abstract states?. The
strings emitted by abstract states are themselves governed by sub-HHMMs, which can be
called recursively. When the sub-HHMM is finished, control is returned to wherever it was
called from; the calling context is memorized using a depth-limited stack.
We illustrate the generative process with Figure 1, which shows the state transition diagram
of an example HHMM which models the regular expression . We start
in the root state, and make a ?vertical transition? to one of its children, say state 0. From
here, we make another vertical transition to state 2. Since state 2 is a production state, it
emits ?a? and then makes a ?horizontal transition? to state 3. State 3 calls its sub-HMM,
which emits x?s and y?s until it enters its end state; then control is returned to the calling
state, in this case state 3. We then make a horizontal transition to state 4, emit ?b?, and
enter the end state, thereby returning control to state 0. Finally, from state 0, we return
control to the root, and optionally start again.
Any HHMM can be converted to an HMM by creating a state for every possible legal stack
configuration ! . If the HHMM transition diagram is a tree, there will be one HMM state
for every HHMM production state. If the HHMM transition diagram has shared substructure (such as the sub-expression
), this structure must be duplicated in the HMM,
generally resulting in a larger model. It is the ability to reuse sub-models in different con-
texts that makes HHMMs more powerful than standard HMMs. In particular, the parameters of such shared sub-models only need to be learned once. (Given segmented data, we
can train the sub-HMMs separately, and then ?glue them together?, but it is also possible
to train the HHMM on unsegmented data; see the full version of this paper for details.)
3 Cubic-time inference
The inference algorithm for HHMMs presented in [FST98] runs in
time and is
based on the Inside-Outside algorithm [LY90], an exact inference algorithm for stochastic
context-free grammars (SCFGs) which we now describe.
In an SCFG, sequences of observations are generated by a set of stochastic production
into either
rules. Each production rule stochastically rewrites a non-terminal symbol
) or a pair of nonterminal symbols (
a symbol of the alphabet (
).
Observation strings are generated by starting with the distinguished ?start? nonterminal
, and continually re-writing all non-terminals using stochastic production rules until,
finally, only symbols of the alphabet remain.
! !
The Inside-Outside algorithm computes
, where ! !
!
!
!
is a subsequence. This can then be used to compute the expected
sufficient statistics needed by the EM algorithm to learn the parameters of the model. If
there are non-terminals in the grammar and the training sequence is of length , then the
time. To see why, note that we must compute
Inside-Outside algorithm requires
! !
for all end points and
, and for all midpoints such that
!
generates ! !
and
generates !
? the three degrees for freedom ,
and gives rise to the
term. The
term arises because we must consider all
,
and
.
!
The inference algorithm for HHMMs presented in [FST98] is based upon a straightforward adaptation of the Inside-Outside algorithm.
The algorithm computes
in state at time ! !
by assuming that sub-state
generates ! !
, that a
!
!
transition to state occurs, and that
generates
. Hence the algorithm
takes
time, where is the total number of states.
"$#
%
" #
" #
We can always ?flatten? an HHMM into a regular HMM and hence do inference in
. Unfortunately, this flat model cannot represent the hierarchical structure, yet
alone learn it. In the next section, we show how to represent the HHMM as a DBN, and
thereby get the best of both worlds: low time complexity without losing hierarchical structure.
4 Representing the HHMM as a DBN
We can represent the HHMM as a dynamic Bayesian network (DBN) as shown in Figure 2.
(We assume for simplicity that all production states are at the bottom of the hierarchy;
this restriction is lifted in the full version of this paper.) The state of the HMM at level
and time is represented by ! . The state of the whole HHMM is encoded by the vector
!
!
!
; intuitively, this encodes the contents of the stack, that specifies the
complete ?path? to take from the root to the leaf state.
'
(
) #
!
&#
) #+
*
) # ,*
is an indicator variable that is ?on? if the HMM at level and time has just ?finished?
(i.e., is about to enter an end state), otherwise it is off. Note that if !
, then !
)
)
)
)
)
)
)
*
)
)
Figure 2: An HHMM represented as a DBN.
is the state at time , level ;
if the
. Shaded nodes are observed;
HMM at level has finished (entered its exit state), otherwise
the remaining nodes are hidden. We may optionally clamp
, where is the length of the
observation sequence, to ensure all models have finished by the end of the sequence. (A similar trick
was used in [Zwe97].)
)
nodes that are ?off? represents the effective height
for all
; hence the number of
of the ?context stack?, i.e., which level of the hierarchy we are currently on.
)
The downward going arcs between the variables represent the fact that a state ?calls? a
sub-state. The upward going arcs between the variables enforce the fact that a higherlevel HMM can only change state when the lower-level one is finished. This ensures proper
nesting of the parse trees, and is the key difference between an HHMM and a hidden
Markov decision tree [JGS96].
We will define the conditional probability distributions (CPDs) of each of the node types
below, which will complete the definition of the model. We consider the bottom, middle
and top layers of the hierarchy separately (since they have different local topology), as well
as the first, middle and last time slices.
4.1 Definition of the CPDs
follows a Markov chain with parameters
Consider the bottom level of the hierarchy.
determined by its position in the automaton, which is encoded by the vector of higher-up
!
!
state variables !
, which we will represent by the integer . When
(
%
If the topology is sparse, this distribution will be 0 for many values of . This will be discussed
in Section 4.2.
)
enters its end state, it will ?turn on?
, to mean it is finished; this will be a signal
that higher-level HMMs can now change state. In addition, it will be a signal that the next
value of
should be drawn from its prior distribution (representing a vertical transition),
instead of its transition matrix (representing a horizontal transition). Formally, we can write
this as follows:
)
% if
if *
where we have assumed
end. is the transition matrix for level given that the
parent variables are in state % , and is just a rescaled version of . Similarly, is
the initial distribution for level given that the parent variables are in state % . The equation
)
for
is simply
) *
% end
Now consider the intermediate levels. As before, # follows a Markov chain with param)
eters determined by
#
, and # specifies whether we should use the) transition matrix
or the prior. The difference is that we now also get a signal from below, #
, specifying
!
!
!
!
!
!
!
whether the sub-model has finished or not; if it has, we are free to change state, otherwise
we must remain in the same state. Formally, we can write this as follows:
if
)
)
*
#
#
#
%
# #
# # ifif * and
and *
) # should ?turn on? only if # is ?allowed? to enter a final state, the probability of which
depends on the current context
#
. Formally, we can write this as follows:
) # * #
#
% ) #
# end ifif *
!
!
!
!
!
!
!
!
!
The top level differs from the intermediate levels in that the node has no parent to
specify which distribution to use. The equations are the same as above, except we eliminate
the conditioning on !
. (Equivalently, we can imagine a dummy top layer HMM,
which is always in state 1: !
. This is often how HHMMs are represented, so that this
top-level state is the root of the overall parse tree, as in Figure 1.)
#
% *
.
for the top
'
If the observations are discrete symbols, we may represent
as a multinomial
(i.e., using a table), or by using any of the more parsimonious representations discussed
in Section' 4.2. If the observations are real-valued vectors, we can use a Gaussian for each
value of , or a mixture of a smaller number of Gaussians, as in [GJ97].
Unlike the automaton representation, the DBN never actually enters an end state (i.e., can
never taken on the value ?end?), because if it did, it would not be able to emit the symbol . Instead,
on, and then enters a new (non-terminal) state at time . This means
causes
andto turn
that
HHMM
are not identical, but satisfy the following
"the!$# &DBN
% ('*) "+ transition
"!$# ,! wherematrices
represents
relation:
the automaton transition matrix, represents
.-0 /$1 "! end is the probability of terminating from state .
the DBN transition matrix, and )
The equations holds because the probability of each
transition in the DBN gets multiplied
; thishorizontal
by the probability that
, which is 2'3)
product should match the original probability.
#
#
% #
The CPDs for the nodes in the first slice are as follows:
level and
, for
!
!
It is easy to see that the new matrix is also stochastic, as required.
!
4.2 Parsimonious representations of the CPDs
# &#
#
%
!
!
!
The number of parameters needed to represent
as a multinomial
is
. If the state-transition diagram of the hierarchical automaton is sparse, many of
the entries in this table will be 0. However, when we are learning a model, we do not know
the structure of the state-transition diagram, and must therefore adopt a representation with
fewer parameters. There are at least three possibilities: decision trees [BFGK96], softmax
!
!
!
as a mixture of smaller transition matrices
nodes, or representing
at different depths c.f. [SJ99]. See the full version of this paper for details.
#
&# #
#
%
5 Linear-time inference
parents
We define inference to be computing
for all sets of nodes
in the DBN. These ?family? marginals are needed by EM. The simplest way to do this is to
merge all the hidden nodes in each slice into a single ?mega node?, ! , with
possible values. (The
term arises from the binary nodes.) We can then apply the
forwards-backwards algorithm for HMMs, which takes
time.
)
Unfortunately, converting the DBN to an HMM in this way will not be tractable for reasonably large or . (Even storing the transition matrix is likely to consume too much
space.) Fortunately, we can do better by exploiting the structure of the model. In [Mur01],
we present a way of applying the junction tree (jtree) algorithm to variable-length DBNs;
we give a brief sketch here. The algorithm works by performing a forwards-backwards
sweep through a chain of jtrees. Each jtree is formed from a ? -slice DBN?; this is a
DBN that contains all the nodes in slice 1 but only the interface nodes from slice 2. The
interface nodes are those nodes in slice 2 that have an incoming temporal arc, plus parents
of nodes that have incoming temporal arcs. In the case of an HHMM, the interface is all
the nodes.
*
The cost of doing inference in each jtree depends on the sizes of the cliques. Minimizing
the maximal clique size is NP-hard, so we used a standard one-step look-ahead (greedy)
algorithm [Kja90]. The resulting cliques are hard to interpret, but we can still analyze the
complexity. Let
be the number of
nodes in clique , let
be the
number of nodes, and let
be the number of cliques. Then the cost of inference in
a jtree is proportional to
)
!
"$#&%('
! * *
) *
) +
#&%('
Empirically
we find that, for a wide2 range of ,
, ,-/.
10
2
3 4 and ,5-/.
6
73 4 . Hence a crude upper bound on the cost of in"0
ference in each jtree is
, yielding an overall time and space complexity of
. We remind readers that the original algorithm has
time complexity, since there can be up to
states in the HHMM. The advantage of
the new algorithm in practice is clearly illustrated in Figure 3.
We can reduce the time (and space) complexity from
to
by
using approximate DBN inference techniques such as the ?factored frontier (FF) algorithm?
[MW01], which is equivalent to applying ?loopy belief propagation? to the DBN using a
left-right scheduling of the messages. (It is still exponential in because of the high fan-in
of the nodes.) We can get a further speedup by using a mixture representation of the CPDs
running time (seconds) vs sequence length
40
35
30
25
20
15
10
5
0
10
linear
cubic
15
20
25
30
35
40
, .
Figure 3: Running time vs. sequence length. Both algorithms were implemented in Matlab. The
HHMM has
(see Section 4.2). In this case, we can exploit the form of the CPD to compute the required
messages efficiently [Mur99], bringing the overall complexity down to
.
We remark that all of the above algorithms can also be used for online filtering. In addition,
by replacing the sum operator with max, we can do Viterbi segmentation in the usual way.
6 Related work
)
Hidden Markov decision trees (HMDT) [JGS96] are DBNs with a structure similar to Figure 2, but they lack the nodes and the upward going arcs; hence they are not able to
represent the call-return semantics of the HHMM. Embedded HMMs [NI00] are a special
case of HHMMs in which the ending ?time? of the sub-HMMs is known in advance (e.g.,
the sub-HMM models exactly one row of pixels). ([Hoe01] calls these models ?hierarchical mixture of Markov chains?.) A variable-duration HMM [Rab89] is a special case of a
2-level HHMM, where the bottom level counts how long we have been in a certain state;
when the counter expires, the node turns on, and the parent can change state.
)
[BVW00] describes the ?Abstract HMM? (AHMM), which is very closely related to HHMMs. These authors are interested in inferring what abstract policy an agent is following
by observing its effects in the world. An AHMM is equivalent to an HHMM if we consider
! to represent the (abstract) policy being followed at level and time ;
!
represents
the concrete action, which causes the observation. We also need to add a hidden global
nodes.
state variable ! , which is a parent of the ! node, all the ! nodes and all the !
( ! is hidden to us as observers, but not to the agent performing the actions.) [BVW00]
consider abstract policies of the ?options? kind [SPS99], which is equivalent to assuming
that there are no horizontal transitions. (HAMs [PR97] generalize this by allowing horizontal transitions (i.e., internal state) within a controller.) In addition, they assume that !
only depends on its immediate parent, ! , but not its whole context, !
, so the
nodes become connected by a chain. This enables them to use Rao-Blackwellized particle
filtering for approximate online inference: conditioned on the nodes, the distribution
over the nodes can be represented as a product of marginals, so they can be efficiently
marginalized out.
#
#
)
)
#
#
Acknowledgements
I would like to thank Dr Christopher Schlick for giving me his Matlab implementation of
the
algorithm, which was used to create part of Figure 3.
References
[BFGK96] C. Boutilier, N. Friedman, M. Goldszmidt, and D. Koller. Context-Specific Independence in Bayesian Networks. In
UAI, 1996.
[BVW00]
H. Bui, S. Venkatesh, and G. West. On the recognition of abstract Markov policies. In AAAI, 2000.
[BVW01]
H. Bui, S. Venkatesh, and G. West. Tracking and surveillance in wide-area spatial environments using the Abstract
Hidden Markov Model. Intl. J. of Pattern Rec. and AI, 2001.
[FST98]
Shai Fine, Yoram Singer, and Naftali Tishby. The hierarchical Hidden Markov Model: Analysis and applications.
Machine Learning, 32:41, 1998.
[GJ97]
Z. Ghahramani and M. Jordan. Factorial hidden Markov models. Machine Learning, 29:245?273, 1997.
[HIM 00] M. Hu, C. Ingram, M.Sirski, C. Pal, S. Swamy, and C. Patten. A Hierarchical HMM Implementation for Vertebrate
Gene Splice Site Prediction. Technical report, Dept. Computer Science, Univ. Waterloo, 2000.
[Hoe01]
J. Hoey. Hierarchical unsupervised learning of facial expression categories. In ICCV Workshop on Detection and
Recognition of Events in Video, 2001.
[IB00]
Y. Ivanov and A. Bobick. Recognition of visual activities and interactions by stochastic parsing. IEEE Trans. on
Pattern Analysis and Machine Intelligence, 22(8):852?872, 2000.
[JGS96]
M. I. Jordan, Z. Ghahramani, and L. K. Saul. Hidden Markov decision trees. In NIPS, 1996.
[Kja90]
U. Kjaerulff. Triangulation of graphs ? algorithms giving small total state space. Technical Report R-90-09, Dept. of
Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.
[LY90]
K. Lari and S. J. Young. The estimation of stochastic context-free grammars using the Inside-Outside algorithm.
Computer Speech and Language, 4:35?56, 1990.
[ME01]
D. Moore and I. Essa. Recognizing multitasked activities using stochastic context-free grammar. In CVPR Workshop
on Models vs Exemplars in Computer Vision, 2001.
[Mur99]
K. Murphy. Pearl?s algorithm and multiplexer nodes. Technical report, U.C. Berkeley, Dept. Comp. Sci., 1999.
[Mur01]
K. Murphy. Applying the junction tree algorithm to variable-length DBNs. Technical report, Comp. Sci. Div., UC
Berkeley, 2001.
[MW01]
K. Murphy and Y. Weiss. The Factored Frontier Algorithm for Approximate Inference in DBNs. In UAI, 2001.
[NI00]
A. Nefian and M. Hayes III. Maximum likelihood training of the embedded HMM for face detection and recognition.
In IEEE Intl. Conf. on Image Processing, 2000.
[PR97]
R. Parr and S. Russell. Reinforcement learning with hierarchies of machines. In NIPS, 1997.
[Rab89]
L. R. Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proc. of the
IEEE, 77(2):257?286, 1989.
[SJ99]
L. Saul and M. Jordan. Mixed memory markov models: Decomposing complex stochastic processes as mixture of
simpler ones. Machine Learning, 37(1):75?87, 1999.
[SPS99]
R.S. Sutton, D. Precup, and S. Singh. Between MDPs and semi-MDPs: A framework for temporal abstraction in
reinforcement learning. Artificial Intelligence, 112:181?211, 1999.
[TRM01]
G. Theocharous, K. Rohanimanesh, and S. Mahadevan. Learning Hierarchical Partially Observed Markov Decision
Process Models for Robot Navigation. In ICRA, 2001.
[Zwe97]
G. Zweig. Speech Recognition with Dynamic Bayesian Networks. PhD thesis, U.C. Berkeley, Dept. Comp. Sci.,
1997.
| 2050 |@word middle:2 version:5 glue:1 hu:1 multitasked:1 thereby:4 solid:1 recursively:1 initial:1 configuration:1 contains:1 current:1 yet:1 must:5 parsing:1 cpds:5 enables:1 designed:1 v:3 alone:1 generative:1 leaf:1 fewer:1 greedy:1 intelligence:2 selected:1 math:1 node:30 simpler:3 height:1 blackwellized:1 become:1 inside:5 expected:1 themselves:1 terminal:4 ivanov:1 param:1 vertebrate:1 bounded:1 what:1 kind:2 string:4 impractical:1 temporal:3 berkeley:5 every:2 exactly:1 returning:1 murphyk:1 control:4 continually:1 before:1 local:1 theocharous:1 sutton:1 path:1 merge:1 plus:1 specifying:1 shaded:1 hmms:9 limited:1 range:1 practice:1 differs:1 area:1 flatten:1 regular:3 get:4 cannot:1 operator:1 scheduling:1 context:9 applying:3 writing:1 restriction:1 equivalent:3 straightforward:1 starting:1 duration:1 automaton:6 simplicity:2 factored:2 rule:3 nesting:1 his:1 handle:1 dbns:6 hierarchy:8 imagine:1 exact:1 losing:1 trick:1 recognition:7 rec:1 bottom:4 observed:2 enters:4 ensures:1 connected:1 counter:1 rescaled:1 russell:1 ham:1 environment:1 complexity:6 dynamic:4 terminating:1 singh:1 rewrite:1 upon:1 exit:1 expires:1 easily:1 represented:4 alphabet:2 train:2 univ:2 describe:2 effective:1 artificial:1 kevin:1 outside:5 encoded:2 larger:1 valued:1 cvpr:1 say:1 drawing:2 otherwise:3 consume:1 grammar:5 ability:1 statistic:1 final:1 online:3 higherlevel:1 sequence:10 advantage:1 essa:1 clamp:1 interconnected:1 product:2 maximal:1 adaptation:1 interaction:1 bobick:1 entered:1 exploiting:1 parent:8 intl:2 derive:2 illustrate:1 exemplar:1 nonterminal:2 implemented:1 c:1 closely:1 stochastic:10 enable:2 memorized:1 generalization:1 ly90:2 extension:1 frontier:2 hold:1 considered:1 viterbi:1 parr:1 adopt:1 omitted:1 estimation:1 proc:1 currently:1 waterloo:1 him:2 create:1 clearly:1 always:2 gaussian:1 rather:1 lifted:1 surveillance:1 quence:1 likelihood:1 inference:23 abstraction:1 typically:1 eliminate:1 hidden:13 relation:1 koller:1 going:3 subroutine:1 interested:1 semantics:1 upward:2 overall:3 pixel:1 plan:1 spatial:2 special:3 smoothing:1 softmax:1 ference:1 uc:1 once:2 never:2 identical:1 represents:4 look:1 unsupervised:1 patten:1 np:1 cpd:1 report:4 aalborg:1 murphy:4 recogntion:1 friedman:1 freedom:1 detection:2 message:2 possibility:1 navigation:2 mixture:5 yielding:1 chain:5 emit:5 facial:1 tree:10 hmdt:1 re:1 rao:1 ence:1 loopy:1 cost:3 entry:1 recognizing:1 too:1 tishby:1 pal:1 off:2 together:1 precup:1 concrete:1 again:1 aaai:1 thesis:1 dr:1 stochastically:1 creating:1 conf:1 return:2 multiplexer:1 converted:1 satisfy:1 depends:3 root:5 observer:1 doing:1 analyze:1 observing:1 start:3 option:1 complicated:2 shai:1 substructure:1 contribution:1 formed:1 efficiently:2 rabiner:1 generalize:1 bayesian:5 eters:1 comp:4 definition:2 handwriting:1 con:1 emits:2 duplicated:1 segmentation:1 actually:1 higher:2 specify:1 wei:1 furthermore:2 just:2 until:2 sketch:1 horizontal:6 parse:2 christopher:1 expressive:1 replacing:1 unsegmented:1 propagation:2 lack:1 effect:1 hence:5 moore:1 illustrated:1 naftali:1 generalized:1 complete:2 interface:3 image:1 multinomial:2 empirically:2 conditioning:2 discussed:2 marginals:2 interpret:1 enter:3 ai:1 dbn:18 similarly:1 particle:1 language:2 robot:1 add:1 triangulation:1 termed:1 certain:1 binary:1 fortunately:1 somewhat:1 converting:1 signal:3 semi:1 full:4 segmented:1 technical:4 match:1 faster:1 long:1 zweig:1 hhmm:24 prediction:1 controller:1 vision:1 represent:14 addition:3 separately:2 fine:1 diagram:5 unlike:1 bringing:1 jordan:3 emitted:2 call:4 integer:1 backwards:2 intermediate:2 iii:1 easy:1 mahadevan:1 independence:1 topology:2 reduce:1 whether:2 expression:4 reuse:1 returned:2 speech:3 cause:2 action:3 matlab:2 remark:1 boutilier:1 generally:1 se:1 factorial:1 category:1 dna:1 simplest:1 specifies:2 tutorial:1 dotted:1 dummy:1 mega:1 discrete:2 write:3 shall:1 key:1 drawn:1 graph:1 sum:1 run:1 letter:1 powerful:1 family:1 reader:1 parsimonious:2 decision:5 layer:2 bound:1 followed:1 fan:1 activity:2 ahead:1 scfgs:2 flat:1 encodes:1 calling:2 generates:4 speed:1 performing:2 speedup:1 department:1 remain:2 smaller:2 em:4 describes:1 making:1 wherever:1 intuitively:1 iccv:1 hoey:1 taken:1 legal:1 equation:3 lari:1 discus:4 turn:4 count:1 needed:4 know:1 singer:1 tractable:1 end:17 junction:3 gaussians:1 decomposing:1 multiplied:1 apply:1 hierarchical:14 enforce:1 distinguished:1 scfg:1 swamy:1 original:5 top:5 remaining:1 ensure:1 running:2 marginalized:1 exploit:1 giving:2 yoram:1 ghahramani:2 icra:1 sweep:1 occurs:1 usual:1 div:1 thank:1 sci:4 hmm:19 me:1 denmark:1 assuming:2 length:10 remind:1 minimizing:1 optionally:2 equivalently:1 unfortunately:4 rise:1 implementation:2 proper:1 policy:4 perform:1 allowing:1 upper:1 vertical:4 observation:11 markov:15 arc:5 immediate:1 stack:4 venkatesh:2 pair:1 required:2 connection:2 california:1 learned:2 pearl:1 nip:2 trans:1 able:2 below:3 pattern:2 max:1 memory:1 video:1 belief:2 event:1 natural:1 indicator:1 representing:5 xml:1 brief:1 mdps:2 finished:7 text:1 prior:2 acknowledgement:1 embedded:2 fully:1 mixed:1 hhmms:14 filtering:3 proportional:1 kjaerulff:1 degree:1 agent:2 sufficient:1 storing:1 production:8 row:1 last:1 free:5 offline:1 wide:1 saul:2 face:1 midpoint:1 sparse:2 ingram:1 slice:7 depth:4 transition:25 world:2 ending:1 computes:2 forward:2 author:1 reinforcement:2 approximate:4 bui:2 gene:1 clique:5 global:1 incoming:2 uai:2 hayes:1 assumed:1 ahmm:2 subsequence:1 continuous:1 why:1 table:2 learn:3 reasonably:1 rohanimanesh:1 ca:1 complex:1 domain:2 ifif:2 did:1 main:1 whole:2 child:1 allowed:1 site:1 west:2 ff:1 cubic:2 sub:12 position:1 inferring:1 exponential:1 governed:1 crude:1 jtree:5 young:1 splice:1 down:1 specific:1 paskin:2 symbol:9 workshop:2 phd:1 downward:1 conditioned:1 easier:1 simply:1 likely:1 visual:2 tracking:1 partially:1 conditional:1 exposition:1 shared:2 content:1 change:4 hard:2 determined:2 except:2 called:3 total:2 formally:3 internal:1 mark:1 goldszmidt:1 arises:2 dept:4 |
1,153 | 2,051 | A General Greedy Approximation Algorithm
with Applications
Tong Zhang
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598
tzhang@watson.ibm.com
Abstract
Greedy approximation algorithms have been frequently used to obtain
sparse solutions to learning problems. In this paper, we present a general
greedy algorithm for solving a class of convex optimization problems.
We derive a bound on the rate of approximation for this algorithm, and
show that our algorithm includes a number of earlier studies as special
cases.
1 Introduction
The goal of machine learning is to obtain a certain input/output functional relationship from
a set of training examples. In order to do so, we need to start with a model of the functional
relationship. In practice, it is often desirable to find the simplest model that can explain the
data. This is because simple models are often easier to understand and can have significant
computational advantages over more complicated models. In addition, the philosophy of
Occam?s Razor implies that the simplest solution is likely to be the best solution among all
possible solutions,
In this paper, we are interested in composite models that can be expressed as linear combinations of basic models. In this framework, it is natural to measure the simplicity of a
composite model by the number of its basic model components. Since a composite model
in our framework corresponds to a linear weight over the basic model space, therefore our
measurement of model simplicity corresponds to the sparsity of the linear weight representation.
In this paper, we are interested in achieving sparsity through a greedy optimization algorithm which we propose in the next section. This algorithm is closely related to a number of
previous works. The basic idea was originated in [5], where Jones observed that if a target
vector in a Hilbert space is a convex combination of a library of basic vectors, then using
with basic library vecgreedy approximation, one can achieve an error rate of
tors. The idea has been refined in [1] to analyze the approximation property of sigmoidal
functions including neural networks.
The above methods can be regarded as greedy sparse algorithms for functional approximation, which is the noise-free case of regression problems. A similar greedy algorithm
can also be used to solve general regression problems under noisy conditions [6]. In addition to regression, greedy approximation can also be applied to classification problems.
The resulting algorithm is closely related to boosting [2] under the additive model point of
view [3]. This paper shows how to generalize the method in [5, 1] for analyzing greedy
algorithms (in their case, for functional approximation problems) and apply it to boosting.
Detailed analysis will be given in Section 4. Our method can also be used to obtain sparse
kernel representations for regression problems. Such a sparse representation is what support vector regression machines try to achieve. In this regard, the method given in this
paper complements some recently proposed greedy kernel methods for Gaussian processes
such as [9, 10].
The proposed greedy approximation method can also be applied to other prediction problems with different loss functions. For example, in density estimation, the goal is to find a
model that has the smallest negative log-likelihood. A greedy algorithm was analyzed in
[7]. Similar approximation bounds can be directly obtained under the general framework
proposed in this paper.
We proceed as follows. Section 2 formalizes the general class of problems considered in
this paper, and proposes a greedy algorithm to solve the formulation. The convergence rate
of the algorithm is investigated in Section 3. Section 4 includes a few examples that can be
obtained from our algorithm. Some final concluding remarks are given in Section 5.
2 General Algorithm
In machine learning, our goal is often to predict an unobserved output value based on an
observed input vector . This requires us to estimate a functional relationship
from a set of example pairs of
. Usually the quality of the predictor
can be
measured by a loss function
that is problem dependent.
In this paper, we are interested
in the following scenario: given a family
of basic predictors
parameterized
by , we want to obtain a good predictor that lies in the convex
with the fewest possible terms:
, where are nonhull of
. This family of models can be regarded as additive
negative weights so that
can be regarded as a vector
models in statistics [4]. Formally, each basic model
in a linear functional space.
Our problem in its
most
general form can thus be described
to minimize a functional of that
as to find a vector in the convex hull of
measures the quality of . This functional of plays the role of loss function for learning
problems.
More formally, we consider a linear vector space , and a subset ! . Denote by "$#
the convex hull of :
"$# %
'&)+* (-, +/.0+21 , +2354 +* (6, +
.7+ 8 9:; =<?>A@B
where we use < > to denote the set of positive integers.
We consider the following optimization problem on "$# :
CDFE Q $R
(1)
GIHKJML$NOBP
In this paper, we assume that is a differentiable convex function on "/# .
We propose the following algorithm to approximately solved (1).
Algorithm 2.1 (Sparse greedy approximation)
Q
8 "/ # R R/ R
Q = and 4 that minimize
Q
, Q M Q
, ,
Q Q
let
given
for
find
,
end
,
For simplicity, we assume that the minimization of
in Algorithm 2.1 can be exactly
achieved at each step. This assumption is not essential, and can be easily removed using a
slightly more refined analysis. However due to the space limitation, we shall not consider
this generalization.
Q %
Q G HKCD7JML/E NOBP Q $R
For convenience, we introduce the following quantity
Q
4
In
next
as
section, we show that under appropriate regularity conditions,
the
, where
is computed from Algorithm 2.1. In addition, the convergence rate
can be bounded as
.
Q
3 Approximation bound
Given any convex function , we have the following proposition, which is a direct consequence of the definition of convexity. In convex analysis, The gradient
can be replaced
by the concept of subgradient, which we do not consider in this paper for simplicity.
Proposition 3.1 Consider a convex function
, and two vectors and , we have
where
Q
Q Q 3 Q Q Q
is the gradient of .
Q
Q
The following lemma is the main theoretical result of the paper, which bounds the performance of each greedy solution step in Algorithm 2.1. We assume that is second order
differentiable.
Lemma 3.1 Let
"!
Q
G G KH JML$NOBP
Q 3
MQ
CDFE
H G HO
Q Q
$
#
$ of exists everywhere in
where we
that
:assume
&the% Hessian
if
, we have
,
+
+
' )( #
* #
&%
if
, we have
.,
+
+
/
' )( #
* #
Q !"$#
Q
H
CDFE
G HO
Q 8"$# Q !
4
,+
MQ
+BQ Q
MQ
Q
Q
"$# . For all vectors
Q $ R
Q
Proof. Using Taylor expansion+ and the21 definition of
for all
,
, and 0
,
Q
+ 3
, we have the following inequality
Q Q
+
$
%
$R
Q + =
Q+
,
* ( + ,+
Q
+ Q + Q + * (
+ ,
+ ,
+2354Q
It is easy to see that this implies the inequality
CDF+ E
+ MQ
+
Q + Q
+
+( +
,
R
R R/R :
Now, consider two sequences
and
(
Multiply the above inequality (with replaced by ) by
+
), such that
.
, and sum over , we obtain
+
$
,
+ Q + Q ) Q
A * (
+ Q + Q Q
+ ,
+
$
R
Using Proposition 3.1, we obtain
CDF+ E +
MQ
+ Q+ Q + * ( + Q+ Q
+ $ R
+ ,
Q
Q
Since in the above, + and + are arbitrary, therefore +( + + can be used to express
Q
,
,
any vector 8"$# . This implies
CDFE ,+
MQ
+ Q Q
+ CDF E Q Q
+ $ R
G H O
G
+
Q
CDFE Q %
in the above inequality, we obtain
C
D
G
Now by setting
the lemma.
Q
Using the above lemma and note that
, it is easy to obtain the following
theorem by induction. For space limitation, we skip the proof.
3
Q
Theorem 3.1 Under the assumptions of Lemma 3.1, Algorithm 2.1 approximately solves
(1), and the rate of convergence for
is given by
/
&%
If
, then we also have
/
Q
R
Q
N G
P R
4 Examples
In this section, we discuss the application of Algorithm 2.1 in some learning problems.
We show that the general formulation considered in this paper includes some previous
formulations as special cases. We will also compare our results with similar results in the
literature.
4.1 Regression
so that the expected loss of
# $
In regression, we would like to approximate as
is small, where we use the squared loss for simplicity (this choice is obviously not crucial
# is the expectation over and , which often corresponds to the
in our framework).
empirical distribution of
pairs. It may also represent the true distribution for some
other engineering applications. Given a set of basis functions
with
, we may
consider the following regression formulation that is slightly different from (1):
CDF E # *
$
*
s.t.
(2)
where
is a positive regularization parameter which is used to control the size of the
weight vector . The above formulation can be readily converted into (1) by considering
the following set of basic vectors:
&
1
@ R
4 (Q
4 ) in Algorithm 2.1. Since the quantity in Lemma 3.1
We may start with
can be bounded as
"! $
$ R
Q
This
implies
that
the sparse solution in Algorithm 2.1, represented as weight
/
R
R
R
and (
), satisfies the following inequality:
# *
$
CDFE
# * ( +
+ $
$ 3!
$
#
+
for all 3
. This leads to the original functional approximation results in [1, 5] and its
generalization in [6].
& /R R R 6@
The sparse regression algorithm studied in this section can also be applied to kernel methods. In this case, corresponds to the input training data space
, and the basis
. Clearly, this corresponds to a special case of
predictors are of the form
(2). A sparse kernel representation can be obtained easily from Algorithm 2.1 which leads
to provably good approximation rate. Our sparse kernel regression formulation is related
to Gaussian processes, where greedy style algorithms have also been proposed [9, 10]. The
bound given here is comparable to the bound given in [10] where a sparse approximation
rate of the form
was obtained.
4.2 Binary classification and Boosting
& @
In binary classification, the output value
is a discrete variable. Given a continuous model
, we consider the following prediction rule:
3 4 R
5
4
4 , which is assumed to occur
The classification error (we shall ignore the point
rarely) can be given by
if 54 R
4 if 4
if
if
Unfortunately, this classification error function is not convex, which cannot be handled in
our formulation. In fact, even in many other popular methods, such as logistic regression
and support vector machines, some kind of convex formulations have to be employed.
Although it is possible for us to analyze their formulations, in this section, we only consider
the following form of loss that is closely related to Adaboost [2]:
!
#
(3)
D
where is a scaling factor.
0 21 , which are often called weak
Again, we consider a set of basis predictors
learners in the boosting literature. We would like to find a strong learner as a convex
combination of weak learners to approximately minimize the above loss:
CD7E D # ! *
(4)
* 354 R
s.t.
(5)
&
1 4
This can be written as formulation (1) with
@R
$ R
$
HKJML$NOBP
4 in Algorithm 2.1. Theorem 3.1 implies that the sparse solution Q ,
We start with
R R/R ), satisfies the following inequality:
represented as weight and (
# ! *
C DF E
# ! * ( +
+
/
$
#
+
Using simple algebra, it is easy to verify that
!
#
"!
$
!
#
#
.
for all 3
(6)
4
Weight in the above inequality is non-negative. Now we consider the
special situation that there exists
such that
+ 354
+
*
C DF E #
# ! + ( +
+ ! $R
4
(7)
* ( +
+ 3 R
+
This condition
will be satisfied in the large margin linearly separable case where there exists
and
such that
and for all data
,
Now, under (7), we obtain from (6) that
*
!
/
$ $R
to obtain
Fix any 3 , we can choose
*
! $
R
(8)
This implies that the misclassification error rate decays exponentially. The exponential decay of misclassification error is the original motivation of Adaboost [2]. Boosting was later
viewed as greedy approximation in the additive model framework [3]. From the learning
theory perspective, the good generalization ability of boosting is related to its tendency to
improve the misclassification error under a positive margin [8]. From this point of view,
inequality (8) gives a much more explicit margin error bound (which decreases exponentially) than a related result in [8].
In the framework of additive models, Adaboost corresponds to the exponential loss (3)
analyzed in this section. As pointed out in [3], other loss functions can also be used.
Using our analysis, we may also obtain sparse approximation bounds for these different
loss functions. However, it is also easy to observe that they will not lead to the exponential
decay of classification error in the separable case. Although the exponential loss in (3) is
attractive for separable problems due to the exponential decay of margin error, it is very
sensitive to outliers in the non-separable case.
We shall mention that an interesting aspect of boosting is the concept of adaptive resampling or sample reweighting. Although this idea has dominated the interpretation of boosting algorithms, it has been argued in [3] that adaptive resampling is only a computational
by-product.
The idea corresponds to a Newton step approximation in the sparse greedy
solution of
in Algorithm 2.1 under the additive model framework which we consider
here. Our analysis further confirmed that the greedy sparse solution of an additive model
in (1), rather than reweighting itself is the key component in boosting. In our framework,
it is also much easier to related the idea of boosting to the greedy function approximation
method outlined in [1, 5].
4.3 Mixture density estimation
In mixture density estimation, the output is the probability density function of the input
vector at . The following negative log-likelihood is commonly used as loss function:
D
where 3 4 is a probability density function.
, which are often called mixture comAgain, we consider a set of basis predictors
ponents. We would like to find a mixture probability density model as a convex combination of mixture components to approximately minimize the negative log-likelihood:
CDFE D *
(9)
*
%3 4 R
s.t.
(10)
This problem was studied in [7]. The quantity
defined in Lemma 3.1 can be computed
as:
"!
"!
$
$
#
#
$
$
$
$
An approximation bound can now be directly obtained from Theorem 3.1. It has a form
similar to the bound given in [7].
R
N P N P HKJML/NOBP
5 Conclusion
This paper studies a formalization of a general class of prediction problems in machine
learning, where the goal is to approximate the best model as a convex combination of
a family of basic models. The quality of the approximation can be measured by a loss
function which we want to minimize.
We proposed a greedy algorithm to solve the problem, and we have shown that for a variety
of loss functions, a convergence rate of
can be achieved using a convex combination of basic models. We have illustrated the consequence of this general algorithm in
regression, classification and density estimation, and related the resulting algorithms to
previous methods.
References
[1] A.R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory, 39(3):930?945, 1993.
[2] Y. Freund and R.E. Schapire. A decision-theoretic generalization of on-line learning
and an application to boosting. J. Comput. Syst. Sci., 55(1):119?139, 1997.
[3] Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Additive logistic regression:
A statistical view of boosting. The Annals of Statistics, 28(2):337?407, 2000. With
discussion.
[4] T. J. Hastie and R. J. Tibshirani. Generalized additive models. Chapman and Hall
Ltd., London, 1990.
[5] Lee K. Jones. A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training. Ann.
Statist., 20(1):608?613, 1992.
[6] Wee Sun Lee, P.L. Bartlett, and R.C. Williamson. Efficient agnostic learning of
neural networks with bounded fan-in. IEEE Transactions on Information Theory,
42(6):2118?2132, 1996.
[7] Jonathan Q. Li and Andrew R. Barron. Mixture density estimation. In S.A. Solla, T.K.
Leen, and K.-R. M?uller, editors, Advances in Neural Information Processing Systems
12, pages 279?285. MIT Press, 2000.
[8] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the
margin: a new explanation for the effectiveness of voting methods. Ann. Statist.,
26(5):1651?1686, 1998.
[9] Alex J. Smola and Peter Bartlett. Sparse greedy Gaussian process regression. In
Advances in Neural Information Processing Systems 13, pages 619?625, 2001.
[10] Tong Zhang. Some sparse approximation bounds for regression problems. In The
Eighteenth International Conference on Machine Learning, pages 624?631, 2001.
| 2051 |@word concept:2 skip:1 implies:6 true:1 verify:1 regularization:1 strong:1 closely:3 quantity:3 hull:2 illustrated:1 attractive:1 gradient:2 mention:1 razor:1 argued:1 yorktown:1 sci:1 originated:1 fix:1 generalization:4 generalized:1 proposition:3 theoretic:1 induction:1 com:1 considered:2 hall:1 relationship:3 obp:5 recently:1 written:1 readily:1 predict:1 unfortunately:1 additive:8 functional:9 robert:2 tor:1 smallest:1 negative:5 exponentially:2 estimation:5 resampling:2 interpretation:1 greedy:22 superposition:1 significant:1 measurement:1 sensitive:1 outlined:1 minimization:1 uller:1 mit:1 boosting:13 clearly:1 gaussian:3 pointed:1 situation:1 sigmoidal:2 rather:1 zhang:2 arbitrary:1 height:1 direct:1 complement:1 pair:2 perspective:1 likelihood:3 introduce:1 scenario:1 certain:1 inequality:8 expected:1 binary:2 watson:2 frequently:1 dependent:1 usually:1 employed:1 sparsity:2 considering:1 interested:3 including:1 bounded:3 provably:1 desirable:1 agnostic:1 among:1 classification:7 what:1 explanation:1 kind:1 proposes:1 misclassification:3 natural:1 special:4 hkc:1 unobserved:1 improve:1 formalizes:1 chapman:1 library:2 prediction:3 jones:2 voting:1 basic:11 regression:16 expectation:1 exactly:1 df:7 kernel:5 control:1 few:1 represent:1 achieved:2 literature:2 wee:2 addition:3 positive:3 want:2 engineering:1 freund:2 loss:14 replaced:2 consequence:2 jml:1 crucial:1 interesting:1 limitation:2 analyzing:1 friedman:1 approximately:4 multiply:1 studied:2 effectiveness:1 integer:1 analyzed:2 mixture:6 editor:1 explicit:1 occam:1 ibm:2 easy:4 variety:1 free:1 hastie:2 practice:1 idea:5 understand:1 bq:1 taylor:1 sparse:16 universal:1 empirical:1 handled:1 bartlett:3 composite:3 projection:1 theoretical:1 ltd:1 regard:1 earlier:1 peter:2 commonly:1 adaptive:2 convenience:1 cannot:1 proceed:1 yoav:1 hessian:1 remark:1 transaction:2 approximate:2 subset:1 detailed:1 ignore:1 predictor:6 center:1 eighteenth:1 statist:2 convex:15 simplest:2 assumed:1 simplicity:5 schapire:2 rule:1 continuous:1 density:8 regarded:3 international:1 tibshirani:2 mq:6 lee:3 discrete:1 shall:3 annals:1 target:1 play:1 express:1 squared:1 again:1 satisfied:1 key:1 achieving:1 choose:1 expansion:1 williamson:1 investigated:1 main:1 linearly:1 motivation:1 noise:1 style:1 subgradient:1 li:1 syst:1 sum:1 converted:1 observed:2 role:1 parameterized:1 everywhere:1 solved:1 includes:3 tzhang:1 family:3 ny:1 sun:2 tong:2 formalization:1 solla:1 decrease:1 removed:1 later:1 view:3 try:1 decision:1 analyze:2 scaling:1 convexity:1 start:3 comparable:1 bound:12 complicated:1 exponential:5 comput:1 lie:1 fan:1 theorem:4 solving:1 minimize:5 algebra:1 occur:1 alex:1 decay:4 learner:3 basis:4 easily:2 generalize:1 weak:2 dominated:1 aspect:1 represented:2 concluding:1 essential:1 exists:3 separable:4 fewest:1 confirmed:1 london:1 margin:5 easier:2 explain:1 combination:6 refined:2 trevor:1 slightly:2 definition:2 likely:1 solve:3 expressed:1 ability:1 statistic:2 proof:2 outlier:1 noisy:1 itself:1 final:1 corresponds:7 obviously:1 popular:1 advantage:1 differentiable:2 sequence:1 satisfies:2 discus:1 goal:4 propose:2 hilbert:2 viewed:1 product:1 ann:2 ponents:1 end:1 philosophy:1 pursuit:1 adaboost:3 achieve:2 apply:1 observe:1 formulation:10 leen:1 barron:2 kh:1 appropriate:1 lemma:8 called:2 smola:1 tendency:1 jerome:1 convergence:5 regularity:1 ho:2 rarely:1 original:2 formally:2 support:2 reweighting:2 jonathan:1 derive:1 andrew:1 logistic:2 newton:1 quality:3 measured:2 dfe:6 solves:1 |
1,154 | 2,052 | Variance Reduction Techniques for Gradient
Estimates in Reinforcement Learning
Evan Greensmith
Australian National University
evan@csl.anu.edu.au
Peter L. Bartlett?
BIOwulf Technologies
Peter.Bartlett@anu.edu.au
Jonathan Baxter?
WhizBang! Labs, East
jbaxter@whizbang.com
Abstract
We consider the use of two additive control variate methods to reduce
the variance of performance gradient estimates in reinforcement learning problems. The first approach we consider is the baseline method,
in which a function of the current state is added to the discounted value
estimate. We relate the performance of these methods, which use sample paths, to the variance of estimates based on iid data. We derive the
baseline function that minimizes this variance, and we show that the variance for any baseline is the sum of the optimal variance and a weighted
squared distance to the optimal baseline. We show that the widely used
average discounted value baseline (where the reward is replaced by the
difference between the reward and its expectation) is suboptimal. The
second approach we consider is the actor-critic method, which uses an
approximate value function. We give bounds on the expected squared
error of its estimates. We show that minimizing distance to the true value
function is suboptimal in general; we provide an example for which the
true value function gives an estimate with positive variance, but the optimal value function gives an unbiased estimate with zero variance. Our
bounds suggest algorithms to estimate the gradient of the performance of
parameterized baseline or value functions. We present preliminary experiments that illustrate the performance improvements on a simple control
problem.
1 Introduction, Background, and Preliminary Results
In reinforcement learning problems, the aim is to select a controller that will maximize
the average reward in some environment. We model the environment as a partially observable Markov decision process (POMDP). Gradient ascent methods (e.g., [7, 12, 15])
estimate the gradient of the average reward, usually using Monte Carlo techniques to cal?
Most of this work was performed while the authors were with the Research School of Information
Sciences and Engineering at the Australian National University.
culate an average over a sample path of the controlled POMDP. However such estimates
tend to have a high variance, which means many steps are needed to obtain a good estimate. GPOMDP [4] is an algorithm for generating an estimate of the gradient in this way.
Compared with other approaches, it is suitable for large systems, when the time between
visits to a state is large but the mixing time of the controlled POMDP is short. However, it
can suffer from the problem of producing high variance estimates. In this paper, we investigate techniques for variance reduction in GPOMDP. One generic approach to reducing the
variance of Monte Carlo estimates of integrals is to use an additive control variate (see, for
example, [6]). Suppose we wish to estimate the integral
ofR f : X ? R, Rand we know the
R
integral of another function ? : X ? R. Since X f = X (f ? ?) + X ?, the integral
of f ? ? can be estimated instead. Obviously if ? = f then the variance is zero. More
generally, Var(f ? ?) = Var(f ) ? 2Cov(f, ?) + Var(?), so that if ? and f are strongly
correlated, the variance of the estimate is reduced.
In this paper, we consider two approaches of this form. The first (Section 2) is the technique
of adding a baseline. We find the optimal baseline and we show that the additional variance
of a suboptimal baseline can be expressed as a weighted squared distance from the optimal
baseline. Constant baselines, which do not depend on the state or observations, have been
widely used [13, 15, 9, 11]. In particular, the expectation over all states of the discounted
value of the state is a popular constant baseline (where, for example, the reward at each
step is replaced by the difference between the reward and the expected reward). We give
bounds on the estimation variance that show that, perhaps surprisingly, this may not be the
best choice.
The second approach (Section 3) is the use of an approximate value function. Such actorcritic methods have been investigated extensively [3, 1, 14, 10]. Generally the idea is
to minimize some notion of distance between the fixed value function and the true value
function. In this paper we show that this may not be the best approach: selecting the fixed
value function to be equal to the true value function is not always the best choice. Even
more surprisingly, we give an example for which the use of a fixed value function that is
different from the true value function reduces the variance to zero, for no increase in bias.
We give a bound on the expected squared error (that is, including the estimation variance)
of the gradient estimate produced with a fixed value function. Our results suggest new
algorithms to learn the optimum baseline, and to learn a fixed value function that minimizes
the bound on the error of the estimate. In Section 5, we describe the results of preliminary
experiments, which show that these algorithms give performance improvements.
POMDP with Reactive, Parameterized Policy
A partially observable Markov decision process (POMDP) consists of a state space, S,
a control space, U, an observation space, Y, a set of transition probability matrices
{P(u) : u ? U}, each with components pij (u) for i, j ? S, u ? U, an observation process ? : S ? PY , where PY is the space of probability distributions over Y, and a
reward function r : S ? R. We assume that S, U, Y are finite, although all our results extend easily to infinite U and Y, and with more restrictive assumptions can be
extended to infinite S. A reactive, parameterized policy for a POMDP is a set of mappings {?(?, ?) : Y ? PU |? ? RK }. Together with the POMDP, this defines the controlled POMDP (S, U, Y, P , ?, r, ?). The joint state, observation and control process,
{Xt , Yt , Ut }, is Markov.
The state process, {Xt }, is also Markov, with transition probP
abilities pij (?) = y?Y,u?U ?y (i)?u (y, ?)pij (u), where ?y (i) denotes the probability of
observation y given the state i, and ?u (y, ?) denotes the probability of action u given parameters ? and observation y. The Markov chain M(?) = (S, P(?)) then describes the
behaviour of the process {Xt }.
Assumption 1 The controlled POMDP (S, U, Y, P , ?, r, ?) satisfies:
For all ? ? RK there exists a unique stationary distribution satisfying ? 0 (?) P(?) = ? 0 (?).
There is an R < ? such that, for all i ? S, |r(i)| ? R.
There is a B < ? such that, for all u ? U, y ? Y and ? ? RK the derivatives ??u (y, ?)/??k (1 ? k ? K) exist, and the vector of these derivatives satisfies
k??u (y, ?)/?u (y, ?)k ? B, where k ? k denotes the Euclidean norm on RK .
h P
i
def
T ?1
We consider the average reward, ?(?) = limT ?? E T1 t=0 r(Xt ) . Assumption 1
implies that this limit exists, and does not depend on the start state X0 . The aim is to
def
select a policy
quantity.i Define the discounted value function, J ? (i, ?) =
h Pto maximize this
T ?1 t
limT ?? E
t=0 ? r(Xt ) X0 = i . Throughout the rest of the paper, dependences
upon ? are assumed,
and dropped
in the notation. For a random vector A, we denote
h
i
Var(A) = E (A ? E [A])2 , where a2 denotes a0 a, and a0 denotes the transpose of the
column vector a.
GPOMDP Algorithm
The GPOMDP
h algorithm i[4] uses a sample path to estimate the gradient approximation
def
u(y)
?? ? = E ??
?u(y) J? (j) . As ? ? 1, ?? ? approaches the true gradient ??, but the
def P2T
variance increases. We consider a slight modification [2]: with Jt = s=t ? s?t r(Xs ),
def
?T =
T ?1
1 X ??Ut (Yt )
Jt+1 .
T t=0 ?Ut (Yt )
(1)
Throughout this paper the process {Xt , Yt , Ut , Xt+1 } is generally understood to be generated by a controlled POMDP satisfying Assumption 1, with X0 ?? (ie the initial state
distributed according to the stationary distribution). That is, before computing the gradient
estimates, we wait until the process has settled down to the stationary distribution.
Dependent Samples
Correlation terms arise in the variance quantities to be analysed. We show here that considering iid samples gives an upper bound on the variance of the general case. The mixing
time of a finite ergodic Markov chain M = (S, P ) is defined as
def
? = min t > 1 : max dT V P t i , P t j ? e?1 ,
i,j
t
t
where
[P ]i denotes the ith row of P and dT V is the total variation distance, dT V (P, Q) =
P
|P
(i)
? Q(i)|.
i
Theorem 1 Let M = (S, P ) be a finite ergodic Markov chain, p
with mixing time ? , and
2|S|e and 0 ? ? <
let ? be its stationary distribution. There are constants L <
exp(?1/(2?
)),
which
depend
only
on
M
,
such
that,
for
all
f
:
S ? R and all t,
Cov?f (t) ? L?t Var? (f), where Var? (f) is the variance of f under ?, and Cov?f (t) is
the auto-covariance of the process {f(Xt )}, where the process {Xt } is generated by M
with initial distribution ?. Hence, for some constant ?? ? 4L? ,
!
T ?1
1 X
??
Var
Var? (f).
f(Xt ) ?
T t=0
T
We call (L, ? ) the mixing constants of the Markov chain M (or of the controlled POMDP
D; ie the Markov chain (S, P )). We omit the proof (all proofs
are in the full version [8]).
Briefly, we show that for a finite ergodic Markov chain M , Cov?f (t) ? Rt (M )Var? (f),
2
for some Rt (M ). We then show that Rt (M ) < 2|S| exp(? ?t ). In fact, for a reversible
chain, we can choose L = 1 and ? = |?2 |, the second largest magnitude eigenvalue of P .
2 Baseline
We consider an alteration of (1),
def
?T =
T ?1
1 X ??Ut (Yt )
(Jt+1 ? Ar (Yt )) .
T t=0 ?Ut (Yt )
(2)
For any baseline Ar : Y ? R, it is easy to show that E [?T ] = E [?T ]. Thus, we select
Ar to minimize variance. The following theorem shows that this variance is bounded by a
variance involving iid samples, with Jt replaced by the exact value function.
Theorem 2 Suppose that D = (S, U, Y, P , ?, r, ?) is a controlled POMDP satisfying
Assumption 1, D has mixing constants (L, ? ), {Xt , Yt , Ut , Xt+1 } is a process generated
by D with X0 ?? ,Ar P
: Y ? R is a baseline that is uniformly bounded by M, and J (j)
s
has the distribution of ?
s=0 ? r(Xt ), where the states Xt are generated by D starting in
X0 = j. Then there are constants C ? 5B2 R(R + M) and ? ? 4L? ln(eT ) such that
!
T ?1
1 X ??Ut (Yt )
?
??u (y)
Var
(Jt+1 ?Ar (Yt )) ? Var?
(J? (j)?Ar (y))
T t=0 ?Ut (Yt )
T
?u (y)
2
?
??u (y)
?
C
+ E
(J (j) ? J? (j)) +
+1
?T ,
T
?u (y)
T
(1 ? ?)2
where, as always, (i, y, u, j) are generated iid with i??, y??(i), u??(y) and j?P i (u).
The proof uses Theorem 1 and [2, Lemma 4.3]. Here we have bounded the variance of (2)
with the variance of a quantity we may readily analyse. The second term on the right hand
side shows the error associated with replacing an unbiased, uncorrelated estimate of the
value function with the true value function. This quantity is not dependent on the baseline.
The final term on the right hand side arises from the truncation of the discounted reward?
and is exponentially decreasing. We now concentrate on minimizing the variance
??u (y)
2
(J? (j) ? Ar (y)) ,
(3)
? r = Var?
?u (y)
which the following lemma relates to the variance ? 2 without a baseline,
??u (y)
J? (j) .
? 2 = Var?
?u (y)
Lemma 3 Let D = (S, U, Y, P , ?, r, ?) be a controlled POMDP satisfying Assumption 1.
For any baseline Ar : Y ? R, and for i??, y??(i), u??(y) and j?Pi (u),
##
"
"
"
2 #
2
??u (y)
??u (y)
2
2
2
? r = ? + E Ar (y) E
J? (j) y .
y ? 2Ar (y)E
?u (y)
?u (y)
From Lemma 3 it can be seen that the task of finding the optimal baseline is in effect that
of minimizing a quadratic for each observation y ? Y. This gives the following theorem.
Theorem 4 For the controlled POMDP as in Lemma 3,
?
#!2
"
#?
"
2
2
??
(y)
??
(y)
u
min ? 2r = ? 2 ? E ? E
J? (j) y
/E ? u(y) y ? ,
Ar
?u (y)
u
and this minimum is attained with the baseline
#
"
#
"
2
2
??
(y)
??
(y)
u
u
?
J? (j) y /E
Ar (y) = E
y .
?u (y)
?u (y)
Furthermore, the optimal constant baseline is
"
#
2
2
??u (y)
??u (y)
?
Ar = E
J? (j) /E
.
?u (y)
?u (y)
The following theorem shows that the variance of an estimate with an arbitrary baseline
can be expressed as the sum of the variance with the optimal baseline and a certain squared
weighted distance between the baseline function and the optimal baseline function.
Theorem 5 If Ar : Y ? R is a baseline function, A?r is the optimal baseline defined in
Theorem 4, and ? 2r? is the variance of the corresponding estimate, then
"
#
2
??
(y)
u
2
? 2r = ? 2r? + E
(Ar (y) ? A?r (y)) ,
?u (y)
where i??, y ??(i), and u??(y). Furthermore, the same result is true for the case of
constant baselines, with Ar (y) replaced by an arbitrary constant baseline Ar , and A?r (y)
replaced by A?r , the optimum constant baseline defined in Theorem 4.
For the constant baseline Ar = E i?? [J? (i)], Theorem 5 implies that ? 2r is equal to
"
#!2
2
2
2
??u (y)
??u (y)
??u (y)
2
min ? r + E
E [J? (j)] ? E
J? (j)
/E ? (y) .
Ar ?R
?u (y)
?u (y)
u
Thus, its performance depends on the random variables (??u (y)/?u (y))2 and J? (j); if
they are nearly independent, E [J? ] is a good choice.
3 Fixed Value Functions: Actor-Critic Methods
We consider an alteration of (1),
T ?1
1 X ??Ut (Yt ) ?
? T def
?
=
J? (Xt+1 ),
T t=0 ?Ut (Yt )
?? : S ? R. Define
for some fixed value function J
" ?
#
X
def
k
A? (j) = E
? d(Xk , Xk+1 ) X0 = j ,
(4)
k=0
def
?? (j) ? J
?? (i) is the temporal difference. Then it is easy to show
where d(i, j) = r(i) + ? J
that the estimate (4) has a bias of
h i
? T = E ??u (y) A? (j) .
?? ? ? E ?
?u (y)
The following theorem gives a bound on the expected squared error of (4). The main tool
in the proof is Theorem 1.
Theorem 6 Let D = (S, U, Y, P , ?, r, ?) be a controlled POMDP satisfying Assumption 1. For a sample path from D, that is, {X0??, Yt??(Xt ), Ut??(Yt ), Xt+1?PXt (Ut )},
2
2 ? ?
??u (y) ?
??u (y)
? T ? ?? ?
Var?
J? (j) + E
A? (j)
,
E ?
?
T
?u (y)
?u (y)
where the second expectation is over i??, y??(i), u??(y), and j?P i (u).
?? (j) = J? (j) + v(j), then by selecting v = (v(1), . . . , v(|S|))0 from the right
If we write J
def P
0
null space of the K ? |S| matrix G, where G =
i,y,u ?i ?y (i)??u (y)Pi (u), (4) will
produce an unbiased estimate of ?? ?. An obvious example of such a v is a constant
vector, (c, c, . . . , c)0 : c ? R. We can use this to construct a trivial example where (4)
produces an unbiased estimate with zero variance. Indeed, let D = (S, U, Y, P , ?, r, ?) be
a controlled POMDP satisfying Assumption 1, with r(i) = c, for some 0 < c ? R. Then
J? (j) = c/(1 ? ?) and ?? ? = 0. If we choose v = (?c/(1 ? ?), . . . , ?c/(1 ? ?))0 and
?? (j) = J? (j) + v(j), then ??u(y) J
?
J
?u(y) ? (j) = 0 for all y, u, j, and so (4) gives an unbiased
estimate of ?? ?, with zero variance. Furthermore, for any D for which there exists a pair
y, u such that ?u (y) > 0 and ??u (y) 6= 0, choosing ?
J? (j) = J? (j) gives a variance
greater than zero?there is a non-zero probability event, (Xt = i, Yt = y, Ut = u, Xt+1 =
u(y)
j), such that ??
?u(y) J? (j) 6= 0.
4 Algorithms
Given a parameterized class of baseline functions Ar (?, ?) : Y ? R ? ? RL , we can
use Theorem 5 to bound the variance of our estimates. Computing the gradient of this
bound with respect to the parameters ? of the baseline function allows a gradient optimization of the baseline. The GDORB Algorithm produces an estimate ? S of these gradients
from a sample path of length S. Under the assumption that the baseline function and its
gradients are uniformly bounded, we can show that these estimates converge to the gradient
of ? 2r . We omit the details (see [8]).
GDORB Algorithm:
Given: Controlled POMDP (S, U, Y, P , ?, r, ?), parameterized baseline Ar .
set z0 = 0 (z0 ? RL ), ?0 = 0 (?0 ? RL )
for all {is , ys , us , is+1 , ys+1 } generated by the POMDP do
?? s(ys ) 2
zs+1 = ?zs + ?Ar (ys ) ?uu(y
)
s
s
?s+1 = ?s +
end for
1
s+1
((Ar (ys ) ? ?Ar (ys+1 ) ? r(xs+1 )) zs+1 ? ?s )
?? (?, ?) : S ? R ? ? RL }, we can
For a parameterized class of fixed value functions {J
use Theorem 6 to bound the expected squared error of our estimates, and compute the
gradient of this bound with respect to the parameters ? of the baseline function. The GBTE
Algorithm produces an estimate ?S of these gradients from a sample path of length S.
Under the assumption that the value function and its gradients are uniformly bounded, we
can show that these estimates converge to the true gradient.
GBTE Algorithm:
Given: Controlled POMDP (S, U, Y, P , ?, r, ?), parameterized fixed value function ?
J? .
set z0 = 0 (z0 ? RK ), ?A0 = 0 (?A0 ? R1?L ), ?B 0 = 0 (?B 0 ? RK ), ?C 0 = 0
(?C 0 ? RK ) and ?D0 = 0 (?D0 ? RK?L )
for all {is , ys , us , is+1 , is+2 } generated by the POMDP do
?? s(ys )
zs+1 = ?zs + ?uu(y
s s)
0
0
??us(ys ) ?
??us(ys )
1
?
?As+1 = ?As + s+1
?J? (is+1 )
? ?As
?us(ys ) J? (is+1 )
?us(ys )
??us(ys ) ?
1
J? (is+1 ) ? ?B s
?B s+1 = ?B s + s+1
?us(ys )
1
?? (is+2 ) ? J
?? (is+1 ) zs+1 ? ?C s
r(is+1 ) + ? J
?C s+1 = ?C s + s+1
0
??us(ys )
1
?
?Ds+1 = ?Ds + s+1 ?u (ys ) ?J? (is+1 ) ? ?D s
s
0
?
??
?
?s+1 = T ?As+1 ? T ?B 0s+1 ?D s+1 ? ?C 0s+1 ?Ds+1
end for
5 Experiments
Experimental results comparing these GPOMDP variants for a simple three state MDP
(described in [5]) are shown in Figure 1. The exact value function plots show how different choices of baseline and fixed value function compare when all algorithms have access
to the exact value function J? . Using the expected value function as a baseline was an
improvement over GPOMDP. Using the optimum, or optimum constant, baseline was a
further improvement, each performing comparably to the other. Using the pre-trained fixed
value function was also an improvement over GPOMDP, showing that selecting the true
value function was indeed not the best choice in this case. The trained fixed value function
was not optimal though, as J? (j) ? A?r is a valid choice of fixed value function.
The optimum baseline, and fixed value function, will not normally be known. The online
plots show experiments where the baseline and fixed value function were trained using online gradient descent whilst the performance gradient was being estimated, using the same
data. Clear improvement over GPOMDP is seen for the online trained baseline variant. For
the online trained fixed value function, improvement is seen until T becomes?given the
simplicity of the system?very large.
References
[1] L. Baird and A. Moore. Gradient descent for general reinforcement learning. In
Advances in Neural Information Processing Systems 11, pages 968?974. MIT Press,
1999.
[2] P. L. Bartlett and J. Baxter. Estimation and approximation bounds for gradient-based
reinforcement learning. Journal of Computer and Systems Sciences, 2002. To appear.
[3] A. G. Barto, R. S. Sutton, and C. W. Anderson. Neuronlike adaptive elements that
can solve difficult learning control problems. IEEE Transactions on Systems, Man,
and Cybernetics, SMC-13:834?846, 1983.
[4] J. Baxter and P. L. Bartlett. Infinite-horizon gradient-based policy search. Journal of
Artificial Intelligence Research, 15:319?350, 2001.
[5] J. Baxter, P. L. Bartlett, and L. Weaver. Infinite-horizon gradient-based policy search:
II. Gradient ascent algorithms and experiments. Journal of Artificial Intelligence Research, 15:351?381, 2001.
[6] M. Evans and T. Swartz. Approximating integrals via Monte Carlo and deterministic
methods. Oxford University Press, 2000.
Exact Value Function?Mean Error
Exact Value Function?One Standard Deviation
0.4
0.4
GPOMDP-J?
BL- [J? ]
BL-A?r (y)
BL-A?r
FVF-pretrain
0.3
0.25
GPOMDP-J?
BL- [J? ]
BL-A?r (y)
BL-A?r
FVF-pretrain
0.35
Relative Norm Difference
Relative Norm Difference
0.35
0.2
0.15
0.1
0.05
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
10
100
1000
10000
100000
1e+06
1e+07
1
10
100
1000
T
Online?Mean Error
100000
1e+06
1e+07
Online?One Standard Deviation
1
1
GPOMDP
BL-online
FVF-online
0.8
Relative Norm Difference
Relative Norm Difference
10000
T
0.6
0.4
0.2
0
GPOMDP
BL-online
FVF-online
0.8
0.6
0.4
0.2
0
1
10
100
1000
10000
100000
1e+06
1e+07
1
10
100
T
1000
10000
100000
1e+06
1e+07
T
Figure 1: Three state experiments?relative norm error k? est ? ??k / k??k. Exact value function plots compare mean error and standard deviations for gradient estimates (with knowledge of
J? ) computed by: GPOMDP [GPOMDP-J? ]; with baseline Ar = [J? ] [BL- [J? ]]; with optimum baseline [BL-A?r (y)]; with optimum constant baseline [BL-A?r ]; with pre-trained fixed value
function [FVF-pretrain]. Online plots do a similar comparison of estimates computed by: GPOMDP
[GPOMDP]; with online trained baseline [BL-online]; with online trained fixed value function [FVFonline]. The plots were computed over 500 runs (1000 for FVF-online), with ? = 0.95. ?? /T was
set to 0.001 for FVF-pretrain, and 0.01 for FVF-online.
[7] P. W. Glynn. Likelihood ratio gradient estimation for stochastic systems. Communications of the ACM, 33:75?84, 1990.
[8] E. Greensmith, P. L. Bartlett, and J. Baxter. Variance reduction techniques for gradient
estimates in reinforcement learning. Technical report, ANU, 2002.
[9] H. Kimura, K. Miyazaki, and S. Kobayashi. Reinforcement learning in POMDPs
with function approximation. In D. H. Fisher, editor, Proceedings of the Fourteenth
International Conference on Machine Learning (ICML?97), pages 152?160, 1997.
[10] V. R. Konda and J. N. Tsitsiklis. Actor-Critic Algorithms. In Advances in Neural
Information Processing Systems 12, pages 1008?1014. MIT Press, 2000.
[11] P. Marbach and J. N. Tsitsiklis. Simulation-Based Optimization of Markov Reward
Processes. Technical report, MIT, 1998.
[12] R. Y. Rubinstein. How to optimize complex stochastic systems from a single sample
path by the score function method. Ann. Oper. Res., 27:175?211, 1991.
[13] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press,
Cambridge MA, 1998. ISBN 0-262-19398-1.
[14] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy Gradient Methods
for Reinforcement Learning with Function Approximation. In Advances in Neural
Information Processing Systems 12, pages 1057?1063. MIT Press, 2000.
[15] R. J. Williams. Simple Statistical Gradient-Following Algorithms for Connectionist
Reinforcement Learning. Machine Learning, 8:229?256, 1992.
| 2052 |@word briefly:1 version:1 norm:6 simulation:1 covariance:1 reduction:3 initial:2 score:1 selecting:3 current:1 com:1 comparing:1 analysed:1 readily:1 evans:1 additive:2 plot:5 stationary:4 intelligence:2 xk:2 ith:1 short:1 consists:1 x0:7 indeed:2 expected:6 discounted:5 decreasing:1 csl:1 considering:1 becomes:1 notation:1 bounded:5 null:1 miyazaki:1 pto:1 minimizes:2 z:6 whilst:1 finding:1 kimura:1 temporal:1 control:6 normally:1 omit:2 appear:1 producing:1 greensmith:2 positive:1 t1:1 engineering:1 dropped:1 understood:1 before:1 limit:1 kobayashi:1 sutton:3 oxford:1 path:7 au:2 smc:1 unique:1 evan:2 whizbang:2 pre:2 wait:1 suggest:2 cal:1 py:2 optimize:1 deterministic:1 yt:16 williams:1 starting:1 pomdp:20 ergodic:3 simplicity:1 notion:1 variation:1 suppose:2 exact:6 us:3 element:1 satisfying:6 environment:2 reward:11 trained:8 depend:3 singh:1 upon:1 easily:1 joint:1 describe:1 monte:3 artificial:2 rubinstein:1 choosing:1 widely:2 solve:1 ability:1 cov:4 analyse:1 final:1 online:16 obviously:1 eigenvalue:1 isbn:1 mixing:5 optimum:7 r1:1 produce:4 generating:1 derive:1 illustrate:1 school:1 implies:2 australian:2 uu:2 concentrate:1 stochastic:2 mcallester:1 behaviour:1 preliminary:3 exp:2 mapping:1 a2:1 estimation:4 largest:1 tool:1 weighted:3 mit:5 always:2 aim:2 barto:2 improvement:7 likelihood:1 pretrain:4 baseline:48 dependent:2 a0:4 equal:2 construct:1 icml:1 nearly:1 pxt:1 report:2 connectionist:1 national:2 replaced:5 neuronlike:1 investigate:1 chain:7 integral:5 euclidean:1 re:1 column:1 ar:25 deviation:3 international:1 ie:2 together:1 squared:7 settled:1 choose:2 derivative:2 oper:1 alteration:2 b2:1 baird:1 depends:1 performed:1 lab:1 start:1 actorcritic:1 minimize:2 variance:37 produced:1 iid:4 comparably:1 carlo:3 pomdps:1 cybernetics:1 glynn:1 obvious:1 proof:4 associated:1 popular:1 knowledge:1 ut:14 probp:1 attained:1 dt:3 rand:1 though:1 strongly:1 anderson:1 furthermore:3 until:2 correlation:1 hand:2 d:3 replacing:1 reversible:1 defines:1 perhaps:1 mdp:1 effect:1 true:10 unbiased:5 hence:1 moore:1 rl:4 exponentially:1 extend:1 slight:1 cambridge:1 marbach:1 access:1 actor:3 pu:1 certain:1 ofr:1 seen:3 minimum:1 additional:1 greater:1 converge:2 maximize:2 swartz:1 ii:1 relates:1 full:1 reduces:1 d0:2 technical:2 visit:1 y:16 controlled:13 involving:1 variant:2 controller:1 expectation:3 limt:2 background:1 rest:1 ascent:2 tend:1 gpomdp:16 call:1 easy:2 baxter:5 variate:2 suboptimal:3 reduce:1 idea:1 bartlett:6 suffer:1 peter:2 action:1 generally:3 clear:1 extensively:1 reduced:1 exist:1 estimated:2 write:1 sum:2 run:1 fourteenth:1 parameterized:7 throughout:2 decision:2 def:11 bound:12 quadratic:1 min:3 performing:1 according:1 describes:1 modification:1 ln:1 needed:1 know:1 end:2 generic:1 denotes:6 konda:1 restrictive:1 approximating:1 bl:12 added:1 quantity:4 p2t:1 dependence:1 biowulf:1 rt:3 gradient:31 distance:6 trivial:1 length:2 ratio:1 minimizing:3 difficult:1 relate:1 policy:6 upper:1 observation:7 markov:11 finite:4 descent:2 extended:1 communication:1 mansour:1 arbitrary:2 pair:1 usually:1 max:1 including:1 suitable:1 event:1 weaver:1 technology:1 auto:1 relative:5 var:14 pij:3 editor:1 uncorrelated:1 critic:3 pi:2 row:1 surprisingly:2 transpose:1 truncation:1 tsitsiklis:2 bias:2 side:2 distributed:1 transition:2 valid:1 author:1 reinforcement:10 adaptive:1 transaction:1 approximate:2 observable:2 assumed:1 search:2 learn:2 investigated:1 complex:1 main:1 arise:1 wish:1 rk:8 down:1 theorem:16 z0:4 xt:19 jt:5 showing:1 x:2 exists:3 adding:1 magnitude:1 anu:3 horizon:2 expressed:2 partially:2 satisfies:2 acm:1 ma:1 ann:1 man:1 fisher:1 infinite:4 reducing:1 uniformly:3 lemma:5 total:1 experimental:1 est:1 east:1 select:3 arises:1 jonathan:1 reactive:2 correlated:1 |
1,155 | 2,053 | Rates of Convergence of Performance Gradient
Estimates Using Function Approximation and
Bias in Reinforcement Learning
Gregory Z. Grudic
University of Colorado, Boulder
grudic@cs.colorado.edu
Lyle H. Ungar
University of Pennsylvania
ungar@cis.upenn.edu
Abstract
We address two open theoretical questions in Policy Gradient Reinforcement Learning. The first concerns the efficacy of using function approximation to represent the state action value function, . Theory is presented showing that linear function approximation representations of
can degrade the rate of convergence of performance gradient estimates
by a factor of
relative to when no function approximation of
is used, where
is the number of possible actions and is the number
of basis functions in the function approximation representation. The second concerns the use of a bias term in estimating the state action value
function. Theory is presented showing that a non-zero bias term can
improve the rate of convergence of performance gradient estimates by
, where
is the number of possible actions. Experimental evidence is presented showing that these theoretical results lead to
significant improvement in the convergence properties of Policy Gradient Reinforcement Learning algorithms.
1 Introduction
Policy Gradient Reinforcement Learning (PGRL) algorithms have recently received attention because of their potential usefulness in addressing large continuous reinforcement
Learning (RL) problems. However, there is still no widespread agreement on how PGRL
algorithms should be implemented. In PGRL, the agent?s policy is characterized by a set
of parameters which in turn implies a parameterization of the agent?s performance metric.
Thus if
represents a dimensional parameterization of the agent?s policy and is
a performance metric the agent is meant to maximize, then the performance metric must
have the form
[6]. PGRL algorithms work by first estimating the performance gradient
(PG)
and then using this gradient to update the agent?s policy using:
! "#!
/
!
(1)
$&%('*)+$-,. !0
where . is a small positive step size. If the estimate of ! "#! is accurate, then the agent can
climb the performance gradient in the parameter space, toward locally optimal policies. In
practice, ! 12! is estimated using samples of the state action value function . The PGRL
formulation is attractive because 1) the parameterization of the policy can directly imply
a generalization over the agent?s state space (e.g., can represent the adjustable weights
in a neural network approximation), which suggests that PGRL algorithms can work well
is
on very high dimensional problems [3]; 2) the computational cost of estimating
linear in the number of parameters , which contrasts with the computational cost for most
RL algorithms which grows exponentially with the dimension of the state space; and 3) PG
[6, 5, 4, 2, 1].
algorithms exist which are guaranteed to give unbiased estimates of
!0 "#!
!0 "#!
This paper addresses two open theoretical questions in PGRL formulations. In PGRL formulations performance gradient estimates typically have the following form:
/
0! ) #'
1' #'
(2)
!
where !"
#! is the estimate of the value of executing action $! in state ! (i.e. the
state action value function), # %! the bias subtracted from !&
& #! in state ! , ' is the
number of steps the agent takes before estimating ! 12!0 , and the form of the function
depends on the PGRL algorithm being used (see Section 2, equation (3) for the form
being considered here). The effectiveness of PGRL algorithms strongly depends on how
!"
& #! is obtained and the form of # %! . The aim of this work is to address these
questions.
The first open theoretical question addressed here is concerned with the use of function
approximation (FA) to represent the state action value function , which is in turn used
to estimate the performance gradient. The original formulation of PGRL [6], the REINFORCE algorithm, has been largely ignored because of the slow rate of convergence of the
PG estimate. The use of FA techniques to represent based on its observations has been
suggested as a way of improving convergence properties. It has been proven that specific
linear FA formulations can be incorporated into PGRL algorithms, while still guaranteeing
convergence to locally optimal solutions [5, 4]. However, whether linear FA representations actually improves the convergence properties of PGRL is an open question. We
present theory showing that using linear basis function representations of , rather than
direct observations of it, can slow the rate of convergence of PG estimates by a factor of
(see Theorem 1 in Section 3.1). This result suggests that PGRL formulations
should avoid the use of linear FA techniques to represent . In Section 4, experimental
evidence is presented supporting this conjecture.
#
The second open theoretical question addressed here is can a non-zero bias term
in
(2) improve the convergence properties of PG estimates? There has been speculation
that an appropriate choice of
can improve convergence properties [6, 5], but theoretical support has been lacking. This paper presents theory showing that if
, where
is the number actions, then the rate of convergence of the
PG estimate is improved by
(see Theorem 2 in Section 3.2). This suggests that the convergence properties of PGRL algorithms can be improved by using a bias
term that is the average of values in each state. Section 4 gives experimental evidence
supporting this conjecture.
#)(* +
& 1
#
#
#
)
2 The RL Formulation and Assumptions
, . -
/
0$
12
:9
+
& $ )F G2
$ 43 365 87
$ <; ) '
1
#=
* @ A> $ )B ?C >+- $&%(' )D A&E $
?
"
@
H @* )JIK-> $&%(' E $ )L+
& $ )MG 2 NO+
A 3P
& ;
Q +
& R )S?C>+-% $ )D E $ )S+ RT2 NO U3P
& U;
The RL problem is modeled as a Markov Decision Process (MDP). The agent?s state at time
is given by
,
. At each time step the agent chooses from a finite
set of
actions
and receives a reward
. The dynamics of
the environment are characterized by transition probabilities
and expected rewards
,
. The
policy followed by the agent is characterized by a parameter vector
, and is defined
by the probability distribution
,
. We
)
assume that
Q +
& R is differentiable with respect to .
Q
+
& 1
Q )I $( ' $ > $
/
&Q
+
)I ( (' ' > $&% $ ) +
& $ )
"Q
exact expression for the performance gradient
is:
where
the
. ! Then
) = !GQ +
& ! R +
& !
(3)
!
!
@
!
(
'
E
where ) ( $ $ - $ )
"Q2 and # .
, under
This policy gradient formulation requires that the state-action value function,
the ! current policy be estimated. ! This estimate, , is derived using the observed value
@ +
& ! . We assume that ! @ +
& ! has the following form:
@ /
#! ) +
& #! ,#" +
& #!
where
" +
# ! has zero mean and finite variance $&@!% ' *)( . Therefore,
if +
! is an es
!
timate of
/
#! obtained by averaging * observations of @ +
& #! , then the mean
and variance are given by:
(4)
IB /
#! ! ) +
& #! O
,+B +
& #! ).-013/5 2 4 (
@ /
! are independently distributed. This is consistent
In addition, we assume that
We use the Policy Gradient Theorem of Sutton et al. [5] and limit our analysis to the start
and state action value
state discount reward formulation. Here the reward function
function
are defined as:
with the MDP assumption.
3 Rate of Convergence Results
$679% 8;: ) !@ @ AB' ! <>@C =0' ? 'EDEDEDE' =GF $6@!% ' * (
H$679% I J ) @N@ A' ! @<GC KM' L 'EDEDEDE' =>F $6@!% ' * (
where $ @!% ' * ( is defined in (4) and
O 79I J )QP ( % ( = SR BT @!' W*)(VU W!X %ZY $ 79% I J
@
!M('
O 798;: )QP ( % ( = R BT @!R ' W* ( U W!X %ZY $679% 8;:
@
!M(' R
Before stating the convergence theorems, we define the following:
(5)
(6)
3.1 Rate of Convergence of PIFA Algorithms
]`_ ]
/
#! ) *) ( )\] [ *)( ' *)( '
(7)
'&^
_ * (3' ]
]* (3'
where
] are weights and are basis functions defined in + 7 . If the
weights
_ *)( ' ] ^ ^ * (a' are chosen based using the observed ! @ +
& ! , and the basis functions,
, satisfy the conditions defined in [5,= 4], then the performance gradient is given by:
! ) !Q +
& ! R * (
(8)
! cb
!
@
!
( '
The following theorem establishes bounds on the rate of convergence for this representation
Consider the PIFA algorithm [5] which uses a basis function representation for estimated
state action value function,
, of the following form:
of the performance gradient.
RR W b
Theorem 1: Let
be an estimate of (8) obtained using the PIFA algorithm and the basis
function representation (7). Then, given the assumptions defined in Section 2 and equations
(5) and (6), the rate of convergence of a PIFA
algorithm is bounded below and above by:
O 97 I J + !
O 798 :
* ! b
*
where is the number of basis functions,
is the number of possible actions, and
the number of independent estimates of the performance gradient.
*
(9)
is
Proof: See Appendix.
3.2 Rate of Convergence of Direct Sampling Algorithms
! @ /
#!
+
& !
In the previous section, the observed
are used to build a linear basis function
representation of the state action value function,
, which is in turn used to estimate the performance gradient. In this section we establish rate of convergence bounds
for performance gradient estimates that directly use the observed
without the
intermediate step of building the FA representation. These bounds are established for the
conditions
and
in (3).
#
#
)
)
#
/
RR W
( * /
1
@ +
& #!
# )
Theorem 2: Let
be a estimate of (3), be obtained using direct samples of
. Then,
, and given the assumptions defined in Section 2 and equations (5) and (6), the
if
rate of convergence of
*
/
RR W
O 97 I J
+ !0/ O 97 8;:
* !
*
is bounded by:
(10)
where is the number of independent estimates of the performance gradient. If
is defined as:
)
=
+
&
'
then the rate of convergence of the performance
gradient
where
# )
(11)
RR W
is bounded by:
O 97 I J
+ ! O 97 8 :
* 0!
*
(12)
is the number of possible actions.
Proof: See Appendix.
Thus comparing (12) and (10) to (9) one can see that policy gradient algorithms such as
PIFA which build FA representations of converge by a factor of
slower than
algorithms which directly sample . Furthermore, if the bias term is as defined in (11),
the bounds on the variance are further reduced by
. In the next section
experimental evidence is given showing that these theoretical consideration can be used to
improve the convergence properties of PGRL algorithms.
4 Experiments
The Simulated Environment: The experiments simulate an agent episodically interacting
in a continuous two dimensional environment. The agent starts each episode in the same
state , and executes a finite number of steps following a policy to a fixed goal state .
The stochastic policy is defined by a finite set of Gaussians, each associated with a specific
!
0.6
4
2
10
b
V[? ? / ? ?] / V[? ? / ? ? ]
No Bias
0.4
3
10
0.3
Linear FA Q
0.2
2
10
0.1
?0.1
0
1
10
0
20
40
60
80
100
0
10
Number of Policy Updates
a) Convergence of
Algorithms
0
1
10
F
?(?)
10
Biased Q
V[? ? / ? ? ] / V[? ? / ? ?]
0.5
/
/
0
2
4
6
8
10
12
Number of Possible Actions (M)
b)
14
+ RR W b + RR W
10
0
/
/
2
4
6
8
10
12
Number of Possible Actions (M)
c)
Figure 1: Simulation Results
+ RR W + RR W
14
is defined as:
%
) Z?
7
( '
where ) #'
1
87 , is the agents state, *'%
7
7 is the Gaussian center,
and *'
1
is the variance along each state space dimension. The probability of
7 in state is
executing action
Q +
& R ) =
((
'
where ) ''
1
'
' '
1
'
1
1= '
1
1=
7
= '
= 7 defines the policy parameters that dictate the agent?s actions. Action ' directs the agent toward the goals state
, while the remaining actions (for ) 0T
) direct the agent towards the corresponding Gaussian center *'%
.
7
Noise is modeled using a uniform random distribution between $
denoted by $
,
such that the noise in dimension ! @ is given by:
) , @ `)
#$ 2
where 9 is the magnitude of the noise,
is the state the agent observes and uses to
agent.
choose actions, and is the actual state of the
action. The Gaussian associated with action
The agent receives a reward of +1 when it reaches the goal state, otherwise it receives a
reward of:
>
)
)
Z?
7 %
( '
Thus the agent gets negative rewards the closer it gets to the origin of the state space, and a
positive reward whenever it reaches the goal state.
@ +
& !
Implementation of the PGRL algorithms: All the PGRL formulations studied here require observations (i.e. samples) of the state action value function.
is sampled
by executing action in state and thereafter following the policy. In the episodic formulation, where the agent executes a maximum of steps during each episode, at the end of
each episode,
for step can be evaluated as follows:
T!
@ $
& $
@ $
& $
@ '
& '
1
@ Q &
'
,
) ' > $&% E $
) +
$ )
&Q
( '
'
1'
#(
Thus, given that the agent executes a complete episode
following the policy , at the completion of the episode we can calculate
. This gives samples of state action value pairs. Equation (3) tells us that we require a total of
state action value function observations to
estimate a performance gradient (assuming the agent can execute
actions). Therefore,
we can obtain the remaining
observations of
by sending the agent out on
+
'
'
'
@
'
$ U)
! @ $
& Q
'
'
& '
1
'
epsisodes, each time allowing it to follow the policy for all steps, with the
exception that action
is executed when
is being observed. This sampling procedure requires a total of
episodes and gives a complete set of
state action pairs for any path
. For the direct sampling algorithms
in Section 3.2, these observations are directly used to estimate the performance gradient.
For the linear basis function based PGRL algorithm in Section 3.1, these observations are
as defined in [5, 4], and then the performance gradient is
first used to calculate the
calculated using (8).
^
* ( ']
/
@
/
+ ! "#! b + ! 12!0
Experimental Results: Figure 1b shows a plot of average
values
over 10,000 estimates of the performance gradient. For each estimate, the goal state, start
;
state, and Gaussian centers are all chosen using a uniform random distribution
the Gaussian variances are sampled from a uniform distribution
. As predicted by
Theorem 1 in Section 3.1 and Theorem 2 in Section 3.2, as the number of actions
increases, this ratio also increases. Note that Figure 1b plots average variance ratios, not the
bounds in variance given in Theorem 1 and Theorem 2 (which have not been experimentally sampled), so the
ratio predicted by the theorems is supported by the increase in
increases. Figure 1c shows a plot of average
values
the ratio as
over 10,000 estimates of the performance gradient. As above, for each estimate, the goal
state, start state, and Gaussian centers are all chosen using a uniform random distribution
; the Gaussian variances are sampled from a uniform distribution
. This also
follows the predicted trends of Theorem 1 and Theorem 2. Finally, Figure 1a shows the
average reward over 100 runs as the three algorithms converge on a two action problem.
Each algorithm is given the same number of
samples to estimate the gradient before
each update. Because
has the least variance, it allows the policy to converge
. Similarly, because
has the highest variance, its
to the highest reward value
policy updates converge to the worst
. Note that because all three algorithms will converge to the same locally optimal policy given enough samples of
, Figure 1a simply
requires more samples than
, which in turn requires more
demonstrates that
samples than
.
+
$
/
+
/
0! "#!
Q
/
!
/ 12!
! "#!
b
! @
Q
+ ! / 12!0 + ! / "#!
`)
+
/
! 12!0 b
/
! 1#!
Q
! @
5 Conclusion
The theoretical and experimental results presented here indicate that how PGRL algorithms
are implemented can substantially affect the number of observations of the state action
value function ( ) needed to obtain good estimates of the performance gradient. Furthermore, they suggest that an appropriately chosen bias term, specifically the average value of
over all actions, and the direct use of observed values can improve the convergence of
PGRL algorithms. In practice linear basis function representations of can significantly
degrade the convergence properties of policy gradient algorithms. This leaves open the
question of whether any (i.e. nonlinear) function approximation representation of value
functions can be used to improve convergence of such algorithms.
References
[1] Jonathan Baxter and Peter L. Bartlett, Reinforcement learning in pomdp?s via direct
gradient ascent, Proceedings of the Seventeenth International Conference on Machine
Learning (ICML?2000) (Stanford University, CA), June 2000, pp. 41?48.
[2] G. Z. Grudic and L. H. Ungar, Localizing policy gradient estimates to action transitions, Proceedings of the Seventeenth International Conference on Machine Learning,
vol. 17, Morgan Kaufmann, June 29 - July 2 2000, pp. 343?350.
[3]
, Localizing search in reinforcement learning, Proceedings of the Seventeenth
National Conference on Artificial Intelligence, vol. 17, Menlo Park, CA: AAAI Press /
Cambridge, MA: MIT Press, July 30 - August 3 2000, pp. 590?595.
[4] V. R. Konda and J. N. Tsitsiklis, Actor-critic algorithms, Advances in Neural Information Processing Systems (Cambridge, MA) (S. A. Solla, T. K. Leen, and K.-R. Mller,
eds.), vol. 12, MIT Press, 2000.
[5] R. S. Sutton, D. McAllester, S. Singh, and Y. Mansour, Policy gradient methods for
reinforcement learning with function approximation, Advances in Neural Information
Processing Systems (Cambridge, MA) (S. A. Solla, T. K. Leen, and K.-R. Mller, eds.),
vol. 12, MIT Press, 2000.
[6] R. J. Williams, Simple statistical gradient-following algorithms for connectionist reinforcement learning, Machine Learning 8 (1992), no. 3, 229?256.
* ( given in (7). In [5] it is shown that
_ *)( ' ]
= !Q +
& ! R
^ = !Q +
& ! R
I @ !( ' ! * ( @
+
& ! )
!M(' !
(13)
!
Let R W @ be the observation of R W (3) after a single episode. Using (13), we get the
R
R
following:
=
N
@
'
(
B
T
*
!
RR W @ ) ( @ !( ( ' R R W U W!X ! @ /
#! RR W , "
) P ( ( = R BT @!' *Z( U W;X +
& ! Y , "
@
!M= ' BT @!R ' W * ( U W;X
) P ( @ !M( ' R R W *) ( Y ,#"
=
]`_ ]
) P ( @ !M( ' R BT @!R ' W * ( U W;X ] ( [ ' *)( ' *)( ' Y ,#"
^ _ ]
=
=
] ]
]
) P !M( (' ] ( ([ ' *Z( ' ] ( @ R BT @!R ' W * ( U W X *Z( ' Y , " !M( ' ] ( ([ ' *)( ' ! ,#"
^
^
where the basis functions ! have
the form
]
_ ]
! ) @ !GQ /!
! R * ( '
I " ) , with variance
and
= !Q +
& ! R %
!
Y
%
+ " )+ P ! @ ) @ !(' ! $ @!% ' * (
Denoting R W b as the least squares (LS) estimate of (3), its form is given by:
R
! ) = [
(14)
!0 b ( ' ]
where are] LS estimates of the weights *)( ' and
correspond to the basis
! . Then, it can be shown that any linear
^ system of the type given in (14) has a
functions
rate of convergence
given
by:
= !GQ +
& #!R %
!
+ ! b ) * + " ) * @ % !( ' ! $ @!% ' * (
Substituting (5) and (6) into the above equation completes the proof.
Appendix: Proofs of Theorems 1 and 2
*Z( ' ]
Proof of Theorem 1: Consider the definition of
there exist
and
such that:
of the performance
! @ /
#* ! . estimates
These examples are averaged
/
=
!
I ! ) @ !( ' !GQ /!
! R /
!
! @ +
& ! is independently distributed, the variance of the estimate is given
Because each
by
/
= !Q +
& #!R %
!
%
+ ! ) * @ ! $ @N% ' *)(
(15)
!( '
Given (5) the worst
/ rate of convergence= is bounded by:
!
+ ! @ % !( ' !Q +!
& ! R % $ 79% 8 : * ) O 798 : *
A similarly argument applies to the lower bound on convergence completing the proof for
(10). Following the same argument for (12), we have
= !Q +
& ! R %
=
!
+ ! ) * @ % !( ' !
+ +
! ( ' +
Where
=
=
+ +
& ! = ' ( ( ' +
& )+ = = ' +
& ! = ' ( ( ' /
!
=
) =
= '
% $6@!% ' *)( , ( ( ' = '
% $6@N% ' *
!
(16)
+ on the far left of (16) is bounded by:
Given (5) the variance
=
=
=
= '
% $6@!% ' *)( , ( ( ' = '
% $6@!% ' *
) =
= '
% $679% 8;: , ( ( ' = '
% $679% 8 :
!
!
798;:
=
'
%
)
= '
,
= '
% $679% 8;:
O )
=
$679% 8 :
Plugging the above into (16) and inserting * from (6) completes the proof for the upper
*
Proof of Theorem 2: We prove equation (10) first. For
gradient, we get independent samples of each
and therefore:
bound. The proof for the lower bound in the variance follows similar reasoning.
| 2053 |@word build:2 implemented:2 c:1 predicted:3 implies:1 indicate:1 unbiased:1 establish:1 question:7 open:6 mdp:2 km:1 fa:8 simulation:1 stochastic:1 attractive:1 pg:6 mcallester:1 during:1 gradient:35 require:2 reinforce:1 simulated:1 ungar:3 generalization:1 efficacy:1 degrade:2 denoting:1 complete:2 toward:2 assuming:1 current:1 comparing:1 reasoning:1 considered:1 modeled:2 consideration:1 ratio:4 recently:1 must:1 cb:1 mller:2 executed:1 substituting:1 rl:4 qp:2 negative:1 plot:3 exponentially:1 update:4 implementation:1 intelligence:1 leaf:1 policy:26 adjustable:1 parameterization:3 significant:1 allowing:1 cambridge:3 upper:1 observation:10 markov:1 finite:4 establishes:1 similarly:2 supporting:2 mit:3 incorporated:1 gaussian:7 aim:1 gc:1 rather:1 interacting:1 avoid:1 actor:1 mansour:1 along:1 august:1 direct:7 prove:1 derived:1 june:2 pair:2 improvement:1 directs:1 speculation:1 contrast:1 upenn:1 expected:1 established:1 address:3 suggested:1 morgan:1 below:1 typically:1 bt:6 actual:1 converge:5 maximize:1 july:2 estimating:4 bounded:5 denoted:1 substantially:1 characterized:3 jik:1 improve:6 sampling:3 plugging:1 imply:1 represents:1 park:1 icml:1 metric:3 demonstrates:1 t2:1 connectionist:1 represent:5 gf:1 addition:1 positive:2 before:3 national:1 addressed:2 relative:1 completes:2 limit:1 lacking:1 sutton:2 appropriately:1 biased:1 proven:1 ab:1 sr:1 path:1 ascent:1 agent:25 studied:1 climb:1 suggests:3 effectiveness:1 consistent:1 critic:1 intermediate:1 enough:1 seventeenth:3 averaged:1 concerned:1 baxter:1 accurate:1 affect:1 supported:1 lyle:1 practice:2 vu:1 closer:1 pennsylvania:1 tsitsiklis:1 bias:9 procedure:1 episodic:1 whether:2 expression:1 distributed:2 significantly:1 dictate:1 theoretical:8 bartlett:1 dimension:3 calculated:1 transition:2 suggest:1 peter:1 get:4 reinforcement:9 localizing:2 far:1 action:37 cost:2 ignored:1 addressing:1 pgrl:21 uniform:5 usefulness:1 center:4 discount:1 williams:1 attention:1 locally:3 independently:2 l:2 pomdp:1 reduced:1 gregory:1 exist:2 chooses:1 continuous:2 search:1 international:2 estimated:3 ca:2 menlo:1 vol:4 improving:1 thereafter:1 colorado:2 exact:1 aaai:1 us:2 choose:1 origin:1 agreement:1 grows:1 trend:1 noise:3 potential:1 observed:6 run:1 worst:2 calculate:2 satisfy:1 slow:2 episode:7 depends:2 solla:2 highest:2 decision:1 observes:1 appendix:3 environment:3 start:4 bound:8 reward:10 grudic:3 guaranteed:1 followed:1 completing:1 dynamic:1 ib:1 theorem:17 singh:1 specific:2 square:1 showing:6 variance:14 largely:1 kaufmann:1 correspond:1 basis:12 concern:2 evidence:4 simulate:1 argument:2 zy:2 ci:1 magnitude:1 conjecture:2 executes:3 artificial:1 tell:1 reach:2 whenever:1 ed:2 simply:1 definition:1 stanford:1 pp:3 otherwise:1 g2:1 proof:9 associated:2 applies:1 timate:1 sampled:4 boulder:1 equation:6 differentiable:1 rr:10 mg:1 improves:1 turn:4 ma:3 goal:6 needed:1 gq:4 towards:1 actually:1 inserting:1 end:1 sending:1 experimentally:1 gaussians:1 follow:1 specifically:1 improved:2 averaging:1 formulation:11 evaluated:1 execute:1 strongly:1 leen:2 furthermore:2 appropriate:1 total:2 subtracted:1 experimental:6 e:1 convergence:30 slower:1 receives:3 exception:1 original:1 guaranteeing:1 executing:3 nonlinear:1 remaining:2 widespread:1 support:1 defines:1 stating:1 completion:1 meant:1 jonathan:1 konda:1 received:1 building:1 |
1,156 | 2,054 | Adaptive N earest Neighbor Classification
using Support Vector Machines
Carlotta Domeniconi, Dimitrios Gunopulos
Dept. of Computer Science, University of California, Riverside, CA 92521
{ carlotta, dg} @cs.ucr.edu
Abstract
The nearest neighbor technique is a simple and appealing method
to address classification problems. It relies on t he assumption of
locally constant class conditional probabilities. This assumption
becomes invalid in high dimensions with a finite number of examples due to the curse of dimensionality. We propose a technique
that computes a locally flexible metric by means of Support Vector
Machines (SVMs). The maximum margin boundary found by the
SVM is used to determine the most discriminant direction over the
query's neighborhood. Such direction provides a local weighting
scheme for input features. We present experimental evidence of
classification performance improvement over the SVM algorithm
alone and over a variety of adaptive learning schemes, by using
both simulated and real data sets.
1
Introduction
In a classification problem, we are given J classes and l training observations. The
training observations consist of n feature measurements x = (Xl,'" ,Xn)T E ~n
and the known class labels j = 1, ... , J. The goal is to predict the class label of a
given query q.
The K nearest neighbor classification method [4, 13, 16] is a simple and appealing
approach to this problem: it finds the K nearest neighbors of q in the training
set, and then predicts the class label of q as the most frequent one occurring in
the K neighbors. It has been shown [5, 8] that the one nearest neighbor rule has
asymptotic error rate that is at most twice t he Bayes error rate, independent of
the distance metric used. The nearest neighbor rule becomes less appealing with
finite training samples, however. This is due to the curse of dimensionality [2].
Severe bias can be introduced in t he nearest neighbor rule in a high dimensional
input feature space with finite samples. As such, the choice of a distance measure
becomes crucial in determining t he outcome of nearest neighbor classification. The
commonly used Euclidean distance implies that the input space is isotropic, which
is often invalid and generally undesirable in many practical applications.
Several techniques [9, 10, 7] have been proposed to try to minimize bias in high dimensions by using locally adaptive mechanisms. The "lazy learning" approach used
by these methods, while appealing in many ways, requires a considerable amount of
on-line computation, which makes it difficult for such techniques to scale up to large
data sets. The feature weighting scheme they introduce, in fact , is query based and
is applied on-line when the test point is presented to the "lazy learner" . In this paper we propose a locally adaptive metric classification method which, although still
founded on a query based weighting mechanism, computes off-line the information
relevant to define local weights.
Our technique uses support vector machines (SVMs) as a guidance for the process of
defining a local flexible metric. SVMs have been successfully used as a classification
tool in a variety of areas [11, 3, 14], and the maximum margin boundary they provide
has been proved to be optimal in a structural risk minimization sense. The solid
theoretical foundations that have inspired SVMs convey desirable computational
and learning theoretic properties to the SVM's learning algorithm, and therefore
SVMs are a natural choice for seeking local discriminant directions between classes.
The solution provided by SVMs allows to determine locations in input space where
class conditional probabilities are likely to be not constant, and guides the extraction
of local information in such areas. This process produces highly stretched neighborhoods along boundary directions when the query is close to the boundary. As a
result, the class conditional probabilities tend to be constant in the modified neighborhoods, whereby better classification performance can be achieved. The amount
of elongation-constriction decays as the query moves further from the boundary
vicinity.
2
Feature Weighting
SVMs classify patterns according to the sign(f(x)), where f(x)
L:~=l (XiYiK(Xi, x) - b, K(x , y) = cpT(x). cp(y) (kernel junction), and cp: 3(n -+ 3(N
is a mapping of the input vectors into a higher dimensional feature space. Here
we assume Xi E 3(n, i = I, . . . ,l, and Yi E {-I,I}. Clearly, in the general case
of a non-linear feature mapping cp, the SVM classifier gives a non-linear boundary
f(x) = 0 in input space. The gradient vector lld = "Vdj, computed at any point
d of the level curve f(x) = 0, gives the perpendicular direction to the decision
boundary in input space at d. As such, the vector lld identifies the orientation in
input space on which the projected training data are well separated, locally over
d's neighborhood. Therefore, the orientation given by lld, and any orientation close
to it, is highly informative for the classification task at hand , and we can use such
information to define a local measure of feature relevance.
Let q be a query point whose class label we want to predict. Suppose q is close
to the boundary, which is where class conditional probabilities become locally non
uniform, and therefore estimation of local feature relevance becomes crucial. Let
d be the closest point to q on the boundary f(x) = 0: d = argminp Ilq - pll,
subject to the constraint f(p) = O. Then we know that the gradient lld identifies a
direction along which data points between classes are well separated.
As a consequence, the subspace spanned by the orientation lld, locally at q, is
likely to contain points having the same class label as q . Therefore, when applying
a nearest neighbor rule at q, we desire to stay close to q along the lld direction,
because that is where it is likely to find points similar to q in terms of class posterior probabilities. Distances should be constricted (large weight) along lld and
along directions close to it. The farther we move from the lld direction, the less
discriminant the correspondent orientation becomes. This means that class labels
are likely not to change along those orientations, and distances should be elongated
(small weight) , thus including in q's neighborhood points which are likely to be
similar to q in terms of the class conditional probabilities.
Formally, we can measure how close a direction t is to lld by considering the dot
product lla ?t. In particular, by denoting with Uj the unit vector along input feature
j, for j = 1, . .. , n, we can define a measure of relevance for feature j, locally at q
(and therefore at d), as Rj(q) == Iu] . lldl = Ind,j l, where lld = (nd,l,'" ,nd,n)T.
The measure of feature relevance, as a weighting scheme, can then be given by the
following exponential weighting scheme: Wj(q) = exp(ARj(q))1 2::7=1 exp(ARi(q)),
where A is a parameter that can be chosen to maximize (minimize) the influence of
R j on Wj' When A = 0 we have Wj = lin, thereby ignoring any difference between
the Rj's. On the other hand, when A is large a change in R j will be exponentially reflected in Wj' The exponential weighting scheme conveys stability to the method by
preventing neighborhoods to extend infinitely in any direction. This is achieved by
avoiding zero weights, which would instead be allowed by linear or quadratic weightings. Thus, the exponential weighting scheme can be used as weights associated with
features for weighted distance computation D(x, y) = )2::7=1 Wi(Xi - Yi)2. These
weights enable the neighborhood to elongate less important feature dimensions, and,
at the same time, to constrict the most influential ones. Note that the technique is
query-based because weightings depend on the query.
3
Local Flexible Metric Classification based on SVMs
To estimate the orientation of local boundaries, we move from the query point along
the input axes at distances proportional to a given small step (whose initial value
can be arbitrarily small, and doubled at each iteration till the boundary is crossed).
We stop as soon as the boundary is crossed along an input axis i, i.e. when a point
P i is reached that satisfies the condition sign(f(q)) x sign(f(pi)) = -1. Given Pi,
we can get arbitrarily close to the boundary by moving at (arbitrarily) small steps
along the segment that joins Pi to q.
Let us denote with d i the intercepted point on the boundary along direction i. We
then approximate lld with the gradient vector lld i = \7 d i f, computed at d i .
We desire that the parameter A in the exponential weighting scheme increases as the
distance of q from the boundary decreases. By using the knowledge that support
vectors are mostly located around the boundary surface, we can estimate how close
a query point q is to the boundary by computing its distance from the closest non
bounded support vector: Bq = minsi Ilq - si ll, where the minimum is taken over
the non bounded (0 < D:i < C) support vectors Si. Following the same principle,
in [1] the spatial resolution around the boundary is increased by enlarging volume
elements locally in neighborhoods of support vectors. Then, we can achieve our goal
by setting A = D - B q , where D is a constant input parameter of the algorithm. In
our experiments we set D equal to the approximated average distance between the
training points Xk and the boundary: D = 2::xk {minsi Ilxk - sill}. If A becomes
negative it is set to zero.
t
By doing so the value of A nicely adapts to each query point according to its location
with respect to the boundary. The closer q is to the decision boundary, the higher
the effect of the Rj's values will be on distances computation.
We observe that this principled guideline for setting the parameters of our technique
takes advantage of the sparseness representation of the solution provided by the
SVM. In fact, for each query point q, in order to compute Bq we only need to
consider the support vectors, whose number is typically small compared to the
Input: Decision boundary f(x)
point q and parameter K.
=
a produced by a
SVM; query
1. Compute the approximated closest point d i to q on the bound-
ary;
2. Compute the gradient vector ndi = \l dJ;
3. Set feature relevance values Rj(q) = Indi,jl for j = 1, . . . ,n;
4. Estimate the distance of q from the boundary as: Bq =
minsi Ilq - sill;
5. Set A = D - B q , where D = EXk {minsi Ilxk - sill};
t
6. Set Wj(q)
=
exp(ARj(q))/ E~=l exp(ARi(q)), for j
1, ... ,n;
7. Use the resulting w for K-nearest neighbor classification at
the query point q.
Figure 1: The LFM-SVM algorithm
total number of training examples. Furthermore, the computation of D's value is
carried out once and off-line.
The resulting local flexible metric technique based on SVMs (LFM-SVM) is summarized in Figure 1. The algorithm has only one adjustable tuning parameter, namely
the number K of neighbors in the final nearest neighbor rule. This parameter is
common to all nearest neighbor classification techniques.
4
Experimental Results
In the following we compare several classification methods using both simulated
and real data. We compare the following classification approaches: (1) LFM-SVM
algorithm described in Figure 1. SV Mlight [12] with radial basis kernels is used
to build the SVM classifier; (2) RBF-SVM classifier with radial basis kernels. We
used SV Mlight [12], and set the value of"( in K(Xi' x) = e-r llxi-xI12 equal to the
optimal one determined via cross-validation. Also the value of C for the soft-margin
classifier is optimized via cross-validation. The output of this classifier is the input
of LFM-SVM; (3) ADAMENN-adaptive metric nearest neighbor technique [7]. It
uses the Chi-squared distance in order to estimate to which extent each dimension
can be relied on to predict class posterior probabilities; (4) Machete [9]. It is a
recursive partitioning procedure, in which the input variable used for splitting at
each step is the one that maximizes the estimated local relevance. Such relevance
is measured in terms of the improvement in squared prediction error each feature is
capable to provide; (5) Scythe [9]. It is a generalization of the machete algorithm, in
which the input variables influence each split in proportion to their estimated local
relevance; (6) DANN-discriminant adaptive nearest neighbor classification [10]. It
is an adaptive nearest neighbor classification method based on linear discriminant
analysis. It computes a distance metric as a product of properly weighted within
and between sum of squares matrices; (7) Simple K-NN method using the Euclidean
distance measure; (8) C4.5 decision tree method [15].
In all the experiments, the features are first normalized over the training data to
have zero mean and unit variance, and the test data features are normalized using
the corresponding training mean and variance. Procedural parameters (including
K) for each method were determined empirically through cross-validation.
4.1
Experiments on Simulated Data
For all simulated data, 10 independent training samples of size 200 were generated.
For each of these, an additional independent test sample consisting of 200 observations was generated. These test data were classified by each competing method
using the respective training data set. Error rates computed over all 2,000 such
classifications are reported in Table 1.
The Problems. (1) Multi-Gaussians. The data set consists of n = 2 input
features, l = 200 training data, and J = 2 classes. Each class contains two spherical bivariate normal subclasses, having standard deviation 1. The mean vectors
for one class are (-3/4, -3) and (3/4,3); whereas for the other class are (3, -3)
and (-3,3). For each class, data are evenly drawn from each of the two normal
subclasses. The first column of Table 1 shows the results for this problem. The
standard deviations are: 0.17, 0.01, 0.01, 0.01, 0.01 0.01, 0.01 and 1.50, respectively. (2) Noisy-Gaussians. The data for this problem are generated as in the
previous example, but augmented with four predictors having independent standard Gaussian distributions. They serve as noise. Results are shown in the second
column of Table 1. The standard deviations are: 0.18, 0.01, 0.02, 0.01, 0.01, 0.01,
0.01 and 1.60, respectively.
Results. Table 1 shows that all methods have similar performances for the MultiGaussians problem, with C4.5 being the worst performer. When the noisy predictors are added to the problem (NoisyGaussians), we observe different levels of
deterioration in performance among the eight methods. LFM-SVM shows the most
robust behavior in presence of noise. K-NN is instead the worst performer. In
Figure 2 we plot the performances of LFM-SVM and RBF-SVM as a function of an
increasing number of noisy features (for the same MultiGaussians problem). The
standard deviations for RBF-SVM (in order of increasing number of noisy features)
are: 0.01, 0.01 , 0.03, 0.03, 0.03 and 0.03. The standard deviations for LFM-SVM
are: 0.17,0.18,0.2,0.3,0.3 and 0.3. The LFM-SVM technique shows a considerable
improvement over RBF -SVM as the amount of noise increases.
Table 1: Average classification error rates for simulated and real data.
LFM-SVM
RBF-SVM
ADAMENN
Machete
Scythe
DANN
K-NN
C4.5
4.2
MultiGauss NoisyGauss Iris Sonar Liver Vote Breast OQ Pima
3.3
3.4
4.0 11.0 28.1 2.6
3.0
3.5 19.3
3.3
4.1
4.0 12.0 26.1 3.0
3.1
3.4 21.3
3.4
4.1
3.0 9.1
30.7 3.0
3.2
3.1 20.4
3.4
4.3
5.0 21.2 27.5 3.4
3.5
7.4 20.4
4.0 16.3 27.5 3.4
4.8
2.7
3.4
5.0 20.0
2.2
4.0 22.2
4.7
3.7
6.0 1.1 30.1 3.0
3.3
7.0
6.0 12.5 32.5 7.8
2.7
5.4 24.2
5.0
5.1
8.0 23.1 38.3 3.4
4.1
9.2 23.8
Experiments on Real Data
In our experiments we used seven different real data sets. They are all taken from
DCI Machine Learning Repository at http://www.cs.uci.edu/,,,-,mlearn/
MLRepository.html. For a description of the data sets see [6]. For the Iris, Sonar,
Liver and Vote data we perform leave-one-out cross-validation to measure performance, since the number of available data is limited for these data sets. For the
36'--'--'---r--'--~--'--'--~--.--'--~
LFM-SVM --+-RBF-SVM ---)(---
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
~ ~~=='P'O L-~--~--~~--~--~~--~--~~--~
o
10
12
14
16
18
20
22
Number of Noisy Variables
Figure 2: Average Error Rates of LFM-SVM and RBF-SVM as a function of an
increasing number of noisy predictors.
I
I
J. T I
- -? -?
- 1-
""'!""
:E
~
;l
""
..J
:E
>
:z
""'
~
z
zOJ
:E
""
""
Q
i
~
"
1j
~
i
z
z
""
Q
z
z
'"
3
Figure 3: Performance distributions for real data.
Breast, OQ-Ietter and Pima data we randomly generated five independent training
sets of size 200. For each of these, an additional independent test sample consisting
of 200 observations was generated. Table 1 (columns 3-9) shows the cross-validated
error rates for the eight methods under consideration on the seven real data. The
standard deviation values are as follows. Breast data: 0.2, 0.2, 0.2, 0.2, 0.2, 0.9, 0.9
and 0.9, respectively. OQ data: 0.2 , 0.2 , 0.2, 0.3, 0.2 , 1.1 , 1.5 and 2.1 , respectively.
Pima data: 0.4, 0.4, 0.4, 0.4, 0.4, 2.4, 2.1 and 0.7, respectively.
Results. Table 1 shows that LFM-SVM achieves the best performance in 2/7 of the
real data sets; in one case it shows the second best performance, and in the remaining
four its error rate is still quite close to t he best one. Following Friedman [9], we
capture robustness by computing the ratio bm of the error rate em of method m and
the smallest error rate over all methods being compared in a particular example:
bm = emf minl~k~8 ek?
Figure 3 plots the distribution of bm for each method over the seven real data sets.
The dark area represents the lower and upper quartiles of the distribution that are
separated by the median. The outer vertical lines show the entire range of values for
the distribution. The spread of the error distribution for LFM-SVM is narrow and
close to one. The results clearly demonstrate that LFM-SVM (and ADAMENN)
obtained the most robust performance over the data sets.
The poor performance of the machete and C4.5 methods might be due to the greedy
strategy they employ. Such recursive peeling strategy removes at each step a subset
of data points permanently from further consideration. As a result, changes in an
early split, due to any variability in parameter estimates, can have a significant
impact on later splits , thereby producing different terminal regions. This makes
predictions highly sensitive to the sampling fluctuations associated with the random
nature of the process that produces the traning data, thus leading to high variance
predictions. The scythe algorithm, by relaxing the winner-take-all splitting strategy
of the machete algorithm, mitigates the greedy nature of the approach, and thereby
achieves better performance.
In [10], the authors show that the metric employed by the DANN algorithm approximates the weighted Chi-squared distance, given that class densities are Gaussian
and have the same covariance matrix. As a consequence, we may expect a degradation in performance when the data do not follow Gaussian distributions and are
corrupted by noise , which is likely the case in real scenarios like the ones tested
here.
We observe that the sparse solution given by SVMs provides LFM-SVM with principled guidelines to efficiently set the input parameters. This is an important advantage over ADAMENN, which has six tunable input parameters. Furthermore,
LFM-SVM speeds up the classification process since it applies the nearest neighbor
rule only once, whereas ADAMENN applies it at each point within a region centered
at the query. We also observe that the construction of the SVM for LFM-SVM is
carried out off-line only once, and there exist algorithmic and computational results
which make SVM training practical also for large-scale problems [12].
The LFM-SVM offers performance improvements over the RBF-SVM algorithm
alone, for both the (noisy) simulated and real data sets. The reason for such performance gain may rely on the effect of our local weighting scheme on the separability
of classes, and therefore on the margin, as shown in [6]. Assigning large weights to
input features close to the gradient direction, locally in neighborhoods of support
vectors, corresponds to increase the spatial resolution along those orientations, and
therefore to improve the separability of classes. As a consequence, better classification results can be achieved as demonstrated in our experiments.
5
Related Work
In [1], Amari and Wu improve support vector machine classifiers by modifying
kernel functions. A primary kernel is first used to obtain support vectors. The
kernel is then modified in a data dependent way by using the support vectors: the
factor that drives the transformation has larger values at positions close to support
vectors. The modified kernel enlarges the spatial resolution around the boundary
so that the separability of classes is increased.
The resulting transformation depends on the distance of data points from the support vectors , and it is therefore a local transformation, but is independent of the
boundary's orientation in input space. Likewise, our transformation metric depends , through the factor A, on the distance of the query point from the support
vectors. Moreover, since we weight features, our metric is directional, and depends
on the orientation of local boundaries in input space. This dependence is driven
by our measure of feature relevance, which has the effect of increasing the spatial
resolution along discriminant directions around the boundary.
6
Conclusions
We have described a locally adaptive metric classification method and demonstrated
its efficacy through experimental results. The proposed technique offers performance improvements over the SVM alone, and has the potential of scaling up to
large data sets. It speeds up, in fact, the classification process by computing offline the information relevant to define local weights, and by applying the nearest
neighbor rule only once.
Acknowledgments
This research has been supported by the National Science Foundation under grants
NSF CAREER Award 9984729 and NSF IIS-9907477, by the US Department of
Defense, and a research award from AT&T.
References
[1] S. Amari and S. Wu, "Improving support vector machine classifiers by modifying
kernel functions", Neural Networks, 12, pp. 783-789, 1999.
[2] R.E. Bellman, Adaptive Control Processes. Princeton Univ. Press, 1961.
[3] M. Brown, W. Grundy, D. Lin, N. Cristianini, C. Sugnet, T. Furey, M. Ares, and
D. Haussler, "Knowledge-based analysis of microarray gene expressions data using
support vector machines", Tech. Report, University of California in Santa Cruz,
1999.
[4] W.S. Cleveland and S.J. Devlin, "Locally Weighted Regression: An Approach to
Regression Analysis by Local Fitting", J. Amer. Statist. Assoc. 83, 596-610, 1988
[5] T.M. Cover and P.E. Hart, "Nearest Neighbor Pattern Classification", IEEE Trans.
on Information Theory, pp. 21-27, 1967.
[6] C. Domeniconi and D. Gunopulos, "Adaptive Nearest Neighbor Classification using
Support Vector Machines", Tech. Report UCR-CSE-01-04, Dept. of Computer Science, University of California, Riverside, June 200l.
[7] C. Domeniconi, J. Peng, and D. Gunopulos, "An Adaptive Metric Machine for Pattern
Classification", Advances in Neural Information Processing Systems, 2000.
[8] R.O. Duda and P.E. Hart, Pattern Classification and Scene Analysis. John Wiley &
Sons, Inc., 1973.
[9] J.H. Friedman "Flexible Metric Nearest Neighbor Classification", Tech. Report, Dept.
of Statistics, Stanford University, 1994.
[10] T. Hastie and R. Tibshirani, "Discriminant Adaptive Nearest Neighbor Classification", IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 18, No.6, pp.
607-615, 1996.
[11] T. Joachims, "Text categorization with support vector machines", Pmc. of European
Conference on Machine Learning, 1998.
[12] T. Joachims, "Making large-scale SVM learning practical" Advances in Kernel Methods - Support Vector Learning, B. Sch6lkopf and C. Burger and A. Smola (ed.), MITPress, 1999. http://www-ai.cs.uni-dortmund.de/thorsten/svm_light.html
[13] D.G. Lowe, "Similarity Metric Learning for a Variable-Kernel Classifier", Neural Computation 7(1):72-85, 1995.
[14] E. Osuna, R. Freund, and F. Girosi, "Training support vector machines: An application to face detection", Pmc. of Computer Vision and Pattern Recognition, 1997.
[15] J.R. Quinlan, C4.5: Programs for Machine Learning. Morgan-Kaufmann Publishers,
Inc., 1993.
[16] C.J. Stone, Nonparametric regression and its applications (with discussion). Ann.
Statist. 5, 595, 1977.
| 2054 |@word repository:1 proportion:1 duda:1 nd:2 covariance:1 thereby:3 solid:1 initial:1 contains:1 efficacy:1 denoting:1 si:2 assigning:1 john:1 cruz:1 informative:1 girosi:1 remove:1 plot:2 alone:3 greedy:2 intelligence:1 isotropic:1 xk:2 farther:1 provides:2 cse:1 location:2 five:1 adamenn:5 along:13 become:1 consists:1 fitting:1 introduce:1 peng:1 behavior:1 multi:1 chi:2 terminal:1 inspired:1 bellman:1 spherical:1 curse:2 considering:1 increasing:4 becomes:6 provided:2 cleveland:1 bounded:2 moreover:1 maximizes:1 furey:1 burger:1 transformation:4 subclass:2 classifier:8 assoc:1 partitioning:1 unit:2 grant:1 control:1 producing:1 local:17 gunopulos:3 consequence:3 fluctuation:1 might:1 twice:1 relaxing:1 limited:1 sill:3 perpendicular:1 range:1 practical:3 acknowledgment:1 recursive:2 carlotta:2 procedure:1 lla:1 area:3 radial:2 mitpress:1 get:1 doubled:1 undesirable:1 close:12 risk:1 applying:2 influence:2 www:2 elongated:1 demonstrated:2 resolution:4 splitting:2 rule:7 haussler:1 spanned:1 stability:1 construction:1 suppose:1 ilq:3 us:2 element:1 approximated:2 recognition:1 located:1 predicts:1 exk:1 capture:1 worst:2 wj:5 region:2 decrease:1 principled:2 grundy:1 cristianini:1 depend:1 segment:1 serve:1 learner:1 basis:2 separated:3 univ:1 query:17 neighborhood:9 outcome:1 whose:3 xi12:1 quite:1 larger:1 stanford:1 amari:2 enlarges:1 statistic:1 noisy:7 final:1 advantage:2 propose:2 product:2 frequent:1 relevant:2 uci:1 till:1 achieve:1 adapts:1 indi:1 description:1 pll:1 correspondent:1 produce:2 categorization:1 leave:1 liver:2 measured:1 nearest:20 c:3 implies:1 direction:14 modifying:2 quartile:1 centered:1 enable:1 generalization:1 around:4 normal:2 exp:4 mapping:2 predict:3 algorithmic:1 achieves:2 early:1 smallest:1 estimation:1 label:6 sensitive:1 successfully:1 tool:1 weighted:4 minimization:1 clearly:2 gaussian:3 modified:3 ndi:1 ax:1 validated:1 june:1 joachim:2 improvement:5 properly:1 sugnet:1 tech:3 sense:1 dependent:1 nn:3 typically:1 entire:1 iu:1 classification:29 flexible:5 orientation:10 among:1 html:2 spatial:4 equal:2 once:4 extraction:1 having:3 elongation:1 nicely:1 sampling:1 represents:1 argminp:1 report:3 employ:1 randomly:1 dg:1 national:1 dimitrios:1 consisting:2 friedman:2 detection:1 highly:3 severe:1 closer:1 capable:1 respective:1 bq:3 tree:1 euclidean:2 guidance:1 theoretical:1 increased:2 classify:1 soft:1 column:3 cover:1 deviation:6 subset:1 uniform:1 predictor:3 reported:1 constriction:1 corrupted:1 sv:2 traning:1 density:1 stay:1 off:3 squared:3 ek:1 leading:1 potential:1 de:1 summarized:1 inc:2 dann:3 depends:3 crossed:2 later:1 try:1 lowe:1 doing:1 reached:1 bayes:1 relied:1 minimize:2 square:1 variance:3 kaufmann:1 efficiently:1 likewise:1 directional:1 produced:1 dortmund:1 drive:1 ary:1 classified:1 mlearn:1 ed:1 pp:3 conveys:1 associated:2 stop:1 gain:1 proved:1 tunable:1 knowledge:2 dimensionality:2 higher:2 follow:1 reflected:1 arj:2 amer:1 furthermore:2 smola:1 hand:2 effect:3 contain:1 normalized:2 brown:1 vicinity:1 ind:1 ll:1 whereby:1 mlrepository:1 iris:2 stone:1 theoretic:1 demonstrate:1 cp:3 consideration:2 ari:2 common:1 empirically:1 winner:1 exponentially:1 volume:1 jl:1 extend:1 he:5 approximates:1 measurement:1 significant:1 ai:1 stretched:1 tuning:1 dj:1 dot:1 moving:1 similarity:1 surface:1 mlight:2 closest:3 posterior:2 driven:1 scenario:1 arbitrarily:3 yi:2 morgan:1 minimum:1 additional:2 performer:2 employed:1 minl:1 determine:2 maximize:1 ii:1 desirable:1 rj:4 cross:5 offer:2 lin:2 hart:2 award:2 impact:1 prediction:3 emf:1 regression:3 breast:3 vision:1 metric:15 iteration:1 kernel:10 deterioration:1 achieved:3 whereas:2 want:1 median:1 microarray:1 crucial:2 publisher:1 subject:1 tend:1 oq:3 structural:1 presence:1 split:3 variety:2 hastie:1 competing:1 devlin:1 six:1 expression:1 defense:1 svm_light:1 riverside:2 cpt:1 generally:1 santa:1 amount:3 nonparametric:1 dark:1 locally:12 statist:2 svms:10 http:2 exist:1 nsf:2 sign:3 estimated:2 tibshirani:1 vol:1 four:2 procedural:1 drawn:1 sum:1 wu:2 decision:4 ucr:2 scaling:1 bound:1 quadratic:1 constraint:1 scene:1 speed:2 influential:1 department:1 according:2 poor:1 lld:12 em:1 separability:3 wi:1 appealing:4 son:1 osuna:1 making:1 thorsten:1 taken:2 mechanism:2 know:1 junction:1 gaussians:2 available:1 elongate:1 observe:4 eight:2 robustness:1 permanently:1 remaining:1 quinlan:1 uj:1 build:1 seeking:1 move:3 added:1 strategy:3 primary:1 dependence:1 gradient:5 subspace:1 distance:18 constricted:1 simulated:6 outer:1 evenly:1 seven:3 extent:1 discriminant:7 reason:1 ratio:1 pmc:2 difficult:1 mostly:1 pima:3 dci:1 negative:1 guideline:2 adjustable:1 perform:1 sch6lkopf:1 upper:1 vertical:1 observation:4 finite:3 defining:1 variability:1 lfm:18 introduced:1 namely:1 optimized:1 c4:5 california:3 narrow:1 trans:2 address:1 pattern:6 ilxk:2 program:1 including:2 natural:1 rely:1 scheme:9 improve:2 identifies:2 axis:1 carried:2 text:1 determining:1 asymptotic:1 freund:1 expect:1 proportional:1 validation:4 foundation:2 principle:1 pi:3 supported:1 soon:1 offline:1 bias:2 guide:1 neighbor:23 face:1 sparse:1 boundary:27 dimension:4 xn:1 curve:1 scythe:3 computes:3 preventing:1 author:1 commonly:1 adaptive:12 projected:1 bm:3 founded:1 approximate:1 uni:1 gene:1 xi:4 machete:5 sonar:2 table:7 nature:2 robust:2 ca:1 career:1 ignoring:1 improving:1 european:1 spread:1 noise:4 allowed:1 convey:1 augmented:1 join:1 wiley:1 position:1 exponential:4 xl:1 weighting:12 peeling:1 enlarging:1 mitigates:1 minsi:4 decay:1 svm:37 evidence:1 bivariate:1 consist:1 occurring:1 sparseness:1 margin:4 likely:6 infinitely:1 lazy:2 desire:2 applies:2 corresponds:1 satisfies:1 relies:1 conditional:5 goal:2 llxi:1 ann:1 invalid:2 rbf:8 considerable:2 change:3 determined:2 degradation:1 total:1 domeniconi:3 experimental:3 vote:2 formally:1 support:21 relevance:9 dept:3 princeton:1 tested:1 avoiding:1 |
1,157 | 2,055 | Infinite Mixtures of Gaussian Process Experts
Carl Edward Rasmussen and Zoubin Ghahramani
Gatsby Computational Neuroscience Unit
University College London
17 Queen Square, London WC1N 3AR, England
edward,zoubin@gatsby.ucl.ac.uk
http://www.gatsby.ucl.ac.uk
Abstract
We present an extension to the Mixture of Experts (ME) model, where
the individual experts are Gaussian Process (GP) regression models. Using an input-dependent adaptation of the Dirichlet Process, we implement a gating network for an infinite number of Experts. Inference in this
model may be done efficiently using a Markov Chain relying on Gibbs
sampling. The model allows the effective covariance function to vary
with the inputs, and may handle large datasets ? thus potentially overcoming two of the biggest hurdles with GP models. Simulations show
the viability of this approach.
1 Introduction
Gaussian Processes [Williams & Rasmussen, 1996] have proven to be a powerful tool for
regression. They combine the flexibility of being able to model arbitrary smooth functions
if given enough data, with the simplicity of a Bayesian specification that only requires inference over a small number of readily interpretable hyperparameters ? such as the length
scales by which the function varies along different dimensions, the contributions of signal
and noise to the variance in the data, etc. However, GPs suffer from two important limitacovariance matrix where is
tions. First, because inference requires inversion of an
the number of training data points, they are computationally impractical for large datasets.
Second, the covariance function is commonly assumed to be stationary, limiting the modeling flexibility. For example, if the noise variance is different in different parts of the input
space, or if the function has a discontinuity, a stationary covariance function will not be
adequate. Goldberg et al [1998] discussed the case of input dependent noise variance.
Several recent attempts have been aimed at approximate inference in GP models [Williams
& Seeger 2001, Smola & Bartlett 2001]. These methods are based on selecting a projection
of the covariance matrix onto a smaller subspace (e.g. a subset of the data points) reducing
the overall computational complexity. There have also been attempts at deriving more
complex covariance functions [Gibbs 1997] although it can be difficult to decide a priori
on a covariance function of sufficient complexity which guarantees positive definiteness.
In this paper we will simultaneously address both the problem of computational complexity
and the deficiencies in covariance functions using a divide and conquer strategy inspired
by the Mixture of Experts (ME) architecture [Jacobs et al, 1991]. In this model the input
space is (probabilistically) divided by a gating network into regions within which specific
separate experts make predictions. Using GP models as experts we gain the double advantage that computation for each expert is cubic only in the number of data point in its region,
rather than in the entire dataset, and that each GP-expert may learn different characteristics
of the function (such as lengths scales, noise variances, etc). Of course, as in the ME, the
learning of the experts and the gating network are intimately coupled.
Unfortunately, it may be (practically and statistically) difficult to infer the appropriate number of experts for a particular dataset. In the current paper we sidestep this difficult problem
by using an infinite number of experts and employing a gating network related to the Dirichlet Process, to specify a spatially varying Dirichlet Process. An infinite number of experts
may also in many cases be more faithful to our prior expectations about complex real-word
datasets. Integrating over the posterior distribution for the parameters is carried out using
a Markov Chain Monte Carlo approach.
Tresp [2001] presented an alternative approach to mixtures of GPs. In his approach both the
experts and the gating network were implemented with GPs; the gating network being
a softmax of
GPs. Our new model avoids several limitations of the previous approach,
which are covered in depth in the discussion.
2 Infinite GP mixtures
The traditional ME likelihood does not apply when the experts are non-parametric. This is
because in a normal ME model the data is assumed
to
be
iid given the
model parameters:
"!
#
where and are inputs and outputs (boldface denotes vectors),
are the parameters
of expert , ! are the parameters of the gating network and are the discrete indicator
variables assigning data points to experts.
This iid assumption is contrary to GP models which solely model the dependencies in the
joint distribution (given the hyperparameters). There is a joint distribution corresponding
to every possible assignment of data points to experts; therefore the likelihood is a sum
over (exponentially many) assignments:
$
%&' ()
*
+,(-
"!
65
5
6
&/. 01324 )78924 )7*
;:<,(-
"!
#=
(1)
Given the configuration (>?,@ 4=A=A=4 $<BC
, the distribution factors into the product, over experts, of the joint Gaussian distribution of all data points assigned to each expert. Whereas
the original ME formulation used expectations of assignment variables called responsibili
ties, this is inadequate for inference
in the mixture of GP experts. Consequently, we directly
represent the indicators, , and Gibbs sample for them to capture their dependencies.
In Gibbs sampling we need the posterior conditional distribution for each indicator given
indicators
all the remaining
and the data:
, D (FE $
G $ C "!
H% I J "(FE $
D (FE $
"!
#
where (CE denotes all indicators except number K . We defer discussion of the second term
defining the gating network to the next section. As discussed, the first term being the
likelihood given the indicators factors into independent terms for each expert. For Gibbs
5MON
sampling we therefore need the probability
of output 5 under GP number :
924L
K $PL C7* <2Q CL PL )7
#=
For a GP model, this conditional density is the well known Gaussian [Williams & Rasmussen, 1996]:
E @ E
6
$
E @
E
$
#
(2)
where the covariance matrix
depends on the parameters . Thus, for the GP expert, we
compute the above conditional density by simply evaluating the GP on the data assigned
to it. Although this equation looks computationally expensive, we can keep track of the
inverse covariance matrices and reuse them for consecutive Gibbs updates by performing
rank one updates (since Gibbs sampling changes at most one indicator at a time).
We are free to choose any valid covariance function for the experts. In our simulations we
employed the following Gaussian covariance function:
$
& $
$+-,
G
"#! %$I 0
('()*
@ . K K0/
(3)
1 controlling the signal variance, @ controlling the noise variance,
$
with hyperparameters
)
and
controlling the length scale or (inverse) relevance of the 2 -th dimension of in
relation to predicting ; . is the Kronecker delta function (i.e. . K K /
! if K K / , o.w. 0).
3 The Gating network
The gating network assigns probability to different experts based entirely on the input. We
will derive a gating network based on the Dirichlet Process which can be defined as the
limit of a Dirichlet distribution when the number of classes tends to infinity. The standard Dirichlet Process is not input dependent, but we will modify it to serve as a gating
mechanism. We start from a symmetric Dirichlet distribution on proportions:
43 @ A= = = 365C 76
- 8:9&;9=<?>A@& BQC7 '%D
G
E 4 76
5 G3HJI 5 E @
E 4 7 F' D
2000] that
where 7 is the (positive) concentration parameter. It can be shown [Rasmussen,
3
the conditional probability of a single indicator when integrating over the
variables and
K
letting D tend to infinity is given by:
E
components where
all other components
4K combined:
&0R
K MLGN
1
( E
:
N &
N
for all K / K (CE
1
76
76
6E ,
O ! 7
7 ,
O ! 7
(4)
where GE
( QP
S . , 8
) is the occupation number of expert excluding observation
K , and is the total number of data points. This shows that the probabilities are proportional
to the occupation numbers. To make the gating network input dependent, we will simply
employ a local estimate 1 for this occupation
classifier:
a kernel
R number
JUWV using
E
4K
T
!
P
S R F W
U V
. ,& 8
P S
U
(5)
where the delta function selects data points assigned to class , and is the kernel function
parametrized by ! . As an example we use a Gaussian kernel function:
UOV
1
$
$
$+
G
X1 Y
#! $
' !
(6)
this local estimate won?t generally be an integer, but this doesn?t have any adverse consequences
$
parameterized by length scales ! for each dimension. These length scales allow dimensions of space to be more or less relevant to the gating network classification.
We Gibbs sample from the indicator variables by multiplying the input-dependent Dirichlet
process prior eq. (4) and (5) with the GP conditional density eq. (2). Gibbs sampling in an
infinite model requires that the indicator variables can take on values that no other indicator
variable has already taken, thereby creating new experts. We use the auxiliary variable
approach of Neal [1998] (algorithm 8 in that paper). In this approach hyperparameters for
new experts are sampled from their prior and the likelihood is evaluated based on these.
This requires finding the likelihood of a Gaussian process with no data. Fortunately, for, the
covariance function eq. (3) this likelihood is Gaussian with zero mean and variance @ .
If all data points are assigned to a single GP, the likelihood calculation will still be cubic
in the number of data points (per Gibbs sweep over all indicators). We can reduce the computational complexity by introducing the constraint that no GP expert can have more than
2
max data points assigned to it. This is easily implemented by modifying the conditionals
in the Gibbs sampler.
The hyperparameter 7 controls the prior probability of assigning a data point to a new
expert, and therefore influences the total number of experts used to model the data. As in
Rasmussen [2000], we give a vague inverse gamma prior to 7 , and sample from its posterior
using Adaptive Rejection Sampling (ARS) [Gilks & Wild, 1992]. Allowing 7 to vary gives
the model more freedom to infer the number of GPs to use for a particular dataset.
Finally we need to do inference for the parameters of the gating function. Given a set of
indicator variables one could use standard methods from kernel classification to optimize
the kernel widths in different directions. These methods typically optimize the leave-oneout pseudo-likelihood (ie the product of the conditionals), since computing the likelihood
in a model defined purely from conditional distributions as in eq. (4), (5) & (6) is generally
difficult (and as pointed out in the discussion section there may not even be a single likelihood). In our model we multiply the pseudo-likelihood by a (vague) prior and sample from
the resulting pseudo-posterior.
4 The Algorithm
The individual GP experts are given a stationary Gaussian covariance function,, with
a sin# (where
gle length scale per dimension, a signal variance and a noise variance, i.e.
is the dimension of the input) hyperparameters per expert, eq. (3). The signal and noise
variances are given inverse gamma priors with hyper-hypers and (separately for the
two variances). This serves to couple the hyperparameters between experts, and allows the
priors on and *@ (which are used when evaluating auxiliary classes) to adapt. Finally we
give vague independent log normal priors to the lenght scale paramters ) and ! .
The algorithm for learning an infinite mixture of GP experts consists of the following steps:
1.
2.
3.
4.
5.
2
Initialize indicator variables
to a single value (or a few values if individual GPs
are to be kept small for computational reasons).
Do a Gibbs sampling sweep over all indicators.
$
Do Hybrid Monte Carlo (HMC) [Duane
et al, 1987] for hyperparameters of the
@
)
GP covariance function,
, for each expert in turn. We used 10 leapfrog
iterations with a stepsize small enough that rejections were rare.
Optimize the hyper-hypers, & , for each of the variance parameters.
Sample the Dirichlet process concentration parameter, 7 using ARS.
We simply set the conditional probability of joining a class which has been deemed full to zero.
100
50
50
Acceleration (g)
Acceleration (g)
100
0
?50
?50
?100
?100
iMGPE
stationary GP
?150
0
0
10
20
30
40
Time (ms)
50
60
?150
0
10
20
30
40
Time (ms)
50
60
Figure 1: The left hand plot shows the motorcycle impact data (133 points) together with
the median of the model?s predictive distribution, and for comparison the mean of a staNJN
tionary covariance
GP model (with optimized hyperparameters). On the right hand plot
we show !
samples from the posterior distribution for the iMGPE of the (noise free)
function evaluated intervals of 1 N ms.
We have jittered the points in the plot along the time
= # ms noise, so that the density can be seen more easily.
dimension by
adding
uniform
# std error ( ) confidence interval for the (noise free) function predicted by
Also, the
a stationary GP is plotted (thin lines).
6. Sample the gating kernel widths, ! ; we use the Metropolis method to sample from
the pseudo-posterior with a Gaussian proposal fit at the current ! 3
7. Repeat from 2 until the Markov chain has adequately sampled the posterior.
5 Simulations on a simple real-world data set
To illustrate our algorithm, we used the motorcycle dataset, fig. 1, discussed in Silverman
[1985]. This dataset is obviously non-stationary and has# input-dependent noise. We noticed
that the raw data is discretized# into bins of size ! = g; accordingly we cut off the prior
for the noise variance at ' ! .
The model is able to capture the general shape of the function and also the input-dependent
N the right hand plot in fig. 1, where the
nature of the noise (fig. 1). This can be seen from
uncertainty of the function is very low for
!
owing to a small inferred noise level
in this region. For comparison, the predictions from a stationary GP has been superimposed
in fig. 1. Whereas the medians of the predictive distributions agree to a large extent (left
N (right hand). The hohand plot), we see a huge difference in the predictive distributions
L
J
N
moscedastic GP cannot capture the very tight distribution for
! ms offered by iMGPE.
Also for large
ms, the iMGPE model predicts with fairly high probability that the
signal could be very close to zero. Note that the predictive distribution of the function is
multimodal, for example, around time 35 ms. Multimodal predictive distributions could
in principle be obtained from an ordinary GP by integrating over hyperparameters, howN
ever, in a mixture of GP?s model they can arise naturally.
The predictive distribution of the
function appears to have significant mass around g which seems somewhat artifactual.
We explicitly did not normalize or center the data, which has a large range in output. The
3
The Gaussian fit uses the derivative and Hessian of the log posterior wrt the log length scales.
Since this is an asymmetric proposal the acceptance probabilities must be modified accordingly. This
scheme has the advantage of containing no tunable parameters; however when the dimension is
large, it may be computationally more efficient to use HMC, to avoid calculation of the Hessian.
14
100
12
90
20
80
70
60
60
50
80
10
frequency
40
8
6
40
4
30
100
20
120
10
0
20
40
60
80
100
120
2
0
5
10
15
20
25
number of occupied experts
30
Figure 2: The left hand plot shows the number of times, out of 100 samples, that the
indicator variables for two data points were equal. The data have been sorted from left-toright according to the value of the time variable (since the data is not equally spaced in time
the axis of this matrix cannot be aligned with the plot in fig.1). The right hand plot shows
a histogram over the 100 samples of the number of GP experts used to model the data.
Gaussian processes had zero mean a priori, which coupled with the concentration of data
around zero may explain the posterior mass at zero. It would be more natural to treat the
GP means as separate hyperparameters controlled by a hyper-hyperparameter (centered at
zero) and do inference on them, rather than fix them all at 0. Although the scatter of data
from the predictive distribution for iMGPE looks somewhat messy with multimodality etc,
it is important to note that it assigns high density to regions that seem probable.
The motorcycle data appears to have roughly three regions: a flat low-noise region, followed by a curved region, and a flat high noise region. This intuition is bourne out by the
model. We can see this in two ways. Fig 2. (left) shows the number of times two data
points were assigned to the same expert. A N clearly defined block captures the initial flat
region and a few other points that lie near the g line; the middle block captures the curved
region, with a more gradual transition to the last flat region. A histogram of the number of
GP experts used showsN that the posterior distribution of number of needed GPs has a broadN
peak between and ! , where less than 3 occupied experts is very unlikely, and above !
becoming progressively less likely. Note that it never uses just a single GP to model the
data which accords with the intuition that a single stationary covariance function would be
inadequate. We should point out that the model is not trying to do model selection between
finite GP mixtures, but rather always assumes that there are infinitely many available, most
of which contribute with small mass 4 to a diffuse density in the background.
In figure 3 we assessed the convergence rate of the Markov Chain by plotting the autoNJN%N mixing time is around
correlation function for several parameters. We conclude that the
J
N
%
N
%
N
N
5N
100 iterations . Consequently, we run the chain for a total of !J!
iterations, discarding
the initial !
(burn-in) and keeping every ! ?th after that. The total computation time
was around 1 hour (1 GHz Pentium).
The right hand panel of figure 3 shows the distribution of the gating function kernel width
The total mass of the non-represented experts is
, where the posterior for in this
experiment is peaked between and
(see figure 3, bottom right panel), corresponding to about
of the total mass
5
the sum of the auto-correlation coefficients from to is an estimate of the mixing time
4
0.8
5
0.6
0
0.4
10
frequency
auto correlation coefficient
log number of occupied experts
log gating kernel width
log Dirichlet concentration
frequency
10
1
0.2
0
0
50
100
150
time lag in iterations
200
?1
?0.5
0
0.5
log (base 10) gating function kernel width
5
0
?0.5
0
0.5
1
log (base 10) Dirichlet process concentration
NJN%left
N hand plot shows the auto-correlation for various parameters of the model
Figure 3: The
N%N of the (log) kernel
based on !%!
iterations. The right hand plots show the distribution
width ! and (log) Dirichlet concentration parameter 7 , based on !
samples from the
posterior.
N
and the concentration
parameter of the Dirichlet process. The posterior 6 kernel width !
=
lies between ! and ; comparing to the scale of the inputs these are quite short distances,
corresponding to rapid transitions between experts (as opposed to lengthy intervals with
multiple active experts). This corresponds well with our visual impression of the data.
!
6 Discussion and Conclusions
We now return to Tresp [2000]. There are four ways in which the infinite mixture of GP
experts differs from, and we believe, improves upon the model presented by Tresp. First,
in his model, although a gating network divides up the input space, each GP expert predicts
on the basis of all of the data. Data that was not assigned to a GP expert can therefore spill
over into predictions of a GP, which will lead to bias near region boundaries especially
for experts with long length scales. Second, if there are
experts, Tresp?s model has
GPs (the experts, noise models, and separate gating functions) each of which requires
4
computations. In our model
computations over the entire dataset resulting in
since the experts divide up the data points, if there are
experts equally dividing the data
computations (each of Gibbs updates requires a rank-one
an iteration takes '
for each of the experts andN the Hybrid Monte Carlo takes
computation '
'
times
). Even for modest
(e.g.
! ) this can be a significant saving.
Inference for the gating length scale parameters is
if the full Hessian is used,
but can be reduced to
for a diagonal approximation, or Hybrid Monte Carlo if the
input dimension is large. Third, by going to the Dirichlet process infinite limit, we allow
the model to infer the number of components required to capture the data. Finally, in our
model the GP hyperparameters are not fixed but are instead inferred from the data.
We have defined the gating network prior implicitly in terms of the conditional distribution
of an indicator variable given all the other indicator variables. Specifically, the distribution
of this indicator variable is an input-dependent Dirichlet process with counts given by local
estimates of the data density in each class eq. (5). We have not been able to prove that
these conditional distributions are always consistent with a single joint distribution over
for comparison
the (vague) prior on the kernel width is log normal with of the mass between
and , corresponding to very short (sub sample) distances upto distances comparable to the
entire input range
6
the indicators. If indeed there does not exist a single consistent joint distribution the Gibbs
sampler may converge to different distributions depending on the order of sampling.
We are encouraged by the preliminary results obtained on the motorcycle data. Future work
should also include empirical comparisons with other state-of-the-art regression methods
on multidimensional benchmark datasets. We have argued here that single iterations of the
MCMC inference are computationally tractable even for large data sets, experiments will
show whether mixing is sufficiently rapid to allow practical application. We hope that the
extra flexibility of the effective covariance function will turn out to improve performance.
Also, the automatic choice of the number of experts may make the model advantageous for
practical modeling tasks.
Finally, we wish to come back to the modeling philosophy which underlies this paper. The
computational problem in doing inference and prediction using Gaussian Processes arises
out of the unrealistic assumption that a single covariance function captures the behavior of
the data over its entire range. This leads to a cumbersome matrix inversion over the entire
data set. Instead we find that by making a more realistic assumption, that the data can be
modeled by an infinite mixture of local Gaussian processes, the computational problem
also decomposes into smaller matrix inversions.
References
Gibbs, M. N. (1997). Bayesian Gaussian Processes for Regression and Classification. PhD
thesis. University of Cambridge.
Goldberg, P. W., Williams, C. K. I., & Bishop C. M. (1998). Regression with Inputdependent Noise, NIPS 10.
Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D. (1987). Hybrid Monte Carlo,
Physics letters B, vol. 55, pp. 2774?2777.
Gilks, W. R. & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied
Statistics 41, 337?348.
Jacobs, R. A., Jordan, M. I., Nowlan, S. J. & Hinton, G. E. (1991). Adaptive mixture of
local experts. Neural Computation, vol 3, pp 79?87.
Neal, R. M. (1998). Markov chain sampling methods for Dirichlet process mixture models.
Technical Report 4915, Department of Statistics, University of Toronto.
http://www.cs.toronto.edu/ radford/mixmc.abstract.html.
Rasmussen, C. E. (2000). The Infinite Gaussian Mixture Model, NIPS 12, S.A. Solla, T.K.
Leen and K.-R. M?uller (eds.), pp. 554?560, MIT Press.
Silverman, B. W. (1985). Some aspects of the spline smoothing approach to non-parametric
regression curve fitting. J. Royal Stat. Society. Ser. B, vol. 47, pp. 1?52.
Smola A. J. and Bartlett, P. (2001). Sparse Greedy Gaussian Process Regression, NIPS 13.
Tresp V. (2001). Mixtures of Gaussian Process, NIPS 13.
Williams, C. K. I. and Seeger, M. (2001). Using the Nystr?om Method to Speed Up Kernel
Machines, NIPS 13.
Williams, C. K. I. and C. E. Rasmussen (1996). Gaussian Processes for Regression, in
D. S. Touretzky, M. C. Mozer and M. E. Hasselmo (editors), NIPS 8, MIT Press.
| 2055 |@word middle:1 inversion:3 advantageous:1 proportion:1 seems:1 simulation:3 gradual:1 covariance:18 jacob:2 thereby:1 nystr:1 initial:2 configuration:1 selecting:1 bc:1 current:2 comparing:1 nowlan:1 assigning:2 scatter:1 must:1 readily:1 realistic:1 shape:1 plot:10 interpretable:1 update:3 progressively:1 stationary:8 greedy:1 accordingly:2 short:2 contribute:1 toronto:2 along:2 consists:1 prove:1 combine:1 wild:2 fitting:1 multimodality:1 indeed:1 rapid:2 behavior:1 roughly:1 discretized:1 inspired:1 relying:1 panel:2 mass:6 finding:1 impractical:1 guarantee:1 pseudo:4 every:2 multidimensional:1 tie:1 classifier:1 uk:2 control:1 unit:1 ser:1 positive:2 local:5 modify:1 tends:1 limit:2 consequence:1 treat:1 joining:1 solely:1 becoming:1 burn:1 range:3 statistically:1 faithful:1 gilks:2 practical:2 block:2 implement:1 differs:1 silverman:2 empirical:1 projection:1 word:1 integrating:3 confidence:1 zoubin:2 onto:1 cannot:2 close:1 selection:1 influence:1 www:2 optimize:3 center:1 williams:6 simplicity:1 assigns:2 deriving:1 his:2 handle:1 limiting:1 controlling:3 carl:1 gps:8 goldberg:2 us:2 expensive:1 std:1 asymmetric:1 cut:1 predicts:2 bottom:1 capture:7 region:12 solla:1 intuition:2 mozer:1 complexity:4 messy:1 tight:1 predictive:7 serve:1 purely:1 upon:1 basis:1 vague:4 easily:2 joint:5 multimodal:2 k0:1 represented:1 various:1 effective:2 london:2 monte:5 hyper:3 pendleton:1 mon:1 lag:1 quite:1 statistic:2 gp:35 obviously:1 advantage:2 ucl:2 product:2 adaptation:1 relevant:1 motorcycle:4 aligned:1 mixing:3 flexibility:3 normalize:1 convergence:1 double:1 leave:1 tions:1 derive:1 illustrate:1 ac:2 stat:1 depending:1 eq:6 edward:2 dividing:1 implemented:2 auxiliary:2 predicted:1 come:1 c:1 direction:1 owing:1 modifying:1 centered:1 bin:1 argued:1 fix:1 preliminary:1 probable:1 extension:1 pl:2 practically:1 around:5 sufficiently:1 normal:3 vary:2 consecutive:1 hasselmo:1 tool:1 hope:1 uller:1 mit:2 clearly:1 gaussian:20 always:2 modified:1 rather:3 occupied:3 avoid:1 varying:1 probabilistically:1 leapfrog:1 rank:2 likelihood:11 superimposed:1 seeger:2 pentium:1 inference:10 dependent:8 entire:5 typically:1 unlikely:1 relation:1 going:1 selects:1 overall:1 classification:3 html:1 priori:2 art:1 softmax:1 initialize:1 fairly:1 smoothing:1 equal:1 never:1 saving:1 sampling:11 encouraged:1 look:2 thin:1 bourne:1 peaked:1 future:1 report:1 spline:1 employ:1 few:2 simultaneously:1 gamma:2 individual:3 attempt:2 freedom:1 huge:1 acceptance:1 multiply:1 mixture:15 wc1n:1 chain:6 bq:1 modest:1 divide:3 plotted:1 roweth:1 modeling:3 ar:3 queen:1 assignment:3 ordinary:1 introducing:1 subset:1 rare:1 uniform:1 inadequate:2 dependency:2 varies:1 jittered:1 combined:1 density:7 peak:1 ie:1 off:1 physic:1 together:1 thesis:1 containing:1 choose:1 opposed:1 creating:1 expert:56 sidestep:1 derivative:1 return:1 coefficient:2 explicitly:1 depends:1 doing:1 start:1 defer:1 contribution:1 om:1 square:1 variance:13 characteristic:1 efficiently:1 spaced:1 inputdependent:1 bayesian:2 raw:1 iid:2 carlo:5 multiplying:1 kennedy:1 explain:1 cumbersome:1 touretzky:1 ed:1 lengthy:1 c7:2 frequency:3 pp:4 naturally:1 couple:1 gain:1 sampled:2 dataset:6 tunable:1 improves:1 back:1 appears:2 specify:1 formulation:1 done:1 evaluated:2 leen:1 just:1 smola:2 until:1 correlation:4 hand:9 believe:1 adequately:1 assigned:7 spatially:1 symmetric:1 neal:2 sin:1 width:8 won:1 m:7 trying:1 impression:1 qp:1 exponentially:1 discussed:3 significant:2 cambridge:1 gibbs:16 automatic:1 pointed:1 had:1 specification:1 etc:3 base:2 posterior:13 recent:1 seen:2 fortunately:1 somewhat:2 employed:1 converge:1 signal:5 full:2 multiple:1 infer:3 smooth:1 technical:1 england:1 calculation:2 adapt:1 long:1 divided:1 equally:2 controlled:1 impact:1 prediction:4 underlies:1 regression:8 expectation:2 iteration:7 represent:1 kernel:13 histogram:2 accord:1 proposal:2 whereas:2 hurdle:1 conditionals:2 separately:1 interval:3 background:1 median:2 extra:1 tend:1 contrary:1 seem:1 jordan:1 integer:1 near:2 viability:1 enough:2 fit:2 architecture:1 reduce:1 whether:1 bartlett:2 reuse:1 suffer:1 hessian:3 adequate:1 generally:2 covered:1 aimed:1 gle:1 reduced:1 http:2 exist:1 neuroscience:1 delta:2 track:1 per:3 hji:1 paramters:1 discrete:1 hyperparameter:2 vol:3 four:1 ce:2 kept:1 sum:2 run:1 inverse:4 parameterized:1 powerful:1 uncertainty:1 letter:1 decide:1 toright:1 comparable:1 entirely:1 followed:1 kronecker:1 deficiency:1 infinity:2 constraint:1 flat:4 diffuse:1 aspect:1 speed:1 performing:1 department:1 according:1 smaller:2 intimately:1 g3:1 metropolis:1 making:1 taken:1 computationally:4 equation:1 agree:1 turn:2 count:1 mechanism:1 wrt:1 needed:1 letting:1 ge:1 tractable:1 serf:1 available:1 apply:1 appropriate:1 upto:1 stepsize:1 alternative:1 original:1 denotes:2 dirichlet:16 remaining:1 assumes:1 include:1 spill:1 ghahramani:1 especially:1 conquer:1 society:1 sweep:2 noticed:1 already:1 strategy:1 parametric:2 concentration:7 traditional:1 diagonal:1 subspace:1 distance:3 separate:3 parametrized:1 me:6 extent:1 reason:1 boldface:1 length:9 modeled:1 difficult:4 unfortunately:1 hmc:2 fe:3 potentially:1 oneout:1 allowing:1 observation:1 markov:5 datasets:4 benchmark:1 finite:1 curved:2 defining:1 hinton:1 excluding:1 ever:1 arbitrary:1 overcoming:1 inferred:2 required:1 optimized:1 hour:1 discontinuity:1 nip:6 address:1 able:3 max:1 royal:1 unrealistic:1 natural:1 hybrid:4 predicting:1 indicator:20 scheme:1 improve:1 axis:1 carried:1 deemed:1 coupled:2 auto:3 tresp:5 prior:12 occupation:3 limitation:1 proportional:1 proven:1 offered:1 sufficient:1 consistent:2 principle:1 plotting:1 editor:1 course:1 repeat:1 last:1 rasmussen:7 free:3 keeping:1 bias:1 allow:3 sparse:1 ghz:1 curve:1 boundary:1 dimension:9 depth:1 evaluating:2 avoids:1 valid:1 doesn:1 world:1 transition:2 commonly:1 adaptive:3 employing:1 approximate:1 implicitly:1 keep:1 active:1 assumed:2 conclude:1 decomposes:1 hown:1 learn:1 nature:1 complex:2 cl:1 did:1 noise:18 hyperparameters:11 arise:1 fig:6 biggest:1 cubic:2 gatsby:3 definiteness:1 sub:1 wish:1 lie:2 third:1 specific:1 discarding:1 bishop:1 gating:23 adding:1 phd:1 tionary:1 rejection:3 artifactual:1 simply:3 likely:1 infinitely:1 visual:1 radford:1 duane:2 corresponds:1 conditional:9 sorted:1 consequently:2 acceleration:2 change:1 adverse:1 infinite:11 except:1 reducing:1 specifically:1 sampler:2 called:1 total:6 andn:1 college:1 arises:1 assessed:1 relevance:1 philosophy:1 mcmc:1 |
1,158 | 2,056 | Associative memory in realistic neuronal
networks
P.E. Latham*
Department of Neurobiology
University of California at Los Angeles
Los Angeles, CA 90095
pel@ucla.edu
Abstract
Almost two decades ago , Hopfield [1] showed that networks of
highly reduced model neurons can exhibit multiple attracting fixed
points, thus providing a substrate for associative memory. It is still
not clear, however, whether realistic neuronal networks can support
multiple attractors. The main difficulty is that neuronal networks
in vivo exhibit a stable background state at low firing rate, typically a few Hz. Embedding attractor is easy; doing so without
destabilizing the background is not. Previous work [2, 3] focused
on the sparse coding limit, in which a vanishingly small number of
neurons are involved in any memory. Here we investigate the case
in which the number of neurons involved in a memory scales with
the number of neurons in the network. In contrast to the sparse
coding limit, we find that multiple attractors can co-exist robustly
with a stable background state. Mean field theory is used to understand how the behavior of the network scales with its parameters,
and simulations with analog neurons are presented.
One of the most important features of the nervous system is its ability to perform
associative memory. It is generally believed that associative memory is implemented
using attractor networks - experimental studies point in that direction [4- 7], and
there are virtually no competing theoretical models. Perhaps surprisingly, however,
it is still an open theoretical question whether attractors can exist in realistic neuronal networks. The "realistic" feature that is probably hardest to capture is the
steady firing at low rates - the background state - that is observed throughout the
intact nervous system [8- 13]. The reason it is difficult to build an attractor network
that is stable at low firing rates, at least in the sparse coding limit, is as follows
[2,3]:
Attractor networks are constructed by strengthening recurrent connections among
sub-populations of neurons. The strengthening must be large enough that neurons
within a sub-population can sustain a high firing rate state, but not so large that the
sub-population can be spontaneously active. This implies that the neuronal gain
functions - the firing rate of the post-synaptic neurons as a function of the average
? http) / culture.neurobio.ucla.edu/ "'pel
firing rate of the pre-synaptic neurons - must be sigmoidal: small at low firing rate
to provide stability, high at intermediate firing rate to provide a threshold (at an
unstable equilibrium), and low again at high firing rate to provide saturation and
a stable attractor. In other words, a requirement for the co-existence of a stable
background state and multiple attractors is that the gain function of the excitatory
neurons be super linear at the observed background rates of a few Hz [2,3]. However
- and this is where the problem lies - above a few Hz most realistic gain function
are nearly linear or sublinear (see, for example, Fig. Bl of [14]).
The superlinearity requirement rests on the implicit assumption that the activity
of the sub-population involved in a memory does not affect the other neurons in
the network. While this assumption is valid in the sparse coding limit , it breaks
down in realistic networks containing both excitatory and inhibitory neurons. In
such networks, activity among excitatory cells results in inhibitory feedback. This
feedback, if powerful enough, can stabilize attractors even without a saturating
nonlinearity, essentially by stabilizing the equilibrium (above considered unstable)
on the steep part of the gain function. The price one pays, though, is that a
reasonable fraction of the neurons must be involved in each of the memories, which
takes us away from the sparse coding limit and thus reduces network capacity [15].
1
The model
A relatively good description of neuronal networks is provided by synaptically coupled, conductance-based neurons. However, because communication is via action
potentials, such networks are difficult to analyze. An alternative is to model neurons by their firing rates. While this is unlikely to capture the full temporal network
dynamics [16], it is useful for studying equilibria. In such simplified models, the
equilibrium firing rate of a neuron is a function of the firing rates of all the other
neurons in the network. Letting VEi and VIi denote the firing rates of the excitatory and inhibitory neurons, respectively, and assuming that synaptic input sums
linearly, the equilibrium equations may be written
?Ei
(~Af;EVEj' ~Af;'V'j)
(la)
?;;
(~AifVEj, ~ Ai!V,j) .
(lb)
Here ?E and ?I are the excitatory and inhibitory gain functions and Aij determines
the connection strength from neuron j to neuron i. The gain functions can, in
principle, be derived from conductance-based model equations [17].
Our goal here is to determine under what conditions Eq. (1) allows both attractors
and a stable state at low firing rate. To accomplish this we will use mean field
theory. While this theory could be applied to the full set of equations, to reduce
complexity we make a number of simplifications. First, we let the inhibitory neurons
be completely homogeneous (?Ii independent of i and connectivity to and from
inhibitory neurons all-to-all and uniform). In that case, Eq. (lb) becomes simply
VI = ?(VE' VI) where VE and VI are the average firing rates of the excitatory and
inhibitory neurons. Solving for VI and inserting the resulting expression into Eq. (la)
results in the expression VEi = ?Ei(LjAijEVEj,AEIVI(VE)) where A EI == LjAijI.
Second, we let cP Ei have the form cP Ei (u, v) = cP E( Xi + bu - ev) where Xi is a Gaussian
random variable, and similarly for cPT (except with different constants band e and
no dependence on i). Finally, we assume that cPT is threshold linear and the network
operates in a regime in which the inhibitory firing rate is above zero. With these
simplifications, and a trivial redefinition of constants, Eq. (la) becomes
(2)
We have dropped the sub and superscript E, since Eq. (2) refers exclusively to
excitatory neurons, defined v to be the average firing rate, v == N-1 Li Vi, and
rescaled parameters. We let the function cP be 0(1), so f3 can be interpreted as the
gain. The parameter p is the number of memories. The reduction from Eq. (1) to
Eq. (2) was done solely to simplify the analysis; the techniques we will use apply
equally well to the general case, Eq. (1).
Note that the gain function in Eq. (2) decreases with increasing average firing rate,
since it's argument is -(1 + a)v and a is positive. This negative dependence on v
arises because we are working in the large coupling regime in which excitation and
inhibition are balanced [18,19]. The negative coupling to firing rate has important
consequences for stability, as we will see below.
We let the connectivity matrix have the form
Here N is the number of excitatory neurons; Cij , which regulates the degree of
connectivity, is lie with probability e and and 0 with probability (1 - e) (except
Cii = 0, meaning no autapses); g(z) is an 0(1) clipping function that keeps weights
from falling below zero or getting too large; (g) is the mean value of g(z), defined
in Eq. (4) below; W i j , which corresponds to background connectivity, is a random
matrix whose elements are Gaussian distributed with mean 1 and variance 8w 2 ; and
J ij produces the attractors. We will follow the Hopfield prescription and write J ij
as
(3)
where f is the coupling strength among neurons involved in the memories, and the
patterns TJ",i determine which neurons participate in each memory. The TJ",i are a
set of uncorrelated vectors with zero mean and unit variance. In simulations we
use TJ",i = [(1 - 1)11]1/2 with probability 1 and -(f 1(1 - IW /2 with probability
1 - I, so a fraction 1 of the neurons are involved in each memory. Other choices
are unlikely to significantly change our results.
2
Mean field equations
The main difficulty in deriving the mean field equations from Eq. (2) is separating
the signal from the noise. Our first step in this endeavor is to analyze the noise
associated with the clipped weights. To do this we break Cijg(Wij
pieces: Cijg(Wij + Jij) = (g) + (g')Jij + bCij where
+ J ij ) into two
The angle brackets around 9 represent an average over the distributions of W ij and
Jij, and a prime denotes a derivative. In the large p limit, bCij can be treated as a
random matrix whose main role is to increase the effective noise [20]. The mean of
bCij is zero and its variance normalized to (g)2 / c, which we denote (Y2, is given by
For large p, the elements of Jij are Gaussian with zero mean and variance
the averages involving 9 can be written
E2,
so
(4)
where k can be either an exponent or a prime and the "I" in g(1 + z) corresponds
to the mean of W ij . In our simulations we use the clipping function g(z) = z if z is
between 0 and 2, 0 if z ::::; 0 and 2 if z ;::: 2.
Our main assumptions in the development of a mean field theory are that
L;#i bCijvj is a Gaussian random variable, and that bCij and Vj are independent.
Consequently,
where (v 2 ) == N- 1 L;i v; is the second moment of the firing rate. Letting 8i be a
zero mean Gaussian random variable with variance 82 == (Y2 (v 2)/ cN, we can use the
above assumptions along with the definition of Jij , Eq. (3), to write Eq. (20) as
(5)
We have defined the clipped memory strength, Ee , as Ee == E(g')/(g). While it is
not totally obvious from the above equations, it can be shown that both (Y2 and
Ee become independent of E for large E. This makes network behavior robust to
changes in E, the strength of the memories, so long as E is large.
Derivation ofthe mean field equations from Eq. (5) follow standard methods [21,22].
For definiteness we take ?(x) to be threshold linear: ?(x) = max(O, x). For the case
of one active memory, the mean field equations may then be written in the form
{3Ec
w
+
q
)
(6a)
1]
(32E~ [1J 2
a(l-r)2 CE~+(1-q)2 [F2(z)+jflF2(w ,z)]
1
r
(
1- r flF1 w,z
(6b)
{32B 2a2/x2
(1 ~ r)2 a [Fl (z) + j flFl (w, zW
a{3Ecq
1-q
(3E~
1+a
(6c)
[Fo(z)
+ jflFo(w,z)]
(6d)
Ec
where a == piN is the load parameter, Xo and B6/P are the mean and variance of
of Xi (see Eq. (2)), and, recall, j is the fraction of neurons that participate in each
memory. The functions Fk and flFk are defined by
1
00
-z
Fdw
d~
k
2
(27r )1/2 (z +~) exp( -~ /2)
+ z) -
Fk( Z) .
For large negative z, Fk(z) vanishes as exp(-z2/2) , while for large positive z,
Fk(Z) --+ zk /k!.
The average firing rate, v, and strength of the memory, m == N- 1 2:: i rJljVj (taken
without loss of generality to be the overlap with pattern 1), are given in terms of z
and was
v
Xo
m
3
Results
The mean field equations can be understood by examining Eqs. (6a) and (6b). The
first of these, Eq. (6a), is a rescaled form of the equation for the overlap, m. (From
the definition of flFt given above, it can be seen that m is proportional to w for
small w). This equation always has a solution at w = 0 (and thus m = 0) , which
corresponds to a background state with no memories active. If {3Ec is large enough,
there is a second solution with w (and thus m) greater than zero. This second
solution corresponds to a memory. The other relevant equation, Eq. (6b), describes
the behavior of the mean firing rate. This equation looks complicated only because
the noise - the variation in firing rate from neuron to neuron - must be determined
self-consistently.
The solutions to Eqs. (6a) and (6b) are plotted in Fig. 1 in the z-w plane. The solid
lines, including the horizontal line at w = 0, represents the solution to Eq. (6a), the
w
,
~
',.:
...
t
t
...
w=o
z
Figure 1: Graphical solution of Eqs. (6a)
and (6b). Solid lines, including the one at
w = 0: solution to Eq. (6a). Dashed line:
solution to Eq. (6b). The arrows indicate
approximate flow directions: vertical arrows indicate time evolution of w at fixed
z; horizontal arrows indicate time evolution of z at fixed w. The black squares
show potentially stable fixed points. Note
the exchange of stability to the right of
the solid curve, indicating that intersections too far to the right will be unstable.
dashed line the solution to Eq. (6b), and their intersections solutions to both. While
stability cannot be inferred from the equilibrium equations, a reasonable assumption
is that the evolution equations for the firing rates , at least near an equilibrium, have
the form Tdvi/dt = ?i - Vi. In that case, the arrows represent flow directions, and
we see that there are potentially stable equilibria at the intersections marked by
the solid squares.
Note that in the sparse coding limit, f ---+ 0, z is independent of w, meaning that the
mean firing rate, v , is independent of the overlap, m. In this limit there can be no
feedback to inhibitory neurons , and thus no chance for stabilization. In terms of Fig.
1, the effect of letting f ---+ 0 is to make the dashed line vertical. This eliminates the
possibility of the upper stable equilibrium (the solid square at w > 0), and returns
us to the situation where a superlinear gain function is required for attractors to be
embedded, as discussed in the introduction.
Two important conclusions can be drawn from Fig. 1. First, the attractors can be
stable even though the gain functions never saturate (recall that we used thresholdlinear gain functions). The stabilization mechanism is feedback to inhibitory neurons, via the -(1 + a)v term in Eq. (2). This feedback is what makes the dashed
line in Fig. 1 bend, allowing a stable equilibrium at w > O. Second, if the dashed
line shifts to the right relative to the solid line, the background becomes destabilized. This is because there is an exchange of stability, as indicated by the arrows.
Thus, there is a tradeoff: w, and thus the mean firing rate of the memory neurons,
can be increased by shifting the dashed line up or to the right , but eventually the
background becomes destabilized. Shifting the dashed line to the left, on the other
hand, will eventually eliminate the solution at w > 0, destroying all attractors but
the background.
For fixed load parameter Ct, fraction of neurons involved in a memory, f, and degree
of connectivity, c, there are three parameters that have a large effect on the location
of the equilibria in Fig. 1: the gain, {3, the clipped memory strength, fe, and the
degree of heterogeneity in individual neurons, Bo. The effect of the first two can
be seen in Fig. 2, which shows a stability plot in the f-{3 plane, determined by
numerically solving the the equations Tdvi/dt = ?i - Vi (see Eq. (2)). The filled
circles indicate regions where memories were embedded without destabilizing the
background, open circles indicate regions where no memories could be embedded,
and xs indicate regions where the background was unstable. As discussed above,
fe becomes approximately independent of the strength of the memories, f, when
f becomes large.
This is seen in Fig. 2A, in which network behavior stabilizes
when f becomes larger than about 4; increasing f beyond 8 would, presumably,
produce no surprises. There is some sensitivity to gain, (3: when f > 4, memories
co-existed with a stable background for (3 in a ?15% range. Although not shown,
the same was true of eo: increasing it by about 20% eliminated the attractors;
decreasing it by the same amount destabilized the background. However, more
detailed analysis indicates that the stability region gets larger as the number of
neurons in the network, N, increases. This is because fluctuations destabilize the
background, and those fluctuations fall off as N - 1 / 2 .
A
E:o
70
11111",1 11
000000000000000
o
2
B
'.2[\momo
N
!:S 35
~
I
o
background
????
4
0
0
~
4
8
E
Figure 2: A. Stability diagram, found by solving the set of equations Tdv;/dt =
cPi - Vi with the argument of cPi given in Eq. (2). Filled circles: memories co-exist
with a stable background (also outlined with solid lines); open circles: memories
could not be embedded; x s: background was unstable. The average background
rate, when the background was stable, was around 3 Hz. The network parameters
were eo = 6, Xo = 1.5, a = 0.5, c = 0.3, 0: = 2.5%, and 8w = 0.3. 2000 neurons
were used in the simulations. These parameters led to an effective gain, pl /2 (3f c , of
about 10, which is consistent with the gain in large networks in which each neuron
receives "-'5-10,000 inputs. B . Plot of firing rate of memory neurons , m, when the
memory was active (upper trace) and not active (lower trace) versus f at (3 = 2.
4
Discussion
The main outcome of this analysis is that attractors can co-exist with a stable
background when neurons have generic threshold-linear gain functions, so long as
the sparse coding limit is avoided. The parameter regime for this co-existence is
much larger than for attractor networks that operate in the sparse coding limit
[2,23]. While these results are encouraging, they do not definitively establishing
t hat attractors can exist in realistic networks. Future work must include inhibitory
neurons , incorporate a much larger exploration of parameter space to ensure that
the results are robust , and ultimately involve simulations with spiking neurons.
5
Acknowledgements
This work was supported by NIMH grant #R01 MH62447.
References
[1] J.J. Hopfield. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci ., 79:2554- 2558, 1982.
[2] N. BruneI. Persistent activity and the single-cell frequency-current curve in a cortical
network model. Network: Computation in Neural Systems, 11:261- 280, 2000.
[3] P.E. Latham and S.N. Nirenberg. Intrinsic dynamics in cultured neuronal networks.
Soc . Neuroscience Abstract, 25:2259, 1999.
[4] J.M. Fuster and G.E. Alexander.
Science, 173:652- 654, 1971.
Neuron activity related to short-term memory.
[5] Y. Miyashita. Inferior temporal cortex: where visual perception meets memory. Annu
R ev Neurosci, 16:245- 263 , 1993.
[6] P.S. Goldman-Rakic. Cellular basis of working memory. Neuron, 14:477- 485 , 1995.
[7] R Romo, C.D. Brody, A. Hernandez , and L. Lemus. Neuronal correlates of parametric
working memory in the prefrontal cortex. Nature , 399:470- 473, 1999.
[8] C.D. Gilbert. Laminar differences in receptive field properties of cells in cat primary
visual cortex. J. Physiol. , 268:391- 421 , 1977.
[9] Y. Lamour, P. Dutar, and A. Jobert. Cerebral neorcortical neurons in the aged rat:
spontaneous activity, properties of pyramidal tract neurons and effect of acetylcholine
and cholinergic drugs. N euroscience, 16:835- 844, 1985.
[10] M.B. Szente, A. Baranyi, and C.D. Woody. Intracellular injection of apamin reduces
a slow potassium current mediating afterhyperpolarizations and IPSPs in neocortical
neurons of cats. Brain Res. , 461:64- 74, 1988.
[11] I. Salimi, H.H. Webster, and RW. Dykes. Neuronal activity in normal and deafferented forelimb somatosensory cortex of the awake cat . Brain Res., 656:263- 273,
1994.
[12] J.F. Herrero and P.M. Headley. Cutaneous responsiveness of lumbar spinal neurons
in awake and halothane-anesthetized sheep. J. N europhysiol. , 74:1549- 1562, 1997.
[13] K. Ochi and J.J. Eggermont. Effects of quinine on neural activity in cat primary
auditory cortex. Hear. Res., 105:105- 18, 1997.
[14] P.E. Latham, B.J. Richmond, P.G. Nelson, and S.N. Nirenberg. Intrinsic dynamics
in neuronal networks. I. Theory. J. Neurophysiol., 83:808- 827, 2000.
[15] M.V. Tsodyks and M.V. Feigel'man. The enhanced storage capacity in neural networks with low activity level. Europhys. Lett. , 6:101- 105, 1988.
[16] A. Treves. Mean-field analysis of neuronal spike dynamics. Network, 4:259- 284, 1993.
[17] O. Shriki, D. Hansel , and H. Sompolonski. Modeling neuronal networks in cortex
by rate models using the current-frequency response properties of cortical cells. Soc .
Neurosci ence Abstract, 24:143 , 1998.
[18] C. van Vreeswijk and H. Sompolinsky. Chaos in neuronal networks with balanced
excitatory and inhibitory activity. Science, 274: 1724- 1726, 1996.
[19] C. van Vreeswijk and H. Sompolinsky. Chaotic balanced state in a model of cortical
circuits. Neural Comput., 10:1321- 1371 , 1998.
[20] H. Sompolinsky. Neural networks with nonlinear synapses and a static noise. Phys.
Rev. A, 34:2571- 2574, 1986.
[21] J. Hertz , A. Krogh, and RG. Palmer. Introduction to the th eory of neural computation. Addison Wesley, Redwood City, CA, 1991.
[22] A.N. Burkitt. Retrieval properties of attractor neural that obey Dale's law using
a self-consistent signal-to-noise analysis. Network: Computation in Neural Systems,
7:517- 531 , 1996.
[23] D.J. Amit and N. BruneI. Dynamics of a recurrent network of spiking neurons before
and following learning. Network, 8:373- 404, 1997.
| 2056 |@word open:3 simulation:5 solid:7 reduction:1 moment:1 exclusively:1 neurobio:1 z2:1 current:3 must:5 written:3 physiol:1 realistic:7 webster:1 plot:2 nervous:2 plane:2 short:1 location:1 sigmoidal:1 along:1 constructed:1 become:1 persistent:1 behavior:4 brain:2 decreasing:1 goldman:1 encouraging:1 increasing:3 becomes:7 provided:1 totally:1 circuit:1 feigel:1 pel:2 what:2 interpreted:1 temporal:2 deafferented:1 unit:1 grant:1 positive:2 before:1 dropped:1 understood:1 limit:10 consequence:1 acad:1 establishing:1 meet:1 firing:27 solely:1 approximately:1 fluctuation:2 black:1 hernandez:1 co:6 dyke:1 palmer:1 range:1 spontaneously:1 chaotic:1 drug:1 destabilizing:2 significantly:1 pre:1 word:1 refers:1 get:1 cannot:1 superlinear:1 bend:1 storage:1 gilbert:1 destroying:1 romo:1 focused:1 stabilizing:1 deriving:1 embedding:1 population:4 stability:8 variation:1 cultured:1 spontaneous:1 enhanced:1 substrate:1 homogeneous:1 element:2 observed:2 role:1 capture:2 tsodyks:1 region:4 sompolinsky:3 decrease:1 rescaled:2 balanced:3 vanishes:1 complexity:1 nimh:1 dynamic:5 jobert:1 ultimately:1 solving:3 f2:1 completely:1 basis:1 neurophysiol:1 hopfield:3 emergent:1 cat:4 derivation:1 effective:2 outcome:1 europhys:1 whose:2 larger:4 nirenberg:2 ability:2 superscript:1 associative:4 vanishingly:1 strengthening:2 jij:5 inserting:1 relevant:1 description:1 getting:1 los:2 potassium:1 requirement:2 produce:2 tract:1 coupling:3 recurrent:2 ij:5 eq:27 krogh:1 soc:2 implemented:1 salimi:1 implies:1 indicate:6 somatosensory:1 direction:3 stabilization:2 exploration:1 exchange:2 quinine:1 pl:1 around:2 considered:1 normal:1 exp:2 presumably:1 equilibrium:11 shriki:1 stabilizes:1 a2:1 proc:1 iw:1 hansel:1 autapses:1 city:1 halothane:1 gaussian:5 always:1 super:1 acetylcholine:1 derived:1 consistently:1 indicates:1 lemus:1 richmond:1 contrast:1 typically:1 unlikely:2 eliminate:1 wij:2 among:3 exponent:1 development:1 field:10 f3:1 never:1 eliminated:1 represents:1 hardest:1 look:1 nearly:1 future:1 simplify:1 destabilized:3 few:3 ve:3 individual:1 attractor:20 conductance:2 brunei:2 highly:1 investigate:1 possibility:1 cholinergic:1 sheep:1 bracket:1 tj:3 natl:1 culture:1 filled:2 circle:4 plotted:1 re:3 theoretical:2 increased:1 modeling:1 ence:1 clipping:2 uniform:1 examining:1 too:2 accomplish:1 sensitivity:1 bu:1 off:1 connectivity:5 again:1 containing:1 prefrontal:1 derivative:1 return:1 li:1 potential:1 vei:2 coding:8 stabilize:1 vi:8 piece:1 break:2 doing:1 analyze:2 complicated:1 vivo:1 b6:1 square:3 variance:6 ofthe:1 ago:1 synapsis:1 fo:1 phys:1 synaptic:3 definition:2 frequency:2 involved:7 obvious:1 e2:1 associated:1 static:1 gain:16 auditory:1 recall:2 wesley:1 dt:3 follow:2 sustain:1 response:1 done:1 though:2 generality:1 implicit:1 working:3 hand:1 horizontal:2 receives:1 ei:5 nonlinear:1 indicated:1 perhaps:1 effect:5 normalized:1 y2:3 true:1 evolution:3 rakic:1 self:2 inferior:1 steady:1 excitation:1 rat:1 neocortical:1 latham:3 cp:4 meaning:2 chaos:1 spiking:2 physical:1 regulates:1 woody:1 spinal:1 cerebral:1 analog:1 discussed:2 numerically:1 ai:1 fk:4 outlined:1 similarly:1 nonlinearity:1 stable:15 cortex:6 attracting:1 inhibition:1 showed:1 prime:2 seen:3 responsiveness:1 greater:1 cii:1 eo:2 determine:2 dashed:7 signal:2 ii:1 multiple:4 full:2 reduces:2 af:2 believed:1 long:2 retrieval:1 prescription:1 post:1 equally:1 ipsps:1 involving:1 redefinition:1 essentially:1 represent:2 synaptically:1 cell:4 background:22 diagram:1 aged:1 pyramidal:1 zw:1 rest:1 eliminates:1 operate:1 probably:1 hz:4 virtually:1 flow:2 ee:3 near:1 intermediate:1 easy:1 enough:3 affect:1 competing:1 reduce:1 cn:1 tradeoff:1 angeles:2 shift:1 whether:2 expression:2 action:1 cpt:2 generally:1 useful:1 clear:1 detailed:1 involve:1 amount:1 band:1 rw:1 reduced:1 http:1 eory:1 exist:5 inhibitory:12 neuroscience:1 write:2 threshold:4 destabilize:1 falling:1 drawn:1 ce:1 fraction:4 sum:1 angle:1 powerful:1 europhysiol:1 clipped:3 almost:1 throughout:1 reasonable:2 brody:1 fl:1 pay:1 ct:1 simplification:2 existed:1 laminar:1 activity:9 strength:7 awake:2 x2:1 ucla:2 argument:2 injection:1 relatively:1 department:1 hertz:1 describes:1 rev:1 thresholdlinear:1 cutaneous:1 xo:3 taken:1 equation:17 pin:1 eventually:2 mechanism:1 vreeswijk:2 addison:1 letting:3 studying:1 apply:1 obey:1 away:1 generic:1 robustly:1 alternative:1 hat:1 existence:2 denotes:1 include:1 ensure:1 graphical:1 build:1 eggermont:1 amit:1 r01:1 bl:1 question:1 spike:1 parametric:1 receptive:1 dependence:2 primary:2 exhibit:2 separating:1 capacity:2 sci:1 participate:2 nelson:1 unstable:5 trivial:1 reason:1 cellular:1 assuming:1 providing:1 difficult:2 steep:1 cij:1 fe:2 potentially:2 mediating:1 trace:2 negative:3 collective:1 perform:1 allowing:1 upper:2 vertical:2 neuron:50 situation:1 neurobiology:1 communication:1 heterogeneity:1 redwood:1 lb:2 treves:1 inferred:1 required:1 connection:2 california:1 miyashita:1 beyond:1 below:3 pattern:2 ev:2 perception:1 regime:3 hear:1 saturation:1 max:1 memory:34 including:2 shifting:2 overlap:3 difficulty:2 treated:1 coupled:1 acknowledgement:1 relative:1 law:1 embedded:4 loss:1 sublinear:1 proportional:1 versus:1 degree:3 consistent:2 principle:1 uncorrelated:1 excitatory:9 surprisingly:1 supported:1 aij:1 understand:1 fall:1 definitively:1 sparse:8 anesthetized:1 distributed:1 van:2 feedback:5 curve:2 cortical:3 valid:1 lett:1 dale:1 simplified:1 avoided:1 ec:3 far:1 correlate:1 approximate:1 keep:1 active:5 xi:3 fuster:1 decade:1 euroscience:1 nature:1 zk:1 robust:2 ca:2 vj:1 main:5 linearly:1 arrow:5 neurosci:2 noise:6 intracellular:1 neuronal:13 fig:8 cpi:2 burkitt:1 definiteness:1 slow:1 sub:5 comput:1 lie:2 down:1 saturate:1 annu:1 load:2 x:1 intrinsic:2 surprise:1 vii:1 rg:1 intersection:3 led:1 simply:1 visual:2 saturating:1 bo:1 corresponds:4 determines:1 chance:1 goal:1 endeavor:1 marked:1 consequently:1 price:1 man:1 change:2 determined:2 except:2 operates:1 experimental:1 la:3 intact:1 indicating:1 support:1 arises:1 alexander:1 incorporate:1 |
1,159 | 2,057 | Matching Free Trees with Replicator Equations
Marcello Pelillo
Dipartimento di Informatica
Universit`a Ca? Foscari di Venezia
Via Torino 155, 30172 Venezia Mestre, Italy
E-mail: pelillo@dsi.unive.it
Abstract
Motivated by our recent work on rooted tree matching, in this paper we
provide a solution to the problem of matching two free (i.e., unrooted)
trees by constructing an association graph whose maximal cliques are
in one-to-one correspondence with maximal common subtrees. We then
solve the problem using simple replicator dynamics from evolutionary
game theory. Experiments on hundreds of uniformly random trees are
presented. The results are impressive: despite the inherent inability of
these simple dynamics to escape from local optima, they always returned
a globally optimal solution.
1 Introduction
Graph matching is a classic problem in computer vision and pattern recognition, instances
of which arise in areas as diverse as object recognition, motion and stereo analysis [1]. In
many problems (e.g., [2, 11, 19]) the graphs at hand have a peculiar structure: they are
connected and acyclic, i.e. they are free trees. Note that, unlike ?rooted? trees, in free
trees there is no distinguished node playing the role of the root, and hence no hierarchy is
imposed on them. Standard graph matching techniques, such as [8], yield solutions that are
not constrained to preserve connectedness and hence cannot be applied to free trees.
A classic approach to solving the graph matching problem consists of transforming it into
the equivalent problem of finding a maximum clique in an auxiliary graph structure, known
as the association graph [1]. This framework is attractive because it casts graph matching
as a pure graph-theoretic problem, for which a solid theory and powerful algorithms have
been developed. Note that, although the maximum clique problem is known to be
hard, powerful heuristics exist which efficiently find good approximate solutions [4].
Motivated by our recent work on rooted tree matching [15], in this paper we propose a
solution to the free tree matching problem by providing a straightforward way of deriving an association graph from two free trees. We prove that in the new formulation there
is a one-to-one correspondence between maximal (maximum) cliques in the derived association graph and maximal (maximum) subtree isomorphisms. As an obvious corollary,
the computational complexity of finding a maximum clique in such graphs is therefore the
same as the subtree isomorphism problem, which is known to be polynomial in the number
of nodes [7].
Following [13, 15], we use a recent generalization of the Motzkin-Straus theorem [12] to
formulate the maximum clique problem as a quadratic programming problem. To (approximately) solve it we employ replicator equations, a class of simple continuous- and discretetime dynamical systems developed and studied in evolutionary game theory [10, 17].
We illustrate the power of the approach via experiments on hundreds of (uniformly) random
trees. The results are impressive: despite the counter-intuitive maximum clique formulation
of the tree matching problem, and the inherent inability of these simple dynamics to escape
from local optima, they always found a globally optimal solution.
2 Subtree isomorphisms and maximal cliques
"!#
$% if & !
"! ' otherwise (
The degree of a node
, denoted )+*-,./
0 , is the number of nodes% adjacent to it. A path is
any sequence of distinct nodes
012
435(2(-(6
87 such that for all 9:
(2(2(; ,
=< 3>?
; in this
case, the length of the path is . If
01@A
B7 the path is called a cycle. A graph is said
to be connected if any two nodes are joined by a path. The distance between two nodes
and , denoted by C8/
5;+ , is the length of the shortest path joining them (by convention
C8/
5;+ DFE , if there is no such path). Given a subset of nodes GIHJ , the induced
subgraph LK GNM is the graph having G as its node set, and two nodes are adjacent in LK GNM
if and only if they are adjacent in . A connected graph with no cycles is called a free tree,
Let
be a graph, where is the set of nodes and is the set of (undirected)
edges. The order of is the number of nodes in , while its size is the number of edges.
Two nodes
are said to be adjacent (denoted
) if they are connected by an
edge. The adjacency matrix of is the
symmetric matrix
defined as
or simply a tree. Trees have a number of interesting properties. One which turns out to
be very useful for our characterization is that in a tree any two nodes are connected by a
unique path.
J.QS
TQ# be two trees. Any bijection UWVYX 3[Z X\Q , with
called a subtree isomorphism if it preserves both the adjacency
Xrelationships
[3 OH]3 8J3 and 3 between
XP Q 3 H?and Q the, O0isQRnodes
and the connectedness of the matched subgraphs. Formally, this means that, given
5;[^X_3 , we have
W if and only if U5/
0 ]U= and, in
addition, the induced subgraphs O 3 K X 3 M and O0Q`K XaQ2M are connected. A subtree isomorphism
is maximal if there is no other subtree isomorphism U4bcV0X^3 b Z X^Q b with XR3 a strict subset of X3 b , and maximum if XR3 has largest cardinality. The maximal (maximum) subtree
isomorphism problem is to find a maximal (maximum) subtree isomorphism between two
Let
trees. A word of caution about terminology is in order here. Despite name similarity, we
are not addressing the so-called subtree isomorphism problem, which consists of determining whether a given tree is isomorphic to a subtree of a larger one. In fact, we are dealing
with a generalization thereof, the maximum common subtree problem, which consists of
determining the largest isomorphic subtrees of two given trees. We shall continue to use
our own terminology, however, as it emphasizes the role of the isomorphism .
U
The free tree association graph (FTAG) of two trees Od3 eB3fP N3- and O Q g
h Q
Q is
the graph i
j where
W 3 @.Q
(1)
and, for any two nodes /
5;kN and /8
l in , we have
(2)
=
kN /8
l nmoC.=
pWC8/kqPlr (
Note that this definition of the association graph is stronger than the standard one used for
matching arbitrary relational structures [1].
A subset of vertices of is said to be a clique if all its nodes are mutually adjacent. A
maximal clique is one which is not contained in any larger clique, while a maximum clique
is a clique having largest cardinality. The maximum clique problem is to find a maximum
clique of .
The following theorem, which is the basis of the work reported here, establishes a one-toone correspondence between the maximum subtree isomorphism problem and the maximum clique problem.
Theorem 1 Any maximal (maximum) subtree isomorphism between two trees induces a
maximal (maximum) clique in the corresponding FTAG, and vice versa.
be a maximal subtree isomorphism between trees On3
U&h:VYPX_ 3 denote
Z X Q the
Q
O
corresponding FTAG. Let G H be defined as
GU maps
r=the
U5path
/
0 ; >between
Vr
X any
the definition of a subtree isomorphism it follows that
3 . From
two nodes
X 3 onto the path joining U5=
B and U= .
This clearly implies that C.=
\C.hU=
B PU5/+ ; for all
? X_3 , and therefore G is a
clique. Trivially, G is a maximal clique because U is maximal, and this proves the first
Proof (outline). Let
and , and let
part of the theorem.
maximal clique of , and let X 3
7 HW isQ . aDefine
for
-
43#all9dY ;
B% 7 (- (2H&(; .BGFrom
3 andr the
X =
Q 3 definition
;k -3 k> 3f4 of#0/;
thek 7 7 ;kFTAG
UVrXR3 Z X Q as U5/
pWk ,
and the hypothesis that G is a clique, it
Suppose now that
U
X 3 U XaQ
G
is simple to see that is a one-to-one and onto correspondence between
and , which
trivially preserves the adjacency relationships between nodes. The fact that is a maximal
isomorphism is a straightforward consequence of the maximality of .
X 3 XaQ
O 3 K X 3 M O0Q K XaQ2M
X[3 OY3
On3n3(-(2K XL(
3 M g
!
1
B
% (2(-(
a
X
1 3n(2(2( A k !
k
!
!
OQ
C.=k ;k ! >
r
2C8 /
G
NH C8/k ;k
C.=
: C8/
5
8 C8
;+
To conclude the proof we have to show that the subgraphs that we obtain when we restrict
ourselves to
and , i.e.
and
, are trees, and this is equivalent to showing
that they are connected. Suppose by contradiction that this is not the case, and let
be two nodes which are not joined by a path in
. Since both and are nodes
of , however, there must exist a path
joining them in . Let
, for some
, be a node on this path which is not in
. Moreover, let
be the -th node on the path
which joins
and
in
(remember that
, and hence
). It is easy to show
that the set
is a clique, thereby contradicting the hypothesis that is
a maximal clique. This can be proved by exploiting the obvious fact that if is a node on
the path joining any two nodes and , then
.
!
O 3
!
3 k k!
G
The ?maximum? part of the statement is proved similarly.
h:P
! ^
C !
The FTAG is readily derived by using a classical representation for graphs, i.e., the socalled distance matrix which, for an arbitrary graph
of order , is the
matrix
where
, the distance between nodes and . Efficient,
classical algorithms are available for obtaining such a matrix [6]. Note also that the distance
matrix of a graph can easily be constructed from its adjacency matrix
. In fact, denoting
by
the
-th entry of the matrix
, the -th power of
, we have that
equals
the least for which
(there must be such an since a tree is connected).
= C "!
C "! ?C8/
!
7 ! /9 ! 7 '
7
!#"
3 Matching free trees with replicator dynamics
7
j be an arbitrary graph of order , and let $Y7 denote the standard simplex of
$07L% & IR7 (V ' b &^ % and
* ) ' Y9d % (2(2(;
Let
IR :
'
G
&
where is the vector whose components equal 1, and a prime denotes transposition. Given
a subset of vertices of , we will denote by its characteristic vector which is the point
in
defined as
$7
$ % G if p9 @G
' otherwise
where G denotes the cardinality of G .
Now, consider the following quadratic function
%
&Y & b & & b &
(3)
where g= ! is the adjacency matrix of . The following theorem, recently proved
by Bomze [3], expands on the Motzkin-Straus theorem [12], a remarkable result which es
tablishes a connection between the maximum clique problem and quadratic programming.
G G $ 7
&&
Theorem 2 Let be a subset of vertices of a graph , and let be its characteristic
vector. Then,
is a maximal (maximum) clique of if and only if is a local
(global)
maximizer
in . Moreover, all local (and hence global) maximizers of
in
are
strict and are characteristic vectors of maximal cliques of .
$7
$07
Unlike the original Motzkin-Straus formulation, which is plagued by the presence
of ?spuon
rious? solutions [14], the previous result guarantees us that all maximizers of
are strict, and are characteristic vectors of maximal/maximum cliques in . In a formal
sense, therefore,
a one-to-one correspondence exists between maximal cliques and local
maximizers of
in
on the one hand, and maximum cliques and global maximizers on
the other hand.
$7
A/kN "!
7
. P
; .
P
;
; ! 3
+!
; !r
;
@
We now turn our attention to a class of simple dynamical systems that we use for solving
our quadratic optimization problem. Let
be a non-negative real-valued
matrix, and consider the following continuous-time dynamical system:
(4)
where a dot signifies derivative with respect to time, and its discrete-time counterpart:
where
%
8
;
P
;
8
: !7 3
! ; !
;
7
; : ! 3 k !
!
; (
(5)
(6)
Both (4) and (5) are called replicator equations in evolutionary game theory, since they
are used to model evolution over time of relative frequencies of interacting, self-replicating
entities [10, 17]. It is readily seen that the simplex
is invariant under these dynamics,
which means that every trajectory starting in
will remain in
for all future times, and
their stationary points coincide.
$7
$7
$47
We are now interested in the dynamical properties of replicator dynamics; it is these properties that will allow us to solve our original tree matching problem. The following result
is known in mathematical biology as the fundamental theorem of natural selection [10, 17]
and, in its original form, traces back to R. A. Fisher.
ib
&b &
Theorem 3 If
is strictly increasing along any non
then the function
constant trajectory under both continuous-time (4) and discrete-time (5) replicator dynamics. Furthermore, any such trajectory converges to a stationary point. Finally, a vector
is asymptotically stable under (4) and (5) if and only if is a strict local maximizer
of
on .
& &4b $ 7 & $07
&
In light of their dynamical properties, replicator equations naturally suggest themselves
as a simple heuristic for solving the maximal subtree isomorphism problem. Indeed, let
and
be two free trees, and let
denote the adjacency
matrix of their FTAG . By letting
O 3 Ih 3
3
O0Q I.Q`P TQ-
W
%
(7)
where is the identity matrix, we know that the replicator dynamical systems (4)
and (5),
starting from an arbitrary initial state, will iteratively maximize the function
defined
in (3) over the simplex and will eventually converge with probability 1 to a strict local
maximizer which, by virtue of Theorem 2, will then correspond to the characteristic vector
of a maximal clique in the association graph. As stated in Theorem 1, this will in turn
induce a maximal subtree isomorphism between
and . Clearly, in theory there is no
guarantee that the converged solution will be a global maximizer of , and therefore that
it will induce a maximum isomorphism between the two original trees, but see below.
O3
O0Q
Recently, there has been much interest around the following exponential version of replicator equations, which arises as a model of evolution guided by imitation [9, 10, 17]:
;
;
%
!7 3
!
;
(8)
where is a positive constant. As tends to 0, the orbits of this dynamics approach those
of the standard, ?first-order? replicator model (4), slowed down by the factor ; moreover,
for large values of the model approximates the so-called ?best-reply? dynamics [9, 10]. A
customary way of discretizing equation (8) is given by the following difference equations:
% : !7
3
; !
;
(
(9)
From a computational perspective, exponential replicator dynamics are particularly attractive as they may be considerably faster and even more accurate than the standard, first-order
model (see [13] and the experiments reported in the next section).
4 Results and conclusions
We tested our algorithms over large random trees. Random structures represent a useful
benchmark not only because they are not constrained to any particular application, but also
because it is simple to replicate experiments and hence to make comparisons with other
algorithms.
In this series of experiments, the following protocol was used. A hundred 100-node free
trees were generated uniformly at random using a procedure described by Wilf in [18].
Then, each such tree was subject to a corruption process which consisted of randomly
deleting a fraction of its nodes (in fact, the to-be-deleted nodes were constrained to be
the terminal ones, otherwise the resulting graph would have been disconnected), thereby
obtaining a tree isomorphic to a proper subtree of the original one. Various levels of corruption (i.e., percentage of node deletion) were used, namely 2%, 10%, 20%, 30% and 40%.
This means that the order of the pruned trees ranged from 98 to 60. Overall, therefore, 500
pairs of trees were obtained, for each of which the corresponding FTAG was constructed
as described in Section 2. To keep the order of the association graph as low as possible, its
vertex set was constructed as follows:
%r=
kN @ b @ b V)+*2,8/
0 ) *2,B/kN
b , the edge set being defined as in (2). It is straightforward to see
assuming b
that when the first tree is isomorphic to a subtree of the second, Theorem 1 continues to
hold. This simple heuristic may significantly reduce the dimensionality of the search space.
We also performed some experiments with unpruned FTAG?s but no significant difference
in performance was noticed apart, of course, heavier memory requirements.
% '
Both the discrete-time first-order dynamics (5) and its exponential counterpart (9) (with
) were used. The algorithms were started from the simplex
barycenter and stopped
) was found or the distance
when either a maximal clique (i.e., a local maximizer of
between two successive points was smaller than a fixed threshold. In the latter case the
converged vector was randomly perturbed, and the algorithms restarted from the perturbed
point. Note that this situation corresponds to convergence to a saddle point.
After convergence, we calculated the proportion of matched nodes, i.e., the ratio between
the cardinality of the clique found and the order of the smaller subtree, and then we averaged. Figure 1(a) shows the results obtained using the linear dynamics (5) as a function of
the corruption level. As can be seen, the algorithm was always able to find a correct maximum isomorphism, i.e. a maximum clique in the FTAG. Figure 1(b) plots the corresponding (average) CPU time taken by the processes, with corresponding error bars (simulations
were performed on a machine equipped with a 350MHz AMDK6-2 processor).
In Figure 2, the results pertaining to the exponential dynamics (8) are shown. In terms of
solution?s quality the algorithm performed exactly as its linear counterpart, but this time it
was dramatically faster. This confirms earlier results reported in [13].
Before concluding, we note that our approach can easily be extended to tackle the problem
of matching attributed (free) trees. In this case, the attributes result in weights being placed
on the nodes of the association graph, and a conversion of the maximum clique problem to a
maximum weight clique problem [15, 5]. Moreover, it is straightforward to formulate errortolerant versions of our framework along the lines suggested in [16] for rooted attributed
trees, where many-to-many node correspondences are allowed. All this will be the subject
of future investigations.
Finally, we think that the results presented in this paper (together with those reported in [13,
15]) raise intriguing questions concerning the connections between (standard) notions of
computational complexity and the ?elusiveness? of global optima in continuous settings.
Acknowledgments. The author would like to thank M. Zuin for his support in performing
the experiments.
References
[1] D. H. Ballard and C. M. Brown. Computer Vision. Prentice-Hall, Englewood Cliffs, NJ, 1982.
[2] H. Blum and R. N. Nagel. Shape description using weighted symmetric axis features. Pattern
Recognition, 10:167?180, 1978.
[3] I. M. Bomze. Evolution towards the maximum clique. J. Glob. Optim., 10:143?164, 1997.
[4] I. M. Bomze, M. Budinich, P. M. Pardalos, and M. Pelillo. The maximum clique problem. In
D.-Z. Du and P. M. Pardalos, editors, Handbook of Combinatorial Optimization (Suppl. Vol. A),
pages 1?74. Kluwer, Boston, MA, 1999.
[5] I. M. Bomze, M. Pelillo, and V. Stix. Approximating the maximum weight clique using replicator dynamics. IEEE Trans. Neural Networks, 11(6):1228?1241, 2000.
Figure 1: Results obtained over 100-node random trees with various levels of corruption, using the
first-order dynamics (5). Top: Percentage of correct matches. Bottom: Average computational time
taken by the replicator equations.
[6] T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press,
Cambridge, MA, 1990.
[7] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman, San Francisco, CA, 1979.
[8] S. Gold and A. Rangarajan. A graduated assignment algorithm for graph matching. IEEE Trans.
Pattern Anal. Machine Intell. 18:377-388, 1996.
[9] J. Hofbauer. Imitation dynamics for games. Collegium Budapest, preprint, 1995.
[10] J. Hofbauer and K. Sigmund. Evolutionary Games and Population Dynamics. Cambridge
University Press, Cambridge, UK, 1998.
[11] T.-L. Liu, D. Geiger, and R. V. Kohn. Representation and self-similarity of shapes. In Proc.
ICCV?98?6th Int. Conf. Computer Vision, pages 1129?1135, Bombay, India, 1998.
500
1000
1500
95
2500
100
2000
Percentage
Average CPU
of correct
time (inmatches
secs)
3000
[12] T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of Tur?an.
Canad. J. Math., 17:533?540, 1965.
Figure 2: Results obtained over 100-node random trees with various levels of corruption, using
the exponential dynamics (9) with
. Top: Percentage of correct matches. Bottom: Average
computational time taken by the replicator equations.
100
200
95
100
300
Percentage
Average CPU
of correct
time (inmatches
secs)
400
[13] M. Pelillo. Replicator equations, maximal cliques, and graph isomorphism. Neural Computation, 11(8):2023?2045, 1999.
[14] M. Pelillo and A. Jagota. Feasible and infeasible maxima in a quadratic program for maximum
clique. J. Artif. Neural Networks, 2:411?420, 1995.
[15] M. Pelillo, K. Siddiqi, and S. W. Zucker. Matching hierarchical structures using association
graphs. IEEE Trans. Pattern Anal. Machince Intell., 21(11):1105?1120, 1999.
[16] M. Pelillo, K. Siddiqi, and S. W. Zucker. Many-to-many matching of attributed trees using
association graphs and game dynamics. In C. Arcelli, L. P. Cordella, and G. Sanniti di Baja,
editors, Visual Form 2001, pages 583?593. Springer, Berlin, 2001.
[17] J. W. Weibull. Evolutionary Game Theory. MIT Press, Cambridge, MA, 1995.
[18] H. Wilf. The uniform selection of free trees. J. Algorithms, 2:204?207, 1981.
[19] S. C. Zhu and A. L. Yuille. FORMS: A flexible object recognition and modeling system. Int. J.
Computer Vision, 20(3):187?212, 1996.
| 2057 |@word version:2 polynomial:1 proportion:1 stronger:1 replicate:1 hu:1 confirms:1 simulation:1 thereby:2 gnm:2 solid:1 initial:1 liu:1 series:1 denoting:1 optim:1 intriguing:1 must:2 readily:2 shape:2 plot:1 stationary:2 transposition:1 characterization:1 math:1 node:34 bijection:1 successive:1 mathematical:1 along:2 constructed:3 consists:3 prove:1 indeed:1 themselves:1 terminal:1 freeman:1 globally:2 p9:1 cpu:3 equipped:1 cardinality:4 increasing:1 matched:2 moreover:4 rivest:1 weibull:1 developed:2 caution:1 finding:2 nj:1 machince:1 guarantee:2 remember:1 bomze:4 every:1 expands:1 tackle:1 exactly:1 universit:1 uk:1 glob:1 unive:1 positive:1 before:1 local:8 tends:1 consequence:1 despite:3 joining:4 cliff:1 path:13 approximately:1 connectedness:2 studied:1 averaged:1 unique:1 acknowledgment:1 procedure:1 area:1 significantly:1 matching:17 word:1 induce:2 suggest:1 cannot:1 onto:2 selection:2 prentice:1 equivalent:2 imposed:1 map:1 straightforward:4 attention:1 starting:2 formulate:2 pure:1 subgraphs:3 gihj:1 q:1 contradiction:1 deriving:1 oh:1 his:1 classic:2 population:1 notion:1 hierarchy:1 suppose:2 programming:2 hypothesis:2 recognition:4 particularly:1 continues:1 bottom:2 role:2 preprint:1 connected:8 cycle:2 counter:1 tur:1 transforming:1 complexity:2 dynamic:19 raise:1 solving:3 yuille:1 basis:1 gu:1 easily:2 various:3 distinct:1 pertaining:1 venezia:2 whose:2 heuristic:3 larger:2 solve:3 valued:1 otherwise:3 think:1 sequence:1 propose:1 maximal:26 budapest:1 subgraph:1 gold:1 intuitive:1 description:1 exploiting:1 convergence:2 optimum:3 requirement:1 rangarajan:1 converges:1 object:2 illustrate:1 pelillo:8 auxiliary:1 implies:1 convention:1 guided:1 correct:5 attribute:1 pardalos:2 adjacency:6 generalization:2 investigation:1 dipartimento:1 strictly:1 hold:1 around:1 hall:1 plagued:1 proc:1 combinatorial:1 largest:3 vice:1 establishes:1 weighted:1 mit:2 clearly:2 always:3 thek:1 corollary:1 derived:2 sense:1 interested:1 overall:1 flexible:1 denoted:3 socalled:1 constrained:3 equal:2 having:2 biology:1 marcello:1 future:2 simplex:4 inherent:2 escape:2 employ:1 randomly:2 preserve:3 intell:2 ourselves:1 tq:2 interest:1 englewood:1 leiserson:1 light:1 subtrees:2 accurate:1 peculiar:1 edge:4 tree:44 orbit:1 toone:1 stopped:1 instance:1 earlier:1 modeling:1 bombay:1 mhz:1 assignment:1 signifies:1 vertex:4 addressing:1 subset:5 entry:1 hundred:3 kq:1 uniform:1 johnson:1 reported:4 kn:5 perturbed:2 considerably:1 fundamental:1 mestre:1 together:1 conf:1 derivative:1 sec:2 wk:1 int:2 bg:1 performed:3 root:1 ir:1 characteristic:5 efficiently:1 yield:1 correspond:1 emphasizes:1 trajectory:3 corruption:5 processor:1 converged:2 definition:3 frequency:1 obvious:2 straus:4 thereof:1 proof:3 di:3 naturally:1 attributed:3 garey:1 proved:3 dimensionality:1 back:1 y7:1 formulation:3 furthermore:1 reply:1 hand:3 maximizer:5 quality:1 artif:1 name:1 consisted:1 ranged:1 brown:1 counterpart:3 evolution:3 hence:5 symmetric:2 iteratively:1 attractive:2 adjacent:5 game:7 self:2 rooted:4 o3:1 outline:1 theoretic:1 motion:1 sigmund:1 recently:2 common:2 replicator:16 nh:1 association:11 approximates:1 kluwer:1 significant:1 versa:1 cambridge:4 trivially:2 similarly:1 replicating:1 dot:1 stable:1 zucker:2 impressive:2 similarity:2 own:1 recent:3 perspective:1 italy:1 apart:1 prime:1 discretizing:1 continue:1 jagota:1 seen:2 xaq:2 unrooted:1 converge:1 shortest:1 maximize:1 faster:2 match:2 concerning:1 j3:1 vision:4 represent:1 suppl:1 addition:1 unlike:2 strict:5 induced:2 subject:2 nagel:1 undirected:1 oq:1 presence:1 easy:1 b7:1 graduated:1 restrict:1 plr:1 reduce:1 whether:1 motivated:2 heavier:1 kohn:1 isomorphism:20 stereo:1 returned:1 dramatically:1 useful:2 u5:3 induces:1 siddiqi:2 informatica:1 discretetime:1 exist:2 percentage:5 vy:1 diverse:1 discrete:3 shall:1 vol:1 terminology:2 threshold:1 blum:1 deleted:1 graph:33 asymptotically:1 fraction:1 powerful:2 geiger:1 collegium:1 correspondence:6 quadratic:5 foscari:1 n3:1 c8:7 pruned:1 concluding:1 performing:1 disconnected:1 cormen:1 remain:1 smaller:2 slowed:1 invariant:1 iccv:1 taken:3 equation:10 mutually:1 turn:3 eventually:1 know:1 letting:1 available:1 hierarchical:1 distinguished:1 customary:1 original:5 denotes:2 top:2 hofbauer:2 prof:1 approximating:1 classical:2 torino:1 noticed:1 question:1 barycenter:1 canad:1 said:3 evolutionary:5 distance:5 thank:1 berlin:1 entity:1 mail:1 uwvyx:1 assuming:1 length:2 relationship:1 providing:1 ratio:1 fe:1 statement:1 trace:1 negative:1 stated:1 anal:2 proper:1 conversion:1 on3:1 benchmark:1 situation:1 relational:1 extended:1 interacting:1 arbitrary:4 cast:1 namely:1 pair:1 connection:2 deletion:1 trans:3 able:1 bar:1 suggested:1 dynamical:6 pattern:4 below:1 program:1 memory:1 deleting:1 power:2 natural:1 zhu:1 maximality:1 lk:2 started:1 axis:1 determining:2 relative:1 dsi:1 interesting:1 acyclic:1 remarkable:1 degree:1 unpruned:1 editor:2 intractability:1 playing:1 course:1 placed:1 free:14 infeasible:1 formal:1 allow:1 guide:1 india:1 calculated:1 author:1 coincide:1 san:1 approximate:1 keep:1 clique:40 dealing:1 global:5 handbook:1 conclude:1 francisco:1 imitation:2 continuous:4 search:1 ballard:1 ca:2 obtaining:2 du:1 constructing:1 protocol:1 arise:1 contradicting:1 allowed:1 join:1 vr:1 exponential:5 xl:1 ib:1 hw:1 theorem:13 down:1 showing:1 npcompleteness:1 virtue:1 maximizers:4 exists:1 subtree:20 boston:1 simply:1 saddle:1 visual:1 motzkin:4 contained:1 joined:2 restarted:1 springer:1 corresponds:1 ma:3 identity:1 towards:1 fisher:1 feasible:1 hard:1 wilf:2 uniformly:3 called:6 isomorphic:4 e:1 formally:1 support:1 latter:1 inability:2 arises:1 tested:1 |
1,160 | 2,058 | K-Local Hyperplane and Convex Distance
Nearest Neighbor Algorithms
Pascal Vincent and Yoshua Bengio
Dept. IRO, Universit?e de Montr?eal
C.P. 6128, Montreal, Qc, H3C 3J7, Canada
vincentp,bengioy @iro.umontreal.ca
http://www.iro.umontreal.ca/ vincentp
Abstract
Guided by an initial idea of building a complex (non linear) decision
surface with maximal local margin in input space, we give a possible
geometrical intuition as to why K-Nearest Neighbor (KNN) algorithms
often perform more poorly than SVMs on classification tasks. We then
propose modified K-Nearest Neighbor algorithms to overcome the perceived problem. The approach is similar in spirit to Tangent Distance, but
with invariances inferred from the local neighborhood rather than prior
knowledge. Experimental results on real world classification tasks suggest that the modified KNN algorithms often give a dramatic improvement over standard KNN and perform as well or better than SVMs.
1 Motivation
The notion of margin for classification tasks has been largely popularized by the success
of the Support Vector Machine (SVM) [2, 15] approach. The margin of SVMs has a nice
geometric interpretation1: it can be defined informally as (twice) the smallest Euclidean
distance between the decision surface and the closest training point. The decision surface
produced by the original SVM algorithm is the hyperplane that maximizes this distance
while still correctly separating the two classes. While the notion of keeping the largest possible safety margin between the decision surface and the data points seems very reasonable
and intuitively appealing, questions arise when extending the approach to building more
complex, non-linear decision surfaces.
Non-linear SVMs usually use the ?kernel trick? to achieve their non-linearity. This conceptually corresponds to first mapping the input into a higher-dimensional feature space
with some non-linear transformation and building a maximum-margin hyperplane (a linear
decision surface) there. The ?trick? is that this mapping is never computed directly, but implicitly induced by a kernel. In this setting, the margin being maximized is still the smallest
Euclidean distance between the decision surface and the training points, but this time measured in some strange, sometimes infinite dimensional, kernel-induced feature space rather
than the original input space. It is less clear whether maximizing the margin in this new
space, is meaningful in general (see [16]).
1
for the purpose of this discussion, we consider the original hard-margin SVM algorithm for two
linearly separable classes.
A different approach is to try and build a non-linear decision surface with maximal distance to the closest data point as measured directly in input space (as proposed in [14]). We
could for instance restrict ourselves to a certain class of decision functions and try to find
the function with maximal margin among this class. But let us take this even further. Extending the idea of building a correctly separating non-linear decision surface as far away
as possible from the data points, we define the notion of local margin as the Euclidean distance, in input space, between a given point on the decision surface and the closest training
point. Now would it be possible to find an algorithm that could produce a decision surface
which correctly separates the classes and such that the local margin is everywhere maximal
along its surface? Surprisingly, the plain old Nearest Neighbor algorithm (1NN) [5] does
precisely this!
So why does 1NN in practice often perform worse than SVMs? One typical explanation, is
that it has too much capacity, compared to SVM, that the class of function it can produce is
too rich. But, considering it has infinite capacity (VC-dimension), 1NN is still performing
quite well... This study is an attempt to better understand what is happening, based on
geometrical intuition, and to derive an improved Nearest Neighbor algorithm from this
understanding.
2 Fixing a broken Nearest Neighbor algorithm
2.1 Setting and definitions
(the input space).
We are given
corresponding
set
atraining
of points
! #"$
%" '&(
and
)!their
*
)!*
class label
where
is the
+
,
number of different classes.
The
pairs
are
assumed
to
be
samples
drawn
from
an
./
0
unknown
. Barring
76 class labels associated to
1 distribution 1 inputs,
4 5the
* duplicate
define a partition of : let
32
each
.
"
97:
The problem is to find
that will generalize well on new
.=a
>decision
0 9 function 8 <;
points drawn from . 8 should ideally minimize the expected classification .=
error,
0
M O$P where ?A@ denotes the expectation with respect to i.e. minimize ?A@CB DFG'E HJILKNF
T
S
&
9
M R denotes the indicator function, whose value is if 8
and DFG(E HJQKN
and U otherwise.
The setting is that of a classical classification problem in
In the previous and following discussion, we often refer to the concept of decision surface,
9
also known as decision
The function 8 corresponding to a given
de6 #boundary.
W
"
* algorithm
2
fines for any class
two regions of the input space: the region V
6
*
9 5X6
and its complement Y V . The decision
surface
for
class
is
the
?boundary?
8
&
*
between those two regions, i.e. the contour of V , and can be seen as a Z Y
dimensional
manifold (a ?surface? in ) possibly made of several disconnected components. For simplicity, when we mention the decision surface in our discussion we consider only the case
of two class discrimination, in which there is a single decision surface.
1
that
When we mention a test point, we mean a point
not belong to the training
9 does
set and for which the algorithm is to decide on a class 8
.
By distance, we mean the usual Euclidean
in input-space
distance
tween two points [ and \ will be written ] [ \ or alternatively ^ [ Y
The distance between
ba<cJdfe point
g'h +
and
+
a `single
i a set of points
point of the set: ]
_
]
.
The
K-neighborhood jLk
to is smallest.
of a test point
*
The
K-c-neighborhood jmk
to is smallest.
of a test point
_
is the set of the l
is the set of l
.
\(^ .
The distance be-
is the distance to the closest
points of whose distance
points of
*
whose distance
By Nearest
Neighbor algorithm (1NN) we mean the following algorithm: the class of a test
point is decided to be the same as the class of its closest neighbor in _ .
By K-Nearest
Neighbor algorithm (KNN) we mean the following algorithm: the class of
a test point is decided
to be the same as the class appearing most frequently among the
K-neighborhood of .
2.2 The intuition
Figure 1: A local view of the decision surface produced by the Nearest Neighbor (left) and
SVM (center) algorithms, and how the Nearest Neighbor solution gets closer to the SVM
solution in the limit, if the support for the density of each class is a manifold which can be
considered locally linear (right).
Figure 1 illustrates a possible intuition about why SVMs outperforms 1NNs when we have
a finite number of samples. For classification tasks where the classes are considered to
be mostly separable,2 we often like to think of each class as residing close to a lowerdimensional manifold (in the high dimensional input space) which can reasonably be considered locally linear3 . In the case of a finite number of samples, ?missing? samples would
appear as ?holes? introducing artifacts in the decision surface produced by classical Nearest Neighbor algorithms. Thus the decision surface, while having the largest possible local
margin with regard to the training points, is likely to have a poor small local margin with
respect to yet unseen samples falling close to the locally linear manifold, and will thus result in poor generalization performance. This problem fundamentally remains with the K
Nearest Neighbor (KNN) variant of the algorithm, but, as can be seen on the figure, it does
not seem to affect the decision surface produced by SVMs (as the surface is constrained to
a particular smooth form, a straight line or hyperplane in the case of linear SVMs). It is
interesting to notice, if the assumption of locally linear class manifolds holds, how the 1NN
solution approaches the SVM solution in the limit as we increase the number of samples.
To fix this problem, the idea is to somehow fantasize the missing points, based on a local linear approximation of the manifold of each class. This leads to modified Nearest
Neighbor algorithms described in the next sections.4
2
By ?mostly separable? we mean that the Bayes error is almost zero, and the optimal decision
surface has not too many disconnected components.
3
i.e. each class has a probability density with a ?support? that is a lower-dimensional manifold,
and with the probability quickly fading, away from this support.
4
Note that although we never generate the ?fantasy? points explicitly, the proposed algorithms are
really equivalent to classical 1NN with fantasized points.
2.3 The basic algorithm
Given a test point , we are really interested in finding the closest neighbor, not among
the training set , but among an abstract, virtually enriched training set that would contain
all the fantasized ?missing? points of the manifold of each6 class, locally approximated by
an affine subspace. We shall thus consider, for each class
, the local affine subspace that
passes& through the l points of the K-c neighborhood of . This affine subspace is typically
l Y dimensional or less, and we will somewhat abusively call it the ?local hyperplane?.5
Formally, the local hyperplane can be defined as
* k 5
i i k
)
N
N
)
>) *
.
j k
where
(1)
&
, ) is to
Another way to define this hyperplane, that gets rid of the constraint
take
point within the hyperplane as an origin, for instance the centroid 6
a reference
)
. This same hyperplane can then be expressed as
k N
k
* k 5
;Y ) )
Y
where
i i
) k
;Y
N
.
(2)
6
Our modified nearest
then associates
a<c d point
whose
9 Aa $test
* neighbor algorithm
* g ] +to
the * kclass
k
is
closest
to
.
Formally
, where
hyperplane
8
F
* ,
]
k
is logically called K-local Hyperplane Distance, hence the name K-local
Hyperplane Distance Nearest Neighbor algorithm (HKNN in short).
Computing, for each class
]
6
F
* ,
k
e g !a!#c "d H QK ^ Y i ^
$
a<cJd ) k ;Y +
(3)
Y
% 'g & (*),+++ Y
++
N
++
++
amounts to solving a linear system in , that can be easily expressed in matrix form as:
.- /0 / 1- / )
(4)
Y
)
where and are Z dimensional column vectors,
;Y
matrix whose columns are the vectors defined earlier.7
k
-
, and
is a Z2
l
Strictly speaking a hyperplane in an 3 dimensional input space is an 35476 affine subspace, while
our ?local hyperplanes? can have fewer dimensions.
6
We could be using one of the 8 neighbors as the reference point, but this formulation with the
centroid will prove useful later.
94 :7;
7
Actually there is an infinite number of solutions to this system since the
are linearly dependent: remember that the initial formulation had an equality constraint and thus only 8<4=6 effective
degrees of freedom. But we are interested
94 :7; in >@?BADCFEHGJK I ?BA LFL not in M so any solution will do. Alternatively, we can remove one of the
from the system so that it has a unique solution.
5
2.4 Links with other paradigms
The proposed
HKNN
* algorithm is very similar in spirit to the Tangent Distance Algok
rithm [13].
can be seen as a tangent hyperplane representing a set of local di;Y
rections of transformation (any linear combination of the vectors) that do not affect
the class identity. These are invariances. The main difference is that in HKNN these invariances are inferred directly from the local neighborhood in the training set, whereas in
Tangent Distance, they are based on prior knowledge. It should be interesting (and relatively easy) to combine both approaches for improved performance when prior knowledge
is available.
Previous work on nearest-neighbor variations based on other locally-defined metrics can
be found in [12, 9, 6, 7], and is very much related to the more general paradigm of Local
Learning Algorithms [3, 1, 10].
We should also mention close similarities between our approach and the recently proposed
Local Linear Embedding [11] method for dimensionality reduction.
The idea of fantasizing points around the training points in order to define the decision
surface is also very close to methods based on estimating the class-conditional input density [14, 4].
Besides, it is interesting to look at HKNN from a different, less geometrical angle: it can be
understood as choosing the class that achieves the best reconstruction (the smallest reconstruction error) of the test pattern through a linear combination of l particular prototypes
of that class (the l neighbors). From this point of view, the algorithm is very similar to
the Nearest Feature Line (NFL) [8] method. They differ in the fact that NFL considers all
pairs of points for its search rather than the local l neighbors, thus looking
at many ( )
&
lines (i.e. 2 dimensional affine subspaces), rather than at a single l Y
dimensional one.
3 Fixing the basic HKNN algorithm
3.1 Problem arising for large K
One problem with the basic HKNN algorithm, as previously described, arises as we increase the value of l , i.e. the number of points considered in the neighborhood of the test
point. In a typical high dimensional setting, exact colinearities between input patterns are
rare, which means that as soon as l Z , any pattern of (including nonsensical ones)
can be produced by a linear combination of the l neighbors. The ?actual? dimensionality
of the manifold may be much less than l . This is due to ?near-colinearities?
di - /0 that producing
rections associated to small eigenvalues of the covariance matrix
are but noise,
that can lead the algorithm to mistake those noise directions for ?invariances?, and may
hurt its performance even for smaller values of l . Another related issue is that the linear
approximation of the class manifold by a hyperplane is valid only locally, so we might
want to restrict the ?fantasizing? of class members to a smaller region of the hyperplane.
We considered two ways of dealing with these problems.8
3.2 The convex hull solution
One way to avoid the above mentioned problems is to restrict ourselves to considering
the convex hull of the neighbors, rather than the whole hyperplane they support (of
which
the convex hull is a subset). This corresponds to adding a constraint of U
to
equation (1). Unlike the problem of computing the distance to the hyperplane, the distance
to the convex hull cannot be found by solving a simple linear system, but typically requires
solving a quadratic programming problem (very similar to the one of SVMs). While this
8
A third interesting avenue, which : we did not have time to explore, would be to keep only the
most relevant principal components of , ignoring those corresponding to small eigenvalues.
is more complex to implement, it should be mentioned that the problems to be solved are
of a relatively small dimension of order l , and that the time of the whole algorithm will
very likely still be dominated by the search of the l nearest neighbors within each class.
This algorithm will be referred to as K-local Convex Distance Nearest Neighbor Algorithm
(CKNN in short).
3.3 The ?weight decay? penalty solution
This consists in incorporating a penalty term to equation (3) to penalize large values of
(i.e. it penalizes moving away from the centroid, especially in non essential directions):
- +
* k a<g'cJ& (0d )+ Y ) Y k ;Y + k
(5)
%
++
++
N
N
++
- / ++ / - / )
The solution for is given by solving the linear system
D
Y
2 Z identity matrix. This is equation (4) with an additional diagonal term.
where D is the Z7T
]
The resulting algorithm is a generalization of HKNN (basic HKNN corresponds to
U
).
4 Experimental results
We performed a number of experiments, to highlight different properties of the algorithms:
A first 2D toy example (see Figure 2) graphically illustrates the qualitative differences in
the decision surfaces produced by KNN, linear SVM and CKNN.
Table 1 gives quantitative results on two real-world digit OCR tasks, allowing to compare
the performance of the different old and new algorithms.
Figure 3 illustrates the problem arising with large l , mentioned in Section 3, and shows
that the two proposed solutions: CKNN and HKNN with an added weight decay , allow
to overcome it.
In our final experiment, we wanted to see if the good performance of the new algorithms
absolutely depended on having all the training points at hand, as this has a direct impact
on speed. So we checked what performance we could get out of HKNN and CKNN when
using only a small but representative subset of the training points, namely the set of support
vectors found by a Gaussian Kernel SVM. The results obtained for MNIST are given in
Table 2, and look very encouraging. HKNN appears to be able to perform as well or better
than SVMs without requiring more data points than SVMs.
Figure 2: 2D illustration of the decision surfaces produced by KNN (left, K=1), linear SVM
(middle), and CKNN (right, K=2). The ?holes? are again visible in KNN. CKNN doesn?t
suffer from this, but keeps the objective of maximizing the margin locally.
5 Conclusion
From a few geometrical intuitions, we have derived two modified versions of the KNN
algorithm that look very promising. HKNN is especially attractive: it is very simple to
implement on top of a KNN system, as it only requires the additional step of solving a
small and simple linear system, and appears to greatly boost the performance of standard
KNN even above the level of SVMs. The proposed algorithms share the advantages of
KNN (no training required, ideal for fast adaptation, natural handling of the multi-class
case) and its drawbacks (requires large memory, slow testing). However our latest result
also indicate the possibility of substantially reducing the reference set in memory without
loosing on accuracy. This suggests that the algorithm indeed captures essential information
in the data, and that our initial intuition on the nature of the flaw of KNN may well be at
least partially correct.
References
[1] C. G. Atkeson, A. W. Moore, and S. Schaal. Locally weighted learning. Artificial Intelligence
Review, 1996.
[2] B. Boser, I. Guyon, and V. Vapnik. An algorithm for optimal margin classifiers. In Fifth Annual
Workshop on Computational Learning Theory, pages 144?152, Pittsburgh, 1992.
[3] L. Bottou and V. Vapnik. Local learning algorithms. Neural Computation, 4(6):888?900, 1992.
[4] Olivier Chapelle, Jason Weston, L?eon Bottou, and Vladimir Vapnik. Vicinal risk minimization.
In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing
Systems, volume 13, pages 416?422, 2001.
[5] T.M. Cover and P.E. Hart. Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13(1):21?27, 1967.
[6] J. Friedman. Flexible metric nearest neighbor classification. Technical Report 113, Stanford
University Statistics Department, 1994.
[7] Trevor Hastie and Robert Tibshirani. Discriminant adaptive nearest neighbor classification and
regression. In David S. Touretzky, Michael C. Mozer, and Michael E. Hasselmo, editors, Advances in Neural Information Processing Systems, volume 8, pages 409?415. The MIT Press,
1996.
[8] S.Z. Li and J.W. Lu. Face recognition using the nearest feature line method. IEEE Transactions
on Neural Networks, 10(2):439?443, 1999.
[9] J. Myles and D. Hand. The multi-class measure problem in nearest neighbour discrimination
rules. Pattern Recognition, 23:1291?1297, 1990.
[10] D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. In S. A. Solla,
T. K. Leen, and K-R. Mller, editors, Advances in Neural Information Processing Systems, volume 12. MIT Press, 2000.
[11] Sam Roweis and Lawrence Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323?2326, Dec. 2000.
[12] R. D. Short and K. Fukunaga. The optimal distance measure for nearest neighbor classification.
IEEE Transactions on Information Theory, 27:622?627, 1981.
[13] P. Y. Simard, Y. A. LeCun, J. S. Denker, and B. Victorri. Transformation invariance in pattern
recognition ? tangent distance and tangent propagation. Lecture Notes in Computer Science,
1524, 1998.
[14] S. Tong and D. Koller. Restricted bayes optimal classifiers. In Proceedings of the 17th National
Conference on Artificial Intelligence (AAAI), pages 658?664, Austin, Texas, 2000.
[15] V.N. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, 1995.
[16] Bin Zhang. Is the maximal margin hyperplane special in a feature space? Technical Report
HPL-2001-89, Hewlett-Packards Labs, 2001.
Table 1: Test-error obtained on the USPS and MNIST digit classification tasks by KNN,
SVM (using a Gaussian Kernel), HKNN and CKNN. Hyper parameters were tuned on
a separate validation set. Both HKNN and CKNN appear to perform much better than
original KNN, and even compare favorably to SVMs.
Data Set
USPS
(6291 train,
1000 valid.,
2007 test points)
MNIST
(50000 train,
10000 valid.,
10000 test points)
Algorithm
KNN
SVM
HKNN
CKNN
KNN
SVM
HKNN
CKNN
Test Error
4.98%
4.33%
3.93%
3.98%
2.95%
1.30%
1.26%
1.46%
Parameters used
l
l
l
l
l
&
& U(U
&f
U
U
&
& U'U
U
U
l
0.032
CKNN
basic HKNN
HKNN, lambda=1
HKNN, lambda=10
0.03
0.028
error rate
0.026
0.024
0.022
0.02
0.018
0.016
0.014
0.012
0
20
40
60
80
100
120
K
Figure 3: Error rate on MNIST as a function of l for CKNN, and HKNN with different
values of . As can be seen the basic HKNN algorithm performs poorly for large values of
l . As expected, CKNN is relatively unaffected by this problem, and HKNN can be made
robust through the added ?weight decay? penalty controlled by .
Table 2: Test-error obtained on MNIST with HKNN and CKNN when using a reduced
training set made of the 16712 support vectors retained by the best Gaussian Kernel SVM.
This corresponds to 28% of the initial 60000 training patterns. Performance is even better
than when using the whole dataset. But here, hyper parameters l and were chosen with
the test set, as we did not have a separate validation set in this setting. It is nevertheless
remarkable that comparable performances can be achieved with far fewer points.
Data Set
MNIST (16712 train s.v.,
10000 test points)
Algorithm
HKNN
CKNN
Test Error
1.23%
1.36%
U
Parameters
used
&
l
l
U
| 2058 |@word version:1 middle:1 seems:1 nonsensical:1 covariance:1 dramatic:1 mention:3 reduction:2 myles:1 initial:4 tuned:1 outperforms:1 yet:1 written:1 visible:1 partition:1 shape:1 wanted:1 remove:1 discrimination:2 intelligence:2 fewer:2 short:3 hyperplanes:1 zhang:1 along:1 direct:1 qualitative:1 prove:1 consists:1 combine:1 indeed:1 expected:2 frequently:1 multi:2 actual:1 encouraging:1 considering:2 estimating:1 linearity:1 maximizes:1 what:2 fantasy:1 substantially:1 vicinal:1 finding:1 transformation:3 remember:1 quantitative:1 nf:1 universit:1 classifier:2 appear:2 producing:1 safety:1 understood:1 local:23 limit:2 mistake:1 depended:1 might:1 twice:1 suggests:1 decided:2 unique:1 lecun:1 testing:1 practice:1 implement:2 digit:2 suggest:1 get:3 cannot:1 close:4 risk:1 www:1 equivalent:1 center:1 maximizing:2 missing:3 graphically:1 latest:1 lfl:1 convex:6 qc:1 simplicity:1 rule:1 embedding:2 notion:3 variation:1 hurt:1 exact:1 programming:1 olivier:1 hjilk:1 origin:1 trick:2 associate:1 approximated:1 recognition:3 solved:1 capture:1 region:4 solla:1 mentioned:3 intuition:6 mozer:1 broken:1 ideally:1 solving:5 usps:2 easily:1 train:3 fast:1 effective:1 artificial:2 hyper:2 neighborhood:7 choosing:1 quite:1 whose:5 stanford:1 otherwise:1 statistic:1 knn:17 unseen:1 think:1 h3c:1 final:1 advantage:1 eigenvalue:2 propose:1 reconstruction:2 maximal:5 adaptation:1 relevant:1 poorly:2 achieve:1 roweis:1 extending:2 produce:2 derive:1 montreal:1 fixing:2 measured:2 nearest:25 indicate:1 differ:1 direction:2 guided:1 drawback:1 correct:1 hull:4 vc:1 bin:1 fix:1 generalization:2 really:2 strictly:1 hold:1 around:1 considered:5 residing:1 cb:1 mapping:2 lawrence:1 mller:1 achieves:1 smallest:5 purpose:1 perceived:1 label:2 largest:2 hasselmo:1 weighted:1 minimization:1 mit:2 j7:1 gaussian:3 modified:5 rather:5 avoid:1 derived:1 schaal:1 improvement:1 logically:1 greatly:1 centroid:3 flaw:1 dependent:1 nn:6 typically:2 koller:1 interested:2 issue:1 classification:11 among:4 pascal:1 flexible:1 constrained:1 special:1 never:2 barring:1 having:2 look:3 yoshua:1 report:2 fundamentally:1 duplicate:1 few:1 neighbour:1 national:1 dfg:2 ourselves:2 attempt:1 freedom:1 friedman:1 montr:1 possibility:1 hewlett:1 closer:1 euclidean:4 old:2 penalizes:1 instance:2 eal:1 column:2 earlier:1 cover:1 introducing:1 subset:2 rare:1 too:3 nns:1 density:3 michael:2 quickly:1 again:1 aaai:1 possibly:1 worse:1 lambda:2 simard:1 toy:1 li:1 de:1 explicitly:1 later:1 try:2 view:2 performed:1 jason:1 lab:1 bayes:2 minimize:2 accuracy:1 largely:1 maximized:1 conceptually:1 generalize:1 vincent:1 produced:7 lu:1 straight:1 unaffected:1 touretzky:1 checked:1 trevor:1 definition:1 associated:2 di:2 dataset:1 knowledge:3 dimensionality:3 cj:1 actually:1 appears:2 higher:1 x6:1 improved:2 formulation:2 leen:2 hpl:1 hand:2 nonlinear:1 propagation:1 somehow:1 artifact:1 name:1 dietterich:1 building:4 concept:1 contain:1 requiring:1 hence:1 equality:1 moore:1 attractive:1 performs:1 geometrical:4 recently:1 umontreal:2 volume:3 belong:1 refer:1 fantasized:2 had:1 moving:1 chapelle:1 similarity:1 surface:26 closest:7 certain:1 success:1 seen:4 additional:2 lowerdimensional:1 somewhat:1 paradigm:2 smooth:1 technical:2 hart:1 controlled:1 impact:1 variant:1 basic:6 regression:2 expectation:1 metric:2 sometimes:1 kernel:7 achieved:1 dec:1 penalize:1 whereas:1 want:1 fine:1 victorri:1 unlike:1 pass:1 induced:2 virtually:1 member:1 spirit:2 seem:1 call:1 near:1 abusively:1 ideal:1 bengio:1 easy:1 affect:2 hastie:2 restrict:3 idea:4 prototype:1 avenue:1 texas:1 nfl:2 whether:1 penalty:3 suffer:1 speaking:1 york:1 useful:1 clear:1 informally:1 amount:1 locally:10 svms:13 reduced:1 http:1 generate:1 notice:1 arising:2 correctly:3 tibshirani:1 shall:1 nevertheless:1 falling:1 drawn:2 angle:1 everywhere:1 almost:1 reasonable:1 strange:1 decide:1 guyon:1 decision:26 comparable:1 quadratic:1 annual:1 fading:1 precisely:1 constraint:3 dominated:1 speed:1 fukunaga:1 performing:1 separable:3 relatively:3 department:1 popularized:1 combination:3 disconnected:2 poor:2 smaller:2 sam:1 appealing:1 intuitively:1 restricted:1 jlk:1 equation:3 remains:1 previously:1 available:1 denker:1 ocr:1 away:3 appearing:1 original:4 denotes:2 top:1 eon:1 build:1 especially:2 classical:3 objective:1 question:1 added:2 usual:1 diagonal:1 subspace:5 distance:23 separate:3 link:1 separating:2 capacity:2 manifold:10 considers:1 discriminant:1 iro:3 besides:1 retained:1 illustration:1 vladimir:1 mostly:2 robert:1 favorably:1 cknn:15 ba:2 unknown:1 perform:5 allowing:1 finite:2 looking:1 rections:2 canada:1 inferred:2 david:1 complement:1 pair:2 namely:1 required:1 boser:1 boost:1 able:1 usually:1 pattern:7 vincentp:2 including:1 memory:2 explanation:1 natural:1 indicator:1 ormoneit:1 representing:1 kclass:1 tresp:1 prior:3 nice:1 geometric:1 tangent:6 understanding:1 review:1 lecture:1 highlight:1 interesting:4 remarkable:1 validation:2 degree:1 affine:5 editor:3 share:1 austin:1 surprisingly:1 keeping:1 soon:1 allow:1 understand:1 neighbor:29 saul:1 face:1 fifth:1 regard:1 overcome:2 plain:1 dimension:3 world:2 boundary:2 rich:1 contour:1 valid:3 doesn:1 made:3 adaptive:1 atkeson:1 far:2 transaction:3 implicitly:1 keep:2 dealing:1 rid:1 pittsburgh:1 assumed:1 alternatively:2 search:2 why:3 table:4 promising:1 nature:2 reasonably:1 robust:1 ca:2 ignoring:1 bottou:2 complex:3 tween:1 did:2 main:1 linearly:2 motivation:1 noise:2 arise:1 whole:3 enriched:1 referred:1 representative:1 rithm:1 slow:1 tong:1 bengioy:1 third:1 jmk:1 decay:3 svm:14 incorporating:1 essential:2 mnist:6 vapnik:4 adding:1 workshop:1 illustrates:3 hole:2 margin:16 likely:2 explore:1 happening:1 expressed:2 partially:1 springer:1 corresponds:4 weston:1 conditional:1 identity:2 loosing:1 hard:1 infinite:3 typical:2 reducing:1 hyperplane:19 principal:1 called:1 invariance:5 experimental:2 meaningful:1 formally:2 support:7 arises:1 absolutely:1 dept:1 handling:1 |
1,161 | 2,059 | Kernel Logistic Regression and the Import
Vector Machine
Trevor Hastie
Department of Statistics
Stanford University
Stanford, CA 94305
hastie@stat.stanford.edu
Ji Zhu
Department of Statistics
Stanford University
Stanford, CA 94305
jzhu@stat.stanford.edu
Abstract
The support vector machine (SVM) is known for its good performance in
binary classification, but its extension to multi-class classification is still
an on-going research issue. In this paper, we propose a new approach
for classification, called the import vector machine (IVM), which is built
on kernel logistic regression (KLR). We show that the IVM not only performs as well as the SVM in binary classification, but also can naturally
be generalized to the multi-class case. Furthermore, the IVM provides an
estimate of the underlying probability. Similar to the ?support points? of
the SVM, the IVM model uses only a fraction of the training data to index
kernel basis functions, typically a much smaller fraction than the SVM.
This gives the IVM a computational advantage over the SVM, especially
when the size of the training data set is large.
1 Introduction
In standard
classification problems,
we are given a set of training data
,
,
, where the output is qualitative and assumes values in a finite set . We
wish to find a classfication rule from the training data, so that when given a new input ,
we can assign a class from to it. Usually it is assumed that the training data are an
independently
and identically distributed sample from an unknown probability distribution
.
"!$#&%'()
The support vector machine (SVM) works well in binary classification, i.e.
, but
its appropriate extension to the multi-class case is stillan
on-going research issue. Another
3$(54687
9
weakness of the SVM is that it only
estimates
, while the probability 2
*+-,/.10 2
9
:
;:<(/= >:?
is often of interest itself,( where 2 @:A
is the conditional probability
of a point being in class given
. In this paper, we propose a new approach, called
the import vector machine (IVM), to address the classification problem. We show that the
IVM not only performs as well as the SVM in binary classification, but also can naturally
be generalized
to the multi-class case. Furthermore, the IVM provides an estimate of the
probability 2
. Similar to the ?support points? of the SVM, the IVM model uses only a
fraction of the training data to index the kernel basis functions.
We call these training data
C$D&
&E8
E cost of the SVM is B
E
import points. The
computational
, while the computational
cost
C
of the IVM is B
, where is the number of import points. Since does not tend to
C
increase as increases, the IVM can be faster than the SVM, especially for large training
data sets. Empirical results show that the number of import points is usually much less than
the number of support points.
In section (2), we briefly review some results of the SVM for binary classification and
compare it with kernel logistic regression (KLR). In section (3), we propose our IVM
algorithm. In section (4), we show some simulation results. In section (5), we generalize
the IVM to the multi-class case.
2 Support vector machines and kernel logistic regression
The standard SVM produces a non-linear classification boundary in the original input space
by constructing a linear boundary in a transformed version of the original input space.
The dimension of the transformed space can be very large, even infinite in some cases.
This seemingly prohibitive computation is achieved through a positive definite reproducing
kernel , which gives the inner product in the transformed space.
Many people have noted the relationship between the SVM and regularized function estimation in the reproducing kernel Hilbert spaces (RKHS). An overview can be found in
Evgeniou et al. (1999), Hastie et al. (2001) and Wahba (1998). Fitting an SVM is equivalent to minimizing:
(
(1)
C
:
!
(
3$
!
with
.
7
classification rule is given by *+-,/.10 .
is the RKHS generated by the kernel
By the representer theorem (Kimeldorf et al (1971)), the optimal
(2)
:
C
9
. The
has the form:
It often happens that a sizeable fraction of the
values of can be zero. This is a
consequence of the truncation property of the first part of criterion (1). This seems to be an
attractive property, because only the points on the wrong side of the classification boundary,
and those on the right side but near the boundary have
an influence in determining
the
9
position of the boundary, and hence have non-zero ?s. The corresponding ?s are called
support points.
( 3
&
Notice that (1) has the form !*&* 2#"&. %$ . The loss function
is plotted in Figure
1, along with several traditional loss functions. As we can see, the negative log-likelihood
(NLL)
of the binomial distribution
has
a similar shape to that of the SVM. If we replace
( 3
'
(*
",+.-0/
in (1) with ()
, the NLL of the binomial distribution, the problem
becomes a KLR problem. We expect that the fitted function performs similarly to the SVM
for binary classfication.
There are two immediate advantages of making such a replacement: (a) Besides giving
9
:
a classification
rule, the KLR also offers a natural estimate
of the probability 2
4 (*
3
(54687
"1/
"1/
, while the SVM only estimates *+-, .10 2
; (b) The KLR can naturally be generalized to the multi-class case through kernel multi-logit regression, whereas
this is not the case for the SVM. However, because the KLR compromises the hinge loss
function
of the SVM, it no longer has the ?support points? property; in other words, all the
?s
in
(2)
are non-zero.
KLR is a well studied problem; see Wahba et al. (1995) and references there; see also
Green et al. (1985) and Hastie et al. (1990).
3.0
0.0
0.5
1.0
1.5
Loss
2.0
2.5
Binomial NLL
Squared Error
Support Vector
-3
-2
-1
0
1
2
3
yf(x)
Figure 1: Several loss functions,
C
D5
The computational cost of the KLR is B
; to save the computational cost, the IVM
algorithm will find a sub-model to approximate the full model (2) given by the KLR. The
sub-model has the form:
1:
(3)
# 8
)
where is a subset of the training data
, and the data in are called import
points. The advantage of this sub-model is that the computational cost is reduced, especially for large training data sets, while not jeopardizing the performance in classification.
Several other researchers have investigated techniques in selecting the subset . Lin et al.
(1998) divide the training data into several clusters, then randomly select a representative
from each clusterE to make up . Smola et al. (2000) develope
a greedy technique to se &
7
E
0
, such that the span of
quentially
select columns of the kernel matrix
7
E
these columns approximates the span of 0
well in the Frobenius norm.
Williams et al. (2001) propose randomly selecting points of the training data, then
using
the Nystrom
method to approximate the eigen-decomposition
of the kernel matrix
&
7
C
0
, and
expanding
the
results
back
up
to
dimensions.
None of these
meth
ods uses the output in selecting the/subset
(i.e.,
the
procedure
only
involves
).
The
IVM algorithm uses both the output and the input
to select the subset , in such a
way that the resulting fit approximates the full model well.
3 Import vector machine
/
!
#5% ()
Following the tradition of logistic regression, we let
for the rest of this paper.
For notational simplicity, the constant term in the fitted function is ignored.
In the KLR, we want to minimize:
:
3
0
3
(
( )
7
6
From (2), it can be shown that this is equivalent to the finite dimensional form:
(4)
:
! #"
3
$
(
()
(
%
! #"
6
'&
:
where
tion matrix
8
'&
:
'"
; the regressor matrix
#" .
:
0
7
; and the regulariza-
To find , we set the derivative of
with respect to equal to 0, and use the NewtonRaphson method to iteratively solve the score equation. It can be shown that the NewtonRaphson step is a weighted least squares step:
(5)
"
:
#"
#&
'"
:
where
is:
the
valueof
( in3 the -
th
7 step,
matrix is
.
+ ,90 2
2
"
+
+
+
3
#& 8
2
. The weight
)
As mentioned in section 2, we want to find a subset of
, such that the
sub-model (3) is a good approximation of the full model (2). Since it is impossible to search
for every subset , we use the following greedy forward strategy:
3.1 Basic algorithm
(
(
(
6
:
) Let
,
) For each
:
#&98
!
to minimize
(6)
( '&
.
3
:
3
:
0
#
"
3
"
(
:
# )
Let
) Repeat steps (
We call the points in
6
:$#
,
,
) and ( ' ) until
"
&
&
(
()
7
7
:
0
"!
:
6
,
&
&
!
7
6
# % )
:
(
( )
$
where the regressor matrix
0
!
# )
E
;
the
regularization
matrix
# )
:
= =
;
.
) Let
:
argmin
(
(
:
Find
:
,
, let
9
)
#&98
&
,
)
!
,
:
,
(
.
converges.
import points.
3.2 Revised algorithm
6
The above algorithm is computationally feasible, but in step ( ) we need to useE the
Newton-Raphson method to find iteratively. When the number of import points becomes large, the Newton-Raphson computation can be expensive. To reduce this computation, we use a further approximation.
Instead of iteratively computing
until it converges, we can just do a one-step iteration,
and use it as an approximation to the converged one. To get a good approximation, we
E
take
advantage
of the fitted result from the current ?optimal? , i.e., the sub-model when
= =:
, and use it as the initial value. This one-step update is similar to the score test in
E
generalized linear models (GLM); but the latter does not have a penalty
term. The updating
C
formula allows the weighted regression (5) to be computed in B
time.
Hence, we have the revised step (
6
) for the basic algorithm:
6
(
"
!
&
) For each
, correspondingly augment
with a column, and
with a
column and a row. Use the updating formula to find in (5). Compute (6).
3.3 Stopping rule for adding point to
In step ( '& ) of the basic algorithm, we need to decide whether
has converged. A
8
be the sequence
natural stopping rule is to look at the regularized NLL. Let
,
of regularized NLL?s obtained in step ( & ). At each step , we compare with
+
:
(
+
where is a pre-chosen small integer, for example
. If the ratio
is less
:
% %%'(
than some pre-chosen small number , for example,
, we stop adding new import
points to .
3.4 Choosing the regularization paramter
So far, we have assumed that
the regularization parameter is fixed. In practice, we also
need to choose an ?optimal? . We can randomly split all the data into a training set and a
tuning
set, and use the misclassification error on the tuning set as a criterion for choosing
. To reduce the computation,
we take advantage of the fact that the regularized NLL
converges
faster for a larger . Thus, instead of running the entire revised algorithm for
procedure, which combines both adding import points to
each , we propose the following
and choosing the optimal :
(
(
6
(
(
(
&
) Start with a large regularization parameter .
) Let
:
,
:
6
#&
)
,
:
(
.
) Run steps (
), ( ) and
( & ) of the
revised
algorithm, until the stopping cri:
#
) . Along the way, also compute the
terion is satisfied at
misclassfication error on the tuning set.
) Decrease
&
to a smaller value.
( ) Repeat steps (
) and (
&
), starting with
:A#&
&
)
.
We choose the optimal as the one that corresponds to the minimum misclassification error
on the tuning set.
4 Simulation
In this section, we use a simulation to illustrate the IVM method. The data in each class
are generated from a mixture of Gaussians (Hastie et al. (2001)). The simulation results
are shown in Figure 2.
4.1 Remarks
The support points of the SVM are those which are
close to
the classification boundary or
9
( 3
misclassified and usually have large weights [2
]. The import points of the
2
IVM are those that decrease the regularized NLL the most, and can be either close to or
far from the classification boundary. This 9difference
is natural, because the SVM is only
3
(84687
concerned with the classification
, while the IVM also focuses on the
*+ , .10 2
unknown probability 2
. Though points away from the classification boundary do not
contribute to determining the position ofthe
classification boundary, they may contribute
to estimating the unknown probability 2
. Figure 3 shows
a comparison of the SVM and
C$D&
&E8
E
the IVM. The total computational
cost ofE the SVM is B
, while the computational
cost
C
of the IVM method is B
, where is the number of import points. Since does not
240
Misclassification rate for different lambda?s Regularized NLL for the optimal lambda
?
?
?
0.34
250
Regularized NLL for different lambda?s
???????
?
?
0
220
??? ?? ?????????????????????????????????????????
???????????????????????
?
??
0.26
??
?? ?
??
??
50
100
150
200
No. of import points added
0
?
?
?
?
?
?
???
??
???
??????
??????????????????????????????????????????????????????????????????????????????
50
100
150
200
No. of import points added
!" #
Figure 2: Radial kernel is used.
,
,
the optimal .
rate
is found to correspond to
stopping criterion is satisfied when
?
180
0.24
??
?
??
????
?
??? ??
?
?? ??
????????
0.22
100
150
?
????
???
??
????
??
??
??
?????????????
??????????????????????????????????????????????????
200
0.28
??
?
?
?
160
0.32
?
200
????
0.30
?
0
50
100
150
200
No. of import points added
. The left and middle panels illustrate how to choose
decreases from
to
. The minimum misclassification
. The right panel is for the optimal
. The
.
C
tend to increase as increases, the computational cost of the IVM can be smaller than that
of the SVM, especially for large training data sets.
5 Multi-class case
$
&%(' &%*) $
$
In this section, we briefly
describe a generalization of the IVM to multi-class classification.
(
Suppose there are
classes. We can write the response as an -vector , with each
:
component
being
either
0 or 1, indicating which class the observation is in. Therefore
(
:
%
:
:
%'
indicates
the response is in the th class,
and
A(
(
indicates the response is in the
th 4 class.
Using the
th class
as the basis,
the
:
%
:
4
:
( )
2
2
( )
2
2
,
,
.
multi-logit can be written as
Hence the Bayes classification rule is given by:
$
.-
:
%
argmax
-$
0 1032 3240
(7)
where
:
:
3
0
5
-
3
(
()
,
" /
1:
:
-
" /
8
:
+
65 $ 87977 6:
-
/- .-
We use + to index the observations, to index the classes, i.e.
Then the regularized negative log-likelihood is
(
7
-
6
-
C
%
,
:
%
(
$
.
, and
:
Using the representer theorem (Kimeldorf et al. (1971)), the th element of
which minimizes has the form
(8)
+%,) $
,
,
SVM - with 107 support points
IVM - with 21 import points
. . . . . . . . . . . . . . +++
++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++
.. ... ... ... ... ... ... ... ... ... ... ... ... ... ++++
+++++++++++++++++++++++++++++++++++++++++++++++++++
++++
+++++++++++++++++++++++++++++++++++++++++++++++++++
++++
. . . . . . . . . . . . .. +++++
+++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++
+++++
++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++
. ... ... ... ... ... ... ... ... ... ... ... ... .. ++++++
+++++++++++++++++++++++++++++++++++++++++++++++++
++++++
.
+++++++++++++++++++++++++++++++++++++++++++++++++
+++++++
+++++++++++++++++++++++++++++++++++++++++++++++++
.. ... ... ... ... ... ... ... ... ... ... ... ... .. +++++++
++++++++++++++++++++++++++++++++++++++++++++++++
+++++++
++++++++++++++++++++++++++++++++++++++++++++++++
++++++++
++++++++++++++++++++++++++++++++++++++++++++++++
.. ... ... ... ... ... ... ... ... ... ... ... ... ... ++++++++
+++++++++++++++++++++++++++++++++++++++++++++++
++++++++
+++++++++++++++++++++++++++++++++++++++++++++++
. . . . . . . . . . . . . ++++++++
+++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++
+++++++++
++++++++++++++++++++++++++++++++++++++++++++++
. . . . . . . . . . . . . . +++++++++
++++++++++++++++++++++++++++++++++++++++++++++
+++++++++
++++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
... ... ... ... ... ... ... ... ... ... ... ... ... ... ++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
. . . . . . . . . . . . . . ++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
+++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
+++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++++ . . . . . . . . . . . . +++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
.....
. . . . . . . . . . . +++++++++++
.
+++++++++++++++++++++++++++++
+++++++++++++++++++++++++++. .. .. .. .. .. .. ++++++++++++++
+++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
++++++++++++++++++++++++++
. . . . . . . . .. +++++++++++++
. ++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
+++++++++++++++++++++++++
+++++++++ ..
+++++++++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. ++++++++++++
. ++++++++++++ .. .. .. .. .. .. .. .. .. .. .. +++++++++
++++++++++++++++++++++++
+++++++++ ..
+++++++++++++++++++++++
. ... ... ... ... ... ... ... ... ... ... ... ... ... ++++++++++++
. .. .. .. .. .. .. .. .. .. .. +++++++++
. +++++++ .. ..
++++++++++++++++++++++
++++++++++++
.
.
++++++++++++++++++++++
.. +++++++++++ .. .. .. .. .. .. .. .. .. .. .. ... +++++++
+++++++++++++++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++++
+++++++. .. ..
++++++++++++++++++++
+++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. ++++++
.. +++++. .. .. ..
+++++++++++++++++++
+++++++++++++++++++
+++++++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++++++++. . . . . . . . . . . . . . . . . . . . . ++++++++++
++++++++++ . . . . . . . . . . . .. .. ++++
++++
. ++ . .. .. .. ..
+++++++++++++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++++++
++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++. .. .. .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ... ...
+++++++++++++++
+++++++++++++++
++++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++++
.. +++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ...
+++++++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++
.
.
++++++++++++
+++
++++++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++++++++
++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++++++++
............................................................
+++++++++
+++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++++++
++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++++
..............................................................
+++++++
++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++++
.
.
.
.
+++++ . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++
.................................................................
++++
++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++
++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++
..................................................................
+++
+++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++
.
+++ . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++
..................................................................
+++
+++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++
+++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++
.. .. .. .. .. .. Error:
. . . . . . 0.160
.....................................................
Training
++++
++++
. . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++
++++ . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++
.. .. .. .. .. .. .. .. .. .. 0.218
.....................................................
Test. Error:
+++++
++++++
. .. .. .. .. .. .. .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
++++++
+++++++
. . . . . . . . 0.210
.....................................................
+++++++
Bayes. Error:
.. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
.
++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++ . . . . . . . . . . . . . . . ++++++++++
++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
+++++++++++
++++++++++++++++++++++++++++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. +++++++++++
.
++++++++++++++++++++++++++++++++++++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
. . . . . . . . . . . . . ++++++++++ ...
+++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
+++++++++++++++++++++++++++++++++++++++++++++
. . . . . . . . . . . . . ++++++++++ ...
+++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++
.. +++++++++. ..
++++++++++++++++++++++++++++++++++++++++++++++
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++++++++++++++++++++++++++++++++++++
++++++++ . .
+++++++++++++++++++++++++++++
. . .. ++++++++++++++++
. . +++++++++++++++. ... ... ... ... ... ... ... ... ... ... ... ... ++++++++
+++++++++++++++++++++++++++
++++++++ .. ..
++++++++++++++++++++++++++
. .. .. .. .. .. .. .. ++++++++++++++
. +++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. ++++++++
+++++++++++++++++++++++++
++++++++ .. ..
+++++++++++++++++++++++++
.. ... ... ... ... ... ... ... ... ... +++++++++++++
.. ++++++++++++ . .. .. .. .. .. .. .. .. .. .. .. ++++++++
.. ..
++++++++++++++++++++++++
++++++++++++++++++++++++
+++++++++++++ . . . . . . . . . . . ++++++++
++++++++
+++++++++++++++++++++++
+++++++ .. .. ..
.. ... ... ... ... ... ... ... ... ... ... ... ... +++++++++++++
.. ++++++++++++ .. .. .. .. .. .. .. .. .. .. .. +++++++
++++++++++++++++++++++
...
++++++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++++
. .. .. .. .. .. .. .. .. .. .. ... ++++++
+++++++++++++++++++++
++++++++++++
++++++ .. .. ..
.
.
+++++++++++++++++++++
++++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. ++++++
.. .. ..
++++++++++++++++++++
++++++
.. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... +++++++++++
+++++++++++++++++++
+++++++++++
++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
+++++++++++++++++++
+++++++++++
++++ . . .. ..
++++++++++++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++ ... ... ... ... ... ... ... ... ... ... ... ... ... ++++
.. .. .. ..
+++++++++++++++++
++++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++++
++++++++++. . . . . . . . . . . . . . ++++
.. +++
++++++++++++++++
++ . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .. ++++++++
+++++++++++++++
.. ++++++
.. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
++++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++++++++
.
.
.
.
++++++++++++++
++
+++++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++++++++++
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
++++++++++++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ..
++++++++++++
..........................................................
+++++++++++
+++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
+++++++++++
++++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..
++++++++++
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .+
+++++++++
+++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++
+++++++++
++
.
.
.
++++++++
.. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ++
++++++++
++
++++++++
++++++++ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++
++
++++++++
++++++++. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++
++
+++++++
++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++
+++++++
+++++++
++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ++
++++++++
++++++++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ++
++++++++
+
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... +
++++++++
++++++++
. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. +
++++++++
+++++++++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +
+
+++++++++
.. .. .. .. .. .. 0.150
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .+
Training.Error:
+++++++++
+
++++++++++
+
. .. .. .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... +
++++++++++
+++++++++++
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. +
++++++++++++
+
Test Error: .. ... ... ... 0.219
++++++++++++
+
+++++++++++++
+
. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... +
++++++++++++++
+++++++++++++++
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .+
++++++++++++++++
+
Bayes Error: 0.210
?? ?
? ?
o ?o
?
? ?
???? ? ? ?
?? ? ??
o
?
? ?
o
o
o
?
? ?o ??? ? ? ?? ??o
o
?
?? o o ?o? ?? ? ? ? o ? oo o
?o?????? o ?? ? ???? o oooo
?
o ooo o o ?? ?
? oo
?
? o?o ?
?
? ?o oooo o ????? ?? o o ?
o
o o
oo? o ???
o
? oo
oooo o ooo oo? ?
o
?
? o?oo? oo
o
?
?
??
o
?? ?
? ?
o ?o
?
? ?
???? ? ? ?
?? ? ??
o
?
? ?
o
o
o
?
? ?o ??? ? ? ?? ??o
o
?
?? o o ?o? ?? ? ? ? o ? oo o
?o?????? o ?? ? ???? o oooo
?
o ooo o o ?? ?
? oo
?
? o?o ?
?
? ?o oooo o ????? ?? o o ?
o
o o
oo? o ???
o
? oo
oooo o ooo oo? ?
o
?
? o?oo? oo
o
o
o
o
o o o o
o
oo oo o
o
ooo
o o o oo o
o
oo
o
o
o
o
o
o
o
o
o o o o
o
oo oo o
o
ooo
o o o oo o
o
oo
o
o
o
o
o
?
?
??
o
Figure 3: The solid lines are the classification boundaries; the dotted lines are the Bayes rule
(
boundaries. For the SVM, the dashed lines are the edges of the margin. For the IVM, the dashed lines
are the
and lines.
Hence, (7) becomes
(9)
where
case; and
$
:
:
"
3
0
+
-
"
:
+
, '"
3
is the + th row of
()
and
".
(
'&
(
"
0
7
6
&
are defined in the same way as in the binary
E8
The multi-class
IVM procedure is similar to the binary case, and the computational cost is
C
. Figure 4 is a simulation of the multi-class IVM. The data in each class are
B
generated from a mixture of Gaussians (Hastie et al. (2001)).
Multi-class IVM - with 32 import points
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
++
++
++
++
++
++
++
++
++
++
++
++
++
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxx
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xx
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++++++xxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+++++++++xxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
++
++
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
++
+
+
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+..........................................
++......
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxx...............
++
++
++
++
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+...........................................
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
.......
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
..............+++++++++++++++++++++++.............................................
............................................
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
...................
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
.......................++++++++++++++++++...............................................
..............................................
xxxxxxxxxxxxxxxxxxxxxxxxxxx
..........................
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxxxxxxxxx
.............................+++++++++++++++++................................................
...............................................
xxxxxxxxxxxxxxxxxxxxx
................................
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxxxxxxxxxxxxxxx
....................................
................................................
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.................................................
xxxxxxxxxxxx
...........................................
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
xxxx
..............................................
.................................................
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+..................................................
..............................................
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
..............................................
..................................................
++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
...............................................
....................................................
+++
+
.................................................
++
++
++
++
++
++
++
++
++
++
++......................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
Training Error: 0.237
..................................................................................................................
..................................................................................................................
..................................................................................................................
Test Error:
0.259
..................................................................................................................
..................................................................................................................
..................................................................................................................
..................................................................................................................
o o
o o ooo oo
oo o
o
o
oo
o oo
o ooooooo o o o o o
o
o oooo o
o
ooo o ooo o o o
o oo
o
oo
o oo
o
oo
o
o o ooo o o
o o o oo oo
o
oo o o
oo o
o o
o o
oo
o o
o o
oo
o
Bayes Error:
0.251
Figure 4: Radial kernel is used.
( ( (
! #
,
,
,
.
6 Conclusion
We have discussed the import vector machine (IVM) method in both binary and multi-class
E8
classification. We showed that it 9not
only performs as well as the SVM, but alsoC provides
E8
2
E the IVM is B
an estimate of the probability
.
The
computational
cost
of
for
C
the binary case and B
for the multi-class case, where is the number of import
points.
$
Acknowledgments
We thank Dylan Small, John Storey, Rob Tibshirani, and Jingming Yan for their helpful
comments. Ji Zhu is partially supported by the Stanford Graduate Fellowship. Trevor
Hastie is partially supported by grant DMS-9803645 from the National Science Foundation, and grant ROI-CA-72028-01 from the National Institutes of Health. Thanks to Grace
Wahba and Chris Williams for pointing out several interesting and important references.
We also want to thank the anonymous NIPS referees who helped improve this paper.
References
[1] Burges, C.J.C. (1998) A tutorial on support vector machines for pattern recognition. In Data
Mining and Knowledge Discovery. Kluwer Academic Publishers, Boston. (Volume 2)
[2] Evgeniou, T., Pontil, M., & Poggio., T. (1999) Regularization networks and support vector machines. In A.J. Smola, P. Bartlett, B. Sch?olkopf, and C. Schuurmans, editors, Advances in Large
Margin Classifiers. MIT Press.
[3] Green, P. & Yandell, B. (1985) Semi-parametric generalized linear models. Proceedings 2nd
International GLIM Conference, Lancaster, Lecture notes in Statistics No. 32 44-55 Springer-Verlag,
New York.
[4] Hastie, T. & Tibshirani, R. (1990) Generalized Additive Models, Chapman and Hall.
[5] Hastie, T., Tibshirani, R., & Friedman, J.(2001) The elements of statistical learning. In print.
[6] Lin, X., Wahba, G., Xiang, D., Gao, F., Klein, R. & Klein B. (1998), Smoothing spline ANOVA
models for large data sets with Bernoulli observations and the randomized GACV. Technical Report
998, Department of Statistics, University of Wisconsin, Madison WI.
[7] Kimeldorf, G. & Wahba, G. (1971) Some results on Tchebycheffian spline functions. J. Math.
Anal. Applic. 33, 82-95.
[8] Smola, A. & Sch?olkopf, B. (2000) Sparse Greedy Matrix Approximation for Machine Learning.
In Proceedings of the Seventeenth International Conference on Machine Learning. Morgan Kaufmann Publishers.
[9] Wahba, G. (1998) Support Vector Machine, Reproducing Kernel Hilbert Spaces and the Randomized GACV. Technical Report 984rr, Department of Statistics, University of Wisconsin, Madison
WI.
[10] Wahba, G., Gu, C., Wang, Y., & Chappell, R. (1995) Soft Classification, a.k.a. Risk Estimation, via Penalized Log Likelihood and Smoothing Spline Analysis of Variance. In D.H. Wolpert,
editor, The Mathematics of Generalization. Santa Fe Institute Studies in the Sciences of Complexity.
Addison-Wesley Publisher.
[11] Williams, C. & Seeger, M (2001) Using the Nystrom Method to Speed Up Kernel Machines.
In T. K. Leen, T. G. Diettrich, and V. Tresp, editors, Advances in Neural Information Processing
Systems 13. MIT Press.
| 2059 |@word version:1 briefly:2 middle:1 seems:1 norm:1 logit:2 nd:1 simulation:5 decomposition:1 usee:1 solid:1 initial:1 score:2 selecting:3 rkhs:2 terion:1 current:1 od:1 import:21 written:1 john:1 additive:1 shape:1 update:1 greedy:3 prohibitive:1 provides:3 math:1 contribute:2 along:2 qualitative:1 fitting:1 combine:1 multi:15 becomes:3 estimating:1 underlying:1 xx:1 panel:2 kimeldorf:3 argmin:1 minimizes:1 every:1 wrong:1 classifier:1 grant:2 positive:1 consequence:1 diettrich:1 studied:1 graduate:1 seventeenth:1 acknowledgment:1 practice:1 definite:1 ofe:1 procedure:3 pontil:1 empirical:1 yan:1 word:1 pre:2 radial:2 get:1 close:2 risk:1 influence:1 impossible:1 yandell:1 equivalent:2 williams:3 starting:1 independently:1 simplicity:1 rule:7 d5:1 suppose:1 us:4 element:2 referee:1 expensive:1 recognition:1 updating:2 wang:1 decrease:3 e8:5 mentioned:1 complexity:1 compromise:1 basis:3 gacv:2 gu:1 describe:1 choosing:3 lancaster:1 stanford:7 solve:1 larger:1 statistic:5 itself:1 seemingly:1 nll:9 advantage:5 sequence:1 rr:1 propose:5 product:1 frobenius:1 olkopf:2 cluster:1 produce:1 converges:3 illustrate:2 oo:36 stat:2 involves:1 assign:1 generalization:2 anonymous:1 extension:2 hall:1 roi:1 pointing:1 estimation:2 weighted:2 mit:2 focus:1 notational:1 bernoulli:1 likelihood:3 indicates:2 seeger:1 tradition:1 helpful:1 cri:1 stopping:4 typically:1 entire:1 going:2 transformed:3 misclassified:1 issue:2 classification:24 augment:1 smoothing:2 equal:1 evgeniou:2 chapman:1 look:1 representer:2 report:2 spline:3 randomly:3 national:2 argmax:1 replacement:1 friedman:1 interest:1 mining:1 weakness:1 mixture:2 edge:1 poggio:1 divide:1 plotted:1 fitted:3 column:4 soft:1 cost:10 subset:6 thanks:1 international:2 randomized:2 regressor:2 squared:1 satisfied:2 choose:3 lambda:3 derivative:1 sizeable:1 tion:1 helped:1 start:1 bayes:5 regulariza:1 minimize:2 square:1 kaufmann:1 who:1 variance:1 correspond:1 generalize:1 none:1 researcher:1 converged:2 trevor:2 nystrom:2 naturally:3 dm:1 stop:1 knowledge:1 oooo:7 hilbert:2 back:1 glim:1 wesley:1 response:3 ooo:10 leen:1 though:1 furthermore:2 just:1 smola:3 until:3 xxxx:2 logistic:5 yf:1 hence:4 regularization:5 iteratively:3 attractive:1 noted:1 criterion:3 generalized:6 performs:4 classfication:2 ji:2 overview:1 volume:1 discussed:1 approximates:2 kluwer:1 tuning:4 mathematics:1 similarly:1 longer:1 showed:1 verlag:1 binary:10 morgan:1 minimum:2 dashed:2 semi:1 full:3 technical:2 faster:2 academic:1 offer:1 raphson:2 lin:2 regression:7 basic:3 iteration:1 kernel:16 achieved:1 whereas:1 want:3 fellowship:1 publisher:3 sch:2 rest:1 comment:1 tend:2 call:2 integer:1 near:1 split:1 identically:1 concerned:1 fit:1 hastie:9 wahba:7 inner:1 reduce:2 whether:1 bartlett:1 penalty:1 york:1 remark:1 ignored:1 se:1 santa:1 reduced:1 tutorial:1 notice:1 dotted:1 tibshirani:3 klein:2 write:1 tchebycheffian:1 chappell:1 anova:1 fraction:4 run:1 decide:1 speed:1 span:2 department:4 smaller:3 wi:2 rob:1 making:1 happens:1 glm:1 computationally:1 equation:1 paramter:1 addison:1 gaussians:2 away:1 appropriate:1 save:1 eigen:1 original:2 binomial:3 assumes:1 running:1 hinge:1 newton:2 madison:2 giving:1 especially:4 added:3 print:1 strategy:1 parametric:1 traditional:1 grace:1 thank:2 chris:1 besides:1 index:4 relationship:1 ratio:1 minimizing:1 fe:1 negative:2 anal:1 unknown:3 observation:3 revised:4 finite:2 immediate:1 reproducing:3 applic:1 nip:1 address:1 in3:1 usually:3 pattern:1 built:1 green:2 misclassification:4 natural:3 regularized:8 zhu:2 meth:1 improve:1 health:1 tresp:1 review:1 discovery:1 determining:2 xiang:1 wisconsin:2 loss:5 expect:1 lecture:1 interesting:1 foundation:1 editor:3 klr:10 row:2 penalized:1 repeat:2 supported:2 truncation:1 side:2 burges:1 institute:2 correspondingly:1 sparse:1 distributed:1 boundary:11 dimension:2 forward:1 far:2 approximate:2 assumed:2 search:1 storey:1 ca:3 expanding:1 schuurmans:1 investigated:1 constructing:1 ooooooo:1 representative:1 sub:5 position:2 wish:1 dylan:1 theorem:2 formula:2 svm:28 adding:3 margin:2 boston:1 wolpert:1 gao:1 partially:2 springer:1 corresponds:1 ivm:29 conditional:1 replace:1 feasible:1 infinite:1 called:4 total:1 indicating:1 select:3 support:14 people:1 newtonraphson:2 latter:1 |
1,162 | 206 | 590
Atiya and Abu-Mostafa
A Method for the Associative Storage
of Analog Vectors
Amir Atiya (*) and Yaser Abu-Mostafa (**)
(*) Department of Electrical Engineering
(**) Departments of Electrical Engineering and Computer Science
California Institute Technology
Pasadena, Ca 91125
ABSTRACT
A method for storing analog vectors in Hopfield's continuous feedback model is proposed. By analog vectors we mean vectors whose
components are real-valued. The vectors to be stored are set as
equilibria of the network. The network model consists of one layer
of visible neurons and one layer of hidden neurons. We propose
a learning algorithm, which results in adjusting the positions of
the equilibria, as well as guaranteeing their stability. Simulation
results confirm the effectiveness of the method .
1 INTRODUCTION
The associative storage of binary vectors using discrete feedback neural nets has
been demonstrated by Hopfield (1982). This has attracted a lot of attention, and
a number of alternative techniques using also the discrete feedback model have
appeared. However, the problem of the distributed associative storage of analog
vectors has received little attention in literature. By analog vectors we mean vectors whose components are real-valued. This problem is important because in a
variety of applications of associative memories like pattern recognition and vector
quantization the patterns are originally in analog form and therefore one can save
having the costly quantization step and therefore also save increasing the dimension
of the vectors. In dealing with analog vectors, we consider feedback networks of the
continuous-time graded-output variety, e.g. Hopfield's model (1984):
du
dt = -u + Wf(u) + a,
x
= f(u),
(1)
where u = (Ul, ... , UN)T is the vector of neuron potentials, x = (x!, ... , XN)T is the
vector of firing rates, W is the weight matrix, a is the threshold vector, and f(u)
means the vector (f( uI), ... , f( UN)) T, where f is a sigmoid-shaped function.
The vectors to be stored are set as equilibria of the network. Given a noisy version
of any of the stored vectors as the initial state of the network, the network state has
A Method for the Associative Storage of Analog Vectors
to reach eventually the equilibrium state corresponding to the correct vector. An
important requirement is that these equilibria be asymtotically stable, otherwise
the attraction to the equilibria will not be guaranteed. Indeed, without enforcing
this requirement, our numerical simulations show mostly unstable equilibria.
2 THE MODEL
It can be shown that there are strong limitations on the set of memory vectors
which can be stored using Hopfield's continuous model (Atiya and Abu-Mostafa
1990). To relieve these limitations, we use an architecture consisting of both visible
and hidden units. The outputs of the visible units correspond to the components of
the stored vector. Our proposed architecture will be close to the continuous version
of the BAM (Kosko 1988). The model consists of one layer of visible units and
another layer of hidden units (see Figure 1). The output of each layer is fed as an
input to the other layer. No connections exist within each of the layers. Let y and
x be the output vectors of the hidden layer and the visible layer respectively. Then,
in our model,
du
dt
= -u + Wf(z) + a
= e,
y
= f(u)
(2a)
dz
= -z + Vf(u) + b = h,
x = f(z)
= [Wij] and V = [Vij] are the weight matrices, a
dt
(2b)
where W
and b are the threshold
vectors, and f is a sigmoid function (monotonically increasing) in the range from
-1 to 1, for example
f(u) = tanh(u).
x
x
hld~n
vlSlbl.
l~y.,.
l~y.,.
Figure 1: The model
591
592
Atiya and Abu?Mostafa
As we mentioned before, for a basin of attraction to exist around a given memory vector, the corresponding equilibrium has to be asymtotically stable. For the
proposed architecture a condition for stability is given by the following theorem.
Theorem: An equilibrium point (u*, z*) satisfying
J'l/2( un 2:IWij If'l/2(zj) < 1
(3a)
j
J'l/\Z;) 2:I Vij l!,l/2(uj) < 1
(3b)
j
for all i is asymptotically stable.
Proof: We linearize (2a), (2b) around the equilibrium. We get
dq
-=Jq,
du
where
if i
if i
= 1, ... , Nl
= Nl + 1, ..., Nl + N 2,
Nl and N2 are the number of units in the hidden layer and the visible layer respectively, and J is the Jacobian matrix, given by
~
aUl
J=
~
fu
~
ae~l
ae~l
ae~l
aUl
aUNl
ah
aZ 1
aeN1
aZN'J
~hl
~
ahNa
ahNa
aZN'J
aUNl
ghl
Ul
8U";1
ahNa
ahNa
aUNl
aUl
aZ 1
Zl
lhl
aZN'J
aZN'J
the partial derivatives evaluated at the equilibrium point. Let Al and A2 be respectively the Nl x Nl and N2 x N2 diagonal matrices with the ith diagonal element
being respectively f'(un and f'(z;). Furthermore, let
The Jacobian is evaluated as
where IL means the L x L identity matrix. Let
(
A-
_A- l
1
V
A Method for the Associative Storage of Analog Vectors
Then,
J=AA.
Eigenvalues of AA are identical to the eigenvalues of A 1/2 AA 1/2 because if ). is an
eigenvalue of AA corresponding to eigenvector v, then
AAv = ).v,
and hence
Now, we have
Al/2AAI/2 _ (-INl
A~/2V A~/2
A~/2WA~/2)
-IN2
.
By Gershgorin's Theorem (Franklin 1968), an eigenvalue of J has to satisfy at least
one of the inequalities:
I). + 11 ::;
f'1/2( un 2:IWii 1f'1/2(zi)
i = 1, ... ,N1
i
I). + 11::;
f'1/2(zn2:lvjil!,1/2(uj)
i = 1, ... ,N2'
i
It follows that under conditions (3a), (3b) that the eigenvalues of J will have negative real parts, and hence the equilibrium of the original system (2a), (2b) will be
asymptotically stable.
Thus, if the hidden unit values are driven far enough into the saturation region
(i.e . with values close to 1 or -1), then the corresponding equilibrium will be stable
because then, 1'(
will be very small, causing Inequalities (3) to be satisfied.
Although there is nothing to rule out the existence of spurious equilibria and limit
cycles, if they occur then they would be far away from the memory vectors because
each memory vector has a basin of attraction around it. In our simulations we have
never encountered limit cycles.
un
3 TRAINING ALGORITHM
=
Let xm, m 1, ... , M be the vectors to be stored. Each xm should correspond to the
visible layer component of one of the asymptotically stable equilibria. We design
the network such that the hidden layer component of the equilibrium corresponding
to xm is far into the saturation region. The target hidden layer component ym can
be taken as a vector of l's and -1 's, chosen arbitrarily for example by generating
the components randomly. Then, the weights have to satisfy
yj = !(2:Wi/X, + aj),
/
xi = ![2:Vjj!(2:Wj/x/
j
/
+ aj) + b;].
593
594
Atiya and Abu-Mostafa
Training is performed in two steps. In the first step we train the weights of the
hidden layer. We use steepest descent on the error function
El
= Lllyj -
f(LWjlX; + aj )11 2 .
I
m,j
In the second step we train the weights of the visible layer, using steepest descent
on the error function
E2
=L
II xi - ![LVij!(LWj/x; + aj) + bd 112.
m,i
j
I
We remark that in the first step convergence might be slow since the targets are lor
-1. A way to have fast convergence is to stop if the outputs are within some constant
(say 0.2) from the targets. Then we multiply the weights and the thresholds of the
hidden layer by a big positive constant, so as to force the outputs of the hidden
layer to be close to 1 or -1.
4 IMPLEMENTATION
We consider a network with 10 visible and 10 hidden units. The memory vectors are
randomly generated (the components are from -0.8 to 0.8 rather than the full range
to have a faster convergence). Five memory vectors are considered. After learning,
the memory is tested by giving memory vectors plus noise (100 vectors for a given
variance). Figure 2 shows the percentage correct recall in terms of the signal to
noise ratio. Although we found that we could store up to 10 vectors, working close
to the full capacity is not recommended, as the recall accuracy dc>teriorates.
/. correct
100
-r--.......--~~---------::::_----->
80
60
40
20
O.f...o.-----------............-----I
-6
6
10
-2
2
snr (db)
Figure 2: Recall accuracy versus signal to noise ratio
A Method for the Associative Storage of Analog Vectors
Acknowledgement
This work is supported by the Air Force Office of Scientific Research under grant
AFO SR-88-0231 .
References
J. Hopfield (1982), "Neural networks and physical systems with emergent collective
computational abilities", Proc. Nat. Acad. Sci. USA, vol. 79, pp. 2554-2558.
J. Hopfield (1984), "Neurons with graded response have collective computational
properties like those of two state neurons", Proc. Nat. Acad. Sci. USA, vol. 81, p.
3088-3092.
A. Atiya and Y. Abu-Mostafa (1990), "An analog feedback associative memory",
to be submitted.
B. Kosko (1988), "Bidirectional associative memories", IEEE Trans. Syst. Man
Cybern., vol. SMC-18, no. 1, pp. 49-60.
J. Franklin (1968) Matrix Theory, Prentice-Hall, Englewood Cliffs, New Jersey.
595
PART VII:
EMPIRICAL ANALYSES
| 206 |@word uj:2 graded:2 version:2 hence:2 correct:3 simulation:3 costly:1 diagonal:2 sci:2 capacity:1 initial:1 unstable:1 franklin:2 enforcing:1 around:3 considered:1 hall:1 ratio:2 attracted:1 bd:1 equilibrium:16 aul:3 sigmoid:2 numerical:1 visible:9 mostafa:6 physical:1 mostly:1 a2:1 negative:1 analog:11 proc:2 design:1 implementation:1 collective:2 tanh:1 amir:1 neuron:5 kosko:2 ith:1 steepest:2 descent:2 dc:1 rather:1 stable:6 lor:1 five:1 afo:1 office:1 consists:2 connection:1 driven:1 store:1 california:1 inequality:2 indeed:1 binary:1 wf:2 arbitrarily:1 trans:1 el:1 lwj:1 pattern:2 bam:1 xm:3 appeared:1 little:1 spurious:1 pasadena:1 hidden:12 increasing:2 jq:1 wij:1 monotonically:1 signal:2 ii:1 full:2 recommended:1 saturation:2 memory:11 force:2 faster:1 eigenvector:1 never:1 having:1 shaped:1 technology:1 ahna:4 identical:1 ae:3 zl:1 unit:7 grant:1 literature:1 randomly:2 acknowledgement:1 before:1 positive:1 engineering:2 limit:2 acad:2 consisting:1 limitation:2 n1:1 cliff:1 sr:1 versus:1 firing:1 might:1 plus:1 englewood:1 multiply:1 db:1 usa:2 basin:2 effectiveness:1 dq:1 vij:2 nl:6 storing:1 smc:1 range:2 enough:1 asymtotically:2 variety:2 supported:1 zi:1 yj:1 fu:1 architecture:3 partial:1 institute:1 empirical:1 distributed:1 feedback:5 dimension:1 xn:1 ul:2 yaser:1 get:1 close:4 aai:1 remark:1 storage:6 prentice:1 cybern:1 far:3 snr:1 demonstrated:1 dz:1 confirm:1 dealing:1 attention:2 atiya:6 stored:6 xi:2 exist:2 percentage:1 zj:1 rule:1 attraction:3 continuous:4 un:6 stability:2 discrete:2 ca:1 ym:1 vol:3 abu:6 target:3 du:3 threshold:3 satisfied:1 element:1 recognition:1 satisfying:1 big:1 derivative:1 ghl:1 asymptotically:3 noise:3 n2:4 syst:1 nothing:1 potential:1 electrical:2 relieve:1 region:2 wj:1 cycle:2 satisfy:2 slow:1 position:1 performed:1 lot:1 mentioned:1 vf:1 ui:1 layer:18 jacobian:2 guaranteed:1 theorem:3 encountered:1 il:1 air:1 accuracy:2 occur:1 variance:1 correspond:2 azn:4 vjj:1 hopfield:6 emergent:1 quantization:2 jersey:1 train:2 nat:2 fast:1 gershgorin:1 department:2 ah:1 submitted:1 zn2:1 vii:1 reach:1 whose:2 valued:2 wi:1 say:1 pp:2 otherwise:1 tested:1 ability:1 e2:1 proof:1 hl:1 inl:1 noisy:1 stop:1 aa:4 associative:9 adjusting:1 taken:1 eigenvalue:5 recall:3 net:1 eventually:1 propose:1 identity:1 lhl:1 fed:1 causing:1 man:1 bidirectional:1 originally:1 dt:3 response:1 away:1 evaluated:2 furthermore:1 az:2 save:2 alternative:1 convergence:3 working:1 requirement:2 existence:1 original:1 in2:1 generating:1 guaranteeing:1 linearize:1 aj:4 aav:1 scientific:1 received:1 giving:1 strong:1 |
1,163 | 2,060 | The 9 Factor: Relating Distributions on
Features to Distributions on Images
James M. Coughlan and A. L. Yuille
Smith-Kettlewell Eye Research Institute,
2318 Fillmore Street ,
San Francisco, CA 94115, USA.
Tel. (415) 345-2146/2144. Fax. (415) 345-8455.
Email: coughlan@ski.org.yuille@ski.org
Abstract
We describe the g-factor, which relates probability distributions
on image features to distributions on the images themselves. The
g-factor depends only on our choice of features and lattice quantization and is independent of the training image data. We illustrate
the importance of the g-factor by analyzing how the parameters of
Markov Random Field (i.e. Gibbs or log-linear) probability models
of images are learned from data by maximum likelihood estimation.
In particular, we study homogeneous MRF models which learn image distributions in terms of clique potentials corresponding to feature histogram statistics (d. Minimax Entropy Learning (MEL)
by Zhu, Wu and Mumford 1997 [11]) . We first use our analysis
of the g-factor to determine when the clique potentials decouple
for different features . Second, we show that clique potentials can
be computed analytically by approximating the g-factor. Third,
we demonstrate a connection between this approximation and the
Generalized Iterative Scaling algorithm (GIS), due to Darroch and
Ratcliff 1972 [2], for calculating potentials. This connection enables us to use GIS to improve our multinomial approximation,
using Bethe-Kikuchi[8] approximations to simplify the GIS procedure. We support our analysis by computer simulations.
1
Introduction
There has recently been a lot of interest in learning probability models for vision.
The most common approach is to learn histograms of filter responses or, equivalently, to learn probability distributions on features (see right panel of figure (1)).
See, for example, [6], [5], [4]. (In this paper the features we are considering will be
extracted from the image by filters - hence we use the terms "features" and "filters"
synonymously. )
An alternative approach, however , is to learn probability distributions on the images
themselves. The Minimax Entropy Learning (MEL) theory [11] uses the maximum
entropy principle to learn MRF distributions in terms of clique potentials determined by the feature statistics (i.e. histograms of filter responses). (We note that
the maximum entropy principle is equivalent to performing maximum likelihood estimation on an MRF whose form is determined by the choice of feature statistics.)
When applied to texture modeling it gives a way to unify the filter based approaches
(which are often very effective) with the MRF distribution approaches (which are
theoretically attractive).
)
\
Figure 1: Distributions on images vs. distributions on features. Left and center
panels show a natural image and its image gradient magnitude map , respectively.
Right panel shows the empirical histogram (i.e. a distribution on a feature) of
the image gradient across a dataset of natural images. This feature distribution
can be used to create a MRF distribution over images[10]. This paper introduces
the g-factor to examine connections between the distribution over images and the
distribution over features.
As we describe in this paper (see figure (1)), distributions on images and on features can be related by a g-factor (such factors arise in statistical physics, see [3]) .
Understanding the g-factor allows us to approximate it in a form that helps explain
why the clique potentials learned by MEL take the form that they do as functions
of the feature statistics. Moreover, the MEL clique potentials for different features
often seem to be decoupled and the g-factor can explain why, and when, this occurs.
(I.e. the two clique potentials corresponding to two features A and B are identical
whether we learn them jointly or independently).
The g-factor is determined only by the form of the features chosen and the spatial
lattice and quantization of the image gray-levels. It is completely independent of
the training image data. It should be stressed that the choice of image lattice,
gray-level quantization and histogram quantization can make a big difference to the
g-factor and hence to the probability distributions which are the output of MEL.
In Section (2), we briefly review Minimax Entropy Learning. Section (3) introduces
the g-factor and determines conditions for when clique potentials are decoupled.
In Section (4) we describe a simple approximation which enables us to learn the
clique potentials analytically, and in Section (5) we discuss connections between
this approximation and the Generalized Iterative Scaling (GIS) algorithm.
2
Minimax Entropy Learning
Suppose we have training image data which we assume has been generated by an
(unknown) probability distribution PT(X) where x represents an image. Minimax
Entropy Learning (MEL) [11] approximates PT(X) by selecting the distribution with
maximum entropy constrained by observed feature statistics i(X) = ;fobs. This gives
-
P(xIA) =
>:. ?( ?)
e Z [>:]
-
-
,where A is a parameter chosen such that Lx P(xIA)?>(X) = 'l/Jobs?
Or equivalently, so that <910;{[>:] = ;fobs.
i
We will treat the special case where the statistics are the histogram of a shiftinvariant filter {fi(X) : i = 1, ... , N} , where N is the total number of pixels in the image. So 'l/Ja = ?>a(x) = -tv L~l ba,' i(X) where a = 1, ... , Q indicates the (quantized)
Q
N
filter response values. The potentials become A??>(X) = -tv La=l Li=l A(a)ba,fi(X) =
-tv L~l A(fi(X)). Hence P(xl,X) becomes a MRF distribution with clique potentials
given by A(fi (x)). This determines a Markov random field with the clique structure
given by the filters {fd.
~
~
MEL also has a feature selection stage based on Minimum Entropy to determine
which features to use in the Maximum Entropy Principle. The features are evaluated by computing the entropy - Lx P(xl,X) log P(xl,X) for each choice of features
(with small entropies being preferred). A filter pursuit procedure was described to
determine which filters/features should be considered (our approximations work for
this also).
3
The g-Factor
This section defines the g-factor and starts investigating its properties in subsection (3.1). In particular, when, and why, do clique potentials decouple? More
precisely, when do the potentials for filters A and B learned simultaneously differ
from the potentials for the two filters when they are learned independently?
We address these issues by introducing the g-factor g(;f) and the associated distribution Po (;f):
(1)
x space
-----+ iii space
GG
g(ijiJ = number of images
with histogram iii
x
Figure 2: The g-factor g(;f) counts the number of images x that have statistics ;f.
Note that the g-factor depends only on the choice of filters and is independent of
the training image data.
Here L is the number of grayscale levels of each pixel, so that LN is the total number
of possible images. The g-factor is essentially a combinational factor which counts
the number of ways that one can obtain statistics ;f, see figure (2). Equivalently,
Po is the default distribution on ;f if the images are generated by white noise (i.e.
completely random images).
We can use the g-factor to compute the induced distribution P(~I'x) on the statistics
determined by MEL:
A
~~
P(1/1 I'\)
=
L
X
6;;:
~~
2(-)P(xl'\)
'j','j'
x
=
g( ~)eX.,j;
~
~, Z[,\]
Z[,\]
=
L
~
X,j;
g(1/1)e? .
(2)
,j;
Observe that both P(~I'x) and log Z[,X] are sufficient for computing the parameters
X. The ,X can be found by solving either of the following two (equivalent) equations:
~ ~ ~
~
8 10 zrXl
~
L:,j; P(1/1 I,\) 1/1 = 1/1obs, or
= 1/1obs, which shows that knowledge of the g-factor
A
;X
and eX. ,j; are all that is required to do MEL.
Observe from equation (2) that we have P(~I'x = 0) = Po(~) . In other words ,
setting ,X = 0 corresponds to a uniform distribution on the images x.
3.1
Decoupling Filters
We now derive an important property of the minimax entropy approach. As mentioned earlier, it often seems that the potentials for filters A and B decouple.
In other words, if one applies MEL to two filters A, B simultaneously bv letting
...
....A . . . B......
....A -B
...
"'""'A
. . .B
.
:..t
...
1/1 = (1/1 ,1/1 ), '\ = (,\ ,'\ ), and 1/1obs = (1/1obs ' 1/1obs)' then the solutIOns'\ A , '\ B to
the equations:
LP(xl,XA , ,XB)(iA(x) , iB(x)) = (~:bs'~!s)'
x
(3)
are the same (approximately) as the solutions to the equations L: x p(xl,XA )iA(x) =
~!s and L: x P(xl,XB)iB(x) = ~!s, see figure (3) for an example.
Figure 3: Evidence for decoupling of features. The left and right panels show the
clique potentials learned for the features I and I respectively. The solid lines
give the potentials when they are learned individually. The dashed lines show the
potentials when they are learned simultaneously. Figure courtesy of Prof. Xiuwen
Liu, Florida State University.
a ax
a ay
We now show how this decoupling property arises naturally if the g-factor for the
two filters factorizes. This factorization, of course, is a property only of the form
of the statistics and is completely independent of whether the statistics of the two
filters are dependent for the training data.
Property I: Suppose we have two sufficient statistics iA(x), iB (x) which are independent on the lattice in the sense that g(~A,~B ) = gA (~A )gB(~B) , then
logZ[,XA,,XB] = logZA[,XA] + logZB[,XB] and p(~A,~B ) = pA(~A)pB(~B ).
This implies that the parameters XA, XB can be solved from the independent
81ogZ A[XA] _ -A
8 1ogZ B [XB ] _
-B
A -A -A
-A
.
- 'ljJobs or L.,j;A P ('ljJ)'ljJ = 'ljJobs'
equatwns
8XA
- 'ljJobs'
8XB
L.,j;B pB(;fB );fB = ;f~s '
A
Moreover, the resulting distribution PUC) can be obtained by multiplying the distributions (l/Z A )e XA .,j;A(x) and (l/ZB) eXB.,j;B(x) together.
The point here is that the potential terms for the two statistics ;fA,;fB decouple if
the phase factor g(;fA,;fB) can be factorized. We conjecture that this is effectively
the case for many linear filters used in vision processing. For example, it is plausible
that the g- factor for features 0/
and 0/
factorizes - and figure (3) shows
that their clique potentials do decouple (approximately). Clearly, if factorization
between filters occurs then it gives great simplification to the system.
ox
4
oy
Approximating the g-factor for a Single Histogram
We now consider the case where the statistic is a single histogram. Our aim is to
understand why features whose histograms are of stereotypical shape give rise to
potentials of the form given by figure (3). Our results , of course, can be directly
extended to multiple histograms if the filters decouple, see subsection (3.1). We
first describe the approximation and then discuss its relevance for filter pursuit.
We rescale the
Xvariables by N
so that we have:
eNX.?(x)
eNX .,j;
P(X'I-\) = Z[X] , P('ljJ I-\) = g('ljJ) Z[X] ,
A
_
_
(4)
We now consider the approximation that the filter responses {Ii} are independent
of each other when the images are uniformly distributed. This is the multinomial
approximation. (We attempted a related approximation [1] which was less successful.) It implies that we can express the phase factor as being proportional to a
multinomial distribution:
(nt:)
<P
=
LN
N!
N1/Jl
N1/JQ
(N'ljJd!. .. (N'ljJQ)!o
... 0Q '
n (nt:) _
N!
N1/Jl
N1/JQ
(N'ljJd!. .. (N'ljJQ)!Ol "'OQ
(5)
where L.~= 1 'ljJa = 1 (by definition) and the {o a} are the means of the components
Na } with respect to the distribution Po (;f). As we will describe later , the {oa}
will be determined by the filters {fi}. See Coughlan and Yuille, in preparation, for
details of how to compute the {oa}.
9
TO
<p
-
This approximation enables us to calculate MEL analytically.
Theorem With the multinomial approximation the log partition function is:
Q
log Z[X] = N log L
+ N log{~= e " a +1og aa } ,
(6)
a=l
and the "potentials" P a} can be solved in t erms of the observed data {'ljJobs ,a} to
be:
\ -- Iog--,
'ljJobs,a a = 1, .. .,Q.
Aa
(7)
Oa
Figure 4: Top row: the multinomial approximation. Bottom row: full implementation of MEL (see text). (Left panels) the potentials, (center panels) synthesized
images, and (right panels) the difference between the observed histogram (dashed
line) and the histogram of the synthesized images (bold line). Filters were d/dx
and d/dy.
We note that there is an ambiguity Aa r-+ Aa + K where K is an arbitrary number
(recall that L~=l 'IjJ(a) = 1). We fix this ambiguity by setting X= 0 if a. = "Jobs.
Proof. Direct calculation.
Our simulation results show that this simple approximation gives the typical potential forms generated by Markov Chain Monte Carlo (MCMC) algorithms for
Minimax Entropy Learning. Compare the multinomial approximation results with
those obtained from a full implementation of MEL by the algorithm used in [11],
see figure (4).
Filter pursuit is required to determine which filters carry most information.
MEL [11] prefers filters (statistics) which give rise to low entropy distributions (this is the "Min" part of Minimax). The entropy is given by H(P) =
- L xP(xIX) log P(xIX) = log Z[X] - L~=l Aa'IjJa ? For the multinomial approximation this can be computed to be N log L - N L~=l 'ljJa log ~. This gives an
intuitive interpretation of feature pursuit: we should prefer filters whose statistical
response to the image training data is as large as possible from their responses to
uniformly distributed images. This is measured by the Kullback-Leibler divergence
L~= l 'ljJa log ~. Recall that if the multinomial approximation is used for multiple
filters then we should simply add together the entropies of different filters.
5
Connections to Generalized Iterative Scaling
In this section we demonstrate a connection between the multinomial approximation and Generalized Iterative Scaling (GIS)[2]. GIS is an iterative procedure for
calculating clique potentials that is guaranteed to converge to the maximum likelihood values of the potentials given the desired empirical filter marginals (e.g. filter
histograms). We show that estimating the potentials by the multinomial approximation is equivalent to the estimate obtained after performing the first iteration
of GIS. We also outline an efficient procedure that allows us to continue additional
GIS iterations to improve upon the multinomial approximation.
The GIS procedure calculates a sequence of distributions on the entire image
(and is guaranteed to converge to the correct maximum likelihood distribution), with an update rule given by p(t+1)(x) ex P(O)(x)Il~=l{ :F; } <pa(x), where
'lfJit ) =< <Pa(X)
>P(t)(x) is the expected histogram for the distribution at time t.
This implies that the corresponding clique potential update equation is given by:
t +1) =
t ) + log 'lfJ~bs - log 'lfJit ).
>.i
>.i
If we initialize GIS so that the initial distribution is the uniform distribution ,
i.e. p(O) (x) = L -N, then the distribution after one iteration is p(1) (x) ex
e2::a <Pa(X) log(1j;~bs /aa) . In other words, the distribution after one iteration is the
MEL distribution with clique potential given by the multinomial approximation.
(The result can be adapted to the case of multiple filters, as explained in Coughlan
and Yuille, in preparation.)
We can iterate GIS to improve the estimate of the clique potentials beyond the
accuracy of the multinomial approximation. The main difficulty lies in estimating
'lfJit ) for t > 0 (at t = 0 this expectation is just the mean histogram with respect
to the uniform distribution, <l:a, which may be calculated efficiently as described in
Coughlan and Yuille, in preparation). One way to approximate these expectations is
to apply a Bethe-Kikuchi approximation technique [8], used for estimating marginals
on Markov Random Fields, to our MEL distribution. Our technique, which was
inspired by the Unified Propagation and Scaling Algorithm [7], consists of writing
the Bethe free energy [8] for our 2-d image lattice, simplifying it using the shift
invariance of the lattice (which enables the algorithm to run swiftly), and using the
Convex-Concave Procedure (CCCP) [9] procedure to obtain an iterative update
equation to estimate the histogram expectations. The GIS algorithm is then run
using these histogram expectations (the results were accurate and did not improve
appreciably by using the higher-order Kikuchi free energy approximation). See
Coughlan and Yuille, in preparation, for details of this procedure.
6
Discussion
This paper describes the g-factor, which depends on the lattice and quantization
and is independent of the training image data. Alternatively it can be thought of as
being proportional to the distribution of feature responses when the input images
are uniformly distributed.
We showed that the g-factor can be used to relate probability distributions on
features to distributions on images. In particular, we described approximations
which, when valid, enable MEL to be computed analytically. In addition, we can
determine when the clique potentials for features decouple, and evaluate how informative each feature is. Finally, we establish a connection between the multinomial
approximation and GIS, and outline an efficient procedure based on Bethe-Kikuchi
approximations that allows us to continue additional GIS iterations to improve upon
the multinomial approximation.
Acknowledgements
We would like to thank Michael Jordan and Yair Weiss for introducing us to Generalized Iterative Scaling and related algorithms. We also thank Anand Rangarajan,
Xiuwen Liu, and Song Chun Zhu for helpful conversations. Sabino Ferreira gave useful feedback on the manuscript. This work was supported by the National Institute
of Health (NEI) with grant number R01-EY 12691-01.
References
[1] J.M. Coughlan and A.L. Yuille. "A Phase Space Approach to Minimax Entropy
Learning and The Minutemax approximation". In Proceedings NIPS '98. 1998.
[2] J. N. Darroch and D. Ratcliff. "Generalized Iterative Scaling for Log-Linear
Models". The Annals of Mathematical Statistics. 1972. Vol. 43, No.5, 14701480.
[3] C. Domb and M.S. Green (Eds). Phase Transitions and Critical Phenomena. Vol. 2. Academic Press. London. 1972.
[4] S. M. Konishi, A.L. Yuille, J.M. Coughlan and Song Chun Zhu. "Fundamental Bounds on Edge Detection: An Information Theoretic Evaluation of Different Edge Cues." In Proceedings Computer Vision and Pattern Recognition
CVPR'99. Fort Collins, Colorado. June 1999.
[5] A.B. Lee, D.B. Mumford, and J. Huang. "Occlusion Models of Natural Images: A Statistical Study of a Scale-Invariant Dead Leaf Model". International
Journal of Computer Vision. Vol. 41, No.'s 1/2. January/February 2001.
[6] J. Portilla and E. P. Simoncelli. "Parametric Texture Model based on Joint
Statistics of Complex Wavelet Coefficients" . International Journal of Computer
Vision. October 2000.
[7] Y. W. Teh and M. Welling. "The Unified Propagation and Scaling Algorithm."
In Proceedings NIPS'01. 2001.
[8] J.S. Yedidia, W.T. Freeman, Y. Weiss, "Generalized Belief Propagation." In
Proceedings NIPS'OO. 2000.
[9] A.L. Yuille. "CCCP Algorithms to Minimize the Bethe and Kikuchi Free Energies," Neural Computation. In press. 2002.
[10] S.C. Zhu and D. Mumford. "Prior Learning and Gibbs Reaction-Diffusion."
PAMI vo1.19, no.11, pp1236-1250, Nov. 1997.
[11] S.C. Zhu, Y. Wu, and D. Mumford. "Minimax Entropy Principle and Its Application to Texture Modeling". Neural Computation. Vol. 9. no. 8. Nov. 1997.
| 2060 |@word briefly:1 seems:1 simulation:2 simplifying:1 solid:1 carry:1 initial:1 liu:2 selecting:1 reaction:1 nt:2 erms:1 dx:1 partition:1 informative:1 shape:1 enables:4 update:3 v:1 cue:1 leaf:1 coughlan:8 smith:1 quantized:1 lx:2 org:2 mathematical:1 direct:1 become:1 kettlewell:1 consists:1 combinational:1 theoretically:1 expected:1 themselves:2 examine:1 ol:1 inspired:1 freeman:1 considering:1 becomes:1 estimating:3 moreover:2 panel:7 factorized:1 unified:2 concave:1 ferreira:1 grant:1 treat:1 analyzing:1 approximately:2 pami:1 factorization:2 procedure:9 logz:1 empirical:2 thought:1 word:3 ga:1 selection:1 writing:1 equivalent:3 map:1 center:2 courtesy:1 independently:2 convex:1 unify:1 rule:1 stereotypical:1 konishi:1 annals:1 pt:2 suppose:2 colorado:1 enx:2 homogeneous:1 us:1 pa:4 recognition:1 observed:3 bottom:1 solved:2 calculate:1 mentioned:1 solving:1 yuille:9 upon:2 completely:3 po:4 joint:1 describe:5 effective:1 monte:1 london:1 whose:3 plausible:1 cvpr:1 statistic:17 gi:14 jointly:1 sequence:1 intuitive:1 rangarajan:1 kikuchi:5 help:1 illustrate:1 derive:1 oo:1 measured:1 rescale:1 job:2 implies:3 differ:1 correct:1 lfj:1 filter:34 enable:1 ja:1 fix:1 considered:1 great:1 estimation:2 individually:1 appreciably:1 create:1 clearly:1 aim:1 factorizes:2 og:1 ax:1 june:1 likelihood:4 ratcliff:2 indicates:1 sense:1 helpful:1 dependent:1 entire:1 jq:2 pixel:2 issue:1 spatial:1 special:1 constrained:1 initialize:1 field:3 ijij:1 identical:1 represents:1 synonymously:1 simplify:1 simultaneously:3 divergence:1 national:1 phase:4 occlusion:1 n1:4 detection:1 interest:1 fd:1 evaluation:1 introduces:2 xb:7 chain:1 accurate:1 edge:2 decoupled:2 desired:1 modeling:2 earlier:1 lattice:7 introducing:2 uniform:3 successful:1 minutemax:1 swiftly:1 fundamental:1 international:2 lee:1 physic:1 michael:1 together:2 na:1 ambiguity:2 huang:1 dead:1 li:1 potential:32 domb:1 bold:1 coefficient:1 depends:3 later:1 lot:1 start:1 minimize:1 il:1 accuracy:1 efficiently:1 carlo:1 multiplying:1 explain:2 ed:1 email:1 definition:1 energy:3 james:1 e2:1 naturally:1 associated:1 proof:1 dataset:1 recall:2 subsection:2 knowledge:1 conversation:1 manuscript:1 higher:1 response:7 wei:2 evaluated:1 ox:1 xa:8 stage:1 just:1 propagation:3 defines:1 gray:2 usa:1 analytically:4 hence:3 leibler:1 white:1 attractive:1 mel:17 generalized:7 gg:1 ay:1 outline:2 demonstrate:2 theoretic:1 image:40 recently:1 fi:5 common:1 multinomial:15 jl:2 interpretation:1 approximates:1 relating:1 synthesized:2 marginals:2 gibbs:2 add:1 showed:1 continue:2 minimum:1 additional:2 ey:1 determine:5 converge:2 dashed:2 ii:1 relates:1 multiple:3 full:2 simoncelli:1 academic:1 calculation:1 cccp:2 iog:1 calculates:1 mrf:6 vision:5 essentially:1 expectation:4 histogram:18 iteration:5 addition:1 induced:1 anand:1 oq:1 seem:1 jordan:1 iii:2 iterate:1 gave:1 shift:1 whether:2 darroch:2 gb:1 fob:2 song:2 prefers:1 useful:1 vol:4 express:1 pb:2 diffusion:1 run:2 wu:2 ob:5 dy:1 scaling:8 prefer:1 bound:1 guaranteed:2 simplification:1 bv:1 adapted:1 precisely:1 min:1 performing:2 conjecture:1 tv:3 across:1 describes:1 lp:1 b:3 explained:1 invariant:1 ln:2 equation:6 discus:2 count:2 letting:1 pursuit:4 yedidia:1 apply:1 observe:2 alternative:1 yair:1 florida:1 top:1 calculating:2 prof:1 establish:1 approximating:2 february:1 r01:1 occurs:2 mumford:4 fa:2 parametric:1 gradient:2 thank:2 oa:3 street:1 equivalently:3 october:1 relate:1 rise:2 ba:2 implementation:2 ski:2 unknown:1 teh:1 xiuwen:2 markov:4 january:1 extended:1 portilla:1 arbitrary:1 nei:1 fort:1 required:2 connection:7 learned:7 nip:3 address:1 beyond:1 pattern:1 green:1 belief:1 ia:3 critical:1 natural:3 difficulty:1 zhu:5 minimax:10 improve:5 eye:1 health:1 fax:1 ljj:4 text:1 review:1 understanding:1 acknowledgement:1 prior:1 oy:1 proportional:2 sufficient:2 xp:1 principle:4 row:2 course:2 supported:1 free:3 understand:1 institute:2 distributed:3 xix:2 xia:2 default:1 calculated:1 valid:1 feedback:1 transition:1 fb:4 san:1 welling:1 approximate:2 nov:2 preferred:1 kullback:1 clique:19 investigating:1 francisco:1 alternatively:1 grayscale:1 iterative:8 why:4 learn:7 bethe:5 ca:1 decoupling:3 tel:1 complex:1 did:1 main:1 big:1 noise:1 arise:1 fillmore:1 xl:7 lie:1 ib:3 third:1 wavelet:1 theorem:1 chun:2 evidence:1 quantization:5 effectively:1 importance:1 texture:3 magnitude:1 entropy:19 simply:1 ijj:1 applies:1 aa:6 corresponds:1 determines:2 extracted:1 determined:5 typical:1 uniformly:3 vo1:1 decouple:7 zb:1 total:2 invariance:1 la:1 attempted:1 shiftinvariant:1 support:1 stressed:1 arises:1 collins:1 relevance:1 preparation:4 evaluate:1 mcmc:1 phenomenon:1 ex:4 |
1,164 | 2,061 | Small-World Phenomena and the
Dynamics of Information
Jon Kleinberg
Department of Computer Science
Cornell University
Ithaca NY 14853
1
Introduction
The problem of searching for information in networks like the World Wide Web can
be approached in a variety of ways, ranging from centralized indexing schemes to
decentralized mechanisms that navigate the underlying network without knowledge
of its global structure. The decentralized approach appears in a variety of settings:
in the behavior of users browsing the Web by following hyperlinks; in the design of
focused crawlers [4, 5, 8] and other agents that explore the Web?s links to gather
information; and in the search protocols underlying decentralized peer-to-peer systems such as Gnutella [10], Freenet [7], and recent research prototypes [21, 22, 23],
through which users can share resources without a central server.
In recent work, we have been investigating the problem of decentralized search
in large information networks [14, 15]. Our initial motivation was an experiment
that dealt directly with the search problem in a decidedly pre-Internet context:
Stanley Milgram?s famous study of the small-world phenomenon [16, 17]. Milgram
was seeking to determine whether most pairs of people in society were linked by
short chains of acquaintances, and for this purpose he recruited individuals to try
forwarding a letter to a designated ?target? through people they knew on a firstname basis. The starting individuals were given basic information about the target
? his name, address, occupation, and a few other personal details ? and had to
choose a single acquaintance to send the letter to, with goal of reaching the target
as quickly as possible; subsequent recipients followed the same procedure, and the
chain closed in on its destination. Of the chains that completed, the median number
of steps required was six ? a result that has since entered popular culture as the
?six degrees of separation? principle [11].
Milgram?s experiment contains two striking discoveries ? that short chains are
pervasive, and that people are able to find them. This latter point is concerned
precisely with a type of decentralized navigation in a social network, consisting of
people as nodes and links joining pairs of people who know each other. From an
algorithmic perspective, it is an interesting question to understand the structure
of networks in which this phenomenon emerges ? in which message-passing with
purely local information can be efficient.
Networks that Support Efficient Search. A model of a ?navigable? network
requires a few basic features. It should contain short paths among all (or most)
pairs of nodes. To be non-trivial, its structure should be partially known and
partially unknown to its constituent nodes; in this way, information about the known
parts can be used to construct paths that make use of the unknown parts as well.
This is clearly what was taking place in Milgram?s experiments: participants, using
the information available to them, were estimating which of their acquaintances
would lead to the shortest path through the full social network. Guided by these
observations, we turned to the work of Watts and Strogatz [25], which proposes a
model of ?small-world networks? that very concisely incorporates these features. A
simple variant of their basic model can be described as follows. One starts with
a p-dimensional lattice, in which nodes are joined only to their nearest neighbors.
One then adds k directed long-range links out of each node v, for a constant k; the
endpoint of each link is chosen uniformly at random. Results from the theory of
random graphs can be used to show that with high probability, there will be short
paths connecting all pairs of nodes (see e.g. [3]); at the same time, the network will
locally retain a lattice-like structure. Asymptotically, our criterion for ?shortness?
of paths is what one obtains from this and similar random constructions: there
should be paths whose lengths are bounded by a polynomial in log n, where n is
the number of nodes. We will refer to such a function as polylogarithmic.
This network model, a superposition of a lattice and a set of long-range links, is
a natural one in which to study the behavior of a decentralized search algorithm.
The algorithm knows the structure of the lattice; it starts from a node s, and is told
the coordinates of a target node t. It successively traverses links of the network so
as to reach the target as quickly as possible; but, crucially, it does not know the
long-range links out of any node that it has not yet visited. In addition to moving
forward across directed links, the algorithm may travel in reverse across any link
that it has already followed in the forward direction; this allows it to back up when
it does not want to continue exploring from its current node. One can view this
as hitting the ?back button? on a Web browser ? or returning the letter to its
previous holder in Milgram?s experiments, with instructions that he or she should
try someone else. We say that the algorithm has search time Y (n) if, on a randomly
generated n-node network with s and t chosen uniformly at random, it reaches the
target t in at most Y (n) steps with probability at least 1 ? ?(n), for a function ?(?)
going to 0 with n.
The first result in [15] is that no decentralized algorithm can achieve a polylogarithmic search time in this network model ? even though, with high probability,
there are paths of polylogarithmic length joining all pairs of nodes. However, if we
generalize the model slightly, then it can support efficient search. Specifically, when
we construct a long-range link (v, w) out of v, we do not choose w uniformly at
random; rather, we choose it with probability proportional to d??, where d is the
lattice distance from v to w. In this way, the long-range links become correlated to
the geometry of the lattice. We show in [15] that when ? is equal to p, the dimension of the underlying lattice, then a decentralized greedy algorithm achieves search
time proportional to log2 n; and for any other value of ?, there is no decentralized
algorithm with a polylogarithmic search time.
Recent work by Zhang, Goel, and Govindan [26] has shown how the distribution of
links associated with the optimal value of ? may lead to improved performance for
decentralized search in the Freenet peer-to-peer system. Adamic, Lukose, Puniyani,
and Huberman [2] have recently considered a variation of the decentralized search
problem in a network that has essentially no known underlying structure; however,
when the number of links incident to nodes follows a power-law distribution, then
a search strategy that seeks high-degree nodes can be effective. They have applied
their results to the Gnutella system, which exhibits such a structure. In joint
work with Kempe and Demers [12], we have studied how distributions that are
inverse-polynomial in the distance between nodes can be used in the design of
gossip protocols for spreading information in a network of communicating agents.
The goal of the present paper is to consider more generally the problem of decentralized search in networks with partial information about the underlying structure.
While a lattice makes for a natural network backbone, we would like to understand
the extent to which the principles underlying efficient decentralized search can be
abstracted away from a lattice-like structure. We begin by considering networks
that are generated from a hierarchical structure, and show that qualitatively similar results can be obtained; we then discuss a general model of group structures,
which can be viewed as a simultaneous generalization of lattices and hierarchies.
We refer to k, the number of out-links per node, as the out-degree of the model.
The technical details of our results ? both in the statements of the results and the
proofs ? are simpler when we allow the out-degree to be polylogarithmic, rather
than constant. Thus we describe this case first, and then move on to the case in
which each node has only a constant number of out-links.
2
Hierarchical Network Models
In a number of settings, nodes represent objects that can be classified according to
a hierarchy or taxonomy; and nodes are more likely to form links if they belong to
the same small sub-tree in the hierarchy, indicating they are more closely related.
To construct a network model from this idea, we represent the hierarchy using a
complete b-ary tree T , where b is a constant. Let V denote the set of leaves of T ; we
use n to denote the size of V , and for two leaves v and w, we use h(v, w) to denote the
height of the least common ancestor of v and w in T . We are also given a monotone
non-increasing function f(?) that will determine link probabilities. For each node
v ? V , we create a random link to w with probability proportional P
to f(h(v, w)); in
other words, the probability of choosing w is equal to f(h(v, w))/ x6=v f(h(v, x)).
We create k links out of each node v this way, choosing the endpoint w each time
independently and with repetition allowed. This results in a graph G on the set V .
For the analysis in this section, we will take the out-degree to be k = c log2 n, for
a constant c. It is important to note that the tree T is used only in the generation
process of G; neither the edges nor the non-leaf nodes of T appear in G. (By way
of contrast, the lattice model in [15] included both the long-range links and the
nearest-neighbor links of the lattice itself.) When we use the term ?node? without
any qualification, we are referring to nodes of G, and hence to leaves of T ; we will
use ?internal node? to refer to non-leaf nodes of T .
We refer to the process producing G as a hierarchical model with exponent ? if the
function f(h) grows asymptotically like b??h:
00
f(h)
b?? h
0
= 0 for all ?00 > ?.
0 h = 0 for all ? < ? and lim
??
h?? b
h?? f(h)
lim
There are several natural interpretations for a hierarchical network model. One is in
terms of the World Wide Web, where T is a topic hierarchy such as www.yahoo.com.
Each leaf of T corresponds to a Web page, and its path from the root specifies an increasingly fine-grained description of the page?s topic. Thus, a particular leaf may be associated with Science/Computer Science/Algorithms or with
Arts/Music/Opera. The linkage probabilities then have a simple meaning ? they
are based on the distance between the topics of the pages, as measured by the
height of their least common ancestor in the topic hierarchy. A page on Sci-
ence/Computer Science/Algorithms may thus be more likely to link to one on Science/Computer Science/Machine Learning than to one on Arts/Music/Opera. Of
course, the model is a strong simplification, since topic structures are not fully hierarchical, and certainly do not have uniform branching and depth. It is worth noting
that a number of recent models for the link structure of the Web, as well as other
relational structures, have looked at different ways in which similarities in content
can affect the probability of linkage; see e.g. [1, 6, 9].
Another interpretation of the hierarchical model is in terms of Milgram?s original
experiment. Studies performed by Killworth and Bernard [13] showed that in choosing a recipient for the letter, participants were overwhelmingly guided by one of two
criteria: similarity to the target in geography, or similarity to the target in occupation. If one views the lattice as forming a simple model for geographic factors,
the hierarchical model can similarly be interpreted as forming a ?topic hierarchy?
of occupations, with individuals at the leaves. Thus, for example, the occupations
of ?banker? and ?stock broker? may belong to the same small sub-tree; since the
target in one of Milgram?s experiments was a stock broker, it might therefore be
a good strategy to send the letter to a banker. Independently of our work here,
Watts, Dodds, and Newman have recently studied hierarchical structures for modeling Milgram?s experiment in social networks [24].
We now consider the search problem in a graph G generated from a hierarchical
model: A decentralized algorithm has knowledge of the tree T , and knows the
location of a target leaf that it must reach; however, it only learns the structure of
G as it visits nodes. The exponent ? determines how the structures of G and T
are related; how does this affect the navigability of G? In the analysis of the lattice
model [15], the key property of the optimal exponent was that, from any point, there
was a reasonable probability of a long-range link that halved the distance to the
target. We make use of a similar idea here: when ? = 1, there is always a reasonable
probability of finding a long-range link into a strictly smaller sub-tree containing
the target. As mentioned above, we focus here on the case of polylogarithmic outdegree, with the case of constant out-degree deferred until later.
Theorem 2.1 (a) There is a hierarchical model with exponent ? = 1 and polylogarithmic out-degree in which a decentralized algorithm can achieve search time
O(log n).
(b) For every ? 6= 1, there is no hierarchical model with exponent ? and polylogarithmic out-degree in which a decentralized algorithm can achieve polylogarithmic
search time.
Due to space limitations, we omit proofs from this version of the paper. Complete
proofs may be found in the extended version, which is available on the author?s
Web page (http://www.cs.cornell.edu/home/kleinber/).
To prove (a), we show that when the search is at a node v whose least common
ancestor with the target has height h, there is a high probability that v has a link
into the sub-tree of height h?1 containing the target. In this way, the search reaches
the target in logarithmically many steps. To prove (b), we exhibit a sub-tree T 0
containing the target such that, with high probability, it takes any decentralized
algorithm more than a polylogarithmic number of steps to find a link into T 0 .
3
Group Structures
The analysis of the search problem in a hierarchical model is similar to the analysis of the lattice-based approach in [15], although the two types of models seem
superficially quite different. It is natural to look for a model that would serve as a
simultaneous generalization of each.
Consider a collection of individuals in a social network, and suppose that we know
of certain groups to which individuals belong ? people who live in the same town,
or work in the same profession, or have some other affiliation in common. We could
imagine that people are more likely to be connected if they both belong to the
same small group. In a lattice-based model, there may be a group for each subset
of lattice points contained in a common ball (grouping based on proximity); in a
hierarchical model, there may be a group for each subset of leaves contained in a
common sub-tree. We now discuss the notion of a group structure, to make this
precise; we follow a model proposed in joint work with Kempe and Demers [12],
where we were concerned with designing gossip protocols for lattices and hierarchies.
A technically different model of affiliation networks, also motivated by these types
of issues, has been studied recently by Newman, Watts, and Strogatz [18].
A group structure consists of an underlying set V of nodes, and a collection of
subsets of V (the groups). The collection of groups must include V itself; and it
must satisfy the following two properties, for constants ? < 1 and ? > 1.
(i) If R is a group of size q ? 2 containing a node v, then there is a group R0 ? R
containing v that is strictly smaller than R, but has size at least ?q.
(ii) If R1, R2, R3, . . . are groups that all have size at most q and all contain a
common node v, then their union has size at most ?q.
The reader can verify that these two properties hold for the collection of balls in a
lattice, as well as for the collection of sub-trees in a hierarchy. However, it is easy
to construct examples of group structures that do not arise in this way from lattices
or hierarchies.
Given a group structure (V, {Ri}), and a monotone non-increasing function f(?), we
now consider the following process for generating a graph on V . For two nodes v
and w, we use q(v, w) to denote the minimum size of a group containing both v and
w. (Note that such a group must exist, since V itself is a group.) For each node
v ? V , we create a random link to w with probability proportional to f(q(v, w));
repeating this k times independently yields k links out of v. We refer to this as a
group-induced model with exponent ? if f(q) grows asymptotically like q ?? :
00
lim
h??
f(q)
q??
0
=
0
for
all
?
<
?
and
lim
= 0 for all ?00 > ?.
h?? f(q)
q??0
A decentralized search algorithm in such a network is given knowledge of the full
group structure, and must follow links of G to a designated target t. We now state
an analogue of Theorem 2.1 for group structures.
Theorem 3.1 (a) For every group structure, there is a group-induced model with
exponent ? = 1 and polylogarithmic out-degree in which a decentralized algorithm
can achieve search time O(log n).
(b) For every ? < 1, there is no group-induced model with exponent ? and polylogarithmic out-degree in which a decentralized algorithm can achieve polylogarithmic
search time.
Notice that in a hierarchical model, the smallest group (sub-tree) containing two
nodes v and w has size bh(v,w) , and so Theorem 3.1(a) implies Theorem 2.1(a).
Similarly, on a lattice, the smallest group (ball) containing two nodes v and w at
lattice distance d has size ?(dp ), and so Theorem 3.1(a) implies a version of the
result from [15], that efficient search is possible in a lattice model when nodes form
links with probability d?p . (In the version of the lattice result implied here, there
are no nearest-neighbor links at all; but each node has a polylogarithmic number
of out-links.)
The proof of Theorem 3.1(a) closely follows the proof of Theorem 2.1(a). We consider a node v ? the current point in the search ? for which the smallest group
containing v and the target t has size q. Using group structure properties (i) and
(ii), we show there is a high probability that v has a link into a group containing t of
size between ?2 q and ?q. In this way, the search passes through groups containing
t of sizes that diminish geometrically, and hence it terminates in logarithmic time.
Note that Theorem 3.1(b) only considers exponents ? < 1. This is because there
exist group-induced models with exponents ? > 1 in which decentralized algorithms
can achieve polylogarithmic search time. For example, consider an undirected graph
G? in which each node has 3 neighbors, and each pair of nodes can be connected
by a path of length O(log n). It is possible to define a group structure satisfying
properties (i) and (ii) in which each edge of G? appears as a 2-node group; but then,
a graph G generated from a group-induced model with a very large exponent ? will
contain all edges of G? with high probability, and a decentralized search algorithm
will be able to follow these edges directly to construct a short path to the target.
However, a lower bound for the case ? > 1 can be obtained if we place one additional
restriction on the group structure. Give a group structure (V, {Ri}), and a cut-off
value q, we define a graph H(q) on V by joining any two nodes that belong to a
common group of size at most q. Note that H(q) is not a random graph; it is defined
simply in terms of the group structure and q. We now argue that if many pairs of
nodes are far apart in H(q), for a suitably large value of q, then a decentralized
algorithm cannot be efficient when ? > 1.
Theorem 3.2 Let (V, {Ri }) be a group structure. Suppose there exist constants
?, ? > 0 so that a constant fraction of all pairs of nodes have shortest-path distance
?(n? ) in H(n? ). Then for every ? > 1, there is no group-induced model on (V, {R i})
with exponent ? and a polylogarithmic number of out-links per node in which a
decentralized algorithm can achieve polylogarithmic search time.
Notice this property holds for group structures arising from both lattices and hierarchies; in a lattice, a constant fraction of all pairs in H(n1/2p) have distance
?(n1/2p), while in a hierarchy, the graph H(n? ) is disconnected for every ? < 1.
4
Nodes with a Constant Number of Out-Links
Thus far, by giving each node more than a constant number of out-links, we have
been able to design very simple search algorithms in networks generated according
to the optimal exponent ?. From each node, there is a way to make progress toward
the target node t, and so the structure of the graph G funnels the search towards
its destination. When the out-degree is constant, however, things get much more
complicated. First of all, with high probability, many nodes will have all their
links leading ?away? from the target in the hierarchy. Second, there is a constant
probability that the target t will have no in-coming links, and so the whole task
of finding t becomes ill-defined. This indicates that the statement of the results
themselves in this case will have to be somewhat different.
In this section, we work with a hierarchical model, and construct graphs with con-
stant out-degree k; the value of k will need to be sufficiently large in terms of other
parameters of the model. It is straightforward to formulate an analogue of our
results for group structures, but we do not go into the details of this here.
To deal with the problem that t itself may have no incoming links, we relax the
search problem to that of finding a cluster of nodes containing t. In a topic-based
model of Web pages, for example, we can consider t as a representative of a desired
type of page, with goal being to find any page of this type. Thus, we are given a
complete b-ary tree T , where b is a constant; we let L denote the set of leaves of
T , and m denote the size of L. We place r nodes at each leaf of T , forming a set
V of n = mr nodes total. We then define a graph G on V as in Section 2: for a
non-increasing function f(?), we create k links out of each node v ? V , choosing w
as an endpoint with probability proportional to f(h(v, w)). As before, we refer to
this process as a hierarchical model with exponent ?, for the appropriate value of
?. We refer to each set of r nodes at a common leaf of T as a cluster, and define
the resolution of the hierarchical model to be the value r.
A decentralized algorithm is given knowledge of T , and a target node t; it must
reach any node in the cluster containing t. Unlike the previous algorithms we have
developed, which only moved forward across links, the algorithm we design here will
need to make use of the ability to travel in reverse across any link that it has already
followed in the forward direction. Note also that we cannot easily reduce the current
search problem to that of Section 2 by collapsing clusters into ?super-nodes,? since
there are not necessarily links joining nodes within the same cluster.
The search task clearly becomes easier as the resolution of the model (i.e. the size of
clusters) becomes larger. Thus, our goal is to achieve polylogarithmic search time
in a hierarchical model with polylogarithmic resolution.
Theorem 4.1 (a) There is a hierarchical model with exponent ? = 1, constant
out-degree, and polylogarithmic resolution in which a decentralized algorithm can
achieve polylogarithmic search time.
(b) For every ? 6= 1, there is no hierarchical model with exponent ?, constant outdegree, and polylogarithmic resolution in which a decentralized algorithm can achieve
polylogarithmic search time.
The search algorithm used to establish part (a) operates in phases. It begins each
phase j with a collection of ?(log n) nodes all belonging to the sub-tree Tj that contains the target t and whose root is at depth j. During phase j, it explores outward
from each of these nodes until it has discovered a larger but still polylogarithmicsized set of nodes belonging to Tj . From among these, there is a high probability
that at least ?(log n) have links into the smaller sub-tree Tj+1 that contains t and
whose root is at depth j + 1. At this point, phase j + 1 begins, and the process
continues until the cluster containing t is found.
Acknowledgments
My thinking about models for Web graphs and social networks has benefited greatly
from discussions and collaboration with Dimitris Achlioptas, Avrim Blum, Duncan Callaway, Michelle Girvan, John Hopcroft, David Kempe, Ravi Kumar, Tom
Leighton, Mark Newman, Prabhakar Raghavan, Sridhar Rajagopalan, Steve Strogatz, Andrew Tomkins, Eli Upfal, and Duncan Watts. The research described here
was supported in part by a David and Lucile Packard Foundation Fellowship, an
ONR Young Investigator Award, NSF ITR/IM Grant IIS-0081334, and NSF Faculty
Early Career Development Award CCR-9701399.
References
[1] D. Achlioptas, A. Fiat, A. Karlin, F. McSherry, ?Web search via hub synthesis,? Proc.
42nd IEEE Symp. on Foundations of Computer Science, 2001.
[2] L. Adamic, R. Lukose, A. Puniyani, B. Huberman, ?Search in Power-Law Networks,?
Phys. Rev. E, 64 46135 (2001)
[3] B. Bollob?
as, F. Chung, ?The diameter of a cycle plus a random matching,? SIAM
J. Disc. Math. 1(1988).
[4] S. Chakrabarti, M. van den Berg, B. Dom, ?Focused crawling: A new approach to
topic-specific Web resource discovery,? Proc. 8th Intl. World Wide Web Conf., 1999.
[5] J. Cho, H. Garcia-Molina, L. Page, ?Efficient Crawling Through URL Ordering,? Proc.
7th Intl. World Wide Web Conf., 1998.
[6] D. Cohn and T. Hofmann, ?The Missing Link ? A Probabilistic Model of Document
Content and Hypertext Connectivity,? Adv. Neural Inf. Proc. Sys. (NIPS) 13, 2000.
[7] I. Clarke, O. Sandberg, B. Wiley, T. Hong, ?Freenet: A Distributed Anonymous Information Storage and Retrieval System,? International Workshop on Design Issues in
Anonymity and Unobservability, 2000.
[8] M. Diligenti, F.M. Coetzee, S. Lawrence, C.L. Giles, M. Gori, ?Focused Crawling Using
Context Graphs,? Proc. 26th Intl. Conf. on Very Large Databases (VLDB), 2000.
[9] L. Getoor, N. Friedman, D. Koller, and B. Taskar. ?Learning Probabilistic Models of
Relational Structure,? Proc. 18th International Conference on Machine Learning, 2001.
[10] Gnutella. http://gnutella.wego.com.
[11] J. Guare, Six Degrees of Separation: A Play (Vintage Books, New York, 1990).
[12] D. Kempe, J. Kleinberg, A. Demers. ?Spatial gossip and resource location protocols,?
Proc. 33rd ACM Symp. on Theory of Computing, 2001.
[13] P. Killworth, H. Bernard, ?Reverse small world experiment,? Social Networks 1(1978).
[14] J. Kleinberg. ?Navigation in a Small World.? Nature 406(2000).
[15] J. Kleinberg. ?The small-world phenomenon: An algorithmic perspective.? Proc. 32nd
ACM Symposium on Theory of Computing, 2000. Also appears as Cornell Computer
Science Technical Report 99-1776 (October 1999).
[16] M. Kochen, Ed., The Small World (Ablex, Norwood, 1989).
[17] S. Milgram, ?The small world problem,? Psychology Today 1(1967).
[18] M. Newman, D. Watts, S. Strogatz, ?Random graph models of social networks,? Proc.
Natl. Acad. Sci., to appear.
[19] A. Oram, editor, Peer-to-Peer: Harnessing the Power of Disruptive Technologies
O?Reilly and Associates, 2001.
[20] A. Puniyani, R. Lukose, B. Huberman, ?Intentional Walks on Scale Free Small
Worlds,? HP Labs Information Dynamics Group, at http://www.hpl.hp.com/shl/.
[21] S. Ratnasamy, P. Francis, M. Handley, R. Karp, S. Shenker, ?A Scalable ContentAddressable Network,? Proc. ACM SIGCOMM, 2001
[22] A. Rowstron, P. Druschel, ?Pastry: Scalable, distributed object location and routing
for large-scale peer-to-peer systems,? Proc. 18th IFIP/ACM International Conference
on Distributed Systems Platforms (Middleware 2001), 2001.
[23] I. Stoica, R. Morris, D. Karger, F. Kaashoek, H. Balakrishnan, ?Chord: A Scalable
Peer-to-peer Lookup Service for Internet Applications,? Proc. ACM SIGCOMM, 2001
[24] D. Watts, P. Dodds, M. Newman, personal communication, December 2001.
[25] D. Watts, S. Strogatz, ?Collective dynamics of small-world networks,? Nature
393(1998).
[26] H. Zhang, A. Goel, R. Govindan, ?Using the Small-World Model to Improve Freenet
Performance,? Proc. IEEE Infocom, 2002.
| 2061 |@word faculty:1 version:4 polynomial:2 leighton:1 nd:2 suitably:1 instruction:1 vldb:1 seek:1 crucially:1 initial:1 contains:3 karger:1 document:1 current:3 com:3 yet:1 crawling:3 must:6 john:1 subsequent:1 hofmann:1 greedy:1 leaf:13 sys:1 short:5 math:1 node:63 location:3 traverse:1 simpler:1 zhang:2 height:4 become:1 symposium:1 chakrabarti:1 prove:2 consists:1 symp:2 behavior:2 themselves:1 nor:1 considering:1 increasing:3 becomes:3 begin:3 estimating:1 underlying:7 bounded:1 what:2 backbone:1 interpreted:1 navigability:1 developed:1 finding:3 every:6 returning:1 grant:1 omit:1 appear:2 producing:1 before:1 service:1 local:1 qualification:1 acad:1 joining:4 path:11 might:1 plus:1 studied:3 forwarding:1 someone:1 callaway:1 range:8 directed:2 acknowledgment:1 union:1 procedure:1 matching:1 reilly:1 pre:1 word:1 get:1 cannot:2 bh:1 storage:1 context:2 live:1 www:3 restriction:1 missing:1 send:2 straightforward:1 go:1 starting:1 independently:3 focused:3 formulate:1 resolution:5 communicating:1 his:1 searching:1 notion:1 coordinate:1 variation:1 target:24 construction:1 hierarchy:13 user:2 suppose:2 imagine:1 play:1 today:1 designing:1 associate:1 logarithmically:1 satisfying:1 anonymity:1 continues:1 cut:1 database:1 taskar:1 hypertext:1 connected:2 cycle:1 adv:1 ordering:1 chord:1 mentioned:1 rowstron:1 dynamic:3 personal:2 dom:1 ablex:1 purely:1 serve:1 technically:1 basis:1 easily:1 joint:2 hopcroft:1 stock:2 druschel:1 effective:1 describe:1 approached:1 newman:5 choosing:4 harnessing:1 peer:10 whose:4 quite:1 larger:2 say:1 relax:1 ability:1 browser:1 itself:4 karlin:1 coming:1 turned:1 entered:1 achieve:10 description:1 moved:1 constituent:1 cluster:7 r1:1 prabhakar:1 intl:3 generating:1 object:2 andrew:1 measured:1 nearest:3 progress:1 strong:1 c:1 implies:2 direction:2 guided:2 closely:2 raghavan:1 routing:1 generalization:2 geography:1 anonymous:1 im:1 exploring:1 strictly:2 hold:2 proximity:1 sufficiently:1 considered:1 diminish:1 intentional:1 lawrence:1 algorithmic:2 achieves:1 early:1 smallest:3 purpose:1 proc:13 travel:2 spreading:1 superposition:1 visited:1 repetition:1 create:4 pastry:1 clearly:2 always:1 super:1 reaching:1 rather:2 cornell:3 karp:1 overwhelmingly:1 pervasive:1 focus:1 she:1 indicates:1 greatly:1 contrast:1 koller:1 ancestor:3 going:1 issue:2 among:2 ill:1 exponent:16 yahoo:1 proposes:1 development:1 art:2 spatial:1 kempe:4 platform:1 equal:2 construct:6 shl:1 outdegree:2 dodds:2 look:1 jon:1 thinking:1 report:1 few:2 randomly:1 individual:5 geometry:1 consisting:1 phase:4 n1:2 friedman:1 centralized:1 message:1 lucile:1 certainly:1 deferred:1 navigation:2 tj:3 mcsherry:1 natl:1 chain:4 edge:4 partial:1 culture:1 tree:14 walk:1 desired:1 modeling:1 giles:1 ence:1 lattice:26 subset:3 uniform:1 my:1 cho:1 referring:1 explores:1 siam:1 international:3 ifip:1 retain:1 destination:2 told:1 off:1 probabilistic:2 connecting:1 synthesis:1 quickly:2 connectivity:1 central:1 town:1 successively:1 choose:3 containing:14 collapsing:1 conf:3 book:1 acquaintance:3 chung:1 leading:1 lookup:1 satisfy:1 stoica:1 performed:1 try:2 closed:1 view:2 linked:1 root:3 later:1 start:2 francis:1 participant:2 complicated:1 lab:1 opera:2 holder:1 who:2 yield:1 dealt:1 generalize:1 famous:1 disc:1 worth:1 ary:2 classified:1 simultaneous:2 reach:5 phys:1 milgram:9 ed:1 associated:2 proof:5 coetzee:1 con:1 oram:1 navigable:1 demers:3 popular:1 knowledge:4 emerges:1 lim:4 stanley:1 fiat:1 vintage:1 profession:1 sandberg:1 back:2 appears:3 steve:1 x6:1 follow:3 tom:1 improved:1 though:1 achlioptas:2 until:3 hpl:1 web:14 adamic:2 cohn:1 grows:2 name:1 contain:3 geographic:1 verify:1 hence:2 deal:1 during:1 branching:1 criterion:2 hong:1 complete:3 ranging:1 meaning:1 recently:3 common:9 endpoint:3 belong:5 he:2 interpretation:2 shenker:1 refer:7 rd:1 similarly:2 hp:2 had:1 moving:1 similarity:3 add:1 halved:1 recent:4 showed:1 perspective:2 inf:1 apart:1 reverse:3 certain:1 server:1 affiliation:2 continue:1 onr:1 molina:1 minimum:1 additional:1 somewhat:1 mr:1 goel:2 r0:1 determine:2 shortest:2 ii:4 full:2 technical:2 long:8 retrieval:1 visit:1 award:2 sigcomm:2 variant:1 basic:3 scalable:3 essentially:1 stant:1 represent:2 addition:1 want:1 fine:1 fellowship:1 else:1 median:1 ithaca:1 unlike:1 pass:1 induced:6 recruited:1 undirected:1 thing:1 december:1 balakrishnan:1 incorporates:1 seem:1 noting:1 easy:1 concerned:2 variety:2 affect:2 psychology:1 reduce:1 idea:2 prototype:1 itr:1 whether:1 six:3 motivated:1 url:1 linkage:2 passing:1 york:1 generally:1 outward:1 repeating:1 locally:1 morris:1 diameter:1 http:3 specifies:1 exist:3 nsf:2 notice:2 arising:1 per:2 ccr:1 group:43 key:1 killworth:2 blum:1 neither:1 ravi:1 graph:15 asymptotically:3 button:1 monotone:2 geometrically:1 fraction:2 inverse:1 letter:5 eli:1 striking:1 place:3 reasonable:2 reader:1 separation:2 home:1 duncan:2 clarke:1 bound:1 internet:2 followed:3 simplification:1 precisely:1 ri:3 kleinberg:4 kumar:1 department:1 designated:2 according:2 watt:7 ball:3 disconnected:1 belonging:2 across:4 slightly:1 increasingly:1 smaller:3 terminates:1 rev:1 den:1 indexing:1 resource:3 discus:2 r3:1 mechanism:1 know:5 available:2 decentralized:27 hierarchical:20 away:2 appropriate:1 original:1 recipient:2 gori:1 include:1 completed:1 tomkins:1 log2:2 music:2 giving:1 establish:1 society:1 seeking:1 move:1 implied:1 question:1 already:2 looked:1 strategy:2 exhibit:2 dp:1 distance:7 link:46 sci:2 topic:8 argue:1 extent:1 considers:1 trivial:1 toward:1 length:3 bollob:1 disruptive:1 october:1 statement:2 taxonomy:1 design:5 collective:1 unknown:2 observation:1 govindan:2 relational:2 extended:1 precise:1 communication:1 discovered:1 david:2 pair:9 required:1 concisely:1 polylogarithmic:23 nip:1 address:1 able:3 middleware:1 firstname:1 dimitris:1 hyperlink:1 rajagopalan:1 packard:1 analogue:2 power:3 getoor:1 natural:4 decidedly:1 scheme:1 improve:1 technology:1 discovery:2 occupation:4 law:2 girvan:1 fully:1 interesting:1 generation:1 proportional:5 limitation:1 infocom:1 funnel:1 foundation:2 upfal:1 norwood:1 agent:2 degree:14 gather:1 incident:1 principle:2 editor:1 share:1 collaboration:1 course:1 supported:1 free:1 allow:1 understand:2 wide:4 neighbor:4 taking:1 michelle:1 van:1 distributed:3 dimension:1 depth:3 world:15 superficially:1 forward:4 qualitatively:1 author:1 collection:6 far:2 social:7 obtains:1 abstracted:1 global:1 investigating:1 incoming:1 knew:1 search:41 nature:2 career:1 necessarily:1 protocol:4 motivation:1 whole:1 arise:1 sridhar:1 allowed:1 benefited:1 representative:1 gossip:3 ny:1 wiley:1 sub:10 learns:1 grained:1 young:1 theorem:11 specific:1 navigate:1 hub:1 r2:1 grouping:1 lukose:3 workshop:1 avrim:1 browsing:1 easier:1 logarithmic:1 garcia:1 simply:1 explore:1 likely:3 forming:3 hitting:1 contained:2 strogatz:5 partially:2 joined:1 corresponds:1 determines:1 acm:5 goal:4 viewed:1 banker:2 towards:1 kochen:1 kaashoek:1 content:2 broker:2 included:1 specifically:1 uniformly:3 huberman:3 operates:1 total:1 bernard:2 indicating:1 berg:1 internal:1 people:7 support:2 latter:1 mark:1 crawler:1 investigator:1 phenomenon:4 correlated:1 |
1,165 | 2,062 | Estimating Car Insurance Premia:
a Case Study in High-Dimensional Data
Inference
Nicolas Chapados, Yoshua Bengio, Pascal Vincent, Joumana
Ghosn, Charles Dugas, Ichiro Takeuchi, Linyan Meng
University of Montreal, dept. IRQ, CP 6128, Succ. Centre-Ville, Montreal, Qc, Canada, H3C3J7
{chapadosJbengioy,vincentp,ghosnJdugas,takeuchi,mengl}~iro.umontreal.ca
Abstract
Estimating insurance premia from data is a difficult regression
problem for several reasons: the large number of variables, many of
which are .discrete, and the very peculiar shape of the noise distribution, asymmetric with fat tails, with a large majority zeros and a
few unreliable and very large values. We compare several machine
learning methods for estimating insurance premia, and test them
on a large data base of car insurance policies. We find that function approximation methods that do not optimize a squared loss,
like Support Vector Machines regression, do not work well in this
context. Compared methods include decision trees and generalized
linear models. The best results are obtained with a mixture of
experts, which better identifies the least and most risky contracts,
and allows to reduce the median premium by charging more to the
most risky customers.
1
Introduction
The main mathematical problem faced by actuaries is that of estimating how much
each insurance contract is expected to cost. This conditional expected claim amount
is called the pure premium and it is the basis of the gross premium charged to the
insured. This expected value is conditionned on information available about the
insured and about the contract, which we call input profile here. This regression
problem is difficult for several reasons: large number of examples, -large number
variables (most of which are discrete and multi-valued), non-stationarity of the
distribution, and a conditional distribution of the dependent variable which is very
different from those usually encountered in typical applications .of machine learning
and function approximation. This distribution has a mass at zero: the vast majority
of the insurance contracts do not yield any claim. This distribution is also strongly
asymmetric and it has fat tails (on one side only, corresponding to the large claims).
In this paper we study and compare several learning algorithms along with methods
traditionally used by actuaries for setting insurance premia. The study is performed
on a large database of automobile insurance policies. The methods that were tried
are the following: the constant (unconditional) predictor as a benchmark, linear
regression, generalized linear models (McCullagh and NeIder, 1989), decision tree
models (CHAID (Kass, 1980)), support vector machine regression (Vapnik, 1998),
multi-layer neural networks, mixtures of neural network experts, and the current
premium structure of the insurance company.
In a variety of practical applications, we often find data distributions with an asymmetric heavy tail extending out towards more positive values. Modeling data with
such an asymmetric heavy-tail distribution is essentially difficult because outliers, which are sampled from the tail of the distribution, have a strong influence
on parameter estimation. When the distribution is symmetric (around the mean),
the problems caused by outliers can be reduced using robust estimation techniques
(Huber, 1982; F.R.Hampel et al., 1986; Rousseeuw and Leroy, 1987) which basically
intend to ignore or downweight outliers. Note that these techniques do not work
for an asymmetric distribution: most outliers are on the same side of the mean,
so downweighting them introduces a strong bias on its estimation: the conditional
expectation would be systematically underestimated.
There is another statistical difficulty, due to the large number of variables (mostly
discrete) and the fact that many interactions exist between them. Thus the traditional actuarial methods based on tabulating average claim amounts for combinations of values are quickly hurt by the curse of dimensionality, unless they
make hurtful independence assumptions (Bailey and Simon, 1960). Finally, there
is a computational difficulty: we had access to a large database of ~ 8 x 106 examples, and the training effort and numerical stability of some algorithms can be
burdensome for such a large number of training examples.
This paper is organized as follows: we start by describing the mathematical criteria
underlying insurance premia estimation (section 2), followed by a brief review of the
learning algorithms that we consider in this study, including our best-performing
mixture of positive-output neural networks (section 3). We then highlight our most
important experimental results (section 4), and in view of them conclude with an examination of the prospects for applying statistical learning algorithms to insurance
modeling (section 5).
2
Mathematical Objectives
The first goal of insurance premia modeling is to estimate the expected claim amount
for a given insurance contract for a future one-year period (here we consider that the
amount is 0 when no claim is filed). Let X E Rm denote the customer and contract
input profile, a vector representing all the information known about the customer
and the proposed insurance policy before the beginning of the contract. Let A E R+
denote the amount that the customer claims during the contract period; we shall
assume that A is non-negative. Our objective is to estimate this claim amount,
which is the pure premium Ppure of a given contract x: 1
Ppure(X)
== E[AIX == x].
(1)
The Precision Criterion. In practice, of course, we have no direct access to the
quantity (1), which we must estimate. One possible criterion is to seek the most
precise estimator, which minimizes the mean-squared error (MSE) over a data set
D == {(xl,a?)}r=l. Let P == {p(?;8)} be a function class parametrized by the
IThe pure premium is distinguished from the premium actually charged to the customer, which must account for the risk remaining with the insurer, the administrative
overhead, desired profit, and other business costs.
parameter vector (). The MSE criterion produces the most precise function (on
average) within the class, as measured with respect to D:
L
()*
= argm:n ~ L(P(Xi; (}) -
ai)2.
(2)
i=1
Is it an appropriate criterion and why? First one should note that if PI and P2 are
two estimators of E[AIX]' then the MSE criterion is a good indication of how close
they are to E[AIX], since by the law of iterated expectations,
E[(PI(X) - A)2] - E[(P2(X) - A)2]
== E[(PI(X) - E[AIX])2]
-E[(P2(X) - E[AIX])2],
and of course the expected MSE is minimized when p(X) == E[AIX].
The Fairness Criterion. However, in insurance policy pricing, the precision criterion is not the sole part of the picture; just as important is that the estimated
premia do not systematically discriminate against specific segments of the population. We call this objective the fairness criterion. We define the bias of the premia
b(P) to be the difference between the average premium and the average incurred
amount, in a given population P:
1
(3)
b(P) = 1FT
p(Xi) - ai,
L
(xi,ai)EP
where IPI denotes the cardinality of the set P, and p(.) is some premia estimation
function. A possible fairness criterion would be based on minimizing the norm of
the bias over every subpopulation Q of P. From a practical standpoint, such a
minimization would be extremely difficult to carry out. Furthermore, the bias over
small subpopulations is hard to estimate with statistical significance. We settle
instead for an approximation that gives good empirical results. After training a
model to minimize the MSE criterion (2), we define a finite number of disjoint
subsets (subpopulations) of the test set P, PkC P, P k n Pj:f;k == 0, and verify that
the absolute bias is not significantly different from zero. The subsets Pk can be
chosen at convenience; in our experiments, we considered 10 subsets of equal-size
delimited by the deciles of the test set premium distribution. In this way, we verify
that, for example, for the group of contracts with a premium between the 5th and
the 6th decile, the average premium matches the average claim amount.
3
Models Evaluated
An important requirement for any model of insurance premia is that it should produce positive premia: the company does not want to charge negative money to its
customers! To obtain positive outputs neural networks we have considered
using an exponential activation function at the output layer but this created numerical difficulties (when the argument of the exponential is large, the gradient is
huge). fustead, we have successfully used the "softplus" activation function (Dugas
et al., 2001):
softplus(s) == log(1 + e 8 )
where s is the weighted sum of an output neuron, and softplus(s) is the corresponding predicted premium. Note that this function is convex, monotone increasing, and
can be considered as a smooth version of the "positive part" function max(O, x).
The best model that we obtained is a mixture of experts in which the experts
are positive outputs neural networks. The gater network (Jacobs et al., 1991)
has softmax outputs to obtain positive w~ights summing to one.
X
10-3
Distribution of (claim - prediction) in each prediction quintile
2
1.8
1.6
1.4
1.2
0.8
0.6
0.4
0.2
oL-.._..I....=~-L-~-l...-_----L_----l-==:::=:~::=::::::r:::===:?==~
-3000
-2000
-1000
1000
2000
claim - prediction
3000
4000
5000
6000
Proportion of non-zero claims in each prediction quintile
0.25 r - - - r - - - - - - - , r - - - - - - - - - - , - - - - - - , . - - - - - - - - , - - - ,
0.15
0.1
0.05
3
quintile
Figure 1: A view of the conditional distribution of the claim amounts in the out-ofsample test set. Top: probability density of (claim amount - conditional expectation) for
5 quintiles of the conditional expectation, excluding zero-claim records. The mode moves
left for increasing conditional expectation quintiles. Bottom: proportion of non-zero claim
records per quintile of the prediction.
The mixture model was compared to other models. The constant model only
has intercepts as free parameters. The linear model corresponds to a ridge linear
regression (with weight decay chosen with the validation set). Generalized linear
models (GLM) estimate the conditional expectation from j(x) == eb+w1x with
parameters b and w. Again weight decay is used and tuned on the validation set.
There are many variants of GLMs and they are popular for building insurance
models, since they provide positive outputs, interpretable parameters, and can be
associated to parametric models of the noise.
Decision trees are also used by practitioners in the insurance industry, in particular
the CHAID-type models (Kass, 1980; Biggs, Ville and Suen, 1991), which use
statistical criteria for deciding how to split nodes and when to stop growing the tree.
We have compared our models with a CHAID implementation based on (Biggs, Ville
and Suen, 1991), adapted for regression purposes using a MANOVA analysis. The
threshold parameters were selected based on validation set MSE.
Regression Support Vector Machines (SVM) (Vapnik, 1998) were also evaluated
Mean-Squared Error
67.1192
.......................................................................................... :;....:-.:--'* .....
67.0851
....
-*"------
~~--:-~-:-~~: :.~:-.:- - -~-~--:.- -.~~--'---~ :
.. ..
. . -..
. . _..
..
Test
.
56.5744
Validation
56.5416
56.1108
.
Training
56.0743
Figure 2: MSE results for eight models. Models have been sorted in ascending order
of test results. The training, validation and test curves have been shifted closer together
for visualization purposes (the significant differences in MSE between the 3 sets are due
to "outliers"). The out-of-sample test performance of the Mixture model is significantly
better than any of the other. Validation based model selection is confirmed on test results.
CondMean is a constructive greedy version of GLM.
but yielded disastrous results for two reasons: (1) SVM regression optimizes an L 1 like criterion that finds a solution close to the conditional median, whereas the
MSE criterion is minimized for the conditional mean, and because the distribution
is highly asymmetric the conditional median is far from the conditional mean; (2)
because the output variable is difficult to predict, the required number of support
vectors is huge, also yielding poor generalization. Since the median is actually 0
for our data, we tried to train the SVM using only the cases with positive claim
amounts, and compared the performance to that obtained with the GLM and the
neural network. The SVM is still way off the mark because of the above two reasons.
Figure 1 (top) illustrates the fat tails and asymetry of the conditional distribution
of the claim amounts.
.
Finally, we compared the best statistical model with a proprietary table-based and
rule-based premium estimation method that was provided to us as the benchmark
against which to judge improvements.
4
Experimental Results
Data from five kinds of losses were included in the study (Le. a sub-premium was
estimated for each type of loss), but we report mostly aggregated results showing
the error on the total estimated premium. The input variables contain information
about the policy (e.g., the date to deal with inflation, deductibles and options), the
car, and the driver (e.g., about past claims, past infractions, etc...). Most variables
are subject to discretization and binning. Whenever possible, the bins are chosen
such that they contain approximately the same number of observations. For most
models except CHAID, the discrete variables are one-hot encoded. The number of
input random variables is 39, all discrete except one, but using one-hot encoding this
results in an input vector x of length m == 266. An overall data set containing about
Table 1: Statistical comparison of the prediction accuracy difference between several
individual learning models and the best Mixture model. The p-value is given under the
null hypothesis oino difference between Model #1 and the best Mixture model. Note that
all differences are statistically significant.
Model #1
Model #2
Constant
Mixture
CHAID
Mixture
GLM
Mixture
Softplus NN Mixture
Linear
Mixture
Mixture
NN
Mean MSE Diff.
3.40709e-02
2.35891e-02
7.54013e-03
6.71066e-03
5.82350e-03
5.23885e-03
Std. Error
3.32724e-03
2.57762e-03
1.15020e-03
1.09351e-03
1.32211e-03
1.41112e-03
Z
p-value
10.2400
9.1515
6.5555
6.1368
4.4047
3.7125
0
0
2.77e-ll
4.21e-l0
5.30e-06
1.02e-04
Table 2: MSE difference between benchmark and Mixture models across the 5 claim
categories (kinds of losses) and the total claim amount. In all cases except category 1, the
IvIixture model is statistically significantly (p < 0.05) more precise than the benchmark
model.
Claim Category
(Kind of Loss)
Category 1
Category 2
Category 3
Category 4
Category 5
Total claim amount
MSE Difference
Benchmark minus Mixture
20669.53
1305.57
244.34
1057.51
1324.31
60187.60
95% Confidence Interval
Lower
Higher
(-4682.83 - 46021.89 )
(1032.76 1578.37 )
(6.12 482.55 )
(623.42 1491.60 )
(1077.95 1570.67 )
( 7743.96 - 112631.24)
8 million examples is randomly permuted and split into a training set, validation
set and test set, respectively of size 50%, 25% and 25% of the total. The validation
set is used to select among models (includi~g the choice of capacity), and th~ test
set is used for final statistical comparisons. Sample-wise paired statistical tests are
used to reduce the effect of huge per-sample variability.
Figure 1 is an attempt at capturing the shape of the conditional distribution of claim
amounts given input profiles, by considering the distributions of claim amounts in
different quantiles of the prediction (pure premium), on the test set. The top figure
excludes the point mass of zero claims and rather shows the difference between the
claim amount and the estimated conditional expectation (obtained with the mixture
model). The bottom histogram shows that the fraction of claims increases nicely
for the higher predicted pure premia.
Table 1 and Figure 2 summarize the comparison between the test MSE of the different tested models. NN is a neural network with linear output activation whereas
Softplus NNhas the softplus output activations. The Mixture is the mixture of softplus neural networks. This result identifies the mixture model with softplus neural
networks as the best-performing of the tested statistical models. Our conjecture is
that the mixture model works better because it is more robust to the effect of "outliers" (large claims). Classical robust regression methods (Rousseeuw and Leroy,
1987) work by discarding or downweighting outliers: they cannot be applied here
because the claims distribution is highly asymmetric (the extreme values are always
large ones, the claims being all non-negative). Note that the capacity of each model
has been tuned on the validation set. Hence, e.g. CHAID could have easily yielded
lower training error, but at the price of worse generalization.
x10
4
Rule-Based minus UdeM Mixture
2,...-------,------,-----r-------.-------,-----.---------.----.......,
Mean = -1.5993e-1 a
Median = 37.5455
... Stddev = 154.65
-
1.5
~
o
c(])
:::l
0(])
u:
0.5
OL.-..----L.----L----.L.----..L---~~
-3000
-2500
-2000
-1500
-1000
-500
Difference between premia ($)
o
500
1000
Figure 3: The premia difference distribution is negatively skewed, but has a positive
m~dian for a mean of zero. This implies that the benchmark model (current pricing)
undercharges risky customers, while overcharging typical customers.
Table 2 shows a comparison of this model against the rule-based benchmark. The
improvements are shown across the five types of losses. In all cases the mixture
improves, and the improvement is significant in four out of the five as well as across
the sum of the five.
A qualitative analysis of the resulting predicted premia shows that the mixture
model has smoother and more spread-out premia than the benchmark. The analysis (figure 3) also reveals that the difference between the mixture premia and the
benchmark premia is negatively skewed, with a positive median, i.e., the typical customer will pay less under the new mixture model, but the "bad" (risky) customers
will pay much more.
To evaluate fairness, as discussed in the previous section, the distribution of premia computed by the best model is analyzed, splitting the contracts in 10 groups
according to their premium level. Figure 4 shows that the premia charged are fair
for each sub-population.
5
Conclusion
This paper illustrates a successful data-mining application in the insurance industry.
It shows that a specialized model (the mixture model), that was designed taking
into consideration the specific problem posed by the data (outliers, asymmetric distribution, positive outputs), performs significantly better than existing and popular
learning algorithms. It also shows that such models can significantly improve over
the current practice, allowing to compute premia that are lower for less risky contracts and higher for more risky contracts, thereby reducing the cost of the median
contract.
Future work should investigate in more detail the role of temporal pon-stationarity,
how to optimize fairness (rather than just test for it afterwards), and how to further
increase the robustness of the model with respect to large claim amounts.
Difference with incurred claims (sum of all KOL-groups)
200
............ 0
?
..
..
.
0
0
.'
..
..
~
C/}
E
~
0
"'C
~
:5
"~ -200
-5
I
:?
:
:1'
:
?1????????? .\.
.
\
. .
.
.
.......:?
?
\
\
"?
CD
~ -400
CD
~
o
-600
-B- Mixture Model (normalized premia)
-* - Rule-Based Model (normalized premia)
2
4
6
8
10
Decile
Figure 4: We ensure fairness by comparing the average incurred amount and premia
within each decile of the premia distribution; both models are generally fair to subpopu1ations. The error bars denote 95% confidence intervals. The comparisqn is for the sum
of claim amounts over all 5 kinds of losses (KOL).
References
Bailey, R. A. and Simon, L. (1960). Two studies in automobile insurance ratemaking. ASTIN Bulletin, 1(4):192-217.
Biggs, D., Ville, B., and Suen, E. (1991). A method of choosing multiway partitions
for classification and decision trees. Journal of Applied Statistics, 18(1):49-62.
Dugas, C., Bengio, Y., Belisle, F., and Nadeau, C. (2001). Incorporating second
order functional? knowledge into learning algorithms. In Leen, T., Dietterich,
T., and Tresp, V., editors, Advances in Neural Information Processing Systems,
volume 13, pages 472-478.
F.R.Hampel, E.M.Ronchetti, P.J.Rousseeuw, and W.A.Stahel (1986). Robust
Statistics, The Approach based on Influence Functions. John Wiley & Sons.
Huber, P. (1982). Robust Statistics. John Wiley & Sons Inc.
Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive
mixture of local experts. Neural Computation, 3:79-87.
Kass, G. (1980). An exploratory technique for investigating large quantities of
categorical data. Applied Statistics, 29(2):119-127.
McCullagh, P. and NeIder, J. (1989). Generalized Linear Models. Chapman and
Hall, London.
'
Rousseeuw, P. and Leroy, A. (1987). Robust Regression and Outlier Detection. John
Wiley & Sons Inc.
Vapnik, V. (1998). Statistical Learning Theory. Wiley, Lecture Notes in Economics
and Mathematical Systems, volume 454.
| 2062 |@word version:2 norm:1 proportion:2 seek:1 tried:2 jacob:2 ronchetti:1 thereby:1 profit:1 minus:2 carry:1 tuned:2 past:2 existing:1 ka:3 current:3 discretization:1 comparing:1 nowlan:1 activation:4 must:2 john:3 numerical:2 partition:1 shape:2 designed:1 interpretable:1 greedy:1 selected:1 beginning:1 argm:1 stahel:1 record:2 node:1 ipi:1 five:4 mathematical:4 along:1 direct:1 driver:1 qualitative:1 overhead:1 huber:2 expected:5 growing:1 multi:2 ol:2 company:2 curse:1 cardinality:1 increasing:2 considering:1 provided:1 estimating:4 underlying:1 mass:2 null:1 kind:4 minimizes:1 temporal:1 every:1 charge:1 fat:3 rm:1 decile:4 positive:12 before:1 local:1 encoding:1 meng:1 approximately:1 eb:1 statistically:2 practical:2 practice:2 gater:1 empirical:1 significantly:5 confidence:2 subpopulation:3 cannot:1 close:2 selection:1 convenience:1 context:1 influence:2 applying:1 risk:1 intercept:1 optimize:2 customer:10 charged:3 pon:1 economics:1 convex:1 qc:1 splitting:1 pure:5 estimator:2 rule:4 stability:1 population:3 exploratory:1 traditionally:1 hurt:1 hypothesis:1 asymmetric:8 std:1 database:2 binning:1 ep:1 ft:1 bottom:2 role:1 prospect:1 gross:1 segment:1 ithe:1 negatively:2 basis:1 biggs:3 easily:1 succ:1 train:1 actuarial:1 london:1 choosing:1 quintiles:2 encoded:1 posed:1 valued:1 statistic:4 final:1 indication:1 dian:1 interaction:1 date:1 requirement:1 extending:1 produce:2 montreal:2 measured:1 sole:1 p2:3 strong:2 predicted:3 judge:1 implies:1 settle:1 bin:1 generalization:2 around:1 considered:3 inflation:1 hall:1 deciding:1 predict:1 claim:34 purpose:2 estimation:6 successfully:1 weighted:1 minimization:1 suen:3 always:1 rather:2 l0:1 improvement:3 burdensome:1 inference:1 dependent:1 nn:3 overall:1 among:1 classification:1 pascal:1 softmax:1 equal:1 nicely:1 chapman:1 fairness:6 future:2 minimized:2 yoshua:1 report:1 few:1 randomly:1 individual:1 attempt:1 detection:1 stationarity:2 huge:3 highly:2 mining:1 investigate:1 insurance:20 w1x:1 introduces:1 mixture:29 extreme:1 analyzed:1 yielding:1 unconditional:1 peculiar:1 closer:1 unless:1 tree:5 desired:1 industry:2 modeling:3 insured:2 cost:3 subset:3 predictor:1 successful:1 quintile:4 density:1 filed:1 contract:14 off:1 aix:6 together:1 quickly:1 squared:3 again:1 containing:1 actuary:2 worse:1 expert:5 account:1 inc:2 caused:1 performed:1 view:2 neider:2 ichiro:1 start:1 option:1 astin:1 simon:2 minimize:1 takeuchi:2 accuracy:1 chapados:1 yield:1 vincent:1 iterated:1 basically:1 confirmed:1 whenever:1 against:3 associated:1 sampled:1 stop:1 popular:2 knowledge:1 car:3 dimensionality:1 improves:1 organized:1 actually:2 delimited:1 higher:3 leen:1 evaluated:2 strongly:1 furthermore:1 just:2 glms:1 ghosn:1 mode:1 pricing:2 tabulating:1 building:1 effect:2 dietterich:1 verify:2 contain:2 normalized:2 hence:1 symmetric:1 deal:1 ll:1 during:1 skewed:2 criterion:14 generalized:4 ridge:1 performs:1 cp:1 wise:1 consideration:1 charles:1 umontreal:1 specialized:1 permuted:1 functional:1 volume:2 million:1 tail:6 discussed:1 significant:3 ai:3 centre:1 multiway:1 had:1 access:2 money:1 etc:1 base:1 pkc:1 optimizes:1 irq:1 aggregated:1 period:2 smoother:1 afterwards:1 x10:1 smooth:1 match:1 dept:1 paired:1 prediction:7 variant:1 regression:11 essentially:1 expectation:7 histogram:1 whereas:2 want:1 interval:2 underestimated:1 median:7 standpoint:1 subject:1 jordan:1 call:2 practitioner:1 bengio:2 split:2 variety:1 independence:1 reduce:2 effort:1 proprietary:1 generally:1 amount:20 rousseeuw:4 category:8 reduced:1 exist:1 shifted:1 estimated:4 disjoint:1 per:2 discrete:5 shall:1 group:3 four:1 threshold:1 downweighting:2 pj:1 kol:2 vast:1 excludes:1 ville:4 monotone:1 year:1 sum:4 fraction:1 decision:4 capturing:1 layer:2 pay:2 followed:1 encountered:1 yielded:2 leroy:3 adapted:1 argument:1 extremely:1 performing:2 downweight:1 conjecture:1 according:1 combination:1 poor:1 across:3 son:3 outlier:9 glm:4 visualization:1 describing:1 ascending:1 available:1 eight:1 appropriate:1 bailey:2 distinguished:1 robustness:1 denotes:1 remaining:1 include:1 top:3 ensure:1 classical:1 objective:3 intend:1 move:1 quantity:2 parametric:1 traditional:1 gradient:1 capacity:2 majority:2 parametrized:1 iro:1 reason:4 length:1 minimizing:1 difficult:5 mostly:2 disastrous:1 negative:3 implementation:1 policy:5 allowing:1 neuron:1 observation:1 hurtful:1 benchmark:9 finite:1 dugas:3 hinton:1 excluding:1 precise:3 variability:1 canada:1 required:1 bar:1 usually:1 ofsample:1 vincentp:1 summarize:1 including:1 max:1 charging:1 hot:2 difficulty:3 examination:1 hampel:2 business:1 representing:1 improve:1 brief:1 risky:6 picture:1 identifies:2 created:1 nadeau:1 categorical:1 tresp:1 faced:1 review:1 law:1 loss:7 lecture:1 highlight:1 validation:9 incurred:3 editor:1 systematically:2 pi:3 heavy:2 cd:2 course:2 free:1 l_:1 side:2 bias:5 taking:1 bulletin:1 absolute:1 curve:1 adaptive:1 ights:1 far:1 belisle:1 ignore:1 unreliable:1 reveals:1 investigating:1 summing:1 conclude:1 xi:3 why:1 table:5 robust:6 nicolas:1 ca:1 mse:13 automobile:2 significance:1 main:1 pk:1 spread:1 noise:2 profile:3 fair:2 quantiles:1 wiley:4 precision:2 sub:2 exponential:2 xl:1 administrative:1 bad:1 specific:2 discarding:1 showing:1 decay:2 svm:4 incorporating:1 vapnik:3 illustrates:2 manova:1 corresponds:1 conditional:15 goal:1 sorted:1 towards:1 price:1 hard:1 mccullagh:2 included:1 typical:3 except:3 diff:1 reducing:1 called:1 total:4 discriminate:1 experimental:2 premium:17 select:1 support:4 softplus:8 mark:1 constructive:1 evaluate:1 tested:2 |
1,166 | 2,063 | Thomas L . Griffiths & Joshua B. Tenenbaum
Department of Psychology
Stanford University, Stanford, CA 94305
{gruffydd,jbt}?psych. stanford. edu
Abstract
Estimating the parameters of sparse multinomial distributions is
an important component of many statistical learning tasks. Recent
approaches have used uncertainty over the vocabulary of symbols
in a multinomial distribution as a means of accounting for sparsity.
We present a Bayesian approach that allows weak prior knowledge,
in the form of a small set of approximate candidate vocabularies,
to be used to dramatically improve the resulting estimates. We
demonstrate these improvements in applications to text compression and estimating distributions over words in newsgroup data.
1
Introduction
Sparse multinomial distributions arise in many statistical domains, including natural language processing and graphical models. Consequently, a number of approaches to parameter estimation for sparse multinomial distributions have been
suggested [3]. These approaches tend to be domain-independent: they make little
use of prior knowledge about a specific domain. In many domains where multinomial distributions are estimated there is often at least weak prior knowledge about'
the potential structure of distributions, such as a set of hypotheses about restricted
vocabularies from which the symbols might be generated. Such knowledge can be
solicited from experts or obtained from unlabeled data. We present a method for
Bayesian_parameter estimation in sparse discrete domains that exploits this weak
form of prior knowledge to improve estimates over knowledge-free approaches.
1.1
Bayesian parameter estimation for multinomial distributions
Following the presentation in [4], we consider a language ~ containing L distinct symbols. A multinomial distribution is specified by a parameter vector
f) == (Ol, ... ,f)L), where f)i is the probability of an observation being symbol i.
Consequently, we have the constraints that Ef==l f)i == 1 and (h ~ 0, i == 1, ... ,L.
The task of multinomial estimation is to take a data set D and produce a'vector f)
that results in a good approximation to the distribution that produced D. In this
case, D consists of N independent observations Xl, ... x N drawn from the distribution to be estimated, which can be summarized by the statistics N i specifying the
number of times the ith symbol occurs in the data. D also determines the set ~o
of symbols that occur in the data.
Stated in this way, multinomial estimation involves predicting the next observation
based on the data. Specifically, we wish to calculate P(XN+1ID). The Bayesian
estimate for this probability is given by
PL(xN+lID)
=
I
p(XN+1IB)P(BID)dB
where P(X N + 1 10) follows from the multinomial distribution corresponding to O. The
posterior probability P(OID) can be obtained via Bayes rule
L
P(OID) oc P(DIO)P(O) == P(8}
II ONi
i==l
where P(O) is the prior probability of a given O.
Laplace used this method with a uniform prior over 0 to give the famous "law of
succession" [6J. A more general approach is to assume a Dirichlet prior over (),
which is conjugate to the multinomial distribution and gives
N i +LCY.i
P(XN+l = ilD) =
N
(1)
+ l:j==l O!.j
where the ai are the hyperparameters of the Dirichlet distribution. Different estimates are obtained for different choices of the ai, with most approaches making the
simplifying assumption that ai == O!. for all i. Laplace's law results from a == 1. The
case with a == 0.5 is the Jeffreys-Perks law or Expected Likelihood Estimation [2]
[5J [9J, while using arbitrary O!. is Lidstone's law [7].
1.2
EstiIllating sparse Illultinomial distributions
Several authors have extended the Bayesian approach to sparse multinomial distributions, in which only a restricted vocabulary of symbols are used, by maintaining
uncertainty over these vocabularies. In [10], Ristad uses assumptions about the
probability of strings based upon different vocabularies to give the estimate
PR (X N +1
== ilD) ==
(Ni + l)/(N +L)
(Ni + l)(N + 1 - kO)/(N 2 + N + 2kO)
{ kO(kO + l)/(L - kO)(N 2 + N + 2kO)
if kO == L
if kO < L 1\ N i
otherwise
>0
where kO == I~o I is the size of the smallest vocabulary consistent with the data.
A different approach is taken by Friedman and Singer in [4], who point out that
Ristad's method is a special case of their framework. Friedman and Singer consider
the vocabulary V ~ :E to be a random variable, allowing them to write
p.(X N +1 == ilD) ==
L p(X
N
+1 == ilV, D)P(VID)
(2)
v
where P{X N + 1 == ilV, D) results from a Dirichlet prior over the symbols in V,
p(X N +1 == ilV, D) ==
{~it'ja
o
v:
if i E
otherWIse
(3)
and by Bayes' rule and the properties of Dirichlet priors
P(VID)
oc
P(DIV)P(V)
{ ~fJ~~(a) niE~O r(~?t~a) P(V)
EO ~ V
otherwise
( )
4
Friedman and Singer assume a hierarchical prior over V, such that all vocabularies
of cardinality k are given equal probability, namely P(S == k)/(t), where P(S == k)
is the probability that the size of the vocabulary (IVI) is k. It follows that if i E ~o,
p(X N + I == ilD) == Lk :+1~P(S == kiD). If i ? ~o, it is necessary to estimate the
proportion of V that contain i for a given k. The simplified result is
PF(X N +1
== ilD) == { %tt~aC(D,L)
L-k O (1-
C(D,L))
if i E ~o
otherwise
(5)
where
.h
WIt
2
mk
P(S == k) (k-kO)!?
- k!
==.
r(ka:)
r(N+ka:) .
Ivlaking use of weak prior knowiedge
Friedman and Singer assume a prior that gives equal probability to all vocabularies
of a given cardinality. However, many real-world tasks provide limited knowledge
about the structure of distributions that we can build into our methods for parameter estimation. In the context of sparse multinomial estimation, one instance of
such knowledge the importance of specific vocabularies. For example, in predicting
the next character in a file, our predictions could be facilitated by considering the
fact that most files either use a vocabulary consisting of ASCII printing characters
(such as text files), or all possible characters (suc~ as object files). This kind of
structural knowledge about a domain is typically easier to solicit from experts than
accurate distributional information, and forms a valuable informational resource.
If we have this kind of prior knowledge, we can restrict our attention to a subset
of the 2L possible vocabularies. fu particular, we can specify a set of vocabularies
V which we consider as hypotheses for the vocabulary used in producing D, where
the elements of V are specified by our knowledge of the domain. This stands as
a compromise between Friedman and Singer's approach, in which V consists of
all vocabularies, and traditional Bayesian parameter estimation as represented by
Equation 1, in which V consists of only the vocabulary containing all words. To
do this, we explicitly evaluate the sum given in Equation 2, where the sum over V
includes all V E V. This sum remains tractable when V is a small subset of the
possible vocabularies, and the efficiency is aided by the fact that P(DIV) shares
common terms across all V which can cancel in normalization.
The intuition behind this approach is that it attempts to classify the target distribution as using one of a known set of vocabularies, where the vocabularies are obtained
either from experts or from unlabeled data. Applying standard Bayesian multinomial estimation within this vocabulary gives enough flexibility for the method to
capture a range of distributions, while making use of our weak prior knowledge.
2.1
An illustration: Text compression
Text compression is an effective test of methods for multinomial estimation. Adaptive coding can be performed by specifying a method for calculating a distribution
over the probability of the next byte in a file based upon the preceding bytes [1].
The extent to which the file is compressed depends upon the quality of these predictions. To illustrate the utility of including prior knowledge, we follow Ristad in
using the Calgary text compression corpus [1]. This corpus consists of 19 files of
Table 1: Text compression lengths (in bytes) on the Calgary corpus
file
bib
book1
book2
geo
nellS
obj1
obj2
paper1
paper2
paper3
paper4
paper5
paper6
pic
progc
progl
progp
trans
size
kO
NH(Ni/ N )
Pv
PF
PR
PL
PJ
111261
768771
610856
102400
377109
21504
246814
53161
82199
46526
13286
11954
38105
513216
39611
71646
49379
93695
81
82
96
256
98
256
256
95
91
84
80
91
93
159
92
87
89
99
72330
435043
365952
72274
244633
15989
193144
33113
47280
27132
7806
7376
23861
77636
25743
42720
30052
64800
18
219
94
161
89
126
182
71
75
70
58
57
68
205
68
74
71
169
89
105
115
162
113
127
184
94
94
85
72
79
90
92
116
124
165
116
129
190
100
105
92
79
83
95
216
91
97
94
105
269
352
329
165
304
129
189
236
259
238
190
181
223
323
222
253
236
252
174
219
212
161
201
126
182
156
167
154
126
122
149
205
150
164
155
169
16~
- 89
91
89
101
several different types, each using some subset of 256 possible characters (L == 256).
The files include Bib'IEXsource (bib), formatted English text (book*, paper*), geological data (geo), newsgroup articles (news), object files (obj*), a bit-mapped
picture (pic), programs in three different languages (prog*) and a terminal transcript (trans). The task was to estimate the distribution from which characters in
the file were drawn based upon the first N characters and thus predict the N + 1st
character. Performance was measured in terms of the length of the resulting file,
where the contribution of the N + 1st character to the length is log2 P(XN+lID).
The results are expressed as the number of bytes required to encode the file relative
to the empirical entropy NH(Ni/N) as assessed by Ristad [10].
Results are shown in Table 1. P v is the restricted vocabulary model outlined above,
with V consisting of just two hypotheses: one corresponding to binary files, containing all 256 characters, and one consisting of a 107 character vocabulary representing
formatted English. The latter vocabulary was estimated from 5MB of English text,
C code, Bib'IEXsource, and newsgroup data from outside the Calgary corpus. PF
is Friedman and Singer's method. For both of these approaches, a was set to 0.5,
to allow direct comparison to the Jeffreys-Perks law, PJo PR and PL are Ristad's
and Laplace's laws respectively. P y outperformed the other methods on all files
based upon English text, bar bookl, and all files using all 256 symbols l . The high
performance followed from rapid classification of these files as using the appropriate
vocabulary in V. When the vocabulary included all symbols Py performed as PJ,
which gave the best predictions for these files.
1 A number of excellent techniques for? text compression exist that outperform all of
those presented here. We have not included these techniques for comparison because our
interest is in using text compression as a means of assessing estimation procedures, rather
than as an end in itself. We thus consider only methods for multinomial estimation as our
comparison. group.
2.2
Maintaining uncertainty in vocabularies
The results for book1 illustrate a weakness of the approach outlined above. The file
length for P y is higher than those for PF and PR , despite the fact that the file uses a
text-based vocabulary. This file contains two characters that were not encountered
in the data used to construct V. These characters caused P y to default to the
unrestricted vocabulary of all 256 characters. From that point P y corresponded to
PJ, which gave poor results on this file.
This behavior results from the assumption that the candidate vocabularies in V are
completely accurate. Since in many cases the knowledge that informs the vocabularies in V may be imperfect, it is desirable to allow for uncertainty in vocabularies.
This uncertainty will be reflected in the fact that symbols outside V are expected
to occur with a vocabulary-specific probability ty,
p(XN+1 == ilV, D) == { (1 - (L -IVI)ty) N~~t~la
ty
where Ny ==
I:iEY
P(DIV)
if i E V
otherwise
N i ? It follows that
= (1 -
r(Ni + a)
r(a:)
r(IVla)
+ 1V1a:)
v
(L -IVJ)?V)NV ?t"-N r(N
y
which replaces Equations 3-4 in specifying P y
iE:EonY
.
When V is determined by the judgments of domain experts, ty is the probability
that an unmentioned word actually belongs to a particular vocabulary. While it
may not be the most efficient use of such data, the V E V can also be estimated from
some form of unlabeled data. In this case, Friedman and Singer's approach provides
a means of setting ty. Friedman and Singer explicitly calculate the probability that
an unseen word is in V based upon a dataset: from the second condition of Equation
5, we find that we should set ty == L_1IYI (1- C(D, L)). We use this method below.
3
Bayesian parameter estimation in natural language
Statistical natural language processing often uses sparse multinomial distributions
over large vocabularies of words. In different contexts, different vocabularies will be
used. By specifying a basis set of vocabularies, we can perform parameter estimation
by classifying distributions according to their vocabulary. This idea was examined
using data from 20 different Usenet newsgroups. This dataset is commonly used
in testing text classification algorithms (eg. [8]). Ten newsgroups were used to
estimate a set of vocabularies V with corresponding ty. These vocabularies were
used in estimating multinomial distributions on these newsgroups and ten others.
The dataset was 20news-18827, which consists of the 20newsgroups data with
headers and duplicates removed, and was preprocessed to remove all punctuation,
capitalization, and distinct numbers. The articl~s in each of the 20 newsgroups were
then divided into three sets. The first 500 articles from ten newsgroups were used to
estimate the candidate vocabularies V and uncertainty parameters ty. Articles 501700 for all 20 newsgroups were used as training data for multinomial estimation.
Articles 701-900 for all 20 newgroups were used as testing data. Following [8],
a dictionary was built up by running over the 13,000 articles resulting from this
division, and all words that occurred only once were mapped to an "unknown"
word. The resulting dictionary contained L == 54309 words.
As before, the restricted vocabulary method (Py), Friedman and Singer's method
(PF ), and Ristad's (PR ), Laplace's (PL ) and the Jeffreys-Perks (PJ ) laws were ap-
alt.atheism
talk.politics.mideast
18
....
11
"
talk.politics. mise -
18
.
17
'.
16
scLspace
rec.motorcycles
~~~
~~~
~~~
~~~
soc.religion.christian
talk.politics.guns
comp.sys.ibm.pc.hardware
rec.sport.hockey
scLelectronics
comp.windows.x
rec.autos
rec.sport.baseball
scLcrypt
scLmed
comp.os.ms-windows.misc
misc.forsale
~::=:= ~-;""",
r.?';'~;:;"
~~~
comp.sys.mac.hardware
100
10000
Number of words
50000
talk.religion.misc
comp.graphics
~~~
100
10000
F~gure 1: Cross-entropy of predictions on newsgroup data as a function of the
logarithm of the number of words. The abscissa is at the empirical entropy of the
test distribution. The top ten panels (talk.polities.mideast and those to its
right) are for the newsgroups with unknown vocabularies. The bottom ten are for
those that contributed vocabularies to V, trained and tested on novel data. PL and
P J are both indicated with dotted lines, but P J always performs better than PL.
The box on talk.polities.mideast indicates the point at which Pv defaults to
the full vocabulary, as the number of unseen words makes this vocabulary more
likely. At this point, the line for Pv joins the line for P J , since both methods give
the same estimates of the distribution.
plied to the task. Both P v and PF used a == 0.5 to facilitate comparison with
P J . 'V featured one vocabulary that contained all words in the dictionary, and
ten vocabularies each corresponding to the words used in the first 500 articles of
one of the newsgroups designated for this purpose. ?y was estimated as outlined
above. Testing for each newsgroup consisted of taking words from the 200 articles
assigned for training purposes, estimating a. distribution using each method, and
then computing the cross-entropy between that distribution and an empirical estimate of the true distribution. The cross-entropy is H{Q; P) == Ei Qi log2 Pi, where
Q is the true distribution and P is the distribution produced by the estimation
method. Q was given by the maximum likelihood estimate formed from the word
frequencies in all 200 articles assigned for testing purposes. The testing procedure
was conducted with just 100 words, and then in increments of 450 up to a total
of 10000 words. Long-run performance was examined on talk.polities.mideast
and talk.polities.mise, each trained with 50000 words.
The results are shown in Figure 1. As expected, P y consistently outperformed
the other methods on the newsgroups that contributed to V. However, performance on novel newsgroups was also greatly improved. As can be seen in Figure
2, the novel newsgroups were classified to appropriate vocabularies - for example
all words
rec.autos
I-----------------rec.motorcycles
rec.sport.baseball
scLcrypt
scLmed
ta1k.politics.guns
talk.politics.mideast
r l \ - - _ f - - - - - - - T - - - - ' r - - - - - - - - - - - alt.atheism
f r t - ' \ . ; : : : : : : : : : : : ; f : = . = ' = f - - t - - - - - - - - - - soc.religion.christian
r--+--
talk. politics. misc
,-------T---+-----------
talk. religion. misc
l-+-if-Hf-------t----------
,--
misc.forsale
scLspace
~~?.;~~~~~~g~ey
comp.sys.mac.hardware
comp.os.ms-windows.m~c'-----------------com~sy&ibm.p~hardware
comp.graphics
o
10000
Number of words
Figure 2: Classification of newsgroup vocabularies. The lines illustrate the vocabulary which had maximum posterior probability for each of the ten test newsgroups
after exposure to differing numbers of words. The vocabularies in V are listed along
the left hand side of the axis, and the lines are identified with newsgroups by the
labels on the right hand side. Lines are offset to facilitate identification.
talk.religion.misc had the highest posterior probability for alt.atheism and
soc. religion. christian, while rec. autos had highest posterior probability for
rec .motorcycles. The proportion of word types occurring in the test data but
not the vocabulary to which the novel newsgroups were classified ranged between
30.5% and 66.2%, with a mean of 42.2%. This illustrates that even approximate
knowledge can facilitate predictions: the basis set of vocabularies allowed the high
frequency words in the data to be modelled effectively, without excess mass being
attributed to the low frequency novel word tokens.
Long-run performance on talk.politics .mideast illustrates the same defaulting
behavior that was shown for text classification: when the data become more probable under the vocabulary containing all words than under a restricted vocabulary
the method defaults to the Jeffreys-Perks law. This guarantees that the method will
tend to perform no worse than P J when unseen words are encountered in sufficient
proportions. This is desirable, since PJ gives good estimates once N becomes large.
4
Discussion
Bayesian approaches to parameter estimation for sparse multinomial distributions
have employed the notion of a restricted vocabulary from which symbols are produced. In many domains where such distributions are estimated; there is often at
least limited knowledge about the structure of these vocabularies. By considering
just the vocabularies suggested by such knowledge, together with some uncertainty
concerning those vocabularies, we can achieve very good estimates of distributions in
these domains. We have presented a Bayesian approach that employs limited prior
knowledge, and shown that it outperforms a range of approaches to multinomial
estimation for both text compression and a task involving natural language.
While our applications in this paper estimated approximate vocabularies from data,
the real promise of this approach lies with domain knowledge solicited from experts.
Experts are typically better at providing qualitative structural information than
quantitative distributional information, and our approach provides a way of using
this information in parameter estimation. For example, in the context of parameter estimation for graphical models to be used in medical diagnosis, distinguishing
classes of symptoms might be informative in determining the parameters governing
their relationship to diseases. This form of knowledge is naturally translated into
a set of vocabularies to be considered for each such distribution. More complex
applications to natural language lllay also be possible, such as using syntactic information in estimating probabilities for n-gram models. The approach we have
presented in this paper provides a simple way to allow this kind of limited domain
knowledge to be useful in Bayesian parameter estimation.
References
[1] T. C. Bell, J. G. Cleary, and 1. H. Witten. Text compression. Prentice Hall, 1990.
[2] G. E. P. Box and G. C. Tiao. Bayesian Inference in Statistical Analysis. AddisonWesley, 1973.
[3] S. F. Chen and J. Goodman. An empirical study of smoothing techniques for language
modeling. Technical Report TR-10-98, Center for Research in Computing Technology,
Harvard University, 1998.
[4] N. Friedman and Y. Singer. Efficient Bayesian parameter estimation in large discrete
domains. In Neural Information Processing Systems, 1998.
[5] H. Jeffreys. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society A, 186:453-461, 1946.
[6] P.-S. Laplace. Philosophical Essay on Probabilities. Springer-Verlag, 1995. Originally
published 1825.
[7] G. Lidstone. Note on the general case of the Bayes-Laplace formula for inductive or
a posteriori probabilities. Transactions of the Faculty of Actuaries, 8:182-192, 1920.
[8] K. Nigam, A. K. Mccallum, S. Thrun, and T. Mitchell. Text classification fro'in labeled
and unlabeled documents using EM. Machine Learning, 39:103-134, 2000.
[9] W. Perks. Some observations on inverse probability, including a new indifference rule.
Journal of the Institute of Actuaries, 73:285-312, 1947.
[10] E. S. Ristad. A natural law ?of succession. Technical Report CS-TR-895-95, Department of Computer Science, Princeton University, 1995.
| 2063 |@word faculty:1 compression:9 proportion:3 essay:1 accounting:1 simplifying:1 tr:2 cleary:1 contains:1 document:1 outperforms:1 ka:2 com:1 informative:1 christian:3 remove:1 mccallum:1 sys:3 ith:1 gure:1 provides:3 along:1 direct:1 become:1 qualitative:1 consists:5 expected:3 rapid:1 behavior:2 abscissa:1 ol:1 terminal:1 informational:1 little:1 pf:6 cardinality:2 considering:2 window:3 becomes:1 estimating:5 panel:1 mass:1 kind:3 psych:1 string:1 differing:1 guarantee:1 quantitative:1 medical:1 producing:1 before:1 book2:1 despite:1 usenet:1 id:1 ap:1 might:2 examined:2 specifying:4 limited:4 range:2 testing:5 polity:4 procedure:2 featured:1 empirical:4 bell:1 word:26 griffith:1 unlabeled:4 prentice:1 context:3 applying:1 py:2 center:1 exposure:1 attention:1 wit:1 rule:3 notion:1 increment:1 laplace:6 target:1 us:3 distinguishing:1 hypothesis:3 harvard:1 element:1 rec:9 distributional:2 labeled:1 bottom:1 capture:1 calculate:2 news:2 removed:1 highest:2 valuable:1 disease:1 lidstone:2 intuition:1 nie:1 trained:2 ristad:7 compromise:1 upon:6 division:1 efficiency:1 baseball:2 completely:1 basis:2 translated:1 vid:2 represented:1 talk:13 distinct:2 effective:1 corresponded:1 oni:1 outside:2 header:1 stanford:3 otherwise:5 compressed:1 obj2:1 statistic:1 unseen:3 syntactic:1 itself:1 mb:1 motorcycle:3 flexibility:1 achieve:1 paper2:1 assessing:1 produce:1 object:2 illustrate:3 informs:1 ac:1 measured:1 transcript:1 soc:3 c:1 involves:1 paper5:1 bib:4 dio:1 ja:1 paper1:1 probable:1 pl:6 considered:1 hall:1 predict:1 dictionary:3 forsale:2 smallest:1 ilv:4 purpose:3 estimation:24 outperformed:2 label:1 always:1 rather:1 progc:1 encode:1 kid:1 improvement:1 consistently:1 likelihood:2 indicates:1 greatly:1 posteriori:1 inference:1 typically:2 classification:5 smoothing:1 special:1 equal:2 construct:1 once:2 cancel:1 others:1 report:2 duplicate:1 employ:1 consisting:3 friedman:10 attempt:1 progp:1 interest:1 paper3:1 weakness:1 punctuation:1 pc:1 behind:1 accurate:2 fu:1 necessary:1 solicited:2 addisonwesley:1 logarithm:1 mk:1 instance:1 classify:1 book1:2 newgroups:1 modeling:1 geo:2 mac:2 subset:3 uniform:1 conducted:1 graphic:2 st:2 ie:1 together:1 ivj:1 containing:4 actuary:2 worse:1 book:1 expert:6 potential:1 summarized:1 coding:1 includes:1 explicitly:2 caused:1 depends:1 performed:2 bayes:3 hf:1 contribution:1 formed:1 ni:5 who:1 succession:2 judgment:1 sy:1 weak:5 bayesian:12 famous:1 identification:1 modelled:1 produced:3 comp:8 solicit:1 published:1 classified:2 ty:8 frequency:3 naturally:1 attributed:1 paper6:1 dataset:3 mise:2 mitchell:1 knowledge:21 actually:1 higher:1 originally:1 follow:1 reflected:1 specify:1 improved:1 box:2 symptom:1 just:3 governing:1 hand:2 ei:1 ild:5 o:2 quality:1 indicated:1 facilitate:3 contain:1 consisted:1 true:2 ranged:1 inductive:1 assigned:2 jbt:1 misc:7 eg:1 oc:2 m:2 tt:1 demonstrate:1 oid:2 performs:1 fj:1 ef:1 novel:5 common:1 witten:1 multinomial:21 nh:2 occurred:1 ai:3 outlined:3 language:8 had:3 posterior:4 recent:1 belongs:1 verlag:1 formatted:2 binary:1 joshua:1 seen:1 unrestricted:1 preceding:1 eo:1 ey:1 employed:1 ii:1 full:1 desirable:2 technical:2 cross:3 long:2 divided:1 concerning:1 qi:1 prediction:5 involving:1 ko:11 normalization:1 goodman:1 ivi:2 file:22 nv:1 capitalization:1 tend:2 db:1 obj:1 structural:2 enough:1 bid:1 newsgroups:15 psychology:1 gave:2 restrict:1 identified:1 imperfect:1 idea:1 politics:7 defaulting:1 utility:1 dramatically:1 useful:1 paper4:1 listed:1 tenenbaum:1 ten:7 hardware:4 outperform:1 exist:1 dotted:1 estimated:7 v1a:1 diagnosis:1 discrete:2 write:1 promise:1 group:1 drawn:2 preprocessed:1 pj:5 sum:3 run:2 facilitated:1 inverse:1 uncertainty:7 prog:1 perk:5 bit:1 followed:1 replaces:1 encountered:2 occur:2 constraint:1 progl:1 gruffydd:1 department:2 designated:1 according:1 poor:1 conjugate:1 across:1 em:1 character:13 lid:2 making:2 jeffreys:5 restricted:6 pr:5 invariant:1 taken:1 resource:1 equation:4 remains:1 singer:10 tractable:1 end:1 hierarchical:1 appropriate:2 thomas:1 top:1 dirichlet:4 include:1 running:1 graphical:2 log2:2 maintaining:2 calculating:1 exploit:1 build:1 society:1 occurs:1 traditional:1 div:3 mapped:2 geological:1 thrun:1 gun:2 extent:1 length:4 code:1 relationship:1 illustration:1 providing:1 stated:1 unknown:2 perform:2 allowing:1 contributed:2 observation:4 extended:1 arbitrary:1 obj1:1 calgary:3 pic:2 namely:1 required:1 specified:2 philosophical:1 trans:2 suggested:2 bar:1 below:1 sparsity:1 program:1 built:1 including:3 royal:1 natural:6 predicting:2 representing:1 improve:2 technology:1 picture:1 lk:1 axis:1 fro:1 auto:3 text:17 prior:17 byte:4 determining:1 relative:1 law:9 sufficient:1 consistent:1 article:8 ascii:1 share:1 classifying:1 ibm:2 pi:1 token:1 free:1 english:4 side:2 allow:3 institute:1 taking:1 sparse:9 default:3 vocabulary:63 xn:6 world:1 stand:1 gram:1 author:1 commonly:1 adaptive:1 simplified:1 transaction:1 excess:1 approximate:3 corpus:4 table:2 hockey:1 ca:1 nigam:1 excellent:1 complex:1 suc:1 domain:13 arise:1 hyperparameters:1 atheism:3 allowed:1 join:1 ny:1 pv:3 wish:1 xl:1 candidate:3 lie:1 ib:1 printing:1 mideast:6 formula:1 specific:3 symbol:12 offset:1 alt:3 effectively:1 importance:1 illustrates:2 occurring:1 chen:1 easier:1 entropy:5 likely:1 expressed:1 contained:2 religion:6 indifference:1 sport:3 springer:1 determines:1 presentation:1 consequently:2 aided:1 included:2 specifically:1 determined:1 total:1 la:1 newsgroup:6 tiao:1 latter:1 assessed:1 evaluate:1 princeton:1 tested:1 |
1,167 | 2,064 | Learning from Infinite Data
in Finite Time
Pedro Domingos
Geoff H ulten
Department of Computer Science and Engineering
University of Washington
Seattle, WA 98185-2350, U.S.A.
{pedrod, ghulten} @cs.washington.edu
Abstract
We propose the following general method for scaling learning
algorithms to arbitrarily large data sets. Consider the model
Mii learned by the algorithm using ni examples in step i (ii =
(nl , ... ,nm )) , and the model Moo that would be learned using infinite examples. Upper-bound the loss L(Mii' M oo ) between them
as a function of ii, and then minimize the algorithm's time complexity f(ii) subject to the constraint that L(Moo , M ii ) be at most
f with probability at most 8. We apply this method to the EM
algorithm for mixtures of Gaussians. Preliminary experiments on
a series of large data sets provide evidence of the potential of this
approach.
1
An Approach to Large-Scale Learning
Large data sets make it possible to reliably learn complex models. On the other
hand , they require large computational resources to learn from. While in the past
the factor limit ing the quality of learnable models was typically the quantity of data
available, in many domains today data is super-abundant, and the bottleneck is t he
t ime required to process it. Many algorithms for learning on large data sets have
been proposed, but in order to achieve scalability they generally compromise the
quality of the results to an unspecified degree. We believe this unsatisfactory state
of affairs is avoidable, and in this paper we propose a general method for scaling
learning algorithms to arbitrarily large databases without compromising the quality
of the results. Our method makes it possible to learn in finite time a model that
is essentially indistinguishable from the one that would be obtained using infinite
data.
Consider the simplest possible learning problem: estimating the mean of a random
variable x. If we have a very large number of samples, most of them are probably
superfluous. If we are willing to accept an error of at most f with probability at most
8, Hoeffding bounds [4] (for example) tell us that, irrespective of the distribution of
x, only n = ~(R/f)2 1n (2/8) samples are needed, where R is x's range. We propose
to extend this type of reasoning beyond learning single parameters, to learning
complex models. The approach we propose consists of three steps:
1. Derive an upper bound on the relative loss between the finite-data and
infinite-data models, as a function of the number of samples used in each
step of the finite-data algorithm.
2. Derive an upper bound on the time complexity of the learning algorithm ,
as a function of the number of samples used in each step.
3. Minimize the time bound (via the number of samples used in each step)
subject to target limits on the loss.
In this paper we exemplify this approach using the EM algorithm for mixtures of
Gaussians. In earlier papers we applied it (or an earlier version of it) to decision
tree induction [2J and k-means clustering [3J. Despite its wide use, EM has long
been criticized for its inefficiency (see discussion following Dempster et al. [1]), and
has been considered unsuitable for large data sets [8J. Many approaches to speeding
it up have been proposed (see Thiesson et al. [6J for a survey) . Our method can be
seen as an extension of progressive sampling approaches like Meek et al. [5J: rather
than minimize the total number of samples needed by the algorithm , we minimize
the number needed by each step, leading to potentially much greater savings; and
we obtain guarantees that do not depend on unverifiable extrapolations of learning
curves.
2
A Loss Bound for EM
In a mixture of Gaussians model, each D-dimensional data point Xj is assumed to
have been independently generated by the following process: 1) randomly choose a
mixture component k; 2) randomly generate a point from it according to a Gaussian
distribution with mean f-Lk and covariance matrix ~k. In this paper we will restrict
ourselves to the case where the number K of mixture components and the probability of selection P(f-Lk) and covariance matrix for each component are known. Given
a training set S = {Xl, ... , XN }, the learning goal is then to find the maximumlikelihood estimates of the means f-Lk. The EM algorithm [IJ accomplishes this by,
starting from some set of initial means , alternating until convergence between estimating the probability p(f-Lk IXj) that each point was generated by each Gaussian (the
Estep), and computing the ML estimates of the means ilk = 2::;':1 WjkXj / 2::f=l Wjk
(the M step), where Wjk = p(f-Lklxj) from the previous E step. In the basic EM
algorithm, all N examples in the training set are used in each iteration. The goal
in this paper is to speed up EM by using only ni < N examples in the ith iteration, while guaranteeing that the means produced by the algorithm do not differ
significantly from those that would be obtained with arbitrarily large N.
Let Mii = (ill , . . . , ilK) be the vector of mean estimates obtained by the finite-data
EM algorithm (i.e., using ni examples in iteration i), and let Moo = (f-L1, ... ,f-LK) be
the vector obtained using infinite examples at each iteration. In order to proceed,
we need to quantify the difference between Mii and Moo . A natural choice is the
sum of the squared errors between corresponding means, which is proportional to
the negative log-likelihood of the finite-data means given the infinite-data ones:
K
L(Mii' Moo ) =
L
k=l
K
Ililk - f-Lkl12 =
D
LL
lilkd - f-Lkdl 2
(1)
k=ld=l
where ilkd is the dth coordinate of il, and similarly for f-Lkd.
After any given iteration of EM, lilkd - f-Lkdl has two components. One, which we call
the sampling error, derives from the fact that ilkd is estimated from a finite sample,
while J-Lkd is estimated from an infinite one. The other component, which we call
the weighting error, derives from the fact that , due to sampling errors in previous
iterations, the weights Wjk used to compute the two estimates may differ. Let J-Lkdi
be the infinite-data estimate of the dth coordinate of the kth mean produced in
iteration i, flkdi be the corresponding finite-data estimate, and flkdi be the estimate
that would be obtained if there were no weighting errors in that iteration. Then
the sampling error at iteration i is Iflkdi - J-Lkdi I, the weighting error is Iflkdi - flkdi I,
and the total error is Iflkdi - J-Lkdi 1 :::; Iflkdi - flkdi 1 + Iflkdi - J-Lkdi I?
Given bounds on the total error of each coordinate of each mean after iteration i-I,
we can derive a bound on the weighting error after iteration i as follows. Bounds
on J-Lkd ,i-l for each d imply bounds on p(XjlJ-Lki ) for each example Xj, obtained by
substituting the maximum and minimum allowed distances between Xjd and J-Lkd ,i-l
into the expression of the Gaussian distribution. Let P}ki be the upper bound on
P(XjlJ-Lki) , and Pjki be the lower bound. Then the weight of example Xj in mean J-Lki
can be bounded from below by
by
W}ki
W (+) -jki
=
min{p}kiP(J-Lk)/
wjki
=
PjkiP(J-Lk)/
~~= l P}k'iP(J-LU, and from above
~~=l Pjk'iP(J-LU, I}. Let w;t: = W}ki if Xj ::::: 0 and
(- ) -- W jki
-'f
> 0 an d W jki
(- ) -- W jki
+ 0 th erWlse.
.
W jki
1 Xj _
'
ot h
erWlse,
an d 1et
W-
jki
Then
Iflkdi -
flkdi
,
IJ-Lkdi
1
< max
-
~7~1 Wjk i Xj
I
",ni
uj=l
{I
,
J-Lkdi -
Wjki
uj =l
" , ni W jki
(+) Xj
",ni
_
uj=l
w jki
II
,
,J-Lkdi -
uj =l
",ni
( - ) Xj
W jki
",ni
uj=l
+
I}
(2)
w jki
A corollary of Hoeffding's [4] Theorem 2 is that, with probability at least 1 - 8, the
sampling error is bounded by
Iflkdi -
J-Lkdi
1
:::;
(3)
where Rd is the range of the dth coordinate of the data (assumed known 1 ). This
bound is independent of the distribution of the data, which will ensure that our
results are valid even if the data was not truly generated by a mixture of Gaussians,
as is often the case in practice. On the other hand, the bound is more conservative than distribution-dependent ones, requiring more samples to reach the same
guarantees.
The initialization step is error-free, assuming the finite- and infinite-data algorithms are initialized with the same means. Therefore the weighting error in
the first iteration is zero, and Equation 3 bounds the total error. From this
we can bound the weighting error in the second iteration according to Equation 2, and therefore bound the total error by the sum of Equations 2 and 3,
and so on for each iteration until the algorithms converge. If the finite- and
infinite-data EM converge in the same number of iterations m, the loss due to
finite data is L(Mii" Moo ) = ~f= l ~~= llflkdm - J-Lkdml 2 (see Equation 1). Assume that the convergence criterion is ~f= l IIJ-Lki - J-Lk ,i- 111 2 :::; f. In general
1 Although a normally distributed variable has infinite range, our experiments show
that assuming a sufficiently wide finite range does not significantly affect the results.
(with probability specified below), infinite-data EM converges at one of the iterations for which the minimum possible change in mean positions is below ,,/, and
is guaranteed to converge at the first iteration for which the maximum possible
change is below "(. More precisely, it converges at one of the iterations for which
~~= l ~~= l (max{ IPkd ,i- l - Pkdil-IPkd,i - l - ftkd,i - ll-IPkdi - ftkdil, O})2 ::; ,,/, and
is guaranteed to converge at the first iteration for which ~~=l ~~=l (IPkd,i-l Pkdil + IPkd ,i-l - ftkd ,i-ll + IPkdi - ftkdil)2 ::; "/. To obtain a bound for L(Mn, Moo),
finite-data EM must be run until the latter condition holds. Let I be the set of
iterations at which infinite-data EM could have converged. Then we finally obtain
where m is the total number of iterations carried out. This bound holds if all of
the Hoeffding bounds (Equation 3) hold. Since each of these bounds fails with
probability at most 8, the bound above fails with probability at most 8* = K Dm8
(by the union bound). As a result, the growth with K, D and m of the number
of examples required to reach a given loss bound with a given probability is only
O(v'lnKDm).
The bound we have just derived utilizes run-time information, namely the distance
of each example to each mean along each coordinate in each iteration. This allows it
to be tighter than a priori bounds. Notice also that it would be trivial to modify the
treatment for any other loss criterion that depends only on the terms IPkdm - ftkdm I
(e.g., absolute loss) .
3
A Fast EM Algorithm
We now apply the previous section's result to reduce the number of examples used
by EM at each iteration while keeping the loss bounded. We call the resulting
algorithm VFEM. The goal is to learn in minimum time a model whose loss relative
to EM applied to infinite data is at most f* with probability at least 1 - 8*. (The
reason to use f* instead of f will become apparent below.) Using the notation of the
previous section, if ni examples are used at each iteration then the running time of
EM is O(KD ~::l ni) , and can be minimized by minimizing ~::l ni. Assume for
the moment that the number of iterations m is known. Then, using Equation 1, we
can state the goal more precisely as follows.
Goal: Minimize ~::l ni, subject to the constraint that ~~=l IIPkm - ftkml12 ::; f*
with probability at least 1 - 8* .
A sufficient condition for ~~=l IIPkm - ftkml12 ::; f* is that Vk IIPkm - ftkmll ::;
Jf*/K. We thus proceed by first minimizing ~::l ni subject to IIPkm - ftkmll ::;
f* / K separately for each mean. 2 In order to do this, we need to express IIPkm ftkm II as a function of the ni 'so By the triangle inequality, IIPki - ftki II ::; IIPki - ftki II +
J
Ilftki - ftk& By Equation 3, Ilftki - ftki II::;
~R2ln(2/8) ~;~ l w;kd(~;~ l Wjki)2,
where R2 = ~~=l RJ and 8 = 8* / K Dm per the discussion following Equation 4.
The (~;~ l Wjki)2 / ~;~ l W;ki term is a measure of the diversity of the weights ,
2This will generally lead to a suboptimal solution; improving it is a matter for future
work.
being equal to 1 -
l/Gini(W~i)'
where
W~i
is the vector of normalized weights
wjki = wjkd 2:j,i=l Wjl ki. It attains a minimum of! when all the weights but one are
zero, and a maximum of ni when all the weights are equal and non-zero. However,
we would like to have a measure whose maximum is independent of ni, so that it
remains approximately constant whatever the value of ni chosen (for sufficiently
large ni). The measure will then depend only on the underlying distribution of the
data. Thus we define f3ki = (2:7~1 Wjki)2 /(ni 2:7~1 W]ki) ' obtaining IliLki - ILkill :::;
JR 2 ln (2/8)/(2f3ki n i). Also, IIP-ki-iLkill =
J2:~= llP-kdi -
iLkdil 2, with lP-kdi-iLkdil
bounded by Equation 2. To keep the analysis tractable, we upper-bound this term
by a function proportional to IIP-kd,i-1 - ILkd,i-111. This captures the notion than the
weighting error in one iteration should increase with the total error in the previous
one. Combining this with the bound for IliLki - ILkill, we obtain
R 2 l n (2/8)
2f3kini
(5)
where CXki is the proportionality constant. Given this equation and IIP-kO - ILkO II = 0,
it can be shown by induction that
m
IIP-km - ILkmll :::;
~~
(6)
where
(7)
The target bound will thus be satisfied by minimizing 2:: 1 ni subject to
2::1 (rkd,;niJ = J E* / K. 3 Finding the n/s by the method of Lagrange multipliers yields
ni =
~ (f ~rkir%j)
2
(8)
)=1
This equation will produce a required value of ni for each mean. To guarantee the
desired E*, it is sufficient to make ni equal to the maximum of these values.
The VFEM algorithm consists of a sequence of runs of EM, with each run using
more examples than the last, until the bound L(Mii' Moo) :::; E* is satisfied, with
L(Mii' Moo) bounded according to Equation 4. In the first run, VFEM postulates a
maximum number of iterations m, and uses it to set 8 = 8* / K Dm. If m is exceeded,
for the next run it is set to 50% more than the number needed in the current run.
(A new run will be carried out if either the 8* or E* target is not met.) The number
of examples used in the first run of EM is the same for all iterations, and is set to
1.1(K/2)(R/E*)2ln(2/8). This is 10% more than the number of examples that would
theoretically be required in the best possible case (no weighting errors in the last
3This may lead to a suboptimal solution for the ni's, in the unlikely case that
II increases with them.
Jtkm
Ilflkm -
iteration, leading to a pure Hoeffding bound, and a uniform distribution of examples
among mixture components). The numbers of examples for subsequent runs are set
according to Equation 8. For iterations beyond the last one in the previous run ,
the number of examples is set as for the first run. A run of EM is terminated
when L~= l L~= l (Iflkd ,i- l - flkdi 1+ Iflkd ,i-l - ILkd ,i-l l + Iflkdi - ILkdi 1)2 :s: "( (see
discussion preceding Equation 4), or two iterations after L~=l IIILki - ILk,i-1 11 2 :s:
"( 13, whichever comes first. The latter condition avoids overly long unproductive
runs. If the user target bound is E, E* is set to min{ E, "( 13}, to facilitate meeting the
first criterion above. When the convergence threshold for infinite-data EM was not
reached even when using the whole training set, VFEM reports that it was unable
to find a bound; otherwise the bound obtained is reported.
VFEM ensures that the total number of examples used in one run is always at least
twice the number n used in the previous run. This is done by, if L ni < 2n, setting
the ni's instead to n~ = 2n(nil L ni). If at any point L ni > mN, where m is the
number of iterations carried out and N is the size of the full training set, Vi ni = N
is used. Thus, assuming that the number of iterations does not decrease with the
number of examples, VFEM's total running time is always less than three times the
time taken by the last run of EM. (The worst case occurs when the one-but-last
run is carried out on almost the full training set.)
The run-time information gathered in one run is used to set the n/s for the next
run. We compute each Ctki as Ilflki - Pkill/llflk ,i-l - ILk ,i-lll. The approximations
made in the derivation will be good, and the resulting ni's accurate, if the means'
paths in the current run are similar to those in the previous run. This may not
be true in the earlier runs , but their running time will be negligible compared to
that of later runs , where the assumption of path similarity from one run to the next
should hold.
4
Experiments
We conducted a series of experiments on large synthetic data sets to compare VFEM
with EM. All data sets were generated by mixtures of spherical Gaussians with
means ILk in the unit hypercube. Each data set was generated according to three
parameters: the dimensionality D , the number of mixture components K , and
the standard deviation (Y of each coordinate in each component. The means were
generated one at a time by sampling each dimension uniformly from the range
(2(Y,1 - 2(Y). This ensured that most of the data points generated were within the
unit hypercube. The range of each dimension in VFEM was set to one. Rather
than discard points outside the unit hypercube, we left them in to test VFEM's
robustness to outliers. Any ILk that was less than (vD1K)(Y away from a previously
generated mean was rejected and regenerated, since problems with very close means
are unlikely to be solvable by either EM or VFEM. Examples were generated by
choosing one of the means ILk with uniform probability, and setting the value of
each dimension of the example by randomly sampling from a Gaussian distribution
with mean ILkd and standard deviation (Y. We compared VFEM to EM on 64 data
sets of 10 million examples each, generated by using every possible combination of
the following parameters: D E {4, 8,12, 16}; K E {3, 4, 5, 6} ; (Y E {.01 , .03, .05, .07}.
In each run the two algorithms were initialized with the same means, randomly
selected with the constraint that no two be less than vD1(2K) apart. VFEM was
allowed to converge before EM's guaranteed convergence criterion was met (see
discussion preceding Equation 4). All experiments were run on a 1 GHz Pentium
III machine under Linux, with "( = O.OOOlDK, 8* = 0.05, and E* = min{O.Ol, "(} .
Table 1: Experimental results. Values are averages over the number of runs shown.
Times are in seconds, and #EA is the total number of example accesses made by
the algorithm, in millions.
Runs
Bound
No bound
All
Algorithm
VFEM
EM
VFEM
EM
VFEM
EM
#Runs
40
40
24
24
64
64
Time
217
3457
7820
4502
3068
3849
#EA
1.21
19.75
43.19
27.91
16.95
22.81
Loss
2.51
2.51
1.20
1.20
2.02
2.02
D
10.5
10.5
9.1
9.1
10
10
K
4.2
4.2
4.9
4.9
4.5
4.5
rr
0.029
0.029
0.058
0.058
0.04
0.04
The results are shown in Table 1. Losses were computed relative to the true means,
with the best match between true means and empirical ones found by greedy search.
Results for runs in which VFEM achieved and did not achieve the required E* and
8* bounds are reported separately. VFEM achieved the required bounds and was
able to stop early on 62.5% of its runs. When it found a bound, it was on average
16 t imes faster than EM. When it did not, it was on average 73% slower. The losses
of the two algorithms were virtually identical in both situations. VFEM was more
likely to converge rapidly for higher D's and lower K's and rr's. When achieved ,
the average loss bound for VFEM was 0.006554, and for EM it was 0.000081. In
other words, the means produced by both algorithms were virtually identical to
those that would be obtained with infinite data. 4
We also compared VFEM and EM on a large real-world data set, obtained by
recording a week of Web page requests from the entire University of Washington
campus. The data is described in detail in Wolman et al. [7], and the preprocessing
carried out for these experiments is described in Domingos & Hulten [3]. The goal
was to cluster patterns of Web access in order to support distributed caching. On a
dataset with D = 10 and 20 million examples, with 8* = 0.05, I = 0.001, E* = 1/3,
K = 3, and rr = 0.01, VFEM achieved a loss bound of 0.00581 and was two orders
of magnitude faster than EM (62 seconds vs. 5928), while learning essentially the
same means.
VFEM's speedup relative to EM will generally approach infinity as the data set
size approaches infinity. The key question is thus: what are the data set sizes at
which VFEM becomes worthwhile? The tentative evidence from these experiments
is that they will be in the millions. Databases of this size are now common, and
their growth continues unabated , auguring well for the use of VFEM.
5
Conclusion
Learning algorithms can be sped up by minimizing the number of examples used in
each step, under the constraint that the loss between the resulting model and the
one that would be obtained with infinite data remain bounded. In this paper we
applied this method to the EM algorithm for mixtures of Gaussians, and observed
the resulting speedups on a series of large data sets.
4The much higher loss values relative to the true means, however, indicate that infinitedata EM would often find only local optima (unless the greedy search itself only found a
suboptimal match).
Acknowledgments
This research was partly supported by NSF CAREER and IBM Faculty awards to
the first author, and by a gift from the Ford Motor Company.
References
[1] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from
incomplete data via the EM algorithm. Journal of the Royal Statistical Society,
Series B, 39:1- 38, 1977.
[2] P. Domingos and G. Hulten. Mining high-speed data streams. In Proceedings
of the Sixth ACM SIGKDD International Conference on Knowledge Discovery
and Data Mining, pp. 71- 80, Boston, MA, 2000. ACM Press.
[3] P. Domingos and G. Hulten. A general method for scaling up machine learning
algorithms and its application to clustering. In Proceedings of the Eighteenth International Conference on Machine Learning, pp. 106-113, Williamstown, MA,
2001. Morgan Kaufmann.
[4] W. Hoeffding. Probability inequalities for sums of bounded random variables.
Journal of the American Statistical Association, 58:13- 30, 1963.
[5] C. Meek, B. Thiesson, and D. Heckerman. The learning-curve method applied
to clustering. Technical Report MSR-TR-01-34, Microsoft Research, Redmond,
WA,2000.
[6] B. Thiesson, C. Meek, and D. Heckerman. Accelerating EM for large databases.
Technical Report MSR-TR-99-31, Microsoft Research, Redmond, WA, 2001.
[7] A. Wolman, G. Voelker, N. Sharma, N. Cardwell, M. Brown, T. Landray, D. Pinnel, A. Karlin, and H. Levy. Organization-based analysis of Web-object sharing
and caching. In Proceedings of the Second USENIX Conference on Internet
Technologies and Systems, pp. 25- 36, Boulder, CO, 1999.
[8] T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH: An efficient data clustering method for very large databases. In Proceedings of the 1996 A CM SIGMOD International Conference on Management of Data, pp. 103- 114, Montreal,
Canada, 1996. ACM Press.
| 2064 |@word msr:2 faculty:1 version:1 proportionality:1 willing:1 km:1 covariance:2 tr:2 ld:1 moment:1 initial:1 inefficiency:1 series:4 past:1 current:2 ixj:1 moo:9 must:1 subsequent:1 motor:1 v:1 greedy:2 selected:1 affair:1 ith:1 zhang:1 along:1 become:1 consists:2 theoretically:1 ol:1 spherical:1 company:1 lll:1 becomes:1 gift:1 estimating:2 bounded:7 notation:1 underlying:1 campus:1 what:1 cm:1 unspecified:1 livny:1 finding:1 guarantee:3 every:1 growth:2 ensured:1 whatever:1 normally:1 unit:3 before:1 negligible:1 engineering:1 local:1 modify:1 limit:2 despite:1 path:2 approximately:1 twice:1 initialization:1 jki:10 co:1 range:6 wjl:1 acknowledgment:1 practice:1 union:1 empirical:1 significantly:2 word:1 close:1 selection:1 eighteenth:1 starting:1 independently:1 survey:1 pure:1 notion:1 coordinate:6 target:4 today:1 user:1 us:1 domingo:4 continues:1 database:4 observed:1 capture:1 worst:1 imes:1 ensures:1 ftki:3 decrease:1 dempster:2 complexity:2 depend:2 compromise:1 triangle:1 geoff:1 derivation:1 fast:1 gini:1 tell:1 outside:1 choosing:1 whose:2 apparent:1 voelker:1 otherwise:1 itself:1 ford:1 ip:2 laird:1 sequence:1 rr:3 karlin:1 propose:4 j2:1 combining:1 rapidly:1 achieve:2 scalability:1 wjk:4 seattle:1 convergence:4 cluster:1 optimum:1 produce:1 guaranteeing:1 converges:2 object:1 oo:1 derive:3 montreal:1 ij:2 c:1 come:1 indicate:1 quantify:1 differ:2 met:2 compromising:1 require:1 pjk:1 preliminary:1 tighter:1 extension:1 hold:4 sufficiently:2 considered:1 week:1 substituting:1 early:1 gaussian:4 always:2 super:1 rather:2 caching:2 hulten:3 corollary:1 derived:1 vk:1 unsatisfactory:1 likelihood:2 pentium:1 sigkdd:1 attains:1 dependent:1 typically:1 unlikely:2 accept:1 entire:1 among:1 ill:1 priori:1 xjd:1 equal:3 saving:1 washington:3 sampling:7 identical:2 progressive:1 future:1 minimized:1 report:3 regenerated:1 randomly:4 ime:1 ourselves:1 microsoft:2 organization:1 mining:2 mixture:10 truly:1 nl:1 superfluous:1 accurate:1 unless:1 tree:1 incomplete:1 initialized:2 abundant:1 desired:1 nij:1 criticized:1 earlier:3 deviation:2 uniform:2 conducted:1 reported:2 synthetic:1 international:3 linux:1 squared:1 postulate:1 nm:1 satisfied:2 iip:4 choose:1 management:1 hoeffding:5 american:1 leading:2 potential:1 diversity:1 matter:1 depends:1 vi:1 stream:1 later:1 extrapolation:1 lkd:4 reached:1 minimize:5 lkl12:1 ni:30 il:1 kaufmann:1 yield:1 gathered:1 produced:3 lu:2 converged:1 reach:2 sharing:1 sixth:1 pp:4 dm:2 stop:1 dataset:1 treatment:1 birch:1 exemplify:1 knowledge:1 dimensionality:1 ea:2 exceeded:1 higher:2 done:1 just:1 rejected:1 until:4 hand:2 web:3 quality:3 believe:1 facilitate:1 brown:1 multiplier:1 true:4 requiring:1 normalized:1 alternating:1 indistinguishable:1 ll:3 criterion:4 l1:1 reasoning:1 common:1 thiesson:3 sped:1 million:4 extend:1 he:1 association:1 rd:1 similarly:1 access:2 similarity:1 rkd:1 discard:1 apart:1 inequality:2 arbitrarily:3 meeting:1 morgan:1 seen:1 minimum:4 greater:1 preceding:2 accomplishes:1 converge:6 sharma:1 ii:11 full:2 rj:1 ing:1 technical:2 match:2 faster:2 long:2 award:1 basic:1 ko:1 essentially:2 iteration:33 achieved:4 separately:2 ot:1 probably:1 subject:5 recording:1 virtually:2 call:3 iii:1 xj:8 affect:1 restrict:1 suboptimal:3 reduce:1 bottleneck:1 expression:1 accelerating:1 vd1:1 proceed:2 generally:3 simplest:1 generate:1 nsf:1 notice:1 estimated:2 overly:1 per:1 express:1 key:1 threshold:1 sum:3 run:33 almost:1 utilizes:1 mii:8 decision:1 scaling:3 bound:41 ki:7 internet:1 meek:3 guaranteed:3 ilk:7 constraint:4 precisely:2 infinity:2 ftk:1 unverifiable:1 speed:2 min:3 estep:1 department:1 speedup:2 according:5 combination:1 request:1 kd:3 jr:1 remain:1 heckerman:2 em:38 lp:1 outlier:1 boulder:1 taken:1 ln:2 resource:1 equation:15 remains:1 previously:1 kdi:2 needed:4 tractable:1 whichever:1 available:1 gaussians:6 apply:2 worthwhile:1 away:1 robustness:1 slower:1 clustering:4 ensure:1 running:3 unsuitable:1 sigmod:1 uj:5 hypercube:3 society:1 question:1 quantity:1 occurs:1 kth:1 ilkd:5 distance:2 unable:1 trivial:1 reason:1 induction:2 assuming:3 minimizing:4 potentially:1 lki:4 negative:1 reliably:1 upper:5 finite:13 situation:1 usenix:1 canada:1 namely:1 required:6 specified:1 kip:1 tentative:1 learned:2 beyond:2 dth:3 llp:1 below:5 able:1 pattern:1 redmond:2 max:2 royal:1 natural:1 solvable:1 mn:2 pedrod:1 technology:1 imply:1 lk:8 irrespective:1 carried:5 speeding:1 discovery:1 relative:5 loss:17 proportional:2 degree:1 sufficient:2 rubin:1 ibm:1 supported:1 last:5 free:1 keeping:1 wide:2 absolute:1 distributed:2 ghz:1 curve:2 dimension:3 xn:1 valid:1 avoids:1 world:1 author:1 made:2 preprocessing:1 wolman:2 keep:1 ml:1 assumed:2 search:2 table:2 learn:4 erwlse:2 career:1 obtaining:1 improving:1 complex:2 domain:1 did:2 terminated:1 whole:1 allowed:2 iij:1 fails:2 position:1 xl:1 levy:1 weighting:8 theorem:1 learnable:1 r2:1 evidence:2 derives:2 magnitude:1 boston:1 likely:1 lagrange:1 pedro:1 williamstown:1 ramakrishnan:1 avoidable:1 acm:3 ma:2 goal:6 jf:1 change:2 infinite:17 uniformly:1 conservative:1 total:10 nil:1 partly:1 experimental:1 support:1 maximumlikelihood:1 latter:2 |
1,168 | 2,065 | Probabilistic Abstraction Hierarchies
Eran Segal
Computer Science Dept.
Stanford University
eran@cs.stanford.edu
Daphne Koller
Computer Science Dept.
Stanford University
koller@cs.stanford.edu
Dirk Ormoneit
Computer Science Dept.
Stanford University
ormoneit@cs.stanford.edu
Abstract
Many domains are naturally organized in an abstraction hierarchy or taxonomy,
where the instances in ?nearby? classes in the taxonomy are similar. In this paper, we provide a general probabilistic framework for clustering data into a set
of classes organized as a taxonomy, where each class is associated with a probabilistic model from which the data was generated. The clustering algorithm
simultaneously optimizes three things: the assignment of data instances to clusters, the models associated with the clusters, and the structure of the abstraction
hierarchy. A unique feature of our approach is that it utilizes global optimization
algorithms for both of the last two steps, reducing the sensitivity to noise and the
propensity to local maxima that are characteristic of algorithms such as hierarchical agglomerative clustering that only take local steps. We provide a theoretical
analysis for our algorithm, showing that it converges to a local maximum of the
joint likelihood of model and data. We present experimental results on synthetic
data, and on real data in the domains of gene expression and text.
1 Introduction
Many domains are naturally associated with a hierarchical taxonomy, in the form of a tree,
where instances that are close to each other in the tree are assumed to be more ?similar? than
instances that are further away. In biological systems, for example, creating a taxonomy of
the instances is often one of the first steps in understanding the system. In particular, much
of the work on analyzing gene expression data [3] has focused on creating gene hierarchies.
Similarly, in text domains, creating a hierarchy of documents is a common task [12, 7].
In many of these applications, the hierarchy is unknown; indeed, discovering the hierarchy is often a key part of the analysis. The standard algorithms applied to the problem
typically use an agglomerative bottom-up approach [3] or a divide-and-conquer top-down
approach [8]. Although these methods have been shown to be useful in practice, they suffer from several limitations: First, they proceed via a series of local improvements, making
them particularly prone to local maxima. Second, at least the bottom-up approaches are
typically applied to the raw data; models (if any), are constructed as a post-processing
step. Thus, domain knowledge about the type of distribution from which data instances are
sampled is rarely used in the formation of the hierarchy.
In this paper, we present probabilistic abstraction hierarchies (PAH), a probabilistically principled general framework for learning abstraction hierarchies from data which
overcomes these difficulties. We use a Bayesian approach, where the different models
correspond to different abstraction hierarchies. The prior is designed to enforce our intuitions about taxonomies: nearby classes have similar data distributions. More specifically,
a model in a PAH is a tree, where each node in the tree is associated with a class-specific
probabilistic model (CPM). Data is generated only at the leaves of the tree, so that a model
basically defines a mixture distribution whose components are the CPMs at the leaves of
the tree. The CPMs at the internal nodes are used to define the prior over models: We
prefer models where the CPM at a child node is close to the CPM at its parent, relative to
some distance function between CPMs. Our framework allows a wide range of notions of
distance between models; we essentially require only that the distance function be convex
in the parameters of the two CPMs. For example, if a CPM is a Gaussian distribution, we
might use a simple squared Euclidean distance between the parameters of the two CPMs.
We present a novel algorithm for learning the model parameters and the tree structure
in this framework. Our algorithm is based on the structural EM (SEM) approach of [4],
but utilizes ?global? optimization steps for learning the best possible hierarchy and CPM
parameters (see also [5, 13] for similar global optimization steps within SEM). Each step in
our procedure is guaranteed to increase the joint probability of model and data, and hence
(like SEM) our procedure is guaranteed to converge to a local optimum.
Our approach has several advantages. (1) It provides principled probabilistic semantics
for hierarchical models. (2) It is model based, which allows us to exploit domain structural
knowledge more easily. (3) It utilizes global optimization steps, which tend to avoid local
maxima and help make the model less sensitive to noise. (4) The abstraction hierarchy
tends to pull the parameters of one model closer to those of nearby ones, which naturally
leads to a form of parameter smoothing or shrinkage [12].
We present experiments for PAH on synthetic data and on two real data sets: gene
expression and text. Our results show that the PAH approach produces hierarchies that are
more robust to noise in the data, and that the learned hierarchies generalize better to test
data than those produced by hierarchical agglomerative clustering.
2 Probabilistic Abstraction Hierarchy
Let be the domain of some random observation, e.g., the set of possible assignments
to a set of features. Our goal is to take a set of instances in , and to cluster them into
some set of classes. Standard ?flat? clustering approaches ? for example, Autoclass [1]
or the -means algorithm ? are special cases of a generative mixture model. In such
models, each data instance belongs to one of the classes, each of which is associated
with a different class-specific probabilistic model (CPM). Each data instance is sampled
independently by first selecting one of the classes according to a multinomial distribution,
and then randomly selecting the data instance itself from the CPM of the chosen class.
In standard clustering models, there is no relation between the individual CPMs, which
can be arbitrarily different. In this paper, we propose a model where the different classes
are related to each other via an abstraction hierarchy, such that classes that are ?nearby? in
the hierarchy have similar probabilistic models. More precisely, we define:
"
!
#
# %
#
$
Our framework does not, in principle, place restrictions on the form of the CPMs; we
can use any probabilistic model that defines a probability distribution over . For example,
# may be a Bayesian network, in which case its specification would include
may bethea parameters, and perhaps also the network structure; in a different setting, #
hidden
Markov model. In practice, however, the choice of CPMs has ramifications both for the
Definition 2.1 A probabilistic abstraction hierarchy (PAH) is a tree with nodes
and undirected edges , such that has exactly leaves
. Each
node ,
, is associated with a CPM
, which defines a distribution over
; we use
to denote
. We also have a multinomial distribution over the
leaves
; we use to denote the parameters of this distribution.
overall hierarchical model and the algorithm.
As discussed above, we assume that data is generated only from the leaves of the tree.
Thus, we augment with an additional hidden class variable for each data item, which
takes the values
denoting the leaf that was chosen to generate this item. Given
a PAH , an element
, and a value for , we define
, where
is the multinomial distribution over the leaves and
is the conditional density of the data item given the CPM at leaf . The induced
%+-718,.+-',9/':# /;< #=1 <>1
+-'),9( &?@*A/;%1
* &
&
+-,.' *0/2123+-,.&45*6/
*
M5 (=M4)
f(g2)
M5
g1
g3
g2
M2
f(g2)
f(g2)
M4
M1
M4
M3
M4
g1
g2
g3
g1
g4
g2
g3
M2
f(g2)
M5 (=M4)
M3
M2
M1
M2
g2
g3
g1
g2
g3
M6 (=M4)
M4
M1
g4
M3
g4
f(g2)
g1
M6 (=M3)
g4
g4
M6 (=M3)
M1
M3
(a)
(b)
Figure 1: (a) A PAH with 3 leaves over a 4-dimensional continuous state space, along
with a visualization of the Gaussian distribution for the 3rd dimension. (b) Two different
weight-preserving transformations for a tree with 4 leaves
.
#
#
' +-,9 ' * /
21
+-,.'-/ 21
*
#
#
distribution of given , from which the data are generated, is simply
, where
is summed out from
.
As we mentioned, the role of the internal nodes in the tree is to enforce an intuitive
interpretation of the model as an abstraction hierarchy, by enforcing similarity between
CPMs at nearby leaves. We achieve this goal by defining a prior distribution over aband
straction hierarchies that penalizes the distance between neighboring CPMs
using a distance function
. Note that we do not require that be a distance in
the mathematical sense; instead, we only require that it be symmetric (as we chose to use
undirected trees), non-negative, and that
iff
.1 One obvious choice
is to define
IDKL
IDKL
, where IDKL
is the KLdistance between the distributions that
and
define over . This distance measure
has the advantage of being applicable to any pair of CPMs over the same space, even if
their parameterization is different. Given a definition of , we define the prior over PAHs
as
, where represents the extent to which differences in distances are penalized (larger represents a larger penalty). 2
Given a set of data instances with domain , our goal is to find a PAH that maximizes
or equivalently,
. By
maximizing this expression, we are trading off the fit of the mixture model over the leaves
to the data , and the desire to generate a hierarchy in which nearby models are similar.
Fig. 1(a) illustrates a typical PAH with Gaussian CPM distributions, where a CPM close
to the leaves of the tree is more specialized and thus has fairly peaked distributions. Conversely, CPMs closer to the root of the tree, acting to bridge between their neighbors, are
expected to have less peaked distributions and peak only around parts of the distribution
which are common to an entire subtree.
7 ,.# # 1
7 ,.# # 1- ,.# # 1 # ,9#
+-,.21
, ! ,9# # 11
"
+-,. /#" 1$
!+-, 21+-," / 21
# 1 # #
,# .# > # 1
9, #># 1
+-, 21 )%*&+( +-," / 21
%'&(
"
3 Learning the Models
# "
#
- ,/. 0 1,2. 3 0
%
"
Our goal in this section is to learn a PAH from a data set
. This
learning task is fairly complex, as many aspects are unknown: the structure of the tree ,
the CPMs
at the nodes of , the parameters , and the assignment of the
instances in to leaves of . Hence, the likelihood function has multiple local maxima,
and no general method exists for finding the global maximum. In this section, we provide
an efficient algorithm for finding a locally optimal .
1
; =KJLB
4!5 687:9<; =5?>A@CBEDF; =5 >A@HGIB
Two models are considered identical if
.
Care must be taken to ensure that
is a proper probability distribution, but this will always
be the case for the choice of we use in this paper. We also note that, if desired, we can modify this
prior to incorporate a prior over the parameters of the
?s.
2
M
@:N
#
#
#
To simplify the algorithm, we assume that the structure of the CPMs
is
fixed. This reduces the choice of each
to a pure numerical optimization problem. The
general framework of our algorithm extends to cases where we also have to solve the model
selection problem for each
, but the computational issues are somewhat different.
We first discuss the case of complete data, where for each data instance
, we
are given the leaf from which it was generated. For this case, we show how to learn the
structure of the tree and the setting of the parameters and
. This problem, of constructing a tree over a set of points that is not fixed, is very closely related to the Steiner
tree problem [10], virtually all of whose variants are NP-hard. We propose a heuristic approach that decouples the joint optimization problem into two subproblems: optimizing the
CPM parameters given the tree structure, and learning a tree structure given a set of CPMs.
Somewhat surprisingly, we show that our careful choice of additive prior allows each of
these subproblems to be tackled very effectively using global optimization techniques.
We begin with the task of learning the CPMs. Thus, assume that we are given both the
data instance
to one of the
structure of the tree and the assignment of each
leaves, denoted
. It remains to find
that minimize
. Substituting the definitions into , we get that
#
2 1 %'&( +-,.21 &
D
%
. " 0
; =
$
,/. " 0 ( "
% $
/> !B
,/. 0 ( "
N
!" $#
N
: %'&( +-, " /
; = />-@ N B
&%'
%
)(
%
*
N
M =@ N @ B
2
43
,+ -/.01
- /5
(1)
The first term, involving the multinomial parameters , separates from the rest, so that
the optimization of relative to reduces to straightforward maximum likelihood estimation. To optimize the CPM parameters, the key property turns out to be the convexity of
the function, which holds in a wide variety of choices of CPMs and ; in particular, it
holds for the models used in our experiments. The convexity property allows us to find the
global minimum of using a simple iterative procedure. In each iteration, we optimize
the
76
parameters of one of the
?s, fixing the parameters of the remaining CPMs
(
).
This procedure is repeated for each of the
?s in a round robin fashion, until convergence.
By the joint convexity of , this iterative procedure is guaranteed to converge to the global
inminimum of . An examination of (1) shows that the optimization of each CPM
volves only the data cases assigned to
(if is a leaf) and the parameters of the CPMs
that are neighbors of
in the tree, thereby simplifying the computation substantially.
We now turn our attention to the second subproblem, of learning the structure of the
tree given the learned CPMs. We first consider an empty tree containing only the (unconnected) leaf nodes
, and find the optimal parameter settings for each leaf CPM as
described above. Note that these CPMs are unrelated, and the parameters of each one are
computed independently of other CPMs. Given this initial set of CPMs for the leaf nodes
, the algorithm tries to learn a good tree structure relative to these CPMs.
The goal is to find the lowest weight tree, subject to the restriction that the tree structure
must keep the same set of leaves
. Due to the decomposability of
, the
. This probpenalty of the tree can be measured via the sum of the edge weights
lem is also a variant of the Steiner tree problem. As a heuristic substitute, we follow the
lines of [5] and use a minimum spanning tree (MST) algorithm for constructing low-weight
trees.
At each iteration, the algorithm starts out with a tree over some set of nodes
.
of this tree, and constructs an MST over them. Of course,
It takes the leaves
in the resulting tree, some of the
are no longer leaves. This problem is corrected by a
transformation that ?pushes? a leaf down the tree, duplicating its model; this transformation
preserves the weight (score) of the tree. By using only
, the algorithm simply
?throws away? the entire structure of the previous tree. However, we can also construct
new MSTs built from all nodes
of the previous tree. For all nodes for 98
:8
which end up as internal nodes, we perform the same transformation described
above. In both cases, this transformation is not unique, as it depends on the order in which
the steps are executed; see Fig. 1(b). The algorithm therefore generates an entire pool of
#
#
#
#
#
#
#
#
,9# # '% &1 ( +-, 21
candidate trees (from both
and
), generated using different random
resolutions of ambiguities in the weight-preserving transformation. For each such tree, the
CPM learning algorithm is used to find an optimal setting of the parameters. The trees are
evaluated relative to our score (
), and the highest scoring tree is kept.
The tree just constructed has a new set of CPMs, so we can repeat this process. To
detect termination, the algorithm also keeps the tree from the previous iteration, and terminates when the score of all trees in the newly constructed pool is lower than the score of
the best tree from previous iteration.
Finally, we address the fact that the data we have is incomplete, in that the assignments
of data instances to classes is not determined. We address the problem of
incomplete data using the standard Expectation Maximization (EM) algorithm [2] and the
structural EM algorithm [4] which extends EM to the problem of model selection. Starting
from an initial model, the algorithm iterates the following two steps: The E-step computes the distribution over the unobserved variables given the observed data and the current
model. In our case, the distribution over the unobserved variables is computed by evalu8
ating
for all 8
. The M-step learns new models that
increase the expected log likelihood of the data, relative to the distribution computed in
the E-step. In our case, the M-step is precisely the algorithm for complete data described
above, but using a soft assignment of data instances to nodes in the tree. The full algorithm
is shown in Fig. 2.
A simple analysis along the lines of [4] can be used to show that the log-probability
increases at every M-step. We therefore obtain the following theorem:
%'&(
+-,. / " 1
& ." 0
+-,.& . " 0 2/ ,/. " 0 2 1
%'&(
" 3/ ")/
+-,.5/<" 1
Theorem 3.1 The algorithm in Fig. 2 converges to a local maximum of
1. Initialize
J D = @
3 5 5 5 3
2. Repeat until convergence:
%*&+(
+-, /<" 1 .
@ B and the models at the leaves. Randomly initialize .
3
M =@ N @ -B
-step:
3
i. Choose an MST over some subset of
3 5 5 5 3 , using
as edge weights.
ii. Transform the MST so that
3 5 5 5 3
become leaves.
3 5 5 5 3
(b) -step: For
, compute the posterior probabilities for the indicator variable
. For
:
(a)
D
=
!D B 9 DF; =
D >
@ -step: Update the CPMs and . Let = B
N 9D " =
# 9 D $ % & $ ' IE(
(c)
3
3
-
JLB F; =
>-@ NB N
&%'
45
D
! " . Then:
D B
; = = B # >) B
%' 3
3
3
-
,5
Figure 2: Abstraction Hierarchy Learning Algorithm
4 Experimental Results
We focus our experimental results on genomic expression data, although we also provide
some results on a text dataset. In gene expression data, the level of mRNA transcript of
every gene in the cell is measured simultaneously, using DNA microarray technology. This
genomic expression data provides researchers with much insight towards understanding
the overall cellular behavior. The most commonly used method for analyzing this data is
clustering, a process which identifies clusters of genes that share similar expression patterns (e.g., [3]), and which are therefore also often involved in similar cellular processes.
, in which case
We apply PAH to this data, using CPMs of the
form
KL-distance is simply: IDKL
, which is simply the sum of
,.# ># 1
35476
+* ,- , / .102 1
#
89
, - 8 - 8 1 0
7,.#
# 1
,9# # 1
squared distances between the means of the corresponding Gaussian components, normalized by their variance. We therefore define
IDKL
.
The most popular clustering method for genomic expression data to date is hierarchical
agglomerative clustering (HAC) [3], which builds a hierarchy among the genes by iteratively merging the closest genes relative to some distance metric. We use the same distance
metric for HAC. (Note that in HAC the metric is used as the distance between data cases
whereas in our algorithm it is used as the distance between models.) To perform a direct
comparison between PAH and HAC, we often need to obtain a probabilistic model from
HAC. To do so, we create CPMs from the genes that HAC assigned to each internal node.
In both PAH and HAC, we then assign each gene (in the training set or the test set) to
the hierarchy by choosing the best (highest likelihood) CPM among all the nodes in the
tree (including internal nodes) and recording the probability
that this CPM
assigns to the gene.
Structure Recovery. A good algorithm for learning abstraction hierarchies should recover
the true hierarchy as well as possible. To test this, we generated a synthetic data set, and
measured the ability of each method to recover the distances between pairs of instances
(genes) in the generating model, where distance here is the length of the path between two
genes in the hierarchy.
We generated the data set by sampling from the leaves of a PAH; to make the data realistic, we sampled from a PAH that we learned from a real gene expression data set. To
allow a comparison with HAC, we generated one data instance from each leaf. We generated data for 80 (imaginary) genes and 100 experiments, for a total of 8000 measurements.
For robustness, we generated 5 different such data sets and ran PAH and HAC for each data
set.
We used the correlation and the
error between the pairwise distances
in the original
for PAH,
and the learned tree as measures
of similiarity. The correlation was
for HAC. The average
compared to a
much worse
error was
for HAC. These results show that PAH recovers an abstraction
for PAH and
hierarchy much better than HAC.
Generalization. We next tested the ability of the different methods to generalize to unobserved (test) data, measuring the extent to which each method captures the underlying
structure in the data. We ran these tests on the yeast data set of [6]. We selected 953 genes
with significant changes in expression, using their full set of 93 experiments.
Again, we ran PAH and HAC and evaluated performance using 5 fold cross validation.
For PAH we also used different settings for (the coefficient of the penalty term in
),
) and greatly
which explores the performance in the range of only fitting the data (
favoring hierarchies in which nearby models are similar (large ). In both cases, we learned
a model using training data, and evaluated the log-likelihood of test instances as described
above. The results, summarized in Fig. 3(a), clearly show that PAH generalizes much better
to previously unobserved data than HAC and that PAH works best at some tradeoff between
fitting the data and generating a hierarchy in which nearby models are similar.
Robustness. Our goal in constructing a hierarchy is to extract meaningful biological conclusions from the data. However, data is invariably partial and noisy. If our analysis produces very different results for slightly different training data, the biological conclusions
are unlikely to be meaningful. Thus, we want genes that are assigned to nearby nodes in
the tree, to be close together also in hierarchies learned from perturbed data sets.
We tested robustness to noise by learning a model from the original data set and from
perturbed data sets in which we permuted a varying percentage of the expression measuments. We then compared the distances (the path length in the tree) between the nodes
assigned to every pair of genes in trees learned from the original data and trees learned
from perturbed data sets. The results are shown in Fig. 3(b),
demonstrating that PAH pre
serves the pairwise distances extremely well even when of the data is perturbed (and
performs
reasonably well for permutation), while HAC completely deteriorates
when of the data is permuted.
+-, /#
+
1
:
+-, 21
-90
1
PAH
HAC
0.8
Correlationcoefficient
Average log probability
-92
0.9
-94
-96
-98
-100
0.7
PAH
HAC
0.6
0.5
0.4
0.3
0.2
-102
0.1
-104
0
0
2
4
6
8
10
12
14
16
18
20
Lambda
0
20
40
60
80
100
Datapermuted(% )
(a)
(b)
Figure 3: (a) Generalization to test data (b) Robustness to noise
methodolog
sigmoid
feedforward
spline
mont
nonLinear carlo
mcmc
causal
causal
sample
vector
gradient
Model
PAH
HAC
PAH
HAC
PAH
HAC
p
90%
80%
70%
Training set
avg. L1 difference
5
5
5
5
5
5
5 5
5 5
5 5
Test set
avg. L1 difference
5
5
5
5
5
5
5 5
5 5
5
5
boltzmann
machin
causat
causat
influenc
influenc
featur
Em
maxim
algorithm
counterfactu pearl
train
maximum
likelihood
bayesian
network
method
markov
forward
global
variant
hidden
learn
artificial
intellig
hmm
graphic
model
statist
graph
parameter
acyclic
(a)
(b)
Figure 4: (a) Robustness of PAH and HAC to different subsets of training instances. (b)
Word hierarchy learned on Cora data.
A second important test is robustness to our particular choice of training data: a particular training set reflects only a subset of the experiments that we could have performed.
In this experiment, we used the Yeast Compendium data of [9], which measures the expression profiles triggered by specific gene mutations. We selected 450 genes and all 298
arrays, focusing on genes that changed significantly. For each of three values of ranging
from to , we generated ten different training sets by sampling (without replacement) percent of the 450 genes, the rest of which form a test set.
We then placed both training and test genes within the hierarchy. For each data set, every
pair of genes either appear together in the training set, the test set, or do not appear together
(i.e., one appears in the training set and the other in the test set). We compared, for each pair
of genes, their distances in training sets in which they appear together and their distances
in test sets in which they appear together. The results are summarized in Fig. 4(a). Our
results on the training data show that PAH consistently constructs very similar hierarchies,
even from very different subsets of the data. By contrast, the hierarchies constructed by
HAC are much less consistent. The results on the test data are even more striking. PAH
is very consistent about its classification into the hierachy even of test instances ? ones
not used to construct the hierarchy. In fact, there is no significant difference between its
performance on the training data and the test data. By contrast, HAC places test instances
in very different configurations in different trees, reducing our confidence in the biological
validity of the learned structure.
Intuitiveness.
To get qualitative insight into the hierarchies produced, we ran
PAH on 350 documents from the Probabilistic Methods category in the Cora dataset
(cora.whizbang.com) and learned hierarchies among the (stemmed) words. We constructed a vector for each word with an entry for each document whose value is the TFIDF-
+
weighted frequency of the word within the document. Fig. 4(b) shows parts of the learned
hierarchy, consisting of 441 nodes, where we list high confidence words for each node.
PAH organized related words into the same region of the tree. Within each region, many
words were arranged in a way which is consistent with our intuitive notion of abstraction.
5 Discussion
We presented probabilistic abstraction hierarchies, a general framework for learning abstraction hierarchies from data, which relates different classes in the hierarchy by a
tree whose nodes correspond to class-specific probability models (CPMs). We utilize a
Bayesian approach, where the prior favors hierarchies in which nearby classes have similar
data distributions, by penalizing the distance between neighboring CPMs.
A unique feature of PAH is the use of global optimization steps for constructing the
hierarchy and for finding the optimal setting of the entire set of parameters. This feature
differentiates us from many other approaches that build hierarchies by local improvements
of the objective function or approaches that optimize a fixed hierarchy [7]. The global optimization steps help in avoiding local maxima and in reducing sensitivity to noise. Our
approach leads naturally to a form of parameter smoothing, and provides much better generalization for test data and robustness to noise than other clustering approaches.
In principle, we can use any probabilistic model for the CPM as long as it defines a
probability distribution over the state space. We have recently [14] applied this approach
to the substantially more complex problem of clustering proteins based on their amino acid
sequence using profile HMMs [11].
Acknowledgements. We thank Nir Friedman for useful comments. This work was supported by NSF Grant ACI-0082554 under the NSF ITR program, and by the Sloan Foundation. Eran Segal was also supported by a Stanford Graduate Fellowship (SGF).
References
[1] P. Cheeseman and J. Stutz. Bayesian Classification (AutoClass): Theory and Results. AAAI
Press, 1995.
[2] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via
the EM algorithm. Journal of the Royal Statistical Society, B 39:1?39, 1977.
[3] M. Eisen, P. Spellman, P. Brown, and D. Botstein. Cluster analysis and display of genome-wide
expression patterns. PNAS, 95:14863?68, 1998.
[4] N. Friedman. The Bayesian structural EM algorithm. In Proc. UAI, 1998.
[5] N. Friedman, M. Ninio, I. Pe?er, and T. Pupko. A structural EM algorithm for phylogentic
inference. In Proc. RECOMB, 2001.
[6] A.P. Gasch et al. Genomic expression program in the response of yeast cells to environmental
changes. Mol. Bio. Cell, 11:4241?4257, 2000.
[7] T. Hofmann. The cluster-abstraction model: Unsupervised learning of topic hierarchies from
text data. In Proc. IJCAI, 1999.
[8] T. Hofmann. The cluster-abstraction model: Unsupervised learning of topic hierarchies from
text data. In Proc. International Joint Conference on Artificial Intelligence, 1999.
[9] T. R. Hughes et al. Functional discovery via a compendium of expression profiles. Cell,
102(1):109?26, 2000.
[10] F.K. Hwang, D.S.Richards, and P. Winter. The Steiner Tree Problem. Annals of Discrete
Mathematics, Vol. 53, North-Holland, 1992.
[11] A. Krogh, M. Brown, S. Mian, K. Sjolander, and D. Haussler. Hidden markov models in
computational biology: Applications to protein modeling. Mol. Biology, 235:1501?1531, 1994.
[12] A. McCallum, R. Rosenfeld, T. Mitchell, and A. Ng. Improving text classification by shrinkage
in a hierarchy of classes. In Proc. ICML, 1998.
[13] M. Meila and M.I. Jordan. Learning with mixtures of trees. Machine Learning, 1:1?48, 2000.
[14] E. Segal and D. Koller. Probabilistic hierarchical clustering for biological data. In RECOMB,
2002.
| 2065 |@word hierachy:1 sgf:1 termination:1 simplifying:1 thereby:1 initial:2 configuration:1 series:1 score:4 selecting:2 denoting:1 document:4 imaginary:1 steiner:3 current:1 com:1 stemmed:1 must:2 mst:5 additive:1 numerical:1 realistic:1 hofmann:2 designed:1 update:1 generative:1 discovering:1 leaf:28 item:3 parameterization:1 selected:2 intelligence:1 mccallum:1 provides:3 iterates:1 node:22 daphne:1 mathematical:1 along:2 constructed:5 direct:1 become:1 qualitative:1 fitting:2 g4:5 pairwise:2 expected:2 indeed:1 behavior:1 begin:1 unrelated:1 underlying:1 maximizes:1 lowest:1 substantially:2 finding:3 transformation:6 unobserved:4 duplicating:1 every:4 exactly:1 decouples:1 bio:1 grant:1 appear:4 local:11 modify:1 tends:1 analyzing:2 path:2 might:1 chose:1 conversely:1 hmms:1 range:2 graduate:1 unique:3 practice:2 hughes:1 procedure:5 significantly:1 whizbang:1 pre:1 word:7 confidence:2 protein:2 get:2 close:4 selection:2 restriction:2 optimize:3 maximizing:1 straightforward:1 attention:1 starting:1 independently:2 convex:1 focused:1 resolution:1 mrna:1 recovery:1 assigns:1 pure:1 m2:4 insight:2 haussler:1 array:1 pull:1 notion:2 pupko:1 annals:1 hierarchy:51 machin:1 element:1 intuitiveness:1 particularly:1 richards:1 featur:1 bottom:2 role:1 subproblem:1 observed:1 capture:1 region:2 highest:2 ran:4 principled:2 intuition:1 mentioned:1 convexity:3 dempster:1 idkl:5 completely:1 easily:1 joint:5 train:1 artificial:2 formation:1 choosing:1 whose:4 heuristic:2 stanford:7 larger:2 solve:1 ability:2 favor:1 g1:5 rosenfeld:1 transform:1 itself:1 noisy:1 laird:1 advantage:2 triggered:1 sequence:1 propose:2 neighboring:2 ramification:1 date:1 iff:1 achieve:1 intuitive:2 parent:1 cluster:7 optimum:1 convergence:2 empty:1 produce:2 generating:2 ijcai:1 converges:2 help:2 fixing:1 measured:3 transcript:1 krogh:1 throw:1 c:3 trading:1 closely:1 require:3 assign:1 generalization:3 tfidf:1 biological:5 hold:2 around:1 considered:1 substituting:1 estimation:1 proc:5 applicable:1 cpm:19 propensity:1 sensitive:1 bridge:1 create:1 reflects:1 weighted:1 cora:3 clearly:1 genomic:4 gaussian:4 always:1 avoid:1 shrinkage:2 varying:1 probabilistically:1 focus:1 improvement:2 consistently:1 likelihood:8 greatly:1 contrast:2 sense:1 detect:1 inference:1 abstraction:19 typically:2 entire:4 unlikely:1 hidden:4 koller:3 relation:1 favoring:1 semantics:1 overall:2 issue:1 among:3 classification:3 augment:1 denoted:1 sjolander:1 smoothing:2 special:1 summed:1 fairly:2 initialize:2 construct:4 ng:1 sampling:2 identical:1 represents:2 biology:2 unsupervised:2 icml:1 peaked:2 np:1 spline:1 simplify:1 randomly:2 winter:1 simultaneously:2 preserve:1 individual:1 m4:7 consisting:1 replacement:1 friedman:3 invariably:1 mixture:4 edge:3 closer:2 partial:1 stutz:1 tree:58 incomplete:3 divide:1 euclidean:1 penalizes:1 desired:1 causal:2 theoretical:1 instance:22 soft:1 modeling:1 measuring:1 assignment:6 maximization:1 ating:1 decomposability:1 subset:4 entry:1 graphic:1 perturbed:4 synthetic:3 density:1 peak:1 sensitivity:2 explores:1 ie:1 international:1 probabilistic:16 off:1 pool:2 together:5 autoclass:2 squared:2 ambiguity:1 again:1 aaai:1 containing:1 choose:1 worse:1 lambda:1 creating:3 segal:3 summarized:2 north:1 coefficient:1 sloan:1 depends:1 performed:1 root:1 try:1 start:1 recover:2 mutation:1 minimize:1 variance:1 characteristic:1 acid:1 correspond:2 generalize:2 raw:1 bayesian:6 produced:2 basically:1 cpms:30 carlo:1 researcher:1 definition:3 frequency:1 involved:1 obvious:1 naturally:4 associated:6 recovers:1 sampled:3 newly:1 pah:35 dataset:2 popular:1 mitchell:1 knowledge:2 organized:3 focusing:1 appears:1 follow:1 botstein:1 response:1 arranged:1 evaluated:3 just:1 mian:1 until:2 correlation:2 nonlinear:1 defines:4 perhaps:1 yeast:3 hwang:1 validity:1 normalized:1 true:1 brown:2 hence:2 assigned:4 symmetric:1 iteratively:1 round:1 m5:3 complete:2 performs:1 l1:2 percent:1 ranging:1 novel:1 recently:1 common:2 sigmoid:1 specialized:1 permuted:2 multinomial:4 functional:1 discussed:1 interpretation:1 m1:4 measurement:1 significant:2 rd:1 meila:1 mathematics:1 similarly:1 specification:1 similarity:1 longer:1 posterior:1 closest:1 optimizing:1 optimizes:1 belongs:1 arbitrarily:1 scoring:1 preserving:2 minimum:2 additional:1 care:1 somewhat:2 converge:2 ii:1 relates:1 multiple:1 full:2 pnas:1 reduces:2 cross:1 long:1 post:1 variant:3 involving:1 essentially:1 expectation:1 df:1 metric:3 iteration:4 cell:4 whereas:1 want:1 fellowship:1 microarray:1 rest:2 hac:23 comment:1 induced:1 tend:1 virtually:1 undirected:2 thing:1 subject:1 recording:1 jordan:1 structural:5 feedforward:1 m6:3 variety:1 fit:1 itr:1 tradeoff:1 expression:16 penalty:2 suffer:1 proceed:1 useful:2 locally:1 statist:1 ten:1 category:1 dna:1 generate:2 percentage:1 nsf:2 deteriorates:1 discrete:1 vol:1 key:2 demonstrating:1 penalizing:1 kept:1 utilize:1 graph:1 sum:2 striking:1 place:2 extends:2 utilizes:3 prefer:1 guaranteed:3 tackled:1 display:1 fold:1 precisely:2 flat:1 nearby:10 generates:1 aspect:1 compendium:2 extremely:1 according:1 terminates:1 slightly:1 em:8 g3:5 making:1 lem:1 intellig:1 taken:1 visualization:1 remains:1 previously:1 discus:1 turn:2 differentiates:1 end:1 serf:1 generalizes:1 apply:1 hierarchical:7 away:2 enforce:2 robustness:7 substitute:1 original:3 top:1 clustering:12 include:1 ensure:1 remaining:1 exploit:1 build:2 conquer:1 society:1 objective:1 eran:3 gradient:1 distance:23 separate:1 thank:1 hmm:1 topic:2 agglomerative:4 extent:2 cellular:2 gasch:1 spanning:1 enforcing:1 length:2 equivalently:1 executed:1 taxonomy:6 subproblems:2 negative:1 proper:1 boltzmann:1 unknown:2 perform:2 observation:1 markov:3 defining:1 unconnected:1 dirk:1 pair:5 kl:1 learned:12 pearl:1 address:2 pattern:2 program:2 built:1 including:1 royal:1 difficulty:1 examination:1 ormoneit:2 indicator:1 cheeseman:1 spellman:1 technology:1 identifies:1 extract:1 nir:1 text:7 prior:8 understanding:2 acknowledgement:1 discovery:1 relative:6 permutation:1 limitation:1 recomb:2 acyclic:1 validation:1 foundation:1 consistent:3 rubin:1 principle:2 share:1 prone:1 course:1 penalized:1 changed:1 surprisingly:1 last:1 repeat:2 mont:1 placed:1 supported:2 allow:1 wide:3 neighbor:2 volves:1 dimension:1 genome:1 computes:1 eisen:1 forward:1 commonly:1 avg:2 gene:26 overcomes:1 keep:2 global:11 uai:1 assumed:1 aci:1 continuous:1 iterative:2 robin:1 learn:4 reasonably:1 robust:1 sem:3 mol:2 improving:1 complex:2 constructing:4 domain:8 noise:7 profile:3 child:1 repeated:1 amino:1 fig:8 fashion:1 similiarity:1 candidate:1 pe:1 learns:1 down:2 theorem:2 specific:4 showing:1 er:1 list:1 exists:1 merging:1 effectively:1 maxim:1 subtree:1 illustrates:1 push:1 simply:4 desire:1 g2:10 holland:1 environmental:1 conditional:1 goal:6 careful:1 towards:1 hard:1 change:2 specifically:1 typical:1 reducing:3 corrected:1 acting:1 determined:1 total:1 experimental:3 m3:6 meaningful:2 rarely:1 internal:5 incorporate:1 dept:3 mcmc:1 tested:2 avoiding:1 |
1,169 | 2,066 | Rao-Blackwellised Particle Filtering
Data Augmentation
.
VIa
Christophe Andrieu
N ando de Freitas
Arnaud Doucet
Statistics Group
University of Bristol
University Walk
Bristol BS8 1TW, UK
Computer Science
UC Berkeley
387 Soda Hall, Berkeley
CA 94720-1776, USA
EE Engineering
University of Melbourne
Parkville, Victoria 3052
Australia
C.Andrieu @bristol.ac.uk
jfgf@cs.berkeley.edu
doucet@ee .mu.oz.au
Abstract
In this paper, we extend the Rao-Blackwellised particle filtering
method to more complex hybrid models consisting of Gaussian latent variables and discrete observations. This is accomplished by
augmenting the models with artificial variables that enable us to
apply Rao-Blackwellisation. Other improvements include the design of an optimal importance proposal distribution and being able
to swap the sampling an selection steps to handle outliers. We focus
on sequential binary classifiers that consist of linear combinations
of basis functions , whose coefficients evolve according to a Gaussian
smoothness prior. Our results show significant improvements.
1
Introduction
Sequential Monte Carlo (SMC) particle methods go back to the first publically
available paper in the modern field of Monte Carlo simulation (Metropolis and
Ulam 1949) ; see (Doucet, de Freitas and Gordon 2001) for a comprehensive review.
SMC is often referred to as particle filtering (PF) in the context of computing
filtering distributions for statistical inference and learning. It is known that the
performance of PF often deteriorates in high-dimensional state spaces. In the past,
we have shown that if a model admits partial analytical tractability, it is possible
to combine PF with exact algorithms (Kalman filters, HMM filters , junction tree
algorithm) to obtain efficient high dimensional filters (Doucet, de Freitas, Murphy
and Russell 2000, Doucet, Godsill and Andrieu 2000). In particular, we exploited
a marginalisation technique known as Rao-Blackwellisation (RB).
Here, we attack a more complex model that does not admit immediate analytical
tractability. This probabilistic model consists of Gaussian latent variables and binary observations. We show that by augmenting the model with artificial variables,
it becomes possible to apply Rao-Blackwellisation and optimal sampling strategies.
We focus on the problem of sequential binary classification (that is, when the data
arrives one-at-a-time) using generic classifiers that consist of linear combinations
of basis functions, whose coefficients evolve according to a Gaussian smoothness
prior (Kitagawa and Gersch 1996). We have previously addressed this problem in
the context of sequential fault detection in marine diesel engines (H0jen-S0rensen,
de Freitas and Fog 2000). This application is of great importance as early detection
of incipient faults can improve safety and efficiency, as well as, help to reduce downtime and plant maintenance in many industrial and transportation environments.
2
Model Specification and Estimation Objectives
Let us consider the following binary classification model. Given at time t = 1,2, .. .
an input Xt we observe Zt E {O, I} such that
Pr( Zt = llxt ,.8t ) = CP(f(xl, .8t}),
(1)
vk J::oo
where CP (u) =
exp (_a 2 /2) da is the cumulative function of the standard
normal distribution. This is the so-called pro bit link. By convention, researchers
tend to adopt a logistic (sigmoidal) link function 'P (u) = (1 + exp (_U)) -1 . However, from a Bayesian computational point of view, the probit link has many advantages and is equally valid. The unknown function is modeled as
K
!(Xt, .8t) = L .8t,k\[ldxt) = \[IT (Xt).8t ,
k=1
where we have assumed that the basis functions \[I (Xt) ? (\[11 (Xt) , ... , \[I K (Xt)/ do
not depend on unknown parameters; see (Andrieu, de Freitas and Doucet 1999) for
the more general case . .8t ? (.8t,1,' .. ,.8t,K )T E ~K is a set of unknown time-varying
regression coefficients. To complete the model , we assume that they satisfy
.8t = At.8t-1
+ BtVt, .80'" N
(rna, Po)
(2)
where Vt i?~.:f N (0 , In.) and A and B control model correlations and smoothing
(regularisation). Typically K is rather large, say 10 or 100, and the bases \[Ik (.) are
multivariate splines, wavelets or radial basis functions (Holmes and Mallick 1998).
2.1
Augmented Statistical Model
We augment the probabilistic model artificially to obtain more efficient sampling
algorithms, as will be detailed in the next section. In particular, we introduce the
set of independent variables Yt , such that
Yt =! (Xt,.8t) + nt,
(3)
h
i.i.d. N (0 1)
d d fi
were nt '"
"an
e ne Zt
that one has Pr ( Zt =
11 Xt, .8t ) =
= {I0
if Yt > 0,
otherwise.
It is then easy to check
CP (f (Xl, .8t)) .
This data augmentation strategy was first introduced in econometrics by economics
Nobel laureate Daniel McFadden (McFadden 1989). In the MCMC context, it has
been used to design efficient samplers (Albert and Chib 1993). Here, we will show
how to take advantage of it in an SMC setting.
2.2
Estimation objectives
Given, at time t , the observations Ol:t ? (Xl:t, Zl:t), any Bayesian inference is based
on the posterior distribution 1 P (d.8o:tl Ol:t)' We are, therefore, interested in estimating sequentially in time this distribution and some of its features , such as
IFor any B, we use P (dBo,tl au) to denote the distribution and p (Bo,tl au) to denote
the density, where P (dBo,tl au) = p (Bo,tl au) dBo,t. Also, Bo,t ~ {Bo, BI , ... , Bd .
lE ( f (xt, ,Bt) I Ol:t) or the marginal predictive distribution at time t for new input
data Xt+1, that is Pr (Zt+1 = 11 01:t, xHd. The posterior density satisfies a time
recursion according to Bayes rule, but it does not admit an analytical expression
and, consequently, we need to resort to numerical methods to approximate it.
3
Sequential Bayesian Estimation via Particle Filtering
A straightforward application of SMC methods to the model (1)-(2) would focus
on sampling from the high-dimensional distribution P (d,Bo:t I 01:t) (H0jen-S0rensen
et al. 2000). A substantially more efficient strategy is to exploit the augmentation
of the model to sample only from the low-dimensional distribution P ( dY1:t I01:t).
The low-dimensional samples allow us then to compute the remaining estimates
analytically, as shown in the following subsection.
3.1
Augmentation and Rao-Blackwellisation
Consider the extended model defined by equations (1)-(2)-(3). One has
p(,Bo:tlo1:t) =
J
p( ,Bo:tl x 1:t,Y1:t)p(Y1:tl o1:t)dY1:t?
Thus if we have a Monte Carlo approximation of P (dY1:t I ol:d of the form
then P (,Bo:tl 01:t) can be approximated via
N
PN (,Bo:tl 01:t) =
L w~i)p ( ,Bo:tl x1:t,yi:O '
i=l
that is a mixture of Gaussians.
From this approximation, one can estimate
lE(,Btlxl:t,Yl:t) and lE(,Bt-Llxl:t,Yl:t). For example, an estimate of the predictive
distribution is given by
PrN(Zt+1 = Il ol:t,XH1) =
J
Pr( Zt+1 = lIYH1)PN(dYl:t+1 lol:t, xt+1)
(4)
N
,,
(i) )
= ~ W t(i) ][(0,+00) ( YHl
,
i=l
where Y~21 ~ P ( dYHll Xl:t+1, Yi~~). This shows that we can restrict ourselves to
the estimation of P (Y1:t1 Ol:t) for inference purposes.
In the SMC framework, we must estimate the "target" density P (Y1:t1 Ol:t)
pointwise up to a normalizing constant. By standard factorisation, one has
t
p(Yl:tlol:t)
IT
Pr( zk IYk)p(Ykl xl:k,Yl:k-l), wherep(YIIY1:0,Xl:0) ,@,p(Yll xd?
k=l
Since Pr (Zk I Yk) is known, we only need to estimate P (Yk I Xl:k, Yl:k-d up to a norIX
malizing constant. This predictive density can be computed using the Kalman filter.
Given (Xl:k' Yl:k-l), the Kalman filter equations are the following. Set ,Bo lo = mo
and
~o l o
= ~o, then for t = 1, ... , k - 1 compute
,Bt lt-1
= At,Bt- 1It- 1
+ BtBI
(xt} ~tlt - 1 \[I (Xt) + 1
Yt lt-1 = \[IT (Xt) ,Bt lt-1
,Btlt = ,Bt lt -1 + ~tlt- 1 \[I (Xt) St- 1 (Yt - Yt lt - t)
~tit = ~t l t-1 - ~t l t-1 \[I (xt} St- 1\[lT (Xt) ~t l t-1'
~t l t-1 = At~t-1 I t-1AI
St =
\[IT
(5)
where ,Bt lt - 1 ~ 1E(,BtIXl:t-1,Yl:t-d, ,Btlt ~ 1E(,Btlxl:t,Yl:t), Ytlt - 1
IE(Ytlxl:t,Yl:t - d, ~t l t-1 ~ cov(,BtIXl:t- 1,Y1:t- 1), ~t lt ~ cov(,Btlxl:t,Yl:t) and
St ~ cov (Ytl Xl:t,Y1:t-1). One obtains
P (Yk I X1:k, Y1:k-d =
3.2
(6)
N (Yk;Y klk- 1' Sk) .
Sampling Algorithm
In this section, we briefly outline the PF algorithm for generating samples
from p(dYl:tlol:t). (For details, please refer to our extended technical report at
http://www . cs. berkeley. edu/ '" jfgf /publications . html.) Assume that at time t - 1
we have N particles {Yi~Ld~l distributed according to P (dYl:t - 11 ol:t- d from
which one can get the following empirical distribution approximation
1
PN (dYl:t-11 ol:t-d
=N
N
L JYi~;_l (dYl:t-d .
i= l
Various SMC methods can be used to obtain N new paths {Yi~~}~l distributed
approximately according to P (dYl:t1 Ol:t)' The most successful of these methods
typically combine importance sampling and a selection scheme. Their asymptotic
convergence (N --t 00) is satisfied under mild conditions (Crisan and Doucet 2000).
Since the selection step is standard (Doucet et al. 2001), we shall concentrate on
describing the importance sampling step. To obtain samples from P( dYl:t IOl:t), we
can sample from a proposal distribution Q(dYl:t) and weight the samples appropriately. Typically, researchers use the transition prior as proposal distribution (Isard
and Blake 1996). Here, we implement an optimal proposal distribution, that is one
that minimizes the variance of the importance weights W (Yl:t) conditional upon not
modifying the path Y1:t-1' In our case, we have
( I
)
P Yt X1:t,Yl:t-1,Zt ex:
{p(YtIXl:t ,Y1:t-dlI[o,+ oo) (Yt)
p(Ytlxl:t ,Yl:t-dlI(- oo,o) (Yt)
if Zt = 1
if Zt = 0 '
which is a truncated Gaussian version of (6) of and consequently
W (Yl:t) ex: Pr (Zt I Xl:t, Y1:t - d =
(1 _
<I> ( _
Y$,l ) ) z,
<I> ( _
Y$,l ) 1-z,
(7)
The algorithm is shown in Figure 1. (Please refer to our technical report for convergence details.)
Remark 1 When we adopt the optimal proposal distribution, the importance weight
Wt ex: Pr (Zt I X1:t, Y1:t - d does not depend on Yt. It is thus possible to carry out
the selection step before the sampling step. The algorithm is then similar to the
auxiliary variable particle filter of (Pitt and Shephard 1999). This modification to
the original algorithm has important implications. It enables us to search for more
Sequential importance sampling step
? For
.
t
? For i
-(i)
h.
(i)
:::{i)
= 1, ... , N, (3t lt-1 = (3t lt-1 and sample Yt
= 1, ... , N,
~ P
(
(i)
)
dYtl Xl:t, Yl:t-1 ' Zt .
evaluate the importance weights using (7).
Selection step
? Multiply/Discard particles {~i ),,B~i~ _l}~l with respect to high/low impor. hts W (i) to 0 b
' N partlc
. Ies { Yt(i) , (3(i)
}N
? tance welg
tam
t lt- 1 i=l '
t
Updatmg step
? Compute ~t+1 I t given ~t l t - 1'
? For i
= 1, ... , N,
C)
use one step of the Kalman recursion (5) to compute {,B~i~ l l t }
-C)
given {y/ ,(3 ti t-1 } and ~t l t-1'
Figure 1: RBPF for semiparametric binary classification.
likely regions of the posterior at time t-1 using the information at time t to generate
better samples at time t. In practice, this increases the robustness of the algorithm
to outliers and allows us to apply it in situations where the distributions are very
peaked (e.g., econometrics and almost deterministic sensors and actuators).
Remark 2 Th e covariance updates of the Kalman jilter are outside the loop over
particles. This results in substantial computational savings.
4
Simulations
To compare our model , using the RBPF algorithm, to standard logistic and probit
classification with PF, we generated data from clusters that change with time as
shown in Figure 2. This data set captures the characteristics of a fault detection
problem that we are currently studying. (For some results of applying PF to fault
detection in marine diesel engines, please refer to (H0jen-S0rensen et al. 2000).
More results will become available once permission is granted.) This data cannot
be easily separated with an algorithm based on a time-invariant model.
For the results presented here, we set the initial distributions to: (30 '" N(O , 51)
and Yo '" N(O, 51). The process matrices were set to A = I and B = JI, where
82 = 0.1 is a smoothing parameter. The number of bases (cubic splines with random
locations) was set to 10. (It is of course possible, when we have some data already,
to initialise the bases locations so that they correspond to the input data. This trick
for efficient classification in high dimensional input spaces is used in the support
vector machines setting (Vapnik 1995).) The experiment was repeated with the
number of particles varying between 10 and 400. Figure 3 shows the "value for
money" summary plot. The new algorithm has a lower computational cost and
shows a significant reduction in estimation variance. Note that the computation
of the RBPF stays consistently low even for small numbers of particles. This has
enabled us to apply the technique to large models consisting of hundreds of Bases
using a suitable regulariser. Another advantage of PF algorithms for classification
is that they yield entire probability estimates of class membership as shown in
Figure 4.
0'<>0
-:5' - - - -o:--- ---::5
-:5: ---'-'-- -0
: ------::5
Data from t=1 to t=100
Data from t=1 00 to t=200
-5'---------5
0
- 5 '-------"-----
5
-5
Data from t=200 to t=300
0
5
Data from t=1 to t=300
Figure 2: Time-varying data.
35
!... . 'j
r,.
I
I
' . ~ , ...... ... .
I
i
T
L -
...
-
I
-n. ... "',i ... T L~
.
?
,-
1'''''
~ ... ~.......... i
-,...
.
50'----~--L-~--~8-~,0~-~,2--,~
4-~,6-~,8
Computation (flops)
,10'
Figure 3: Number of classification errors as the number of particles varies between
10 and 400 (different computational costs). The algorithm with the augmentation
trick (RBPF) is more efficient than standard PF algorithms.
5
Conclusions
In this paper, we proposed a dynamic Bayesian model for time-varying binary classification and an efficient particle filtering algorithm to perform the required computations. The efficiency of our algorithm is a result of data augmentation, RaoBlackwellisation, adopting the optimal importance distribution, being able to swap
the sampling and selection steps and only needing to update the Kalman filter means
in the particles loop. This extends the realm of efficient particle filtering to the ubiquitous setting of Gaussian latent variables and binary observations. Extensions to
n-ary observations, different link functions and estimation of the hyper-parameters
can be carried out in the same framework.
50
-1
Figure 4: Predictive density.
References
Albert , J. and Chib , S. (1993) . Bayesian analysis of binary and polychotomous response
data, Journal of the American Statistical Association 88(422): 669- 679.
Andrieu, C., de Freitas, N. and Doucet , A. (1999) . Sequential Bayesian estimation and
model selection applied to neural networks, Technical Report CUED/F-INFENG/TR
341, Cambridge University Engineering Department.
Crisan, D. and Doucet, A. (2000) . Convergence of sequential Monte Carlo methods, Technical Report CUED/F-INFENG/TR 381, Cambridge University Engineering Department.
Doucet , A. , de Freitas, N. and Gordon, N. J. (eds) (2001) . Sequential Monte Carlo Methods
in Practice, Springer-Verlag.
Doucet , A. , de Freitas, N. , Murphy, K. and Russell, S. (2000). Rao blackwellised particle
filtering for dynamic Bayesian networks, in C. Boutilier and M. Godszmidt (eds),
Uncertainty in Artificial Intelligence, Morgan Kaufmann Publishers, pp . 176- 183.
Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling
methods for Bayesian filtering , Statistics and Computing 10(3): 197- 208.
H0jen-S0rensen, P. , de Freitas, N. and Fog, T. (2000). On-line probabilistic classification
with particle filters, IEEE N eural Networks for Signal Processing, Sydney, Australia.
Holmes, C. C. and Mallick, B. K. (1998). Bayesian radial basis functions of variable
dimension, Neural Computation 10(5): 1217- 1233.
Isard, M. and Blake, A. (1996) . Contour tracking by stochastic propagation of conditional
density, European Conference on Computer Vision, Cambridge, UK, pp. 343- 356.
Kitagawa, G. and Gersch, W. (1996). Smoothn ess Priors Analysis of Tim e Series, Vol.
116 of Lecture Notes In Statistics, Springer-Verlag.
McFadden, D. (1989). A method of simulated momemts for estimation of discrete response
models without numerical integration, Econometrica 57: 995- 1026.
Metropolis, N. and Uiam, S. (1949). The Monte Carlo method, Journal of th e American
Statistical Association 44(247): 335- 341.
Pitt , M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters ,
Journal of the American Statistical Association 94(446): 590- 599.
Vapnik , V. (1995). Th e Nature of Statistical Learning Th eory, Springer-Verlag, New York.
| 2066 |@word mild:1 version:1 briefly:1 simulation:3 covariance:1 tr:2 klk:1 carry:1 ld:1 reduction:1 initial:1 series:1 daniel:1 past:1 freitas:9 nt:2 bd:1 must:1 numerical:2 enables:1 plot:1 hts:1 update:2 isard:2 intelligence:1 es:1 marine:2 location:2 attack:1 sigmoidal:1 welg:1 become:1 ik:1 consists:1 combine:2 introduce:1 ol:10 pf:8 becomes:1 estimating:1 substantially:1 minimizes:1 blackwellised:3 berkeley:4 ti:1 xd:1 classifier:2 uk:3 control:1 zl:1 safety:1 t1:3 engineering:3 before:1 path:2 approximately:1 au:5 smc:6 bi:1 practice:2 implement:1 empirical:1 downtime:1 radial:2 get:1 cannot:1 selection:7 context:3 applying:1 www:1 deterministic:1 transportation:1 yt:12 go:1 economics:1 straightforward:1 factorisation:1 rule:1 holmes:2 initialise:1 enabled:1 handle:1 target:1 exact:1 trick:2 approximated:1 econometrics:2 capture:1 region:1 russell:2 prn:1 yk:4 substantial:1 environment:1 mu:1 econometrica:1 dynamic:2 depend:2 tit:1 predictive:4 upon:1 efficiency:2 swap:2 basis:5 po:1 dyl:8 easily:1 various:1 iol:1 separated:1 monte:7 artificial:3 hyper:1 outside:1 whose:2 say:1 otherwise:1 statistic:3 cov:3 advantage:3 analytical:3 loop:2 oz:1 convergence:3 cluster:1 ulam:1 generating:1 help:1 oo:3 cued:2 ac:1 augmenting:2 tim:1 shephard:2 sydney:1 auxiliary:2 c:2 convention:1 concentrate:1 filter:9 modifying:1 stochastic:1 australia:2 enable:1 kitagawa:2 extension:1 hall:1 blake:2 normal:1 exp:2 great:1 mo:1 pitt:2 early:1 adopt:2 purpose:1 estimation:8 currently:1 bs8:1 sensor:1 gaussian:6 rna:1 rather:1 pn:3 crisan:2 varying:4 publication:1 ytl:1 focus:3 yo:1 improvement:2 vk:1 consistently:1 check:1 industrial:1 inference:3 publically:1 i0:1 membership:1 typically:3 bt:7 entire:1 interested:1 classification:9 html:1 augment:1 smoothing:2 integration:1 uc:1 marginal:1 field:1 once:1 saving:1 sampling:11 peaked:1 report:4 spline:2 gordon:2 modern:1 chib:2 comprehensive:1 murphy:2 consisting:2 ourselves:1 ando:1 detection:4 multiply:1 mixture:1 arrives:1 fog:2 implication:1 partial:1 tree:1 impor:1 walk:1 melbourne:1 rao:7 tractability:2 cost:2 hundred:1 successful:1 llxl:1 varies:1 st:4 density:6 ie:1 stay:1 probabilistic:3 yl:15 polychotomous:1 augmentation:6 satisfied:1 ykl:1 admit:2 tam:1 resort:1 american:3 de:9 coefficient:3 satisfy:1 view:1 bayes:1 il:1 variance:2 characteristic:1 kaufmann:1 correspond:1 yield:1 bayesian:9 carlo:7 researcher:2 dbo:3 bristol:3 ary:1 ed:2 i01:1 pp:2 subsection:1 realm:1 ubiquitous:1 back:1 response:2 correlation:1 propagation:1 llxt:1 logistic:2 h0jen:4 usa:1 andrieu:6 analytically:1 arnaud:1 please:3 outline:1 complete:1 cp:3 pro:1 fi:1 dy1:3 ji:1 extend:1 association:3 significant:2 refer:3 cambridge:3 ai:1 smoothness:2 particle:18 lol:1 specification:1 money:1 base:4 multivariate:1 posterior:3 discard:1 verlag:3 binary:8 christophe:1 fault:4 accomplished:1 exploited:1 yi:4 vt:1 morgan:1 signal:1 needing:1 technical:4 jfgf:2 equally:1 y:1 infeng:2 regression:1 maintenance:1 vision:1 albert:2 adopting:1 proposal:5 semiparametric:1 addressed:1 publisher:1 appropriately:1 marginalisation:1 jyi:1 tend:1 ee:2 easy:1 restrict:1 reduce:1 expression:1 granted:1 york:1 remark:2 boutilier:1 detailed:1 ifor:1 eory:1 http:1 generate:1 deteriorates:1 rb:1 discrete:2 shall:1 vol:1 group:1 incipient:1 uncertainty:1 soda:1 extends:1 almost:1 bit:1 department:2 according:5 combination:2 gersch:2 tw:1 metropolis:2 modification:1 outlier:2 invariant:1 pr:8 equation:2 previously:1 describing:1 studying:1 available:2 junction:1 gaussians:1 victoria:1 apply:4 observe:1 actuator:1 generic:1 permission:1 robustness:1 original:1 remaining:1 include:1 exploit:1 objective:2 already:1 strategy:3 link:4 simulated:1 hmm:1 nobel:1 kalman:6 o1:1 modeled:1 pointwise:1 godsill:2 design:2 regulariser:1 zt:13 unknown:3 perform:1 observation:5 truncated:1 immediate:1 situation:1 extended:2 flop:1 y1:11 introduced:1 required:1 dli:2 engine:2 yll:1 able:2 tance:1 mallick:2 suitable:1 parkville:1 hybrid:1 recursion:2 scheme:1 improve:1 iyk:1 ne:1 carried:1 prior:4 review:1 evolve:2 regularisation:1 asymptotic:1 plant:1 probit:2 lecture:1 mcfadden:3 filtering:10 rbpf:4 lo:1 course:1 summary:1 blackwellisation:4 allow:1 distributed:2 dimension:1 valid:1 cumulative:1 transition:1 contour:1 tlt:2 approximate:1 obtains:1 laureate:1 doucet:13 sequentially:1 assumed:1 search:1 latent:3 sk:1 nature:1 zk:2 ca:1 complex:2 artificially:1 european:1 da:1 repeated:1 x1:4 augmented:1 eural:1 referred:1 tl:10 wherep:1 cubic:1 xl:11 wavelet:1 xh1:1 xt:17 admits:1 normalizing:1 consist:2 vapnik:2 sequential:10 importance:9 lt:11 likely:1 tracking:1 bo:11 springer:3 satisfies:1 yhl:1 conditional:2 consequently:2 change:1 sampler:1 wt:1 called:1 s0rensen:4 support:1 evaluate:1 mcmc:1 ex:3 |
1,170 | 2,067 | Perceptual Metamers
in Stereoscopic Vision
Benjamin T. Backus*
Department of Psychology
University of Pennsylvania
Philadelphia, PA 19104-6196
backus@psych.upenn.edu
Abstract
Theories of cue combination suggest the possibility of constructing
visual stimuli that evoke different patterns of neural activity in
sensory areas of the brain, but that cannot be distinguished by any
behavioral measure of perception. Such stimuli, if they exist,
would be interesting for two reasons. First, one could know that
none of the differences between the stimuli survive past the
computations used to build the percepts.
Second, it can be
difficult to distinguish stimulus-driven components of measured
neural activity from top-down components (such as those due to
the interestingness of the stimuli). Changing the stimulus without
changing the percept could be exploited to measure the stimulusdriven activity. Here we describe stimuli in which vertical and
horizontal disparities trade during the construction of percepts of
slanted surfaces, yielding stimulus equivalence classes.
Equivalence class membership changed after a change of vergence
eye posture alone, without changes to the retinal images. A formal
correspondence can be drawn between these ?perceptual metamers?
and more familiar ?sensory metamers? such as color metamers.
1
Introduction
Two types of perceptual process might, in principle, map physically different visual
stimuli onto the same percept. First, the visual system has a host of constancy
mechanisms that extract information about the visual environment across
uninteresting changes in the proximal stimulus. Some of these mechanisms could
be ?leak-proof,? leaving no trace of the original differences between the stimuli.
Second, the visual system must combine information from redundant cues if it is to
build percepts robustly. Recent cue conflict experiments have shown that the visual
system?s estimate of a scene parameter, as evinced in a visual percept, is often
simply a weighted average of the parameter as specified by each cue separately [1][2]. Thus, a properly balanced cue-conflict stimulus might come to evoke the same
percept as a ?natural? or cue-concordant stimulus.
*
http://psych.upenn.edu/~backus
Here, random-dot stereograms will be used to argue that leak-proof versions of both
types of process exist. When a vertical magnifier is placed before one eye, a truly
frontoparallel surface appears slanted. Adding horizontal magnification in the same
eye restores frontoparallel appearance. The original stimulus and the magnified
stimulus therefore have different patterns of binocular disparity but give rise to
similar judgments of surface slant [3]. We show here that such stimuli are
perceptually indistinguishable to practiced observers in a psychophysical
discrimination task, which implies the loss of some disparity information.
This loss could occur, first, in a well-studied constancy mechanism that uses vertical
disparity to correct the depth relief pattern associated with horizontal disparity [4].
However, the amount of horizontal magnification needed to null vertical
magnification is less than would be predicted from use of this constancy mechanism
alone; a second constancy mechanism exists that corrects horizontal disparities by
using felt eye position, not vertical disparity [5]. Adding vertical magnification
without changing eye position therefore creates a cue conflict stimulus. We show
here that the amount of horizontal magnification needed to null the vertical
magnification changes with the vergence posture of the eyes, which implies that
both types of process (constancy and cue combination) are leak-proof across certain
ranges of variation (magnifications) in these stereoscopic stimuli.
2
Stereoscopic slant perception: review of theory
The stereo component of the perceived slant of a random-dot surface can be
modeled as the visual system?s weighted average of two stereo slant estimates [5][ 6]. Horizontal disparity is ambiguous because it depends not only on surface slant,
but also on surface patch location relative to the head. One stereo estimator
resolves this ambiguity using vertical disparity (images are vertically larger in the
closer eye), and the other resolves it using felt eye position. Vertical magnification
in one eye thus creates a cue-conflict because it affects only the estimator that uses
vertical disparity.
The two stereo estimators have different relative reliability at different distances, so
the weights assigned to them by the visual system changes as a function of distance
[7]. Since vergence eye posture is a cue to distance [8], one might predict that
?perceptually metameric? stereo stimuli, if they exist, will lose their metameric
status after a pure change of vergence eye posture that preserves the metamers?
retinal images [9].
We shall now briefly describe the two stereoscopic slant estimators. This theory is
covered elsewhere in greater detail [5]. Although surface slant has two components
(slant and tilt [10]), we will consider only slant about a vertical axis. The arguments
can be extended to slant about axes of arbitrary orientation [5].
The visual signals used in stereoscopic slant perception can be conveniently
parameterized by four numbers [5]. Each can be considered a signal. A surface
patch typically gives rise to all four signals. Two signals are the horizontal gradient
of horizontal disparity, and the vertical gradient of vertical disparity, which we
parameterize as horizontal size ratio (HSR) and vertical size ratio (VSR),
respectively, in the manner of Rogers and Bradshaw [11]. They are defined as the
horizontal (or vertical) size of the patch in the left eye, divided by the horizontal (or
vertical) size in the right eye. These two signals must be measured from the retinal
images. The two remaining signals are the headcentric azimuth and vergence of the
surface patch. These signals can be known either by measuring the eyes? version
and vergence, respectively, or from the retinal images [12].
A very good approximation that relates surface slant to horizontal disparity and
VSR is:
S HSR,VSR = -tan-1 [ 1 ln HSR
?
VSR
]
Equation 1
where ? is the vergence of the surface patch in radians. We call this method of slant
estimation slant from HSR and VSR.
A very good approximation that relates surface slant to horizontal disparity and
azimuth is:
S HSR,EP = -tan-1 [
1
ln HSR - tan?
?
]
Equation 2
where ? is the azimuth of the surface patch. We call this method of slant estimation
slant from HSR and eye position on the supposition that azimuth per se is known to
the visual system primarily through measurement of the eyes? version.
Each estimator uses three of the four signals available to estimate surface slant from
horizontal disparity. Nonstereo slant estimates can be rendered irrelevant by the
choice of task, in which case perceived slant is a weighted average of the slants
predicted from these two stereoscopic slant estimates [5, 6]. In principle, the
reliability of slant estimation by HSR and eye position is limited at short viewing
distances (large ?) by error in the measurement of ?. Slant from HSR and VSR, on
the other hand, continues to become more reliable as viewing distance decreases. If
one assumes that the visual system knows how reliable each estimator is, one would
predict that greater weight is given to the HSR and VSR estimate at near than at far
distances, and this is in fact the case [7].
Whether each estimate is separately computed in its own neural process, and then
given a weight, is not known. A maximum a posteriori Bayesian scheme that
simply estimates the most likely slant given the observed signals behaves in a
similar fashion as the weighted estimates model, though actual likelihood density
(probability per deg of slant) is extraordinarily small in the case of stimuli that
contain large cue conflicts [9]. The real visual system does not flinch, but instead
produces a slant estimate that looks for all the world like a weighted average. It
remains a possibility therefore that optimal slant estimation is implemented as a
weighted combination of separate estimates.
We have now developed the theory to explain why HSR and VSR trade with each
other at the ?constancy? level of a single estimator (Equation 1), and why natural
stimuli might appear the same as cue conflict stimuli (weighted averaging of
estimates derived from exploitation of Equations 1 and 2, respectively). We next
describe experiments that tested whether magnified (cue conflict) stimuli are
distinguishable from natural (concordant) stimuli.
3
Existence of stereoscopic metamers
Stimuli were sparse random dot stereograms (RDS) on a black background, 28 deg
in diameter, presented directly in front of the head using a haploscope. Observers
performed a forced choice task with stimuli that contained different amounts of
unilateral vertical and horizontal magnification. Vertical magnification was zero for
the ?A? stimuli, and 2% in the right eye for the ?B? stimuli (1% minification in the
left eye and 1% magnification in the right eye). Horizontal magnification was set at
the value that nulled apparent slant in ?A? stimuli (i.e. approximately 0%), and took
on a range of values in ?B? stimuli. Each trial consisted of two ?A? stimuli and one
?B? stimulus. The observer?s task was to determine whether the three stimuli were
presented in AAB or BAA order [13], i.e., whether the stimulus with vertical
magnification was first or last of the three stimuli. Each stimulus was presented for
0.5 sec. Each stimulus was generated using a fresh set of 200 randomly positioned
dots. Each dot had a circular raised cosine luminance profile that was 30 arcmin in
diameter. Three observers participated, including the author. Results are shown in
Figure 1.
vMags = 0% and 2%
N = 40 trials per datapoint
Percent Correct
100
BTB
MJN
JRF
80
60
40
-3 -2 -1
0
-1.4
-1 -0.6
-2
0
2
Horiz mag in left eye in stimulus B (%)
Figure 1. Observers are unable to distinguish 0% and 2% unilateral
vertical magnification when unilateral horizontal magnification is
added as well. Open squares show the horizontal magnification
that evoked zero perceived slant under 2% vertical magnification.
For each observer, there was a value of horizontal magnification that, when added to
the ?B? stimulus, rendered it indistinguishable from the ?A? stimulus. This is
shown in Figure 1 by the fact that performance drops to chance (50%) at some value
of horizontal magnification. From this experiment it is evident that stimuli with very
different disparity patterns can be made perceptually indistinguishable in a forcedchoice task with well-practiced observers.
3.1
Experimental conditions necessary for stereo metamers
Several properties of the experiment were essential to the effect. First, the vertical
magnification must not be to large. At large vertical magnifications it is still
possible to null apparent slant, but the stimuli are distinguishable because the dots
themselves look different (they look as though blurred in the vertical direction).
Two out of three observers were able to distinguish the ?A? and ?B? stimuli 100%
of the time when the vertical magnification was increased from 2% to 5%. Second,
observers must be instructed to maintain fixation. If left and right saccades are
allowed, the ?B? stimulus appears slanted in the direction predicted by its horizontal
magnification. This is a rather striking effect?the surface appears to change slant
simply because one starts looking about. This effect was not found previously [14]
but is predicted as a consequence of sequential stereopsis [15]. Finally, if the
stimuli are shown for more than about 1 sec it is possible to distinguish ?A? and ?B?
stimuli by making vertical saccades from the top to the bottom of the stimulus, by
taking advantage of the fact that in forward gaze, vertical saccades have equal
amplitude in the two eyes [16]. For ?B? stimuli only, the dots are diplopic (seen in
double vision) immediately after a saccade to the top (or bottom) of the stimulus.
An automatic vertical vergence eye movement then brings the dots into register after
about 0.5 sec. At that point a saccade to the bottom (or top) of the stimulus again
causes diplopia.
4
Breaking metamerization though change of vergence eye posture
In the haploscope it was possible to present unchanged retinal images across a range
of vergence eye postures. Stimuli that were metameric to each other with the eyes
verged at 100 cm were presented again with the eyes verged at 20 cm. For three out
of four observers, the images were then distinguishable. Figure 2 illustrates this
effect schematically, and Figure 3 quantifies it by plotting the amount of horizontal
magnification that was needed to null apparent slant at each of the two vergence
angles for one observer (left panel) and all four observers (right panel).
HSR & VSR
HSR & eye pos
Percept
Vmag
Hmag
Figure 2. Schematic illustration of the effect of distance in the
slant-nulling task. First panel: both stereoscopic methods of
estimating slant indicate that the surface is frontoparallel, and it
appears so. Second panel: a vertical magnifier is placed before one
eye, changing the estimate that uses vertical disparity, but not the
estimate that uses eye position. The resulting percept is a weighted
average of the two. Third panel: horizontal magnification is added
until the surface appears frontoparallel again. At this point the two
stereo estimates have opposite sign. Fourth panel: increasing the
apparent distance to the stimulus (by decreasing the vergence)
scales up both estimates by the same factor. The surface no longer
appears frontoparallel because the weighting of the estimates has
changed.
Horiz magnification
to null slant (%)
Vertical magnification: ?2%
2.8
20
cm
100
cm
2.4
20
cm
2.0
1.6
1.2
100
cm
100
cm
0 5 10 15 20 25 30 35 40
Trial
MJ
N
BT
CS
JRF
B
Subject
Figure 3. When the eyes were verged at 100 or 20 cm distance,
different amounts of horizontal magnification were needed to null
the slant induced by vertical magnification. Left: 10 settings that
nulled slant at 100 cm, followed by 20 settings at 20 cm, followed
by 10 at 100 cm (observer BTB). Right: three out of four observers
show an effect of vergence per se. Error bars are SEs of the mean.
5
Comparison of perceptual and sensory metamers
The stimuli described here appear the same as a result of perceptual computations
that occur well after transduction of light energy by the photoreceptors. Physically
different stimuli that are transduced identically might be dubbed sensory metamers.
One example of a sensory metamer is given by the trade between intensity and
duration for briefly flashed lights (Bloch?s Law [17]): two flashes containing the
same number of photons are indistinguishable if their durations are both less than 10
msec. Another example of sensory metamerization, that we will now consider in
greater detail, is the traditional color metamer. The three cone photoreceptor types
can support color vision because they are sensitive to different wavelengths of light.
However, each cone type responds to a range of wavelengths, and two lights with
different spectra may activate the three cone types identically. From that point on,
the lights will be indistinguishable within the nervous system. (See [18] for a review
of color metamers).
Table 1 summarizes several properties of color metamers, and analogous properties
of our new stereo metamers. We can approximate the visible spectrum of a light by
sampling its power within N different wavelength intervals, where N is large. Thus
light t can be represented by an Nx1 vector. Light t? is metameric to t if Bt? = Bt,
where B is the 3xN matrix whose rows represent the spectral sensitivities of the
three cone mechanisms [19]. The transformation that maps one stereo metamer to
another is simply a scaling of one eyes? image in the vertical and horizontal
directions, with less scaling typically needed in the horizontal than vertical
direction. Let u and v represent the x and y disparity, respectively, so that [u v] is a
function of location (x,y) within the cyclopean image. Then two random-dot image
pairs (representing flat surfaces slanted about a vertical axis) will be metameric if
their disparity patterns, [u? v?] and [u v], are related to each other by [u? v?] =
[u(1+m) v(1+n)], where m and n are small (on the order of 0.01), with m/n equal to
the weight of SHSR,VSR in the final slant estimate.
Table 1: properties of color and stereo metamers
PROPERTY
COLOR METAMERS
STEREO METAMERS.
Metamer type:
Sensory
Perceptual
Site of loss:
Peripheral
Central (two places)
Loss process:
Transduction
Metameric
class
formation:
Lights t? and t are metameric iff Bt? = Bt, where
B is the 3xN matrix of
cone spectral sensitivities
Computation
Disparity map [u v] is
metameric to [u? v?] iff
[u? v?] = [u(1+m) v(1+n)]
where m and n are small and
in the proper ratio
Dimensionality
reduction:
N? 3
loss of 1 degree of freedom
Etiology:
Capacity limit
Recovery of scene parameter
Computation of surface slant removes one dimension from the set of all physical
stimuli. Depending how the problem is framed, this is a reduction from 2
dimensions (HSR and VSR) to one (slant), or from many dimensions (all physical
stimuli that represent slanted surfaces) to one fewer dimensions.
While color and stereo metamers can be described as sensory and perceptual,
respectively, the boundary between these categories is fuzzy, as is the boundary
between sensation and perception. Would motion metamers based on ?early?
motion detectors be sensory or perceptual? What of stimuli that look identical to
retinal ganglion cells, after evoking different patterns of photoreceptor activity?
While there is a real distinction to be made between sensory and perceptual
metamers, but not all metamers need be easily categorized as one or the other.
5.1 The metamer hierarchy
Loftus [20] makes a distinction reminiscent of the one made here, between ?memory
metamers? and ?perceptual metamers,? with memory metamers being stimuli that
evoke distinguishable percepts during live viewing, but that become
indistinguishable after mnemonic encoding. Thus, Loftus classified as ?perceptual?
both our perceptual and sensory metamers. Figure 4 suggests how the three
concepts are related. In this framework, color and stereo metamers are both
perceptual metamers, but only color metamers are sensory metamers.
Memory
metamers
Perceptual metamers
(e.g. stereo)
Sensory metamers
(e.g. color)
Figure 5. The metamer hierarchy.
6
Conclusions
At each vergence eye posture it was possible to create stereoscopic stimuli with
distinct disparity patterns that were nonetheless indistinguishable in a forced choice
task. Stimuli that were metamers with the eyes in one position became
distinguishable after a change of vergence eye posture alone, without changes to the
retinal images. We can conclude that horizontal disparity per se is lost to the visual
system after combination with the other signals that are used to interpret it as depth.
Presumably, stereo metamers have distinguishable representations in primary visual
cortex?one suspects this would be evident in evoked potentials or fMRI. The loss
of information that renders these stimuli metameric probably occurs in two places.
First, there appears to be a leak-proof ?constancy? computation in which vertical
disparity is used to correct horizontal disparity (Equation 1). The output of this
computation is unaffected if equal amounts of horizontal and vertical magnification
are added to one eyes? image. However, the estimator that uses felt eye position can
distinguish these stimuli, because their horizontal size ratios differ. Thus a second
leak-proof step must occur, in which slant estimates are combined in a weighted
average. It seems reasonable to call these stimuli ?perceptual metamers,? by
analogy with, and to distinguish them from, the traditional ?sensory?
metamerization of colored lights.
Acknowledgments
This work was supported by startup funds provided to the author by the University of
Pennsylvania. The author thanks Mark Nolt for help conducting the experiments, Rufus
Frazer for serving as an observer, and Jack Nachmias and David Brainard for comments
on an earlier draft of this paper.
References
1.
Clark, J.J. and A.L. Yuille, Data fusion for sensory information processing systems.
1990, Boston: Kluwer.
2.
Landy, M.S., et al., Measurement and modeling of depth cue combination: in
defense of weak fusion. Vision Research, 1995. 35(3): p. 389-412.
3.
Ogle, K.N., Induced size effect. I. A new phenomenon in binocular space
perception associated with the relative sizes of the images of the two eyes. Archives
of Ophthalmology, 1938. 20: p. 604-623.
4.
G?rding, J., et al., Stereopsis, vertical disparity and relief transformations. Vision
Res, 1995. 35(5): p. 703-22.
5.
Backus, B.T., et al., Horizontal and vertical disparity, eye position, and
stereoscopic slant perception. Vision Res, 1999. 39(6): p. 1143-70.
6.
Banks, M.S. and B.T. Backus, Extra-retinal and perspective cues cause the small
range of the induced effect. Vision Res, 1998. 38(2): p. 187-94.
7.
Backus, B.T. and M.S. Banks, Estimator reliability and distance scaling in
stereoscopic slant perception. Perception, 1999. 28(2): p. 217-42.
8.
Foley, J.M., Binocular distance perception. Psychol Rev, 1980. 87(5): p. 411-34.
9.
Backus, B.T. and M.J. Nolt, Analysis of stereoscopic metamers. Journal of Vision
(Vision Sciences conference supplement), 2001. 1: p. in press.
10.
Stevens, K.A., Slant-tilt: the visual encoding of surface orientation. Biol Cybern,
1983. 46(3): p. 183-95.
11.
Rogers, B.J. and M.F. Bradshaw, Vertical disparities, differential perspective and
binocular stereopsis. Nature, 1993. 361(6409): p. 253-5.
12.
Mayhew, J.E. and H. Longuet-Higgins, C, A computational model of binocular
depth perception. Nature, 1982. 297(5865): p. 376-378.
13.
Calkins, D.J., J.E. Thornton, and E.N. Pugh, Jr., Monochromatism determined at a
long-wavelength/middle-wavelength cone- antagonistic locus. Vision Res, 1992.
32(12): p. 2349-67.
14.
van Ee, R. and C.J. Erkelens, Temporal aspects of binocular slant perception.
Vision Res, 1996. 36(1): p. 43-51.
15.
Enright, J.T., Sequential stereopsis: a simple demonstration. Vision Res, 1996.
36(2): p. 307-12.
16.
Schor, C.M., J. Gleason, and D. Horner, Selective nonconjugate binocular
adaptation of vertical saccades and pursuits. Vision Res, 1990. 30(11): p. 1827-44.
17.
Barlow, H.B., Temporal and spatial summation in human vision at different
backgound intensities. Journal of Physiology, 1958. 141: p. 337-350.
18.
Wandell, B.A., Foundations of vision. 1995, Sunderland, MA: Sinauer Associates.
19.
Baylor, D.A., B.J. Nunn, and J.L. Schnapf, Spectral sensitivity of cones of the
monkey Macaca fascicularis. J Physiol, 1987. 390: p. 145-60.
20.
Loftus, G.R. and E. Ruthruff, A theory of visual information acquisition and visual
memory with special application to intensity-duration trade-offs. J Exp Psychol
Hum Percept Perform, 1994. 20(1): p. 33-49.
| 2067 |@word trial:3 exploitation:1 briefly:2 version:3 middle:1 seems:1 open:1 reduction:2 disparity:27 mag:1 practiced:2 past:1 must:5 slanted:5 reminiscent:1 physiol:1 visible:1 remove:1 drop:1 fund:1 discrimination:1 alone:3 cue:15 fewer:1 nervous:1 short:1 metamerization:3 colored:1 draft:1 location:2 become:2 diplopia:1 differential:1 fixation:1 combine:1 behavioral:1 manner:1 rding:1 upenn:2 themselves:1 brain:1 decreasing:1 resolve:2 actual:1 increasing:1 provided:1 estimating:1 panel:6 transduced:1 null:6 what:1 cm:11 psych:2 monkey:1 fuzzy:1 developed:1 magnified:2 dubbed:1 transformation:2 temporal:2 appear:2 evoking:1 interestingness:1 before:2 vertically:1 limit:1 consequence:1 encoding:2 approximately:1 might:5 black:1 studied:1 equivalence:2 evoked:2 suggests:1 limited:1 range:5 jrf:2 forcedchoice:1 acknowledgment:1 lost:1 area:1 physiology:1 nulled:2 suggest:1 cannot:1 onto:1 pugh:1 live:1 cybern:1 map:3 duration:3 recovery:1 immediately:1 pure:1 estimator:9 aab:1 higgins:1 variation:1 analogous:1 antagonistic:1 construction:1 tan:3 hierarchy:2 us:6 pa:1 associate:1 magnification:30 continues:1 ep:1 constancy:7 observed:1 bottom:3 parameterize:1 trade:4 backus:7 decrease:1 movement:1 balanced:1 benjamin:1 environment:1 leak:5 stereograms:2 yuille:1 creates:2 po:1 easily:1 represented:1 forced:2 distinct:1 describe:3 activate:1 startup:1 rds:1 formation:1 extraordinarily:1 apparent:4 whose:1 larger:1 s:1 final:1 thornton:1 advantage:1 took:1 adaptation:1 iff:2 verged:3 macaca:1 arcmin:1 double:1 produce:1 help:1 depending:1 brainard:1 measured:2 mayhew:1 implemented:1 predicted:4 c:1 come:1 implies:2 indicate:1 frontoparallel:5 direction:4 sensation:1 differ:1 stevens:1 correct:3 human:1 viewing:3 rogers:2 summation:1 considered:1 exp:1 presumably:1 predict:2 early:1 perceived:3 estimation:4 lose:1 sensitive:1 create:1 weighted:9 offs:1 rather:1 schnapf:1 ax:1 derived:1 properly:1 likelihood:1 posteriori:1 membership:1 typically:2 bt:5 sunderland:1 selective:1 orientation:2 restores:1 raised:1 spatial:1 special:1 equal:3 sampling:1 identical:1 look:4 survive:1 fmri:1 stimulus:63 primarily:1 mjn:1 randomly:1 preserve:1 familiar:1 relief:2 maintain:1 freedom:1 possibility:2 vsr:11 circular:1 truly:1 yielding:1 light:10 bloch:1 closer:1 necessary:1 re:7 increased:1 earlier:1 modeling:1 measuring:1 uninteresting:1 azimuth:4 front:1 proximal:1 combined:1 thanks:1 density:1 sensitivity:3 corrects:1 gaze:1 again:3 ambiguity:1 central:1 containing:1 horiz:2 potential:1 photon:1 retinal:8 sec:3 blurred:1 register:1 depends:1 performed:1 observer:15 start:1 square:1 became:1 conducting:1 percept:11 judgment:1 rufus:1 weak:1 bayesian:1 none:1 unaffected:1 classified:1 explain:1 datapoint:1 detector:1 energy:1 nonetheless:1 acquisition:1 proof:5 associated:2 radian:1 gleason:1 color:11 dimensionality:1 amplitude:1 positioned:1 appears:7 nonconjugate:1 though:3 binocular:7 until:1 hand:1 horizontal:32 brings:1 effect:8 concept:1 consisted:1 barlow:1 contain:1 assigned:1 flashed:1 indistinguishable:7 during:2 ambiguous:1 cosine:1 btb:2 evident:2 motion:2 percent:1 image:13 jack:1 wandell:1 behaves:1 physical:2 tilt:2 kluwer:1 interpret:1 measurement:3 slant:46 framed:1 automatic:1 had:1 dot:9 reliability:3 longer:1 surface:22 cortex:1 own:1 recent:1 perspective:2 irrelevant:1 driven:1 certain:1 exploited:1 seen:1 greater:3 fascicularis:1 determine:1 redundant:1 signal:10 relates:2 mnemonic:1 long:1 divided:1 host:1 schematic:1 baa:1 vision:15 physically:2 represent:3 cell:1 background:1 schematically:1 separately:2 participated:1 interval:1 leaving:1 extra:1 archive:1 probably:1 comment:1 subject:1 induced:3 suspect:1 call:3 ee:1 near:1 identically:2 affect:1 psychology:1 pennsylvania:2 opposite:1 whether:4 defense:1 etiology:1 stereo:15 render:1 unilateral:3 cause:2 covered:1 se:3 amount:6 schor:1 bradshaw:2 category:1 diameter:2 http:1 exist:3 stereoscopic:12 sign:1 per:5 serving:1 cyclopean:1 metameric:9 shall:1 four:6 drawn:1 loftus:3 changing:4 luminance:1 frazer:1 cone:7 angle:1 parameterized:1 fourth:1 striking:1 place:2 reasonable:1 patch:6 summarizes:1 scaling:3 followed:2 distinguish:6 correspondence:1 evinced:1 activity:4 occur:3 scene:2 flat:1 felt:3 aspect:1 argument:1 rendered:2 department:1 peripheral:1 combination:5 nachmias:1 jr:1 across:3 ophthalmology:1 rev:1 making:1 ln:2 equation:5 remains:1 previously:1 mechanism:6 needed:5 know:2 locus:1 available:1 pursuit:1 spectral:3 distinguished:1 robustly:1 existence:1 original:2 top:4 remaining:1 assumes:1 landy:1 build:2 unchanged:1 psychophysical:1 added:4 posture:8 occurs:1 hum:1 primary:1 traditional:2 responds:1 gradient:2 distance:11 separate:1 unable:1 capacity:1 argue:1 reason:1 fresh:1 modeled:1 illustration:1 ratio:4 demonstration:1 baylor:1 difficult:1 trace:1 rise:2 proper:1 perform:1 vertical:41 extended:1 looking:1 head:2 arbitrary:1 intensity:3 david:1 pair:1 specified:1 conflict:7 distinction:2 horner:1 able:1 bar:1 pattern:7 perception:11 reliable:2 including:1 memory:4 power:1 natural:3 representing:1 scheme:1 eye:39 axis:2 psychol:2 extract:1 philadelphia:1 foley:1 review:2 relative:3 law:1 sinauer:1 loss:6 interesting:1 analogy:1 clark:1 foundation:1 degree:1 principle:2 plotting:1 bank:2 row:1 elsewhere:1 changed:2 placed:2 last:1 supported:1 formal:1 taking:1 sparse:1 van:1 boundary:2 depth:4 xn:2 world:1 dimension:4 sensory:15 author:3 made:3 instructed:1 forward:1 far:1 approximate:1 status:1 evoke:3 deg:2 photoreceptors:1 conclude:1 stereopsis:4 spectrum:2 quantifies:1 vergence:15 why:2 table:2 mj:1 nature:2 longuet:1 constructing:1 profile:1 allowed:1 categorized:1 site:1 fashion:1 transduction:2 position:9 msec:1 hsr:14 perceptual:15 breaking:1 third:1 weighting:1 down:1 supposition:1 fusion:2 exists:1 essential:1 adding:2 sequential:2 supplement:1 metamers:34 perceptually:3 illustrates:1 boston:1 distinguishable:6 simply:4 appearance:1 likely:1 wavelength:5 ganglion:1 visual:18 conveniently:1 contained:1 saccade:6 chance:1 ma:1 flash:1 change:10 determined:1 averaging:1 experimental:1 concordant:2 photoreceptor:2 nx1:1 support:1 mark:1 phenomenon:1 tested:1 biol:1 |
1,171 | 2,068 | Learning hierarchical structures with
Linear Relational Embedding
Alberto Paccanaro
Geoffrey E. Hinton
Gatsby Computational Neuroscience Unit
UCL, 17 Queen Square, London, UK
alberto,hinton @gatsby.ucl.ac.uk
Abstract
We present Linear Relational Embedding (LRE), a new method of learning a distributed representation of concepts from data consisting of instances of relations between given concepts. Its final goal is to be able
to generalize, i.e. infer new instances of these relations among the concepts. On a task involving family relationships we show that LRE can
generalize better than any previously published method. We then show
how LRE can be used effectively to find compact distributed representations for variable-sized recursive data structures, such as trees and lists.
1 Linear Relational Embedding
Our aim is to take a large set of facts about a domain expressed as tuples of arbitrary symbols in a simple and rigid syntactic format and to be able to infer other ?common-sense?
facts without having any prior knowledge about the domain. Let us imagine a situation in
which we have a set of concepts and a set of relations among these concepts, and that our
data consists of few instances of these relations that hold among the concepts. We want
to be able to infer other instances of these relations. For example, if the concepts are the
people in a certain family, the relations are kinship relations, and we are given the facts
?Alberto has-father Pietro? and ?Pietro has-brother Giovanni?, we would like to be able to
infer ?Alberto has-uncle Giovanni?. Our approach is to learn appropriate distributed representations of the entities in the data, and then exploit the generalization properties of the
distributed representations [2] to make the inferences. In this paper we present a method,
which we have called Linear Relational Embedding (LRE), which learns a distributed representation for the concepts by embedding them in a space where the relations between
concepts are linear transformations of their distributed representations.
Let us consider the case in which all the relations are binary, i.e. involve two concepts.
In this case our data consists of triplets
, and the problem
we are trying to solve is to infer missing triplets when we are given only few of them.
Inferring a triplet is equivalent to being able to complete it, that is to come up with one of
its elements, given the other two. Here we shall always try to complete the third element
of the triplets 1 . LRE will then represent each concept in the data as a learned vector in a
1
Methods analogous to the ones presented here that can be used to complete any element of a
triplet can be found in [4].
Euclidean space and each relationship between the two concepts as a learned matrix that
maps the first concept into an approximation to the second concept. Let us assume that
our data consists of such triplets containing distinct concepts and
binary relations.
will denote the set of -dimensional
We shall call this set of triplets ;
vectors corresponding to the
concepts, and
the set of
matrices corresponding to the
relations. Often we shall need to indicate the vectors
and the matrix which correspond to the concepts and the relation in a certain triplet . In
this case we shall denote the vector corresponding to the first concept with , the vector
corresponding to the second concept with and the matrix corresponding to the relation
with . We shall therefore write the triplet as
where
and
.
The operation that relates a pair
to a vector
is the matrix-vector multiplication,
, which produces an approximation to . If for every triplet
we think of
as a noisy version of one of the concept vectors, then one way to learn an embedding
is to maximize the probability that it is a noisy version of the correct completion, . We
imagine that a concept has an average location in the space, but that each ?observation? of
the concept is a noisy realization of this average location. Assuming spherical Gaussian
noise with a variance of
on each dimension, the discriminative goodness function that
corresponds to the log probability of getting the right completion, summed over all training
triplets is:
!"
%$
$ &
#
!
'
()+*
, . 1 ( 24365 . 6798:8 ;
<>= ?<@7BAC<@8:E 8 D F
(1)
0/ E+F0GIH 798:8 ;
<>= ?<J7 8:8 D
1
where is the number of triplets in having the first two terms equal to the ones of , but
differing in the third term .
,
Learning based on maximizing with respect to all the vector and matrix components has
given good results, and has proved successful
, in generalization as well [5]. However, when
we learn an embedding by maximizing , we are not making use of exactly the information
that we have in the triplets. For each triplet , we are making the vector representing
the
$ B , while
correct completion K more probable than any other concept vector given
,
the triplet states that $ & must be equal to L . The numerator of does exactly this,
but
we also have the denominator, which is necessary in order to stay away from the trivial
M solution
. We noticed however that the denominator is critical at the beginning of the
learning, but- as the vectors and matrices differentiate we could gradually lift this burden,
$ &QPRS O to become the real goal of the learning. To do this we
allowing N
O
0
/
modify the discriminative function to include a parameter T , which is annealed from ( to U
V during learning :
. 1 ( 24365 Y . 679W079;
W0<J;
= ?<J<J= ?7B<>A%7 <XE W0F D W0DX[]\
(2)
0/
E+FZGIH
We would like our system to assign equal probability to each of the correct completions.
de4f eBk Thel m
l tokQapproximate
discrete probability distribution that we want
is therefore: ^_a`cd b N
b6g6hji the
w@ykzl"u@{Xm in n . Our system implements
where is the discrete delta function and rangesu>vx
qJsIt h over the vectorswhere
b
discreteg probability distribution: o _ ` pr
is the normalization factor.
|
The }X~ ? factor in eq.1 ensures that we are minimizing the Kullback-Leibler divergence between ^
and o .
v ? w y ? approach
The obvious
an embedding would be to minimize the sum of squared distances
? overto find
between
and
all the triplets, with respect to all the vector and matrix components.
i
Unfortunately this minimization
(almost) always causes all of the vectors and matrices to collapse to
the trivial ? solution.
one-to-many relations we must not decrease the value of ? all the way to ? , because this
v ? wJFor
y ? cause
would
some concept vectors to become coincident. This is because the only way to make
equal to ? different vectors, is by collapsing them onto a unique vector.
2
3
4
2
3
4
V
V
During learning this function (for Goodness) is maximized with respect to all the vector
and matrix components. This gives a much better generalization performance than the one
obtained by just maximizing . The results presented in the next sections were obtained by
maximizing using gradient ascent. All the vector and matrix components were updated
simultaneously at each iteration. One effective method of performing the optimization
is conjugate gradient. Learning was fast, usually requiring only a few hundred updates.
It is worth pointing out that, in general, different initial configurations and optimization
algorithms caused the system to arrive at different solutions, but these solutions were almost
always very similar in terms of generalization performance.
,
2 LRE results
Here we present the results obtained applying LRE to the Family Tree Problem [1]. In this
problem, the data consists of people and relations among people belonging to two families, one Italian and one English, shown in fig.1 (left) 5 . All the information in these trees
can be represented in simple propositions of the form
. Using the relations father, mother, husband, wife, son, daughter, uncle, aunt, brother, sister,
nephew, niece there are 112 such triplets in the two trees. Fig.1 (right) shows the embedding obtained after training with LRE. Notice how the Italians are linearly separable from
the English people. From the Hinton diagram, we can see that each member of a family is
symmetric to the corresponding member in the other family. The sign of the third component of the vectors is (almost) a feature for the nationality. When testing the generalization
1
Margaret = Arthur
9
8
2
7
Christopher = Penelope
Andrew = Christine
10 Victoria = James 4
3
Colin
6
13
Charlotte
19
Jennifer = Charles
2
0
12
14
Aurelio = Maria
5
11
?2
20
?5
Bortolo = Emma
0
?5
0
5
English
Italians
5
Grazia = Pierino
21
22
15
18
Giannina = Pietro
Alberto
Doralice = Marcello
16
23
17
Mariemma
24
Figure 1: Left: Two isomorphic family trees. The symbol ?=? means ?married to?. Right
Top: layout of the vectors representing the people obtained for the Family Tree Problem in
3D. Vectors end-points are indicated by *, the ones in the same family tree are connected
to each other. All
triplets were used for training. Right Bottom: Hinton diagrams of
the 3D vectors shown above. The vector of each person is a column, ordered according to
the numbering on the tree diagram on the left.
(6(*
$ &
performance,
for each triplet in the test set , we chose as completion the concepts
according to their probability, given
. The system was generally able to complete
correctly all
triplets even when of them, picked at random, had been left out during
training. These results on the Family Tree Problem are much better than the ones obtained
using any other method on the same problem: Quinlan?s FOIL [7] could generalize on
triplets, while Hinton (1986) and O?Reilly (1996) made one or more errors when only
test cases were held out during training.
(6(*
*
5
The names of the Italian family have been altered from those originally used in Hinton (1986) to
match those of one of the author?s family.
For most problems there exist triplets which cannot be completed. This is the case, for
example, of (Christopher, father, ?) in the Family Tree Problem. Therefore, here we
argue that it is not sufficient to test generalization by merely testing the completion of
those complete-able triplets which have not been used for training. The proper
test for
generalization is to see how the system completes any triplet of the kind where
ranges over the concepts and R over the relations. We cannot assume to have knowledge
of which triplets admit a completion, and which do not. To our knowledge this issue has
never been analyzed before (even though FOIL handles this problem correctly). To do this
the system needs a way to indicate when a triplet does not admit a completion. Therefore,
once the maximization of
is terminated, we build a new probabilistic model around
the solution which has been found. This new model is constituted, for each relation, of
a mixture of
identical spherical Gaussians, each centered on a concept vector, and a
Uniform distribution. The Uniform distribution will take care of the ?don?t know? answers,
and will be competing with all the other Gaussians, each representing a concept vector. For
each relation the Gaussians have different variances and the Uniform a different height.
The parameters of this probabilistic model are, for each relation , the variances of the
Gaussians
and the relative density under the Uniform distribution, which we shall write
as
. These parameters are learned using a validation set, which will be the
union of a set of complete-able (positive) triplets and a set of pairs which
cannot be
and
completed (negative); that is
where
indicates the fact that the result of applying relation to does not belong to . This is
done by maximizing the following discriminative goodness function over the validation
set :
V
ZP ; ;I) * ;
>
/
c
Z/
P D D
.
2
4
6
3
5
. ZP O $ P O
P D
/
* ;
D E+F0GIH
P W0;
= ?
. 1 ( $ 2 3I5
7B
A IW D
D $ " P
(3)
.
O
O
D
Z
P
!
Z
P
/
* ;
D E+F GIH
with respect to the
> else
is kept fixed. Having
; and ; parameters, while everything
we compute the
learned these parameters, in order to complete any triplet
probability distribution over each of the
Gaussians and the Uniform distribution given $ . The
system then chooses a vector
or the ?don?t know? answer according to those probabili
ties, as the completion to the triplet.
(*
We used this method on the Family Tree Problem using a train, test and validation sets
built in the following way. The test set contained
positive triplets chosen at random,
but such that there was a triplet per relation. The validation set contained a group of
positive and a group of
negative triplets, chosen at random and such that each group
had a triplet per relation. The train set contained the remaining positive triplets. After
learning a distributed representation for the entities in the data by maximizing over the
training set, we learned the parameters of the probabilistic model by maximizing over
the validation set. The resulting system was able to correctly complete all the possible
triplets . Figure 2 shows the distribution of the probabilities when completing one
complete-able and one uncomplete-able triplet in the test set.
(*
V
(*
*
>
LRE seems to scale up well to problems of bigger size. We have used it on a much bigger
version of the Family Tree Problem, where the family tree is a branch of the real family
# people over $ generations. Using the same set
tree of one of the authors containing "
of
relations used in the Family Tree Problem, there is a total of % positive triplets.
After learning using a training set of $ positive triplets, and a validation set constituted
(*
I*
Charlotte uncle
Emma aunt
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Figure 2: Distribution of the probabilities assigned to each concept for one complete-able
(left) and one uncomplete-able (right) triplet written above each diagram. The completeable triplet has two correct completions but neither of the triplets had been used for training.
Black bars from to are the probabilities of the people ordered according to the numbering in fig.1. The last grey bar on the right, is the probability of the ?don?t know? answer.
( *
IU
IU
by
positive and
negative triplets, the system is able to complete correctly almost
all the possible triplets. When many completions are correct, a high probability is always
assigned to each one of them. Only in few cases is a non-negligible probability assigned to
some wrong completions. Almost all the generalization errors are of a specific form. The
system appears to believe that ?brother/sister of? means ?son/daughter of parents of?. It
fails to model the extra restriction that people cannot be their own brother/sister. On the
other hand, nothing in the data specifies this restriction.
3 Using LRE to represent recursive data structures
In this section, we shall show how LRE can be used effectively to find compact distributed
representations for variable-sized recursive data structures, such as trees and lists. Here we
discuss binary trees, but the same reasoning applies to trees of any valence. The approach
is inspired by Pollack?s RAAM architecture [6]. A RAAM is an auto-encoder which is
trained using backpropagation. Figure 3 shows the architecture of the network for binary
trees. The system can be thought as being composed of two networks. The first one, called
~
l
~r
Reconstructor
~
l
~r
R1
R1
R2
R2
C1
C2
Verb Phrase
Noun Phrase
R2
C
Compressor
l
r
l
R2
R1
r
C1
Adjective
a
R1
C2
C1
Noun
b
Verb
c
C2
Noun
d
Figure 3: Left: the architecture of a RAAM for binary trees. The layers are fully connected. Adapted from [6]. Center: how LRE can be used to learn a representation for
binary trees in a RAAM-like fashion. Right: the binary tree structure of the sentences used
in the experiment.
compressor encodes two fixed-width patterns into a single pattern of the same size. The
second one, called reconstructor, decodes a compressed pattern into facsimiles of its parts,
and determines when the parts should be further decoded. To encode a tree the network
must learn as many auto-associations as the total number of non-terminal nodes in the tree.
The codes for the terminal nodes are supplied, and the network learns suitable codes for
the other nodes. The decoding procedure must decide whether a decoded vector represents
a terminal node or an internal node which should be further decoded. This is done by
using binary codes for the terminal symbols, and then fixing a threshold which is used for
checking for ?binary-ness? during decoding.
The RAAM approach can be cast as an LRE problem, in which concepts are trees, subtrees or leaves, or pairs of trees, sub-trees or leaves, and there exist relationships:
implementing the compressor, and and which jointly implement the reconstructor
(see fig.3). We can then learn a representation for all the trees, and the matrices by maximizing in eq.2. This formulation, which we have called Hierarchical LRE (HLRE),
solves two problems encountered in RAAMs. First, one does not need to supply codes for
the leaves of the trees, since LRE will learn an appropriate distributed representation for
them. Secondly, one can also learn from the data when to stop the decoding process. In
fact, the problem of recognizing whether a node needs to be further decoded, is similar
to the problem of recognizing that a certain triplet does not admit a completion, that we
solved in the previous section. While before we built an outlier model for the ?don?t know?
answers, now we shall build one for the non-terminal nodes. This can be done by learning
appropriate values of and for relations and maximizing in eq.3. The set of
triplets
where
is not a leaf of the tree, will play the role of the set which
appears in eq.3.
V
!B
We have applied this method to the problem of encoding binary trees which correspond
to sentences of words from a small vocabulary. Sentences had a fixed structure: a noun
phrase, constituted of an adjective and a noun, followed by a verb phrase, made of a verb
and a noun (see fig.3). Thus each sentence had a fixed grammatical structure, to which we
added some extra semantic structure in the following way. Words of each grammatical category were divided into two disjoint sets. Nouns were in
girl, woman, scientist or
in
dog, doctor, lawyer ; adjectives were in
pretty, young or in
ugly,
old ; verbs were in
help, love or in
hurt, annoy . Our training set was consentences of the type:
stituted by
and of the
type
, where the suffix indicates the set to which
each word type belongs. In this way, sentences of the kind ?pretty girl annoy scientist?
were not allowed in the training set, and there were possible sentences that satisfied
the constraints which were implicit in the training set.
We used HLRE to learn a distributed representation for all the nodes in the trees, maximizing using the sentences in the training set. In 7D, after having built the outlier model
for the non-terminal symbols, given any root or internal node the system would reconstruct
its children, and if they were non-terminal symbols would further decode each of them.
The decoding process would always halt providing the correct reconstruction for all the
sentences in the training set. The top row of fig.4 shows the distributed representations
found for each word in the vocabulary. Notice how the and sets of adjectives and verbs
are almost symmetric with respect to the origin; the difference between the and sets
is less evident for the nouns, due to the fact that while there exists a restriction on which
nouns can be used in position
, there is no restriction on the nouns appearing in position
in the training sentences (see fig.3, right). We tested how well this system could generalize
beyond the training set using the same procedure used by Pollack to enumerate the set of
trees that RAAMs are able to represent [6]: for every pair of patterns for trees, first we
encoded them into a pattern for a new higher level tree, and then we decoded this tree back
into the patterns of the two sub-trees. If the norm of the difference between the original
, then the tree
and the reconstructed sub-trees was within a tolerance, which we set to
could be considered to be well formed. The system shows impressive generalization performance: after training using the sentences, the four-word sentences it generates are all
the well formed sentences, and only those. It does not generate sentences which are either grammatically wrong, like ?dog old girl annoy?, nor sentences which violate semantic
constraints, like ?pretty girl annoy scientist?. This is striking when compared to the poor
generalization performance obtained by the RAAM on similar problems. As recognized by
V
\
(U
\
\ \
\
\ \
(
( U
\
*IU
*+U
(
T
*+U
T
U (
Nouns
Adjectives
Verbs
C1 ? C1 ? girl
C1 ? C2 ? girl
C2 ? C1 ? girl
C2 ? C2 ? girl
R1 ? R1 ? C1 ? C2 ? Nouns
R1 ? R1 ? C1 ? C1 ? Adjectives
R1 ? R1 ? C2 ? C2 ? Nouns
R1 ? R1 ? C2 ? C1 ? Verbs
Figure 4: For Hinton diagrams with multiple rows, each row relates to a word, in the following order - Adjectives: 1=pretty; 2=young; 3=ugly; 4=old ; Nouns: 1=girl; 2=woman;
3=scientist; 4=dog; 5=doctor; 6=lawyer ; Verbs: 1=help; 2=love; 3=hurt; 4=annoy ; .
, (higher), from ,
, (lower). Top row: The disBlack bars separate ,
tributed representation of the words in the sentences found after learning. Center row:
The different contributions given to the root of the tree by the word ?girl? when placed in
position ,
, and in the tree. Bottom row: The contribution of each leaf to the reconstruction of , when adjectives, nouns, verbs and nouns are applied in positions ,
, and
respectively.
\ \ \
Pollack [6], this was almost certainly due to the fact that for the RAAMs the representation
for the leaves was too similar, a problem that the HLRE formulation solves, since it learns
their distributed representations.
Let us try to explain why HLRE can generalize so well. The matrix can be decomposed
into two sub-matrices, and , such that for any two children of a given node, and , we
have:
, where ?;? denotes the concatenation operator. Therefore
we have a pair of matrices, either or , associated to each link in the
graph. Once the system has learned an embedding, finding a distributed representation
for
a given tree amounts to multiplying the representation of its leaves by all the
matrices
found on all the paths from the leaves to the root, and adding them up. Luckily matrix
multiplication is non-commutative, and therefore every sequence of words on its leaves
can generate a different representation at the root node. The second row of fig.4 makes
this point clear showing the different contributions given to the root of the tree by the word
?girl? , depending on its position in the sentence. A tree can be ?unrolled?
from the root to
its leaves by multiplying its distributed representation using the
matrices. We can now
analyze how a particular leaf is reconstructed. Leaf , for example, is reconstructed as:
$Y [
$ R $
$ $
$ $ $
$ $ $
$ $
The third row of fig.4 shows the contribution of each leaf to the reconstruction of , when
adjectives, nouns, verbs and nouns are placed on leaves ,
, and respectively. We can
see that the contributions from the adjectives, match very closely their actual distributed
representations, while the contributions from the nouns in position are negligible. This
means that any adjective placed on will tend to be reconstructed correctly, and that its
reconstruction is independent of the noun we have in position . On the other hand, the
contributions from nouns and verbs in positions
and are non-negligible, and notice how
those given by words belonging to the subsets are almost symmetric to those given by
words in the subsets. In this way the system is able to enforce the semantic agreement
between words in positions ,
and . Finally, the reconstruction of , when adjectives,
nouns, verbs and nouns are not placed on leaves ,
, and respectively, assigns a very
low probability to any word, and thus the system does not generate sentences which are not
well formed.
T
4 Conclusions
Linear Relational Embedding is a new method for learning distributed representations of
concepts and relations from data consisting of instances of relations between given concepts. It finds a mapping from the concepts into a feature-space by imposing the constraint
that relations in this feature-space are modeled by linear operations. LRE shows excellent
generalization performance. The results on the Family Tree Problem are far better than
those obtained by any previously published method. Results on other problems are similar.
Moreover we have shown elsewhere [4] that, after learning a distributed representation for
a set of concepts and relations, LRE can easily modify these representations to incorporate
new concepts and relations and that it is possible extract logical rules from the solution
and to couple LRE with FOIL [7]. Learning is fast and LRE rarely converges to solutions
with poor generalization. We began introducing LRE for binary relations, and then we saw
how these ideas can be easily extended to higher arity relation by simply concatenating
concept vectors and using rectangular matrices for the relations. The compressor relation
for binary trees is a ternary relation; for trees of higher valence the compressor relation will
have higher arity. We have seen how HLRE can be used to find distributed representations
for hierarchical structures, and its generalization performance is much better than the one
obtained using RAAMs on similar problems.
It is easy to prove that, when all the relations are binary, given a sufficient number of
dimensions, there always exists an LRE-type of solution that satisfies any set of triplets
[4]. However, due to its linearity, LRE cannot represent some relations of arity greater
than . This limitation can be overcome by adding an extra layer of non-linear units for
representing the relations. This new method, called Non-Linear Relational Embedding
(NLRE) [4], can represent any relation and has given good generalization results.
*
References
[1] Geoffrey E. Hinton. Learning distributed representations of concepts. In Proceedings of the
Eighth Annual Conference of the Cognitive Science Society, pages 1?12. Erlbaum, NJ, 1986.
[2] Geoffrey E. Hinton, James L. McClelland, and David E. Rumelhart. Distributed representations.
In David E. Rumelhart, James L. McClelland, and the PDP research Group, editors, Parallel
Distributed Processing, volume 1, pages 77?109. The MIT Press, 1986.
[3] Randall C. O?Reilly. The LEABRA model of neural interactions and learning in the neocortex.
PhD thesis, Department of Psychology, Carnegie Mellon University, 1996.
[4] Alberto Paccanaro. Learning Distributed Representations of Relational Data using Linear Relational Embedding. PhD thesis, Computer Science Department, University of Toronto, 2002.
[5] Alberto Paccanaro and Geoffrey E. Hinton. Learning distributed representations by mapping
concepts and relations into a linear space. In Pat Langley, editor, Proceedings of ICML2000,
pages 711?718. Morgan Kaufmann, Stanford University, 2000.
[6] Jordan B. Pollack. Recursive distributed representations. Artificial Intelligence, 46:77?105,
1990.
[7] J. R. Quinlan. Learning logical definitions from relations. Machine Learning, 5:239?266, 1990.
| 2068 |@word niece:1 version:3 seems:1 norm:1 grey:1 initial:1 configuration:1 must:4 written:1 update:1 intelligence:1 leaf:15 beginning:1 node:11 location:2 toronto:1 lawyer:2 penelope:1 height:1 c2:11 become:2 supply:1 consists:4 prove:1 emma:2 love:2 annoy:5 nor:1 terminal:7 inspired:1 spherical:2 decomposed:1 actual:1 moreover:1 linearity:1 kinship:1 kind:2 differing:1 finding:1 transformation:1 nj:1 every:3 tie:1 exactly:2 wrong:2 uk:2 qpr:1 unit:2 before:2 positive:7 negligible:3 scientist:4 modify:2 encoding:1 tributed:1 path:1 black:1 chose:1 married:1 raams:4 collapse:1 range:1 unique:1 thel:1 testing:2 ternary:1 recursive:4 union:1 implement:2 backpropagation:1 procedure:2 langley:1 thought:1 reilly:2 word:14 onto:1 cannot:5 operator:1 applying:2 restriction:4 equivalent:1 map:1 missing:1 maximizing:10 annealed:1 layout:1 center:2 rectangular:1 assigns:1 rule:1 embedding:13 handle:1 hurt:2 analogous:1 updated:1 imagine:2 play:1 decode:1 origin:1 agreement:1 element:3 rumelhart:2 bottom:2 role:1 probabili:1 solved:1 ensures:1 connected:2 decrease:1 trained:1 girl:11 easily:2 represented:1 train:2 distinct:1 fast:2 effective:1 london:1 artificial:1 lift:1 encoded:1 stanford:1 solve:1 reconstruct:1 compressed:1 encoder:1 syntactic:1 think:1 noisy:3 jointly:1 final:1 differentiate:1 sequence:1 ucl:2 reconstruction:5 interaction:1 kzl:1 realization:1 margaret:1 getting:1 parent:1 zp:2 r1:12 produce:1 converges:1 help:2 depending:1 andrew:1 ac:1 fixing:1 completion:13 eq:4 solves:2 come:1 indicate:2 closely:1 correct:6 luckily:1 centered:1 vx:1 everything:1 implementing:1 assign:1 generalization:14 proposition:1 probable:1 secondly:1 hold:1 around:1 considered:1 mapping:2 pointing:1 iw:1 saw:1 minimization:1 mit:1 j7:1 always:6 gaussian:1 aim:1 encode:1 maria:1 indicates:2 sense:1 inference:1 rigid:1 suffix:1 relation:42 italian:4 iu:3 issue:1 among:4 noun:23 summed:1 ness:1 equal:4 once:2 never:1 having:4 identical:1 represents:1 marcello:1 few:4 composed:1 simultaneously:1 divergence:1 consisting:2 certainly:1 analyzed:1 mixture:1 nephew:1 held:1 subtrees:1 necessary:1 arthur:1 tree:47 euclidean:1 old:3 pollack:4 instance:5 column:1 goodness:3 queen:1 maximization:1 phrase:4 introducing:1 subset:2 hundred:1 uniform:5 father:3 successful:1 recognizing:2 erlbaum:1 too:1 answer:4 chooses:1 person:1 density:1 stay:1 probabilistic:3 decoding:4 squared:1 thesis:2 satisfied:1 containing:2 woman:2 charlotte:2 collapsing:1 admit:3 cognitive:1 caused:1 try:2 picked:1 root:6 reconstructor:3 analyze:1 doctor:2 parallel:1 contribution:7 minimize:1 square:1 formed:3 variance:3 kaufmann:1 maximized:1 correspond:2 generalize:5 decodes:1 multiplying:2 worth:1 published:2 explain:1 husband:1 definition:1 james:3 obvious:1 icml2000:1 associated:1 couple:1 stop:1 proved:1 logical:2 knowledge:3 back:1 appears:2 originally:1 higher:5 formulation:2 done:3 though:1 just:1 implicit:1 hand:2 christopher:2 indicated:1 believe:1 name:1 concept:38 requiring:1 assigned:3 symmetric:3 leibler:1 semantic:3 numerator:1 during:5 aunt:2 width:1 paccanaro:3 trying:1 evident:1 complete:11 christine:1 reasoning:1 charles:1 began:1 common:1 lre:22 volume:1 belong:1 association:1 mellon:1 imposing:1 mother:1 nationality:1 had:5 impressive:1 own:1 belongs:1 certain:3 binary:13 xe:1 raam:6 seen:1 morgan:1 greater:1 care:1 recognized:1 maximize:1 colin:1 relates:2 branch:1 violate:1 multiple:1 infer:5 match:2 alberto:7 divided:1 bigger:2 halt:1 involving:1 denominator:2 iteration:1 represent:5 normalization:1 c1:11 want:2 diagram:5 completes:1 else:1 extra:3 ascent:1 tend:1 member:2 grammatically:1 jordan:1 call:1 easy:1 psychology:1 architecture:3 competing:1 idea:1 whether:2 cause:2 enumerate:1 generally:1 clear:1 involve:1 amount:1 neocortex:1 gih:1 bac:1 category:1 mcclelland:2 generate:3 specifies:1 supplied:1 exist:2 notice:3 brother:4 sign:1 neuroscience:1 delta:1 correctly:5 sister:3 per:2 disjoint:1 write:2 discrete:2 shall:8 carnegie:1 group:4 four:1 threshold:1 neither:1 kept:1 graph:1 pietro:3 merely:1 sum:1 wife:1 i5:1 striking:1 arrive:1 family:19 almost:8 decide:1 layer:2 completing:1 followed:1 encountered:1 annual:1 adapted:1 constraint:3 encodes:1 generates:1 performing:1 separable:1 format:1 department:2 numbering:2 according:4 leabra:1 poor:2 conjugate:1 belonging:2 son:2 making:2 randall:1 outlier:2 gradually:1 pr:1 previously:2 jennifer:1 discus:1 know:4 end:1 operation:2 gaussians:5 victoria:1 hierarchical:3 away:1 appropriate:3 enforce:1 appearing:1 original:1 top:3 remaining:1 include:1 denotes:1 completed:2 quinlan:2 exploit:1 build:2 society:1 noticed:1 added:1 gradient:2 distance:1 valence:2 separate:1 link:1 entity:2 concatenation:1 w0:2 argue:1 trivial:2 assuming:1 code:4 modeled:1 relationship:3 providing:1 minimizing:1 unrolled:1 unfortunately:1 negative:3 daughter:2 proper:1 allowing:1 observation:1 coincident:1 pat:1 situation:1 hinton:10 relational:8 extended:1 pdp:1 arbitrary:1 verb:13 david:2 pair:5 cast:1 dog:3 sentence:17 learned:6 able:16 bar:3 beyond:1 usually:1 pattern:6 xm:1 eighth:1 adjective:12 built:3 critical:1 suitable:1 representing:4 altered:1 auto:2 extract:1 prior:1 checking:1 multiplication:2 relative:1 fully:1 generation:1 limitation:1 geoffrey:4 validation:6 sufficient:2 editor:2 cd:1 foil:3 row:8 elsewhere:1 placed:4 last:1 english:3 ugly:2 distributed:24 grammatical:2 tolerance:1 dimension:2 vocabulary:2 giovanni:2 overcome:1 author:2 made:2 far:1 facsimile:1 reconstructed:4 compact:2 kullback:1 tuples:1 discriminative:3 don:4 triplet:46 pretty:4 why:1 learn:9 uncle:3 excellent:1 domain:2 constituted:3 linearly:1 aurelio:1 terminated:1 noise:1 nothing:1 allowed:1 child:2 fig:9 fashion:1 gatsby:2 fails:1 inferring:1 decoded:5 sub:4 position:9 concatenating:1 third:4 learns:3 young:2 specific:1 showing:1 arity:3 symbol:5 list:2 r2:4 burden:1 exists:2 adding:2 effectively:2 phd:2 commutative:1 simply:1 expressed:1 ordered:2 contained:3 compressor:5 applies:1 corresponds:1 determines:1 satisfies:1 goal:2 sized:2 called:5 total:2 isomorphic:1 rarely:1 internal:2 people:8 incorporate:1 tested:1 |
1,172 | 2,069 | Thin Junction Trees
Francis R. Bach
Computer Science Division
University of California
Berkeley, CA 94720
fbach@cs.berkeley.edu
Michael I. Jordan
Computer Science and Statistics
University of California
Berkeley, CA 94720
jordan@cs.berkeley.edu
Abstract
We present an algorithm that induces a class of models with thin junction
trees?models that are characterized by an upper bound on the size of
the maximal cliques of their triangulated graph. By ensuring that the
junction tree is thin, inference in our models remains tractable throughout
the learning process. This allows both an efficient implementation of
an iterative scaling parameter estimation algorithm and also ensures that
inference can be performed efficiently with the final model. We illustrate
the approach with applications in handwritten digit recognition and DNA
splice site detection.
Introduction
Many learning problems in complex domains such as bioinformatics, vision, and information retrieval involve large collections of interdependent variables, none of which has a
privileged status as a response variable or class label. In such problems, the goal is generally that of characterizing the principal dependencies in the data, a problem which is often
cast within the framework of multivariate density estimation. Simple models are often preferred in this setting, both for their computational tractability and their relative immunity
to overfitting. Thus models involving low-order marginal or conditional probabilities?
e.g., naive independence models, trees, or Markov models?are in wide use. In problems
involving higher-order dependencies, however, such strong assumptions can be a serious
liability.
A number of methods have been developed for selecting models of higher-order dependencies in data, either within the maximum entropy setting?in which features are selected [9, 16]?and the graphical model setting?in which edges are selected [8]. Simplicity also plays an important role in the design of these algorithms; in particular, greedy
methods that add or subtract a single feature or edge at a time are generally employed. The
model that results at each step of this process, however, is often not simple, and this is
problematic both computationally and statistically in large-scale problems.
In the current paper we describe a methodology that can be viewed as a generalization of
the Chow-Liu algorithm for constructing tree models [2]. Note that tree models have the
property that their junction trees have no more than two nodes in any clique?the treewidth
of tree models is one. In our generalization, we allow the treewidth to be a larger, but still
controlled, value. We fit data within the space of models having ?thin? junction trees.
Models with thin junction trees are tractable for exact inference, indeed the complexity of
any type of inference (joint, marginal, conditional) is controlled by the upper bound that
is imposed on the treewidth. This makes it possible to achieve some of the flexibility that
is often viewed as a generic virtue of generative models, but is not always achievable in
practice. For example, in the classification setting we are able to classify partially observed
data (e.g., occluded digits) in a simple and direct way?we simply marginalize away the
unobserved variables, an operation which is tractable in our models. We illustrate this
capability in a study of handwritten digit recognition in Section 4.2, where we compare thin
junction trees and support vector machines (SVMs), a discriminative technique which does
not come equipped with a simple and principled method for handling partially observed
data. As we will see, thin junction trees are quite robust to missing data in this domain.
There are a number of issues that need to be addressed in our framework. In particular, tree
models come equipped with particularly efficient algorithms for parameter estimation and
model selection?algorithms which do not generalize readily to non-tree models, including
thin junction tree models. It is important to show that efficient algorithms can nonetheless
be found to fit such models. We show how this can be achieved in Sections 1, 2 and 3.
Empirical results using these algorithms are presented in Section 4.
1 Feature induction
We assume an input space with
variables and a target probability distribution . Our
goal is to find a probability distribution that minimizes the Kullback-Leibler divergence
. Consider a vector-valued ?feature? or ?sufficient statistic?
, where
is the dimensionality of the feature space. The feature can also be thought in terms
of its components as a set of real-valued features
. We focus on exponential family
distributions (also known as ?Gibbs? or ?maximum entropy? distributions) based on these
features:
where
is a parameter vector,
is a base-measure (typically uniform), and is the normalizing constant. (Section 3
considers the closely-related problem of inducing edges rather than features).
#
!#"!$ & %' (
()+*
8:9:;=<>; ? ?+?@; BA
*
% ,%
.- /
02143657
Each feature is a function of a certain subset of variables, and we let
index the subset of variables referred to by feature . Let us consider
the undirected graphical model
, where the set of edges is the set of all pairs
are the maximal cliques of the graph
included in at least one . With this definition the
and, if is decomposable in this graph, the exponential family distribution with features
and reference distribution
is also decomposable in this graph. We assume without
loss of generality that the graph is connected. For each possible triangulation of the graph,
we can define a junction tree [4], where for all there exists a maximal clique containing
. The complexity of exact inference depends on the size of the maximal clique of the
triangulated graph. We define the treewidth of our original graph to be one less than the
minimum possible value of this maximal clique size for all possible triangulations. We say
that a graphical model has a thin junction tree if its treewidth is small.
01
#
0G1
CD 5 (; E
+
01
E :1
H
F
H
Our basic feature induction algorithm is a constrained variant of that proposed by [9]. Given
a set of available features, we perform a greedy search to find the set of features that enables
the best possible fit to , under the constraint of having a thin junction tree. At each step,
candidates are ranked according to the gain in KL divergence, with respect to the empirical
distribution, that would be achieved by their addition to the current set of features. Features
that would generate a graphical cover with treewidth greater than a given upper bound
are removed from the ranking.
%
H
The parameter values are held fixed during each step of the feature ranking process. Once
a set of candidate features are chosen, however, we reestimate all of the parameters (using
the algorithm to be described in Section 2) and iterate.
F EATURE I NDUCTION
.
;%
,
, a set of available features
1. Initialization:
2. Repeat steps (a) to (d) until no further progress is made with respect to a model
selection criterion (e.g., MDL or cross-validation)
(a) Ranking: generate samples from and rank feature candidates according to
the KL gain
(b) Elimination: remove all candidates that would generate a model with
treewidth greater than
and add them to
(c) Selection: select the best features
(d) Parameter Estimation: Estimate using the junction tree implementation of
Iterative Scaling (see Section 2)
H
%
; ?+? ?#;
Freezing the parameters during the feature ranking step is suboptimal, but it yields an
essential computational efficiency. In particular, as shown by [9], under these conditions we
can rank a new feature by solving a polynomial equation whose degree is the number of
values can take minus one, and whose coefficients are expectations under of functions
of . This equation has only one root and can be solved efficiently by Newton?s method.
When the feature is binary the process is even more efficient?the equation is linear and
can be solved directly. Consequently, with a single set of samples from , we can rank
many features very cheaply.
For the feature elimination operation, algorithms exist that determine in time linear in the
number of nodes whether a graph has a treewidth smaller than , and if so output a triangulation in which all cliques are of size less than [1]. These algorithms are super-exponential
in , however, and thus are applicable only to problems with small treewidths. In practice
we have had success using fast heuristic triangulation methods [11] that allow us to guarantee the existence of a junction tree with a maximal clique no larger than for a given
model. (This is a conservative technique that may occasionally throw out models that in
fact have small treewidth).
H
H
H
H
A critical bottleneck in the algorithm is the parameter estimation step, and it is important
to develop a parameter estimation algorithm that exploits the bounded treewidth property.
We now turn to this problem.
2 Iterative Scaling using the junction tree
Fitting an exponential family distribution under expectation constraints is a well studied
problem; the basic technique is known as Iterative Scaling. A generalization of Iterative
Proportional Fitting (IPF), it updates the parameters
sequentially [5]. Algorithms that
update the parameters in parallel have also been proposed; in particular the Generalized
Iterative Scaling algorithm [6], which imposes the constraint that the features sum to one,
and the Improved Iterative Scaling algorithm [9], which removes this constraint. These
algorithms have an important advantage in our setting in that, for each set of parameter
updates, they only require computations of expectation that can all be estimated with a
single set of samples from the current distribution.
%
When the input dimensionality is large, however, we would like to avoid sampling algorithms altogether. To do so we exploit the bounded treewidth of our models. We present
a novel algorithm that uses the junction tree and the structure of the problem to speed up
parameter estimation. The algorithm generalizes to Gibbs distributions the ?effective IPF?
algorithm of [10].
When working with a junction tree, a efficient way of performing Iterative Scaling is to
update parameters block by block so that each update is performed for a relatively small
number of features on a small number of variables. Each block can be fit with any parameter estimation algorithm, in particular Improved Iterative Scaling (IIS). The following
algorithm exploits this idea by grouping the features whose supports are in the same clique
of the triangulated graph. Thus, parameter estimation is done in spaces of dimensions at
most
, and all the needed expectations can be evaluated cheaply.
H 9
2.1 Notation
&
1 ;+? ?+?#; : 11
Let be our -dimensional feature. Let
denote the maximal cliques of the
triangulated graph, with potentials
. We assign each feature to one of the cliques
that contains . For each clique
we denote
as the set of features
assigned to .
G0 1
2.2 Algorithm
E FFICIENT I TERATIVE S CALING
80 1
1
A
1. Initialization:
supp
?Construct a junction tree associated with the subsets
?Assign each
to one , such that
(equivalent to determining
for all )
?Set
and decompose onto the junction tree
?Set
1 ; ?+? ?#; 1
1
% ,% ; ?+? ? ; %
01 3
%
2. Loop until convergence: Repeat step (3) until convergence of the ?s
3. Loop through all cliques: Repeat steps (a) to (c) for all cliques
(a) Define the root of the junction tree to be
(b) Collect evidence from the leaves to the root of the junction tree and normalize
potential
(c) Calculate the maximum likelihood
-dimensional exponential family distribution with features
and reference distribution
, using IIS. Replace
by this distribution and add the resulting parameters (one for each feature
in ) to the corresponding ?s:
.
%
&% 1 ; ?+? ?+; % 1
After step (b), the potential
is exactly marginalized to , so that performing IIS for
can be done using
instead of the full distribution . Moreover, each
the features
pass through all the cliques is equivalent to one pass of Iterative Scaling and therefore this
algorithm converges to the maximum likelihood distribution.
3 Edge induction
Thus far we have emphasized the exponential family representation. Our algorithm can,
however, be adapted readily to the problem of learning the structure of a graphical model.
This is achieved by using features that are indicators of subsets of variables, ensuring that
there is one such indicator for every combination of values of the variables in a clique. In
this case, Iterative Scaling reduces to Iterative Proportional Fitting.
We generally employ a further approximation when ranking and selecting edges. In particular, we evaluate an edge only in terms of the two variables associated directly with the
edge. The clique formed by the addition of the edge, however, may involve additional
higher-order dependencies, which can be parameterized and incorporated in the model.
Evaluating edges in this way thus underestimates the potential gain in KL divergence.
20
15
10
5
0
0
10
20
30
Figure 1: (Left) Circular Boltzmann machine of treewidth 4. (Right) Proportion (in ) of
edges not in common between the fitted model and the generating model vs the number of
available training examples (in thousands).
We should not expect to be able to find an exact edge-selection method?recent work by
Srebro [15] has shown that the related problem of finding the maximum likelihood graphical model with bounded treewidth is NP-hard.
4 Empirical results
4.1 Small graphs with known generative model
In this experiment we generate samples from a known graphical model and fit our model
to the data. We consider circular Boltzmann machines of known treewidth equal to 4 as
shown in Figure 1. Our networks
all
have 32 nodes and the weights were selected from a
?so that each edge is significant. For an increasing
uniform distribution in
number of training samples, ten replications were performed for each case using our feature
induction algorithm with maximum treewidth equal to 4. Figure 1 shows
that with enough
samples we are able to recover the structure almost exactly (up to
of the original
edges).
< 9
9 =<
?
4.2 MNIST digit dataset
In this section we study the performance of the thin junction tree method on the MNIST
dataset of handwritten digits. While discriminative methods outperform generative methods in this high-dimensional setting [12], generative methods offer capabilities that are not
provided by discriminative classifiers; in particular, the ability to deal with large fractions
of missing pixels and the ability to to reconstruct images from partial data. It is of interest
to see how much performance loss we incur and how much robustness we gain by using a
sophisticated generative model for this problem.
9 9
<
<
The MNIST training set
is composed
of
4-bit grayscale pixels that have been
resized and cropped to
binary images (an example is provided in the leftmost plots
in Figure 2). We used thin junction trees as density estimators in the 256-dimensional pixel
space by training ten different models, one for each of the ten classes. We used binary
features of the form
. No vision-based techniques such as de-skewing or
virtual examples were used. We utilized ten percent fractions of the training data for crossvalidation and test.
9
Density estimation: The leftmost plot in Figure 3 shows how increasing the maximal allowed treewidth, ranging from 1 (trees) to 15, enables a better fit to data.
Classification: We built classifiers from the bank of ten thin junction tree (?TJT?) models
using one of the following strategies: (1) take the maximum likelihood among the ten
Figure 2: Digit from the MNIST database. From left to right, original digit, cropped and
resized digits used in our experiments, 50% of missing values, 75% of missing values,
occluded digit.
100
70
80
65
60
60
40
55
50
20
0
5
10
15
0
0
50
100
Figure 3: (Left) Negative log likelihood for the digit 2 vs maximal allowed treewidth.
(Right) Error rate as a function of the percentage of erased pixels for the TJT classifier
(plain) and a support vector machine (dotted). See text for details.
models (TJT-ML), or (2) train a discriminative model using the outputs of the ten models.
We used softmax regression (TJT-Softmax) and the support vector machine (TJT-SVM) in
the latter case.
The classification error rates were as follows: LeNet 0.7, SVM 0.8, Product of experts, 2.0,
TJT-SVM 3.8, TJT-Softmax 4.2, TJT-ML 5.3, Chow-Liu 8.5, and Linear classifier 12.0. (See
[12] and [13] for further details on the non-TJT models).
It is important to emphasize that our models are tractable for full joint inference; indeed,
the junction trees have a maximal clique size of 10 in the largest models we used on the ten
classes. Thus we can use efficient exact calculations to perform inference. The following
two sections demonstrate the utility of this fact.
Missing pixels: We ran an experiment in which pixels were chosen uniformly at random and
erased, as shown in Figure 2. In our generative model, we treat them as hidden variables
that were marginalized out. The rightmost plot in Figure 3 shows the error rate on the
testing set as a function of the percentage of unknown pixels, for our models and for a
SVM. In the case of the SVM, we used a polynomial kernel of degree four [7] and we tried
various heuristics to fill in the value of the non-observed pixels, such as the average of that
pixel over the training set or the value of a blank pixel. Best classification performance
was achieved with replacing the missing value by the value of a blank pixel. Note that very
little performance decrement is seen for our classifier even with up to 50 percent of the
pixels missing, while for the SVM, although performance is better for small percentages,
performance degrades more rapidly as the percentage of erased digits increases.
Reconstruction: We conducted an additional experiment in which the upper halves of images were erased. We ran the junction tree inference algorithm to fill in these missing
values, choosing the maximizing value of the conditional probability (max-propagation).
Figure 3 shows the results. For each line, from left to right, we show the original digit, the
digit after erasure, reconstructions based on the model having the maximum likelihood, and
0
0
2
6
5
5
8
3
1
1
9
7
6
6
0
2
2
2
6
5
7
7
9
4
3
3
5
8
8
8
6
3
4
4
7
9
9
7
9
3
Figure 4: Reconstructions of images whose upper halves have been deleted. See text for
details.
reconstruction based on the model having the second and third largest values of likelihood.
4.3 SPLICE Dataset
The task in this dataset is to classify splice junctions in DNA sequences. Splice junctions
can either be an exon/intron (EI) boundary, an intron/exon (IE) boundary, or no boundary. (Introns are the portions of genes that are spliced out during transcription; exons are
retained in the mRNA).
Each sample is a sequence of 60 DNA bases (where each base can take one of four values,
A,G,C, or T). The three different classes are: EI exactly at the middle (between the 30th
and the 31st bases), IE exactly at the middle (between the 30th and the 31st bases), no
splice junction. The dataset is composed of 3175 training samples. In order to be able to
compare to previous experiments using this dataset, performance is assessed by picking
2000 training data points at random and testing on the 1175 others, with 20 replications.
We treat classification as a density estimation problem in this case by treating the class
variable as another variable. We classify by choosing the value of that maximizes the
conditional probability
. We tested both feature induction and edge induction; in the
were
former case only binary features that are products of features of the form
tested and induced. MDL was used to pick the number of features or edges.
?
?
?9
Our feature induction algorithm, with a maximum treewidth equal to 5, gave an error rate
of
, while the edge induction algorithm gave an error rate of
. This is better than
the best reported results in the literature; in particular, neural networks have an error rate
of
and the Chow and Liu algorithm has an error rate of
[14].
?
5 Conclusions
We have described a methodology for feature selection, edge selection and parameter estimation that can be viewed as a generalization of the Chow-Liu algorithm. Drawing on the
feature selection methods of [9, 16], our method is quite general, building an exponential
family model from the general vocabulary of features on overlapping subsets of variables.
By maintaining tractability throughout the learning process, however, we build this flexible
representation of a multivariate density while retaining many of the desirable aspects of the
Chow-Liu algorithm.
Our methodology applies equally well to feature or edge selection. In large-scale, sparse
domains in which overfitting is of particular concern, however, feature selection may be the
preferred approach, in that it provides a finer-grained search in the space of simple models
than is allowed by the edge selection approach.
Acknowledgements
We wish to acknowledge NSF grant IIS-9988642 and ONR MURI N00014-00-1-0637. The
results presented here were obtained using Kevin Murphy?s Bayes Net Matlab toolbox and
SVMTorch [3].
References
[1] H. Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth,
Siam J. Computing, 25, 105-1317, 1996.
[2] C.K. Chow and C.N. Liu, Approximating discrete probability distributions with dependence
trees, IEEE Trans. Information Theory, 42, 393-405, 1990.
[3] R. Collobert and S. Bengio, SVMTorch: support vector machines for large-scale regression
problems, Journal of Machine Learning Research, 1, 143-160, 2001.
[4] R.G. Cowell, A.P. Dawid, S.L. Lauritzen, and D.J. Spiegelhalter, Probabilistic Networks and
Expert Systems, Springer-Verlag, New York, 1999.
[5] I. Csisz?ar, I-divergence geometry of probability distributions and minimization problems, Annals of Probability, 3, 146-158, 1975.
[6] J.N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models, Ann. Math.
Statist., 43, 1470-1480, 1972.
[7] D. DeCoste and B. Sch?olkopf, Training invariant support vector machines, Machine Learning,
46, 1-3, 2002.
[8] D. Heckerman, D. Geiger, and D.M. Chickering, Learning Bayesian networks: The combination of knowledge and statistical data, Machine Learning, 20, 197-243, 1995.
[9] S. Della Pietra, V. Della Pietra, and J. Lafferty, Inducing features of random fields, IEEE Trans.
PAMI, 19, 380-393, 1997.
[10] R. Jirousek and S. Preucil, On the effective implementation of the iterative proportional fitting
procedure, Computational Statistics and Data Analysis, 19, 177-189, 1995.
[11] U. Kjaerulff, Triangulation of graphs?algorithms giving small total state space, Technical
Report R90-09, Dept. of Math. and Comp. Sci., Aalborg Univ., Denmark, 1990.
[12] Y. Le Cun, http://www.research.att.com/?yann/exdb/mnist/index.html
[13] G. Mayraz and G. Hinton, Recognizing hand-written digits using hierarchical products of experts, Adv. NIPS 13, MIT Press, Cambridge, MA, 2001.
[14] M. Meila and M.I. Jordan, Learning with mixtures of trees, Journal of Machine Learning Research, 1, 1-48, 2000.
[15] N. Srebro, Maximum likelihood bounded tree-width Markov networks, in UAI 2001.
[16] S.C. Zhu, Y.W. Wu, and D. Mumford, Minimax entropy principle and its application to texture
modeling, Neural Computation, 9, 1997.
| 2069 |@word middle:2 polynomial:2 achievable:1 proportion:1 tried:1 decomposition:1 pick:1 minus:1 liu:6 contains:1 att:1 selecting:2 rightmost:1 current:3 blank:2 com:1 mayraz:1 written:1 readily:2 enables:2 remove:2 plot:3 treating:1 update:5 v:2 greedy:2 selected:3 generative:6 leaf:1 half:2 provides:1 math:2 node:3 direct:1 replication:2 fitting:4 indeed:2 little:1 decoste:1 equipped:2 increasing:2 provided:2 bounded:4 notation:1 moreover:1 maximizes:1 minimizes:1 developed:1 unobserved:1 finding:2 guarantee:1 berkeley:4 every:1 exactly:4 classifier:5 grant:1 treat:2 pami:1 initialization:2 studied:1 collect:1 statistically:1 testing:2 practice:2 block:3 digit:14 procedure:1 erasure:1 empirical:3 thought:1 onto:1 marginalize:1 selection:10 www:1 equivalent:2 imposed:1 missing:8 maximizing:1 mrna:1 simplicity:1 decomposable:2 estimator:1 fill:2 annals:1 target:1 play:1 exact:4 us:1 dawid:1 recognition:2 particularly:1 utilized:1 muri:1 database:1 jirousek:1 observed:3 role:1 solved:2 calculate:1 thousand:1 ensures:1 connected:1 adv:1 spliced:1 removed:1 ran:2 principled:1 complexity:2 occluded:2 solving:1 incur:1 division:1 efficiency:1 exon:3 joint:2 various:1 train:1 univ:1 fast:1 describe:1 effective:2 kevin:1 choosing:2 quite:2 whose:4 larger:2 valued:2 heuristic:2 say:1 drawing:1 reconstruct:1 ability:2 statistic:3 g1:1 final:1 advantage:1 sequence:2 net:1 reconstruction:4 maximal:10 product:3 loop:2 rapidly:1 flexibility:1 achieve:1 inducing:2 csisz:1 normalize:1 olkopf:1 crossvalidation:1 convergence:2 generating:1 converges:1 illustrate:2 develop:1 lauritzen:1 progress:1 strong:1 throw:1 c:2 treewidth:19 come:2 triangulated:4 closely:1 elimination:2 virtual:1 require:1 assign:2 generalization:4 decompose:1 tjt:9 estimation:12 applicable:1 label:1 largest:2 minimization:1 mit:1 always:1 super:1 rather:1 avoid:1 resized:2 focus:1 rank:3 likelihood:8 ratcliff:1 inference:8 typically:1 chow:6 hidden:1 pixel:12 issue:1 classification:5 among:1 flexible:1 html:1 retaining:1 constrained:1 softmax:3 marginal:2 equal:3 once:1 construct:1 having:4 field:1 sampling:1 thin:13 np:1 others:1 report:1 serious:1 employ:1 aalborg:1 composed:2 divergence:4 murphy:1 pietra:2 geometry:1 detection:1 interest:1 circular:2 mdl:2 mixture:1 held:1 edge:20 partial:1 tree:37 fitted:1 classify:3 modeling:1 cover:1 ar:1 tractability:2 subset:5 uniform:2 recognizing:1 conducted:1 reported:1 dependency:4 st:2 density:5 siam:1 ie:2 probabilistic:1 picking:1 michael:1 containing:1 expert:3 supp:1 potential:4 de:1 coefficient:1 ranking:5 depends:1 collobert:1 performed:3 root:3 francis:1 portion:1 recover:1 bayes:1 capability:2 parallel:1 formed:1 efficiently:2 yield:1 generalize:1 handwritten:3 bayesian:1 none:1 comp:1 finer:1 reestimate:1 definition:1 underestimate:1 nonetheless:1 associated:2 gain:4 dataset:6 knowledge:1 dimensionality:2 sophisticated:1 higher:3 methodology:3 response:1 improved:2 done:2 evaluated:1 generality:1 until:3 working:1 hand:1 freezing:1 replacing:1 ei:2 overlapping:1 propagation:1 ipf:2 building:1 former:1 lenet:1 assigned:1 leibler:1 deal:1 during:3 width:1 criterion:1 generalized:2 leftmost:2 exdb:1 demonstrate:1 percent:2 image:4 ranging:1 novel:1 common:1 significant:1 cambridge:1 gibbs:2 meila:1 had:1 add:3 base:5 multivariate:2 r90:1 recent:1 triangulation:5 occasionally:1 certain:1 n00014:1 verlag:1 binary:4 success:1 onr:1 seen:1 minimum:1 greater:2 additional:2 employed:1 determine:1 ii:4 full:2 desirable:1 reduces:1 technical:1 characterized:1 calculation:1 bach:1 cross:1 retrieval:1 offer:1 equally:1 privileged:1 controlled:2 ensuring:2 involving:2 basic:2 variant:1 regression:2 vision:2 expectation:4 kernel:1 achieved:4 addition:2 cropped:2 addressed:1 sch:1 fbach:1 induced:1 undirected:1 lafferty:1 jordan:3 bengio:1 enough:1 iterate:1 independence:1 fit:6 gave:2 suboptimal:1 idea:1 bottleneck:1 whether:1 utility:1 darroch:1 york:1 skewing:1 matlab:1 generally:3 involve:2 ten:8 statist:1 induces:1 svms:1 dna:3 generate:4 http:1 outperform:1 exist:1 percentage:4 problematic:1 nsf:1 dotted:1 estimated:1 discrete:1 four:2 deleted:1 graph:13 fraction:2 sum:1 parameterized:1 throughout:2 family:6 almost:1 yann:1 wu:1 geiger:1 scaling:11 bit:1 bound:3 adapted:1 constraint:4 aspect:1 speed:1 performing:2 relatively:1 according:2 combination:2 smaller:1 heckerman:1 cun:1 invariant:1 computationally:1 equation:3 remains:1 turn:1 needed:1 tractable:4 junction:29 operation:2 available:3 generalizes:1 hierarchical:1 away:1 generic:1 robustness:1 altogether:1 bodlaender:1 existence:1 original:4 graphical:7 marginalized:2 newton:1 maintaining:1 exploit:3 giving:1 build:1 approximating:1 g0:1 mumford:1 strategy:1 degrades:1 dependence:1 sci:1 evaluate:1 considers:1 induction:8 denmark:1 index:2 retained:1 negative:1 ba:1 implementation:3 design:1 boltzmann:2 unknown:1 perform:2 upper:5 markov:2 acknowledge:1 hinton:1 incorporated:1 treewidths:1 cast:1 pair:1 kl:3 toolbox:1 immunity:1 eature:1 california:2 nip:1 trans:2 able:4 built:1 including:1 max:1 critical:1 ranked:1 indicator:2 zhu:1 minimax:1 spiegelhalter:1 naive:1 text:2 interdependent:1 literature:1 acknowledgement:1 determining:1 relative:1 loss:2 expect:1 proportional:3 srebro:2 validation:1 kjaerulff:1 degree:2 sufficient:1 imposes:1 principle:1 bank:1 cd:1 repeat:3 liability:1 allow:2 wide:1 characterizing:1 sparse:1 boundary:3 dimension:1 plain:1 evaluating:1 vocabulary:1 collection:1 made:1 far:1 emphasize:1 preferred:2 status:1 kullback:1 gene:1 clique:18 ml:2 transcription:1 overfitting:2 sequentially:1 uai:1 discriminative:4 grayscale:1 search:2 iterative:14 terative:1 robust:1 ca:2 complex:1 constructing:1 domain:3 decrement:1 allowed:3 site:1 referred:1 wish:1 exponential:7 candidate:4 chickering:1 third:1 grained:1 splice:5 emphasized:1 intron:3 svm:6 virtue:1 normalizing:1 grouping:1 exists:1 essential:1 evidence:1 mnist:5 concern:1 texture:1 subtract:1 entropy:3 simply:1 cheaply:2 partially:2 applies:1 cowell:1 springer:1 ma:1 conditional:4 goal:2 viewed:3 consequently:1 ann:1 replace:1 erased:4 hard:1 included:1 uniformly:1 principal:1 conservative:1 total:1 pas:2 select:1 support:6 latter:1 assessed:1 bioinformatics:1 svmtorch:2 dept:1 tested:2 della:2 handling:1 |
1,173 | 207 | 524
Fablman and Lebiere
The Cascade-Correlation Learning Architecture
Scott E. Fahlman and Christian Lebiere
School of Computer Science
Carnegie-Mellon University
Pittsburgh, PA 15213
ABSTRACT
Cascade-Correlation is a new architecture and supervised learning algorithm for artificial neural networks. Instead of just adjusting the weights
in a network of fixed topology. Cascade-Correlation begins with a minimal network, then automatically trains and adds new hidden units one
by one, creating a multi-layer structure. Once a new hidden unit has
been added to the network, its input-side weights are frozen. This unit
then becomes a permanent feature-detector in the network, available for
producing outputs or for creating other, more complex feature detectors. The Cascade-Correlation architecture has several advantages over
existing algorithms: it learns very quickly, the network .determines its
own size and topology, it retains the structures it has built even if the
training set changes, and it requires no back-propagation of error signals
through the connections of the network.
1 DESCRIPTION OF CASCADE?CORRELATION
The most important problem preventing the widespread application of artificial neural
networks to real-world problems is the slowness of existing learning algorithms such as
back-propagation (or "backprop"). One factor contributing to that slowness is what we
call the moving target problem: because all of the weights in the network are changing
at once, each hidden units sees a constantly changing environment. Instead of moving
quickly to assume useful roles in the overall problem solution, the hidden units engage in
a complex dance with much wasted motion. The Cascade-Correlation learning algorithm
was developed in an attempt to solve that problem. In the problems we have examined,
it learns much faster than back-propagation and solves some other problems as well.
The Cascade-Correlation Learning Architecture
Outputs
o
0
Output Units
Hidden Unit 2
Hidden unit 1
~~--------~~--.----
o--------~&_-------mr_--------------~----~--~~
Inpu~
O--------~~----~H-------------~.---~~--~
o--------~~------~--------------------------~
+1
Figure 1: The Cascade architecture, after two hidden units have been added. The
vertical lines sum all incoming activation. Boxed connections are frozen, X connections
are trained repeatedly.
Cascade-Correlation combines two key ideas: The first is the cascade architecture, in
which hidden units are added to the network one at a time and do not change after they
have been added. The second is the learning algorithm, which creates and installs the
new hidden units. For each new hidden unit, we attempt to maximize the magnitude of
the correlation between the new unit's output and the residual error signal we are trying
to eliminate.
The cascade architecture is illustrated in Figure 1. It begins with some inputs and one or
more output units, but with no hidden units. The number of inputs and outputs is dictated
by the problem and by the I/O representation the experimenter has chosen. Every input
is connected to every output unit by a connection with an adjustable weight. There is
also a bias input, permanently set to +1.
The output units may just produce a linear sum of their weighted inputs, or they may
employ some non-linear activation function. In the experiments we have run so far, we
use a symmetric sigmoidal activation function (hyperbolic tangent) whose output range
is -1.0 to + 1.0. For problems in which a precise analog output is desired, instead of a
binary classification, linear output units might be the best choice, but we have not yet
studied any problems of this kind.
We add hidden units to the network one by one. Each new hidden unit receives a
connection from each of the network's original inputs and also from every pre-existing
hidden unit. The hidden unit's input weights are frozen at the time the unit is added to
the net; only the output connections are trained repeatedly. Each new unit therefore adds
525
526
Fahlman and Lebiere
a new one-unit "layer" to the network, unless some of its incoming weights happen to be
zero. This leads to the creation of very powerful high-order feature detectors; it also may
lead to very deep networks and high fan-in to the hidden units. There are a number of
possible strategies for minimizing the network depth and fan-in as new units are added,
but we have not yet explored these strategies.
The learning algorithm begins with no hidden units. The direct input-output connections
are trained as well as possible over the entire training set. With no need to back-propagate
through hidden units, we can use the Widrow-Hoff or "delta" rule, the Perceptron learning
algorithm, or any of the other well-known learning algorithms for single-layer networks.
In our simulations, we use Fahlman's "quickprop" algorithm [Fahlman, 1988] to train the
output weights. With no hidden units, this acts essentially like the delta rule, except that
it converges much faster.
At some point, this training will approach an asymptote. When no significant error
reduction has occurred after a certain number of training cycles (controlled by a "patience"
parameter set by the operator), we run the network one last time over the entire training
set to measure the error. If we are satisfied with the network's performance, we stop; if
not, we attempt to reduce the residual errors further by adding a new hidden unit to the
network. The unit-creation algorithm is described below. The new unit is added to the
net, its input weights are frozen, and all the output weights are once again trained using
quickprop. This cycle repeats until the error is acceptably small (or until we give up).
To create a new hidden unit, we begin with a candidate unit that receives trainable input
connections from all of the network's external inputs and from all pre-existing hidden
units. The output of this candidate unit is not yet connected to the active network. We run
a number of passes over the examples of the training set, adjusting the candidate unit's
input weights after each pass. The goal of this adjustment is to maximize S, the sum over
all output units 0 of the magnitude of the correlation (or, more precisely, the covariance)
between V, the candidate unit's value, and Eo, the residual output error observed at unit
o. We define S as
S=
L: L:(Vp o
V) (Ep,o - Eo)
p
where 0 is the network output at which the error is measured and p is the training pattern.
The quantities V and Eo are the values of V and Eo averaged over all patterns.
In order to maximize S, we must compute 8Sj8wi, the partial derivative of S with
respect to each of the candidate unit's incoming weights, Wi. In a manner very similar
to the derivation of the back-propagation rule in [Rumelhart, 1986], we can expand and
differentiate the fonnula for S to get
8Sj8Wj
=L: uo(Ep,o -
Eo)J;,lj,p
p,o
where U o is the sign of the correlation between the candidate's value and output
o,ff, is
The Cascade-Correlation Learning Architecture
the derivative for pattern p of the candidate unit's activation function with respect to the
sum of its inputs, and li,p is the input the candidate unit receives from unit i for pattern
p.
After computing 8 S/ 8Wi for each incoming connection, we can perform a gradient ascent
to maximize S. Once again we are training only a single layer of weights. Once again
we use the quickprop update rule for faster convergence. When S stops improving, we
install the new candidate as a unit in the active network, freeze its input weights, and
continue the cycle as described above.
Because of the absolute value in the formula for S, a candidate unit cares only about the
magnitude of its correlation with the error at a given output, and not about the sign of
the correlation. As a rule, if a hidden unit correlates positively with the error at a given
unit, it will develop a negative connection weight to that unit, attempting to cancel some
of the error; if the correlation is negative, the output weight will be positive. Since a
unit's weights to different outputs may be of mixed sign, a unit can sometimes serve two
purposes by developing a positive correlation with the error at one output and a negative
correlation with the error at another.
Instead of a single candidate unit. it is possible to use a pool of candidate units, each
with a different set of random initial weights. All receive the same input signals and see
the same residual error for each pattern and each output. Because they do not interact
with one another or affect the active network during training, all of these candidate units
can be trained in parallel; whenever we decide that no further progress is being made,
we install the candidate whose correlation score is the best. The use of this pool of
candidates is beneficial in two ways: it greatly reduces the chance that a useless unit will
be permanently installed because an individual candidate got stuck during training, and
(on a parallel machine) it can speed up the training because many parts of weight-space
can be explored simultaneously.
The hidden and candidate units may all be of the same type, for example with a sigmoid
activation function. Alternatively, we might create a pool of candidate units with a
mixture of nonlinear activation functions-some sigmoid, some Gaussian, some with
radial activation functions. and so on-and let them compete to be chosen for addition
to the active network. To date, we have explored the all-sigmoid and all-Gaussian cases,
but we do not yet have extensive simulation data on networks with mixed unit-types.
One final note on the implementation of this algorithm: While the weights in the output
layer are being trained, the other weights in the active network are frozen. While the
candidate weights are being trained, none of the weights in the active network are changed.
In a machine with plenty of memory. it is possible to record the unit-values and the output
errors for an entire epoch, and then to use these cached values repeatedly during training.
rather than recomputing them repeatedly for each training case. This can result in a
tremendous speedup as the active network grows large.
527
528
Fahlman and Lebiere
Figure 2: Training points for the two-spirals problem, and output pattern for one network
trained with Cascade-Correlation.
2 BENCHMARK RESULTS
2.1
THE TWO-SPIRALS PROBLEM
The "two-spirals" benchmark was chosen as the primary benchmark for this study because
it is an extremely hard problem for algorithms of the back-propagation family to solve.
n was first proposed by Alexis Wieland of MImE Corp. The net has two continuousvalued inputs and a single output. The training set consists of 194 X-Y values, half of
which are to produce a +1 output and half a -1 output. These training points are arranged
in two interlocking spirals that go around the origin three times, as shown in Figure 2a.
The goal is to develop a feed-forward network with sigmoid units that properly classifies
all 194 training cases. Some hidden units are obviously needed, since a single linear
separator cannot divide two sets twisted together in this way.
Wieland (unpublished) reported that a modified version of backprop in use at MITRE
required 150,000 to 200,000 epochs to solve this problem, and that they had never
obtained a solution using standard backprop. Lang and Witbrock [Lang, 1988] tried the
problem using a 2-5-5-5-1 network (three hidden layers of five units each). Their network
was unusual in that it provided "shortcut" connections: each unit received incoming
connections from every unit in every earlier layer, not just from the immediately preceding
layer. With this architecture, standard backprop was able to solve the problem in 20,000
epochs, backprop with a modified error function required 12,000 epochs, and quickprop
required 8000. This was the best two-spirals performance reported to date. Lang and
Witbrock also report obtaining a solution with a 2-5-5-1 net (only ten hidden units in
all), but the solution required 60,000 quickprop epochs.
We ran the problem 100 times with the Cascade-Correlation algorithm using a Sigmoidal
activation function for both the output and hidden units and a pool of 8 candidate units.
All trials were successful, requiring 1700 epochs on the average. (This number counts
The Cascade-Correlation Learning Architecture
both the epochs used to train output weights and the epochs used to train candidate units.)
The number of hidden units built into the net varied from 12 to 19, with an average of
15.2 and a median of 15. Here is a histogram of the number of hidden units created:
Hidden
Units
12
13
14
15
16
17
18
19
Number of
Trials
4 ####
#########
9
24 ########################
19 ###################
24 ########################
13 #############
5 #####
2 ##
In terms of training epochs, Cascade-Correlation beats quickprop by a factor of 5 and
standard backprop by a factor of 10, while building a network of about the same complexity (15 hidden units). In terms of actual computation on a serial machine, however,
the speedup is much greater than these numbers suggest In backprop and quickprop,
each training case requires a forward and a backward pass through all the connections in
the network; Cascade-Correlation requires only a forward pass. In addition, many of the
Cascade-Correlation epochs are run while the network is much smaller than its final size.
Finally, the cacheing strategy described above makes it possible to avoid re-computing
the unit values for parts of the network that are not changing.
Suppose that instead of epochs, we measure learning time in connection crossings, defined
as the number of multiply-accumulate steps necessary to propagate activation values
forward through the network and error values backward. This measure leaves out some
computational steps, but it is a more accurate measure of computational complexity
than comparing epochs of different sizes or comparing runtimes on different machines.
The Lang and Witbrock result of 20,000 backprop epochs requires about 1.1 billion
connection crossings. Their solution using 8000 quickprop epochs on the same network
requires about 438 million crossings. An average Cascade-Correlation run with a pool of
8 candidate units requires about 19 million crossings-a 23-fold speedup over quickprop
and a 50-fold speedup over standard backprop. With a smaller pool of candidate units the
speedup (on a serial machine) would be even greater, but the resulting networks might
be somewhat larger.
Figure 2b shows the output of a 12-hidden-unit network built by Cascade-Correlation
as the input is scanned over the X-V field. This network properly classifies all 194
training points. We can see that it interpolates smoothly for about the first 1.5 turns of
the spiral, but becomes a bit lumpy farther out, where the training points are farther apart.
This "receptive field" diagram is similar to that obtained by Lang and Witbrock using
backprop, but is somewhat smoother.
529
530
Fahlman and Lebiere
2.2
N-INPUT PARITY
Since parity has been a popular benchmark among other researchers, we ran CascadeCorrelation on N-input parity problems with N ranging from 2 to 8. The best results
were obtained with a sigmoid output unit and hidden units whose output is a Gaussian
function of the sum of weighted inputs. Based on five trials for each value of N, our
results were as follows:
N
Cases
2
3
4
5
6
7
8
4
8
16
32
64
128
256
Hidden
Units
1
1
2
2-3
3
4-5
4-5
Average
Epochs
24
32
66
142
161
292
357
For a rough comparison, Tesauro and Janssens [Tesauro, 1988] report that standard backprop takes about 2000 epochs for 8-input parity. In their study, they used 2N hidden units.
Cascade-Correlation can solve the problem with fewer than N hidden units because it uses
short-cut connections.
As a test of generalization, we ran a few trials of Cascade-Correlation on the lO-input
parity problem, training on either 50% or 25% of the 1024 patterns and testing on the
rest. The number of hidden units built varied from 4 to 7 and training time varied from
276 epochs to 551. When trained on half of the patterns, perfonnance on the test set
averaged 96% correct; when trained on one quarter of the patterns, test-set performance
averaged 90% correct Note that the nearest neighbor algorithm would get almost all of
the test-set cases wrong.
3 DISCUSSION
We believe that that Cascade-Correlation algorithm offers the following advantages over
network learning algorithms currently in use:
? There is no need to guess the size, depth, and connectivity pattern of the network
in advance. A reasonably small (though not optimal) net is built automatically,
perhaps with a mixture of unit-types .
? Cascade-Correlation learns fast In backprop, the hidden units engage in a complex
dance before they settle into distinct useful roles; in Cascade-Correlation, each unit
sees a fixed problem and can move decisively to solve that problem. For the
problems we have investigated to date, the learning time in epochs grows roughly
as NlogN, where N is the number of hidden units ultimately needed to solve the
problem.
The Cascade-Correlation Learning Architecture
? Cascade-Correlation can build deep nets (high-order feature detectors) without the
dramatic slowdown we see in deep back-propagation networks.
? Cascade-Correlation is useful for incremental learning. in which new infonnation is
added to an already-trained net. Once built. a feature detector is never cannibalized.
It is available from that time on for producing outputs or more complex features.
? At any given time. we train only one layer of weights in the network. The rest of
the network is constant. so results can be cached.
? There is never any need to propagate error signals backwards through network
connections. A single residual error signal can be broadcast to all candidates.
The weighted connections transmit signals in only one direction. eliminating one
difference between these networks and biological synapses.
? The candidate units do not interact. except to pick a winner. Each candidate sees the
same inputs and error signals. This limited communication makes the architecture
attractive for parallel implementation.
4
RELATION TO OTHER WORK
The principal differences between Cascade-Correlation and older learning architectures
are the dynamic creation of hidden units. the way we stack the new units in multiple
layers (with a fixed output layer). the freezing of units as we add them to the net. and
the way we train new units by hill-climbing to maximize the unit's correlation with the
residual error. The most interesting discovery is that by training one unit at a time instead
of training the whole network at once. we can speed up the learning process considerably.
while still creating a reasonably small net that generalizes well.
A number of researchers [Ash. 1989.Moody. 1989] have investigated networks that add
new units or receptive fields within a single layer in the course of learning. While
single-layer systems are well-suited for some problems. these systems are incapable of
creating higher-order feature detectors that combine the outputs of existing units. The
idea of building feature detectors and then freezing them was inspired in part by the
work of Waibel on modular networks [Waibel. 19891. but in his model the structure of
the sub-networks must be fixed before learning begins.
We know of only a few attempts to build up multi-layer networks as the learning progresses. Our decision to look at models in which each unit can see all pre-existing units
was inspired to some extent by work on progressively deepening threshold-logic models
by Merrick Furst and Jeff Jackson at Carnegie Mellon. (They are not actively pursuing
this line at present.) Gallant [Gallant. 1986] briefly mentions a progressively deepening
perceptron model (his "inverted pyramid model) in which units are frozen after being
installed. However. he has concentrated most of his research effort on models in which
new hidden units are generated at random rather than by a deliberate training process.
The SONN model of Tenorio and Lee [Tenorio, 1989] builds a multiple-layer topology
U
531
532
Fahlman and Lebiere
to suit the problem at hand. Their algorithm places new -two-input units at randomly selected locations, using a simulated annealing search to keep only the most useful ones-a
very different approach from ours.
Acknowledgments
We would like to thank Merrick Furst, Paul Gleichauf, and David Touretzlcy for asking
good questions that helped to shape this work. This research was sponsored in part by
the National Science Foundation (Contract EET-8716324) and in part by the Defense
Advanced Research Projects Agency (Contract F3361S-87-C-1499).
References
[Ash, 1989]
Ash, T. (1989) "Dynamic Node Creation in Back-Propagation Networks", Technical Report 8901, Institute for Cognitive Science, University of California, San Diego.
[Fahlman, 1988]
Fahlman, S. E. (1988) "Faster-Learning Variations on BackPropagation: An Empirical Study" in Proceedings of the 1988 Connectionist Models Summer School, Morgan Kaufmann.
[Gallant, 1986]
Gallant, S. I. (1986) "Three Constructive Algorithms for Network
Learning" in Proceedings. 8th Annual Conference of the Cognitive
Science Society.
[Lang, 1988]
Lang, K. J. and Witbrock, M. J. (1988) "Learning to Tell Two Spirals
Apart" in Proceedings of the 1988 Connectionist Models Summer
School, Morgan Kaufmann.
[Moody, 1989]
Moody, J. (1989) "Fast Learning in Multi-Resolution Hierarchies" in
D. S. Touretzky (ed.), Advances in Neural Information Processing
Systems 1, Morgan Kaufmann.
[Rumelhart, 1986] Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986) "Learning
Internal Representations by Error Propagation" in Rumelhart, D. E.
and McClelland, J. L.,Parallel Distributed Processing: Explorations
in the Microstructure of Cognition, MIT Press.
[Tenorio, 1989]
Tenorio, M. E, and Lee, W. T. (1989) "Self-Organizing Neural Nets
for the Identification Problem" in D. S. Touretzky (ed.), Advances in
Neural Information Processing Systems 1, Morgan Kaufmann.
[Tesauro, 1988]
Tesauro, G. and Janssens, B. (1988) "Scaling Relations in BackPropagation Learning" in Complex Systems 2 39-44.
[Waibel, 1989]
Waibel, A. (1989) "Consonant Recognition by Modular Construction
of Large Phonemic Time-Delay Neural Networks" in D. S. TouretzlcY
(ed.), Advances in Neural Information Processing Systt ms 1, Morgan
Kaufmann.
| 207 |@word trial:4 briefly:1 version:1 eliminating:1 simulation:2 propagate:3 quickprop:9 covariance:1 tried:1 pick:1 dramatic:1 mention:1 reduction:1 initial:1 score:1 ours:1 existing:6 comparing:2 activation:9 yet:4 lang:7 must:2 merrick:2 happen:1 shape:1 christian:1 asymptote:1 sponsored:1 update:1 progressively:2 half:3 leaf:1 fewer:1 guess:1 selected:1 short:1 farther:2 record:1 node:1 location:1 sigmoidal:2 five:2 direct:1 install:2 consists:1 combine:2 manner:1 roughly:1 multi:3 inspired:2 continuousvalued:1 automatically:2 actual:1 becomes:2 begin:5 classifies:2 provided:1 project:1 twisted:1 what:1 kind:1 developed:1 every:5 act:1 wrong:1 unit:101 uo:1 acceptably:1 producing:2 positive:2 before:2 installed:2 might:3 studied:1 examined:1 limited:1 range:1 averaged:3 acknowledgment:1 testing:1 backpropagation:2 nlogn:1 empirical:1 cascade:29 hyperbolic:1 got:1 pre:3 radial:1 suggest:1 get:2 cannot:1 operator:1 fonnula:1 interlocking:1 go:1 williams:1 resolution:1 immediately:1 rule:5 jackson:1 his:3 variation:1 transmit:1 target:1 suppose:1 diego:1 engage:2 hierarchy:1 construction:1 us:1 alexis:1 origin:1 pa:1 crossing:4 rumelhart:4 recognition:1 cut:1 observed:1 role:2 ep:2 connected:2 cycle:3 ran:3 environment:1 agency:1 complexity:2 touretzlcy:2 dynamic:2 ultimately:1 trained:11 creation:4 serve:1 creates:1 derivation:1 train:6 cacheing:1 fast:2 distinct:1 artificial:2 tell:1 whose:3 modular:2 larger:1 solve:7 final:2 obviously:1 differentiate:1 advantage:2 frozen:6 net:11 date:3 organizing:1 description:1 billion:1 convergence:1 produce:2 cached:2 incremental:1 converges:1 widrow:1 develop:2 measured:1 nearest:1 school:3 received:1 progress:2 phonemic:1 sonn:1 solves:1 direction:1 correct:2 exploration:1 settle:1 backprop:12 microstructure:1 generalization:1 biological:1 around:1 cognition:1 furst:2 purpose:1 currently:1 infonnation:1 create:2 weighted:3 rough:1 mit:1 gaussian:3 modified:2 rather:2 avoid:1 properly:2 greatly:1 eliminate:1 entire:3 lj:1 hidden:42 relation:2 expand:1 overall:1 classification:1 among:1 hoff:1 field:3 once:7 never:3 runtimes:1 look:1 cancel:1 plenty:1 report:3 connectionist:2 employ:1 few:2 randomly:1 simultaneously:1 national:1 individual:1 suit:1 attempt:4 decisively:1 multiply:1 mixture:2 accurate:1 partial:1 necessary:1 perfonnance:1 unless:1 divide:1 desired:1 re:1 minimal:1 recomputing:1 earlier:1 asking:1 retains:1 witbrock:5 delay:1 successful:1 reported:2 considerably:1 lee:2 contract:2 pool:6 together:1 quickly:2 moody:3 connectivity:1 again:3 satisfied:1 deepening:2 broadcast:1 external:1 creating:4 cognitive:2 derivative:2 li:1 actively:1 permanent:1 mime:1 helped:1 parallel:4 kaufmann:5 climbing:1 vp:1 identification:1 none:1 researcher:2 lumpy:1 detector:7 synapsis:1 touretzky:2 whenever:1 ed:3 lebiere:6 stop:2 experimenter:1 adjusting:2 popular:1 back:8 janssens:2 feed:1 higher:1 supervised:1 arranged:1 though:1 just:3 correlation:36 until:2 hand:1 receives:3 freezing:2 nonlinear:1 propagation:8 widespread:1 perhaps:1 believe:1 grows:2 building:2 requiring:1 symmetric:1 illustrated:1 attractive:1 during:3 self:1 m:1 trying:1 hill:1 motion:1 ranging:1 sigmoid:5 quarter:1 winner:1 million:2 analog:1 occurred:1 he:1 accumulate:1 mellon:2 significant:1 freeze:1 had:1 moving:2 add:5 own:1 dictated:1 apart:2 tesauro:4 slowness:2 certain:1 corp:1 incapable:1 binary:1 continue:1 inverted:1 morgan:5 greater:2 care:1 somewhat:2 preceding:1 eo:5 maximize:5 signal:7 smoother:1 multiple:2 reduces:1 technical:1 faster:4 offer:1 serial:2 inpu:1 controlled:1 essentially:1 histogram:1 sometimes:1 pyramid:1 receive:1 addition:2 annealing:1 diagram:1 median:1 rest:2 pass:1 ascent:1 call:1 backwards:1 spiral:7 affect:1 gleichauf:1 architecture:13 topology:3 reduce:1 idea:2 defense:1 effort:1 interpolates:1 repeatedly:4 deep:3 useful:4 ten:1 concentrated:1 mcclelland:1 wieland:2 deliberate:1 sign:3 delta:2 carnegie:2 key:1 threshold:1 changing:3 backward:2 wasted:1 sum:5 run:5 compete:1 powerful:1 place:1 family:1 almost:1 decide:1 pursuing:1 decision:1 patience:1 scaling:1 bit:1 layer:15 summer:2 fan:2 fold:2 annual:1 scanned:1 precisely:1 speed:2 extremely:1 attempting:1 speedup:5 developing:1 waibel:4 beneficial:1 smaller:2 cascadecorrelation:1 wi:2 turn:1 count:1 needed:2 know:1 unusual:1 available:2 generalizes:1 permanently:2 original:1 build:3 society:1 move:1 added:8 quantity:1 already:1 question:1 strategy:3 primary:1 receptive:2 gradient:1 thank:1 simulated:1 extent:1 useless:1 minimizing:1 negative:3 implementation:2 adjustable:1 perform:1 gallant:4 vertical:1 benchmark:4 beat:1 hinton:1 communication:1 precise:1 varied:3 stack:1 david:1 unpublished:1 required:4 trainable:1 extensive:1 connection:18 california:1 tremendous:1 able:1 below:1 pattern:10 scott:1 built:6 memory:1 residual:6 advanced:1 older:1 created:1 epoch:18 discovery:1 tangent:1 contributing:1 mixed:2 interesting:1 ash:3 foundation:1 lo:1 course:1 changed:1 repeat:1 fahlman:9 last:1 parity:5 slowdown:1 side:1 bias:1 perceptron:2 institute:1 neighbor:1 absolute:1 distributed:1 depth:2 world:1 preventing:1 stuck:1 made:1 forward:4 san:1 far:1 correlate:1 eet:1 logic:1 keep:1 active:7 incoming:5 pittsburgh:1 consonant:1 alternatively:1 search:1 reasonably:2 obtaining:1 improving:1 interact:2 boxed:1 investigated:2 complex:5 separator:1 mitre:1 whole:1 paul:1 positively:1 ff:1 sub:1 candidate:26 learns:3 formula:1 explored:3 adding:1 magnitude:3 suited:1 smoothly:1 tenorio:4 adjustment:1 determines:1 constantly:1 chance:1 goal:2 jeff:1 shortcut:1 change:2 hard:1 except:2 principal:1 pas:3 internal:1 constructive:1 dance:2 |
1,174 | 2,070 | Latent Dirichlet Allocation
David M. Blei, Andrew Y. Ng and Michael I. Jordan
University of California, Berkeley
Berkeley, CA 94720
Abstract
We propose a generative model for text and other collections of discrete data that generalizes or improves on several previous models
including naive Bayes/unigram, mixture of unigrams [6], and Hofmann's aspect model , also known as probabilistic latent semantic
indexing (pLSI) [3]. In the context of text modeling, our model
posits that each document is generated as a mixture of topics,
where the continuous-valued mixture proportions are distributed
as a latent Dirichlet random variable. Inference and learning are
carried out efficiently via variational algorithms. We present empirical results on applications of this model to problems in text
modeling, collaborative filtering, and text classification.
1
Introduction
Recent years have seen the development and successful application of several latent
factor models for discrete data. One notable example, Hofmann's pLSI/aspect
model [3], has received the attention of many researchers, and applications have
emerged in text modeling [3], collaborative filtering [7], and link analysis [1]. In
the context of text modeling, pLSI is a "bag-of-words" model in that it ignores the
ordering of the words in a document. It performs dimensionality reduction, relating
each document to a position in low-dimensional "topic" space. In this sense, it is
analogous to PCA, except that it is explicitly designed for and works on discrete
data.
A sometimes poorly-understood subtlety of pLSI is that, even though it is typically
described as a generative model , its documents have no generative probabilistic
semantics and are treated simply as a set of labels for the specific documents seen
in the training set. Thus there is no natural way to pose questions such as "what is
the probability of this previously unseen document?". Moreover, since each training
document is treated as a separate entity, the pLSI model has a large number of
parameters and heuristic "tempering" methods are needed to prevent overfitting.
In this paper we describe a new model for collections of discrete data that provides
full generative probabilistic semantics for documents. Documents are modeled via a
hidden Dirichlet random variable that specifies a probability distribution on a latent,
low-dimensional topic space. The distribution over words of an unseen document is
a continuous mixture over document space and a discrete mixture over all possible
topics.
2
2.1
Generative models for text
Latent Dirichlet Allocation (LDA) model
To simplify our discussion, we will use text modeling as a running example throughout this section, though it should be clear that the model is broadly applicable to
general collections of discrete data.
In LDA, we assume that there are k underlying latent topics according to which
documents are generated, and that each topic is represented as a multinomial distribution over the IVI words in the vocabulary. A document is generated by sampling
a mixture of these topics and then sampling words from that mixture.
More precisely, a document of N words w = (W1,'" ,WN) is generated by the
following process. First, B is sampled from a Dirichlet(a1,'" ,ak) distribution.
This means that B lies in the (k - I)-dimensional simplex: Bi 2': 0, 2: i Bi = 1.
Then, for each of the N words, a topic Zn E {I , ... , k} is sampled from a Mult(B)
distribution p(zn = ilB) = Bi . Finally, each word Wn is sampled, conditioned on
the znth topic, from the multinomial distribution p(wl zn). Intuitively, Bi can be
thought of as the degree to which topic i is referred to in the document . Written
out in full, the probability of a document is therefore the following mixture:
p(w) =
Ie (11 z~/(wnlzn; ,8)P( Zn IB?) p(B; a)dB,
(1)
where p(B ; a) is Dirichlet , p(znIB) is a multinomial parameterized by B, and
p( Wn IZn;,8) is a multinomial over the words. This model is parameterized by the kdimensional Dirichlet parameters a = (a1,' .. ,ak) and a k x IVI matrix,8, which are
parameters controlling the k multinomial distributions over words. The graphical
model representation of LDA is shown in Figure 1.
As Figure 1 makes clear, this model is not a simple Dirichlet-multinomial clustering
model. In such a model the innermost plate would contain only W n ; the topic
node would be sampled only once for each document; and the Dirichlet would be
sampled only once for the whole collection. In LDA, the Dirichlet is sampled for
each document, and the multinomial topic node is sampled repeatedly within the
document. The Dirichlet is thus a component in the probability model rather than
a prior distribution over the model parameters.
We see from Eq. (1) that there is a second interpretation of LDA. Having sampled
B, words are drawn iid from the multinomial/unigram model given by p(wIB) =
2::=1 p(wl z)p(z IB). Thus, LDA is a mixture model where the unigram models
p(wIB) are the mixture components, and p(B ; a) gives the mixture weights. Note
that unlike a traditional mixture of unigrams model, this distribution has an infinite
o
1'0 '.
Zn
Wn
Nd I
D
Figure 1: Graphical model representation of LDA. The boxes are plates representing
replicates. The outer plate represents documents, while the inner plate represents
the repeated choice of topics and words within a document.
Figure 2: An example distribution on unigram models p(wIB) under LDA for three
words and four topics. The triangle embedded in the x-y plane is the 2-D simplex
over all possible multinomial distributions over three words. (E.g. , each of the
vertices of the triangle corresponds to a deterministic distribution that assigns one
of the words probability 1; the midpoint of an edge gives two of the words 0.5
probability each; and the centroid of the triangle is the uniform distribution over
all 3 words). The four points marked with an x are the locations of the multinomial
distributions p(wlz) for each of the four topics , and the surface shown on top of the
simplex is an example of a resulting density over multinomial distributions given
by LDA.
number of continuously-varying mixture components indexed by B. The example
in Figure 2 illustrates this interpretation of LDA as defining a random distribution
over unigram models p(wIB).
2.2
Related models
The mixture of unigrams model [6] posits that every document is generated by a
single randomly chosen topic:
(2)
This model allows for different documents to come from different topics, but fails to
capture the possibility that a document may express multiple topics. LDA captures
this possibility, and does so with an increase in the parameter count of only one
parameter: rather than having k - 1 free parameters for the multinomial p(z) over
the k topics, we have k free parameters for the Dirichlet.
A second related model is Hofmann's probabilistic latent semantic indexing
(pLSI) [3], which posits that a document label d and a word ware conditionally
independent given the hidden topic z :
p(d, w) = L~=l p(wlz)p(zld)p(d).
(3)
This model does capture the possibility that a document may contain multiple topics
since p(zld) serve as the mixture weights of the topics. However, a subtlety of pLSIand the crucial difference between it and LDA-is that d is a dummy index into
the list of documents in the training set. Thus, d is a multinomial random variable
with as many possible values as there are training documents, and the model learns
the topic mixtures p(zld) only for those documents on which it is trained. For this
reason, pLSI is not a fully generative model and there is no clean way to use it
to assign probability to a previously unseen document. Furthermore, the number
of parameters in pLSI is on the order of klVl + klDI, where IDI is the number of
documents in the training set. Linear growth in the number of parameters with the
size of the training set suggests that overfitting is likely to be a problem and indeed,
in practice, a "tempering" heuristic is used to smooth the parameters of the model.
3
Inference and learning
Let us begin our description of inference and learning problems for LDA by examining the contribution to the likelihood made by a single document. To simplify
our notation, let w~ = 1 iff Wn is the jth word in the vocabulary and z~ = 1
iff Zn is the ith topic. Let j3ij denote p(w j = Ilzi = 1), and W = (WI, ... ,WN),
Z = (ZI, ... ,ZN). Expanding Eq. (1), we have:
(4)
This is a hypergeometric function that is infeasible to compute exactly [4].
Large text collections require fast inference and learning algorithms and thus we
have utilized a variational approach [5] to approximate the likelihood in Eq. (4).
We use the following variational approximation to the log likelihood:
logp(w; a, 13)
log
r :Ep(wlz; j3)p(zIB)p(B; a) qq~:,,Z",
z:" ~~ dB
le
z
> Eq[logp(wlz;j3) +logp(zIB) +logp(B;a) -logq(B,z; , ,?)],
where we choose a fully factorized variational distribution q(B, z;" ?)
q(B; ,) fIn q(Zn; ?n) parameterized by , and ?n, so that q(B; ,) is Dirichlet({), and
q(zn; ?n) is MUlt(?n). Under this distribution, the terms in the variational lower
bound are computable and differentiable, and we can maximize the bound with
respect to, and ? to obtain the best approximation to p(w;a,j3).
Note that the third and fourth terms in the variational bound are not straightforward to compute since they involve the entropy of a Dirichlet distribution, a
(k - I)-dimensional integral over B which is expensive to compute numerically. In
the full version of this paper, we present a sequence of reductions on these terms
which use the log r function and its derivatives. This allows us to compute the
integral using well-known numerical routines.
Variational inference is coordinate ascent in the bound on the probability of a single
document. In particular, we alternate between the following two equations until the
objective converges:
(5)
,i
+ 2:~=1 ?ni
derivative of the log r function.
ai
(6)
where \]i is the first
Note that the resulting variational parameters can also be used and interpreted as an approximation of the
parameters of the true posterior.
In the current paper we focus on maximum likelihood methods for parameter estimation. Given a collection of documents V = {WI' ... ' WM}, we utilize the EM
algorithm with a variational E step, maximizing a lower bound on the log likelihood:
M
logp(V) 2::
l:= Eqm [logp(B, z, w)]- Eqm [logqm(B, z)].
(7)
m=l
The E step refits qm for each document by running the inference step described
above. The M step optimizes Eq. (7) with respect to the model parameters a
and (3. For the multinomial parameters (3ij we have the following M step update
equation:
M
(3ij ex:
Iwml
l:= l:=
?>mniwtnn?
(8)
m=l n=l
The Dirichlet parameters ai are not independent of each other and we apply
Newton-Raphson to optimize them:
The variational EM algorithm alternates between maximizing Eq. (7) with respect
to qm and with respect to (a, (3) until convergence.
4
Experiments and Examples
We first tested LDA on two text corpora. 1 The first was drawn from the TREC AP
corpus, and consisted of 2500 news articles, with a vocabulary size of IVI = 37,871
words. The second was the CRAN corpus, consisting of 1400 technical abstracts,
with IVI = 7747 words.
We begin with an example showing how LDA can capture multiple-topic phenomena
in documents. By examining the (variational) posterior distribution on the topic
mixture q(B; ')'), we can identify the topics which were most likely to have contributed
to many words in a given document; specifically, these are the topics i with the
largest ')'i. Examining the most likely words in the corresponding multinomials can
then further tell us what these topics might be about. The following is an article
from the TREC collection.
The William R andolph Hearst Foundation will give $1.25 million to Lincoln Center,
Metropolitan Opera Co., New York Philharmonic and Juilliard School.
"Our board felt that we had a real opportunity to make a mark on the future of the
performing arts with these grants an act every bit as important as our traditional areas of support in health , medical research, education and the social services," Hearst
Foundation President Randolph A. Hearst said Monday in announcing the grants.
Lincoln Center's share will be $200,000 for its new building, which will house young
artists and provide new public facilities. The Metropolitan Opera Co. and New York
Philharmonic will receive $400 ,000 each. The Juilliard School, where music and the
performing arts are taught, will get $250,000 .
The Hearst Foundation, a leading supporter of the Lincoln Center Consolidated Corporate Fund, will make its usual annual $100,000 donation, too.
Figure 3 shows the Dirichlet parameters of the corresponding variational distribution for those topics where ')'i > 1 (k = 100) , and also lists the top 15 words (in
iTo enable repeated large scale comparison of various models on large corpora, we
implemented our variational inference algorithm on a parallel computing cluster. The
(bottleneck) E step is distributed across nodes so that the qm for different documents are
calculated in parallel.
I"
Topic 1
Topic 2
Topic 3
Topic 4
Topic 5
SCHOOL
SAID
STUDENTS
BOARD
SCHOOLS
STUDENT
TEACHER
POLICE
PROGRAM
TEACHERS
MEMBERS
YEAROLD
GANG
DEPARTMENT
MILLION
YEAR
SAID
SALES
BILLION
TOTAL
SHARE
EARNINGS
PROFIT
QUARTER
ORDERS
LAST
DEC
REVENUE
SAID
AIDS
HEALTH
DISEASE
VIRUS
CHILDREN
BLOOD
PATIENTS
TREATMENT
STUDY
IMMUNE
CANCER
PEOPLE
PERCENT
SAID
NEW
PRESIDENT
CHIEF
CHAIRMAN
EXECUTIVE
VICE
YEARS
COMPANY
YORK
SCHOOL
TWO
TODAY
COLUMBIA
SAID
NEW
MUSIC
YEAR
THEATER
MUSICAL
BAND
PLAY
WON
TWO
AVAILABLE
AWARD
OPERA
BEST
Figure 3: The Dirichlet parameters where Ii > 1 (k = 100), and the top 15 words
from the corresponding topics, for the document discussed in the text.
__ LDA
-x- pLSI
..
~
pLSI(00 lemper)
MIx1Un;grams
v ? ram
woo '
.~
4500
',.. _
.l\
!
-><---------------
k (number of topics)
k (number of topiCS)
Figure 4: Perplexity results on the CRAN and AP corpora for LDA, pLSI, mixture
of unigrams, and t he unigram model.
order) from these topics. This document is mostly a combination of words about
school policy (topic 4) and music (topic 5). The less prominent topics reflect other
words about education (topic 1) , finance (topic 2), and health (topic 3).
4.1
Formal evaluation: Perplexity
To compare the generalization performance of LDA with other models, we computed the perplexity of a test set for the AP and CRAN corpora. The perplexity, used by convention in language modeling, is monotonically decreasing in the
likelihood of the test data, and can be thought of as the inverse of the per-word
likelihood. More formally, for a test set of M documents, perplexity(Vtest ) =
exp (-l:m logp(wm)/ l:m Iwml}.
We compared LDA to both the mixture of unigrams and pLSI described in Section 2.2. We trained the pLSI model with and without tempering to reduce overfitting. When tempering, we used part of the test set as the hold-out data, thereby
giving it a slight unfair advantage. As mentioned previously, pLSI does not readily
generate or assign probabilities to previously unseen documents; in our experiments,
we assigned probability to a new document d by marginalizing out the dummy training set indices 2 : pew ) = l: d( rr : =1l:z p(w n lz)p(z ld))p(d) .
2 A second natural method, marginalizing out d and z to form a unigram model using
the resulting p(w)'s, did not perform well (its performance was similar to the standard
unigram model).
1-:- ~Dc.~UrUg,ams I
W'
? M"
~
x
NaiveBaes
k (number of topics)
k (number of topics)
Figure 5: Results for classification (left) and collaborative filtering (right)
Figure 4 shows the perplexity for each model and both corpora for different values
of k. The latent variable models generally do better than the simple unigram model.
The pLSI model severely overfits when not tempered (the values beyond k = 10
are off the graph) but manages to outperform mixture of unigrams when tempered.
LDA consistently does better than the other models. To our knowledge, these are
by far the best text perplexity results obtained by a bag-of-words model.
4.2
Classification
We also tested LDA on a text classification task. For each class c, we learn a separate
model p(wlc) of the documents in that class. An unseen document is classified by
picking argmaxcp(Clw) = argmaxcp(wlc)p(c). Note that using a simple unigram
distribution for p(wlc) recovers the traditional naive Bayes classification model.
Using the same (standard) subset of the WebKB dataset as used in [6], we obtained
classification error rates illustrated in Figure 5 (left). In all cases, the difference
between LDA and the other algorithms' performance is statistically significant (p <
0.05).
4.3
Collaborative filtering
Our final experiment utilized the EachMovie collaborative filtering dataset. In this
dataset a collection of users indicates their preferred movie choices. A user and
the movies he chose are analogous to a document and the words in the document
(respectively) .
The collaborative filtering task is as follows. We train the model on a fully observed set of users. Then, for each test user, we are shown all but one of the
movies that she liked and are asked to predict what the held-out movie is. The
different algorithms are evaluated according to the likelihood they assign to the
held-out movie. More precisely define the predictive perplexity on M test users
to be exp( - ~~=llogP(WmNd lwml' ... ,Wm(Nd-l))/M) . With 5000 training users,
3500 testing users, and a vocabulary of 1600 movies, we find predictive perplexities
illustrated in Figure 5 (right).
5
Conclusions
We have presented a generative probabilistic framework for modeling the topical
structure of documents and other collections of discrete data. Topics are represented
explicitly via a multinomial variable Zn that is repeatedly selected, once for each
word, in a given document. In this sense, the model generates an allocation of
the words in a document to topics. When computing the probability of a new
document, this unknown allocation induces a mixture distribution across the words
in the vocabulary. There is a many-to-many relationship between topics and words
as well as a many-to-many relationship between documents and topics.
While Dirichlet distributions are often used as conjugate priors for multinomials in
Bayesian modeling, it is preferable to instead think of the Dirichlet in our model as
a component of the likelihood. The Dirichlet random variable e is a latent variable
that gives generative probabilistic semantics to the notion of a "document" in the
sense that it allows us to put a distribution on the space of possible documents.
The words that are actually obtained are viewed as a continuous mixture over this
space, as well as being a discrete mixture over topics. 3
The generative nature of LDA makes it easy to use as a module in more complex
architectures and to extend it in various directions. We have already seen that
collections of LDA can be used in a classification setting. If the classification variable
is treated as a latent variable we obtain a mixture of LDA models, a useful model for
situations in which documents cluster not only according to their topic overlap, but
along other dimensions as well. Another extension arises from generalizing LDA to
consider Dirichlet/multinomial mixtures of bigram or trigram models, rather than
the simple unigram models that we have considered here. Finally, we can readily
fuse LDA models which have different vocabularies (e.g., words and images); these
models interact via a common abstract topic variable and can elegantly use both
vocabularies in determining the topic mixture of a given document.
Acknowledgments
A. Ng is supported by a Microsoft Research fellowship. This work was also supported by a grant from Intel Corporation, NSF grant IIS-9988642, and ONR MURI
N00014-00-1-0637.
References
[1] D. Cohn and T. Hofmann. The missing link- A probabilistic model of document
content and hypertext connectivity. In Advances in Neural Information Processing
Systems 13, 2001.
[2] P.J. Green and S. Richardson. Modelling heterogeneity with and without the Dirichlet
process. Technical Report, University of Bristol, 1998.
[3] T. Hofmann. Probabilistic latent semantic indexing. Proceedings of th e Twenty-Second
Annual International SIGIR Conference, 1999.
[4] T. J. Jiang, J. B. Kadane, and J. M. Dickey. Computation of Carlson's multiple
hypergeometric functions r for Bayesian applications. Journal of Computational and
Graphical Statistics, 1:231- 251 , 1992.
[5] M. I. Jordan , Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. Introduction to variational methods for graphical models. Machine Learning, 37:183- 233, 1999.
[6] K. Nigam, A. Mccallum, S. Thrun, and T. Mitchell. Text classification from labeled
and unlabeled documents using EM. Machine Learning, 39(2/3):103- 134, 2000.
[7] A. Popescul, L. H. Ungar, D. M. Pennock, and S. Lawrence. Probabilistic models for
unified collaborative and content-based recommendation in sparse-data environments.
In Uncertainty in Artificial Intelligence, Proceedings of the Seventeenth Conference,
2001.
3These remarks also distinguish our model from the Bayesian Dirichlet/Multinomial
allocation model (DMA)of [2], which is a finite alternative to the Dirichlet process . The
DMA places a mixture of Dirichlet priors on p(wl z ) and sets O i = 00 for all i .
| 2070 |@word version:1 bigram:1 proportion:1 nd:2 innermost:1 profit:1 thereby:1 ld:1 reduction:2 document:58 current:1 virus:1 written:1 readily:2 numerical:1 hofmann:5 designed:1 update:1 fund:1 generative:9 selected:1 intelligence:1 plane:1 mccallum:1 ith:1 randolph:1 blei:1 provides:1 node:3 location:1 idi:1 monday:1 earnings:1 along:1 indeed:1 decreasing:1 company:1 begin:2 webkb:1 moreover:1 underlying:1 notation:1 factorized:1 what:3 interpreted:1 consolidated:1 unified:1 corporation:1 berkeley:2 every:2 act:1 growth:1 finance:1 exactly:1 preferable:1 qm:3 sale:1 grant:4 medical:1 service:1 understood:1 severely:1 ak:2 jiang:1 ware:1 ap:3 might:1 chose:1 suggests:1 co:2 bi:4 statistically:1 seventeenth:1 acknowledgment:1 testing:1 practice:1 area:1 empirical:1 mult:2 thought:2 word:36 get:1 unlabeled:1 put:1 context:2 optimize:1 deterministic:1 center:3 maximizing:2 missing:1 straightforward:1 attention:1 sigir:1 assigns:1 theater:1 notion:1 coordinate:1 analogous:2 qq:1 president:2 controlling:1 today:1 play:1 user:7 expensive:1 utilized:2 muri:1 logq:1 labeled:1 ep:1 observed:1 module:1 capture:4 hypertext:1 news:1 ordering:1 disease:1 mentioned:1 environment:1 asked:1 trained:2 predictive:2 serve:1 triangle:3 represented:2 various:2 llogp:1 train:1 fast:1 describe:1 artificial:1 tell:1 emerged:1 heuristic:2 valued:1 statistic:1 unseen:5 richardson:1 think:1 final:1 sequence:1 differentiable:1 advantage:1 rr:1 propose:1 iff:2 poorly:1 lincoln:3 description:1 billion:1 convergence:1 cluster:2 liked:1 converges:1 donation:1 andrew:1 pose:1 ij:2 school:6 received:1 eq:6 implemented:1 come:1 convention:1 direction:1 posit:3 announcing:1 enable:1 public:1 education:2 require:1 wlc:3 assign:3 ungar:1 generalization:1 extension:1 hold:1 considered:1 exp:2 lawrence:1 predict:1 trigram:1 estimation:1 applicable:1 bag:2 label:2 largest:1 wl:3 vice:1 metropolitan:2 zib:2 rather:3 varying:1 jaakkola:1 focus:1 she:1 consistently:1 modelling:1 likelihood:9 indicates:1 centroid:1 sense:3 am:1 inference:7 typically:1 hidden:2 semantics:3 classification:9 development:1 art:2 once:3 having:2 ng:2 sampling:2 represents:2 future:1 simplex:3 report:1 ilb:1 simplify:2 randomly:1 consisting:1 william:1 microsoft:1 possibility:3 evaluation:1 replicates:1 mixture:27 held:2 dickey:1 edge:1 integral:2 indexed:1 modeling:8 zn:10 logp:7 vertex:1 subset:1 uniform:1 successful:1 examining:3 too:1 kadane:1 teacher:2 density:1 international:1 ie:1 probabilistic:9 off:1 picking:1 kdimensional:1 michael:1 continuously:1 w1:1 connectivity:1 reflect:1 choose:1 derivative:2 leading:1 student:2 notable:1 explicitly:2 unigrams:6 overfits:1 wm:3 bayes:2 parallel:2 collaborative:7 contribution:1 opera:3 ni:1 musical:1 efficiently:1 identify:1 bayesian:3 artist:1 manages:1 iid:1 researcher:1 bristol:1 classified:1 recovers:1 sampled:8 dataset:3 treatment:1 mitchell:1 knowledge:1 improves:1 dimensionality:1 routine:1 actually:1 evaluated:1 though:2 box:1 furthermore:1 until:2 cran:3 cohn:1 lda:27 building:1 contain:2 true:1 wlz:4 consisted:1 facility:1 assigned:1 semantic:3 illustrated:2 conditionally:1 won:1 prominent:1 plate:4 performs:1 percent:1 image:1 variational:14 common:1 multinomial:19 quarter:1 million:2 extend:1 interpretation:2 discussed:1 relating:1 numerically:1 he:2 slight:1 significant:1 ai:2 pew:1 language:1 had:1 immune:1 vtest:1 surface:1 posterior:2 plsi:15 recent:1 optimizes:1 perplexity:9 n00014:1 onr:1 wib:4 tempered:2 seen:3 maximize:1 eqm:2 monotonically:1 ii:2 full:3 multiple:4 corporate:1 eachmovie:1 smooth:1 technical:2 raphson:1 award:1 a1:2 j3:3 patient:1 sometimes:1 dec:1 receive:1 fellowship:1 crucial:1 unlike:1 ivi:4 ascent:1 pennock:1 db:2 member:1 jordan:2 easy:1 wn:6 zi:1 architecture:1 inner:1 reduce:1 computable:1 supporter:1 bottleneck:1 pca:1 york:3 repeatedly:2 remark:1 generally:1 useful:1 clear:2 involve:1 band:1 induces:1 generate:1 specifies:1 outperform:1 nsf:1 dummy:2 per:1 broadly:1 discrete:8 express:1 taught:1 four:3 blood:1 tempering:4 drawn:2 prevent:1 clean:1 utilize:1 ram:1 graph:1 fuse:1 clw:1 year:4 inverse:1 parameterized:3 fourth:1 uncertainty:1 place:1 throughout:1 bit:1 bound:5 distinguish:1 annual:2 gang:1 precisely:2 felt:1 generates:1 aspect:2 performing:2 department:1 according:3 alternate:2 combination:1 conjugate:1 across:2 em:3 wi:2 intuitively:1 indexing:3 equation:2 previously:4 count:1 needed:1 generalizes:1 available:1 apply:1 alternative:1 top:3 clustering:1 dirichlet:25 running:2 graphical:4 opportunity:1 newton:1 music:3 carlson:1 giving:1 ghahramani:1 objective:1 question:1 already:1 chairman:1 usual:1 traditional:3 said:6 link:2 separate:2 thrun:1 entity:1 outer:1 topic:55 reason:1 modeled:1 index:2 relationship:2 izn:1 mostly:1 popescul:1 refit:1 policy:1 contributed:1 perform:1 unknown:1 twenty:1 fin:1 finite:1 defining:1 situation:1 heterogeneity:1 trec:2 dc:1 topical:1 hearst:4 police:1 david:1 california:1 hypergeometric:2 zld:3 beyond:1 program:1 including:1 green:1 overlap:1 treated:3 natural:2 representing:1 movie:6 carried:1 woo:1 naive:2 health:3 columbia:1 text:14 prior:3 marginalizing:2 determining:1 embedded:1 fully:3 allocation:5 filtering:6 revenue:1 foundation:3 executive:1 degree:1 article:2 share:2 cancer:1 supported:2 last:1 free:2 jth:1 infeasible:1 formal:1 saul:1 midpoint:1 sparse:1 distributed:2 calculated:1 vocabulary:7 gram:1 dimension:1 ignores:1 collection:10 made:1 lz:1 far:1 social:1 approximate:1 preferred:1 overfitting:3 corpus:7 continuous:3 latent:12 chief:1 learn:1 nature:1 ca:1 expanding:1 nigam:1 interact:1 complex:1 elegantly:1 did:1 whole:1 dma:2 repeated:2 child:1 referred:1 intel:1 board:2 aid:1 fails:1 position:1 lie:1 house:1 unfair:1 ib:2 third:1 ito:1 learns:1 young:1 unigram:11 specific:1 showing:1 list:2 conditioned:1 illustrates:1 entropy:1 generalizing:1 simply:1 likely:3 subtlety:2 recommendation:1 corresponds:1 marked:1 viewed:1 content:2 infinite:1 except:1 specifically:1 total:1 formally:1 mark:1 support:1 people:1 arises:1 tested:2 phenomenon:1 ex:1 |
1,175 | 2,071 | Distribution of Mutual Information
Marcus Hutter
IDSIA, Galleria 2, CH-6928 Manno-Lugano, Switzerland
marcus@idsia.ch
http://www.idsia.ch/- marcus
Abstract
The mutual information of two random variables z and J with joint
probabilities {7rij} is commonly used in learning Bayesian nets as
well as in many other fields. The chances 7rij are usually estimated
by the empirical sampling frequency nij In leading to a point estimate J(nij In) for the mutual information. To answer questions
like "is J (nij In) consistent with zero?" or "what is the probability
that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian
framework one can answer these questions by utilizing a (second
order) prior distribution p( 7r) comprising prior information about
7r. From the prior p(7r) one can compute the posterior p(7rln), from
which the distribution p(Iln) of the mutual information can be calculated. We derive reliable and quickly computable approximations
for p(Iln). We concentrate on the mean, variance, skewness, and
kurtosis , and non-informative priors. For the mean we also give an
exact expression. Numerical issues and the range of validity are
discussed.
1
Introduction
The mutual information J (also called cross entropy) is a widely used information
theoretic measure for the stochastic dependency of random variables [CT91, SooOO] .
It is used, for instance, in learning Bayesian nets [Bun96, Hec98] , where stochastically dependent nodes shall be connected. The mutual information defined in (1)
can be computed if the joint probabilities {7rij} of the two random variables z and J
are known. The standard procedure in the common case of unknown chances 7rij is
to use the sample frequency estimates n~; instead, as if they were precisely known
probabilities; but this is not always appropriate. Furthermore, the point estimate
J (n~; ) gives no clue about the reliability of the value if the sample size n is finite.
For instance, for independent z and J, J(7r) =0 but J(n~;) = O(n- 1 / 2 ) due to noise
in the data. The criterion for judging dependency is how many standard deviations
J(":,;) is away from zero. In [KJ96, Kle99] the probability that the true J(7r) is
greater than a given threshold has been used to construct Bayesian nets. In the
Bayesian framework one can answer these questions by utilizing a (second order)
prior distribution p(7r),which takes account of any impreciseness about 7r. From the
prior p(7r) one can compute the posterior p(7rln), from which the distribution p(Iln)
of the mutual information can be obtained.
The objective of this work is to derive reliable and quickly computable analytical
expressions for p(1ln). Section 2 introduces the mutual information distribution,
Section 3 discusses some results in advance before delving into the derivation. Since
the central limit theorem ensures that p(1ln) converges to a Gaussian distribution
a good starting point is to compute the mean and variance of p(1ln). In section
4 we relate the mean and variance to the covariance structure of p(7rln). Most
non-informative priors lead to a Dirichlet posterior. An exact expression for the
mean (Section 6) and approximate expressions for t he variance (Sections 5) are
given for the Dirichlet distribution. More accurate estimates of the variance and
higher central moments are derived in Section 7, which lead to good approximations
of p(1ln) even for small sample sizes. We show that the expressions obtained in
[KJ96, Kle99] by heuristic numerical methods are incorrect. Numerical issues and
the range of validity are briefly discussed in section 8.
2
Mutual Information Distribution
We consider discrete random variables Z E {l, ... ,r} and J E {l, ... ,s} and an i.i.d.
random process with samples (i,j) E {l ,... ,r} x {l, ... ,s} drawn with joint probability
7rij. An important measure of the stochastic dependence of z and J is the mutual
information
T
1(7r) =
S
7rij
L L 7rij log ~
= L 7rij log 7rij - L 7ri+ log7ri+ - L 7r +j log7r
i=1 j = 1
H
+J
ij
i
+j'
(1)
j
log denotes the natural logarithm and 7ri+ = Lj7rij and 7r +j = L i 7rij are marginal
probabilities. Often one does not know the probabilities 7rij exactly, but one has a
sample set with nij outcomes of pair (i,j). The frequency irij := n~j may be used as
a first estimate of the unknown probabilities. n:= L ijnij is the total sample size.
This leads to a point (frequency) estimate 1(ir) = Lij n~j logn:~:j for the mutual
informat ion (per sample).
Unfortunately the point estimation 1(ir) gives no information about its accuracy.
In the Bayesian approach to this problem one assumes a prior (second order) probability density p( 7r) for the unknown probabilities 7rij on the probability simplex.
From this one can compute the posterior distribution p( 7rln) cxp( 7r) rr ij7r~;j (the nij
are multinomially distributed). This allows to compute the posterior probability
density of the mutual information.1
p(Iln) =
2The
80
f
8(1(7r) - I)p(7rln)d TS 7r
(2)
distribution restricts the integral to 7r for which 1(7r) =1. For large sam-
1 I(7r) denotes the mutual information for the specific chances 7r, whereas I in the context
above is just some non-negative real number. I will also denote the mutual information
random variable in the expectation E [I] and variance Var[I]. Expectaions are always w.r.t.
to the posterior distribution p(7rln).
2Since O~I(7r) ~Imax with sharp upper bound Imax :=min{logr,logs}, the integral may
be restricted to
which shows that the domain of p(Iln) is [O,Imax] .
J:mam,
pIe size n ---+ 00, p(7rln) is strongly peaked around 7r = it and p(Iln) gets strongly
peaked around the frequency estimate I = I(it). The mean E[I] = fooo Ip(Iln) dI =
f I(7r)p(7rln)dTs 7r and the variance Var[I] =E[(I - E[I])2] = E[I2]- E[Ij2 are of central interest.
3
Results for I under the Dirichlet P (oste )rior
Most 3 non-informative priors for p(7r) lead to a Dirichlet posterior distribution
nij -1
h
p (7r I)
n ex: IT ij 7rij
WI?th?IIIt erpre t a t?IOn nij - n,
+ n,ij , were
n 'ij are th e numb er
ij ,
of samples (i,j), and n~j comprises prior information (1 for the uniform prior, ~ for
Jeffreys' prior, 0 for Haldane's prior, -?:s for Perks' prior [GCSR95]). In principle
this allows to compute the posterior density p(Iln) of the mutual information. In
sections 4 and 5 we expand the mean and variance in terms of n- 1 :
E[I]
~ nij I
nijn
L...J og - - .. n
ni+n+j
'J
+
(r - 1)(8 - 1)
2n
+
O( -2)
n
,
(3)
Var[I]
The first term for the mean is just the point estimate I(it). The second term is
a small correction if n ? r? 8. Kleiter [KJ96, Kle99] determined the correction by
Monte Carlo studies as min {T2~1 , 8;;;,1 }. This is wrong unless 8 or rare 2. The
expression 2E[I]/n they determined for the variance has a completely different
structure than ours. Note that the mean is lower bounded by co~st. +O(n- 2 ), which
is strictly positive for large, but finite sample sizes, even if z and J are statistically
independent and independence is perfectly represented in the data (I (it) = 0). On
the other hand, in this case, the standard deviation u= y'Var(I) '" ~ ",E[I] correctly
indicates that the mean is still consistent with zero.
Our approximations (3) for the mean and variance are good if T~8 is small. The
central limit theorem ensures that p(Iln) converges to a Gaussian distribution with
mean E[I] and variance Var[I]. Since I is non-negative it is more appropriate to
approximate p(II7r) as a Gamma (= scaled X2 ) or log-normal distribution with mean
E[I] and variance Var[I], which is of course also asymptotically correct.
A systematic expansion in n -1 of the mean, variance, and higher moments is possible
but gets arbitrarily cumbersome. The O(n - 2) terms for the variance and leading
order terms for the skewness and kurtosis are given in Section 7. For the mean it
is possible to give an exact expression
1
E[I] = -
L nij[1jJ(nij + 1) -1jJ(ni+ + 1) -1jJ(n+j + 1) + 1jJ(n + 1)]
n ..
(4)
'J
with 1jJ(n+1)=-,),+L~= lt=logn+O(~) for integer n. See Section 6 for details
and more general expressions for 1jJ for non-integer arguments.
There may be other prior information available which cannot be comprised in a
Dirichlet distribution. In this general case, the mean and variance of I can still be
3But not all priors which one can argue to be non-informative lead to Dirichlet posteriors. Brand [Bragg] (and others), for instance, advocate the entropic prior p( 7r) ex e-H(rr).
related to the covariance structure of p(7fln), which will be done in the following
Section.
4
Approximation of Expectation and Variance of I
In the following let fr ij := E[7fij]. Since p( 7fln) is strongly peaked around 7f = fr for
large n we may expand J(7f) around fr in the integrals for the mean and the variance.
With I:::..ij :=7fij -frij and using L: ij7fij = 1 = L:ijfrij we get for the expansion of (1)
fr .. )
1:::.. 2 .
1:::.. 2
1:::.. 2 .
J(7f) = J(fr) + 2)og ( ~ I:::..ij +
----}J-~~+O(1:::..3). (5)
..
7fi+7f+j
.. 27fij
. 27fi+
. 27f+j
2J
2J
2
J
L
L
L
Taking the expectation, the linear term E[ I:::.. ij ] = a drops out. The quadratic terms
E[ I:::..ij I:::..kd = Cov( 7fij ,7fkl) are the covariance of 7f under distribution p(7fln) and are
proportional to n- 1 . It can be shown that E[1:::..3] ,,-,n- 2 (see Section 7).
1 " , (bikbjl
bik - -Abjl) COV7fij,7fkl
(
) +On
(-2) .
( A) +-~
[ ] = J7f
EJ
- A - - -A2 ijkl
7fij
7fi+
7f+j
The Kronecker delta bij is 1 for i = j and
order in n - 1 is
a otherwise.
(6)
The variance of J in leading
(7)
where :t means = up to terms of order n -2. So the leading order variance and
the leading and next to leading order mean of the mutual information J(7f) can be
expressed in terms of the covariance of 7f under the posterior distribution p(7fln).
5
The Second Order Dirichlet Distribution
Noninformative priors for p(7f) are commonly used if no additional prior information
is available. Many non-informative choices (uniform, Jeffreys' , Haldane's, Perks',
prior) lead to a Dirichlet posterior distribution:
II
1
n;j - 1 (
N(n)
.. 7fij
b 7f++ - 1)
with normalization
2J
N(n)
(8)
where r is the Gamma function, and nij = n~j + n~j, where n~j are the number of
samples (i,j), and n~j comprises prior information (1 for the uniform prior, ~ for
Jeffreys' prior, a for Haldane's prior, -!s for Perks' prior) . Mean and covariance of
p(7fln) are
[ ] =nij
7fij:= E7fij
-,
A
n
(9)
Inserting this into (6) and (7) we get after some algebra for the mean and variance
of the mutual information I(7r) up to terms of order n- 2 :
E[I]
J
+
(r - 1)(8 - 1)
2(n + 1)
~1 (K - J2) +
Var[I]
n+
+
O( -2)
n
,
(10)
(11)
0(n-2),
(12)
(13)
J and K (and L, M, P, Q defined later) depend on 7rij = ":,j only, i.e. are 0(1)
in n. Strictly speaking we should expand n~l = ~+0(n-2), i.e. drop the +1, but
the exact expression (9) for the covariance suggests to keep the +1. We compared
both versions with the exact values (from Monte-Carlo simulations) for various
parameters 7r. In most cases the expansion in n~l was more accurate, so we suggest
to use this variant.
6
Exact Value for E[I]
It is possible to get an exact expression for the mean mutual information E[I] under
the Dirichlet distribution. By noting that xlogx= d~x,6I,6= l' (x = {7rij,7ri+ ,7r+j}),
one can replace the logarithms in the last expression of (1) by powers. From (8) we
see that E[ (7rij ),6] = ~i~:~ t~~~;l. Taking the derivative and setting ,8 = 1 we get
E[7rij log 7rij]
d
1
= d,8E[(7rij) ,6],6=l = ;;: 2:::: nij[1j!(nij + 1) -1j!(n + 1)].
"J
The 1j! function has the following properties (see [AS74] for details)
1j!(z)
=
dlogf(z)
dz
=
n- l
1j!(n) = -"( +
L
f'(z)
f(z)'
1
k'
1j!(z + 1)
= log z +
1
1
1
2z - 12z2 + O( Z4)'
1j!(n +~) = -"( + 2log2 + 2
k=l
n
L
1
2k _ l'
(14)
k=l
The value of the Euler constant "( is irrelevant here, since it cancels out. Since the
marginal distributions of 7ri+ and 7r+j are also Dirichlet (with parameters ni+ and
n+j) we get similarly
-n1 L. n+j[1j!(n+j + 1) -1j!(n + 1)].
J
Inserting this into (1) and rearranging terms we get the exact expression 4
E[I]
1
=-
L nij[1j! (nij + 1) -1j!(ni+ + 1) -1j!(n+j + 1) + 1j!(n + 1)]
n ..
4This expression has independently been derived in [WW93].
(15)
For large sample sizes, 'Ij;(z + 1) ~ logz and (15) approaches the frequency estimate
I(7r) as it should be. Inserting the expansion 'Ij;(z + 1) = logz + + ... into (15) we
also get the correction term (r - 11~s - 1) of (3).
2\
The presented method (with some refinements) may also be used to determine an
exact expression for the variance of I(7f). All but one term can be expressed in terms
of Gamma functions. The final result after differentiating w.r.t. (31 and (32 can be
represented in terms of 'Ij; and its derivative 'Ij;' . The mixed term E[( 7fi+ )131 (7f +j )132]
is more complicated and involves confluent hypergeometric functions, which limits
its practical use [WW93] .
7
Generalizations
A systematic expansion of all moments of p(Iln) to arbitrary order in n -1 is possible,
but gets soon quite cumbersome. For the mean we already gave an exact expression
(15), so we concentrate here on the variance, skewness and the kurtosis of p(Iln).
The 3rd and 4th central moments of 7f under the Dirichlet distribution are
(
)2(
) [27ra7rb7rc - 7ra7rbc5bc - 7rb7rcc5ca - 7rc7rac5ab
n+l n+2
+ 7rac5abc5bc]
(16)
~2 [37ra7rb7rc7rd -
jrc!!d!!a c5 ab - A7rbjrdA7rac5ac - A7rbA7rcA7rac5ad
-7fa7fd7fbc5bc - 7fa7fc7fbc5bd - 7fa7fb7fcc5cd
(17)
+7ra7rcc5abc5cd + 7ra7rbc5acc5bd + 7ra7rbc5adc5bc] + O(n- 3)
with a=ij, b= kl, ... E {1, ... ,r} x {1, ... ,8} being double indices, c5 ab =c5ik c5 jl ,... 7rij = n~j ?
Expanding D.. k = (7f_7r)k in E[D..aD..b ... ] leads to expressions containing E[7fa7fb ... ],
which can be computed by a case analysis of all combinations of equal/unequal
indices a,b,c, ... using (8). Many terms cancel leading to the above expressions. They
allow to compute the order n- 2 term of the variance of I(7f). Again, inspection of
(16) suggests to expand in [(n+l)(n+2)]-1, rather than in n- 2 . The variance in
leading and next to leading order is
Var[I]
M
K - J2 + M + (r - 1)(8 - 1)(~ - J) - Q + O(n - 3)
(n + l)(n + 2)
n+ 1
(18)
L
(19)
ij
(~-
_1_ _ _ 1_
nij
ni+
n+j
+~) nij log nijn ,
n
ni+n+j
2
l-L~?
Q
ij
ni+n+j
(20)
J and K are defined in (12) and (13). Note that the first term ~+f also contains
second order terms when expanded in n -1. The leading order terms for the 3rd and
4th central moments of p(Iln) are
L
.-
'""" nij
~-
j
I og--nij n
n
32 [K - J 2
n
ni+n+j
F+ O(n -
3 ),
from which the skewness and kurtosis can be obtained by dividing by Var[Ij3/2
and Var[IF respectively. One can see that the skewness is of order n- 1 / 2 and the
kurtosis is 3 + 0 (n - 1). Significant deviation of the skewness from a or the kurtosis
from 3 would indicate a non-Gaussian I. They can be used to get an improved
approximation for p(Iln) by making, for instance, an ansatz
and fitting the parameters b, c, jJ" and (j-2 to the mean, variance, skewness, and
kurtosis expressions above. Po is the Normal or Gamma distribution (or any other
distribution with Gaussian limit). From this, quantiles p(I>I*ln):= fI:'p(Iln) dI,
needed in [KJ96, Kle99], can be computed. A systematic expansion of arbitrarily
high moments to arbitrarily high order in n- 1 leads, in principle, to arbitrarily
accurate estimates.
8
Numerics
There are short and fast implementations of'if;. The code of the Gamma function in
[PFTV92], for instance, can be modified to compute the 'if; function. For integer and
half-integer values one may create a lookup table from (14) . The needed quantities
J, K, L, M, and Q (depending on n) involve a double sum, P only a single sum,
and the r+s quantities J i + and J+ j also only a single sum. Hence, the computation
time for the (central) moments is of the same order O(r?s) as for the point estimate
(1). "Exact" values have been obtained for representative choices of 7rij, r, s,
and n by Monte Carlo simulation. The 7rij := Xij / x++ are Dirichlet distributed,
if each Xij follows a Gamma distribution. See [PFTV92] how to sample from a
Gamma distribution. The variance has been expanded in T~S, so the relative error
Var [I]app" o.-Var[I] .. act of the approximation (11) and (18) are of the order of T'S and
Var[Il e? act
n
(T~S)2 respectively, if z and J are dependent. If they are independent the leading
term (11) drops itself down to order n -2 resulting in a reduced relative accuracy
O( T~S) of (18). Comparison with the Monte Carlo values confirmed an accurracy in
the range (T~S)1...2. The mean (4) is exact. Together with the skewness and kurtosis
we have a good description for the distribution of the mutual information p(Iln) for
not too small sample bin sizes nij' We want to conclude with some notes on useful
accuracy. The hypothetical prior sample sizes n~j = {a, -!S' ~,1} can all be argued to
be non-informative [GCSR95]. Since the central moments are expansions in n- 1 ,
the next to leading order term can be freely adjusted by adjusting n~j E [0 ... 1]. So
one may argue that anything beyond leading order is free to will, and the leading
order terms may be regarded as accurate as we can specify our prior knowledge.
On the other hand, exact expressions have the advantage of being safe against
cancellations. For instance, leading order of E [I] and E[I2] does not suffice to
compute the leading order of Var[I].
Acknowledgements
I want to thank Ivo Kwee for valuable discussions and Marco Zaffalon for encouraging me to investigate this topic. This work was supported by SNF grant 200061847.00 to Jiirgen Schmidhuber.
References
[AS74]
M. Abramowitz and 1. A. Stegun, editors. Handbook of mathematical functions.
Dover publications, inc., 1974.
[Bra99]
M. Brand. Structure learning in conditional probability models via an entropic
prior and parameter extinction. N eural Computation, 11(5):1155- 1182, 1999.
[Bun96]
W. Buntine. A guide to the literature on learning probabilistic networks from
data. IEEE Transactions on Knowledge and Data Engineering, 8:195- 210,
1996.
[CT91]
T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley Series
in Telecommunications. John Wiley & Sons, New York, NY, USA, 1991.
[GCSR95] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis.
Chapman, 1995.
Learnig in
[Hec98]
D. Heckerman. A tutorial on learning with Bayesian networks.
Graphical Models, pages 301-354, 1998.
[KJ96]
G. D. Kleiter and R. Jirousek. Learning Bayesian networks under the control
of mutual information. Proceedings of the 6th International Conference on
Information Processing and Management of Uncertainty in Knowledge-Based
Systems (IPMU-1996), pages 985- 990, 1996.
[Kle99]
G. D. Kleiter. The posterior probability of Bayes nets with strong dependences.
Soft Computing, 3:162- 173, 1999.
[PFTV92] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical
R ecipes in C: Th e Art of Scientific Computing. Cambridge University Press,
Cambridge, second edition, 1992.
[SooOO]
E. S. Soofi. Principal information theoretic approaches. Journal of the American Statistical Association, 95:1349- 1353, 2000.
[WW93]
D. R. Wolf and D. H. Wolpert. Estimating functions of distributions from A
finite set of samples, part 2: Bayes estimators for mutual information, chisquared, covariance and other statistics. Technical Report LANL-LA-UR-93833, Los Alamos National Laboratory, 1993. Also Santa Fe Insitute report
SFI-TR-93-07-047.
| 2071 |@word version:1 briefly:1 extinction:1 simulation:2 covariance:7 tr:1 moment:8 contains:1 series:1 ours:1 z2:1 john:1 numerical:4 informative:6 noninformative:1 drop:3 half:1 ivo:1 inspection:1 dover:1 short:1 node:1 mathematical:1 multinomially:1 incorrect:1 advocate:1 fitting:1 encouraging:1 estimating:1 bounded:1 suffice:1 what:1 skewness:8 hypothetical:1 act:2 exactly:1 wrong:1 scaled:1 control:1 grant:1 before:1 positive:1 engineering:1 limit:4 suggests:2 co:1 abramowitz:1 range:3 statistically:1 practical:1 procedure:1 logz:2 snf:1 empirical:1 suggest:1 get:11 cannot:1 gelman:1 context:1 www:1 dz:1 go:1 starting:1 independently:1 oste:1 estimator:1 utilizing:2 regarded:1 imax:3 exact:13 element:1 idsia:3 jirousek:1 rij:22 ensures:2 connected:1 jrc:1 valuable:1 ij2:1 depend:1 algebra:1 completely:1 manno:1 po:1 joint:3 represented:2 various:1 derivation:1 fast:1 insitute:1 monte:4 outcome:1 quite:1 heuristic:1 larger:1 widely:1 otherwise:1 cov:1 statistic:1 itself:1 ip:1 final:1 advantage:1 rr:2 net:4 kurtosis:8 analytical:1 rln:8 fr:5 inserting:3 j2:2 zaffalon:1 description:1 los:1 double:2 converges:2 derive:2 depending:1 ij:17 strong:1 dividing:1 involves:1 indicate:1 switzerland:1 concentrate:2 fij:7 safe:1 correct:1 stochastic:2 bin:1 argued:1 generalization:1 adjusted:1 strictly:2 correction:3 marco:1 around:4 normal:2 entropic:2 numb:1 a2:1 estimation:1 create:1 always:2 gaussian:4 modified:1 rather:1 ej:1 og:3 publication:1 derived:2 indicates:1 dependent:2 vetterling:1 expand:4 comprising:1 issue:2 logn:2 art:1 mutual:22 marginal:2 field:1 construct:1 haldane:3 equal:1 sampling:1 chapman:1 cancel:2 peaked:3 simplex:1 t2:1 others:1 report:2 gamma:7 national:1 n1:1 ab:2 interest:1 investigate:1 introduces:1 accurate:4 integral:3 unless:1 logarithm:2 nij:21 hutter:1 instance:6 soft:1 cover:1 deviation:3 rare:1 euler:1 uniform:3 comprised:1 alamo:1 too:1 buntine:1 dependency:2 answer:3 cxp:1 st:1 density:3 international:1 systematic:3 probabilistic:1 ansatz:1 together:1 quickly:2 again:1 central:8 management:1 containing:1 stochastically:1 american:1 derivative:2 leading:16 logr:1 account:1 lookup:1 inc:1 ad:1 later:1 bayes:2 complicated:1 il:1 ir:2 accuracy:3 ni:8 variance:26 bayesian:9 ijkl:1 carlo:4 confirmed:1 app:1 cumbersome:2 against:1 frequency:6 jiirgen:1 di:2 galleria:1 adjusting:1 knowledge:3 confluent:1 higher:2 specify:1 improved:1 done:1 strongly:3 furthermore:1 just:2 hand:2 scientific:1 usa:1 validity:2 fooo:1 true:2 hence:1 sfi:1 laboratory:1 i2:2 iln:15 anything:1 criterion:1 theoretic:2 fi:5 common:1 jl:1 discussed:2 he:1 association:1 significant:1 cambridge:2 rd:2 z4:1 similarly:1 cancellation:1 reliability:1 posterior:12 irrelevant:1 schmidhuber:1 arbitrarily:4 iiit:1 greater:1 additional:1 freely:1 determine:1 ii:1 technical:1 cross:1 variant:1 expectation:3 normalization:1 ion:2 whereas:1 want:2 bik:1 integer:4 noting:1 independence:1 carlin:1 gave:1 perfectly:1 computable:2 expression:19 speaking:1 york:1 jj:7 useful:1 santa:1 involve:1 reduced:1 http:1 xij:2 restricts:1 tutorial:1 judging:1 estimated:1 delta:1 correctly:1 per:1 discrete:1 shall:1 threshold:1 drawn:1 chisquared:1 asymptotically:1 sum:3 uncertainty:1 telecommunication:1 perk:3 informat:1 bound:1 quadratic:1 precisely:1 kronecker:1 ri:4 x2:1 argument:1 min:2 expanded:2 combination:1 kd:1 heckerman:1 son:1 sam:1 ur:1 wi:1 making:1 jeffreys:3 restricted:1 ln:5 discus:1 needed:2 know:1 available:2 away:1 appropriate:2 thomas:1 denotes:2 dirichlet:12 assumes:1 mam:1 graphical:1 log2:1 objective:1 question:3 already:1 quantity:2 dependence:2 thank:1 me:1 topic:1 argue:2 marcus:3 code:1 index:2 pie:1 unfortunately:1 fe:1 relate:1 negative:2 numerics:1 implementation:1 stern:1 unknown:3 upper:1 finite:3 t:1 sharp:1 arbitrary:1 pair:1 kl:1 lanl:1 unequal:1 hypergeometric:1 dts:1 beyond:2 usually:1 reliable:2 power:1 natural:1 lij:1 prior:28 literature:1 acknowledgement:1 relative:2 mixed:1 proportional:1 var:14 consistent:2 rubin:1 principle:2 editor:1 course:1 supported:1 last:1 soon:1 free:1 rior:1 allow:1 guide:1 taking:2 differentiating:1 distributed:2 calculated:1 stegun:1 commonly:2 clue:1 refinement:1 c5:3 transaction:1 approximate:2 keep:1 handbook:1 conclude:1 table:1 delving:1 rearranging:1 expanding:1 expansion:7 domain:1 noise:1 edition:1 eural:1 representative:1 quantiles:1 ny:1 wiley:2 ct91:2 comprises:2 lugano:1 bij:1 theorem:2 down:1 specific:1 er:1 entropy:1 flannery:1 lt:1 wolpert:1 expressed:2 ch:3 ipmu:1 wolf:1 chance:3 teukolsky:1 conditional:1 replace:1 fkl:2 determined:2 principal:1 called:1 total:1 la:1 brand:2 ex:2 |
1,176 | 2,072 | Sampling Techniques for Kernel Methods
Dimitris Achlioptas
Microsoft Research
optas@microsoft.com
Frank McSherry
University of Washington
mcsherry@cs.washington.edu
Bernhard Sch?olkopf
Biowulf Technologies NY
bs@conclu.de
Abstract
We propose randomized techniques for speeding up Kernel Principal
Component Analysis on three levels: sampling and quantization of the
Gram matrix in training, randomized rounding in evaluating the kernel
expansions, and random projections in evaluating the kernel itself. In all
three cases, we give sharp bounds on the accuracy of the obtained approximations. Rather intriguingly, all three techniques can be viewed as
instantiations of the following idea: replace the kernel function by a
?randomized kernel? which behaves like in expectation.
1 Introduction
Given a collection of training data
, techniques such as linear SVMs [13] and
PCA extract features from by computing linear functions of this data. However, it is
often the case that the structure present in the training data is not simply a linear function
of the data representation. Worse, many data sets do not readily support linear operations
such as addition and scalar multiplication (text, for example).
In a ?kernel method? is first mapped into some dot product space using .
The dimension of can be very large, even infinite, and therefore it may not be practical
(or possible) to work with the mapped data explicitly. Nonetheless, in many cases the dot
products
! " can be evaluated efficiently using a positive definite kernel for
, ?.e. a function so that #$ % '&()*+ ! " .
Any algorithm whose operations can be expressed in terms of dot products can be generalized to an algorithm which operates on , , simply by presenting the Gram matrix
-
. /&
$
as the input covariance matrix. Note that at no point is the function explicitly computed;
the kernel implicitly performs the dot product calculations between mapped points.
While this ?kernel trick? has- been extremely successful, a problem common to all kernel
is a dense matrix, making the input size scale as 021 . For
methods is that, in general,
example, in Kernel PCA such a matrix has to be diagonalized, while in SVMs a quadratic
program of size 0 1 must be solved. As the size of training sets in practical applications
increases, the growth of the input size rapidly poses severe computational limitations.
Various methods have been proposed to deal with this issue, such as decomposition methods for SVM training (e.g., [10]), speedup methods for Kernel PCA [12], and other kernel
methods [2, 14]. Our research is motivated by the need for such speedups that are also
accompanied by strong, provable performance guarantees.
In this paper we give three such speedups for Kernel PCA. We start by simplifying the Gram
matrix via a novel matrix sampling/quantization scheme, motivated by spectral properties
of random matrices. We then move on to speeding up classification, by using randomized
rounding in evaluating kernel expansions. Finally, we consider the evaluation of kernel
functions themselves and show how many popular kernels can be approximated efficiently.
Our first technique relates matrix simplification to the stability of invariant subspaces. The
other two are, in fact, completely general and apply to all kernel methods. What is more,
our techniques suggest the notion of randomized kernels, whereby each evaluation of the
kernel is replaced by an evaluation of a randomized function (on the same input pair).
The idea is to use a function which for every input pair behaves like in expectation (over
its internal coin-flips), yet confers significant computational benefits compared to using .
In fact, each one of our three techniques can be readily cast as an appropriate randomized
kernel, with no other intervention.
2 Kernel PCA
Given 0 training
points recall that is an 0 0 matrix with . & . For some
method
choice
[11] computes the largest eigenvalues,
of
0 , the Kernel PCA (KPCA)
, and eigenvectors, of . Then, given an input point , the method
computes the value of nonlinear feature extraction functions
&
1
$# 2
There are several methods for computing the principal components of a symmetric matrix.
The choice depends on the properties of the matrix and on how many components one is
seeking. In particular, if relatively few principal components are required, as is the case in
KPCA, Orthogonal Iteration is a commonly used method.1
Orthogonal Iteration
matrix with orthonormal columns.
1. Let be a random 0
2. While not converged, do
(a)
(b) Orthonormalize
3. Return
It is worth looking closely at the complexity of performing Orthogonal Iteration on a matrix
the computational bottleneck. The
. Step 1 can be done in 0 steps, making
stepis 2overwhelmed
orthonormalization step 2b takes time 0 1 and
by the cost of comput
ing in step 2a which, generally, takes 0 1 . The number of iterations of the while
loop is a somewhat complicated issue, but one can prove that the ?error? in (with respect
to the true principal components) decreases exponentially with the number of iterations.
All in all, the running time of Orthogonal Iteration scales linearly with the cost of the matrix multiplication . If is sparse, ?.e., if roughly one
out of every entries of is
non-zero, then the matrix multiplication costs 0 1 .
-
As mentioned earlier, the matrix used in Kernel PCA is almost never sparse. In the next
section, we will show how to sample and quantize the entries of , obtaining a matrix
which is sparser and whose entries have simpler data representation,
yet has essentially the
same spectral structure, i.e. eigenvalues/eigenvectors, as .
1
Our discussion applies equally well to Lanczos Iteration which, while often preferable, is a more
complicated method. Here we focus on Orthogonal Iteration to simplify exposition.
3 Sampling Gram Matrices
In this section we describe two general ?matrix simplification? techniques and discuss their
implications for Kernel PCA. In particular, under natural assumptions on the spectral structure of , we will prove that applying KPCA to the simplified
matrix yields subspaces
which are very close to those that KPCA would find in . As a result, when we project
vectors onto these spaces (as performed by the feature extractors) the results are provably
close to the original ones.
First, our sparsification process works by randomly omitting entries in
we let the matrix be described entrywise as
with probability
-
/ &
$
-
&
.
-
. Precisely stated,
, where
with probability - .
2
with probability
-
Second, - our quantization process rounds each entry in
to one of
. , thus reducing the representation of each entry to a single bit.
- .
with probability
-
/ &
$
&
&
Sparsification greatly accelerates the computation of eigenvectors by accelerating multiplication by . Moreover, both approaches greatly reduce the space required to store the
matrix (and they can be readily combined), allowing for much bigger training sets to fit
in main memory. Finally, we note that i) sampling also speeds up the construction of the
that remain in , while ii)
Gram matrix since we need only compute those values of
quantization allows us to replace exact kernel evaluations by coarse unbiased estimators,
which can be more efficient to compute.
While the two processes above are quite different, they share one important commonality:
&
in each case, . &
,
/ . Moreover, the entries of the error matrix,
are independent random variables, having expectation zero and bounded variance. Large
deviation extensions [5] of Wigner?s famous semi-circle law, imply that with very high
probability such matrices have small L2 norm (denoted by
throughout).
!
Theorem 1 (Furedi and Komlos [5]) Let
be an 0 0 symmetric matrix whose entries are independent random variables with mean 0, variance bounded above
, by 1 , and
magnitude bounded by
0
0 . With probability
1 0
"$# &% '()
1
2"#
0
+ * - , ./" 0
"
"
It is worth noting that this upper bound is within a constant factor of the lower bound on
the L2 norm of any matrix where the mean squared entry equals 1 . More precisely, it is
0 .
easy to show that every matrix with Frobenius norm #0 $1 has L2 norm at least
Therefore, we see that the L2 error introduced
by is within a factor of 4 of the L2 error
associated with any modification to that has the same entrywise mean squared error.
3"
"4#
We will analyze three different cases of spectral stability, corresponding to progressively
stronger assumptions. At the heart of these results is the stability of invariant subspaces
in the presence of additive noise. This stability is very strong, but can be rather technical
to express. In stating each of these results, it is important to note that the eigenvectors
correspond exactly to the feature extractors associated with Kernel PCA. For an input point
, let denote the vector whose th coordinate is #$ and recall that
5
6
# &
1
5%87
.5
"2
-
Recall that in KPCA we associate
the largest eigenvalues
of , where is
features
with
. First, we
typically chosen by requiring
,
for
some
threshold
consider what
is not large. Observe that in this case we cannot hope
happens when
to equate
all # and , as the th feature is very sensitive to small changes in . However,
far from are treated consistently in - and - .
we can show that all features with
Theorem 2 Let be any matrix- whose columns form an orthonormal
basis
for the
space of features (eigenvectors) in
whose eigenvalue is at least . Let be any
matrix whose columns form an orthonormal basis for the orthogonal complement of .
Let and be the analogous matrices for . For any
,
1
1
and
-
1
If we use the threshold for the eigenvalues of - , the first equation asserts that the features
.
KPCA recovers are not among the features of
whose eigenvalues are less than
Similarly, the second equation
that KPCA will recover all the features of whose
asserts
eigenvalues are larger than
.
Proof: We employ the techniques of Davis and Kahan [4]. Observe that
-
$
&
-
&
-
-
&
&
and are diagonal matrices whose entries (the eigenvalues of where
, respectively. Therefore
least and at most
1
1
1
1
and
-
) are at
which implies the first stated result. The second proof is essentially identical.
In our second result we will still not be able to isolate individual
as the error
and
features,
. However,
matrix can reorder their importance by, say, interchanging
we can
show that any such interchange will occur consistently in all test vectors. Let be the
-dimensional vector whose th coordinate is 1
# , ?.e., here we do not normalize
features to ?equal importance?. Recall that is the vector whose th coordinate is # .
5
6
1 1 for some . There is an orthonormal
# & !
# 1 5
and & .
Proof: Instantiate Theorem 2 with &
Theorem 3 Assume that
rotation matrix such that for all
Note that the rotation matrix becomes completely irrelevant if we are only concerned with
differences, angles, or inner products of feature vectors.
Finally, we prove that in the special
- case where a feature is well separated from its neighboring features in the spectrum of , we get a particularly strong bound.
Theorem 4 If
,
1
# +
1 , and
1 , then
#
5
-
-
Proof:(sketch) As before, we specialize Theorem 2, but first shift
both
and by
in isolation.
This does not change the eigenvectors, and allows us to consider
.
4 Approximating Feature Extractors Quickly
Having determined eigenvalues and eigenvectors, given an input point , the value of on
each feature reduces to evaluating, for some unit vector , a function
&
$ 2
!
1 . Assume that .
where we dropped the subscript , as
by
well as the scaling
take values in an interval of width and let
be any unit vector. We will devise a
fast, unbiased, small-variance estimator for , by sampling and rounding the expansion
coefficients .
Fix
1 1
6 ( then let & ; if let
with probability !
. For each : if
&
otherwise.
That is, after potentially keeping some large coefficients deterministically, we proceed to
perform ?randomized rounding? on the (remaining) coefficients of . Let
Clearly, we have &
bound the behavior of #
ing. In particular, this gives
# &
$*
# . Moreover, using Hoeffding?s inequality [7], we can
# arising from the terms subjected to probabilistic round +1
+* -,
# &
0 1
(1)
Note now that
in Kernel PCA we typically expect
# 0 , i.e., dense eigenvectors.
This makes # 0 the natural scale for measuring # and suggests that using far fewer
than 0 kernel evaluations we can get good
approximations of
chosen (fixed) value of let us say that # is trivial if
% '( 0
. In particular, for a
Having picked some threshold (for
SVM expansions is related to the classification
offset) we want to determine whether is non-trivial and, if so, we want to get a good
relative error estimate for it.
and
2
0
1. There are fewer than
)
Theorem 5 For any
probability at least
2. Either both
0
% #
0
0 set &
#
0
. With
non-zero .
are trivial or
%
and
+%1'(
. Note that 0
Proof: Let 0 denote the number
of non-zero and let & 6
equals plus the sum of 0 independent Bernoulli trials. It is not hard to show that the
probability that the event in 1 fails is bounded by the corresponding probability for the case
where all coordinates of are equal.
In that case, 0 is a Binomial random variable
0 with
0 trials and probability of success # 0 and, by our choice of , 0 &
) .
The Chernoff bound now implies that the event in 1 fails to occur with probability ! 0
is at least .
For
the
enent
in
2
it
suffices
to
observe
that
failure
occurs
only
if
% '( 0 . By (1), this also occurs with probability ! 0
) .
" #
5 Quick batch approximations of Kernels
In this section we devise fast approximations of the kernel function itself. We focus on kernels sharing the following two characteristics: i) they map -dimensional Euclidean space,
and, ii) the mapping depends only on the distance and/or inner product of the considered
points. We note that this covers some of the most popular kernels, e.g., RBFs and polynomial kernels. To simplify exposition we focus on the following task: given a sequence
of (test) vectors
determine
for each of a fixed set of (training) vectors
1
, where 0 .
To get a fast batch approximdition, the idea is that rather than evaluating distances and
inner products directly, we will use a fast, approximately correct oracle for these quantities
offering the following guarantee: it will answer all queries with small relative error.
A natural approach for creating such an oracle is to pick of the coordinates in input
space and use the projection onto these coordinates to determine distances
and inner products. The problem with this approach is that if &
,
any coordinate sampling scheme is bound to do poorly. On the other hand, if we knew that
all coordinates contributed ?approximately equally? to , then coordinate sampling
would be much more appealing. We will do just this, using the technique of random projections [8], which can be viewed as coordinate sampling preceded by a random rotation.
Imagine that we applied a spherically random rotation to ! (before training)
and then applied the same random rotation to each input point as it became available.
Clearly, all distances and inner products would remain the same and we would get exactly
the same results as without the rotation. The interesting part is that any fixed vector that
was a linear combination of training and/or input vectors,
e.g.
, after being rotated
becomes a spherically random vector of length . As a result, the coordinates of are
! , enabling coordinate sampling.
i.i.d. random variables, in fact
$
#
Our oracle amounts to multiplying
%1'( each training and input point by the same
projection
matrix , where &
, and using the resulting -dimensional points to estimate
rotation matrix
distances and inner products. (Think of as the result of taking a
and keeping the first columns (sampling)). Before describing the choice of and the
quality of the resulting approximations, let us go over the computational savings.
%1'(
. Note that
1. Rotating the 0 training vectors takes 0
This cost will be amortized over the sequence of input vectors.
This rotation can be performed in the training phase.
2. The kernel evaluations for each now take 0
instead of 0 .
3. Rotating takes time
which is dominated by 0
.
%1'(
% '(
%1'(
Having motivated our oracle as a spherically random rotation followed by coordinate sampling, we will actually employ a simpler method to perform the projection. Namely, we
will rely on a recent result of [1], asserting that we can do at least as well by taking
* where the */ are i.i.d. with
/
, each case having probabil / & .
until the end,
ity
. Thus, postponing the scaling by
each of the new coordinates
is formed as follows: split the coordinates randomly into two groups; sum the coordinates
in each group; take the difference of the two sums.
#
#
Regarding the quality of approximations we get
Theorem 6 Consider
sets of points
any
and for given
let
&
and #% in . Let & 0
2 %
1
)
Let
.
be a random matrix defined by .
, each case having probability
the
& .*For/ # any where
let
pair of points #
, for every
1 .
1
/
1
"& " 1
With probability at least
and
1
(2)
(3)
,
+
*
, )
Proof: We use Lemma 5 of [1], asserting that for any
1
1
. are i.i.d. with
denote .
and any
1
1 (4)
1 . Thus, by the
By our choice of , the
r.h.s.
of
(4)
is
union bound, with prob the lengths of all 0 0
vectors
ability at least
corresponding to
1
a factor of . This
and , & , &
0 , are maintained
within
1 1 and thus if
readily
we observe that " & 1
1 yields
1 and(2). For (3)
1 are
within of 1 1 and
1 , then (3) holds.
6
6 Conclusion
We have described three techniques for speeding up kernel methods through the use of randomization. While the discussion has focused on Kernel PCA, we feel that our techniques
have potential for further development and empirical evaluation in a more general setting.
Indeed, the methods for sampling kernel expansions and for speeding up the kernel evaluation are universal; also, the Gram matrix sampling is readily applicable to any kernel
technique based on the eigendecomposition of the Gram matrix [3]. Furthermore, it might
enable us to speed up SVM training by sparsifying the Hessian and then applying a sparse
QP solver, such as the ones described in [6, 9].
Our sampling and quantization techniques, both in training and classification, amount to
repeatedly replacing single kernel evaluations with independent random variables that have
appropriate expectations. Note, for example, that while we have represented the sampling
of the kernel expansion as randomized rounding of coefficients, this rounding is also equivalent to the following process: consider each coefficients as is, but replace every kernel
invocation ! with an invocation of a randomized kernel function, distributed as
$
$
&
with probability
otherwise.
Similarly, the process of sampling in training can be thought of as replacing
&
with probability
with probability
with
while an analogous randomized kernel is the obvious choice for quantization.
We feel that this approach suggests a notion of randomized kernels, wherein kernel evaluations are no longer considered as deterministic but inherently random, providing unbiased
estimators for the corresponding inner products. Given bounds on the variance of these estimators, it seems that algorithms which reduce to computing weighted sums of kernel evaluations can exploit concentration of measure. Thus, randomized kernels appear promising
as a general tool for speeding up kernel methods, warranting further investigation.
Acknowledgments. BS would like to thank Santosh Venkatesh for detailed discussions
on sampling kernel expansions.
References
[1] D. Achlioptas, Database-friendly random projections, Proc. of the 20th Symposium
on Principle of Database Systems (Santa Barbara, California), 2001, pp. 274?281.
[2] C. J. C. Burges, Simplified support vector decision rules, Proc. of the 13th International Conference on Machine Learning, Morgan Kaufmann, 1996, pp. 71?77.
[3] N. Cristianini, J. Shawe-Taylor, and H. Lodhi, Latent semantic kernels, Proc. of the
18th International Conference on Machine Learning, Morgan Kaufman, 2001.
[4] C. Davis and W. Kahan, The rotation of eigenvectors by a perturbation 3, SIAM
Journal on Numerical Analysis 7 (1970), 1?46.
[5] Z. F?uredi and J. Koml?os, The eigenvalues of random symmetric matrices, Combinatorica 1 (1981), no. 3, 233?241.
[6] N. I. M. Gould, An algorithm for large-scale quadratic programming, IMA Journal
of Numerical Analysis 11 (1991), no. 3, 299?324.
[7] W. Hoeffding, Probability inequalities for sums of bounded random variables, Journal
of the American Statistical Association 58 (1963), 13?30.
[8] W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert
space, Conference in modern analysis and probability (New Haven, Conn., 1982),
American Mathematical Society, 1984, pp. 189?206.
[9] R. H. Nickel and J. W. Tolle, A sparse sequential quadratic programming algorithm,
Journal of Optimization Theory and Applications 60 (1989), no. 3, 453?473.
[10] E. Osuna, R. Freund, and F. Girosi, An improved training algorithm for support vector
machines, Neural Networks for Signal Processing VII, 1997, pp. 276?285.
[11] B. Sch?olkopf, A. J. Smola, and K.-R. M?uller, Nonlinear component analysis as a
kernel eigenvalue problem, Neural Computation 10 (1998), 1299?1319.
[12] A. J. Smola and B. Sch?olkopf, Sparse greedy matrix approximation for machine
learning, Proc. of the 17th International Conference on Machine Learning, Morgan
Kaufman, 2000, pp. 911?918.
[13] V. Vapnik, The nature of statistical learning theory, Springer, NY, 1995.
[14] C. K. I. Williams and M. Seeger, Using the Nystrom method to speed up kernel machines, Advances in Neural Information Processing Systems 2000, MIT Press, 2001.
| 2072 |@word trial:2 polynomial:1 seems:1 norm:4 stronger:1 lodhi:1 covariance:1 decomposition:1 simplifying:1 pick:1 offering:1 diagonalized:1 com:1 yet:2 must:1 readily:5 numerical:2 additive:1 girosi:1 progressively:1 greedy:1 instantiate:1 fewer:2 tolle:1 coarse:1 simpler:2 mathematical:1 symposium:1 prove:3 specialize:1 indeed:1 behavior:1 themselves:1 roughly:1 solver:1 becomes:2 project:1 moreover:3 bounded:5 what:2 kaufman:2 sparsification:2 guarantee:2 every:5 friendly:1 growth:1 preferable:1 exactly:2 unit:2 intervention:1 appear:1 positive:1 before:3 dropped:1 subscript:1 approximately:2 might:1 plus:1 warranting:1 suggests:2 practical:2 acknowledgment:1 union:1 definite:1 orthonormalization:1 universal:1 empirical:1 thought:1 projection:6 suggest:1 get:6 onto:2 close:2 cannot:1 applying:2 confers:1 map:1 quick:1 equivalent:1 deterministic:1 go:1 williams:1 focused:1 estimator:4 rule:1 orthonormal:4 ity:1 stability:4 notion:2 coordinate:16 analogous:2 feel:2 construction:1 imagine:1 exact:1 programming:2 trick:1 associate:1 amortized:1 approximated:1 particularly:1 database:2 solved:1 decrease:1 mentioned:1 complexity:1 cristianini:1 completely:2 basis:2 various:1 represented:1 separated:1 fast:4 describe:1 query:1 whose:12 quite:1 larger:1 say:2 otherwise:2 ability:1 kahan:2 think:1 itself:2 komlos:1 sequence:2 eigenvalue:11 propose:1 product:11 neighboring:1 loop:1 rapidly:1 poorly:1 frobenius:1 asserts:2 normalize:1 olkopf:3 probabil:1 rotated:1 stating:1 pose:1 uredi:1 strong:3 c:1 implies:2 closely:1 correct:1 enable:1 fix:1 suffices:1 investigation:1 randomization:1 extension:2 hold:1 considered:2 mapping:2 commonality:1 proc:4 applicable:1 sensitive:1 largest:2 tool:1 weighted:1 hope:1 uller:1 mit:1 clearly:2 rather:3 focus:3 consistently:2 bernoulli:1 greatly:2 seeger:1 typically:2 provably:1 issue:2 classification:3 among:1 denoted:1 development:1 special:1 equal:4 santosh:1 never:1 intriguingly:1 washington:2 sampling:18 extraction:1 having:6 identical:1 chernoff:1 saving:1 interchanging:1 simplify:2 haven:1 few:1 employ:2 modern:1 randomly:2 individual:1 ima:1 replaced:1 phase:1 microsoft:2 evaluation:11 severe:1 mcsherry:2 implication:1 orthogonal:6 euclidean:1 taylor:1 rotating:2 circle:1 column:4 earlier:1 cover:1 lanczos:1 measuring:1 kpca:7 cost:4 deviation:1 entry:10 successful:1 rounding:6 johnson:1 answer:1 combined:1 international:3 randomized:13 siam:1 probabilistic:1 quickly:1 squared:2 hoeffding:2 worse:1 creating:1 american:2 return:1 potential:1 de:1 accompanied:1 coefficient:5 explicitly:2 depends:2 performed:2 picked:1 analyze:1 start:1 recover:1 complicated:2 rbfs:1 formed:1 accuracy:1 became:1 variance:4 characteristic:1 efficiently:2 equate:1 yield:2 correspond:1 kaufmann:1 famous:1 multiplying:1 worth:2 converged:1 sharing:1 failure:1 nonetheless:1 pp:5 obvious:1 nystrom:1 associated:2 proof:6 recovers:1 popular:2 recall:4 hilbert:1 actually:1 wherein:1 improved:1 entrywise:2 done:1 evaluated:1 furthermore:1 just:1 smola:2 achlioptas:2 until:1 sketch:1 hand:1 replacing:2 nonlinear:2 o:1 quality:2 omitting:1 requiring:1 true:1 unbiased:3 symmetric:3 spherically:3 semantic:1 deal:1 round:2 width:1 davis:2 whereby:1 maintained:1 generalized:1 presenting:1 performs:1 wigner:1 novel:1 common:1 rotation:10 behaves:2 preceded:1 qp:1 exponentially:1 association:1 significant:1 similarly:2 shawe:1 dot:4 longer:1 recent:1 irrelevant:1 barbara:1 store:1 inequality:2 success:1 devise:2 morgan:3 somewhat:1 determine:3 signal:1 semi:1 relates:1 ii:2 reduces:1 ing:2 technical:1 calculation:1 equally:2 bigger:1 essentially:2 expectation:4 iteration:8 kernel:55 addition:1 want:2 interval:1 sch:3 isolate:1 noting:1 presence:1 split:1 easy:1 concerned:1 fit:1 isolation:1 reduce:2 idea:3 inner:7 regarding:1 shift:1 bottleneck:1 whether:1 motivated:3 pca:11 accelerating:1 proceed:1 hessian:1 repeatedly:1 generally:1 detailed:1 eigenvectors:9 santa:1 amount:2 svms:2 arising:1 express:1 group:2 sparsifying:1 conn:1 threshold:3 sum:5 angle:1 prob:1 almost:1 throughout:1 decision:1 scaling:2 bit:1 accelerates:1 bound:9 followed:1 simplification:2 quadratic:3 oracle:4 occur:2 precisely:2 dominated:1 speed:3 extremely:1 performing:1 relatively:1 gould:1 speedup:3 combination:1 remain:2 osuna:1 appealing:1 b:2 making:2 modification:1 happens:1 invariant:2 heart:1 equation:2 discus:1 describing:1 flip:1 subjected:1 end:1 koml:1 available:1 operation:2 apply:1 observe:4 appropriate:2 spectral:4 batch:2 coin:1 original:1 binomial:1 running:1 remaining:1 exploit:1 approximating:1 society:1 seeking:1 move:1 quantity:1 occurs:2 concentration:1 biowulf:1 diagonal:1 subspace:3 distance:5 thank:1 mapped:3 trivial:3 provable:1 length:2 providing:1 postponing:1 potentially:1 frank:1 stated:2 perform:2 allowing:1 upper:1 contributed:1 enabling:1 looking:1 orthonormalize:1 perturbation:1 sharp:1 introduced:1 complement:1 pair:3 cast:1 required:2 namely:1 venkatesh:1 california:1 able:1 dimitris:1 program:1 memory:1 event:2 natural:3 treated:1 rely:1 scheme:2 technology:1 imply:1 extract:1 speeding:5 text:1 l2:5 multiplication:4 relative:2 law:1 freund:1 expect:1 interesting:1 limitation:1 nickel:1 eigendecomposition:1 principle:1 share:1 keeping:2 burges:1 taking:2 sparse:5 benefit:1 distributed:1 dimension:1 gram:7 evaluating:5 lindenstrauss:1 computes:2 asserting:2 interchange:1 collection:1 commonly:1 simplified:2 far:2 implicitly:1 bernhard:1 instantiation:1 reorder:1 knew:1 spectrum:1 latent:1 promising:1 nature:1 inherently:1 obtaining:1 expansion:7 quantize:1 dense:2 main:1 linearly:1 noise:1 ny:2 fails:2 deterministically:1 comput:1 invocation:2 extractor:3 theorem:8 offset:1 svm:3 quantization:6 vapnik:1 sequential:1 importance:2 magnitude:1 overwhelmed:1 sparser:1 vii:1 simply:2 expressed:1 scalar:1 applies:1 springer:1 viewed:2 exposition:2 replace:3 lipschitz:1 change:2 hard:1 infinite:1 determined:1 operates:1 reducing:1 principal:4 lemma:1 internal:1 support:3 combinatorica:1 |
1,177 | 2,073 | A Natural Policy Gradient
Sham Kakade
Gatsby Computational Neuroscience Unit
17 Queen Square, London, UK WC1N 3AR
http: //www.gatsby.ucl.ac.uk
sham @gatsby.ucl.ac.uk
Abstract
We provide a natural gradient method that represents the steepest
descent direction based on the underlying structure of the parameter space. Although gradient methods cannot make large changes
in the values of the parameters, we show that the natural gradient is moving toward choosing a greedy optimal action rather than
just a better action. These greedy optimal actions are those that
would be chosen under one improvement step of policy iteration
with approximate, compatible value functions, as defined by Sutton et al. [9]. We then show drastic performance improvements in
simple MDPs and in the more challenging MDP of Tetris.
1
Introduction
There has been a growing interest in direct policy-gradient methods for approximate
planning in large Markov decision problems (MDPs). Such methods seek to find
a good policy 7r among some restricted class of policies, by following the gradient
of the future reward. Unfortunately, the standard gradient descent rule is noncovariant. Crudely speaking, the rule !:l.()i = oJ] f / a()i is dimensionally inconsistent
since the left hand side has units of ()i and the right hand side has units of l/()i
(and all ()i do not necessarily have the same dimensions).
In this paper, we present a covariant gradient by defining a metric based on the
underlying structure of the policy. We make the connection to policy iteration
by showing that the natural gradient is moving toward choosing a greedy optimal
action. We then analyze the performance of the natural gradient in both simple
and complicated MDPs. Consistent with Amari's findings [1], our work suggests
that the plateau phenomenon might not be as severe using this method.
2
A Natural Gradient
A finite MDP is a tuple (S, So, A, R, P) where: S is finite set of states, So is a start
state, A is a finite set of actions, R is a reward function R : S x A --+ [0, Rmax], and
P is the transition model. The agent 's decision making procedure is characterized
by a stochastic policy 7r(a; s) , which is the probability of taking action a in state
s (a semi-colon is used to distinguish the random variables from the parameters of
the distribution). We make the assumption that every policy 7r is ergodic, ie has a
well-defined stationary distribution p7f. Under this assumption, the average reward
(or undiscounted reward) is 1]( 7r) == 2:: s ,a p7f (s )7r(a; S)R(s, a), the state-action value
is Q7f(S, a) == E7f{2:::oR(st,at) -1](7r)lso = s,ao = a} and the value function is
J7f(s) == E7f(a' ;s) {Q7f(s, a')}, where and St and at are the state and action at time t.
We consider the more difficult case where the goal of the agent is to find a policy that
maximizes the average reward over some restricted class of smoothly parameterized
policies, fr = {7rO : 8 E ~m}, where tro represents the policy 7r(a; S, 8).
The exact gradient of the average reward (see [8, 9]) is:
\11](8)
= Lp7f(s)\17r(a;s, 8)Q7f(s ,a)
(1)
s,a
where we abuse notation by using 1](8) instead of 1](7ro). The steepest descent
direction of 1](8) is defined as the vector d8 that minimizes 1](8 + d8) under
the constraint that the squared length Id812 is held to a small constant. This
squared length is defined with respect to some positive-definite matrix G(8), ie
Id812 == 2::ij Gij (8)d8 i d8j = d8 T G(8)d8 (using vector notation). The steepest descent direction is then given by G- 1 \11](8) [1]. Standard gradient descent follows
the direction \11](8) which is the steepest descent under the assumption that G(8)
is the identity matrix, I. However, this as hoc choice of a metric is not necessarily
appropriate. As suggested by Amari [1], it is better to define a metric based not
on the choice of coordinates but rather on the manifold (ie the surface) that these
coordinates parameterize. This metric defines the natural gradient.
Though we slightly abuse notation by writing 1](8), the average reward is technically
a function on the set of distributions {7rO : 8 E ~m}. To each state s, there
corresponds a probability manifold, where the distribution 7r(a; S, 8) is a point on
this manifold with coordinates 8. The Fisher information matrix of this distribution
7r(a; s,8) is
F (8) = E
s
-
7f(a;s,O)
[81o g 7r(a; s,8)
08 i
olog 7r(a;
s,8)]
08
'
j
(2)
and it is clearly positive definite. As shown by Amari (see [1]), the Fisher information matrix, up to a scale, is an invariant metric on the space of the parameters
of probability distributions. It is invariant in the sense that it defines the same
distance between two points regardless of the choice of coordinates (ie the parameterization) used, unlike G = I.
Since the average reward is defined on a set of these distributions , the straightforward choice we make for the metric is:
(3)
where the expectation is with respect to the stationary distribution of 7ro. Notice
that although each Fs is independent of the parameters of the MDP's transition
model, the weighting by the stationary distribution introduces dependence on these
parameters. Intuitively, Fs (8) measures distance on a probability manifold corresponding to state sand F(8) is the average such distance. The steepest descent
direction this gives is:
(4)
3
The Natural Gradient and Policy Iteration
We now compare policy improvement under the natural gradient to policy iteration.
For an appropriate comparison, consider the case in which Q7r (s, a) is approximated
by some compatible function approximator r(s ,a;w) parameterized by w [9, 6].
3.1
Compatible Function Approximation
For vectors (), w E
~m,
we define:
'IjJ (s , a)7r = \7logn(a;s,()),
r(s,a;w) = wT 'ljJ7r(s,a)
(5)
where [\7logn(a ;s, ())]i = 8logn(a;s, ())!8()i. Let w minimize the squared error
f(W, n) == L,s ,a p7r (s )n(a; s, ())(r (s, a; w) _Q7r (s, a))2. This function approximator is
compatible with the policy in the sense that if we use the approximations f7r (s, a; w)
in lieu of their true values to compute the gradient (equation 1), then the result
would still be exact [9, 6] (and is thus a sensible choice to use in actor-critic schemes).
Theorem 1. Let w minimize the squared error f(W, no). Then
w=
~1}(()) .
Proof. Since w minimizes the squared error, it satisfies the condition 8f!8wi = 0,
which implies:
LP7r(s)n(a;s,())'ljJ7r (s,a)('ljJ7r (s,a?w - Q7r(s,a)) = O.
s,a
or equivalently:
s,a
s,a
By definition of 'ljJ7r, \7n(a;s,()) = n(a;s,())'ljJ7r(s,a) and so the right hand side is
equal to \71}. Also by definition of 'ljJ7r, F( ()) = L,s ,a p7r (s )n( a; s, ()) 'ljJ7r (s, a)'ljJ7r (s, a) T.
Substitution leads to:
F(())w = \71}(()) .
Solving for w gives w = F(()) - l\71}(()), and the result follows from the definition of
the natural gradient.
D
Thus, sensible actor-critic frameworks (those using f7r(s , a; w)) are forced to use the
natural gradient as the weights of a linear function approximator. If the function approximation is accurate, then good actions (ie those with large state-action values)
have feature vectors that have a large inner product with the natural gradient.
3.2
Greedy Policy Improvement
A greedy policy improvement step using our function approximator would choose
action a in state s if a E argmaxa, f7r (s, a'; w). In this section, we show that the
natural gradient tends to move toward this best action, rather than just a good
action.
Let us first consider policies in the exponential family (n(a ;s, ()) IX exp(()T?sa)
where ?sa is some feature vector in ~m). The motivation for the exponential family
is because it has affine geometry (ie the flat geometry of a plane), so a translation of
a point by a tangent vector will keep the point on the manifold. In general, crudely
speaking, the probability manifold of 7r(a; s, 0) could be curved, so a translation of
a point by a tangent vector would not necessarily keep the point on the manifold
(such as on a sphere). We consider the general (non-exponential) case later.
We now show, for the exponential family, that a sufficiently large step in the natural
gradient direction will lead to a policy that is equivalent to a policy found after a
greedy policy improvement step.
Theorem 2. For 7r(a; s, 0) ex: exp(OT 1>sa), assume that ~'TJ(O) is non-zero and that
w minimizes the approximation error. Let7roo (a;s) =lima-+oo7r(a;s , O+a~'TJ(O)).
Then 7r 00 (a; s) 1- 0 if and only if a E argmaxa, F' (s, a'; w).
Proof. By the previous result, F'(s,a ;w) = ~'TJ(O)T'lj;7r(s,a). By definition of
7r(a; s, 0) , 'lj;7r (s, a) = 1>sa - E 7r (a';s ,O) (1)sa'). Since E 7r (a';s,O) (1)sa') is not a function
of a, it follows that
argmax a , r(s, a'; w) = argmax a , ~'TJ(Of 1>sa' .
After a gradient step, 7r(a; s, 0 + a~'TJ(O)) ex: exp(OT 1>sa + a~'TJ(O)T 1>sa). Since
~'TJ(O) 1- 0, it is clear that as a -+ 00 the term ~'TJ(O)T 1>sa dominates , and so
D
7r 00 (a, s) = 0 if and only if a f{. argmax a , ~ 'TJ( 0) T 1>sa' .
It is in this sense that the natural gradient tends to move toward choosing the best
action. It is straightforward to show that if the standard non-covariant gradient
rule is used instead then 7r oo (a; s) will select only a better action (not necessarily
the best), ie it will choose an action a such that F'(s ,a;w) > E 7r (a';s){F'(s,a';w)}.
Our use of the exponential family was only to demonstrate this point in the extreme
case of an infinite learning rate.
Let us return to case of a general parameterized policy. The following theorem shows
that the natural gradient is locally moving toward the best action, determined by
the local linear approximator for Q7r (s, a).
Theorem 3. Assume that w minimizes the approximation error and let the update
to the parameter be 0' = 0 + a~'TJ(O). Then
7r(a; s, 0') = 7r(a; s, 0)(1 + r(s , a; w)) + 0(a 2 )
Proof. The change in 0, ,6.0, is a~'TJ(O), so by theorem 1, ,6.0 = aw. To first order,
7r(a; s, 0')
7r(a; s, 0)
+ fJ7r(a~;, O)T ,6.0 + 0(,6.0 2 )
7r(a; s, 0)(1
7r(a; s, 0)(1
7r(a;s,O)(l
+ 'lj;(s, af ,6.0) + 0(,6.0 2 )
+ a'lj;(s, af w) + 0(a 2 )
+ ar(s,a;w)) + 0(a 2 ) ,
where we have used the definition of 'lj; and
f.
D
It is interesting to note that choosing the greedy action will not in general improve
the policy, and many detailed studies have gone into understanding this failure [3].
However, with the overhead of a line search, we can guarantee improvement and
move toward this greedy one step improvement. Initial improvement is guaranteed
since F is positive definite.
4
Metrics and Curvatures
Obviously, our choice of F is not unique and the question arises as to whether or
not there is a better metric to use than F. In the different setting of parameter
estimation, the Fisher information converges to the Hessian, so it is asymptotically
efficient [1], ie attains the Cramer-Rao bound. Our situation is more similar to
the blind source separation case where a metric is chosen based on the underlying
parameter space [1] (of non-singular matrices) and is not necessarily asymptotically
efficient (ie does not attain second order convergence). As argued by Mackay [7],
one strategy is to pull a metric out of the data-independent terms of the Hessian (if
possible), and in fact, Mackay [7] arrives at the same result as Amari for the blind
source separation case.
Although the previous sections argued that our choice is appropriate, we would like
to understand how F relates to the Hessian V 2 TJ(B), which, as shown in [5], has the
form:
sa
(6)
Unfortunately, all terms in this Hessian are data-dependent (ie are coupled to stateaction values) . It is clear that F does not capture any information from these last
two terms, due to their VQ7r dependence. The first term might have some relation
to F due the factor of V 2 7f. However, the Q values weight this curvature of our
policy and our metric is neglecting such weighting.
Similar to the blind source separation case, our metric clearly does not necessarily
converge to the Hessian and so it is not necessarily asymptotically efficient (ie does
not attain a second order convergence rate). However, in general, the Hessian will
not be positive definite and so the curvature it provides could be of little use until
B is close to a local maxima. Conjugate methods would be expected to be more
efficient near a local maximum.
5
Experiments
We first look at the performance of the natural gradient in a few simple MDPs
before examining its performance in the more challenging MDP of Tetris. It is
straightforward to estimate F in an online manner, since the derivatives V log 7f
must be computed anyway to estimate VTJ(B). If the update rule
f
f-
f + V log 7f(at; St,B)Vlog7f(at; St,Bf
is used in a T-Iength trajectory, then fiT is a consistent estimate of F. In our
first two examples, we do not concern ourselves with sampling issues and instead
numerically integrate the exact derivative (B t = Bo + J~ VTJ(BddB). In all of our
simulations, the policies tend to become deterministic (V log 7f -+ 0) and to prevent
F from becoming singular, we add about 10- 3 1 at every step in all our simulations.
We simulated the natural policy gradient in a simple I-dimensional linear quadratic
regulator with dynamics x(t + 1) = .7x(t) + u(t) + E(t) and noise distribution
E ~ G(O,l).
The goal is to apply a control signal u to keep the system at
x = 0, (incurring a cost of X(t)2 at each step). The parameterized policy used
was 7f(u; x, B) ex exp(Blx 2 + B2X). Figure lA shows the performance improvement
when the units of the parameters are scaled by a factor of 10 (see figure text). Notice that the time to obtain a score of about 22 is about three orders of magnitude
-
'--''''' '~''? ~......
unsealed
-
$=10 s=1 ......
1
2
-'::',$,=1 $2=10 _. - ":.
\: .:-.
W
~2:=':3=::l4'
--"-,-,
L
_-=-2 --':
-'':::::;0:::=:::'
I09 10 (time)
(' ,\2
~R=O)
~
rl
I
"E 1 Ir
h
"::>:"
20
B 8'
~a
C 21
i
D
-11
______
',05
..
~
~0 '::0 --0~5C------:-'-----:-':
'5C------::'
2
~21
~,
0
7
time x 10
L------------
a
0.5
1
/:--------1.
-
1.5
time
2
2.5
3
"
Q-
"\"
.\';
L---::-,::::;?7J~========-~
5
8.,
10
15
Figure 1: A) The cost Vs. 10glo(time) for an LQG (with 20 time step trajectories).
The policy used was 7f(u; x, ()) ex: exp(()lslX2 + ()2S2X) where the rescaling constants,
Sl and S2, are shown in the legend. Under equivalent starting distributions (()lSl =
()2S2 = -.8) , the right-most three curves are generated using the standard gradient
method and the rest use the natural gradient. B) See text. C top) The average
reward vs. time (on a 107 scale) of a policy under standard gradient descent using
the sigmoidal policy parameterization (7f(I; s, ()i) ex: exp(()i)/(1 + exp(()i)), with the
initial conditions 7f(i , 1) = .8 and 7f(j, 1) = .1. C bottom) The average reward vs.
time (unscaled) under standard gradient descent (solid line) and natural gradient
descent (dashed line) for an early window of the above plot. D) Phase space plot
for the standard gradient case (the solid line) and the natural gradient case (dashed
line) .
faster. Also notice that the curves under different rescaling are not identical. This
is because F is not an invariant metric due to the weighting by Ps.
The effects of the weighting by p(s) are particularly clear in a simple 2-state MDP
(Figure IB), which has self- and cross-transition actions and rewards as shown.
Increasing the chance of a self-loop at i decreases the stationary probability of j.
Using a sigmoidal policy parameterization (see figure text) and initial conditions
corresponding to p(i) = .8 and p(j) = .2, both self-loop action probabilities will initially be increased under a gradient rule (since one step policy improvement chooses
the self-loop for each state). Since the standard gradient weights the learning to
each parameter by p(s) (see equation 1), the self-loop action at state i is increased
faster than the self loop probability at j, which has the effect of decreasing the effective learning-rate to state j even further. This leads to an extremely fiat plateau
with average reward 1 (shown in Figure lC top), where the learning for state j is
thwarted by its low stationary probability. This problem is so severe that before the
optimal policy is reached p(j) drops as low as 10- 7 from its initial value of .2, which
is disastrous for sampling methods. Figure 1 C bottom shows the performance of
the natural gradient (in a very early time window of Figure lC top). Not only is
the time to the optimal policy decreased by a factor of 107 , the stationary distribution of state i never drops below .05. Note though the standard gradient does
increase the average reward faster at the start, but only to be seduced by sticking
at state i. The phase space plot in Figure ID shows the uneven learning to the
different parameters, which is at the heart of the problem. In general, if a table
lookup Boltzmann policy is used (ie 7f( a; s , ()) ex: exp( () sa)), it is straightforward to
show that the natural gradient weights the components of ~'fJ uniformly (instead of
using p(s)), thus evening evening out the learning to all parameters.
The game of Tetris provides a challenging high dimensional problem. As shown in
[3], greedy policy iteration methods using a linear function approximator exhibit
drastic performance degradation after providing impressive improvement (see [3]
for a description of the game, methods , and results). The upper curve in Figure2A
replicates these results. Tetris provides an interesting case to test gradient methods,
A
5000, - - - - - - - - - - - - - ,
B
7000, - - - - - - - - - - , - - - - ,
C
6000
4000
5000
~3000
~4000
?0
a... 2000
&3000
2000
1000
1000
1 I09,O( lteralions)
2
500
1000
Iterations
1500
2000
Figure 2: A) Points vs. 10g(Iterations) . The top curve duplicates the same results
in [3] using the same features (which were simple functions of the heights of each
column and the number of holes in the game). We have no explanation for this performance degradation (nor does [3]). The lower curve shows the poor performance
of the standard gradient rule. B) The curve on the right shows the natural policy
gradient method (and uses the biased gradient method of [2] though this method
alone gave poor performance). We found we could obtain faster improvement and
higher asymptotes if the robustifying factor of 10- 3 I that we added to F was more
carefully controlled (we did not carefully control the parameters). C) Due to the
intensive computational power required of these simulations we ran the gradient in a
smaller Tetris game (height of 10 rather than 20) to demonstrate that the standard
gradient updates (right curve) would eventually reach the same performance of the
natural gradient (left curve).
which are guaranteed not to degrade the policy. We consider a policy compatible
with the linear function approximator used in [3] (ie 7f(a ;s, (}) ex: exp((}T?sa) where
?sa are the same feature vectors). The features used in [3] are the heights of each
column, the differences in height between adjacent columns, the maximum height,
and the number of 'holes' . The lower curve in Figure 2A shows the particularly
poor performance of the standard gradient method. In an attempt to speed learning, we tried a variety of more sophisticated methods to no avail, such as conjugate
methods, weight decay, annealing, the variance reduction method of [2], the Hessian in equation 6, etc. Figure 2B shows a drastic improvement using the natural
gradient (note that the timescale is linear). This performance is consistent with our
theoretical results in section 3, which showed that the natural gradient is moving
toward the solution of a greedy policy improvement step. The performance is somewhat slower than the greedy policy iteration (left curve in Figure 2B) which is to be
expected using smaller steps. However, the policy does not degrade with a gradient
method. Figure 2 shows that the performance of the standard gradient rule (right
curve) eventually reaches the the same performance of the natural gradient, in a
scaled down version of the game (see figure text).
6
Discussion
Although gradient methods cannot make large policy changes compared to greedy
policy iteration, section 3 implies that these two methods might not be that disparate, since a natural gradient method is moving toward the solution of a policy
improvement step. With the overhead of a line search, the methods are even more
similar. The benefit is that performance improvement is now guaranteed, unlike in
a greedy policy iteration step.
It is interesting, and unfortunate, to note that the F does not asymptotically converge to the Hessian, so conjugate gradient methods might be more sensible asymptotically. However, far from the converge point, the Hessian is not necessarily
informative, and the natural gradient could be more efficient (as demonstrated in
Tetris). The intuition as to why the natural gradient could be efficient far from the
maximum, is that it is pushing the policy toward choosing greedy optimal actions.
Often, the region (in parameter space) far from from the maximum is where large
performance changes could occur. Sufficiently close to the maximum, little performance change occurs (due to the small gradient), so although conjugate methods
might converge faster near the maximum, the corresponding performance change
might be negligible. More experimental work is necessary to further understand the
effectiveness of the natural gradient.
Acknowledgments
We thank Emo Todorov and Peter Dayan for many helpful discussions. Funding is
from the NSF and the Gatsby Charitable Foundation.
References
[I] S. Amari. Natural gradient works efficiently in learning.
10(2):251- 276, 1998.
Neural Computation,
[2] J. Baxter and P. Bartlett. Direct gradient-based reinforcement learning. Technical
report, Australian National University, Research School of Information Sciences and
Engineering, July 1999.
[3] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific,
1996.
[4] P. Dayan and G. Hinton. Using em for reinforcement learning. Neural Computation,
9:271- 278 , 1997.
[5] S. Kakade. Optimizing average reward using discounted reward. COLT. in press.,
200l.
[6] V. Konda and J. Tsitsiklis. Actor-critic algorithms. Advances in N eural Information
Processing Systems, 12, 2000.
[7] D . MacKay. Maximum likelihood and covariant algorithms for independent component analysis. Technical report , University of Cambridge, 1996.
[8] P. Marbach and J . Tsitsiklis. Simulation-based optimization of markov reward processes. Technical report, Massachusetts Institute of Technology, 1998.
[9] R. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for
reinforcement learning with function approximation. Neural Information Processing
Systems, 13, 2000.
[10] L. Xu and M. 1. Jordan. On convergence properties of the EM algorithm for gaussian
mixtures. Neural Computation, 8(1):129- 151, 1996.
| 2073 |@word version:1 q7f:3 bf:1 simulation:4 seek:1 tried:1 solid:2 reduction:1 initial:4 substitution:1 score:1 must:1 informative:1 lqg:1 asymptote:1 plot:3 drop:2 update:3 v:4 stationary:6 greedy:14 alone:1 parameterization:3 plane:1 steepest:5 provides:3 sigmoidal:2 height:5 direct:2 become:1 overhead:2 manner:1 expected:2 planning:1 growing:1 nor:1 discounted:1 decreasing:1 little:2 window:2 increasing:1 underlying:3 notation:3 maximizes:1 rmax:1 minimizes:4 finding:1 guarantee:1 every:2 stateaction:1 ro:4 scaled:2 uk:3 control:2 unit:4 bertsekas:1 positive:4 negligible:1 before:2 local:3 engineering:1 tends:2 sutton:2 id:1 becoming:1 abuse:2 might:6 suggests:1 challenging:3 gone:1 unique:1 acknowledgment:1 definite:4 procedure:1 attain:2 argmaxa:2 cannot:2 close:2 writing:1 www:1 equivalent:2 deterministic:1 demonstrated:1 straightforward:4 regardless:1 starting:1 ergodic:1 rule:7 pull:1 lso:1 coordinate:4 anyway:1 lima:1 lsl:1 exact:3 programming:1 us:1 approximated:1 particularly:2 bottom:2 capture:1 parameterize:1 region:1 decrease:1 ran:1 intuition:1 reward:16 dynamic:2 singh:1 solving:1 technically:1 unsealed:1 forced:1 effective:1 london:1 q7r:4 choosing:5 amari:5 timescale:1 online:1 obviously:1 hoc:1 ucl:2 vtj:2 product:1 fr:1 loop:5 sticking:1 description:1 convergence:3 undiscounted:1 p:1 converges:1 oo:1 ac:2 avail:1 ij:1 school:1 sa:15 implies:2 australian:1 direction:6 stochastic:1 mcallester:1 sand:1 argued:2 ao:1 sufficiently:2 cramer:1 s2x:1 exp:9 early:2 estimation:1 clearly:2 gaussian:1 rather:4 improvement:17 likelihood:1 attains:1 sense:3 colon:1 helpful:1 dependent:1 dayan:2 lj:5 initially:1 relation:1 issue:1 among:1 colt:1 logn:3 mackay:3 equal:1 never:1 sampling:2 identical:1 represents:2 blx:1 look:1 future:1 report:3 duplicate:1 few:1 national:1 geometry:2 argmax:3 ourselves:1 phase:2 attempt:1 interest:1 severe:2 replicates:1 introduces:1 mixture:1 extreme:1 arrives:1 tj:12 held:1 wc1n:1 accurate:1 tuple:1 neglecting:1 necessary:1 theoretical:1 increased:2 column:3 rao:1 ar:2 queen:1 cost:2 examining:1 aw:1 chooses:1 st:4 ie:13 squared:5 choose:2 d8:5 derivative:2 return:1 rescaling:2 lookup:1 blind:3 later:1 analyze:1 reached:1 start:2 complicated:1 minimize:2 square:1 ir:1 variance:1 efficiently:1 trajectory:2 plateau:2 reach:2 definition:5 failure:1 proof:3 massachusetts:1 fiat:1 carefully:2 sophisticated:1 e7f:2 higher:1 though:3 just:2 crudely:2 until:1 hand:3 defines:2 olog:1 scientific:1 mdp:5 effect:2 true:1 adjacent:1 game:5 self:6 demonstrate:2 fj:1 tro:1 funding:1 rl:1 numerically:1 cambridge:1 marbach:1 moving:5 actor:3 impressive:1 surface:1 glo:1 add:1 etc:1 curvature:3 showed:1 optimizing:1 dimensionally:1 p7r:2 somewhat:1 converge:4 dashed:2 signal:1 semi:1 relates:1 july:1 sham:2 technical:3 faster:5 characterized:1 af:2 cross:1 sphere:1 controlled:1 neuro:1 expectation:1 metric:13 iteration:10 i09:2 decreased:1 annealing:1 singular:2 source:3 biased:1 ot:2 rest:1 unlike:2 tend:1 legend:1 inconsistent:1 effectiveness:1 jordan:1 near:2 baxter:1 variety:1 todorov:1 fit:1 gave:1 inner:1 intensive:1 whether:1 bartlett:1 f:2 peter:1 speaking:2 hessian:9 action:22 clear:3 detailed:1 locally:1 http:1 sl:1 nsf:1 notice:3 neuroscience:1 prevent:1 asymptotically:5 parameterized:4 family:4 separation:3 decision:2 bound:1 guaranteed:3 distinguish:1 quadratic:1 occur:1 constraint:1 flat:1 regulator:1 speed:1 robustifying:1 extremely:1 poor:3 conjugate:4 smaller:2 slightly:1 em:2 wi:1 kakade:2 making:1 intuitively:1 restricted:2 invariant:3 heart:1 equation:3 emo:1 eventually:2 drastic:3 lieu:1 incurring:1 apply:1 appropriate:3 slower:1 top:4 unfortunate:1 pushing:1 konda:1 move:3 question:1 added:1 occurs:1 strategy:1 dependence:2 exhibit:1 gradient:63 distance:3 thank:1 simulated:1 athena:1 sensible:3 degrade:2 manifold:7 toward:9 length:2 providing:1 equivalently:1 difficult:1 unfortunately:2 disastrous:1 disparate:1 policy:49 boltzmann:1 upper:1 markov:2 finite:3 descent:10 curved:1 defining:1 situation:1 thwarted:1 hinton:1 mansour:1 required:1 connection:1 suggested:1 below:1 oj:1 explanation:1 power:1 natural:33 scheme:1 improve:1 technology:1 mdps:4 coupled:1 iength:1 text:4 understanding:1 tangent:2 interesting:3 approximator:7 foundation:1 integrate:1 agent:2 affine:1 consistent:3 charitable:1 critic:3 unscaled:1 translation:2 compatible:5 last:1 tsitsiklis:3 side:3 understand:2 institute:1 taking:1 benefit:1 curve:11 dimension:1 transition:3 reinforcement:3 far:3 approximate:2 keep:3 search:2 evening:2 why:1 table:1 necessarily:8 did:1 motivation:1 noise:1 s2:2 eural:1 xu:1 gatsby:4 lc:2 exponential:5 ib:1 weighting:4 ix:1 theorem:5 down:1 showing:1 b2x:1 decay:1 dominates:1 concern:1 magnitude:1 hole:2 smoothly:1 ijj:1 bo:1 covariant:3 corresponds:1 satisfies:1 chance:1 goal:2 identity:1 fisher:3 change:6 infinite:1 determined:1 uniformly:1 wt:1 degradation:2 gij:1 tetri:6 experimental:1 la:1 l4:1 select:1 uneven:1 arises:1 phenomenon:1 ex:7 |
1,178 | 2,074 | Contextual Modulation of Target Saliency
Antonio Torralba
Dept. of Brain and Cognitive Sciences
MIT, Cambridge, MA 02139
torralba@ai. mit. edu
Abstract
The most popular algorithms for object detection require the use of
exhaustive spatial and scale search procedures. In such approaches,
an object is defined by means of local features. fu this paper we
show that including contextual information in object detection procedures provides an efficient way of cutting down the need for
exhaustive search. We present results with real images showing
that the proposed scheme is able to accurately predict likely object
classes, locations and sizes.
1
Introduction
Although there is growing evidence of the role of contextual information in human
perception [1], research in computational vision is dominated by object-based representations [5,9,10,15]. In real-world scenes, intrinsic object information is often
degraded due to occlusion, low contrast, and poor resolution. In such situations, the
object recognition problem based on intrinsic object representations is ill-posed. A
more comprehensive representation of an object should include contextual information [11,13]: Obj. representatian == {intrisic obj. model, contextual obj. model}.
In this representation, an object is defined by 1) a model of the intrinsic properties of the object and 2) a model of the typical contexts in which the object is
immersed. Here we show how incorporating contextual models can enhance target
object saliency and provide an estimate of its likelihood and intrinsic properties.
2
Target saliency and object likelihood
Image information can be partitioned into two sets of features: local features, VL,
that are intrinsic to an object, and contextual features, rUe which encode structural
properties of the background. In a statistical framework, object detection requires
evaluation of the likelihood function (target saliency function): P(O IVL, va) which
provides the probability of presence of the object 0 given a set of local and contextual measurements. 0 is the set of parameters that define an object immersed in a
scene: 0 == {on, x, y, i} with on==object class, (x,y)==location in image coordinates
and bobject appearance parameters. By applying Bayes rule we can write:
P(O IVL, va)
=
P(vL11 va) P(VL
10, va)P(O Iva)
(1)
Those three factors provide a simplified framework for representing three levels of attention guidance when looking for a target: The normalization factor, 1/P(VL I va),
does not depend on the target or task constraints, and therefore is a bottom-up factor. It provides a measure of how unlikely it is to find a set of local measurements VL
within the context va. We can define local saliency as S(x,y) == l/P(vL(x,y) Iva).
Saliency is large for unlikely features ina' scene. The second factor, P(VL 10, va),
gives the likelihood of the local measurements VL when the object is present at such
location in a particular context. We can write P(VL 10, va) ~ P(VL 10), which is a
convenient approximation when the aspect of the target object is fully determined
by the parameters given by the description O. This factor represents the top-down
knowledge of the target? appearance and how it contributes to the search. Regions
of the image with features unlikely to belong to the target object are vetoed. and
regions with attended features are enhanced. The third factor, the PDF P(O I va),
provides context-based priors on object class, location and scale. It is of capital
importance for insuring reliable inferences in situations where the local image measurements VL produce ambiguous interpretations. This factor does not depend on
local measurements and target models [8,13]. Therefore, the term P(O Iva) modulates the saliency of local image properties when looking for an object of the class
On. Contextual priors become more evident if we apply Bayes rule successively in
order to split the PDF P( 0 I va) into three factors that model three kinds of context
priming on object search:
(2)
According to this decomposition of the PDF, the contextual modulation of target
saliency is a function of three main factors:
Object likelihood: P(on Iva) provides the probability of presence of the object class
in the scene. If P( On Iva) is very small, then object search need not be initiated
(we do not need to look for cars in a living room).
On
Contextual control of focus of attention: P(x, y I On, va)? This PDDF gives the
most likely locations for the presence of object On given context information, and
it allocates computational resources into relevant scene regions.
Contextual selection of local target appearance: P(tl.va, on). This gives the likely
(prototypical) shapes (point of views, size, aspect ratio, object aspect) of the object
On in the context Va- Here t == {a, p}, with a==scale and p==aspect ratio. Other
parameters describing the appearance of an object in an image can be added.
The image features most commonly used for describing local structures are the
energy outputs of oriented band-pass filters, as they have been shown to be relevant
for the task of object detection [9,10] and scene recognition [2,4,8,12]~ Therefore,
the local image representation at the spatial location (x) is given by the vector
VL(X) == {v(X,k)}k==l,N with:
(3)
1
1
",.-.....
1
",.-.....
'0
;>
u
;>
-a
0
u
;>
-a
0
'0:
-a
0
P:
o1
2 3 4
0:
o1
2 3 4
2 3 4
Figure 1: Contextual object prImIng of four objects categories (I-people, 2furniture, 3-vehicles and 4-trees)
where i(x) is the input image and gk(X) are oriented band-pass filters defined by
gk(i) == e-llxI12/u~e27fj<f~,x>. In such a representation [8], v(i,k) is the output
magnitude- at the location i of a complex Gabor filter tuned to the spatial frequency f~. The variable k indexes filters tuned to different spatial frequencies and
orientations.
On the other ,hand, contextual features have to summarize the structure of the
whole image. It has been shown that a holistic low-dimensional encoding of the
local image features conveys enough information for a semantic categorization of
the scene/context [8] and can be used for contextual priming in object recognition
tasks [13]. Such a representation can be achieved by decomposing the image features
into the basis functions provided by PCA:
an ==
L L v{x, k) 1/ln{x, k)
x
k
N
v(x, k) ~
L an1/ln(x, k)
(4)
n=l
We propose to use the decomposition coefficients vc == {a n }n=l,N as context features. The functions 1/ln are the eigenfunctions of the covariance operator given by
v(x, k). By using only a reduced set of components (N == 60 for the rest of the
paper), the coefficients {a n }n=l,N encode the main spectral characteristics of the
scene with a coarse description of their spatial arrangement. In essence, {a n }n=l,N
is a holistic representation as all the regions of the image contribute to all the coefficients, and objects are not encoded individually [8]. In the rest of the paper we
show the efficacy of this set of features in context modeling for object detection
tasks.
3
Contextual object priming
The PDF P( On Iva) gives the probability of presence of the object class On given
contextual information. In other words, the PDF P{on Ive) evaluates the consistency of the object On with the context vc. For instance, a car has a high
probability of presence in a highway scene but it is inconsistent with an indoor
environment. The goal of P(on Ive) is to cut down the number of possible object categories to deal with before- expending computational resources in the object
recognition process. The learning of the PDF P(on Ive) == P(ve IOn)P(on)/p(ve)
with p(vo) == P(vc IOn)P{on) + P(vc l-,on)P(-,on) is done by approximating the
in-class and out-of-class PDFs by a mixture of Gaussians:
L
P(ve IOn) ==
L bi,nG(VC;Vi,n, Vi,n)
i=l
(5)
Figure 2: Contextual control of focus of attention when the algorithm is looking for
cars (upper row) or heads (bottom row).
The model parameters (bi,n, Vi,n, Vi,n) for the object class On are obtained using the
EM algorithm [3]. The learning requires the use of few Gaussian clusters (L == 2
provides very good performances). For the learning, the system is trained with
a set of examples manually annotated with the .presence/absence of four objects
categories (i-people, 2-furniture, 3-vehicles and 4-trees). Fig. 1 shows some typical
results from the priming model on the four superordinate categories of objects
defined. Note. that the probability function P(on Ive) provides information about
the probable presence of one object without scanning the picture. If P( On Ive) > 1th then we can predict that the target is present. On the other hand, if P( On Ive) <
th we can predict that the object is likely to be absent before exploring the image.
The number of scenes in which the system may be able to take high confidence
decisions will depend on different factors such as: the strength of the relationship
between the target object and its context and the ability of ve for efficiently characterizing the context. Figure 1 shows some typical results from the priming model for
a set of super-ordinate categories of objects. When forcing the model to take binary
decisions in all the images (by selecting an acceptance threshold of th == 0.5) the
presence/absence of the objects was correctly predicted by the model on 81 % of the
scenes of the test set. For each object category, high confidence predictions (th == .1)
were made in at least 50% of the tested scene pictures and the presence/absence
of each object class was correctly predicted by the model on 95% of those images.
Therefore, for those images, we do not need to use local image analysis to decide
about the presence/absence of the object.
4
Contextual control of focus of attention
One of the strategies that biological visual systems use to deal with the analysis
of real-world scenes is to focus attention (and, therefore, computational resources)
onto the important image regions while neglecting others. Current computational
models of visual attention (saliency maps anQ target detection) rely exclusively on
local information or intrinsic object models [6,7,9,14,16]. The control of the focus
of attention by contextual information that we propose. here is both task driven
(looking for object on) and context driven (given global context information: ve).
However, it does riot include any model of the target object at this stage. In our
framework, the problem of contextual control of the focus of attention involves the
S??
10 .~ ...
1
~
CARS
o
???
P;
~
_
-
filii
:
?? fIlIIe?':
..
?? \
\.:
~
.\.
tI':,._.: ??
, ??-=- ??
.-
0: ?
S:
.~
I
? ?
"'0
, -: ?
?
fill
":I':?.?
1
CARS.
~
t. ,.,:-.,,, ?
E
1~ ,
Q.)
~
.~.~.
"'0
~
100
HEADS
???? 11 1.8
?? ~
Q.)
100 ]
~
? ?\
II
Real scale
10 pixels 100
0.4
1 ,---~_"""""""-----R_eal_sc_al--..Je
1
10 pixels 100
tre.
?
.tto
?
.:
0.4
oReal pose
1
Figure 3: Estimation results of object scale and pose based on contextual features.
evaluation of the PDF P(xlon,vo). For the learning, the joint PDF is modeled
as a sum of gaussian clusters. Each cluster is decomposed into the product of
two gaussians modeling respectively the distribution of object locations and the
distribution of contextual features for each cluster:
L
P(x, vol on) == L bi,n G(x; Xi,n, Xi,n)G(VO; Vi,n, Vi,n)
(6)
i==l
The training set used for the learning of the PDF P(x, vol on) is a subset of'the
pictures that contain the object On. The training data is {Vt}t==l,Nt and {Xt}t==l,Nt
where Vt are the contextual features of the picture t of the training set and Xt is
the location of object On in the image. The model parameters are obtained using
the EM algorithm [3,13]. We used 1200 pictures for training and a separate set of
1200 pictures for testing. The success of the PDF in narrowing the region of the
focus of attention will depend on the consistency of the relationship between the
object and the context. Fig. 2 shows several examples of images and the selected
regions based on contextual features when looking for cars and faces. From the
PDF P(x, Vo IOn) we selected the region with the highest probability (33% of the
image size on average). 87% of the heads present in the test pictures were inside
the selected regions.
5
Contextual selection of object appearance models
One major problem for computational approaches to object detection is the large
variability in object appearance. The classical solution is to explore the space of
possible shapes looking for the best match. The main sources of variability in object
appearance are size, pose and intra-class shape variability (deformations, style, etc.).
We show here that including contextual information can reduce at le.ast the first
two sources of variability. For instance, the expected size of people in an image
differs greatly between an indoor environment and a perspective view of a street.
Both environments produce different patterns of contextual features vo [8]. For
the second factor, pose, in the case of cars, there is a strong relationship between
the possible orientations of the object and the scene configuration. For instance,
looking down a highway, we expect to see the back of the cars, however, in a street
view, looking towards the buildings, lateral views of cars are more likely.
The expected scale and pose of the target object can be estimated by a regression
procedure. The training database used for building the regression is a set of 1000
images in which the target object On is present. For each training image the target
Figure 4: Selection of prototypical object appearances based on contextual cues.
object was selected by cropping a rectangular window. For faces and cars we define
the u == scale as the height of the selected window and the P == pose as the ratio between the horizontal and vertical dimensions of the window (~y/ ~x). On average,
this definition of pose provides a good estimation of the orientation for cars but not
for heads. Here we used regression using a mixture of gaussians for estimating the
conditional PDFs between scale, pose and contextual features: P(u I Va, on) and
PCP I va, on). This yields the next regression procedures [3]:
(j
== Ei Ui,nbi,n G (Va; Vi,n, Vi,n)
Ei bi,nG(vO; Vi,n, Vi,n)
_ EiPi,nbi,nG(VO;Vi,n, Vi,n)
P == Ei bi,nG(VC;Vi,n, Vi,n)
(7)
The results summarized in fig. 3 show that context is a strong cue for scale selection for the face detection task but less important for the car detection task. On
the other hand, context introduces strong constraints on the prototypical point of
views of cars but not at all for heads. Once the two parameters (pose and scale)
have been estimated, we can build a prototypical model of the target object. In the
case of a view-based object representation, the model of the object will consist of
a collection of templates that correspond to the possible aspects of the target. For
each image the system produces a collection of views, selected among a database
of target examples that have the scale and pose given by eqs. (7). Fig. 4 shows
some results from this procedure. In the statistical framework, the object detection requires the evaluation of the function P(VL 10, va). We can approximate
Input image
(target = cars)
Object priming and
Contextual control
Target model selection of focus of attention
Integration of
local features
Target saliency
1
Figure 5: Schematic layout of the model for object detection (here cars) by integration of contextual and local information. The bottom example is an error in
detection due to incorrect context identification.
P(VL 10, va) ~ P(VL IOn' (J", p). Fig. 5 and 6 show the complete chain of operations and some detection results using a simple correlation technique between the
image and the generated object models (100 exemplars) at only one scale. The last
image of each row shows the total object likelihood obtained by multiplying the
object saliency maps (obtained by the correlation) and the contextual control of
the focus of attention. The result shows how the use of context helps reduce false
alarms. This results in good detection performances despite the simplicity of the
matching procedure used.
6
Conclusion
The contextual schema of a scene provides the likelihood of presence, typical locations and appearances of objects within the scene. We have proposed a model for
incorporating such contextual cues in the task of object detection. The main aspects
of our approach are: 1) Progressive reduction of the window of focus of attention:
the system reduces the size of the focus of attention by first integrating contextual
information and then local information. 2) Inhibition of target like patterns that
are in inconsistent locations. 3) Faster detection of correctly scaled targets that
have a pose in agreement with the context. 4) No requirement of parsing a scene
into individual objects. Furthermore, once one object has been detected, it can
introduce new contextual information for analyzing the rest of the scene.
Acknowledglllents
The author wishes to thank Dr. Pawan Sinha, Dr. Aude Oliva and Prof. Whitman
Richards for fruitful discussions.
References
[1] Biederman, I., Mezzanotte, R.J., & Rabinowitz, J.C. (1982). Scene perception: detecting and judging objects undergoing relational violations. Cognitive Psychology, 14:143177.
Feature maps
\
I
V t---HXJ---+l
..
..
~
Figure 6: Schema for object detection (e.g. cars) integrating local and giobal information.
[2] Carson, C., Belongie, S., Greenspan, H., and Malik, J. (1997). Region-based image
querying. Proc. IEEE W. on Content-Based Access of Image and Video Libraries, pp:
42-49.
[3] Gershnfeld, N. The nature of mathematical modeling. Cambridge university press, 1999.
[4] Gorkani, M. M., Picard, R. W. (1994). Texture orientation for sorting photos 'at a
glance'. Proc. Int. Conf. Pat. Rec., Jerusalem, Vol. I: 459-464.
[5] Heisle, B., T. Serre, S. Mukherjee and T. Poggio. (2001) Feature Reduction and Hierarchy of Classifiers for Fast Object Detection in Video Images. In: Proceedings of 2001
IEEE Computer Society Conference on Computer Vision and Pattern Recognition, IEEE
Computer Society Press, Jauai, Hawaii.
[6] Itti, L., Koch, C., & Niebur, E. (1998). A model of saliency-based visual attention for
rapid scene analysis. IEEE Trans. Pattern Analysis and Machine Vision, 20(11):1254.
[7] Moghaddam, B., & Pentland, A. (1997). Probabilistic Visual Learning for Object Representation. IEEE Trans. Pattern Analysis and Machine Vision, 19(7):696-710.
[8] Oliva, A., & Torralba, A. (2001). Modeling the Shape of the Scene: A holistic representation of the spatial envelope. Int. Journal of Computer Vision, 42(3):145-175.
[9] Rao, R.P.N., Zelinsky, G.J., Hayhoe, M.M., & Ballard, D.H. (1996). Modeling saccadic
targeting in visual search. NIPS 8. Cambridge, MA: MIT Press.
[10] Schiele, B., Crowley, J. L. (2000) Recognition without Correspondence using Multidimensional Receptive Field Histograms, Int. Journal of Computer Vision, Vol. 36(1):31-50.
[11] Strat, T. M., & Fischler, M. A. (1991). Context-based vision: recognizing objects
using information from both 2-D and 3-D imagery. IEEE trans. on Pattern Analysis and
Machine Intelligence, 13(10): 1050-1065.
[12] Szummer, M., and Picard, R. W. (1998). Indoor-outdoor image classification. In
IEEE intl. workshop on Content-based Access of Image and Video Databases, 1998.
[13] Torralba, A., & Sinha, P. (2001). Statistical context priming for object detection.
IEEE Proc. Of Int. Conf in Compo Vision.
[14] Treisman, A., & Gelade, G. (1980). A feature integration theory of attention. Cognitive Psychology, Vol. 12:97-136.
[15] Viola, P. and Jones, M. (2001). Rapid object detection using a boosted cascade
of simple features. In: Proceedings of 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2001), IEEE Computer Society Press, Jauai,
Hawaii.
[16] Wolfe, J. M. (1994). Guided search 2.0. A revised model of visual search. Psychonomic Bulletin and Review, 1:202-228
| 2074 |@word decomposition:2 covariance:1 attended:1 reduction:2 configuration:1 efficacy:1 selecting:1 exclusively:1 tuned:2 current:1 contextual:37 nt:2 pcp:1 parsing:1 shape:4 cue:3 selected:6 intelligence:1 compo:1 coarse:1 provides:9 contribute:1 location:11 detecting:1 height:1 mathematical:1 become:1 incorrect:1 inside:1 introduce:1 expected:2 rapid:2 growing:1 brain:1 decomposed:1 window:4 provided:1 estimating:1 kind:1 iva:6 multidimensional:1 ti:1 scaled:1 classifier:1 control:7 intrisic:1 before:2 local:19 despite:1 encoding:1 analyzing:1 initiated:1 modulation:2 bi:5 testing:1 differs:1 procedure:6 gabor:1 cascade:1 convenient:1 matching:1 word:1 confidence:2 integrating:2 onto:1 targeting:1 selection:5 operator:1 context:23 applying:1 ast:1 fruitful:1 map:3 layout:1 attention:15 jerusalem:1 rectangular:1 resolution:1 simplicity:1 rule:2 crowley:1 fill:1 coordinate:1 target:27 enhanced:1 hierarchy:1 agreement:1 wolfe:1 recognition:7 rec:1 richards:1 cut:1 mukherjee:1 database:3 bottom:3 role:1 narrowing:1 region:10 nbi:2 highest:1 environment:3 ui:1 schiele:1 fischler:1 trained:1 depend:4 basis:1 whitman:1 joint:1 fast:1 detected:1 exhaustive:2 encoded:1 posed:1 ive:6 cvpr:1 ability:1 propose:2 product:1 relevant:2 holistic:3 description:2 cluster:4 requirement:1 cropping:1 produce:3 categorization:1 intl:1 object:91 help:1 pose:11 exemplar:1 eq:1 strong:3 predicted:2 involves:1 guided:1 annotated:1 filter:4 vc:6 human:1 require:1 probable:1 biological:1 exploring:1 koch:1 predict:3 major:1 torralba:4 estimation:2 proc:3 individually:1 highway:2 mit:3 gaussian:2 super:1 greenspan:1 boosted:1 encode:2 focus:11 pdfs:2 likelihood:7 ivl:2 contrast:1 greatly:1 inference:1 vl:14 unlikely:3 pixel:2 among:1 ill:1 orientation:4 classification:1 spatial:6 integration:3 field:1 once:2 ng:4 manually:1 represents:1 progressive:1 look:1 jones:1 others:1 few:1 oriented:2 ve:5 comprehensive:1 individual:1 occlusion:1 pawan:1 detection:20 acceptance:1 intra:1 picard:2 evaluation:3 introduces:1 mixture:2 violation:1 chain:1 moghaddam:1 fu:1 neglecting:1 poggio:1 allocates:1 tree:2 guidance:1 deformation:1 riot:1 sinha:2 instance:3 modeling:5 rao:1 subset:1 recognizing:1 scanning:1 probabilistic:1 enhance:1 treisman:1 imagery:1 successively:1 zelinsky:1 dr:2 hawaii:2 cognitive:3 conf:2 style:1 itti:1 summarized:1 coefficient:3 int:4 vi:14 vehicle:2 view:7 schema:2 bayes:2 degraded:1 characteristic:1 efficiently:1 yield:1 saliency:12 correspond:1 identification:1 accurately:1 niebur:1 multiplying:1 definition:1 evaluates:1 energy:1 frequency:2 pp:1 conveys:1 popular:1 knowledge:1 car:16 back:1 strat:1 done:1 furthermore:1 stage:1 correlation:2 hand:3 tre:1 horizontal:1 ei:3 glance:1 rabinowitz:1 aude:1 building:2 serre:1 contain:1 semantic:1 deal:2 ambiguous:1 essence:1 carson:1 pdf:11 evident:1 complete:1 vo:7 image:35 psychonomic:1 belong:1 interpretation:1 measurement:5 cambridge:3 ai:1 consistency:2 access:2 inhibition:1 etc:1 perspective:1 driven:2 forcing:1 binary:1 success:1 vt:2 living:1 ii:1 expending:1 reduces:1 match:1 faster:1 va:18 schematic:1 prediction:1 regression:4 oliva:2 vision:9 histogram:1 normalization:1 achieved:1 ion:5 background:1 source:2 envelope:1 rest:3 eigenfunctions:1 inconsistent:2 obj:3 structural:1 presence:11 split:1 enough:1 psychology:2 reduce:2 absent:1 pca:1 antonio:1 band:2 category:6 reduced:1 judging:1 estimated:2 correctly:3 write:2 vol:5 four:3 threshold:1 capital:1 immersed:2 sum:1 decide:1 decision:2 furniture:2 correspondence:1 tto:1 strength:1 constraint:2 scene:21 dominated:1 aspect:6 according:1 poor:1 em:2 partitioned:1 ln:3 resource:3 describing:2 photo:1 decomposing:1 gaussians:3 operation:1 apply:1 spectral:1 top:1 include:2 build:1 prof:1 approximating:1 classical:1 society:4 malik:1 added:1 arrangement:1 strategy:1 saccadic:1 receptive:1 separate:1 thank:1 lateral:1 street:2 anq:1 o1:2 index:1 relationship:3 modeled:1 ratio:3 gk:2 upper:1 vertical:1 revised:1 pentland:1 pat:1 situation:2 relational:1 looking:8 head:5 variability:4 viola:1 hxj:1 biederman:1 ordinate:1 nip:1 trans:3 gelade:1 able:2 hayhoe:1 superordinate:1 perception:2 pattern:7 indoor:3 summarize:1 including:2 reliable:1 video:3 rely:1 representing:1 scheme:1 vetoed:1 library:1 picture:7 prior:2 review:1 ina:1 fully:1 expect:1 prototypical:4 querying:1 row:3 last:1 template:1 characterizing:1 face:3 bulletin:1 dimension:1 world:2 author:1 insuring:1 commonly:1 made:1 simplified:1 collection:2 approximate:1 cutting:1 global:1 belongie:1 xi:2 search:8 an1:1 nature:1 ballard:1 contributes:1 priming:8 complex:1 rue:1 main:4 whole:1 alarm:1 fig:5 je:1 tl:1 wish:1 outdoor:1 third:1 down:4 xt:2 showing:1 undergoing:1 evidence:1 intrinsic:6 incorporating:2 consist:1 false:1 workshop:1 importance:1 modulates:1 texture:1 magnitude:1 sorting:1 likely:5 appearance:9 explore:1 visual:6 ma:2 conditional:1 goal:1 towards:1 room:1 absence:4 content:2 typical:4 determined:1 total:1 pas:2 people:3 szummer:1 dept:1 tested:1 |
1,179 | 2,075 | Modeling the Modulatory Effect of
Attention on Human Spatial Vision
Laurent Itti
Computer Science Department, Hedco Neuroscience Building HNB-30A,
University of Southern California, Los Angeles, CA 90089-2520, U.S.A.
J oehen Braun
nstitute of Neuroscience and School of Computing,
University of Plymouth, Plymouth Devon PL4 8AA, U.K.
Christof Koch
Computation and Neural Systems Program, MC 139-74,
California Institute of Technology, Pasadena, CA 91125 , U.S.A.
Abstract
We present new simulation results , in which a computational model
of interacting visual neurons simultaneously predicts the modulation of spatial vision thresholds by focal visual attention, for five
dual-task human psychophysics experiments. This new study complements our previous findings that attention activates a winnertake-all competition among early visual neurons within one cortical
hypercolumn. This "intensified competition" hypothesis assumed
that attention equally affects all neurons, and yielded two singleunit predictions: an increase in gain and a sharpening of tuning
with attention. While both effects have been separately observed
in electrophysiology, no single-unit study has yet shown them simultaneously. Hence, we here explore whether our model could still
predict our data if attention might only modulate neuronal gain,
but do so non-uniformly across neurons and tasks. Specifically, we
investigate whether modulating the gain of only the neurons that
are loudest, best-tuned, or most informative about the stimulus,
or of all neurons equally but in a task-dependent manner, may account for the data. We find that none of these hypotheses yields
predictions as plausible as the intensified competition hypothesis,
hence providing additional support for our original findings.
1
INTRODUCTION
Psychophysical studies as well as introspection indicate that we are not blind outside the focus of attention, and that we can perform simple judgments on objects
not being attended to [1], though those judgments are less accurate than in the
presence of attention [2, 3]. While attention thus appears not to be mandatory for
early vision, there is mounting experimental evidence from single-neuron electrophysiology [4, 5, 6, 7, 8, 9, 10], human psychophysics [11 , 12, 13, 14,3, 2, 15, 16] and
human functional imaging experiments [17, 18, 19, 20, 21, 22, 23] that focal visual
attention modulates, top-down, activity in early sensory processing areas. In the
visual domain, this modulation can be either spatially-defined (i.e., neuronal activity only at the retinotopic location attended to is modulated) or feature-based (i.e.,
neurons with stimulus preference matching the stimulus attended to are enhanced
throughout the visual field), or a combination of both [7, 10, 24].
Computationally, the modulatory effect of attention has been described as enhanced
gain [8, 10], biased [4] or intensified [14, 2] competition, enhanced spatial resolution
[3], sharpened neuronal tuning [5, 25] or as modulated background activity [19],
effective stimulus strength [26] or noise [15]. One theoretical difficulty in trying
to understand the modulatory effect of attention in computational terms is that,
although attention profoundly alters visual perception, it is not equally important
to all aspects of vision. While electrophysiology demonstrates "increased firing
rates" with attention for a given task, psychophysics show "improved discrimination
thresholds" on some other tasks, and functional magnetic resonance imaging (fMRI)
reports "increased activation" for yet other tasks, the computational mechanism at
the origin of these observations remains largely unknown and controversial.
While most existing theories are associated to a specific body of data, and a specific experimental task used to engage attention in a given experiment, we have
recently proposed a unified computational account [2] that spans five such tasks (32
thresholds under two attentional conditions, i.e., 64 datapoints in total). This theory predicts that attention activates a winner-take-all competition among neurons
tuned to different orientations within a single hyper column in primary visual cortex
(area VI). It is rooted in new information-theoretic advances [27], which allowed us
to quantitatively relate single-unit activity in a computational model to human psychophysical thresholds. A consequence of our "intensified competition hypothesis"
is that attention both increases the gain of early visual neurons (by a factor 3.3),
and sharpens their tuning for the orientation (by 40%) and spatial frequency (by
30%). While gain modulation has been observed in some of the single-unit studies
mentioned above [8, 10] (although much smaller effects are typically reported, on the
order of 10-15%, probably because these studies do not use dual-task paradigms and
thus poorly engage the attention of the animal towards or away from the stimulus
of interest), and tuning modulation has been observed in other single-unit studies
[5, 25], both gain and tuning modulation have not been simultaneously observed in
a single electrophysiological set of experiments [10].
In the present study, we thus investigate alternatives to our intensified competition hypothesis which only involve gain modulation. Our previous results [2] have
shown that both increased gain and sharper tuning were necessary to simultaneously account for our five pattern discrimination tasks, if those modulatory effects
were to equally affect all visual neurons at the location of the stimulus and to be
equal for all tasks. Thus, we here extend our computational search space under two
new hypotheses: First, we investigate whether attention might only modulate the
gain of selected sub-populations of neurons (responding the loudest, best tuned , or
most informative about the stimulus) in a task-independent manner. Second, we
investigate whether attention might equally modulate the gain of all visual neurons
responding to the stimulus, but in a task-dependent manner. Thus, the goal of
the present study is to determine, using new computational simulations, whether
the modulatory effect of attention on early visual processing might be explained by
gain-only modulations, if such modulations are allowed to be sufficiently complex
(affecting only select visual neurons , or task-dependent). Although attention certainly affects most stages of visual processing, we here continue to focus on early
vision, as it is widely justified by electrophysiological and fMRI evidence that some
modulation does happen very early in the processing hierarchy [5, 8, 9, 23].
2
PSYCHOPHYSICAL DATA
Our recent study [2] measured psychophysical thresholds for three pattern discrimination tasks (contrast, orientation and spatial frequency discriminations), and two
spatial masking tasks (32 thresholds) . We used a dual-task paradigm to measure
thresholds either when attention was fully available to the task of interest (presented
in the near periphery), or when it was poorly available because engaged elsewhere
by a concurrent attention-demanding task (a letter discrimination task at the center
of the display). The results are summarized in Fig. 1 and [2].
'I'
c::
A
B
0
4::
c::
.g
.S
.g
~
E
.;:
.Q...
.S
~
.?
~
c::
.S
.~
.~
~
... 8
Q
'"
E
c::
~
-
10.1
W
~
['
? Fully Attended
0 Poorly Attended
I
t ;
:~
f-
10-2
~
c::
CU
::s
:
t~
I
~
;
a4::
!!
/1
",0.3
Q)
>
oJ
~ 0.1
00
~
10'2
10"
Mask contrast
0
~
1:3 0.2
0- ?-
I
C
~
....
II I
Q
c::
.g
.sc::
II I
0.2
8
.~
0.4 0.6
Contrast
0.8
1
.~
0
6
I
~ 4j
2
00
!
0.2
0.4
0.6
0.8
1
Contrast
'I'
D
E
I't
I:J\
I:J\
.S
.S
.lC
'"
.lC
0.4
~
...
:E
1;;
l!! 0.3
'" C8 0.2
E
c:: W
~ f-~ 0.1
0
h
? Fully Attended
o Poorly Attended
0.4
~
...
1;;
0
"
c:: W
I
20
40
I
60
80
Mask e - Target
e C)
I
I
'"
l!! 0.3
'" c
E
1+
0
'"
:E
~ f-~
0.2
0.1
0
0.5
Mask w / Target w
2
Figure 1: Psychophysical data from Lee et ai. Central targets appeared at 0 - 0.8?
eccentricity and measured 0.4? across. Peripheral targets appeared for 250 ms at 4?
eccentricity, in a circular aperture of 1.5 0 ? They were either sinusoidal gratings (B,
C) or vertical stripes whose luminance profile was given by the 6th derivative of a
Gaussian (A, D, E) . Mask patterns were generated by superimposing 100 Gabor
filters , positioned randomly within the circular aperture (A, D, E). Thresholds
were established with an adaptive staircase method (80 trials per block). A complex
pattern of effects is observed, with a strong modulation of orientation and spatial
frequency discriminat ions (B, C) , smaller modulation of contrast discriminations
(A) , and modulation of contrast masking that depends on stimulus configurations
(D, E). These complex observations can be simultaneously accounted for by our
computational model of one hypercolumn in primary visual cortex.
3
COMPUTATIONAL MODEL
The model developed to quantitatively account for this data comprises
three successive stages [14, 27]. In the
first stage, a bank of Gabor-like linear
filters (12 orientations and 5 spatial
scales) analyzes a given visual location, similarly to a cortical hyper column. In the second stage, filters nonlinearly interact through both a selfexcitation component, and a divisive
inhibition component that is derived
from a pool of similarly-tuned units. With E)."o being the linear response from a
unit tuned to spatial period A and orientation (), the response R)."o after interactions
is given by (see [27] for additional details):
Linear filters
Divisive inhibition
R
Decision
).,,0 -
(S)O
L
+
+B
(A.E)."o) '
W)."o(A',()') (A.E)..!,O')O
(1)
'
()..! ,O') EA x 8
where:
W
(A' ()') =
)., ,0 ,
exp
(_ (log(A') -log(A))2 _ (()' - ())2)
2A2
2A2
).,
(2)
0
is a 2D Gaussian weighting function centered around (A, ()) whose widths are determined by the scalars Ao and A).,. The neurons are assumed to be noisy, with noise
variance V{o given by a generalized Poisson model: V{o = (3(R)."o + <:).
The third stage relates activity in the population of interacting noisy units to behavioral discrimination performance. To allow us to quantitatively predict thresholds
from neural activity for any task, our decision stage assumes that observers perform
close to an unbiased efficient statistic, that is, the best possible estimator (in the
statistical estimation sense) of the characteristics of the stimulus given the noisy
neuronal responses. This methodology (described further in [27]) allows us to quantitatively compute thresholds in any behavioral situation, and eliminates the need
for task-dependent assumptions about the decision strategy used by the observers.
4
RESULTS and DISCUSSION
The 10 free model parameters (Fig. 2) were automatically adjusted to best fit
the psychophysical data from all experiments, using a multidimensional downhill simplex with simulated annealing overhead (see [27]) , running on our 16CPU Linux Beowulf system (16 x 733 MHz, 4 GB RAM, 0.5 TB disk; see
http://iLab . usc. edu/beo/). Parameters were simultaneously adjusted for both
attentional conditions; that is, the total fit error was the sum of the error obtained
with the baseline set of parameters on the poorly attended data, and of the error
obtained with the same parameters plus some attentional perturbation on the fully
attended data. Thus, no bias was given to any of the two attentional conditions.
For the "separate fits" (Fig. 2), all parameters were allowed to differ with attention [2], while only the interaction parameters b, 8) could differ in the "intensified
competition" case. The "loudest filter" was the one responding loudest to the entire visual pattern presented (stimulus + mask), the "best-tuned filter" was that
responding best to the stimulus component alone, and the "most informative filter" was that for which the Fisher information about the stimulus was highest (see
Model
Parameters
y,o:
S, ?:
11
S:
y
-
Interaction strength
~: ~~; ~::~:~ ~~~~g
Attentional
manipulations
Top-down Attention
Dark noise
Light noise
Semisaturation
~ Ul
r Q (" )
:!: ,,(oct)
B/ A
S/A
" ,,(oct)
(I)
11
en
? / Rmu,
-.- .---
0.01
0.11
- very good fit overall
- all parameters biologically
plausible
- attention significantly
modulates interactions
and noise
~
BI A
*****
Top-down Attention
Parameters
(I)
m
0.65
0.01
Parameters
7.03
0.24
3.00
1.5e-9
8635.52
?
c
-
-
-........./
0.31
3~:!~
~::
I: 9(")
I: .. (OCI)
BIA
S IA
~
EI R"""
Parameters
-
Poorly
~:~~~
\~"\
..~
'\,
9 (" )
o .. (OCI)
... QQQ ...
'y".... -
Fully
1.45 ~05
Top-down Attention
.
- no modulation of contrast
detection threshold
- no modulation of orientation thresholds
- no modulation of period
thresholds
- poor prediction of masking
- filter tuning too narrow
- gain modulation too large
- no modulation of orientation th resholds
- no modulation of period
thresholds
- contrast discrimination
and masking well fit
- only fit predicting broad
pooling in spatial period
- noise parameters unrealistic
****
Parameters
[affects all filters, but differently for each task]
.~
I
~.
,,,,,,.
V-"\.
14.71
~~ Q~Q
(I)
Poorly
16:: '\\
[stimulus-dependent; only affects filter
most infonnative a~out target stimulus]
1-8.
*
Fully
1.40 ~~
Top-down Attention
c
\.'\
1.09
0.21
12.16
~~ Q~~
(I)
~
\'""i.
6.62
1 ~:~
[stimulus-dependent; only affects filter
beslluned to/arget stimulus]
!=
UlLL
Poorly
1.03
~. Q ~~
...
Ul"
Fully
2.22 ~56
[stimulus-dependent; only affects filter responding
most to whole (targr+maSk) stimulus]
Top-down Attention
~(I)
0.96
5.39
0.56
0.02
S/A
P
:::l=
OLL
-I
c
- very good fit overall
- all parameters biologically
plausible
- modulation of orientation
thresholds slightly underestimated
- contrast masking with
variable mask orientation
not perfectly predicted
-
8. 92
0.41
20.80
0.31
EI R"""
",S
-
*****
Poorly
3.941.40
3.551.00
0 0 (0)
O,,(OCI)
I: 9 (")
I: ,,(oct)
(I)
- ...
CU
1. 10
8.05
0.17
0.03
Fully
co.
,SE
c 0
-0
:::l
0.30
10.12
Parameters -
(1).-
(I)
1.85
-
*****
Top-down Attention
"5
.-
"c
Poorly
3.78
3.421 .80
9.6913.19
0.440.36
23.01 23.90
0.810.18
0 0 (0)
Q.LL
Ul
Fully
Parameters -
(I)
CU:!::
Discussion
-..........-
(1" 9( ")
(I" .,(oct)
I: 9(")
r .,(oct)
BIA
S/A
,
?/R""",
3.4<-4
0.01
*
Fully
- no contrast discrimination
"dipper"
- power-law rather than
sigmoidal contrast response (S=O)
- modulation of orientation
thresholds slightly underestimated
- noise parameter unrealistic
***
Poorl y
- very good fit overall
- gain modulation unrealistically high , especially
for orientation discrimination (filter gain when
attending to orientation
is > 20x poorly attended)
Figure 2: Attentional modulation hypot heses and corresponding model parameters.
See next page for the corresponding model predictions on our five tasks, for the
hypot heses shown. The middle column shows which parameters were allowed to
differ with attention, and t he best-fit values for both attentional conditions.
Contrast
Increment
Spatial Frequency
Discrimination
Orientation
Discrimination
Contrast Masking,
Variable Mask e
R
lM~
,.~
C
!
0.2
~
.0.1
810 "
~"
10 -,
0.2
0.4 0.6
Contrast
0.8
1
R02
I .
~ 0.1
???
"----'--,--"'7----l
o
10 .2
?0
10 "
0.2
Mask contrast
****
- Fully attended
Poorly attended
10"2
10"
Mask contrast
0.2
0.4 0.6
Contrast
0.8
1
*****
0.4 0.6
Contrast
C
?
0.8
- Fully attended
Poorly attended
8'0"
0.4
0.6
0.8
1
!:~~
0.2
0.4 0.6
Contrast
0.8
Poorly attended
o
0.2
0.4
0.6
f
?
? Fullyattended
POOrly attended
0.8
1
c:B I
%4
'..
10'>
00
10-'
0.2
0.4 0.6
Contrast
0.8
1
00
0.2
0.4 0.6
Contrast
1
****
0;
?
l"
- Fully attended
Poorly attended
?10"
o
f.3~ r
i
j
~ 01
'.
?
~
00
**
4 .
0.4 0.6
Contrast
0.8
1
0.3
~
~
~0.1 . ~
l"
10'>
10.2
10"
Mask contrast
***
00
00
0.2
0.4 0.6
Contrast
0.8
*****
0.2
0.4 0.6
Contrast
0.8
1
1
8~
t
6
i,
2
00
I
0.4 0.6
Contrast
?
0.5
1
MaskwfTarget
*****
2
****
0.4[ 8
~
03
,
8
0.2
'.
I
?
*****
*****
? Fully attended
Poorly attended
+
(J)
o
I
0.8
2
t
0020406080
Mask O- TargetOr}
I
o
0.5
1
MaskwfTarget
+
2
(J)
0.4[ 8
t
0.3
0.2
~
' ? ?
0.2
2
!?
0.5
1
MaskwlTargetw
$ 0.1
?
***
!o,\~ ~
~'0"
0.5
1
MaskwlTargetw
Qj
2
.
I
0.4~
I .
o
?020406080
Mask O- TargetOr}
? Fully attended
Poorly attended
*****
~
~
0.2
.
0. 3
8 02
$ 0.1
10 ~
10"
10 "
Mask contrast
j
t
Qj
*****
R0.2
0.1
*
? Fully attended
Poorly attended
?
0.8
2
0.4~
0.2
$" 0.1
60 80
Mask6 - Target6(O)
?
2
0.5
1
MaskwlTargetw
****
t:0.4~
I . I
10~
Mask contrast
+
0.3
o
?020406080
***
Contrast
f3~
1!!
~
.~
~
I I
1
?
Mask6 - Target6(O)
Contrast
.~
~
I
~ 0.1
!!'-
0.2
***
?0
L
*****
? Fullyattended
?
****
R0.2
~
???
?0
\
I
'
0.2
o
?020406080
Mask6 - Target6(O)
****
I
6
2
1
0.4[ 8
0.3
~
~
***
~
?0
*****
~
8
C
~ 0.1
lM~ ,.~I
~
? Fully attended
Poorly attended
2
?0
****
!
I
6
Contrast Masking,
Variable Mask 00
?
?
+
~ 0.1
1
0020406080
MaskO - TargetO(O}
o
*****
*****
0.5
1
MaskwlTargetw
2
Figure 3: Model predictions for t he different attentional modulation hypotheses
studied. The different rows correspond to t he different attentional manipulations
studied, as labeled in t he previous figure. Ratings (stars below t he plots) were
derived from t he residual error of t he fits .
[14, 27]). Finally, in the "task-dependent" case, the gain of all filters was affected
equally (parameter ')'), but with three different values for the contrast (discrimination and masking), orientation and spatial frequency tasks. Overall, very good
fits were obtained in the "separate fits" and "intensified competition" conditions
(as previously reported) , as well as in the "most informative filter" and "taskdependent" conditions (Fig. 3) , while the two remaining hypotheses yielded very
poor predictions of orientation and spatial frequency discriminations. In the "most
informative filter" case, the dipper in the contrast increment thresholds was missing
because the nonlinear response function of the neurons converged to a power law
rather than the usually observed sigmoid [27]; thus, this hypothesis lost some of
its appeal because of its lower biological plausibility. More importantly, a careful
analysis of the very promising results for the "task-dependent" case also revealed
their low biological plausibility, with a gain modulation in excess of 20-fold being
necessary to explain the orientation discrimination data (Fig. 2).
In summary, we found that none of the simpler (gain only) attentional manipulations studied here could explain as well the psychophysical data as our previous manipulation, "intensified competition," which implied that attention both increases
the gain and sharpens the tuning of early visual neurons. Two of the four new
manipulations studied yielded good quantitative model predictions: affecting the
gain of the filter most informative about the target stimulus, and affecting the gain
of all filters in a task-dependent manner. In both cases, however, some of the internal model parameters associated with the fits were biologically unrealistic, thus
reducing the plausibility of these two hypotheses. In all manipulations studied, the
greatest difficulty was in trying to account for the orientation and spatial frequency
discrimination data without unrealistically high gain changes (greater than 20-fold).
Our results hence provide additional evidence for the hypothesis that sharpening
of tuning may be necessary to account for these thresholds, as was originally suggested by our separate fits and our intensified competition hypothesis and has been
recently supported by new investigations [16].
Acknowledgements
This research was supported by the National Eye Institute, the National Science
Foundation, the NSF-supported ERC center at Caltech, the National Institutes for
Mental Health, and startup funds from the Charles Lee Powell Foundation and the
USC School of Engineering.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Braun J & Sagi D. P ercept Psychophys , 1990;48(1):45- 58.
Lee DK , Itti L, Koch C et al. Nat Neurosci, 1999;2(4):375-81.
Yeshurun Y & Carrasco M. Nature, 1998;396(6706) :72- 75 .
Moran J & Desimone R . Science , 1985 ;229(4715) :782- 4.
Spitzer H, Desimone R & Moran J. Sci ence, 1988 ;240(4850):338- 40 .
Chelazzi L, Miller EK, Duncan J et al. Nature , 1993;363(6427):345- 7.
Motter BC. J Neurosci, 1994;14(4):2178-89.
Treue S & Maunsell JH. Nature, 1996;382(6591):539- 41.
Luck SJ, Chelazzi L, Hillyard SA et al. J Neurophysiol, 1997;77(1) :24- 42.
Treue S & Trujillo JCM . Nature, 1999 ;399(6736) :575- 579 .
Nakayama K & Mackeben M. Vision Res, 1989 ;29(11) :1631- 47.
Bonnel AM , Stein JF & Bertucci P. Q J Exp Psychol A, 1992 ;44(4):601- 26.
Lee DK , Koch C & Braun J. Vision R es, 1997;37(17):2409- 18 .
[14] Itti L, Braun J, Lee DK et al. In NIPS*ll. MIT Press, 1999; pp. 789- 795.
[15] Dosher BA & Lu ZL. Vision Res, 2000;40(10-12):1269- 1292.
[16] Carrasco M, Penpeci-Talgar C & Eckstein M. Vision Res, 2000;40(10-12):1203- 1215.
[17] Corbett a M, Miezin FM, Dobmeyer S et al. Science, 1990;248(4962):1556- 9.
[18] Rees G, Frackowiak R & Frith C. Science, 1997;215(5301):835- 8.
[19] Chawla D, Rees G & Friston KJ. Nat Neurosci, 1999;2(7):671- 676.
[20] Brefczynski JA & DeYoe EA. Nat Neurosci, 1999;2(4):370- 374.
[21] Corbetta M, Kincade JM, Ollinger JM et al. Nat Neurosci, 2000;3(3):292- 297.
[22] Kanwisher N & Wojciulik E. Nat Rev Neurosci, 2000;1:91- 100.
[23] Ress D, Backus BT & Heeger DJ. Nat Neurosci, 2000;3(9):940- 945.
[24] Barcelo F, Suwazono S & Knight RT. Nat Neurosci, 2000;3(4) :399- 403.
[25] Desimone R & Duncan J . Annu Rev Neurosci, 1995 ;18:193- 222 .
[26] Reynolds JH, Pasternak T & Desimone R. Neuron, 2000;26(3):703- 714 .
[27] Itti L, Koch C & Braun J. J Opt Soc Am A, 2000;11(11):1899- 1917.
| 2075 |@word trial:1 cu:3 middle:1 sharpens:2 disk:1 simulation:2 attended:26 configuration:1 tuned:6 bc:1 reynolds:1 existing:1 activation:1 yet:2 happen:1 informative:6 plot:1 fund:1 mounting:1 discrimination:16 alone:1 selected:1 mental:1 location:3 preference:1 successive:1 sigmoidal:1 simpler:1 five:4 overhead:1 behavioral:2 manner:4 kanwisher:1 mask:16 automatically:1 cpu:1 jm:2 retinotopic:1 spitzer:1 developed:1 unified:1 finding:2 sharpening:2 quantitative:1 multidimensional:1 braun:5 demonstrates:1 zl:1 unit:7 maunsell:1 christof:1 engineering:1 sagi:1 consequence:1 laurent:1 firing:1 modulation:24 might:4 plus:1 studied:5 co:1 bi:1 lost:1 block:1 powell:1 area:2 gabor:2 significantly:1 matching:1 close:1 center:2 missing:1 attention:34 resolution:1 estimator:1 attending:1 importantly:1 datapoints:1 population:2 increment:2 enhanced:3 hierarchy:1 target:6 engage:2 hypothesis:12 origin:1 qqq:1 stripe:1 carrasco:2 predicts:2 labeled:1 observed:6 highest:1 luck:1 backus:1 knight:1 mentioned:1 arget:1 neurophysiol:1 yeshurun:1 differently:1 frackowiak:1 effective:1 sc:1 startup:1 hyper:2 outside:1 whose:2 widely:1 plausible:3 statistic:1 noisy:3 interaction:4 poorly:20 competition:11 los:1 r02:1 eccentricity:2 object:1 measured:2 school:2 sa:1 strong:1 soc:1 grating:1 predicted:1 indicate:1 differ:3 filter:18 centered:1 human:5 ja:1 ao:1 discriminat:1 investigation:1 opt:1 biological:2 adjusted:2 koch:4 sufficiently:1 around:1 exp:2 predict:2 lm:2 early:8 a2:2 estimation:1 modulating:1 concurrent:1 mit:1 activates:2 gaussian:2 rather:2 treue:2 derived:2 focus:2 contrast:33 baseline:1 sense:1 am:2 dependent:10 typically:1 entire:1 bt:1 pasadena:1 overall:4 dual:3 among:2 orientation:18 resonance:1 spatial:14 animal:1 psychophysics:3 field:1 equal:1 f3:1 broad:1 introspection:1 fmri:2 report:1 stimulus:20 quantitatively:4 simplex:1 randomly:1 simultaneously:6 national:3 loudest:4 usc:2 detection:1 interest:2 investigate:4 circular:2 certainly:1 light:1 accurate:1 desimone:4 plymouth:2 necessary:3 re:3 theoretical:1 increased:3 column:3 modeling:1 ence:1 mhz:1 infonnative:1 too:2 reported:2 rees:2 lee:5 pool:1 linux:1 sharpened:1 central:1 bonnel:1 ek:1 derivative:1 itti:4 account:6 sinusoidal:1 star:1 summarized:1 blind:1 vi:1 depends:1 observer:2 masking:8 variance:1 largely:1 characteristic:1 miller:1 yield:1 judgment:2 correspond:1 mc:1 none:2 lu:1 converged:1 explain:2 frequency:7 pp:1 associated:2 gain:22 electrophysiological:2 singleunit:1 positioned:1 ea:2 appears:1 originally:1 methodology:1 response:5 improved:1 though:1 stage:6 ei:2 nonlinear:1 building:1 effect:8 staircase:1 unbiased:1 hence:3 spatially:1 ll:2 width:1 rooted:1 m:1 generalized:1 trying:2 theoretic:1 recently:2 charles:1 sigmoid:1 functional:2 ilab:1 winner:1 extend:1 he:9 trujillo:1 ai:1 tuning:9 focal:2 similarly:2 erc:1 winnertake:1 dj:1 hillyard:1 cortex:2 inhibition:2 recent:1 mandatory:1 periphery:1 manipulation:6 continue:1 caltech:1 dosher:1 analyzes:1 additional:3 greater:1 r0:2 determine:1 paradigm:2 period:4 ii:2 relates:1 plausibility:3 equally:6 target6:3 prediction:7 vision:9 poisson:1 ion:1 justified:1 background:1 affecting:3 separately:1 unrealistically:2 annealing:1 underestimated:2 biased:1 eliminates:1 probably:1 pooling:1 dipper:2 near:1 presence:1 revealed:1 affect:7 fit:14 perfectly:1 fm:1 angeles:1 qj:2 whether:5 gb:1 ul:3 modulatory:5 se:1 involve:1 dark:1 stein:1 http:1 nsf:1 alters:1 neuroscience:2 per:1 profoundly:1 affected:1 motter:1 four:1 threshold:18 luminance:1 imaging:2 ram:1 corbetta:1 sum:1 bia:2 letter:1 throughout:1 decision:3 duncan:2 display:1 fold:2 chelazzi:2 yielded:3 activity:6 strength:2 aspect:1 span:1 c8:1 department:1 peripheral:1 combination:1 poor:2 across:2 smaller:2 slightly:2 deyoe:1 rev:2 biologically:3 explained:1 hypot:2 computationally:1 remains:1 previously:1 mechanism:1 available:2 away:1 magnetic:1 chawla:1 pasternak:1 alternative:1 original:1 top:7 responding:5 assumes:1 running:1 remaining:1 a4:1 res:1 especially:1 psychophysical:7 implied:1 strategy:1 primary:2 rt:1 southern:1 attentional:10 separate:3 simulated:1 intensified:9 sci:1 providing:1 sharper:1 relate:1 ba:1 unknown:1 perform:2 vertical:1 neuron:18 observation:2 situation:1 interacting:2 perturbation:1 rating:1 complement:1 nonlinearly:1 hypercolumn:2 eckstein:1 devon:1 california:2 narrow:1 established:1 nip:1 suggested:1 below:1 perception:1 pattern:5 usually:1 psychophys:1 appeared:2 program:1 tb:1 oj:1 ia:1 unrealistic:3 demanding:1 difficulty:2 power:2 greatest:1 predicting:1 friston:1 residual:1 technology:1 eye:1 pl4:1 psychol:1 health:1 kj:1 acknowledgement:1 law:2 fully:17 oci:3 foundation:2 controversial:1 bank:1 row:1 jcm:1 elsewhere:1 summary:1 accounted:1 supported:3 free:1 bias:1 allow:1 understand:1 jh:2 institute:3 cortical:2 sensory:1 adaptive:1 excess:1 sj:1 aperture:2 assumed:2 corbett:1 search:1 promising:1 nature:4 ca:2 nakayama:1 frith:1 interact:1 complex:3 domain:1 neurosci:9 whole:1 noise:7 profile:1 allowed:4 body:1 neuronal:4 fig:5 en:1 lc:2 sub:1 comprises:1 downhill:1 heeger:1 weighting:1 third:1 down:7 annu:1 miezin:1 specific:2 moran:2 appeal:1 dk:3 evidence:3 oll:1 modulates:2 taskdependent:1 nat:7 electrophysiology:3 explore:1 visual:18 beo:1 scalar:1 aa:1 oct:5 modulate:3 goal:1 careful:1 towards:1 jf:1 fisher:1 change:1 specifically:1 determined:1 uniformly:1 reducing:1 total:2 engaged:1 experimental:2 hnb:1 superimposing:1 divisive:2 e:1 select:1 internal:1 support:1 modulated:2 |
1,180 | 2,076 | Exact differential equation population
dynamics for Integrate-and-Fire neurons
Julian Eggert *
HONDA R&D Europe (Deutschland) GmbH
Future Technology Research
Carl-Legien-StraBe 30
63073 Offenbach/Main, Germany
julian. eggert@hre-ftr.f.rd.honda.co.jp
Berthold Bauml
Institut fur Robotik und Mechatronik
Deutsches Zentrum fur Luft und Raumfahrt (DLR)
oberpfaffenhofen
Berthold.Baeuml@dlr.de
Abstract
Mesoscopical, mathematical descriptions of dynamics of populations of spiking neurons are getting increasingly important for the
understanding of large-scale processes in the brain using simulations. In our previous work, integral equation formulations for
population dynamics have been derived for a special type of spiking neurons. For Integrate- and- Fire type neurons , these formulations were only approximately correct. Here, we derive a mathematically compact, exact population dynamics formulation for
Integrate- and- Fire type neurons. It can be shown quantitatively
in simulations that the numerical correspondence with microscopically modeled neuronal populations is excellent.
1
Introduction and motivation
The goal of the population dynamics approach is to model the time course of the collective activity of entire populations of functionally and dynamically similar neurons
in a compact way, using a higher descriptionallevel than that of single neurons and
spikes. The usual observable at the level of neuronal populations is the populationaveraged instantaneous firing rate A(t), with A(t)6.t being the number of neurons
in the population that release a spike in an interval [t, t+6.t). Population dynamics
are formulated in such a way, that they match quantitatively the time course of a
given A(t), either gained experimentally or by microscopical, detailed simulation.
At least three main reasons can be formulated which underline the importance
of the population dynamics approach for computational neuroscience. First, it
enables the simulation of extensive networks involving a massive number of neurons
and connections, which is typically the case when dealing with biologically realistic
functional models that go beyond the single neuron level. Second, it increases the
analytical understanding of large-scale neuronal dynamics , opening the way towards
better control and predictive capabilities when dealing with large networks. Third,
it enables a systematic embedding of the numerous neuronal models operating at
different descriptional scales into a generalized theoretic framework, explaining the
relationships, dependencies and derivations of the respective models.
Early efforts on population dynamics approaches date back as early as 1972, to the
work of Wilson and Cowan [8] and Knight [4], which laid the basis for all current
population-averaged graded-response models (see e.g. [6] for modeling work using
these models). More recently, population-based approaches for spiking neurons were
developed, mainly by Gerstner [3, 2] and Knight [5]. In our own previous work [1],
we have developed a theoretical framework which enables to systematize and simulate a wide range of models for population-based dynamics. It was shown that
the equations of the framework produce results that agree quantitatively well with
detailed simulations using spiking neurons, so that they can be used for realistic
simulations involving networks with large numbers of spiking neurons. Nevertheless, for neuronal populations composed of Integrate-and-Fire (I&F) neurons, this
framework was only correct in an approximation. In this paper, we derive the exact
population dynamics formulation for I&F neurons. This is achieved by reducing
the I&F population dynamics to a point process and by taking advantage of the
particular properties of I&F neurons.
2
2.1
Background: Integrate-and-Fire dynamics
Differential form
We start with the standard Integrate- and- Fire (I&F) model in form of the wellknown differential equation [7]
(1)
which describes the dynamics of the membrane potential Vi of a neuron i that is
modeled as a single compartment with RC circuit characteristics. The membrane
relaxation time is in this case T = RC with R being the membrane resistance and C
the membrane capacitance. The resting potential v R est is the stationary potential
that is approached in the no-input case. The input arriving from other neurons is
described in form of a current ji.
In addition to eq. (1), which describes the integrate part of the I&F model, the
neuronal dynamics are completed by a nonlinear step. Every time the membrane
potential Vi reaches a fixed threshold () from below, Vi is lowered by a fixed amount
Ll > 0, and from the new value of the membrane potential integration according to
eq. (1) starts again.
if Vi(t) = () (from below) .
(2)
At the same time, it is said that the release of a spike occurred (i.e., the neuron
fired), and the time ti = t of this singular event is stored. Here ti indicates the
time of the most recent spike. Storing all the last firing times , we gain the sequence
of spikes {t{} (spike ordering index j, neuronal index i).
2.2
Integral form
Now we look at the single neuron in a neuronal compound. We assume that the
input current contribution ji from presynaptic spiking neurons can be described
using the presynaptic spike times tf, a response-function ~ and a connection weight
W? .
',J
ji(t) =
Wi ,j
~(t - tf)
(3)
l:
l:
j
f
Integrating the I&F equation (1) beginning at the last spiking time tT, which determines the initial condition by Vi(ti) = vi(ti - 0) - 6., where vi(ti - 0) is the
membrane potential just before the neuron spikes, we get 1
Vi(t) = v Rest
+ fj(t - t:) + l: Wi ,j l: a(t - t:; t - tf) ,
j
f
-
Vi(t:)) e- S / T
(4)
with the refractory function
fj(s) = - (v Rest
(5)
and the alpha-function
r ds"
JSI_ S
Sf
a(s; s') =
e-[sf -S"J/T
~(s") .
(6)
If we start the integration at the time ti* of the spike before the last spike, then for
ti* :::; t < ti the membrane potential is given by an expression like eq. (4), where ti
is replaced by t:i* . Especially we can now express v( ti - 0) and therefore the initial
condition for an integration starting at tT in terms of ti* and v(ti* - 0). In this
way, we can proceed repetitively and move back into the past. After some simple
algebra this results in
Vi(t) = v Rest
+ l:ry(t-t{)+ l:Wi,j l:a(oo ;t - tf) ,
f
~
vfef(t)
(7)
j
f
~-------y~------~
v~yn(t)
with a refractory function wich differs in the scale factor from that in eq. (5)
ry(s) = -6. e- S / T
?
(8)
The components vref(t) and v?n(t) to the membrane potential indicate refractory
and synaptic components to the neuron i, respectively, as normally used in the
Spike- Response- Model (SRM) notation [2].
Both equations (4) and (7) formulate the neuronal dynamics using a refractory
component, which depends on the own spike releases of a neuron, and a synaptic
component, which comprises the integrated input contribution to the membrane
potential by arrival of spikes from other neurons 2. The synaptic component is based
on the alpha-function characteristic of isolated arriving spikes, with an increase of
the membrane potential after spike arrival and a subsequent exponential decrease.
1 Strictly speaking, the constants vRest, T, () and ,6, and the function 1]( s) may vary for
each neuron, so that they should be written with a subindex i [similarly for n(s; s') , which
may vary for each connection j -+ i so that we should write it with subindices i, j]. For
the sake of clarity, we omit these indices here .
2S0 the I&F model can be formulated as a special case of the Spike- Response- Model,
which defines the neuronal dynamics in the integral formulation.
The comparison of the equivalent expressions eq. (4) and eq. (7) reveals an interesting property of the I&F model. They look very similar, but in eq. (4), the refractory
component depends only on the time elapsed since the last spike (thus reflecting a
renewal property, sometimes also called a short term memory for refractory properties), whereas in eq. (7), it depends on a sum of the contributions of all past spikes.
The simpler form of the refractory contribution in eq. (4) is achieved at the cost
of an alpha-function that now depends on the time t - ti elapsed since the last
own spike in addition to the times t - tf elapsed since the release of spikes at the
presynaptic neurons j that provide input to i. In eq. (7) , we have a more complex
refractory contribution, but an alpha-function that does not depend on the last own
spike time any more.
2.3
Probabilistic spike release
Probabilistic firing is introduced into the I&F model eq. (4) resp. (7) by using
threshold noise. The spike release of each neuron is controlled by a hazard function
>.(v), so that
>.(v)dt = Prob. that a neuron with membrane potential v spikes in [t , t + dt)
(9)
When a neuron spikes, we proceed as usual: The membrane potential is reset by a
fixed amount 6. and the I&F dynamics continues.
Population dynamics
3
3.1
Density description
Descriptions of neuronal populations usually assume a neuronal density function
p(t; X) which depends on the state variables X of the neurons. The density quantifies the likelihood that a neuron picked out of the population will be found in the
vicinity of the point X in state space,
p(t; X) dX
= Portion of neurons
at time t with state in [X, X
+ dX)
(10)
If we know p(t; X) , the population activity A(t) can be easily calculated. Using the
hazard function >'(t; X), the instantaneous population activity (spikes per time) can
be calculated by computing the spike release averaged over the population,
A(t) =
J
dX >.(t; X) p(t; X)
(11)
The population dynamics is then given by the time course of the neuronal density function p(t; X), which changes because each neuron evolves according to its
own internal dynamics, e.g. after a spike release and the subsequent reset of the
membrane potential.
The main challenge for the formulation of a population dynamics resides in selecting
a low-dimensional state space [for an easy calculation of A(t)] and a suitable form
for gtp(t; X).
As an example, for the population dynamics for I&F neurons it would be straightforward to use the membrane potential v from eq. (1) as the state variable X. But
this leads to a complicated density dynamics, because the dynamics for v(t) consist
of a continuous (differential equation (1)) and a discrete part (spike generation).
Therefore, here we concentrate on an alternative description that allows a compact
formulation of the desired I&F density dynamics.
3.2
Exact population dynamics for I&F neurons
Which is the best state space for a population dynamics of I&F neurons? For the
formulation of a population dynamics, it is usually assumed that the synaptic contributions to the membrane potential are identical for all neurons. This is the case
if we group all neurons of the same dynamical type and with identical connectivity
patterns into one population. That is, we say that neurons i and i' belong to the
same population if Wi,j = Wi',j for all j (for simulations of realistic networks of
spiking neurons, this will of course never be exactly the case, but it is reasonable
to assume that a grouping of neurons into populations can be achieved to a good
approximation) .
In our case, looking at eq. (4), we see that , since o:(s, s') depends on s = t - ti and
therefore on the own last spike time, the synaptic contribution to the membrane
potential differs according to the state of the neuron. Thus we regard eq. (7). Here,
we see that for identical connectivity patterns Wi,j, the synaptic contributions are
the same for all neurons, because 0:(00, s') does not depend on the own spike time
any more. Which are then the state variables of eq. (1) for the density description?
We see that, for a fixed synaptic contribution, the membrane potential Vi is fully
determined by the set of the own past spiking times {tf}. But this would mean
an infinite-dimensional density for the state description of a population, and, accordingly, a computationally overly expensive calculation of the population activity
A(t) according to eq. (11).
To avoid this we take advantage of a particular property of the I&F model. According to eq. (8), the single spike refractory contributions 'TJ(s) are exponential. Since
any sum of exponential functions with common relaxation constant T can be again
expressed as as an exponential function with the same T , we can write instead of
vrf(t) from eq. (7)
(12)
Now the membrane potential Vi(t) only depends on the time of the last own spike
ti and the refractory contribution amplitude modulation factor at the last spike 'TJi .
That is, we have transferred the effect of all spikes previous to the last one into 'TJi.
In addition, we have to care about updating of ti and 'TJi when a neuron spikes so
that we get 3
'TJi --+ 'TJi = 1 + 'TJie -(t-tn!T ,
(13)
ti --+ ti = t .
The effect of taking into account more than the most recent spike ti in the refractory
component vief(t) leads to a modulation factor 'TJi greater than 1, in particular if
spikes come in a rapid succession so that refractory contributions can accumulate.
Instead of using a modulation factor 'TJi the effect of previous spikes can also be
taken into account by introducing an effective last spiking time ii.
(14)
vi"f(t) = 'TJ(t - in = 'TJi'TJ(t - tn ,
where ii and 'TJi are connected by
i; = t;
+ TIn'TJi
(15)
The effect of i* is sort of funny. Because of 'TJi ::::: 1 it holds for the effective last
spiking time ii ::::: ti. This means, that , while at a given time t it is allways ti :::; t,
it happens that ii ::::: t, meaning the neurons behave as if they would spike in the
future.
3Here, the order of reemplacement matters; first we have to reemplace
1]:,
then
ti.
For the membrane potential we get now instead of eq. (7)
Vi(t) = v Rest
+ ry(t -
tn
+ 2..: Wi ,j 2..: 0:(00; t -
t;)
(16)
f
j
and for the update rule for the effective last spiking time
*) '
* = f (t 'tA
tAi* - +tA
i
i
with
t; follows
(17)
(18)
Therefore we can regard the dimensionality of the state space of the I&F dynamics
as 1-dimensional in the description of eq. (16). The dynamics of the single I&F
neurons now turns out to be very simple: Calculate the membrane potential Vi(t)
using eq. (16) together with the state variable t;, and check if Vi(t) exceeds the
threshold. If not, move forward in time and calculate again. If the membrane
potential exceeds threshold, update according to eq. (17) and then proceed with
the calculation of Vi(t) as normal.
t;
Using this single neuron dynamics , we can now proceed to gain a population dynamics using a density p(t; t*). The time t is here the explicit time dependence,
whereas t* denote the state variable of the population. By fixing t* and the synaptic contribution vsyn(t) to the membrane potential, the state of a neuron is fully
determined and the hazard function can be written as ,X[vsyn(t); t*].
The dynamics of the density p(t; t*) is then calculated as follows. Changes of p(t; t*)
occur when neurons spike and t* is updated according to eq. (17). The hazard
function controls the spike release, and, therefore, the change of p(t; t*). For chosen
state variables, p(t; t*) decreases due to spiking of the neurons with the fixed t*,
and increases because neurons with other t'* spike and get updated in just that
way that after updating their state variable falls around t*. This occurs according
to the reemplacement rule eq. (17) when
f(t, t'*) = t* .
(19)
Taking all together the dynamics of the density p(t; t*) is given by
decrease due to same state t* spiking
A
-ftp(t;t*)
+
1
=
'-,X[vsyn(t); t*]p(t; t*)'
(20)
+ 00
-00
dt'* 8[J(t, t'*) - t*] ,X[vsyn(t); t'*] p(t; t'*)
increase due to spiking of neurons with other states t'*
The population activity can then be calculated using the density according to
eq. (11) as follows
1
+00
A(t) =
- 00
dt* ,X[vsyn(t); t*] p(t; t*)
(21)
Remark that the expression for the density dynamics (eq. 20) automatically conserves the norm of the density, so that
1
+00
- 00
dt* p( t ; t*) = const ,
(22)
which is a necessary condition because the number of neurons participating in the
dynamics must remain constant.
4
Simulations
The dynamics of a population of I&F neurons , represented by the time course
of their joint activity, can now be easily calculated in terms of the differential
equation (20) , if the neuronal state density of the neuronal population p(t; i*) and
the synaptic input vsyn(t) are known. This means that all we have to store is the
density p(t; i*) for past and future effective last spiking times i* 4 . Favorably for
numerical simulations, only a limited time window of i* around the actual time t
is needed for the dynamics. The activity A(t) only appears as an auxiliary variable
that is calculated with the help of the neuronal density.
In figure 1 the simulation results for populations of of spiking neurons are shown.
The neurons are uncoupled and a hazard function
A(V) = ~ e2,B(v-e)
(23)
,
TO
with spike rate at threshold liTO = 1.0ms- 1 , a kind of inverse temperature (3 = 2.0,
which controls the noise level, and the threshold = 1.0. The other parameters of
the model in eq. (1) are: resting potential v Rest = 0, jump in membrane potential
after spike release ~ = 1 and time constant T = 20ms. This parameters are chosen
to be biologicaly plausible.
e
A (spikes/ms)
0.14
0.12
0.1
0.08
0.06
0.04
r-------
0.02
:
b)
=
II
""
n
""::
::
!l
:~
I
1\
*:\
! \_ .. ----2 '-1
1l
! \_, .. ----:
_______ J:
! . . . . ..'!
~,'
j
~
I:,
r-------
vsyr'i-'_ _1_00_ _15
_0_ _
20_0 _ _2_50_ _
30_0 ----,1
(ms)
o~
I
c)
100
150
200
250
300
I
(ms)
Figure 1: Activity A(t) of simulated populations of neurons. The neurons are
uncoupled and to each neuron the same synaptic field vsyn(t), ploted in c) and d),
is applied. a) shows the activity A(t) for a population of I&F neurons simulated
on the one hand as N = 10000 single neurons (solid line) using eq. (7) and on the
other hand using the density dynamics eq. (20) (dashed line). In b) the activity
A(t) of a population ofI&F neurons (dashed line) and a population of SRM neurons
with renewal (solid line) are compared. For all simulations the same parameters as
specified in the text were used.
The simulations show that the density dynamics eq. (20) reproduces the activity
A( t) of a population of single I&F neurons almost perfect, with the exception of the
noise in the single neuron simulations due to the finite size effects. This holds even
for the peaks occuring at the steps of the applied synaptic field v syn (t), although the
density dynamics is entirely based on differential equations and one would therefore
not expect such an excellent match for fast changes in activity.
S ll (t) only appears as a scalar in the dynamics, so that no integration over time takes
4V Y
place here.
The simulations also show that there can be a big difference between I&F and SRM
neurons with renewal. Because of the accumulation of the refractory effects of all
former spikes in the case of I&F neurons the activity A(t) is generaly lower than
for the SRM neurons with renewal and the higher the absolute actitvity level the
bigger is the difference between both.
5
Conclusions
In this paper we derived an exact differential equation density dynamics for a population of I&F neurons starting from the microscopical equations for a single neuron.
This density dynamics allows a compuationaly efficient simulation of a whole population of neurons.
In future work we want to simulate a network of connected neuronal populations.
In such a network of populations (indexed e.g. by x) , a self-consistent system of
differential equations based on the population's p(x, t; i*) and A(x, t) emerges if
we constrain ourselves to neuronal populations connected synaptically according to
the constraints given by the pool definition [2]. In this case, two neurons i and j
belong to pools x and y, if Wi,j = W(x, y). This allows us to write for the synaptic
component of the membrane potential
v syn(x,t)
= 2: W (x , y)
y
1
00
ds'a(oo;s')A(y,t-s')
(24)
0
Using the alpha-function a(oo ; s') as introduced in (6), and a "nice" responsefunction ~ for the input current time course after a spike, we can write eq. (24)
using differential equations that use A(y, t) as input. This results in a system that
is based entirely on differential equations and is very cheap to compute.
References
[1] J. Eggert and J.L. van Hemmen. Modeling neuronal assemblies: Theory and implementation. N eural Computation, 13(9):1923- 1974, 200l.
[2] W. Gerstner. Population dynamics of spiking neurons: Fast transients, asynchronous
states and locking. Neural Computation, 12:43- 89 , 2000.
[3] W . Gerstner and J . L. van Hemmen. Associative memory in a network of 'spiking'
neurons. Network, 3:139- 164, 1992.
[4] B. W. Knight . Dynamics of encoding in a populations of neurons. J. Gen. Physiology,
59:734- 766 , 1972.
[5] B. W. Knight. Dynamics of Encoding in Neuron Populations: Some General Mathematical Features. Neural Comput., 12:473- 518, 2000.
[6] Z. Li. A neural model of contour integration in the primary visual cortex. Neural
Comput. , 10(4):903- 940, 1998.
[7] H. C. Tuckwell. Introduction to Th eoretical N eurobiology. Cambridge University Press,
Cambridge, 1988.
[8] H. R. Wilson and J. D. Cowan. Excitatory and inhibitory interactions in localized
populations of model neurons. Biophys. J ., 12:1- 24, 1972.
| 2076 |@word norm:1 underline:1 simulation:15 solid:2 initial:2 selecting:1 past:4 current:4 dx:3 written:2 must:1 realistic:3 subsequent:2 numerical:2 enables:3 cheap:1 update:2 stationary:1 accordingly:1 beginning:1 short:1 honda:2 simpler:1 mathematical:2 rc:2 differential:10 rapid:1 ry:3 brain:1 automatically:1 actual:1 window:1 notation:1 circuit:1 vref:1 kind:1 developed:2 every:1 ti:22 exactly:1 control:3 normally:1 omit:1 yn:1 before:2 encoding:2 firing:3 modulation:3 approximately:1 dynamically:1 co:1 limited:1 range:1 averaged:2 differs:2 physiology:1 integrating:1 get:4 accumulation:1 equivalent:1 go:1 straightforward:1 starting:2 wich:1 formulate:1 vsyn:7 rule:2 embedding:1 population:56 updated:2 resp:1 massive:1 exact:5 carl:1 expensive:1 conserve:1 updating:2 continues:1 vrf:1 calculate:2 connected:3 ordering:1 decrease:3 knight:4 und:2 locking:1 dynamic:48 depend:2 algebra:1 predictive:1 basis:1 easily:2 joint:1 represented:1 derivation:1 fast:2 effective:4 approached:1 plausible:1 say:1 associative:1 advantage:2 sequence:1 analytical:1 interaction:1 reset:2 date:1 gen:1 fired:1 subindices:1 description:7 participating:1 getting:1 produce:1 perfect:1 ftp:1 derive:2 oo:3 help:1 fixing:1 eq:31 auxiliary:1 indicate:1 come:1 concentrate:1 vrest:1 correct:2 tji:11 transient:1 mathematically:1 strictly:1 hold:2 around:2 normal:1 vary:2 early:2 tf:6 avoid:1 wilson:2 derived:2 release:10 fur:2 indicates:1 mainly:1 likelihood:1 check:1 entire:1 typically:1 integrated:1 germany:1 special:2 integration:5 renewal:4 eurobiology:1 field:2 never:1 identical:3 look:2 future:4 quantitatively:3 opening:1 composed:1 zentrum:1 microscopically:1 replaced:1 ourselves:1 fire:6 tj:3 integral:3 necessary:1 respective:1 institut:1 indexed:1 desired:1 isolated:1 theoretical:1 modeling:2 cost:1 introducing:1 srm:4 stored:1 dependency:1 density:22 peak:1 systematic:1 probabilistic:2 pool:2 together:2 connectivity:2 again:3 li:1 account:2 potential:25 de:1 matter:1 vi:17 depends:7 picked:1 portion:1 start:3 sort:1 capability:1 complicated:1 contribution:13 compartment:1 characteristic:2 succession:1 reach:1 synaptic:12 definition:1 e2:1 gain:2 emerges:1 dimensionality:1 amplitude:1 syn:2 back:2 reflecting:1 appears:2 higher:2 dt:5 ta:2 response:4 formulation:8 just:2 d:2 hand:2 nonlinear:1 defines:1 effect:6 former:1 vicinity:1 tuckwell:1 ll:2 self:1 m:5 generalized:1 theoretic:1 tt:2 occuring:1 eggert:3 tn:3 temperature:1 fj:2 meaning:1 instantaneous:2 recently:1 common:1 functional:1 spiking:19 microscopical:2 ji:3 refractory:13 jp:1 belong:2 occurred:1 resting:2 functionally:1 eoretical:1 accumulate:1 cambridge:2 rd:1 similarly:1 lowered:1 europe:1 cortex:1 operating:1 own:9 recent:2 wellknown:1 compound:1 store:1 greater:1 care:1 dashed:2 ii:5 exceeds:2 match:2 calculation:3 repetitively:1 hazard:5 bigger:1 controlled:1 ftr:1 involving:2 sometimes:1 achieved:3 synaptically:1 background:1 addition:3 whereas:2 want:1 interval:1 singular:1 rest:5 cowan:2 easy:1 dlr:2 expression:3 effort:1 resistance:1 proceed:4 speaking:1 remark:1 subindex:1 detailed:2 amount:2 inhibitory:1 neuroscience:1 overly:1 per:1 write:4 discrete:1 express:1 group:1 nevertheless:1 threshold:6 clarity:1 relaxation:2 sum:2 prob:1 inverse:1 place:1 laid:1 reasonable:1 almost:1 funny:1 entirely:2 correspondence:1 activity:13 occur:1 constraint:1 constrain:1 sake:1 simulate:2 transferred:1 according:10 membrane:25 describes:2 remain:1 increasingly:1 wi:8 evolves:1 biologically:1 happens:1 taken:1 computationally:1 equation:14 agree:1 tai:1 turn:1 needed:1 know:1 alternative:1 assembly:1 completed:1 const:1 especially:1 graded:1 move:2 capacitance:1 spike:48 occurs:1 primary:1 dependence:1 usual:2 said:1 simulated:2 presynaptic:3 reason:1 modeled:2 relationship:1 index:3 julian:2 favorably:1 implementation:1 collective:1 neuron:78 finite:1 behave:1 looking:1 introduced:2 specified:1 extensive:1 connection:3 elapsed:3 uncoupled:2 beyond:1 below:2 usually:2 dynamical:1 pattern:2 challenge:1 memory:2 gtp:1 event:1 suitable:1 technology:1 numerous:1 text:1 nice:1 understanding:2 fully:2 expect:1 interesting:1 generation:1 localized:1 integrate:7 consistent:1 s0:1 storing:1 course:6 excitatory:1 last:14 asynchronous:1 arriving:2 explaining:1 wide:1 taking:3 fall:1 absolute:1 van:2 regard:2 berthold:2 calculated:6 resides:1 contour:1 forward:1 jump:1 alpha:5 compact:3 observable:1 dealing:2 reproduces:1 reveals:1 assumed:1 continuous:1 quantifies:1 excellent:2 gerstner:3 complex:1 main:3 motivation:1 noise:3 big:1 arrival:2 whole:1 eural:1 gmbh:1 neuronal:19 hemmen:2 comprises:1 explicit:1 sf:2 exponential:4 comput:2 third:1 tin:1 grouping:1 consist:1 gained:1 importance:1 biophys:1 visual:1 expressed:1 lito:1 scalar:1 determines:1 goal:1 formulated:3 towards:1 experimentally:1 change:4 determined:2 infinite:1 reducing:1 hre:1 called:1 est:1 ofi:1 exception:1 internal:1 |
1,181 | 2,077 | Constructing Distributed Representations
Using Additive Clustering
Wheeler Ruml
Division of Engineering and Applied Sciences
Harvard University
33 Oxford Street, Cambridge, MA 02138
ruml@eecs.harvard.edu
Abstract
If the promise of computational modeling is to be fully realized in higherlevel cognitive domains such as language processing, principled methods
must be developed to construct the semantic representations used in such
models. In this paper, we propose the use of an established formalism
from mathematical psychology, additive clustering, as a means of automatically constructing binary representations for objects using only pairwise similarity data. However, existing methods for the unsupervised
learning of additive clustering models do not scale well to large problems. We present a new algorithm for additive clustering, based on a
novel heuristic technique for combinatorial optimization. The algorithm
is simpler than previous formulations and makes fewer independence assumptions. Extensive empirical tests on both human and synthetic data
suggest that it is more effective than previous methods and that it also
scales better to larger problems. By making additive clustering practical,
we take a significant step toward scaling connectionist models beyond
hand-coded examples.
1 Introduction
Many cognitive models posit mental representations based on discrete substructures. Even
connectionist models whose processing involves manipulation of real-valued activations
typically represent objects as patterns of 0s and 1s across a set of units (Noelle, Cottrell,
and Wilms, 1997). Often, individual units are taken to represent specific features of the
objects and two representations will share features to the degree to which the two objects
are similar. While this arrangement is intuitively appealing, it can be difficult to construct
the features to be used in such a model. Using random feature assignments clouds the
relationship between the model and the objects it is intended to represent, diminishing
the model?s value. As Clouse and Cottrell (1996) point out, hand-crafted representations
are tedious to construct and it can be difficult to precisely justify (or even articulate) the
principles that guided their design. These difficulties effectively limit the number of objects
that can be encoded, constraining modeling efforts to small examples. In this paper, we
investigate methods for automatically synthesizing feature-based representations directly
from the pairwise object similarities that the model is intended to respect. This automatic
Table 1: An 8-feature model derived from consonant confusability data. With c = 0.024,
the model accounts for 91.8% of the variance in the data.
Wt. Objects with feature
Interpretation
.350
f?
front unvoiced fricatives
.243
dg
back voiced stops
.197 p k
unvoiced stops (without t)
.182
b v?
front voiced
.162 ptk
unvoiced stops
.127
mn nasals
.075
dgv?z?
z
voiced (without b)
.049 ptkf?s?
s
unvoiced
approach eliminates the manual burden of selecting and assigning features while providing
an explicit design criterion that objectively connects the representations to empirical data.
After formalizing the problem, we will review existing algorithms that have been proposed
for solving it. We will then investigate a new approach, based on combinatorial optimization. When using a novel heuristic search technique, we find that the new approach, despite
its simplicity, performs better than previous algorithms and that, perhaps more important,
it maintains its effectiveness on large problems.
1.1 Additive Clustering
We will formalize the problem of constructing discrete features from similarity information
using the additive clustering model of Shepard and Arabie (1979). In this framework,
abbreviated A DCLUS, clusters represent arbitrarily overlapping discrete features. Each of
the k features has a non-negative real-valued weight wk , and the similarity between two
objects i and j is just the sum of the weights of the features they share. If f ik is 1 if object
i has feature k and 0 otherwise, and c is a real-valued constant, then the similarity of i and
j is modeled as
X
s?ij =
wk fik fjk + c .
k
This class of models is very expressive, encompassing non-hierarchical as well as hierarchical arrangements of clusters. An example model, derived using the ewindclus-klb
algorithm described below, is shown in Table 1. The representation of each object is simply
the binary column specifying its membership or absence in each cluster. Additive clustering is asymmetric in the sense that only the shared features of two objects contribute to
their similarity, not the ones they both lack. (This is the more general formulation, as an
additional feature containing the set complement of the original feature could always be
used to produce such an effect.)
With a model formalism in hand, we can then phrase the problem of constructing feature
assignments as simply finding the A DCLUS model that best matches the given similarity
data using the desired number of features. The fit of a model (comprising F , W , and c) to
a matrix, S, can be quantified by the variance accounted for (VAF), which compares the
model?s accuracy to merely predicting using the mean similarity:
P
?ij )2
i,j (sij ? s
VAF = 1 ? P
?)2
i,j (sij ? s
A VAF of 0 can always be achieved by setting all wk to 0 and c to s?.
2 Previous Algorithms
Additive clustering is a difficult 0-1 quadratic programming problem and only heuristic
methods, which do not guarantee an optimal model, have been proposed. Many different
approaches have been taken:
Subsets: Shepard and Arabie (1979) proposed an early algorithm based on subset analysis that was clearly superseded by Arabie?s later work below. Hojo (1983) also
proposed an algorithm along these lines. We will not consider these algorithms
further.
Non-discrete Approximation: Arabie and Carroll (1980) and Carroll and Arabie (1983)
proposed the two-stage indclus algorithm. In the first stage, cluster memberships are treated as real values and optimized for each cluster in turn by gradient
descent. At the same time, a penalty term for non-0-1 values is gradually increased. Afterwards, a combinatorial clean-up stage tries all possible changes to
1 or 2 cluster memberships. Experiments reported below use the original code,
modified slightly to handle large instances. Random initial configurations were
used.
Asymmetric Approximation: In the sindclus algorithm, Chaturvedi and Carroll
(1994) optimize anP
asymmetric model with two sets of cluster memberships, having the form s?ij = k wk fik gjk + c. By considering each cluster in turn, this formulation allows a fast method for determining each of F , G, and w given the other
two. In practice, F and G often become identical, yielding an A DCLUS model.
Experiments reported below use both a version of the original implementation that
has been modified to handle large instances and a reimplemented version (resindclus) that differs in its behavior at boundary cases (handling 0 weights,
empty clusters, ties). Models from runs in which F and G did not converge were
each converted into several A DCLUS models by taking only F , only G, their intersection, or their union. The weights and constants of each model were optimized
using constrained least-squares linear regression (Stark and Parker, 1995), ensuring non-negative cluster weights, and the one with the highest VAF was used.
Alternating Clusters: Kiers (1997) proposed an element-wise simplified sindclus algorithm, which we abbreviate as ewindclus. Like sindclus, it considers
each cluster in turn, alternating between the weights and the cluster memberships,
although only one set of clusters is maintained. Weights are set by a simple regression and memberships are determined by a gradient function that assumes
object independence and fixed weights. The experiments reported below use a
new implementation, similar to the reimplementation of sindclus.
Expectation Maximization: Tenenbaum (1996) reformulated A DCLUS fitting in probabilistic terms as a problem with multiple hidden factorial causes, and proposed
a combination of the EM algorithm, Gibbs sampling, and simulated annealing to
solve it. The experiments below use a modified version of the original implementation which we will notate as em-indclus. It terminates early if 10 iterations
of EM pass without a change in the solution quality. (A comparison with the original code showed this modification to give equivalent results using less running
time.)
Unfortunately, it is not clear which of these approaches is the best. Most published comparisons of additive clustering algorithms use only a small number of test problems (or
only artificial data) and report only the best solution found within an unspecified amount
of time. Because the algorithms use random starting configurations and often return solutions of widely varying quality even when run repeatedly on the same problem, this leaves
it unclear which algorithm gives the best results on a typical run. Furthermore, different
Table 2: The performance of several previously proposed algorithms on data sets from
psychological experiments.
indclus
sindclus
re-sindclus
ewindclus
Name VAF
IQR
VAF
IQR
r VAF
IQR
r VAF
IQR
r
animals-s
77
75?80
66
65?65
8
78
79 ?80 12
64
60?69
4
numbers 83
81?86
84
82 ?86 5
78
75?81
7
82
79?85
5
workers
83
82?85
81
79?83
9
84
82?85
7
67
63?72
2
consonants 89
89?90
88
87?89
6
81
80?82
5
51
44?57
1
animals
71
69?74
66
66?66
9
66
66?66
13
72
71 ?73 26
80
80?80
78
78?79
7
68
65?72
5
74
73?75
17
letters
Table 3: The performance of indclus and em-indclus on the human data sets.
indclus
em-indclus
Name
n
k VAF
IQR
r VAF
IQR
animals-s 10
3
80
80?80
23
80
80?80
numbers 10
8
91
90?91 157
90
89?90
89
88?89
89
87
87?89
workers 14
7
consonants 16
8
91
91?91 291
91
91?91
71
69?74
1
N/A
animals 26 12
letters 30
5
82
82?83 486
82
82?83
algorithms require very different running times, and multiple runs of a fast algorithm with
high variance in solution quality may produce a better result in the same time as a single run
of a more predictable algorithm. The next section reports on a new empirical comparison
that addresses these concerns.
2.1 Evaluation of Previous Algorithms
We compared indclus, both implementations of sindclus, ewindclus, and emindclus on 3 sets of problems. The first set is a collection of 6 typical data sets from
psychological experiments that have been used in previous additive clustering work (originally by Shepard and Arabie (1979), except for animals-s, Mechelen and Storms (1995),
and animals, Chaturvedi and Carroll (1994)). The number of objects (n) and the number of
features used (k) are listed for each instance as part of Table 3. The second set of problems
contains noiseless synthetic data derived from A DCLUS models with 8, 16, 32, 64, and 128
objects. In a rough approximation of the human data, the number of clusters was set to
2 log2 (n), and as in previous A DCLUS work, each object was inserted in each cluster with
probability 0.5. A single similarity matrix was generated from each model using weights
and constants uniformly distributed between 1 and 6. The third set of problems was derived from the second by adding gaussian noise with a variance of 10% of the variance of
the similarity data and enforcing symmetry. Each algorithm was run at least 50 times on
each data set. Runs that crashed or resulted in a VAF < 0 were ignored. To avoid biasing
our conclusions in favor of methods requiring more computation time, those results were
then used to derive the distribution of results that would be expected if all algorithms were
run simultaneously and those that finished early were re-run repeatedly until the slowest
algorithm finished its first run, with any re-runs in progress at that point discarded. 1
1
Depending as it does on running time, this comparison remains imprecise due to variations in
the degree of code tuning and the quality of the compilers used, and the need to normalize timings
between the multiple machines used in the tests.
Summaries of the time-equated results produced by each algorithm on each of the human
data sets are shown in Table 2. (em-indclus took much longer than the other algorithms
and its performance is shown separately in Table 3.) The mean VAF for each algorithm
is listed, along with the inter-quartile range (IQR) and the mean number of runs that were
necessary to achieve time parity with the slowest algorithm on that data set (r). On most
instances, there is remarkable variance in the VAF achieved by each algorithm. 2 Overall,
despite the variety of approaches that have been brought to bear over the years, the original indclus algorithm appears to be the best. (Results in which another algorithm was
superior to indclus are marked with a box.) Animals-s is the only data set on which its
median performance was not the best, and its overall distribution of results is consistently
competitive. It is revealing to note the differences in performance between the original and
reimplemented versions of sindclus. Small changes in the handling of boundary cases
make a large difference in the performance of the algorithm.
Surprisingly, on the synthetic data sets (not shown), the relative performance of the algorithms was quite different, and almost the same on the noisy data as on the noise-free data.
(This suggests that the randomly generated data sets that are commonly used to evaluate
A DCLUS algorithms do not accurately reflect the problems of interest to practitioners.)
ewindclus performed best here, although it was only occasionally able to recover the
original models from the noise-free data.
Overall, it appears that current methods of additive clustering are quite sensitive to the
type of problem they are run on and that there is little assurance that they can recover the
underlying structure in the data, even for small problems. To address these problems, we
turn now to a new approach.
3 A Purely Combinatorial Approach
One common theme in indclus, sindclus, and ewindclus is their computation
of each cluster and its weight in turn, at each step fitting only the residual similarity not
accounted for by the other clusters. This forces memberships to be considered in a predetermined order and allows weights to become obsolete. Inspired in part by recent work of Lee
(in press), we propose an orthogonal decomposition of the problem. Instead of computing
the elements and weight of each cluster in succession, we first consider all the memberships and then derive all the weights using constrained regression. And where previous
algorithms recompute all the memberships of one cluster simultaneously (and therefore independently), we will change memberships one by one in a dynamically determined order
using simple heuristic search techniques, recomputing the weights after each step. (An
incremental bounded least squares regression algorithm that took advantage of the previous solution would be ideal, but the algorithms tested below did not incorporate such an
improvement.) From this perspective, one need only focus on changing the binary membership variables, and A DCLUS becomes a purely combinatorial optimization problem.
We will evaluate three different algorithms based on this approach, all of which attempt to
improve a random initial model. The first, indclus-hc, is a simple hill-climbing strategy
which attempts to toggle individual memberships in an arbitrary order and the first change
resulting in an improved model is accepted. The algorithm terminates when no membership
can be changed to give an improvement. This strategy is reminiscent of a proposal by
Clouse and Cottrell (1996), although here we are using the A DCLUS model of similarity.
In the second algorithm, indclus-pbil, the PBIL algorithm of Baluja (1997) is used
2
Table 3 shows one anomaly: no em-indclus run on animals resulted in a VAF ? 0. This also
occurred on all synthetic problems with 32 or more objects (although very good solutions were found
on the smaller problems). Tenenbaum (personal communication) suggests that the default annealing
schedule in the em-indclus code may need to be modified for these problems.
Table 4: The performance of the combinatorial algorithms on human data sets.
indclus-hc
Name
animals-s
numbers
workers
consonants
animals
letters
VAF
IQR
80
90
88
86
71
70
80?80
90?91
88?89
85?87
70?72
69?71
r
44
24
16
11
8
3
ind-pbil
ewind-klb
VAF
IQR
VAF
IQR
74
87
86
80
66
66
71?74
85?88
84?87
76?82
65?69
64?68
80
91
89
92
74
76
80?80
91?91
89?89
92?92
74?74
74?78
indclus
r
74
18
13
9
6
2
VAF
IQR
80
90
88
91
74
82
80?80
89?91
88?89
91?91
74?74
81?82
r
47
59
53
61
36
57
to search for appropriate memberships. This is a simplification of the strategy suggested
by Lee (in press), whose proposal also includes elements concerned with automatically
controlling model complexity. We use the parameter settings he suggests but only allow
the algorithm to generate 10,000 solutions.
3.1 KL Break-Out: A New Optimization Heuristic
While the two approaches described above do not use any problem-specific information beyond solution quality, the third algorithm uses the gradient function from the ewindclus
algorithm to guide the search. The move strategy is a novel combination of gradient ascent
and the classic method of Kernighan and Lin (1970) which we call ?KL break-out?. It proceeds by gradient ascent, changing the entry in F whose ewindclus gradient points most
strongly to the opposite of its current value. When the ascent no longer results in an improvement, a local maximum has been reached. Motivated by results suggesting that good
maxima tend to cluster (Boese, Kahng, and Muddu, 1994; Ruml et al., 1996), the algorithm tries to break out of the current basin of attraction and find a nearby maximum rather
than start from scratch at another random model. It selects the least damaging variable to
change, using the gradient as in the ascent, but now it locks each variable after changing
it. The pool of unlocked variables shrinks, thus forcing the algorithm out of the local maximum and into another part of the space. To determine if it has escaped, a new gradient
ascent is attempted after each locking step. If the ascent surpasses the previous maximum,
the current break-out attempt is abandoned and the ascent is pursued. If the break-out procedure changes all variables without any ascent finding a better maximum, the algorithm
terminates. The procedure is guaranteed to return a solution at least as good as that found
by the original KL method (although it will take longer), and it has more flexibility to
follow the gradient function. This algorithm, which we will call ewindclus-klb, surpassed the original KL method in time-equated tests. It is also conceptually simple and has
no parameters that need to be tuned.
3.2 Evaluation of the Combinatorial Algorithms
The time-equated performance of the combinatorial algorithms on the human data sets is
shown in Table 4, with indclus, the best of the previous algorithms, shown for comparison. As one might expect, adding heuristic guidance to the search helps it enormously: ewindclus-klb surpasses the other combinatorial algorithms on every problem. It performs better than indclus on three of the human data sets (top panel), equals
its performance on two, and performs worse on one data set, letters. (Results in which
ewindclus-klb was not the best are marked with a box.) The variance of indclus on
letters is very small, and the full distributions suggest that ewindclus-klb is the better
choice on this data set if one can afford the time to take the best of 20 runs. (Experiments
Table 5: ewindclus-klb and indclus on noisy synthetic data sets of increasing size.
ewindclus-klb
indclus
n VAF
IQR
VAF
IQR
r
8
97
96?97
95
93?97
1
16
91
90?92
86
85?87
4
32
90
88?92
83
82?84
22
64
91
90?91
84
84?85 100
128
91
91?91
88
87?90 381
using 7 additional human data sets found that letters represented the weakest performance
of ewindclus-klb.)
Performance of a plain KL strategy (not shown) surpassed or equaled indclus on all
but two problems (consonants and letters), indicating that the combinatorial approach, in
tandem with heuristic guidance, is powerful even without the new ?KL break-out? strategy.
While we have already seen that synthetic data does not predict the relative performance
of algorithms on human data very well, it does provide a test of how well they scale to
larger problems. On noise-free synthetic data, ewindclus-klb reliably recovered the
original model on all data sets. It was also the best performer on the noisy synthetic data (a
comparison with indclus is presented in Table 5. These results show that, in addition to
performing best on the human data, the combinatorial approach retains its effectiveness on
larger problems.
In addition to being able to handle larger problems than previous methods, it is important to
note that the higher VAF of the models induced by ewindclus-klb often translates into
increased interpretability. In the model shown in Table 1, for instance, the best previously
published model (Tenenbaum, 1996), whose VAF is only 1.6% worse, does not contain s?
in the unvoiced cluster.
4 Conclusions
We formalized the problem of constructing feature-based representations for cognitive
modeling as the unsupervised learning of A DCLUS models from similarity data. In an
empirical comparison sensitive to variance in solution quality and computation time, we
found that several recently proposed methods for recovering such models perform worse
than the original indclus algorithm of Arabie and Carroll (1980). We suggested a purely
combinatorial approach to this problem that is simpler than previous proposals, yet more
effective. By changing memberships one at a time, it makes fewer independence assumptions. We also proposed a novel variant of the Kernighan-Lin optimization strategy that
is able to follow the gradient function more closely, surpassing the performance of the
original.
While this work has extended the reach of the additive clustering paradigm to large problems, it is directly applicable to feature construction of only those cognitive models whose
representations encode similarity as shared features. (The cluster weights can be represented by duplicating strong features or by varying connection weights.) However, the
simplicity of the combinatorial approach should make it straightforward to extend to models in which the absence of features can enhance similarity. Other future directions include
using the output of one algorithm as the starting point for another, and incorporating measures of model complexity(Lee, in press).
5 Acknowledgments
Thanks to Josh Tenenbaum, Michael Lee, and the Harvard AI Group for stimulating discussions; to Josh, Anil Chaturvedi, Henk Kiers, J. Douglas Carroll, and Phipps Arabie for
providing source code for their algorithms; Josh, Michael, and Phipps for providing data
sets; and Michael for sharing unpublished work. This work was supported in part by the
NSF under grants CDA-94-01024 and IRI-9618848.
References
Arabie, Phipps and J. Douglas Carroll. 1980. MAPCLUS: A mathematical programming
approach to fitting the adclus model. Psychometrika, 45(2):211?235, June.
Baluja, Shumeet. 1997. Genetic algorithms and explicit search statistics. In Michael C.
Mozer, Michael I. Jordan, and Thomas Petsche, editors, NIPS 9.
Boese, Kenneth D., Andrew B. Kahng, and Sudhakar Muddu. 1994. A new adaptive
multi-start technique for combinatorial global optimizations. Operations Research Letters,
16:101?113.
Carroll, J. Douglas and Phipps Arabie. 1983. INDCLUS: An individual differences
generalization of the ADCLUS model and the MAPCLUS algorithm. Psychometrika,
48(2):157?169, June.
Chaturvedi, Anil and J. Douglas Carroll. 1994. An alternating combinatorial optimization
approach to fitting the INDCLUS and generalized INDCLUS models. Journal of Classification, 11:155?170.
Clouse, Daniel S. and Garrison W. Cottrell. 1996. Discrete multi-dimensional scaling. In
Proceedings of the 18th Annual Conference of the Cognitive Science Society, pp. 290?294.
Hojo, Hiroshi. 1983. A maximum likelihood method for additive clustering and its applications. Japanese Psychological Research, 25(4):191?201.
Kernighan, B. and S. Lin. 1970. An efficient heuristic procedure for partitioning graphs.
The Bell System Technical Journal, 49(2):291?307, February.
Kiers, Henk A. L. 1997. A modification of the SINDCLUS algorithm for fitting the
ADCLUS and INDCLUS models. Journal of Classification, 14(2):297?310.
Lee, Michael D. in press. A simple method for generating additive clustering models with
limited complexity. Machine Learning.
Mechelen, I. Van and G. Storms. 1995. Analysis of similarity data and Tversky?s contrast
model. Psychologica Belgica, 35(2?3):85?102.
Noelle, David C., Garrison W. Cottrell, and Fred R. Wilms. 1997. Extreme attraction: On
the discrete representation preference of attractor networks. In M. G. Shafto and P. Langley,
eds, Proceedings of the 19th Annual Conference of the Cognitive Science Society, p. 1000.
Ruml, Wheeler, J. Thomas Ngo, Joe Marks, and Stuart Shieber. 1996. Easily searched encodings for number partitioning. Journal of Optimization Theory and Applications, 89(2).
Shepard, Roger N. and Phipps Arabie. 1979. Additive clustering: Representation of
similarities as combinations of discrete overlapping properties. Psychological Review,
86(2):87?123, March.
Stark, Philip B. and Robert L. Parker. 1995. Bounded-variable least-squares: An algorithm
and applications. Computational Statistics, 10:129?141.
Tenenbaum, Joshua B. 1996. Learning the structure of similarity. In D. S. Touretzky,
M. C. Mozer, and M. E. Hasselmo, editors, NIPS 8.
| 2077 |@word version:4 tedious:1 pbil:3 decomposition:1 initial:2 configuration:2 contains:1 selecting:1 daniel:1 tuned:1 genetic:1 existing:2 current:4 recovered:1 activation:1 assigning:1 yet:1 must:1 reminiscent:1 cottrell:5 additive:16 predetermined:1 pursued:1 fewer:2 leaf:1 assurance:1 obsolete:1 mental:1 recompute:1 contribute:1 preference:1 simpler:2 mathematical:2 along:2 become:2 ik:1 fitting:5 pairwise:2 inter:1 expected:1 behavior:1 gjk:1 multi:2 inspired:1 automatically:3 little:1 considering:1 increasing:1 becomes:1 tandem:1 psychometrika:2 underlying:1 formalizing:1 bounded:2 panel:1 unspecified:1 developed:1 finding:2 guarantee:1 duplicating:1 every:1 vaf:22 tie:1 partitioning:2 unit:2 grant:1 engineering:1 shumeet:1 timing:1 local:2 phipps:5 limit:1 despite:2 encoding:1 oxford:1 might:1 quantified:1 dynamically:1 specifying:1 suggests:3 limited:1 range:1 practical:1 acknowledgment:1 practice:1 union:1 reimplementation:1 differs:1 procedure:3 wheeler:2 langley:1 empirical:4 bell:1 revealing:1 imprecise:1 suggest:2 optimize:1 equivalent:1 straightforward:1 iri:1 starting:2 independently:1 simplicity:2 formalized:1 fik:2 attraction:2 classic:1 handle:3 variation:1 mapclus:2 controlling:1 construction:1 anomaly:1 programming:2 us:1 harvard:3 element:3 asymmetric:3 cloud:1 inserted:1 highest:1 principled:1 mozer:2 predictable:1 complexity:3 locking:1 arabie:11 personal:1 tversky:1 solving:1 purely:3 division:1 easily:1 represented:2 fast:2 effective:2 hiroshi:1 artificial:1 whose:5 heuristic:8 larger:4 valued:3 encoded:1 solve:1 widely:1 otherwise:1 quite:2 objectively:1 favor:1 statistic:2 noisy:3 ptk:1 higherlevel:1 advantage:1 took:2 propose:2 chaturvedi:4 flexibility:1 achieve:1 normalize:1 cluster:23 empty:1 produce:2 generating:1 incremental:1 object:17 help:1 derive:2 depending:1 andrew:1 ij:3 progress:1 strong:1 recovering:1 involves:1 direction:1 posit:1 guided:1 closely:1 shafto:1 quartile:1 human:10 require:1 generalization:1 articulate:1 considered:1 predict:1 early:3 klb:11 applicable:1 combinatorial:15 sensitive:2 hasselmo:1 noelle:2 rough:1 clearly:1 brought:1 always:2 gaussian:1 modified:4 rather:1 avoid:1 fricative:1 varying:2 encode:1 derived:4 focus:1 june:2 improvement:3 consistently:1 likelihood:1 slowest:2 contrast:1 equaled:1 sense:1 membership:15 typically:1 diminishing:1 hidden:1 comprising:1 selects:1 overall:3 classification:2 animal:10 constrained:2 equal:1 construct:3 having:1 sampling:1 identical:1 stuart:1 unsupervised:2 future:1 connectionist:2 report:2 randomly:1 dg:1 simultaneously:2 resulted:2 individual:3 intended:2 connects:1 attractor:1 attempt:3 interest:1 investigate:2 evaluation:2 extreme:1 yielding:1 worker:3 necessary:1 orthogonal:1 desired:1 re:3 guidance:2 cda:1 psychological:4 increased:2 formalism:2 modeling:3 column:1 instance:5 recomputing:1 retains:1 assignment:2 maximization:1 phrase:1 surpasses:2 subset:2 entry:1 front:2 reported:3 eec:1 synthetic:8 notate:1 thanks:1 probabilistic:1 lee:5 pool:1 enhance:1 michael:6 reflect:1 containing:1 worse:3 cognitive:6 return:2 stark:2 account:1 converted:1 suggesting:1 wk:4 includes:1 later:1 try:2 performed:1 break:6 compiler:1 competitive:1 recover:2 maintains:1 reached:1 start:2 substructure:1 voiced:3 square:3 accuracy:1 variance:8 succession:1 climbing:1 conceptually:1 accurately:1 produced:1 published:2 reach:1 touretzky:1 manual:1 sharing:1 ed:1 pp:1 storm:2 stop:3 formalize:1 schedule:1 back:1 appears:2 originally:1 higher:1 follow:2 improved:1 formulation:3 box:2 strongly:1 shrink:1 furthermore:1 just:1 stage:3 roger:1 until:1 hand:3 expressive:1 overlapping:2 lack:1 kernighan:3 quality:6 perhaps:1 name:3 effect:1 requiring:1 contain:1 alternating:3 semantic:1 ind:1 maintained:1 criterion:1 generalized:1 toggle:1 hill:1 performs:3 wise:1 novel:4 recently:1 superior:1 common:1 shepard:4 extend:1 interpretation:1 occurred:1 he:1 surpassing:1 significant:1 cambridge:1 gibbs:1 ai:1 automatic:1 tuning:1 language:1 similarity:18 carroll:9 longer:3 showed:1 recent:1 perspective:1 forcing:1 manipulation:1 occasionally:1 binary:3 arbitrarily:1 joshua:1 seen:1 additional:2 performer:1 converge:1 determine:1 paradigm:1 afterwards:1 multiple:3 full:1 technical:1 match:1 iqr:13 lin:3 escaped:1 coded:1 ensuring:1 variant:1 regression:4 noiseless:1 expectation:1 surpassed:2 iteration:1 represent:4 achieved:2 proposal:3 addition:2 separately:1 annealing:2 median:1 source:1 eliminates:1 anp:1 ascent:8 induced:1 tend:1 effectiveness:2 jordan:1 practitioner:1 call:2 ngo:1 ideal:1 constraining:1 kiers:3 concerned:1 variety:1 independence:3 fit:1 psychology:1 opposite:1 translates:1 motivated:1 effort:1 penalty:1 reformulated:1 cause:1 afford:1 repeatedly:2 ignored:1 clear:1 listed:2 nasal:1 factorial:1 amount:1 tenenbaum:5 generate:1 nsf:1 discrete:7 promise:1 group:1 changing:4 douglas:4 clean:1 kenneth:1 graph:1 merely:1 sum:1 year:1 run:15 letter:8 powerful:1 almost:1 scaling:2 guaranteed:1 simplification:1 quadratic:1 annual:2 precisely:1 nearby:1 performing:1 wilms:2 combination:3 march:1 across:1 slightly:1 em:8 terminates:3 smaller:1 appealing:1 making:1 modification:2 intuitively:1 gradually:1 sij:2 taken:2 previously:2 remains:1 abbreviated:1 turn:5 operation:1 hierarchical:2 appropriate:1 petsche:1 original:13 abandoned:1 assumes:1 clustering:16 running:3 top:1 include:1 thomas:2 log2:1 lock:1 february:1 society:2 move:1 arrangement:2 realized:1 already:1 strategy:7 unclear:1 gradient:10 simulated:1 street:1 philip:1 considers:1 toward:1 enforcing:1 code:5 modeled:1 relationship:1 providing:3 difficult:3 unfortunately:1 robert:1 negative:2 synthesizing:1 design:2 implementation:4 kahng:2 reliably:1 perform:1 unvoiced:5 discarded:1 descent:1 extended:1 communication:1 arbitrary:1 david:1 complement:1 crashed:1 unpublished:1 kl:6 extensive:1 optimized:2 connection:1 adclus:3 established:1 nip:2 address:2 beyond:2 able:3 suggested:2 below:7 pattern:1 reimplemented:2 proceeds:1 biasing:1 interpretability:1 confusability:1 difficulty:1 treated:1 force:1 predicting:1 abbreviate:1 residual:1 mn:1 improve:1 shieber:1 finished:2 superseded:1 fjk:1 review:2 determining:1 relative:2 fully:1 encompassing:1 bear:1 expect:1 clouse:3 remarkable:1 degree:2 basin:1 principle:1 editor:2 share:2 summary:1 accounted:2 surprisingly:1 parity:1 free:3 changed:1 supported:1 guide:1 allow:1 taking:1 distributed:2 van:1 boundary:2 default:1 plain:1 fred:1 equated:3 collection:1 commonly:1 adaptive:1 simplified:1 global:1 consonant:5 search:6 table:13 symmetry:1 hc:2 japanese:1 constructing:5 domain:1 ruml:4 did:2 noise:4 henk:2 crafted:1 enormously:1 parker:2 garrison:2 theme:1 explicit:2 psychologica:1 third:2 anil:2 specific:2 concern:1 weakest:1 burden:1 incorporating:1 joe:1 adding:2 effectively:1 intersection:1 simply:2 josh:3 ma:1 stimulating:1 marked:2 shared:2 absence:2 change:7 determined:2 typical:2 except:1 uniformly:1 justify:1 wt:1 baluja:2 pas:1 accepted:1 attempted:1 indicating:1 damaging:1 mark:1 searched:1 incorporate:1 evaluate:2 tested:1 scratch:1 handling:2 |
1,182 | 2,078 | A Generalization of Principal Component
Analysis to the Exponential Family
Michael Collins
Sanjoy Dasgupta
Robert E. Schapire
AT&T Labs Research
180 Park Avenue, Florham Park, NJ 07932
mcollins, dasgupta, schapire @research.att.com
Abstract
Principal component analysis (PCA) is a commonly applied technique
for dimensionality reduction. PCA implicitly minimizes a squared loss
function, which may be inappropriate for data that is not real-valued,
such as binary-valued data. This paper draws on ideas from the Exponential family, Generalized linear models, and Bregman distances, to give a
generalization of PCA to loss functions that we argue are better suited to
other data types. We describe algorithms for minimizing the loss functions, and give examples on simulated data.
1 Introduction
Principal component analysis (PCA) is a hugely popular dimensionality reduction technique that attempts to find a low-dimensional subspace passing close to a given set of
points
. More specifically, in PCA, we find a lower dimensional subspace
that
minimizes the sum of the squared distances from the data points to their projections
in the subspace, i.e.,
(1)
This turns out to be equivalent to choosing a subspace that maximizes the sum of the
squared lengths of the projections , which is the same as the (empirical) variance of
these projections if the data happens to be centered at the origin (so that ).
PCA also has another convenient interpretation that is perhaps less well known. In this
probabilistic interpretation, each point is thought of as a random draw from some un
known distribution "! , where denotes a unit Gaussian with mean #$ . The purpose
then of PCA is to find the set of parameters %
that maximizes the likelihood of the
data, subject to the condition that these parameters all lie in a low-dimensional subspace. In
are considered to be noise-corrupted versions of some true points
other
words,
& which lie in a subspace; the goal is to find these true points, and the main assumption is that the noise is Gaussian. The equivalence of this interpretation to the ones
given above follows simply from the fact that negative log likelihood under this Guassian
model is equal (ignoring constants) to Eq. (1).
This Gaussian assumption may be inappropriate, for instance if data is binary-valued, or
integer-valued, or is nonnegative. In fact, the Gaussian is only one of the canonical distributions that make up the exponential family, and it is a distribution tailored to real-valued
data. The Poisson is better suited to integer data, and the Bernoulli to binary data. It seems
natural to consider variants of PCA which are founded upon these other distributions in
place of the Gaussian.
be any parameterized set
We extend PCA to the rest of the exponential family.
Let
of distributions from the exponential family, where is the natural parameter of a distribution. For instance, a one-dimensional Poisson distribution can be parameterized by ,
and distribution
corresponding to mean
Given data
& , the goal is now to find parameters
!
which lie in a low-dimensional subspace and for which the log-likelihood
is maximized.
Our unified approach effortlessly permits hybrid dimensionality reduction schemes in
which different types of distributions can be used for different attributes of the data. If
the data have a few binary attributes
and a few integer-valued attributes, then some co
ordinates of the corresponding can be parameters of binomial distributions while others
are parameters of Poisson distributions. (However, for simplicity of presentation, in this
abstract we assume all distributions are of the same type.)
The dimensionality reduction schemes for non-Gaussian distributions are substantially different from PCA. For instance, in PCA the parameters , which are means of Gaussians,
lie in a space which coincides with that of the data . This is not the case in general, and
therefore, although the parameters lie in a linear subspace, they typically correspond to
a nonlinear surface in the space of the data.
The discrepancy and interaction between the space of parameters and the space of the
data is a central preoccupation in the study of exponential families, generalized linear
models (GLM?s), and Bregman distances. Our exposition is inevitably woven around these
three intimately related subjects. In particular, we show that the way in which we generalize
PCA is exactly analogous to the manner in which regression is generalized by GLM?s. In
this respect, and in others which will be elucidated later, it differs from other variants of
PCA recently proposed by Lee and Seung [7], and by Hofmann [4].
We show that the optimization problem we derive can be solved quite naturally by an algorithm that alternately minimizes over the components of the analysis and their coefficients;
thus, the algorithm is reminiscent of Csisz?ar and Tusn?ady?s alternating minization procedures [2]. In our case, each side of the minimization is a simple convex program that can
be interpreted as a projection with respect to a suitable Bregman distance; however, the
overall program is not generally convex. In the case of Gaussian distributions, our algorithm coincides exactly with the power method for computing eigenvectors; in this sense it
is a generalization of one of the oldest algorithms for PCA. Although omitted for lack of
space, we can show that our procedure converges in that any limit point of the computed
coefficients is a stationary point of the loss function. Moreover, a slight modification of the
optimization criterion guarantees the existence of at least one limit point.
!
#"
$ &%
Some comments on notation: All vectors in this paper are row vectors. If
we denote its ?th row by and its ?th element by .
is a matrix,
2 The Exponential Family, GLM?s, and Bregman Distances
2.1 The Exponential Family and Generalized Linear Models
In the exponential family of distributions the conditional probability of a value
parameter value takes the following form:
' ( ) +* ' -, '. 0/
'
given
(2)
'
/
/
'
( ) * '
We use to denote the domain of ' . The
sum is replaced by an integral in the continuous
case, where defines a density over . * is a term that depends only on ' , and can
usually be ignored as a constant during estimation.
difference between different
. We willTheseemain
that almost all of the concepts of
members of the family is the form of /
the PCA algorithms in this paper stem directly from the definition of / .
A first example is a normal distribution,
and unit variance, which has a density
' with( )mean
that
that is usually written as
' . It can be verified
this is a member
of
the
exponential
family
with
,
*
'
'
,
. Another
and /
common
case
is
a
Bernoulli
distribution
for
the
case
of
binary
. The probability of ' is usually written ' +
case
outcomes.
-
In thiswhere
. This is a member of the exponential family
is a parameter in
,
.
with * ' , ( ) , and /
/ , which we will denote as throughout this
A critical function is the derivative
/ * ' , it is easily verified
paper.
that
'
,Bythedifferentiating
expectation
of
under
.
In
the
normal
distribution,
,
and
in
'
'
'
. In the general case, '
is referred to as the ?expectation
the Bernoulli case '
parameter?, and defines a function from the natural parameter values to the expectation
Here, is the ?natural parameter? of the distribution, and can usually take any value in the
reals.
is a function that ensures that the sum (integral) of
over the domain of
is 1. From this it follows that
.
parameter values.
Our generalization of PCA is analogous to the manner in which generalized linear models (GLM?s) [8] provide a unified treatment of regression for the exponential family by
generalizing least-squares regression to loss functions that are more appropriate for other
members of this family. The regression set-up assumes a training sample of pairs,
where is a vector of attributes, and is some response variable. The pa . The dot product
is taken to be an
rameters of the model are a vector
are set to be
approximation of . In least squares regression the optimal parameters
& .
!" $#&%(' *) +
In GLM?s, , - is taken to approximate the expectation parameter of the exponential model, where , is the inverse of the ?link function? [8]. A natural choice is to use
the ?canonical link?, where , , being the derivative /.
. In this case the natural
/
parameters are directly
approximated
by
,
and
the
log-likelihood
'
,
/ 2
. In the case of a normal distribution
is simply ( ) * 01
/
and it follows easily that the maximum-likelihood criwith fixed variance,
terion is equivalent
to the least
squares
Another interesting case is logistic re
,
, andcriterion.
gression3 where /
the
negative
log-likelihood for parameters is
! 7 8 !:9
,
where if ,
if .
) 546
2.2 Bregman Distances and the Exponential Family
;2<>=@? be a differentiable and strictly convex function defined on a closed, convex
=BA . The Bregman distance associated with ; is defined for
C D= to be
EGF H JI C ; H + ; C LK C M C
where K ' ; ' . It can be shown that, in general, every Bregman distance is nonnegative and is equal to zero if and only if its two arguments are equal.
' is directly related to a Bregman
For the exponential family the log-likelihood ( )
Let
set
normal
$
, . -/
56.-/87 $39 ' .-3
<>= ?A@"B0
-
< = I-A@
.-
$ 0J
Poisson
!#
! #
.-/
0
1
?
B
?
0 GD FE
1
-N
! # 1
0
1
-
C?
Bernoulli
"!#
%
&('*) %
-3
-
/
.+ -/
0
21
0 41
;' 9 :
'
?
?/
:
0 ED FG
H1
0 ' 99 D F
-LKH7
"! 9 : # where
M
1
B0
1
0
1
6
Table 1: Various functions of interest for three members of the exponential family
/ and :
; , / ; through
(3)
K
It can be shown under fairly general conditions that ' ' . Application of these
identities implies that the negative log-likelihood of a point can be expressed through a
Bregman distance [1, 3]:
(4)
( ) ' ( ) +* ' ; ' , E F ' I
distance. Specifically, [1, 3] define a ?dual? function
In other words, negative log-likelihood can always be written as a Bregman distance plus
a term that is constant with respect to and which therefore can be ignored. Table 1
summarizes various functions of interest for examples of the exponential family.
We will find it useful to extend the idea of Bregman distances to divergences between vectheT notation
tors and matrices. If , O are vectors, and P , Q are matrices, then we overload
SR
U
and
. (The
as
O
P
Q
notion of Bregman distance as well as our generalization of PCA can be extended to vectors in a more general manner; here, for simplicity, we restrict our attention to Bregman
distances and PCA problems of this particular form.)
EF I
E F ' I
EF I
%E F % I %
3 PCA for the Exponential Family
We now generalize PCA to other members of the exponential family. We wish to find ?s
that are ?close? to the ?s and which belong to a lower dimensional subspace of parameter
space. Thus, our approach is to find a basis
V %
R
Y VXW Y in and to represent each as
the linear combination of these elements ZY V that is ?closest? to
.
]\_^
\b^
Let [ be the
matrix
matrix whose
d\ whose ?th row is . LetR ` be the a
d\fc ^ ?th row
is V Y , and let P be the
matrix
a matrix with elements Y . Then e PG` is an
whose ?th row is as above. This is a matrix of natural parameter values which define
the probability of each point in [ .
`
[
' &% &%
%
,
' &% % , / %
%
Following the discussion in Section 2, we consider the loss function taking the form
g
P
P
`
Zh
where h is a constant term which will be dropped from here on. The loss function varies
depending on which member of the exponential family is taken, which simply changes the
form of . For example, if [ is a matrix
of real values, and the normal distribution is
appropriate for the data, then
and the loss criterion is the usual squared loss
for PCA. For the Bernoulli distribution,
. If we define
,
!
i
!
i
g
then ` P
.
/
/ ,
/ )
,
%- 6
' & % ' &%
From the relationship between log-likelihood and Bregman distances (see Eq. (4)), the loss
can also be written as
g
`
P
E F ' &% I %
%
E F I
(where we allow to be applied to vectors and matrices in a pointwise manner). Once `
and P have been found for the data points, the ?th data point can be represented
W
as the vector in the lower dimensional space . Then are the coefficients which
define a Bregman projection of the vector :
" &# % ' E F I 3`
(5)
The generalized form of PCA can also be considered to be search for a low dimensional
basis (matrix ` ) which defines a surface
that is close to all the data points . We define the
3`
W . The optimal value for ` then minset of points ` to be `
imizes the sum of projection distances: `
.
and the Bregman distance is Euclidean disNote that for the normal distribution
tance
so that the projection operation in Eq. (5) is a simple linear projection ("
` ).
is also simplified in the normal case, simply being the hyperplane whose basis is ` .
`
/
#&%(' E F I
" # %('
To summarize, once a member of the exponential family ? and by implication a convex
function
? is chosen, regular PCA is generalized in the following way:
The loss function is negative log-likelihood,
constant.
( ) '
' , / -,
The matrix e PG` is taken to be a matrix of natural parameter values.
of / defines a matrix of expectation parameters, PG` .
The derivative
E F is derived from ; .
A function ; is derived from / and . A Bregman distance
The loss is a sum of Bregman distances from the elements ' % to values % % .
PCA can also be thought of as search for a matrix `
is ?close? to all the data points.
`
that defines a surface
which
, and
The normal distribution is a simple case because
the divergence is Euclidean
distance. The projection operation is a linear operation, and ` is the hyperplane which
has ` as its basis.
4 Generic Algorithms for Minimizing the Loss Function
We now describe a generic algorithm for minimization of the loss function. First, we concentrate on the simplest
case where there is just a single component so that a
. (We drop
R
the c subscript from Y and Y .) The method is iterative, with an initial random choice for
the value of ` . Let `
,P
, etc. denote the values at the ?th
iteration, and let
g ` be the
P
`
P
initial random
choice.
We
propose
the
iterative
updates
g
g
and `
.
Thus
is
alternately
minimized
with
respective
to
`
P
its two arguments, each time optimizing one argument while keeping the other one fixed,
reminiscent of Csisz?ar and Tusn?ady?s alternating minization procedures [2].
%
" # %('
" # %('
3
9
% E E F F 3 ' &% I I S*R R % 9
' % .
It is useful to write these minimization problems as follows:
R
For
,
^
For
,
"
!"
! " $#&%('
%
$ #&%('
*
,
^
We can then see that there are
optimization problems, and that each one is essentially identical to a GLM regression problem (a very simple one, where there is a single
parameter being optimized over). These sub-problems are easily solved, as the functions
are convex in the argument being optimized over, and the large literature on maximumlikelihood estimation in GLM?s can be directly applied to the problem.
These
updates
take
a
simple
form
for
the
nor
P
mal distribution:
, and `
. It
`
[
P
P
[ `
`
[ [ h , where h is a scalar value. The method is then equivfollows that `
alent to the power method (see Jolliffe [5]) for finding the eigenvector of [ [ with the
largest eigenvalue, which is the best single component solution for ` . Thus the generic
algorithm generalizes one of the oldest algorithms for solving the regular PCA problem.
The loss is convex in either of its arguments with the other fixed, but in general is not
convex in the two arguments together. This makes it very difficult to prove convergence to
the global minimum. The normal distribution is an interesting special case in this respect ?
the power method is known to converge to the optimal solution, in spite of the non-convex
nature of the loss surface. A simple proof of this comes from properties of eigenvectors
(Jolliffe [5]). It can also be explained by analysis of the Hessian : for any stationary point
is not positive semi-definite. Thus these stationary
which is not the global minimum,
points are saddle points rather than local minima. The Hessian for the generalized loss
function is more complex; it remains an open problem whether it is also not positive semidefinite at stationary points other than the global minimum. It is also open to determine
under which conditions this generic algorithm will converge to a global minimum. In
preliminary numerical studies, the algorithm
well behaved in this respect.
seems to be
P `
Moreover, any limit point of the sequence e
will be a stationary point.
However, it is possible for this sequence to diverge since the optimum may be at infinity.
To avoid such degenerate choices of e , we can use a modified loss
E F ' &% I % , E F * I %
%
where is a small
positive constant, and +* is any value in the range of (and therefore
for which * is finite). This is roughly equivalent to adding a conjugate prior and
finding the maximum
solution. It can be proved, for this modified loss, that the
remainsa posteriori
sequence e
in a bounded region and hence always has at least one limit point
which must be a stationary point. (All proofs omitted for lack of space.)
There are various ways to optimize the loss function when there is more than one component. We give one algorithm which cycles through the a components, optimizing each in
turn while the others are held fixed:
!
/
10
//Initialization
7
7
Set
,
//Cycle through
components
times
7
7
0 ,
0 :
For
//Now optimize & the ?th component with other components fixed
7
Initialize
+ randomly, and set
7
For
0 convergence
&
& 9 '
7
7
@ $
,
For
< =
+
+
&
7
7
@ $ &
< =
8 ,
For
+
+
3
9
##"" $ % '&)(#*,+ - 3 " #" . 9
$ % '&)(#*32 -
.
The modified Bregman projections now include a term 4 &% representing the contribution of
the a fixed components. These sub-problems are again a standard optimization problem
regarding Bregman distances, where the terms 4 % form a ?reference prior?.
90
500
data
PCA
exp
80
data
PCA
exp
450
70
400
60
350
300
50
Y
Y
40
250
200
30
150
20
100
10
50
0
0
10
20
30
40
50
60
70
80
0
0
90
20
40
60
X
80
100
X
Figure 1: Regular PCA vs. PCA for the exponential distribution.
B
B
1
1
0.8
0.8
B?
0.6
0.4
0.4
0.2
0.2
0
1
0
1
A
B?
0.6
C?
C
0.8
A
1
0.6
D
C
0.8
1
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0
0.8
0.6
0.4
0.4
0.2
C?
D?
0.2
0
0
0
E
Figure 2: Projecting from 3- to 1-dimensional space, via Bernoulli PCA. Left: the three
points h are projected onto a one-dimensional curve. Right: point is added.
5 Illustrative examples
Exponential distribution. Our generalization of PCA behaves rather differently for different members of the exponential family. One interesting example is that of the exponential
distributions on nonnegative reals. For one-dimensional data, these densities are usually
written as , where is the mean. In the uniform system of notation we have been
using, we would instead index each distribution by a single natural parameter
, where
(basically,
.
), and write the density as
The link function in this case is
, the mean of the distribution.
'
/
'
and want to find the best one-dimensional
apSuppose we are given data [
*R
V
proximation: a vector V and coefficients such that the approximation
& has minimum loss. The alternating minimization procedure of the previous section has a simple closed form in this case, consisting of the iterative update rule
V
^
[
[b V
Here the shorthand denotes a componentwise reciprocal, i.e.,
. Notice the
[ [bV . Once V is found,
similarity to the update rule of the power
for PCA: V
^ method
R
The points V lie on a line through
we can recover the coefficients
toSR alsoR lie on a straight line;
the origin. Normally, we would not expect the points
however, in this case they do, because any point of the form
V
, can be written
as and so must lie in the direction .
)
Therefore, we can reasonably ask how the lines found under this exponential assumption
differ from those found under a Gaussian assumption (that is, those found by regular PCA),
provided all data is nonnegative. As a very simple illustration, we conducted two toy experiments with twenty data points in (Figure 1). In the first, the points all lay very close
to a line, and the two versions of PCA produced similar results. In the second experiment,
a few of the points were moved farther afield, and these outliers had a larger effect upon
regular PCA than upon its exponential variant.
Bernoulli distribution. For the Bernoulli distribution, a linear subspace of the space of
parameters is typically a nonlinear surface in the
space of the data. In Figure 2 (left),
three points in the three-dimensional hypercube are mapped via our PCA to a onedimensional curve. The curve passes through one of the points ( ); the projections of the
h ) are indicated. Notice that the curve is symmetric about
two other (
and h
the center of the hypercube,
. In Figure 2 (right), another point (D) is added,
and causes the approximating one-dimensional curve to swerve closer to it.
E ? E
?
6 Relationship to Previous Work
Lee and Seung [6, 7] and Hofmann [4] also describe probabilistic alternatives to PCA,
tailored to data types that are not gaussian. In contrast to our method, [4, 6, 7] approximate mean parameters underlying the generation of the data points, with constraints on
the matrices P and ` ensuring that the elements of PG` are in the correct domain. By
instead choosing to approximate the natural parameters, in our method the matrices P and
` do not usually need to be constrained?instead, we rely on the link function to give a
transformed matrix PG`
which lies in the domain of the data points.
% ' &% ( ) % , &%
More specifically, Lee and Seung [6] use the loss function
R
(ignoring constant factors, and again defining $ Y Y Y ). This is optimized with the
constraint that P and ` should be positive. This method has a probabilistic interpretation,
where each data point is generated from a Poisson distribution with mean
parameter ! i .
For the Poisson distribution, our method uses the loss function
,
but without any constraints on the matrices P and ` . The algorithm in Hofmann [4] uses
, where the matrices P and ` are constrained such that
a loss function
all the ?s are positive, and also such that
.
' &%
%
% ' &% ( ) %
&%
%
%
% ' % &% ,
% %
Bishop and Tipping [9] describe probabilistic variants of the gaussian case. Tipping [10]
discusses a model that is very similar to our case for the Bernoulli family.
Acknowledgements. This work builds upon intuitions about exponential families and
Bregman distances obtained largely from interactions with Manfred Warmuth, and from
his papers. Thanks also to Andreas Buja for several helpful comments.
References
[1] Katy S. Azoury and M. K. Warmuth. Relative loss bounds for on-line density estimation with
the exponential family of distributions. Machine Learning, 43:211?246, 2001.
[2] I. Csisz?ar and G. Tusn?ady. Information geometry and alternating minimization procedures.
Statistics and Decisions, Supplement Issue, 1:205?237, 1984.
[3] J?urgen Forster and Manfred Warmuth. Relative expected instantaneous loss bounds. Journal of
Computer and System Sciences, to appear.
[4] Thomas Hofmann. Probabilistic latent semantic indexing. In Proceedings of the 22nd Annual
International ACM SIGIR Conference on Research and Development in Information Retrieval,
1999.
[5] I. T. Jolliffe. Principal Component Analysis. Springer-Verlag, 1986.
[6] D. D. Lee and H. S. Seung. Learning the parts of objects with nonnegative matrix factorization.
Nature, 401:788, 1999.
[7] Daniel D. Lee and H. Sebastian Seung. Algorithms for non-negative matrix factorization. In
Advances in Neural Information Processing Systems 13, 2001.
[8] P. McCullagh and J. A. Nelder. Generalized Linear Models. CRC Press, 2nd edition, 1990.
[9] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the
Royal Statistical Society, Series B, 61(3):611?622, 1999.
[10] Michael E. Tipping. Probabilistic visualisation of high-dimensional binary data. In Advances
in Neural Information Processing Systems 11, pages 592?598, 1999.
| 2078 |@word version:2 seems:2 nd:2 open:2 pg:5 reduction:4 initial:2 series:1 att:1 daniel:1 terion:1 com:1 reminiscent:2 written:6 must:2 numerical:1 hofmann:4 afield:1 drop:1 update:4 v:1 stationary:6 warmuth:3 oldest:2 reciprocal:1 farther:1 manfred:2 vxw:1 prove:1 shorthand:1 manner:4 expected:1 roughly:1 nor:1 inappropriate:2 provided:1 moreover:2 notation:3 maximizes:2 bounded:1 underlying:1 interpreted:1 minimizes:3 substantially:1 eigenvector:1 unified:2 finding:2 nj:1 guarantee:1 every:1 alent:1 exactly:2 unit:2 normally:1 appear:1 positive:5 dropped:1 local:1 limit:4 subscript:1 plus:1 initialization:1 equivalence:1 co:1 factorization:2 range:1 definite:1 differs:1 procedure:5 empirical:1 thought:2 projection:11 convenient:1 word:2 regular:5 spite:1 onto:1 close:5 optimize:2 equivalent:3 center:1 attention:1 hugely:1 convex:9 sigir:1 simplicity:2 rule:2 his:1 imizes:1 notion:1 analogous:2 mcollins:1 us:2 origin:2 pa:1 element:5 approximated:1 lay:1 solved:2 mal:1 region:1 ensures:1 cycle:2 intuition:1 seung:5 woven:1 solving:1 upon:4 basis:4 easily:3 differently:1 various:3 represented:1 guassian:1 describe:4 choosing:2 outcome:1 quite:1 whose:4 larger:1 valued:6 florham:1 statistic:1 sequence:3 differentiable:1 eigenvalue:1 propose:1 interaction:2 product:1 degenerate:1 moved:1 csisz:3 convergence:2 optimum:1 converges:1 object:1 derive:1 depending:1 b0:2 eq:3 implies:1 come:1 differ:1 concentrate:1 direction:1 correct:1 attribute:4 centered:1 crc:1 generalization:6 preliminary:1 strictly:1 effortlessly:1 around:1 considered:2 normal:9 exp:2 tor:1 omitted:2 purpose:1 estimation:3 largest:1 minimization:5 gaussian:10 always:2 modified:3 rather:2 avoid:1 derived:2 bernoulli:9 likelihood:11 contrast:1 sense:1 posteriori:1 helpful:1 letr:1 typically:2 visualisation:1 transformed:1 overall:1 dual:1 issue:1 development:1 constrained:2 special:1 fairly:1 initialize:1 urgen:1 equal:3 ady:3 once:3 identical:1 park:2 discrepancy:1 minimized:1 others:3 few:3 randomly:1 divergence:2 replaced:1 geometry:1 consisting:1 attempt:1 interest:2 semidefinite:1 held:1 implication:1 bregman:21 integral:2 closer:1 respective:1 euclidean:2 re:1 instance:3 ar:3 uniform:1 conducted:1 varies:1 corrupted:1 gd:1 thanks:1 density:5 international:1 probabilistic:7 lee:5 diverge:1 michael:2 together:1 squared:4 central:1 again:2 derivative:3 toy:1 coefficient:5 depends:1 later:1 h1:1 lab:1 closed:2 recover:1 contribution:1 square:3 variance:3 largely:1 maximized:1 correspond:1 generalize:2 zy:1 basically:1 produced:1 straight:1 sebastian:1 ed:1 definition:1 naturally:1 associated:1 proof:2 proved:1 treatment:1 popular:1 ask:1 dimensionality:4 tipping:4 response:1 just:1 nonlinear:2 lack:2 defines:5 logistic:1 indicated:1 perhaps:1 behaved:1 effect:1 concept:1 true:2 hence:1 alternating:4 symmetric:1 semantic:1 during:1 illustrative:1 coincides:2 criterion:2 generalized:9 instantaneous:1 ef:2 recently:1 common:1 behaves:1 ji:1 extend:2 interpretation:4 slight:1 belong:1 onedimensional:1 had:1 dot:1 similarity:1 surface:5 etc:1 closest:1 optimizing:2 verlag:1 binary:6 minimum:6 converge:2 determine:1 semi:1 stem:1 retrieval:1 ensuring:1 variant:4 regression:6 essentially:1 expectation:5 poisson:6 iteration:1 represent:1 tailored:2 want:1 rest:1 sr:1 pass:1 comment:2 subject:2 member:9 integer:3 restrict:1 andreas:1 idea:2 regarding:1 avenue:1 whether:1 pca:37 passing:1 hessian:2 cause:1 ignored:2 generally:1 useful:2 eigenvectors:2 simplest:1 schapire:2 canonical:2 notice:2 write:2 dasgupta:2 verified:2 sum:6 inverse:1 parameterized:2 place:1 family:26 almost:1 throughout:1 draw:2 decision:1 summarizes:1 bound:2 nonnegative:5 elucidated:1 annual:1 bv:1 infinity:1 constraint:3 argument:6 combination:1 conjugate:1 intimately:1 modification:1 happens:1 explained:1 projecting:1 outlier:1 indexing:1 glm:7 taken:4 remains:1 turn:2 discus:1 jolliffe:3 generalizes:1 gaussians:1 operation:3 permit:1 appropriate:2 generic:4 alternative:1 existence:1 thomas:1 denotes:2 binomial:1 assumes:1 include:1 build:1 approximating:1 hypercube:2 society:1 tusn:3 added:2 usual:1 forster:1 subspace:10 distance:22 link:4 mapped:1 simulated:1 argue:1 length:1 pointwise:1 relationship:2 index:1 illustration:1 minimizing:2 difficult:1 robert:1 fe:1 negative:6 ba:1 twenty:1 finite:1 inevitably:1 defining:1 extended:1 buja:1 ordinate:1 pair:1 optimized:3 componentwise:1 alternately:2 usually:6 summarize:1 program:2 royal:1 tance:1 power:4 suitable:1 critical:1 natural:10 hybrid:1 rely:1 representing:1 scheme:2 lk:1 prior:2 literature:1 swerve:1 acknowledgement:1 zh:1 relative:2 loss:26 expect:1 interesting:3 rameters:1 generation:1 row:5 keeping:1 side:1 allow:1 taking:1 fg:1 curve:5 commonly:1 projected:1 simplified:1 founded:1 approximate:3 implicitly:1 global:4 proximation:1 nelder:1 un:1 continuous:1 search:2 iterative:3 latent:1 table:2 nature:2 reasonably:1 ignoring:2 complex:1 domain:4 main:1 azoury:1 noise:2 edition:1 referred:1 sub:2 wish:1 exponential:29 lie:9 bishop:2 adding:1 preoccupation:1 supplement:1 suited:2 generalizing:1 fc:1 simply:4 saddle:1 expressed:1 katy:1 scalar:1 springer:1 acm:1 conditional:1 goal:2 presentation:1 identity:1 exposition:1 change:1 mccullagh:1 specifically:3 hyperplane:2 minization:2 principal:5 sanjoy:1 egf:1 maximumlikelihood:1 collins:1 overload:1 |
1,183 | 2,079 | Improvisation and Learning
Judy A. Franklin
Computer Science Department
Smith College
Northampton, MA 01063
jfranklin@cs.smith.edu
Abstract
This article presents a 2-phase computational learning model and application. As a demonstration, a system has been built, called CHIME for
Computer Human Interacting Musical Entity. In phase 1 of training, recurrent back-propagation trains the machine to reproduce 3 jazz melodies.
The recurrent network is expanded and is further trained in phase 2 with a
reinforcement learning algorithm and a critique produced by a set of basic
rules for jazz improvisation. After each phase CHIME can interactively
improvise with a human in real time.
1 Foundations
Jazz improvisation is the creation of a jazz melody in real time. Charlie Parker, Dizzy Gillespie, Miles Davis, John Coltrane, Charles Mingus, Thelonious Monk, and Sonny Rollins et
al. were the founders of bebop and post bop jazz [9] where drummers, bassists, and pianists
keep the beat and maintain harmonic structure. Other players improvise over this structure
and even take turns improvising for 4 bars at a time. This is called trading fours.
Meanwhile, artificial neural networks have been used in computer music [4, 12]. In particular, the work of (Todd [11]) is the basis for phase 1 of CHIME, a novice machine improvisor that learns to trade fours. Firstly, a recurrent network is trained with back-propagation
to play three jazz melodies by Sonny Rollins [1], as described in Section 2. Phase 2 uses
actor-critic reinforcement learning and is described in Section 3. This section is on jazz basics.
1.1 Basics: Chords, the ii-V-I Chord Progression and Scales
The harmonic structure mentioned above is a series of chords that may be reprated and that
are often grouped into standard subsequences. A chord is a group of notes played simultaneously. In the chromatic scale, C-Db-D-Eb-E-F-Gb-G-Ab-A-Bb-B-C, notes are separated
by a half step. A flat (b) note is a half step below the original note; a sharp (#) is a half above.
Two half steps are a whole step. Two whole steps are a major third. Three half steps are a
minor third.
A major triad (chord) is the first or tonic note, then the note a major third up, then the note a
minor third up. When F is the tonic, F major triad is F-A-C. A minor triad (chord) is the tonic
www.cs.smith.edu/?jfrankli
then a minor third, then a major third. F minor triad is F-Ab-C. The diminished triad is the
tonic, then a minor third, then a minor third. F diminished triad is F-Ab-Cb. An augmented
triad is the tonic, then a major third, then a major third. The F augmented triad is F-A-Db.
A third added to the top of a triad forms a seventh chord. A major triad plus a major third is
the major seventh chord. F-A-C-E is the F major seventh chord (Fmaj7). A minor triad plus
a minor third is a minor seventh chord. For F it is F-Ab-C-Eb (Fm7). A major triad plus a
minor third is a dominant seventh chord. For F it is F-A-C-Eb (F7). These three types of
chords are used heavily in jazz harmony. Notice that each note in the chromatic scales can
be the tonic note for any of these types of chords.
A scale, a subset of the chromatic scale, is characterized by note intervals. Let W be a whole
step and H be a half. The chromatic scale is HHHHHHHHHHHH. The major scale or ionian
mode is WWHWWWH. F major scale is F-G-A-Bb-C-D-E-F. The notes in a scale are degrees; E is the seventh degree of F major. The first, third, fifth, and seventh notes of a major
scale are the major seventh chord. The first, third, fifth, and seventh notes of other modes
produce the minor seventh and dominant seventh chords. Roman numerals represent scale
degrees and their seventh chords. Upper case implies major or dominant seventh and lower
case implies minor seventh [9]. The major seventh chord starting at the scale tonic is the I
(one) chord. G is the second degree of F major, and G-Bb-D-F is Gm7, the ii chord, with respect to F. The ii-V-I progression is prevalent in jazz [9], and for F it is Gm7-C7-Fmaj7. The
minor ii-V-i progression is obtained using diminished and augmented triads, their seventh
chords, and the aeolian mode. Seventh chords can be extended by adding major or minor
thirds, e.g. Fmaj9, Fmaj11, Fmaj13, Gm9, Gm11, and Gm13. Any extension can be raised
or lowered by 1 step [9] to obtain, e.g. Fmaj7#11, C7#9, C7b9, C7#11.
Most jazz compositions are either the 12 bar blues or sectional forms (e.g. ABAB, ABAC,
or AABA) [8]. The 3 Rollins songs are 12 bar blues. ?Blue 7? has a simple blues form. In
?Solid? and ?Tenor Madness?, Rollins adds bebop variations to the blues form [1]. ii-V-I
and VI-II-V-I progressions are added and G7+9 substitutes for the VI and F7+9 for the V
(see section 1.2 below); the II-V in the last bar provides the turnaround to the I of the first bar
to foster smooth repetition of the form. The result is at left and in Roman numeral notation
Bb7 Bb7 Bb7
Bb7
I
I
I
I
Eb7
Eb7 Bb7
G7+9
IV IV I
VI
at right:
Cm7 F7
Bb7 G7+9 C7 F7+9
ii
V
I VI II V
1.2 Scale Substitutions and Rules for Reinforcement Learning
First note that the theory and rules derived in this subsection are used in Phase 2, to be described in Section 3. They are presented here since they derive from the jazz basics immediately preceding. One way a novice improvisor can play is to associate one scale with each
chord and choose notes from that scale when the chord is presented in the musical score.
Therefore, Rule 1 is that an improvisor may choose notes from a ?standard? scale associated with a chord. Next, the 4th degree of the scale is often avoided on a major or dominant
seventh chord (Rule 3), unless the player can resolve its dissonance. The major 7th is an
avoid note on a dominant seventh chord (Rule 4) since a dominant seventh chord and its
scale contain the flat 7th, not the major 7th.
Rule 2 contains many notes that can be added. A brief rationale is given next. The C7 in
Gm7-C7-Fmaj7 may be replaced by a C7#11, a C7+ chord, or a C7b9b5 or C7alt chord [9].
The scales for C7+ and C7#11 make available the raised fourth (flat 5), and flat 6 (flat 13)
for improvising. The C7b9b5 and C7alt (C7+9) chords and their scales make available the
flat9, raised 9, flat5 and raised 5 [1]. These substitutions provide the notes of Rule 2. These
rules (used in phase 2) are stated below, using for reinforcement values very bad (-1.0), bad
(-0.5), a little bad (-0.25), ok (0.25), good (0.5), and very good (1.0). The rules are discussed
further in Section 4.
The Rule Set:
1) Any note in the scale associated with the chord is ok (except as noted in rule 3).
2) On a dominant seventh, hip notes 9, flat9, #9, #11, 13 or flat13 are very good. One hip
note 2 times in a row is a little bad. 2 hip notes more than 2 times in a row is a little bad.
3) If the chord is a dominant seventh chord, a natural 4th note is bad.
4) If the chord is a dominant seventh chord, a natural 7th is very bad.
5) A rest is good unless it is held for more than 2 16th notes and then it is very bad.
6) Any note played longer than 1 beat (4 16th notes) is very bad.
7) If two consecutive notes match the human?s, that is good.
2 CHIME Phase 1
In Phase 1, supervised learning is used to train a recurrent network to reproduce the three
Sonny Rollins melodies.
2.1 Network Details and Training
The recurrent network?s output units are linear. The hidden units are nonlinear (logistic
function). Todd [11] used a Jordan recurrent network [6] for classical melody learning and
generation. In CHIME, a Jordan net is also used, with the addition of the chord as input
(Figure 1. 24 of the 26 outputs are notes (2 chromatic octaves), the 25th is a rest, and the
26th indicates a new note. The output with the highest value above a threshold is the next
note, including the rest output. The new note output
if this is a new note, or if it
indicates
is the same note being held for another time step (
note resolution).
The 12 chord inputs (12 notes in a chromatic scale), are 1 or 0. A chord is represented as
its first, third, fifth, and seventh notes and it ?wraps around? within the 12 inputs. E.g., the
Fm7 chord F-Ab-C-Eb is represented as C, Eb, F, Ab or 100101001000. One plan input per
song enables distinguishing between songs. The 26 context inputs use eligibility traces, giving the hidden units a decaying history of notes played. CHIME (as did Todd) uses teacher
forcing [13], wherein the target outputs for the previous step are used as inputs (so erroneous outputs are not used as inputs). Todd used from 8 to 15 hidden units; CHIME uses
50. The learning rate is 0.075 (Todd used 0.05). The eligibility rate is 0.9 (Todd used 0.8).
Differences in values perhaps reflect contrasting styles of the songs and available computing
power.
Todd used 15 output units and assumed a rest when all note units are ?turned off.? CHIME
uses 24 output note units (2 octaves). Long rests in the Rollins tunes require a dedicated
output unit for a rest. Without it, the note outputs learned to turn off all the time. Below
are results of four representative experiments. In all experiments, 15,000 presentations of
the songs were made. Each song has 192 16th note events. All songs are played at a fixed
tempo. Weights are initialized to small random values. The squared error is the average
squared error over one complete presentation of the song. ?Finessing? the network may
improve these values. The songs are easily recognized however, and an exact match could
impair the network?s ability to improvise. Figure 2 shows the results for ?Solid.?
Experiment 1. Song: Blue Seven. Squared error starts at 185, decreases to 2.67.
Experiment 2. Song: Tenor Madness. Squared error starts at 218, decreases to 1.05.
Experiment 3. Song: Solid. Squared error starts at 184, decreases to 3.77.
Experiment 4. Song: All three songs: Squared error starts at 185, decreases to 36.
Figure 1: Jordan recurrent net with addition of chord input
2.2 Phase 1 Human Computer Interaction in Real Time
In trading fours with the trained network, human note events are brought in via the MIDI
interface [7]. Four bars of human notes are recorded then given, one note event at a time
to the context inputs (replacing the recurrent inputs). The plan inputs are all 1. The chord
inputs follow the ?Solid? form. The machine generates its four bars and they are played in
real time. Then the human plays again, etc. An accompaniment (drums, bass, and piano),
produced by Band-in-a-Box software (PG Music), keeps the beat and provides chords for
the human.
Figure 3 shows an interaction. The machine?s improvisations are in the second and fourth
lines. In bar 5 the flat 9 of the Eb7 appears; the E. This note is used on the Eb7 and Bb7
chords by Rollins in ?Blue 7?, as a ?passing tone.? D is played in bar 5 on the Eb7. D is the
natural 7 over Eb7 (with its flat 7) but is a note that Rollins uses heavily in all three songs,
and once over the Eb7. It may be a response to the rest and the Bb played by the human in
bar 1. D follows both a rest and a Bb in many places in ?Tenor Madness? and ?Solid.? In
bar 6, the long G and the Ab (the third then fourth of Eb7) figure prominently in ?Solid.? At
the beginning of bar 7 is the 2-note sequence Ab-E that appears in exactly the same place
in the song ?Blue 7.? The focus of bars 7 and 8 is jumping between the 3rd and 4th of Bb7.
At the end of bar 8 the machine plays the flat 9 (Ab) then the flat 3 (Bb), of G7+9. In bars
13-16 the tones are longer, as are the human?s in bars 9-12. The tones are the 5th, the root,
the 3rd, the root, the flat 7, the 3rd, the 7th, and the raised fourth. Except for the last 2, these
are chord tones.
3 CHIME Phase 2
In Phase 2, the network is expanded and trained by reinforcement learning to improvise
according to the rules of Section 1.2 and using its knowledge of the Sonny Rollins songs.
3.1 The Expanded Network
Figure 4 shows the phase 2 network. The same inputs plus 26 human inputs brings the total
to 68. The weights obtained in phase 1 initialize this network. The plan and chord weights
Figure 2: At left ?Solid? played by a human; at right the song reproduced by the ANN.
are the same. The weights connecting context units to the hidden layer are halved. The same
weights, halved, connect the 26 human inputs to the hidden layer. Each output unit gets the
100 hidden units? outputs as input. The original 50 weights are halved and used as initial
values of the two sets of 50 hidden unit weights to the output unit.
3.2 SSR and Critic Algorithms
Using actor-critic reinforcement learning ([2, 10, 13]), the actor chooses the next note to
play. The critic receives a ?raw? reinforcement signal from the critique made by the
rules of Section 1.2. For output j, the SSR (actor) computes mean
. A
Gaussian distribution with mean and standard deviation chooses the output . is
generated, the critic modifies and produces . is further modified by a self-scaling
algorithm that tracks, via moving average, the maximum and minimum reinforcement and
uses them to scale the signal to produce !#"$%'&(*)+", -%/.102"$3) .
%2&4 ! 56
%2&4 ! 56
)78%'&49 %'&4 -5:)3;: <
)7A@B%2&4 !56 )6A ",@C)D
%/.10= -56
%/.10= -56
)= %2.>0+9 %/.10= !5:)?;: 3<
)=E@B %2.10= !5F6 )6A ",@C)D
The goal is to make small gains in reinforcement more noticeable and to scale the values
"A !%2.>0H" %2&4*)I
"AJ and if
between -1 and 1. If G%'&4 , then
8%2.10 , then 8 !%2.>02"I%'&4K)+" J7" . If JMLNLHO , the extremes of -1 and
1 are approached. The weight and standard deviation updates use :
P !5F6 )=Q
4 !5:)F6SRT !M"UK )DVKKWXVK
-5F6 )=8%'&49 CY[Z3\;]%/.10+9_^ !5:)6 "I^ )`a !%'&4b"U %2.10F)3;cBY
< < ;cOdLH^IL
If the difference between the max and min reinforcement stays large, over time will increase (to a max of Y[Z3\ ) and allow more exploration. When rmax-rmin is small, over
time e will shrink (to a min of Y
). The actor?s hidden units are updated using back ) as ?error.? See [3, 5, 13] for more details on the
propagation as before, using f`T ! "g
SSR algorithm and its precursors.
The critic inputs are the outputs of the hidden layer of the actor network; it ?piggy-backs? on
the actor and uses its learned features (see Figure 5). This also alleviates the computational
burden so it can run in real time. There are delays in reward, e.g. in that a note played too
many times in a row may result in punishment, and if 2 notes in a row coincide with the
human?s it is rewarded. If !;:5:) is the prediction of future reward [10] for state x at time
t,
Figure 3: Phase1 trading 4 bars: 4 human, 4 machine, 4 human, 4 machine
-56 )= e !5:)F6,^ ` ! -5F6 )3;:5:) "2 - -5:)3;:5:) for O
The critic is a linear function of its inputs: ! -5:)3;]5:)T
updated incrementally using the value of :
LHY ^ L .
! 5:)D
!5:)
The weights are
-5F6 )=Q
-5:)F6 V4 *WVK
is in effect an error signal, a difference between consecutive predicted rewards [10]. The
critic also uses eligibility traces of the inputs, so !5:) is actually !5:)a !5:)76 \ !5=" )
where \ -5:) !5:) . While this is all experimental, initial results show that the system
with both the self-scalar and the critic performs better than with just one or without either
one. A more systematic study is planned.
4 Results and Comments
Recall the rules of Section 1.2. Rules 1-4 are based on discussions with John Payne, a professional jazz musician and instructor of 25 years 1 . The rules by no means encompass all
of jazz theory or practice but are a starting point. The notes in rule 2 were cast as good in
a ?hip? situation. The notion of hip requires human sophistication so for now these notes
are reinforced if played sporadically on the dominant seventh. Rule 5 was added to discourage not playing any notes. Rules 5 and 6 focus on not allowing an output of one note
for too long. Each chord is assigned a scale for rule 1. C is limited, O O
$L egL O ,
providing stability, and deliberate action uncertainty so different notes are played, for the
same network state. Generally the goal of reinforcement learning is to find the best action
for a given state, with uncertainty used for further exploration. Here, reinforcement learning
finds the best set of actions for a given state. In a typical example using the phase 1 network
prior to phase 2 improvement, the average reinforcement value according to the rule set is
-.37 (on a scale from -1 to 1). After Phase 2, the average reinforcement value is .28 after
30-100 off-line presentations of the human solo of 1800 note events.
Figure 6 shows 12 bars of a human solo and 12 bars of a machine solo. The note durations
1
The rules are not meant to represent John Payne
Figure 4: Recurrent reinforcement learning network with human input used in phase 2.
Figure 5: Phase 2 network with critic ?piggy-backing? on hidden layer.
are shortened, reflecting the rules to prevent settling onto one note. The machine plays chord
tones, such as Bb and D in bars 1 and 2. The high G is the 13 of Bb7, a hip note. In bars 3
and 4 it plays C sharp, a hip note (the #9 of Bb7) and high G. These notes are played in bars
9 and 11 on Bb7. In bars 5 and 6 the 9 and 13 (F and C) are played on Eb7. The natural 7
(D) reflects its heavy use in Rollins? melodies. Hip notes show up in bar 9 on Cm7: the 13
(G) and the 9 (D). In bar 11 G is played on G7+9 as is the hip flat 9 (the Ab). In bar 12, the
Eb (flat 7 chord tone) is played on the F7+9. In bars 2, 4, 7, 9, and 10 the machine starts at
the G at the top of the staff and descends through several chord tones, producing a recurring
motif, an artifact of a ?good? jazz solo. The phase 2 network has been used to interact with a
human in real time while still learning. It keeps its recurrence since the human has a separate
set of inputs.
A limitation to be addressed for CHIME is to move beyond one chord at a time. To achieve
this, it must use more context, over more time. There are plenty of improvisation rules for
chord progressions [8]. Because CHIME employs reinforcement learning, it has a stochastic
element that allows it to play ?outside the chord changes.? A research topic is to understand
how to enable it to do this more pointedly.
Figure 6: At left, 12 bars of human solo. At right, 12 bars the machine plays in response.
References
[1] J. Aebersold. You can play Sonny Rollins. A New Approach to Jazz Improvisation Vol 8. Jamey
Aebersold, New Albany, IND., 1976.
[2] A. G. Barto, R. S. Sutton, and C. W. Anderson. Neuronlike adaptive element that can solve difficult learning control problems. IEEE Transactions on Systems, Man, and Cybernetics, SMC13:834?846, 1983.
[3] H. Benbrahim and J. Franklin. Biped walking using reinforcement learning. Robotics and Autonomous Systems, 22:283?302, 1997.
[4] N. Griffith and P. Todd. Musical Networks: Parallel Distributed Perception and Performance.
MIT Press, Cambridge MA, 1999.
[5] V. Gullapalli, J. Franklin, and H. Benbrahim. Acquiring robot skills via reinforcement learning.
IEEE Control Systems Magazine, 1994.
[6] M. Jordan. Attractor dynamics and parallelism in a connectionist sequential machine. In Proceedings of the Eighth Annual Conference of the Cognitive Science Society, 1986.
[7] P. Messick. Maximum MIDI. Manning Publications, Greenwich, CT, 1988.
[8] S. Reeves. Creative Jazz Improvisation. 2nd Ed. Prentice Hall, Upper Saddle River NJ, 1995.
[9] M. A. Sabatella. Whole Approach to Jazz Improvisation. A.D.G. Productions, Lawndale CA,
1996.
[10] R. Sutton. Learning to predict by the methods of temporal differences. Machine Learning, 3:9?
44, 1988.
[11] P. M. Todd. A connectionist approach to algorithmic composition. In P. M. Todd and e. D. Loy,
editors, Music and Connectionism. MIT Press, Cambridge MA, 1991.
[12] P. M. Todd and e. D. Loy. Music and Connectionism. MIT Press, Cambridge, MA, 1991.
[13] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement
learning. Machine Learning, 8:229?256, 1992.
| 2079 |@word nd:1 pg:1 solid:7 initial:2 substitution:2 series:1 score:1 contains:1 accompaniment:1 franklin:3 must:1 john:3 enables:1 update:1 half:6 tone:7 monk:1 beginning:1 smith:3 provides:2 firstly:1 resolve:1 little:3 precursor:1 notation:1 benbrahim:2 rmax:1 contrasting:1 nj:1 temporal:1 exactly:1 control:2 unit:14 producing:1 before:1 todd:11 sutton:2 shortened:1 critique:2 improvisation:8 plus:4 eb:6 limited:1 g7:5 practice:1 instructor:1 griffith:1 get:1 onto:1 prentice:1 context:4 www:1 musician:1 modifies:1 williams:1 starting:2 duration:1 resolution:1 immediately:1 rule:25 stability:1 notion:1 variation:1 autonomous:1 updated:2 target:1 play:10 heavily:2 magazine:1 exact:1 distinguishing:1 us:8 associate:1 element:2 walking:1 cy:1 triad:13 bass:1 trade:1 highest:1 chord:52 decrease:4 mentioned:1 reward:3 dynamic:1 trained:4 creation:1 basis:1 easily:1 represented:2 train:2 separated:1 artificial:1 approached:1 outside:1 solve:1 ability:1 reproduced:1 sequence:1 net:2 interaction:2 turned:1 payne:2 alleviates:1 achieve:1 aebersold:2 produce:3 derive:1 recurrent:9 minor:15 noticeable:1 descends:1 c:2 predicted:1 trading:3 implies:2 stochastic:1 exploration:2 human:23 enable:1 melody:6 numeral:2 require:1 connectionism:2 extension:1 around:1 hall:1 cb:1 algorithmic:1 predict:1 chime:11 major:24 consecutive:2 f7:5 albany:1 jazz:17 harmony:1 grouped:1 repetition:1 reflects:1 brought:1 mit:3 j7:1 gaussian:1 modified:1 avoid:1 chromatic:6 barto:1 publication:1 derived:1 focus:2 vk:1 improvement:1 prevalent:1 indicates:2 motif:1 hidden:10 reproduce:2 backing:1 plan:3 raised:5 initialize:1 once:1 dissonance:1 plenty:1 future:1 connectionist:3 roman:2 employ:1 simultaneously:1 replaced:1 phase:21 attractor:1 maintain:1 ab:10 neuronlike:1 extreme:1 held:2 solo:5 jumping:1 unless:2 iv:2 initialized:1 hip:9 planned:1 deviation:2 subset:1 delay:1 seventh:25 too:2 connect:1 teacher:1 chooses:2 punishment:1 river:1 stay:1 v4:1 off:3 systematic:1 connecting:1 again:1 squared:6 reflect:1 interactively:1 recorded:1 choose:2 cognitive:1 style:1 f6:7 rollins:11 vi:4 root:2 sporadically:1 start:5 decaying:1 parallel:1 il:1 musical:3 reinforced:1 lhy:1 raw:1 produced:2 cybernetics:1 history:1 ed:1 c7:11 associated:2 gain:1 recall:1 subsection:1 knowledge:1 actually:1 back:4 reflecting:1 appears:2 ok:2 supervised:1 follow:1 wherein:1 response:2 box:1 shrink:1 anderson:1 just:1 receives:1 replacing:1 nonlinear:1 propagation:3 incrementally:1 dvk:1 ionian:1 brings:1 aj:1 artifact:1 perhaps:1 mode:3 logistic:1 effect:1 contain:1 assigned:1 mile:1 ind:1 self:2 eligibility:3 recurrence:1 davis:1 noted:1 octave:2 complete:1 performs:1 dedicated:1 interface:1 harmonic:2 charles:1 discussed:1 composition:2 cambridge:3 rd:3 reef:1 biped:1 lowered:1 improvise:4 actor:7 longer:2 moving:1 robot:1 etc:1 add:1 dominant:10 halved:3 forcing:1 rewarded:1 minimum:1 preceding:1 staff:1 recognized:1 signal:3 ii:9 encompass:1 smooth:1 match:2 characterized:1 long:3 post:1 prediction:1 basic:4 represent:2 robotics:1 addition:2 interval:1 addressed:1 bop:1 rest:8 comment:1 db:2 jordan:4 drum:1 gullapalli:1 gb:1 song:18 passing:1 action:3 generally:1 tune:1 band:1 deliberate:1 notice:1 per:1 track:1 blue:8 vol:1 group:1 four:6 threshold:1 prevent:1 year:1 run:1 fourth:4 uncertainty:2 you:1 place:2 improvising:2 scaling:1 layer:4 ct:1 played:15 annual:1 rmin:1 flat:12 software:1 generates:1 min:2 expanded:3 department:1 according:2 creative:1 manning:1 founder:1 turn:2 end:1 available:3 progression:5 tempo:1 professional:1 substitute:1 original:2 top:2 charlie:1 music:4 giving:1 classical:1 society:1 move:1 added:4 abab:1 gradient:1 wrap:1 separate:1 entity:1 seven:1 topic:1 z3:2 providing:1 demonstration:1 loy:2 difficult:1 trace:2 stated:1 allowing:1 upper:2 beat:3 situation:1 extended:1 tonic:7 ssr:3 interacting:1 sharp:2 cast:1 learned:2 impair:1 bar:29 recurring:1 below:4 beyond:1 perception:1 parallelism:1 eighth:1 built:1 including:1 max:2 gillespie:1 power:1 event:4 natural:4 settling:1 improve:1 brief:1 pianist:1 prior:1 piano:1 rationale:1 generation:1 limitation:1 foundation:1 degree:5 article:1 foster:1 editor:1 playing:1 critic:10 heavy:1 production:1 row:4 aaba:1 gl:1 last:2 allow:1 understand:1 fifth:3 distributed:1 computes:1 made:2 reinforcement:19 coincide:1 avoided:1 adaptive:1 novice:2 transaction:1 bb:7 skill:1 midi:2 keep:3 assumed:1 subsequence:1 tenor:3 ca:1 interact:1 meanwhile:1 discourage:1 did:1 whole:4 augmented:3 representative:1 parker:1 judy:1 prominently:1 third:19 learns:1 erroneous:1 bad:9 burden:1 adding:1 sequential:1 madness:3 sophistication:1 saddle:1 sectional:1 scalar:1 acquiring:1 ma:4 piggy:2 goal:2 presentation:3 ann:1 man:1 change:1 diminished:3 abac:1 except:2 typical:1 called:2 total:1 experimental:1 player:2 college:1 meant:1 gm13:1 |
1,184 | 208 | 550
Ackley and Littman
Generalization and scaling in reinforcement
learning
David H. Ackley
Michael L. Littman
Cognitive Science Research Group
Bellcore
Morristown, NJ 07960
ABSTRACT
In associative reinforcement learning, an environment generates input
vectors, a learning system generates possible output vectors, and a reinforcement function computes feedback signals from the input-output
pairs. The task is to discover and remember input-output pairs that
generate rewards. Especially difficult cases occur when rewards are
rare, since the expected time for any algorithm can grow exponentially
with the size of the problem. Nonetheless, if a reinforcement function
possesses regularities, and a learning algorithm exploits them, learning
time can be reduced below that of non-generalizing algorithms. This
paper describes a neural network algorithm called complementary reinforcement back-propagation (CRBP), and reports simulation results
on problems designed to offer differing opportunities for generalization.
1
REINFORCEMENT LEARNING REQUIRES SEARCH
Reinforcement learning (Sutton, 1984; Barto & Anandan, 1985; Ackley, 1988; Allen,
1989) requires more from a learner than does the more familiar supervised learning
paradigm. Supervised learning supplies the correct answers to the learner, whereas
reinforcement learning requires the learner to discover the correct outputs before
they can be stored. The reinforcement paradigm divides neatly into search and
learning aspects: When rewarded the system makes internal adjustments to learn
the discovered input-output pair; when punished the system makes internal adjustments to search elsewhere.
Generalization and Scaling in Reinforcement Learning
1.1
MAKING REINFORCEMENT INTO ERROR
Following work by Anderson (1986) and Williams (1988), we extend the backpropagation algorithm to associative reinforcement learning. Start with a "garden variety" backpropagation network: A vector i of n binary input units propagates
through zero or more layers of hidden units, ultimately reaching a vector 8 of m
sigmoid units, each taking continuous values in the range (0,1). Interpret each 8j
as the probability that an associated random bit OJ takes on value 1. Let us call
the continuous, deterministic vector 8 the search vector to distinguish it from the
stochastic binary output vector o.
Given an input vector, we forward propagate to produce a search vector 8, and
then perform m independent Bernoulli trials to produce an output vector o. The
i - 0 pair is evaluated by the reinforcement function and reward or punishment
ensues. Suppose reward occurs. We therefore want to make 0 more likely given i.
Backpropagation will do just that if we take 0 as the desired target to produce an
error vector (0 - 8) and adjust weights normally.
Now suppose punishment occurs, indicating 0 does not correspond with i. By choice
of error vector, backpropagation allows us to push the search vector in any direction;
which way should we go? In absence of problem-specific information, we cannot pick
an appropriate direction with certainty. Any decision will involve assumptions. A
very minimal "don't be like 0" assumption-employed in Anderson (1986), Williams
(1988), and Ackley (1989)-pushes s directly away from 0 by taking (8 - 0) as the
error vector. A slightly stronger "be like not-o" assumption-employed in Barto &
Anandan (1985) and Ackley (1987)-pushes s directly toward the complement of 0
by taking ((1 - 0) - 8) as the error vector. Although the two approaches always
agree on the signs of the error terms, they differ in magnitudes. In this work,
we explore the second possibility, embodied in an algorithm called complementary
reinforcement back-propagation ( CRBP).
Figure 1 summarizes the CRBP algorithm. The algorithm in the figure reflects three
modifications to the basic approach just sketched. First, in step 2, instead of using
the 8j'S directly as probabilities, we found it advantageous to "stretch" the values
using a parameter v. When v < 1, it is not necessary for the 8i'S to reach zero or
one to produce a deterministic output. Second, in step 6, we found it important
to use a smaller learning rate for punishment compared to reward. Third, consider
step 7: Another forward propagation is performed, another stochastic binary output vector 0* is generated (using the procedure from step 2), and 0* is compared
to o. If they are identical and punishment occurred, or if they are different and
reward occurred, then another error vector is generated and another weight update
is performed. This loop continues until a different output is generated (in the case
of failure) or until the original output is regenerated (in the case of success). This
modification improved performance significantly, and added only a small percentage
to the total number of weight updates performed.
551
552
Ackley and Littman
O. Build a back propagation network with input dimensionality n and output
dimensionality m. Let t = 0 and te = O.
1. Pick random i E 2n and forward propagate to produce a/s.
2. Generate a binary output vector o. Given a uniform random variable ~ E [0,1]
and parameter 0 < v < 1,
OJ
=
{1,
0,
if(sj - !)/v+! ~ ~j
otherwise.
3. Compute reinforcement r = f(i,o). Increment t. If r < 0, let te = t.
4. Generate output errors ej. If r > 0, let tj = OJ, otherwise let tj = 1- OJ. Let
ej = (tj - sj)sj(l- Sj).
5. Backpropagate errors.
6. Update weights. 1:::..Wjk = 1]ekSj, using 1] = 1]+ if r ~ 0, and 1] = 1]- otherwise,
with parameters 1]+,1]- > o.
7. Forward propagate again to produce new Sj's. Generate temporary output
vector 0*. If (r > 0 and 0* #- 0) or (r < 0 and 0* = 0), go to 4.
8. If te ~ t, exit returning te, else go to 1.
Figure 1: Complementary Reinforcement Back Propagation-CRBP
2
ON-LINE GENERALIZATION
When there are many possible outputs and correct pairings are rare, the computational cost associated with the search for the correct answers can be profound.
The search for correct pairings will be accelerated if the search strategy can effectively generalize the reinforcement received on one input to others. The speed of
an algorithm on a given problem relative to non-generalizing algorithms provides a
measure of generalization that we call on-line generalization.
O. Let z be an array of length 2n. Set the z[i] to random numbers from 0 to
2m - 1. Let t = te = O.
1. Pick a random input i E 2n.
2. Compute reinforcement r = f(i, z[i]). Increment t.
3. If r < 0 let z[i] = (z[i] + 1) mod 2m , and let te = t.
4. If te <t:: t exit returning t e, else go to 1.
Figure 2: The Table Lookup Reference Algorithm Tref(f, n, m)
Consider the table-lookup algorithm Tref(f, n, m) summarized in Figure 2. In this
algorithm, a separate storage location is used for each possible input. This prevents
the memorization of one i - 0 pair from interfering with any other. Similarly,
the selection of a candidate output vector depends only on the slot of the table
corresponding to the given input. The learning speed of T ref depends only on the
input and output dimensionalities and the number of correct outputs associated
Generalization and Scaling in Reinforcement Learning
with each input. When a problem possesses n input bits and n output bits, and
there is only one correct output vector for each input vector, Tre{ runs in about 4n
time (counting each input-output judgment as one.) In such cases one expects to
take at least 2n - 1 just to find one correct i - 0 pair, so exponential time cannot be
avoided without a priori information. How does a generalizing algorithm such as
CRBP compare to Trer?
3
SIMULATIONS ON SCALABLE PROBLEMS
We have tested CRBP on several simple problems designed to offer varying degrees
and types of generalization. In all of the simulations in this section, the following
details apply: Input and output bit counts are equal (n). Parameters are dependent
on n but independent of the reinforcement function f. '7+ is hand-picked for each
n,l 11- = 11+/10 and II = 0.5. All data points are medians of five runs. The stopping
criterion te ~ t is interpreted as te +max(2000, 2n+l) < t. The fit lines in the figures
are least squares solutions to a x bn , to two significant digits.
As a notational convenience, let c = ~
3.1
n
E ij
;=1
-
the fraction of ones in the input.
n-MAJORlTY
Consider this "majority rules" problem: [if c > ~ then 0 = In else 0 = on]. The i-o
mapping is many-to-l. This problem provides an opportunity for what Anderson
(1986) called "output generalization": since there are only two correct output states,
every pair of output bits are completely correlated in the cases when reward occurs.
G)
'iii
u
rn
C)
0
::::.
G)
E
;
10 7
10 6
10 5
10 4
x
Table
D
CRBP n-n-n
+ CRBP n-n
10 3
10 2
10 1
10 0
0
1
2
3
456
78
91011121314
n
Figure 3: The n-majority problem
Figure 3 displays the simulation results. Note that although Trer is faster than
CRBP at small values of n, CRBP's slower growth rate (1.6n vs 4.2n ) allows it to
cross over and begin outperforming Trer at about 6 bits. Note also--in violation of
1 For n = 1 to 12. we used '1+
0.219. 0.170. 0.121}.
= {2.000. 1.550. 1.130.0.979.0.783.0.709.0.623.0.525.0.280.
553
554
Ackley and Littman
some conventional wisdom-that although n-majority is a linearly separable problem, the performance of CRBP with hidden units is better than without. Hidden
units can be helpful--even on linearly separable problems-when there are opportunities for output generalization.
3.2
n-COPY AND THE 2k -ATTRACTORS FAMILY
As a second example, consider the n-copy problem: [0 = i]. The i-o mapping is now
1-1, and the values of output bits in rewarding states are completely uncorrelated,
but the value of each output bit is completely correlated with the value of the
corresponding input bit. Figure 4 displays the simulation results. Once again, at
G)
'ii
tA
Q
0
::::.
G)
-
.5
10 7
10 6
10 5
10 4
x
150*2.0I\n
D
10 3
10 2
12*2.2I\n
+
Table
CRBP n-n-n
CRBP n-n
10 1
10 0
0
1
2
3
4
5
6
7
8
9
10 1112
n
Figure 4: The n-copy problem
low values of n, Trer is faster, but CRBP rapidly overtakes Trer as n increases. In
n-copy, unlike n-majority, CRBP performs better without hidden units.
The n-majority and n-copy problems are extreme cases of a spectrum. n-majority
can be viewed as a "2-attractors" problem in that there are only two correct
outputs-all zeros and all ones-and the correct output is the one that i is closer
to in hamming distance. By dividing the input and output bits into two groups
and performing the majority function independently on each group, one generates
a "4-aUractors" problem. In general, by dividing the input and output bits into
1 ~ Ie ~ n groups, one generates a "2i:-attractors" problem. When Ie = 1, nmajority results, and when Ie n, n-copy results.
=
Figure 5 displays simulation results on the n = 8-bit problems generated when Ie is
varied from 1 to n. The advantage of hidden units for low values of Ie is evident,
as is the advantage of "shortcut connections" (direct input-to-output weights) for
larger values of Ie. Note also that combination of both hidden units and shortcut
connections performs better than either alone.
Generalization and Scaling in Reinforcement Learning
105~--------------------------------~
CASP 8-10-8
-+- CASP 8-8
.... CASP 8-10-Sls
-0-
... Table
3
2
1
5
4
7
6
8
k
Figure 5: The 21:- attractors family at n = 8
3.3
n-EXCLUDED MIDDLE
All of the functions considered so far have been linearly separable. Consider this
"folded majority" function: [if
< c < then 0 on else 0 In]. Now, like
n-majority, there are only two rewarding output states, but the determination of
which output state is correct is not linearly separable in the input space. When
n = 2, the n-excluded middle problem yields the EQV (i.e., the complement of
XOR) function, but whereas functions such as n-parity [if nc is even then 0
on
else 0 = In] get more non-linear with increasing n, n-excluded middle does not.
i
i
=
=
=
107~------------------------------~~
-
10 6
10 5
D)
10 4
10 3
I)
'ii
u
f)
.2
I)
E
:::
x
c
17oo*1.6"n
Table
CRSP n-n-n/s
10 2
10 1
10 0
0
1
2
3
4
5
6
7
8
9
10 1112
n
Figure 6: The n-excluded middle problem
Figure 6 displays the simulation results. CRBP is slowed somewhat compared to
the linearly separable problems, yielding a higher "cross over point" of about 8 bits.
555
556
Ackley and Littman
4
STRUCTURING DEGENERATE OUTPUT SPACES
All of the scaling problems in the previous section are designed so that there is
a single correct output for each possible input. This allows for difficult problems
even at small sizes, but it rules out an important aspect of generalizing algorithms
for associative reinforcement learning: If there are multiple satisfactory outputs
for given inputs, a generalizing algorithm may impose structure on the mapping it
produces.
We have two demonstrations of this effect, "Bit Count" and "Inverse Arithmetic."
The Bit Count problem simply states that the number of I-bits in the output should
equal the number of I-bits in the input. When n = 9, Tref rapidly finds solutions
involving hundreds of different output patterns. CRBP is slower--especially with
relatively few hidden units-but it regularly finds solutions involving just 10 output
patterns that form a sequence from 09 to 19 with one bit changing per step.
0+Ox4=0
1+0x4=1
2+0x4=2
3+0x4=3
0+2x4=8
1+2x4=9
2 + 2 x 4 = 10
3+2x4=11
4+0x4=4 4+ 2 x 4 =
5+0x4=5 5 + 2 x 4 =
6+0x4=6 6 + 2 x 4 =
7+0x4=7 7 + 2 x 4 =
12
13
14
15
2+2-4=0 2+2+4=8
3+2-4=1 3+2+4=9
2+2+4=2 2 + 2 x 4 = 10
3+2+4=3 3+2x4=1l
6+2-4=4
7+2-4=5
6+2+4=6
7+2-.;-4=7
6+
7+
6+
7+
2+ 4 =
2+ 4 =
2x4=
2x4=
0+4 x 4 = 16 0+6 x 4 =
1+4x4=17 1 + 6 x 4 =
2 + 4 x 4 = 18 2 + 6 x 4 =
3 +4 x 4 = 19 3 + 6 x 4 =
24
25
26
27
4+4
5+ 4
6+ 4
7+ 4
=
=
=
=
28
29
30
31
24
25
26
27
x
x
x
x
4=
4=
4=
4=
6+ 6 + 4 =
7+6+4=
2+ 4 x 4 =
3+ 4 x 4=
12 4 x 4 +
13 5 + 4 x
14 6 + 4 x
15 7 +4 x
4=
4=
4
4=
=
20 4 + 6 x
21 5 + 6 x
22 6 + 6 x
23 7 + 6 x
4
4
4
4
16
17
18
19
0+6 x
1+ 6 x
2+ 6x
3+ 6x
4=
4=
4=
4=
20
21
22
23
4+
5+
6+
7+
4 = 28
4 = 29
4 30
4 = 31
6
6
6
6
x
x
x
x
=
Figure 7: Sample CRBP solutions to Inverse Arithmetic
The Inverse Arithmetic problem can be summarized as follows: Given i E 25 , find
:1:, y, z E 23 and 0, <> E {+(OO)' -(01)' X (10)' +(11)} such that :I: oy<>z = i. In all there are
13 bits of output, interpreted as three 3-bit binary numbers and two 2-bit operators,
and the task is to pick an output that evaluates to the given 5-bit binary input
under the usual rules: operator precedence, left-right evaluation, integer division,
and division by zero fails.
As shown in Figure 7, CRBP sometimes solves this problem essentially by discovering positional notation, and sometimes produces less-globally structured solutions,
particularly as outputs for lower-valued i's, which have a wider range of solutions.
Generalization and Scaling in Reinforcement Learning
5
CONCLUSIONS
Some basic concepts of supervised learning appear in different guises when the
paradigm of reinforcement learning is applied to large output spaces. Rather than
a "learning phase" followed by a "generalization test," in reinforcement learning
the search problem is a generalization test, performed simultaneously with learning.
Information is put to work as soon as it is acquired.
The problem of of "overfitting" or "learning the noise" seems to be less of an issue,
since learning stops automatically when consistent success is reached. In experiments not reported here we gradually increased the number of hidden units on
the 8-bit copy problem from 8 to 25 without observing the performance decline
associated with "too many free parameters."
The 2 k -attractors (and 2 k -folds-generalizing Excluded Middle) families provide
a starter set of sample problems with easily understood and distinctly different
extreme cases.
In degenerate output spaces, generalization decisions can be seen directly in the
discovered mapping. Network analysis is not required to "see how the net does it."
The possibility of ultimately generating useful new knowledge via reinforcement
learning algorithms cannot be ruled out.
References
Ackley, D.H. (1987) A connectionist machine for genetic hillclimbing. Boston, MA: Kluwer
Academic Press.
Ackley, D.H. (1989) Associative learning via inhibitory search. In D.S. Touretzky (ed.),
Advances in Neural Information Processing Systems 1, 20-28. San Mateo, CA: Morgan
Kaufmann.
Allen, R.B. (1989) Developing agent models with a neural reinforcement technique. IEEE
Systems, Man, and Cybernetics Conference. Cambridge, MA.
Anderson, C.W. (1986) Learning and problem solving with multilayer connectionist systems. University of Mass. Ph.D. dissertation. COINS TR 86-50. Amherst, MA.
Barto, A.G. (1985) Learning by statistical cooperation of self-interested neuron-like computing elements. Human Neurobiology, 4:229-256.
Barto, A.G., & Anandan, P. (1985) Pattern recognizing stochastic learning automata.
IEEE Transactions on Systems, Man, and Cybernetics, 15, 360-374.
Rumelhart, D.E., Hinton, G.E., & Williams, R.J. (1986) Learning representations by backpropagating errors. Nature, 323, 533-536.
Sutton, R.S. (1984) Temporal credit assignment in reinforcement learning. University of
Mass. Ph.D. dissertation. COINS TR 84-2. Amherst, MA.
Williams, R.J. (1988) Toward a theory of reinforcement-learning connectionist systems.
College of Computer Science of Northeastern University Technical Report NU-CCS-88-3.
Boston, MA.
557
| 208 |@word trial:1 middle:5 seems:1 stronger:1 advantageous:1 simulation:7 propagate:3 bn:1 pick:4 tr:2 genetic:1 designed:3 update:3 v:1 alone:1 discovering:1 dissertation:2 provides:2 location:1 casp:3 five:1 direct:1 supply:1 profound:1 pairing:2 acquired:1 expected:1 globally:1 automatically:1 increasing:1 begin:1 discover:2 notation:1 mass:2 what:1 interpreted:2 differing:1 nj:1 certainty:1 remember:1 every:1 temporal:1 morristown:1 growth:1 returning:2 unit:10 normally:1 appear:1 before:1 understood:1 sutton:2 mateo:1 range:2 backpropagation:4 digit:1 procedure:1 significantly:1 get:1 cannot:3 convenience:1 selection:1 operator:2 storage:1 put:1 memorization:1 conventional:1 deterministic:2 williams:4 go:4 independently:1 automaton:1 rule:3 array:1 increment:2 target:1 suppose:2 element:1 rumelhart:1 particularly:1 continues:1 ackley:10 environment:1 reward:7 littman:5 ultimately:2 solving:1 division:2 exit:2 learner:3 completely:3 easily:1 larger:1 valued:1 otherwise:3 associative:4 advantage:2 sequence:1 net:1 loop:1 rapidly:2 degenerate:2 wjk:1 regularity:1 produce:8 generating:1 wider:1 oo:2 ij:1 received:1 solves:1 dividing:2 differ:1 direction:2 correct:13 stochastic:3 human:1 generalization:15 precedence:1 stretch:1 considered:1 credit:1 mapping:4 punished:1 reflects:1 always:1 reaching:1 rather:1 ej:2 varying:1 barto:4 structuring:1 notational:1 bernoulli:1 helpful:1 dependent:1 stopping:1 hidden:8 interested:1 sketched:1 issue:1 bellcore:1 priori:1 equal:2 once:1 identical:1 x4:14 regenerated:1 report:2 others:1 connectionist:3 few:1 simultaneously:1 familiar:1 phase:1 attractor:5 possibility:2 evaluation:1 adjust:1 violation:1 extreme:2 yielding:1 tj:3 closer:1 necessary:1 divide:1 desired:1 ruled:1 minimal:1 increased:1 assignment:1 cost:1 rare:2 expects:1 uniform:1 hundred:1 recognizing:1 too:1 stored:1 ensues:1 reported:1 answer:2 punishment:4 amherst:2 ie:6 rewarding:2 michael:1 again:2 cognitive:1 lookup:2 summarized:2 depends:2 performed:4 picked:1 observing:1 reached:1 start:1 square:1 xor:1 kaufmann:1 correspond:1 judgment:1 wisdom:1 yield:1 generalize:1 cc:1 cybernetics:2 reach:1 touretzky:1 ed:1 failure:1 evaluates:1 nonetheless:1 associated:4 hamming:1 stop:1 knowledge:1 dimensionality:3 back:4 ta:1 higher:1 supervised:3 improved:1 evaluated:1 anderson:4 just:4 until:2 hand:1 tre:1 propagation:5 effect:1 concept:1 excluded:5 satisfactory:1 self:1 backpropagating:1 criterion:1 evident:1 performs:2 allen:2 sigmoid:1 exponentially:1 extend:1 occurred:2 kluwer:1 interpret:1 significant:1 cambridge:1 similarly:1 neatly:1 rewarded:1 binary:6 success:2 outperforming:1 seen:1 morgan:1 anandan:3 somewhat:1 impose:1 employed:2 paradigm:3 signal:1 arithmetic:3 ii:3 multiple:1 technical:1 faster:2 determination:1 academic:1 offer:2 cross:2 involving:2 basic:2 scalable:1 multilayer:1 essentially:1 sometimes:2 whereas:2 want:1 else:5 grow:1 median:1 unlike:1 posse:2 regularly:1 mod:1 call:2 integer:1 counting:1 iii:1 variety:1 fit:1 decline:1 useful:1 involve:1 ph:2 reduced:1 generate:4 sl:1 percentage:1 inhibitory:1 sign:1 per:1 group:4 changing:1 fraction:1 run:2 inverse:3 family:3 decision:2 summarizes:1 scaling:6 bit:23 layer:1 followed:1 distinguish:1 display:4 fold:1 occur:1 generates:4 aspect:2 speed:2 performing:1 separable:5 relatively:1 tref:3 structured:1 developing:1 combination:1 describes:1 slightly:1 smaller:1 making:1 modification:2 slowed:1 gradually:1 agree:1 count:3 apply:1 away:1 appropriate:1 coin:2 slower:2 original:1 opportunity:3 exploit:1 especially:2 build:1 added:1 occurs:3 strategy:1 usual:1 distance:1 separate:1 majority:9 toward:2 length:1 demonstration:1 nc:1 difficult:2 perform:1 neuron:1 neurobiology:1 hinton:1 discovered:2 rn:1 overtakes:1 varied:1 david:1 complement:2 pair:7 required:1 connection:2 temporary:1 nu:1 starter:1 below:1 pattern:3 oj:4 max:1 garden:1 embodied:1 relative:1 oy:1 degree:1 agent:1 consistent:1 propagates:1 uncorrelated:1 interfering:1 elsewhere:1 cooperation:1 parity:1 copy:7 soon:1 free:1 taking:3 distinctly:1 feedback:1 eqv:1 computes:1 forward:4 reinforcement:29 san:1 avoided:1 far:1 transaction:1 sj:5 overfitting:1 don:1 spectrum:1 search:11 continuous:2 table:7 learn:1 nature:1 ca:1 linearly:5 noise:1 complementary:3 ref:1 fails:1 guise:1 exponential:1 candidate:1 third:1 northeastern:1 specific:1 effectively:1 magnitude:1 te:9 push:3 boston:2 backpropagate:1 generalizing:6 simply:1 likely:1 explore:1 positional:1 hillclimbing:1 prevents:1 adjustment:2 ma:5 slot:1 viewed:1 absence:1 shortcut:2 man:2 folded:1 called:3 total:1 indicating:1 college:1 internal:2 accelerated:1 tested:1 correlated:2 |
1,185 | 2,080 | The Emergence of Multiple Movement Units in
the Presence of Noise and Feedback Delay
Michael Kositsky
Andrew G. Barto
Department of Computer Science
University of Massachusetts
Amherst, MA 01003-4610
kositsky,barto @cs.umass.edu
Abstract
Tangential hand velocity profiles of rapid human arm movements often appear as sequences of several bell-shaped acceleration-deceleration
phases called submovements or movement units. This suggests how the
nervous system might efficiently control a motor plant in the presence of
noise and feedback delay. Another critical observation is that stochasticity in a motor control problem makes the optimal control policy essentially different from the optimal control policy for the deterministic case.
We use a simplified dynamic model of an arm and address rapid aimed
arm movements. We use reinforcement learning as a tool to approximate
the optimal policy in the presence of noise and feedback delay. Using
a simplified model we show that multiple submovements emerge as an
optimal policy in the presence of noise and feedback delay. The optimal
policy in this situation is to drive the arm?s end point close to the target by
one fast submovement and then apply a few slow submovements to accurately drive the arm?s end point into the target region. In our simulations,
the controller sometimes generates corrective submovements before the
initial fast submovement is completed, much like the predictive corrections observed in a number of psychophysical experiments.
1 Introduction
It has been consistently observed that rapid human arm movements in both infants and
adults often consist of several submovements, sometimes called ?movement units? [21].
The tangential hand velocity profiles of such movements show sequences of several bellshaped acceleration-deceleration phases, sometimes overlapping in the time domain and
sometimes completely separate. Multiple movement units are observed mostly in infant
reaching [5, 21] and in reaching movements by adult subjects in the face of difficult timeaccuracy requirements [15]. These data provide clues about how the nervous system efficiently produces fast and accurate movements in the presence of noise and significant
feedback delay. Most modeling efforts concerned with movement units have addressed
only the kinematic aspects of movement, e.g., [5, 12].
We show that multiple movement units might emerge as the result of a control policy that is
optimal in the face of uncertainty and feedback delay. We use a simplified dynamic model
of an arm and address rapid aimed arm movements. We use reinforcement learning as a
tool to approximate the optimal policy in the presence of noise and feedback delay.
An important motivation for this research is that stochasticity inherent in the motor control
problem has a significant influence on the optimal control policy [9]. We are following the
preliminary work of Zelevinsky [23] who showed that multiple movement units emerge
from the stochasticity of the environment combined with a feedback delay. Whereas he
restricted attention to a finite-state system to which he applied dynamic programming, our
model has a continuous state space and we use reinforcement learning in a simulated realtime learning framework.
2 The model description
The model we simulated is sketched in Figure 1. Two main parts of this model are the
?RL controller? (Reinforcement Learning controller) and the ?plant.? The controller here
represents some functionality of the central nervous system dealing with the control of
reaching movements. The plant represents a simplified arm together with spinal circuitry.
The controller generates the control signal, , which influences how the state, , of the plant
changes over time. To simulate delayed
feedback the state of the plant is made available
to
the controller after a delay period , so at time the controller can only observe
.
To introduce stochasticity, we disturbed by adding noise to it, to produce a corrupted
control . The controller learns to move the plant state as quickly as possible into a small
region about a target state
. The reward structure block in Figure 1 provides a negative
unit reward when the plant?s state is out of the target area of the state space, and it provides
zero reward when the plant state is within the target area. The reinforcement learning
controller tries to maximize the total cumulative reward for each movement. With the above
mentioned reward structure, the faster the plant is driven into the target region, the less
negative reward is accumulated during the movement. Thus this reward structure specifies
the minimum time-to-goal criterion.
r
reward
RL controller
s
u efferent copy
u
r reward target state s T
structure
state
s
delay
noise
~
u
target
plant
Figure 1: Sketch of the model used in our simulations. ?RL controller? stands for a Reinforcement Learning controller.
2.1 The plant
To model arm dynamics together with the spinal reflex mechanisms we used a fractionalpower damping dynamic model [22]. The simplest model that captures the most critical
dynamical features is a spring-mass system with a nonlinear damping:
"!$#
Here, is the position of the mass attached to the spring, and are respectively the
velocity and the acceleration of the object, is the mass of the object (the mass of the
spring is assumed equal to zero), is the damping coefficient, is the stiffness coefficient,
and is the control signal which determines the resting, or equilibrium, position. Later
in this paper, we call activation, referring to the activation level of a muscle pair. The
Table 1: Parameter values used in the simulations.
description
the basic simulation time step
the feedback delay,
initial position
initial velocity
target position
target velocity
target position radius
value
1 ms
200 ms
0 cm
0 cm/s
5 cm
5 cm
0.5 cm
description
threshold velocity radius
standard deviation of the noise
value function learning rate
preferences learning rate
discount factor,
bootstrapping factor,
value
0.1 cm/s
1 cm
0.5
1
0.9
0.9
values for the mass, the damping coefficient, and the stiffness coefficient were taken from
!
Barto et al. [3]:
kg,
,
. These values provide movement
trajectories qualitatively similar to those observed in human wrist movements [22].
The fractional-power damping in this model is motivated by both biological evidence
[8, 14] and computational considerations. Controlling a system with such a concave damping function is an easier control problem than for a system with apparently simpler linear
damping. Fractional-power damping creates a qualitatively novel dynamical feature called
a stiction region, a region in the position space around the equilibrium position consisting
of pseudo-stable states, where the velocity of the plant remains very close to zero. Such
states are stable states for all practical purposes. For the parameter magnitudes used in our
simulations, the stiction region is a region of radius 2.5 cm about the true equilibrium in
the position space.
Another essential feature of the neural signal transmission can be accounted for by using a
cascade of low-pass temporal filters on the activation level [16]. We used a second-order
low-pass filter with the time constant of 25 ms.
2.2 The reinforcement learning controller
We used the version of the actor-critic algorithm described by Sutton and Barto [20]. A possible model of how an actor-critic architecture might be implemented in the nervous system
was suggested by Barto [2] and Houk et al. [10]. We implemented the actor-critic algorithm
for a continuous state space and a finite set of actions, i.e., activation level magnitudes
evenly spaced every 1 cm between 0 cm and 10 cm. To represent functions defined over
the continuous state space we used a CMAC representation [1] with 10 tilings, each tiling
spans all three dimensions of the state space and has 10 tiles per dimension. The tilings
have random offsets drawn from the uniform distribution. Learning is done in episodes. At
the beginning of each episode the plant is at a fixed initial state, and the episode is complete
when the plant reaches the target region of the state space. Table 1 shows the parameter
values used in the simulations. Refer to ref. [20] for algorithm details.
2.3 Clocking the control signal
For the controller to have sufficient information about the current state of the plant, the
controller internal representation of the state should be augmented by a vector of all the
actions selected during the last delay period. To keep the dimension of the state space at
a feasible level, we restrict the set of available policies and make the controller select a
new activation level, , in a clocked manner at time intervals equal to the delay period.
One step of the reinforcement learning controller is performed once a delay period, which
corresponds to many steps of the underlying plant simulation. To simulate a stochastic
plant we added Gaussian noise to . A new Gaussian disturbance was drawn every time a
new activation level was selected.
Apart from the computational motivation, there is evidence of intermittent motor control
by human subjects [13]. In our simulations we use an oversimplified special kind of intermittent control with a piecewise constant control signal whose magnitude changes at equal
time intervals, but this is done for the sake of acceleration of the simulations and overall
clarity. Intermittent control does not necessarily assume this particular kind of the control
signal; the most important feature is that control segments are selected at particular points
in time, and each control segment determines the control signal for an extended time interval. The time interval until selection of the next control segment can itself be one of the
parameters [11].
3 Results
The model learned to move the mass quickly and accurately to the target in approximately
1,000 episodes. Figure 2 shows the corresponding learning curve. Figure 3 shows a typical
movement accomplished by the controller after learning. The movement shown in Figure 3
has two acceleration-deceleration phases called movement units or submovements.
4000
3500
time per episode, ms
3000
2500
2000
1500
1000
500
0
0
100
200
300
400
500
episode #
600
700
800
900
1000
Figure 2: The learning curve averaged over 100 trials. The performance is measured in
time-per-episode.
Corrective submovements may occur before the plant reaches zero velocity. The controller
generates this corrective submovement ?on the fly,? i.e., before the initial fast submovement is completed. Figure 4 shows a sample movement accomplished by the controller
after learning where such overlapping submovements occur. This can be seen clearly in
panel (b) of Figure 4 where the velocity profile of the movement is shown. Each of the
submovements appears as a bell-shaped unit in the tangential velocity plot.
Sometimes the controller accomplishes a movement with a single smooth submovement.
A sample of such a movement is shown in Figure 5.
4 Discussion
The model learns to produce movements that are fast and accurate in the presence of noise
and delayed sensory feedback. Most of the movements consist of several submovements.
The first submovement is always fast and covers most of the distance from the initial po-
(d)
25
4
20
2
0
0
200
400
600
t, ms
800
1000
1200
(b)
velocity, cm/s
position, cm
(a)
6
15
10
5
velocity, cm/s
30
0
20
10
?5
0
?10
0
200
400
600
t, ms
800
1000
1200
800
1000
1200
0
1
2
3
position, cm
4
5
6
(c)
activation, cm
15
10
5
0
0
200
400
600
t, ms
Figure 3: A sample movement accomplished by the controller after learning. Panels (a) and
(b) show the position and velocity time course respectively. Panel (c) shows the activation
time courses. The thin solid line shows the activation selected by the controller. The
thick solid line shows the disturbed activation which is sent as the control signal to the
plant. The dashed line shows the activation after the temporal filtering is applied. Panel (d)
shows the phase trajectory of the movement. The thick bar at the lower-right corner is the
target region.
sition to the target. All of the subsequent submovements are much slower and cover much
shorter segments in the position space.
This feature stands in good agreement with the dual control model [12, 17], where the initial part of a movement is conducted in a ballistic manner, and the final part is conducted
under closed-loop control. Some evidence for this kind of dual control strategy comes from
experiments in which subjects were given visual feedback only during the initial stage of
movement. Subjects did not show significant improvement under these conditions compared to trials in which they were deprived of visual feedback during the entire movement
[4, 6]. In another set of experiments, proprioceptive feedback was altered by stimulations
of muscle tendons. Movement accuracy decreased only when the stimulation was applied
at the final stages of movement [18]. Note, however, that the dual control strategy though is
not explicitly designed into our model, but naturally emerges from the existing constraints
and conditions.
The reinforcement learning controller is encouraged by the reward structure to accomplish
each movement as quickly as possible. On the other hand, it faces high uncertainty in
the plant behavior. In states with low velocities the information available to the controller
determines the actual state of the plant quite accurately as opposed to states with high
(a)
(d)
20
4
15
2
0
0
200
400
600
800
1000
t, ms
velocity, cm/s
position, cm
6
10
5
(b)
velocity, cm/s
20
0
15
10
?5
5
0
0
200
400
600
800
1000
600
800
1000
0
1
2
3
position, cm
4
5
6
t, ms
(c)
activation, cm
15
10
5
0
0
200
400
t, ms
Figure 4: A sample movement accomplished by the controller after learning with a well
expressed predictive correction.
velocities. If the controller were to adopt a policy in which it attempts to directly hit the
target in one fast submovement, then very often it would miss the target and spend long
additional time to accomplish the task. The optimal policy in this situation is to move the
arm close to the target by one fast submovement and then apply a few slow submovements
to accurately move arm into the target region.
The model learns to produce control sequences consisting of pairs of high activation steps
followed by low activation steps. This feature stands in good agreement with pulse-step
models of motor control [7, 19]. Each of the pulse-step combinations produces a submovement characterized by a bell-shaped unit in the velocity profile.
In biological motor control corrective submovements are observed very consistently, including both the overlapping and separate submovements. In the case of overlapping
submovements, the corrective movement is called a predictive correction. Multiple submovements are observed mostly in infant reaching [5]. Adults perform routine everyday
reaching movements ordinarily with a single smooth submovement, but in case of tight
time constraints or accuracy requirements they revert to multiple submovements [15]. The
suggested model sometimes accomplishes movements with a single smooth submovement
(see Figure 5), but in most cases it produces multiple submovements much like an infant or
an adult subject trying to move quickly and accurately.
The suggested model is also consistent with theories of basal ganglia information processing for motor control [10]. Some of these theories suggest that dopamine neurons in
the basal ganglia carry information similar to the secondary reinforcement (or temporal
difference) in the actor-critic controller, i.e., information about how the expected perfor-
(a)
(d)
20
4
15
2
0
0
200
400
t, ms
600
800
velocity, cm/s
position, cm
6
10
5
(b)
velocity, cm/s
20
0
15
10
?5
5
0
0
200
400
t, ms
600
800
600
800
0
1
2
3
position, cm
4
5
6
(c)
activation, cm
15
10
5
0
0
200
400
t, ms
Figure 5: A sample movement accomplished by the controller after learning with a single
smooth submovement.
mance (time-to-target) changes over time during a movement. A possible use of this kind
of information is for initiating corrective submovements before the current movement is
completed. This kind of behavior is exhibited by our model (Figure 4).
Acknowledgments
This work was supported by NIH Grant MH 48185?09. We thank Andrew H. Fagg and
Michael T. Rosenstein for helpful comments.
References
[1] J. S. Albus. A new approach to manipulator control: the cerebellar model articulation controller
(CMAC). Journal of Dynamics, Systems, Measurement and Control, 97:220?227, 1975.
[2] A. G. Barto. Adaptive critics and the basal ganglia. In J. C. Houk, J. L. Davis, and D. G. Beiser,
editors, Models of Information Processing in the Basal Ganglia, pages 215?232. MIT Press,
Cambridge, MA, 1995.
[3] A. G. Barto, A. H. Fagg, N. Sitkoff, and J. C. Houk. A cerebellar model of timing and prediction
in the control of reaching. Neural Computation, 11:565?594, 1999.
[4] D. Beaubaton and L. Hay. Contribution of visual information to feedforward and feedback
processes in rapid pointing movements. Human Movement Science, 5:19?34, 1986.
[5] N. E. Berthier. Learning to reach: a mathematical model. Developmental Psychology, 32:811?
832, 1996.
[6] L. G. Carlton. Processing of visual feedback information for movement control. Journal of
Experimental Psychology: Human Perception and Performance, 7:1019?1030, 1981.
[7] C. Ghez. Contributions of central programs to rapid limb movement in the cat. In H. Asanuma
and V. J. Wilson, editors, Integration in the Nervous System, pages 305?320. Igaku-Shoin,
Tokyo, 1979.
[8] C. C. A. M. Gielen and J. C. Houk. A model of the motor servo: incorporating nonlinear spindle
receptor and muscle mechanical properties. Biological Cybernetics, 57:217?231, 1987.
[9] C. M. Harris and D. M. Wolpert. Signal-dependent noise determines motor planning. Nature,
394:780?784, 1998.
[10] J. C. Houk, J. L. Adams, and A. G. Barto. A model of how the basal ganglia generates and
uses neural signals that predict reinforcement. In J. C. Houk, J. L. Davis, and D. G. Beiser,
editors, Models of Information Processing in the Basal Ganglia, pages 249?270. MIT Press,
Cambridge, MA, 1995.
[11] M. Kositsky. Motor Learning and Skill Acquisition by Sequences of Elementary Actions. PhD
thesis, The Weizmann Institute of Science, Israel, October 1998.
[12] D. E. Meyer, S. Kornblum, R. A. Abrams, C. E. Wright, and J. E. K. Smith. Optimality in
human motor performance: ideal control of rapid aimed movements. Psychological Review,
95(3):340?370, 1988.
[13] R. C. Miall, D. J. Weir, and J. F. Stein. Intermittency in human manual tracking tasks. Journal
of Motor Behavior, 25:53?63, 1993.
[14] L. E. Miller. Reflex stiffness of the human wrist. Master?s thesis, Department of Physiology,
Northwestern University, Evanston, IL, 1984.
[15] K. E. Novak, L. E. Miller, and J. C. Houk. Kinematic properties of rapid hand movements in a
knob turning task. Experimental Brain Research, 132:419?433, 2000.
[16] L. D. Partridge. Integration in the central nervous system. In J. H. U. Brown and S. S. Gann,
editors, Engineering Principles in physiology, pages 47?98. Academic Press, New York, 1973.
[17] R. Plamondon and A. M. Alimi. Speed/accuracy trade-offs in target-directed movements. Behavioral and Brain Science, 20:279?349, 1997.
[18] C. Redon, L. Hay, and J.-L. Velay. Proprioceptive control of goal directed movements in man
studied by means of vibratory muscle tendon stimulation. Journal of Motor Behavior, 23:101?
108, 1991.
[19] D. A. Robinson. Oculomotor control signals. In G. Lennerstrand and P. B. y Rita, editors, Basic
Mechanisms of Ocular Mobility and Their Clinical Implications, pages 337?374. Pergamon
Press, Oxford, 1975.
[20] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge,
MA, 1998.
[21] C. von Hofsten. Structuring of early reaching movements: A longitudinal study. Journal of
Motor Behavior, 23:280?292, 1991.
[22] C. H. Wu, J. C. Houk, K. Y. Young, and L. E. Miller. Nonlinear damping of limb motion. In
J. M. Winters and S. L.-Y. Woo, editors, Multiple Muscle Systems: Biomechanics and Movement
Organization, pages 214?235. Springer-Verlag, New York, 1990.
[23] L. Zelevinsky. Does time-optimal control of a stochastic system with sensory delay produce
movement units? Master?s thesis, University of Massachusetts, Amherst, 1998.
| 2080 |@word trial:2 version:1 pulse:2 simulation:9 solid:2 carry:1 initial:8 uma:1 longitudinal:1 existing:1 current:2 activation:15 subsequent:1 berthier:1 motor:14 plot:1 designed:1 infant:4 selected:4 nervous:6 beginning:1 smith:1 provides:2 preference:1 simpler:1 mathematical:1 novak:1 asanuma:1 behavioral:1 lennerstrand:1 manner:2 introduce:1 expected:1 rapid:8 behavior:5 planning:1 brain:2 oversimplified:1 initiating:1 actual:1 underlying:1 panel:4 mass:6 israel:1 kg:1 cm:26 kind:5 bootstrapping:1 pseudo:1 temporal:3 every:2 concave:1 hit:1 evanston:1 control:39 unit:12 grant:1 appear:1 before:4 engineering:1 timing:1 sutton:2 receptor:1 oxford:1 approximately:1 might:3 studied:1 suggests:1 averaged:1 weizmann:1 directed:2 practical:1 acknowledgment:1 wrist:2 block:1 cmac:2 area:2 bell:3 cascade:1 physiology:2 suggest:1 close:3 selection:1 influence:2 disturbed:2 deterministic:1 attention:1 deceleration:3 target:21 controlling:1 programming:1 us:1 rita:1 agreement:2 velocity:20 observed:6 fly:1 capture:1 region:10 episode:7 hofsten:1 movement:55 trade:1 servo:1 mentioned:1 environment:1 developmental:1 reward:10 dynamic:6 tight:1 segment:4 predictive:3 creates:1 completely:1 po:1 mh:1 cat:1 corrective:6 revert:1 fast:8 whose:1 quite:1 spend:1 emergence:1 itself:1 final:2 sequence:4 loop:1 albus:1 description:3 everyday:1 bellshaped:1 requirement:2 transmission:1 produce:7 adam:1 object:2 andrew:2 measured:1 implemented:2 c:1 come:1 radius:3 thick:2 tokyo:1 functionality:1 filter:2 stochastic:2 human:9 rosenstein:1 preliminary:1 biological:3 elementary:1 correction:3 around:1 wright:1 houk:8 equilibrium:3 predict:1 circuitry:1 pointing:1 adopt:1 early:1 purpose:1 ballistic:1 tool:2 mit:3 clearly:1 offs:1 gaussian:2 always:1 reaching:7 barto:9 wilson:1 knob:1 structuring:1 improvement:1 consistently:2 helpful:1 dependent:1 accumulated:1 entire:1 sketched:1 overall:1 dual:3 special:1 integration:2 equal:3 once:1 shaped:3 encouraged:1 represents:2 thin:1 piecewise:1 inherent:1 tangential:3 few:2 winter:1 delayed:2 zelevinsky:2 phase:4 consisting:2 attempt:1 organization:1 kinematic:2 clocking:1 implication:1 accurate:2 shorter:1 mobility:1 damping:9 psychological:1 modeling:1 cover:2 deviation:1 uniform:1 delay:15 conducted:2 corrupted:1 accomplish:2 combined:1 referring:1 spindle:1 amherst:2 michael:2 together:2 quickly:4 ghez:1 thesis:3 central:3 von:1 opposed:1 tile:1 corner:1 coefficient:4 explicitly:1 later:1 try:1 performed:1 closed:1 apparently:1 contribution:2 il:1 accuracy:3 who:1 efficiently:2 miller:3 spaced:1 accurately:5 trajectory:2 drive:2 cybernetics:1 reach:3 manual:1 acquisition:1 ocular:1 naturally:1 efferent:1 massachusetts:2 fractional:2 emerges:1 routine:1 appears:1 done:2 though:1 stage:2 until:1 hand:4 sketch:1 nonlinear:3 overlapping:4 manipulator:1 brown:1 true:1 proprioceptive:2 during:5 davis:2 clocked:1 criterion:1 m:13 trying:1 complete:1 motion:1 consideration:1 novel:1 nih:1 stimulation:3 rl:3 spinal:2 attached:1 he:2 resting:1 significant:3 refer:1 measurement:1 cambridge:3 stochasticity:4 stable:2 actor:4 showed:1 driven:1 apart:1 hay:2 verlag:1 carlton:1 accomplished:5 muscle:5 seen:1 minimum:1 additional:1 accomplishes:2 maximize:1 period:4 signal:11 dashed:1 beiser:2 multiple:9 smooth:4 faster:1 characterized:1 academic:1 clinical:1 long:1 biomechanics:1 prediction:1 basic:2 controller:30 essentially:1 sition:1 dopamine:1 represent:1 sometimes:6 cerebellar:2 whereas:1 decreased:1 addressed:1 interval:4 exhibited:1 comment:1 subject:5 sent:1 call:1 presence:7 ideal:1 feedforward:1 concerned:1 psychology:2 architecture:1 restrict:1 motivated:1 effort:1 york:2 action:3 aimed:3 discount:1 stein:1 simplest:1 specifies:1 per:3 basal:6 threshold:1 drawn:2 clarity:1 fagg:2 uncertainty:2 master:2 wu:1 realtime:1 tendon:2 followed:1 occur:2 constraint:2 sake:1 generates:4 aspect:1 simulate:2 speed:1 span:1 spring:3 optimality:1 department:2 combination:1 deprived:1 restricted:1 taken:1 remains:1 mechanism:2 end:2 tiling:3 available:3 mance:1 stiffness:3 apply:2 observe:1 limb:2 slower:1 completed:3 psychophysical:1 move:5 pergamon:1 added:1 strategy:2 abrams:1 distance:1 separate:2 thank:1 simulated:2 evenly:1 difficult:1 mostly:2 october:1 negative:2 ordinarily:1 policy:11 perform:1 observation:1 neuron:1 finite:2 situation:2 extended:1 intermittency:1 intermittent:3 pair:2 mechanical:1 learned:1 robinson:1 address:2 adult:4 suggested:3 bar:1 dynamical:2 perception:1 articulation:1 oculomotor:1 program:1 including:1 power:2 critical:2 disturbance:1 turning:1 arm:12 altered:1 woo:1 review:1 plant:21 northwestern:1 filtering:1 sufficient:1 consistent:1 principle:1 editor:6 critic:5 course:2 accounted:1 supported:1 last:1 copy:1 institute:1 face:3 emerge:3 feedback:16 dimension:3 curve:2 stand:3 cumulative:1 sensory:2 made:1 reinforcement:12 clue:1 simplified:4 qualitatively:2 adaptive:1 miall:1 approximate:2 skill:1 keep:1 dealing:1 assumed:1 continuous:3 table:2 nature:1 necessarily:1 domain:1 did:1 main:1 motivation:2 noise:12 profile:4 ref:1 augmented:1 slow:2 position:16 meyer:1 learns:3 young:1 offset:1 evidence:3 consist:2 essential:1 incorporating:1 adding:1 phd:1 magnitude:3 easier:1 wolpert:1 gielen:1 ganglion:6 visual:4 expressed:1 tracking:1 reflex:2 springer:1 corresponds:1 determines:4 harris:1 ma:4 goal:2 acceleration:5 weir:1 man:1 feasible:1 change:3 plamondon:1 vibratory:1 typical:1 miss:1 called:5 total:1 pas:2 secondary:1 experimental:2 select:1 perfor:1 internal:1 |
1,186 | 2,081 | Fragment completion in humans and machines
David Jacobs
NEC Research Institute
4 Independence Way, Princeton, NJ 08540
dwj@research.nj.nec.com
Archisman Rudra
CS Department at NYU
251 Mercer St., New York, NY 10012
archi@cs.nyu.edu
Bas Rokers
Psychology Department at UCLA
PO Box 951563, Los Angeles, CA 90095
rokers@psych.ucla.edu
Zili Liu
Psychology Department at UCLA
PO Box 951563, Los Angeles CA 90095
zili@psych.ucla.edu
Abstract
Partial information can trigger a complete memory. At the same time,
human memory is not perfect. A cue can contain enough information to
specify an item in memory, but fail to trigger that item. In the context
of word memory, we present experiments that demonstrate some basic
patterns in human memory errors. We use cues that consist of word fragments. We show that short and long cues are completed more accurately
than medium length ones and study some of the factors that lead to this
behavior. We then present a novel computational model that shows some
of the flexibility and patterns of errors that occur in human memory. This
model iterates between bottom-up and top-down computations. These
are tied together using a Markov model of words that allows memory to
be accessed with a simple feature set, and enables a bottom-up process to
compute a probability distribution of possible completions of word fragments, in a manner similar to models of visual perceptual completion.
1 Introduction
This paper addresses the problem of retrieving items in memory from partial information.
Human memory is remarkable for its flexibility in handling a wide range of possible retrieval cues. It is also very accurate, but not perfect; some cues are more easily used than
others. We hypothesize that memory errors occur in part because a trade-off exists between
memory accuracy and the complexity of neural hardware needed to perform complicated
memory tasks. If this is true, we can gain insight into mechanisms of human memory
by studying the patterns of errors humans make, and we can model human memory with
systems that produce similar patterns as a result of constraints on computational resources.
We experiment with word memory questions of the sort that arise in a game called superghost. Subjects are presented with questions of a form: ?*p*l*c*?. They must find a
valid English word that matches this query, by replacing each ?*? with zero or more letters.
So for this example, ?place?, ?application?, and ?palace? would all be valid answers. In ef-
fect, the subject is given a set of letters and must think of a word that contains all of those
letters, in that order, with other letters added as needed.
Most of the psychological literature on word completion involves the effects of priming
certain responses with recent experience (Shacter and Tulving[18]). However, priming is
only able to account for about five percent of the variance in a typical fragment completion
task (Olofsson and Nyberg[13], Hintzman and Hartry[6]). We describe experiments that
show that the difficulty of a query depends on what we call its redundancy. This measures
the extent to which all the letters in the query are needed to find a valid answer. We show
that when we control for the redundancy of queries, we find that the difficulty of answering
questions increases with their length; queries with many letters tend to be easy only because
they tend to be highly redundant. We then describe a model that mimics these and other
properties of human memory.
Our model is based on the idea that a large memory system can gain efficiency by keeping
the comparison between input and items in memory as simple as possible. All comparisons
use a small, fixed set of features. To flexibly handle a range of queries, we add a bottomup process that computes the probability that each feature is present in the answer, given
the input and a generic, Markov model of words. So the complexity of the bottom-up
computation does not grow with the number of items in memory. Finally, the system is
allowed to iterate between this bottom up and a top down process, so that a new generic
model of words is constructed based on a current probability distribution over all words in
memory, and this new model is combined with the input to update the probability that each
feature is present in the answer.
Previous psychological research has compared performance of word-stem and wordfragment completion. In the former a number of letters (i.e. a fragment) is given beginning
with the first letter(s) of the word. In the latter, the string of letters given may begin at any
point in the word, and adjacent letters in the fragment do not need, but may, be adjacent
in the completed word. For example, for stem completion the fragment ?str? may be completed into ?string?, but for fragment completion also into ?satire?. Performance for wordfragment completion is lower than word-stem completion (Olofsson and Nyberg[12]). In
addition words, for which the ending fragment is given, show performance closer to wordstem completion than to word-fragment completion (Olofsson and Nyberg[13]).
Seidenberg[17] proposed a model based on tri-grams. Srinivas et al.[21] indicate that assuming orthographic encoding is in most cases sufficient to describe word completion performance in humans. Orthographic Markov models of words have often been used computationally, as, for example, in Shannon?s[19] famous work. Following this work, our model
is also orthographic. We find that a bigram rather than a trigram representation is sufficient,
and leads to a simpler model.
Contradicting evidence exists for the influence of fragment length on word completion.
Oloffsson and Nyberg [12] failed to find a difference between two and three letter fragments
on words of length of five to eight letters. However this might have been due to the fact
that in their task, each fragment has a unique completion.
Many recurrent neural networks have been proposed as models of associative memory
(Anderson[1] contains a review). Perhaps most relevant to our work are models that use
an input query to activate items from a complete dictionary in memory, and then use these
items to alter the activations of the input. For example, in the Interactive Activation model
of Rumelhart and McClelland[16], the presence of letters activates words, which boost
the activity of the letters they contain. In Adaptive Resonance models (Carpenter and
Grossberg[3]) activated memory items are compared to the input query and de-activated
if they do not match. Also similar in spirit to our approach is the bidirectional model of
Kosko[10] (for more recent work see, eg., Sommer and Palm[20]). Other models iteratively
combine top-down and bottom-up information (eg., Hinton et al.[5], Rao and Ballard[14]),
although these are not used as part of a memory system with complete items stored in
memory.
Our model differs from all of these in using a Markov model as an intermediate layer between the input and the dictionary. This allows the model to answer superghost queries, and
leads to different computational mechanisms that we will detail. We find that superghost
queries seem more natural to people than associative memory word problems (compare the
superghost query ?think of a word with an a? to the associative memory query ?think of a
word whose seventh letter is an a?). However, it is not clear how to extend most models of
associative memory to handle superghost problems.
Our use of features is more related to feedforward neural nets, and especially the ?information bottleneck? approach of Tishby, Pereira and Bialek[22] (see also Baum, et al.[2]). Our
work differs from feedforward methods in that our method is iterative, and uses features
symmetrically to relate the memory to input in both directions.
Our approach is also related to work on visual object recognition that combines perceptual organization and top-down knowledge (see Ullman[23]). Our model is inspired
by Mumford?s[11] and Williams and Jacobs?[24] use of Markov models of contours for
bottom-up perceptual completion.
Especially relevant to our work is that of Grimes and Mozer[4]. Simultaneous with our
work ([8]) they use a bigram model to solve anagram problems, in which letters are unscrambled to match words in a dictionary. They also use a Markov model to find letter
orderings that conform with the statistics of English spelling. Their model is quite different
in how this is done, due to the different nature of the anagram problem. They view anagram
solving as a mix of low-level processing and higher level cognitive processes, while it is
our goal to focus just on lower level memory.
2 Experiments with Human Subjects
In our experiments, fragments and matching words were drawn from a large standard corpus of English text. The frequency of a word is the number of times it appears in this
corpus. The frequency of a fragment is the sum of the frequency of all words that the fragment matches. We used fragments of length two to eight, discarding any fragments with
frequency lower than one thousand.
Fragments selected for an experiment were presented in random order. In our first experiment we systematically varied the length of the fragments, but otherwise selected them
from a uniform, random distribution. Consequently, shorter fragments tended to match
more words, with greater total frequency. In the second experiment, fragments were selected so that a uniform distribution of frequencies was ensured over all fragment lengths.
For example, we used length two fragments that matched unusually few words. As a result
the average frequency in experiment two is also much lower than in experiment one.
A fragment was presented on a computer screen with spaces interspersed, indicating the
possibility of letter insertion. The subject was required to enter a word that would fit the
fragment. A subject was given 10 seconds to produce a completion, with the possibility to
give up. For each session 50 fragments were presented, with a similar number of fragments
of each length.
Reaction times were recorded by measuring the time elapsed between the fragment first
appearing on screen and the subject typing the first character of a matching word. Words
that did not match the fragment or did not exist in the corpus were marked as not completed.
Each experiment was completed by thirty-one subjects. The subjects were undergraduate
students at Rutgers University, participating in the experiment for partial credit. Total time
1
0.8
Fraction Completed
Fraction Completed
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
1
2
3
4
5
6
7
8
0
1
9
Fragment Length
2
3
4
5
6
Fragment Length
7
8
9
1
Fraction Completed
0.8
R1
0.6
R0
R4
0.4
R3
R2
0.2
0
1
2
3
4
5
6
7
8
9
Fragment Length
Figure 1: Fragment completion as a function of fragment length for randomly chosen cues
(top-left) and cues of equal frequency (top-right). On the bottom, the equal frequency cues
are divided into five groups, from least redundancy (R0) to most (R5) .
spent on the task varied from 15 minutes to close to one hour.
Results
For each graph we plot the number of fragments completed divided
by the number
of
, where is the
fragments presented (Figure 1). Error
bars are calculated as
percent correct in the sample, and is the number
of trials. This assumes that all decisions
are independent and correct with probability ; more precise results can be obtained by
accounting for between-subject variance, but roughly the same results hold.
For random, uniformly chosen fragments, there is a U-shaped dependence of performance
on length. Controlling for frequency reduces performance because on average lower frequency fragments are selected. The U-shaped curve is flattened, but persists; hence Ushaped performance is not just due to frequency
Finally, we divide the fragments from the two experiments into five groups, according to
their redundancy. This is a rough measure of how important each letter is in finding a correct answer to the overall question. It is the probability that if we randomly delete a letter
from the fragment and find a matching word, that this word will match the full fragment.
Specifically, let denote the frequency of a query fragment of length (total frequency
of words that match it). Let denote the frequency of the fragment that results
when we
delete the ?th letter from the query (note, ). Then redundancy is: ! "# .
In all cases where there is a significant difference, greater redundancy leads to better performance. In almost all cases, when we control for redundancy performance decreases with
length. We will discuss the implications of these experiments after describing corresponding experiments with our model.
3 Using Markov Models for Word Retrieval
We now describe a model of word memory in which matching between the query and
memory is mediated by a simple set of features. Specifically, we use bigrams (adjacent
pairs of letters) as our feature set. We denote the beginning and end of a word using
the symbols ?0? and ?1?, respectively, so that bigram probabilities also indicate how often
individual letters begin or end a word. Bottom up processing of a cue is done using this as
a Markov model of words. Then bigram probabilities are used to trigger words in memory
that might match the query.
Our algorithm consists of three steps. First, we compute a prior distribution on how likely
each word in memory is to match our query. In our simulations, we just use a uniform
distribution. However, this distribution could reflect the frequency with which each word
occurs in English. It could also be used to capture priming phenomena; for example, if a
word has been recently seen, its prior probability could increase, making it more likely that
the model would retrieve this word. Then, using these we compute a probability that each
bigram will appear if we randomly select a bigram from a word selected according to our
prior distribution.
Second, we use these bigram probabilities as a Markov model, and compute the expected
number of times each bigram will occur in the answer, conditioned on the query. That is,
as a generic model of words we assume that each letter in the word depends on the adjacent
letters, but is conditionally independent of all others. This conditional independence allows
us to decompose our problem into a set of small, independent problems. For example,
consider the query ?*p*l*c*?. Implicitly, each query begins with ?0? and ends with ?1?, so
the expected number of times any bigram will appear in the completed word is the sum of
the number of times it appears in the completions of the fragments: ?0*p?, ?p*l?, ?l*c?, and
?c*1?.
To compute this, we assume a prior distribution on the number of letters that will replace
a
?*? in the completed word. We use an
exponential
model,
setting
the
probability
of
letters
to be
(in practice we truncate at 5 and normalize the probabilities). A similar model
is used in the perceptual completion of contours ([11, 24]). Using these priors, it becomes
straightforward to compute
a probability distribution on the bigrams that will
appear
in the
bigrams,
completed cue. For a fixed , we structure this problem as a belief net with
and each bigram depending on only its neighbors. The conditional probability of each
bigram given its neighbor comes from the Markov model, and we can solve the problem
with belief propagation.
Beginning the third step of the algorithm, we know the expected number of times that each
bigram appears in the completed cue. Each bigram then votes for all words containing that
bigram. The weight of this vote is the expected number of times each bigram appears in
the completed cue, divided by the prior probability of each bigram, computed in step 1. We
combine these votes multiplicatively. We update the prior for each word as the product of
these votes with the previous probability. We can view this an approximate computation of
the probability of each word being the correct answer, based on the likelihood that a bigram
appears in the completed cue, and our prior on each word being correct.
After the third step, we once again have a probability that each word is correct, and can
iterate, using this probability to initialize step one. After a small number of iterations, we
terminate the algorithm and select the most probable word as our answer. Empirically, we
find that the answer the algorithm produces often changes in the first one or two iterations,
and then generally remains the same. The answer may or may not actually match the input
cue, and by this we judge whether it is correct or incorrect.
We can view this algorithm as an approximate computation of the probability that each
1
0.9
0.9
Fraction Completed
Fraction Completed
1
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
1
2
3
4
5
6
Fragment Length
7
8
0.4
1
9
2
3
4
5
6
7
8
9
Fragment Length
1
R4
R2
0.8
R1
Fraction Completed
0.9
R3
0.7
0.6
R0
0.5
0.4
1
2
3
4
5
6
7
8
9
Fragment Length
Figure 2: Performance as a function of cue length, for cues of frequency between 4 and 22
(top-left) and between 1 and 3 (top-right). On the bottom, we divide the first set of cues
into five groups ranging from the least redundant (R0) to the most (R4).
word matches the cue, where the main approximation comes from using a small set of features to bring the cue into contact with items in memory. Denote the number of features
), the number of features in each word by
by (with a bigram representation,
(ie., the word length plus
one), the number of words by , and the maximum number of
blanks replacing a ?*? by . Then steps one and three require O(mw) computation, and step
two requires O(Fn) computation. In a neural network, the primary requirement would be
bidirectional connections between each feature (bigram) and each item in memory. Therefore, computational simplicity is gained by using a small feature set, at the cost of some
approximation in the computation.
Experiments
We have run experiments to compare the performance of this model to that of human subjects. For simplicity, we used a memory of 6,040 words, each with eight characters. First,
we simulated the conditions described in Olofsson and Nyberg[12] comparing word stem
and word fragment completion. To match their experiments, we used a modified algorithm
that handled cues in which the number of missing letters can be specified. We used cues
that specified the first three letters of a word, the last three letters, or three letters scattered
throughout the word. The algorithm achieved accuracy of 95% in the first case, 87% in the
second, and 80% in the third. This qualitatively matches the results for human subjects.
Note that our algorithm treats the beginning and end of words symmetrically. Therefore,
the fact that it performs better when the first letters of the word are given than when the last
are given is due to regularities in English spelling, and is not built into the algorithm.
Next we simulated conditions comparable to our own experiments on human subjects, using superghost cues. First we selected cues of varying length that match between four
and twenty-two words in the dictionary. Figure 2-top-left shows the percentage of queries
the algorithm correctly answered, for cues of lengths two to seven. This figure shows a
U-shaped performance curve qualitatively similar to that displayed by human subjects.
We also ran these experiments using cues that matched one to three words (Figure 2-topright). These very low frequency cues did not display this U-shaped behavior. The algorithm performs differently on fragments with very low frequency because in our corpus
the shorter of these cues had especially low redundancy and the longer fragments had especially high redundancy, in comparison to fragments with frequencies between 4 and 22.
Next (Figure 2-bottom) we divided the cues into five groups of equal size, according to
their redundancy. We can see that performance increases with redundancy and decreases
with cue length.
Discussion
Our experiments indicate two main effects in human word memory that our model also
shares. First, performance improves with the redundancy of cues. Second, when we control
for this, performance drops with cue length. Since redundancy tends to increase with cue
length, this creates two conflicting tendencies that result in a U-shaped memory curve. We
conjecture that these factors may be present in many memory tasks, leading to U-shaped
memory curves in a number of domains.
In our model, the fact that performance drops with cue length is a result of our use of
a simple feature set to mediate matching the cue to words in memory. This means that
not all the information present in the cue is conveyed to items in memory. When the
length of a cue increases, but its redundancy remains low, all the information in the cue
remains important in getting a correct answer, but the amount of information in the cue
increases, making it harder to capture it all with a limited feature set. This can account for
the performance of our model; similar mechanisms may account for human performance
as well. On the other hand, the extent to which redundancy grows with cue length is really
a product of the specific words in memory and the cues chosen. Therefore, the exact shape
of the performance curve will also depend on these factors. This may partly explain some
of the quantitative differences between our model and human performance.
Finally, we also point out that our measure of redundancy is rather crude. In particular, it
tends to saturate at very high or very low levels. So, for example, if we add a letter to a cue
that is already highly redundant, the new letter may not be needed to find a correct answer,
but that is not reflected by much of an increase in the cue?s redundancy.
4 Conclusions
We have proposed superghost queries as a domain for experimenting with word memory,
because it seems a natural task to people, and requires models that can flexibly handle
somewhat complicated questions. We have shown that in human subjects, performance on
superghost improves with the redundancy of a query, and otherwise tends to decrease with
word length. Together, these effects results in a U-shaped performance curve.
We have proposed a computational model that uses a simple, generic model of words to
map a superghost query onto a simple feature set of bigrams. This means that somewhat
complicated questions can be answered while keeping comparisons between the fragments
and words in memory very simple. Our model displays the two main trends we have found
in human memory. It also does better at word stem completion than word fragment completion, which agrees with previous work on human memory. Future work will investigate
the modification of our model to account for priming effects in memory.
References
[1] J. Anderson. An Introduction to Neural Networks, MIT Press, Cambridge MA. 1995.
[2] E. Baum, J. Moody and F. Wilczek. ?Internal Representations for Associative Memory,? Biological Cybernetics, 59:217-228, 1988.
[3] G. Carpenter, and S. Grossberg. ?ART 2: Self-Organization of Stable Category Recognition
Codes for Analog Input Patterns,? Applied Optics, 26:4919-4930, 1987.
[4] D. Grimes and M. Mozer. ?The interplay of symbolic and subsymbolic processes in anagram
problem solving,? NIPS, 2001.
[5] G. Hinton, P. Dayan, B. Frey, and R. Neal. ?The ?Wake-Sleep? Algorithm for Unsupervised
Neural Networks,? Science, 268:1158-1161, 1995.
[6] D.L. Hintzman and A.L. Hartry. Item effects in recognition and fragment completion: Contingency relations vary for different sets of words. JEP: Learning, Memory and Cognition, 17:
341-345, 1990.
[7] J. Hopfield. ?Neural networks and Physical Systems with Emergent Collective Computational
Abilities.? Proc. of the Nat. Acad. of Science, 79:2554-2558, 1982.
[8] D. Jacobs and A. Rudra. ?An Iterative Projection Model of Memory,? NEC Research Institute
Technical Report, 2000.
[9] G.V. Jones. Fragment and schema models for recall. Memory and Cognition, 12(3):250-63,
1984.
[10] B. Kosko. ?Adaptive Bidirectional Associative Memory?, Applied Optics, 26(23):4947-60,
1987.
[11] D. Mumford. ?Elastica and Computer Vision.? C. Bajaj (Ed), Algebraic Geometry and its Applications New York: Springer-Verlag. 1994.
[12] U. Olofsson and L. Nyberg. Swedish norms for completion of word stems and unique word
fragments. Scandinavian Journal of Psychology, 33(2):108-16, 1992.
[13] U. Olofsson and L. Nyberg. Determinants of word fragment completion. Scandinavian Journal
of Psychology, 36(1):59-64, 1995.
[14] R. Rao and D. Ballard. ?Dynamic Model of Visual Recognition Predicts Neural Response Properties in the Visual Cortex,? Neural Computation, 9(4):721-763, 1997.
[15] R.H. Ross and G.H. Bower. Comparisons of models of associative recall. Memory and Cognition, 9(1):1-16, 1981.
[16] D. Rumelhart and J. McClelland. ?An interactive activation model of context effects in letter
perception: part 2. The contextual enhancement effect and some tests and extensions of the
model?, Psychological Review, 89:60-94, 1982.
[17] M.S. Seidenberg. Sublexical structures in visual word recognition: Access units or orthographic
redundancy? In M. Coltheart (Ed.), Attention and performance XII, 245-263. Hillsdale, NJ:
Erlbaum. 1987.
[18] D.L. Shacter and E. Tulving. Memory systems. Cambridge, MA: MIT Press. 1994.
[19] C. Shannon. ?Prediction and Entropy of Printed English,? Bell Systems Technical Journal,
30:50-64, 1951.
[20] Sommer, F., and Palm, G., 1997, NIPS:676-681.
[21] K. Srinivas, H.L. Roediger 3d and S. Rajaram. The role of syllabic and orthographic properties
of letter cues in solving word fragments. Memory and Cognition, 20(3):219-30, 1992.
[22] N. Tishby, F. Pereira and W. Bialek. ?The Information Bottleneck Method,? 37th Allerton Conference on Communication, Control, and Computing. 1999.
[23] S. Ullman. High-level Vision, MIT Press, Cambridge, MA. 1996.
[24] L. Williams & D. Jacobs. ?Stochastic Completion Fields: A Neural Model of Illusory Contour
Shape and Salience?. Neural Computation, 9:837?858, 1997.
Acknowledgements
The authors would like to thank Nancy Johal for her assistance in conducting the psychological
experiments presented in this paper.
| 2081 |@word trial:1 determinant:1 bigram:23 seems:1 norm:1 simulation:1 jacob:4 accounting:1 harder:1 liu:1 contains:2 fragment:59 reaction:1 current:1 com:1 blank:1 comparing:1 contextual:1 activation:3 must:2 fn:1 shape:2 enables:1 hypothesize:1 plot:1 drop:2 update:2 cue:43 selected:6 item:13 beginning:4 short:1 iterates:1 allerton:1 simpler:1 accessed:1 five:6 constructed:1 retrieving:1 incorrect:1 consists:1 shacter:2 combine:3 manner:1 anagram:4 expected:4 roughly:1 behavior:2 inspired:1 str:1 becomes:1 begin:3 matched:2 medium:1 what:1 psych:2 string:2 finding:1 nj:3 quantitative:1 unusually:1 interactive:2 coltheart:1 ensured:1 control:4 unit:1 appear:3 persists:1 frey:1 treat:1 tends:3 acad:1 encoding:1 might:2 plus:1 r4:3 limited:1 range:2 grossberg:2 unique:2 thirty:1 practice:1 orthographic:5 differs:2 jep:1 bell:1 printed:1 matching:5 projection:1 word:87 symbolic:1 onto:1 close:1 context:2 influence:1 map:1 missing:1 baum:2 williams:2 straightforward:1 flexibly:2 attention:1 simplicity:2 insight:1 retrieve:1 handle:3 controlling:1 trigger:3 exact:1 us:2 trend:1 rumelhart:2 recognition:5 predicts:1 bottom:10 role:1 capture:2 thousand:1 ordering:1 trade:1 decrease:3 ran:1 mozer:2 complexity:2 insertion:1 dynamic:1 depend:1 solving:3 creates:1 efficiency:1 po:2 easily:1 hopfield:1 differently:1 emergent:1 describe:4 activate:1 query:24 olofsson:6 whose:1 quite:1 solve:2 nyberg:7 otherwise:2 ability:1 statistic:1 think:3 associative:7 interplay:1 net:2 product:2 elastica:1 relevant:2 flexibility:2 participating:1 normalize:1 getting:1 los:2 regularity:1 requirement:1 r1:2 enhancement:1 produce:3 perfect:2 object:1 spent:1 depending:1 recurrent:1 completion:27 c:2 involves:1 indicate:3 come:2 judge:1 direction:1 correct:9 stochastic:1 human:21 hillsdale:1 require:1 really:1 decompose:1 probable:1 biological:1 extension:1 hold:1 credit:1 cognition:4 trigram:1 dictionary:4 vary:1 proc:1 ross:1 agrees:1 rough:1 mit:3 activates:1 modified:1 rather:2 varying:1 focus:1 likelihood:1 experimenting:1 dayan:1 her:1 relation:1 overall:1 resonance:1 art:1 initialize:1 equal:3 once:1 field:1 shaped:7 r5:1 jones:1 unsupervised:1 alter:1 mimic:1 future:1 others:2 rokers:2 report:1 few:1 palace:1 randomly:3 individual:1 geometry:1 organization:2 highly:2 possibility:2 investigate:1 grime:2 activated:2 implication:1 accurate:1 rudra:2 closer:1 partial:3 experience:1 shorter:2 divide:2 delete:2 psychological:4 rao:2 measuring:1 cost:1 uniform:3 seventh:1 erlbaum:1 tishby:2 stored:1 answer:13 combined:1 st:1 ie:1 off:1 together:2 moody:1 fect:1 reflect:1 recorded:1 again:1 containing:1 cognitive:1 leading:1 ullman:2 account:4 de:1 student:1 depends:2 view:3 schema:1 sort:1 complicated:3 accuracy:2 variance:2 conducting:1 rajaram:1 famous:1 accurately:1 cybernetics:1 simultaneous:1 explain:1 tended:1 ed:2 frequency:20 gain:2 illusory:1 recall:2 knowledge:1 nancy:1 improves:2 actually:1 appears:5 bidirectional:3 higher:1 reflected:1 specify:1 response:2 swedish:1 done:2 box:2 anderson:2 just:3 hand:1 replacing:2 wilczek:1 propagation:1 perhaps:1 grows:1 effect:7 contain:2 true:1 former:1 hence:1 iteratively:1 neal:1 eg:2 conditionally:1 adjacent:4 assistance:1 game:1 self:1 complete:3 demonstrate:1 performs:2 bring:1 percent:2 ranging:1 novel:1 ef:1 recently:1 physical:1 empirically:1 interspersed:1 extend:1 analog:1 significant:1 cambridge:3 enter:1 session:1 had:2 stable:1 scandinavian:2 longer:1 cortex:1 access:1 add:2 own:1 recent:2 certain:1 verlag:1 seen:1 greater:2 somewhat:2 r0:4 redundant:3 full:1 mix:1 reduces:1 stem:6 technical:2 match:15 long:1 retrieval:2 divided:4 prediction:1 basic:1 vision:2 rutgers:1 iteration:2 achieved:1 addition:1 wake:1 grow:1 tri:1 subject:14 tend:2 spirit:1 seem:1 call:1 mw:1 presence:1 symmetrically:2 intermediate:1 feedforward:2 enough:1 easy:1 iterate:2 independence:2 fit:1 psychology:4 zili:2 idea:1 angeles:2 bottleneck:2 whether:1 handled:1 algebraic:1 york:2 generally:1 clear:1 amount:1 hardware:1 mcclelland:2 category:1 exist:1 percentage:1 correctly:1 conform:1 xii:1 group:4 redundancy:19 four:1 drawn:1 graph:1 fraction:6 sum:2 run:1 letter:36 place:1 almost:1 throughout:1 decision:1 comparable:1 layer:1 display:2 sleep:1 activity:1 occur:3 optic:2 constraint:1 archi:1 ucla:4 answered:2 conjecture:1 department:3 palm:2 according:3 truncate:1 character:2 making:2 modification:1 satire:1 computationally:1 resource:1 remains:3 discus:1 r3:2 fail:1 mechanism:3 needed:4 describing:1 tulving:2 know:1 end:4 studying:1 eight:3 generic:4 appearing:1 top:9 assumes:1 sommer:2 completed:18 especially:4 contact:1 question:6 added:1 occurs:1 mumford:2 already:1 primary:1 dependence:1 spelling:2 bialek:2 thank:1 simulated:2 seven:1 dwj:1 extent:2 assuming:1 length:30 code:1 multiplicatively:1 relate:1 ba:1 collective:1 twenty:1 perform:1 kosko:2 markov:10 displayed:1 hinton:2 communication:1 precise:1 varied:2 david:1 pair:1 required:1 specified:2 connection:1 elapsed:1 conflicting:1 boost:1 hour:1 nip:2 address:1 able:1 bar:1 pattern:5 perception:1 built:1 memory:57 belief:2 difficulty:2 natural:2 typing:1 mediated:1 text:1 review:2 literature:1 prior:8 acknowledgement:1 remarkable:1 syllabic:1 contingency:1 conveyed:1 sufficient:2 mercer:1 systematically:1 share:1 last:2 keeping:2 english:6 salience:1 srinivas:2 institute:2 wide:1 neighbor:2 curve:6 calculated:1 valid:3 ending:1 gram:1 computes:1 contour:3 author:1 qualitatively:2 adaptive:2 approximate:2 implicitly:1 corpus:4 bottomup:1 iterative:2 seidenberg:2 ballard:2 nature:1 terminate:1 ca:2 roediger:1 priming:4 domain:2 did:3 main:3 arise:1 mediate:1 bajaj:1 contradicting:1 allowed:1 carpenter:2 screen:2 scattered:1 ny:1 pereira:2 exponential:1 crude:1 answering:1 tied:1 perceptual:4 third:3 bower:1 subsymbolic:1 down:4 minute:1 saturate:1 discarding:1 specific:1 symbol:1 nyu:2 r2:2 evidence:1 consist:1 exists:2 undergraduate:1 flattened:1 gained:1 nec:3 nat:1 conditioned:1 entropy:1 likely:2 visual:5 failed:1 springer:1 ma:3 hintzman:2 conditional:2 goal:1 marked:1 consequently:1 replace:1 change:1 typical:1 specifically:2 uniformly:1 called:1 total:3 partly:1 tendency:1 shannon:2 vote:4 indicating:1 select:2 internal:1 people:2 latter:1 princeton:1 phenomenon:1 handling:1 |
1,187 | 2,082 | Global Coordination of Local Linear Models
Sam Roweis , Lawrence K. Saul , and Geoffrey E. Hinton
Department of Computer Science, University of Toronto
Department of Computer and Information Science, University of Pennsylvania
Abstract
High dimensional data that lies on or near a low dimensional manifold can be described by a collection of local linear models. Such a description, however, does
not provide a global parameterization of the manifold?arguably an important
goal of unsupervised learning. In this paper, we show how to learn a collection
of local linear models that solves this more difficult problem. Our local linear
models are represented by a mixture of factor analyzers, and the ?global coordination? of these models is achieved by adding a regularizing term to the standard
maximum likelihood objective function. The regularizer breaks a degeneracy
in the mixture model?s parameter space, favoring models whose internal coordinate systems are aligned in a consistent way. As a result, the internal coordinates change smoothly and continuously as one traverses a connected path on
the manifold?even when the path crosses the domains of many different local
models. The regularizer takes the form of a Kullback-Leibler divergence and
illustrates an unexpected application of variational methods: not to perform approximate inference in intractable probabilistic models, but to learn more useful
internal representations in tractable ones.
1 Manifold Learning
Consider an ensemble of images, each of which contains a face against a neutral background. Each image can be represented by a point in the high dimensional vector space
of pixel intensities. This representation, however, does not exploit the strong correlations
between pixels of the same image, nor does it support many useful operations for reasoning
about faces. If, for example, we select two images with faces in widely different locations
and then average their pixel intensities, we do not obtain an image of a face at their average
location. Images of faces lie on or near a low-dimensional, curved manifold, and we can
represent them more usefully by the coordinates on this manifold than by pixel intensities. Using these ?intrinsic coordinates?, the average of two faces is another face with the
average of their locations, poses and expressions.
To analyze and manipulate faces, it is helpful to imagine a ?magic black box? with levers
or dials corresponding to the intrinsic coordinates on this manifold. Given a setting of the
levers and dials, the box generates an image of a face. Given an image of a face, the box
deduces the appropriate setting of the levers and dials. In this paper, we describe a fairly
general way to construct such a box automatically from an ensemble of high-dimensional
vectors. We assume only that there exists an underlying manifold of low dimensionality
and that the relationship between the raw data and the manifold coordinates is locally linear
and smoothly varying. Thus our method applies not only to images of faces, but also to
many other forms of highly distributed perceptual and scientific data (e.g., spectrograms of
speech, robotic sensors, gene expression arrays, document collections).
2 Local Linear Models
The global structure of perceptual manifolds (such as images of faces) tends to be highly
nonlinear. Fortunately, despite their complicated global structure, we can usually characterize these manifolds as locally linear. Thus, to a good approximation, they can be
represented by collections of simpler models, each of which describes a locally linear
neighborhood[3, 6, 8]. For unsupervised learning tasks, a probabilistic model that nicely
captures this intuition is a mixture of factor analyzers (MFA)[5]. The model is used to
describe high dimensional data that lies on or near a lower dimensional manifold. MFAs
parameterize a joint distribution over observed and hidden variables:
(1)
where the observed variable,
, represents the high dimensional data; the discrete
hidden variables,
, indexes different neighborhoods on the manifold; and
the continuous hidden variables,
, represent low dimensional local coordinates.
The model assumes that data is sampled from different neighborhoods on the manifold
with prior probabilities
, and that within each neighborhood, the data?s local
coordinates are normally distributed1 as:
!
" $# % &')()(*(+-,
. / 0 !1
243
256&87.:9 1;<
=?>A@ BDC &%
*E
F (
(2)
G
H
I
J &7 I 8 9K ;< =?>A@LBDC & % M C G C H :
N E I 9
K M C G C H :
N F ( (3)
O , is obtained by summing/integrating out the model?s
The marginal data distribution,
Finally, the model assumes that the data?s high and low dimensional coordinates are related
by linear processes parameterized by centers , loading matrices
and noise levels :
discrete and continuous latent variables. The result is a mixture of Gaussian distributions
with parameterized covariance matrices of the form:
2QP R3 - &872 H H "E S I ) 9
K ;< =)>T@ B C L&% M C G N E H H E S I 9
K M C G N:FU(
(4)
, transformations H , and
The learning problem
for
MFAs
is
to
estimate
the
centers
G
3 of sampling
noise levels I of these linear processes, as well as the prior probabilities
data from different parts of the manifold. Parameter estimation in MFAs can be handled
by an Expectation-Maximization (EM) algorithm[5] that attempts to maximize the logprobability,
, averaged over training examples.
\
VXWY Z
H -[ H :\/
\ \ E ba
H [ H \
,
Note that the parameter space of this model exhibits an invariance: taking
where
are
orthogonal matrices (
), does not change the marginal dis. The transformations
correspond to arbitrary rotations and
tribution,
reflections of the local coordinates in each linear model. The objective function for the EM
algorithm is unchanged by these transformations. Thus, maximum likelihood estimation
in MFAs does not favor any particular alignment; instead, it produces models whose internal representations change unpredictably as one traverses connected paths on the manifold.
Can we encourage models whose local coordinate systems are aligned in a consistent way?
2]_ ^`]
3 Global Coordination
Suppose the data lie near a smooth manifold with a locally flat (developable) structure.
Then there exist a single set of ?global coordinates? which parametrize the manifold
c
1
Although in principle each neighborhood could have a different prior on its local coordinates,
without loss of generality we have made the standard assumption that
is the same for all
settings of and absorbed the shape of each local Gaussian model into the matrices .
k
d/egfih*j k?l
m2h
hidden
variables s,z
Figure 1: Graphical model for globally coordinated MFAs. Al-
g
though global coordinates are unobserved, they affect the learning through a regularization term. After learning, inferences about
the global variables are made by computing posterior distributions,
. Likewise, data can easily be generated by sampling from
. All these operations are particthe conditional distribution,
ularly tractable due to the conditional independencies of the model.
d/e j .l
x
global
coordinates
data
d/e Zj l
everywhere. Furthermore, to a good approximation, these global coordinates can be related
to the local coordinates of different neighborhoods (in their region of validity) by linear 2
transformations:
(5)
What does it mean to say that the coordinates
provide a global parameterization
of the manifold? Intuitively, if a data point belongs to overlapping neighborhoods, then the
global coordinates computed from their local coordinate systems, given by eq. (5), should
agree. We can formalize this ?global coordination? of different local models by treating the
coordinates as unobserved variables and incorporating them into the probabilistic model:
c 6
c
S (
c
?
c
$ A c C
C )
(6)
(Here we posit a deterministic relationship between local and global coordinates, although
it is possible to add noise to this mapping as well.) The globally coordinated MFA is represented by the graphical model in Fig. 1. We can appeal to its conditional independencies
to make other useful inferences. In particular:
c R:
c '
(7)
]
R R c
'
P 6 ' c R: ?(
(8)
with
Now, if two or more mixture components?say, K and < ?explain a data point
non-negligible probability, then the posterior distributions for the global
coordinates
of
this
induced by eq. (8), should be nearly identical: that is,
c R: K
c data
point,
< . Toasenforce
we need to penalize models whose
this
'criterion
given byofeq.agreement,
posterior distributions c
(8) are multimodal, since multiple modes only
arise when different mixture components
to inconsistent global coordinates. While
give
' rise
directly penalizing multimodality of c
is difficult, a penalty which encourages consistency can be easily incorporated into the learning algorithm. We introduce
c a family
T ' , ofto
unimodal distributions over both
and
,
and
encourage
the
true
posteriors,
c
: ' , of this family.
be close to some member, c
Developing this idea further, we introduce a new objective function for unsupervised learning in MFAs. The new objective function incorporates a regularizer to encourage the global
consistency of local models:
P VXWY ' C
P
c
c T i XV WY c : T i
c
(9)
The first term in this objective function computes the log-probability of the data. The
second term computes a sum of Kullback-Leibler (KL) divergences; these are designed to
h
2
Without loss of generality, the matrices can be taken to be symmetric and positive-definite,
by exploiting the polar factorization and absorbing reflection and rotation into the local coordinate
systems. (In practice, though, it may be easier to optimize the objective function without constraining
the matrices to be of this form.) In the experiments reported below, we have further restricted them to
be diagonal. Together, then, the coordination matrices and vectors
account for an axis-aligned
scaling and uniform translation between the global and local coordinate systems.
h
h
penalize MFAs whose posterior distributions over global coordinates are not unimodal. The
twin goals of density estimation and manifold learning in MFAs
are pursued by attempting
to balance these terms in the objective function.
The factor controls the tradeoff between
density modeling and global coordination: as
only strict invariances (which do not
affect likelihood)
are
exploited
in
order
to
achieve
submodel
agreement. In what follows
arbitrarily; further optimization is possible.
we have set
[
%
The most convenient way to parameterize the family of unimodal distributions is a factorized form involving a Gaussian density and a multinomial:
c ' T ' c ' c '
6T '2 (10)
T ' in eq. (10) factorizes over and c , implying that?
Note that the distribution c
according to this family of models?the
coordinate
is independent of the mixture
. global
c isc Gaussian,
component given the data point
Also,
andc :thus
T i unimodal.
These are exactly the constraints we wish to impose on the
posterior
, and mixture weights. At eachare
iteration of learning, the means c , covariance
matrices
determined separately for each data point,
so as to maximize the
in
:T objective
, best function
c
eq. (9): this amounts to computing
the
unimodal
distributions,
matched
to
: ' .
the true posterior distributions, c
c T '2
4 Learning Algorithm
Latent variable models are traditionally estimated by maximum likelihood or Bayesian
methods whose objective functions do not reward the interpretability of their internal representations. Note how the goal of developing more useful internal representations has
changed the learning problem in a fundamental
way. Now we have additional ?coordina
tion? parameters?the offsets
and weights ?that must also be learned from examples.
We also
have auxiliary parameters
for each data point?the means
, covariance
matri
, and mixture weights
?that determine the target distributions,
. All
ces
these parameters, as well as the MFA model parameters
, must be chosen
to ?stitch together? the local coordinates systems in a smooth way and to learn internal
representations easily coordinated by the local-to-global mapping in eq. (6).
c
T '
#3 H G I , c
Optimization of the objective function in eq. (9) is reminiscent of so-called ?variational?
methods for approximate learning[7]. In these methods, an approximation to an exact
(but intractable) posterior distribution is fitted by minimizing a KL divergence between the
two distributions. The auxiliary parameters of the approximating distribution are known
as variational parameters. Our objective function illustrates an unexpected application of
such variational methods: not to perform approximate inference in intractable probabilistic
models, but to learn more useful internal representations
in tractable ones. We introduce the
unimodal
and factorized distributions
to regularize the multimodal distributions
. Penalizing the KL divergence between these distributions lifts a degeneracy in
the model?s parameter space and favors local linear models that can be globally aligned.
c T '
c T
4.1 Computing and optimizing the objective function
Evaluating the objective function in eq. (9) requires a sum and integral over the latent variables of the model. These operations are simplified by rewriting the objective function as:
c : ' M C V W Y c T ' S VXWY R c N (
(11)
:T ' makes it straightforward to perform the
The factored form of the distributions c
and
required sums and integrals. The final result is a simple form in terms of entropies
P
c
energies
O
C
% C S
& VXWY
V W Y
associated with the th data point:
P
(12)
]& V W Y &7.
(13)
%
%
%
& c E c S & E I 9
K C c E E H E I 9
K S & M
N
S &% VXWY I 8 S V W Y C VXWY 3R S &S ] VXWY 6&87.?
(14)
C
G
where we haveC introduced simplifying notation for the vector differences
a
9K S H E I 9
K H 9
K .
c
and c
and the local precision matrices
Iteratively maximizing the objective function by coordinate ascent now leads to a learning
algorithm of the same general style as EM.
# c R ,
%
P c P % $ 9 !9 #" &" $ ( (15)
" S
9
K
E E
where
43 . NoticeH thatI and. These
equations can be solved by iteration with
initialization
only need to be computed once before iterating the fixed point equations.
The objective
function is completely invariant
C to translation
E ). To
and rescaling of c and (since , and c appear only in the form c
4.2 E-step
Maximizing
the objective function, eq. (9), with
respect to the regularizing parameters
(and subject to the constraint
) leads to the fixed point equations:
9K
remove this degeneracy, after solving the equations above we further constrain the global
coordinates to have mean zero and unit variance in each direction. These constraints are
enforced without changing the value of the objective
function by simply translating the
offsets ' and rescaling the diagonal matrices .
4.3 M-step
The M-step consists of maximizing the objective function, eq. (9), with respect to
the generative
parameters. Let us denote the updated parameter estimates by
( model
(
)( + * ,*
(
. Letting
, the M-step updates for the first three
of these are:
( .!/
*
$
* ,(16)
%$
# 3R G H I * ,
3 * P
9
K P c
G 9K P Z R(
The remaining updates,
shown, are given in terms of
* to beC performed
* 0* in theC order
*
. updated
* +*
difference vectors
,
the
correlations
G
c
c
1
c E ,
E
* *
N
S
M
c c .
and the variances 2
(
9
K
H
1 2
(17)
3 (
I 465 - 9
K P B 3 * C H ( 9
K c * 4 <5 S 3 H ( 9K E H ( E 465?F (18)
( 9
K a S H E I 9
K H 9
8K 7 E * S H E I ( 9
K 1 + 9 2 9K
(19)
At the optimum, the coordination weights
satisfy an algebraic Riccati equation which
can be solved by iterating the update shown above. (Such equations can also be solved
by much more sophisticated methods well known in the engineering community. Most
approaches involve inverting the previous value of
which may be expensive for full
matrices but is fast in our diagonal implementation.)
Figure 2: Global coordination of local linear models. (left) A model trained using maximum likelihood, with the arrows indicating the
direction of increase for each factor analyzer?s
local coordinate system. (right) A coordinated
model; arrows indicate the direction in the data
space corresponding to increasing the global
coordinate as inferred by the algorithm. The
ellipses show the one standard deviation contour of the density of each analyzer.
5 Experiments
We have tested our model on simple synthetic manifolds whose structure is known as well
as on collections of images of handwritten digits and faces. Figure 2 illustrates the basic
concept of coordination, as achieved by our learning rule. In the coordinated model, the
global coordinate always points in the same direction
along the data manifold, as defined
by the composition of the transformations
and
. In the model trained with maximum likelihood, the density is well captured but each local latent variable has a random
orientation along the manifold.
H
We also applied the algorithm to collections of images of handwritten digits and of faces.
The representation of was an unprocessed vector
of raw 8-bit grayscale pixel
intensities
for each image (of dimensionality 256 for the
digits and 560 for the
faces.)
The MFAs had 64 local models and the global coordinates were two dimensional. After
training, the coordinated MFAs had learned a smooth, continuous mapping from the plane
to images of digits or of faces. This allows us both to infer a two-dimensional location given
any image by computing
and to generate new images from any point in the plane
. (Precisely what we wanted from the magic box.) In general, both
by computing
of these conditional distributions have the form of a mixture of Gaussians. Figure 3 shows
the inferred global coordinates
(i.e. the means of the unimodal distributions
)
of the training points after the last iteration of training as well as examples of new images
from the generative model, created by evaluating the mean of
along straight line
paths in the global coordinate space. In the case of digits, it seems as though our models
have captured tilt/shape and identity and represented them as the two axes of the space; in
the case of the faces the axes seem to capture pose and expression. (For the faces, the final
space was rotated by hand to align interpretable directions with the coordinate axes.)
c
% ^ %
& ^ &
c 2
c
c
c
c
c
As with all EM algorithms, the coordinated MFA learning procedure is susceptible to local optima. Crucial to the success of our experiments is a good initialization, which was
provided by the Locally Linear Embedding
algorithm[9]. We clamped
equal to the embedding coordinate provided by LLE and
to a small value and trained until convergence
(typically 30-100 iterations). Then we proceeded with training using the full EM equations
to update , again until convergence (usually 5-10 more iterations). Note, however, that
LLE and other embedding algorithms such as Isomap[10] are themselves unsupervised, so
the overall procedure, including this initial phase, is still unsupervised.
c
c
6 Discussion
Mixture models provide a simple way to approximate the density of high dimensional data
that lies on or near a low dimensional manifold. However, their hidden representations
do not make explicit the relationship between dissimilar data vectors. In this paper, we
have shown how to learn global coordinates that can act as an encapsulating interface, so
that other parts of a learning system do not need to interact with the individual components of a mixture. This should improve generalization as well as facilitate the propagation
and exchange of information when these models are incorporated into a larger (perhaps
Figure
3:
Automatically constructed two
dimensional global parameterizations of manifolds
of digits and faces. Each
plot shows the global
coordinate space discovered by the unsupervised
algorithm; points indicate
the inferred means for
each training item at the
end of learning. The image
stacks on the borders are
not from the training set
but are generated from the
model itself and represent
the mean of the predictive
distribution
at the
corresponding open circles
(sampled along the straight
lines in the global space).
d/e Zj l
The models provide both a
two degree-of-freedom generator for complex images
via
as well as a
pose/slant recognition system via .
d/e Zj l
d/e j .l
For the handwritten digits,
the training set consisted
of 1100 examples of the
digit ?2? (shown as crosses
above) mixed with 1100 examples of ?3?s (shown as
triangles). The digits are
from the NIST dataset, digitized at 16x16 pixels. For
the faces, we used 2000 images of a single person with
various poses and expressions taken from consecutive frames of a video digitized at 20x20 pixels. Brendan Frey kindly provided
the face data.
hierarchical) architecture for probabilistic reasoning.
Two variants of our purely unsupervised proposal are possible. The first is to use an embedding algorithm (such as LLE or Isomap) not only as an initialization step but to provide
clamped values for the global coordinates. While this supervised approach may work in
practice, unsupervised coordination makes clear the objective function that is being opti-
Figure 4: A situation in which an un-coordinated mixture model?trained to do density estimation?cannot be ?postcoordinated?. Noise has caused one of the local density models to orient orthogonal to the manifold. In globally coordinated learning, there is an additional pressure to align with
neighbouring models which would force the local model to
lie in the correct subspace.
mized, which unifies the goals of manifold learning and density estimation. Another variant
is to train an unsupervised mixture model (such as a MFA) using a traditional maximum
likelihood objective function and then to ?post-coordinate? its parameters by applying local
reflections/rotations and translations to create global coordinates. As illustrated in figure 4,
however, this two-step procedure can go awry because of noise in the original training set.
When both density estimation and coordination are optimized simultaneously there is extra
pressure for local experts to fit the global structure of the manifold.
Our work can be viewed as a synthesis of two long lines of research in unsupervised
learning. In the first are efforts at learning the global structure of nonlinear manifolds
[1, 4, 9, 10]; in the second are efforts at developing probabilistic graphical models for reasoning under uncertainty[5, 6, 7]. Our work proposes to model the global coordinates on
manifolds as latent variables, thus attempting to combine the representational advantages
of both frameworks. It differs from embedding by providing a fully probabilistic model
valid away from the training set, and from work in generative topographic mapping[2] by
not requiring a uniform discretized gridding of the latent space. Moreover, by extending
the usefulness of mixture models,it further develops an architecture that has already proved
quite powerful and enormously popular in applications of statistical learning.
Acknowledgements
We thank Mike Revow for sharing his unpublished work (at the University of Toronto) on coordinating mixtures, and Zoubin Ghahramani, Peter Dayan, Jakob Verbeek and two anonymous reviewers
for helpful comments and corrections.
References
[1] D. Beymer & T. Poggio. Image representations for visual learning. pringerScience 272 (1996).
[2] C. Bishop, M. Svensen, and C. Williams. GTM: The generative topographic mapping.
Neural Computation 10 (1998).
[3] C. Bregler & S. Omohundro. Nonlinear image interpolation using manifold learning.
Advances in Neural Information Processing Systems 7 (1995).
[4] D. DeMers & G.W. Cottrell. Nonlinear dimensionality reduction.
Advances in Neural Information Processing Systems 5 (1993).
[5] Ghahramani, Z. and Hinton, G. The EM algorithm for mixtures of factor analyzers.
University of Toronto Technical Report CRG-TR-96-1 (1996).
[6] Hinton, G., Dayan, P., and Revow, M. Modeling the manifolds of images of handwritten digits.
IEEE Transactions on Neural Networks 8 (1997).
[7] M. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. An introduction to variational methods for
graphical models. Machine Learning 37(2) (1999).
[8] N. Kambhatla and T. K. Leen. Dimension reduction by local principal component analysis.
Neural Computation 9 (1997).
[9] S. T. Roweis & L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding.
Science 290 (2000).
[10] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear
dimensionality reduction. Science 290 (2000).
| 2082 |@word proceeded:1 loading:1 seems:1 open:1 covariance:3 simplifying:1 pressure:2 tr:1 reduction:4 initial:1 contains:1 document:1 must:2 reminiscent:1 cottrell:1 shape:2 wanted:1 remove:1 treating:1 designed:1 update:4 interpretable:1 plot:1 implying:1 pursued:1 generative:4 item:1 parameterization:2 plane:2 awry:1 parameterizations:1 toronto:3 traverse:2 location:4 simpler:1 along:4 vxw:7 constructed:1 consists:1 combine:1 multimodality:1 introduce:3 themselves:1 nor:1 discretized:1 globally:4 automatically:2 increasing:1 provided:3 underlying:1 matched:1 notation:1 factorized:2 moreover:1 what:3 unobserved:2 transformation:5 act:1 usefully:1 exactly:1 control:1 normally:1 unit:1 appear:1 arguably:1 positive:1 negligible:1 before:1 frey:1 local:33 tends:1 engineering:1 xv:1 despite:1 opti:1 path:4 interpolation:1 black:1 initialization:3 factorization:1 averaged:1 practice:2 tribution:1 definite:1 differs:1 digit:10 procedure:3 convenient:1 integrating:1 zoubin:1 cannot:1 close:1 applying:1 optimize:1 deterministic:1 reviewer:1 center:2 maximizing:3 straightforward:1 go:1 williams:1 factored:1 rule:1 array:1 submodel:1 regularize:1 his:1 embedding:6 coordinate:45 traditionally:1 updated:2 imagine:1 suppose:1 target:1 exact:1 neighbouring:1 agreement:2 expensive:1 recognition:1 bec:1 observed:2 mike:1 solved:3 capture:2 parameterize:2 region:1 connected:2 intuition:1 reward:1 trained:4 solving:1 predictive:1 purely:1 completely:1 triangle:1 easily:3 joint:1 multimodal:2 represented:5 various:1 gtm:1 regularizer:3 train:1 fast:1 describe:2 lift:1 neighborhood:7 whose:7 quite:1 widely:1 larger:1 say:2 favor:2 topographic:2 itself:1 final:2 advantage:1 aligned:4 deduces:1 riccati:1 achieve:1 roweis:2 representational:1 description:1 exploiting:1 convergence:2 optimum:2 extending:1 produce:1 rotated:1 svensen:1 pose:4 eq:9 solves:1 strong:1 auxiliary:2 indicate:2 direction:5 posit:1 correct:1 translating:1 exchange:1 generalization:1 anonymous:1 bregler:1 crg:1 correction:1 lawrence:1 mapping:5 kambhatla:1 consecutive:1 estimation:6 polar:1 coordination:11 create:1 sensor:1 gaussian:4 always:1 varying:1 factorizes:1 jaakkola:1 ax:3 likelihood:7 brendan:1 helpful:2 inference:4 dayan:2 typically:1 hidden:5 favoring:1 pixel:7 overall:1 orientation:1 proposes:1 fairly:1 marginal:2 equal:1 construct:1 once:1 nicely:1 sampling:2 identical:1 represents:1 unsupervised:10 nearly:1 report:1 ofto:1 develops:1 simultaneously:1 divergence:4 individual:1 phase:1 attempt:1 freedom:1 highly:2 alignment:1 mixture:17 isc:1 egfih:1 fu:1 encourage:3 integral:2 poggio:1 orthogonal:2 circle:1 fitted:1 modeling:2 maximization:1 deviation:1 neutral:1 uniform:2 usefulness:1 characterize:1 reported:1 synthetic:1 person:1 density:10 fundamental:1 probabilistic:7 together:2 continuously:1 synthesis:1 again:1 lever:3 expert:1 style:1 rescaling:2 account:1 de:1 twin:1 coordinated:9 satisfy:1 caused:1 tion:1 break:1 performed:1 analyze:1 complicated:1 variance:2 likewise:1 ensemble:2 correspond:1 raw:2 bayesian:1 handwritten:4 unifies:1 straight:2 explain:1 sharing:1 against:1 energy:1 associated:1 degeneracy:3 sampled:2 demers:1 dataset:1 proved:1 popular:1 dimensionality:5 formalize:1 sophisticated:1 supervised:1 leen:1 box:5 though:3 generality:2 furthermore:1 correlation:2 until:2 hand:1 langford:1 nonlinear:6 overlapping:1 propagation:1 thati:1 mode:1 perhaps:1 scientific:1 facilitate:1 validity:1 requiring:1 true:2 isomap:2 concept:1 consisted:1 regularization:1 symmetric:1 leibler:2 iteratively:1 illustrated:1 encourages:1 criterion:1 omohundro:1 reflection:3 interface:1 silva:1 reasoning:3 image:23 variational:5 regularizing:2 rotation:3 absorbing:1 multinomial:1 qp:1 tilt:1 composition:1 slant:1 consistency:2 analyzer:5 had:2 add:1 align:2 posterior:8 optimizing:1 belongs:1 arbitrarily:1 success:1 exploited:1 captured:2 fortunately:1 additional:2 impose:1 spectrogram:1 determine:1 maximize:2 multiple:1 unimodal:7 full:2 infer:1 smooth:3 technical:1 cross:2 long:1 post:1 manipulate:1 ellipsis:1 verbeek:1 involving:1 basic:1 variant:2 expectation:1 iteration:5 represent:3 achieved:2 penalize:2 proposal:1 background:1 separately:1 crucial:1 extra:1 strict:1 ascent:1 induced:1 subject:1 comment:1 member:1 inconsistent:1 incorporates:1 seem:1 jordan:1 near:5 constraining:1 affect:2 fit:1 pennsylvania:1 architecture:2 idea:1 tradeoff:1 unprocessed:1 expression:4 handled:1 effort:2 penalty:1 peter:1 algebraic:1 speech:1 logprobability:1 useful:5 iterating:2 clear:1 involve:1 amount:1 locally:6 tenenbaum:1 generate:1 exist:1 zj:3 estimated:1 coordinating:1 discrete:2 independency:2 changing:1 penalizing:2 ce:1 rewriting:1 sum:3 enforced:1 orient:1 parameterized:2 everywhere:1 uncertainty:1 powerful:1 family:4 scaling:1 bit:1 matri:1 constraint:3 precisely:1 constrain:1 flat:1 generates:1 attempting:2 mized:1 bdc:1 department:2 developing:3 according:1 describes:1 em:6 sam:1 intuitively:1 restricted:1 invariant:1 taken:2 equation:7 agree:1 r3:1 encapsulating:1 letting:1 tractable:3 end:1 parametrize:1 operation:3 gaussians:1 hierarchical:1 away:1 appropriate:1 gridding:1 original:1 assumes:2 remaining:1 graphical:4 dial:3 exploit:1 ghahramani:3 approximating:1 unchanged:1 objective:21 already:1 diagonal:3 traditional:1 exhibit:1 subspace:1 thank:1 manifold:32 index:1 relationship:3 providing:1 balance:1 minimizing:1 difficult:2 susceptible:1 x20:1 rise:1 ba:1 magic:2 implementation:1 perform:3 nist:1 curved:1 situation:1 hinton:3 incorporated:2 digitized:2 frame:1 discovered:1 stack:1 jakob:1 arbitrary:1 community:1 intensity:4 inferred:3 introduced:1 inverting:1 unpublished:1 required:1 kl:3 optimized:1 learned:2 usually:2 below:1 interpretability:1 including:1 video:1 mfa:15 force:1 improve:1 axis:1 created:1 prior:3 geometric:1 acknowledgement:1 loss:2 fully:1 mixed:1 geoffrey:1 generator:1 degree:1 consistent:2 principle:1 translation:3 changed:1 last:1 dis:1 lle:3 saul:3 face:21 taking:1 distributed:1 dimension:1 evaluating:2 valid:1 contour:1 computes:2 collection:6 made:2 simplified:1 transaction:1 approximate:4 kullback:2 gene:1 global:40 robotic:1 summing:1 grayscale:1 continuous:3 latent:6 un:1 learn:5 interact:1 complex:1 domain:1 kindly:1 arrow:2 border:1 noise:5 arise:1 fig:1 enormously:1 x16:1 precision:1 wish:1 explicit:1 lie:6 clamped:2 perceptual:2 bishop:1 appeal:1 offset:2 intractable:3 intrinsic:2 exists:1 incorporating:1 adding:1 illustrates:3 easier:1 smoothly:2 entropy:1 simply:1 beymer:1 absorbed:1 visual:1 unexpected:2 stitch:1 applies:1 conditional:4 goal:4 identity:1 viewed:1 revow:2 change:3 determined:1 unpredictably:1 principal:1 called:1 invariance:2 indicating:1 select:1 internal:8 support:1 dissimilar:1 tested:1 |
1,188 | 2,083 | The Method of Quantum Clustering
David Horn and Assaf Gottlieb
School of Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv 69978, Israel
Abstract
We propose a novel clustering method that is an extension of ideas inherent to scale-space clustering and support-vector clustering. Like the latter, it associates every data point with a vector in Hilbert space, and like
the former it puts emphasis on their total sum, that is equal to the scalespace probability function. The novelty of our approach is the study of
an operator in Hilbert space, represented by the Schr?odinger equation of
which the probability function is a solution. This Schr?odinger equation
contains a potential function that can be derived analytically from the
probability function. We associate minima of the potential with cluster
centers. The method has one variable parameter, the scale of its Gaussian
kernel. We demonstrate its applicability on known data sets. By limiting
the evaluation of the Schr?odinger potential to the locations of data points,
we can apply this method to problems in high dimensions.
1 Introduction
Methods of data clustering are usually based on geometric or probabilistic considerations
[1, 2, 3]. The problem of unsupervised learning of clusters based on locations of points in
data-space, is in general ill defined. Hence intuition based on other fields of study may be
useful in formulating new heuristic procedures. The example of [4] shows how intuition
derived from statistical mechanics leads to successful results. Here we propose a model
based on tools that are borrowed from quantum mechanics.
We start out with the scale-space algorithm of [5] that uses a Parzen-window estimator of
the probability distribution based on the data. Using a Gaussian kernel, one generates from
the data points in a Euclidean space of dimension a probability distribution given by,
up to an overall normalization, the expression
(1)
where are the data points. It seems quite natural [5] to associate maxima of this function
with cluster centers.
The same kind of Gaussian kernel was the basis of another method, Support Vector Clustering (SVC) [6], associating the data-points with vectors in an abstract Hilbert space.
Here we will also consider a Hilbert space, but, in contradistinction with kernel methods
where the Hilbert space is implicit, here we work with a Schr?odinger equation that serves
as the basic framework of the Hilbert space. Our method was introduced in [7] and is
further expanded in this presentation. Its main emphasis is on the Schr?odinger potential,
whose minima will determine the cluster centers. This potential is part of the Schr?odinger
equation that is a solution of.
2 The Schr?odinger Potential
We define[7] the Schr?odinger equation
(2)
is a solution, or eigenstate.1 The simplest case is that of a single Gaussian,
for which
Then it turns out that
. This
when represents a single point at .
quadratic function, whose center lies at , is known
as the harmonic potential
in quantum
is the lowest possible eigenvalue of ,
mechanics (see, e.g., [8]). Its eigenvalue
hence the Gaussian function is said to describe the ground state of
.
and one searches for solutions,
Conventionally, in
quantum
mechanics, one is given
or eigenfunctions,
we have already
, asdetermined
by the data points, we
. Here,
whose
. This can
ask therefore for the
solution is the given
be easily obtained
through
is still left undefined. For this purpose we require
This sets the value of
and determines
"!$#
(3)
to be positive definite, i.e. min =0.
(4)
uniquely. Using Eq. 3 it is easy to prove that
%"& ('
(5)
3 2D Examples
3.1 Crab Data
To show the power of our new method we discuss the crab data set taken from Ripley?s book
[9]. This data set is defined over a five-dimensional parameter space. When analyzed in
terms of the 2nd and 3rd principal components of the correlation matrix one observes a nice
separation of the 200 instances into their four classes. We start therefore with this problem
as our first test case. In Fig. 1 we show the data as well as the Parzen probability distribution
using the width parameter
. It is quite obvious that this width is not small
enough to deduce the correct clustering according to the approach of [5]. Nonetheless, the
potential displayed in Fig. 2 shows the required four minima for the same width parameter.
Thus we conclude that the necessary information is already available. One needs, however,
the quantum clustering approach, to bring it out.
1
+
+
,
*)
- .
(the Hamiltonian) and (potential energy) are conventional quantum mechanical operators,
rescaled so that
depends on one parameter, .
is a (rescaled) energy eigenvalue in quantum
mechanics.
2
1
2
1
0
0
?1
?2
?1
?3
?2
PC3
PC2
Figure 1: A plot of Roberts? probability distribution for Ripley?s crab data [9] as defined
over the 2nd and 3rd principal components of the correlation matrix. Using a Gaussian
width of
we observe only one maximum. Different symbols label the four
classes of data.
)
1.2
1
V/E
0.8
0.6
0.4
0.2
2
0
2
1
1
0
0
?1
PC2
?2
?1
?3
?2
PC3
Figure 2: A plot of the Schr?odinger potential for the same problem as Fig. 1. Here we
clearly see the required four minima. The potential is plotted in units of .
Note in Fig. 2 that the potential grows quadratically outside the domain over which the
data are located. This is a general property of Eq. 3. sets the relevant scale over which
one may look for structure
potential. If the width is decreased more structure is to
of the, two
be expected. Thus, for
more minima appear, as seen in Fig. 3. Nonetheless,
they lie high and contain only a few data points. The major minima are the same as in Fig.
2.
3.2 Iris Data
Our second example consists of the iris data set [10], which is a standard benchmark obtainable from the UCI repository [11]. Here we use the first two principal components to
define
the
two dimensions in which we apply our method. Fig. 4, which shows the case for
, provides an almost perfect separation of the 150 instances into the three classes
into which they should belong.
%
4
Application of Quantum Clustering
The examples displayed in the previous section show that, if the spatial representation of
the data allows for meaningful clustering using geometric information, quantum clustering
(QC) will do the job. There remain, however, several technical questions to be answered:
What is the preferred choice of ? How can QC be applied in high dimensions? How does
one choose the appropriate space, or metric, in which to perform the analysis? We will
confront these issues in this section.
4.1 Varying
In the crabs-data we find that as is decreased to , the previous minima of
get
deeper and two new minima are formed. However the latter are insignificant, in the sense
that they lie at high values (of order ), as shown in Fig. 3. Thus, if we
data-points
classify
, roughly
same
to clusters according to their topographic location
on
the
surface
of
as for . By the way, the wave the
clustering assignment is expected for
function
. As is being further decreased, more
acquires only one additional maximum
at
and more maxima are expected in and an ever increasing number of minima (limited by
) in .
The one parameter of our problem, , signifies the distance that we probe. Accordingly we
expect to find clusters relevant to proximity information of the same order of magnitude.
One may therefore vary continuously and look for stability of cluster solutions, or limit
oneself to relatively high values of and decide to stop the search once a few clusters are
being uncovered.
4.2 Higher Dimensions
In the iris problem we obtained excellent clustering results using the first two principal
components, whereas in the crabs problem, clustering that depicts correctly the classification necessitates components 2 and 3. However, once this is realized, it does not harm to
add the 1st component. This requires
in a 3-dimensional space, spanned by the
onworking
three leading PCs. Calculating
a fine computational grid becomes a heavy task
in high dimensions. To cut down complexity, we propose using the analytic expression of
Eq. 3 and evaluating the potential on data points only. This should be good enough to give
a close estimate of where the minima lie, and it reduces the complexity to
irrespective
of dimension. In the gradient-descent algorithm described below, we will require further
computations, also restricted to well defined locations in space.
1.2
1
V/E
0.8
0.6
0.4
0.2
2
1
0
2
0
1
0
?1
?1
?2
?2
?3
PC3
PC2
Figure 3: The potential for the crab data with
displays two additional, but insignificant, minima. The four deep minima are roughly at the same locations as in Fig.
2.
1.5
1
0.5
PC2
0
?0.5
?1
?1.5
?2
?2.5
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
PC1
%
in a space spanned by the first
Figure 4: Quantum clustering of the iris data for
two principal components.
symbols represent the three classes. Equipotential
Different
lines are drawn at
When
locations of data points, we evaluate on a discrete
restricted
. We tocanthethen
set of
points
express in terms of the distance matrix
as
with chosen appropriately so that min =0.
(6)
All problems that we have used as examples were such that data were given in some space,
and we have exercised our freedom to define a metric, using the PCA approach, as the basis
for distance calculations. The previous analysis tells us that QC can also be applied to data
for which only the distance information is known.
4.3 Principal Component Metrics
The QC algorithm starts from distance information. The question how the distances are calculated is another - very important - piece of the clustering procedure. The PCA approach
defines a metric that is intrinsic to the data, determined by their second order statistics. But
even then, several possibilities exist, leading to non-equivalent results.
Principal component decomposition can be applied both to the correlation matrix and to the
covariance matrix. Moreover, whitening normalization may be applied. The PCA approach
that we have used is based on a whitened correlation matrix. This turns out to lead to the
good separation of crab-data in PC2-PC3 and of iris-data in PC1-PC2. Since our aim was to
convince the reader that once a good metric is found, QC conveys the correct information,
we have used the best preprocessing before testing QC.
5 The Gradient Descent Algorithm
After discovering the
cluster
centers we are faced with the problem of allocating the data
points to the different clusters.
propose using a gradient descent algorithm for this
weWe
purpose. Defining
define the
process
%
(7)
letting the points reach an asymptotic fixed value coinciding with a cluster center. More
sophisticated minimum search algorithms, as given in chapter 10 of [12], may be used for
faster convergence.
To demonstrate the results of this algorithm, as well as the application of QC to higher
dimensions, we analyze the iris data in 4 dimensions. We use the original data space with
only one modification: all axes are normalized to lie within a unified range of variation.
The results are displayed in Fig. 5. Shown here are different windows for the four different
axes, within which we display the values of the points after descending the potential surface
and reaching its minima, whose values are shown in the fifth window. These results are
very satisfactory, having only 5 misclassifications. Applying QC to data space without
normalization of the different axes, leads to misclassifications of the order of 15 instances,
similar to the clustering quality of [4].
6 Discussion
In the literature of image analysis one often looks for the curve on which the Laplacian of
the Gaussian filter of an image vanishes[13]. This is known as zero-crossing and serves as
dim 1
1.5
1
dim 2
0.5
0
50
100
150
0
50
100
150
0
50
100
150
0
50
100
150
1
dim 3
2
1
0
dim 4
2
1
0
V/E
0.2
0.1
0
20
40
60
80
serial number
100
120
140
Figure 5: The fixed points of the four-dimensional iris problem following the gradientdescent algorithm. The
show almost perfect clustering into the three families of 50
results
instances each for
.
%
%
a measure
of the image. Its analogue in the scale-space approach is where
of .segmentation
Clearly each such contour can also be viewed as surrounding maxima of the
probability function, and therefore representing some kind of cluster boundary, although
different from the conventional one [5]. It is known that the number of such boundaries
[13] is a non-decreasing function of . Note that such
can be read off Fig. 4.
contours
contours on the periphery of
Comparison with Eq. 3 tells us that they are the
this figure. Clearly they surround the data but do not give a satisfactory
indication of where
the clusters are. Cluster cores are better defined by
curves in this figure. One
may therefore speculate that equipotential levels of may serve as alternatives to
curves in future applications to image analysis.
%
%
Image analysis is a 2-dimensional problem, in which differential operations have to be
formulated and followed on a fine grid. Clustering is a problem that may occur in any
number of dimensions. It is therefore important to develop a tool that can deal with it
accordingly. Since the Schr?odinger potential, the function that plays the major role in our
analysis, has minima that lie in the neighborhood of data points, we find that it suffices
to evaluate it at these points. This enables us to deal with clustering in high dimensional
spaces. The results, such as the iris problem of Fig. 5, are very promising. They show that
the basic idea, as well as the gradient-descent algorithm of data allocation to clusters, work
well.
Quantum clustering does not presume any particular shape or any specific number of clusters. It can be used in conjunction with other clustering methods. Thus one may start
with SVC to define outliers which will be excluded from the construction of the QC potential. This would be one example where not all points are given the same weight in the
construction of the Parzen probability distribution.
It may seem strange to see the Schr?odinger equation in the context of machine learning. Its
usefulness here is due to the fact that the two different terms of Eq. 2 have opposite effects
on the wave-function. The potential represents the attractive force that tries to concentrate
the distribution around its minima. The Laplacian has the opposite effect of spreading the
wave-function. In a clustering analysis we implicitly assume that two such effects exist.
QC models them with the Schr?odinger equation. Its success proves that this equation can
serve as the basic tool of a clustering method.
References
[1] A.K. Jain and R.C. Dubes. Algorithms for clustering data. Prentice Hall, Englewood
Cliffs, NJ, 1988.
[2] K. Fukunaga. Introduction to Statistical Pattern Recognition. Academic Press, San
Diego, CA, 1990.
[3] R.O. Duda, P.E. Hart and D.G. Stork. Pattern Classification. Wiley-Interscience, 2nd
ed., 2001.
[4] M. Blat, S. Wiseman and E. Domany. Super-paramagnetic clustering of data. Phys.
Rev. Letters 76:3251-3255, 1996.
[5] S.J. Roberts. Non-parametric unsupervised cluster analysis. Pattern Recognition,
30(2):261?272, 1997.
[6] A. Ben-Hur, D. Horn, H.T. Siegelmann, and V. Vapnik. A Support Vector Method for
Clustering. in Advances in Neural Information Processing Systems 13: Proceedings
of the 2000 Conference Todd K. Leen, Thomas G. Dietterich and Volker Tresp eds.,
MIT Press 2001, pp. 367?373.
[7] David Horn and Assaf Gottlieb. Algorithm for Data Clustering in Pattern Recognition
Problems Based on Quantum Mechanics. Phys. Rev. Lett. 88 (2002) 018702.
[8] S. Gasiorowicz. Quantum Physics. Wiley 1996.
[9] B. D. Ripley Pattern Recognition and Neural Networks. Cambridge University Press,
Cambridge UK, 1996.
[10] R.A. Fisher. The use of multiple measurements in taxonomic problems. Annual
Eugenics, 7:179?188, 1936.
[11] C.L. Blake and C.J. Merz. UCI repository of machine learning databases, 1998.
[12] W. H. Press, S. A. Teuklosky, W. T. Vetterling and B. P. Flannery. Numerical Recipes
- The Art of Scientific Computing 2nd ed. Cambridge Univ. Press, 1992.
[13] A. L. Yuille and T. A. Poggio. Scaling theorems for zero crossings. IEEE Trans. Pattern Analysis and Machine Intelligence PAMI-8, 15-25, 1986.
| 2083 |@word repository:2 faculty:1 seems:1 duda:1 nd:4 decomposition:1 covariance:1 contains:1 uncovered:1 paramagnetic:1 numerical:1 shape:1 analytic:1 enables:1 plot:2 intelligence:1 discovering:1 accordingly:2 hamiltonian:1 core:1 provides:1 location:6 five:1 differential:1 prove:1 consists:1 assaf:2 interscience:1 expected:3 roughly:2 mechanic:6 decreasing:1 window:3 increasing:1 becomes:1 moreover:1 lowest:1 israel:1 what:1 kind:2 unified:1 astronomy:1 nj:1 every:1 uk:1 unit:1 appear:1 positive:1 before:1 todd:1 limit:1 cliff:1 pami:1 emphasis:2 limited:1 range:1 horn:3 testing:1 definite:1 procedure:2 get:1 close:1 operator:2 put:1 context:1 applying:1 prentice:1 descending:1 conventional:2 equivalent:1 center:6 qc:10 estimator:1 spanned:2 stability:1 variation:1 limiting:1 construction:2 play:1 diego:1 exact:1 us:1 associate:3 crossing:2 recognition:4 located:1 cut:1 database:1 role:1 rescaled:2 observes:1 intuition:2 vanishes:1 complexity:2 serve:2 yuille:1 basis:2 necessitates:1 easily:1 represented:1 chapter:1 surrounding:1 univ:1 jain:1 describe:1 tell:2 outside:1 neighborhood:1 quite:2 heuristic:1 whose:4 statistic:1 topographic:1 eigenvalue:3 indication:1 propose:4 relevant:2 uci:2 recipe:1 convergence:1 cluster:17 perfect:2 ben:1 develop:1 dubes:1 school:1 borrowed:1 eq:5 job:1 concentrate:1 correct:2 filter:1 require:2 suffices:1 extension:1 proximity:1 crab:7 around:1 ground:1 hall:1 blake:1 major:2 vary:1 purpose:2 label:1 spreading:1 exercised:1 tool:3 mit:1 clearly:3 gaussian:7 aim:1 super:1 reaching:1 varying:1 volker:1 conjunction:1 derived:2 ax:3 sense:1 dim:4 vetterling:1 overall:1 issue:1 ill:1 classification:2 spatial:1 art:1 equal:1 field:1 once:3 having:1 represents:2 look:3 unsupervised:2 future:1 inherent:1 few:2 freedom:1 englewood:1 possibility:1 evaluation:1 analyzed:1 undefined:1 pc:1 allocating:1 necessary:1 poggio:1 euclidean:1 plotted:1 instance:4 classify:1 wiseman:1 assignment:1 signifies:1 applicability:1 usefulness:1 successful:1 convince:1 st:1 probabilistic:1 physic:2 off:1 parzen:3 continuously:1 choose:1 book:1 leading:2 potential:19 speculate:1 depends:1 piece:1 try:1 analyze:1 start:4 wave:3 formed:1 beverly:1 presume:1 reach:1 phys:2 ed:3 nonetheless:2 energy:2 pp:1 obvious:1 conveys:1 stop:1 ask:1 hur:1 hilbert:6 segmentation:1 obtainable:1 sophisticated:1 higher:2 coinciding:1 leen:1 implicit:1 correlation:4 defines:1 quality:1 scientific:1 aviv:2 grows:1 effect:3 dietterich:1 contain:1 normalized:1 former:1 analytically:1 hence:2 read:1 excluded:1 satisfactory:2 deal:2 attractive:1 width:5 uniquely:1 acquires:1 iris:8 equipotential:2 demonstrate:2 scalespace:1 bring:1 image:5 harmonic:1 consideration:1 novel:1 svc:2 stork:1 belong:1 measurement:1 surround:1 cambridge:3 rd:2 grid:2 surface:2 whitening:1 deduce:1 add:1 periphery:1 success:1 seen:1 minimum:16 additional:2 novelty:1 determine:1 multiple:1 reduces:1 technical:1 faster:1 academic:1 calculation:1 hart:1 serial:1 laplacian:2 basic:3 whitened:1 confront:1 metric:5 kernel:4 normalization:3 represent:1 whereas:1 fine:2 decreased:3 appropriately:1 pc3:4 eigenfunctions:1 seem:1 easy:1 enough:2 misclassifications:2 associating:1 opposite:2 idea:2 domany:1 oneself:1 expression:2 pca:3 deep:1 useful:1 simplest:1 exist:2 correctly:1 discrete:1 express:1 four:7 drawn:1 sum:1 letter:1 taxonomic:1 almost:2 reader:1 decide:1 family:1 strange:1 separation:3 pc2:6 scaling:1 sackler:1 followed:1 display:2 quadratic:1 annual:1 occur:1 generates:1 answered:1 min:2 formulating:1 fukunaga:1 expanded:1 relatively:1 according:2 remain:1 rev:2 modification:1 outlier:1 restricted:2 taken:1 equation:8 turn:2 discus:1 letting:1 serf:2 available:1 operation:1 apply:2 observe:1 probe:1 appropriate:1 alternative:1 original:1 thomas:1 clustering:28 calculating:1 siegelmann:1 prof:1 already:2 question:2 realized:1 parametric:1 said:1 gradient:4 distance:6 robert:2 perform:1 benchmark:1 descent:4 displayed:3 defining:1 ever:1 schr:12 pc1:2 david:2 introduced:1 required:2 mechanical:1 quadratically:1 trans:1 eugenics:1 usually:1 below:1 pattern:6 analogue:1 power:1 natural:1 force:1 representing:1 irrespective:1 conventionally:1 tresp:1 raymond:1 faced:1 nice:1 geometric:2 literature:1 asymptotic:1 expect:1 allocation:1 heavy:1 deeper:1 fifth:1 curve:3 dimension:10 calculated:1 evaluating:1 boundary:2 contour:3 quantum:13 lett:1 preprocessing:1 san:1 preferred:1 implicitly:1 harm:1 conclude:1 ripley:3 search:3 promising:1 ca:1 tel:2 excellent:1 domain:1 main:1 fig:12 depicts:1 wiley:2 lie:6 down:1 gradientdescent:1 theorem:1 specific:1 symbol:2 insignificant:2 intrinsic:1 vapnik:1 magnitude:1 flannery:1 contradistinction:1 determines:1 viewed:1 presentation:1 formulated:1 fisher:1 determined:2 gottlieb:2 principal:7 total:1 odinger:12 merz:1 meaningful:1 support:3 latter:2 evaluate:2 |
1,189 | 2,084 | Intransitive Likelihood-Ratio Classifiers
Jeff Bilmes
and
Gang Ji
Department of Electrical Engineering
University of Washington
Seattle, WA 98195-2500
bilmes,gji @ee.washington.edu
Marina Meil?a
Department of Statistics
University of Washington
Seattle, WA 98195-4322
mmp@stat.washington.edu
Abstract
In this work, we introduce an information-theoretic based correction term
to the likelihood ratio classification method for multiple classes. Under
certain conditions, the term is sufficient for optimally correcting the difference between the true and estimated likelihood ratio, and we analyze
this in the Gaussian case. We find that the new correction term significantly improves the classification results when tested on medium vocabulary speech recognition tasks. Moreover, the addition of this term
makes the class comparisons analogous to an intransitive game and we
therefore use several tournament-like strategies to deal with this issue.
We find that further small improvements are obtained by using an appropriate tournament. Lastly, we find that intransitivity appears to be a good
measure of classification confidence.
1 Introduction
An important aspect of decision theory is multi-way pattern classification whereby one
must determine the class for a given data vector that minimizes the overall risk:
argmin
where is the loss in choosing when the true class is . This decision rule is
provably optimal for the given loss function [3]. For the 0/1-loss functions, it is optimal to
simply use the posterior probability to determine the optimal class
argmax
This procedure may equivalently be specified using a tournament style game-playing strat(egy. In this case, there is an implicit class ordering "!#!$!# %& , and a class-pair (' and
) scoring function for an unknown sample :
)+*-,
*-,
(
*.,
0/21
*.,
*.,
(
43#576 8 '9 :; 8 is the log-likelihood)8ratio
43$56 '<=:> is
such that
<?<
@ and 1
the ) log
odds. The strategy
proceeds
by
evaluating
which
if
positive
is followed
?
prior
A
A
)
@
by
and otherwise by
. This continues until a ?winner? is found. Of course, the
order of the classes does not matter, as the same winner is found for all permutations. In
any event, this style of classification can be seen as a transitive game [5] between players
who correspond to the individual classes.
In this work we extend the likelihood-ratio based classification with a term, based on the
Kullback-Leibler divergence [2], that expresses the inherent posterior confusability between the underlying likelihoods being compared for a given pair of players. We find that
by including this term, the results of a classification system significantly improve, without
changing or increasing the quantity of the estimated free model parameters. We also show
how, under certain assumptions, the term can be seen as an optimal correction between the
estimated model likelihood ratio and the true likelihood ratio, and gain further intuition by
examining the case when the likelihoods 8 '9 are Gaussians. Furthermore, we observe
that the new strategy leads to an intransitive game [5], and we investigate several strategies for playing such games. This results in further (but small) improvements. Finally, we
consider the instance of intransitivity as a confidence measure, and investigate an iterative
approach to further improve the correction term.
Section 2 first motivates and defines our approach, and shows the conditions under which it
is optimal. Section 2.1 then reports experimental results which show significant improvements where the likelihoods are hidden Markov models trained on speech data. Section 3
then recasts the procedure as intransitive games, and evaluates a variety of game playing
strategies yielding further (small) error reductions. Section 3.1 attempts to better understand our results via empirical analysis, and evaluates additional classification strategies.
Section 4 explores an iterative strategy for improving our technique, and finally Section 5
concludes and discusses future work.
2 Extended Likelihood-Ratio-based Classification
The Kullback-Leibler (KL) divergence[2], an asymmetric measure of the distance between
two probability densities, is defined as follows:
83$56
where and are probability densities over the same sample space. The KL-divergence is
also called the average (under ) information for discrimination in favor of over . For our
purposes, we are interested in KL-divergence between class-conditional likelihoods 8 '9
where ' is the class number:
(
'
3$56
(
8 '<
( 8 '<7
8
(
One intuitive way of viewing ' is as follows: if ' ( is small, then (samples of
to be( erroneously classified as class than
when ' is large.
class ' are more likely
(
(
Comparing ' and >'< should tell us which of ' and is more likely
( to have
( its
samples mis-classified by the other model. Therefore,
the
difference
'
'9 ,
(
when positive, indicates that samples of class are more likely( to be mis-classified as
class ' than samples of class ' are to be mis-classified as class
(and vice-versa
when
(
(
the difference is negative). In other words, ' ?steals? from ( more than steals from '
when the difference is positive, thereby suggesting that class should receive aid in this
case. This difference can be viewed as a form of posterior (i.e., based on the data)
?bias?
*.,
indicating which class should receive favor over the other.1 We can adjust
(the log(likelihood ratio) with this posterior bias, to obtain a new function comparing classes ' and
as follows:
) *-,
*-,
*-,
*.,
1
/
1
!
Note that this is not the normal notion of statistical bias as in
model parameters.
where
is an estimate of
where
*-,
(
'
(
(
>'<
-* ,
The
is positive, and in favor of ' when
*., likelihood ratio is adjusted
) *-, in favor of when
is negative. We then use
, and when it is positive, choose class ' .
The above
*-, intuition does
*-, not explain why such a correction factor should be used, since
along with 1
is already optimal. In practice, however, we do not have access
using
to the true likelihood ratios but instead
to an approximation
that has been estimated from
*.,
(
$
3
5
6
training
data.
Let
the
variable
be
the true log-likelihood ratio,
8
<
'
=
>
:
*.,
(
and 3#576 8 '9 : 8 be the model-based log ratio. Furthermore, let
(
'
8 '<
( 8 '<7
8
3$56
be the modified KL-divergence between the class
*., conditional
( models, measured *.modulo
,
(
the true distribution '< , and let
(resp.
'
>'< . Finally, let 1
1
*.,
) be the true (resp. estimated) log prior odds. Our (usable) scoring function becomes:
)+*.,
*-,
0/
1
*-,
*-,
!
(1)
which has an intuitive explanation similar to the above.
There are certain conditions under which the above approach
is theoretically justifiable.
Let
(
(
.
us assume for now a two-class problem where ' and are the two classes, so '9;/
A sufficient condition
is
for the estimated quantities above to yield optimal performance
(
for /
1 / 1 for all .2 Since this is not the case in practice, an ' -dependent
constant term may be added correcting for any differences as best as possible. This
yields / 1 / 1 /
. We can define an -dependent cost function
/21
1
which, when minimized, yields
/ 1
1 stating that the optimal
under this cost function is just the mean of the difference
of the
Note that
remaining terms.
'< ' ( ( ( '< and
'< ' ( ( ( '9 . Several
additional assumptions lead to Equation 1. First, let us assume that the prior probabilities
are equal
(so '< ! ) and that the estimated and true priors are negligibly different (i.e.,
1 1 ). Secondly, if we assume that
, this implies that
(
(
(
'< :;
>'<=:
'
(
' under equal priors. While KL-divergence is not sym-
(
which means that >'<
metric in general, we can see that if this
holds (or is approximately
true for a given problem)
*-,
exactly
then the remaining correction is
yielding
in Equation 1.
To gain further insight, we can examine
* , the case when* the, likelihoods are Gaussian univariate distributions, with means and variances . In this case,
*.,
*
,
,
* /
,
*
! , *# "$
*-,
*
(2)
,
It is easy to see that for the value of
is zero for any . By computing the
@
*-,
*.,
*
derivative %'&)(+* we can show that
is monotonically increasing with . Hence,
is
*
%', *. ( -
positive iff
2
,
,
and therefore it penalizes the distribution (class) with higher variance.
Note that we have dropped the
/
argument for notational simplicity.
*-,
VOCAB SIZE
75
150
300
600
)+*-,
WER
2.33584
3.31072
5.22513
7.39268
WER
1.91561
2.89833
4.51365
6.18517
*.,
Table
1: Word error rates (WER) for likelihood ratio
and augmented likelihood ratio
)+*-,
based classification for various numbers of classes (VOCAB SIZE).
Similar relations hold for multivariate Gaussians with means
*-,
*
,
,
*
/
*
,
*
,
,
and variances
*
*
,
*
,
.
(3)
The above is zero when the two covariance
( matrices are
( equal. This implies that for Gaussians with equal covariance matrices, $ '<
' $ and our correction term is optimal.
This is the same
analysis (LDA). Moreover,
* as the
, condition for Fisher?s linear*-discriminant
, we have that , for and *-, for
in the case
with *. ,
which again implies that
penalizes the class that has larger covariance.
2.1 Results
*-,
*.,
1
) on a medium vocabulary speech
We tried this method (assuming that 1
recognition task. In our case the likelihood functions '< are hidden Markov model
(HMM) scores3 . The task we chose is NYNEX PHONEBOOK[4], an isolated word speech
corpus. Details of the experimental setup, training/test sets, and model topologies, are
described in [1]4 .
*-,
In general, there are a number of ways to compute
. These include 1) analytically, using
estimated model parameters (possible, for example, with Gaussian densities), 2) computing
the KL-divergences on training data using a law-of-large-numbers-like average of likelihood ratios and using training-data estimated model parameters, 3) doing
the same as 2 but
*.,
using test data where hypothesized answers come from a first pass -based classification,
and 4) Monte-Carlo methods where again the same procedure as 2 is used, but the data is
sampled from the training-data estimated distributions. For HMMs, method 1 above is not
possible. Also, the data set we used (PHONEBOOK) uses different classes for the training and test sets. In other words, the training and test vocabularies are different. During
training, phone models are constructed that are pieced together for the test vocabularies.
Therefore, method 2 above is also not possible for this data.
Either method 3 or 4 can be used in our case, and we used method 3 in all our experiments.
Of course, using the true test labels in method 3 would be the ideal measure of the degree
of confusion between models, but these are of course not available (see Figure 2, however,
showing the results of a cheating
*-, experiment). Therefore, we use the hypothesized labels
.
from a first stage to compute
The procedure thus is as follows: 1) obtain '< using maximum likelihood EM training,
*-,
2) classify the test set using only
and record the error
*., rate, 3) using the hypothesized
class labels (answers
with
errors)
to
step
2,
compute
, 4) re-classify the test set using
*-, *.,
) *-,
) *-,
(
and record the new error rate.
is used if either one of '#
the score
3
Using 4 state per phone, 12 Gaussian mixtures per state HMMs, totaling 200k free model parameters for the system.
4
Note, however, that error results here are reported on the development set, i.e., PHONEBOOK
lists a,b,c,d o,y
VOCAB
75
150
300
600
*-,
2.33584
3.31072
5.22513
7.39268
RAND1
1.87198
2.88505
4.41428
6.15828
RAND500
1.82047
2.71881
4.34608
6.13085
RAND1000
1.91467
2.72809
4.28930
5.91440
WORLD CUP
2.12777
2.79516
3.81583
5.93883
Table 2: The WER under different tournament strategies
(
or $ '< is below a threshold (i.e., when a likely confusion exists), otherwise
for classification.
*-,
is used
Table 1 shows the result of this experiment. The first column shows the vocabulary size
of the system (identical to the*-, number of classes)5 . The second column shows
the word
) *.,
error rate (WER) using just , and the third column shows WER using
. As can be
seen, the WER decreases significantly with this approach. Note also that no additional free
parameters are used to obtain these improvements.
3 Playing Games
*-,
) *.,
(
We may view either
or
as providing
a score of class ' over ? when positive,
(
class ' wins, and when negative, class wins. In general, the classification procedure
may be viewed as a tournament-style game, where for a given sample , different classes
correspond to different players. Players pair together and play each other, and the winner
goes on to play another match with a different player. The strategy leading to table 1
required a particular class presentation order ? in that case the order was just the numeric
ordering of the arbitrarily assigned integer classes (corresponding to words in this case).
*-,
alone is used, the order of the comparisons do not matter, leading to
Of course when
a transitive
game
[5]
(the order of player pairings do not change the final winner). The
) *-,
quantity
, however, is not guaranteed to be transitive, and when used in a tournament
it results in what is called an intransitive game[5]. This means, for example, that might
win over who might win over who then might win over . Games may be depicted
as directed graphs, where an edge between two players point towards the winner. In an
intransitive game, the graph contains directed cycles. There has been very little research on
intransitive game strategies ? there are in fact a number of philosophical issues relating
to if such games are valid or truly exist. Nevertheless, we derived a number of tournament strategies for playing such intransitive games and evaluated their performance in the
following.
Broadly, there are two tournament types that we considered. Given a particular ordering of
the classes "!#!$!$= % , we define a sequential tournament when plays , the winner
plays , the winner plays and so on. We also define a tree-based tournament when
plays , plays , and so on. The tree-based tournament is then applied recursively on
the resulting : winners until a final winner is found.
Based on the above, we investigated several intransitive game playing strategies. For
RAND1, we just choose a single random tournament order in a sequential tournament. For
RAND500, we run 500 sequential tournaments, each one with a different random order.
The ultimate winner is taken to be the player who wins the most tournaments. The third
strategy plays 1000 rather than 500 tournaments. The final strategy is inspired by worldcup soccer tournaments: given a randomly generated permutation, the class sequence is
5
The 75-word case is an average result of 8 experiments, the 150-word case is an average of 4
cases, and the 300-word case is an average of 2 cases. There are 7291 separate test samples in the
600-word case, and on average about 911 samples per 75-word test case.
vocabulary
75
150
300
600
1.0047
1.0061
1.0241
1.0319
var
0.0071
0.0126
0.0551
0.0770
max
2.7662
3.6539
4.0918
5.0460
1.0285
1.0118
1.0170
1.0533
var
0.0759
0.0263
0.0380
0.1482
max
3.8230
3.8724
3.9072
5.5796
Table 3: The statistics of winners. Columns 2-4: 500 random tournaments, Columns 5-7:
1000 random tournaments.
separated into 8 groups. We pick the winner of each group using a sequential tournament
(the ?regionals?). Then a tree-based tournament is used on the group winners.
60
70
50
60
probability of error (%)
probability of error (%)
Table 1 compares these different strategies. As can be seen, the results get slightly better
(particularly with a larger number of classes) as the number of tournaments increases. Finally, the single word cup strategy does surprisingly well for the larger class sizes. Note that
the improvements are statistically significant over the baseline (0.002 using a difference of
proportions significance test) and the improvements are more dramatic for increasing vocabulary size. Furthermore, the it appears that the larger vocabulary sizes benefit more
from the larger number (1000 rather than 500) of random tournaments.
40
30
20
10
0
50
40
30
20
10
1
2
3
4
length of cycle
5
6
0
0
1
2
3
4
number of cycles detected
5
Figure 1: 75-word vocabulary case. Left: probability of error given that there exists a cycle
of at least the given length (a cycle length of one means no cycle found). Right:probability
of error given that at least the given number of cycles exist.
3.1 Empirical Analysis
In order to better understand our results, this section analyzes the 500 and 1000 random
tournament strategies described above. Each set of random tournaments
produces a set
of winners which may be described by a histogram. The entropy
of that histogram
describes its spread, and the number of typical winners is approximately
. This is of
), variance,
course relative to each sample so we may look at the average (
and
maximum of this number (the minimum is 1.0 in every case). This is given in Table 3 for
the 500 and 1000 cases.
is approximately 1
The table indicates that there is typically only one winner since
and the variances are small. This shows further that the winner is typically not in a cycle, as
the existence of a directed cycle in the tournament graph would probably lead to different
winners for each random tournament. The relationship between properties of cycles and
WER is explored below.
When the tournament is intransitive (and therefore the graph possess a cycle), our second
analyses shows that the probability of error tends to increase. This is shown in Figure 1
showing that the error probability increases both as the detected cycle length and the num-
vocabulary
75
150
300
600
*.,
2.33584
3.31072
5.22513
7.39268
skip WER
1.90237
2.76814
4.46296
6.50117
#cycles(%)
13.89
19.6625
22.38
31.96
break WER
1.90223
2.67814
4.46296
6.50117
#cycles(%)
9.34
16.83
21.34
31.53
Table 4: WER results using two strategies (skip and break)
that utilize
information
about
*.,
cycles in the tournament graphs, compared to baseline . The
and
columns show
the number of cycles detected relative to the number of samples in each case.
ber of detected cycles increases. 6 This property suggests that the existence of intransitivity
could be used as a confidence measure, or could be used to try to reduce errors.
As an attempt at the latter, we evaluated two very simple heuristics that try to eliminate
cycles as detected during classification. In the first method (skip), we run a sequential
tournament (using a random class ordering) until either a clear winner is found (a transitive
game), or a cycle is detected. If a cycle is detected, we select two players not in the
cycle, effectively jumping out of the cycle, and continue playing until the end of the class
ordering. If winner*-, cannot be determined (because there are too few players remaining), we
backoff and use
to select the winner. In a second method (break), if a cycle is detected,
we eliminate the class having the smallest likelihood from that cycle, and then continue
playing as before. Neither method detects all the cycles in the graph (their number can be
exponentially large).
As can be seen, the WER results still provide significant improvements over the baseline,
but are no better than earlier results. Because the tournament strategy is coupled with cycle
detection, the cycles detected are different in each case (the second method detecting fewer
cycles presumably because the eliminated class is in multiple cycles). In any case, it is
apparent that further work is needed to investigate the relationship between the existence
and properties of cycles and methods to utilize this information.
4 Iterative Determination of KL-divergence
In all of our experiments so far, KL-divergence is calculated according to the initial hypothesized answers. We would expect that using the true answers to determine the KLdivergence would improve our results further. The top horizontal lines in Figure 2 shows
the original baseline results, and the bottom lines show the results using the true answers (a
cheating experiment) to determine the KL-divergence.
As can be seen, the improvement is
*-,
significant thereby confirming that using
can significantly improve classification performance. Note also that the relative improvement stays about constant with increasing
vocabulary size.
This further indicates that an iterative strategy for determining KL-divergence might fur*-,
ther improve our results. In this case,
is used to determine the answers to compute the
) *-
,
first set of KL-divergences used in
. This is then used to compute a new set of an-
) *-,
swers which then is used to compute a new scores
and so on. The remaining plots in
Figure 2 show the results of this strategy for the 500 and 1000 random trials case (i.e., the
answers used to compute the KL-divergences in each case are obtained from the previous
set of random tournaments using the histogram peak procedure described earlier). Rather
surprisingly, the results show that iterating in this fashion does not influence the results in
6
Note that this shows a lower bound on the number of cycles detected. This is saying that if we
find, for example, four or more cycles then the chance of error is high.
75 classes
150 classes
3.5
word error rate (%)
word error rate (%)
2.5
2
1.5
0
2
4
6
8
number of iterations
3
2.5
2
10
0
2
300 classes
10
600 classes
5.4
7.5
word error rate (%)
5.2
word error rate (%)
4
6
8
number of iterations
5
4.8
4.6
4.4
baseline
cheating
500 trials
1000 trials
7
6.5
6
4.2
4
0
2
4
6
8
number of iterations
10
5.5
0
2
4
6
8
number of iterations
10
Figure 2: Baseline using likelihood ratio (top lines), cheating results using correct answers
for KL-divergence (bottom lines), and the iterative determination of KL-distance using
hypothesized answers from previous iteration (middle lines).
any appreciable way ? the WERs seem to decrease only slightly from their initial drop. It
is the case, however, that as the number of random tournaments increases, the results become closer to the ideal as the vocabulary size increases. We are currently studying further
such iterative procedures for recomputing the KL-divergences.
5 Discussion and Conclusion
We have introduced a correction term to the likelihood ratio classification method that is
justified by the difference between the
*-, estimated and true class conditional probabilities
8 '<> '< . The correction term
is an estimate of the classification
bias that would
*-,
makes the class comoptimally compensate for these differences. The presence of
parisons intransitive and we*., introduce several tournament-like strategies to compensate.
While the introduction of
consistently improves the classification results, further improvements are obtained by the selection of the comparison strategy. Further details and
results of our methods will appear in forthcoming publications and technical reports.
References
[1] J. Bilmes. Natural Statistic Models for Automatic Speech Recognition. PhD thesis, U.C. Berkeley, Dept. of EECS, CS Division, 1999.
[2] T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley, 1991.
[3] R.O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification. John Wiley and Sons, Inc., 2000.
[4] J. Pitrelli, C. Fong, S.H. Wong, J.R. Spitz, and H.C. Lueng. PhoneBook: A phonetically-rich isolated-word telephone-speech database. In Proc.
IEEE Intl. Conf. on Acoustics, Speech, and Signal Processing, 1995.
[5] P.D. Straffin. Game Theory and Strategy. The Mathematical ASsociation of America, 1993.
| 2084 |@word trial:3 middle:1 proportion:1 duda:1 tried:1 covariance:3 pick:1 dramatic:1 thereby:2 recursively:1 reduction:1 initial:2 contains:1 score:3 comparing:2 must:1 john:1 confirming:1 plot:1 drop:1 discrimination:1 nynex:1 alone:1 fewer:1 steal:2 record:2 num:1 detecting:1 mathematical:1 along:1 constructed:1 become:1 pairing:1 introduce:2 theoretically:1 examine:1 multi:1 vocab:3 inspired:1 detects:1 little:1 increasing:4 becomes:1 moreover:2 underlying:1 medium:2 what:1 argmin:1 minimizes:1 kldivergence:1 berkeley:1 every:1 recasts:1 exactly:1 classifier:1 appear:1 positive:7 before:1 engineering:1 dropped:1 tends:1 meil:1 approximately:3 tournament:34 chose:1 might:4 suggests:1 hmms:2 statistically:1 directed:3 practice:2 procedure:7 empirical:2 significantly:4 confidence:3 word:18 get:1 cannot:1 selection:1 risk:1 influence:1 wong:1 go:1 simplicity:1 correcting:2 rule:1 insight:1 notion:1 analogous:1 resp:2 play:8 modulo:1 us:1 element:1 recognition:3 particularly:1 continues:1 asymmetric:1 database:1 bottom:2 negligibly:1 electrical:1 pieced:1 cycle:32 ordering:5 decrease:2 intuition:2 trained:1 division:1 various:1 america:1 separated:1 monte:1 detected:10 tell:1 choosing:1 apparent:1 heuristic:1 larger:5 otherwise:2 favor:4 statistic:3 final:3 sequence:1 iff:1 intuitive:2 seattle:2 intl:1 produce:1 stating:1 stat:1 measured:1 c:1 skip:3 implies:3 come:1 correct:1 viewing:1 secondly:1 adjusted:1 correction:9 hold:2 considered:1 normal:1 presumably:1 smallest:1 purpose:1 proc:1 label:3 currently:1 vice:1 gaussian:4 modified:1 rather:3 totaling:1 publication:1 derived:1 improvement:10 notational:1 fur:1 likelihood:25 indicates:3 consistently:1 baseline:6 dependent:2 typically:2 eliminate:2 hidden:2 relation:1 interested:1 provably:1 issue:2 classification:19 overall:1 development:1 equal:4 having:1 washington:4 eliminated:1 identical:1 look:1 future:1 minimized:1 report:2 inherent:1 few:1 randomly:1 divergence:15 individual:1 argmax:1 attempt:2 detection:1 investigate:3 intransitive:11 adjust:1 mixture:1 truly:1 yielding:2 wers:1 edge:1 closer:1 jumping:1 tree:3 penalizes:2 re:1 isolated:2 instance:1 classify:2 column:6 earlier:2 recomputing:1 phonebook:4 cover:1 cost:2 examining:1 too:1 optimally:1 reported:1 answer:9 fong:1 eec:1 density:3 explores:1 peak:1 stay:1 together:2 again:2 thesis:1 choose:2 conf:1 usable:1 style:3 derivative:1 leading:2 suggesting:1 matter:2 inc:1 view:1 break:3 try:2 analyze:1 doing:1 phonetically:1 variance:5 who:4 correspond:2 yield:3 carlo:1 bilmes:3 justifiable:1 classified:4 explain:1 evaluates:2 spitz:1 mi:3 gain:2 sampled:1 improves:2 appears:2 strat:1 higher:1 evaluated:2 furthermore:3 just:4 implicit:1 lastly:1 stage:1 until:4 horizontal:1 defines:1 lda:1 hypothesized:5 true:13 hence:1 analytically:1 assigned:1 leibler:2 deal:1 game:19 during:2 whereby:1 soccer:1 theoretic:1 confusion:2 ji:1 stork:1 winner:21 exponentially:1 extend:1 association:1 relating:1 significant:4 versa:1 cup:2 automatic:1 access:1 posterior:4 multivariate:1 phone:2 certain:3 arbitrarily:1 continue:2 scoring:2 seen:6 analyzes:1 additional:3 minimum:1 determine:5 monotonically:1 signal:1 multiple:2 technical:1 match:1 determination:2 compensate:2 hart:1 marina:1 rand1:2 metric:1 histogram:3 iteration:5 receive:2 addition:1 justified:1 posse:1 probably:1 seem:1 odds:2 integer:1 ee:1 presence:1 ideal:2 easy:1 variety:1 forthcoming:1 topology:1 reduce:1 ultimate:1 speech:7 iterating:1 clear:1 backoff:1 exist:2 estimated:11 per:3 broadly:1 express:1 group:3 four:1 threshold:1 nevertheless:1 changing:1 neither:1 utilize:2 graph:6 run:2 wer:12 saying:1 decision:2 bound:1 followed:1 guaranteed:1 gang:1 erroneously:1 aspect:1 argument:1 department:2 according:1 describes:1 slightly:2 em:1 son:1 taken:1 equation:2 discus:1 needed:1 end:1 studying:1 available:1 gaussians:3 observe:1 appropriate:1 existence:3 original:1 thomas:1 top:2 remaining:4 include:1 already:1 quantity:3 added:1 strategy:24 win:6 distance:2 separate:1 hmm:1 discriminant:1 assuming:1 length:4 relationship:2 gji:1 ratio:18 providing:1 equivalently:1 setup:1 negative:3 motivates:1 unknown:1 markov:2 extended:1 introduced:1 pair:3 cheating:4 specified:1 kl:15 required:1 philosophical:1 intransitivity:3 acoustic:1 ther:1 proceeds:1 below:2 pattern:2 including:1 max:2 explanation:1 confusability:1 event:1 natural:1 improve:5 concludes:1 transitive:4 coupled:1 prior:5 determining:1 relative:3 law:1 loss:3 expect:1 permutation:2 var:2 degree:1 sufficient:2 playing:8 course:5 surprisingly:2 free:3 sym:1 bias:4 understand:2 ber:1 benefit:1 calculated:1 vocabulary:12 evaluating:1 world:1 numeric:1 valid:1 rich:1 far:1 kullback:2 corpus:1 iterative:6 why:1 table:9 improving:1 investigated:1 significance:1 spread:1 augmented:1 fashion:1 aid:1 wiley:2 mmp:1 third:2 showing:2 list:1 explored:1 exists:2 sequential:5 effectively:1 phd:1 egy:1 entropy:1 depicted:1 simply:1 likely:4 univariate:1 chance:1 conditional:3 viewed:2 presentation:1 towards:1 appreciable:1 jeff:1 fisher:1 change:1 typical:1 determined:1 telephone:1 called:2 pas:1 experimental:2 player:10 indicating:1 select:2 latter:1 dept:1 tested:1 |
1,190 | 2,085 | Multiplicative Updates for Classification
by Mixture Models
Lawrence K. Saul and Daniel D. Lee
Department
of Computer and Information Science
Department of Electrical Engineering
University of Pennsylvania, Philadelphia, PA 19104
Abstract
We investigate a learning algorithm for the classification of nonnegative data by
mixture models. Multiplicative update rules are derived that directly optimize
the performance of these models as classifiers. The update rules have a simple
closed form and an intuitive appeal. Our algorithm retains the main virtues of
the Expectation-Maximization (EM) algorithm?its guarantee of monotonic improvement, and its absence of tuning parameters?with the added advantage of
optimizing a discriminative objective function. The algorithm reduces as a special case to the method of generalized iterative scaling for log-linear models. The
learning rate of the algorithm is controlled by the sparseness of the training data.
We use the method of nonnegative matrix factorization (NMF) to discover sparse
distributed representations of the data. This form of feature selection greatly
accelerates learning and makes the algorithm practical on large problems. Experiments show that discriminatively trained mixture models lead to much better
classification than comparably sized models trained by EM.
1
Introduction
Mixture models[11] have been widely applied to problems in classification. In these problems, one must learn a decision rule mapping feature vectors ( ) to class labels ( ) given
labeled examples. Mixture models are typically used to parameterize class-conditional dis
, and then to compute posterior probabilities,
, from Bayes rule.
tributions,
Parameter estimation in these models is handled by an Expectation-Maximization (EM)
algorithm[3],
procedure that monotonically increases the joint log likelihood,
a learning
, summed over training examples (indexed by ). A virtue of this algorithm is that it does not require the setting of learning rates or other tuning parameters.
A weakness of the above approach is that the model parameters are optimized by maximum likelihood estimation, as opposed to a discriminative criterion more closely related
to classification error[14]. In this paper, we derive multiplicative update rules for the parameters
models that directly maximize the discriminative objective function,
of mixture
. This objective function measures the conditional log likelihood that
the training examples are correctly classified. Our update rules retain the main virtues of
the EM algorithm?its guarantee of monotonic improvement, and its absence of tuning
parameters?with the added advantage of optimizing a discriminative cost function. They
also have a simple closed form and appealing intuition. The proof of convergence combines ideas from the EM algorithm[3] and methods for generalized and improved iterative
scaling[2, 4].
The approach in this paper is limited to the classification of nonnegative data, since from
the constraint of nonnegativity emerges an especially simple learning algorithm. This limitation, though, is not too severe. An abundance of interesting data occurs naturally in
this form: for example, the pixel intensities of images, the power spectra of speech, and
the word-document counts of text. Real-valued data can also be coerced into this form by
addition or exponentiation. Thus we believe the algorithm has broad applicability.
2
Mixture models as generative models
Mixture models are typically used
as generative models to parameterize probability distributions over feature vectors . Different mixture models are used to model different
classes of data. The parameterized distributions take the form:
(1)
rows
the
where
of the nonnegative weight
matrix
are constrained to sum to unity,
, and the basis functions are properly normalized distributions, such
that
for all . The model can be interpreted as the latent variable model,
(2)
is used
where the discrete latent variable indicates which mixture component
to gen
erate the observed variable . In this setting, one identifies
and
. The basis functions, usually chosen from the exponential family,
define ?bumps? of high probability in the feature space. A popular choice is the multivariate
Gaussian distribution:
!
3 40"6 5 # 3 7
(3)
" $#&%('*),+.-0/21
1
1
3
"
with means and covariance matrices . Gaussian distributions are extremely versatile,
but not always the most appropriate. For sparse nonnegative data, a more natural choice is
the exponential distribution:
98;: < :>= 5?@BA
CA
<
with parameter vectors . Here, the value of 3
<
indexes the elements of
parameters of these basis functions must be estimated from data.
(4)
and . The
Generative models can be viewed as a prototype method for classification, with the parameters of each mixture component defining a particular basin of attraction in the feature
space. Intuitively, patterns are labeled by the most similar prototype, chosen from among
all possible classes. Formally, unlabeled examples are classified by computing posterior
probabilities from Bayes? rule,
EDGF
H
IH
(5)
where
denote the prior probabilities of each class. Examples are classified by the
label with the highest posterior probability.
An Expectation-Maximization (EM) algorithm can be used to estimate the parameters of
mixture models. The EM algorithm optimizes the joint log likelihood,
(6)
summed over training examples. If basis functions are not shared across different classes,
can be done independently for each class label .
then the parameter estimation for
This has the tremendous advantage of decomposing the original learning problem into several smaller problems. Moreover, for many types of basis functions, the EM updates have a
simple closed form and are guaranteed to improve the joint log likelihood at each iteration.
These properties account for the widespread use of mixture models as generative models.
3
Mixture models as discriminative models
Mixture models can also be viewed as purely discriminative models. In this view, their
purpose is simply to provide a particular way of parameterizing the posterior distribution,
. In this paper, we study posterior distributions of the form:
(7)
.a valid
The right hand side of this equation defines
posterior distribution provided that the
are nonnegative. Note that for this intermixture weights
and basis functions
pretation, the mixture weights and basis functions do not need to satisfy the more stringent
normalization constraints of generative models. We will deliberately exploit this freedom,
an idea that distinguishes our approach from previous work on discriminatively trained
mixture models[6] and hidden Markov models[5, 12]. In particular, the unnormalized basis
functions we use are able to parameterize ?saddles? and ?valleys? in the feature space, as
well as the ?bumps? of normalized basis functions. This makes them more expressive than
their generative counterparts: examples can not only be attracted to prototypes, but also
repelled by opposites.
The posterior distributions in eq. (7) must be further specified by parameterizing the basis
as a function of . We study basis functions of the form
functions
= @
(8)
denotes a real-valued vector and
where
denotes a nonnegative and possibly ?expanded? representation[14] of the original feature vector. The exponential form in eq. (8)
allows us to recover certain generative models as a special case. For example, consider
the multivariate Gaussian distribution in eq. (3). By defining the ?quadratically expanded?
feature vector:
(9)
# '
# # # '
5#
(8) by choosing the parameter
vectors
we can equate the basis functions in eqs. (3) and
to act on in the same way that the means 3 and covariance matrices " act on . The
exponential distributions in eq. (4) can be recovered in a similar way. Such generative
models provide a cheap way to initialize discriminative models for further training.
4
Learning algorithm
Our learning algorithm directly optimizes the performance of the models in eq. (7) as classifiers. The objective function we use for discriminative training is the conditional log
likelihood,
(10)
9
summed over training examples. Let
denote the binary matrix whose th element
denotes whether the th training example belongs to
Then we can write the
the th class., where:
objective function as the difference of two terms,
5
1
5
=
@
E
= @
(11)
(12)
5
The competition between these terms give rise to a scenario of contrastive learning. It is
the subtracted term,
, which distinguishes the conditional log likelihood optimized by
discriminative training from the joint log likelihood optimized by EM.
Our learning algorithm works by alternately updating the mixture weights and the basis
function parameters. Here we simply present the update rules for these parameters; a
derivation and proof of convergence are given in the appendix. It is easiest to write the
basis function updates in terms of the nonnegative parameters
. The updates then take
the simple multiplicative form:
= @A
/
= @A
/
= @BA
5
7
+
:
5 : 7 where
:
: :
(13)
(14)
It is straightforward to compute the gradients in these ratios and show that they are always
nonnegative.
(This is a consequence of the nonnegativity constraint on the feature vectors:
for all
examples and feature components .) Thus, the nonnegativity constraints
on the mixture weights and basis functions are enforced by these multiplicative udpates.
3
The updates have a simple intuition[9] based on balancing opposing terms in the gradient of the conditional log likelihood. In particular, note that the fixed points of this
at stationary
update
rule occur
points of the conditional log likelihood?that
is, where
and
, or equivalently, where
and
. The learning rate is controlled by the ratios
of these gradients and?additionally, for the basis function updates?by the exponent , which measures the sparseness of the training data. The value of is the maximum sum of features
that occurs in the training data. Thus, sparse feature vectors leads to faster learning, a
crucial point to which we will return shortly.
:
5
: :
5
It is worth comparing these multiplicative updates to others in the literature. Jebara and
Pentland[6] derived similar updates for mixture weights, but without emphasizing the special form of eq. (13). Others have investigated multiplicative updates by the method of
exponentiated gradients (EG)[7]. Our updates do not have the same form as EG updates:
in particular, note that the gradients in eqs. (13?14) are not exponentiated. If we use one
basis function per class and an identity matrix for the mixture weights, then the updates reduce to the method of generalized iterative scaling[2] for logistic or multinomial regression
(also known as maximum entropy modeling). More generally, though, our multiplicative
updates can be used to train much more powerful classifiers based on mixture models.
5
Feature selection
As previously mentioned, the learning rate for the basis function parameters is controlled
by the sparseness of the training data. If this data is not intrinsically sparse, then the multiplicative upates in eqs. (13?14) can be impractically slow (just as the method of iterative
pixel image
NMF basis vectors
01-10
11-20
21-30
31-40
NMF feature vector
10
41-50
51-60
5
61-70
71-80
0
20
40
60
80
Figure 1: Left: nonnegative basis vectors for handwritten digits discovered by NMF. Right: sparse
feature vector for a handwritten ?2?. The basis vectors are ordered by their contribution to this image.
scaling). In this case, it is important to discover sparse distributed representations of the
data that encode the same information. On large problems, such representations can accelerate learning by several orders of magnitude.
The search for sparse distributed representations can be viewed as a form of feature selection. We have observed that suitably sparse representations can be discovered by the
method of nonnegative matrix factorization (NMF)[8]. Let the raw nonnegative (and posmatrix , where is its raw dimensibly nonsparse) data be represented by the
sionality and is the number of training examples. Algorithms
a factor for NMF yield
ization
, where is a
nonnegative marix and is a
nonnegative
matrix. In this
factorization, the columns of are interpreted as basis vectors, and the
columns of as coefficients (or new feature vectors). These coefficients are typically very
sparse, because the nonnegative basis vectors can only be added in a constructive way to
approximate the original data.
The effectiveness of NMF is best illustrated by example. We used the method to discover
sparse distributed representations of the MNIST data set of handwritten digits[10]. The
data set has 60000 training and 10000 test examples that were deslanted and cropped to
training data was therefore represented by
form
grayscale pixel images.
The raw
of Fig. 1 shows the
a
matrix, with
and
. The left plot
basis vectors discovered by NMF, each plotted as a
image. Most of these basis
vectors resemble strokes, only a fraction of which are needed to reconstruct any particular
image in the training set. For example, only about twenty basis vectors make an appreciable
contribution to the handwritten ?2? shown in the right plot of Fig. 1. The method of NMF
thus succeeds in discovering a highly sparse representation of the original images.
6
Results
Models were evaluated on the problem of recognizing handwritten digits from the MNIST
data set. From the grayscale pixel images, we generated two
vectors: one
sets ; ofthefeature
by NMF, with nonnegative features and dimensionality
other, by principal
. These
components analysis (PCA), with real-valued features and dimensionality
reduced dimensionality feature vectors were used for both training and testing.
Baseline mixture models for classification were trained by EM algorithms. Gaussian mixture models with diagonal covariance marices were trained on the PCA features, while
exponential mixture models (as in eq. (4)) were trained on the NMF features. The mixture
models were trained for up to 64 iterations of EM, which was sufficient to ensure a high
degree of convergence. Seven baseline classifiers were
trained
set, with
on each
feature
different numbers of mixture components per digit (
). The error
rates of these models, indicated by EM-PCA40 and EM-NMF80, are shown in Table 1.
Half as many PCA features were used as NMF features so as to equalize the number of
fitted parameters in different basis functions.
Mixture models on the NMF features were also trained discriminatively by the multiplicative updates
in
Models with varying numbers of mixture components per
eqs.
(13?14).
) were trained by 1000 iterations of these updates. Again, this was
digit (
sufficient to ensure a high degree of convergence; there was no effort at early stopping.
for randomly selected feaThe models were initialized by setting
and
ture vectors. The results of these experiments, indicated by DT-NMF80, are also shown
in Table 1. The results show that the discriminatively trained models classify much better
than comparably sized models trained by EM. The ability to learn more compact classifiers
appears to be the major advantage of discriminative training. A slight disadvantage is that
the resulting classifiers are more susceptible to overtraining.
model
EM-PCA40
1
2
4
8
16
32
64
10.2
8.5
6.8
5.3
4.0
3.1
1.9
10.1
8.3
6.4
5.1
4.4
3.6
3.1
EM-NMF80
15.7
12.3
9.3
7.8
6.2
5.0
3.9
DT-NMF80
14.7
10.7
8.2
7.0
5.7
5.1
4.2
5.5
4.0
2.8
1.7
1.0
5.8
4.4
3.5
3.2
3.4
Table 1: Classification error rates (%) on the training set (
and the test set (
) for mixture models
with different numbers of mixture components per digit (
the same number of fitted parameters.
). Models in the same row have roughly
It is instructive to compare our results to other benchmarks on this data set[10]. Without
making use of prior knowledge,
better error rates on the test
been obtained by sup
set have
), and fully connected
port vector machines (
neighbor (
), k-nearest
multilayer neural networks (
). These results, however, either required storing
large numbers of training examples or training significantly larger models. For example, the
nearest neighbor and support vector classifiers required storing tens of thousands of training examples (or support vectors), while the neural network had over 120,000
By
weights.
contrast, the
discriminatively trained mixture model (with
) has less
than 6500 iteratively adjusted parameters, and most of its memory footprint is devoted to
preprocessing by NMF.
We conclude by describing the problems best suited to the mixture models in this paper.
These are problems with many classes, large amounts of data, and little prior knowledge
of symmetries or invariances. Support vector machines and nearest neighbor algorithms
do not scale well to this regime, and it remains tedious to train large neural networks with
unspecified learning rates. By contrast, the compactness of our models and the simplicity
of their learning algorithm make them especially attractive.
A
Proof of convergence
In this appendix, we show that the multiplicative updates from section 4 lead to monotonic
improvement in the conditional log likelihood. This guarantee of convergence (to a stationary point) is proved by computing a lower bound on the conditional log likelihood for
updated estimates of the mixture weights and basis function parameters. We indicate these
updated estimates by
and , and we indicate the resulting values of the conditional
, and . The proof of convergence rests
log likelihood and its component terms by ,
on three simple inequalities applied to .
H
H
H
H H
H5
The first term in the conditional log likelihood can be lower bounded by Jensen?s inequality.
The same bound is used here as in the derivation of the EM algorithm[3, 13] for maximum
likelihood estimation:
= @F
F
=
H
H 9
H @
(15)
The right hand side of this inequality introduces an auxiliary probability distribution
bound holds for arbitrary distributions, provided
The
for each example in the training
set.
they are properly normalized:
for all .
The second term in the conditional log likelihood occurs with a minus sign, so for this term
we require an upper bound. The same bounds can be used here as in derivations
ofiterative
scaling[1, 2, 4, 13]. Note that the logarithm function is upper bounded by:
for all
. We can therefore write:
= @F
H =
1 (16)
H5 1 5
To further bound =the
hand side of eq. (16), we make the following observation: though
@F
right
H : ,
H
the exponentials
are convex functions of the parameter vector
with
elements
F
=
= F
@
@BA ,
they are concave functions of the ?warped? parameter vector
with elements
where is defined by eq. (14).
(The validity of this observation hinges on the nonnegativity of the = feature
= F set,= the
F
vectors .) It follows that for any example in the training
@BA ,
@
@BA
is upper bounded by its linearized expansion around
exponential
given by:
.
:
=
F
=
F
=
@
: = @BA 1 = @ A "! = @@BA #
@
(17)
=
= @BA in eq. (17) is the derivative of @
with respect to the
The last term in parentheses
, computed by the chain rule. Tighter bounds are possible than
independent variable
eq. (17), but at the expense of more complicated update rules.
Combining the above inequalities with a judicious choice for the auxiliary parameters
,
we obtain a proof of convergence for the multiplicative updates in eqs. (13?14). Let:
=
=
5 #
@
(18)
=
=
#
5
@
5
(19)
Eq. (18) sets the auxiliary parameters
, while eq. (19) defines an analogous distribu 5
= F
H = @
1
tion
for the opposing term in the conditional log likelihood. (This will prove to be a
useful notation.) Combining these definitions with eqs. (15?17) and rearranging terms, we
obtain the following inequality:
H1
H
/
H
H
5
1
1
1
7
= F 5 @BA
@BA 1
# (20)
Both sides of the inequality vanish (yielding an equality) if
H
and H . We
derive the update rules by maximizing the right hand side of this inequality. Maximizing
the right hand side with respect to
H while holding the basis function parameters fixed
yields the update, eq. (13). Likewise, maximizing the right hand side with respect to H
the mixture weights fixed yields the update, eq. (14). Since these choices
while holding
H
and H lead to positive values on the right hand side of the inequality (except at
for
H :
:
!
fixed points), it follows that the multiplicative updates in eqs. (13?14) lead to monotonic
improvement in the conditional log likelihood.
References
[1] M. Collins, R. Schapire, and Y. Singer (2000). Logistic regression, adaBoost, and Bregman distances. In Proceedings of the Thirteenth Annual Conference on Computational Learning Theory.
[2] J. N. Darroch and D. Ratcliff (1972). Generalized iterative scaling for log-linear models. Annals
of Mathematical Statistics 43:1470?1480.
[3] A. P. Dempster, N. M. Laird, and D. B. Rubin (1977). Maximum likelihood from incomplete
data via the EM algorithm. J. Royal Stat. Soc. B 39: 1?37.
[4] S. Della Pietra, V. Della Pietra, and J. Lafferty (1997). Inducing features of random fields. IEEE
Transactions on Pattern Analysis and Machine Intelligence 19(4): 380?393.
[5] P.S. Gopalakrishnan, D. Kanevsky, A. Ndas and D. Nahamoo (1991). An inequality for rational functions with applications to some statistical estimation problems. IEEE Transactions on
Information Theory 37: 107?113.
[6] T. Jebara and A. Pentland (1998). Maximum conditional likelihood via bound maximization and
the CEM algorithm. In M. Kearns, S. Solla, and D. Cohn (eds.). Advances in Neural Information
Processing Systems 11, 494?500. MIT Press: Cambridge, MA.
[7] J. Kivinen and M. Warmuth (1997). Additive versus exponentiated gradient updates for linear
prediction. Journal of Information and Computation 132: 1?64.
[8] D. D. Lee and H. S. Seung (1999). Learning the parts of objects with nonnegative matrix factorization. Nature 401: 788?791.
[9] D. D. Lee and H. S. Seung (2000). Algorithms for nonnegative matrix factorization. In T. Leen,
T. Dietterich, and V. Tresp (eds.). Advances in Neural Information Processing Systems 13. MIT
Press: Cambridge, MA.
[10] Y.LeCun, L. Jackel, L.Bottou, A.Brunot, C.Cortes, J. Denker, H.Drucker, I.Guyon, U. Muller,
E.Sackinger, P.Simard, and V.Vapnik (1995). A comparison of learning algorithms for handwritten digit recognition. In F.Fogelman and P.Gallinari (eds.). International Conference on Artificial
Neural Networks, 1995, Paris: 53?60.
[11] G. McLachlan and K. Basford (1988). Mixture Models: Inference and Applications to Clustering. Marcel Dekker.
[12] Y. Normandin (1991). Hidden Markov Models, Maximum Mutual Information Estimation and
the Speech Recognition Problem. Ph.D. Thesis, McGill University, Montreal.
[13] J. A. O?Sullivan (1998). Alternating minimization algorithms: from Blahut-Arimoto to
Expectation-Maximization. In A. Vardy (ed.). Codes, Curves, and Signals: Common Threads
in Communications. Kluwer: Norwell, MA.
[14] V. Vapnik (1999). The Nature of Statistical Learning Theory. Springer Verlag.
| 2085 |@word erate:1 suitably:1 tedious:1 dekker:1 linearized:1 covariance:3 contrastive:1 minus:1 versatile:1 daniel:1 document:1 recovered:1 comparing:1 must:3 attracted:1 additive:1 cheap:1 plot:2 update:29 stationary:2 generative:8 discovering:1 half:1 selected:1 intelligence:1 warmuth:1 mathematical:1 prove:1 combine:1 roughly:1 little:1 provided:2 discover:3 moreover:1 bounded:3 notation:1 easiest:1 interpreted:2 unspecified:1 guarantee:3 act:2 concave:1 classifier:7 gallinari:1 positive:1 engineering:1 consequence:1 factorization:5 limited:1 practical:1 lecun:1 testing:1 della:2 footprint:1 sullivan:1 digit:7 procedure:1 significantly:1 word:1 unlabeled:1 selection:3 valley:1 optimize:1 maximizing:3 straightforward:1 independently:1 convex:1 simplicity:1 rule:12 attraction:1 parameterizing:2 analogous:1 updated:2 annals:1 mcgill:1 pa:1 element:4 recognition:2 updating:1 tributions:1 labeled:2 observed:2 electrical:1 parameterize:3 thousand:1 connected:1 solla:1 highest:1 mentioned:1 intuition:2 equalize:1 dempster:1 seung:2 trained:13 purely:1 basis:29 accelerate:1 joint:4 represented:2 derivation:3 train:2 artificial:1 choosing:1 whose:1 widely:1 valued:3 larger:1 reconstruct:1 ability:1 statistic:1 laird:1 advantage:4 combining:2 gen:1 intuitive:1 inducing:1 competition:1 convergence:8 object:1 derive:2 montreal:1 stat:1 nearest:3 eq:22 soc:1 auxiliary:3 resemble:1 indicate:2 marcel:1 closely:1 stringent:1 require:2 tighter:1 adjusted:1 hold:1 around:1 lawrence:1 mapping:1 bump:2 major:1 early:1 purpose:1 estimation:6 label:3 jackel:1 mclachlan:1 minimization:1 mit:2 gaussian:4 always:2 varying:1 encode:1 derived:2 improvement:4 properly:2 likelihood:20 indicates:1 ratcliff:1 greatly:1 contrast:2 baseline:2 inference:1 stopping:1 typically:3 compactness:1 hidden:2 pixel:4 fogelman:1 classification:9 among:1 exponent:1 constrained:1 special:3 summed:3 initialize:1 mutual:1 field:1 broad:1 others:2 distinguishes:2 randomly:1 pietra:2 blahut:1 opposing:2 freedom:1 investigate:1 highly:1 severe:1 weakness:1 introduces:1 mixture:36 yielding:1 devoted:1 chain:1 norwell:1 bregman:1 indexed:1 incomplete:1 logarithm:1 initialized:1 plotted:1 fitted:2 column:2 modeling:1 classify:1 disadvantage:1 retains:1 maximization:5 cost:1 applicability:1 recognizing:1 too:1 international:1 retain:1 lee:3 deslanted:1 again:1 thesis:1 opposed:1 possibly:1 warped:1 derivative:1 simard:1 return:1 account:1 coefficient:2 satisfy:1 multiplicative:13 view:1 tion:1 closed:3 h1:1 sup:1 bayes:2 recover:1 complicated:1 contribution:2 equate:1 likewise:1 yield:3 handwritten:6 raw:3 comparably:2 worth:1 classified:3 stroke:1 overtraining:1 ed:4 definition:1 naturally:1 proof:5 basford:1 rational:1 proved:1 popular:1 intrinsically:1 knowledge:2 emerges:1 dimensionality:3 appears:1 dt:2 adaboost:1 improved:1 leen:1 done:1 though:3 evaluated:1 just:1 hand:7 expressive:1 cohn:1 sackinger:1 widespread:1 defines:2 logistic:2 indicated:2 believe:1 dietterich:1 validity:1 normalized:3 deliberately:1 counterpart:1 equality:1 ization:1 alternating:1 iteratively:1 illustrated:1 eg:2 attractive:1 nonsparse:1 unnormalized:1 criterion:1 generalized:4 image:8 common:1 multinomial:1 arimoto:1 slight:1 kluwer:1 cambridge:2 tuning:3 had:1 posterior:7 multivariate:2 optimizing:2 optimizes:2 belongs:1 scenario:1 certain:1 verlag:1 inequality:9 binary:1 muller:1 maximize:1 monotonically:1 signal:1 reduces:1 faster:1 controlled:3 parenthesis:1 prediction:1 coerced:1 regression:2 multilayer:1 expectation:4 iteration:3 normalization:1 addition:1 cropped:1 thirteenth:1 crucial:1 rest:1 lafferty:1 effectiveness:1 ture:1 kanevsky:1 pennsylvania:1 opposite:1 reduce:1 idea:2 prototype:3 drucker:1 whether:1 thread:1 handled:1 pca:3 darroch:1 effort:1 speech:2 generally:1 useful:1 amount:1 ten:1 ph:1 reduced:1 schapire:1 sign:1 estimated:1 correctly:1 per:4 discrete:1 write:3 fraction:1 sum:2 enforced:1 exponentiation:1 parameterized:1 powerful:1 family:1 guyon:1 decision:1 appendix:2 scaling:6 accelerates:1 bound:8 guaranteed:1 nonnegative:18 annual:1 nahamoo:1 occur:1 constraint:4 extremely:1 expanded:2 department:2 across:1 smaller:1 em:18 unity:1 appealing:1 making:1 brunot:1 intuitively:1 equation:1 previously:1 remains:1 describing:1 count:1 needed:1 singer:1 decomposing:1 denker:1 appropriate:1 subtracted:1 shortly:1 original:4 denotes:3 clustering:1 ensure:2 hinge:1 exploit:1 especially:2 objective:5 added:3 occurs:3 diagonal:1 gradient:6 distance:1 seven:1 gopalakrishnan:1 code:1 index:1 ratio:2 equivalently:1 susceptible:1 holding:2 expense:1 repelled:1 rise:1 ba:10 twenty:1 upper:3 observation:2 markov:2 benchmark:1 pentland:2 defining:2 communication:1 discovered:3 arbitrary:1 jebara:2 intensity:1 nmf:14 upates:1 required:2 specified:1 paris:1 optimized:3 quadratically:1 tremendous:1 alternately:1 able:1 usually:1 pattern:2 regime:1 royal:1 memory:1 power:1 natural:1 kivinen:1 improve:1 identifies:1 philadelphia:1 tresp:1 text:1 prior:3 literature:1 fully:1 discriminatively:5 interesting:1 limitation:1 versus:1 degree:2 basin:1 sufficient:2 rubin:1 port:1 storing:2 balancing:1 row:2 last:1 dis:1 distribu:1 side:8 exponentiated:3 saul:1 neighbor:3 sparse:11 distributed:4 curve:1 valid:1 preprocessing:1 transaction:2 approximate:1 compact:1 cem:1 conclude:1 discriminative:10 spectrum:1 grayscale:2 search:1 iterative:6 latent:2 table:3 additionally:1 learn:2 nature:2 rearranging:1 symmetry:1 expansion:1 investigated:1 bottou:1 main:2 fig:2 slow:1 nonnegativity:4 exponential:7 vanish:1 abundance:1 emphasizing:1 jensen:1 appeal:1 cortes:1 virtue:3 ih:1 mnist:2 vapnik:2 magnitude:1 sparseness:3 suited:1 entropy:1 simply:2 saddle:1 ordered:1 monotonic:4 springer:1 ma:3 conditional:14 sized:2 viewed:3 identity:1 appreciable:1 shared:1 absence:2 judicious:1 except:1 impractically:1 principal:1 kearns:1 invariance:1 succeeds:1 formally:1 support:3 collins:1 constructive:1 h5:2 instructive:1 |
1,191 | 2,086 | Estimating the Reliability of leA
Projections
F. Meinecke l ,2, A. Ziehe l , M. Kawanabe l and K.-R. Miiller l ,2*
1 Fraunhofer FIRST.IDA, Kekuh~str. 7, 12489 Berlin, Germany
2University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany
{meinecke,ziehe,nabe,klaus}?first.fhg.de
Abstract
When applying unsupervised learning techniques like ICA or temporal decorrelation, a key question is whether the discovered projections are reliable. In other words: can we give error bars or can
we assess the quality of our separation? We use resampling methods to tackle these questions and show experimentally that our
proposed variance estimations are strongly correlated to the separation error. We demonstrate that this reliability estimation can
be used to choose the appropriate ICA-model, to enhance significantly the separation performance, and, most important, to mark
the components that have a actual physical meaning. Application
to 49-channel-data from an magneto encephalography (MEG) experiment underlines the usefulness of our approach.
1
Introduction
Blind source separation (BSS) techniques have found wide-spread use in various
application domains , e.g. acoustics , telecommunication or biomedical signal processing. (see e.g. [9, 5, 6, 1, 2, 4, 14, 8]).
BSS is a statistical technique to reveal unknown source signals when only mixtures
of them can be observed. In the following we will only consider linear mixtures; the
goal is then to estimate those projection directions, that recover the source signals.
Many different BSS algorithms have been proposed, but to our knowledge, so far,
no principled attempts have been made to assess the reliability of BSS algorithms,
such that error bars are given along with the resulting projection estimates. This
lack of error bars or means for selecting between competing models is of course a
basic dilemma for most unsupervised learning algorithms. The sources of potential
unreliability of unsupervised algorithms are ubiquous , i.e. noise, non-stationarities,
small sample size or inadequate modeling (e.g. sources are simply dependent instead of independent). Unsupervised projection techniques like PCA or BSS will
always give an answer that is found within their model class, e.g. PCA will supply
an orthogonal basis even if the correct modeling might be non-orthogonal. But how
can we assess such a miss-specification or a large statistical error?
Our approach to this problem is inspired by the large body of statistics literature on
? To whom correspondence should be addressed.
resampling methods (see [12] or [7] for references), where algorithms for assessing
the stability of the solution have been analyzed e.g. for peA [3].
We propose reliability estimates based on bootstrap resampling. This will enable
us to select a good BSS model, in order to improve the separation performance and
to find potentially meaningful projection directions. In the following we will give
an algorithmic description of the resampling methods, accompanied by some theoretical remarks (section 2) and show excellent experimental results (sections 3 and
4). We conclude with a brief discussion.
2
2.1
Resampling Techniques for BSS
The leA Model
In blind source separation we assume that at time instant t each component Xi(t)
of the observed n-dimensional data vector, x(t) is a linear superposition of m ::::: n
statistically independent signals:
m
Xi(t) = LAijSj(t)
j=l
(e.g. [8]). The source signals Sj(t) are unknown, as are the coefficients Aij of the
mixing matrix A. The goal is therefore to estimate both unknowns from a sample
of the x(t), i.e. y(t) = s(t) = Wx(t), where W is called the separating matrix.
Since both A and s(t) are unknown, it is impossible to recover the scaling or the
order of the columns of the mixing matrix A. All that one can get are the projection
directions. The mixing/ demixing process can be described as a change of coordinates. From this point of view the data vector stays the same, but is expressed
in different coordinate systems (passive transformation). Let {ed be the canonical basis of the true sources s = 'E eiSi. Analogous, let {fj} be the basis of the
estimated leA channels: y = 'E fjYj. Using this, we can define a component-wise
separation error Ei as the angle difference between the true direction of the source
and the direction of the respective leA channel:
Ei = arccos
("e~i: ~ifill) .
To calculate this angle difference, remember that component-wise we have Yj
'E WjkAkisi. With Y = s, this leads to: fj = 'E ei(WA)ij1, i.e. fj is the j-th
column of (WA) - l.
In the following, we will illustrate our approach for two different source separation
algorithms (JADE, TDSEP). JADE [4] using higher order statistics is based on
the joint diagonalization of matrices obtained from 'parallel slices' of the fourth
order cumulant tensor. TDSEP [14] relies on second order statistics only, enforcing
temporal decorrelation between channels.
2.2
About Resampling
The objective of resampling techniques is to produce surrogate data sets that
eventually allow to approximate the 'separation error' by a repeated estimation of
the parameters of interest. The underlying mixing should of course be independent
of the generation process of the surrogate data and therefore remain invariant
under resampling.
Bootstrap R esampling
The most popular res amp ling methods are the Jackknife and the Bootstrap
(see e.g. [12, 7]) The Jackknife produces surrogate data sets by just deleting one
datum each time from the original data. There are generalizations of this approach
like k-fold cross-validation which delete more than one datum at a time. A more
general approach is the Bootstrap. Consider a block of, say, N data points. For
obtaining one bootstrap sample, we draw randomly N elements from the original
data, i.e. some data points might occur several times, others don't occur at all in
the bootstrap sample. This defines a series {at} with each at telling how often
the data point x(t) has been drawn. Then, the separating matrix is computed on
the full block and repeatedly on each of the N -element bootstrap samples. The
variance is computed as the squared average difference between the estimate on
the full block and the respective bootstrap unmixings. (These resampling methods
have some desirable properties, which make them very attractive; for example, it
can be shown that for iid data the bootstrap estimators of the distributions of
many commonly used statistics are consistent.) It is straight forward to apply this
procedure to BSS algorithms that do not use time structure; however , only a small
modification is needed to take time structure into account. For example, the time
lagged correlation matrices needed for TDSEP, can be obtained from {ad by
1
Cij(T) = N
N
2: at 'Xi(t)Xj(t+T)
t= l
with
L
at = N and at E {O, 1, 2, ... }.
Other resampling methods
Besides the Bootstrap, there are other res amp ling methods like the Jackknife or cross-validation which can be understood as special cases of Bootstrap.
We have tried k-fold cross-validation, which yielded very similar results to the ones
reported here.
2.3
The Resampling Algorithm
After performing BSS, the estimated ICA-projections are used to generate surrogate data by resampling. On the whitened l surrogate data, the source separation
algorithm is used again to estimate a rotation that separates this surrogate data.
In order to compare different rotation matrices, we use the fact that the matrix
representation of the rotation group SO(N) can be parameterized by
r5~r5t - r5~r5b , where the matrices Mij are generators of the group
with (Mab)ij
and the aij are the rotation parameters (angles) of the rotation matrix R. Using
this parameterization we can easily compare different N-dimensional rotations by
comparing the rotation parameters aij. Since the sources are already separated,
the estimated rotation matrices will be in the vicinity of the identity matrix.2 .
IThe whitening transformation is defined as x' = Vx with V = E[xxTtl/2.
21t is important to perform the resampling when the sources are already separated, so
that the aij are distributed around zero, because SO(N) is a non-Abelian group; that
means that in general R(a)R?(3 ) of- R?(3) R(a) .
Var(aij) measures the instability of the separation with respect to a rotation in
the (i, j)-plane. Since the reliability of a projection is bounded by the maximum
angle variance of all rotations that affect this direction, we define the uncertainty of
the i-th ICA-Projection as Ui := maxj Var(aij). Let us summarize the resampling
algorithm:
1. Estimate the separating matrix W with some ICA algorithm.
Calculate the ICA-Projections y = Wx
2. Produce k surrogate data sets from y and whiten these data sets
3. For each surrogate data set: do BSS, producing a set of rotation matrices
4. Calculate variances of rotation parameters (angles) aij
5. For each ICA component calculate the uncertainty Ui = maxVar(aij).
J
2.4
Asymptotic Considerations for Resampling
Properties of res amp ling methods are typically studied in the limit when the number
of bootstrap samples B -+ 00 and the length of signal T -+ 00 [12]. In our case, as
B -+ 00, the bootstrap variance estimator Ut(B) computed from the aiJ's converge
to Ut(oo) := maxj Varp[aij] where aij denotes the res amp led deviation and F
denotes the distribution generating it. Furthermore, if F -+ F, Ut (00) converges to
the true variance Ui = maxj VarF[aij ] as T -+ 00. This is the case, for example, if
the original signal is i.i.d. in time. When the data has time structure, F does not
necessarily converge to the generating distribution F of the original signal anymore.
Although we cannot neglect this difference completely, it is small enough to use our
scheme for the purposes considered in this paper, e.g. in TDSEP, where the aij
depend on the variation of the time-lagged covariances Cij(T) of the signals, we can
show that their estimators Ctj (T) are unbiased:
Furthermore, we can bound the difference t:.ijkl(T,V) =
COV p
[Ctj ( T), Ckl (v)]
COVF
[Cij(T),Ckl(V)]
between the covariance of the real matrices and their boot-
strap estimators as
if :3a < 1, M ;::: 1, Vi: ICii (T) I :S M aJLICii(O) I. In our experiments, however, the
bias is usually found to be much smaller than this upper bound.
3
3.1
Experiments
Comparing the separation error with the uncertainty estimate
To show the practical applicability of the resampling idea to ICA, the separation
error Ei was compared with the uncertainty Ui . The separation was performed on
different artificial 2D mixtures of speech and music signals and different iid data
sets of the same variance. To achieve different separation qualities, white gaussian
noise of different intensity has been added to the mixtures.
0.7 , - - - - - - - - - - - - - - - - - - - - - - - _
0.6
ur
Uj
~ 0.5
=0.015
? 0.4
~
.
~0.3
c
~ 0.2
U. = 0.177
' - - -' j
0.1
o L---~~~~~~~~--~
o
0.2
0.4
0.6
separation error E j
o L-----~----~----~--~
0.05
0.8
0.15
0.25
0.35
0.45
Figure 1: (a) The probability distribution for the separation error for a small uncertainty
is close to zero, for higher uncertainty it spreads over a larger range. (b) The expected
error increases with the uncertainty.
Figure 1 relates the uncertainty to the separation error for JADE (TDSEP results
look qualitatively the same) . In Fig.1 (left) we see the separation error distribution
which has a strong peak for small values of our uncertainty measure, whereas for
large uncertainties it tends to become flat, i.e. - as also seen from Fig.1 (right) the uncertainty reflects very well the true separation error.
3.2
Selecting the appropriate BSS algorithm
As our variance estimation gives a high correlation to the (true) separation error,
the next logical step is to use it as a model selection criterion for: (a) selecting
some hyperparameter of the BSS algorithm, e.g. choosing the lag values for
TDSEP or (b) choosing between a set of different algorithms that rely on different
assumptions about the data, i.e. higher order statistics (e.g. JADE, INFO MAX,
FastICA, ... ) or second order statistics (e.g. TDSEP). It could, in principle, be
much better to extract the first component with one and the next with another
assumption/ algorithm. To illustrate the usefulness of our reliability measure, we
study a five-channel mixture of two channels of pure white gaussian noise, two audio
signals and one channel of uniformly distributed noise. The reliability analysis for
higher order statistics (JADE)
0.3
0.25
0.25
~- 0.2
E
0.15
:rg
0.1
::J
g
::J
TDSEP 3
9.17.10- 5
~- 0.2
E
:rg
temporal decorrelation (TDSEP)
0.3
0.15
g
0.05
TDSEP 4
1.29.10-5
,----
,----
,---0 .1
0.05
3
ICA Channel i
3
ICA Channel i
Figure 2: Uncertainty of leA projections of an artificial mixture using JADE and TDSEP.
Resampling displays the strengths and weaknesses of the different models
JADE gives the advice to rely only on channels 3,4,5 (d. Fig.2 left). In fact , these
are the channels that contain the audio signals and the uniformly distributed noise.
The same analysis applied to the TDSEP-projections (time lag = 0, ... ,20) shows,
that TDSEP can give reliable estimates only for the two audio sources (which is
to be expected; d. Fig.2 right). According to our measure, the estimation for the
audio sources is more reliable in the TDSEP-case. Calculation of the separation
error verifies this: TDSEP separates better by about 3 orders of magnitude (JADE:
E3 = 1.5 . 10- 1 , E4 = 1.4 . 10- 1 , TDSEP: E 3 = 1.2 . 10- 4 , E4 = 8.7? 10- 5 ). Finally,
in our example, estimating the audio sources with TDSEP and after this applying
JADE to the orthogonal subspace, gives the optimal solution since it combines the
small separation errors E 3, E4 for TDSEP with the ability of JADE to separate
the uniformly distributed noise.
3.3
Blockwise uncertainty estimates
For a longer time series it is not only important to know which ICA channels are
reliable, but also to know whether different parts of a given time series are more
(or less) reliable to separate than others. To demonstrate these effects, we mixed
two audio sources (8kHz, lOs - 80000 data points) , where the mixtures are partly
corrupted by white gaussian noise. Reliability analysis is performed on windows of
length 1000, shifted in steps of 250; the resulting variance estimates are smoothed.
Fig.3 shows again that the uncertainty measure is nicely correlated with the true
separation error, furthermore the variance goes systematically up within the noisy
part but also in other parts of the time series that do not seem to match the
assumptions underlying the algorithm. 3 So our reliability estimates can eventually
Figure 3: Upper panel: mixtures, partly corrupted by noise. Lower panel: the blockwise
variance estimate (solid line) vs the true separation error on this block (dotted line) .
be used to improve separation performance by removing all but the 'reliable' parts
of the time series. For our example this reduces the overall separation error by 2
orders of magnitude from 2.4.10- 2 to 1.7.10- 4 .
This moving-window resampling can detect instabilities of the projections in two
different ways: Besides the resampling variance that can be calculated for each
window, one can also calculate the change of the projection directions between two
windows. The later has already been used successfully by Makeig et. al. [10].
4
Assigning Meaning: Application to Biomedical Data
We now apply our reliability analysis to biomedical data that has been produced
by an MEG experiment with acoustic stimulation. The stimulation was achieved
by presenting alternating periods of music and silence, each of 30s length, to the
subjects right ear during 30 min. of total recording time (for details see [13]). The
measured DC magnetic field values, sampled at a frequency of 0.4 Hz , gave a total number of 720 sample points for each of the 49 channels. While previously
3For example, the peak in the last third of the time series can be traced back to the
fact that the original time series are correlated in this region.
[13] analysing the data, we found that many of the ICA components are seemingly
meaningless and it took some medical knowledge to find potential meaningful projections for a later close inspection. However, our reliability assessment can also
be seen as indication for meaningful projections, i.e. meaningful components should
have low variance. In the experiment, BSS was performed on the 23 most powerful
principal components using (a) higher order statistics (JADE) and (b) temporal
decorrelation (TDSEP, time lag 0 .. 50). The results in Fig.4 show that none of
higher order statistics (JADE)
temporal decorrelation (TDSEP)
0.35
0.35
0.3
0.3
0.25
0 .25
-
::J
::J
~ 0.2
~ 0.2
ig
~
g0. 15
0.1 5
::J
::J
0.1
0 .1
0.05
0.05
10
15
leA-Channel i
,~
20
10
15
leA-Channel i
20
Figure 4: Resampling on the biomedical data from MEG experiment shows: (a) no JADE
projection is reliable (has low uncertainty) (b) TDSEP is able to identify three sources
with low uncertainty.
the JADE-projections (left) have small variance whereas TDSEP (right) identifies
three sources with a good reliability. In fact , these three components have physical
meaning: while component 23 is an internal very low frequency signal (drift) that
is always present in DC-measurements, component 22 turns out to be an artifact of
the measurement; interestingly component 6 shows a (noisy) rectangular waveform
that clearly displays the 1/308 on/off characteristics of the stimulus (correlation to
stimulus 0.7; see Fig.5) . The clear dipole-structure of the spatial field pattern in
0.5
~
~O
In
~
stimulUS
-0.5
1
234
5
6
7
t[min)
Figure 5: Spatial field pattern, frequency content and time course of TDSEP channel 6.
Fig.5 underlines the relevance of this projection. The components found by JADE
do not show such a clear structure and the strongest correlation of any component
to the stimulus is about 0.3, which is of the same order of magnitude as the strongest
correlated PCA-component before applying JADE.
5
Discussion
We proposed a simple method to estimate the reliability of ICA projections based on
res amp ling techniques. After showing that our technique approximates the separation error, several directions are open(ed) for applications. First, we may like to use
it for model selection purposes to distinguish between algorithms or to chose appropriate hyperparameter values (possibly even component-wise). Second, variances
can be estimated on blocks of data and separation performance can be enhanced
by using only low variance blocks where the model matches the data nicely. Finally
reliability estimates can be used to find meaningful components. Here our assumption is that the more meaningful a component is, the more stably we should be able
to estimate it. In this sense artifacts appear of course also as meaningful, whereas
noisy directions are discarded easily, due to their high uncertainty.
Future research will focus on applying res amp ling techniques to other unsupervised
learning scenarios. We will also consider Bayesian modelings where often a variance
estimate comes for free, along with the trained model.
Acknowledgments K-R.M thanks Guido Nolte and the members of the Oberwolfach
Seminar September 2000 in particular Lutz Dumbgen and Enno Mammen for helpful
discussions and suggestions. K -R. M and A. Z. acknowledge partial funding by the EU
project (IST-1999-14190 - BLISS). We thank the Biomagnetism Group of the PhysikalischTechnische Bundesanstalt (PTB) for providing the MEG-DC data.
References
[1] S. Amari, A. Cichocki, and H. H. Yang. A new learning algorithm for blind signal
separation. In D .S. Touretzky, M.C. Mozer, and M.E . Hasselmo , editors, Advances
in Neural Information Processing Systems (NIPS 95), volume 8, pages 882-893. The
MIT Press, 1996.
[2] A. J. Bell and T. J. Sejnowski. An information maximisation approach to blind
separation and blind deconvolution. N eural Computation, 7:1129- 1159, 1995.
[3] R. Beran and M.S. Srivastava. Bootstrap tests and confidence regions for functions
of a covariance matrix. Annals of Statistics, 13:95- 115, 1985.
[4] J.-F. Cardoso and A. Souloumiac. Blind beamforming for non Gaussian signals. IEEE
Proceedings-F, 140(6):362- 370, December 1994.
[5] P. Comon. Independent component analysis, a new concept ?
36(3):287-314, 1994.
Signal Processing,
[6] G. Deco and D. Obradovic. An information-theoretic approach to neural computing.
Springer, New York, 1996.
[7] B. Efron and R.J. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall,
first edition, 1993.
[8] A. Hyviirinen, J. Karhunen , and E. Oja. Independent Compon ent Analysis. Wiley,
200l.
[9] Ch. Jutten and J. Herault. Blind separation of sources, part I: An adaptive algorithm
based on neuromimetic architecture . Signal Processing, 24:1- 10, 1991.
[10] S. Makeig, S. Enghoff, T.-P. Jung, and T. Sejnowski. Moving-window ICA decomposition of EEG data reveals event-related changes in oscillatory brain activity. In Proc.
2nd Int. Workshop on Independent Component Analysis and Blind Source Separation
(ICA '2000), pages 627- 632 , Helsinki, Finland, 2000.
[11] F . Meinecke, A. Ziehe, M. Kawanabe, and K-R. Muller. Assessing reliability of ica
projections - a resampling approach. In ICA '01. T.-W. Lee, Ed., 200l.
[12] J. Shao and D. Th. The Jackknife and Bootstrap. Springer, New York, 1995.
[13] G. Wubbeler, A. Ziehe, B.-M. Mackert, K-R. Muller, L. Trahms, and G. Curio.
Independent component analysis of non-invasively recorded cortical magnetic dc-fields
in humans. IEEE Transactions on Biomedical Engineering, 47(5):594-599 , 2000.
[14] A. Ziehe and K-R. Muller. TDSEP - an efficient algorithm for blind separation using
time structure. In L. Niklasson , M. Boden, and T. Ziemke, editors , Proc. Int. Conf.
on Artificial Neural N etworks (ICANN'9S) , pages 675 - 680 , Skiivde, Sweden, 1998.
Springer Verlag.
| 2086 |@word underline:2 nd:1 open:1 tried:1 covariance:3 decomposition:1 solid:1 series:7 selecting:3 r5t:1 interestingly:1 amp:6 ida:1 comparing:2 assigning:1 ij1:1 wx:2 resampling:21 v:1 parameterization:1 plane:1 inspection:1 five:1 along:2 become:1 supply:1 ziemke:1 combine:1 expected:2 ica:17 brain:1 ptb:1 inspired:1 actual:1 str:1 window:5 project:1 estimating:2 underlying:2 bounded:1 panel:2 transformation:2 temporal:5 remember:1 stationarities:1 tackle:1 makeig:2 medical:1 unreliability:1 producing:1 appear:1 before:1 understood:1 engineering:1 tends:1 limit:1 might:2 chose:1 studied:1 range:1 statistically:1 practical:1 acknowledgment:1 yj:1 maximisation:1 block:6 bootstrap:16 procedure:1 bell:1 significantly:1 projection:22 word:1 confidence:1 get:1 cannot:1 close:2 selection:2 applying:4 impossible:1 instability:2 go:1 rectangular:1 pure:1 dipole:1 estimator:4 stability:1 coordinate:2 variation:1 analogous:1 annals:1 enhanced:1 guido:1 element:2 observed:2 calculate:5 region:2 eu:1 principled:1 mozer:1 ui:4 trained:1 depend:1 ithe:1 dilemma:1 icii:1 basis:3 completely:1 shao:1 easily:2 joint:1 various:1 separated:2 sejnowski:2 artificial:3 klaus:1 jade:16 choosing:2 lag:3 larger:1 say:1 amari:1 ability:1 statistic:10 cov:1 noisy:3 seemingly:1 indication:1 took:1 propose:1 ckl:2 mixing:4 achieve:1 description:1 los:1 ent:1 tdsep:24 assessing:2 produce:3 generating:2 converges:1 illustrate:2 oo:1 measured:1 ij:1 strong:1 come:1 direction:9 waveform:1 correct:1 pea:1 vx:1 human:1 enable:1 generalization:1 mab:1 around:1 considered:1 hall:1 algorithmic:1 finland:1 enno:1 purpose:2 estimation:5 proc:2 superposition:1 hasselmo:1 eisi:1 successfully:1 reflects:1 mit:1 clearly:1 always:2 gaussian:4 ctj:2 focus:1 obradovic:1 am:1 detect:1 sense:1 helpful:1 dependent:1 typically:1 fhg:1 germany:2 overall:1 herault:1 arccos:1 spatial:2 special:1 neuen:1 field:4 nicely:2 chapman:1 r5:2 look:1 unsupervised:5 future:1 others:2 stimulus:4 randomly:1 oja:1 maxj:3 attempt:1 interest:1 weakness:1 mixture:8 analyzed:1 partial:1 respective:2 sweden:1 orthogonal:3 re:6 theoretical:1 delete:1 column:2 modeling:3 applicability:1 deviation:1 usefulness:2 fastica:1 inadequate:1 reported:1 answer:1 corrupted:2 thanks:1 peak:2 stay:1 meinecke:3 off:1 lee:1 enhance:1 squared:1 again:2 ear:1 deco:1 recorded:1 choose:1 possibly:1 conf:1 account:1 potential:2 de:1 accompanied:1 coefficient:1 bliss:1 int:2 blind:9 ad:1 vi:1 performed:3 view:1 later:2 recover:2 parallel:1 encephalography:1 ass:3 variance:17 characteristic:1 identify:1 bayesian:1 ijkl:1 produced:1 iid:2 none:1 straight:1 oscillatory:1 strongest:2 touretzky:1 ed:3 frequency:3 sampled:1 popular:1 logical:1 knowledge:2 ut:3 efron:1 back:1 higher:6 varf:1 strongly:1 furthermore:3 just:1 biomedical:5 varp:1 correlation:4 ei:4 assessment:1 lack:1 defines:1 jutten:1 stably:1 artifact:2 reveal:1 quality:2 effect:1 contain:1 true:7 unbiased:1 concept:1 vicinity:1 alternating:1 white:3 attractive:1 during:1 mammen:1 whiten:1 oberwolfach:1 criterion:1 palais:1 presenting:1 theoretic:1 demonstrate:2 passive:1 fj:3 meaning:3 wise:3 consideration:1 funding:1 rotation:12 niklasson:1 stimulation:2 physical:2 khz:1 volume:1 approximates:1 measurement:2 reliability:15 moving:2 specification:1 longer:1 whitening:1 scenario:1 verlag:1 muller:3 seen:2 converge:2 period:1 signal:17 relates:1 full:2 desirable:1 enghoff:1 reduces:1 match:2 calculation:1 cross:3 basic:1 whitened:1 achieved:1 lea:7 whereas:3 addressed:1 source:21 meaningless:1 subject:1 recording:1 hz:1 member:1 beamforming:1 december:1 seem:1 yang:1 enough:1 xj:1 affect:1 gave:1 nolte:1 competing:1 nabe:1 architecture:1 idea:1 whether:2 pca:3 miiller:1 speech:1 e3:1 york:2 remark:1 repeatedly:1 clear:2 cardoso:1 generate:1 canonical:1 shifted:1 dotted:1 estimated:4 tibshirani:1 hyperparameter:2 group:4 key:1 ist:1 traced:1 drawn:1 angle:5 parameterized:1 fourth:1 telecommunication:1 uncertainty:17 powerful:1 separation:34 draw:1 scaling:1 bound:2 datum:2 display:2 correspondence:1 fold:2 distinguish:1 yielded:1 activity:1 strength:1 occur:2 helsinki:1 flat:1 min:2 performing:1 jackknife:4 according:1 remain:1 smaller:1 ur:1 modification:1 comon:1 invariant:1 covf:1 previously:1 etworks:1 turn:1 eventually:2 needed:2 know:2 neuromimetic:1 apply:2 kawanabe:2 appropriate:3 magnetic:2 anymore:1 original:5 denotes:2 instant:1 neglect:1 music:2 uj:1 tensor:1 objective:1 g0:1 question:2 already:3 added:1 surrogate:8 september:1 subspace:1 separate:4 thank:1 berlin:1 separating:3 whom:1 enforcing:1 meg:4 besides:2 length:3 providing:1 cij:3 abelian:1 potentially:1 blockwise:2 info:1 lagged:2 unknown:4 perform:1 upper:2 boot:1 discarded:1 acknowledge:1 dc:4 discovered:1 smoothed:1 intensity:1 drift:1 acoustic:2 potsdam:2 nip:1 able:2 bar:3 usually:1 pattern:2 summarize:1 reliable:7 max:1 deleting:1 event:1 decorrelation:5 rely:2 scheme:1 improve:2 brief:1 identifies:1 fraunhofer:1 extract:1 cichocki:1 lutz:1 literature:1 asymptotic:1 mixed:1 generation:1 suggestion:1 var:2 generator:1 validation:3 consistent:1 principle:1 editor:2 systematically:1 beran:1 course:4 jung:1 last:1 free:1 aij:13 bias:1 allow:1 silence:1 telling:1 wide:1 distributed:4 slice:1 bs:13 calculated:1 souloumiac:1 cortical:1 forward:1 made:1 commonly:1 qualitatively:1 ig:1 adaptive:1 far:1 transaction:1 sj:1 approximate:1 reveals:1 conclude:1 xi:3 don:1 hyviirinen:1 channel:16 correlated:4 obtaining:1 eeg:1 boden:1 excellent:1 necessarily:1 domain:1 icann:1 spread:2 r5b:1 noise:8 ling:5 edition:1 verifies:1 repeated:1 body:1 eural:1 fig:8 advice:1 wiley:1 seminar:1 third:1 e4:3 removing:1 showing:1 invasively:1 demixing:1 deconvolution:1 workshop:1 curio:1 diagonalization:1 magnitude:3 karhunen:1 rg:2 led:1 simply:1 expressed:1 springer:3 mij:1 ch:1 relies:1 goal:2 identity:1 content:1 experimentally:1 change:3 analysing:1 uniformly:3 miss:1 principal:1 strap:1 called:1 total:2 partly:2 experimental:1 meaningful:7 select:1 ziehe:5 internal:1 mark:1 cumulant:1 relevance:1 audio:6 srivastava:1 |
1,192 | 2,087 | Face Recognition Using Kernel Methods
Ming-Hsuan Yang
Honda Fundamental Research Labs
Mountain View, CA 94041
myang@hra.com
Abstract
Principal Component Analysis and Fisher Linear Discriminant
methods have demonstrated their success in face detection, recognition, and tracking. The representation in these subspace methods
is based on second order statistics of the image set, and does not
address higher order statistical dependencies such as the relationships among three or more pixels. Recently Higher Order Statistics
and Independent Component Analysis (ICA) have been used as informative low dimensional representations for visual recognition.
In this paper, we investigate the use of Kernel Principal Component Analysis and Kernel Fisher Linear Discriminant for learning
low dimensional representations for face recognition, which we call
Kernel Eigenface and Kernel Fisherface methods. While Eigenface
and Fisherface methods aim to find projection directions based on
the second order correlation of samples, Kernel Eigenface and Kernel Fisherface methods provide generalizations which take higher
order correlations into account. We compare the performance of
kernel methods with Eigenface, Fisherface and ICA-based methods for face recognition with variation in pose, scale, lighting and
expression. Experimental results show that kernel methods provide better representations and achieve lower error rates for face
recognition.
1
Motivation and Approach
Subspace methods have been applied successfully in numerous visual recognition
tasks such as face localization, face recognition, 3D object recognition, and tracking.
In particular, Principal Component Analysis (PCA) [20] [13] ,and Fisher Linear Discriminant (FLD) methods [6] have been applied to face recognition with impressive
results. While PCA aims to extract a subspace in which the variance is maximized
(or the reconstruction error is minimized), some unwanted variations (due to lighting, facial expressions, viewing points, etc.) may be retained (See [8] for examples).
It has been observed that in face recognition the variations between the images of
the same face due to illumination and viewing direction are almost always larger
than image variations due to the changes in face identity [1]. Therefore, while the
PCA projections are optimal in a correlation sense (or for reconstruction" from a
low dimensional subspace), these eigenvectors or bases may be suboptimal from the
classification viewpoint.
Representations of Eigenface [20] (based on PCA) and Fisherface [6] (based on FLD)
methods encode the pattern information based on the second order dependencies,
i.e., pixelwise covariance among the pixels, and are insensitive to the dependencies
among multiple (more than two) pixels in the samples. Higher order dependencies
in an image include nonlinear relations among the pixel intensity values, such as
the relationships among three or more pixels in an edge or a curve, which can capture important information for recognition. Several researchers have conjectured
that higher order statistics may be crucial to better represent complex patterns.
Recently, Higher Order Statistics (HOS) have been applied to visual learning problems. Rajagopalan et ale use HOS of the images of a target object to get a better
approximation of an unknown distribution. Experiments on face detection [16] and
vehicle detection [15] show comparable, if no better, results than other PCA-based
methods.
The concept of Independent Component Analysis (ICA) maximizes the degree
of statistical independence of output variables using contrast functions such as
Kullback-Leibler divergence, negentropy, and cumulants [9] [10]. A neural network algorithm to carry out ICA was proposed by Bell and Sejnowski [7], and was
applied to face recognition [3]. Although the idea of computing higher order moments in the ICA-based face recognition method is attractive, the assumption that
the face images comprise of a set of independent basis images (or factorial codes)
is not intuitively clear. In [3] Bartlett et ale showed that ICA representation outperform PCA representation in face recognition using a subset of frontal FERET
face images. However, Moghaddam recently showed that ICA representation does
not provide significant advantage over PCA [12]. The experimental results suggest
that seeking non-Gaussian and independent components may not necessarily yield
better representation for face recognition.
In [18], Sch6lkopf et ale extended the conventional PCA to Kernel Principal Component Analysis (KPCA). Empirical results on digit recognition using MNIST data
set and object recognition using a database of rendered chair images showed that
Kernel PCA is able to extract nonlinear features and thus provided better recognition results. Recently Baudat and Anouar, Roth and Steinhage, and Mika et
ale applied kernel tricks to FLD and proposed Kernel Fisher Linear Discriminant
(KFLD) method [11] [17] [5]. Their experiments showed that KFLD is able to extract the most discriminant features in the feature space, which is equivalent to
extracting the most discriminant nonlinear features in the original input space.
In this paper we seek a method that not only extracts higher order statistics of
samples as features, but also maximizes the class separation when we project these
features to a lower dimensional space for efficient recognition. Since much of the
important information may be contained in the high order dependences among
the pixels of a: face image, we investigate the use of Kernel PCA and Kernel FLD
for face recognition, which we call Kernel Eigenface and Kernel Fisherface methods,
and compare their performance against the standard Eigenface, Fisherface and ICA
methods. In the meanwhile, we explain why kernel methods are suitable for visual
recognition tasks such as face recognition.
2
Kernel Principal Component Analysis
Given a set of m centered (zero mean, unit variance) samples Xk, Xk ==
[Xkl, ... ,Xkn]T ERn, PCA aims to find the projection directions that maximize
the variance, C, which is equivalent to finding the eigenvalues from the covariance
matrix
AW=CW
(1)
for eigenvalues A ~ 0 and eigenvectors W E Rn. In Kernel PCA, each vector x is
projected from the input space, Rn, to a high dimensional feature space, Rf, by a
nonlinear mapping function: <t> : Rn -+ Rf, f ~ n. Note that the dimensionality
of the feature space can be arbitrarily large. In Rf, the corresponding eigenvalue
problem is
"AW4> = C4>w4>
(2)
where C4> is a covariance matrix. All solutions weI> with A =I- 0 lie in the span of
<t> (x1), ..., <t> (Xm ), and there exist coefficients ai such that
m
w4> =
E ai<t>(xi)
(3)
i=l
Denoting an m x m matrix K by
K??
x?) -- <t>(x?)?
<t>(x?)
~1 - k(x?~'1
~
1
(4)
, the Kernel PCA problem becomes
mAKa =K2 a
(5)
(6)
mAa =Ka
where a denotes a column vector with entries aI, ... , am. The above derivations
assume that all the projected samples <t>(x) are centered in Rf. See [18] for a ~ethod
to center the vectors <t>(x) in Rf.
Note that conventional PCA is a special case of Kernel PCA with polynomial kernel
of first order. In other words, Kernel PCA is a generalization of conventional PCA
since different kernels can be utilized for different nonlinear projections.
We can now project the vectors in Rf to a lower dimensional space spanned by the
eigenvectors weI>, Let x be a test sample whose projection is <t>(x) in Rf, then the
projection of <t>(x) onto the eigenvectors weI> is the nonlinear principal components
corresponding to <t>:
m
w4> . <t>(x)
= E ai (<t> (Xi) . <t>(x))
m
=
i=l
E aik(xi, x)
(7)
i=l
In other words, we can extract the first q (1 ~ q ~ m) nonlinear principal components (Le., eigenvectors w4? using the kernel function without the expensive
operation that explicitly projects the samples to a high dimensional space Rf" The
first q components correspond to the first q non-increasing eigenvalues of (6). For
face recognition where each x encodes a face image, we call the extracted nonlinear
principal components Kernel Eigenfaces.
3
Kernel Fisher Linear Discriminant
Similar to the derivations in Kernel PCA, we assume the projected samples <t>(x)
are centered in Rf (See [18] for a method to center the vectors <t>(x) in Rf), we
formulate the equations in a way that use dot products for FLD only. Denoting the
within-class and between-class scatter matrices by S~ and SiJ, and applying FLD
in kernel space, we need to find eigenvalues A and eigenvectors weI> of
AS~WeI> = siJweI>
(8)
, which can be obtained by
<P
WOPT
I(W<P)T S~W<P I
= argw;x I(Wq,)TS~Wq,1 =
[<P
Wl
<P
W2
...
w;.]
(9)
where {w[ Ii == 1, 2, ... ,m} is the set of generalized eigenvectors corresponding to
the m largest generalized eigenvalues {Ai Ii == 1,2, ... ,m}.
For given classes t and u and their samples, we define the kernel function by
Let K be a m x m matrix defined by the elements (Ktu)~1;:::,cc, where K tu is a
matrix composed of dot products in the feature space Rf, Le.,
K == (Ktu )=l,
u=l, ,c,c where K tu == (k rs )r=l,
s=l,
,lt
,I'U
(11)
Note K tu is a It x Iu matrix, and K is a m x m symmetric matrix. We also define
a matrix Z:
(12)
where (Zt) is a It x It matrix with terms all equal to ~, Le., Z is a m x m block
diagonal matrix. The between-class and within-class scatter matrices in a high
dimensional feature space Rf are defined as
c
siJ == L
liJ.ti (p/f)T
(13)
ep(Xij )~(Xij)T
(14)
i=l
C
Ii
i=l
j=l
LL
S~ ==
where pi is the mean of class i in Rf, Ii is the number of samples belonging to class
i. From the theory of reproducing kernels, any solution w<P E Rf must lie in the
span of all training samples in Rf, Le.,
c
w<P
==
Ip
LL
cy'pqep(xpq )
(15)
p=lq=l
It follows that we can get the solution for (15) by solving:
AKKa==KZKa
(16)
Consequently, we can write (9) as
<P
I(WifJ)T sifJwifJl
WOPT == argmaxwifJ I(WifJ)TS!WifJ I
== argmaxw?p
== [wi ... w~]
laKZKal
laKKal
(17)
We can project ~(x) to a lower dimensional space spanned by the eigenvectors w<P
in a way similar to Kernel PCA (See Section 2). Adopting the same technique in
the Fisherface method (which avoids singularity problems in computing W6PT) for
face recognition [6], we call the extracted eigenvectors in (17) Kernel Fisherfaces.
4
Experiments
We test both kernel methods against standard rCA, Eigenface, and Fisherface methods using the publicly available AT&T and Yale databases. The face images in
these databases have several unique characteristics. While the images in the AT&T
database contain the facial contours and vary in pose as well scale, the face images
in the Yale database have been cropped and aligned. The face images in the AT&T
database were taken under well controlled lighting conditions whereas the images
in the Yale database were acquired under varying lighting conditions. We use the
first database as a baseline study and then use the second one to evaluate face
recognition methods under varying lighting conditions.
4.1
Variation in Pose and Scale
The AT&T (formerly Olivetti) database contains 400 images of 40 subjects. To
.reduce computational complexity, each face image is downsampled to 23 x 28 pixels. We represent each image by a raster scan vector of the intensity values, .and
then normalize them to be zero-mean vectors. The mean and standard deviation
of Kurtosis of the face images are 2.08 and 0.41, respectively (the Kurtosis of a
Gaussian distribution is 3). Figure 1 shows images of two subjects. In contrast to
images of the Yale database, the images include the facial contours, and variation
in pose as well as scale. However, the lighting conditions remain constant.
Fig~re 1:
Face images in the AT&T database (Left) and the Yale database (Right).
The experiments are performed using the "leave-one-out" strategy: To classify an
image of person, that image is removed from the training set of (m - 1) images and
the projection matrix is computed. All the m images in the training set are projected
to a reduced space using the computed projection matrix w or weI> and recognition
is performed based on a nearest neighbor classifier. The number of principal components or independent components are empirically determined to achieve the lowest
error rate by each method. Figure 2 shows the experimental results. Among all the
methods, the Kernel Fisherface method with Gaussian kernel and second degree
polynomial kernel achieve the lowest error rate. Furthermore, the kernel methods
perform better than standard rCA, Eigenface and Fisherface methods. Though our
experiments using rCA seem to contradict to the good empirical results reported in
[3] [4] [2]' a close look at the data sets reveals a significant difference in pose and
scale variation of the face images in the AT&T database, whereas a subset of frontal
FERET face images with change of expression was used in [3] [2]. Furthermore, the
comparative study on classification with respect to PCA in [4] (pp. 819, Table 1)
and the errors made by two rCA algorithms in [2] (pp. 50, Figure 2.18) seem to
suggest that lCA methods do not have clear advantage over other approaches in
recognizing faces with pose and scale variation.
4.2
Variation in Lighting and Expression
The Yale database contains 165 images of 11 subjects that includes variation in
both facial expression and lighting. For computational efficiency, each image has
been downsampled to 29 x 41 pixels. Likewise, each face image is represented by a
Method
I rCA
Eigenface
Fisherface
Kernel Eigenface, d==2
Kernel Eigenface, d==3
Kernel Fisherface (P)
Kernel Fisherface (G)
30
14
50
50
14
14
2.75 (11/400)
1.50 (6/400)
2.50 (10/400)
2.00 (8/400)
1.25 (5/400)
1.25 (5/400)
Figure 2: Experimental results on AT&T database.
centered vector of normalized intensity values. The mean and standard deviation
of Kurtosis of the face images are 2.68 and 1.49, respectively. Figure 1 shows 22
closely cropped images of two subjects which include internal facial structures such
as the eyebrow, eyes, nose, mouth and chin, but do not contain the facial contours.
Using the same leave-one-out strategy, we experiment with the number of principal components and independent components to achieve the lowest error rates for
Eigenface and Kernel Eigenface methods. For Fisherface and Kernel Fisherface
methods, we project all the samples onto a subspace spanned by the c - 1 largest
eigenvectors. The experimental results are shown in Figure 3. Both kernel methods
perform better than standard ICA, Eigenface and Fisherface methods. Notice that
the improvement by the kernel methods are rather significant (more than 10%). Notice also that kernel methods consistently perform better than conventional methods
for both databases. The performance achieved by the ICA method indicates that
face representation using independent sources is not effective when the images are
taken under varying lighting conditions.
Method
35
30
29.09
28..49
27.27
24.24
~ 25
I lCA
~ 20
~
8 15
~ 10
o.
-<
~
<
ul:l-.
~~
~s
-<,-..
~&
Q
....:l
~
Q,-..
~&
?
Q
S
~
Eigenface
Fisherface
Kernel Eigenface, d==2
Kernel Eigenface, d==3
Kernel Fisherface (P)
Kernel Fisherface (G)
30
14
80
60
14
14
28.48 (47/165)
8.48 (14/165)
27.27 (45/165)
24.24 (40/165)
6.67 (11/165)
6.06 (10/165)
Figure 3: Experimental results on Yale database.
Figure 4 shows the training samples of the Yale database projected onto the first two
eigenvectors extracted by the Kernel Eigenface and Kernel Fisherface methods. The
projected samples of different classes are smeared by the Kernel Eigenface method
whereas the samples projected by the Kernel Fisherface are separated quite welL
In fact, the samples belonging to the same class are projected to the same position
by the largest two eigenvectors. This example provides an explanation to the good
results achieved by the Kernel Fisherface method.
The experimental results show that Kernel Eigenface and Fisherface methods are
able to extract nonlinear features and achieve lower error rate. Instead of using a
nearest neighbor classifier, the performance can potentially be improved by other
classifiers (e.g., k-nearest neighbor and perceptron). Another potential improvement
is to use all the extracted nonlinear components as features (Le., without projecting
to a lower dimensional space) and use a linear Support Vector Machine (SVM)
to construct a decision surface. Such a two-stage approach is, in spirit, similar
to nonlinear SVMs in which the samples are first projected to a high dimensional
feature space where a hyperplane with largest hyperplane is constructed. In fact,
one important factor of the recent success in SVM applications for visual recognition
is due to the use of kernel methods.
r1-o+.
?
+
1-
:..
.?it'CIlIl''''''~
IX'';:)
:
;
?
?
~
:.'"
?
:.u??.. ?
;.,,? .. ? .. ?
~ -21-
~
,
~
:
,
,
?,??? .. ???1
~
~ ~:::: ~
1-
,
;.
:
:
;
?, .. ??
y
V
A
1iiI=-*$*_~"
1-
<l
:*
:
:
;
:
:
;
:
:
;
:
:
:
;
?..
c1ass4
0
class?
class8
class9
I-
;
0
;:
class13
1-
,
,
."
01-
,
:
-:
E
? ~i:~~H
1
?
I-
*
1
class 1
~::::~
:
1- .......... :? ..~
...... ?~~:-~"'-e>"0~O?"'....???:-?-?.-(I?? .. ? .. ;??????O?.?? .. ; .... ??????-:?? ..........:.. ?? ...... ?1
~~~~::~::~~~ ~
~
:
1>
~
:
,
,
:.?
*
,
?
?;?
1
? .. 1
?' .. ??
,
* !
1
'-'
<l
t>
~*
~
5
--=.,,,0.:- -1
0
* *
'*
o
2
4
(a) Kernel Eigenface method.
:0.08
-0.06
-0.04
-0.02
0.02
0.04
0.)6
O.DB
(b) Kernel Fisherface method.
Figure 4: Samples projected by Kernel PCA and Kernel Fisher methods.
5
Discussion and Conclusion
The representation in the conventional Eigenface and Fisherface approaches is based
on second order statistics of the image set, Le., covariance matrix, and does not use
high order statistical dependencies such as the relationships among three or more
pixels. For face recognition, much of the important information may be contained
in the high order statistical relationships among the pixels. Using the kernel tricks
that are often used in SVMs, we extend the conventional methods to kernel space
where we can extract nonlinear features among three or more pixels. We have investigated Kernel Eigenface and Kernel Fisherface methods, and demonstrate that
they provide a more effective representation for face recognition. Compared to
other techniques for nonlinear feature extraction, kernel methods have the advantages that they do not require nonlinear optimization, but only the solution of an
eigenvalue problem. Experimental results on two benchmark databases show that
Kernel Eigenface and Kernel Fisherface methods achieve lower error rates than the
ICA, Eigenface and Fisherface approaches in face recognition. The performance
achieved by the ICA method also indicates that face representation using independent basis images is not effective when the images contain pose, scale or lighting
variation. Our future work will focus on analyzing face recognition methods using other kernel methods in high dimensional space. We plan to investigate and
compare the performance of other face recognition methods [14] [12] [19].
References
[1] Y. Adini, Y. Moses, and S. Ullman. Face recognition: The problem of compensating for changes in illumination direction. IEEE PAMI, 19(7):721-732,
1997.
[2] M. S. Bartlett. Face Image Analysis by Unsupervised Learning and Redundancy
Reduction. PhD thesis, University of California at San Diego, 1998.
[3] M. S. Bartlett, H. M. Lades, and T. J. Sejnowski. Independent component
representations for face recognition. In Proc. of SPIE, volume 3299, pages
528-539, 1998.
[4] M. S. Bartlett and T. J. Sejnowski. Viewpoint invariant face recognition using
independent component analysis and attractor networks. In NIPS 9, page 817,
1997.
[5] G. Baudat and F. Anouar. Generalized discriminant analysis using a kernel
approach. Neural Computation, 12:2385-2404,2000.
[6] P. Belhumeur, J. Hespanha, and D. Kriegman. Eigenfaces vs. Fisherfaces:
Recognition using class specific linear projection. IEEE PAMI, 19(7):711-720,
1997.
[7] A. J. Bell and T. J. Sejnowski. An information - maximization approach to
blind separation and blind deconvolution. Neural Computation, 7(6):11291159, 1995.
[8] C. 1\1. Bishop. fleural fretworks for .J.Dattern Recognition. Oxford University
Press, 1995.
[9] P. Comon. Independent component analysis: A new concept? Signal Processing, 36(3):287-314-, 1994.
[10] A. Hyviirinen, J. Karhunen, and E. Oja. Independent Component Analysis.
Wiley-Interscience, 2001.
[11] S. Mika, G. Riitsch, J. Weston, B. Sch6lkopf, A. Smola, and K.-R. Muller.
Invariant feature extraction and classification in kernel spaces. In NIPS 12,
pages 526-532, 2000.
[12] B. Moghaddam. Principal manifolds and bayesian subspaces for visual recognition. In Proc. IEEE Int'l Conf. on Computer Vision, pages 1131-1136,1999.
[13] B. Moghaddam and A. Pentland. Probabilistic visual learning for object recognition. IEEE PAMI, 19(7):696-710, 1997.
[14] P. J. Phillips. Support vector machines applied to face recognition. In NIPS
11, pages 803-809, 1998.
[15] A. N. Rajagopalan, P. Burlina, and R. Chellappa. Higher order statistical
learning for vehicle detection in images. In Proc. IEEE Int'l Con!. on Computer
Vision, volume 2, pages 1204-1209,1999.
[16] A. N. Rajagopalan, K. S. Kumar, J. Karlekar, R. Manivasakan, and M. M.
Patil. Finding faces in photographs. In Proc. IEEE Int'l Conf. on Computer
Vision, pages 640-645, 1998.
[17] V. Roth and V. Steinhage. Nonlinear discriminant analysis using kernel functions. In NIPS 12, pages 568-574,2000.
[18] B. Sch6lkopf, A. Smola, and K.-R. Muller. Nonlinear component analysis as a
kernel eigenvalue problem. Neural Computation, 10(5):1299-1319,1998.
[19] Y. W. Teh and G. E. Hinton. Rate-coded restricted Boltzmann machines for
face recognition. In NIPS 13, pages 908-914, 2001.
[20] M. Turk and A. Pentland. Eigenfaces for recognition. J. of Cognitive Neuroscience, 3(1):71-86, 1991.
| 2087 |@word polynomial:2 r:1 seek:1 covariance:4 xkn:1 carry:1 reduction:1 moment:1 contains:2 denoting:2 riitsch:1 ka:1 com:1 negentropy:1 scatter:2 must:1 informative:1 v:1 xk:2 provides:1 honda:1 constructed:1 fld:6 interscience:1 acquired:1 ica:12 compensating:1 ming:1 increasing:1 becomes:1 provided:1 project:5 maximizes:2 lowest:3 mountain:1 finding:2 anouar:2 ti:1 unwanted:1 k2:1 classifier:3 unit:1 analyzing:1 oxford:1 pami:3 mika:2 unique:1 block:1 digit:1 w4:4 empirical:2 bell:2 projection:9 word:2 downsampled:2 suggest:2 get:2 onto:3 close:1 baudat:2 applying:1 conventional:6 equivalent:2 demonstrated:1 roth:2 center:2 formulate:1 hsuan:1 spanned:3 variation:11 target:1 diego:1 aik:1 trick:2 element:1 recognition:43 expensive:1 utilized:1 database:19 observed:1 ep:1 capture:1 cy:1 removed:1 complexity:1 kriegman:1 solving:1 localization:1 efficiency:1 basis:2 represented:1 derivation:2 separated:1 effective:3 chellappa:1 sejnowski:4 whose:1 quite:1 larger:1 statistic:6 ip:1 advantage:3 eigenvalue:8 kurtosis:3 reconstruction:2 product:2 tu:3 aligned:1 achieve:6 normalize:1 r1:1 comparative:1 leave:2 object:4 pose:7 nearest:3 kzka:1 direction:4 closely:1 centered:4 viewing:2 eigenface:26 require:1 generalization:2 singularity:1 mapping:1 vary:1 proc:4 largest:4 wl:1 successfully:1 smeared:1 always:1 gaussian:3 aim:3 rather:1 varying:3 encode:1 focus:1 improvement:2 consistently:1 indicates:2 contrast:2 baseline:1 sense:1 am:1 relation:1 pixel:11 iu:1 among:10 classification:3 plan:1 special:1 equal:1 comprise:1 construct:1 extraction:2 look:1 unsupervised:1 future:1 minimized:1 oja:1 composed:1 divergence:1 attractor:1 detection:4 investigate:3 moghaddam:3 edge:1 hra:1 facial:6 re:1 column:1 classify:1 cumulants:1 maximization:1 kpca:1 deviation:2 subset:2 entry:1 recognizing:1 pixelwise:1 reported:1 dependency:5 aw:1 person:1 fundamental:1 probabilistic:1 thesis:1 conf:2 cognitive:1 ullman:1 account:1 potential:1 includes:1 coefficient:1 int:3 explicitly:1 blind:2 vehicle:2 view:1 performed:2 lab:1 publicly:1 variance:3 characteristic:1 likewise:1 maximized:1 yield:1 correspond:1 bayesian:1 lighting:10 researcher:1 cc:1 explain:1 against:2 raster:1 pp:2 turk:1 spie:1 con:1 dimensionality:1 higher:9 wei:6 improved:1 though:1 furthermore:2 stage:1 smola:2 correlation:3 nonlinear:16 concept:2 xpq:1 contain:3 normalized:1 lades:1 symmetric:1 leibler:1 attractive:1 ll:2 generalized:3 chin:1 xkl:1 demonstrate:1 image:41 recently:4 empirically:1 insensitive:1 volume:2 extend:1 significant:3 ai:5 phillips:1 fisherface:29 dot:2 impressive:1 surface:1 etc:1 base:1 showed:4 recent:1 olivetti:1 conjectured:1 success:2 arbitrarily:1 muller:2 belhumeur:1 maximize:1 ale:4 ii:4 signal:1 multiple:1 myang:1 coded:1 controlled:1 vision:3 kernel:75 represent:2 adopting:1 achieved:3 cropped:2 whereas:3 source:1 crucial:1 w2:1 subject:4 steinhage:2 db:1 spirit:1 seem:2 call:4 extracting:1 yang:1 iii:1 independence:1 suboptimal:1 reduce:1 idea:1 expression:5 pca:21 bartlett:4 ul:1 clear:2 eigenvectors:12 factorial:1 svms:2 reduced:1 outperform:1 exist:1 xij:2 notice:2 moses:1 neuroscience:1 write:1 redundancy:1 karlekar:1 almost:1 separation:2 decision:1 comparable:1 wopt:2 yale:8 encodes:1 chair:1 span:2 kumar:1 rendered:1 ern:1 lca:2 belonging:2 remain:1 wi:1 feret:2 argmaxw:1 comon:1 intuitively:1 projecting:1 sij:2 rca:5 invariant:2 restricted:1 taken:2 equation:1 nose:1 available:1 operation:1 ho:2 original:1 denotes:1 include:3 patil:1 seeking:1 strategy:2 dependence:1 diagonal:1 subspace:6 cw:1 manifold:1 discriminant:9 code:1 retained:1 relationship:4 potentially:1 hespanha:1 ethod:1 zt:1 boltzmann:1 unknown:1 fisherfaces:2 sch6lkopf:3 perform:3 teh:1 benchmark:1 t:2 pentland:2 extended:1 hinton:1 rn:3 reproducing:1 intensity:3 c4:2 california:1 nip:5 address:1 able:3 pattern:2 xm:1 rajagopalan:3 eyebrow:1 rf:15 explanation:1 mouth:1 suitable:1 eye:1 numerous:1 extract:7 lij:1 formerly:1 degree:2 viewpoint:2 pi:1 perceptron:1 neighbor:3 eigenfaces:3 face:50 ktu:2 curve:1 avoids:1 contour:3 made:1 projected:10 san:1 contradict:1 kullback:1 reveals:1 xi:3 why:1 table:1 hyviirinen:1 ca:1 investigated:1 complex:1 necessarily:1 meanwhile:1 motivation:1 x1:1 fig:1 wiley:1 position:1 lq:1 lie:2 ix:1 specific:1 bishop:1 svm:2 deconvolution:1 mnist:1 phd:1 illumination:2 karhunen:1 lt:1 photograph:1 visual:7 contained:2 tracking:2 maa:1 extracted:4 weston:1 identity:1 consequently:1 fisher:6 change:3 determined:1 hyperplane:2 principal:11 experimental:8 wq:2 internal:1 support:2 scan:1 frontal:2 evaluate:1 |
1,193 | 2,088 | 1
Bayesian morphometry of hippocampal cells
suggests same-cell somatodendritic repulsion
Giorgio A. Ascoli *
Alexei Samsonovich
Krasnow Institute for Advanced Study at George Mason University
Fairfax, VA 22030-4444
ascoli@gmu.edu
asamsono@gmu.edu
Abstract
Visual inspection of neurons suggests that dendritic orientation may be
determined both by internal constraints (e.g. membrane tension) and by
external vector fields (e.g. neurotrophic gradients). For example, basal
dendrites of pyramidal cells appear nicely fan-out. This regular
orientation is hard to justify completely with a general tendency to
grow straight, given the zigzags observed experimentally. Instead,
dendrites could (A) favor a fixed (?external?) direction, or (B) repel
from their own soma. To investigate these possibilities quantitatively,
reconstructed hippocampal cells were subjected to Bayesian analysis.
The statistical model combined linearly factors A and B, as well as the
tendency to grow straight. For all morphological classes, B was found
to be significantly positive and consistently greater than A. In addition,
when dendrites were artificially re-oriented according to this model, the
resulting structures closely resembled real morphologies. These results
suggest that somatodendritic repulsion may play a role in determining
dendritic orientation. Since hippocampal cells are very densely packed
and their dendritic trees highly overlap, the repulsion must be cellspecific. We discuss possible mechanisms underlying such specificity.
1 I n t r od uc t i on
The study of brain dynamics and development at the cellular level would greatly benefit
from a standardized, accurate and yet succinct statistical model characterizing the
morphology of major neuronal classes. Such model could also provide a basis for
simulation of anatomically realistic virtual neurons [1]. The model should accurately
distinguish among different neuronal classes: a morphological difference between classes
would be captured by a difference in model parameters and reproduced in generated
virtual neurons. In addition, the model should be self-consistent: there should be no
statistical difference in model parameters measured from real neurons of a given class
and from virtual neurons of the same class. The assumption that a simple statistical model
of this sort exists relies on the similarity of average environmental and homeostatic
conditions encountered by individual neurons during development and on the limited
amount of genetic information that underlies differentiation of neuronal classes.
Previous research in computational neuroanatomy has mainly focused on the topology
and internal geometry of dendrites (i.e., the properties described in ?dendrograms?) [2,3].
Recently, we attempted to include spatial orientation in the models, thus generating
2
virtual neurons in 3D [4]. Dendritic growth was assumed to deviate from the straight
direction both randomly and based on a constant bias in a given direction, or ?tropism?.
Different models of tropism (e.g. along a fixed axis, towards a plane, or away from the
soma) had dramatic effects on the shape of virtual neurons [5]. Our current strategy is to
split the problem of finding a statistical model describing neuronal morphology in two
parts. First, we maintain that the topology and the internal geometry of a particular
dendritic tree can be described independently of its 3D embedding (i.e., the set of local
dendritic orientations). At the same time, one and the same internal geometry (e.g., the
experimental dendrograms obtained from real neurons) may have many equally plausible
3D embeddings that are statistically consistent with the anatomical characteristics of that
neuronal class. The present work aims at finding a minimal statistical model describing
local dendritic orientation in experimentally reconstructed hippocampal principal cells.
Hippocampal neurons have a polarized shape: their dendrites tend to grow from the soma
as if enclosed in cones. In pyramidal cells, basal and apical dendrites invade opposite
hemispaces (fig. 1A), while granule cell dendrites all invade the same hemispace. This
behavior could be caused by a tendency to grow towards the layers of incoming fibers to
establish synapses. Such tendency would correspond to a tropism in a direction roughly
parallel to the cell main axis. Alternatively, dendrites could initially stem in the
appropriate (possibly genetically determined) directions, and then continue to grow
approximately in a radial direction from the soma. A close inspection of pyramidal
(basal) trees suggests that dendrites may indeed be repelled from their soma (Fig. 1B). A
typical dendrite may reorient itself (arrow) to grow nearly straight along a radius from the
soma. Remarkably, this happens even after many turns, when the initial direction is lost.
Such behavior may be hard to explain without tropism. If the deviations from straight
growth were random, one should be able to ?remodel?th e trees by measuring and
reproducing the statistics of local turn angles, assuming its independence of dendritic
orientation and location. Figure 1C shows the cell from 1A after such remodeling. In this
case basal and apical dendrites retain only their initial (stemming) orientations from the
original data. The resulting ?cotton ball?s uggests that dendritic turns are not in dependent
of dendritic orientation. In this paper, we use Bayesian analysis to quantify the dendritic
tropism.
2 M e t h od s
Digital files of fully reconstructed rat hippocampal pyramidal cells (24 CA3 and 23 CA1
neurons) were kindly provided by Dr. D. Amaral. The overall morphology of these cells,
as well as the experimental acquisition methods, were extensively described [6]. In these
files, dendrites are represented as (branching) chains of cylindrical sections. Each section
is connected to one other section in the path to the soma, and may be connected on the
other extremity to two other sections (bifurcation), one other section (continuation point),
or no other section (terminal tip). Each section is described in the file by its ending point
coordinates, its diameter and its "parent", i.e., the attached section in the path to the soma
[5,7]. In CA3 cells, basal dendrites had an average of 687(?216) continuation points and
72(?17) bifurcations per cell, while apical dendrites had 717(?156) continuation points
and 80(?21) bifurcations per cell. CA1 cells had 462(?138) continuation points and
52(?12) bifurcations (basal), 860(?188) continuation points and 120(?22) bifurcations
(apical). In the present work, basal and apical trees of CA3 and CA1 pyramidal cells were
treated as 4 different classes. Digital data of rat dentate granule cells [8] are kindly made
available by Dr. B. Claiborne through the internet (http://cascade.utsa.edu/bjclab). Only
36 of the 42 cells in this archive were used: in 6 cases numerical processing was not
accomplished due to minor inconsistencies in the data files. The ?rejected? cells were
1208875, 3319201, 411883, 411884A, 411884B, 803887B. Granule dendrites had
3
549(?186) continuation points and 30(?6) bifurcations per cell. Cells in these or similar
formats can be rendered, rotated, and zoomed with a java applet available through the
internet (www.cns.soton.ac.uk) [7].
Figure 1: A: A pyramidal cell (c53063) from Amaral?s archive. B: A zoom-in from panel
A (arrows point to the same basal tree location). Dotted dendrites are behind the plane. C:
Same cell (c53063) with its dendritic orientation remodeled assuming zero tropism and
same statistics of all turn angles (see Results).
In agreement with the available format of morphological data (described above), the
process of dendritic growth1 can be represented as a discrete stochastic process consisting
of sequential attachment of new sections to each growing dendrite. Here we keep the
given internal geometry of the experimental data while remodeling the 3D embedding
geometry (dendritic orientation). The task is to make a remodeled geometry statistically
consistent with the original structure. The basic assumption is that neuronal development1
is a Markov process governed by local rules [4]. Specifically, we assume that (i) each
step in dendritic outgrowth only depends on the preceding step and on current local
conditions; and (ii) dendrites do not undergo geometrical or topological modification
after their formation (see, however, Discussion). In this Markov approximation, a
plausible 3D embedding can be found by sequentially orienting individual sections,
starting from the soma and moving toward the terminals. We are implementing this
procedure in two-step iterations (1). First, at a given node i with coordinates ri we select a
section i+1, disregard its given orientation, and calculate its most likely expected
direction n'i+1 based on the model (here section i+1 connects nodes i and i+1, and n
stands for a unit vector). For a continuation point, the most likely direction n'i+1 is
computed as the direction of the vector sum ni + vi. The first term is the direction of the
parent section ni , and reflects the tendency dendrites exhibit to grow relatively straight
due to membrane tension, mechanical properties of the cytoskeleton, etc. The second
term is a local value of a vector field: vi = v(ri ), which comprises the influence of external
local conditions on the direction of growth (as specified below). Finally, we generate a
perturbation of the most likely direction n'i+1 to produce a particular plausible instance of
a new direction. In summary, the new direction ni+1 is generated as:
1
Although we resort to a developmental metaphor, our goal is to describe accurately
the result of development rather than the process of development.
4
n i +1 = Ti n'i +1 ,
(1)
n'i +1 || n i + v i .
Here Ti is an operator that deflects n'i+1 into a random direction. If we view each
deflection as a yaw of angle ?i, then the corresponding rolling angle (describing rotation
around the axis of the parent dendrite) is distributed uniformly between 0 and 2?. The
probability distribution function for deflections as a function of ?i is taken in a form that,
as we found, well fits experimental data:
P(Ti ) ? e
?
?i
?
,
(2)
where ? << 1 is a parameter of the model. At bifurcation points, the same rule (1), (2) is
applied for each daughter independently. A more accurate and plausible description of
dendritic orientation at bifurcations might require a more complex model. However, our
simple choice yields surprisingly good results (see below). The model (1), (2) can be used
in the simulation of virtual neuronal morphology. In this case one would first need to
generate the internal geometry of the dendrites [1-5]. Most importantly, model (1), (2)
can be used to quantitatively assess the significance of the somatocentric (radial) tropism
of real dendrites. Assuming that there is a significant preferential directionality of growth
in hippocampal dendrites, the two main alternatives are (see Introduction):
HA: The dominating tropic factor is independent of the location of the soma.
HB: The dominating tropic factor is radial with respect to the soma.
The simplest model for the vector field v that discriminates between these alternative
hypotheses includes both factors, A and B, linearly:
v i = a + bn ir .
(3)
Here a = (ax, ay, az) is a constant vector representing global directionality of cellindependent environmental factors (chemical gradients, density of neurites, etc.)
influencing dendritic orientation. nri is the unit vector in the direction connecting the
soma to node i, thus representing a somatocentric tropic factor. In summary, ax, ay, az, b
and ? are the parameters of the model. Finding that the absolute value a = |a| is
significantly greater than b would support HA. On the contrary, finding that b is greater
than a would support HB. Based on a Bayesian approach, we compute the most likely
values of a, b and ? by maximization of the likelihood of all experimentally measured
orientations (taken at continuation points only) of a given dendritic tree:
(a*, b *) = arg {max
?
? P(Ti ) = arg min
{a , b} i
a , b ,? }
,
(4)
i
where ?i is given by (1)-(3) with experimental section orientations substituted for ni, ni+1,
asterisk denotes most likely values, and the average is over all continuation points. Given
a* and b*, the value of ?* can be found from the average value of ?i computed with a =
a* and b = b*. The relation results from differentiation of (4) by ?. The same relation
holds for the average value of ? computed based on the probability distribution function
(2) with ? = ?*. Therefore, <?i> computed from the neurometric data with a = a* and b
= b* is equal to <?> based on (2) with ? = ?*. The model is thus self-consistent: the
measured value of ?* in a remodeled neuron is guaranteed to coincide on average with
the input parameter ? used for simulation. In addition, our numerical analysis indicates
self-consistency of the model with respect to a and b, when their values are within a
practically meaningful range.
5
3 R e s ul t s
Results of the Bayesian analysis are presented in Table 1. Parameters a and b were
optimized for each cell individually, then the absolute value a = |a| was taken for each
cell. The mean value and the standard deviation of a in Table 1 were computed based on
the set of the individual absolute values, while each individual value of b was taken with
its sign (which was positive in all cases but one). The most likely direction of a varied
significantly among cells, i.e., no particular fixed direction was generally preferred.
Table 1: Results from Bayesian analysis (mean ? standard deviation). ? is the minimized
deflection angle, a and b are parameters of the model (1)-(3) computed according to (4).
Dataset
?
CA3-bas 16.4 ? 2.3
CA3-apic 15.2 ? 1.9
CA1-bas 16.6 ? 1.6
CA1-apic 19.1 ? 2.0
Granule 19.1 ? 2.7
Original data
B
a
0.49 ? 0.17 0.08 ? 0.05
0.36 ? 0.16 0.12 ? 0.07
0.49 ? 0.26 0.14 ? 0.10
0.30 ? 0.20 0.16 ? 0.15
1.01 ? 0.64 0.17 ? 0.11
Z coordinate set to zero
b
A
?
12.0 ? 2.4 0.42 ? 0.15 0.06 ? 0.05
12.0 ? 2.9 0.29 ? 0.23 0.10 ? 0.14
14.2 ? 1.9 0.48 ? 0.31 0.16 ? 0.12
17.3 ? 2.4 0.22 ? 0.17 0.11 ? 0.10
11.0 ? 1.9 0.36 ? 0.16 0.07 ? 0.05
The key finding is that a is not significantly different from zero, while b is significantly
positive. The slightly higher coefficient of variation obtained for granule cells could be
due to a larger experimental error in the z coordinate (orthogonal to the slice). In several
granule cells (but in none of the pyramidal cells) the greater noise in z was apparent upon
visual inspection of the rendered structures. Therefore, we re-ran the analysis discarding
the z coordinate (right columns). Results changed only minimally for pyramidal cells, and
the granule cell parameters became more consistent with the pyramidal cells.
The measured average values of the model parameters were used for remodeling of
experimental neuronal shapes, as described above. In particular, b was set to 0.5, while a
was set to zero. We kept the internal geometry and the initial stemming direction of each
tree from the experimental data, and simulated dendritic orientation at all nodes separated
by more than 2 steps from the soma. A typical result is shown in Figure 2. Generally, the
artificially re-oriented dendrites looked better than one could expect for a model as
simple as (1) ? (3). This result may be compared with figure 1C, which shows an
example of remodeling based on the same model in the absence of tropism (a = b = 0).
Although in this case the shape can be improved by reducing ?, the result never gets as
close to a real shape as in Fig. 2 C, D, even when random, uncorrelated local distortions
("shuffling") are applied to the generated geometry. Thus, although the tendency to grow
straight represents the dominant component of the model (i.e., b<1), somatocentric
tropism may exert a dramatic effect on dendritic shape. Surprisingly, even the asymmetry
of the dendritic spread (compare front and side views) is preserved after remodeling.
However, two details are difficult to reproduce with this model: the uniform distribution
of dendrites in space and other subtle medium-distance correlations among dendritic
deflections. In order to account for these properties, we may need to consider spatially
correlated inhomogeneities of the medium and possible short range dendrodendritic
interactions.
6
4 D i s c us s i on
The key results of this work is that, according to Bayesian analysis, dendrites of
hippocampal principal cells display a significant radial tropism. This means that the
spatial orientation of these neuronal trees can be statistically described as if dendrites
were repelled from their own soma. This preferential direction is stronger than any
tendency to grow along a fixed direction independent of the location of the soma. These
results apply to all dendritic classes, but in general pyramidal cell basal trees (and granule
cell dendrites) display a bigger somatocentric tropism than apical trees.
Figure 2: Dendritic remodeling with somatocentric tropism. A, B: front and side views
of cell 10861 from Amaral' s archive. C, D: Same views after remodeling with parameters
! #" $ #
a = 0, b = 0.5, ? = 0.15 (corresponding to <?> = 17
stem were taken in their original orientations; all subsequent experimental orientations
were disregarded and regenerated from scratch according to the model.
Assuming that dendrites are indeed repelled from their soma during development, what
could be a plausible mechanism? Principal cells are very densely packed in the
hippocampus, and their dendrites highly overlap. If repulsion were mediated by a
diffusible chemical factor, in order for dendrites to be repelled radially from their own
soma, each neuron should have its own specific chemical marker (a fairly unlikely
possibility). If the same repulsive factor were released by all neurons, each dendrite
would be repelled by hundreds of somata, and not just by their own. The resulting
tropism would be perpendicular to the principal cell layer, i.e. each dendrite would be
pushed approximately in the same direction, independent of the location of its soma. This
scenario is in clear contrast with the result of our statistical analysis. Thus, how can a
growing dendrite sense the location of its own soma? One possibility involves the
spontaneous spiking activity of neurons during development. A cell that spikes becomes
unique in its neighborhood for a short period of time. The philopodia of dendritic growth
7
cones could possess voltage-gated receptors to sense transient chemical gradients (e.g.,
pH) created by the spiking cell. Only dendrites that are depolarized during the transient
chemical gradient (i.e., those belonging to the same spiking cell) would be repelled by it.
Alternatively, depolarized philopodia could be sensitive to the small voltage difference
created by the spike in the extracellular space (a voltage that can be recorded by tetrodes).
The main results obtained with the simple model presented in this work are independent
of the z coordinate in the morphometric files, i.e. the most error-prone measurement in
the experimental reconstruction. However, it is important to note that any observed
deviation of dendritic path from a straight line, including that due to measurement errors,
causes an increase in the most likely values of parameters a and b. Another possibility is
that dendrites do grow almost precisely in straight lines, and the measured values of a and
b reflect distortions of dendritic shapes after development. In order to assess the effect of
these factors on a and b, we pre-processed the experimental data by adding a gradually
increasing noise to all coordinates of dendritic sections. Then we were able to extrapolate
the dependence of a*, b* and <?>* on the amplitude of noise in order to estimate the
parameter values in the absence of the experimental error (which was conservatively
taken to be of 0.5 ?m). For basal trees of CA3 pyramidal cells, this analysis yielded an
estimated ?corrected? value of b between 0.14 and 0.25, with a remaining much smaller
than b. Interestingly, our analysis based on extrapolation shows that, regardless of the
assumed amount of distortion present in the experimental data, given the numbers
measured for CA3 basal trees, positive initial <?> implies positive initial b. In other
words, not only measurement errors, but also possible biological distortions of the
dendritic tree may not be capable of accounting for the observed positivity of the
parameter b. Although these factors affect our results quantitatively, they do not change
the statistical significance nor the qualitative trends. However, a more rigorous analysis
needs to be carried out. Nevertheless, artificially reoriented dendrites according to our
simple model appear almost as realistic as the original structures, and we could not
achieve the same result with any choice of parameters in models of distortion without a
somatocentric tropism. In conclusion, whether the present Bayesian analysis reveals a
phenomenon of somatodendritic repulsion remains an (experimentally testable) open
question. What is unquestionable is that the presented model is a significant step forward
in the formulation of an accurate statistical description of dendritic morphology.
A c k n ow l e d g me n t s
This work was supported in part by Human Brain Project Grant R01 NS39600, funded jointly by
NINDS and NIMH.
References
[1] Ascoli G.A. (1999) Progress and perspectives in computational neuroanatomy. Anat. Rec.
257(6):195-207.
[2] van Pelt J. (1997) Effect of pruning on dendritic tree topology. J. Theor. Biol. 186(1):17-32.
[3] Burke R.E., W. Marks, B. Ulfhake (1992) A parsimonious description of motoneurons dendritic
morphology using computer simulation. J. Neurosci. 12(6):2403-2416.
[4] Ascoli G.A., J. Krichmar (2000) L-Neuron: a modeling tool for the efficient generation and
parsimonious description of dendritic morphology. Neurocomputing 32-33:1003-1011.
[5] Ascoli G.A., J. Krichmar, S. Nasuto, S. Senft (2001) Generation, description and storage of
dendritic morphology data. Phil. Trans. R. Sci. B, In Press.
[6] Ishizuka N., W. Cowan, D. Amaral (1995) A quantitative analysis of the dendritic organization
of pyramidal cells in the rat hippocampus. J. Comp. Neurol. 362(1):17-45.
[7] Cannon R.C., D. Turner, G. Pyapali, H. Wheal (1998) An on-line archive of reconstructed
hippocampal neurons. J Neurosci. Meth. 84(1-2):49-54.
[8] Rihn L.L., B. Claiborne (1990) Dendritic growth and regression in rat dentate granule cells
during late postnatal development. Dev. Brain Res. 54(1):115-124
| 2088 |@word cylindrical:1 stronger:1 hippocampus:2 open:1 simulation:4 bn:1 accounting:1 dramatic:2 initial:5 genetic:1 interestingly:1 current:2 od:2 yet:1 must:1 stemming:2 numerical:2 subsequent:1 realistic:2 shape:7 inspection:3 plane:2 postnatal:1 short:2 node:4 location:6 along:3 qualitative:1 expected:1 indeed:2 roughly:1 behavior:2 samsonovich:1 growing:2 morphology:9 brain:3 terminal:2 nor:1 metaphor:1 increasing:1 becomes:1 provided:1 project:1 underlying:1 panel:1 medium:2 what:2 ca1:5 finding:5 differentiation:2 quantitative:1 ti:4 growth:6 uk:1 unit:2 grant:1 appear:2 positive:5 giorgio:1 influencing:1 local:8 receptor:1 extremity:1 path:3 approximately:2 might:1 exert:1 minimally:1 suggests:3 limited:1 range:2 statistically:3 perpendicular:1 unique:1 lost:1 procedure:1 significantly:5 cascade:1 java:1 pre:1 radial:4 regular:1 specificity:1 word:1 suggest:1 get:1 close:2 operator:1 storage:1 influence:1 www:1 phil:1 regardless:1 starting:1 independently:2 focused:1 rule:2 importantly:1 dendrograms:2 embedding:3 coordinate:7 variation:1 spontaneous:1 play:1 hypothesis:1 agreement:1 trend:1 rec:1 observed:3 role:1 calculate:1 connected:2 morphological:3 ran:1 discriminates:1 developmental:1 nimh:1 dynamic:1 upon:1 completely:1 basis:1 represented:2 fiber:1 separated:1 describe:1 formation:1 neighborhood:1 apparent:1 larger:1 plausible:5 dominating:2 distortion:5 favor:1 statistic:2 jointly:1 itself:1 inhomogeneity:1 reproduced:1 reconstruction:1 interaction:1 zoomed:1 krichmar:2 achieve:1 description:5 az:2 parent:3 asymmetry:1 produce:1 generating:1 remodeled:3 rotated:1 ac:1 measured:6 minor:1 progress:1 involves:1 implies:1 quantify:1 direction:23 radius:1 closely:1 stochastic:1 human:1 transient:2 virtual:6 implementing:1 require:1 dendritic:36 biological:1 theor:1 hold:1 practically:1 around:1 burke:1 dentate:2 major:1 dendrodendritic:1 released:1 sensitive:1 individually:1 tool:1 reflects:1 aim:1 rather:1 cannon:1 voltage:3 ax:2 consistently:1 likelihood:1 mainly:1 indicates:1 greatly:1 contrast:1 rigorous:1 sense:2 dependent:1 repulsion:5 unlikely:1 initially:1 relation:2 reproduce:1 reorient:1 overall:1 among:3 orientation:20 arg:2 development:8 spatial:2 bifurcation:8 uc:1 fairly:1 field:3 equal:1 never:1 nicely:1 represents:1 nearly:1 yaw:1 minimized:1 regenerated:1 quantitatively:3 oriented:2 randomly:1 densely:2 zoom:1 individual:4 neurocomputing:1 geometry:9 consisting:1 cns:1 connects:1 maintain:1 organization:1 investigate:1 possibility:4 alexei:1 highly:2 behind:1 chain:1 zigzag:1 accurate:3 capable:1 preferential:2 orthogonal:1 tree:15 gmu:2 re:4 minimal:1 instance:1 column:1 modeling:1 dev:1 measuring:1 maximization:1 ca3:7 apical:6 deviation:4 rolling:1 uniform:1 hundred:1 front:2 combined:1 density:1 retain:1 tip:1 connecting:1 reflect:1 recorded:1 possibly:1 positivity:1 dr:2 external:3 resort:1 account:1 utsa:1 includes:1 coefficient:1 caused:1 depends:1 vi:2 view:4 extrapolation:1 applet:1 sort:1 parallel:1 ass:2 ni:5 ir:1 became:1 characteristic:1 correspond:1 yield:1 bayesian:8 accurately:2 none:1 comp:1 straight:9 explain:1 synapsis:1 acquisition:1 soton:1 dataset:1 radially:1 subtle:1 amplitude:1 neurotrophic:1 higher:1 tension:2 improved:1 formulation:1 rejected:1 just:1 correlation:1 marker:1 invade:2 orienting:1 effect:4 chemical:5 spatially:1 during:5 self:3 branching:1 rat:4 hippocampal:9 ay:2 geometrical:1 recently:1 rotation:1 spiking:3 attached:1 significant:3 neurites:1 measurement:3 shuffling:1 consistency:1 had:5 funded:1 moving:1 similarity:1 etc:2 dominant:1 own:6 perspective:1 rihn:1 scenario:1 continue:1 inconsistency:1 accomplished:1 captured:1 motoneuron:1 george:1 greater:4 preceding:1 neuroanatomy:2 period:1 ii:1 stem:2 equally:1 bigger:1 va:1 underlies:1 basic:1 regression:1 iteration:1 cell:45 preserved:1 morphometry:1 remarkably:1 addition:3 pyramidal:12 grow:10 depolarized:2 posse:1 archive:4 file:5 tend:1 undergo:1 cowan:1 contrary:1 diffusible:1 split:1 embeddings:1 hb:2 independence:1 fit:1 affect:1 topology:3 opposite:1 whether:1 ul:1 cause:1 generally:2 clear:1 amount:2 extensively:1 ph:1 processed:1 diameter:1 simplest:1 continuation:9 http:1 generate:2 dotted:1 sign:1 estimated:1 per:3 anatomical:1 discrete:1 basal:11 key:2 soma:20 nevertheless:1 kept:1 cone:2 sum:1 deflection:4 angle:5 almost:2 parsimonious:2 polarized:1 ninds:1 pushed:1 layer:2 internet:2 guaranteed:1 distinguish:1 display:2 fan:1 topological:1 encountered:1 yielded:1 activity:1 hemispace:1 constraint:1 precisely:1 ri:2 min:1 remodeling:7 rendered:2 format:2 relatively:1 extracellular:1 pelt:1 according:5 ball:1 belonging:1 membrane:2 smaller:1 slightly:1 modification:1 happens:1 unquestionable:1 anatomically:1 gradually:1 taken:6 remains:1 discus:1 describing:3 mechanism:2 turn:4 subjected:1 repulsive:1 available:3 nri:1 apply:1 away:1 appropriate:1 alternative:2 original:5 standardized:1 denotes:1 include:1 remaining:1 testable:1 establish:1 granule:9 r01:1 question:1 looked:1 spike:2 strategy:1 dependence:1 exhibit:1 gradient:4 ow:1 distance:1 fairfax:1 simulated:1 sci:1 me:1 cellular:1 toward:1 neurometric:1 assuming:4 difficult:1 tropic:3 repelled:6 daughter:1 ba:2 packed:2 gated:1 neuron:17 ascoli:5 markov:2 perturbation:1 reproducing:1 homeostatic:1 varied:1 mechanical:1 specified:1 repel:1 optimized:1 cotton:1 trans:1 able:2 below:2 genetically:1 max:1 including:1 overlap:2 treated:1 advanced:1 turner:1 representing:2 meth:1 axis:3 attachment:1 created:2 carried:1 mediated:1 deviate:1 determining:1 fully:1 expect:1 generation:2 krasnow:1 enclosed:1 digital:2 asterisk:1 consistent:5 morphometric:1 uncorrelated:1 prone:1 summary:2 changed:1 surprisingly:2 supported:1 bias:1 side:2 institute:1 characterizing:1 absolute:3 benefit:1 distributed:1 slice:1 van:1 ending:1 stand:1 conservatively:1 forward:1 made:1 coincide:1 amaral:4 reconstructed:4 pruning:1 claiborne:2 preferred:1 keep:1 global:1 sequentially:1 incoming:1 reveals:1 assumed:2 alternatively:2 table:3 scratch:1 correlated:1 dendrite:37 complex:1 artificially:3 substituted:1 kindly:2 significance:2 main:3 spread:1 linearly:2 arrow:2 neurosci:2 noise:3 succinct:1 neuronal:9 fig:3 comprises:1 governed:1 late:1 anat:1 resembled:1 discarding:1 specific:1 mason:1 tropism:14 neurol:1 tetrode:1 exists:1 sequential:1 adding:1 disregarded:1 likely:7 visual:2 environmental:2 relies:1 goal:1 towards:2 absence:2 change:1 hard:2 experimentally:4 directionality:2 determined:2 typical:2 specifically:1 uniformly:1 justify:1 reducing:1 corrected:1 principal:4 tendency:7 experimental:13 attempted:1 disregard:1 meaningful:1 select:1 internal:7 support:2 mark:1 phenomenon:1 biol:1 extrapolate:1 |
1,194 | 2,089 | Convolution Kernels for Natural Language
Michael Collins
AT&T Labs?Research
180 Park Avenue, New Jersey, NJ 07932
mcollins@research.att.com
Nigel Duffy
Department of Computer Science
University of California at Santa Cruz
nigeduff@cse.ucsc.edu
Abstract
We describe the application of kernel methods to Natural Language Processing (NLP) problems. In many NLP tasks the objects being modeled
are strings, trees, graphs or other discrete structures which require some
mechanism to convert them into feature vectors. We describe kernels for
various natural language structures, allowing rich, high dimensional representations of these structures. We show how a kernel over trees can
be applied to parsing using the voted perceptron algorithm, and we give
experimental results on the ATIS corpus of parse trees.
1 Introduction
Kernel methods have been widely used to extend the applicability of many well-known algorithms, such as the Perceptron [1], Support Vector Machines [6], or Principal Component
Analysis [15]. A key property of these algorithms is that the only operation they require
is the evaluation of dot products between pairs of examples. One may therefore replace
into a new
the dotproduct with a Mercer kernel, implicitly mapping feature vectors in
space
, and applying the original algorithm in this new feature space. Kernels provide
an efficient way to carry out these calculations when is large or even infinite.
This paper describes the application of kernel methods to Natural Language Processing
(NLP) problems. In many NLP tasks the input domain cannot be neatly formulated as a sub
set of . Instead, the objects being modeled are strings, trees or other discrete structures
which require some mechanism to convert them into feature vectors. We describe kernels
for various NLP structures, and show that they allow computationally feasible representations in very high dimensional feature spaces, for example a parse tree representation that
tracks all subtrees. We show how a tree kernel can be applied to parsing using the perceptron algorithm, giving experimental results on the ATIS corpus of parses. The kernels we
describe are instances of ?Convolution Kernels?, which were introduced by Haussler [10]
and Watkins [16], and which involve a recursive calculation over the ?parts? of a discrete
structure. Although we concentrate on NLP tasks in this paper, the kernels should also be
useful in computational biology, which shares similar problems and structures.
1.1 Natural Language Tasks
Figure 1 shows some typical structures from NLP tasks. Each structure involves an ?observed? string (a sentence), and some hidden structure (an underlying state sequence or
tree). We assume that there is some training set of structures, and that the task is to learn
a) Lou Gerstner is chairman of IBM
[S [NP Lou Gerstner ] [VP is [NP chairman [PP of [NP IBM ] ] ] ] ]
b) Lou Gerstner is chairman of IBM Lou/SP Gerstner/CP is/N chairman/N of/N IBM/SC
c) Lou/N Gerstner/N is/V chairman/N of/P IBM/N
Figure 1: Three NLP tasks where a function is learned from a string to some hidden structure. In (a), the hidden structure is a parse tree. In (b), the hidden structure is an underlying sequence of states representing named entity boundaries (SP = Start person, CP =
Continue person, SC = Start company, N= No entity). In (c), the hidden states represent
part-of-speech tags (N = noun, V = verb, P = preposition,).
the mapping from an input string to its hidden structure. We refer to tasks that involve trees
as parsing problems, and tasks that involve hidden state sequences as tagging problems.
In many of these problems ambiguity is the key issue: although only one analysis is plausible, there may be very many possible analyses. A common way to deal with ambiguity
is to use a stochastic grammar, for example a Probabilistic Context Free Grammar (PCFG)
for parsing, or a Hidden Markov Model (HMM) for tagging. Probabilities are attached to
rules in the grammar ? context-free rules in the case of PCFGs, state transition probabilities and state emission probabilities for HMMs. Rule probabilities are typically estimated
using maximum likelihood estimation, which gives simple relative frequency estimates.
Competing analyses for the same sentence are ranked using these probabilities. See [3] for
an introduction to these methods.
This paper proposes an alternative to generative models such as PCFGs and HMMs. Instead
of identifying parameters with rules of the grammar, we show how kernels can be used to
form representations that are sensitive to larger sub-structures of trees or state sequences.
The parameter estimation methods we describe are discriminative, optimizing a criterion
that is directly related to error rate.
While we use the parsing problem as a running example in this paper, kernels over NLP
structures could be used in many ways: for example, in PCA over discrete structures, or
in classification and regression problems. Structured objects such as parse trees are so
prevalent in NLP that convolution kernels should have many applications.
2 A Tree Kernel
The previous section introduced PCFGs as a parsing method. This approach essentially
counts the relative number of occurences of a given rule in the training data and uses these
counts to represent its learned knowledge. PCFGs make some fairly strong independence
assumptions, disregarding substantial amounts of structural information. In particular, it
does not appear reasonable to assume that the rules applied at level in the parse tree are
unrelated to those applied at level .
As an alternative we attempt to capture considerably more structural information by considering all tree fragments that occur in a parse tree. This allows us to capture higher order
dependencies between grammar rules. See figure 2 for an example. As in a PCFG the new
representation tracks the counts of single rules, but it is also sensitive to larger sub-trees.
Conceptually we begin by enumerating all tree fragments that occur in the training data
. Note that this is done only implicitly. Each tree is represented by an dimensional vector where the ?th component counts the number of occurences of the ?th tree
fragment. Let us define the function
to be the number of occurences of the ?th tree
fragment in tree , so that is now represented as !
" #$ .
a)
S
NP
b)
VP
N
V
Jeff
ate
NP
D
N
the
apple
NP
NP
D
N
the
apple
D
N
D
N
NP
the
apple
D
NP
N
D
the
N
apple
Figure 2: a) An example tree. b) The sub-trees of the NP covering the apple. The tree in
(a) contains all of these sub-trees, and many others. We define a sub-tree to be any subgraph which includes more than one node, with the restriction that entire (not partial) rule
productions must be included. For example, the fragment [NP [D the ]] is excluded
because it contains only part of the production NP
D N.
Note that will be huge (a given tree will have a number of subtrees that is exponential in
its size). Because of this we would like design algorithms whose computational complexity
does not depend on .
Representations of this kind have been studied extensively by Bod [2]. However, the work
in [2] involves training and decoding algorithms that depend computationally on the num
ber of subtrees involved. The parameter estimation techniques described in [2] do not
correspond to maximum-likelihood estimation or a discriminative criterion: see [11] for
discussion. The methods we propose show that the score for a parse can be calculated in
polynomial time in spite of an exponentially large number of subtrees, and that efficient parameter estimation techniques exist which optimize discriminative criteria that have been
well-studied theoretically.
Goodman [9] gives an ingenious conversion of the model in [2] to an equivalent PCFG
whose number of rules is linear in the size of the training data, thus solving many of the
computational issues. An exact implementation of Bod?s parsing method is still infeasible,
but Goodman gives an approximation that can be implemented efficiently. However, the
method still suffers from the lack of justification of the parameter estimation techniques.
The key to our efficient use of this high dimensional representation is the definition of an
appropriate kernel. We begin by examining the inner product between two trees and
under this representation, # # . To compute we first define
the set of nodes in trees and as and respectively. We define the indicator
function
to be if sub-tree is seen rooted at node and 0 otherwise. It follows
that
#
and
. The first step to efficient
computation of the inner product is the following property (which can be proved with some
simple algebra):
#
#
#
#
"!$#&% '(% ) *!
! !2#43 6573 658 9 :!
!
65-=>5 -
+ -,/.0 + 1,/.
+ 9,/.0 + ,;.2$<
where we define ? @
A
. Next, we note that ? can be
computed in polynomial time, due to the following recursive definition:
B
B
If the productions at and are different ?
DC .
If the productions at
? .
and
are the same, and
and are pre-terminals, then
In training, a parameter is explicitly estimated for each sub-tree. In searching for the best parse,
calculating the score for a parse in principle requires summing over an exponential number of deriva underlying a tree, and in practice is approximated using Monte-Carlo techniques.
tions
Pre-terminals are nodes directly above words in the surface string, for example the N, V, and D
B
Else if the productions at and are the same and and are not pre-terminals,
?
$"?
!
$
where
is the number of children of in the tree; because the productions at /
are the same, we have
. The ?th child-node of is
$ .
To see that this recursive definition is correct, note that ? simply counts the number
of common subtrees that are found rooted at both and . The first two cases are trivially
correct. The last, recursive, definition follows because a common subtree for and can
be formed by taking the production at / , together with a choice at each child of simply
taking the non-terminal at that child, or any one of the common sub-trees at that child.
$$ possible choices at the ?th child. (Note
Thus there are "?
that a similar recursion is described by Goodman [9], Goodman?s application being the
conversion of Bod?s model [2] to an equivalent PCFG.)
It is clear from the identity # ? , and the recursive definition
of ? , that # can be calculated in
time: the matrix of
? values can be filled in, then summed. This can be a pessimistic estimate of
the runtime. A more useful characterization is that it runs in time linear in the number of
members such that the productions at and are the same. In our
data we have found a typically linear number of nodes with identical productions, so that
most values of ? are 0, and the running time is close to linear in the size of the trees.
This recursive kernel structure, where a kernel between two objects is defined in terms
of kernels between its parts is quite a general idea. Haussler [10] goes into some detail
describing which construction operations are valid in this context, i.e. which operations
maintain the essential Mercer conditions. This paper and previous work by Lodhi et al. [12]
examining the application of convolution kernels to strings provide some evidence that
convolution kernels may provide an extremely useful tool for applying modern machine
learning techniques to highly structured objects. The key idea here is that one may take
a structured object and split it up into parts. If one can construct kernels over the parts
then one can combine these into a kernel over the whole object. Clearly, this idea can be
extended recursively so that one only needs to construct kernels over the ?atomic? parts of
a structured object. The recursive combination of the kernels over parts of an object retains
information regarding the structure of that object.
Several issues remain with the kernel we describe over trees and convolution kernels in
general. First, the value of $
will depend greatly on the size of the trees .
# $
One may normalize the kernel by using
which also satisfies the essential Mercer conditions. Second, the value of the kernel when
applied to two copies of the same tree can be extremely large (in our experiments on the
order of C! ) while the value of the kernel between two different trees is typically much
smaller (in our experiments the typical pairwise comparison is of order 100). By analogy
with a Gaussian kernel we say that the kernel is very peaked. If one constructs a model
which is a linear combination of trees, as one would with an SVM [6] or the perceptron,
the output will be dominated by the most similar tree and so the model will behave like
a nearest neighbor rule. There are several possible solutions to this problem. Following
Haussler [10] we may radialize the kernel, however, it is not always clear that the result is
still a valid kernel. Radializing did not appear to help in our experiments.
These problems motivate two simple modifications to the tree kernel. Since there will
be many more tree fragments of larger size ? say depth four versus depth three ? and
symbols in Figure 2.
consequently less training data, it makes sense to downweight the contribution of larger
tree fragments to the kernel. The first method for doing this is to simply restrict the depth
of the tree fragments we consider. The second method is to scale the relative importance of
,
tree fragments with their size. This can be achieved by introducing a parameter C
and modifying the base case and recursive case of the definitions of ? to be respectively
and ?
?
"?
!
$$
# $
, where
This corresponds to a modified kernel #
is the number of rules in the ?th fragment. This kernel downweights the contribution
of tree fragments exponentially with their size.
It is straightforward to design similar kernels for tagging problems (see figure 1) and for
another common structure found in NLP, dependency structures. See [5] for details. In the
tagging kernel, the implicit feature representation tracks all features consisting of a subsequence of state labels, each with or without an underlying word. For example, the paired sequence Lou/SP Gerstner/CP is/N chairman/N of/N IBM/SC would include features such as SP CP , SP Gerstner/CP N , SP CP is/N N of/N
and so on.
3 Linear Models for Parsing and Tagging
This section formalizes the use of kernels for parsing and tagging problems. The method
is derived by the transformation from ranking problems to a margin-based classification
problem in [8]. It is also related to the Markov Random Field methods for parsing suggested
in [13], and the boosting methods for parsing in [4]. We consider the following set-up:
B
Training data is a set of example input/output pairs. In parsing we would have training
examples
where each
is a sentence and each
is the correct tree for that sentence.
B
We assume some way of enumerating a set of candidates for a particular sentence. We
use
to denote the ?th candidate for the ?th sentence in training data, and
to denote the set of candidates for
.
B
Without loss of generality we take
to be the correct parse for
(i.e.,
).
B Each candidate
is represented by a feature vector
in the space . The param . We then define the ?ranking score? of each
eters of the model are also a vector
example as
. This score is interpreted as an indication of the plausibility of the
.
candidate. The output of the model on a training or test example is
!
!
"$#&%(')"$*
+ $, !
!
When considering approaches to training the parameter vector , note that a ranking function that correctly ranked the correct parse above all competing candidates would satisfy
4C
. It is simple to modify the Perceptron
the conditions $
and Support Vector Machine algorithms to treat this problem. For example, the SVM optimization problem (hard margin version) is to find the
which minimizes subject to
the constraints
. Rather than explicitly calculating
, the perceptron algorithm and Support Vector Machines can be formulated as a search
!
!
!
- /.
- :.
10 32 42 576
9! 8
;5 <2 =2 5>6
!
? This can be achieved using a modified dynamic programming table where 65 =>5 =@
stores
5)-=(58 of depth @ or less. The recursive <definition of can
the number of common subtrees at nodes
<
be modified
appropriately.
A A context-free
grammar ? perhaps taken straight from the training examples ? is one way of
enumerating candidates.
5 Another choice is to use a hand-crafted grammar (such as the LFG grammar
in [13]) or to take the most probable parses from an existing probabilistic parser (as in [4]).
-
- :.
-
Define:
Initialization: Set dual parameters
DC
For
If
do nothing, Else
- 10
-
6
Figure 3: The perceptron algorithm for ranking problems.
Depth
Score
Improvement
1
2
3
4
5
6
!
Table 1: Score shows how the parse score varies with the maximum depth of sub-tree
considered by the perceptron. Improvement is the relative reduction in error in comparison
to the PCFG, which scored 74%. The numbers reported are the mean and standard deviation
over the 10 development sets.
for ?dual parameters?
which determine the optimal weights
! 8
!
! 8
-
:.
$
(1)
(we use
as shorthand for
). It follows that the score of a parse can be
calculated using the dual parameters, and inner products between feature vectors, without
having to explicitly deal with feature or parameter vectors in the space
:
! 8
!
.
For example, see figure 3 for the perceptron algorithm applied to this problem.
4 Experimental Results
To demonstrate the utility of convolution kernels for natural language we applied our tree
kernel to the problem of parsing the Penn treebank ATIS corpus [14]. We split the treebank
randomly into a training set of size 800, a development set of size 200 and a test set of size
336. This was done 10 different ways to obtain statistically significant results. A PCFG
was trained on the training set, and a beam search was used to give a set of parses, with
PCFG probabilities, for each of the sentences. We applied a variant of the voted perceptron
algorithm [7], which is a more robust version of the original perceptron algorithm with
performance similar to that of SVMs. The voted perceptron can be kernelized in the same
way that SVMs can but it can be considerably more computationally efficient.
We generated a ranking problem by having the PCFG generate its top 100 candidate parse
trees for each sentence. The voted perceptron was applied, using the tree kernel described
previously, to this re-ranking problem. It was trained on 20 trees selected randomly from
the top 100 for each sentence and had to choose the best candidate from the top 100 on the
test set. We tested the sensitivity to two parameter settings: first, the maximum depth of
sub-tree examined, and second, the scaling factor used to down-weight deeper trees. For
each value of the parameters we trained on the training set and tested on the development
set. We report the results averaged over the development sets in Tables 1 and 2.
We report a parse score which combines precision and recall. Define
to be the number
of correctly placed constituents in the ?th test tree, "
to be the number of constituents
Scale
Score
Imp.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
!
Table 2: Score shows how the parse score varies with the scaling factor for deeper sub-trees
is varied. Imp. is the relative reduction in error in comparison to the PCFG, which scored
74%. The numbers reported are the mean and standard deviation over the 10 development
sets.
proposed, and
to be the number of constistuents in the true parse tree. A constituent is
defined by a non-terminal label and its span. The score is then
!
C C
"
6
The precision and recall on the ?th parse are
/"
and
/
respectively. The score is then
the average precision recall, weighted by the size of the trees
. We also give relative
improvements over the PCFG scores. If the PCFG score is
and the perceptron score is ,
$ C C
, i.e., the relative reduction in error.
the relative improvement is C C
.
.
We finally used the development set for cross-validation to choose the best parameter settings for each split. We used the best parameter settings (on the development sets) for each
split to train on both the training and development sets, then tested on the test set. This gave
C with the best choice of maximum depth and a score
a relative goodness score of
of
C with the best choice of scaling factor. The PCFG scored on the test data.
All of these results were obtained by running the perceptron through the training data only
once. As has been noted previously by Freund and Schapire [7], the voted perceptron often
obtains better results when run multiple times through the training data. Running through
the data twice with a maximum depth of 3 yielded a relative goodness score of ,
while using a larger number of iterations did not improve the results significantly.
In summary we observe that in these simple experiments the voted perceptron and an appropriate convolution kernel can obtain promising results. However there are other methods
which perform considerably better than a PCFG for NLP parsing ? see [3] for an overview
? future work will investigate whether the kernels in this paper give performance gains over
these methods.
5 A Compressed Representation
When used with algorithms such as the perceptron, convolution kernels may be even more
computationally attractive than the traditional radial basis or polynomial kernels. The linear
combination of parse trees constructed by the perceptron algorithm can be viewed as a
weighted forest. One may then search for subtrees in this weighted forest that occur more
than once. Given a linear combination of two trees " which contain a common
subtree, we may construct a smaller weighted acyclic graph, in which the common subtree
occurs only once and has weight . This process may be repeated until an arbitrary linear
combination of trees is collapsed into a weighted acyclic graph in which no subtree occurs
more than once. The perceptron may now be evaluated on a new tree by a straightforward
generalization of the tree kernel to weighted acyclic graphs of the form produced by this
procedure.
Given the nature of our data ? the parse trees have a high branching factor, the words are
chosen from a dictionary that is relatively small in comparison to the size of the training
data, and are drawn from a very skewed distribution, and the ancestors of leaves are part
of speech tags ? there are a relatively small number of subtrees in the lower levels of the
parse trees that occur frequently and make up the majority of the data. It appears that the
approach we have described above should save a considerable amount of computation. This
is something we intend to explore further in future work.
6 Conclusions
In this paper we described how convolution kernels can be used to apply standard kernel
based algorithms to problems in natural language. Tree structures are ubiquitous in natural language problems and we illustrated the approach by constructing a convolution kernel
over tree structures. The problem of parsing English sentences provides an appealing example domain and our experiments demonstrate the effectiveness of kernel-based approaches
to these problems. Convolution kernels combined with such techniques as kernel PCA and
spectral clustering may provide a computationally attractive approach to many other problems in natural language processing. Unfortunately, we are unable to expand on the many
potential applications in this short note, however, many of these issues are spelled out in a
longer Technical Report [5].
References
[1] Aizerman, M., Braverman, E., and Rozonoer, L. (1964). Theoretical Foundations of the Potential
Function Method in Pattern Recognition Learning. Automation and Remote Control, 25:821?837.
[2] Bod, R. (1998). Beyond Grammar: An Experience-Based Theory of Language. CSLI Publications/Cambridge University Press.
[3] Charniak, E. (1997). Statistical techniques for natural language parsing. In AI Magazine, Vol. 18,
No. 4.
[4] Collins, M. (2000). Discriminative Reranking for Natural Language Parsing. Proceedings of the
Seventeenth International Conference on Machine Learning. San Francisco: Morgan Kaufmann.
[5] Collins, M. and Duffy, N. (2001). Parsing with a Single Neuron: Convolution Kernels for Natural
Language Problems. Technical report UCSC-CRL-01-01, University of California at Santa Cruz.
[6] Cortes, C. and Vapnik, V. (1995). Support?Vector Networks. Machine Learning, 20(3):273?297.
[7] Freund, Y. and Schapire, R. (1999). Large Margin Classification using the Perceptron Algorithm.
In Machine Learning, 37(3):277?296.
[8] Freund, Y., Iyer, R.,Schapire, R.E., & Singer, Y. (1998). An efficient boosting algorithm for combining preferences. In Machine Learning: Proceedings of the Fifteenth International Conference.
San Francisco: Morgan Kaufmann.
[9] Goodman, J. (1996). Efficient algorithms for parsing the DOP model. In Proceedings of the
Conference on Empirical Methods in Natural Language Processing (EMNLP 96), pages 143-152.
[10] Haussler, D. (1999). Convolution Kernels on Discrete Structures. Technical report, University
of Santa Cruz.
[11] Johnson, M. The DOP estimation method is biased and inconsistent. To appear in Computational Linguistics.
[12] Lodhi, H., Christianini, N., Shawe-Taylor, J., and Watkins, C. (2001). Text Classification using
String Kernels. To appear in Advances in Neural Information Processing Systems 13, MIT Press.
[13] Johnson, M., Geman, S., Canon, S., Chi, S., & Riezler, S. (1999). Estimators for stochastic
?unification-based? grammars. In Proceedings of the 37th Annual Meeting of the Association for
Computational Linguistics. San Francisco: Morgan Kaufmann.
[14] Marcus, M., Santorini, B., & Marcinkiewicz, M. (1993). Building a large annotated corpus of
english: The Penn treebank. Computational Linguistics, 19, 313-330.
[15] Scholkopf, B., Smola, A.,and Muller, K.-R. (1999). Kernel principal component analysis. In B.
Scholkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods ? SV Learning,
pages 327-352. MIT Press, Cambridge, MA.
[16] Watkins, C. (2000). Dynamic alignment kernels. In A.J. Smola, P.L. Bartlett, B. Schlkopf, and
D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 39-50, MIT Press.
| 2089 |@word version:2 polynomial:3 lodhi:2 recursively:1 carry:1 reduction:3 contains:2 att:1 fragment:11 score:20 charniak:1 existing:1 com:1 must:1 parsing:19 cruz:3 generative:1 selected:1 leaf:1 reranking:1 short:1 num:1 characterization:1 boosting:2 cse:1 node:7 provides:1 preference:1 ucsc:2 constructed:1 scholkopf:2 shorthand:1 combine:2 pairwise:1 theoretically:1 tagging:6 frequently:1 terminal:5 chi:1 company:1 param:1 considering:2 begin:2 underlying:4 unrelated:1 kind:1 interpreted:1 string:8 minimizes:1 transformation:1 nj:1 formalizes:1 runtime:1 classifier:1 control:1 penn:2 appear:4 modify:1 treat:1 riezler:1 twice:1 initialization:1 studied:2 examined:1 hmms:2 pcfgs:4 statistically:1 averaged:1 seventeenth:1 atomic:1 recursive:9 practice:1 procedure:1 empirical:1 significantly:1 pre:3 word:3 radial:1 spite:1 cannot:1 close:1 context:4 applying:2 collapsed:1 restriction:1 optimize:1 equivalent:2 go:1 straightforward:2 identifying:1 occurences:3 rule:12 haussler:4 estimator:1 searching:1 justification:1 rozonoer:1 mcollins:1 construction:1 parser:1 magazine:1 exact:1 programming:1 us:1 approximated:1 recognition:1 geman:1 observed:1 capture:2 remote:1 substantial:1 complexity:1 dynamic:2 motivate:1 depend:3 solving:1 trained:3 algebra:1 basis:1 jersey:1 various:2 represented:3 train:1 describe:6 monte:1 sc:3 whose:2 quite:1 widely:1 plausible:1 larger:5 say:2 csli:1 otherwise:1 compressed:1 grammar:10 sequence:5 indication:1 propose:1 product:5 combining:1 subgraph:1 bod:4 normalize:1 constituent:3 spelled:1 object:10 tions:1 help:1 nearest:1 strong:1 implemented:1 involves:2 concentrate:1 correct:5 annotated:1 modifying:1 stochastic:2 require:3 generalization:1 marcinkiewicz:1 pessimistic:1 probable:1 considered:1 mapping:2 dictionary:1 estimation:7 label:2 sensitive:2 tool:1 weighted:6 mit:3 clearly:1 gaussian:1 always:1 modified:3 rather:1 publication:1 derived:1 emission:1 improvement:4 prevalent:1 likelihood:2 greatly:1 sense:1 typically:3 entire:1 hidden:8 kernelized:1 ancestor:1 expand:1 issue:4 classification:4 dual:3 proposes:1 development:8 noun:1 summed:1 fairly:1 field:1 construct:4 once:4 having:2 biology:1 identical:1 park:1 imp:2 peaked:1 future:2 np:12 others:1 report:5 modern:1 randomly:2 consisting:1 maintain:1 attempt:1 huge:1 highly:1 investigate:1 braverman:1 evaluation:1 alignment:1 subtrees:8 partial:1 unification:1 experience:1 tree:68 filled:1 taylor:1 re:1 theoretical:1 instance:1 retains:1 goodness:2 applicability:1 introducing:1 deviation:2 examining:2 johnson:2 reported:2 nigel:1 dependency:2 varies:2 sv:1 considerably:3 combined:1 person:2 international:2 sensitivity:1 probabilistic:2 decoding:1 michael:1 together:1 ambiguity:2 choose:2 emnlp:1 potential:2 includes:1 automation:1 satisfy:1 explicitly:3 ranking:6 lab:1 doing:1 start:2 contribution:2 formed:1 voted:6 kaufmann:3 efficiently:1 correspond:1 vp:2 conceptually:1 eters:1 produced:1 schlkopf:1 carlo:1 apple:5 straight:1 suffers:1 definition:7 pp:1 frequency:1 involved:1 gain:1 proved:1 recall:3 knowledge:1 ubiquitous:1 appears:1 higher:1 done:2 evaluated:1 generality:1 implicit:1 smola:3 until:1 hand:1 parse:21 lack:1 perhaps:1 building:1 contain:1 true:1 excluded:1 illustrated:1 deal:2 attractive:2 skewed:1 branching:1 covering:1 rooted:2 noted:1 criterion:3 demonstrate:2 cp:6 common:8 overview:1 attached:1 exponentially:2 extend:1 association:1 atis:3 refer:1 significant:1 cambridge:2 ai:1 trivially:1 neatly:1 language:14 shawe:1 had:1 dot:2 longer:1 surface:1 base:1 something:1 optimizing:1 store:1 continue:1 meeting:1 muller:1 seen:1 morgan:3 canon:1 determine:1 multiple:1 technical:3 calculation:2 downweights:1 plausibility:1 cross:1 paired:1 variant:1 regression:1 essentially:1 fifteenth:1 iteration:1 kernel:63 represent:2 achieved:2 beam:1 else:2 goodman:5 appropriately:1 biased:1 subject:1 member:1 inconsistent:1 effectiveness:1 structural:2 split:4 independence:1 gave:1 competing:2 restrict:1 inner:3 idea:3 regarding:1 avenue:1 enumerating:3 whether:1 pca:2 utility:1 bartlett:1 speech:2 useful:3 santa:3 involve:3 clear:2 amount:2 extensively:1 svms:2 deriva:1 generate:1 schapire:3 exist:1 estimated:2 track:3 correctly:2 discrete:5 vol:1 key:4 four:1 drawn:1 christianini:1 graph:4 convert:2 run:2 named:1 reasonable:1 scaling:3 yielded:1 annual:1 occur:4 constraint:1 tag:2 dominated:1 extremely:2 span:1 downweight:1 relatively:2 department:1 structured:4 combination:5 describes:1 ate:1 remain:1 smaller:2 appealing:1 modification:1 taken:1 computationally:5 previously:2 describing:1 count:5 mechanism:2 singer:1 operation:3 apply:1 observe:1 appropriate:2 spectral:1 save:1 alternative:2 original:2 top:3 running:4 nlp:12 include:1 clustering:1 linguistics:3 calculating:2 giving:1 intend:1 ingenious:1 occurs:2 chairman:6 traditional:1 lou:6 unable:1 entity:2 hmm:1 majority:1 marcus:1 modeled:2 unfortunately:1 design:2 implementation:1 perform:1 allowing:1 conversion:2 convolution:14 neuron:1 markov:2 behave:1 extended:1 santorini:1 dc:2 varied:1 verb:1 arbitrary:1 introduced:2 pair:2 sentence:10 california:2 learned:2 beyond:1 suggested:1 aizerman:1 pattern:1 natural:13 ranked:2 indicator:1 recursion:1 representing:1 improve:1 text:1 relative:10 freund:3 loss:1 par:3 dop:2 analogy:1 versus:1 acyclic:3 validation:1 foundation:1 mercer:3 principle:1 treebank:3 editor:2 share:1 ibm:6 production:9 preposition:1 summary:1 placed:1 last:1 free:3 copy:1 infeasible:1 english:2 allow:1 deeper:2 perceptron:21 ber:1 neighbor:1 burges:1 taking:2 boundary:1 calculated:3 depth:9 transition:1 valid:2 rich:1 san:3 obtains:1 implicitly:2 corpus:4 summing:1 francisco:3 discriminative:4 subsequence:1 search:3 table:4 promising:1 learn:1 nature:1 robust:1 forest:2 schuurmans:1 gerstner:7 constructing:1 domain:2 sp:6 did:2 whole:1 scored:3 nothing:1 child:6 repeated:1 crafted:1 precision:3 sub:12 exponential:2 candidate:9 watkins:3 down:1 symbol:1 disregarding:1 svm:2 cortes:1 evidence:1 essential:2 vapnik:1 pcfg:13 importance:1 subtree:4 iyer:1 duffy:2 margin:4 simply:3 explore:1 corresponds:1 satisfies:1 ma:1 identity:1 formulated:2 viewed:1 consequently:1 jeff:1 replace:1 crl:1 feasible:1 hard:1 considerable:1 included:1 infinite:1 typical:2 principal:2 experimental:3 support:4 collins:3 tested:3 |
1,195 | 209 | Unsupervised Learning in Neurodynamics
Unsupervised Learning in Neurodynamics Using
the Phase Velocity Field Approach
Michail Zak
Nikzad Toornarian
Center for Space Microelectronics Technology
Jet Propulsion Laboratory
California Institute of Technology
Pasadena, CA 91109
ABSTRACT
A new concept for unsupervised learning based upon examples introduced to the neural network is proposed. Each example is considered as an interpolation node of the velocity field in the phase
space. The velocities at these nodes are selected such that all the
streamlines converge to an attracting set imbedded in the subspace
occupied by the cluster of examples. The synaptic interconnections
are found from learning procedure providing selected field. The
theory is illustrated by examples.
This paper is devoted to development of a new concept for unsupervised learning
based upon examples introduced to an artificial neural network. The neural network
is considered as an adaptive nonlinear dissipative dynamical system described by
the following coupled differential equations:
N
Ui
+ K,Ui =
L 11j g(
Uj )
+ Ii
i=I,2, ... ,N
(I)
j=1
in which U is an N-dimensional vector, function of time, representing the neuron
activity, T is a constant matrix whose elements represent synaptic interconnections
between the neurons, 9 is a monotonic nonlinear function, Ii is the constant exterior
input to each neuron, and K, is a positive constant .
583
584
Zak and Toomarian
Let us consider a pattern vector u represented by its end point in an n-dimensional
phase space, and suppose that this pattern is introduced to the neural net in the
form of a set of vectors - examples u Ck ), k
1,2 ... K (Fig. 1). The difference
between these examples which represent the same pattern can be caused not only
by noisy measurements, but also by the invariance of the pattern to some changes
in the vector coordinates (for instance, to translations, rotations etc.). If the set
of the points u Ck ) is sufficiently dense, it can be considered as a finite-dimensional
approximation of some subspace OCl).
=
Now the goal of this study is formulated as following: find the synaptic interconnections 7ij and the input to the network h such that any trajectory which is
originated inside of OCl) will be entrapped there. In such a performance the subspace OCl) practically plays the role of the basin of attraction to the original pattern
U. However, the position of the attractor itself is not known in advance: the neural
net has to create it based upon the introduced representative examples. Moreover,
in general the attractor is not necessarily static: it can be periodic, or even chaotic.
The achievement of the goal formulated above would allow one to incorporate into
a neural net a set of attractors representing the corresponding clusters of patterns,
where each cluster is imbedded into the basin of its attractor. Any new pattern
introduced to such a neural net will be attracted to the "closest" attractor. Hence,
the neural net would learn by examples to perform content-addressable memory
and pattern recognition.
A
A
\
\
~-
Fig. 1: Two-Dimensional Vectors as Examples, uk, and Formation of Clusters O.
Unsupervised Learning in Neurodynamics
Our approach is based upon the utilization of the original clusters of the example
points u O:) as interpolation nodes of the velocity field in the phase space. The
assignment of a certain velocity to an example point imposes a corresponding constraint upon the synaptic interconnections Tij and the input Ii via Eq. (1). After
these unknowns are found, the velocity field in the phase space is determined by Eq.
(1). Hence, the main problem is to assign velocities at the point examples such that
the required dynamical behavior of the trajectories formulated above is provided.
One possibility for the velocity selection based upon the geometrical center approach
was analyzed by M. Zak, (1989). In this paper a "gravitational attraction" approach
to the same problem will be introduced and discussed.
Suppose that each example-point u(k) is attracted to all the other points u(k')(k' =j:.
k) such that its velocity is found by the same rule as a gravitational force:
v~k) = Vo
,
in which
Vo
u~k')
K
-
u~k)
?; [2:1 (u?') _ u??)2]3/2
(2)
=1
Ir'?Ir
is a constant scale coefficient.
Actual velocities at the same points are defined by Eq. (1) rearranged as:
N
u~k)
= 2: 7ijg( u~,,) -
uod -
IC( u~k)
- Uoi)
j=l
i= 1,2, ... ,N
k=1,2, ... ,J{
(3)
The objective is to find synaptic interconnections Tij and center of gravity Uoi such
that they minimize the distance between the assigned velocity (Eq. 2) and actual
calculated velocities (Eq. 3).
Introducing the energy:
(4)
one can find Tij and
Uoi
from the condition:
E-min
i.e., as the static attractor of the dynamical system:
?
uoi
=
?
T.. ?
2
-(k
8E
--
(5a)
8E
87ij
(5b)
8U oi
2
--(k--
') -
in which (k is a time scale parameter for learning. By appropriate selection of this
parameter the convergence of the dynamical system can be considerably improved
(J. Barhen, S. Gulati, and M. Zak, 1989).
585
586
Zak and Toomarian
Obviously, the static attractor of Eqs. (5) is unique. As follows from Eq. (3)
GU~k)
(k)
GU j
=
dg~k)
Iij (k)'
dU j
(i
i:- j)
(6)
d (Ie>
Since g(u) is a monotonic function, sgn.f.m is constant which in turn implies that
dU j
GU~k)
sgn
-W
=
Gu.
const
(i
i:- j)
(7)
1
Applying this result to the boundary of the cluster one concludes that the velocity
at the boundary is directed inside of the cluster (Fig. 2).
For numerical illustration of the new learning concept developed above, we select
6 points in the two dimensional space, (i.e., two neurons) which constructs two
separated clusters (Fig. 3, points 1-3 and 16-18 (three points are the minimum
to form a cluster in two dimensional space?. Coordinates of the points in Fig.
3 are given in Table 1. The assigned velocity vf calculated based on Eq. 2 and
Vo
0.04 are shown in dotted line. For a random initialization of Tij and Uoi,
the energy decreases sharply from an initial value of 10.608 to less than 0.04 in
about 400 iterations and at about 2000 iterations the final value of 0.0328 has been
achieved, (Fig. 4). To carry out numerical integration of the differential equations,
first order Euler numerical scheme with time step of 0.01 has been used. In this
simulation the scale parameter a 2 was kept constant and set to one. By substituting
the calculated Iij and Uoi into Eq. (3) for point uk, (k = 1,2,3,16,17,18), one
will obtain the calculated velocities at these points (shown as dashed lines in Fig.
3). As one may notice, the assigned and calculated velocities are not exactly the
same. However, this small difference between the velocities are of no importance as
long as the calculated velocities are directed toward the interior of the cluster. This
directional difference of the velocities is one of the reasons that the energy did not
vanish. The other reason is the difference in the value of these velocities, which is
of no importance either, based on the concept developed.
=
Fig. 2: Velocities at Boundaries are directed Toward Inside of the Cluster.
Unsupervised Learning in Neurodynamics
In order to show that for different initial conditions, Eq. 3 will converge to an
attractor which is inside one of the two clusters, this equation was started from different points (4-15,19-29). In all points, the equation converges to either (0.709,0.0)
or (-0.709,0.0). However, the line x
in this case is the dividing line, and all the
points on this line will converge to u o .
=
?
The decay coefficient", and the gain of the hyperbolic tangent were chosen to be
1. However, during the course of this simulation it was observed that the system
is very sensitive to these parameters as well as v o , which calls for further study in
this area.
15
29
14
4
20
9
Fig. 3:.
Cluster 1 (1-3) and Cluster 2 (16-19).
Calculated Velocity (- -)
?
Assigned Velocity ( .. )
?
Activation Dynamics initiated at different points.
7
587
588
Zak and Thomarian
Table 1. - Coordinate of Points in
point
X
Y point
X
1
0.50 0.00
16
-0.50
2
1.00 0.25
17
-1.00
3
1.00 -0.25
18
-1.00
4
1.25 0.25
19
-1.25
5
1.25 -0.25
20
-1.25
6
1.00 0.50
21
-1.00
7
1.00 -0.50
22
-1.00
8
0.75 0.50
23
-0.75
9
0.75 -0.50
24
-0.75
10
0.50 0.25
25
-0.50
11
0.50 -0.25
26
-0.50
12
0.25 0.10
27
-0.25
13
0.25 -0.10
28
-0.25
14
0.02 1.00
29
-0.02
15
0.00 1.00
Figure 4.
Y
0.00
0.25
0.25
0.25
-0.25
0.50
-0.50
0.50
-0.50
-0.25
-0.25
0.10
-0.10
1.00
\0
0
?
1"""'4
?
~
~
~
~
Z
C"1
?
I.I"t
~
?
o
??
..???
?..???
?
:\..........................::::
....~
....~
....=
.....""'. ...------,.-----~
o
100
200
300
ITERATIONS
Fig 4: Profile of Neuromorphic Energy over Time Iterations
Acknowledgement
This research was carried out at the Center for Space Microelectronic Technology,
Jet Propulsion Laboratory, California Institute of Technology. Support for the work
came from Agencies of the U.S. Department of Defense, including the Innovative
Science and Technology Office of the Strategic Defense Initiative Organization and
the Office of the Basic Energy Sciences of the US Dept. of Energy, through an
agreement with the National Aeronautics and Space Administration.
Unsupervised Learning in Neurodynamics
References
M. Zak (1989), "Unsupervised Learning in Neurondynamics Using Example Interaction Approach", Appl. Math. Letters, Vol. 2, No.3, pp. 381- 286.
J. Barhen, S. Gulati, M. Zak (1989), "Neural Learning of Constrained nonlinear
Transformations", IEEE Computer, Vol. 22(6), pp. 67-76.
589
| 209 |@word dividing:1 uj:1 concept:4 implies:1 hence:2 assigned:4 objective:1 laboratory:2 simulation:2 illustrated:1 imbedded:2 sgn:2 during:1 subspace:3 distance:1 carry:1 assign:1 initial:2 propulsion:2 vo:3 toward:2 reason:2 gravitational:2 practically:1 sufficiently:1 considered:3 ic:1 activation:1 geometrical:1 illustration:1 attracted:2 providing:1 rotation:1 numerical:3 substituting:1 discussed:1 selected:2 unknown:1 perform:1 measurement:1 neuron:4 sensitive:1 zak:8 finite:1 create:1 math:1 node:3 incorporate:1 ck:2 occupied:1 attracting:1 etc:1 aeronautics:1 differential:2 office:2 initiative:1 closest:1 introduced:6 required:1 inside:4 certain:1 california:2 came:1 behavior:1 minimum:1 dynamical:4 pattern:8 michail:1 actual:2 pasadena:1 converge:3 provided:1 dashed:1 moreover:1 toomarian:2 ii:3 memory:1 including:1 force:1 development:1 jet:2 constrained:1 integration:1 developed:2 long:1 dept:1 field:5 construct:1 transformation:1 representing:2 scheme:1 technology:5 unsupervised:8 basic:1 started:1 concludes:1 gravity:1 exactly:1 carried:1 coupled:1 iteration:4 uk:2 utilization:1 represent:2 achieved:1 acknowledgement:1 gulati:2 dg:1 tangent:1 positive:1 national:1 phase:5 attractor:8 initiated:1 interpolation:2 organization:1 possibility:1 initialization:1 dissipative:1 basin:2 imposes:1 appl:1 analyzed:1 barhen:2 call:1 translation:1 devoted:1 course:1 directed:3 unique:1 allow:1 chaotic:1 institute:2 procedure:1 addressable:1 administration:1 area:1 hyperbolic:1 defense:2 boundary:3 calculated:7 instance:1 adaptive:1 interior:1 selection:2 assignment:1 neuromorphic:1 applying:1 strategic:1 introducing:1 tij:4 euler:1 center:4 rearranged:1 periodic:1 considerably:1 rule:1 attraction:2 ijg:1 ocl:3 notice:1 dotted:1 ie:1 table:2 neurodynamics:5 learn:1 ca:1 coordinate:3 exterior:1 vol:2 suppose:2 play:1 du:2 necessarily:1 agreement:1 did:1 velocity:23 element:1 recognition:1 kept:1 dense:1 main:1 profile:1 observed:1 role:1 letter:1 fig:10 representative:1 coefficient:2 uoi:6 streamlines:1 caused:1 iij:2 decrease:1 position:1 originated:1 vf:1 agency:1 ui:2 microelectronic:1 vanish:1 dynamic:1 activity:1 minimize:1 oi:1 ir:2 constraint:1 sharply:1 upon:6 decay:1 gu:4 microelectronics:1 directional:1 represented:1 min:1 innovative:1 importance:2 trajectory:2 separated:1 department:1 artificial:1 formation:1 synaptic:5 whose:1 energy:6 pp:2 interconnection:5 static:3 monotonic:2 noisy:1 itself:1 final:1 gain:1 obviously:1 equation:4 net:5 turn:1 goal:2 formulated:3 interaction:1 end:1 content:1 change:1 determined:1 improved:1 appropriate:1 uod:1 invariance:1 achievement:1 entrapped:1 convergence:1 cluster:14 select:1 original:2 nonlinear:3 converges:1 support:1 const:1 ij:2 eq:10 |
1,196 | 2,090 | A Variational Approach to Learning Curves
D?orthe Malzahn
Manfred Opper
Neural Computing Research Group
School of Engineering and Applied Science
Aston University, Birmingham B4 7ET, United Kingdom.
[malzahnd,opperm]@aston.ac.uk
Abstract
We combine the replica approach from statistical physics with a variational approach to analyze learning curves analytically. We apply the
method to Gaussian process regression. As a main result we derive approximative relations between empirical error measures, the generalization error and the posterior variance.
1 Introduction
Approximate expressions for generalization errors for finite dimensional statistical data
models can be often obtained in the large data limit using asymptotic expansions. Such
methods can yield approximate relations for empirical and true errors which can be used
to assess the quality of the trained model see e.g. [1]. Unfortunately, such an approximation scheme does not seem to be easily applicable to popular non-parametric models
like Gaussian process (GP) models and Support Vector Machines (SVMs). We apply the
replica approach of statistical physics to asses the average case learning performance of
these kernel machines. So far, the tools of statistical physics have been successfully applied to a variety of learning problems [2]. However, this elegant method suffers from the
drawback that data averages can be performed exactly only under very idealistic assumptions on the data distribution in the ?thermodynamic? limit of infinite data space dimension.
We try to overcome these limitations by combining the replica method with a variational
approximation. For Bayesian models, our method allows us to express useful data averaged a-posteriori expectations by means of an approximate measure. The derivation of this
measure requires no assumptions about the data density and no assumptions about the input
dimension.
The main focus of this article is Gaussian process regression where we demonstrate the
various strengths of the presented method. It solves some of the problems stated at the end
of our previous NIPS paper [3] which was based on a simpler somewhat unmotivated truncation of a cumulant expansion. For Gaussian process models we show that our method
does not only give explicit approximations for generalization errors but also of their sample
fluctuations. Furthermore, we show how to compute corrections to our theory and demonstrate the possibility of deriving approximate universal relations between average empirical
and true errors which might be of practical interest.
An earlier version of our approach, which was still restricted to the assumption of idealized
data distributions appeared in [4].
2 Setup and Notation
We assume that
a class of elementary predictors (neural networks, regressors etc.) is given
by functions
. In a Bayesian formulation, we have a prior distribution over this class of
functions . Assuming
that a set of observations
is conditionally
independent
given inputs
, we assign a likelihood term of the form
to
each
observation.
Posterior
expectations
(denoted
by
angular
brackets)
of
any
functional
!#" $
are expressed in the form
) + ,.!#" $
, / 2 3 5464
*
(1)
10
*
+
where the partition function normalizes the posterior and denotes
% !#" $ &5the
8 9 expectation
with respect to the prior. We are interested in computing averages
7
of posterior
' "
<$
;
#
expectations over different drawings of training data sets :
% !#" $ &('
were all data examples are independently generated from the same distribution. In the next
section we will show how to derive a measure which enables us to compute analytically
approximate combined data and posterior averages.
3 A Grand-Canonical Approach
We utilize the statistical mechanics approach to>the
of learning. Our aim is to
=1? * analysis
8@9
which serves as a generating
compute the so-called averaged ?free energy? 7
*
function for suitable data averages of posterior expectations. The partition function is
, / . 3 4
(2)
10
=A? * 8B9
=1? * 8B9C'C=AD1EGFHJILK.M NPO Q.F SUR TWV
we use the replica trick 7
,
To perform
K
* F 8B9 the average 7
where 7
is computed for integer X and the continuation is performed at the end [5].
We obtain
F
* FYZ[ ' * F 8B9\' + F^]_G`
, / a . 54Jbc1de fg hi
(3)
7
a0
+ F
*
where
' +
denotes the expectation over the replicated prior measure.
Eq.(3) can be transformed
into a simpler form by introducing the ?grand canonical? partiF kl
tion function j
F kl 'n
/ m Zs r * F Z>t' + F vu F
(4)
0YIpo q
with the Hamiltonian
F
uwFx'y
`
, / l a z4Jbc1de fg
(5)
a0
oq
g
~F
R evaluates all8 c1X de freplicas
of the predictor at the same data
The
density
point
}
is taken with respect to the true data density ?
and theo|{expectation
7W
.
F?kl
The ?grand canonical? partition functionc j g
represents a ?poissonized? version
k of the
original model with fluctuating number of examples. The ?chemical potential? deterZ?' K.M N? R
k?'?=A?Z
mines the expected value of
which yields simply
for X???? . For
K
Z
q
sufficiently large , we can replace theq sum in Eq. (4) by its dominating term
=A? * FY Z>t??=A? FYkll??Zs =1??Z? U?Z?k
)
j
(6)
j
c g
c
g
thereby neglecting relative fluctuations. We recover the original (canonical) free energy as
K M NQ F R ? K6M N? R F M N
.
K
K
4 Variational Approximation
FYkl
For most interesting cases, the partition function j
can not be computed
in closed
u F
form for given X . Hence,
by a different
u F I we use a variational approach to approximate
tractable Hamiltonian
. It is easy to write down the firstu terms
F u inF I an expansion of the
?grand canonical? free energy with respect to the difference
>=A?
j
FYkl ' >=1? + F
%
&zI
o {}
I
R ? % uwFY?u F 5& I ? )
% u#F??u F I &5I; % uwFY?u F I & I
(7)
The brackets
denote
averages with respect to the effective measure which is induced
R and acts in the space of replicated variables. As is well known,
by the prior and
o;{.} terms in Eq.(7) present an upper bound [6] to >=A? j FYkl . Although
the first two leading
differentiating the bound with respect to X will
u F I usually not preserve the inequality, we still
is a sensible thing to do [7].
expect 1 that an optimization with respect to
4.1 Variational Equations
The grand-canonical ensemble was chosen
such that Eq.(5)u can
Z
|?as an" integral
$
F ' berewritten
a
over a local quantity in the input variable , i.e. in the form
with
F
s " a .<$ '
, / l a z4
?
a0
(8)
We will now specialize to Gaussian priors over , for which a local quadratic expression
u FI '
| /
a
) a . a
.l? /
a
%
a a .
&I
(9)
.
is a suitable
. trial Hamiltonian, leading to Gaussian averages
. The functions a
and a
are variational parameters
to be optimized. It is important to have an explicit de
pendence
on the input variable in order to take a non uniform input density into account.
To perform the variation of the first >
two
Eq.(7)
=A? terms
u F I ? we% uwnote
F that
u F I &zthe
I locality of Eq.(8)
+ F inzv
makes the ?variational free energy?
an explicit function
of the first two local moments
a . ' % a . .&5I
Hence, a straightforward variation yields
Z
% s &zI
'
.
a
a
Z
a . ' % a &5I
% ? . &zI
'
.
a
(10)
a
(11)
To extend the variational solutions to non-integer
values
?' of
. X , we assumeathat
.?for
' all. the
a
optimal
parameters
are
replica
symmetric,
ie.
as
well
as
for
'
' I|
notation for
a . and a .
and aa
. We also use a corresponding
1
Guided by the success of the method in physical applications, for instance in polymer physics.
4.2 Interpretation of
u FI
Note, that our approach
uwI is not equivalent to a variational approximation of the original
posterior. In contrast,
contains the full information of the statistics of the training data.
R in order to compute approximate
We can use the distribution induced by the prior and
.
{
}
o
combined data and posterior
averages.
As
an
example,
we
first consider the expected local
% .& % .& 8 9
. '
7
. Following the algebra of the replica
posterior variance
method (see [5]) this is approximated within the variational replica approach as
' |F =AD1H E I % a & I % a
. & I '
I .t
.
(12)
Second, we consider the noisy local mean square prediction error of the posterior mean
' % &
' .t 8@9
predictor
which is given by
7
. In this case
'
'
F =AD1H E I % a . &5I? % a .&5I
U ?? (
(13)
We can also calculate fluctuations with respect to the data average, for example
7
.t
| B
@8 9 ' F =AD1H E I
e ( e B @
0
0a a
I
(14)
5 Regression with Gaussian Processes
'
2 3 ?
, where is
This statistical model assumes that data
are generated as
Gaussian white
noise
t' with
. 58 { . The prior lover
functions
t' has .zero
mean and
+ variance
covariance
7
. Hence, we have
. Using the
definitions Eqs.(12,13), we get
% ? " a .<$ &5I 'y . 1= ? ?? | .?
?
)
X
.
?
?
(15)
{
which yields the set of variational equations (11). They become
particularly easy when the
and the input distribution
regression model uses a translationally invariant kernel
is homogeneous in a finite interval. The variational equations (11) can then be solved in
terms of the eigenvalues of the Gaussian process kernel.
[8, 9] studied learning curves for Gaussian process regression
which are not only averaged
over
the
data
but
also
over
the
data
generating
process
using
a Gaussian process prior on
. Applying these averages
of.[9]
to. our
theory
and
.adapting
? the notation
simply replaces
.
?
by
.
while
in Eq.(15) the term
5.1 Learning Curves and Fluctuations
Practical situations differ from this ?typical case? analysis. The data generating process
is unknown but assumed to be fixed. The resulting learning curve is then conditioned on
this particular ?teacher? . The left panel of Fig.1 shows an example. Displayed are the
mean
square prediction error (circle and solid line) and its sample fluctuations
(error bars)
with respect to the data average (cross and broken line). The target was a random
but fixed realization
p' "from
! az Gaussian
$process
# &% ' prior
' ' with a periodic Radial Basis Function
kernel
,
? ) . We keep the example
' simple,
' ? ?e.g) .
the Gaussian process regression model used the same kernel and noise)(
8
The inputs are one dimensional, independent and uniformly distributed { 7 ?
) . Symbols
represent simulation data. A typical property of our theory (lines) is that it becomes very
accurate for sufficiently large number of example data.
0
0
Theory: Lines
Simulation: Symbols
Correction of Free Energy
Generalization Error ?, Fluctuation ??
10
?1
10
?2
10
?
?3
10
?1
? =0.25
?1
?1
? =0.01
?1
? =0.0001
?2
?3
??
?4
10 0
100
50
150
Number m of Example Data
?4
0
200
20
40
60
80
Number m of Example Data
100
Figure 1: Gaussian
( process8 regression using a periodic Radial Basis Function kernel, input
7 ?
) , and homogeneous
dimension d=1,
density. Left: Generalization error
' ' input
and fluctuations for data noise
? ? ) . Right: Correction of the free energy.
{
Symbols: We subtracted the first two contributions
to Eq.(7) from the true value of the free
energy. The latter was obtained by simulations. Lines show the third contribution of Eq.(7).
decreases from top to bottom. All y-data was set equal
The value of the noise variance
{
to zero.
5.2 Corrections to the Variational Approximation
It is a strength of our method that the quality of the variational approximation Eq.(7) can
be characterized and systematically improved. In this paper, we restrict ourself to a characterization and
consider the case where all -data is set equal to zero. Since the posterior
variance
is independent of the data this is still an interesting model from which the
posterior variance can be estimated. We consider the third term in the expansion to the free
energy Eq.(7). It is a correction to the variational free energy and evaluates to
d ed
I
F & I 'y[) 7 I . I B ;
B58
Z
I
`
b d ed
??
(16)
{
& I
. Eq.(16)
is shown by lines in the
right panel of Fig.1 for different values of the model noise
. We considered a homo{
geneous input density, the input dimension is one and the regression
model uses a periodic
I
FA= DAH E I ) % u F u F & I % u F u
X
Z
?
` =A? ,
4Jb d e d ?
) l?^
{
' =1DAE FH I % a . a p a .
with
RBF kernel. The symbols in Fig.1 show the difference between the true value of the free
energy which is obtained by simulations and the first two terms of Eq.(7). The correction
term is found to be qualitatively accurate and emphasizes a discrepancy between free energy and the first two terms of the expansion Eq.(7) for a medium amount of example data.
The calculated learning curves inherit this behaviour.
5.3 Universal Relations
We can relate the training error and the empirical posterior variance
' ) ` /
Z A 0
b
9
' ) ` /
Z 1 0
; b
9
(17)
0.8
0.8
0.6
2
2
0.6
2
Theory
1d, periodic
2d, periodic
3d, periodic
2
[??(x,y)/(?? (x)+1) ](x,y)
1
[?? (x)/(?? (x)+1)]x
1
0.4
Theory
d=1, periodic
d=2, periodic
d=3, periodic
d=2+2, non-periodic
0.2
0
0
0.2
0.4
??T
2
0.8
0.6
0.4
0.2
0
0
1
0.2
0.4
??T
0.6
0.8
1
Figure 2: Illustration
of relation Eq.(19) (left) and Eq.(20) (right). All error measures are
scaled with . Symbols show simulation results
for Radial Basis Function (RBF) regression
x'
)
dimensions (square, circle, diamond).
and a homogeneous input distribution in
The RBF kernel was periodic. Additionally, the left figure shows an
?
example
were the
'
inputs lie on a quasi two-dimensional manifold which is embedded in
dimensions
(cross). In this case the RBF kernel was non-periodic.
?=A? *
89 '
?
to the free energy 7
. Using Eqs.(6,7) and the stationarity of
the grand-canonical free energy with respect to the variational parameters we obtain the
following relation
| 7
>=1? *
8B9 ?yZ
% ? " a $ &5I
X
|
(18)
We use the fact that the posterior variance is independent of the -data and simply estimate
it from the model where all -data is set equal to zero. In this case, Eq.(18) yields
'
?
)
?^
.
(19)
.
which relates
at
the empirical posterior variance to the local posterior variance
test inputs . Similarly, we can derive an expression for the training error by using
Eqs.(15,18) in combination with Eq.(19)
'
|
??
?
?
.
)
(20)
It is interesting to note, that the relations (19,20) contain no assumptions about the data
generating process. They hold in general for Gaussian process models with a Gaussian
likelihood. An illustration of Eqs.(19,20) is given by Fig.2 for the example of Gaussian
process regression with a Radial Basis Function kernel. In the left panel of Fig.2,
learning
starts in the upper right corner as the rescaled empirical posterior variance is initially
one and decreases with increasing number of example data. For the right panel of Fig.2,
learning starts in the lower left corner. The rescaled training error on the noisy data set
is initially zero and increases to one with increasing number of example data. The theory
(line) holds for a sufficiently large number of example data and its accuracy increases with
the input dimension. Eqs.(19,20) can also be tested on real data. For common benchmark
sets such as Abalone and Boston Housing data we find that Eqs.(19,20) hold well even for
small and medium sizes of the training data set.
6 Outlook
One may question if our approximate universal relations are of any practical use as, for
example, the relation between
training error and generalization error involves also the un .
known posterior variance
. Nevertheless, this relation could be useful for cases, where
a large number of data inputs without output labels are available. Since for regression, the
posterior variance
is independent of the output labels, we could use these extra input points
to estimate
.
The application of our technique to more complicated
# .models
. is possible and technically
) in Eq.(1) and further rescalby
more involved. For
example,
replacing
t' % o {
ing the kernel
of the Gaussian process prior
gives a model for hard
. The condition
margin Support Vector Machine Classification with SVM kernel
of maximum margin classification will be ensured by the limes ? .
Of particular interest is the computation of empirical estimators that can be used in practice
for model selection as well as the calculation of fluctuations (error bars) for such estimators.
A prominent example is an efficient approximate leave-one-out estimator for SVMs.
Work on these issues is in progress.
Acknowledgement
We would like to thank Peter Sollich for may inspiring discussions. The work was supported by EPSRC grant GR/M81601.
References
[1] N. Murata, S. Yoshizawa, S. Amari, IEEE Transactions on Neural Networks 5, p.
865-872, (1994).
[2] A. Engel, C. Van den Broeck, Statistical Mechanics of Learning, Cambridge University Press (2001).
[3] D. Malzahn, M. Opper, Neural Information Processing Systems 13, p. 273, T. K.
Leen, T. G. Dietterich and V. Tresp, eds., MIT Press, Cambridge MA (2001).
[4] D. Malzahn, M. Opper, Lecture Notes in Computer Science 2130, p. 271, G. Dorffner,
H. Bischof and K. Hornik, eds., Springer, Berlin (2001).
[5] M. M?ezard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond, World Scientific,
Singapore, (1987).
[6] R. P. Feynman and A. R. Hibbs, Quantum mechanics and path integrals, Mc GrawHill Inc., (1965).
[7] T. Garel, H. Orland, Europhys. Lett. 6, p. 307 (1988).
[8] P. Sollich, Neural Information Processing Systems 11, p. 344, M. S. Kearns, S. A.
Solla and D. A. Cohn, eds., MIT Press, Cambridge MA (1999).
[9] P. Sollich, Neural Information Processing Systems 14, T. G. Dietterich, S. Becker, Z.
Ghahramani, eds., MIT Press (2002).
| 2090 |@word trial:1 version:2 simulation:5 covariance:1 thereby:1 outlook:1 solid:1 moment:1 contains:1 united:1 partition:4 enables:1 hamiltonian:3 manfred:1 characterization:1 simpler:2 become:1 specialize:1 combine:1 expected:2 mechanic:3 increasing:2 becomes:1 notation:3 panel:4 medium:2 z:2 orland:1 act:1 exactly:1 ensured:1 scaled:1 uk:1 grant:1 engineering:1 local:6 limit:2 fluctuation:8 path:1 might:1 studied:1 averaged:3 practical:3 vu:1 practice:1 universal:3 empirical:7 adapting:1 radial:4 get:1 selection:1 applying:1 equivalent:1 straightforward:1 independently:1 estimator:3 deriving:1 variation:2 target:1 homogeneous:3 approximative:1 us:2 hjilk:1 trick:1 approximated:1 particularly:1 bottom:1 epsrc:1 solved:1 calculate:1 solla:1 decrease:2 rescaled:2 broken:1 mine:1 ezard:1 trained:1 algebra:1 technically:1 basis:4 easily:1 various:1 derivation:1 effective:1 europhys:1 dominating:1 drawing:1 amari:1 statistic:1 gp:1 noisy:2 housing:1 eigenvalue:1 parisi:1 combining:1 realization:1 opperm:1 generating:4 leave:1 derive:3 ac:1 school:1 progress:1 eq:25 solves:1 involves:1 differ:1 guided:1 drawback:1 behaviour:1 assign:1 generalization:6 polymer:1 elementary:1 correction:6 hold:3 sufficiently:3 b9:4 considered:1 idealistic:1 birmingham:1 applicable:1 label:2 engel:1 successfully:1 tool:1 mit:3 gaussian:18 aim:1 focus:1 likelihood:2 contrast:1 glass:1 posteriori:1 a0:3 initially:2 relation:10 transformed:1 quasi:1 interested:1 theq:1 classification:2 issue:1 denoted:1 equal:3 represents:1 discrepancy:1 jb:2 preserve:1 uwi:1 translationally:1 malzahnd:1 stationarity:1 interest:2 possibility:1 homo:1 bracket:2 pendence:1 accurate:2 integral:2 neglecting:1 circle:2 dae:1 virasoro:1 instance:1 earlier:1 introducing:1 predictor:3 uniform:1 gr:1 teacher:1 periodic:12 broeck:1 combined:2 density:6 grand:6 ie:1 physic:4 m81601:1 corner:2 leading:2 account:1 potential:1 de:1 inc:1 idealized:1 performed:2 try:1 tion:1 closed:1 analyze:1 start:2 recover:1 complicated:1 contribution:2 ass:2 square:3 spin:1 accuracy:1 variance:13 ensemble:1 yield:5 murata:1 bayesian:2 emphasizes:1 mc:1 suffers:1 ed:6 definition:1 evaluates:2 energy:13 npo:1 involved:1 yoshizawa:1 popular:1 improved:1 formulation:1 leen:1 furthermore:1 angular:1 replacing:1 cohn:1 quality:2 scientific:1 dietterich:2 contain:1 true:5 analytically:2 hence:3 chemical:1 symmetric:1 orthe:1 twv:1 white:1 conditionally:1 abalone:1 prominent:1 demonstrate:2 variational:17 fi:2 common:1 functional:1 physical:1 b4:1 extend:1 interpretation:1 cambridge:3 z4:1 similarly:1 etc:1 posterior:19 inf:1 inequality:1 success:1 somewhat:1 relates:1 thermodynamic:1 full:1 ing:1 characterized:1 calculation:1 cross:2 prediction:2 regression:11 expectation:7 kernel:12 represent:1 interval:1 extra:1 induced:2 elegant:1 thing:1 oq:1 lover:1 seem:1 integer:2 easy:2 variety:1 zi:3 restrict:1 expression:3 becker:1 peter:1 dorffner:1 hibbs:1 useful:2 amount:1 inspiring:1 svms:2 continuation:1 canonical:7 singapore:1 estimated:1 write:1 express:1 group:1 nevertheless:1 replica:8 utilize:1 sum:1 lime:1 bound:2 hi:1 quadratic:1 replaces:1 strength:2 combination:1 sollich:3 den:1 restricted:1 invariant:1 taken:1 equation:3 tractable:1 feynman:1 end:2 serf:1 available:1 rewritten:1 apply:2 fluctuating:1 subtracted:1 original:3 denotes:2 assumes:1 top:1 ghahramani:1 question:1 quantity:1 parametric:1 thank:1 berlin:1 sensible:1 ourself:1 manifold:1 fy:6 assuming:1 sur:1 illustration:2 kingdom:1 unfortunately:1 setup:1 relate:1 stated:1 unknown:1 perform:2 diamond:1 upper:2 observation:2 benchmark:1 finite:2 displayed:1 situation:1 kl:7 optimized:1 bischof:1 nip:1 malzahn:3 beyond:1 c1d:2 bar:2 usually:1 appeared:1 suitable:2 scheme:1 aston:2 tresp:1 prior:10 acknowledgement:1 asymptotic:1 relative:1 embedded:1 expect:1 lecture:1 interesting:3 limitation:1 article:1 systematically:1 normalizes:1 supported:1 truncation:1 free:13 theo:1 differentiating:1 distributed:1 van:1 curve:6 opper:3 dimension:7 overcome:1 calculated:1 world:1 quantum:1 lett:1 qualitatively:1 regressors:1 replicated:2 far:1 transaction:1 approximate:9 keep:1 assumed:1 un:1 additionally:1 hornik:1 expansion:5 inherit:1 main:2 noise:5 fig:6 explicit:3 lie:1 third:2 down:1 symbol:5 svm:1 conditioned:1 margin:2 boston:1 locality:1 simply:3 expressed:1 springer:1 ma:2 rbf:4 replace:1 hard:1 infinite:1 typical:2 uniformly:1 kearns:1 called:1 support:2 latter:1 cumulant:1 tested:1 |
1,197 | 2,091 | Fast and Robust Classification using Asymmetric
AdaBoost and a Detector Cascade
Paul Viola and Michael Jones
Mistubishi Electric Research Lab
Cambridge, MA
viola@merl.com and mjones@merl.com
Abstract
This paper develops a new approach for extremely fast detection in domains where the distribution of positive and negative examples is highly
skewed (e.g. face detection or database retrieval). In such domains a
cascade of simple classifiers each trained to achieve high detection rates
and modest false positive rates can yield a final detector with many desirable features: including high detection rates, very low false positive rates,
and fast performance. Achieving extremely high detection rates, rather
than low error, is not a task typically addressed by machine learning algorithms. We propose a new variant of AdaBoost as a mechanism for
training the simple classifiers used in the cascade. Experimental results
in the domain of face detection show the training algorithm yields significant improvements in performance over conventional AdaBoost. The
final face detection system can process 15 frames per second, achieves
over 90% detection, and a false positive rate of 1 in a 1,000,000.
1 Introduction
In many applications fast classification is almost as important as accurate classification.
Common examples include robotics, user interfaces, and classification in large databases.
In this paper we demonstrate our approach in the domain of low latency, sometimes called
?real-time?, face detection. An extremely fast face detector is a critical component in
many applications. User-interfaces can be constructed which detect the presence and number of users. Teleconference systems can automatically devote additional bandwidth to
participant?s faces. Video security systems can record facial images of individuals after
unauthorized entry.
Recently we presented a real-time face detection system which scans video images at 15
frames per second [8] yet achieves detection rates comparable with the best published results (e.g. [7]) 1 Face detection is a scanning process, in which a face classifier is evaluated
at every scale and location within each image. Since there are about 50,000 unique scales
1
In order to achieve real-time speeds other systems often resort to skin color filtering in color
images or motion filtering in video images. These simple queues are useful but unreliable. In large
image databases color and motion are often unavailable. Our system detects faces using only static
monochrome information.
and locations in a typical image, this amounts to evaluating the face classifier 750,000 times
per second.
One key contribution of our previous work was the introduction of a classifier cascade.
Each stage in this cascade was trained using AdaBoost until the required detection performance was achieved [2]. In this paper we present a new training algorithm designed
specifically for a classifier cascade called asymmetric AdaBoost. The algorithm is a generalization of that given in Singer and Shapire [6]. Many of the formal guarantees presented
by Singer and Shapire also hold for this new algorithm. The paper concludes with a set
of experiments in the domain of face detection demonstrating that asymmetric AdaBoost
yields a significant improvement in detection performance over conventional boosting.
2 Classifier Cascade
In the machine learning community it is well known that more complex classification functions yield lower training errors yet run the risk of poor generalization. If the main consideration is test set error, structural risk minimization provides a formal mechanism for
selecting a classifier with the right balance of complexity and training error [1].
Another significant consideration in classifier design is computational complexity. Since
time and error are fundamentally different quantities, no theory can simply select the optimal trade-off. Nevertheless, for many classification functions computation time is directly
related to the structural complexity. In this way temporal risk minimization is clearly related to structural risk minimization.
This direct analogy breaks down in domains where the distribution over the class labels
is highly skewed. For example, in the domain of face detection, there are at most a few
dozen faces among the 50,000 sub-windows in an image. Surprisingly in these domains
it is often possible to have the best of both worlds: high detection rates and extremely
fast classification. The key insight is that while it may be impossible to construct a simple
classifier which can achieve a low training/test error, in some cases it is possible to construct
a simple classifier with a very low false negative rate. For example, in the domain of face
detection, we have constructed an extremely fast classifier with a very low false negative
rate (i.e. it almost never misses a face) and a 50% false positive rate. Such a detector might
be more accurately called a classification pre-filter: when an image region is labeled ?nonface? then it can be immediately discarded, but when a region is labeled ?face? then further
classification effort is required. Such a pre-filter can be used as the first stage in a cascade
of classifiers (see Figure 1).
In our face detection application (described in more detail in Section 5) the cascade has
38 stages. Even though there are many stages, most are not evaluated for a typical nonface input window since the early stages weed out many non-faces. In fact, over a large
test set, the average number of stages evaluated is less than 2. In a cascade, computation
time and detection rate of the first few stages is critically important to overall performance.
The remainder of the paper describes techniques for training cascade classifiers which are
efficient yet effective.
3 Using Boosting to Train the Cascade
In general almost any form of classifier can be used to construct a cascade; the key properties are that computation time and the detection rate can be adjusted. Examples include
support vector machines, perceptrons, and nearest neighbor classifiers. In the case of an
SVM computation time is directly related to the number of support vectors and detection
rate is related to the margin threshold [1].
All Sub?windows
T
1
F
T
2
F
T
3
Further
Processing
F
Reject Sub?window
Figure 1: Schematic depiction of a detection cascade. A sequence of classifiers are applied
to every example. The initial classifier eliminates a large number of negative examples
with very little processing. Subsequent stages eliminate additional negatives but require
additional computation. Extremely few negative examples remain after several stages.
In our system each classifier in the cascade is a single layer perceptron whose input is a
set of computationally efficient binary features. The computational cost of each classifier
is then simply the number of input features. The detection rate is adjusted by changing the
threshold (or bias).
Much of the power of our face detection system comes from the very large and varied
set of features available. In our experiments over 6,000,000 different binary features were
available for inclusion in the final classifiers (see Figure 4 for some example features). The
efficiency of each classifier, and hence the efficiency of the cascade, is ensured because a
very small number of features are included in the early stages; the first stage has 1 (!) feature, the second stage 5 features, then 20, and then 50. See Section 5 for a brief description
of the feature set. The main contribution of this paper is the adaptation of AdaBoost for the
task of feature selection and classifier learning.
Though it is not widely appreciated, AdaBoost provides a principled and highly efficient
mechanism for feature selection[2, 6]. If the set of weak classifiers is simply the set of
binary features (this is often called boosting stumps) each round of boosting adds a single
feature to the set of current features.
AdaBoost is an iterative process in which each round selects a weak classifier,
minimizes:
!
$#&%
, which
(1)
is the weight on example
at round " ,
Following
(')*the
'+ notation of Shapire and Singer,
!,
is the target label of the example, is the example, and
is a confidence
rated binary classifier[6]. After every round the weights
are updated as follows:
-/. 01
2
3-465879;: <;=>=
(2)
CBDE5F7>9;: <G@H@
The classifier
takes on two possible values
and
<G@>= ,
<?=A@
where IKJ*L is the weight of the examples given the label which have true label M . These
predictions insure that the weights on the next round are balanced: that the relative weights
of positive and negative examples one each side of the classification boundary are equal.
Minimizing
minimizes the weighted exponential loss
round " . Minimizing
in each
atwhich
round is also a greedy technique for minimizing N
is an upper bound on the
training error of the strong classifier. It has also been observed that the example weights
are directly related to example margin, which leads to a principled argument for AdaBoost?s
generalization capabilities [5].
The key advantage of AdaBoost as a feature selection mechanism, over competitors such
as the wrapper method [3], is the speed of learning. Given the constraint that the search
,
over features is greedy, AdaBoost efficiently selects the feature which minimizes N
a surrogate for overall classification error. The entire dependence on previously selected
features is efficiently and compactly encoded using the example weights. As a result, the
addition of the 100th feature requires no more effort than the selection of the first feature. 2
4 Asymmetric AdaBoost
One limitation of AdaBoost arises in the context of skewed example distributions and cascaded classifiers: AdaBoost minimizes a quantity related to classification error; it does
not minimize the number of false negatives. Given that the final form of the classifier is
a weighted majority of features, the detection and false positive rates are adjustable after
training. Unfortunately feature selection proceeds as if classification error were the only
goal, and the features selected are not optimal for the task of rejecting negative examples.
One naive scheme for ?fixing? AdaBoost is to modify the initial distribution over the training examples. If we hope to minimize false negatives then the weight on positive examples
could be increased so that the minimum error criteria will also have very few false negatives. We can formalize this intuitive approach as follows. Recall that AdaBoost is a
scheme which minimizes:
$ /
A
(3)
Each term in the summation is bounded above by a simple loss function:
>
0$
'
)
if
otherwise
where
is the class assigned by the boosted classifier. As a result, minimizing
minimizes an upper bound on simple loss.
(4)
N
We can introduce a related notion of asymmetric
loss:
'
6
(' )
if
(and
'
1 E' )
01 . if
and
(5)
otherwise
where false negatives cost "
than false positives.
Note that
0 >!* 5879
times
0 . more
If we take the bound in Equation 4 and
by " we obtain a bound on the asymmetric loss:
multiply $# both sides
% .
H
Minimization of this bound
can be achieved using AdaBoost by pre-weighting each ex"
> 5879
ample by
. The derivation is identical to that of Equation 3. Expanding
2
Given that there are millions of features and thousands of examples, the boosting process requires
days of computation. Many other techniques while feasible for smaller problems are likely to be
infeasible for this sort of problem.
Equation 2 repeatedly for
0
-/. 1
B3. 0
we arrive
at,
H 5879
N
#
in terms of
"
)
(6)
where the second term in the numerator arises because of the initial asymmetric weighting.
Noticing that the left hand side must sum to 1 yields the following equality,
$ ! /
A D
F5 7>9
"
(7)
Therefore AdaBoost minimizes the required bound on asymmetric loss.
Unfortunately this naive technique is only somewhat effective. The main reason is AdaBoost?s balanced reweighting scheme. As a result the initially asymmetric example
weights are immediately lost. Essentially the AdaBoost process is too greedy. The first
classifier selected absorbs the entire effect of the initial asymmetric weights. The remaining rounds are entirely symmetric.
We propose a closely related approach that results in the minimization of the same bound,
throughout all rounds. Instead of applywhich nevertheless preserves the asymmetric loss
> 5879
ing the necessary asymmetric multiplier
at the first round of an round
"
. 5F7>9
"
process, the nth root
is applied before each round. Referring to Equation 6 we can see the final effect is the same; this preserves the bound on asymmetric loss.
But the effect on the training process is quite different. In order to demonstrate this approach we generated an artificial data set and learned strong classifiers containing 4 weak
classifiers. The results are shown inFigure 2. In this figure we can see that all but the
first weak classifier learned by the naive rule are poor, since they each balance positive and
negative errors. The final combination of these classifiers cannot yield high detection rates
without introducing many false positives. All the weak classifiers generated by the proposed Asymmetric Adaboost rule are consistent with asymmetric loss and the final strong
classifier yields very high detection rates and modest false positive rates.
One simple reinterpretation of this distributed scheme for asymmetric reweighting
,$ . 5F7>9 is as a
reduction in the positive confidence of each weak classifier
. This
forces each subsequent weak classifier to focus asymmetrically on postive examples.
5 Experiments
We performed two experiments in the domain of frontal face detection to demonstrate the
advantages of asymmetric AdaBoost. Experiments follow the general form, though differ
in details, from those presented in Viola and Jones [8]. In each round of boosting one of
a very large set of binary features are selected. These features, which we call rectangle
features, are briefly described in Figure 4.
In the first experiment a training and test set containing faces and non-faces of a fixed size
were acquired (faces were scaled to a size
pixels). The training set consisted of 1500
face examples and 5000 non-face examples. Test data included 900 faces and 5000 nonfaces. The face examples were manually cropped from a large collection of Web images
while the non-face examples were randomly chosen patches from Web images that were
known not to contain any faces.
Naive asymetric AdaBoost and three parameterizations of Asymmetric AdaBoost were
used to train classifiers with 4 features on this data. Figure 3 shows the ROC curves on
Figure 2: Two simple examples: positive examples are ?x?, negative ?o? and weak classifiers
are linear separators. On the left is the naive asymetric result. The first feature selected is
labelled ?1?. Subsequent features attempt to balance positive and negative errors. Notice
that no linear combination of the 4 weak classifiers can achieve a low false positive and
low false negative rate. On the right is the asymetric boosting result. After learning 4 weak
classifier the positives are well modelled and most of the negative are rejected.
0.995
0.99
0.985
0.98
0.975
NAIVE
T11-F10
T15-F10
T20-F10
0.97
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
Figure 3: ROC curves for four boosted classifier with 4 features. The first is naive asymmetric boosting. The other three results are for the new asymmetric approach, each using
slightly different parameters. The ROC curve has been cropped to show only the region
of interest in training a cascaded detector, the high detection rate regime. Notice that that
at 99% detection asymmetric Adaboost cuts the false positive by about 20%. This will
significantly reduce the work done by later stages in the cascade.
B
A
C
D
Figure 4: Left: Example rectangle features shown relative to the enclosing detection window. The sum of the pixels which lie within the white rectangles are subtracted from
the sum of pixels in the gray rectangles. A threshold operation is then applied to yield a
binary output. Two-rectangle features are shown in (A) and (B). Figure (C) shows a threerectangle feature, and (D) a four-rectangle feature. Right: The first two example feature
selected by the boosting process. Notice that the first feature relies on the fact that the
horizontal region of the eyes is darker than the horizontal region of the cheeks. The second feature, whose selection is conditioned on the first, acts to distinguish horizontal edges
from faces by looking for a strong vertical edge near the nose.
test data for the three classifiers. The key result here is that at high detection rates the false
positive rate can be reduced significantly.
In the second experiment, naive and asymmetric AdaBoost were used to train two different
complete cascaded face detectors. Performance of each cascade was determined on a realworld face detection task, which requires scanning of the cascade across a set of large
images which contain embedded faces.
The cascade training process is complex, and as a result comparing detection results is
useful but potentially risky. While the data used to train the two cascades were identical,
the performance of earlier stages effects the selection of non-faces used to train later stages.
As a result different non-face examples are used to train the corresponding stages for the
Naive and Asymmetric results.
Layers were added to each of the cascades until the number of false positives was reduced below 100 on a validation set. For normal boosting this occurred with 34 layers.
For asymmetric AdaBoost this occurred with 38 layers. Figure 5 shows the ROC curves
for the resulting face detectors on the MIT+CMU [4] test set. 3 Careful examination of
the ROC curves show that the asymmetric cascade reduces the number of false positives
significantly. At a detection rate of 91% the reduction is by a factor of 2.
6 Conclusions
We have demonstrated that a cascade classification framework can be used to achieve fast
classification, high detection rates, and very low false positive rates. The goal for each
classifier in the cascade is not low error, but instead extremely high detection rates and
modest false positive rates. If this is achieved, each classifier stage can be used to filter out
and discard many negatives.
3
Note: the detection and false positive rates for the simple 40 feature experiment and the more
complex cascaded experiment are not directly comparable, since the test sets are quite different.
ROC curves for face detector with different boosting algorithms
correct detection rate
0.95
0.9
0.85
Asymmetric Boosting
Normal Boosting
0.8
0
50
100
150
200
250
300
false positives
Figure 5: ROC curves comparing the accuracy of two full face detectors, one trained using
normal boosting and the other with asymmetric AdaBoost. Again, the detector trained
using asymmetric AdaBoost is more accurate over a wide range of false positive values.
Many modern approaches for classification focus entirely on the minimization of errors.
Questions of relative loss only arise in the final tuning of the classifier. We propose a
new training algorithm called asymmetric AdaBoost which performs learning and efficient
feature selection with the fundamental goal of achieving high detection rates. Asymmetric
AdaBoost is a simple modification of the ?confidence-rated? boosting approach of Singer
and Shapire. Many of their derivations apply to this new approach as well.
Experiments have demonstrated that asymmetric AdaBoost can lead to significant improvements both in classification speed and in detection rates.
References
[1] Corinna Cortes and Vladimir Vapnik. Support-vector networks. Machine Learning, 20, 1995.
[2] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning
and an application to boosting. In Computational Learning Theory: Eurocolt ?95, pages 23?37.
Springer-Verlag, 1995.
[3] G. John, R. Kohavi, and K. Pfleger. Irrelevant features and the subset selection problem. In
Machine Learning Conference, pages 121?129. Morgan Kaufmann, 1994.
[4] H. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. In IEEE Patt. Anal.
Mach. Intell., volume 20, pages 22?38, 1998.
[5] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee. Boosting the margin: a new explanation for
the effectiveness of voting methods. Ann. Stat., 26(5):1651?1686, 1998.
[6] Robert E. Schapire and Yoram Singer. Improved boosting algorithms using confidence-rated
predictions. Machine Learning, 37:297?336, 1999.
[7] H. Schneiderman and T. Kanade. A statistical method for 3D object detection applied to faces
and cars. In Computer Vision and Pattern Recognition, 2000.
[8] Paul Viola and Michael J. Jones. Robust real-time object detection. In Proc. of IEEE Workshop
on Statistical and Computational Theories of Vision, 2001.
| 2091 |@word briefly:1 reduction:2 initial:4 wrapper:1 pfleger:1 selecting:1 current:1 com:2 comparing:2 yet:3 must:1 john:1 subsequent:3 designed:1 greedy:3 selected:6 record:1 provides:2 parameterizations:1 boosting:18 location:2 constructed:2 direct:1 absorbs:1 introduce:1 acquired:1 detects:1 eurocolt:1 automatically:1 little:1 window:5 notation:1 insure:1 bounded:1 minimizes:7 guarantee:1 temporal:1 every:3 act:1 voting:1 ensured:1 classifier:49 scaled:1 positive:26 before:1 modify:1 mach:1 might:1 range:1 unique:1 lost:1 cascade:25 reject:1 significantly:3 pre:3 confidence:4 cannot:1 selection:9 risk:4 impossible:1 context:1 conventional:2 demonstrated:2 immediately:2 insight:1 rule:2 notion:1 updated:1 target:1 user:3 recognition:1 asymmetric:31 cut:1 database:3 labeled:2 observed:1 thousand:1 region:5 trade:1 principled:2 balanced:2 complexity:3 rowley:1 trained:4 reinterpretation:1 efficiency:2 compactly:1 derivation:2 train:6 fast:8 effective:2 artificial:1 whose:2 encoded:1 widely:1 quite:2 otherwise:2 final:8 sequence:1 advantage:2 propose:3 remainder:1 adaptation:1 achieve:5 f10:3 description:1 intuitive:1 object:2 stat:1 fixing:1 nearest:1 strong:4 come:1 differ:1 closely:1 correct:1 filter:3 require:1 generalization:4 summation:1 adjusted:2 hold:1 normal:3 achieves:2 early:2 f7:2 proc:1 label:4 weighted:2 minimization:6 hope:1 mit:1 clearly:1 rather:1 boosted:2 monochrome:1 focus:2 improvement:3 detect:1 typically:1 eliminate:1 entire:2 initially:1 selects:2 pixel:3 overall:2 classification:17 among:1 equal:1 construct:3 never:1 manually:1 identical:2 jones:3 develops:1 fundamentally:1 few:4 modern:1 randomly:1 preserve:2 intell:1 individual:1 attempt:1 detection:45 asymetric:3 interest:1 highly:3 multiply:1 accurate:2 edge:2 necessary:1 facial:1 modest:3 merl:2 increased:1 earlier:1 yoav:1 cost:2 introducing:1 entry:1 subset:1 too:1 scanning:2 referring:1 fundamental:1 lee:1 off:1 michael:2 again:1 containing:2 f5:1 resort:1 stump:1 performed:1 break:1 root:1 lab:1 later:2 sort:1 participant:1 capability:1 contribution:2 minimize:2 accuracy:1 kaufmann:1 efficiently:2 yield:8 weak:10 modelled:1 accurately:1 critically:1 rejecting:1 published:1 detector:10 competitor:1 static:1 recall:1 color:3 car:1 formalize:1 day:1 follow:1 adaboost:33 improved:1 evaluated:3 though:3 done:1 rejected:1 stage:17 until:2 hand:1 horizontal:3 web:2 reweighting:2 gray:1 b3:1 effect:4 consisted:1 true:1 multiplier:1 contain:2 hence:1 assigned:1 equality:1 symmetric:1 white:1 round:13 skewed:3 nonface:2 numerator:1 criterion:1 complete:1 demonstrate:3 theoretic:1 performs:1 motion:2 interface:2 image:12 consideration:2 recently:1 common:1 volume:1 million:1 occurred:2 significant:4 cambridge:1 tuning:1 inclusion:1 depiction:1 add:1 irrelevant:1 discard:1 verlag:1 binary:6 morgan:1 minimum:1 additional:3 somewhat:1 full:1 desirable:1 reduces:1 ing:1 retrieval:1 schematic:1 prediction:2 variant:1 vision:2 essentially:1 cmu:1 sometimes:1 robotics:1 achieved:3 addition:1 cropped:2 addressed:1 kohavi:1 eliminates:1 ample:1 effectiveness:1 call:1 structural:3 near:1 presence:1 bandwidth:1 reduce:1 bartlett:1 effort:2 queue:1 repeatedly:1 useful:2 latency:1 amount:1 reduced:2 schapire:3 notice:3 per:3 patt:1 key:5 four:2 demonstrating:1 nevertheless:2 achieving:2 threshold:3 changing:1 rectangle:6 sum:3 run:1 realworld:1 noticing:1 schneiderman:1 arrive:1 almost:3 throughout:1 cheek:1 patch:1 decision:1 comparable:2 entirely:2 layer:4 bound:8 distinguish:1 constraint:1 speed:3 argument:1 extremely:7 combination:2 poor:2 describes:1 remain:1 smaller:1 slightly:1 across:1 modification:1 ikj:1 computationally:1 equation:4 previously:1 mechanism:4 singer:5 nose:1 infigure:1 available:2 operation:1 apply:1 subtracted:1 corinna:1 remaining:1 include:2 t11:1 yoram:1 cbd:1 shapire:4 skin:1 added:1 quantity:2 question:1 dependence:1 surrogate:1 devote:1 majority:1 reason:1 balance:3 minimizing:4 vladimir:1 unfortunately:2 robert:2 potentially:1 negative:18 design:1 enclosing:1 anal:1 adjustable:1 upper:2 vertical:1 postive:1 discarded:1 viola:4 looking:1 frame:2 varied:1 community:1 required:3 security:1 learned:2 proceeds:1 below:1 pattern:1 regime:1 t20:1 including:1 video:3 explanation:1 power:1 critical:1 force:1 examination:1 cascaded:4 nth:1 scheme:4 rated:3 brief:1 eye:1 risky:1 concludes:1 naive:9 relative:3 embedded:1 loss:10 freund:2 limitation:1 filtering:2 analogy:1 validation:1 consistent:1 surprisingly:1 infeasible:1 appreciated:1 formal:2 bias:1 side:3 perceptron:1 neighbor:1 wide:1 face:41 distributed:1 boundary:1 curve:7 evaluating:1 world:1 collection:1 unreliable:1 search:1 iterative:1 kanade:2 robust:2 expanding:1 unavailable:1 complex:3 separator:1 electric:1 domain:10 main:3 paul:2 arise:1 weed:1 roc:7 darker:1 sub:3 exponential:1 lie:1 weighting:2 dozen:1 down:1 svm:1 cortes:1 workshop:1 false:25 vapnik:1 conditioned:1 margin:3 simply:3 likely:1 springer:1 relies:1 ma:1 goal:3 ann:1 careful:1 labelled:1 feasible:1 included:2 typical:2 specifically:1 determined:1 baluja:1 miss:1 called:5 asymmetrically:1 experimental:1 t15:1 perceptrons:1 select:1 support:3 scan:1 arises:2 frontal:1 ex:1 |
1,198 | 2,092 | On Spectral Clustering:
Analysis and an algorithm
Andrew Y. Ng
CS Division
U.C. Berkeley
ang@cs.berkeley.edu
Michael I. Jordan
CS Div. & Dept. of Stat.
U.C. Berkeley
jordan@cs.berkeley.edu
Yair Weiss
School of CS & Engr.
The Hebrew Univ.
yweiss@cs.huji.ac.il
Abstract
Despite many empirical successes of spectral clustering methodsalgorithms that cluster points using eigenvectors of matrices derived from the data- there are several unresolved issues. First,
there are a wide variety of algorithms that use the eigenvectors
in slightly different ways. Second, many of these algorithms have
no proof that they will actually compute a reasonable clustering.
In this paper, we present a simple spectral clustering algorithm
that can be implemented using a few lines of Matlab. Using tools
from matrix perturbation theory, we analyze the algorithm, and
give conditions under which it can be expected to do well. We
also show surprisingly good experimental results on a number of
challenging clustering problems.
1
Introduction
The task of finding good clusters has been the focus of considerable research in
machine learning and pattern recognition. For clustering points in Rn-a main application focus of this paper- one standard approach is based on generative models, in which algorithms such as EM are used to learn a mixture density. These
approaches suffer from several drawbacks. First, to use parametric density estimators , harsh simplifying assumptions usually need to be made (e.g., that the density
of each cluster is Gaussian) . Second, the log likelihood can have many local minima
and therefore multiple restarts are required to find a good solution using iterative
algorithms. Algorithms such as K-means have similar problems.
A promising alternative that has recently emerged in a number of fields is to use
spectral methods for clustering. Here, one uses the top eigenvectors of a matrix
derived from the distance between points. Such algorithms have been successfully
used in many applications including computer vision and VLSI design [5, 1]. But
despite their empirical successes, different authors still disagree on exactly which
eigenvectors to use and how to derive clusters from them (see [11] for a review).
Also, the analysis of these algorithms, which we briefly review below, has tended to
focus on simplified algorithms that only use one eigenvector at a time.
One line of analysis makes the link to spectral graph partitioning, in which the sec-
ond eigenvector of a graph's Laplacian is used to define a semi-optimal cut. Here,
the eigenvector is seen as a solving a relaxation of an NP-hard discrete graph partitioning problem [3], and it can be shown that cuts based on the second eigenvector
give a guaranteed approximation to the optimal cut [9, 3]. This analysis can be
extended to clustering by building a weighted graph in which nodes correspond to
datapoints and edges are related to the distance between the points. Since the majority of analyses in spectral graph partitioning appear to deal with partitioning the
graph into exactly two parts, these methods are then typically applied recursively
to find k clusters (e.g. [9]). Experimentally it has been observed that using more
eigenvectors and directly computing a k way partitioning is better (e.g. [5, I]).
Here, we build upon the recent work of Weiss [11] and Meila and Shi [6], who
analyzed algorithms that use k eigenvectors simultaneously in simple settings. We
propose a particular manner to use the k eigenvectors simultaneously, and give
conditions under which the algorithm can be expected to do well.
2
Algorithm
Given a set of points S =
{81' ... ,8 n }
in
jRl
that we want to cluster into k subsets:
1. Form the affinity matrix A E R nx n defined by A ij
i # j , and A ii = O.
= exp(-Ilsi
- sjW/2( 2 ) if
2. Define D to be the diagonal matrix whose (i , i)-element is the sum of A's i-th
row, and construct the matrix L = D-l / 2AD-l / 2 . 1
3. Find Xl , X2 , ... , Xk , the k largest eigenvectors of L (chosen to be orthogonal
to each other in the case of repeated eigenvalues), and form the matrix X =
[XIX2 . . . Xk) E R n xk by stacking the eigenvectors in columns.
4. Form the matrix Y from X by renormalizing each of X's rows to have unit length
(i.e. Yij = X ij/CL.j X~)1 / 2).
5. Treating each row of Y as a point in Rk , cluster them into k clusters via K-means
or any other algorithm (that attempts to minimize distortion).
6. Finally, assign the original point Si to cluster j if and only if row i of the matrix
Y was assigned to cluster j.
Here, the scaling parameter a 2 controls how rapidly the affinity Aij falls off with
the distance between 8i and 8j, and we will later describe a method for choosing
it automatically. We also note that this is only one of a large family of possible
algorithms, and later discuss some related methods (e.g., [6]).
At first sight, this algorithm seems to make little sense. Since we run K-means
in step 5, why not just apply K-means directly to the data? Figure Ie shows an
example. The natural clusters in jR2 do not correspond to convex regions, and Kmeans run directly finds the unsatisfactory clustering in Figure li. But once we map
the points to jRk (Y 's rows) , they form tight clusters (Figure lh) from which our
method obtains the good clustering shown in Figure Ie. We note that the clusters
in Figure lh lie at 90 0 to each other relative to the origin (cf. [8]).
lReaders familiar with spectral graph theory [3) may be more familiar with the Laplacian 1- L. But as replacing L with 1- L would complicate our later discussion, and only
changes the eigenvalues (from Ai to 1 - Ai ) and not the eigenvectors, we instead use L .
3
3.1
Analysis of algorithm
Informal discussion: The "ideal" case
To understand the algorithm, it is instructive to consider its behavior in the "ideal"
case in which all points in different clusters are infinitely far apart. For the sake of
discussion, suppose that k = 3, and that the three clusters of sizes n1, n2 and n3
are 8 1 ,82 , and 8 3 (8 = 8 1 U 8 2 U 8 3 , n = n1 +n2 + n3)' To simplify our exposition,
also assume that the points in 8 = {Sl,'" ,Sn} are ordered according to which
cluster they are in, so that the first n1 points are in cluster 8 1 , the next n2 in 8 2 ,
etc. We will also use "j E 8/' as a shorthand for s? E 8 i . Moving the clusters
"infinitely" far apart corresponds to zeroing all the efements Aij corresponding to
points Si and Sj in different clusters. More precisely, define Aij = 0 if Xi and Xj are
in different clusters, and Aij = Aij otherwise. Also let t , D , X and Y be defined
as in the previous algorithm, but starting with A instead of A. Note that A and t
are therefore block-diagonal:
A. =
[
A(ll)
0
0
A(22)
o
o
o
1;
A
L
=
[L(11)
0
o
?(22)
(1)
A~~
0
0
o
where we have adopted the convention of using parenthesized superscripts to index
into subblocks of vectors/matrices, and Lrii) = (D(ii)) - 1/2A(ii) (D(ii)) - 1/2. Here,
A(ii) = A(ii) E jRni xni is the matrix of "intra-cluster" affinities for cluster i. For future use, also define d(i) E jRni to be the vector containing D(ii) 's diagonal elements,
and dE jRn to contain D's diagonal elements.
To construct X, we find t's first k = 3 eigenvectors. Since t is block diagonal, its
eigenvalues and eigenvectors are the union of the ei~envalues and eigenvectors of its
blocks (the latter padded appropriately with zeros). It is straightforward to show
that Lrii) has a strictly positive principal eigenvector xii) E jRni with eigenvalue
1. Also, since A)~) > 0 (j i:- k), the next eigenvalue is strictly less than 1. (See,
e.g., [3]). Thus, stacking t 's eigenvectors in columns to obtain
X=
xi1)
[
0
o
xi 2)
0
0
0
0
xi 3)
X,
we have:
1
E jRnx3.
(2)
Actually, a subtlety needs to be addressed here. Since 1 is a repeated eigenvalue
in t, we could just as easily have picked any other 3 orthogonal vectors spanning
the same subspace as X's columns, and defined them to be our first 3 eigenvectors.
That is, X could have been replaced by XR for any orthogonal matrix R E jR3X3
(RT R = RRT = 1). Note that this immediately suggests that one use considerable
caution in attempting to interpret the individual eigenvectors of L, as the choice
of X's columns is arbitrary up to a rotation, and can easily change due to small
perturbations to A or even differences in the implementation of the eigensolvers.
Instead, what we can reasonably hope to guarantee about the algorithm will be
arrived at not by considering the (unstable) individual columns of X, but instead
the subspace spanned by the columns of X, which can be considerably more stable.
Next, when we renormalize each of X's rows to have unit length, we obtain:
y=
[
y(l)
y(2)
jRni xk
0r 00
1
R
(3)
to denote the i-th subblock of Y. Letting
fiji)
y(3)
where we have used y(i) E
1 [r0
0 0
r
denote the j-th row of17(i) , we therefore see that fjY) is the i-th row ofthe orthogonal
matrix R. This gives us the following proposition.
Proposition 1 Let A's off-diagonal blocks A(i j ) , i =I- j, be zero. Also assume
that each cluster Si is connected. 2 Then there exist k orthogonal vectors 1'1, . .. ,1' k
(1'; l' j = 1 if i = j, 0 otherwise) so that Y's rows satisfy
, (i)
~
( )
4
=G
for all i = 1, ... ,k, j = 1, ... ,ni.
In other words , there are k mutually orthogonal points on the surface of the unit
k-sphere around which Y 's rows will cluster. Moreover, these clusters correspond
exactly to the true clustering of the original data.
3.2
The general case
In the general case, A's off-diagonal blocks are non-zero, but we still hope to recover
guarantees similar to Proposition 1. Viewing E = A - A as a perturbation to the
"ideal" A that results in A = A+E, we ask: When can we expect the resulting rows
of Y to cluster similarly to the rows of Y? Specifically, when will the eigenvectors
of L, which we now view as a perturbed version of L, be "close" to those of L?
Matrix perturbation theory [10] indicates that the stability of the eigenvectors of a
matrix is determined by the eigengap. More precisely, the subspace spanned by L's
first 3 eigenvectors will be stable to small changes to L if and only if the eigengap
8 = IA3 - A41, the difference between the 3rd and 4th eigenvalues of L, is large. As
discussed previously, the eigenvalues of L is the union of the eigenvalues of D11),
D22), and D33), and A3 = 1. Letting Ay) be the j-th largest eigenvalue of Dii), we
therefore see that A4 = maxi A~i). Hence, the assumption that IA3 - A41 be large is
exactly the assumption that maXi A~i) be bounded away from 1.
Assumption AI. There exists 8
> 0 so that, for all i
= 1, ... ,k, A~i)
:s: 1 -
8.
Note that A~i) depends only on Dii), which in turn depends only on A(ii) = A(ii) ,
the matrix of intra-cluster similarities for cluster Si' The assumption on A~i) has a
very natural interpretation in the context of clustering. Informally, it captures the
idea that if we want an algorithm to find the clusters Sl, S2 and S3, then we require
that each of these sets Si really look like a "tight" cluster. Consider an example
in which Sl = S1.1 U S1.2 , where S1.1 and S1.2 are themselves two well separated
clusters. Then S = S1.1 U S1.2 U S2 U S3 looks like (at least) four clusters, and it
would be unreasonable to expect an algorithm to correctly guess what partition of
the four clusters into three subsets we had in mind.
This connection between the eigengap and the cohesiveness of the individual clusters
can be formalized in a number of ways.
Assumption ALl. Define the Cheeger constant [3] of the cluster Si to be
_.
h(S.) - mmI
~lE I, kIi' I A;;.,')
'(.) ~
,(.)}.
. {~
mm
lEI
d
,
kli'I
where the outer minimum is over all index subsets I
there exists 8 > 0 so that (h(Si))2 /2 ~ 8 for all i.
2This condition is satisfied by A.j~)
> 0 (j i- k) , which
(5)
dk
~
{I, ...
,nd.
Assume that
is true in our case.
A standard result in spectral graph theory shows that Assumption Al.l implies
Assumption Al. Recall that d)i) = 2:k A)~) characterizes how "well connected"
or how "similar" point j is to the other points in the same cluster. The term in
the minI{?} characterizes how well (I , I) partitions Si into two subsets, and the
minimum over I picks out the best such partition. Specifically, if there is a partition
of Si'S points so that the weight of the edges across the partition is small, and so
that each of the partitions has moderately large "volume" (sum of dY) 's), then the
Cheeger constant will be small. Thus, the assumption that the Cheeger constants
h(Si) be large is exactly that the clusters Si be hard to split into two subsets.
We can also relate the eigengap to the mixing time of a random walk (as in [6])
defined on the points of a cluster, in which the chance of transitioning from point i
to j is proportional to A ij , so that we tend to jump to nearby-points. Assumption
Al is equivalent to assuming that, for such a walk defined on the points of any
one of the clusters Si , the corresponding transition matrix has second eigenvalue at
most 1- 8. The mixing time of a random walk is governed by the second eigenvalue;
thus, this assumption is exactly that the walks mix rapidly. Intuitively, this will be
true for tight (or at least fairly "well connected") clusters, and untrue if a cluster
consists of two well-separated sets of points so that the random walk takes a long
time to transition from one half of the cluster to the other. Assumption Al can also
be related to the existence of multiple paths between any two points in the same
cluster.
Assumption A2. There is some fixed fl
i l =j:. i 2, we have that
> 0, so that for every iI , i2
E {I, ... ,k} ,
(6)
To gain intuition about this, consider the case of two "dense" clusters il and i2 of
size O(n) each. Since dj measures how "connected" point j is to other points in
the same cluster, it will be dj = O(n) in this case, so the sum, which is over 0(n 2 )
terms , is in turn divided by djdk = O(n 2 ) . Thus, as long as the individual Ajk's
are small, the sum will also be small, and the assumption will hold with small fl.
Whereas dj measures how connected Sj E Si is to the rest of Si, 2:k:k'itSi Ajk
measures how connected Sj is to points in other clusters. The next assumption is
that all points must be more connected to points in the same cluster than to points
in other clusters; specifically, that the ratio between these two quantities be small.
Assumption A3. For some fixed f2 > 0, for every i = 1, ... ,k, j E Si, we have:
(7)
For intuition about this assumption, again consider the case of densely connected
clusters (as we did previously). Here, the quantity in parentheses on the right hand
side is 0(1), so this becomes equivalent to demanding that the following ratio be
small: (2:k:k'it Si Ajk)/dj = (2: k:k'it Si Ajk)/(2:k:kESi A jk ) = 0(f2) .
Assumption A4. There is some constant C >
. _
' (i)
ni
' (i )
J - 1, ... ,ni, we have dj ~ (2: k =l dk )/(Cni).
?
so that for every i = 1, .. . ,k,
This last assumption is a fairly benign one that no points in a cluster be "too much
less" connected than other points in the same cluster.
Theorem 2 Let assumptions Al, A2, A3 and A4 hold. Set f = Jk(k - l)fl
+ kE~.
If 0 > (2 + V2}::, then there exist k orthogonal vectors rl, . .. , rk (rF r j = I if i = j,
o otherwise) so that Y's rows satisfy
(8)
Thus, the rows of Y will form tight clusters around k well-separated points (at 90 0
from each other) on the surface of the k-sphere according to their "true" cluster Si.
4
Experiments
To test our algorithm, we applied it to seven clustering problems. Note that whereas
was previously described as a human-specified parameter, the analysis also suggests a particularly simple way of choosing it automatically: For the right (J2,
Theorem 2 predicts that the rows of Y will form k "tight" clusters on the surface
of the k-sphere. Thus, we simply search over (J2 , and pick the value that, after
clustering Y 's rows, gives the tightest (smallest distortion) clusters. K-means in
Step 5 of the algorithm was also inexpensively initialized using the prior knowledge
that the clusters are about 90 0 apart. 3 The results of our algorithm are shown in
Figure l a-g. Giving the algorithm only the coordinates of the points and k, the
different clusters found are shown in the Figure via the different symbols (and colors, where available). The results are surprisingly good: Even for clusters that do
not form convex regions or that are not cleanly separated (such as in Figure 19) ,
the algorithm reliably finds clusterings consistent with what a human would have
chosen.
(J2
We note that there are other, related algorithms that can give good results on a
subset of these problems, but we are aware of no equally simple algorithm that
can give results comparable to these. For example, we noted earlier how K-means
easily fails when clusters do not correspond to convex regions (Figure Ii). Another
alternative may be a simple "connected components" algorithm that, for a threshold
T, draws an edge between points Si and Sj whenever Iisi - sjl12 :s: T, and takes the
resulting connected components to be the clusters. Here, T is a parameter that can
(say) be optimized to obtain the desired number of clusters k. The result of this
algorithm on the threecircles-j oined dataset with k = 3 is shown in Figure lj.
One of the "clusters" it found consists of a singleton point at (1.5,2). It is clear
that this method is very non-robust.
We also compare our method to the algorithm of Meila and Shi [6] (see Figure lk).
Their method is similar to ours, except for the seemingly cosmetic difference that
they normalize A's rows to sum to I and use its eigenvectors instead of L 's, and do
not renormalize the rows of X to unit length. A refinement of our analysis suggests
that this method might be susceptible to bad clusterings when the degree to which
different clusters are connected (L: j d;il) varies substantially across clusters.
3 Briefiy, we let the first cluster centroid be a randomly chosen row of Y , and then
repeatedly choose as the next centroid the row of Y that is closest to being 90? from
all the centroids (formally, from the worst-case centroid) already picked. The resulting
K-means was run only once (no restarts) to give the results presented. K-means with the
more conventional random initialization and a small number of restarts also gave identical
results. In contrast, our implementation of Meila and Shi 's algorithm used 2000 restarts.
flips,8clusten
o
o
(a)
(b)
(c)
th reeci~es-joiJ\ed,2c1ust 8fS
squigg les, 4 clusteNl
(d)
Ih reecirdes_joined,3clusters
(e)
RowsoJYOittered , rarKlomlysubsa m pled) lorlW<lCirc~
(i)
~nea r.dballs , 3 dus\efs(Meil a and Shi algor1lhm)
o
o q,
o~ 00 0
o~
~&
0
o
o
0
&~llO
~~
o
0
o
(j)
lWo circles, 2 cluSle<S (K_means)
(h)
(g)
threecircles-joined, 3 clusters(conoecled """'l""'enlS)
(f)
(k)
flips, 6 cluste<s (Kannan elal ,aigor;thm)
N
(I)
Figure 1: Clustering examples, with clusters indicated by different symbols (and colors,
where available). (a-g) Results from our algorithm, where the only parameter varied across
runs was k. (h) Rows of Y (jittered, subsampled) for twocircles dataset . (i) K-means.
(j) A "connected components" algorithm. (k) Meila and Shi algorithm. (1) Kannan et al.
Spectral Algorithm I. (See text.)
5
Discussion
There are some intriguing similarities between spectral clustering methods and Kernel peA, which has been empirically observed to perform clustering [7, 2]. The main
difference between the first steps of our algorithm and Kernel PCA with a Gaussian
kernel is the normalization of A (to form L) and X. These normalizations do improve the performance of the algorithm, but it is also straightforward to extend our
analysis to prove conditions under which Kernel PCA will indeed give clustering.
While different in detail , Kannan et al. [4] give an analysis of spectral clustering
that also makes use of matrix perturbation theory, for the case of an affinity matrix
with row sums equal to one. They also present a clustering algorithm based on
k singular vectors , one that differs from ours in that it identifies clusters with
individual singular vectors. In our experiments, that algorithm very frequently
gave poor results (e.g., Figure 11).
Acknowledgments
We thank Marina Meila for helpful conversations about this work. We also thank
Alice Zheng for helpful comments. A. Ng is supported by a Microsoft Research
fellowship. This work was also supported by a grant from Intel Corporation, NSF
grant IIS-9988642, and ONR MURI N00014-00-1-0637.
References
[1] C. Alpert, A. Kahng, and S. Yao. Spectral partitioning: The more eigenvectors, the
better. Discrete Applied Math , 90:3- 26, 1999.
[2] N. Christianini, J. Shawe-Taylor, and J. Kandola. Spectral kernel methods for clustering. In Neural Information Processing Systems 14, 2002.
[3] F. Chung. Spectral Graph Theory. Number 92 in CBMS Regional Conference Series
in Mathematics. American Mathematical Society, 1997.
[4] R. Kannan, S. Vempala, and A. Yetta. On clusterings- good, bad and spectral.
In Proceedings of the 41st Annual Symposium on Foundations of Computer Science,
2000.
[5] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for image
segmentation. In Perceptual Organization for Artificial Vision Systems. Kluwer, 2000.
[6] M. Meila and J. Shi. Learning segmentation by random walks. In N eural Information
Processing Systems 13, 200l.
[7] B. Scholkopf, A. Smola, and K.-R Miiller. Nonlinear component analysis as a kernel
eigenvalue problem. N eural Computation, 10:1299- 1319, 1998.
[8] G. Scott and H. Longuet-Higgins. Feature grouping by relocalisation of eigenvectors
of the proximity m atrix. In Proc. British Machine Vision Conference, 1990.
[9] D. Spielman and S. Teng. Spectral partitioning works: Planar graphs and finite
element meshes. In Proceedings of the 37th Annual Symposium on Foundations of
Computer Science, 1996.
[10] G. W. Stewart and J.-G. Sun. Matrix Perturbation Th eory. Academic Press, 1990.
[11] Y . Weiss. Segmentation using eigenvectors: A unifying view. In International Conf erence on Computer Vision, 1999.
| 2092 |@word version:1 briefly:1 seems:1 nd:1 cleanly:1 llo:1 simplifying:1 pick:2 atrix:1 recursively:1 eigensolvers:1 series:1 ours:2 si:19 intriguing:1 must:1 mesh:1 partition:6 benign:1 treating:1 rrt:1 generative:1 half:1 guess:1 xk:4 math:1 node:1 mathematical:1 symposium:2 scholkopf:1 shorthand:1 consists:2 prove:1 manner:1 indeed:1 expected:2 behavior:1 themselves:1 frequently:1 automatically:2 little:1 considering:1 becomes:1 moreover:1 bounded:1 nea:1 what:3 substantially:1 eigenvector:5 caution:1 finding:1 corporation:1 untrue:1 guarantee:2 berkeley:4 every:3 exactly:6 partitioning:7 unit:4 control:1 grant:2 appear:1 positive:1 local:1 despite:2 meil:1 path:1 might:1 initialization:1 suggests:3 challenging:1 alice:1 acknowledgment:1 ond:1 union:2 block:5 differs:1 xr:1 empirical:2 erence:1 word:1 close:1 context:1 equivalent:2 map:1 conventional:1 shi:7 straightforward:2 starting:1 convex:3 d22:1 ke:1 formalized:1 immediately:1 estimator:1 higgins:1 spanned:2 datapoints:1 stability:1 coordinate:1 suppose:1 us:1 origin:1 element:4 recognition:1 jk:2 particularly:1 cut:3 predicts:1 muri:1 observed:2 capture:1 worst:1 region:3 connected:13 sun:1 cheeger:3 intuition:2 moderately:1 iisi:1 efs:1 engr:1 solving:1 tight:5 upon:1 division:1 f2:2 easily:3 univ:1 separated:4 describe:1 artificial:1 choosing:2 whose:1 emerged:1 distortion:2 say:1 otherwise:3 superscript:1 seemingly:1 eigenvalue:13 propose:1 unresolved:1 j2:3 rapidly:2 mixing:2 normalize:1 cluster:69 renormalizing:1 derive:1 andrew:1 ac:1 stat:1 ij:3 school:1 lwo:1 implemented:1 c:6 implies:1 convention:1 drawback:1 pea:1 human:2 viewing:1 dii:2 require:1 kii:1 assign:1 yweiss:1 really:1 proposition:3 yij:1 strictly:2 mm:1 hold:2 around:2 proximity:1 exp:1 a2:2 smallest:1 proc:1 largest:2 successfully:1 tool:1 weighted:1 hope:2 gaussian:2 sight:1 derived:2 focus:3 unsatisfactory:1 likelihood:1 indicates:1 cosmetic:1 contrast:1 centroid:4 sense:1 helpful:2 leung:1 typically:1 lj:1 vlsi:1 issue:1 fairly:2 field:1 construct:2 once:2 aware:1 ng:2 equal:1 identical:1 look:2 future:1 np:1 simplify:1 few:1 randomly:1 simultaneously:2 densely:1 kandola:1 individual:5 familiar:2 replaced:1 subsampled:1 n1:3 microsoft:1 attempt:1 inexpensively:1 organization:1 intra:2 zheng:1 mixture:1 analyzed:1 d11:1 xni:1 edge:3 lh:2 orthogonal:7 taylor:1 walk:6 initialized:1 desired:1 renormalize:2 circle:1 column:6 earlier:1 stewart:1 stacking:2 subset:6 too:1 perturbed:1 varies:1 jittered:1 considerably:1 st:1 density:3 international:1 huji:1 ie:2 off:3 xi1:1 michael:1 yao:1 again:1 satisfied:1 containing:1 choose:1 conf:1 american:1 chung:1 li:1 jrk:1 de:1 singleton:1 sec:1 satisfy:2 ad:1 depends:2 later:3 view:2 picked:2 analyze:1 characterizes:2 recover:1 minimize:1 il:3 ni:3 who:1 correspond:4 ofthe:1 cni:1 mmi:1 tended:1 whenever:1 complicate:1 ed:1 proof:1 gain:1 dataset:2 ask:1 recall:1 knowledge:1 color:2 conversation:1 segmentation:3 actually:2 cbms:1 restarts:4 planar:1 wei:3 just:2 smola:1 hand:1 replacing:1 ei:1 nonlinear:1 indicated:1 lei:1 building:1 contain:1 true:4 hence:1 assigned:1 i2:2 deal:1 ll:1 noted:1 arrived:1 ay:1 jrl:1 image:1 recently:1 rotation:1 rl:1 empirically:1 volume:1 discussed:1 interpretation:1 extend:1 kluwer:1 interpret:1 ai:3 rd:1 meila:6 mathematics:1 similarly:1 zeroing:1 shawe:1 had:1 dj:5 moving:1 stable:2 similarity:2 surface:3 etc:1 closest:1 recent:1 apart:3 n00014:1 onr:1 success:2 a41:2 seen:1 minimum:3 r0:1 subblock:1 semi:1 ii:12 multiple:2 mix:1 academic:1 sphere:3 long:2 divided:1 equally:1 marina:1 laplacian:2 parenthesis:1 vision:4 kernel:6 normalization:2 whereas:2 want:2 fellowship:1 addressed:1 singular:2 appropriately:1 rest:1 regional:1 comment:1 tend:1 jordan:2 ideal:3 split:1 variety:1 xj:1 gave:2 idea:1 pca:2 eigengap:4 suffer:1 f:1 miiller:1 repeatedly:1 matlab:1 clear:1 eigenvectors:23 informally:1 ang:1 jrn:1 clusten:1 eory:1 sl:3 exist:2 nsf:1 s3:2 sjw:1 correctly:1 xii:1 discrete:2 alpert:1 four:2 threshold:1 christianini:1 graph:10 relaxation:1 padded:1 sum:6 run:4 family:1 reasonable:1 draw:1 dy:1 scaling:1 comparable:1 fl:3 guaranteed:1 annual:2 precisely:2 x2:1 n3:2 sake:1 nearby:1 attempting:1 vempala:1 according:2 yetta:1 poor:1 across:3 slightly:1 em:1 s1:6 intuitively:1 mutually:1 previously:3 discus:1 turn:2 mind:1 letting:2 flip:2 informal:1 adopted:1 available:2 tightest:1 unreasonable:1 apply:1 away:1 spectral:16 v2:1 alternative:2 yair:1 existence:1 original:2 top:1 clustering:24 cf:1 a4:3 unifying:1 giving:1 build:1 society:1 pled:1 malik:1 already:1 quantity:2 parametric:1 rt:1 diagonal:7 div:1 affinity:4 subspace:3 distance:3 link:1 thank:2 majority:1 outer:1 nx:1 seven:1 unstable:1 spanning:1 kannan:4 assuming:1 length:3 index:2 mini:1 ratio:2 hebrew:1 susceptible:1 relate:1 design:1 implementation:2 reliably:1 kahng:1 perform:1 disagree:1 finite:1 extended:1 rn:1 perturbation:6 varied:1 arbitrary:1 thm:1 required:1 specified:1 connection:1 optimized:1 usually:1 pattern:1 below:1 scott:1 rf:1 including:1 demanding:1 natural:2 improve:1 identifies:1 lk:1 harsh:1 sn:1 text:1 review:2 prior:1 relative:1 expect:2 proportional:1 foundation:2 fiji:1 degree:1 consistent:1 row:22 surprisingly:2 last:1 cohesiveness:1 supported:2 aij:5 side:1 understand:1 wide:1 fall:1 transition:2 contour:1 author:1 made:1 jump:1 refinement:1 simplified:1 far:2 sj:4 obtains:1 belongie:1 xi:3 search:1 iterative:1 why:1 promising:1 learn:1 reasonably:1 robust:1 parenthesized:1 longuet:1 cl:1 did:1 main:2 dense:1 s2:2 n2:3 repeated:2 eural:2 intel:1 fails:1 xl:1 lie:1 governed:1 perceptual:1 rk:2 theorem:2 transitioning:1 bad:2 british:1 maxi:2 symbol:2 dk:2 a3:3 grouping:1 exists:2 ih:1 texture:1 simply:1 infinitely:2 ordered:1 joined:1 subtlety:1 corresponds:1 chance:1 xix2:1 kli:1 kmeans:1 exposition:1 ajk:4 considerable:2 hard:2 experimentally:1 change:3 specifically:3 determined:1 except:1 principal:1 teng:1 experimental:1 e:1 formally:1 subblocks:1 latter:1 spielman:1 dept:1 instructive:1 |
1,199 | 2,093 | Sequential noise compensation by
sequential Monte Carlo method
Kaisheng Yao and Satoshi Nakamura
ATR Spoken Language Translation Research Laboratories
2-2-2, Hikaridai Seika-cho, Souraku-gun, Kyoto, 619-0288, Japan
E-mail: {kaisheng.yao, satoshi.nakamura}@slt.atr.co.jp
Abstract
We present a sequential Monte Carlo method applied to additive
noise compensation for robust speech recognition in time-varying
noise. The method generates a set of samples according to the prior
distribution given by clean speech models and noise prior evolved
from previous estimation. An explicit model representing noise effects on speech features is used, so that an extended Kalman filter
is constructed for each sample, generating the updated continuous
state estimate as the estimation of the noise parameter, and prediction likelihood for weighting each sample. Minimum mean square
error (MMSE) inference of the time-varying noise parameter is carried out over these samples by fusion the estimation of samples according to their weights. A residual resampling selection step and
a Metropolis-Hastings smoothing step are used to improve calculation efficiency. Experiments were conducted on speech recognition
in simulated non-stationary noises, where noise power changed artificially, and highly non-stationary Machinegun noise. In all the
experiments carried out, we observed that the method can have significant recognition performance improvement, over that achieved
by noise compensation with stationary noise assumption.
1
Introduction
Speech recognition in noise has been considered to be essential for its real applications. There have been active research efforts in this area. Among many approaches,
model-based approach assumes explicit models representing noise effects on speech
features. In this approach, most researches are focused on stationary or slow-varying
noise conditions. In this situation, environment noise parameters are often estimated before speech recognition from a small set of environment adaptation data.
The estimated environment noise parameters are then used to compensate noise
effects in the feature or model space for recognition of noisy speech.
However, it is well-known that noise statistics may vary during recognition. In
this situation, the noise parameters estimated prior to speech recognition of the
utterances is possibly not relevant to the subsequent frames of input speech if environment changes.
A number of techniques have been proposed to compensate time-varying noise effects. They can be categorized into two approaches. In the first approach, timevarying environment sources are modeled by Hidden Markov Models (HMM) or
Gaussian mixtures that were trained by prior measurement of environments, so
that noise compensation is a task of identification of the underlying state sequences
of the noise HMMs, e.g., in [1], by maximum a posterior (MAP) decision. This approach requires making a model representing different conditions of environments
(signal-to-noise ratio, types of noise, etc.), so that statistics at some states or mixtures obtained before speech recognition are close to the real testing environments.
In the second approach, environment model parameters are assumed to be timevarying, so it is not only an inference problem but also related to environment
statistics estimation during speech recognition. The parameters can be estimated
by Maximum Likelihood estimation, e.g., sequential EM algorithm [2][3][4]. They
can also be estimated by Bayesian methods. In the Bayesian methods, all relevant
information on the set of environment parameters and speech parameters, which are
denoted as ?(t) at frame t, is included in the posterior distribution given observation sequence Y (0 : t), i.e., p(?(t)|Y (0 : t)). Except for a few cases including linear
Gaussian state space model (Kalman filter), it is formidable to evaluate the distribution updating analytically. Approximation techniques are required. For example,
in [5], a Laplace transform is used to approximate the joint distribution of speech
and noise parameters by vector Taylor series. The approximated joint distribution
can give analytical formula for posterior distribution updating.
We report an alternative approach for Bayesian estimation and compensation of
noise effects on speech features. The method is based on sequential Monte Carlo
method [6]. In the method, a set of samples is generated hierarchically from the prior
distribution given by speech models. A state space model representing noise effects
on speech features is used explicitly, and an extended Kalman filter (EKF) is constructed in each sample. The prediction likelihood of the EKF in each sample gives
its weight for selection, smoothing, and inference of the time-varying noise parameter, so that noise compensation is carried out afterwards. Since noise parameter
estimation, noise compensation and speech recognition are carried out frame-byframe, we denote this approach as sequential noise compensation.
2
Speech and noise model
Our work is on speech features derived from Mel Frequency Cepstral Coefficients
(MFCC). It is generated by transforming signal power into log-spectral domain, and
finally, by discrete Cosine transform (DCT) to the cepstral domain. The following
derivation of the algorithm is in log-spectral domain. Let t denote frame (time)
index.
In our work, speech and noise are respectively modeled by HMMs and a Gaussian
mixture. For speech recognition in stationary additive noise, the following formula [4] has been shown to be effective in compensating noise effects. For Gaussian
mixture kt at state st , the Log-Add method transforms the mean vector ?lst kt of
the Gaussian mixture by,
?
?lst kt
=
?lst kt + log(1 + exp(?ln ? ?lst kt ))
(1)
where ?ln is the mean vector in the noise model. st ? {1, ? ? ? , S}, kt ? {1, ? ? ? , M }.
S and M each denote the number of states in speech models and the number of
mixtures at each state. Superscript l indicates that parameters are in the logspectral domain.
After the transformation, the mean vector ?
? lst kt is further transformed by DCT,
and then plugged into speech models for recognition of noisy speech. In case of
time-varying noise, the ?ln should be a function of time, i.e., ?ln (t). Accordingly,
the compensated mean is ?
? lst kt (t).
s0
st ?1
st
sT
k0
kt ?1
kt
kT
? sl 0 k 0 (0)
? sl t ?1kt ?1 (t ? 1)
Y l ( 0)
? nl (0)
? sl t k t (t )
Y l (t ? 1)
? nl (t ? 1)
? sl T kT (T )
Y l (t )
? nl (t )
Y l (T )
? nl (T )
Figure 1: The graphical model representation of the dependences of the speech and
noise model parameters. st and kt each denote the state and Gaussian mixture
at frame t in speech models. ?lst kt (t) and ?ln (t) each denote the speech and noise
parameter. Y l (t) is the noisy speech observation.
The following analysis can be viewed in Figure 1. In Gaussian mixture kt at state st
of speech model, speech parameter ?lst kt (t) is assumed to be distributed in Gaussian
with mean ?lst kt and variance ?lst kt . On the other hand, since the environment
parameter is assumed to be time varying, the evolution of the environment mean
vector can be modeled by a random walk function, i.e.,
?ln (t) = ?ln (t ? 1) + v(t)
(2)
where v(t) is the environment driving noise in Gaussian distribution with zero mean
and variance V .
Then, we have,
p(st , kt , ?lst kt (t), ?ln (t)|st?1 , kt?1 , ?lst?1 kt?1 (t ? 1), ?ln (t ? 1))
=
ast?1 st pst kt N (?lst kt (t); ?lst kt , ?lst kt )N (?ln (t); ?ln (t ? 1), V )
(3)
where ast?1 st is the state transition probability from st?1 to st , and pst kt is the
mixture weight. The above formula gives the prior distribution of the set of speech
and noise model parameter ?(t) = {st , kt , ?lst kt (t), ?ln (t)}.
Furthermore, given observation Y l (t), assume that the transformation by (1) has
modeling and measurement uncertainty in Gaussian distribution, i.e.,
Y l (t) = ?lst kt (t) + log (1 + exp (?ln (t) ? ?lst kt (t))) + wst kt (t)
(4)
where wst kt (t) is Gaussian with zero mean and variance ?lst kt , i.e., N (?; 0, ?lst kt ).
Thus, the likelihood of observation Y l (t) at state st and mixture kt is
p(Y l (t)|?(t)) = N (Y l (t); ?lst kt (t) + log (1 + exp (?ln (t) ? ?lst kt (t))), ?lst kt )
(5)
Refereeing to (3) and (5), the posterior distribution of ?(t) given Y l (t) is
p(st , kt , ?lst kt (t), ?ln (t)|Y l (t)) ?
p(Y l (t)|?(t))ast?1 st pst kt N (?lst kt (t); ?lst kt , ?lst kt )N (?ln (t); ?ln (t ? 1), V ) (6)
The time-varying noise parameter is estimated by MMSE, given as,
Z
XZ
l
l
p(?(t)|Y l (0 : t))d?lst kt (t)d?ln (t)
?n (t)
?
?n (t) =
?ln (t)
st ,kt
?ls
t kt
(7)
(t)
However, it is difficult to obtain the posterior distribution p(?(t)|Y l (0 : t)) analytically, since p(?lst kt (t), ?ln (t)|Y l (t)) is non-Gaussian in ?lst kt (t) and ?ln (t) due to
the non-linearity in (4). It is thus difficult, if possible, to assign conjugate prior
of ?ln (t) to the likelihood function p(Y l (t)|?(t)). Another difficulty is that the
speech state and mixture sequence is hidden in (7). We thus rely on the solution
by computational Bayesian approach [6].
3
Time-varying noise parameter estimation by sequential
Monte Carlo method
We apply the sequential Monte Carlo method [6] for posterior distribution updating. At each frame t, a proposal importance distribution is sampled whose target
is the posterior distribution in (7), and it is implemented by sampling from lower
distributions in hierarchy. The method goes through the sampling, selection, and
smoothing steps frame-by-frame. MMSE inference of the time-varying noise parameter is a by-product of the steps, carried out after the smoothing step.
In the sampling step, the prior distribution given by speech models is
set to the proposal importance distribution, i.e., q(?(t)|?(t ? 1)) =
ast?1 st pst kt N (?lst kt (t); ?lst kt , ?lst kt ). The samples are then generated by sampling
hierarchically of the prior distribution described as follows: set i = 1 and perform
the following steps:
(i)
1. sample st ? as(i)
s
t?1 t
2. sample
(i)
kt
3. sample ?
? ps(i) kt
l(i)
(i) (i)
st kt
t
(t) ? N (; ?l (i)
(i)
st kt
, ?l (i)
(i)
st kt
), and set i = i + 1
4. repeat step 1 to 3 until i = N
where superscript (i) denotes the index of samples and N denotes the number of
samples. Each sample represents certain speech and noise parameter, which is
(i) (i)
l(i)
l(i)
denoted as ?(i) (t) = (st , kt , ? (i) (i) (t), ?n (t)). The weight of each sample is
st kt
Qt
p(?(? )(i) |Y l (? ))
given by ? =1 q(?(?
. Refereeing to (6), the weight is calculated by
)(i) |?(? ?1)(i) )
l(i)
?(i)
? (i) (t) = p(Y l (t)|?(i) (t))N (?l(i)
n (t); ?n (t ? 1), V )? (t ? 1)
(8)
where ??(i) (t ? 1) is the sample weight from previous frame. The remaining part
in the right side of above equation, in fact, represents the prediction likelihood of
the state space model given by (2) and (4) for each sample (i). This likelihood
can be obtained analytically since after linearization of (4) with respect to ?ln (t) at
l(i)
?n (t ? 1), an extended Kalman filter (EKF) can be obtained, where the prediction
likelihood of the EKF gives the weight, and the updated continuous state of EKF
l(i)
gives ?n (t).
In practice, after the above sampling step, the weights of all but several samples may
become insignificant. Given the fixed number of samples, this will results in degeneracy of the estimation, where not only some computational resources are wasted, but
also estimation might be biased because of losing detailed information on some parts
important to the parameter estimation. A selection step by residual resampling [6]
is adopted after the sampling step. The method avoids the degeneracy by discarding those samples with insignificant weights, and in order to keep the number of the
samples constant, samples with significant weights are duplicated. Accordingly, the
weights after the selection step are also proportionally redistributed. Denote the
?
? (i) (t); i = 1 ? ? ? N } with weights
set of samples after the selection step as ?(t)
= {?
? = {??(i) (t); i = 1 ? ? ? N }.
?(t)
After the selection step at frame t, these N samples are distributed approximately
according to the posterior distribution in (7). However, the discrete nature of
the approximation can lead to a skewed importance weights distribution, where
?
the extreme case is all the samples have the same ?(t)
estimated. A MetropolisHastings smoothing [7] step is introduced in each sample where the step involves
? (i) (t) according to the proposal
sampling a candidate ??(i) (t) given the current ?
?
(i)
? (t)). The Markov chain then moves towards
importance distribution q(? (t)|?
?(i)
l
? (i)
?(i)
|Y (t))q(? |?
)
??(i) (t) with acceptance possibility as min{1, p(?
? (i) |Y l (t))q(??(i) |?
? (i) ) }, otherwise it
p(?
? (i) . To simplify calculation, we assume that the importance distriburemains at ?
?
? (i) (t)) is symmetric, and after some mathematical manipulation, it
tion q(? (t)|?
?(i)
(t)
is shown that the acceptance possibility is given by min{1, ???(i) (t)
}. Denote the
(i)
?
?
?
obtained samples as ?(t) = {? (t); i = 1 ? ? ? N } with weights ?(t) = {??(i) (t); i =
1 ? ? ? N }.
Noise parameter ?ln (t) is estimated via MMSE over the samples, i.e.,
?
?ln (t) =
N
X
i=1
l(i)
??(i) (t)
?l(i)
PN ?(j) ?
n (t)
(t)
j=1 ?
where ?
?n (t) is the updated continuous state of the EKF in the sample after the
smoothing step. Once the estimate ?
? ln (t) has been obtained, it is plugged into (1)
to do non-linear transformation of clean speech models.
4
4.1
Experimental results
Experimental setup
Experiments were performed on the TI-Digits database down-sampled to 16kHz.
Five hundred clean speech utterances from 15 speakers and 111 utterances unseen
in the training set were used for training and testing, respectively. Digits and
silence were respectively modeled by 10-state and 3-state whole word HMMs with
4 diagonal Gaussian mixtures in each state.
The window size was 25.0ms with a 10.0ms shift. Twenty-six filter banks were used
in the binning stage. The features were MFCC + ? MFCC. The baseline system
had a 98.7% Word Accuracy under clean conditions.
We compared three systems. The first was the baseline trained on clean speech without noise compensation, and the second was the system with noise compensation by
(1) assuming stationary noise [4]. They were each denoted as Baseline and Stationary Compensation. The sequential method was un-supervised, i.e., without training
transcript, and it was denoted according to the number of samples and variance of
the environment driving noise V . Four seconds of contaminating noise was used in
each experiment to obtain noise mean vector ?ln in (1) for Stationary Compensation. It was also for initialization of ?ln (0) in the sequential method. The initial
l(i)
?n (0) for each sample was sampled from N (?ln (0), 0.01) + N (?ln (0) + ?(0), 10.0),
where ?(0) was flat distribution in [?1.0, 9.0].
4.2
Speech recognition in simulated non-stationary noise
White noise signal was multiplied by a Chirp signal and a rectangular signal, so that
the noise power of the contaminating White noise changed continuously, denoted
as experiment A, and dramatically, denoted as experiment B. As a result, signalto-noise ratio (SNR) of the contaminating noise ranged from 0dB to 20.4dB. We
plotted the noise power in 12th filter bank versus frames in Figure 2, together with
the estimated noise power by the sequential method with number of samples set to
120 and environment driving noise variance set to 0.0001. As a comparison, we also
plotted the noise power and its estimate by the method with the same number of
samples but larger driving noise variance to 0.001.
By Figure 2 and Figure 3, we have the following observations. First, the method
can track the evolution of the noise power. Second, the larger driving noise variance
V will make faster convergence but larger estimation error of the method. In terms
of recognition performance, Table 1 shows that the method can effectively improve
system robustness to the time-varying noise. For example, with 60 samples, and
the environment driving noise variance V set to 0.001, the method can improve
word accuracy from 75.30% achieved by ?Stationary Compensation?, to 94.28% in
experiment A. The table also shows that, the word accuracies can be improved
by increasing number of samples. For example, given environment driving noise
variance V set to 0.0001, increasing number of samples from 60 to 120, can improve
word accuracy from 77.11% to 85.84% in experiment B.
Table 1: Word Accuracy (in %) in simulated non-stationary noises, achieved by
the sequential Monte Carlo method in comparison with baseline without noise compensation, denoted as Baseline, and noise compensation assuming stationary noise,
denoted as Stationary Compensation.
Experiment
Baseline
Stationary
Compensation
A
B
48.19
53.01
75.30
78.01
4.3
# samples = 60
V
0.001
0.0001
94.28
93.98
82.23
77.11
# samples = 120
V
0.001
0.0001
94.28
94.58
85.84
85.84
Speech recognition in real noise
In this experiment, speech signals were contaminated by highly non-stationary Machinegun noise in different SNRs. The number of samples was set to 120, and the
environment driving noise variance V was set to 0.0001. Recognition performances
are shown in Table 2, together with ?Baseline? and ?Stationary Compensation?.
Figure 2: Estimation of the time-varying parameter ?ln (t) by the sequential Monte
Carlo method at 12th filter bank in experiment A. Number of samples is 120.
Environment driving noise variance is 0.0001. Solid curve is the true noise power.
Dash-dotted curve is the estimated noise power.
It is observed that, in all SNR conditions, the method can further improve system performance, compared to that obtained by ?Stationary Compensation?, over
?Baseline?. For example, in 8.86dB SNR, the method can improve word accuracy
from 75.60% by ?Stationary Compensation? to 83.13%. As a whole, the method
can have a relative 39.9% word error rate reduction compared to ?Stationary Compensation?.
Table 2: Word Accuracy (in %) in Machinegun noise, achieved by the sequential
Monte Carlo method in comparison with baseline without noise compensation, denoted as Baseline, and noise compensation assuming stationary noise, denoted as
Stationary Compensation.
SNR (dB)
28.86
14.88
8.86
1.63
5
Baseline
90.36
64.46
56.02
50.0
Stationary Compensation
92.77
76.81
75.60
68.98
#samples = 120, V = 0.0001
97.59
88.25
83.13
72.89
Summary
We have presented a sequential Monte Carlo method for Bayesian estimation of
time-varying noise parameter, which is for sequential noise compensation applied to
robust speech recognition. The method uses samples to approximate the posterior
distribution of the additive noise and speech parameters given observation sequence.
Figure 3: Estimation of the time-varying parameter ?ln (t) by the sequential Monte
Carlo method at 12th filter bank in experiment A. Number of samples is 120.
Environment driving noise variance is 0.001. Solid curve is the true noise power.
Dash-dotted curve is the estimated noise power.
Once the noise parameter has been inferred, it is plugged into a non-linear transformation of clean speech models. Experiments conducted on digits recognition in
simulated non-stationary noises and real noises have shown that the method is very
effective to improve system robustness to time-varying additive noise.
References
[1] A. Varga and R.K. Moore, ?Hidden markov model decomposition of speech and noise,?
in ICASSP, 1990, pp. 845?848.
[2] N.S. Kim, ?Nonstationary environment compensation based on sequential estimation,?
IEEE Signal Processing Letters, vol. 5, no. 3, March 1998.
[3] K. Yao, K. K. Paliwal, and S. Nakamura, ?Sequential noise compensation by a sequential kullback proximal algorithm,? in EUROSPEECH, 2001, pp. 1139?1142, extended
paper submitted for publication.
[4] K. Yao, B. E. Shi, S. Nakamura, and Z. Cao, ?Residual noise compensation by a
sequential em algorithm for robust speech recognition in nonstationary noise,? in
ICSLP, 2000, vol. 1, pp. 770?773.
[5] B. Frey, L. Deng, A. Acero, and T. Kristjansson, ?Algonquin: Iterating laplace?s
method to remove multiple types of acoustic distortion for robust speech recognition,?
in EUROSPEECH, 2001, pp. 901?904.
[6] J. S. Liu and R. Chen, ?Sequential monte carlo methods for dynamic systems,? J.
Am. Stat. Assoc, vol. 93, pp. 1032?1044, 1998.
[7] W. K. Hastings, ?Monte carlo sampling methods using markov chains and their applications,? Biometrika, vol. 57, pp. 97?109, 1970.
| 2093 |@word kristjansson:1 decomposition:1 solid:2 reduction:1 initial:1 liu:1 series:1 mmse:4 current:1 dct:2 subsequent:1 additive:4 remove:1 resampling:2 stationary:23 accordingly:2 five:1 mathematical:1 constructed:2 become:1 seika:1 xz:1 compensating:1 window:1 increasing:2 underlying:1 linearity:1 formidable:1 evolved:1 spoken:1 transformation:4 ti:1 biometrika:1 assoc:1 before:2 frey:1 approximately:1 might:1 initialization:1 chirp:1 co:1 hmms:3 testing:2 practice:1 digit:3 area:1 word:9 close:1 selection:7 acero:1 ast:4 map:1 compensated:1 shi:1 go:1 l:1 focused:1 rectangular:1 laplace:2 updated:3 target:1 hierarchy:1 losing:1 us:1 recognition:21 approximated:1 updating:3 database:1 binning:1 observed:2 environment:22 transforming:1 dynamic:1 trained:2 efficiency:1 icassp:1 joint:2 k0:1 lst:33 derivation:1 snrs:1 effective:2 monte:12 whose:1 larger:3 distortion:1 otherwise:1 statistic:3 unseen:1 transform:2 noisy:3 superscript:2 sequence:4 analytical:1 redistributed:1 product:1 adaptation:1 relevant:2 cao:1 convergence:1 p:1 generating:1 stat:1 qt:1 transcript:1 implemented:1 involves:1 filter:8 assign:1 icslp:1 considered:1 exp:3 driving:10 vary:1 estimation:16 gaussian:13 ekf:6 pn:1 varying:15 timevarying:2 publication:1 derived:1 improvement:1 likelihood:8 indicates:1 baseline:11 kim:1 am:1 inference:4 hidden:3 transformed:1 among:1 denoted:10 smoothing:6 once:2 sampling:8 represents:2 report:1 contaminated:1 simplify:1 few:1 acceptance:2 highly:2 possibility:2 mixture:12 extreme:1 nl:4 chain:2 algonquin:1 kt:62 wst:2 taylor:1 plugged:3 walk:1 plotted:2 modeling:1 snr:4 hundred:1 conducted:2 eurospeech:2 proximal:1 cho:1 st:25 together:2 continuously:1 yao:4 possibly:1 japan:1 coefficient:1 explicitly:1 tion:1 performed:1 square:1 accuracy:7 variance:12 satoshi:2 identification:1 bayesian:5 carlo:12 mfcc:3 submitted:1 frequency:1 pp:6 degeneracy:2 sampled:3 pst:4 duplicated:1 supervised:1 improved:1 furthermore:1 stage:1 until:1 hand:1 hastings:2 effect:7 ranged:1 true:2 evolution:2 analytically:3 symmetric:1 laboratory:1 moore:1 white:2 during:2 skewed:1 speaker:1 mel:1 cosine:1 m:2 khz:1 jp:1 significant:2 measurement:2 refereeing:2 language:1 had:1 etc:1 add:1 posterior:9 contaminating:3 manipulation:1 paliwal:1 certain:1 minimum:1 deng:1 signal:7 afterwards:1 multiple:1 kyoto:1 faster:1 calculation:2 compensate:2 prediction:4 achieved:4 proposal:3 source:1 biased:1 db:4 nonstationary:2 shift:1 six:1 effort:1 speech:47 dramatically:1 varga:1 detailed:1 proportionally:1 iterating:1 transforms:1 sl:4 dotted:2 estimated:11 track:1 discrete:2 vol:4 four:1 clean:6 wasted:1 letter:1 uncertainty:1 decision:1 dash:2 flat:1 generates:1 min:2 according:5 march:1 conjugate:1 em:2 metropolis:1 making:1 ln:32 equation:1 resource:1 adopted:1 multiplied:1 apply:1 spectral:2 alternative:1 robustness:2 assumes:1 denotes:2 remaining:1 graphical:1 hikaridai:1 move:1 kaisheng:2 dependence:1 diagonal:1 atr:2 simulated:4 hmm:1 gun:1 mail:1 assuming:3 kalman:4 modeled:4 index:2 ratio:2 difficult:2 setup:1 twenty:1 perform:1 observation:6 markov:4 compensation:29 situation:2 extended:4 frame:11 inferred:1 introduced:1 required:1 acoustic:1 including:1 power:11 metropolishastings:1 difficulty:1 nakamura:4 rely:1 residual:3 representing:4 improve:7 carried:5 utterance:3 prior:9 relative:1 versus:1 s0:1 bank:4 translation:1 changed:2 summary:1 repeat:1 silence:1 side:1 cepstral:2 slt:1 distributed:2 curve:4 calculated:1 transition:1 avoids:1 approximate:2 kullback:1 keep:1 active:1 assumed:3 continuous:3 un:1 table:5 nature:1 robust:4 artificially:1 domain:4 hierarchically:2 whole:2 noise:98 categorized:1 slow:1 explicit:2 candidate:1 weighting:1 formula:3 down:1 discarding:1 insignificant:2 fusion:1 essential:1 sequential:22 effectively:1 importance:5 linearization:1 chen:1 signalto:1 viewed:1 towards:1 change:1 included:1 except:1 experimental:2 evaluate:1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.