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+ShadowNav: Crater-Based Localization for Nighttime
+and Permanently Shadowed Region Lunar Navigation
+Abhishek Cauligi*
+abhishek.s.cauligi@jpl.nasa.gov
+R. Michael Swan*
+robert.m.swan@jpl.nasa.gov
+Hiro Ono
+masahiro.ono@jpl.nasa.gov
+Shreyansh Daftry
+shreyansh.daftry@jpl.nasa.gov
+John Elliott
+john.o.elliott@jpl.nasa.gov
+Larry Matthies
+lhm@jpl.nasa.gov
+Deegan Atha
+deegan.j.atha@jpl.nasa.gov
+Jet Propulsion Laboratory, California Institute of Technology
+Pasadena, CA 91109, USA
+Abstract—There has been an increase in interest in missions
+that drive significantly longer distances per day than what
+has currently been performed.
+For example, Endurance-A
+proposes driving several kilometers a day in order to reach
+its target traverse of 2000 km in 4 years. Additionally, some
+of these proposed missions, including Endurance-A and rovers
+for Permanently Shadowed Regions (PSRs) of the moon, re-
+quire autonomous driving and absolute localization in darkness.
+Endurance-A proposes to drive 1200 km of its total traverse at
+night. The lack of natural light available during these missions
+limits what can be used as visual landmarks and the range
+at which landmarks can be observed. In order for planetary
+rovers to traverse long-ranges, onboard absolute localization is
+critical to the rover’s ability to maintain its planned trajectory
+and avoid known hazardous regions. Currently, the localization
+performed onboard rovers is relative to the rover’s frame of
+reference and is performed through the integration of wheel
+and visual odometry and inertial measurements. To accomplish
+absolute localization, a “ground-in-the-loop” (GITL) operation
+is performed wherein a human operator matches local maps
+or images from onboard with orbital images and maps. This
+GITL operation places a limit on the distance that can be
+driven in a day to a few hundred meters, which is the distance
+that the rover can maintain acceptable localization error via
+relative methods. Previous work has shown that using craters
+as landmarks is a promising approach for performing absolute
+localization on the moon during the day.
+In this work we
+present a method of absolute localization that utilizes craters
+as landmarks and matches detected crater edges on the surface
+with known craters in orbital maps. We focus on a localization
+method based on a perception system which has an external
+illuminator and a stereo camera. While other methods based
+on lidar exist, lidar is not currently planned for deployment
+on the current proposed nighttime and PSR missions. In this
+paper, we evaluate (1) both monocular and stereo based surface
+crater edge detection techniques, (2) methods of scoring the
+crater edge matches for optimal localization, and (3) localization
+performance on simulated Lunar surface imagery at night. We
+demonstrate that this technique shows promise for maintaining
+absolute localization error of less than 10 m required for most
+planetary rover missions.
+TABLE OF CONTENTS
+1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+1
+2. RELATED WORKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+*Abhishek Cauligi and R. Michael Swan contributed equally to this work.
+978-1-6654-9032-0/23/$31.00 ©2023. California Institute of Technology.
+Government sponsorship acknowledged.
+The research was carried out at the Jet Propulsion Laboratory, California
+Institute of Technology, under a contract with the National Aeronautics and
+Space Administration (80NM0018D0004).
+3. APPROACH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+4. DATASETS OVERVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+5. CRATER DETECTION PERFORMANCE . . . . . . . . . . .
+6
+6. LOCALIZATION PERFORMANCE . . . . . . . . . . . . . . . . .
+8
+7. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+L
+R
+Figure 1: The ShadowNav localization algorithm per-
+forms absolute localization for a Lunar rover mission
+located at the red position in the left image by matching
+known craters from (left) an orbital map against (right)
+detected craters from the rover stereo cameras.
+1. INTRODUCTION
+Long-range Lunar navigation, and specifically navigating
+within darkness, has gained a significant amount of traction
+recently. For example, missions to Permanently Shadowed
+Regions (PSRs) of the moon have been proposed such as
+the VIPER mission [1], [2] and the Lunar Polar Volatiles
+Explorer mission concepts. Furthermore, there are missions
+that have proposed driving during the Lunar night in order
+to traverse longer distances. For example, the new Decadal
+Survey [3] recommends the Endurance-A Lunar rover mis-
+sion should be implemented as a strategic medium-class
+mission as the highest priority of the Lunar Discovery and
+Exploration Program. The Endurance-A rover proposal plans
+to drive 2000 km in the South Pole-Aitken (SPA) Basin to
+collect
+100 kg of samples, which would be delivered to
+Artemis astronauts. This mission concept study [4] identified
+several key capabilities required to complete this mission
+which are: (1) Endurance will need to drive 70% of its total
+distance during the night to enable daytime hours dedicated
+to science and sampling. (2) The mission will require on-
+board autonomy for the majority of its operations, while the
+1
+arXiv:2301.04630v1  [cs.RO]  11 Jan 2023
+
+Orbital Image Generation
+Stereo Images Generation
+Crater Edge 
+Detection
+Local-to-Global 
+Transform
+Particle Filter 
+Step
+Compute Q-Score
+Orbital Image Generation
+Particle Filter
+Disparity 
+Generation
+Disparity Hole 
+Filler
+Figure 2: Schematic of the ShadowNav algorithm proposed to perform absolute localization on the Moon. A particle
+filter is used to match craters detected by the rover stero cameras with known craters from an orbital map.
+ground only handles contingencies. (3) Global localization
+is necessary to maintain an error of <10 m relative to orbital
+maps.
+At present, existing rovers perform onboard localization rel-
+ative to their own reference frame.
+This is accomplished
+by using wheel and visual odometry and inertial measure-
+ments. Absolute localization is performed periodically with
+a “ground-in-the-loop” (GITL) operation. This is acceptable
+for current driving distances which are a few hundred meters
+a day. Existing relative localization has around 2% drift and
+therefore can only drive at most 500 m before the error will
+be larger than 10 m. In order to traverse longer distances, on
+the order of several kilometers a day proposed by missions
+such as Endurance-A, autonomous absolute localization be-
+comes critical. At present the Lunar surface does not have
+continuous communication with Earth. Therefore, having to
+perform several GITL operations for absolute localization in
+a day will significantly reduce the distance that can be driven.
+The lack of frequent absolute localization for the rover would
+lead to errors greater than the maximum 10 m localization
+error which would present significant risks to the mission
+through deviations from the desired trajectory and risk for
+unidentified obstacles.
+Craters as landmarks have been shown to be promising for
+absolute localization on the Moon [5], [6]. However, the lack
+of natural light available while driving within a PSR or during
+the Lunar night limits what can be used as a landmark and the
+range at which the landmarks can be observed. Using craters
+is still promising as the average distance between craters of
+≥10 m in diameter is 100 m on terrain with relatively fresh
+craters and
+10 m on terrain with old craters [7].
+Addi-
+tionally the Lunar Reconnaissance Orbiter Camera (LROC)
+provides digital elevation models (DEMs) with a resolution
+between 0.5 m-5 m per pixel [8] and there are some DEMs
+within PSRs [9].
+In this work, we propose using a stereo camera with an
+illuminator positioned below the stereo camera in order to
+detect crater rims within the darkness. The use of such an
+illuminator is motivated by the Endurance-A mission concept
+study [4], which proposes the use of a stereo camera with
+an illumination source as the perception system for a rover
+operating in darkness.
+Global localization is then accom-
+plished by matching the detected crater rims against known
+craters from an orbital image as shown in Figure 2. To handle
+the uncertainty and nonlinearity of the crater rim detection
+model, we utilize a particle filter with a novel Q-Score metric
+for ranking potential crater matches in order to estimate the
+absolute position of the rover within an orbital map. This
+paper demonstrates the initial results of both crater detection
+within darkness and absolute localization within simulation
+which are the results of the first two years of a planned
+three year effort to validate this approach. Work is ongoing
+to collect and validate this approach in a real-world Lunar
+analogue test location.
+Statement of Contributions: This paper presents an approach
+to absolute localization on the Moon that can be performed
+while a rover is in darkness, such as within a PSR or during
+the Lunar night.
+The main contributions of the work as
+summarized below:
+1. We developed a simulator based on Blender [10] which
+renders simulated surface stereo imagery of the Lunar sur-
+face in darkness located within a known orbital position.
+The rendering process utilizes the Hapke lighting model for
+more accurate surface reflectance as well as DEMs captured
+by LROC for realistic crater distributions.
+2. We evaluated different crater-edge detection techniques
+and demonstrate a method which captures 80% of the leading
+crater arc at 10 m and can detect crater arcs out to 20 m.
+3. We present a method to localize a rover within an orbital
+map using surface crater-edge detections and known orbital
+craters based on a particle filter and a metric we call the Q-
+Score which is detailed in Section 3.
+4. We demonstrate our absolute localization technique can
+achieve less than 2 m absolute error with an assumed odome-
+try drift of 2% and an initial 3-sigma uncertainty of 3 m.
+2. RELATED WORKS
+Absolute localization on planetary surfaces is critical for
+expanding the range rovers can travel in a day and over the
+course of a mission and there have been many previous works
+that investigate this problem. There have been techniques
+proposed for the Martian surface. Works such as [11], [12]
+consider far range and horizon features which are at ranges
+that are beyond what is expected can be seen in the dark.
+2
+
+Figure 3: Figure demonstrating the impact of placement
+of light source on crater rim shadows.
+Left: Sample
+render of a crater with light source even with camera.
+Right Sample render of a crater with light source below
+the camera.
+[13] proposes a technique on the Martian surface for absolute
+localization that uses rocks and DEMs surface features.
+In our work, we focus on the problem of global localization in
+darkness which is relevant for permanently shadowed regions
+of the moon, for which there has been a surge of interest in
+conducting scientific measurements and activities [14]. Our
+solution approach is inspired by a host of recent works that
+seek to leverage orbital maps for global rover localization in
+these shadowed regions. In [13], the authors propose a local-
+ization procedure that matches an observed rover image with
+an orbital map, but this approach neglects the rover motion
+model and yields a deterministic estimate of the robot belief.
+A purely data-driven approach is presented in [15], wherein
+a convolutional neural network is trained on synthetic data to
+match the rover observations with orbital imagery. Closest
+to our approach, [16] presents a particle filtering technique
+to compare rover monocular camera imagery with orbital
+imagery and uses a Siamese neural network approach to
+assign each particle a likelihood weight. The authors in [6]
+propose a similar approach for Lunar absolute localization
+known as LunarNav. However, LunarNav focuses on the day-
+time localization problem and therefore considers different
+methods of crater matching that rely on greater knowledge of
+the surface geometry than available in the nighttime case.
+3. APPROACH
+In this work, we propose an absolute localization approach
+which utilizes a crater’s leading edge as landmarks for local-
+ization. The end result of this approach will be an estimated
+position and uncertainty within the orbital frame. At present,
+this approach only considers position localization.
+Rover
+orientation is assumed to be given by a star tracker which
+can compute orientation in three dimensions from celestial
+measurements. Our approach consists of two primary com-
+ponents:
+1. A leading-edge crater detection methodology for use with
+a Lunar rover equipped with a stereo camera system and
+illumination source.
+2. A particle filter for computing a position belief based on
+a score computed based on the association of crater edges
+and known orbital ground truth craters, which we call the Q-
+Score, and the robot motion model.
+A. Surface Crater Detection
+In order to identify craters on the surface, the system was de-
+signed to be used in conjunction with a perception system that
+contained a stereo camera and an illumination source. This
+perception system was configured where the illumination
+source was beneath the stereo camera. Examples of simulated
+images with the light at the same height as the cameras and
+the light positioned beneath the cameras are in Figure 3. It
+was observed that placing the illumination source below the
+camera results in a shadow at the leading edge of a negative
+obstacle. Furthermore, offsetting the light with the cameras
+reduced the impact of the Hapke model washing out some of
+the surface texture. Further details on the Hapke model and
+its impact on surface terrain are provided in Section 4.
+Here, we first review the three different techniques studied in
+this work for detecting a crater’s leading edge: (1) a method
+of detecting jumps within stero disparities, (2) a Canny edge
+detector used to find the shadow on the leading edge, and (3)
+a convolutional neural network (CNN)-based edge detector
+that uses both the monocular and disparity image as input.
+1. Stereo Disparity Discontinuity Method The first approach
+for leading edge crater detection relies on detecting discon-
+tinuities within the stereo disparity image. To accomplish
+this, the stereo disparity image must first be generated using
+methods such as the JPLV algorithm [17] or the Semi-Global
+Block Matching (SGBM) approach [18], among others. To
+account for the low contrast that may be present in the Lunar
+rover case, Contrast Limited Adaptive Histogram Equaliza-
+tion (CLAHE) is first run on the input images prior to running
+stereo. CLAHE is an adaptive histogram equalization and
+operates on sub-regions of an image which allows more
+consistent equalization across different lighting conditions
+within an image. This is useful for this application as there is
+a light-to-dark gradient from near-to-far within the images.
+The resulting disparity image is then scanned column-by-
+column and, when the difference between any two disparities
+is greater than some pre-defined threshold, the larger column
+index is marked as a crater edge.
+Further, any numerical
+issues stemming from stereo holes are accounted for by
+omitting any pixels with spurious values during comparison.
+2.
+Canny Edge Detector Method For sensor configura-
+tions that contain an illuminator located beneath the stereo
+cameras, shadows appear on the leading edge of negative
+obstacles. In such cases, a Canny edge detector can be used
+to distinguish the stark contrasting dark line along the rim.
+In this work, the Canny edge detector from OpenCV [19] is
+used to find these shadows.
+3.
+CNN-Based Edge Detector Method The Holistically-
+Nested Edge Detection (HED) approach presents a CNN-
+based deep learning based method for leading edge crater
+detection [20].
+This method uses the HED approach and
+can be performed by directly using the publicly released
+neural network weights. HED is capable of performing both
+monocular and stereo depth based edge detection. For HED
+to perform edge detection within a depth image, it generates a
+three channel image that contains horizontal disparity, height
+above ground, and angle of the local surface normal with the
+inferred direction of gravity. The RGB and depth predictions
+of the CNN are then merged to generate the desired output.
+Positive Obstacle False Positive Rejection—One shortcoming
+of the aforementioned leading edge crater detection approach
+is the susceptibility of false positive cases in the presence
+of positive obstacles. In order to account for this positive
+3
+
+Algorithm 1 Q-Score Computation
+Require: Belief bt
+i, set of crater observations {zt
+0,rover, ..., zt
+m,rover},
+set of ground truth craters {ct
+0,world, ..., ct
+ℓ,world}, positive value
+ε
+1:
+Qinc ← ε
+2:
+for i = 1, . . . , m do
+3:
+zt
+0,world ← rover to world(zt
+0,rover)
+4:
+dcr ← min ∥cj,world − zt
+0,world∥
+5:
+Qinc ← Qinc + dcr
+6:
+end for
+7:
+Qscore ← min
+�
+1, ( 1
+mQinc)−1�
+8:
+return Qscore
+Algorithm 2 ShadowNav Particle Filtering Algorithm
+Require: Initial belief distribution (µ0, Σ0), number of particles
+Ns, number of effective particles threshold Neff,thresh
+1:
+{b0
+1, ..., b0
+Ns} ← sample beliefs(µ0, Σ0)
+2:
+{w0
+1, ..., w0
+Ns} ← {1, ..., 1}
+3:
+t ← 1
+4:
+while particle filter running do
+5:
+{zt
+0, ..., zt
+m} ← get observations()
+6:
+{qt
+1, ..., qt
+Ns} ← {0, ..., 0}
+7:
+for i = 1, ..., Ns do
+8:
+bt
+i ← propagate sample(bt−1
+i
+)
+9:
+qt
+i ← log Q score(bt
+i, {zt
+0, ..., zt
+m})
+10:
+end for
+11:
+qt
+min ← min(qt
+1, ..., qt
+Ns)
+12:
+for i = 1, ..., Ns do
+13:
+wt
+i ← wt−1
+i
++ qt
+i − qt
+min
+14:
+end for
+15:
+Neff ← compute Neff(wt
+1, ..., wt
+Ns)
+16:
+if Neff ≤ Neff,thresh then
+17:
+{bt
+1, ..., bt
+Ns} ← resample beliefs({bt
+i}Ns
+i=1, {wt
+i}Ns
+i=1)
+18:
+{wt
+1, ..., wt
+Ns} ← {1, ..., 1}
+19:
+end if
+20:
+t ← t + 1
+21: end while
+obstacle issue, the detected edge points are passed through
+a filter that removes points which have hits on the far side of
+the crater edge with a detected negative or flat slope. Detected
+edge points are kept only if, within the region directly beyond
+the detected edge, there exists a positive slope or if there is
+not enough stereo to accurately compute the slope. Thus, the
+case of a detected positive slope is assumed to correspond
+to the rising edge of the crater under the assumption that
+the detected edge is the leading edge of a negative obstacle.
+Alternatively, a detected edge is also retained if the far edge
+is not captured due to low light conditions, as this is assumed
+to be an indication of the presence of a large crater.
+B. Particle Filter
+Here, we provide an overview of the proposed ShadowNav
+particle filtering approach. First, we provide further details
+on the Q-Score metric that is used in the belief update step.
+Q-Score—The Q-Score provided the measurement probabil-
+ity of some position belief based on rover frame observations
+and an orbital map. The procedure for computing the Q-Score
+is given in Algorithm 1. The algorithm takes as input a given
+belief bt
+i, a set of m observed edges in rover frame, and a set
+of ℓ ground truth crater observations to associate these mea-
+surements with (Line 1). A value Qinc is initialized to some
+negligibly small, positive value ε to later avoid divide-by-zero
+Algorithm 3 Systematic Resampling
+Require: Particles
+{bt
+1, ..., bt
+Ns}
+and
+associated
+weights
+{wt
+1, ..., wt
+Ns}
+1:
+nt = log
+� �Ns
+i=1 exp(bt
+i)
+�
+2:
+{ ˜wt
+0, ..., ˜wt
+Ns} ← {0, ..., 0}
+3:
+for i = 1, ..., Ns do
+4:
+˜wt
+i ← exp(wt
+i − nt)
+5:
+end for
+6:
+{q0, ..., qNs} ← cum sum({ ˜wt
+0, ..., ˜wt
+Ns})
+7:
+n ← 0
+8:
+m ← 0
+9:
+u0 ∼ U(0,
+1
+Ns )
+10: while n ≤ Ns do
+11:
+u = u0 +
+n
+Ns
+12:
+while qm ≤ u do
+13:
+m ← m + 1
+14:
+end while
+15:
+n ← n + 1
+16:
+bt
+n ← bt
+m
+17: end while
+18: return {bt
+0, ..., bt
+Ns}
+issues (Line 1). Next, for each measurement zt
+i in the rover
+frame, the detected edge is converted to world frame (Line 3)
+and the minimum distance to an edge from the ground truth
+map computed (Line 4).
+The Qinc is incremented by the
+distance between the observed edge and its associated ground
+truth observation (Line 5). The Q-Score is computed as the
+reciprocal of Qinc and a min operation is applied to ensure
+that the score provided by any particular run is between 0 and
+1 (Line 7). This implies that observations and belief pairs
+which are less than 1 m away from ground truth will receive
+the same score as those exactly 1m away from ground truth,
+which is seen as acceptable given the orbital DEM resolution
+and mission concept localization requirements.
+In addition to the shortest distance formulation from Line 4,
+additional approaches were also explored for determining
+the Q-Score. One alternate approach investigated included
+fitting a Gaussian normal distribution on the orbital map
+crater edges and the Q-Score value was them computed based
+on the intensity (i.e., distance to the computed mean) of the
+point hit by observations or 0 in cases when no point was
+hit. In practice, it was determined that the shortest distance
+formulation provided the most robust results for use with the
+particle filter and also did not require additional projection
+calculations to project each belief from the orbital frame to
+rover frame.
+Overview—A description of the ShadowNav particle filtering
+algorithm is given in Alg. 2. The algorithm takes as input a
+Gaussian belief distribution (µ0, Σ0) assumed for the initial
+robot position, the number of particles Ns to use in the
+particle filter, and a threshold for the effective number of
+beliefs Neff,thresh used to trigger resampling (Line 2).
+The
+filter is initialized by sampling Ns particles from the initial
+belief distribution and assigning a weight of equal importance
+for each particle (Lines 1-2). As common in particle filtering
+implementations [21], we note that we used the log of the
+weights for improved numerical stability of the weight update
+step [22]. Given a new set of crater observations (Line 5),
+a set of Q-Score measurements is initialized for computing
+for each individual particle (Line 6).
+After applying the
+motion model update to each particle (Line 8), the Q-Score
+for each updated particle is computed using the procedure
+from Alg. 1 by comparing against the current measurements
+4
+
+(a) Sample of terrain
+with 90◦ from cam-
+era.
+(b) Sample opposi-
+tion effect during the
+day.
+(c) Sample effect of
+surface reflectance at
+night with an illumi-
+nator.
+Figure 4: The opposition effect simulated during the day
+and its effect at night with an external illuminator.
+(Line 9). The particle weights are then updated in log-domain
+(Line 13) with a normalization step to ensure non-negative
+weights (Line 11). Next, the number of effective samples Neff
+at the current iteration is calculated (Line 15). A common
+pitfall of particle filters is “degeneracy”, wherein the weights
+{wt
+i} collapse around a handful of particles and computa-
+tional resources are wasted on propagating low likelihood
+particles [21]. If Neff is below the threshold Neff,thresh, then
+this indicates that the filter is degenerating and a resample
+operation is triggered (Line 17).
+Further details on the systematic resampling approach used in
+this work are provided in Algorithm 3. Given a set of particles
+and their associated weights, the weights are first normalized
+to (0, 1] from log-domain (Lines 1-4) and the cumulative sum
+of these normalized weights ˜wt
+i computed (Line 6). The key
+step in systematic resampling is to sample a random value
+u0 from a uniform distribution inversely proportional to Ns
+(Line 9) and then incrementally sample a new particle from
+this “bin” of width
+1
+Ns . This ensures that, after resampling,
+at least one particle is retained from each
+1
+Ns interval from
+the previous belief distribution.
+C. Surface to Orbital Crater Transformation
+For every observation step, rover frame crater edges were
+detected with a stereo camera pair that provided the depth,
+and thus a relative position for the crater edge was saved. This
+relative crater distance was added to each particle’s belief
+position to form an estimate of the observed crater position
+in the world frame for each particle. The orbital map was
+projected to the world frame and then the shortest distance
+metric noted in the Q-Score algorithm was used to determine
+which particle belief positions were most likely and thus
+which observed crater was the most likely one to match the
+known orbital craters.
+Stereo hole filling—As some crater edge detections do not
+rely on depth information, not all pixels in the stero camera
+depth or disparity image will have a detected depth value
+and, in such cases, no relative position would be available
+for matching rover observations to the orbital map. For such
+observations, a simple plane fit can be carried out to fill in the
+depth information. A future area of investigation includes
+carrying out an improved stereo hole filling approach, in
+particular using existing knowledge on what the regional
+terrain looks like.
+Figure 5: The trajectories used for the numerical ex-
+periments are overlaid on the orbital map here with the
+crater numbers in black.
+A red square indicator is at
+the start and a green circle indicator is at the end of
+each trajectory. Trajectory 1 is in blue, trajectory 2 is
+in orange, and trajectory 3 is in purple.
+4. DATASETS OVERVIEW
+A. Simulated Lunar Environment
+At the time of writing, a Lunar dataset with images captured
+in the dark with an illuminator did not exist. Therefore, to
+evaluate the approach, a simulation environment was devel-
+oped using the Blender software [10]. In order to simulate
+images as realistically as possible, the Hapke lighting model
+[23], [24], [25] was implemented. This model approximates
+the Lunar surface reflectance and will simulate the “opposi-
+tion effect”. This effect leads to a focused point of extreme
+saturation at a location within an image where the camera ray
+and light source are at zero phase angle. The Hapke lighting
+model was implemented using the “old highland” parameters
+of the moon provided in [26], as these most closely match the
+poles of the moon where PSRs can be found. The coherent
+backscattering opposition effect (CBOE) was left out of our
+implementation and only the shadow hiding opposition effect
+(SHOE) was implemented as it dominates most or all lighting
+calculations in our use case, while CBOE has a negligible
+or very small effect. Initial implementation was done using
+the Open Shading Language (OSL), however not all rays are
+available for calculation due to optimizations made in OSL,
+so workarounds were needed to implement the Hapke light-
+ing model in Blender using OSL. While this was partially
+successful, it was not very robust and we had numerous
+issues.
+Instead of using OSL, we opted to modify the
+source of Blender to add the Hapke bidirectional reflectance
+distribution function (BRDF) directly into the Blender Cycles
+renderer code which also reduced the render time by greater
+than a factor of 2 through the use of Nvidia CUDA.
+In order to represent a realistic 3D model of the sur-
+face geometry, DEMs produced from LROC were utilized.
+While LROC has enough resolution to resolve craters of
+around 10 m, its resolution is not quite good enough for
+generating smooth surface imagery. In order to have smooth
+surface image renders, the DEMs from LROC were scaled
+down to be 0.25 m resolution. Crater measurements in fu-
+5
+
+10Table 1: Table of crater sizes in crater detection dataset.
+Crater
+Diameter (m)
+Depth (m)
+1
+9.2
+1.0
+2
+9.1
+0.75
+3
+11.3
+0.84
+4
+4.4
+0.55
+5
+3.7
+0.40
+6
+8.3
+0.27
+7
+11.9
+0.44
+8
+3.9
+0.48
+9
+4.1
+0.49
+10
+2.3
+0.25
+ture discussions were based on this scaled resolution. This
+scaled DEM was imported into Blender and a surface texture
+was added.
+The surface texture comprised of two scales
+of fractal Brownian motion, which is a natural noise that
+was added to the DEM in order to simulate Lunar surface
+texture for stereo to utilize.
+Figure 4 demonstrates three
+sample renders, two in the daylight and one at night with
+an illumination source from our simulation. It demonstrates
+what the surface looks like in daytime conditions as well as
+the effect of the Hapke model during the day with the sun
+behind the camera and the effect of the illumination source.
+From this it was observed that the full amount of daytime
+texture is not observed during the night with an illumination
+source.
+B. Simulated Craters for Detection Analysis
+In order to evaluate the performance of different crater de-
+tection techniques, a dataset with different sized craters was
+built.
+This dataset was built using the simulation process
+within Blender and captured stereo pair renders between 5 m
+and 20 m from the front crater rim in increments of 0.1 m.
+This dataset contained 10 different craters with varying sizes
+and depths. The sizes of the craters within this dataset are in
+Table 1 and their locations corresponding to the crater ID in
+our simulated environment are marked in Figure 5.
+C. Simulated Trajectories for Localization Analysis
+In order to evaluate the localization performance, several
+trajectories were run in the simulated environment. These
+trajectories were run to generate an image every 1 m and
+were designed to approach craters in different ways that
+might present challenges to our filtering approach.
+The
+1 m observation delta was used to reduce render times of
+our dataset, as rendering every 0.1 m did not result in a
+significant localization performance change. An overview of
+the trajectories within the orbital environment are displayed
+in Figure 5.
+D. Real Data of Negative Obstacles at Night
+In addition to the simulated data generated, a dataset was
+collected in the Arroyo, which is a dry river bed near the
+NASA Jet Propulsion Laboratory. This dataset contained a
+few different negative obstacles that were imaged at 5, 10,
+and 15 m away from the leading edge.
+This dataset was
+used to validated that the stereo and crater edge detection
+algorithms work on real data collected at night with an
+external illuminator.
+Figure 6:
+Plots of different metrics evaluating crater
+detection performance. Left: Plot that shows image-based
+crater edge detection score versus range for all craters
+evaluated. Right: Plot that shows percent of the crater
+front arc detected for all craters evaluated.
+Figure 7: Sample stereo results using JPLV stereo on a
+sample negative obstacle.
+5. CRATER DETECTION PERFORMANCE
+A. Metrics
+In order to evaluate the performance of surface crater detec-
+tion, the dataset referenced in Section 4 was utilized. Five
+different combinations of algorithms were evaluated. These
+were disparity discontinuity detection within SGBM stereo,
+disparity discontinuity detection within JPLV stereo, HED
+using SGBM stereo, HED using JPLV stereo, and a hybrid
+JPLV disparity discontinuity detection and canny edge detec-
+tion approach. The hybrid discontinuity detection and canny
+approach was implemented so that Canny only ran on the
+portion of the image that was 10 m away or further. This
+was done since it was observed the discontinuity detection
+worked well in the near range but stereo began to degrade
+beyond 10 m.
+These algorithms were evaluated with two different metrics.
+The first was an image based edge scoring method which
+captures an average Gaussian probability that a detected edge
+is on a ground truth crater edge. It utilizes a distance error
+computed in image space as represented in Equation 1 where
+Errordistpx is the pixel error from ground truth to detection,
+rangegt is the known ground truth range, fl is the focal length
+of camera, ss is the sensor size of the camera, and Errordist
+is the error in meters of the detection.
+Errordist = Errordistpx ∗ rangegt
+(fl ∗ ss)
+(1)
+6
+
+5m
+10mImageBasedCraterEdgeScoreversusRange
+1.0
++
+Disparity JPLV
+Disparity SGBM
++
+HED JPLV
+0.8
+HEDSGBM
+Disparity + Canny jPLV
+0.6
+Score
++
++
++
+0.4
+0.2
+0.0
+6
+8
+10
+12
+14
+16
+18
+20
+GT Range (m)Percent Crater Front Arc Detected vs.Range
+100
++
+Disparity JPLV
+Percent of Crater Front Arc Detected (%)
+HED JPLV
+Disparity SGBM
+80
+HED SGBM
+Disparity JPLV + Canny
+60
+++
+40
+20
+0
+6
+8
+10
+12
+14
+16
+18
+20
+Range(m)(a) Ground Truth at 7 m
+(b) Ground Truth at 12 m
+(c) Ground Truth at 17 m
+(d) JPLV Disparity + Canny at 7 m
+(e) JPLV Disparity + Canny at 12 m
+(f) JPLV Disparity + Canny at 17 m
+(g) JPLV HED at 7 m
+(h) JPLV HED at 12 m
+(i) JPLV HED at 17 m
+Figure 8: The efficacy of the JPLV HED approach over JPLV Disparity + Canny is demonstrated in simulations of
+crater rim detection overlay samples for crater 1.
+The distance error was then passed into a Gaussian.
+The
+Gaussian probabilities for all of the detected pixels were
+summed together and normalized by number of detected
+points to obtain a score. This scoring method infused ground
+truth range values to remove the impact of stereo holes
+and stereo range uncertainty on the projection in order to
+better isolate the specific performance of the crater detection
+algorithms. The sigma value for the Gaussian that was used
+in these experiments was 0.25m. This was chosen because
+the resolution of the DEM utilized was 0.25 m. Therefore
+most detections should fall within this boundary if they are
+highly accurate.
+The second metric used was ”percent of
+front arc detected”. In this metric, there is a ground truth
+circle of the orbital crater.
+Depending on the pose of the
+simulated cameras, the half arc of the ground truth circle that
+was nearest the simulated camera was projected into image
+space.
+The crater detection was then matched to the half
+arc and the percentage of the half arc that was successfully
+identified was determined. This metric removes the Gaussian
+component from the first metric; however, it does not capture
+7
+
+(a) Negative obstacle 5 m away
+(b) Negative obstacle 10 m away
+(c) Negative obstacle 10 m away
+(d) Negative obstacle 5 m away
+(e) Negative obstacle 15 m away
+(f) Negative obstacle 10 m away
+Figure 9: Qualitative edge detections using JPLV disparity discontinuity detection and Canny hybrid on negative
+obstacles on a real dataset collected in a dry river bed at night. These results demonstrate the transferability of the
+crater detection algorithms from simulated data to a real environment.
+false positives like the first metric.
+B. Detection Results on Simulated Data
+The results of running the different algorithms on the simu-
+lated dataset are observed in Figure 6. There were several
+notable observations from the results. The first was that the
+algorithms tended to perform the best around 10 m and did
+not improve as craters came closer.
+This was believed to
+be because as the camera gets closer to the crater, more of
+the crater becomes visible and the discontinuities become
+smaller. However, as the crater becomes further than 10 m,
+the stereo began to degrade.
+Additionally, for the hybrid
+stereo and Canny technique, the Canny detection started de-
+tection at 10 m and led to a significant jump in performance.
+In terms of algorithm comparison, JPLV disparity disconti-
+nuity performed better than SGBM disparity discontinuity
+which is likely due JPLV having more holes than SGBM.
+These holes at the boundary helped the disparity discontinuity
+detector find a better edge. However, for HED, it performed
+well with either stereo technique, likely due to its representa-
+tion of depth containing height values. HED was used with
+its out of the box weights from its authors. It likely could be
+improved with finetuning on a Lunar dataset.
+In addition to quantitative results, samples of crater rim de-
+tection overlays are in Figure 8. These results were on crater
+1 which is a nearly 10 m in diameter crater. Both methods
+were able to detect the craters well, but JPLV HED did have
+more falloff at 17 m than the Canny detector.
+However,
+the Canny edge detector was optimized for this environment
+where as HED was a generalized detector.
+Overall the
+generalization of HED was extremely promising as a crater
+rim detection approach.
+C. Detection Results on Real Data
+As described previously, data was collected from a location
+with negative obstacles at night.
+This dataset was used
+to validate the performance of stereo and crater detection
+algorithms.
+Figure 7 presents a sample of 5 m and 10 m
+negative obstacles and the corresponding stereo results from
+JPLV. From this figure is was observed that stereo is dense up
+unto the leading edge of the negative obstacle. Additionally,
+at 5 m, the far edge of the negative obstacle was captured
+in the disparity values. At 10 m, the far edge, did contain
+some disparity values but it was sparse.
+While not fully
+representative of the Lunar surface, this demonstrated that
+current stereo techniques do have the capability to work in
+low light conditions at the ranges necessary. The data was
+also used to evaluate the edge detection techniques. The JPLV
+disparity discontinuity and Canny edge detection hybrid was
+found to be the best on simulation data and therefore it
+was used on the real data.
+Figure 9 demonstrates sample
+detections at different ranges.
+These detection results did
+contain false positives on some of the vegetation as the false
+positive rejection was not run.
+Vegetation is not present
+on the moon, however, objects such as rocks could present
+similar issues. Overall, the negative obstacle edge detection
+qualitatively performs well.
+6. LOCALIZATION PERFORMANCE
+In this section, we provide Monte Carlo results on the per-
+formance of the proposed ShadowNav filtering algorithm.
+For each simulation, we analyzed the performance of the
+ShadowNav filter on the basis of the following metrics:
+8
+
+(a) Ground truth error for traj. 1. (b) Filter covariance for traj. 1. (c) Ground truth error for traj. 2. (d) Filter covariance for traj. 2.
+Figure 10: A comparison of the four proposed resampling schemes demonstrated that systematic resampling empirically
+outperforms the other scheme in terms of relatively lower ground truth error and reduced uncertainty in the filter.
+(a) Ground truth error.
+(b) Filter covariance.
+Figure 11: Monte Carlo simulations for trajectories 1–3
+demonstrated the efficacy of the Q-Score based particle
+filtering approach at accomplishing global rover localiza-
+tion.
+(a) Traj. 1 traverse – case A.
+(b) Traj. 1 traverse – case B.
+Figure 12: Two Monte Carlo trials for trajectory 1 are
+illustrated with the ground truth in red and the weighted
+average belief µt in blue. The comparatively better per-
+formance of the filter in case A (left) was due to false
+positive crater rim measurements in case B (right) that led
+to worse localization.
+Ground truth error: We computed the weighted average
+mean µt = �Ns
+i=1 wt
+ibt
+i at time t for the filter using the particle
+weights and beliefs and compute the ℓ2-distance to the ground
+truth gtt, i.e., ∥µt − gtt∥2.
+Particle filter uncertainty: To capture the uncertainty asso-
+ciated with the current belief, we additionally computed the
+weighted covariance matrix Σt = �Ns
+i=1 ˜wt
+i(bt
+i−µt)(bt
+i−µt),
+where ˜wt
+i are the normalized weights detailed in Alg. 3. The
+metric we report at each time step was the square root of
+the largest eigenvalue
+�
+λmax(Σt), which corresponded to the
+worst case variance of the estimation error [27], [28].
+Mahalanobis distance: The final metric we computed was the
+Mahalanobis distance, which measures the distance between
+and the particle filter distribution and ground truth posi-
+tion. We approximately computed this by fitting a Gaussian
+distribution N(µt, Σt) to the particle filter distribution, for
+which the Mahalnobis distance is simply a weighted ℓ2-norm
+�
+(µt − gtt)T (Σt)−1(µt − gtt).
+A. Resampling Scheme Comparison
+In this section, we compared the baseline systematic resam-
+pling approach detailed in Alg. 3 against three other resam-
+pling methods utilized: multinomial, residual, and stratified
+(we refer the reader to [21], [29], [30] for a thorough review
+of these approaches.) Figure 10 presents the ground truth
+error and filter uncertainty for the four different resampling
+approaches. We saw that, for the two trajectories compared
+in Figure 10, systematic resampling led to comparable ground
+truth error as the other resampling approaches, but that
+systematic resampling outperformed the other approaches in
+terms of the overall uncertainty of the filter. Indeed, we note
+that multinomial resampling, the most commonly employed
+resampling technique, fared quite poorly in terms of the
+variance of the filter uncertainty (Fig. 10b and 10d).
+B. Baseline Performance Evaluation
+Finally, we evaluated the performance of the proposed Shad-
+owNav particle filter approach on three test trajectories. Our
+analysis consisted of Monte Carlos simulations with 25 seeds
+and utilizing 2% odometry noise and initial belief distribution
+with σ0 =3 m. Each simulation was run with Ns = 100
+particles and systematic resampling as the resampling scheme
+with Neff,thresh = 50 as the resampling threshold.
+Figure 11 shows Monte Carlo simulation results for the three
+test trajectories. We saw that the initial uncertainty in the filter
+began at approximately 3 m as expected by sampling from a
+distribution with σ0 =3 m. Thereafter, the filter was able to
+improve the rover position estimate, which led to an absolute
+error reduction of 4 m. Further, we see in Table 2 that the
+metrics computed at the final time step indicate convergence
+of the filter, with an average final error of ≤4 m and an
+absolute error reduction of 4 m.
+As seen in Figure 13, while the filter performed well on
+trajectories 2 and 3, the filter was less performant for the
+trajectory 1 test case. Figure 12 illustrates the performance
+of the filter on trajectory 1 for two different random seeds
+as the rover starts from the northern edge of the orbital map
+and moves southward.
+During the middle portion of this
+traverse, the craters were out-of-sight for the rover and, as we
+9
+
+[x-x| [m]
+6
+5
+[u] Ix->
+4
+X
+3
+Resampling
+2
+Residual
+Systematic
+1
+Stratified
+Multinomial
+0
+0
+20
+40
+60
+80
+100
+120
+Iterationg
+Resampling
+Residual
+4
+Systematic
+Stratified
+Multinomial
+3
+[m]
+2
+1
+0
+0
+20
+40
+60
+80
+100
+120
+Iteration[x-x] [m]
+7
+Resampling
+Residual
+6
+Systematic
+Stratified
+5
+Multinomial
+[m]
+4
+[x-)
+<X
+3
+2
+1
+0
+20
+40
+60
+80
+100
+120
+140
+Iterationg
+Resampling
+3.5
+Residual
+Systematic
+3.0
+Stratified
+Multinomial
+2.5
+1.5
+1.0
+0.5
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+Iteration[x-x] [m]
+Traj_Name
+7
+Traj. 1
+Traj. 2
+6
+Traj. 3
+5
+[u] [x->
+4
+X
+3
+2
+1
+0
+0
+20
+40
+60
+80
+100
+120
+140
+Iterationg
+4.0
+Traj Name
+Traj. 1
+3.5
+Traj. 2
+Traj. 3
+3.0
+2.5
+[w]
+2.0
+1.5
+1.0
+0.5
+0.0
+0
+20
+40
+60
+80
+100
+120
+140
+Iteration(a) Trajectory 1
+(b) Trajectory 2
+(c) Trajectory 3
+Figure 13: The final ground truth error distribution for 25 Monte Carlo simulations showed filter convergence to ≤ 4m
+error in all cases for trajectories 2 and 3 and for the majority of cases for trajectory 1.
+Table 2: The metrics computed at the end of a long-range
+lunar traverse indicate convergence of the particular filter
+on trajectories 2 and 3, but spurious measurements from
+unlabeled crater lead to relatively poor performance on
+trajectory 1.
+Error
+Uncertainty
+Mahalanobis Dist.
+Traj. 1
+3.84 ± 2.78
+1.84 ± 1.12
+8.74 ± 10.03
+Traj. 2
+1.75 ± 0.78
+1.32 ± 0.76
+2.75 ± 1.88
+Traj. 3
+1.68 ± 0.7
+1.39 ± 0.61
+2.92 ± 1.91
+see in Figure 11, false positive observations led to increases
+in the error and uncertainty in the filter.
+As the crater in
+the southern portion of the orbital map became observable
+for the rover, we saw that the estimate quickly improved in
+case A (Fig. 12a), but continues to have a residual error in
+case B (Fig. 12b). This poor convergence behavior was also
+explained by false positive observations, wherein the filter
+had difficulty reconciling the front edge of the rim with the
+back edge, an issue that requires further investigation.
+C. Debugging
+When testing the particle filter, we found it helpful to generate
+“perfect” datasets where ground truth depth was generated
+directly from the simulator as shown in Figure 14b) and crater
+edges were plotted into the rover frame using their exact
+known world coordinates (see Fig. 8a-8c).
+This approach
+uncovered bugs with our perception and projection pipeline
+as well as the particle filter pipeline and it is highly recom-
+mended to build such a dataset for all similar work.
+7. CONCLUSIONS
+In this work we present a system to perform autonomous
+absolute localization on a Lunar rover while it is in darkness.
+This system entails using a stereo camera and illuminator. We
+enhanced a Blender based simulation with a custom Lunar
+texture and an implementation of the Hapke model to model
+surface reflectance as accurately as possible.
+We further
+demonstrate both geometric and learning based techniques
+for detecting the leading edge of a crater with ability to
+detect some craters out to 20 m range. We propose a method
+of matching the detected leading crater rims with known
+craters within an orbital map and using these matches to score
+(a)
+Simulated
+image
+from
+Blender.
+(b) Perfect depth: blue is close,
+red is far
+Figure 14: Crater 1 viewed from 5 m away from front rim.
+observations with our Q-Score. Finally we demonstrate abso-
+lute localization within our simulation environment with less
+than 4 m error, and an absolute error reduction of 4 m upon
+detecting craters.
+These results show promise for further
+investigation in the future on more simulation environments
+as well as on to be collected real analogue datasets.
+D. Future Work
+In the future, we seek to perform several updates and addi-
+tional evaluations. The primary focus is to experimentally
+collect a nighttime dataset using representative hardware in
+an analogue Lunar environment with negative obstacles to
+evaluate the system.
+Additional evaluation is planned to
+evaluate the performance of the proposed approach along
+longer trajectories, on more varied Lunar type locales, and
+for different rover specific parameters such as camera height
+off of the ground. Finally, we plan to validate our proposed
+approach on a flight-like embedded computer (e.g., a Snap-
+dragon) to demonstrate that it is computationally feasible for
+use onboard a Lunar rover.
+ACKNOWLEDGMENTS
+The research was carried out at the Jet Propulsion Labo-
+ratory, California Institute of Technology, under a contract
+with the National Aeronautics and Space Administration
+(80NM0018D0004). The authors would like to thank Yang
+Cheng, Olivier Lamarre, and Scott Tepsuporn for their dis-
+cussions during the development of this work.
+10
+
+14
+12
+10
+Count
+8
+6
+4
+2
+0
+0
+2
+4
+6
+8
+10
+Final GT Error [m]14
+12
+10
+Count
+8
+6
+4
+2
+0
+0
+2
+4
+6
+8
+10
+Final GT Error [m]14
+12
+10
+Count
+8
+6
+4
+2
+0
+0
+2
+4
+6
+8
+10
+Final GT Error [m]REFERENCES
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+11
+
+BIOGRAPHY[
+Abhishek Cauligi is a Robotics Tech-
+nologist in the Robotic Surface Mobility
+Group at NASA Jet Propulsion Labo-
+ratory, California Institute of Technol-
+ogy.
+Abhishek received his B.S. in
+Aerospace Engineering from the Uni-
+versity of Michigan - Ann Arbor and
+his PhD. in Aeronautics and Astronau-
+tics from Stanford University.
+His re-
+search interests lie in leveraging recent
+advances in nonlinear optimization, machine learning, and
+control theory towards planning and control for complex
+robotic systems.
+R. Michael Swan is a Robotics Systems
+Engineer at NASA Jet Propulsion Lab-
+oratory, California Institute of Technol-
+ogy. He received his B.S. in Computer
+Engineering from Walla Walla Univer-
+sity and his M.S. in Computer Science
+from the University of Southern Califor-
+nia. He is interested in robotic surface
+and aerial autonomy, perception, simu-
+lation, and robotic system architecture.
+Hiro Ono is the Group Leader of
+the Robotic Surface Mobility Group at
+NASA Jet Propulsion Laboratory, Cal-
+ifornia Institute of Technology.
+Since
+he joined JPL in 2013, he has led a
+number of research projects on Mars
+rover autonomy, as well as three NIAC
+studies on Enceladus Vent Explorer and
+Comet Hitchhiker. Hiro was a flight soft-
+ware developer of M2020’s Enhanced
+AutoNav and the lead of M2020 Landing Site Traversability
+Analysis. He also led the development of a machine learning-
+based Martian terrain classifier, SPOC (Soil Property and
+Object Classification), which won JPL’s Software of the Year
+Award in 2020.
+Shreyansh Daftry is a Robotics Tech-
+nologist at NASA Jet Propulsion Labora-
+tory, California Institute of Technology.
+He received his M.S. degree in Robotics
+from the Robotics Institute, Carnegie
+Mellon University, and his B.S. degree
+in Electronics and Communication En-
+gineering.
+His research interest lies
+at the intersection of space technology
+and autonomous robotic systems, with
+an emphasis on machine learning applications to percep-
+tion, planning, and decision making.
+At JPL, he is the
+Group Leader of the Perception Systems group, is working
+on the Mars Sample Recovery Helicopter mission, and has
+led/contributed to technology development for autonomous
+navigation of ground, airborne, and subterranean robots.
+John Elliott is a principal engineer
+in JPL’s Mission Concept Systems De-
+velopment group.
+He currently serves
+as Program Engineer for the Planetary
+Science Formulation office. His recent
+tasks have included serving as study
+lead for the Planetary Decadal Survey’s
+three lunar rover mission concept stud-
+ies, Intrepid, INSPIRE, and Endurance,
+and performing systems engineering and
+leadership roles on a number of recent Discovery and New
+Frontiers mission proposals. Mr. Elliott’s past experience
+includes six years in the terrestrial nuclear power industry
+with Bechtel Corporation in addition to 30 years in aerospace
+systems at TRW and JPL.
+Larry Matthies is the technology co-
+ordinator in the Mars Exploration Pro-
+gram Office at JPL. He received B.S.,
+M. Math,and PhD degrees in Computer
+Science from the University of Regina
+(1979), University of Waterloo (1981),
+and Carnegie Mellon University (1989).
+He has been with JPL for more than
+32 years. He has conducted technology
+development in perception systems for
+autonomous navigation of robotic vehicles for land, sea,
+air, and space.
+He supervised the JPL Computer Vision
+group for 21 years. He led development of computer vision
+algorithms for Mars rovers, landers, and helicopters. He is a
+Fellow of the IEEE and a member of the editorial boards of
+Autonomous Robots and the Journal of Field Robotics.
+Deegan Atha is a Robotics Technologist
+within the Perception Systems Group of
+the Mobility and Robotic Systems Sec-
+tion at the Jet Propulsion Laboratory.
+He received his B.S. degree from Purdue
+University in Electrical Engineering and
+his M.S. in Computer Science from the
+Georgia Institute of Technology. He is
+currently the Principal Investigator for
+the ShadowNav task and leading the se-
+mantic perception effort for the DARPA RACER project. His
+interests are in the infusion of machine learning and robotic
+perception into autonomous systems operating in unstruc-
+tured environments.
+12
+

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+Silent Killer: Optimizing Backdoor Trigger Yields a Stealthy and Powerful Data
+Poisoning Attack
+Tzvi Lederer1 , Gallil Maimon2 , Lior Rokach1
+1Ben Gurion University , 2The Hebrew University of Jerusalem
+{tzvil, liorrk}@post.bgu.ac.il
+Abstract
+We propose a stealthy and powerful backdoor attack
+on neural networks based on data poisoning (DP).
+In contrast to previous attacks, both the poison and
+the trigger in our method are stealthy. We are able to
+change the model’s classification of samples from a
+source class to a target class chosen by the attacker.
+We do so by using a small number of poisoned train-
+ing samples with nearly imperceptible perturbations,
+without changing their labels. At inference time,
+we use a stealthy perturbation added to the attacked
+samples as a trigger. This perturbation is crafted as
+a universal adversarial perturbation (UAP), and the
+poison is crafted using gradient alignment coupled
+to this trigger. Our method is highly efficient in craft-
+ing time compared to previous methods and requires
+only a trained surrogate model without additional re-
+training. Our attack achieves state-of-the-art results
+in terms of attack success rate while maintaining
+high accuracy on clean samples.
+1
+Introduction
+Deep learning models have been shown to be susceptible
+to data poising attacks [Chen et al., 2017; Gu et al., 2017;
+Shafahi et al., 2018]. In these attacks, the attacker manipu-
+lates training examples and adds them to the victim’s training
+set. The victim trains the model on the poisoned dataset and
+unknowingly adds unwanted functionality to the model. One
+typical example of such functionalities is a backdoor. The
+goal of backdoor attacks is to be able to change the model’s
+prediction at inference time by adding a trigger to the sam-
+ple (e.g., a small colorful patch in the image), such that the
+model will misclassify it in the way the attacker planned.
+(e.g. replacing stop signs prediction with speed limits [Gu
+et al., 2017]). In a world where many models are trained on
+large datasets scraped from the internet [Ramesh et al., 2022;
+Kuznetsova et al., 2020], the threat of DP becomes more rel-
+evant than ever. A malicious attacker can generate poisoned
+samples and put them on public sites such as Wikipedia or
+even on his own site. In high probability, those samples will
+end up in the training set of a large company and can stealthily
+damage the model.
+Figure 1: Illustration of the Silent-Killer attack. In the first step
+(1), the attacker uses a surrogate model to craft a targeted universal
+adversarial perturbation that will be used as a trigger. (2) The attacker
+creates poisoned samples using gradient alignment. The attacker
+releases the poisoned samples into the victim’s dataset (3), and the
+victim trains the model on the poisoned dataset. After the victim
+deploys the model, the attacker can add the trigger to his own images
+and activate the unwanted behavior of the model (4).
+However, it is difficult to carry out a successful poisoning
+attack on a model without being noticed. Earlier work in this
+field [Gu et al., 2017] has shown that embedding the trigger
+directly into the training set can poison the model, but this
+method is not stealthy because the trigger is visible and can be
+removed. Additionally, these methods typically involve flip-
+ping the labels of the poisoned samples which can be identified
+during re-labeling. Other works [Saha et al., 2020] introduce a
+method to craft poisoned examples with small stealthy pertur-
+bations that do not change their label, but this method, which
+is based on features collision, does not achieve high perfor-
+mance when training the model from scratch, but only in the
+fine-tuning scenario. A recent work, named Sleeper-Agent
+[Souri et al., 2022] has demonstrated a gradient-based op-
+timization method for poison crafting. In their work, they
+show that using the gradient alignment method, they can craft
+stealthy and effective poisoned samples, even in the challeng-
+ing scenario that previous methods struggled to handle. For
+example, this method succeeds under the clean label settings,
+when the attacker is not allowed to change the sample label,
+and when the attacker trains the model from scratch. However,
+this method uses as a trigger a striking, colorful patch, which is
+arXiv:2301.02615v1  [cs.CR]  5 Jan 2023
+
+Trigger
+Poison
+Victim
+Attack
+Crafting
+Crafting
+.Training
+.Realization
+X
+(1)
+(2)
+(3)
+(4)not stealthy and can be detected by the victim at inference time.
+Examples of this trigger can be found in figure 3. Furthermore,
+the patch is arbitrary. As we show in this work, optimizing
+a trigger toward the attacker’s objective, rather than using an
+arbitrary patch, can yield a significant improvement in attacks
+like this.
+Another line of research in the field of adversarial attacks
+is evasion attacks, also known as adversarial examples. In
+these, the attacker uses gradient-based optimization to care-
+fully perturb the input samples at test time and have them
+misclassified. Specifically, some works [Moosavi-Dezfooli
+et al., 2017] show that a single perturbation, which can be
+interpreted as a trigger, can change model predictions on many
+samples. However, as noted in their work, the transferability
+of the attack is not completely reliable. This means that in
+many cases, the attack performance will significantly drop
+when it is applied to other models. In today’s reality, when
+companies often don’t publicly release their models, assuming
+access to the exact model is implausible.
+In this work, we show that a universal adversarial pertur-
+bation (UAP)[Moosavi-Dezfooli et al., 2017] can be used
+as a stealthy and powerful trigger. This trigger, combined
+with the gradient alignment method to craft poisoned sam-
+ples for training [Souri et al., 2022], yields a strong, stealthy
+attack that is more robust and powerful than existing meth-
+ods. Specifically, we show two attacks. In the first attack, we
+craft a universal perturbation that is bounded in magnitude
+and use it as a trigger. The optimization process is inspired
+by UAP [Moosavi-Dezfooli et al., 2017] which iteratively
+updates the perturbation of a set of input samples in the di-
+rection that minimizes the attacker loss on a large number
+of samples. The main differences are that we use a targeted
+attack rather than an untargeted attack, and at each iteration,
+we clip the perturbations up to the upper bound rather than
+projecting it on a l2 norm ball. Examples of this method
+are shown in figure 2. In our second attack, we use a small
+square patch similar to existing methods [Gu et al., 2017;
+Souri et al., 2022] and optimize it without constraints (ex-
+cept patch size). This optimization yields a patch that is sim-
+ilar to triggers used in previous works [Saha et al., 2020;
+Souri et al., 2022].
+Examples can be found in figure 3.
+In both cases, our approach achieves state-of-the-art results
+in terms of attack success rate and robustness under our
+setup. Furthermore, our method does not require retraining
+the model during poison crafting nor using an ensemble of
+models as was done in previous works [Huang et al., 2020;
+Souri et al., 2022]. We demonstrate the success of our method
+in a black-box scenario and its high transferability to unseen
+architectures. The attack scheme is shown in figure 1. Code is
+available at this link.
+Our main contributions are as follows:
+• We demonstrate for the first time a stealthy data-
+poisoning-based black-box backdoor attack on NN image
+classifiers, in which both the poison and the trigger are
+nearly imperceptible, and the label of the poisoned sam-
+ples is not changed (clean-label).
+• Our attack achieves state-of-the-art results in terms of at-
+tack success rate without harming the accuracy on benign
+Figure 2: Samples of our attack on ImageNet. The images on the left
+are poisoned samples sampled from the target class, and the images
+on the right are test samples with the trigger embedded into them. A
+successful attack will result in shark images being classified as cocks.
+samples.
+• This attack is simple and effective. It does not require
+model optimization and does not need ensembles or other
+hand-crafted solutions to achieve effective results.
+2
+Related Work
+Backdoor Attacks.
+The concept of a backdoor attack or
+Trojan attack involves the embedding of unexpected behavior
+into a neural network by an attacker. This behavior is not
+anticipated by the network’s developer and may only mani-
+fest in specific, pre-defined cases. For instance, a classifica-
+tion network may exhibit high accuracy on most samples but
+be designed to misclassify a specific, chosen sample [Geip-
+ing et al., 2020]. Backdoor attacks can be implemented in
+several ways, such as by modifying the victim network di-
+rectly [Gu et al., 2017; Zhang et al., 2021], contaminating the
+pre-trained network used by the victim [Kurita et al., 2020;
+Gu et al., 2017], poisoning the training dataset [Yang et al.,
+2017], or even modifying the training process or loss function
+[Bagdasaryan and Shmatikov, 2021]. In some cases, a com-
+bination of these methods may be used, such as in [Qi et al.,
+2021], where the poisoned training set and network weights
+are learned together. A comprehensive review of backdoor
+attacks against neural networks can be found in [Li et al.,
+2022]. In this work, we focus on backdoor attacks via DP.
+This allows the attacker to control the output of the model by
+using a trigger added to samples at inference time.
+Data Poisoning.
+DP is a type of attack on neural networks,
+in which the attacker manipulates the training dataset in order
+to influence the behavior of models trained on this data [Biggio
+et al., 2011; Geiping et al., 2020; Gu et al., 2017]. As machine
+learning models require a large amount of data for training,
+many organizations gather data from the internet. This data
+may not always be verified and can therefore be exploited by
+attackers. In DP attacks, the attacker alters or adds samples to
+the training dataset. This is in order to change the behavior
+of the model when it is trained on this modified dataset. Cin`a
+et al. categorize DP attacks into three categories based on
+their purpose: (1) indiscriminate attacks, which increase the
+test error on all samples in the test set [Biggio et al., 2011;
+
+Poisoned Training
+Clean Training
+Poison
+Test Samples with
+Test Samples
+Trigger
+Perturbations
+Samples
+Trigger
+without Trigger
+Perturbations
+SamplesCin`a et al., 2022], (2) targeted attacks, which increase the test
+error on specific samples [Geiping et al., 2020; Shafahi et al.,
+2018], and (3) backdoor attacks, which increase the test error
+on samples that contain a specific trigger embedded within
+them [Gu et al., 2017; Souri et al., 2022]. In this work, we
+focus on implementing a DP-based backdoor attack.
+Backdoor Attacks with Data Poisoning.
+Backdoor attacks
+have often been conducted with unrealistic assumptions for
+the attacker, such as the ability to modify the victim model
+or its weights. There are few works that demonstrate the suc-
+cessful backdooring of a model using only a small portion of
+the poisoned training set. Gu et al. blended a patch into a
+portion of the training set and showed that inserting this patch
+into samples in the test set would cause the model to classify
+them as the attacker planned. Subsequent works investigated
+other heuristic approaches, but they did not find a systematic
+optimization-based approach to find stealthy poison. Li et al.
+described an attack that uses a stealthy bounded norm trigger,
+but it does not work under the clean label setting and is not
+applicable to from-scratch training. Saha et al. described a
+method based on feature collusion that crafts a clean label
+stealthy poison, but it does not work on from-scratch training.
+Souri et al. proposed using gradient alignment on a surrogate
+network to craft a poisoned training set, but the trigger in their
+work was noticeable and the poisoning process is hard for the
+attacker, especially in large models, due to the need for retrain-
+ing the model and use of ensemble during crafting. In this
+work, we present a stealthy data-poisoning-based backdoor
+attack that achieves state-of-the-art results.
+Gradient Alignment.
+Gradient alignment, or gradient
+matching, is a relatively recent method that was proven
+suitable to DP [Souri et al., 2022; Geiping et al., 2020;
+Fowl et al., 2021]. The idea was borrowed from the data
+condensation domain, where the goal is to find ”informative
+samples”, samples producing rich gradients during model
+training. In other words, the goal is to find a few samples
+where an optimization upon them minimizes the loss over a
+large number of samples [Zhao et al., 2021]. Recent works
+show that this method can be used for DP [Geiping et al.,
+2020] and backdoor attacks [Souri et al., 2022]. For example,
+in a backdoor attack, the attacker wants the model to classify
+samples with a certain trigger as a specific target class. How-
+ever, the attacker is not able to directly optimize the model to
+do this. The gradient alignment method allows the attacker to
+adjust the gradients of the model’s weights when it is being
+optimized on poisoned samples. This is so that the gradients
+are as close as possible to the gradients of the attacker’s de-
+sired classification loss on the trigger samples. The attacker
+tries to maximize the cosine similarity between the gradients
+of the poisoned samples and the gradients of the attacker’s
+loss during the optimization process. A more comprehensive
+formulation can be found in section 3.3.
+Evasion Attacks.
+Evasion attack, also called adversarial
+examples, is a type of attack at test time on a machine learning
+model, aiming to change the model’s input so that the model
+will predict wrongly on the manipulated example [Goodfellow
+et al., 2014; Kurakin et al., 2018; Madry et al., 2017; Carlini
+and Wagner, 2017]. An important attack for our method is the
+Figure 3: An illustration of our experiment with optimized patches.
+From left to right: pre-defined arbitrary patch [Souri et al., 2022],
+optimized patch (ours), poisoned training samples, and clean training
+samples. The poisoned samples were crafted with the optimized
+patch, but the poison crafted with the pre-defined patch looked very
+similar. By optimizing the trigger, we achieve superior results in
+terms of ASR.
+Algorithm 1 Trigger Crafting
+Input: {xi}M
+i=1 ∈ Ds, Fs(·; θs), lt, Tt(·, ·)
+Parameter: Rt, αt, ϵt, σ2
+Output: δt
+1: Initialize δt ∼ N(0, σ2)
+2: for r = 1, 2, ..., Rt optimization steps do
+3:
+Xt = {Tt(xi, δt)}M
+i=1
+4:
+ˆYt = {Fs(xt
+i; θs) : xt
+i ∈ Xt}M
+i=1
+5:
+∆ = sign( 1
+N
+�
+ˆyt
+i∈ ˆYt ∇δtL(ˆyt
+i, lt))
+6:
+δt ← δt − αt · ∆
+7:
+δt ← clip(δt, −ϵt, ϵt)
+8: end for
+9: return δt
+UAP attack [Moosavi-Dezfooli et al., 2017], which optimizes
+a single perturbation on all samples. Inspired by this method,
+we use a targeted iterative optimization [Kurakin et al., 2018]
+to craft a universal perturbation on several samples from the
+source label and use it as a trigger for the backdoor attack. In
+combination with gradient-alignment-based poison crafting,
+this results in a powerful DP attack.
+3
+Method
+3.1
+Threat Model
+Our threat model is as follows. We conducted experiments
+under two scenarios: gray-box, where the attacker knows the
+victim’s architecture but not the weights, and black-box, where
+the attacker has no information about the architecture and the
+weights. In either case, the attacker has access to data drawn
+from the same distribution as the victim. The attacker can
+train a surrogate model, or use a pre-trained model without the
+need to perform training himself. In addition, the attacker can
+
+pre-defined
+optimized
+poisoned
+clean
+Trigger
+PoisonAlgorithm 2 Poison Crafting
+Input: Fs(·; θs), Tp(·; ·), Da = {(xj, yj)}N
+j=1 ∈ Dt,
+Dδt = {Tt(xk; δt) : xk ∼ Ds}K
+k=1, lt
+Parameter: Rp, αp, ϵp, σ2
+Output: δp
+1: Initialize δp ← {δp
+j ∼ N(0, σ2)}N
+j=1
+2: La = 1
+K
+�
+xt
+k∈Dδt L(Fs(xt
+k, θs), lt)
+3: for r = 1, 2, ..., Rp optimizations steps do
+4:
+for j = 1, 2, ..., N do
+5:
+Lv,j = L(Fs(xj + δp
+j , θs), yj)
+6:
+Aj = 1 −
+∇θsLv,j·∇θsLa
+||∇θsLv,j||·||∇θsLa||
+7:
+δp
+j ← δp
+j − αp · ∂Aj
+∂δj
+8:
+δp
+j ← clip(δp
+j , −ϵp, ϵp)
+9:
+end for
+10: end for
+11: return δp
+poison a small portion (e.g., 1%) of poisoned training samples
+of the victim’s training set. The attack has to be clean label,
+which means that the attacker cannot change the labels of the
+poisoned samples, which is more challenging [Schwarzschild
+et al., 2021]. Finally, the attacker can embed a stealthy trigger
+into a sample and feed it to the model at inference time. The
+attack is successful if the model predicts this sample as the
+target class specified by the attacker. The victim can train a
+model from-scratch, which is more challenging than the case
+when we assume a fine-tuning regime [Schwarzschild et al.,
+2021]. Training a model from-scratch is very challenging for
+some methods based on feature-collision [Saha et al., 2020].
+3.2
+Notation and Problem Setup
+The victim has a model F with parameters θ, a dataset D =
+{(xi, yi)}i and a loss function L (e.g. Cross-entropy loss).
+The attacker has access to a surrogate model Fs(·; θs), which
+can have the same architecture as F (gray-box) or a different
+one (black-box). While preliminary results showed that using
+a different dataset from the same distribution would also work,
+we assume that the attacker has access to the victim dataset
+D, to be compatible with the setup in Souri et al.. We refer to
+the subset containing the poisoned samples as Da. |Da| = N,
+when N is the number of samples that the attacker is allowed
+to modify in the training set. Dc is the portion of the data
+that the attacker cannot modify. Dp = Dc ∪ Da is the dataset
+containing the poisoned samples with the clean samples. The
+victim trains a model to find θ = arg min L(Dp, F(·; θ)).
+The attacker aims to find perturbations δp = {δp
+i }N
+i=1 to
+embed into training samples xp
+i = Tp(xi, δp
+i ), and create the
+poisoned set Da = {(xp
+i , yi)}N
+i=1. In addition, the attacker has
+to find a single trigger δt which will be embedded into samples
+at test time xt = Tt(x, δt); x ∼ Ds which will make the
+model predict the sample as target class F(xt; θ) = lt, when
+ls ̸= lt. We refer to the distribution of samples from the source
+class and the target class as Ds and Dt respectively. ls is the
+source label and lt is the target label. Tp(xi, δp
+i ) and Tt(xi, δt)
+insert δp
+i and δt to xi as a poison and a trigger respectively. In
+our work, Tp is an additive function Tp(xj; δp
+j ) = Clip(xj +
+δp
+j , 0, 1), and Tt can be an additive function, or alternatively
+embed a patch into the sample. The details of these functions
+are described in section 3.3.
+The attacker aims to minimize E(xi,yi)∼DsLa when
+La = L(F(xt
+i; θ(δp)), lt), as the victim minimizes his loss
+1
+|Dp|
+�
+(xi,yi)∼Dp Lv; Lv = L(F(xi; θ), yi). In general, the
+perturbations and the trigger have to solve the following bi-
+level optimization problem:
+arg
+min
+δp
+i |N
+i=1∈∆p,δt∈∆t
+E(x,y)∼DsL(F(Tt(x, δt); θ(δp)), lt)
+s.t. θ(δp) = arg min
+ˆθ
+E(xi,yi)∼DpL(F(xi(δp); ˆθ), yi)
+Where ∆p = {δ : ||δ||∞ < ϵp} and ∆t = {δ : ||δ||∞ < ϵt}.
+ϵp and ϵt control the maximal norm of the perturbation of the
+poison and the trigger respectively. The larger they are, the
+simpler it is to perform the attack, but it makes the attack less
+stealthy. Alternatively, ∆p can be the set of small patches with
+pre-defined size, as we describe later.
+The bilevel optimization problem is known to be difficult to
+solve [Huang et al., 2020]. Our approach to dealing with this
+problem is described in section 3.3.
+Previous methods [Souri et al., 2022; Saha et al., 2020]
+also used hidden poison in their attacks, but their trigger was
+a visible colorful patch. We are the first to use both a stealthy
+poison and a stealthy trigger. Furthermore, in these works,
+only the poison was optimized, but not the trigger. In our
+method, we aim to find an optimized trigger that outperforms
+the use of simple arbitrary triggers.
+3.3
+Our Approach
+Trigger Crafting
+Our attack consists of two stages: trigger crafting and poi-
+son crafting. In the first stage, inspired by [Kurakin et al.,
+2018] and [Moosavi-Dezfooli et al., 2017], we craft a univer-
+sal perturbation δt that tries to change the surrogate model
+Fs(·; θs) prediction of samples from the source class ls to
+the target class lt. This process is described in algorithm
+1. We initialize δt ∈ N(0, σ2) when σ2 = 0.01, but uni-
+form noise or initialization with zeros works as well. In our
+work, we use samples from victim training set, but in general,
+any samples drawn from the source label distribution can be
+used. In our main attack, the trigger is an additive perturbation
+with the same dimensions as the images, added to the samples
+Tt(x; δt) = Clip(x+δt, 0, 1), bounded such that ||δt||∞ ≤ ϵt.
+We follow [Souri et al., 2022] that uses ϵ = 16/255 for the
+poison perturbation and use this threshold for ϵt. Our second
+attack was designed to compare our method to [Souri et al.,
+2022], which used a small patch as a trigger. We optimize a
+trigger of size 8 × 8 pixels instead of selecting a pre-defined
+one. We use algorithm 1 with Tt as a patch insertion function:
+Tt(x, δt)[i, j] =
+�δt[i′, j′]
+(i, j) ∈ R
+x[i, j]
+else
+
+Attack success rate [%]
+Architecture
+ResNet18
+MobileNet-V2
+VGG11
+Hidden Trigger Backdoor [Saha et al., 2020]
+3.50
+3.76
+5.02
+Clean-Label Backdoor [Turner et al., 2019]
+2.78
+3.50
+4.70
+Sleeper Agent [Souri et al., 2022]
+78.84
+75.96
+86.60
+Sleeper Agent (base)
+57.42
+15.00
+25.25
+Silent Killer (patch) (ours)
+95.92
+41.68
+97.35
+Silent Killer (perturbation) (ours)
+89.85
+97.91
+97.86
+Table 1: Silent Killer vs other popular clean-label poisoning attacks, in terms of ASR. The comparison is made in gray-box settings, trained
+from scratch on CIFAR-10, with 1% of poisoned samples. We evaluate our method with both additive stealthy trigger (perturbation), bound by
+l∞ < 16/255, and optimized colorful patch (patch). The results of the three first methods were taken from [Souri et al., 2022]. Sleeper Agent
+(base) is our implementation of Sleeper Agent without retraining and ensemble (to be comparable to our method which does not use it).
+Architecture
+Attack success rate (ASR) [%]
+Clean accuracy [%]
+Clean
+Silent Killer
+Clean
+Silent Killer
+ResNet18
+46.07 ± 20.52
+89.85 ± 6.46
+83.76 ± 0.17
+83.61 ± 0.17
+VGG11
+74.47 ± 20.54
+97.86 ± 2.10
+86.00 ± 0.10
+85.86 ± 0.12
+MobileNetV2
+55.84 ± 20.18
+97.91 ± 01.72
+82.05 ± 0.38
+82.64 ± 1.44
+Table 2: A comparison of the results of our trigger, both with and without data poisoning, on CIFAR10. The table shows the mean and standard
+deviation of the ASR and the clean accuracy. One can notice that the clean accuracy in the poisoned model is very close to the clean model.
+This means that the poison does not harm model performance on clean data.
+When x[i, j] is the pixel value of the image in pixel i, j,
+R = {(i, j) : imin ≤ i ≤ imax, jmin ≤ j ≤ jmax}
+is the set of pixel indexes between the patch boundaries
+(imin, imax, jmin, jmax). In our experiment, we chose a patch
+size of 8 × 8, which means that imax = imin + 8, jmax =
+jmin + 8. Finally, i′ = i − imin, j′ = j − jmin. The location
+of the patch was randomly chosen both during training and
+testing. Souri et al. showed that optimization in this way
+gives better results. During crafting, instead of computing the
+gradients with respect to all the input dimensions, we optimize
+only the pixels in the position of the patch. Additionally, in
+this case, we choose ϵt = 1 which effectively means we do
+not clip the trigger and allow it to have any norm. Examples
+of the two types of triggers can be found in figures 2 and 3.
+Poison Crafting
+The second stage of our attack is poison crafting. We use
+the trigger computed in the previous stage to craft poison
+examples. We follow the method proposed by Souri et al.
+and align the gradients achieved from the victim training loss
+∇θLv which is computed on the poisoned dataset, to the gra-
+dients of the weights computed from the attacker loss ∇θLa.
+First, we compute the predictions of the model on source
+label samples embedded with the trigger ˆyt
+i = Fs(xt
+i; θs)
+when xt
+i = Tt(xi; δt). Next, we compute the loss function of
+this prediction with respect to the target class Li = L(ˆyt
+i, lt)
+and backpropagate it to the weights θs obtaining ∇θsL =
+1
+N
+�
+i ∇θsLi. ∇θsL is the average gradient of the labels with
+the trigger to which we want the gradients of the poisoned
+samples to be aligned. Similarly, we compute the gradients
+of the chosen poison samples xp
+j = Tp(xj; δp
+j ) and obtain
+∇θsLj. In this step, our goal is to ��nd the perturbation δp
+j that
+will make ∇θsLj to be aligned with ∇θsL. To this end, we
+compute the alignment loss:
+Aj = 1 −
+∇θsLj · ∇θsL
+||∇θsLj|| · ||∇θsL||
+(1)
+This term quantifies the distance between the gradients ob-
+tained from the poisoned sample (xp
+j, lt) and the average gra-
+dients obtained from the samples {(xt
+i, lt)}i. To minimize this
+distance, one can derive this term with respect to the poison
+∂Aj
+∂δp
+j and perform a standard gradient descent algorithm to
+{δp
+j }. In our work, we use signed SGD for both poison craft-
+ing and trigger crafting. For the selection of samples to poison,
+we follow [Souri et al., 2022] which shows that selecting sam-
+ples with the largest gradients yields a higher attack success
+rate. The complete method is summarized in Algorithm 2.
+4
+Experiments
+4.1
+Setup
+We follow [Souri et al., 2022] and conducted experiments
+on CIFAR10 using ResNet18, VGG11, and MobileNetV2
+architectures. Unless specified otherwise, we used 500 poison
+samples (1% of the data) in all experiments. The main metric
+for evaluating the attack’s performance was the attack success
+rate (ASR), defined as ASR = #Success
+#T otal , where #Success
+is the number of successful classifications of source samples
+as the target class after the trigger was embedded into them,
+and #Total is the total number of samples. We did not count
+samples where the predicted label was altered to a class other
+than the target class. We also measured the clean accuracy or
+the accuracy of the model on benign samples from the test set.
+
+Architecture
+Attack success rate (ASR) [%]
+Clean Accuracy [%]
+Predefined
+Silent Killer (patch)
+Clean
+Predefined
+Silent Killer (patch)
+ResNet18
+57.42 ± 25.13
+95.92 ± 3.86
+83.76 ± 0.17
+83.19 ± 0.51
+83.31 ± 0.32
+VGG11
+25.25 ± 21.51
+97.35 ± 4.91
+86.00 ± 0.10
+85.97 ± 0.24
+85.69 ± 0.21
+MobileNetV2
+15.00 ± 11.57
+41.68 ± 25.63
+82.05 ± 0.38
+81.97 ± 0.91
+81.89 ± 0.65
+Table 3: Mean and STD of the ASR and the clean accuracy of our second attack (square patch as a trigger) on CIFAR-10 with 1% poisoned
+samples. Predefined means using a random trigger as in Souri et al., whereas Silent Killer (patch) creates it using algorithm 1. The tables show
+the advantages of the optimized patch.
+Surrogate\Target
+ResNet18
+VGG11
+MobileNetV2
+ResNet18
+-
+81.60 ± 14.63
+90.92 ± 11.89
+VGG11
+76.97 ± 20.67
+-
+95.56 ± 6.74
+MobileNetV2
+32.44 ± 18.27
+77.53 ± 14.43
+-
+Table 4: Mean ASR and STD for black-box attacks. The rows
+indicate the surrogate architecture that was used for crafting the
+poison and the trigger, and the columns the target architecture, on
+which we train (using the poisoned dataset) and evaluate the results.
+Each result was evaluated on 24 runs of source-target pairs.
+We use the same 24 source-label pairs for all of our exper-
+iments and report the average results. In the baseline experi-
+ment (perturbation as a trigger) we run the evaluation step on
+the training set, e.g. the victim training phase, five times with
+different seeds ending up with 24 × 5 = 120 runs.
+In the first experiment, we compared our method to a base-
+line without DP in the gray-box setting (surrogate and victim
+architectures are the same) to measure the impact of DP on
+the attack success rate. The results in Table 2 show signifi-
+cantly better ASR with poisoning compared to the baseline
+and almost no change in clean accuracy.
+On the other hand, to confirm that the improvement in
+attack success rate is not only due to the DP but also to the
+optimization of the trigger, we compare our results to gradient
+alignment with a pre-defined arbitrary patch. We perform
+poison crafting with both the optimized and non-optimized
+patches using the same checkpoints of the surrogate models
+for poison crafting. For the trigger, we use a 8 × 8 colorful
+patch. In the baseline, we use the same patch that was applied
+in previous works [Souri et al., 2022; Saha et al., 2020], and
+for the trigger optimization, we start from a random patch
+and optimize it with algorithm 1 for 500 optimization steps.
+We use the trigger to craft poisoned examples with gradient
+alignment (algorithm 2), and evaluate the success of the attack.
+Some samples from this experiment are shown in figure 3.
+As shown in table 3, the optimization of the trigger has a
+significant impact on the attack success rate. On average,
+the ASR with an optimized trigger was significantly higher
+than the ASR without trigger optimization by 45.76%. This
+demonstrates the importance of carefully crafting the trigger
+to improve the effectiveness of the attack.
+Note that the results of MobileNet-V2 are significantly
+lower than those of the other architectures. When we in-
+vestigated it, we noticed that the success of the poisoning
+attack using gradient alignment was highly correlated with the
+success of the trigger crafting using the algorithm 1. This sug-
+Figure 4: Sensitivity analysis of the number of poisoned samples
+added to the training set. The dotted lines represent the results of
+using the trigger without poisoning. We can see that even with a
+small amount of data (e.g. 100, which is 0.2%), reasonable results
+can be obtained.
+gests that the effectiveness of the poisoning attack is closely
+tied to the use of an optimized trigger that performs well
+as an evasion attack on the surrogate model. Therefore we
+estimate that different evasion attacks [Madry et al., 2017;
+Carlini and Wagner, 2017] may yield more effective triggers
+and improve the results on this architecture.
+4.2
+Black-Box
+Our attack is highly effective in black-box settings compared
+to other data-poisoning-based clean-label backdoor attacks.
+To demonstrate this, we compared the efficiency of a poisoned
+dataset crafted using a surrogate model to attack a model with
+a different architecture (Fs ̸= F). The comparison was made
+for all pairs of the three models: ResNet18, VGG11, and
+MobileNetV2. The results are presented in table 4.
+For comparison, Souri et al. achieved ASR of 29.10%
+and 31.96% on MobileNet-V2 and VGG11 with ResNet18 as
+a surrogate model, while we achieved ASR of 90.92% and
+81.60% on the same models. It is worth saying that [Souri et
+al., 2022] retrained the model during poison crafting, whereas
+we did not. Furthermore, even when [Souri et al., 2022] uses
+an ensemble of four different architectures, he does not reach
+our results. This suggests that our attack is more effective
+than previous approaches in black-box settings. Note that,
+as shown in [Souri et al., 2022], the attack can be improved
+even further by using an ensemble of multiple models and
+retraining the models several times during crafting.
+
+mobilenet
+resnet18
+vggllFigure 5: Sensitivity analysis of the magnitude of the perturbation
+(l∞ norm) to the poisoned samples in the training set and of the
+trigger. The units on the x-axis are in the range (0, 255). The larger
+the norm, the better the performance.
+Method
+ASR [%]
+Accuracy [%]
+Activation Clustering
+71.28 ± 20.73
+73.13 ± 2.07
+DPSGD
+91.16 ± 5.34
+84.36 ± 0.32
+MixUp
+94.24 ± 4.44
+81.39 ± 0.57
+Table 5: ASR and clean accuracy after applying defenses. Clean
+accuracy without defense is 82.64%. As we can see, the success of
+the attack does not significantly deteriorate after using these defenses.
+4.3
+Sensitivity Analysis
+One challenge of poisoning attacks in the real world is in-
+serting poisoned samples into the victim’s training set. If the
+attacker can use a small number of samples, it is easier to bring
+these samples to the training set, and it will be difficult for the
+victim to find them. Therefore, we investigated the sensitivity
+of the ASR to the choice of N, the number of poisoned sam-
+ples. In this experiment, we evaluate our perturbation-trigger
+attack when selecting N = {50, 100, 200, 300, 400} and test
+the results on the three architectures. As shown in figure 4,
+our method was successful with as few as 50 samples, which
+is only 0.1% of the size of the training set. This indicates that
+our method is highly efficient and can be successful with a
+limited number of training samples.
+Furthermore, we analyzed ϵt, ϵp, the upper bound of the
+perturbation of the trigger and the poison. We chose ϵt =
+ϵp = ϵ ∈ {4/255, 8/255, 12/255, 16/255}. As we expected,
+there is a tradeoff between the stealthiness of the attack, i.e.
+the value of ϵ, and its performance. The results and some
+perturbation examples are shown in figures 5 and 6.
+4.4
+ImageNet Evaluation
+To evaluate a more realistic scenario of DP, when the images
+have a larger size, we evaluate our method on ImageNet. Due
+to computational limits, we use only the 10 first classes of
+ImageNet to train and evaluate the results. We chose one
+source-target pair, crafted poison to 185 samples from the
+target class, and performed a partial training of ImageNet on
+ResNet18. Some samples are shown in figure 2. We found
+that the ASR was 76% for this preliminary experiment, which
+indicates that the danger is also present in real-world datasets.
+Figure 6: Samples of images with triggers with different l∞ norms.
+4.5
+Defences
+We evaluate the effectiveness of three different types of de-
+fenses against our method, based on three different concepts:
+gradient shaping, data filtering, and data augmentation. We
+chose methods that were found effective at evading gradient-
+alignment-based DP [Souri et al., 2022]. The first, Activation
+Clustering [Chen et al., 2018], filters out samples based on
+their activations in the second to last layer. We follow this
+method, perform K-Means clustering with K = 2 on each of
+the labels in the training set, and filter out the small cluster.
+The second method, named DP-SGD [Hong et al., 2020], is
+based on reshaping the gradients of the weights during training.
+In this method, the victim clips the training gradients and adds
+some noise to them, to mitigate the effect of poisoned samples.
+We tried several values for the clipping rate and the noise taken
+from them to evaluate this defense. We found that gradient
+clipping of 4 and additive gaussian noise with a variance of
+10−5 is reasonable. More strict values make the model not
+converge. The last defense is based on [Borgnia et al., 2021]
+which shows that data augmentations can be a viable defense
+against backdoor attacks. We follow [Souri et al., 2022] and
+evaluate the defense using MixUp augmentation [Zhang et al.,
+2017] during victim training. All of these methods did not
+significantly deteriorate the attack’s effectiveness. The results
+can be found in table 5.
+5
+Conclusion
+In this paper, we present a novel data poisoning attack on
+neural networks that is stealthy and difficult to detect. Our
+proposed method is highly efficient in crafting time and only
+requires a trained surrogate model, without the need for ad-
+ditional retraining. While our attack achieves state-of-the-art
+results in terms of attack success rate, it may not perform as
+well on certain model architectures. In future research, we
+could investigate the impact of more complex triggers on at-
+tack performance. We could also optimize both the poison
+and the trigger together to find a better trigger that is more
+effective at activating the poison. It is also critical to continue
+developing robust defenses against DP attacks and to prioritize
+the security of training data.
+
+16/255
+12/255
+8/255
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+

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@@ -0,0 +1,2064 @@
+Extraction of neutron density distributions from high-statistics coherent elastic neutrino-nucleus
+scattering data
+D. Aristizabal Sierra1, ∗
+1Universidad T´ecnica Federico Santa Mar´ıa - Departamento de F´ısica
+Casilla 110-V, Avda. Espa˜na 1680, Valpara´ıso, Chile
+Forthcoming fixed-target coherent elastic neutrino-nucleus scattering experiments aim at measurements with
+O(tonne)-scale detectors and substantially reduced systematic and statistical uncertainties. With such high
+quality data, the extraction of point-neutron distributions mean-square radii requires a better understanding of
+possible theoretical uncertainties. We quantify the impact of single-nucleon electromagnetic mean-square radii
+on the weak-charge form factor and compare results from weak-charge form factor parametrizations and weak-
+charge form factor decompositions in terms of elastic vector proton and neutron form factors, including nucleon
+form factors Q-dependent terms up to order Q2. We assess as well the differences arising from results derived
+using weak-charge form factor decompositions in terms of elastic vector proton and neutron form factors and a
+model-independent approach based solely on the assumption of spherically symmetric nuclear ground state. We
+demonstrate the impact of the main effects by assuming pseudo-data from a one-tonne LAr detector and find that,
+among the effects and under the assumptions considered in this paper, weak-charge form factor parametrizations
+and weak-charge form factor decompositions in terms of elastic vector proton and neutron form factors enable
+the extraction of the 40Ar point-neutron distribution mean-square radius with a ∼ 15% accuracy. With a sub-
+stantial reduction of the beam-related neutron and steady-state backgrounds a ∼ 1% precision extraction seems
+feasible, using either of the two approaches.
+CONTENTS
+I. Introduction
+1
+II. Nuclear charge and weak-charge form factors
+2
+A. Electromagnetic and weak charge radii
+3
+III. Theoretical and phenomenological uncertainties
+3
+A. Uncertainties due to leading-order point-proton
+distribution mean-square radius and leading-order
+nucleon form factors
+4
+B. Uncertainties due to elastic vector proton and
+neutron form factor parametrizations
+6
+C. Model-independent versus form factor
+parametrizations approaches
+6
+IV. CEνNS cross section and event rate
+8
+V. Extraction of the argon neutron distribution
+root-mean-square radius using different approaches
+11
+VI. Conclusions
+13
+Acknowledgments
+13
+References
+13
+I.
+INTRODUCTION
+Since its first observation by the COHERENT collabora-
+tion in 2017 [1], CEνNS has become a powerful tool for the
+∗ daristizabal@ulg.ac.be
+determination of nuclear and SM properties as well as for con-
+straining new physics. Current datasets involve measurements
+in CsI and LAr detectors with fiducial volumes in the order of
+10kg [1–3]. With such active volumes, the datasets comprise
+about ∼ 100 events, with—to a certain degree—sizable sys-
+tematics and statistical uncertainties. Using these data a wide
+range of analyses have been carried out. They include the
+determination of cesium, iodide and argon point-neutron dis-
+tributions root-mean-square (rms) radii [4–9], measurements
+of the weak mixing angle at renormalization scales of the or-
+der of 50 MeV [5, 6, 9, 10], constraints on new interactions
+in the neutrino sector [11–24], constraints on neutrino elec-
+tromagnetic properties [25–27] and limits on sterile neutrinos
+[10].
+Plans for enhancing statistics by deploying an O(tonne)
+LAr detector in the future second target station (STS) at the
+Oak Ridge National Laboratory have been discussed in Refs.
+[28, 29] (see Ref.
+[30] as well)1.
+With improvements on
+statistical and systematic uncertainties, such detector will de-
+liver high quality data with which improvements on the dif-
+ferent analyses that have been done so far are expected. This
+calls for a better understanding of theoretical uncertainties.
+For instance—depending on the experimental uncertainties—
+the inclusion of one-loop order electroweak corrections [34]
+might become necessary, regardless of the physics case the
+data will be used for.
+As far as we know, the extraction of point-neutron distri-
+butions mean-square radii with existing or forecasted data has
+been done considering only leading order (LO) effects. With
+this in this paper we mean the following: The measured nu-
+clear charge radius and the point-proton distribution rms ra-
+1 The Coherent Captain-Mills Experiment at Los Alamos National Labora-
+tory aims at measurements with an O(tonne) LAr detector as well [31] (see
+e.g. Refs. [32, 33] for discussions of its physics reach).
+arXiv:2301.13249v1  [hep-ph]  30 Jan 2023
+
+2
+dius have been assumed to be equal, and only LO nucleon
+form factor terms (Q independent terms) have been considered
+[6, 35–37]. Furthermore, analyses using nuclear weak-charge
+form factor parametrizations, nuclear weak-charge form fac-
+tor decompositions in terms of elastic vector proton and neu-
+tron form factors and power series expansions of the nuclear
+weak-charge form factor have been used [4, 8, 36–38]. With
+all of them implying—in principle—different precision levels.
+In this paper we quantify the uncertainties implied by con-
+sidering only LO effects—defined as we have done in the pre-
+vious paragraph—in the extraction of point-neutron distribu-
+tions mean-square radii. The analysis is split in two parts. In
+the first part, we quantify uncertainties at the nuclear weak-
+charge level for heavy and light nuclei (cesium and argon,
+taken as representative examples of heavy and light nuclei).
+To do so we compare results from calculations using only
+LO effects with calculations where: (a) The point-proton dis-
+tribution radius is corrected with single-nucleon electromag-
+netic mean-square radii, (b) nucleon form factor Q-dependent
+terms (up to order Q2) are included 2. We then determine the
+uncertainties implied by the most commonly used form fac-
+tor parametrizations and by expansions in terms of even mo-
+ments.
+In the second part, we quantify the precision at which the
+point-neutron distribution rms radius can be extracted, includ-
+ing the main effects found in the first part. In this case we
+proceed by assuming pseudo-data from a one-tonne LAr de-
+tector with assumptions on the beam-related neutron (BRN)
+and steady-state (SS) backgrounds as well as systematic un-
+certainties extrapolated from current COHERENT measure-
+ments.
+The remainder of this paper is organized as follows. In Sec.
+II we provide a general discussion of the nuclear charge and
+weak-charge form factors as well as of the nuclear charge and
+weak-charge radii. In Sec. III we determine nuclear weak-
+charge form factor uncertainties, while in Sec. IV we briefly
+discuss the CEνNS differential cross section and differential
+event rate. In Sec. V, based on the results from Sec. III, we
+extract the 40Ar neutron distribution mean-square radius. Our
+conclusions are presented in Sec. VI
+II.
+NUCLEAR CHARGE AND WEAK-CHARGE FORM
+FACTORS
+The single-nucleon electromagnetic and weak currents can
+be expressed in terms of the Sachs form factors [41]. For a
+nucleon N (N = n, p) they read
+Jµ,N
+EM = us′(p′)
+�
+GN
+Eγµ +
+�GN
+M −GN
+E
+1+τ
+��
+τγµ +iσµν qν
+2m
+��
+us(p) ,
+(1)
+Jµ,N
+NC = us′(p′)
+�
+�GN
+Eγµ +
+� �GN
+M − �GN
+E
+1+τ
+��
+τγµ +iσµν qν
+2m
+��
+us(p) .
+(2)
+where m is a universal nucleon mass, q = p′ − p the trans-
+ferred four momentum and τ = Q2/4/m2 (with Q2 = −q2
+a timelike vector) and us(p) and us′(p′) on-shell nucleon
+spinors. GN
+E,M = GN
+E,M(Q2) and �GN
+E,M = �GN
+E,M(Q2) are the
+single-nucleon form factors for N.
+The weak neutral cur-
+rent (NC) involves as well axial and pseudoscalar form fac-
+tors, which are sensitive to the nucleon spin distribution in the
+nuclear medium. For elastic scattering their contribution can
+then be regarded as a subleading effect (actually vanishing in
+nuclei with even number of protons and neutrons), and so are
+not considered in Eq. (2).
+The nuclear charge and weak-charge form factors follow
+from: (i) Trading the on-shell nucleon spinors to spinor wave-
+functions that account for nucleons in a potential, (ii) assum-
+ing the impulse approximation (i.e.
+assuming that single-
+nucleon form factors are valid as well in the nuclear medium),
+(iii) integration over nuclear volume, (iv) taking into account
+contributions from protons and neutrons (for details see e.g.
+2 Other subleading effects involve as well relativistic Darwin-Foldy and spin-
+orbit corrections [39, 40], which our analysis does not account for.
+[40]). Explicitly one finds
+ZFC = ∑
+N=p,n
+�
+GN
+EFN
+V +
+�GN
+M −GN
+E
+1+τ
+��
+τFN
+V + q
+2mFN
+T
+��
+,
+(3)
+QWFW = ∑
+N=p,n
+�
+�GN
+EFN
+V +
+� �GN
+M − �GN
+E
+1+τ
+��
+τFN
+V + q
+2mFN
+T
+��
+,
+(4)
+with QW determined by the couplings of the proton and neu-
+tron to the Z gauge boson, QW = N gn
+V + Z gp
+V (gp
+V = 1/2 −
+2sin2 θW and gn
+V = −1/2) 3. Note that the form factors writ-
+ten as in Eqs. (3) and (4) satisfy the normalization condition
+FC(Q2 = 0) = FW(Q2 = 0) = 1. These expressions contain in-
+formation on nucleon and nuclear structure, the latter encoded
+in the vector and tensor nuclear form factors. Spin-orbit ef-
+fects governed by FT are subleading compared with nuclear
+3 One-loop corrected proton and neutron electroweak couplings are given by
+gp
+V = 0.0721 and gn
+V = −0.988 [42]. These are the values we use in the
+calculations presented in Secs. III and V.
+
+3
+spin-independent effects controlled by FV [40]. Thus keeping
+only LO effects and taking into account that τ ≪ 1 for the typ-
+ical transferred momentum in neutrino stopped-pion sources
+(Q2 ≲ m2
+µ/2), the charge and weak-charge form factors reduce
+to a rather simple form
+FC(q2) = 1
+Z
+�
+Gp
+E(q2)F p
+V (q2)+Gn
+E(q2)Fn
+V (q2)
+�
+,
+(5)
+FW(q2) =
+1
+QW
+�
+�Gp
+E(q2)F p
+V (q2)+ �Gn
+E(q2)Fn
+V (q2)
+�
+.
+(6)
+Numerically one finds that spin-orbit effects are of the order
+of 0.1% [40], while terms proportional to τ contribute at the
+order of 0.01% in stopped-pion neutrino experiments where
+Q2 ≃ (10MeV)2. Eqs. (5) and (6) are thus precise enough at
+the percent level.
+A.
+Electromagnetic and weak charge radii
+LO expressions for the electromagnetic and weak-charge
+radii of the nucleus follow from Eqs. (5) and (6), as we now
+discuss. Assuming spherical symmetry, the nucleon and spin-
+independent nucleon structure functions can be expanded in
+terms of their moments. For nucleons one has
+Gp,n
+E
+=
+∞
+∑
+i=0,1
+(−1)i
+Q2i
+(2i+1)!r2i
+X ,
+(7)
+�GX
+E = gX
+V
+∞
+∑
+i=0
+(−1)i
+Q2i
+(2i+1)! ˜r2i
+X ,
+(8)
+where the order Q2 terms in the expansion in Eq.
+(7) in-
+volve the single-nucleon electromagnetic mean-square radii,
+r2
+X (X = p,n), while those in Eq. (8) the single-nucleon weak-
+charge mean-square radii, ˜r2
+X. In Eq. (7) the sum starts at
+i = 0 (i = 1) for protons (neutrons). For the spin-independent
+nuclear form factors one instead has (see e.g. [38])
+F p
+V = Z
+∞
+∑
+i=0
+(−1)i
+Q2i
+(2i+1)!R2i
+p ,
+(9)
+Fn
+V = N
+∞
+∑
+i=0
+(−1)i
+Q2i
+(2i+1)!R2i
+n .
+(10)
+In this case—at order Q2—terms involve the point-proton and
+point-neutron distributions mean-square radii, R2
+p and R2
+n (sec-
+ond moment of the distributions)4. The nuclear charge and
+weak-charge radii, R2
+C and R2
+W, follow from
+R2
+C = − 6 dFC
+dQ2
+����
+Q2=0
+,
+R2
+W = −6 dFW
+dQ2
+����
+Q2=0
+,
+(11)
+4 Here following standard conventions we use lower-case for nucleons and
+upper-case for nuclei.
+after expansions in Eqs. (7), (8), (9) and (10) are inserted in
+Eqs. (5) and (6). Their explicit expressions are given by
+R2
+C = R2
+p +r2
+p + N
+Z r2
+n ,
+(12)
+R2
+W = Zgp
+V
+QW
+�
+R2
+p + ˜r2
+p
+�
++ Ngn
+V
+QW
+�
+R2
+n + ˜r2
+n
+�
+.
+(13)
+The single-nucleon weak-charge mean-square radii in RW can
+be reexpressed in terms of the single-nucleon electromagnetic
+mean-square radii as follows [40]
+˜r2
+p = r2
+p + gn
+V
+gp
+V
+r2
+n +ξ(0)
+V r2
+s ,
+˜r2
+n = r2
+p + gp
+V
+gn
+V
+r2
+n +ξ(0)
+V r2
+s , (14)
+where ξ(0)
+V
+= gu
+V + gd
+V + gs
+V (with gq
+V the quark q weak-vector
+charge) and r2
+s the mean-square strange radius. Numerically,
+ξ(0)
+V = −0.988 and r2
+s = −0.00430fm2 with the latter obtained
+from lattice QCD calculations [43]. Thus, the strange quark
+contribution provides per mille corrections to the proton and
+neutron LO expressions in Eq. (14). It can therefore be ne-
+glected in the following analysis.
+With the aid of these equations the weak-charge mean-
+square radius can finally be rewritten as
+R2
+W = Z
+QW
+�
+gp
+VR2
+p +gp
+Vr2
+p +gn
+Vr2
+n
+�
++ N
+QW
+�
+gn
+VR2
+n +gn
+Vr2
+p +gp
+Vr2
+n
+�
+,
+(15)
+which in turn can be recast in terms of the electromagnetic
+charge radius and the neutron skin, R2
+n −R2
+p, [8]
+R2
+W = R2
+C + Ngn
+V
+QW
+�
+(R2
+n −R2
+p)+ Z2 −N2
+N Z
+r2
+n
+�
+.
+(16)
+Due to the small transferred momentum, stopped-pion
+CEνNS experiments are not particularly sensitive to nucleon
+structure. Thus the nucleon form factors in Eqs. (5) and (6)
+can be truncated at order Q2, resulting in the following ex-
+pression for the weak-charge form factor [8]
+FW ≃
+1
+QW
+�
+Z
+�
+gp
+V − gp
+V
+6 r2
+pQ2 − gn
+V
+6 r2
+nQ2
+�
+F p
+V (Q2)
++N
+�
+gn
+V − gn
+V
+6 r2
+pQ2 − gp
+V
+6 r2
+nQ2
+�
+Fn
+V (Q2)
+�
+,
+(17)
+where the nuclear form factors have been normalized,
+F p
+V (Q2 = 0) = 1 and Fn
+V (Q2 = 0) = 1.
+III.
+THEORETICAL AND PHENOMENOLOGICAL
+UNCERTAINTIES
+In this Section we determine the numerical variations to
+which the weak-charge form factor can be subject to. For that
+aim we consider the Helm parametrization [44] for the elastic
+vector proton and neutron form factors (see Sec. III A). We
+
+4
+first quantify FW by considering LO expressions for the point-
+proton distribution mean-square radius and the nucleon form
+factors, which we compare with FW obtained by considering
+the full expression in Eq. (17), including the single-nucleon
+electromagnetic mean-square radii in the determination of Rp.
+We compare as well those results with those obtained assum-
+ing the Helm parametrization for the weak-charge form fac-
+tor. We then proceed to calculate FW by assuming different
+parametrizations for the elastic vector proton and neutron nu-
+clear form factors. Our analysis relies on the Fourier trans-
+form of the symmetrized Fermi function [39] and the Klein-
+Nystrand [45] parametrizations, in addition to the Helm ap-
+proach. In doing so, we quantify the dependence of FW on
+parametrization choice. Finally, we compare the parametriza-
+tion approach with the model-independent treatment based on
+series expansions of the elastic vector proton and neutron form
+factors.
+A.
+Uncertainties due to leading-order point-proton
+distribution mean-square radius and leading-order nucleon
+form factors
+We start by considering three different cases, two limits and
+the full case:
+1) Limit 1 (Case 1):
+LO nucleon form factors (Q-
+independent terms) and point-proton distribution mean-
+square radius equal to the nuclear charge mean-square ra-
+dius, R2
+p = R2
+C.
+2) Limit 2 (Case 2): LO nucleon form factors and full point-
+proton distribution mean-square radius given by Eq. (12).
+3) Full case (Case 3): Full nucleon form factors, up to Q2 as
+given in Eq. (17) and full point-proton distribution mean-
+square radius given by Eq. (12).
+These limits are motivated as follows. In most analyses in
+which the point-neutron distribution rms radius is extracted
+from CEνNS data (see e.g. Ref. [37]) the point-proton distri-
+bution mean-square radius is fixed to be equal to the nuclear
+charge radius. As can be seen from Eq. (12), doing so over-
+estimates Rp. A larger value for Rp means a smaller value
+for Fp at a given Q. Under such simplification, an uncertainty
+in the extraction of R2
+n from CEνNS data is implied. Order
+Q2 corrections in the nucleon form factors are also typically
+ignored and, though suppressed, are potentially a source of
+sizable uncertainties. Ignoring order Q2 terms enhance FW
+[see Eq. (17)], and so the CEνNS event rate, resulting in a
+potential underestimation of R2
+n.
+To proceed, we adopt the Helm form factor parametrization
+for the elastic vector proton and neutron form factors, F p
+V and
+Fn
+V , namely [44]
+FH(Q2) = 3 j1(QR0)
+QR0
+e−(Qs)2/2 .
+(18)
+Here R0 refers to the diffraction radius, related with the point-
+nucleon distribution mean-square radius through the skin
+thickness s = 0.9fm [46] according to
+R0 =
+�
+5
+3
+�
+R2
+X −3s2�
+(X = p,n) .
+(19)
+To avoid mixing nuclear physics effects with detector ef-
+fects we use argon and cesium as target materials. For ar-
+gon this means one can assume the detector to be 100% com-
+posed of 40Ar (up to per mille corrections), while for cesium
+of 133Cs. Parameters used in our calculation are displayed in
+Tab. I, along with the values for the single-nucleon electro-
+magnetic mean-square radii central values [47].
+Isotope
+40Ar
+133Cs
+Abundance (Xi)
+99.6%
+100.0%
+RC [fm]
+3.4274
+4.0414
+Nucleon
+Proton
+Neutron
+r2
+X [fm2]
+0.7071
+-0.1155
+TABLE I. 40Ar and 133Cs relative abundances along with their nu-
+clear charge radii used in our calculation. Nuclear charge radii taken
+from Ref. [48]. Single-nucleon electromagnetic mean-square radii
+central values taken from Ref. [47].
+To quantify deviations implied by the limits in items 1 and
+2 with the full result in item 3 the following percentage differ-
+ence factor is employed
+∆FW [%] = FW|Ci −FW|C j
+FW|Ci
+×100% ,
+(20)
+where FW|Ci > FW|C j and Ci, j refer to the cases used for the
+calculation of the weak-charge form factor. The results are
+shown in Fig. 1. We have fixed the point-neutron distribution
+rms radius according to Rn = Rp+0.1fm and Rn = Rp+0.2fm
+for argon and cesium, respectively. Note that these choices are
+not intended to have any particular meaning, they are chosen
+just to illustrate the deviations implied by the different limits
+we are considering.
+From Fig. 1 (left graph) one can see that by assuming that
+the point-proton rms radius distribution amounts to the mea-
+sured nuclear charge rms radius has a small effect. All over the
+relevant transferred momentum range (Q2 ≲ m2
+µ) deviations
+are at (or below) the per mille level, regardless of nuclide. In-
+cluding the single-nucleon electromagnetic mean-square radii
+in the determination of R2
+p becomes relevant only if experi-
+mental uncertainties reach that level of precision, otherwise
+the R2
+p = R2
+C approximation is fairly accurate. Differences
+due to nucleon form factors Q-dependent terms are instead
+slightly more relevant. Results in Fig. 1 (right graph) show
+that they can reach values up to 3%, basically regardless of
+nuclide. Nucleon form factors effects should then accounted
+for in high-statistics CEνNS experiments with low systematic
+and statistical uncertainties.
+Rather than calculating the weak-charge form factor using
+Eq. (17), one could instead use a form factor parametrization
+for FW—e.g. the Helm parametrization—and fix R2
+X = R2
+W in
+Eq. (19), with R2
+W as given in Eq. (16) (as done e.g. in Ref.
+
+5
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+FW [%]
+Percentage uncertainties from limits 1 and 2
+Argon: Limit 1 - Limit 2
+Cesium: Limit 1 - Limit 2
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+FW [%]
+Percentage uncertainties from limit 1 and NLO nucleon form factor terms
+Argon: Limit 1 - Full
+Cesium: Limit 1 - Full
+FIG. 1. Nuclear weak-charge form factor percentage difference calculated according to Eq. (20) and for: Nucleon form factors LO terms
+(momentum-independent terms only) and R2p = R2
+C (Limit 1, item 1), nucleon form factor LO terms and R2p = R2
+C − r2p − (N/Z)r2n (Limit 2,
+item 2), full nucleon form factors up to order Q2 (Full, item 3) and R2p = R2
+C −r2p −(N/Z)r2n. Left graph: Percentage difference obtained by
+comparing limits 1 and 2, Right graph: percentage difference obtained by comparing limit 1 with full nucleon form factor expressions and
+R2p = R2
+C − r2p − (N/Z)r2n. For the point-neutron distribution rms radii we have taken Rn = Rp + 0.1fm and Rn = Rp + 0.2fm for argon and
+cesium, respectively. These values taken just for the sake of illustrating the deviations implied by the different limits compared with the case
+in which full expressions are considered.
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0.0
+1.5
+3.0
+4.5
+6.0
+7.5
+9.0
+FW [%]
+Argon
+Percentage uncertainties from Helm parametrization and full FW
+FW = FH vs FW: Limit 1
+FW = FH vs FW: Full
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0
+3
+6
+9
+12
+15
+18
+FW [%]
+Cesium
+Percentage uncertainties from Helm parametrization and full FW
+FW = FH vs FW: Limit 1
+FW = FH vs FW: Full
+FIG. 2. Nuclear weak-charge form factor percentage uncertainty for argon (left graph) and cesium (right graph) calculated using the Helm
+parametrization with the diffraction radius R0 fixed through the weak-charge mean-square radius R2
+W and by using Eq. (17). Percentage
+uncertainties are calculated in limit 1 (item 1) and in the full case (item 3). The point-neutron distribution rms radii have been fixed according
+to Rn = Rp +0.1fm and Rn = Rp +0.2fm.
+[8]). Since R2
+W involves the point-neutron distribution mean-
+square radius, from such procedure one could as well extract
+from CEνNS data a range for its value at a certain confidence
+level. To show what are the expected percentage differences
+following this procedure with that dictated by Eq. (17), we
+have calculated FW as we have described above and compared
+with FW calculated using Eq. (17) in limit 1 (item 1) and in
+the full case (item 3). Deviations are quantified with the aid of
+Eq. (20) where in this case Ci refers to FW calculated through
+the Helm parametrization and Cj to FW calculated using Eq.
+(17) in the aforementioned cases.
+The result is displayed in Fig. 2, left (right) graph for ar-
+gon (cesium). One can see that differences between both ap-
+proaches are more pronounced for heavier nuclei. And grow
+depending on whether one considers the most simplified as-
+sumptions (limit 1) or the full weak-charge form factor in-
+cluding nucleon form factors corrections and single-nucleon
+electromagnetic mean-square radii. For argon, the percentage
+uncertainty can raise up to ∼ 9% while for cesium—instead—
+up to ∼ 18%. Some caution, however, is required. Differences
+grow with transferred momentum and so at their peak (in the
+relevant range) the stopped-pion neutrino flux is expected to
+be fading away. Thus, although substantially large, these un-
+certainties might not have the same effect they exhibit here on
+the CEνNS event rate. We will come back to this issue later
+in Sec. V.
+
+6
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+FW [%]
+Argon
+Percentage uncertainties from FF parametrizations
+KN-Helm
+Fermi-Helm
+KN-Fermi
+0
+20
+40
+60
+80
+100
+Q[MeV]
+0.0
+0.5
+1.0
+1.5
+2.0
+FW [%]
+Cesium
+Percentage uncertainties from FF parametrizations
+KN-Helm
+Fermi-Helm
+KN-Fermi
+FIG. 3. Left graph: Weak-charge form factor percentage difference for argon obtained by using for the elastic vector proton and neutron
+form factors the Helm, Klein-Nystrand and the Fourier transform of the symmetrized Fermi distribution parametrizations. The results include
+nucleon form factors Q-dependent terms (up to order Q2) and single-nucleon electromagnetic mean-square radii. The argon point-neutron
+distribution rms radius has been fixed according to Rn = Rp +0.1fm. Right graph: Same as left graph but for cesium, with the point-neutron
+distribution rms radius fixed according to Rn = Rp +0.2fm.
+B.
+Uncertainties due to elastic vector proton and neutron form
+factor parametrizations
+Event rate spectra derived from form factor parametriza-
+tions are expected to depend on the parametrization used, with
+the dependence increasing with increased Q [36]. To deter-
+mine the size of these dependences we calculate F p
+V and Fn
+V
+using as well the Klein-Nystrand form factor [45] and the
+Fourier transform of the symmetrized Fermi distribution [39].
+The Klein-Nystrand form factor follows from folding a
+Yukawa potential (range ak) over a hard sphere distribution
+of radius RA, namely [45]
+FKN = 3 j1(QRA)
+QRA
+1
+1+Q2a2
+k
+.
+(21)
+The range of the potential ak is 0.7 fm [45] and the hard
+sphere radius is determined through the point-proton and
+point-neutron distributions mean-square radii according to
+RA =
+�
+5
+3
+�
+R2
+X −6a2
+k
+�
+(X = p,n) .
+(22)
+The Fourier transform of the symmetrized Fermi distribution
+follows instead from fSF = fF(r)+ fF(−r)−1, where fF(r) is
+the conventional Fermi function
+fF =
+1
+1+e(r−c)/a ,
+(23)
+where c refers to the half-density radius and a (a = 0.52fm
+[49]) to the surface diffuseness. The Fourier transform can be
+analytically integrated resulting in
+FSF = 3
+Qc
+�sin(Qc)
+(Qc)2
+��
+πQa
+tanh(πQa)
+�
+− cos(Qc)
+Qc
+�
+×
+�
+πQa
+sinh(πQa)
+�
+1
+1+(πa/c)2 .
+(24)
+In this case the half-density radius c proceeds from the point-
+proton and point-neutron distributions mean-square radii and
+the surface diffuseness a
+c =
+�
+5
+3R2
+X − 7
+3(πa)2
+(X = p,n) .
+(25)
+We now calculate the weak-charge form factor percentage dif-
+ference obtained by calculating FW according to Eq. (17),
+using for the elastic vector proton and neutron form factors
+the three different parametrizations already discussed. The
+result is displayed in Fig. 3. The left graph shows results
+for argon, while the right graph results for cesium. Differ-
+ences are slightly more pronounced for the latter, thus demon-
+strating they are more relevant for heavy nuclides. As Q in-
+creases, the Helm form factor decreases more steeply. This
+effect is less pronounced for the Fourier transform of the
+symmetrized Fermi distribution and even less for the Klein-
+Nystrand parametrization. This means that event rates cal-
+culated with the Helm form factor will produce slightly less
+events than those calculated with the Fermi parametrization,
+and even less than those obtained with the Klein-Nystrand
+form factor. This translates into uncertainties of the order of
+1% (or below) for argon and 2% (or below) for cesium. In
+summary, uncertainties due to form factor parametrizations of
+the elastic vector proton and neutron form factors are compa-
+rable to those implied by the Q-dependent nucleon form factor
+terms discussed in the previous Section.
+C.
+Model-independent versus form factor parametrizations
+approaches
+Assuming the nucleon distributions to be spherically sym-
+metric (i.e. assuming that the charge density distribution de-
+pends solely on the distribution radius) leads to the series ex-
+pansions of the elastic vector proton and neutron form fac-
+
+7
+tors, Eqs. (9) and (10). These expansions, subject only to
+the spherical symmetry hypothesis, involve the point-nucleon
+mean-square radii (second radial moment) at order Q2. So
+rather than sticking to a form factor parametrization or a nu-
+clear physics model one can use these expansions to fit R2
+n to
+data [38, 50]. The question is of course whether a Q2-order
+description suffices, or whether higher order terms should be
+included to increase convergence. Ref. [38] addressed this
+question and ended up concluding that order Q4 terms are re-
+quired (for argon for which the analysis was done). Following
+this approach, this implies the introduction of a new param-
+eter, the fourth radial moment of the nucleon distributions,
+⟨R4
+X⟩ (X = p,n) 5 .
+To compare the accuracy of the model-independent analysis
+and the form factor parametrization approach, we first calcu-
+late the fourth radial moment for the three parametrizations
+we are using. We do so by taking into account that the series
+expansions in Eqs. (9) and (10) can be compared term by term
+to the Taylor expansions of the F p
+V (Q2) and Fn
+V (Q2) functions.
+Up to order Q4 this reduces to
+R2
+X = −6 dFX
+V
+dQ2
+����
+Q2=0
+and
+⟨R4
+X⟩ = 60 d2FX
+V
+dQ4
+����
+Q2=0
+. (26)
+More generally, the 2i-th moment can be written according to
+⟨R2i
+X ⟩ = (−1)i (2i+1)!
+i!
+diFX
+V
+dQ2i
+����
+Q2=0
+(X = p,n) .
+(27)
+The first relation in Eq. (26) leads to the expressions for the
+diffraction, hard sphere and half density radii (R0, RA and c)
+for the Helm, Klein-Nystrand and Fourier transform of the
+symmetrized Fermi distribution, Eqs. (19), (22) and (25). The
+second relation in Eq. (26) leads instead to
+Helm:
+⟨R4
+X⟩ =3
+7R4
+0 +6R2
+0s2 +15s4 ,
+KN:
+⟨R4
+X⟩ =3
+7R4
+A +12a2
+kR2
+A +120a4
+k ,
+Fermi:
+⟨R4
+X⟩ =3
+7c4 + 18
+7 c2(aπ)2 + 31
+7 (aπ)4 .
+(28)
+Of course the higher the order in Q the expansion extends to,
+the better the convergence and so the reliability of the result.
+Inclusion of terms up to order Q6 require the sixth radial mo-
+ment. From Eq. (27) explicit expressions for each case read
+Helm: ⟨R6
+X⟩ =1
+3R6
+0 +9R2
+0s2 +63R2
+0s4 +105s6 ,
+KN:
+⟨R6
+X⟩ =1
+3R6
+A +18R4
+Aa2
+k +504R2
+ka4
+k +5040a6
+k ,
+Fermi:⟨R6
+X⟩ =1
+3c6 + 11
+3 c4(πa)2 + 239
+15 c2(πa)4 + 127
+5 (πa)6 .
+(29)
+5 Note that we have simplified our notation for the point-nucleon mean-
+square radii R2
+X ≡ ⟨R2
+X⟩. For ⟨R4
+X⟩ we do not do so to avoid confusion.
+Using Eqs. (19), (22) and (25) for R0, RA and c one can write
+the fourth and sixth radial moments for each parametrization
+solely in terms of R2
+X. In doing so one can then compare the
+level of convergence of the Q2, Q4 and Q6 expansions with
+the full expressions for each form factor. Results are shown in
+Figs. 4 and 5 for argon and cesium, respectively (representa-
+tive of light and heavy nuclides).
+Top graphs in Fig. 4 show results for the three form factor
+parametrizations calculated for argon. Bottom graphs show
+percentage uncertainties for each case. One can see that in-
+clusion of only the second moment leads to deviations above
+�� 5% for transferred momenta of the order of 80MeV. At
+100MeV those deviations can reach ∼ 20%. These uncer-
+tainties, however, should be taken with care as those found
+in the previous Section for the same reason. They increase at
+relatively large Q, where the neutrino flux is expected to be
+fading away to a certain degree. They will—of course—have
+an impact when comparing results from both approaches, but
+the best way to understand their actual impact is through the
+calculation of the CEνNS event rate, something that will be
+done and discussed in Sec. V.
+Top and bottom graphs in Fig. 5 show—instead—results
+for cesium. In this case a few percentage convergence requires
+the inclusion of the sixth moment. As can be seen in the bot-
+tom graphs, up to order Q4 the expansion involves uncertain-
+ties that can readily reach ∼ 10% at transferred momenta of
+the order of 80MeV, values for which the neutrino flux is still
+sizable enough for the uncertainty to have an impact on the
+CEνNS event rate. Aiming at few percent level precision thus
+requires the inclusion of the sixth moment, inline with Ref.
+[38].
+The adoption of form factor parametrizations suffers from
+the model dependence implied by the different assumptions
+those parametrizations come along with.
+However, as we
+have discussed in Sec. III B, using one or the other (which
+can be understood as a variation of the underlying nuclear
+physics hypotheses) introduces uncertainties at the few per-
+cent level. The same level of precision can be achieved with
+the model-independent power expansion approach, which re-
+lies only on the assumption of a spherical symmetric nuclear
+ground state. However to achieve that level of precision new
+parameters should be considered. Thus, take for instance the
+case of cesium. Given a CEνNS dataset an analysis relying
+on form factor parametrizations is expected to imply a few
+percent uncertainty (in addition to systematic and statistical
+uncertainties due to e.g. quenching factor, neutrino flux, BRN
+and SS backgrounds). An analysis based on radial moments
+expansions as well, but then the dataset has to be used to fit not
+only the neutron distribution rms radius but also the fourth and
+sixth moments (depending on the nuclide the detector is built
+with). The extraction procedure then becomes a multiparame-
+ter problem, which might worsen the precision with which R2
+n
+can be determined. We will come back to that discussion in
+Sec. V.
+
+8
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Argon
+Weak-charge FF: Full vs moments expansion
+Helm: Full
+Helm: Order Q2
+Helm: Order Q4
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Argon
+Weak-charge FF: Full vs moments expansion
+KN: Full
+KN: Order Q2
+KN: Order Q4
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Argon
+Weak-charge FF: Full vs moments expansion
+Fermi: Full
+Fermi: Order Q2
+Fermi: Order Q4
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+| FW| [%]
+Argon
+Weak-charge FF: Full vs moments expansion
+Helm at Q2
+Helm at Q4
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+| FW| [%]
+Argon
+Weak-charge FF: Full vs moments expansion
+KN at Q2
+KN at Q4
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+| FW| [%]
+Argon
+Weak-charge FF: Full vs moments expansion
+Fermi at Q2
+Fermi at Q4
+FIG. 4. Top graphs: Convergence level for the weak-charge form factor series expansions at order Q2, Q4 and Q6 (in argon) for the Helm,
+Klein-Nystrand and the Fourier transform of the symmetrized Fermi distribution. Expansion at order Qn involve radial moments up to n-th
+order (see text). Bottom graphs: Percentage uncertainty in each case. From these results one can see that if one relies on elastic vector form
+factor power expansions and demands precision below a few percent the fourth radial moment should be included for light nuclides. Inline
+with Ref. [38].
+IV.
+CEνNS CROSS SECTION AND EVENT RATE
+The CEνNS differential cross section follows from a neutral
+current process. In terms of nuclear recoil energy it is given
+by [51–54]
+dσ
+dEr
+= G2
+FmN
+2π
+Q2
+W F2
+W
+�
+2− mNEr
+E2ν
+�
+,
+(30)
+where subleading kinematic terms have been neglected and
+mN here refers to nuclear mass. The strength at which the
+Z boson couples to the nucleus is determined by the Q-
+dependent coupling QW ×FW(Q2). The coupling is such that
+with increasing transferred momentum (increasing incoming
+neutrino energy) the “effective” weak charge decreases and so
+the interaction probability. Uncertainties in FW, as those we
+have discussed in the previous Sections and as those discussed
+
+9
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Cesium
+Weak-charge FF: Full vs moments expansion
+Helm: Full
+Helm: Order Q2
+Helm: Order Q4
+Helm: Order Q6
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Cesium
+Weak-charge FF: Full vs moments expansion
+KN: Full
+KN: Order Q2
+KN: Order Q4
+KN: Order Q6
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0.2
+0.4
+0.6
+0.8
+1.0
+FW
+Cesium
+Weak-charge FF: Full vs moments expansion
+Fermi: Full
+Fermi: Order Q2
+Fermi: Order Q4
+Fermi: Order Q6
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+25
+30
+| FW| [%]
+Cesium
+Weak-charge FF: Full vs moments expansion
+Helm at Q2
+Helm at Q4
+Helm at Q6
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+25
+30
+FW [%]
+Cesium
+Weak-charge FF: Full vs moments expansion
+KN at Q2
+KN at Q4
+KN at Q6
+0
+20
+40
+60
+80
+100
+Q [MeV]
+0
+5
+10
+15
+20
+25
+30
+FW [%]
+Cesium
+Weak-charge FF: Full vs moments expansion
+Fermi at Q2
+Fermi at Q4
+Fermi at Q6
+FIG. 5. Top graphs: Convergence level for the weak-charge form factor series expansions at order Q2, Q4 and Q6 (in cesium) for the Helm,
+Klein-Nystrand and the Fourier transform of the symmetrized Fermi distribution. Expansion at order Qn involve radial moments up to n-th
+order (see text). Bottom graphs: Percentage uncertainty in each case. From these results one can see that if one relies on elastic vector form
+factor power expansions and demands precision below a few percent the sixth radial moment should be included. Inline as well with Ref. [38].
+in Ref. [36] 6, translate into uncertainties in the CEνNS cross
+section. They are indeed responsible for the nuclear physics
+uncertainties the process involves and so are entirely respon-
+sible for the theoretical uncertainties of the CEνNS event
+rate, unless one-loop electroweak corrections are accounted
+for [34].
+6 Uncertainties on the weak-charge form factor using the large-scale nuclear
+shell model for a long list of nuclei of interest have been discussed in Ref.
+[55].
+The CEνNS differential event rate, as any other rate, fol-
+lows from a convolution of the differential cross section in Eq.
+(30) and the incoming neutrino flux. For CEνNS to be used as
+a tool for the extraction of neutron distributions mean-square
+radii the neutrino flux should lie in either the “intermediate”
+or “high” energy windows. The former defined by pion decay
+at rest, while the latter by pion decay in flight. Note that from
+this perspective COHERENT experiments operate in the in-
+termediate energy window and the νBDX-DRIFT experiment
+will in the high energy regime [37, 56]. Because of the large
+statistics our analysis is interested in we focus on the interme-
+
+10
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+Er [keV]
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+Events
+Events in 1 year-ton LAr detector (Helm with RW)
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+Er [keV]
+0
+500
+1000
+1500
+2000
+2500
+3000
+Events
+Events in 1 year-ton LAr detector (with vector elastic form p and n form factors)
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+4.2
+RW [fm]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+2
+95% CL
+90% CL
+Helm with RW in 1 tonne-year LAr detector
+BRN+SS=10×CE NS
+BRN+SS=1.0×CE NS
+BRN+SS=0.1×CE NS
+2.6
+2.8
+3.0
+3.2
+3.4
+3.6
+3.8
+4.0
+4.2
+Rn [fm]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+2
+95% CL
+90% CL
+Weak-charge FF with elastic vector FFs in 1 tonne-year LAr detector
+BRN+SS=10×CE NS
+BRN+SS=1.0×CE NS
+BRN+SS=0.1×CE NS
+FIG. 6. Top graphs: Toy experiment signals calculated by: (i) Using the Helm form factor and fixing the diffraction radius with RW (left
+graph), (ii) Using the weak-charge form factor as given in Eq. (17) (right graph). In both cases Rn = Rp +0.1fm, with Rp calculated from the
+nuclear charge radius taken from Ref. [48] and Eq. (12). Bottom graphs: Least-square function as a function of the weak-charge radius RW
+(left graph) and the point-neutron distribution rms radius (right graph). In both graphs results for two additional analyses with two different
+BRN and SS background hypotheses are shown as well.
+diate energy window, where O(tonne) detectors are expected
+in the near future [28, 29]. In this case the neutrino spec-
+trum consist of a monochromatic component (prompt com-
+ponent) and two continuous spectra (delayed components).
+Their spectral functions follow from the π+ and µ+ energy
+distributions and read
+Φνµ = δ
+�
+Eν − m2
+π −m2
+µ
+2mπ
+�
+,
+Φνµ = 192
+mµ
+� Eν
+mµ
+�2 �1
+2 − Eν
+mµ
+�
+,
+Φνe = 64
+mµ
+� Eν
+mµ
+�2 �3
+4 − Eν
+mµ
+�
+.
+(31)
+Since neutrinos are isotropically produced, normalization of
+the neutrino flux is given by nν = r ×POT/4/π2/L2. We use
+POT = 2.1 × 1023/year, L = 28.0 m and r = 8.0 × 10−2, in-
+spired by recent COHERENT measurements and future plans
+[1–3, 28]. Thus taking into account the three neutrino com-
+ponents, the differential recoil spectrum in a detector with a
+fiducial volume composed of different stable isotopes is writ-
+ten as
+dR
+dEr
+= mdetNA
+mmol,i
+nν∑
+i
+Xi
+∑
+α=νµ,νµ,νe
+� Emax
+ν
+Emin
+ν
+Φα
+dσi
+dEr
+Eν .
+(32)
+Here mdet refers to the detector fiducial volume mass, NA =
+6.022×1023/mol to the Avogadro number, mmol,i to the molar
+mass of the ith isotope and Xi to its relative abundance. Lower
+and upper integration limits are given by Emin
+ν
+=
+�
+ErmN/2
+and Eν = mµ/2, respectively. Assuming uniform bin size ∆Er,
+the total event rate evaluated in the kth bin central value Ek
+r
+follows from integration, namely
+N =
+� Ekr +∆Er/2
+Ekr −∆Er/2
+dR
+dEr
+dEr .
+(33)
+With the results from Secs. II and III, along with those
+discussed in this section, we are now in a position to study
+the precision with which the point-neutron distribution rms
+can be extracted from data under different weak-charge form
+factor approaches. For that aim we consider a one-tonne LAr
+detector as we now discuss.
+
+11
+V.
+EXTRACTION OF THE ARGON NEUTRON
+DISTRIBUTION ROOT-MEAN-SQUARE RADIUS USING
+DIFFERENT APPROACHES
+We now proceed with the determination of the neutron dis-
+tribution rms radius in some of the different approaches we
+have discussed. We focus on three cases: (i) The weak-charge
+form factor parametrized `a la Helm, (ii) the weak-charge form
+factor written as in Eq. (17) with the elastic vector form fac-
+tors parametrized as well `a la Helm, (iii) model-independent
+approach based on expansions in terms of even-moments as
+discussed in Sec. III C.
+Rather than using existing CsI and LAr data [1–3] we in-
+stead assume a one-tonne LAr detector with specifications as
+explained in Sec. IV. This choice provides flexibility with
+background and so enable us to highlight the main differences
+arising in each case. As we have already stressed plans for de-
+ploying an O(tonne)-scale LAr detector at the STS [28] have
+been discussed in Ref. [29].
+To proceed we define a simplistic spectral least-square
+function that accounts only for energy spectral data, but en-
+capsulates the main features of such an experimental envi-
+ronment: Uncertainties from quenching factor, neutrino flux
+and efficiency as well as Beam Related Neutron (BRN) and
+Steady State (SS) backgrounds. We, however, do not consider
+systematic uncertainties due to energy calibration and pulse-
+shape discrimination and assume that the neutrino-induced
+neutron (NIN) background is subdominant compared with the
+BRN and SS backgrounds. Note that such assumptions do not
+aim whatsoever at representing the actual experimental setup,
+they are just choices that enable pointing out the effects we
+are aiming at. The spectral least-square function then reads
+χ2 =
+5
+∑
+i=1
+�
+NExp
+i
+−(1+α)NTh
+i (P)
+σi
+�2
++
+� α
+σα
+�2
+.
+(34)
+Here NExp
+i
+refers to the number of events in the ith bin gen-
+erated in a toy experiment and α a nuisance parameter. The
+least-square function becomes a function of α, with its actual
+value following from minimization over it. NTh
+i
+the number
+of events generated by varying the set P = {R2
+n,⟨R4
+n⟩} over
+a certain range, where ⟨R4
+n⟩ is only present in case (ii). For
+the statistical uncertainty we assume σ2
+i = NTh
+i
++ ∑j B j, with
+∑j B j the BRN and SS related backgrounds which we assume
+to follow the same energy dependence of the signal.
+The
+latter assumption is certainly not the case as can be seen in
+the LAr CEνNS measurement release [3]. However, rather
+than extrapolating that background to the case we are inter-
+ested in (multi-ton LAr detector) we believe this assumption
+represents a better choice. At present BRCEνNS ≃ 3% while
+BRBRN ≃ 14% and BRSS ≃ 83%, but reduction of that back-
+ground to lower levels seems achievable in the future [57].
+We then assume Bj = 10 × NExp
+j
+7. For the systematical un-
+7 Note that at present BRN plus SS backgrounds in the LAr data release
+amount to about 29 times the CEνNS signal [3].
+certainty encoded in σα we take σα = 0.11 from a 10% uncer-
+tainty due to neutrino flux, 5% uncertainty due to efficiency
+and 1% due to quenching factor. For the efficiency, and fol-
+lowing once again the CENNS-10 LAr detector, we assume
+a Heaviside function at 5keVnr and a 10keVnr bin size. To
+generate the toy experiment data, we fix the point-proton dis-
+tribution rms radius with the aid of Eq. (12) with the values for
+the different quantities given in Tab. I. For the point-neutron
+distribution rms radius we use Rn = Rp + 0.1fm, as we have
+done in the previous Sections.
+Results for cases (i) and (ii) are shown in Fig. 6. Top graphs
+display results for the toy experiments, while those at the bot-
+tom the results of the χ2 analyses. Case (i) leads to a fit for
+RW from which we find at the 90%CL the following result
+RW = 3.522+0.449
+−0.474 fm .
+(35)
+The point-neutron distribution rms radius can then be ex-
+tracted from this result with the aid of Eq. (16). To do so
+one should—in principle—take into account uncertainties on
+the single-nucleon electromagnetic mean-square radii as well
+as on the 40Ar nuclear charge radius. These uncertainties are
+given by ∆
+�
+r2p = 4.0 × 10−4 fm, ∆r2
+n = 1.7 × 10−3 fm2 [47]
+and ∆RC = 1.8×10−2 fm [48] and so can be safely neglected
+when compared with those from RW in Eq. (35). Results for
+Rn at the 90%CL thus read
+Rn = 3.428+0.434
+−0.458 fm .
+(36)
+In case (ii), c’est-`a-dire using Eq. (17) for the weak-charge
+form factor, allows fitting Rn directly. In this case the diffrac-
+tion radius in the proton and neutron elastic vector form fac-
+tors are fixed from Rp and Rn. The former fixed from the
+nuclear-charge radius and Eq. (12) (using central values for
+the relevant quantities), while the latter varying over the range
+[2.6,4.1]fm. At the 90%CL we find
+Rn = 3.457+0.492
+−0.611 fm .
+(37)
+Since the systematic uncertainty as well as the BRN and SS
+backgrounds for both analyses have been equally fixed and
+the toy experiments signals have been generated under the as-
+sumptions that define each case, differences in the results in
+Eqs. (36) and (37) can be only attributed to theoretical as-
+sumptions. Parametrizing the weak-charge form factor `a la
+Helm, Rn can be determined with a ∼ 14% precision. Us-
+ing Eq. (17)—including nucleon form factor terms up to or-
+der Q2—allows a determination at the 18% level. We then
+conclude that both procedures seem to produce results with
+comparable levels of precision, though with a slight improve-
+ment if a parametrization of the weak-charge form factor is
+used. In Sec. III we found that numerical deviations from one
+procedure or the other could be as large as ∼ 5% in the rel-
+evant transferred momentum range. We attribute the differ-
+ences found here in the extraction of Rn to that fact.
+There is of course the question of whether precision can be
+improved by improving upon the BRN and SS backgrounds.
+To assess that question we have run to extra analyses, assum-
+ing Bj = 1.0×NExp
+j
+and Bj = 0.1×NExp
+j
+. The results can be
+
+12
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+110
+120
+130
+Er [keV]
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+Events
+Events in 1 year-ton LAr detector (model-independent approach)
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+R2
+n
+1/2 [fm]
+3.6
+3.8
+4.0
+4.2
+4.4
+4.6
+4.8
+5.0
+5.2
+R4
+n
+1/4 [fm]
+95% CL
+90% CL
+90% CL and 95% CL limits in a 1-tonne LAr detector
+FIG. 7. Left graph: Toy experiment data used in the model-independent analysis generated by fixing Rn = Rp +0.1fm and ⟨R4n⟩ = 210.3fm4,
+the latter obtained from the results in Eq. (28) in the Helm parametrization case. Right graph: 90% and 95% CL isocontours in the neutron
+distribution rms radius, Rn =
+�
+⟨R2n⟩, and fourth moment,
+�
+⟨R4n⟩, plane.
+seen in both graphs in Fig. 6. In the case in which the Helm
+parametrization is used for the weak-charge form factor, preci-
+sion in the extraction of Rn improves at the 5% and 1% levels,
+respectively. In the case in which FW is decomposed in terms
+of elastic vector proton and neutron form factors, instead, at
+the 8% and 5% levels.
+One might wonder whether this conclusion holds as well for
+heavy nuclei. Although we have not done an analysis for such
+a case (e.g. for cesium), we find no reason why this should
+not be the case. Of course nuclear parameters will change, but
+given the discussion in the previous Sections we expect the
+trend to be valid in this case too.
+We now turn to case (iii), for which for the calculation we
+do not include nucleon form factors corrections in Eq. (17)
+and expand the elastic vector proton and neutron form factors
+in terms of even moments up to order Q4 (as required from the
+discussion in Sec. III C). Such procedure leads to
+FW =
+1
+QW
+�
+Zgp
+V
+�
+1− Q2
+3! R2
+p + Q4
+5! ⟨R4
+p⟩
+�
++ Ngn
+V
+�
+1− Q2
+3! R2
+n + Q4
+5! ⟨R4
+n⟩
+��
+.
+(38)
+This expression depends—in principle—upon three free pa-
+rameters: The point-neutron distribution mean-square radius
+and the proton and neutron fourth moments. However, the
+contribution controlled by ⟨R4
+p⟩ can be safely neglected. This
+can be readily checked by sticking—for this matter—to the
+Helm parametrization, using then the corresponding expres-
+sion in Eq. (28) and evaluating the proton Q4-to-Q2 ratio. Do-
+ing so one finds 2 × 10−5(Q/MeV)2, which combined with
+the fact that Q ≲ 50MeV and that the cross section is domi-
+nated by the neutron contribution reduces the calculation to a
+two-parameter problem.
+Dropping the proton fourth moment one can then determine
+from data the neutron distribution mean-square radius and its
+fourth moment. To generate the toy experiment data we fix
+⟨R4
+n⟩ with the aid of Eq. (28) assuming the Helm parametriza-
+tion. Results for that pseudo-data are displayed in the left
+graph in Fig. 7. We then proceed with the least-square analysis
+by varying Rn ≡
+�
+⟨R2n⟩ and ⟨R4
+n⟩. Results of this calculation
+in the
+�
+⟨R2n⟩-⟨R4
+n⟩1/4 plane are shown in the right graph in
+Fig. 7. One can see that under the same background and de-
+tector assumptions that those used in the previous two cases,
+only an upper limit on Rn can be derived. At the 90%CL one
+gets
+Rn ≲ 5.9fm .
+(39)
+The result for the fourth moment is instead constrained to
+an interval (at both the 90% and 95% CLs), although wide.
+Fitting Rn through the model-independent approach has—of
+course—the advantage of not relying on particular nuclear
+physics assumptions, but implies fitting two parameters rather
+than one. That is why, with the statistics and background as-
+sumptions we have adopted, only an upper limit on Rn can be
+placed.
+In some respect this result differs from the one reported
+in Ref. [38], in particular in the shape of the available 90%
+and 95% CLs contours. It seems to us these differences are
+expected because of the following reasons: (a) Our analysis
+includes the proton contribution (which might become im-
+portant in regions of parameter space where Rn/⟨R4
+n⟩1/2 ≃
+Q/
+√
+20), (b) our analysis involves a 3.5 less statistics (a 1-
+tonne rather than 3.5-tonne LAr detector), (c) our systematic
+and statistical uncertainties are much larger (in particular the
+statistical uncertainty which includes a background hypothe-
+sis that exceeds the signal by a factor 10), (d) the statistical
+methods employed in the extraction of the relevant quantities.
+Results from the three different approaches we have em-
+ployed here thus seem to favor the inclusion of nucleon
+form factors Q-dependent terms and the decomposition of the
+weak-charge form factor in terms of elastic vector proton and
+neutron form factors, as given in Eq. (17). With such an ap-
+proach a percent determination of the neutron distribution rms
+radius seems achievable, as shown in Eq. (37). Improving the
+statistical uncertainty by improving upon BRN and SS back-
+grounds may lead to ∼ 1% determination of Rn.
+
+13
+VI.
+CONCLUSIONS
+We have quantified uncertainties on the extraction of point-
+neutron distributions mean-square radii from CEνNS data.
+Our analysis is motivated by future O(tonne)-scale CEνNS
+detectors which will deliver thousands of events per year with
+well-controlled systematic and statistical uncertainties. The-
+oretically, uncertainties are encoded in the weak-charge form
+factor, for which a variety of effects and parametrization ap-
+proaches have different impact.
+Here we have quantified at the weak-charge form factor
+level uncertainties due to: (i) The absence/presence of single-
+nucleon electromagnetic mean-square radii in the determina-
+tion of the point-proton distribution mean-square radius, (ii)
+Q-dependent nucleon form factor terms up to order Q2, (iii)
+parametrizations of the elastic vector proton and neutron form
+factors, (iv) parametrizations of the weak-charge form factor,
+(v) model-independent series expansions of the elastic vector
+proton and neutron form factors.
+At the weak-charge level we have found that the inclusion
+of single-nucleon electromagnetic mean-square radii in the
+determination of the point-proton distribution mean-square ra-
+dius has a per mille impact. Thus showing that taking the
+point-proton distribution mean-square radius equal to the nu-
+clear charge radius is precise enough. Inclusion of nucleon
+form factors Q-dependent terms, instead, has a percent ef-
+fect. Comparison of results from decompositions of the weak-
+charge form factor in terms of elastic vector proton and neu-
+tron form factors with weak-charge form factor parametriza-
+tions shows a wider uncertainty, that can rise up to the order
+of ∼ 5% (∼ 10%) in the relevant transferred momentum range
+for light (heavy) nuclei. Comparison of results from decom-
+positions of the weak-charge form factor in terms of elastic
+vector proton and neutron form factors with series expansions
+of the elastic vector proton and neutron form factors shows
+uncertainties of the same order.
+To better understand the impact of these effects we have
+considered pseudo-data from a one-tonne LAr detector with
+substantial BRN and SS backgrounds, though suppressed
+compared with those in current LAr data. Our results show
+that weak-charge form factor parametrizations and decompo-
+sitions of the weak-charge form factor in terms of elastic vec-
+tor proton and neutron form factors lead to the same level of
+accuracy. Though the weak-charge form factor parametriza-
+tion leads to a slightly better accuracy. Suppressed BRN and
+SS backgrounds seem to enable reaching ∼ 1% accuracies in
+the extraction of Rn.
+ACKNOWLEDGMENTS
+The author warmly thank Dimitrios Papoulias and Jorge
+Piekarewicz for very useful comments on the manuscript. He
+thank as well the “Universidad de Antioquia” Physics Depart-
+ment for its hospitality during the completion of this work and
+ANID for financial support through grant “Fondecyt Regular”
+1221445.
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@@ -0,0 +1,1883 @@
+Quantitative Verification of Scheduling Heuristics
+Saksham Goel
+UT Austin
+Benjamin Mikek
+Georgia Tech
+Jehad Aly
+Alexandria University
+Venkat Arun
+MIT CSAIL
+Ahmed Saeed
+Georgia Tech
+Aditya Akella
+UT Austin
+Abstract
+Computer systems use many scheduling heuristics to allo-
+cate resources. Understanding their performance properties is
+hard because it requires a representative workload and exten-
+sive code instrumentation. As a result, widely deployed sched-
+ulers can make poor decisions leading to unpredictable perfor-
+mance. We propose a methodology to study their specification
+using automated verification tools to search for performance
+issues over a large set of workloads, system characteristics and
+implementation details. Our key insight is that much of the
+complexity of the system can be overapproximated without
+oversimplification, allowing system and heuristic developers
+to quickly and confidently characterize the performance of
+their designs. We showcase the power of our methodology
+through four case studies. First, we produce bounds on the
+performance of two classical algorithms, SRPT scheduling
+and work stealing, under practical assumptions. Then, we
+create a model that identifies two bugs in the Linux CFS
+scheduler. Finally, we verify a recently made observation that
+TCP unfairness can cause some ML training workloads to
+spontaneously converge to a state of high network utilization.
+1
+Introduction
+Modern software systems are expected to meet strict opera-
+tional goals, such as high availability and good performance.
+However, when subject to unforeseen workloads or operating
+environments, these systems can be left exposed to corner
+cases where they fall well short of their goals. The overall per-
+formance of the system is critically impacted by scheduling
+algorithms since they decide how system resources are allo-
+cated. As a result, these algorithms have received significant
+attention from academia and industry.
+This paper argues that although scheduling algorithms con-
+trol complex systems and exhibit complex behaviors, they
+often have a simple mathematical description. This opens the
+possibility of using formal verification techniques to guar-
+antee their robustness under arbitrary, pathological, scenar-
+ios. Conventional approaches to ensuring robustness, such
+as testing and fuzzing, require workloads that can trigger all
+potential problems. These can be difficult or even impossi-
+ble to build (see §2.1). On the other hand, theoretical tools
+(e.g., queueing theory and control theory) offer the ability of
+deriving bounds on the performance of such algorithms. How-
+ever, they require making lots of oversimplifying assumptions,
+leading to results that don’t necessarily reflect the algorithm’s
+performance in practice. In contrast, verification can guaran-
+tee performance for all workloads with a simple but accurate
+description of the algorithm under study.
+Formal verification has grown in popularity in recent years,
+with frameworks for verifying key-value stores [7,9,22], net-
+work configurations [5,11,12,27], compilers [24,26,29,30],
+and more. However, most efforts focus on verifying quali-
+tative properties related to correctness, such as safety and
+liveness. In contrast, proving quantitative properties that pro-
+vide guarantees on performance has received less attention.
+To that end, this paper presents a method for encoding
+scheduling heuristics, the systems in which they operate, and
+the performance property to be tested (e.g. work conservation
+or near-optimality) in an SMT solver. The solver can then
+either prove that a property always holds or provide a concrete
+workload where it is violated. Since solvers can quickly and
+efficiently explore a large number of workloads and system
+behaviors, they can help us understand complex behaviors
+that would be difficult or tedious for humans to analyze.
+To limit complexity of the analysis while still maintain-
+ing accuracy, we model only the parts of the surrounding
+system that directly interact with the heuristic. Our models
+overapproximate the system by capturing a superset of system
+behaviors. Thus, if the model does not contain any behaviors
+that cause the algorithm to violate the desired property, we
+can be sure that the real system will not exhibit such behavior
+either. The principles we adopt (see §3) allow us to reason
+about complex systems in a clear and concise way.
+To illustrate these principles concretely and demonstrate
+generality, we include four case-studies. The first two prove
+bounds on the optimality of shortest remaining processing
+time (SRPT) schedulers and work stealing, demonstrating
+that our tool can provide insights for well-studied algorithms
+1
+arXiv:2301.04205v1  [cs.LO]  10 Jan 2023
+
+by relaxing assumptions made in theoretical models. To go
+beyond the classical theoretical results, we study the perfor-
+mance of SRPT when preemption is not allowed and tasks
+can block. Further, we study the performance of work stealing
+algorithms while accounting for context switching costs.
+Our third case study is concerned with the performance
+of the Linux CFS load balancer. In particular, we check if
+the algorithm is work conserving, reproducing a bug that was
+recently identified by the community. Further, we identify a
+new bug that can lead to the violation of the work conserva-
+tion property. Finally, we show that our approach may not
+be limited to CPU schedulers by verifying a recent observa-
+tion [37] that TCP unfairness can cause flows to synchronize
+to a schedule where they efficiently occupy a shared link.
+This paper’s central thesis is that, using our modeling ap-
+proach, it is possible to quickly and easily analyze scheduling
+algorithms and draw useful conclusions about their behavior
+in the real world. The most challenging aspect of scheduler
+analysis is still asking the right questions and interpreting the
+answers. However, automation can make this process easier
+and more reliable. We invite the community to incorporate
+performance verification as part of their workflow when de-
+signing scheduling algorithms.
+2
+Motivation
+2.1
+The Problem of Analyzing Heuristics
+Heuristics, by definition, are imperfect. However, experts de-
+sign them to produce results with reasonable performance.
+The typical way to evaluate the performance of a heuristic
+is through thorough benchmarks and fuzzing, which show
+that the heuristic performs well in a wide range of scenarios.
+These benchmarks can give confidence in the heuristic, but
+they do not provide guarantees on its worst-case performance.
+Without guarantees, heuristics can have corner cases that can
+lead to poor performance, wasted debugging efforts to iden-
+tify the heuristic as the cause of the problem, and decreased
+overall reliability of the system.
+As an example, we will use the load balancer in the Linux
+CFS scheduler to illustrate our argument. The Linux load
+balancer implements complex heuristics to decide which tasks
+should be moved to which CPU in order to balance the load
+between CPUs, distribute idle CPU cycles, give each task a
+fair share of CPU time, avoid moving tasks too much, and
+respect cache and NUMA placement when moving tasks. Due
+to its complexity, it is reasonable to question whether the load
+balancer is work-conserving and fair.
+It has been demonstrated repeatedly that the answer is “no”.
+Many of the identified issues were caused by the algorithm
+itself, not just the implementation. For example, prior work
+found that CPUs may remain idle for a long time, even when
+other cores are overloaded with a large number of active tasks,
+due to an inadequate method for quantifying the load of a
+group of CPUs [31]. While these were later fixed, other per-
+formance bugs remain. For instance, we discovered a new
+bug where CPUs can remain idle even with a large number
+of runnable tasks, because of a spurious check that forces
+the balancer to return early (see §6.2.2). This occurs despite
+mechanisms designed apparently to prevent this scenario.
+Identifying such performance bugs using conventional
+methods is a major undertaking. Typically, experienced users
+or developers observe anomalous behavior when using the
+heuristic over an extended period of time. Such observations
+prompt a more methodical effort by the community to repro-
+duce the bug, leading to efforts that can require hours or even
+days of experimentation. For example, the complete rewrite of
+the Linux load balancer in v5.5 took several months of careful
+manual scrutiny by the community to prevent bugs [15].
+Not all heuristics are based solely on the designers’ intu-
+ition. Many are supported by solid theoretical results that
+prove the optimality of the heuristic or a version of it under
+simplifying assumptions. For example, there are strong theo-
+retical bounds on the performance of work stealing algorithms.
+The models used to prove these bounds ignore practical con-
+siderations like the context switching costs [6]. Translating
+these algorithms to real-world systems results in much more
+variable performance than is guaranteed by theory, and this
+has become a active area of research [18,32,34]. Designers
+need systematic guidance to decide whether work stealing is
+right for them.
+There is a tension between two objectives when evaluating
+heuristics: 1) fidelity of the evaluated heuristics and workloads
+with respect to the real system, and 2) confidence provided
+by the evaluation methodology. Theoretical approaches have
+low fidelity because they rely on simplifications that make
+the problem tractable for human reasoning. Benchmarks and
+fuzzers provide higher fidelity by evaluating the actual imple-
+mentation of the heuristic, but they compromise on confidence
+by not providing formal guarantees. In this work, we aim to
+provide a practical tool that offers high confidence in the
+performance of heuristics without sacrificing fidelity.
+2.2
+Quantitative Verification
+In deploying a scheduler in real-world settings, a system de-
+signer or operator may be interested in several questions per-
+taining to the performance of the scheduling heuristics. We
+identify three broad classes of such questions:
+Q1: Boolean questions about performance. Examples in-
+clude whether a scheduler is work-conserving, or fair. An-
+swers to these questions could either be that the scheduler
+satisfies the property in question, or a counterexample show-
+ing a valid schedule that breaks the property. These types of
+questions help the designer ensure the correctness of their
+design, and identify corner cases missed by the heuristic.
+Q2: Comparison with an Oracular Scheduler. When de-
+signing a heuristic, it’s typically helpful to know how the
+heuristic compares to an optimal, oracular scheduler with
+complete offline knowledge of inputs that can even solve NP-
+2
+
+complete problems. Although such an oracular scheduler is
+impractical, comparing to it helps identify room for improve-
+ment and define bounds on the performance of the heuristic.
+For example, we may desire to define a bound on the perfor-
+mance of the studied heuristic (e.g., it takes at most twice as
+long as optimal to finish all tasks).
+Q3: Precise workload characterization. Heuristics based
+on theoretical results will perform optimally when operating
+in settings or on workloads that match the assumptions made
+in the theoretical model. The same performance might hold
+for a broader class of workloads, but might also break for other
+classes of workloads. Similarly, benchmarking and fuzzing
+can only evaluate a heuristic for specific and finite number of
+workloads. A heuristic designer will typically be interested
+in systematically characterizing workloads (as opposed to
+experimenting with different canonical workloads on a trial-
+and-error basis) for which the heuristic performs poorly.
+Recent advances in verification tools provide an avenue
+for efficient verification of a wide range of properties. In
+our context, given a model of the heuristic in question, we
+interpret “correctness” to mean achieving a quantitative per-
+formance criterion, thereby helping answer questions of type
+Q1. Further, existing verification tools can find assignments
+that optimize a quantitative measure, allowing us to compare
+the performance of an encoded heuristic to that of an oracular
+scheduler. Finally, given a performance metric, we can an-
+swer questions of type Q3 by using verification tools to find
+workloads which perform poorly according to the metric in
+each of several different specified workload classes.
+3
+Methodology
+Our main contribution is defining a clear methodology for
+scheduling heuristic designers to use formal verification tools
+to debug and better understand the performance of their de-
+signs. A designer expresses environment constraints (i.e.,
+CPUs and a task model), a scheduling algorithm (or heuris-
+tic), and a performance query. Queries ask whether some
+performance property is always true for the given scheduling
+algorithm. If a counterexample exists, it helps the user iden-
+tify corner cases that can break the studied algorithm. If there
+is no counterexample, then we have successfully verified the
+behavior encoded by the query. To answer queries we use Z3,
+a Satisfiability Modulo Theories (SMT) solver [10]. To keep
+our models tractable, we only use the theory of linear real
+arithmetic. Table 1 provides a summary of how our method-
+ology is applied to four different scheduling algorithms. We
+adopt three design tenets for our models.
+Overapproximation not oversimplification. Reasoning
+about the complexity of a full system is intractable, whether
+using automated tools or conventional analytical tools. Thus,
+abstraction is a necessary step. Conventional analytical tools
+tackle that abstraction with simplifying assumptions, favoring
+human tractability over soundness. Formal verification, by
+contrast, favors soundness. Formal verification models tend
+to overapproximate systems; i.e., the set of possible model be-
+haviors is a superset of the possible behaviors of the modeled
+system. Hence, any theorems proved in the overapproximated
+model are also true for the real system. However, counterex-
+amples produced by the model require human inspection to
+ensure that they are also possible in the underlying system. In
+our experience, all counterexamples we encountered have a
+real-world counterpart.
+Our methodology is to overapproximate all behavior ex-
+ternal to the scheduling algorithm. For example, all blocking
+calls (e.g., networking, storage, or synchronization) that force
+a thread to yield a CPU core can be treated collectively as one,
+allowing a task’s blocking time to have a wide range of values.
+The solver can then choose any value for the blocking time a
+thread faces. Despite networking and storage calls typically
+having bounded latencies, overapproximation helps us avoid
+making any assumptions about their behavior, allowing us to
+make general conclusions about the scheduling algorithm.
+Overapproximating external behavior makes the system
+easier to model for the user. In particular, it allows users to
+focus on modeling the details of their algorithm while the
+solver picks the behavior of components external to the mod-
+eled system. Further, reducing the volume of details makes
+user–created models more concise. In all of the case studies
+presented here, for example, the SMT constraints are gener-
+ated by less than 1,000 lines of Python.
+Event-based modeling. Schedulers determine the order-
+ing and time of execution of tasks. There are multiple ways
+to encode their behavior. As discussed above, we overapprox-
+imate all behavior not integral to the heurstic. However, this
+leaves an important question: how should we model the de-
+tails of complicated heuristics with many steps and states?
+Capturing every step and state in a heuristic creates intractable
+models. A more tractable approach is to discretize time, al-
+lowing the automatic solver to capture all possible state tran-
+sitions that can happen between two timesteps. This approach
+is intuitive and has been used before to model, for example,
+the performance of congestion control algorithms [2]. How-
+ever, this modeling approach suffers from several downsides.
+First, it limits the time span captured by the model, reducing
+the scalability of the model. Second, it requires introducing
+complex constraints to ensure that only valid state transitions,
+including scheduling decisions, are made between two time
+steps.
+By contrast, our approach relies on identifying key events in
+schedules produced by a heuristic (e.g., the point in time when
+a thread blocks). Where necessary, we represent a state with
+two events: its starting and finishing times. The automated
+solver can assign arbitrary time values to events, enabling it to
+capture arbitrarily large time spans. In addition, an algorithm
+makes scheduling decisions at a particular event, meaning
+that scheduling behaviors need only be encoded relative to
+events, potentially simplifying the model.
+Unconstrained initial conditions. Event-based modeling
+3
+
+Heuristic
+Queries
+Events
+Overapproximation
+Single-core SRPT Scheduling (§4)
+Comparison to an oracle scheduler
+for various objectives
+Tasks changing their states from
+ready to running to blocking
+Causes of blocking and
+task profiles
+Work Stealing (§5)
+Characterizing the impact of
+context switching cost on performance
+Start and end of individual
+tasks in a DAG
+Context switching cost
+Linux CFS Load Balancer (§6)
+Work conservation
+Periodic load balancing ticks
+Changes in the states of
+tasks between ticks
+Ring Reduce Scheduling (§7)
+Achieving high network utilization
+Start and end of flows
+Behavior of congestion
+control algorithms
+Table 1: A summary of the application of the modeling methodology to four scheduling heurisitics.
+limits the number of tasks and events that a model captures,
+leaving a critical question open: How can a small number of
+events provide useful insights about complicated heuristics?
+Testing a scheduling heuristic for violations of a given per-
+formance property using benchmarks or fuzzing requires the
+identification of a concrete task workload under which the
+heuristic fails. Such a workload is likely to be complex and dif-
+ficult to analyze, even if only a few of its events cause the prop-
+erty violation. Our approach solves this problem by letting the
+solver pick arbitrary initial conditions for the system. Com-
+pared to a concrete workload under the testing approach, the
+solver may choose initial conditions corresponding the state
+of the system just before the events which caused the property
+violation. In this sense, unconstraining initial conditions al-
+lows us to jump past irrelevant events which a benchmarking
+or fuzzing approach would have to exhaustively consider.
+4
+Case study: Single Core SRPT Scheduling
+We begin with the simplest case study: shortest remaining
+processing time first (SRPT) on a single processor. It is well
+known that SRPT is optimal with respect to average time to
+completion [40]. However, proofs of optimality assume both
+preemption and no task blocking, for example, to perform
+IO operations. Later work on queueing theory has provided
+bounds on the average running time of tasks in a system with
+general load [3], and investigated the fairness implications
+of SRPT [4]. The performance of non-preemptive schedulers
+has also been investigated [1], but not with blocking.
+We fill this gap for a simple non-preemptive SRPT sched-
+uler by providing a model which can be used to verify per-
+formance properties without prior expertise. For a simple
+system with a single core and plain SRPT scheduling, we try
+to understand SRPT’s performance when tasks can block.
+4.1
+Model
+We focus on the context of a single core where tasks can
+alternate between the states of running and blocking and no
+preemption is allowed. This context is reflective of many RPC
+execution settings where threads run to completion and only
+yield when they make blocking calls [18,34,44]. Within this
+context, a task can be in one of three states: ready, running, or
+blocking, as shown in Figure 1a. Instead of modeling states
+explicitly, we identify two events corresponding to the passage
+of a task through each state: the time when it enters the state
+and the time it leaves. We call one passage through the state
+D
+R
+B
+Start
+Finish
+Blocking
+Running
+Ready
+(a) Modeled state machine of a task
+...
+Time
+Ds1
+Df1=Rs1
+Rf1=Bs1
+Bf1=Ds2
+Bfs
+Waiting in 
+the queue
+Running
+Finished
+Blocking
+(b) Mapping the state machine to an event-based model in time
+Figure 1: Task model, showing the model of a single task
+cycle a step; a step has 6 events, two each for ready, running,
+and blocking as shown in Figure 1b. We overapproximate
+real systems by allowing events to happen at arbitrary points
+in time subject only to constraints enforced by our context
+(i.e., having a single core). We constrain the system such that
+only one task can be running at any point in time. Under this
+model, a schedule is simply an ordering on the time each task
+enters each state.
+Tasks. We consider a model with n tasks T1,...,Tn. We limit
+the number of events in the model by fixing the number of
+steps s per task. A step is one ready–running–blocking cycle
+in Figure 1; each task has six events per step. We define a task
+Ti as a tuple of (Li,Di,Ri,Bi), where Li > 0 is the total length
+of the task (i.e., the sum of all the time it spends running).
+D,R,B ∈ (R ×R)s are sets of pairs that define the start and
+end of ready, running, and blocking periods, respectively. We
+denote these with D =
+��
+Dsj
+i ,Df j
+i
+�
+| 1 ≤ j ≤ s
+�
+, where
+Dsj
+i is the start of the ready event of task i in step j and Df j
+i is
+the end of the same ready event. We define R and B similarly.
+The model includes constraints that ensure the proper tim-
+ing of all events. At the task level, we ensure that the sum of
+the running times per task is equal to its length. Further, tasks
+must always be in one of the three states, and must proceed
+from waiting to running to blocking in that order. Formally:
+∀i
+(Df j
+i = Rsj
+i < Rf j
+i = Bs j
+i ≤ Bf j
+i ),
+1 ≤ j ≤ s
+∀i
+(Bf j
+i = Dsj+1
+i
+),
+1 ≤ j < s
+Scheduler. Since a schedule is simply a valid ordering on
+events, we can represent a scheduling algorithm as a con-
+straint on task variables. SRPT allows a task to run iff that task
+has the least remaining processing time or all tasks with less
+4
+
+remaining time are blocking. Formally, let the remaining pro-
+cessing time of task i’s step j be ej
+i = Li −∑j
+k=0(Rf k
+i −Rsk
+i ).
+SRPT can be defined as on ordering on running times, subject
+to the remaining time of each task
+∀i,l ≤ n ∀ j,k ≤ s
+(Rs j
+i < Rsk
+l ) ⇐⇒
+�
+ej−1
+i
+≤ ek−1
+l
+∧Rf k−1
+l
+≤ Rsj
+i
+�
+∨
+�
+ek−1
+l
+≤ ej−1
+i
+∧Rf k−1
+l
+≤ Rsj
+i
+∧Bsk−1
+l
+≤ Rsj
+i ∧Rs j
+i < Bf k−1
+l
+�
+∨
+�
+R f k−1
+l
+> Rsj
+i
+�
+The first condition checks that by step j −1, task i has less
+remaining time than task l by step k −1. The second condi-
+tion checks that if the first predicate is false that task l will
+be blocking when step j of task i runs. The last condition
+checks that l’s step k comes after i’s step j. We note that the
+SRPT scheduling constraint is an overapproximation because
+it assumes perfect knowledge of tasks’ remaining time. In real
+workloads where remaining time is uncertain, SRPT could
+perform even worse than the results presented in this section.
+Objective. We create two schedules for the same set of tasks,
+one generated by SRPT and the other representing the best
+possible schedule subject to the query. Formally, we define a
+schedule as a set of tasks {T1,...,Tn}. Our queries are repre-
+sented by two schedules SchedSRPT = T and Schedquery = T ′,
+where SchedSRPT follows an SRPT schedule. Both schedules
+have the same set of tasks, or Rf j
+i − Rsj
+i = R′ f j
+i − R′sj
+i and
+Bf j
+i −Bs j
+i = B′ f j
+i −B′sj
+i for all i ≤ n and j ≤ s.
+We specify two queries: 1) comparing the average comple-
+tion time of tasks under the two schedules, and 2) comparing
+the number of tasks that finish within a specific deadline. For
+average completion time, the query fixes a ratio q > 0, and
+asks whether average completion time in SchedSRPT can be
+q times more than in Schedquery: (∑n
+i=0 Bf s
+i ) = q(∑n
+i=0 B′ f s
+i )
+For a deadline, the query specifies a time G and compares the
+number of tasks a and a′ finished by time G in each schedule:
+���
+Ti | B f i
+s ≤ G
+��� = a ∧
+���
+T ′
+i | B′ f i
+s ≤ G
+��� = a′
+Expressiveness and performance. The model is general and
+can be used to evaluate the performance of any valid schedul-
+ing algorithm (i.e., an ordering on the running steps of individ-
+ual tasks) under any objective. For our model, the performance
+of Z3 scales approximately exponentially with the number of
+variables; it is therefore infeasible to make queries with very
+large number of tasks or steps. To maximize the number of
+queries, deadline objective results were generated with a = 1,
+a′ = n, and s = 2; for smaller numbers of tasks n ≤ 4, we
+confirmed that the results hold for s = 3 and s = 4 as well.
+Despite these limitations, our modeling approach remains
+highly expressive; with each query, it checks an infinite class
+of inputs, a task that would be infeasible with a simulator.
+In our evaluation, our model was able to check more than
+500,000 infinite classes of tasks in about a day on a standard
+desktop by scanning different values of a, a′, s, n, and bounds
+on running and blocking times.
+4.2
+Results
+Due to the simplicity of SRPT scheduling, we focus on the Q2
+and Q3 classes of questions. Questions in Q1 are very simple
+to reason about (e.g., SRPT is by definition work conserving).
+Comparison to an oracular scheduler. Our model allows
+us to characterize SRPT’s performance by creating queries
+about the existence of schedules that significantly outperform
+SRPT, for both objectives. We start with a simple question
+to verify existing results. In particular, when no blocking is
+allowed, all queries for both average completion time (i.e., k >
+1) and number of tasks finished (i.e., a′ > a) are unsatisfiable,
+meaning that there is no schedule which performs better than
+SRPT. This matches the known result of SRPT optimality.
+Then, we unconstrain blocking times, allowing them to
+take any value. In that setting, the solver can generate valid
+task sets for which SRPT performs much worse than an ideal
+schedule. In particular, for average completion time, the solver
+generates a task pattern like that shown in Figure 2a with one
+long task, and n − 1 short tasks all of which are forced to
+finish after the long task. Our model can produce schedules
+for which average completion time under SRPT is up to n−ε
+times worse than the query schedule, for any small ε, where
+n is the number of tasks. For example, for queries with n =
+3, q = 3 is infeasible because the average running time in
+SchedSRPT cannot be more than the total running times of all
+tasks, and at least one of the tasks in Schedquery must also take
+this long. Obviously, preemption can mitigate such scenarios
+of poor performance.
+For the deadline query, the solver can generate valid sched-
+ules for a′ >> a, showing that SRPT can finish a very small
+number of tasks when it’s feasible to finish a much larger
+number. For example, the solver was able to generate a sched-
+ule for the values a = 1 and a′ = 10. The relationship does
+not depend on the input choice of deadline; the solver can
+generate similar examples for any reasonable choice of dead-
+line. Figure 2b shows a concrete set of tasks for which SRPT
+finishes only one task, while an optimal scheduler finishes
+five. A limitation of our tool is the need to set concrete values
+of a and a′. Thus, our results are limited by the values we
+scanned. The key insight for the poor performance of SRPT
+under the deadline query is that SRPT can force all tasks to
+block simultaneously, thus wasting processor time. Without
+the SRPT constraint, it is possible to strategically create a task
+ordering which minimizes this wasted time. For example, in
+Figure 2b, delaying the start of T3 allows its running period
+to overlap with the blocking periods of other tasks, thereby
+reducing wasted time.
+Workload characterization. Our next set of questions is
+aimed at understanding the relationship between blocking
+time (i.e., the maximum time that can be spent in a single
+5
+
+5
+10
+15
+20
+25
+T1
+21.75
+T2
+19.25
+T3
+20.5
+T1
+4.5
+T2
+3.25
+T3
+23
+x
+x
+SRPT
+Ideal
+Time
+(a) A set of tasks for which average completion time under SRPT is 2× higher
+than an ideal schedule (10.25 vs 20.5, with units arbitrary). By extending
+the running period marked x, it is possible to achieve any performance gap
+smaller than 3×. The first blocking period of T2 must be longer than T1’s
+running period to force T3 to start running.
+T1
+
+T2
+
+T3
+
+T4
+
+T5
+
+T1
+
+T2
+
+T3
+
+T4
+
+T5
+
+SRPT
+Ideal
+Time
+(b) A concrete set of tasks for which it is feasible to finish 5 times more tasks
+than SRPT. SRPT finishes only T1, while the oracle produces a schedule
+which finishes all 5 tasks. The time scale is arbitrary.
+Figure 2: Solver–generated schedules which for which SRPT
+performs badly.
+blocking call) and the performance of SRPT. We formulate
+our question such that all time values are relative to the mini-
+mum time a task can spend running, making it our unit time
+(i.e., R f j
+i −Rsj
+i ≥ 1 for all i, j). Further, we bound the max-
+imum blocking time to α. Since the time scale is arbitrary,
+α acts as a bound on the ratio of minimum running times to
+maximum blocking times.
+For the average completion time objective, the results fol-
+low the pattern shown in Figure 2a. Whenever α > n−2, it
+is possible for the SRPT schedule to have arbitrarily higher
+average completion time than the query schedule (subject
+to the feasibility bound of q = n). This is the case because
+α > n − 2 allows one short task to block during the entire
+running periods of n−2 other short tasks, thereby forcing a
+single long task to run before any of the short tasks can finish.
+We have confirmed this result with our model up to n = 6. If
+we have α ≤ n−2, then the possible values of r grow with
+the number of tasks, but do not depend on α.
+For the deadline objective, we create queries with fixed
+a = 1 and scan all integer values 1 ≤ a′ ≤ 7 for each integer
+value 1 ≤ α ≤ 7, creating 49 queries. We find a linear rela-
+tionship between α and the worst case performance of SRPT.
+In particular, for every α, an optimal schedule could finish α
+times more tasks than SRPT, but not more. This relationship
+with α is also related to the ability of an ideal scheduler to
+place multiple running periods of one task within the blocking
+time of another, thus reducing wasted time. For example, if
+only two tasks are present and all blocking periods are smaller
+than the smallest running period, wasted time is guaranteed.
+As the maximum blocking time increases, more and more run-
+ning time periods of other tasks can overlap with the blocking
+of another, allowing an ideal scheduler to perform better than
+SRPT. As in Figure 2b, the key is to minimize the time wasted
+when all tasks are blocking.
+Impact of work conservation. The results presented in this
+section, so far, are based on queries that attempt to find a better-
+performing schedule subject to a single constraint on the
+gap in performance between that schedule and SRPT sched-
+ules. Our goal is to better understand the better-performing
+schedules. Our initial queries have no constraints except for
+the performance they achieve. We initially thought that such
+schedules can strategically order tasks by being non-work
+conserving. Thus, we added a constraint on Schedquery to be
+work conserving. However, to our surprise, this constraint
+didn’t change any of our results.
+The key takeaway from this case study is that our methodol-
+ogy allows for exploring different aspects of the performance
+of a heuristic by simply formulating a reasonable query, with-
+out the need for any theoretical background or laborious ex-
+ample checking.
+5
+Case study: Work Stealing
+Work stealing schedulers assign tasks to multiple proces-
+sors. Each processor executes tasks in its local queue in
+order. When a processor becomes idle, it steals the oldest
+task from another processor’s queue. Work stealing has been
+well-studied and has had a significant impact on real-world re-
+source schedulers [18,34]. A well-known theorem guarantees
+that work stealing will find a schedule that finishes all tasks in
+at most twice the time of an offline optimal scheduler [6], but
+does not account for context switching cost between threads.
+We show how our methodology can address this gap.
+A known model exists that over-approximates the sys-
+tem [6]. The algorithm is invariant to the specific operations
+of tasks, so task lengths are represented by a single real num-
+ber. Dependencies between tasks, such as locks and multi-
+threaded channels, are represented by edges in a Directed
+Acyclic Graph (DAG). We encode the DAG as a boolean ad-
+jacency matrix. To add context-switching costs to the model,
+we group tasks into “threads”. Tasks in a thread must be con-
+nected in a straight sequence in the DAG. When a processor
+switches between tasks in the same thread, it incurs no con-
+text switching cost. Otherwise, it incurs a cost that may be
+different for each task. In general, context switching cost can
+depend on the tasks and the processor involved because of ar-
+chitectural features like NUMA, and the status of any caches.
+We adopt a simpler model, and let the solver arbitrarily de-
+cide switching costs for every task. Fig. 3a shows an example
+6
+
+sc = 1
+t1 
+len = 1.75
+sc = 0.5
+sc = 0.75
+t4 
+len = 1
+t3 
+len = 1.75
+t2 
+len = 1
+t5 
+len = 2
+t6 
+len = 1
+(a) An example DAG. The color of each task represents the owning
+thread. Red edges denote inter-task dependencies. The blue dashed lines
+show CPU jumps between tasks belonging to different threads, with the
+switching cost (sc).
+CPU1
+t1
+sc
+t3
+t4
+sc
+t6
+CPU2
+t5
+idle
+sc
+t2
+CPU1
+t1
+t3
+t4
+t6
+CPU2
+t2
+sc
+t5
+OPT:
+WS:
+Time
+(b) Work Stealing and Optimal schedules
+Figure 3: Work Stealing vs Optimal
+DAG with tasks grouped into threads.
+The solver searches every choice of DAG, assignment of
+tasks to threads, task lengths and context switching costs.
+Together these choices specify a “job”. Constraints ensure
+the choices make sense, i.e. costs are positive, the adjacency
+matrix forms a DAG etc. To keep the problem tractable, we
+restrict to a finite number of tasks and processors, though
+there are no bounds on task lengths and costs.
+The solver chooses variables representing two schedules for
+this job. One schedule is constrained to mimic work stealing
+and the other can be arbitrary. The solver is instructed to
+maximize the ratio of the time taken by the work stealing
+schedule to the time taken to finish the arbitrary one. This
+forces it to minimize the completion time for the arbitrary
+schedule, making it optimal. Our queries explore how this
+ratio varies as we impose different constraints on the context
+switching costs.
+A schedule is represented by the start and finish times for
+each task, as well as a boolean matrix that maps tasks to
+CPUs. While the two schedules can be different, constraints
+ensure that both respect the tasks’ properties, such as their
+lengths, dependencies, and switching costs. For example, the
+following constraints ensure that a task is ready to execute
+when all of its dependencies are finished1:
+1tasks is a small, finite set. Hence ∀ and ∃ can be written in quantifier
+free logic. This is easier to solve.
+∀t1,t2 ∈ tasks
+edge(t1,t2) =⇒
+t1.sched.end_time≤ t2.sched.ready_time
+(1)
+∀t2 ∈ tasks
+(∃t1 ∈ tasks,
+edge(t1,t2)
+∧
+t1.sched.end_time= t2.sched.ready_time)
+∨
+t2.sched.ready_time= 0
+(2)
+Fig. 3b shows an example work stealing schedule and com-
+pares it with the optimal schedule.
+5.1
+Queries and Results
+We ask the solver to maximize the ratio between completion
+times in the work stealing and unconstrained schedules:
+maximize(timews/timeopt)
+(3)
+First, we set the costs of context switching to zero and
+queried for the maximal ratio between work stealing and op-
+timal. We found that the bound was not 2, but 2− 1
+P, where
+P is the number of processors. Since we are using an SMT
+solver, the bound it found is exact and matches the known
+theoretical result. Thus, even though we only queried for up
+to 4 processors, 4 threads, and 9 tasks, we believe that the pre-
+cision and consistency of our results allow for extrapolation
+to larger values.
+Next, we introduce switching costs. We parametrize the
+switching cost in two ways. Parameter k caps the maximum
+switching cost and c caps how different the switching costs
+can be from each other. Formally:
+∀t ∈ tasks,
+switch_cost(t) ≤ k ×lenmin
+(4)
+max
+t∈tasks(switch_cost(t))
+≤ c× min
+t∈tasks(switch_cost(t))
+(5)
+Since the unit of time is arbitrary, we pick one so that the
+maximum task length is 1. We fix c = 1 and k = 10. This
+means that the solver can select any context switching cost
+up to 10× the task length, as long as the costs are equal to
+each other. Figure 4 shows the optimality ratio as we vary
+the maximum size of the DAG tasks. For small DAGs, work
+stealing performs close to optimal. The performance worsens
+linearly as the size increases. It is interesting to note that the
+bound grows at a slower rate with increasing number of CPUs
+and the ratio remains bounded even though context switch
+cost can be up to 10× the task length.
+Next, we vary the bound on the maximum allowed switch-
+ing cost while constraining that all costs be equal to each
+other. Figure 5 plots the optimality ratio for different num-
+bers of CPUs and tasks.2 Interestingly, it plateaus to a value
+only slightly larger than in the case without context switching
+cost. For instance, with 2 processors and up to 7 tasks, the
+optimality ratio is smaller than 4.
+2In realistic systems, k should not be much larger than 1.
+7
+
+1
+2
+3
+4
+5
+6
+7
+c
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+Work Stealing vs Optimal
+#CPU=2
+#CPU=3
+Figure 4: Work Stealing performance for different DAG sizes
+(k = 10,c = 1). When the number of tasks is less than the
+number of CPUs, work stealing is trivially optimal.
+0
+2
+4
+6
+8
+10
+k
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Work Stealing vs Optimal
+#CPU=2, #Tasks=7
+#CPU=2, #Tasks=6
+#CPU=3, #Tasks=7
+#CPU=3, #Tasks=6
+Figure 5: Work Stealing performance as switching costs scale
+up. Every pair of task has the same cost (c = 1).
+1
+2
+3
+4
+5
+6
+7
+8
+c
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+Work Stealing vs Optimal
+#CPU=2, #Task=6
+#CPU=2, #Task=5
+#CPU=3, #Task=7
+#CPU=3, #Task=6
+#CPU=2, #Task=4
+#CPU=3, #Task=5
+Figure 6: Work Stealing performance when costs between dif-
+ferent pairs of tasks can vary. (k = 10)
+This sets the stage up for the final result, wherein we we
+allow switching costs to vary between 0 and 10 as long as
+they are within c× of each other, as shown in figure 6. Here
+the ratio grows linearly with c, showing that the variation in
+switching cost matters more than its absolute value.
+What can practitioners take away from this analysis? If
+context switching cost is small, work stealing is near optimal.
+If not, it is still near optimal if the number of CPUs is large
+or if cost variation is small and the job is representable by a
+small DAG. A caveat is that our results are only rigorously
+proven for small numbers of CPUs and tasks. Nevertheless
+the numbers fit so perfectly on a line that we are tempted to
+conjecture and extrapolate.
+6
+Case study: The Linux CFS Load Balancer
+The Completely Fair Scheduler (CFS) scheduler is the de-
+fault process scheduler in Linux for systems with multiple
+processing units. It aims to ensure fairness between running
+threads without sacrificing performance. We studied the load
+balancing logic in the CFS scheduler for Linux v5.5 [14].
+Overview. The load balancer supports multi-core architec-
+tures, including SMT, SMP, and NUMA. To minimize cost,
+Level 0
+Level 1
+Level 2
+CPU1
+CPU2
+CPU3
+CPU4
+Domain11
+Domain12
+Domain21
+Figure 7: Domain hierarchy with 4 CPUs. The smaller sub-
+domains also act as the groups of the larger domain.
+it tries to move tasks between nearby CPUs. To capture how
+close CPUs are to each other, the load balancer divides CPUs
+into scheduling domains. Scheduling domains form a hier-
+archy starting from individual CPUs at bottom level to the
+top-level domain that includes all CPUs. For example, a do-
+main could be the set of CPUs in the same NUMA node.
+Ideally, the algorithm should balance load between all CPUs
+at all domains. However, this task becomes expensive as the
+number of CPUs grows. To reduce the cost of load balancing,
+the algorithm balances the aggregate load between groups
+of CPUs. In particular, each domain is divided into multiple
+groups, each containing one or more CPUs. The algorithm
+attempts to balance load between the groups in each domain.
+Figure 7 shows an example a domain hierarchy.
+Algorithm 1 describes LoadBalance(), a function run by
+CPU c to balance domain sd. Each CPU runs its own in-
+stance of the load balancer at regular intervals. At each tick,
+the balancer traverses the domain hierarchy upwards and calls
+LoadBalance() to balance work between groups within pro-
+gressively larger domains. 3 At any point in time, exactly one
+CPU is responsible to balance the load in a particular domain
+(Line 36). To avoid moving small tasks between multiple
+CPUs, the balancer only moves tasks if the imbalance between
+two groups is larger than a configurable threshold (Line 38).
+To further reduce the balancing cost, a CPU only moves tasks
+from the busiest CPU of the busiest group (Line 43). De-
+pending on workload characteristics, the algorithm selects
+a ‘migration type’ (Line 44). These types define the rules
+to determine the busiest CPU, the imbalance between two
+groups, and the load generated by each task. We describe how
+each decision is made as needed later in the section.
+While we attempt to capture the full complexity of the per-
+tick algorithm, we forego modeling some optional features
+such as task pinning and priorities. We do not model cache-
+aware heuristics. Even with these simplifications, we find our
+model to be expressive enough to discover algorithmic bugs.
+6.1
+Model
+At 10,000+ lines of code [17], the load balancer includes
+several complex heuristics that interact with each other at
+each scheduling tick to quantify a measure of imbalance, and
+3Domains can have their own balancing interval that can change dy-
+namically. Interval calculations are nuanced, and we omit these details for
+simplicity.
+8
+
+Algorithm 1 Linux CFS Load Balancing Algorithm
+1: procedure MIGRATIONTYPE(busiest,local,idle)
+2:
+if local.HasSpare() then
+3:
+if busiest.Overloaded() then
+4:
+has_cap ← local.capacity > local.util
+5:
+if !idle or has_cap then
+6:
+return MIGRATE_UTIL
+7:
+else
+8:
+return MIGRATE_TASK
+9:
+···
+10: end procedure
+11: procedure BUSIESTCPU(CPUs,m_type)
+12:
+key1 ← lambda c : c.util_avg
+13:
+key2 ← lambda c : c.nr_running
+14:
+moreThanOne ← Filter(CPUs,key2)
+▷ added in
+v5.7
+15:
+switch m_type do
+16:
+case MIGRATE_UTIL
+17:
+return argmax(moreThanOne, key1)
+18:
+case MIGRATE_TASK
+19:
+return argmax(CPUs, key2)
+20:
+case ···
+21: end procedure
+22: procedure CALCIMB(busiest,local,m_type)
+23:
+switch m_type do
+24:
+case MIGRATE_UTIL
+25:
+return local.capacity−local.util
+26:
+case MIGRATE_TASK
+27:
+if busiest.Overloaded() then
+28:
+return 1
+29:
+else
+30:
+t1 ← busiest.nr_idle
+31:
+t2 ← local.nr_idle
+32:
+return max(0,(t1 −t2)/2)
+33:
+case ···
+34: end procedure
+35: procedure LOADBALANCE(c,sd)
+36:
+if !IsResponsible(c,sd) then
+37:
+return
+38:
+if !ConsiderableImb(c.idle,sd) then
+39:
+return
+40:
+dst_g ← sd.FindGroup(c)
+41:
+if !sd.GroupAboveAvg(dst_g) then
+42:
+return
+43:
+src_g ← sd.FindBusiestGroup()
+44:
+m_type ← MigrationType(src_g,dst_g,c.idle)
+45:
+src_c ← BusiestCPU(src_g.CPUs,m_type)
+46:
+imb ← CalcImb(src_g,dst_g,m_type)
+47:
+DetachTasks(src_c,c,m_type,imb)
+▷ moves tasks
+based on migration type
+48: end procedure
+decide on how to balance it if at all. While we attempt to
+capture the full complexity of the per-tick algorithm, to keep
+analysis tractable, we make several simplifications. We forego
+modeling some optional features such as task pinning and
+priorities. We also do not model cache-aware heuristics. As
+a result, our model is sound, but not complete. Nevertheless,
+our model captures a large subset of possible Linux behaviors
+and detected several bugs.
+In general, CPUs can load balance asynchronously upon be-
+coming idle. However, we choose to only model the per-tick
+behavior to keep our model tractable. Timesteps are repre-
+sented as an integer vector of fixed size. Each CPU attempts to
+balance every domain that it belongs to at fixed, regular inter-
+vals between successive timesteps. The domains are balanced
+in the order of their level in the domain hierarchy.
+A task is represented as a collection of real variables to
+denote properties such as the weighted moving average of the
+time it was runnable (i.e., runnable average) and its running
+time (i.e., utilization average). Variables representing the prop-
+erties of tasks are defined for each timestep. We do not model
+asynchronous load balancing due to a CPU becoming idle,
+and assume tasks are runnable through all modeled timesteps.
+Nevertheless, the solver is free to choose any initial values.
+For instance, it can pick any initial value of the utilization
+average and the runnable average of tasks as long as the latter
+is larger. Since the choice of initial values is unconstrained,
+this model is akin to simulating a few microseconds start-
+ing from any reachable state of the system, including those
+caused by tasks alternating arbitrarily between runnable and
+non-runnable.
+The domain hierarchy, the number of tasks, and the num-
+ber of timesteps are configurable parameters. Real valued
+variables representing CPUs, groups, and domains are also
+defined for each timestep. For instance, the utilization average
+of each CPU tracks the sum of utilization averages of each
+task in its runqueue (i.e., runnable tasks assigned to that CPU).
+Similarly, a group’s utilization is just the sum of the utilization
+averages of CPUs contained in it. A CPU × Task boolean
+matrix encodes which task is runnin on which CPU at each
+timestep. Finally, we add constraints that connect metrics and
+the schedule at successive timesteps, closely following the
+actual code. As an example, the following constraints rep-
+resent IsResponsible(c, sd) (line 36), the function that
+determines if a CPU c is responsible to balance domain sd at
+timestep t:
+first_idle(t,c,sd)∨first_cpu(t,c,sd)
+9
+
+where,
+first_idle(c,sd) := c.nr_running[t −1] = 0
+∧
+(∀c′ ∈ sd,c′.index< c.index =⇒
+c′.nr_running[t −1] > 0)
+first_cpu(c,sd) := c.index= 0
+∧
+(∀c′ ∈ sd,c′.nr_running[t −1] > 0)
+c.nr_running[t −1] is the number of tasks running in CPU c
+at timestep t −1 and c.index is a statically assigned index to
+each CPU. This constraint is applied to every (CPU, domain)
+pair to find the responsible CPU at each timestep.
+6.2
+Queries and Verification
+For this model, we focus on the Q1 class of questions. In
+particular, we formulate a single query to verify whether the
+algorithm is work conservating property of the algorithm.
+First we set the number of runnable tasks to be greater than
+the number of CPUs and ask whether at the end of T timesteps,
+any CPU is idle: ∃c ∈ CPUs,
+c.nr_running[T] == 0
+For a work conserving scheduler, the answer should be
+“no”, no matter how tasks are distributed initially. For small
+T, imbalance is acceptable and sometimes intended by the
+scheduler to avoid excessive task movement due to sudden
+changes in workload. We want to detect workloads where it
+never balances, no matter how long load is imbalanced. We
+run our model with up to 4 CPUs, 7 tasks, and 6 timesteps.
+All the bugs discovered with this setup were also found with
+just 3 timesteps.
+6.2.1
+Bug 1: Busiest CPU with a single task
+As described earlier, there are multiple migration types
+that are determined based on the state of the system.
+MIGRATE_UTIL is chosen when the busiest group is over-
+loaded, and the current group has spare capacity (Line 6).
+Such a decision can be made in the scenario shown in Fig-
+ure 8 when the utilization average of all tasks in Group 1
+is high (e.g. their sum is greater than 90% of the group’s
+capacity).
+Choosing MIGRATE_UTIL implies that the algorithm’s goal
+is to balance the utilization average amongst groups by steal-
+ing tasks from the busiest CPU. The CPU with the highest
+utilization average in the busiest group is determined to be
+the busiest (Line 17). A CPU’s utilization average is just the
+sum of utilization averages of the tasks in its runqueue. The
+utilization average of a task is defined as the weighted moving
+average of its running time in the past.
+In our example, CPU2 can be the busiest CPU, even though
+it has a single task, if that task has a higher utilization average
+than the sum of the utilization averages of tasks on CPU1.
+However CPU3 steal work from CPU2 since it only has a
+single runnable task. This constraint helps avoid bouncing
+tasks between idle CPUs, but causes the scheduler to not be
+work conserving. One way to fix this ‘bug’ is to define the
+busiest CPU to be the one with the highest utilization average
+CPU1
+CPU2
+CPU3
+CPU4
+Group 1
+Group 2
+Bug 2
+Bug 1
+Figure 8: Example state to showcase the bugs. Except CPU3,
+each CPU’s runqueue has tasks represented as rectangles. CPU3
+is idle. Under different settings, CPU3 may be unable to steal
+any task from Group 1.
+that has more than one runnable task (line 14). After finding
+the bug using this model, we realized that it was also identified
+by the Linux community and fixed in Linux v5.7 [16] .
+6.2.2
+Bug 2: Imbalance with idle CPUs
+With the previous bug fixed, we reran the work conservation
+query, identifying another bug. This bug happens when the
+migration type ‘MIGRATE_TASK’ is selected. Task migration
+is chosen when the the current group has spare capacity, but
+the conditions for the type ‘MIGRATE_UTIL‘ are not satisfied.
+In particular, it can be selected when no group is overloaded.
+Here, the algorithm’s goal is to balance the number of tasks
+between groups. The busiest CPU in the busiest group is
+decided based on the number of tasks in the runqueue of the
+CPU. Imbalance is defined as half of the difference between
+the number of idle CPUs in the busiest and the current group
+(Line 32). At first glance, this seems reasonable and should
+even out the number of idle CPUs. However, Linux can only
+perform integer division, leading to calculating the imbalance
+to be zero when the difference in idle CPUs is 1. This behavior
+is captured by the scenario in Figure 8 when group 1 is not
+overloaded. CPU1 is deemed as the busiest CPU with 3 tasks,
+but CPU3 is still unable to steal any of them.
+It is worth noting that both these issues intricately depend
+on the workload characteristics generated by our solver. Slight
+deviation in these characteristics can lead to a completely dif-
+ferent outcome. For instance, a different migration type may
+apply as the number of tasks or the blocking pattern of ex-
+isting tasks evolves. A different measure of imbalance may
+be able to push tasks to idle CPUs and balance load more
+evenly in general. Finely controlling the workload charac-
+teristics in synthetic benchmarks that highlight these bugs
+is difficult. This makes it hard for fuzzers and other tests to
+detect them. However, it does not preclude them from appear-
+ing in the real world. Our framework is not limited by this,
+and can freely search the workload space to generate intricate
+violating traces.
+7
+Case study: TCP Synchronization in Ring
+Allreduce Training
+The above examples look at well-known schedulers. Perhaps
+the most compelling use-case for our methodology is to un-
+derstand less well-studied systems. As an example, we consid-
+ered a recent paper [37] which examines complex emergent
+10
+
+0
+10
+20
+30
+40
+50
+60
+70
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+Link Utilization (Gbps)
+Time (sec)
+(b) Unfair bandwidth sharing
+0
+10
+20
+30
+40
+50
+60
+70
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+Link Utilization (Gbps)
+Time (sec)
+(a) Fair bandwidth sharing
+J1 Comm. phase
+J2 Comm. phase
+J2 Comm. phase
+J2 Comm. phase
+J1 Comm. phase
+J1 Comm. phase
+J1 Comm. phase
+J2 Comm. phase
+J1
+J1
+J2
+J2
+J1 takes more bandwidth 
+because its congestion control
+algorithm is more aggressive
+Iteration 1
+Iteration 2
+Iteration 3
+Jobs share bandwidth equally
+J1 starts its comm. phase
+for the next iteration before J2
+Unfairness keeps sliding the 
+comm. phase of J2 to the right
+Jobs share bandwidth equally
+Jobs share bandwidth equally
+The sliding continues until there is minimal overlap between the 
+comm. phases of jobs. Subsequent iterations maintain this pattern.
+Jobs share bandwidth equally
+Iteration 4
+J2 Comm. phase
+J1 Comm. phase
+J2 Comm. phase
+J1 Comm. phase
+Iteration 5
+Jobs share bandwidth equally
+Figure 9: Intuition behind why synchronization occurs. Reproduced from [37] with permission.
+Link
+(a)
+Link
+Link
+(b)
+Link
+Link
+Link
+(c)
+Link
+Link
+Link
+(d)
+Figure 10: Configurations of ring reduce jobs sharing links
+behavior in a cluster running multiple distributed neural net-
+work training jobs. Neural network training processes data in
+batches, computing the gradient of the error function for each
+batch to adjust the weights before processing the next batch.
+The computation-communication pattern for each batch is
+similar, making the process predictable and providing oppor-
+tunities for better scheduling [35,43,46].
+A training job consists of n servers connected logically in
+a ring. During training, servers have phases involving intense
+computation followed by communication along the ring. A
+datacenter may run multiple jobs sharing the same physical
+network. As a result, some of the logical links may be shared
+with other training jobs as shown in figure 10. One can maxi-
+mize network utilization by scheduling one job’s computation
+while the other communicates. This way, each job gets the
+full available bandwidth to itself when it is communicating.
+The reference paper [37] observes that jobs spontaneously
+synchronize to this desired schedule due to TCP unfairness.
+In this section, we verify this claim under complex settings
+and find it to be true.
+Preliminaries: We provide a brief overview of data-parallel
+distributed training using the ring-all-reduce method, omitting
+details that are not necessary to understand the scheduling
+problem. Each round of training processes one batch and
+consists of three phases: backpropagation, sum and broadcast.
+The first is purely computation and the rest are primarily
+communication. The model is divided into n pieces. Servers
+transmit data as soon as it is available. When summing, the
+% Completion
+Time
+Backpropagation
+Sum
+Broadcast
+Start of
+round i
+Start of
+round i+1
+100%
+0%
+A
+B
+C
+D
+E
+F
+Figure 11: Example of one server’s view of ring reduce during
+one round. Point A shows that server can send the first 1/nth
+chunk of data as soon as it has computed the gradients for
+that segment of the network. Till point B, it is waiting for the
+previous server to send data. Between B to C, it is sending at
+link rate. Between C and D, it is sharing the link with another
+job, due to which it sends at a lower rate. Once sum finishes,
+broadcast can start. Again, the first chunk can be sent without
+waiting for data from the previous server (point E). After this,
+it sends at less than line rate because it is waiting on data from
+the previous server, which it forwards as soon as possible.
+first piece is available as soon as backpropagation is done
+processing that part of the neural network. After this the
+server must wait for data from the preceeding server before
+forwarding it. The same holds for the broadcast step, except
+that the first piece is immediately available.
+Figure 9 is an experimental result showing the intuition
+behind why synchronization occurs. Suppose job 1 has been
+transmitting on the shared link at full link capacity, and a new
+job 2 starts transmitting over the same link. Both jobs will
+detect the congestion and adjust their sending rates so that
+both transmit at half of the full link capacity. However, it takes
+time for the transmission rates to reach this new equilibrium
+and many congestion control algorithms never reach it. In
+the meantime, job 1 gets more bandwidth than flow 2. As a
+result, the job corresponding to flow 1 will finish its sum and
+broadcast steps earlier than it otherwise would have, which
+will make it start even sooner for the next batch. This will
+continue until the transmissions on the shared link are fully
+separated in time – which is what we ideally want.
+This argument works for when there are just two rings.
+But what if there are more? One shared link could be forcing
+the job to slowly its schedule in one direction, while another
+shared link has an opposite effect. There could even be cycles.
+Figure 10 illustrates representative examples of the topologies
+for which we verify synchronization.
+11
+
+Figure 11 shows some model variables for a single server
+at each timestep. These include the percentage of backprop-
+agation, sum, and broadcast finished. It also includes an in-
+teger indicating the round number. A round is defined as the
+processing of a batch. We keep the state transition function
+between timesteps simple. For example, the number of flows
+transmitting on any link is fixed between two timesteps. How-
+ever, the time gap between timesteps is variable, allowing the
+solver to add a timestep whenever this changes. This way, we
+do not discretize time.
+In several places, we overapproximate. For example, we
+do not model the congestion control algorithm explicitly, but
+instead, constrain that (1) the link is fully utilized and (2)
+whichever job starts transmitting first gets more bandwidth.
+This models a wide range of congestion control behaviors
+while keeping the reasoning simple for both computers and
+humans. Additionally, we allow a server to send the first 1/nth
+chunk of "sum" and "broadcast" data as soon as it is done
+computing, without waiting for data from its predecessor. The
+amount of data it can send depends on the neural network
+architecture and system structure. Hence we let the solver
+arbitrarily pick the how much data is available.
+To verify synchronization, we constrain that the communi-
+cation of each job can fit inside the computation time of its
+neighbors, which is the only case in which the results in our
+reference paper hold. We then ask the solver to find a case
+where overlap in communication increases (or remains con-
+stant). If the solver cannot find such a case, we have proved
+that synchronization will always occur. It is important to note
+that the solver only proves this for a small and finite num-
+ber of events, and the initial state is unconstrained. However,
+this also proves that a larger sequence of events cannot exist
+where communication overlap increases, as if such a sequence
+were to exist, there would be a short sub-sequence where it
+increases, which we have proved is impossible.
+8
+Related Work
+Automated Verification. A long history of automated ver-
+ification of schedulers can be traced back to mechanized
+proofs of real-time scheduling algorithms using proof assis-
+tants like Coq. These include proofs for specific algorithms
+such as Priority Inheritance [45], implementations in systems
+like CertiKOS [21], and frameworks to design them [8,28].
+These focus on qualitative correctness properties such as en-
+suring no task misses a deadline. In contrast, we focus on
+verifying quantitative performance properties and optimality.
+In a similar vein, model checking has achieved incredible
+success in formally verifying large scale systems including
+network stacks [33,36,41], language runtimes [23,25], and
+databases [20]. Large parts of the Linux subsystems have
+also been verified [19, 42]. Our approach has been directly
+inspired from model checking principles of building abstract
+models with specifications to verify it. However, existing
+model checking tools are restricted to identifying correctness
+bugs. We provide a framework to apply these principles to
+identify subtle performance bugs in schedulers.
+The verification work closest to our approach is CCAC
+[2]. CCAC models congestion control algorithms to formally
+verify their performance or generate violating traces. While
+CCAC provides a specialized tool for modeling congestion
+control algorithms, our framework is general and is applicable
+to a larger class of scheduling algorithms.
+Tracing and Benchmarking. All major operating systems
+include specialized tracers to analyze kernel performance.
+ftrace and perf_events are tracers built into Linux. They
+provide hooks to instrument various subsystems including
+the scheduler, and generate performance statistics. More re-
+cently, eBPF [39] and SystemTap [38] extend this function-
+ality by allowing custom user code to run inside the kernel
+to detect specific behavior. These tracers are used in conjunc-
+tion with standard benchmark suites to detect performance
+issues. Hackbench [13] benchmarks the average latency for
+communication between tasks, and has become the defacto
+standard to test improvements on the Linux load balancer.
+While tracers and benchmarks are often useful in detecting
+and explaining unexpected performance behavior, they pro-
+vide no guarantees on absence of bugs. Our framework, on
+the other hand, is an attempt to provide formal guarantees on
+scheduler performance.
+9
+Limitations and Conclusion
+While we believe our methodology to be generally applicable
+to all scheduling algorithms, it is not a panacea for detecting
+all problems in an algorithm. Our methodology is dependent
+on the user’s ingenuity in creating tractable models and formu-
+lating relevant queries. For example, our model of the Linux
+CFS load balancer included several simplifying assumptions
+(e.g., ignoring asynchronous load balancing steps), yet the
+model was useful enough to detect practical bugs. Further,
+our modeling approach can only find performance issues for
+which a user formulates a metric and query; deeper issues
+which are not exposed by user–generated queries are not de-
+tected by our approach. Solver performance limitations mean
+that we can only perform bounded model checking, so our
+models won’t detect problems that happen only in the pres-
+ence of large number of tasks or events. However, we find
+that in many scenarios, it’s possible to generalize our results
+by creating models that focus on critical system components
+and formulating reasonable queries.
+In conclusion, we have demonstrated that formal meth-
+ods can provide a deeper and more rigorous understanding of
+scheduling heuristics used in practice. Further, since we verify
+the specification of these heuristics, and not the code, veri-
+fication effort is minimal. The most time consuming part of
+our method is posing the right queries and interpreting coun-
+terexamples in context. We invite the community to adopt
+quantitaive verification as a part of their worklflow when de-
+veloping scheduling heuristics.
+12
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+

89AyT4oBgHgl3EQfdPcr/content/tmp_files/2301.00297v1.pdf.txt

@@ -0,0 +1,475 @@
+ 
+ 
+Deterministic and Nondeterministic Particle Motion with Interaction 
+Mechanisms  
+Cameron McNamee1,2*, Renee Reijo Pera1 
+1McLaugling Research Institute, Great Falls, MT, USA 
+2Department of Mathematics, California Institute of Technology, Pasadena, CA, USA 
+* Correspondence:  
+cmcnamee@caltech.edu 
+Keywords: particle modeling, migration, interaction mechanisms, deterministic modeling, 
+Beauchemin modeling 
+Abstract 
+Studying systems where many individual bodies in motion interact with one another is a complex and 
+interesting area. Simple mechanisms that may be determined for biological, chemical, or physical 
+reasons can lead to astonishingly complex results that require a further understanding of the moving 
+bodies. With the increasing interaction between computation and various scientific areas, it has 
+become more useful, feasible, and important to create models for these systems. Here, we present two 
+families of models, deterministic and nondeterministic, along with three distinct and realistic 
+interaction mechanisms. These are combined in a unique way to provide the groundwork for particle 
+system models across multiple disciplines. This work has applications that range from biology to 
+chemistry and physics. In addition to describing the motivations and math behind all the models, 
+software is provided that allows researchers to quickly adjust and implement what is described here. 
+1 
+Introduction 
+This work was inspired by consideration of quantum biology (QB) and cell migration and tissue 
+formation as previously described (Vecheck et al., 2022) with the goal of identifying quantitative 
+measures of particle clustering in a scaffold. Identifying such measures involves observing the 
+locations of the particles using image processing and applying these locations to a novel clustering 
+algorithm. The results were some surprising numbers that led to a search for other, more robust 
+methods of cluster analysis and an environment in which to test resulting algorithms. This proved not 
+to be a simple task prompting development of potentially new models to understand how particles 
+may interact under different conditions. From a biological perspective, and in view of the eventual 
+application of this work, this paper explores the coalescing of particles in a space towards one 
+another. It is also important to understand that in the simulation process, the goal is to replicate what 
+is seen in realistic settings. In addition to biology, many systems wherein multiple autonomous 
+particles are driven in a migratory pattern can be seen in physics and chemistry, allowing this work to 
+be expanded to multiple fields (Hansen et al., 2013; E. Courtier et al., 2019). Nanoparticle and ion 
+motion are two distinct areas that this type of computational research can be readily applied and lead 
+to interesting features of such a system being uncovered (Hansen et al., 2013; E. Courtier et al., 
+2019). With the diverse opportunities in mind, this paper seeks to abstract the motion and interaction 
+of individual bodies in such systems to create general models. These general models may then be 
+taken by a researcher and applied with the appropriate parameters to match a particular system, with 
+
+ 
+2 
+the addition of the available software laying the foundation of such a model (Figure 1). This paper is 
+by no means all-encompassing; however, the range of these general models spans many fields with 
+many important implications.  
+2 Methods  
+All software was implemented in MATLAB. Methods are described below as follows: First, different 
+environments in which particles may be generated are discussed. These environments decide what is 
+called the “initial seeding” of the particles. These initial seedings vary in ways that may be more 
+suitable for certain applications. Second, two families of models are explored: deterministic models 
+and nondeterministic models. Differences between these two families of models are described in more 
+detail; however, as the names suggest, the most important difference is that the deterministic model 
+will produce the exact same results given the same initial seedings, while the nondeterministic models 
+have an element of randomness that allows for changes in outcome. The third and last part discussed 
+is the interactions between particles. This is a unique addition to these models and distinguishes this 
+work for specific applications in various fields.  
+2.1 Initial Seeding Environments  
+Initial seeding environments are different environments that serve different purposes and are each 
+suited for their own application. All seedings are centered at the origin.  
+2.1.1 Uniform Seeding 
+Non-random uniform seeding places each particle in a lattice point of a Cartesian coordinate system. 
+This serves as what can be considered an “in vitro” environment, since in most applications, particles 
+are not placed in such an orderly fashion (Peercy and Starz-Gaiano, 2020; Calvillo et al., 2022).   
+2.1.2 Uniform Random Seeding 
+ 
+This seeding generates a designated number of particles within specified dimensions. This is carried 
+out by generating three random numbers within ± the designated range for each particle. These 
+coordinates then determine the placement of each particle, providing uniformly random particles 
+throughout the space with an expected center at the origin.  
+2.1.3 Spherical Random Seeding 
+Like the seeding above, spherical random seeding is a uniform distribution of particles with an 
+expected center at the origin. This differs from uniform random seeding in that the seeding is formed 
+with a given number of particles and a radius. The problem of creating a uniform distribution within 
+a sphere is not trivial; thus, this was accomplished by using polar coordinates.  
+To randomly generate a radius that places particles uniformly, a floating-point number between [0,1] 
+is sampled and the cube root is taken. Then, the maximum radius is multiplied by the resulting value. 
+This allows, for reasons that can be clearly seen from a geometric perspective, for the radius to 
+generate uniform points. Generating the random angles is described in more detail here, along with 
+more extensive reasoning throughout the entire process: (Simon, no date). This random process is 
+done for as many particles as desired, and this determines the initial seeding.  
+2.2 Random Migration Models 
+
+ 
+3 
+This class of models stems from the Beauchemin model for migrating bodies, which has been used 
+previously for biological modeling (Textor, Sinn and de Boer, 2013). All random migration models 
+were taken from Textor et al. (Textor, Sinn and de Boer, 2013).  
+2.2.1 Beauchemin Random Walk 
+To develop a biased migration model for a group of particles, it is helpful to first create a model that 
+has no bias. The Beauchemin random walk model functions under two phases: a pause phase and a 
+free phase. During the pause phase, a particle generates two random angles, 𝜙 and 𝜃, and uses these 
+angles to generate a uniform random direction. The generation of these angles follows the same 
+method mentioned above for uniform sphere generation. After the pause phase, the particle enters the 
+free phase. In this random walk model, every particle has the same, given velocity 𝑣. Using its given 
+direction, a particle travels at the velocity 𝑣 for the entire free phase. The free phase length may be 
+changed, but for most modeling applications, it is 4 times longer than the pause phase. This is 
+arbitrary and may be changed easily.  
+As the name suggests, the Beauchemin random walk method is not biased in any way, so the 
+particles walk uniformly in any direction. This unbiased nature leads to no pattern of particle 
+migration but provides a basis wherein specific factors may be changed. With the unbiased model 
+established, an introduction to different biases is now needed.  
+2.2.2 Simple Phenomenological  
+The Simple Phenomenological model (SP) introduces bias by adding a “drift element” during the 
+pause phase (Textor, Sinn and de Boer, 2013). Instead of the particles remaining stationary during the 
+pause phase, they drift towards some defined point. This causes uniform clumping amongst the 
+particles. In this implementation, the choice was made to make this point the origin. This is of course 
+arbitrary and may be changed to suit whatever environment is most sensible. 
+The drift velocity is defined as the given velocity 𝑣 times the bias variable 𝑝. For 𝑝 = 1, equivalent 
+bias is achieved between random walking and drifting towards the origin as the particles approach the 
+center at the same rate that they walk randomly. Of course, for 𝑝 = 0 there is no bias, so this model 
+would operate the same as the random walk above. 
+The drift vector is defined by using the current 𝑥, 𝑦, and 𝑧 coordinates to calculate the new 
+coordinates. First, 𝑑, the distance from the drift particle, is calculated using the 𝑙� norm 
+𝑑 = �𝑥�
+� + 𝑦�
+� + 𝑧�
+� 
+Then, the drift velocity is used to “shrink” the coordinates towards the origin 
+𝑥����� = 𝑥� ⋅ 𝑑 − 𝑣𝑝
+𝑑
+ 
+𝑦����� = 𝑦� ⋅ 𝑑 − 𝑣𝑝
+𝑑
+ 
+𝑧����� = 𝑧� ⋅ 𝑑 − 𝑣𝑝
+𝑑
+ 
+
+ 
+4 
+In this implementation, there is an added condition that a particle only moves towards the origin if it 
+is farther away than the drift velocity will take it. This prevents the particle from “overshooting” its 
+target.  
+2.2.3 Topotaxis Model  
+The Topotaxis Model (TM) differs from SP by introducing a different form of bias. Instead of 
+moving the particle in the target direction, TM changes the distribution from which the angles are 
+derived. Instead of 𝜙 and 𝜃 being taken from a uniform distribution, they are drawn from a beta 
+distribution. A beta distribution is essentially a truncated normal distribution. The normal distribution 
+cannot be used in this situation because of its infinite tails and the angles are within a strict range, 
+[0, 2𝜋] and [0, 𝜋] respectively. The beta distribution’s shape is determined by two parameters. Here, 
+the two parameters, 𝛼 and 𝛽, and are set equal. Since they are equal, we only refer to 𝛼 from here on. 
+To calculate the shape of the distribution given a single bias variable 𝑝, we need to consider that 
+when 𝑝 = 0, the distribution should be uniform and as 𝑝 approaches 1, the distribution should 
+become more skewed. This is accomplished by decreasing the variance in the distribution of angles 
+while keeping the mean towards the origin with the result that the particles have an increased 
+likelihood of orienting themselves towards the target location. In MATLAB, if 𝑎 =  1, the 
+distribution is defined as uniform and as 𝑎 → ∞, the variance decreases, and a peak centered at the 
+mean value is formed. To calculate 𝑎 given 𝑝, the following is done  
+𝑎 =
+1
+1 − 𝑝 
+From this definition we see that for this model 𝑝 ≠ 1, but we also see lim
+�→� 𝑎 =  ∞, as desired. Thus, 
+for 𝑝 = 0, there is a uniform, or unbiased, distribution, and as 𝑝 → 1, there is an increased bias 
+towards the targeted location.  
+2.3 Deterministic Migration Models 
+The deterministic migration models are a family of migration models that feature no random 
+elements. Within this class of models, there are two different groups: uniform clustering and local 
+clustering. The uniform clustering algorithms behave on a global scale, causing all particles to 
+migrate in a wholistic sense. The local clustering algorithms organize themselves at a much smaller 
+scale.  
+2.3.1 Uniform Clustering, the Naïve Approach 
+The first and simplest way of clustering particles within space is to “shrink” the space they are in. This 
+first requires a clustering rate, 𝑟 ∈ [0,1]. This method is as follows  
+𝑈�𝑐�, 𝑐�, 𝑐�� = �𝑐� ⋅ (1 − 𝑟), 𝑐� ⋅ (1 − 𝑟), 𝑐� ⋅ (1 − 𝑟)�; ∀𝑐 ∈ 𝐶 (Eq. 1) 
+Where C is the set of all particles, c is an individual particle, and 𝑐�, 𝑐� and 𝑐� are the x-, y-, and z-
+coordinates of c, respectively. This function causes all particles to “march” inward, maintaining their 
+relative distances to one another. This does produce a set where each particle is relatively closer to all 
+other particles but does so in a way that is not representative of typical physical settings. 
+2.3.2 Nearest Neighbor Clustering  
+
+ 
+5 
+In nearest neighbor clustering, particles are not clustered into a single lump, but rather orient to their 
+neighbors and move toward them. This approach is much more nuanced compared to Uniform 
+Clustering and reveals samples that are much more reminiscent of certain applications. The method 
+behind this approach assumes that particles can only sense their n nearest neighbors (NN’s), and so 
+they only approach those particles. This method is conducted by first finding the geometric center 
+(GC) described by the n NN’s and then moving the particle towards that at a clustering rate, 𝑟 ∈ [0,1]. 
+Using the same notation as above, we have the set of distances from all particles to a given particle c.  
+𝑁� =  min
+�
+{ 𝑑�: 𝑑� = ||𝑠� − 𝑐||, ∀s� ∈ 𝐶} (Eq. 2) 
+Here, �|𝑠� − 𝑐|� is the 𝑙� normed distance between the particle 𝑠� and the particle c and min
+� (𝑆) 
+represents taking the n smallest elements from the set S. With the NNs now identified, the GC (𝑔�) is 
+calculated for a given 𝑐. This center is calculated without including c itself.  
+𝑔�� = 
+∑
+𝑠�
+�∈��
+𝑛
+ 
+𝑔�� = 
+∑
+𝑠�
+�∈��
+𝑛
+ 
+𝑔�� = 
+∑
+𝑠�
+�∈��
+𝑛
+ 
+𝑔� = �𝑔��, 𝑔��, 𝑔��� (Eq. 3) 
+ 
+In other words, the unweighted mean of x-, y-, and z-coordinates of each NN. This gives us the GC 
+described by the n NN’s. With this, we now have a location to move c towards.  
+ �𝑐� , 𝑐�, 𝑐�� ↦ (𝑐�(1 − 𝑟) + 𝑔�𝑟, 𝑐�(1 − 𝑟) + 𝑔�𝑟, 𝑐�(1 − 𝑟) + 𝑔�𝑟)  (Eq. 4) 
+For a given clustering, this calculation must be carried out for each particle, and it is done in its 
+entirety before any particle moves (i.e., no particle’s location changes until after every particle’s new 
+location has been calculated. Then, all particles move to their new locations at once). The result is 
+clustering in small groups that after enough time steps, form individual groups. 
+2.3.3 Threshold Clustering  
+Threshold (TH) clustering differs from nearest neighbor clustering in that, instead of searching for a 
+certain number of neighbors, the particles have an assumed maximum influence distance. This 
+maximum influence distances, a threshold, is analogous to a particle only being able to sense other 
+particles for a small radius around itself. This method is implemented by collecting the set of particles 
+within the maximum sensing distance, t, calculating the GC described by these particles, and then 
+moving towards this center at the clustering rate r.  
+𝑇� = {𝑠 ∈ 𝐶: �|𝑠 − 𝑐|� <  𝑡} (Eq. 5) 
+
+ 
+6 
+Then, as above, the GC of this set is calculated using the unweighted mean (Eq. 3). This gives 𝑔�, 
+which we then use to calculate the new particle’s position using Eq. 4. This, as with NN, is done for 
+every particle before any particle changes its location. This mode produces several small groups 
+throughout the space whose group size changes with 𝑡. 
+2.3.4 Weighted Clustering 
+This method is the next logical step for simulating clustering. Instead of taking the unweighted mean 
+to calculate the GC, why not weigh the locations based on distance? Weighted clustering is done by 
+first calculating 𝐷�, the set of all distances from each particle to c. Next, the weighted geometric 
+center (WGC) for c, 𝑤�, is calculated as follows: 
+𝑤�� = 
+∑
+𝑠��
+𝑑�
+�∈��
+∑
+� 1
+𝑑��
+�∈��
+ 
+𝑤�� = 
+∑
+𝑠��
+𝑑�
+�∈��
+∑
+� 1
+𝑑��
+�∈��
+ 
+𝑤�� = 
+∑
+𝑠��
+𝑑�
+�∈��
+∑
+� 1
+𝑑��
+�∈��
+ 
+𝑤� = (𝑤��, 𝑤��, 𝑤��) (Eq. 6) 
+Where 𝐼� is the indexing set corresponding to 𝐷�. The reason for using the reciprocals of the weights 
+follows from the idea that as a particle get further, it should affect c less. The new coordinates for c 
+are calculated according to Eq. 4. This is a global clustering on particles, like uniform clustering. This 
+causes no small groups to small, but rather the entire population of particles congregate. This 
+congregation, however, does not appear the same nor does it occur at the same rate at uniform 
+clustering.  
+2.3.5 Weighted Nearest Neighbor Clustering  
+In accordance with the previous modes, this weighted nearest neighbor (WNN) clustering is the next 
+intuitive step. This method is, as the name suggests, a combination of NN clustering and weighted 
+clustering. This combines the logic of only looking towards the closest particles, while also weighing 
+the particles according to their distance. The set of NN’s is calculated according to Eq. 2. 
+𝐼�� = {𝑛 ∈ 𝐼: 𝑠� ∈ 𝑁�} 
+The WGC is calculated according to Eq. 6 with 𝐼�� used in place of 𝐼�. With this WGC, particle c is 
+mapped using Eq. 4. This mode appears very similar to the original NN clustering but tends to form 
+clusters that are skewed in slightly different locations. 
+2.3.6 Weighted Threshold Clustering  
+
+ 
+7 
+Weighted threshold (WT) clustering follows the same intuition. 𝑇� is calculated according to Eq. 5. 
+This defines the limited distance indexing set: 
+𝐼�� = {𝑛 ∈ 𝐼: 𝑠� ∈ 𝑇�}  
+Using Eq. 6 with 𝐼�� in place of 𝐼� gives WGC about the particles within the threshold. Eq. 4 is used 
+to map c. As with WNN clustering, WT clustering appears quite similar to the TH clustering on small 
+scales.  
+2.4 Interaction Mechanics  
+With the addition of interaction, defined here as how two particles behave when they physically hit 
+each other in the migration process, there are many different fields that may provide insights. Some 
+outcomes of physical interaction are cessation of movement, repolarization, or adhesion, to name just 
+a few examples of how particles may interact with one another.  
+2.4.1 Cessation of Movement 
+The simplest self-interaction is cessation of movement, meaning simply that if two particles are 
+within a designated distance of one another, they both stop moving. In simulation, implementation of 
+this outcome simply checks if a particle is close to any other particle. If it is, it stays in the same spot. 
+Otherwise, it keeps moving as it was intended.  
+2.4.2 Repolarization  
+Repolarization means that if two particles hit each other, they both turn around in the opposite 
+direction, much like an elastic collision. Implementation is carried out by taking the two angles that 
+defined the direction that a given particle was traveling, and negating both.  
+2.4.3 Adhesion  
+Adhesion is the clustering or clumping of particles to form a single body that moves as one. This is 
+accomplished by assigning a clump to every particle. If a particle is within a given radius of other 
+particles, it gets put into that clump. Additionally, a clump is a collection of not just the particles that 
+are touching a given particle, but any that are touching its neighbors, the neighbor’s neighbors, and 
+so on. The clumps partition the set of particles and create moving bodies that can obtain more 
+particles. No limit to the size of clumps was designated, however this addition is trivial.  
+3 Discussion  
+The purpose of this discussion is to explore the potential applications of the models described above 
+and how the simulations generated in the linked code can prove useful to other researchers. Many 
+particle or particle-like systems are seen across all scientific fields including physics, chemistry, and 
+biology. Proper implementation of these models requires a good understanding of the system at hand 
+and how the particles behave with one another. An understanding of the system is necessary to select 
+the proper migration model.  
+In highly chaotic systems, nondeterministic models are more suited as randomness (pseudo 
+randomness rather) is a key feature of such a system. In systems where the environment itself does 
+not drive motion of particles themselves, the deterministic models are better approximations. 
+
+ 
+8 
+Implementing different interaction dynamics in the same setup leads to very different results. Thus, it 
+is also very important to understand how individual particles behave upon collision. With this 
+necessity understood and considered, let us explore specific use-cases.  
+The above models offer insight in biological settings. Cell migration is an important component of 
+many cell types, and thus estimating that motion can provide information about cellular mechanisms 
+(Davis et al., 2015; Gopinathan and Gov, 2019; Metzcar et al., 2019; Norden and Lecaudey, 2019; 
+Alert and Trepat, 2020; Guberman, Sherief and Regan, 2020; Peercy and Starz-Gaiano, 2020; 
+SenGupta, Parent and Bear, 2021). In different settings, cellular motion may be estimated by 
+different models. Chemotaxis, which is motion based on the sensing of chemical gradients, is the 
+driving force of many migration patterns at the cellular level (Hu et al., 2010; Gopinathan and Gov, 
+2019; Norden and Lecaudey, 2019; Alert and Trepat, 2020; SenGupta, Parent and Bear, 2021). This 
+mode of migration is akin to the TM as the chemical gradient introduces bias as to what direction the 
+cells will migrate. The stronger the cells’ reaction to the chemical gradient, or the stronger the 
+chemical gradient itself, the greater this bias. Migratory cells have been noted to have a degree of 
+continuous bias, meaning they can have varying levels of directional influence due to chemotaxis, not 
+simply a binary model of directed motion (Hu et al., 2010; Alert and Trepat, 2020). This, along with 
+the mechanistic understanding of chemotaxis lends this type of cellular motion to the Topotaxis 
+model.  
+In addition to chemical sensitivity, further support of the nondeterministic models comes from the 
+observations that cells perform motion that follow the pause phase-free phase mechanism that 
+dictates these models (Gopinathan and Gov, 2019; Alert and Trepat, 2020; Peercy and Starz-Gaiano, 
+2020). This comes from the discontinuation of motion in favor of direction recalibration noted in cell 
+motion, which strongly motivates the use of Beauchemin random walks in favor of other forms of 
+random walks (Textor, Sinn and de Boer, 2013; Gopinathan and Gov, 2019; Alert and Trepat, 2020; 
+Peercy and Starz-Gaiano, 2020). Other than chemical gradients, the mechanical force induced by the 
+microenvironment about the cell can influence motion (Peercy and Starz-Gaiano, 2020; SenGupta, 
+Parent and Bear, 2021). This is the defining feature of the SP model as there is an external force on 
+the cells driving them in a biased direction, much like fluid flowing. SP can be used to describe 
+mechanistic motion of cells during embryonic development and within the neural crest (Giniūnaitė et 
+al., 2020; Peercy and Starz-Gaiano, 2020). Contexts such as these are strong candidates for applying 
+nondeterministic models outlined in this paper in the field on biology.  
+Deterministic models also have niches in biological settings. Tumor growth has many similarities to 
+TH, as does cancer cell migration (Metzcar et al., 2019). This is an especially important area for 
+computational research, as hypotheses must be tested efficiently and accurately. Modeling can also 
+be used to determine metabolism of cancer and stem cells, giving useful information that can be 
+verified experimentally (Metzcar et al., 2019). This is a great example of how computational 
+modeling can uncover additional information and provide meaningful insight in a timely fashion.   
+Cellular mechanics also offer interesting interaction dynamics. All three modes of interaction take 
+place across the field of biology. Adhesion, as described above, occurs in migrating cell groups, 
+cancer metastasis, and embryogenesis (Gopinathan and Gov, 2019; Norden and Lecaudey, 2019; 
+Alert and Trepat, 2020; Guberman, Sherief and Regan, 2020; Peercy and Starz-Gaiano, 2020). This 
+typically occurs in homogenous cell groupings and is seen throughout the immune system (Davis et 
+al., 2015; Norden and Lecaudey, 2019; Alert and Trepat, 2020). Cessation of movement can be 
+observed in tissue growth for both epithelial cells and fibroblast cells (Abercrombie, 1961; 
+Guberman, Sherief and Regan, 2020). In fact, this behavior is the original mechanism that led to the 
+
+ 
+9 
+term “contact inhibition of locomotion,” better known as CIL (Abercrombie, 1961). Notably, CIL 
+also can be described in different contexts as a repolarization effect, meaning within biology any of 
+the above interaction mechanisms may be observed (Davis et al., 2015; Roycroft et al., 2018; Alert 
+and Trepat, 2020; Guberman, Sherief and Regan, 2020).  
+Moving to a different field, in chemistry and physics nanoparticles display many of the features of 
+both clustering and motion. In certain settings, nanoparticles can behave in random walk migration 
+which can be biased due to outside forces to appear as any of the nondeterministic models outlined 
+above (Hansen et al., 2013). In addition, a mode of random potential driven motion along with 
+particle adhesion described as Oswald Ripening can be appropriately described by the TM with 
+particle adhesion (Hansen et al., 2013). Ion motion in solar cells holds many of the key mechanistic 
+contexts outlined above (E. Courtier et al., 2019). Additionally, the dynamic change of chemical 
+gradients, useful for chemistry and biological contexts, can by approximated by SP (Hu et al., 2010).  
+The novelty of the work contained within this paper comes from the diverse applications readily 
+available. It is the addition of particle interactions that leads to results with astounding complexity 
+that are reminiscent of what is captured in scientific settings, while keeping the simulations in a 
+tractable form. While this work was inspired from a biological setting, it is of upmost importance to 
+recognize the applications of such work to other fields, for it is the interaction between the sciences 
+that leads to strides in innovation. The union between computer science and physical science is 
+something that has never been seen before, and we as a scientific community have but scratched the 
+surface.  
+With these diverse and important applications available from the relatively simple models outlined 
+above, it is important to note the limitations of not just these models themselves, but also the current 
+state of computational physical science. First, we can only model what we understand, and this 
+knowledge is limited not by our ability to create models but by our methods in the physical sciences. 
+To aptly describe a complex system, a complex understanding of such system must be had first. 
+However, as far as biology, much of chemistry, and many places of physics goes, we do not have 
+such a deep understanding (SenGupta, Parent and Bear, 2021). The power of predictive models is the 
+ability to extrapolate our current understanding to generate nuanced hypotheses that may be tested in 
+the future. A major pitfall of this ability is, however, that false initial assumptions may lead to a 
+fruitless path.  
+The goal and purpose of this newfound computational power, a power that is only growing as time 
+passes, is to approximate to the best of our ability the real world. This allows us to uncover, with 
+unparalleled speed, extremely important new aspects of physical systems that were previously 
+obscure and serve to inspire experimentalists. Continuing to cautiously push forward with this 
+technology will lead to profound discoveries across academia. 
+The authors declare that the research was conducted in the absence of any commercial or financial 
+relationships that could be construed as a potential conflict of interest. 
+CM contributed to the experimental design, analysis of data and interpretation of the results.  CM and 
+RRP prepared the manuscript and edited the final manuscript for publication. 
+This work was supported by funding from the National Institute of Health (1R01HD096026). 
+Abercrombie, M. (1961) ‘The bases of the locomotory behaviour of fibroblasts’, Experimental Cell 
+Research, 8, pp. 188–198. Available at: https://doi.org/10.1016/0014-4827(61)90348-2. 
+
+ 
+10 
+Alert, R. and Trepat, X. (2020) ‘Physical Models of Collective Cell Migration’, Annual Review of 
+Condensed Matter Physics, 11(1), pp. 77–101. Available at: https://doi.org/10.1146/annurev-
+conmatphys-031218-013516. 
+Calvillo, L. et al. (2022) ‘Quantum Biology Research Meets Pathophysiology and Therapeutic 
+Mechanisms: A Biomedical Perspective’, Quantum Reports, 4(2), pp. 148–172. Available at: 
+https://doi.org/10.3390/quantum4020011. 
+Davis, J.R. et al. (2015) ‘Inter-Cellular Forces Orchestrate Contact Inhibition of Locomotion’, Cell, 
+161(2), pp. 361–373. Available at: https://doi.org/10.1016/j.cell.2015.02.015. 
+E. Courtier, N. et al. (2019) ‘How transport layer properties affect perovskite solar cell performance: 
+insights from a coupled charge transport/ion migration model’, Energy & Environmental Science, 
+12(1), pp. 396–409. Available at: https://doi.org/10.1039/C8EE01576G. 
+Giniūnaitė, R. et al. (2020) ‘Modelling collective cell migration: neural crest as a model paradigm’, 
+Journal of Mathematical Biology, 80(1), pp. 481–504. Available at: https://doi.org/10.1007/s00285-
+019-01436-2. 
+Gopinathan, A. and Gov, N.S. (2019) ‘Cell cluster migration: Connecting experiments with physical 
+models’, Seminars in Cell & Developmental Biology, 93, pp. 77–86. Available at: 
+https://doi.org/10.1016/j.semcdb.2018.09.009. 
+Guberman, E., Sherief, H. and Regan, E.R. (2020) ‘Boolean model of anchorage dependence and 
+contact inhibition points to coordinated inhibition but semi-independent induction of proliferation 
+and migration’, Computational and Structural Biotechnology Journal, 18, pp. 2145–2165. Available 
+at: https://doi.org/10.1016/j.csbj.2020.07.016. 
+Hansen, T.W. et al. (2013) ‘Sintering of Catalytic Nanoparticles: Particle Migration or Ostwald 
+Ripening?’, Accounts of Chemical Research, 46(8), pp. 1720–1730. Available at: 
+https://doi.org/10.1021/ar3002427. 
+Hu, B. et al. (2010) ‘Physical Limits on Cellular Sensing of Spatial Gradients’, Physical Review 
+Letters, 105(4), p. 048104. Available at: https://doi.org/10.1103/PhysRevLett.105.048104. 
+Metzcar, J. et al. (2019) ‘A Review of Cell-Based Computational Modeling in Cancer Biology’, JCO 
+Clinical Cancer Informatics, (3), pp. 1–13. Available at: https://doi.org/10.1200/CCI.18.00069. 
+Norden, C. and Lecaudey, V. (2019) ‘Collective cell migration: general themes and new paradigms’, 
+Current Opinion in Genetics & Development, 57, pp. 54–60. Available at: 
+https://doi.org/10.1016/j.gde.2019.06.013. 
+Peercy, B.E. and Starz-Gaiano, M. (2020) ‘Clustered cell migration: Modeling the model system of 
+Drosophila border cells’, Seminars in Cell & Developmental Biology, 100, pp. 167–176. Available at: 
+https://doi.org/10.1016/j.semcdb.2019.11.010. 
+Roycroft, A. et al. (2018) ‘Redistribution of Adhesive Forces through Src/FAK Drives Contact 
+Inhibition of Locomotion in Neural Crest’, Developmental Cell, 45(5), pp. 565-579.e3. Available at: 
+https://doi.org/10.1016/j.devcel.2018.05.003. 
+
+ 
+11 
+SenGupta, S., Parent, C.A. and Bear, J.E. (2021) ‘The principles of directed cell migration’, Nature 
+Reviews Molecular Cell Biology, 22(8), pp. 529–547. Available at: https://doi.org/10.1038/s41580-
+021-00366-6. 
+Simon, C. (no date) Generating uniformly distributed numbers on a sphere, Mathemathinking. 
+Available at: http://CorySimon.github.io/articles/uniformdistn-on-sphere/ (Accessed: 29 September 
+2022). 
+Textor, J., Sinn, M. and de Boer, R.J. (2013) ‘Analytical results on the Beauchemin model of 
+lymphocyte migration’, BMC Bioinformatics, 14(6), p. S10. Available at: 
+https://doi.org/10.1186/1471-2105-14-S6-S10. 
+Vecheck, A.M. et al. (2022) ‘Quantum Biology in Cellular Migration’. bioRxiv, p. 
+2022.09.09.507322. Available at: https://doi.org/10.1101/2022.09.09.507322. 
+9 Data Availability Statement 
+The software for this study can be found at “Cameron-McNamee/Particle_Modeling” 
+[https://github.com/Cameron-McNamee/Particle_Modeling]. 
+

89AyT4oBgHgl3EQfdPcr/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf,len=488
+page_content='Deterministic and Nondeterministic Particle Motion with Interaction  Mechanisms  Cameron McNamee1,2*, Renee Reijo Pera1  1McLaugling Research Institute, Great Falls, MT, USA  2Department of Mathematics, California Institute of Technology, Pasadena, CA, USA  Correspondence:  cmcnamee@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='edu  Keywords: particle modeling, migration, interaction mechanisms, deterministic modeling,  Beauchemin modeling  Abstract  Studying systems where many individual bodies in motion interact with one another is a complex and  interesting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Simple mechanisms that may be determined for biological, chemical, or physical  reasons can lead to astonishingly complex results that require a further understanding of the moving  bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With the increasing interaction between computation and various scientific areas, it has  become more useful, feasible, and important to create models for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Here, we present two  families of models, deterministic and nondeterministic, along with three distinct and realistic  interaction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' These are combined in a unique way to provide the groundwork for particle  system models across multiple disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This work has applications that range from biology to  chemistry and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In addition to describing the motivations and math behind all the models,  software is provided that allows researchers to quickly adjust and implement what is described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  1  Introduction  This work was inspired by consideration of quantum biology (QB) and cell migration and tissue  formation as previously described (Vecheck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2022) with the goal of identifying quantitative  measures of particle clustering in a scaffold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Identifying such measures involves observing the  locations of the particles using image processing and applying these locations to a novel clustering  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The results were some surprising numbers that led to a search for other, more robust  methods of cluster analysis and an environment in which to test resulting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This proved not  to be a simple task prompting development of potentially new models to understand how particles  may interact under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' From a biological perspective, and in view of the eventual  application of this work, this paper explores the coalescing of particles in a space towards one  another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' It is also important to understand that in the simulation process, the goal is to replicate what  is seen in realistic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In addition to biology, many systems wherein multiple autonomous  particles are driven in a migratory pattern can be seen in physics and chemistry, allowing this work to  be expanded to multiple fields (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Courtier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Nanoparticle and ion  motion are two distinct areas that this type of computational research can be readily applied and lead  to interesting features of such a system being uncovered (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Courtier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With the diverse opportunities in mind, this paper seeks to abstract the motion and interaction  of individual bodies in such systems to create general models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' These general models may then be  taken by a researcher and applied with the appropriate parameters to match a particular system, with  2  the addition of the available software laying the foundation of such a model (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This paper is  by no means all-encompassing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' however, the range of these general models spans many fields with  many important implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2 Methods  All software was implemented in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Methods are described below as follows: First, different  environments in which particles may be generated are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' These environments decide what is  called the “initial seeding” of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' These initial seedings vary in ways that may be more  suitable for certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Second, two families of models are explored: deterministic models  and nondeterministic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Differences between these two families of models are described in more  detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' however, as the names suggest, the most important difference is that the deterministic model  will produce the exact same results given the same initial seedings, while the nondeterministic models  have an element of randomness that allows for changes in outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The third and last part discussed  is the interactions between particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is a unique addition to these models and distinguishes this  work for specific applications in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1 Initial Seeding Environments  Initial seeding environments are different environments that serve different purposes and are each  suited for their own application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' All seedings are centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1 Uniform Seeding  Non-random uniform seeding places each particle in a lattice point of a Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  This serves as what can be considered an “in vitro” environment, since in most applications, particles  are not placed in such an orderly fashion (Peercy and Starz-Gaiano, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Calvillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2 Uniform Random Seeding  This seeding generates a designated number of particles within specified dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is carried  out by generating three random numbers within ± the designated range for each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' These  coordinates then determine the placement of each particle, providing uniformly random particles  throughout the space with an expected center at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3 Spherical Random Seeding  Like the seeding above, spherical random seeding is a uniform distribution of particles with an  expected center at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This differs from uniform random seeding in that the seeding is formed  with a given number of particles and a radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The problem of creating a uniform distribution within  a sphere is not trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' thus, this was accomplished by using polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  To randomly generate a radius that places particles uniformly, a floating-point number between [0,1]  is sampled and the cube root is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Then, the maximum radius is multiplied by the resulting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  This allows, for reasons that can be clearly seen from a geometric perspective, for the radius to  generate uniform points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Generating the random angles is described in more detail here, along with  more extensive reasoning throughout the entire process: (Simon, no date).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This random process is  done for as many particles as desired, and this determines the initial seeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2 Random Migration Models  3  This class of models stems from the Beauchemin model for migrating bodies, which has been used  previously for biological modeling (Textor, Sinn and de Boer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' All random migration models  were taken from Textor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' (Textor, Sinn and de Boer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1 Beauchemin Random Walk  To develop a biased migration model for a group of particles, it is helpful to first create a model that  has no bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The Beauchemin random walk model functions under two phases: a pause phase and a  free phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' During the pause phase, a particle generates two random angles, 𝜙 and 𝜃, and uses these  angles to generate a uniform random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The generation of these angles follows the same  method mentioned above for uniform sphere generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' After the pause phase, the particle enters the  free phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In this random walk model, every particle has the same, given velocity 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Using its given  direction, a particle travels at the velocity 𝑣 for the entire free phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The free phase length may be  changed, but for most modeling applications, it is 4 times longer than the pause phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is  arbitrary and may be changed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  As the name suggests, the Beauchemin random walk method is not biased in any way, so the  particles walk uniformly in any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This unbiased nature leads to no pattern of particle  migration but provides a basis wherein specific factors may be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With the unbiased model  established, an introduction to different biases is now needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2 Simple Phenomenological  The Simple Phenomenological model (SP) introduces bias by adding a “drift element” during the  pause phase (Textor, Sinn and de Boer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Instead of the particles remaining stationary during the  pause phase, they drift towards some defined point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This causes uniform clumping amongst the  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In this implementation, the choice was made to make this point the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is of course  arbitrary and may be changed to suit whatever environment is most sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The drift velocity is defined as the given velocity 𝑣 times the bias variable 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' For 𝑝 = 1, equivalent  bias is achieved between random walking and drifting towards the origin as the particles approach the  center at the same rate that they walk randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Of course, for 𝑝 = 0 there is no bias, so this model  would operate the same as the random walk above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The drift vector is defined by using the current 𝑥, 𝑦, and 𝑧 coordinates to calculate the new  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' First, 𝑑, the distance from the drift particle, is calculated using the 𝑙� norm  𝑑 = �𝑥�  � + 𝑦�  � + 𝑧�  �  Then, the drift velocity is used to “shrink” the coordinates towards the origin  𝑥����� = 𝑥� ⋅ 𝑑 − 𝑣𝑝  𝑑  𝑦����� = 𝑦� ⋅ 𝑑 − 𝑣𝑝  𝑑  𝑧����� = 𝑧� ⋅ 𝑑 − 𝑣𝑝  𝑑  4  In this implementation, there is an added condition that a particle only moves towards the origin if it  is farther away than the drift velocity will take it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This prevents the particle from “overshooting” its  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3 Topotaxis Model  The Topotaxis Model (TM) differs from SP by introducing a different form of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Instead of  moving the particle in the target direction, TM changes the distribution from which the angles are  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Instead of 𝜙 and 𝜃 being taken from a uniform distribution, they are drawn from a beta  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' A beta distribution is essentially a truncated normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The normal distribution  cannot be used in this situation because of its infinite tails and the angles are within a strict range,  [0, 2𝜋] and [0, 𝜋] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The beta distribution’s shape is determined by two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Here,  the two parameters, 𝛼 and 𝛽, and are set equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Since they are equal, we only refer to 𝛼 from here on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  To calculate the shape of the distribution given a single bias variable 𝑝, we need to consider that  when 𝑝 = 0, the distribution should be uniform and as 𝑝 approaches 1, the distribution should  become more skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is accomplished by decreasing the variance in the distribution of angles  while keeping the mean towards the origin with the result that the particles have an increased  likelihood of orienting themselves towards the target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In MATLAB, if 𝑎 =  1, the  distribution is defined as uniform and as 𝑎 → ∞, the variance decreases, and a peak centered at the  mean value is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' To calculate 𝑎 given 𝑝, the following is done  𝑎 =  1  1 − 𝑝  From this definition we see that for this model 𝑝 ≠ 1, but we also see lim  �→� 𝑎 =  ∞, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Thus,  for 𝑝 = 0, there is a uniform, or unbiased, distribution, and as 𝑝 → 1, there is an increased bias  towards the targeted location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3 Deterministic Migration Models  The deterministic migration models are a family of migration models that feature no random  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Within this class of models, there are two different groups: uniform clustering and local  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The uniform clustering algorithms behave on a global scale, causing all particles to  migrate in a wholistic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The local clustering algorithms organize themselves at a much smaller  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1 Uniform Clustering, the Naïve Approach  The first and simplest way of clustering particles within space is to “shrink” the space they are in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  first requires a clustering rate, 𝑟 ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This method is as follows  𝑈�𝑐�, 𝑐�, 𝑐�� = �𝑐� ⋅ (1 − 𝑟), 𝑐� ⋅ (1 − 𝑟), 𝑐� ⋅ (1 − 𝑟)�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' ∀𝑐 ∈ 𝐶 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 1)  Where C is the set of all particles, c is an individual particle, and 𝑐�, 𝑐� and 𝑐� are the x-, y-, and z-  coordinates of c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This function causes all particles to “march” inward, maintaining their  relative distances to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This does produce a set where each particle is relatively closer to all  other particles but does so in a way that is not representative of typical physical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2 Nearest Neighbor Clustering  5  In nearest neighbor clustering, particles are not clustered into a single lump, but rather orient to their  neighbors and move toward them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This approach is much more nuanced compared to Uniform  Clustering and reveals samples that are much more reminiscent of certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The method  behind this approach assumes that particles can only sense their n nearest neighbors (NN’s), and so  they only approach those particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This method is conducted by first finding the geometric center  (GC) described by the n NN’s and then moving the particle towards that at a clustering rate, 𝑟 ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Using the same notation as above, we have the set of distances from all particles to a given particle c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  𝑁� =  min  �  { 𝑑�: 𝑑� = ||𝑠� − 𝑐||, ∀s� ∈ 𝐶} (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 2)  Here, �|𝑠� − 𝑐|� is the 𝑙� normed distance between the particle 𝑠� and the particle c and min  � (𝑆)  represents taking the n smallest elements from the set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With the NNs now identified, the GC (𝑔�) is  calculated for a given 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This center is calculated without including c itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  𝑔�� =  ∑  𝑠�  �∈��  𝑛  𝑔�� =  ∑  𝑠�  �∈��  𝑛  𝑔�� =  ∑  𝑠�  �∈��  𝑛  𝑔� = �𝑔��, 𝑔��, 𝑔��� (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 3)  In other words, the unweighted mean of x-, y-, and z-coordinates of each NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This gives us the GC  described by the n NN’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With this, we now have a location to move c towards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  �𝑐� , 𝑐�, 𝑐�� ↦ (𝑐�(1 − 𝑟) + 𝑔�𝑟, 𝑐�(1 − 𝑟) + 𝑔�𝑟, 𝑐�(1 − 𝑟) + 𝑔�𝑟)  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 4)  For a given clustering, this calculation must be carried out for each particle, and it is done in its  entirety before any particle moves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', no particle’s location changes until after every particle’s new  location has been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Then, all particles move to their new locations at once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The result is  clustering in small groups that after enough time steps, form individual groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3 Threshold Clustering  Threshold (TH) clustering differs from nearest neighbor clustering in that, instead of searching for a  certain number of neighbors, the particles have an assumed maximum influence distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  maximum influence distances, a threshold, is analogous to a particle only being able to sense other  particles for a small radius around itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This method is implemented by collecting the set of particles  within the maximum sensing distance, t, calculating the GC described by these particles, and then  moving towards this center at the clustering rate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  𝑇� = {𝑠 ∈ 𝐶: �|𝑠 − 𝑐|� <  𝑡} (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 5)  6  Then, as above, the GC of this set is calculated using the unweighted mean (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This gives 𝑔�,  which we then use to calculate the new particle’s position using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This, as with NN, is done for  every particle before any particle changes its location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This mode produces several small groups  throughout the space whose group size changes with 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='4 Weighted Clustering  This method is the next logical step for simulating clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Instead of taking the unweighted mean  to calculate the GC, why not weigh the locations based on distance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Weighted clustering is done by  first calculating 𝐷�, the set of all distances from each particle to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Next, the weighted geometric  center (WGC) for c, 𝑤�, is calculated as follows:  𝑤�� =  ∑  𝑠��  𝑑�  �∈��  ∑  � 1  𝑑��  �∈��  𝑤�� =  ∑  𝑠��  𝑑�  �∈��  ∑  � 1  𝑑��  �∈��  𝑤�� =  ∑  𝑠��  𝑑�  �∈��  ∑  � 1  𝑑��  �∈��  𝑤� = (𝑤��, 𝑤��, 𝑤��) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 6)  Where 𝐼� is the indexing set corresponding to 𝐷�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The reason for using the reciprocals of the weights  follows from the idea that as a particle get further, it should affect c less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The new coordinates for c  are calculated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is a global clustering on particles, like uniform clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  causes no small groups to small, but rather the entire population of particles congregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  congregation, however, does not appear the same nor does it occur at the same rate at uniform  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='5 Weighted Nearest Neighbor Clustering  In accordance with the previous modes, this weighted nearest neighbor (WNN) clustering is the next  intuitive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This method is, as the name suggests, a combination of NN clustering and weighted  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This combines the logic of only looking towards the closest particles, while also weighing  the particles according to their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The set of NN’s is calculated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  𝐼�� = {𝑛 ∈ 𝐼: 𝑠� ∈ 𝑁�}  The WGC is calculated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 6 with 𝐼�� used in place of 𝐼�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With this WGC, particle c is  mapped using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This mode appears very similar to the original NN clustering but tends to form  clusters that are skewed in slightly different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='6 Weighted Threshold Clustering  7  Weighted threshold (WT) clustering follows the same intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 𝑇� is calculated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  This defines the limited distance indexing set:  𝐼�� = {𝑛 ∈ 𝐼: 𝑠� ∈ 𝑇�}  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 6 with 𝐼�� in place of 𝐼� gives WGC about the particles within the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 4 is used  to map c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' As with WNN clustering, WT clustering appears quite similar to the TH clustering on small  scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='4 Interaction Mechanics  With the addition of interaction, defined here as how two particles behave when they physically hit  each other in the migration process, there are many different fields that may provide insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Some  outcomes of physical interaction are cessation of movement, repolarization, or adhesion, to name just  a few examples of how particles may interact with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1 Cessation of Movement  The simplest self-interaction is cessation of movement, meaning simply that if two particles are  within a designated distance of one another, they both stop moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In simulation, implementation of  this outcome simply checks if a particle is close to any other particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' If it is, it stays in the same spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Otherwise, it keeps moving as it was intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='2 Repolarization  Repolarization means that if two particles hit each other, they both turn around in the opposite  direction, much like an elastic collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Implementation is carried out by taking the two angles that  defined the direction that a given particle was traveling, and negating both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='3 Adhesion  Adhesion is the clustering or clumping of particles to form a single body that moves as one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is  accomplished by assigning a clump to every particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' If a particle is within a given radius of other  particles, it gets put into that clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Additionally, a clump is a collection of not just the particles that  are touching a given particle, but any that are touching its neighbors, the neighbor’s neighbors, and  so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The clumps partition the set of particles and create moving bodies that can obtain more  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' No limit to the size of clumps was designated, however this addition is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  3 Discussion  The purpose of this discussion is to explore the potential applications of the models described above  and how the simulations generated in the linked code can prove useful to other researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Many  particle or particle-like systems are seen across all scientific fields including physics, chemistry, and  biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Proper implementation of these models requires a good understanding of the system at hand  and how the particles behave with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' An understanding of the system is necessary to select  the proper migration model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  In highly chaotic systems, nondeterministic models are more suited as randomness (pseudo  randomness rather) is a key feature of such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In systems where the environment itself does  not drive motion of particles themselves, the deterministic models are better approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  8  Implementing different interaction dynamics in the same setup leads to very different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Thus, it  is also very important to understand how individual particles behave upon collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' With this  necessity understood and considered, let us explore specific use-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The above models offer insight in biological settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Cell migration is an important component of  many cell types, and thus estimating that motion can provide information about cellular mechanisms  (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Gopinathan and Gov, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Metzcar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Norden and Lecaudey, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Alert and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Guberman, Sherief and Regan, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Peercy and Starz-Gaiano, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  SenGupta, Parent and Bear, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In different settings, cellular motion may be estimated by  different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Chemotaxis, which is motion based on the sensing of chemical gradients, is the  driving force of many migration patterns at the cellular level (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Gopinathan and Gov,  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Norden and Lecaudey, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' SenGupta, Parent and Bear, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  mode of migration is akin to the TM as the chemical gradient introduces bias as to what direction the  cells will migrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The stronger the cells’ reaction to the chemical gradient, or the stronger the  chemical gradient itself, the greater this bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Migratory cells have been noted to have a degree of  continuous bias, meaning they can have varying levels of directional influence due to chemotaxis, not  simply a binary model of directed motion (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert and Trepat, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This, along with  the mechanistic understanding of chemotaxis lends this type of cellular motion to the Topotaxis  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  In addition to chemical sensitivity, further support of the nondeterministic models comes from the  observations that cells perform motion that follow the pause phase-free phase mechanism that  dictates these models (Gopinathan and Gov, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Peercy and Starz-Gaiano,  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This comes from the discontinuation of motion in favor of direction recalibration noted in cell  motion, which strongly motivates the use of Beauchemin random walks in favor of other forms of  random walks (Textor, Sinn and de Boer, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Gopinathan and Gov, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Peercy and Starz-Gaiano, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Other than chemical gradients, the mechanical force induced by the  microenvironment about the cell can influence motion (Peercy and Starz-Gaiano, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' SenGupta,  Parent and Bear, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is the defining feature of the SP model as there is an external force on  the cells driving them in a biased direction, much like fluid flowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' SP can be used to describe  mechanistic motion of cells during embryonic development and within the neural crest (Giniūnaitė et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Peercy and Starz-Gaiano, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Contexts such as these are strong candidates for applying  nondeterministic models outlined in this paper in the field on biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Deterministic models also have niches in biological settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Tumor growth has many similarities to  TH, as does cancer cell migration (Metzcar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is an especially important area for  computational research, as hypotheses must be tested efficiently and accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Modeling can also  be used to determine metabolism of cancer and stem cells, giving useful information that can be  verified experimentally (Metzcar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This is a great example of how computational  modeling can uncover additional information and provide meaningful insight in a timely fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Cellular mechanics also offer interesting interaction dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' All three modes of interaction take  place across the field of biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Adhesion, as described above, occurs in migrating cell groups,  cancer metastasis, and embryogenesis (Gopinathan and Gov, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Norden and Lecaudey, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Alert and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Guberman, Sherief and Regan, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Peercy and Starz-Gaiano, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This  typically occurs in homogenous cell groupings and is seen throughout the immune system (Davis et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Norden and Lecaudey, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert and Trepat, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Cessation of movement can be  observed in tissue growth for both epithelial cells and fibroblast cells (Abercrombie, 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Guberman, Sherief and Regan, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In fact, this behavior is the original mechanism that led to the  9  term “contact inhibition of locomotion,” better known as CIL (Abercrombie, 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Notably, CIL  also can be described in different contexts as a repolarization effect, meaning within biology any of  the above interaction mechanisms may be observed (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Roycroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Alert  and Trepat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Guberman, Sherief and Regan, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Moving to a different field, in chemistry and physics nanoparticles display many of the features of  both clustering and motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In certain settings, nanoparticles can behave in random walk migration  which can be biased due to outside forces to appear as any of the nondeterministic models outlined  above (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' In addition, a mode of random potential driven motion along with  particle adhesion described as Oswald Ripening can be appropriately described by the TM with  particle adhesion (Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Ion motion in solar cells holds many of the key mechanistic  contexts outlined above (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Courtier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Additionally, the dynamic change of chemical  gradients, useful for chemistry and biological contexts, can by approximated by SP (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The novelty of the work contained within this paper comes from the diverse applications readily  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' It is the addition of particle interactions that leads to results with astounding complexity  that are reminiscent of what is captured in scientific settings, while keeping the simulations in a  tractable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' While this work was inspired from a biological setting, it is of upmost importance to  recognize the applications of such work to other fields, for it is the interaction between the sciences  that leads to strides in innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The union between computer science and physical science is  something that has never been seen before, and we as a scientific community have but scratched the  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  With these diverse and important applications available from the relatively simple models outlined  above, it is important to note the limitations of not just these models themselves, but also the current  state of computational physical science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' First, we can only model what we understand, and this  knowledge is limited not by our ability to create models but by our methods in the physical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  To aptly describe a complex system, a complex understanding of such system must be had first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  However, as far as biology, much of chemistry, and many places of physics goes, we do not have  such a deep understanding (SenGupta, Parent and Bear, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' The power of predictive models is the  ability to extrapolate our current understanding to generate nuanced hypotheses that may be tested in  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' A major pitfall of this ability is, however, that false initial assumptions may lead to a  fruitless path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The goal and purpose of this newfound computational power, a power that is only growing as time  passes, is to approximate to the best of our ability the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' This allows us to uncover, with  unparalleled speed, extremely important new aspects of physical systems that were previously  obscure and serve to inspire experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Continuing to cautiously push forward with this  technology will lead to profound discoveries across academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  The authors declare that the research was conducted in the absence of any commercial or financial  relationships that could be construed as a potential conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  CM contributed to the experimental design, analysis of data and interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' CM and  RRP prepared the manuscript and edited the final manuscript for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  This work was supported by funding from the National Institute of Health (1R01HD096026).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  Abercrombie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' (1961) ‘The bases of the locomotory behaviour of fibroblasts’, Experimental Cell  Research, 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' 188–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1016/0014-4827(61)90348-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  10  Alert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
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+page_content=' 529–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1038/s41580-  021-00366-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
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+page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
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+page_content=' (2022) ‘Quantum Biology in Cellular Migration’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' bioRxiv, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='507322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content=' Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='1101/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='507322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='  9 Data Availability Statement  The software for this study can be found at “Cameron-McNamee/Particle_Modeling”  [https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}
+page_content='com/Cameron-McNamee/Particle_Modeling].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89AyT4oBgHgl3EQfdPcr/content/2301.00297v1.pdf'}

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+Biometrical Journal  DOI  
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                           
+ 
+Improving Software Engineering in Biostatistics: Challenges 
+and Opportunities 
+Daniel Sabanés Bové*,1, Heidi Seibold2, Anne-Laure Boulesteix3, Juliane Manitz4, 
+Alessandro Gasparini5,6, Burak K. Günhan7, Oliver Boix8, Armin Schüler7, Sven 
+Fillinger9, Sven Nahnsen9,12, Anna E. Jacob3, Thomas Jaki10,11 
+1 Hoffmann-La Roche Ltd., Product Development Data Sciences, Grenzacherstrasse 124, 4070 Basel, 
+Switzerland 
+2 IGDORE, Elsenheimerstr. 48, 80687 München, Germany 
+3 Institute for Medical Information Processing, Biometry and Epidemiology, LMU Munich, 81377 
+Munich, Germany 
+4 EMD Serono, 45A Middlesex Turnpike, Billerica, MA 01821, USA 
+5 Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, PO Box 281, 17177 
+Stockholm, Sweden 
+6 Red Door Analytics AB, Dianavägen 11A, 11542, Stockholm, Sweden 
+7 Merck Healthcare KGaA, Frankfurter Strasse 250, 64293 Darmstadt, Germany 
+8 Bayer AG, Aprather Weg 18a, 42113 Wuppertal, Germany 
+9 Quantitative Biology Center (QBiC), University of Tübingen, 72076 Tübingen, Germany 
+10 Faculty of Informatics and Data Science, University of Regensburg, Bajuwarenstraße 4 
+93053 Regensburg, Germany 
+11 MRC Biostatistics Unit, University of Cambridge, East Forvie Building, Forvie Site, Robinson Way,  
+Cambridge CB2 0SR, UK 
+12 Biomedical Data Science, Department of Computer Science, University of Tübingen, 72076 Tübingen, 
+Germany 
+ 
+ 
+Received zzz, revised zzz, accepted zzz 
+ 
+Programming is ubiquitous in applied biostatistics; adopting software engineering skills will help 
+biostatisticians do a better job. To explain this, we start by highlighting key challenges for software 
+development and application in biostatistics. Silos between different statistician roles, projects, 
+departments, and organizations lead to the development of duplicate and suboptimal code. Building on 
+top of open-source software requires critical appraisal and risk-based assessment of the used modules. 
+Code that is written needs to be readable to ensure reliable software. The software needs to be easily 
+understandable for the user, as well as developed within testing frameworks to ensure that long term 
+maintenance of the software is feasible. Finally, the reproducibility of research results is hindered by 
+manual analysis workflows and uncontrolled code development. We next describe how the awareness of 
+the importance and application of good software engineering practices and strategies can help address 
+these challenges. The foundation is a better education in basic software engineering skills in schools, 
+universities, and during the work life. Dedicated software engineering teams within academic institutions 
+and companies can be a key factor for the establishment of good software engineering practices and 
+catalyze improvements across research projects. Providing attractive career paths is important for the 
+retainment of talents. Readily available tools can improve the reproducibility of statistical analyses and 
+their use can be exercised in community events. Similarly, tools exist to assess the risk of R packages, 
+and initiatives are developing shared repositories of trusted R packages. Finally, collaboration between 
+software developers from different organizations is key to harness open-source software efficiently and 
+optimally, while building trusted solutions. We illustrate the potential with examples of successful 
+projects. 
+Key words:  Collaboration; Open-source; Programming; Software engineering.  
+ 
+ 
+ 
+*Corresponding author: e-mail: daniel.sabanes_bove@roche.com 
+
+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+1 Introduction 
+As technology and methods advance, one of the key goals in the field of biostatistics is for the 
+statisticians of tomorrow to practice software engineering in a sustainable way. They can do this by 
+making methods and software open source, integrating new software in an established ecosystem, 
+organizing long-term maintenance, and adhering to good software engineering practices (e.g., 
+documentation, code readability, code organization, and version control) (Seibold et al., 2021). In this 
+paper, we highlight opportunities and challenges on the path to this long-term vision – highlighting 
+how we can achieve this desirable future. The content originates from a panel discussion on “Research 
+Software Engineering for Clinical Biostatistics” which took place at the 43rd Annual Conference of the 
+International Society for Clinical Biostatistics (ISCB) in Newcastle in August 2022. 
+To provide a better understanding of what software engineering is, we will use a standardized 
+definition from the Institute of Electrical and Electronics Engineers (IEEE) to aid the interpretation of 
+this discipline. IEEE defines software engineering as “[t]he systematic application of scientific and 
+technological knowledge, methods, and experience to the design, implementation, testing, and 
+documentation of software” (“ISO/IEC/IEEE International Standard - Systems and Software 
+Engineering–Vocabulary,” 2017). From this, one can conclude that software engineering implies the 
+conscious application of methods and practices in the field of software craftsmanship to deliver a 
+software product of certain quality.  In addition, we want to adhere in the following to the definition of 
+open source as stated by the Open Source Initiative (The Open Source Definition, 2007). 
+So far, there have been a few related publications within the statistical literature and beyond. Sanchez 
+et al. (2021) describe key steps for implementing a code quality assurance process that researchers can 
+follow throughout a project to improve their coding practices regarding the quality of the final data, 
+code, analyses, and results. This includes code style, documentation, version control, data management, 
+testing, and code review. Taschuk and Wilson (2017) present ten simple rules to make research 
+software robust enough to be run reproducibly. The rules include version control, documentation, 
+release versioning, build tools, and tests among others. Anzt et al. (2021) identify challenges for 
+research software sustainability in Germany and beyond in terms of motivation, selection, research 
+software engineering personnel, funding, infrastructure, and legal aspects. They recommend strategies 
+and measures to create an environment for sustainable research software, which enables valid, 
+reproducible, and sustainable research.  
+In this paper, we aim to summarize key challenges regarding programming in the field of biostatistics 
+and provide opportunities from the software engineering perspective to address them. In section 2, we 
+describe the challenges faced by software engineering in biostatistics today. In section 3, we highlight 
+important opportunities to overcome these challenges as a call to action. Examples of successful 
+projects implementing good software engineering practices are described in section 4. Finally, in 
+section 5, we discuss the key points. 
+2 Challenges 
+2.1 Silos 
+Silos are a particular challenge for software engineering in biostatistics (de Waal et al., 2019). In the 
+pharmaceutical industry, there are often purpose silos, with statistical analyses prepared for regulatory 
+use being run on a different system with a different programming language, compared to statistical 
+analyses prepared for exploratory use. This is inefficient because as soon as an exploratory analysis 
+becomes relevant for regulatory use, it must be recoded and run on the other system. 
+Another example are different projects within a company or research institution, where project team 
+members may communicate with members of the other projects, but only copy and paste code between 
+each other. This is time consuming because the code then needs to be adapted to the current project 
+manually, which is naturally error prone. It is problematic when there is not enough time to invest into 
+
+Biometrical Journal 63 (2021), ZZZZZZ / DOI 10.1002/bimj.201010000 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+the development of macros or packages that implement more generic code which could be easily used 
+across multiple projects in a sustainable way. With respect to resource efficiency, the situation can 
+worsen if these related code bases grow to more complex applications. This is especially the case when 
+there is a lack of strategic design that would have considered joint development efforts from the 
+beginning.  
+Similarly, often different departments are siloed from each other, e.g., the IT department and the 
+statistics department do not communicate sufficiently. Statistics departments need to rely on the IT 
+departments to provide the computing infrastructure (e.g., laptops, servers, cloud-based containers, etc.) 
+for running the statistical software, being able to develop it, and share it with users (e.g., for web-based 
+interactive applications). Especially in larger companies or research institutions, the IT landscape is 
+often complex, and requests from statistics departments for access to modern tools need to be properly 
+communicated to, received, and implemented by the IT departments. Otherwise, it is a source of 
+frustration for both sides and the statistics department will not be able to use the modern tools it 
+requires. 
+Finally, different organizations in academia or industry often reinvent the wheel for standard analysis 
+pipelines, which is inefficient from the perspective of society. This repeated effort is unnecessary 
+because those standard analyses are not the subject of innovation, research, or the business itself, but 
+only used as means to an end. 
+ 
+2.2 Reliability 
+With the transition of medical practices towards data-driven assessment and the introduction of 
+statistical learning to biomedicine and patient care (for instance in personalized oncology), the notion 
+of software reliability needs to be given exceptional importance. 
+Most software engineering in biostatistics today is done within ecosystems of existing modular open-
+source software packages which augment a foundational programming language. The most prominent 
+example is R, but there is also Python, and more recently Julia. In this situation, it is up to the 
+developer to judge which existing packages are of high enough quality to build upon them with the new 
+software, and which are not. While reliability is also a concern for proprietary software, they are sold 
+with appropriate documentation of quality and accuracy. Therefore, the user can build upon such 
+proprietary software with more confidence in the quality as compared to open-source alternatives. 
+In addition, all newly developed software should be developed using professional workflows and tested 
+sufficiently to ensure that the analyses are performed correctly, and the respective results are accurate. 
+In the clinical trial context, for instance, the ICH E9 guideline states that “software used should be 
+reliable, and documentation of appropriate software testing procedures should be available” (European 
+Medicines Agency, 1998). Risk assessment frameworks have been developed alongside tools to help 
+with this task and will be described in section 3.5. While observational research outside of clinical trials 
+does not fall under this regulation, it is clear that similar quality standards should be followed in order 
+to guarantee reliable analyses and results. 
+Making code open source is a good step to enable contributions from the open-source community. 
+However, this does not guarantee that the code is easy to understand for others (Martin, 2009). The 
+ratio of code being read to code being written is often greater than one and therefore code that is hard to 
+understand can lead to longer debugging time and makes it more difficult to build upon. Since 
+readability of code is a concern that is not domain specific, there is already sophisticated literature 
+available that suggests helpful guidelines to achieve readable code. The challenge is how to bring this 
+knowledge into domain-specific software engineering areas, e.g., statistics, where there might be little 
+awareness. Additionally, assuring its implementation, especially when there is no additional direct 
+value for the developer (e.g., public recognition) or part of a good practice statute within the 
+organization (i.e., lack of extrinsic motivation factors), is important.  
+
+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+2.3 Usability 
+Open-source projects often start from code written by an individual for their own use. Either because of 
+altruistic motives or external incentives the code is published at one point in time. However, handling 
+of the software for others sometimes is difficult due to a lack of documentation, intuitive interface, or 
+naming conventions resulting in researchers sometimes preferring to start from scratch instead of using 
+the available software. At the same time, the usability of the software has a profound influence on its 
+subsequent use and success.  
+Application Program Interface (API) design itself can have a great influence in the usability of software 
+libraries, especially when the intentions of methods are unclear. Even simple elements such as the 
+semantics of method signatures can make a difference. Usability tests can help in designing efficient 
+and effective user interfaces or client APIs but might not have priority or are not part of the 
+development process. 
+2.4 Maintenance 
+In academia, principal investigators often have neither sufficient time to work on complex software 
+themselves nor to keep pace with recent developments in a rapidly evolving ecosystem. Similarly in the 
+industry, statisticians working on projects have too many other duties to be able to write any software 
+themselves. It is also not usually part of the job for statistical programmers to engineer software; 
+instead, the expectation focuses on writing scripts that execute predefined macros.  
+Software engineering thus often relies on the efforts of earlier career statisticians such as PhD students, 
+post-doctoral researchers, or interns, who typically leave the institution and switch to other projects 
+before the end of the lifetime of the packages they developed. This raises major challenges related to 
+maintenance – in terms of expertise (other group members may not have the required expertise), 
+funding (which is typically not available for maintenance years after the package’s primary 
+development), and incentives (taking over another person’s package as a new maintainer is usually not 
+perceived as rewarding). We claim that funding and incentive structures must be increased to ensure 
+the middle- to long-term maintenance of packages developed by academic researchers, see Schönbrodt  
+(2022) for an example proposal. Package longevity should be taken into account from the beginning of 
+a project, for example by distributing competence over several researchers to improve the “bus factor” 
+(Cosentino et al., 2015). 
+Another aspect of the maintenance challenge is that building upon existing open-source packages 
+comes with risks. For example, a package our software depends on might be retired or abandoned by 
+the previous developers. Then all dependent packages are at risk of also becoming unusable. Naturally, 
+however, the more packages that depend on it, the larger the base of developers to inherit (i.e., become 
+the maintainer), update by contributing code changes (i.e., becoming a co-developer), or address 
+problems with this package will be. For example, recently the R package isoband, which is a 
+dependency of the popular package ggplot2, was at risk of being removed from the Comprehensive 
+R Archive Network (CRAN) repository. This could have impacted thousands of other R packages that 
+depend on ggplot2 themselves. Fortunately, the problem could be resolved quickly by the developer 
+community (Szymański, 2022). This is one reason to generally reduce the number of dependencies as 
+much as possible, while not copying code when possible.  
+2.5 Reproducibility 
+The issue of reproducibility in scientific research (or lack thereof) has been heavily discussed in the 
+past years (Begley & Ioannidis, 2015; Cacho & Taghva, 2020; Mullane et al., 2018, p. 1; Niven et al., 
+2018; Stupple et al., 2019). For a certain biomedical study to be reproducible in practice, that is, for 
+independent researchers to be able to replicate the output of a study, data and software ought to be 
+shared by the original authors. Research reproducibility is paramount for the accumulation of scientific 
+
+Biometrical Journal 63 (2021), ZZZZZZ / DOI 10.1002/bimj.201010000 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+evidence (Peng et al., 2006) but it is increasingly complex to achieve in practice due to the increasing 
+complexity of data collected (and used) and more complicated statistical methods, bioinformatics, and 
+analysis pipelines.  
+For analyses using R, reproducibility is a challenge because more and more packages with different 
+versions are available on CRAN and results can differ depending on the exact versions. Theußl et al. 
+(2011) describe general challenges resulting from the increasing number of R packages. 
+Technical tools to streamline and simplify the creation and maintenance of reproducible workflows, 
+such as the targets package in R, have emerged in recent years (Landau, 2021). However, such 
+tools are not widely used in the community, partly because they are still complicated and require a 
+steep learning curve to their adoption. Thus, as alluded to in previous sections, good RSE practices are 
+fundamentally important as they provide basic building blocks to improve research reproducibility.  
+Initiatives such as The Turing Way have been advocating for all stakeholders to understand their roles 
+and responsibility of reproducibility in quantitative research (Arnold et al., 2019). Furthermore, they 
+introduce and promote tools that, while common in the RSE community, are still severely underused in 
+biostatistical settings: version control, containerization, code review, and continuous integration, to 
+name a few. Software engineering in biostatistics (and other research disciplines), for both the creation 
+of bespoke analysis pipelines and reusable software packages, could then greatly benefit from the 
+adoption of modern RSE tools. Finally, the issue of research reproducibility is often too abstract or 
+broad of a problem to be appreciated in practice.  
+3 Opportunities 
+3.1 Education 
+In both academia and industry, we need skilled statistical software engineers. But how do we get them? 
+Some university courses include software engineering practices and coding classes for researchers, 
+such as “The Carpentries”, which has been operating since 1998 (The Carpentries, n.d.). If we look at 
+training strategically, we see three main pillars: (1) software education in schools, (2) undergraduate 
+and graduate training at universities, and (3) life-long learning opportunities for academics and industry 
+personnel.  
+We envision a future where fun coding projects at schools become the norm rather than the notable and 
+often extracurricular exception (see for example Girls Who Code or Jugend Hackt) (Girls Who Code, 
+n.d.; “Jugend hackt – Mit Code die Welt verbessern,” 2019). Bachelor’s and master’s programs in 
+statistics and data science are already incorporating more software skills than they used to, yet, to our 
+knowledge, no specific programs with a focus on statistical software engineering exist. We encourage 
+lecturers to incorporate topics such as good engineering practices (e.g., version control, documenting, 
+modular coding) in current statistics curricula. Further, we suggest that universities cater to the 
+increasing demand in skilled RSEs, particularly in the fields of statistics and data science. As most 
+statisticians have an increasing need for software skills, life-long learning is the third important pillar. 
+Online courses (e.g., Coursera), books (e.g., Software Engineering with Python), RSE workshops, and 
+conferences provide good possibilities to continue learning RSE skills after university studies 
+(Coursera, n.d.; Irving et al., 2021). 
+We note, as an important development, that many university curricula in the life sciences (e.g., biology 
+and biochemistry) introduce mandatory data science education for their students. While this still a 
+rather new development, we consider the formal introduction of software engineering skills to students 
+of application domains to be of paramount importance. 
+3.2 RSE teams 
+
+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+Statistical software engineers can support and train statisticians and researchers with their expertise and 
+build infrastructure for efficient research. RSEs who are integrated into research teams are one part of 
+the puzzle, and dedicated RSE teams are another.  
+In academia, we currently see dedicated RSE teams being established in many institutions in the United 
+Kingdom. While these are not necessarily focused on statistical software, examples include the RSE 
+groups at the University of Sheffield and the Alan Turing Institute. Another example is the Quantitative 
+Biology Center (QBiC) at the University of Tübingen, which operates as a core facility for the life-
+science campus and has offered services for data-management and bioinformatics analysis since 2012. 
+Sustainable software engineering and reproducible analysis became a central pillar over the last years, 
+resulting in a joint effort community for reproducible  bioinformatics analysis (Ewels et al., 2020). At 
+QBiC, the need for sustainable development of business-critical software has been addressed with the 
+implementation of dedicated groups and leadership positions that constantly improve in software 
+engineering practices. This includes coding katas (CodeKata, n.d.), pair programming sessions, peer-
+review of pull-requests, workshops (e.g., in user experience design), and teaching in software 
+architecture principles. Especially for young developers, the resistance towards code sharing with 
+others and fear of receiving negative feedback can be high. However, this can be addressed when they 
+participate in these regular sessions of sharing and learning from each other. The implementation of a 
+research software engineering team as a central unit of an academic institution has been critical for 
+strategic developments in biomedical research projects and is also used as a blueprint in similar settings 
+internationally. 
+Similarly, dedicated RSE teams focused on statistics are being formed in industry settings. One 
+example is the Statistical Engineering team in Roche, which is closely working together with applied 
+biostatisticians, methods experts, and IT professionals (Sabanés Bové, 2022). The team develops 
+business critical R packages, R/Shiny modules, and how-to templates that are used across multiple 
+projects to enable efficient data science solutions. An important part of the strategy is the open-source 
+collaboration with other companies and institutions, for example, see the crmPack example described 
+below. Another example is the Digitalization & Computational Science team in the Bayer Oncology 
+Strategic Business Unit with similar focus as the Roche Statistical Engineering team. The team was 
+recently formed and has grown significantly due to the increasing need for professional data science 
+solutions within statistics and data management, but also outside of the classical data science functions, 
+e.g., in Medical Writing and Clinical Operations to complement and accelerate their current practices 
+and processes using advanced analytics. 
+For statistics there is still plenty of open room for dedicated RSE teams, but with better training (see 
+above) and more attractive career paths (see below) we predict that dedicated statistical software 
+engineering teams will become common soon. 
+3.3 Career paths 
+Statistical software engineers are highly valuable experts as they are knowledgeable in research, 
+statistics, and software engineering. With that skill set one has plenty of opportunities for work in both 
+academia and industry (including self-employment). It is of vital importance for the interested 
+employers to provide an attractive work environment. In academia, the discussion already starts with 
+valuing software as a research work output and incentivizing good research software. Currently, a 
+paper publication is still the most valued output of a research group but we notice increasing demands 
+to make software a first class citizen in research and to give credit to software contributors and 
+maintainers (Kuzak et al., 2018; Taschuk & Wilson, 2017). Of course, not all RSEs want to or can be 
+on the classical academic career path that leads to a professorship. Hence, we need to provide dedicated 
+career paths for RSEs that go beyond the postdoc level, such as RSE group leadership (e.g., 
+implemented in the QBiC) or career paths mirroring those of librarians and lecturers. The UK and 
+France are among the countries who are already addressing the issue of attractive permanent positions 
+for RSEs in academia, but in most countries we see large opportunities for improvement (Society of 
+Research Software Engineering, n.d.).  
+
+Biometrical Journal 63 (2021), ZZZZZZ / DOI 10.1002/bimj.201010000 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+The demand for software engineers fluent in data science languages such as R and Python is also high 
+in the industry (Varney, 2018). This is not restricted to statistics, e.g., chemistry departments are also 
+employing software engineers (Python Success Stories, n.d.). It needs to be accounted for that RSEs are 
+also in demand by tech companies, and hence the competition for such profiles is high. The search and 
+recruitment process including interviews can thus take substantial resources and time. Therefore, it is 
+even more important to retain RSEs once they are found and hired and offering a competitive career 
+path is an important retainment factor. The career paths in the industry are diverse for biostatisticians 
+and software engineers within biostatistics; both contract-based temporary roles as well as permanent 
+roles are available. It will be important to allow the same seniority levels and compensation packages 
+for software engineers compared to, for example, methodology expert roles or managerial roles.  
+3.4 Reproducibility 
+Dedicated software utilities can help to address the reproducibility challenge. To address the 
+dependency challenge for R software, renv (Ushey et al., 2022) allows saving the state of the R 
+library in a single file and later restoring it from there. Another more recent dependency management 
+toolkit for R is the small command line tool Rmageddon which helps to use the R session information 
+and resolve all dependencies against Conda’s bioconda and R package channels (Fillinger, 
+2018/2020). The resulting environment file can then be used to build immutable and portable Docker 
+containers that can be shared with others. A recent review of tools applied to biostatistics is given by 
+Hejblum et al. (2020).  
+Approaches such as that pioneered by ReproHack, where workshop participants “attempt to reproduce 
+published research of their choice from a list of proposed papers with publicly available associated 
+code and data”, are fundamentally important to improve the general awareness and solutions for 
+reproducible research (ReproHack Core, n.d.). Practicing research reproducibility as a learning 
+experience allows critical failure points (such as software dependencies and versioning, remember for 
+instance the abovementioned “isoband” incident) to be fully experienced and appreciated, 
+highlighting the inherent complexities of building and maintaining reproducible research pipelines. 
+3.5 Reliability 
+Reliability is a key requirement for statistical software, and we mention here several reliability 
+initiatives for the R programming language. 
+The R Validation Hub is a collaboration to support the adoption of R within a biopharmaceutical 
+regulatory setting. The group received funding from the R Consortium and has participants from over 
+60 organizations. In early 2020, the R Validation Hub published a white paper introducing "A risk-
+based approach for assessing R package accuracy within a validated infrastructure” (Nicholls et al., 
+2020). The framework addresses concerns raised by statisticians, statistical programmers, informatics 
+teams, executive leadership, quality assurance teams, and others within the pharmaceutical industry 
+about the use of R and selected R packages as a primary tool for statistical analysis for regulatory 
+submission work (Nicholls et al., 2020). In a nutshell, there is minimal risk in using base R and 
+recommended packages as a component in a validated system for regulatory analysis and reporting 
+(The R Foundation for Statistical Computing, 2021).  
+Contributed R packages may differ in popularity and accuracy. Risk assessment criteria can be broken 
+down into four categories: package purpose, good maintenance practices, testing coverage, and 
+community usage. The R Validation Hub has also created tools to facilitate gathering information for 
+risk assessment, including the riskmetric R Package and the Risk Assessment 
+application (R Validation Hub et al., n.d., 2020/2022).  The concept of risk-based R package 
+assessment has been implemented by various companies into their standard processes. Reflection and 
+
+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+learnings, including which aspects were easy to implement into practice and where difficulties 
+occurred, are discussed in  Manitz et al. (2022).  
+In addition to sharing ideas and tools on how to check the reliability of R packages, it would be of great 
+help to have a repository of R packages which are deemed reliable enough to be used for medical data 
+analysis. To this end, the Regulatory R Package Repository working group was established (RC 
+Working Group on Repositories, 2021/2022). The idea is to help users to differentiate easily between 
+high-quality and standard packages and thus make the right decisions when choosing which software to 
+use for a given purpose. 
+3.6 Collaboration 
+While intra-organization collaboration is important, we focus here on the opportunities for inter-
+organization collaboration. The availability of real time video conferencing, document sharing and 
+editing, code reviewing, editing, and execution for data science applications via cloud-based web 
+services is providing an ideal technological infrastructure to seamlessly work together across 
+geographies and organizations. With open-source software being hosted on code sharing platforms such 
+as GitHub, Gitlab, Bitbucket, etc., it is only a matter of creating an account before being 
+able to ask questions to the developers and contribute ideas, documentation, or code to the software 
+project. 
+While this is also possible for sharing code building on top of proprietary software, it is easier to do for 
+packages and modules on top of open-source software, as this allows running of integration checks for 
+new code contributions directly on the code sharing platform. Moreover, every interested reader can 
+install the underlying open-source software together with the extension packages and try it out after 
+finding it online. 
+For both industry and academia, publication of code as open source is important. External stakeholders 
+for pharmaceutical companies, such as the regulatory authorities, health insurance payer institutions, 
+legislators, and the general public rightfully want to know how the data from clinical trials was 
+analyzed and summarized into the final published results. Here the possibility of sharing clinical trial 
+data is a first step (Hopkins et al., 2018), but the availability of the full stack of software used in the 
+data analysis will be an important second step. Interestingly, even for software companies it is 
+increasingly a better business model to publish the base version of the software as open source (Sahu, 
+2022).  
+For academic research, increasingly the software developed for studying new statistical methods is 
+required to be available as online appendices of papers. Here it is a competitive advantage to build on 
+open-source software. Starting from the publication of open-source software, it is then natural to 
+collaborate across organizations on this common public code base. It would not be of high academic 
+value to develop the same functionality in a separate software again, and it would not be good use of 
+resources for a company to internally build a software which is available open source already. It is 
+economical to start with what is readily available and contribute, especially since the software is getting 
+more reliable when the burden of developing, testing, and addressing user requests is shared across 
+more developers. 
+Certain hurdles might need to be overcome, e.g., discussions with legal departments will be needed 
+before publishing the first software modules open source that were previously kept internal in a 
+company. Parts of the initial software might need to be split in separate, internally kept modules when 
+they access internal APIs. In general, smaller software modules can be more easily reused across 
+companies, and therefore loosely coupled software packages should be preferred over tightly connected 
+software packages.  
+
+Biometrical Journal 63 (2021), ZZZZZZ / DOI 10.1002/bimj.201010000 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+Within the pharmaceutical industry, several initiatives drive forward the synergistic collaboration 
+between companies on common open-source software. The R consortium works with and provides 
+support to the R Foundation and to the key organizations developing, maintaining, distributing, and 
+using R software through the identification, development, and implementation of infrastructure 
+projects (R: The R Foundation, n.d.; RConsortium, n.d.). It hosts several working groups that work on 
+specific topics, e.g., submissions, tabulations, certifications, and repositories. The Software 
+Engineering working group (SWE WG) aims to engineer selected R packages to fill gaps in the open-
+source statistical software landscape and to promote good software engineering practices within 
+biostatistics (ASA Biopharmaceutical Section Software Engineering Working Group, 2022). The 
+PHUSE organization is a global community and platform for the discussion of statistical programming 
+topics, and PSI is a community dedicated to leading and promoting the use of statistics within the 
+healthcare industry (PSI Web, 2018; Warren, 2022). Both PHUSE and PSI have working groups 
+focused on programming with open-source software. 
+4 Examples 
+4.1 Reproducible bioinformatics analysis pipelines 
+A comparably recent example for the community-based implementation of RSE principles and the 
+development and maintenance of state-of-the-art scientific software is nf-core (Ewels et al., 2020; 
+Nf-Core, n.d.). Nf-core is a community effort and framework that is built on the basis of Nextflow 
+as a workflow management system. (Data-Driven Computational Pipelines, n.d.; Di Tommaso et al., 
+2017), The idea of nf-core is to transparently develop and provide bioinformatic analysis pipelines 
+with the scientific community and resolve redundant pipeline development due to silos of individual 
+bioinformaticians trying to address similar problems all over the world. The aim is to provide pipelines 
+that are reliable, reproducible, and well documented. Continuous integration and testing are a central 
+part of the development workflow, and containerization of software dependencies promotes numerical 
+stability of results when executing pipelines in different computing environments. What started with a 
+handful of bioinformaticians and an idea in 2017 matured to a world-wide community of 
+bioinformaticians with regular hackathons, trainings, and tutorials. 
+Such approaches to create frameworks can be role models for other domains and analysis types as well. 
+Typically, if designed well, such initiatives can start bottom-up without any heavy-lifting at the 
+beginning. Rmageddon mentioned above is another example. However, these helper tools need 
+support and commitment from motivated RSEs to mature to stable tools that continuously apply 
+methods from the software engineering discipline.    
+4.2 Dose escalation R package  
+Another example of a successful RSE project concerns the design and analysis of dose escalation trials. 
+These trials are often the first experimentation of new drugs in humans with the primary objective of 
+finding the maximum tolerable dose along with the recommended dose for further clinical 
+development.  In these trials, cohorts of patients are added sequentially and decisions regarding the 
+dose for the next cohort are made based on the available data. Dose escalation trials are exploratory by 
+nature and the design depends often on the properties of the investigational drug and the characteristics 
+of the target population. Frequent variations between dose escalation studies can be of operational or 
+methodological nature, e.g., inclusion of different dose levels, varying cohort sizes, flexible rules for 
+stopping the trial, and the underlying statistical model. 
+Designing such a phase I dose escalation trial using state of the art methodology usually includes study 
+simulations to derive operational characteristics. In addition, analysis software is needed to support 
+dose escalation meetings during the conduct of the trial. Often in-house developed software, 
+proprietary/commercially available software, or open-source packages without reliable maintenance are 
+used. Problems with this include the maintenance of in-house software, the inflexibility of proprietary 
+
+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+software, and the lack of validation of open-source software. Due to the continuous research and 
+proposal of new methods to be used in phase I trials, the problem becomes even more pronounced. 
+To overcome these problems, a group of industry, CRO, and academia statisticians came together to 
+evaluate the possibility to collaborate on the open-source R package crmPack (Sabanés Bové et 
+al., 2019). This package was originally developed at Hoffmann-La Roche Ltd. and open sourced in 
+2015. crmPack provides a simple and unified object-oriented framework for model-based dose 
+escalation designs. The package has already been used in some phase I trials in the industry and 
+academia individually, often tailoring it further to specific needs of the study. The group found that 
+crmPack already covers a wide variety of methods and that different companies and institutions 
+extended the package by including additional functions, more documentation, and testing for their 
+needs separately. To avoid duplication and ensure continued maintenance, the group agreed to 
+collaborate on the further development of the package and develop the package using modern software 
+development methods and tools. 
+The current workflow takes place on GitHub to ensure version control and reliable collaboration. The 
+collaborative work is driven by short iteration cycles by working on small tasks and prioritizing review 
+of pull requests. This way of working is fundamentally different from past software development in 
+clinical biostatistics, which typically involved long requirements documents and inefficient 
+collaborative work. Furthermore, the package is extended by unit tests for the functions to prepare for 
+subsequent validation of the package. The collaborative aspects of this collaboration were co-presented 
+by members of the development team at the ISCB conference (Boix & Günhan, 2022).  
+5 Discussion 
+A key element of modern biostatistics is to bring science and data together to generate knowledge. 
+Neither science nor data alone are keys to success. Both must be combined efficiently. The link 
+between both are computer programs. This link must be optimized to extract actionable insights from 
+the data. For example, by making use of data standards (e.g., CDISC) and moving away from one-off 
+analysis scripts to reusable analysis software for standard tasks, researchers would be relieved of some 
+menial work and could instead focus on complex tasks that add value to their organizations.  Analysis 
+methods are getting more complex, and the volume of data is increasing. Thus, special expertise is 
+needed to link both in a strong way. For this, state-of-the art programming methodology must be used 
+and should not merely be conducted alongside methodological research or data management. In our 
+view, implementing Research Software Engineering (RSE) as a recognized and dedicated profession, 
+jointly with a basic RSE skills education for all statisticians, is the way forward in fulfilling these 
+needs. RSE can facilitate cross-functional discussion and would support other functions to implement 
+novel ideas. This would not only include multidisciplinary collaboration within an organization, but 
+also structured cooperation across both academia and industry. 
+Today, we must also acknowledge the needed effort on maintenance and development for open-source 
+projects. The open-source community approach would benefit from RSE by dedicated experts who 
+focus on maintenance and rigor development of statistical software. For example, the current level of 
+interdependencies sometimes seen between R packages is somewhat like the “Spaghetti Crisis” 
+identified 40 years ago (Steele, 1977), leading to the implementation of a more structured code 
+development including dedicated informatic career path. In other words, companies and academia must 
+spend dedicated resources. Having dedicated research engineers would account for this, as the saying 
+goes, “nothing comes from nothing”.  
+An important element to foster RSE is that it must be regarded as a profession of its own with an 
+adequate placement within the job hierarchy of academia and industry. Young researchers must see this 
+as a desirable career opportunity. It is a serious, complex, and important task not to be underestimated. 
+Software Engineering is a little bit like riding a bicycle, in that most of us possess the basic skillset to 
+ride a bike, but that does not necessarily make us experts in designing racing bikes. RSE is more than 
+
+Biometrical Journal 63 (2021), ZZZZZZ / DOI 10.1002/bimj.201010000 
+ 
+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
+coding. Knowing basic programming language syntax is just a start, not the end. It requires objective 
+input from “outside”, collaborating with experts and more experienced peers can significantly boost 
+one’s skills.    
+In conclusion, the way of software development has drastically changed with the introduction of the 
+open-source concept. Similarly, the daily tasks for biostatisticians in academia and industry have 
+drastically changed. We must change our ways to match. It is time to improve software engineering in 
+biostatistics. 
+Acknowledgements: 
+We would like to thank Andy Nicholls and Martin Shaw who also participated in the 
+panel discussion at ISCB 43. 
+Conflict of Interest  
+The authors have declared no conflict of interest. 
+ 
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+Sabanés Bové et al: Improving Software Engineering in Biostatistics 
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+© 2021 Wiley-VCH GmbH, Weinheim   
+ 
+ 
+ 
+ 
+                          www.biometrical-journal.com 
+ 
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+ 
+

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+A general framework for implementing distances for categorical
+variables
+Michel van de Velden
+Alfonso Iodice D’Enza
+Angelos Markos
+Carlo Cavicchia
+January 6, 2023
+Abstract
+The degree to which subjects differ from each other with respect to certain properties measured
+by a set of variables, plays an important role in many statistical methods. For example, classification,
+clustering, and data visualization methods all require a quantification of differences in the observed
+values. We can refer to the quantification of such differences, as distance. An appropriate definition
+of a distance depends on the nature of the data and the problem at hand. For distances between
+numerical variables, there exist many definitions that depend on the size of the observed differences.
+For categorical data, the definition of a distance is more complex, as there is no straightforward
+quantification of the size of the observed differences. Consequently, many proposals exist that can
+be used to measure differences based on categorical variables. In this paper, we introduce a general
+framework that allows for an efficient and transparent implementation of distances between observations
+on categorical variables. We show that several existing distances can be incorporated into the framework.
+Moreover, our framework quite naturally leads to the introduction of new distance formulations and
+allows for the implementation of flexible, case and data specific distance definitions. Furthermore, in
+a supervised classification setting, the framework can be used to construct distances that incorporate
+the association between the response and predictor variables and hence improve the performance of
+distance-based classifiers.
+1
+Introduction
+In many statistical methods, the quantification of dissimilarity, that is, the degree to which objects differ
+from each other, plays an important role. We can refer to such dissimilarity quantification as a distance.
+Classification methods such as K-Nearest Neighbors (KNN, Cover and Hart 1967), but also clustering
+methods as K-means, (MacQueen 1967), partitioning around medoids (PAM, Kaufman and Rousseeuw
+1990) and hierarchical linkage methods (Gordon 1999), and data visualization methods such as multidi-
+mensional scaling (Borg and Groenen 2005) and biplots (Gabriel 1971; Gower, Lubbe, and Le Roux 2011),
+require a definition of distance between subjects and/or objects. The way to select a definition of distance
+depends on the nature of the data and problem at hand.
+Distance measures for numerical data are typically based on the magnitude of the observed differences
+in values (for a list of different distance measures, see, e.g., Mardia 1978). For categorical data, however,
+the situation is more complex, as we do not directly observe, and hence cannot directly quantify, sizes of
+differences. We can only directly establish whether there is a difference or not.
+For distance calculations in multivariate contexts, two cases can be distinguished. First, the distances
+are calculated for each variable independently and then added. Second, the association between the
+variables is taken into account when calculating the distances. For numerical variables, several well-known
+distances, for example, Euclidean or Manhattan distances, implicitly assume independence between the
+variables. Obviously, in such “independent” cases, the measurement scales must be commensurable. For
+categorical variables, there are also several measures that take the sum of dissimilarities per variable when
+considering a multivariate distance. For example, in simple matching, distance between two observations is
+defined as the number of times that the categories of corresponding variables do not match.
+For numerical variables, the association between variables can be accounted for using the Mahalanobis
+distance, where (sample) covariances are used to weigh observed differences to account for correlation
+1
+arXiv:2301.02190v1  [stat.ML]  4 Jan 2023
+
+between the variables. For categorical data, so-called association-based distances exist. In such distances,
+the association between categorical variables is used to quantify differences between observations. The
+question of how to account for associations in a categorical setting is not trivial. Several relatively recent
+new proposals for distances between categorical variables are indeed association-based distances (see, e.g.,
+Le and Ho 2005; Ahmad and Dey 2007; Jia, Cheung, and Liu 2014; Ring et al. 2015).
+The complexity of defining a distance for categorical variables, and recent interest in this topic, is
+illustrated by a wide range of articles that review existing (e.g., Boriah, Chandola, and Kumar 2008; Alves,
+Couceiro, and Napoli 2019) or introduce (new) distances (e.g., Le and Ho 2005; Ahmad and Dey 2007; Jia,
+Cheung, and Liu 2014; Ring et al. 2015; Šulc and ˇRezanková 2019; Bai and Liang 2022). In this paper,
+we propose a general framework for implementing categorical distances. Our framework can be used to
+incorporate existing categorical variable distances, but it also allows researchers to define and implement
+new or customized distances.
+By reformulating existing distances in our framework, it becomes possible to assess the differences
+and similarities between them. Currently, such comparisons are not trivial due to the wide variety of
+notation and research fields (and hence objectives) in which methods have been proposed. In addition, our
+framework makes it possible to construct and define new and highly customizable distances. For example,
+in a supervised classification context, the framework can be used to define a distance that takes into account
+association with the classes of the response variable.
+As our framework is not method- or application-specific, it can be used to calculate distance matrices
+for any method or application requiring distance calculations. For example, multidimensional scaling,
+cluster analysis, or, in a supervised context, K-nearest neighbors. We show that distance calculations
+using the framework are fast, efficient, and transparent and can be a significant improvement over existing
+implementations. In particular, we show that for the distance for categorical variables proposed in Ahmad
+and Dey (2007), our implementation is much faster than existing implementations.
+An important issue with regard to the definition of distance measures is the validation of the measures.
+That is, how does one know that the chosen measure is appropriate? Although the new framework does
+not provide an answer to this question, having one general formulation simplifies both theoretical and
+empirical comparisons between different choices.
+We illustrate our method by applying distance-based data analysis methods to several well-known
+categorical data sets using a selection of known and new, association-based, categorical distances. We
+implemented functions to perform all categorical distance calculations using our general framework in the
+R package catdist, which is available on GitHub1 and is soon to be released on CRAN.
+This paper is organized as follows. After introducing some notation in Section 2, we describe our general
+framework in Section 3. In Section 4, we introduce several common distance measures for categorical
+variables and show how they can be incorporated. Categorical distances based on co-occurrences are
+introduced in Section 5, with particular attention to a distance measure proposed by Ahmad and Dey (2007).
+In Section 6, we show how supervised distances can be constructed and implemented using our framework.
+Tuning of distance definitions is described in Section 7, after which we illustrate our methodology using
+several data sets in Section 8. Section 9 concludes the paper.
+2
+Notation
+Suppose that we have n observations on Q categorical variables and let the number of categories for the
+j ∈ 1...Q-th variable be qj. We can then code the categorical data by using indicator matrices. That is, for
+each categorical variable j ∈ 1...Q, we create an n×qj binary matrix Z j, where the n rows correspond
+to observations and the qj columns to categories. The observed category is indicated by a one, and all
+other categories are assigned zeros. Furthermore, for each observation of a categorical variable, exactly one
+category is observed, and we only include categories that have been observed at least once in the data set.
+Hence, each column of Z j contains at least one element equal to one and Z j1q j = 1n, where, generically, 1i
+denotes an i by 1 vector of ones. That is, the sum over the columns is 1.
+1https://github.com/alfonsoIodiceDE/catdist_package
+2
+
+Using these indicator matrices, we can code data on Q categorical variables in a so-called super-indicator
+matrix by collecting all indicator matrices next to each other. That is,
+Z =
+�
+Z1
+...
+ZQ
+�
+.
+Furthermore, define
+P = 1
+nZ′Z,
+(1)
+and
+Pd = 1
+n
+�
+Z′Z
+�
+⊙IQ∗,
+(2)
+where ⊙ indicates the Hadamard product, that is, element-wise multiplication, and Q∗ = ∑Q
+j=1 qj. Note
+that Pd is a diagonal matrix with as its diagonal elements the observed relative frequencies (within each
+variable) for the categories. Moreover, let
+p = Pd1Q∗
+(3)
+denote the vector of observed relative frequencies, and
+p− = P−1
+d 1Q∗
+(4)
+is the vector of inverse observed relative frequencies.
+Note that the i j-th off-diagonal block of P gives the relative frequencies of co-occurrences for the
+categories of variables i and j. They can be seen as (empirical) joint probability distributions for variables i
+and j. For the calculation of association-based distances in Section 5, we also define
+R = P−1
+d (P−Pd).
+(5)
+The rows of the i j-th off-diagonal block of R give, for the categories of the i-th variable, the distributions
+over the categories of the j-th variable. These can be interpreted as (empirical) conditional distributions.
+3
+Categorical distance calculations based on category dissimilarities
+For a categorical variable, it is not obvious how to quantify differences between different categories. For
+example, suppose that we observe three individuals, one from the Netherlands, one from Italy, and one
+from Greece. Geographically, and perhaps also culturally, Italy and Greece are more similar than the
+Netherlands. How to take such differences into account is, however, not trivial. In our framework, we do so
+by defining category dissimilarities.
+A matrix ∆∆∆ j is the category dissimilarity matrix for variable j. The elements of this matrix, δab, where
+a and b indicate two categories of variable j, quantify the dissimilarities between the categories a and b
+of the j-th variable. We can impose conditions on the dissimilarity matrix that are consistent with typical
+distance definitions. That is, 1) the dissimilarity of a category from itself is zero (δaa = 0, for all categories).
+2) Dissimilarities are symmetric (δab = δba, for all pairs of categories). 3) Dissimilarities satisfy the triangle
+inequality. That is, if a,b and c denote different categories for a variable j, then, for all categories a,b and
+c,
+δac ≤ δab +δbc.
+If all three of these conditions are satisfied, the dissimilarities can be considered as metric distances between
+categories. If they are non-negative and satisfy only the first two conditions, they can be interpreted as
+non-metric distances between categories. However, we refer to them as category dissimilarities and reserve
+the term “distance” for the distances between observations.
+3
+
+If we have Q categorical variables, each with a category dissimilarity matrix ∆∆∆ j, we can construct a
+Q∗ ×Q∗, block diagonal matrix ∆∆∆, with separate category dissimilarity matrices as diagonal blocks.
+The category dissimilarity matrices can be used to calculate a between observations distance matrix
+as follows. First, consider the n by qj indicator matrix Zj corresponding to the j-th categorical variable.
+Furthermore, we have the corresponding category dissimilarity matrix ∆∆∆ j. We can formulate the following
+theorems:
+Theorem 1. The distances between the observations for the categorical variable j are
+Dj = Z j∆∆∆ jZ′
+j.
+Proof. The matrix multiplication of the row i of Zj with ∆∆∆ j selects the row of ∆∆∆ j corresponding to the
+category chosen by the individual i. Similarly, matrix multiplication of this row by the i′-th column of Z′
+j
+(i.e., the i′-th observation) selects the element corresponding to the category chosen by the individual i′.
+Hence, the (i,i′)-th element of D j is the dissimilarity between the categories chosen by individuals i and
+i′.
+■
+Theorem 2. If we define the distance between observations on Q categorical variables as the sum of Q
+dissimilarities for each categorical variable, the n×n distance matrix can be calculated as
+D = Z∆∆∆Z′.
+(6)
+Proof.
+D = Z∆∆∆Z′
+=
+�
+Z1
+...
+ZQ
+�
+�
+�
+�
+∆∆∆1
+...
+∆∆∆Q
+�
+�
+�
+�
+�
+�
+Z′
+1...
+Z′
+Q
+�
+�
+�
+=
+Q
+∑
+j=1
+Zj∆∆∆ jZ′
+j
+=
+Q
+∑
+j=1
+Dj
+■
+From (6), it follows that distances between observations of categorical variables depend on the choices
+of the category dissimilarity matrices ∆∆∆ j. This allows for great flexibility in defining a suitable distance
+measure for a set of categorical variables. In the next section, we briefly review some choices for ∆∆∆, and
+we show how they relate to existing distances.
+Note that associations between categorical variables are not explicitly incorporated in this formulation.
+That is, the differences between the categories observed for one variable are not related to the differences
+in the categories observed for other variables. There are, however, ways to account for such observations.
+For example, rather than creating an indicator matrix for each categorical variable, one could construct an
+indicator matrix for all possible combinations (or subsets thereof) of observations. That is, one can create,
+for each (or a subset of) combination of categories one indicator matrix. The number of columns of such a
+matrix is therefore ∏Q
+j=1 qj and only one category dissimilarity matrix is needed where each category is a
+combination of the categories for all Q variables. However, with several categorical variables, the total
+number of combinations and hence the number of categories of the final indicator matrix quickly becomes
+large. Furthermore, finding an appropriate category dissimilarity matrix for the combinations is not a trivial
+task.
+An alternative way to account for associations between the categorical variables is to use them in
+the construction of the category dissimilarity matrices. That is, by defining the dissimilarities between
+the categories of a variable in ∆∆∆ j, based on the associations with other variables. In Section 5, we give
+examples of such category dissimilarity measures.
+4
+
+3.1
+Distances between sets
+Suppose that we have two separate sets of observations on the same Q categorical variables. Data for these
+two sets can be collected in the n1 ×Q∗ and n2 ×Q∗ super indicator matrices Z(1) and Z(2). Then, for a
+known category dissimilarity matrix ∆∆∆, it is easily verified that the distances between the observations for
+the two sets can be calculated as
+D(12) = Z(1)∆∆∆Z(2)′.
+(7)
+Note that the matrix D(12) is of order n1 ×n2.
+The calculation of distances between sets can be useful when considering distance-based classification
+problems. In KNN, for example, distances between “new” (unlabeled) observations and observations in a
+labeled data set are required. The KNN predictions are based on (usually by majority vote) the labels of
+the K nearest neighbors in the training set. Similarly, in partitioning around medoids, a popular clustering
+algorithm similar to K-means, where instead of considering within-cluster variation around the mean,
+variation around an actual observation, the medoid, is considered. If the medoids are collected in Z(1) and
+“new” data points in Z(2), we can assign the new points to existing clusters considering the distances D(12)
+and selecting the smallest distances.
+4
+Independent category dissimilarity matrices
+We first consider several definitions of dissimilarity for categorical data that do not take into account the
+association between variables. Hence, in a multivariate context, distances are calculated as the sum of
+distances per variable, and for each variable, the category dissimilarities are independent of the observations
+on other variables. However, category dissimilarities may depend on the observed frequencies for a variable.
+We do not aim to be complete with respect to the different definitions of dissimilarity. Instead, we select
+definitions from Šulc and ˇRezanková (2019) (which contain several definitions also reviewed in Boriah,
+Chandola, and Kumar 2008; Alves, Couceiro, and Napoli 2019), and transform them into dissimilarities. We
+show how these dissimilarities can be incorporated into our framework by defining the appropriate category
+dissimilarity matrices ∆∆∆. For a more detailed description, as well as study on the relative performances of
+these dissimilarity definitions in a cluster analysis setting, see Šulc and ˇRezanková (2019).
+4.1
+Overlap or simple matching
+The idea of simple matching (SM) is that the distance between observations is 1 if the categories do not
+match and 0 if they do. Consequently, all different between category dissimilarities are 1. For the j-th
+categorical variable with q j categories, we define
+∆∆∆Mj = 1q j1′
+qj −Iq j.
+That is, the distance between each category is exactly 1. If we have Q categorical variables and want to
+calculate a simple matching for all variables, we simply collect all ∆∆∆Mj in a block diagonal matrix ∆∆∆M.
+∆∆∆M = Kb −IQ∗,
+where Kb is a Q∗ ×Q∗ block diagonal matrix with, for j = 1...Q, qj ×q j matrices (Kb j) of ones as its
+diagonal blocks.
+4.2
+Eskin
+For Eskin distance (Eskin et al. 2002), category dissimilarities depend on the number of categories.
+Dissimilarities for variables with more categories are smaller than dissimilarities for variables with fewer
+categories. In particular, the dissimilarity between different categories for a variable with qj categories is
+5
+
+2/q2
+j. Therefore, for the j-th categorical variable with qj categories, the category dissimilarity matrix is
+defined as
+∆∆∆Ej =
+�
+2/q2
+j
+��
+1qj1′
+q j −Iqj
+�
+.
+Collecting all Q dissimilarity matrices in a block diagonal matrix produces the Eskin category dissimilarity
+matrix ∆∆∆E. If all variables have the same number of categories, Eskin merely re-scales the simple matching
+dissimilarity.
+4.3
+Lin
+Lin (1998) proposed an information-theoretic measurethat gives more weight to matches on frequent values
+and lower weight to mismatches on infrequent values. We implement Lin’s proposal as follows: define
+Pr = p1′
+Q∗ and Pc = 1Q∗p′, where p is as defined in 3. Furthermore, let �P = Pr + Pc − Pd. Then, the
+category dissimilarity matrix can be defined as
+∆∆∆Lin =
+�
+log(Pr)+log(Pc)−2log(�P)
+�
+⊘2log(Pr +Pc),
+where ⊘ indicates the Hadamard division (i.e., element-wise) and log(·) takes the logarithms of the
+elements of the parenthesized object and collects them in an object of the same size.
+Note that Lin’s dissimilarity for a category with itself is zero. Furthermore, in our implementation, for
+each variable, the dissimilarity between different categories, say categories a and b, is
+[log(pa)+log(pb)−2log(pa + pb)]/2log(pa + pb),
+where pa and pb are, respectively, the relative frequencies of categories a and b.
+4.4
+Inverse occurrence frequency
+For inverse occurrence frequency (IOF, Boriah, Chandola, and Kumar 2008), a higher dissimilarity is
+assigned when categories are more frequently observed. In particular, the category dissimilarity matrix is
+defined as
+∆∆∆IOF = [log(np)][log(np)]′ −[log(np)][log(np)]′ ⊙IQ∗.
+It is worth observing that, for each variable, IOF dissimilarity for a category with itself is zero, and the
+dissimilarity between two different categories, say a and b, corresponds to
+log(npa)log(npb).
+The IOF measure is related to the concept of inverse document frequency (TF-IDF) from information
+retrieval, where it is used to account for document relevance for a given term (Spärck Jones 1972). In other
+words, since a rare term contributes more information than a more frequent term, the IOF measure accounts
+for how rare the term is, and a lower IOF dissimilarity corresponds to a rarer term. Log frequency is used
+to reduce the impact of terms of very high frequencies.
+4.5
+Occurrence frequency
+For occurrence frequency (OF) dissimilarity, dissimilarities are higher if the categories are observed less
+frequently. The category dissimilarity matrix is defined as
+∆∆∆OF =
+�
+log
+�
+p−���
+log
+�
+p−��′ −
+�
+log
+�
+p−���
+log
+�
+p−��′ ⊙IQ∗.
+Therefore, OF dissimilarity for a category with itself is zero, and the dissimilarity between two different
+categories a and b is
+log(pa)log(pb).
+6
+
+4.6
+Goodall dissimilarities
+In Boriah, Chandola, and Kumar (2008), four variations of Goodall’s similarity are considered. These are
+based on Goodall’s original proposal (Goodall 1966). After transforming similarities into dissimilarities,
+where dissimilarity is 1− similarity, the four measures have in common that dissimilarities between
+different categories are, as is the case with simple matching, always equal to one. However, the dissimilarity
+of a category with respect to the same category depends on the observed proportions of the categories.
+Below we provide the category dissimilarity matrices for Goodall 3 and Goodall 4. For Goodall 1 and 2,
+we can also construct such matrices. However, these definitions require conditional sums of proportions.
+In particular, for Goodall 1, the dissimilarity for category a with itself, is defined as the sum of squared
+observed proportions that are smaller or equal to the observed proportion of category a. For Goodall 2, it is
+the sum of squared observed proportions that are larger or equal to the observed proportion of category a.
+The Goodall 3 and 4 measures do not require the calculation of a (conditional) sum and have the
+squared proportion and one minus the squared proportion of a category, respectively, on the diagonal blocks
+of ∆∆∆. That is,
+∆∆∆G3 = Kb −IQ∗ +P2
+d,
+and
+∆∆∆G4 = Kb −P2
+d.
+In these definitions, dissimilarity of a category with the same category is not zero. Consequently, the
+resulting “distances” do not satisfy the typical requirements of a distance. Note that for the Goodall 1 and 3
+measures, a higher dissimilarity is assigned when the matching categories are frequent, whereas for the
+Goodall 2 and 4 measures a higher dissimilarity is assigned when the matching categories are infrequent.
+4.7
+Variable Entropy and Variable Mutability dissimilarities
+Šulc and ˇRezanková (2019) proposed two variability-based dissimilarity measures that are related to
+Goodall 1 and 2, respectively. These dissimilarities are equal to one if the categories do not match, while,
+if they do match, the Variable Entropy (VE) measure uses the entropy and the Variable Mutability (VM)
+measure uses the Gini coefficient to quantify “dissimilarity”. In particular, for the j-th categorical variable
+with qj categories, the category dissimilarity matrices are defined as
+∆∆∆VEj = Kb j +
+�
+1
+logqj
+q j
+∑
+l=1
+pl log pl
+�
+Iqj
+and
+∆∆∆VMj = Kb j −
+�
+qj
+qj −1
+�
+1−
+q j
+∑
+l=1
+p2
+l
+��
+Iq j,
+respectively. Collecting all Q dissimilarity matrices in a block diagonal matrix returns the VE and VM
+category dissimilarity matrices ∆∆∆VE and ∆∆∆VM.
+4.8
+Ordered categories
+If categories are ordered, the order can be reflected in the dissimilarities. A simple choice would be to
+define the dissimilarities as the difference in category numbers. That is, the dissimilarity between categories
+a and b is simply b −a. If the data are rank order data or rating (e.g., Likert) scale data, this definition
+would imply treating the data as interval data. However, implementation of alternative, custom, definitions
+of ordered between-category distances is also straightforward. For example, if the categories correspond to
+bins on a numerical scale (e.g., age or income groups), differences between the midpoints of the bins can
+7
+
+be used to define dissimilarities that better reflect the underlying values. More generally, let ∆∆∆i
+o denote the
+i-th diagonal block of ∆∆∆∗
+o, and δ i
+ab its ab-th element. Then, dissimilarities between ordered categories can
+be imposed by letting δ i
+ab ≤ δ i
+ab∗, for a,b ∈ 1...iqi, b∗ ̸= b and b∗ > a. Note that this definition does not
+guarantee that the triangle inequality holds. That is, without additional constraints, it may be the case that
+the direct distances between two categories are larger than the indirect distances between those categories.
+5
+Association-based category dissimilarity matrices
+Several authors (e.g., Ahmad and Dey 2007; Jia, Cheung, and Liu 2014; Le and Ho 2005; Ring et al.
+2015) have proposed distance measures for categorical variables that take into account the association
+between categorical variables. The general idea is that, similar to the case of the Mahalanobis distance for
+numerical variables, differences that are in line with the association between variables are less informative
+(i.e., should correspond to smaller dissimilarity values) than differences that are not in line with the general
+association. How to exactly implement this idea depends on the calculation of the association between
+categorical variables, and how to incorporate this association in the category dissimilarities.
+Here, we present a general form to calculate and collect association-based dissimilarities that can be
+directly implemented in our general framework in Section 3. We then present some specific variants and
+link them to recent proposals.
+5.1
+A general form for association-based dissimilarities
+Fundamental in the calculation of association-based distances is the matrix of proportions of co-occurrences
+P and the corresponding profile matrix R as defined in Equations (1) and (5). In particular, recall that
+the off-diagonal blocks of P and R can be interpreted as (empirical) joint and conditional probability
+distributions, respectively. By considering different ways to quantify the dissimilarities between the
+conditional distributions (i.e., the rows of the off-diagonal blocks of R) we can construct different category
+dissimilarity matrices ∆∆∆ that, by applying Equation (6) can be used to obtain the between-observation
+distances.
+Let Ri j denote the i j-th off-diagonal block of R, and let ri j
+a denote its a-th row. Note that the elements
+of each row of Rij add up to 1. Hence, these elements can be seen as (empirical) conditional probabilities.
+We define the dissimilarities between categories for all pairs of categories of variable i (for i = 1...Q)
+based on the association with variable j (with j ̸= i), as
+δ ij(a,b) = Φi j �
+ri j
+a ,ri j
+b
+�
+,
+(8)
+where, generically, a and b indicate categories of variable i and Φi j is the dissimilarity function that
+quantifies the differences between profiles based on the association between variables i and j. The overall
+between category dissimilarities for all pairs of categories of variable i can be defined as
+δ i(a,b) =
+Q
+∑
+j̸=i
+wi jδ i j(a,b) =
+Q
+∑
+j̸=i
+wi jΦi j �
+ri j
+a ,ri j
+b
+�
+.
+(9)
+The weights wij in Equation (9) allow flexibility with respect to the importance of different variables in the
+calculation of association-based category dissimilarities, as defined by Φi j. By collecting, for each variable
+i, the elements δ i(a,b) in a category dissimilarity matrix ∆∆∆i
+Φ, and organizing them on the diagonal of a
+block diagonal matrix, we obtain a dissimilarity matrix ∆∆∆Φ, which can be used to calculate the distances
+between the observations using Equation (6).
+Equation 9 provides a very general way to define category dissimilarities using pair-specific weights
+and dissimilarity functions.
+For the association-based dissimilarity functions Φi j, any function that quantifies the difference between
+two distributions can be used. A brief overview of 46 different functions and their implementation in the
+R package philentropy is described in Drost (2018), and a more comprehensive overview of those
+functions, dividing them into different types and classes, can be found in Cha (2007).
+8
+
+Concerning the choice of weights wij, we distinguish two options. In the first, all weights are equal.
+Usually 1 or 1/(Q−1), so that either sums or averages are obtained. Alternatively, different weights can
+be used for different pairs. These weights can either be selected using expert knowledge (e.g., based on
+the experience and preferences of the researcher) or by using a data-driven approach. For example, one
+could set certain weights to zero and others to some constant based on some predetermined data dependent
+criterion (e.g., a measure of association like Cramér’s V). Pairs with non-zero weights can then be referred
+to as “context” variables. Approaches using such context-based dissimilarities are described in Ienco,
+Pensa, and Meo (2009), Jia, Cheung, and Liu (2014), and Ring et al. (2015).
+If an objective measure of overall fit of a solution is available, one could consider wi j and Φi j as tuning
+parameters, and search combinations of these parameters to make a choice. In Section 7 we shall further
+explore the tuning of wij and Φij.
+In the next subsections, we describe some specific choices of dissimilarity functions. In particular,
+we provide definitions for category dissimilarities between categories a and b of variable i, based on the
+association between variables i and j. That is, we present specific choices of Φi j in Equation (8). Inserting
+these definitions into Equation (9) results in a dissimilarity matrix ∆∆∆Φ that can be used to calculate the
+between-observation distances. For ease of notation, we now drop the superscripts ij.
+5.2
+Total variation distance between profiles
+The total variation distance (TVD) between two discrete probability distributions can be defined as 1/2
+times the L1 norm between the distributions. We can implement this in our framework by defining the
+category dissimilarity function Φ as
+Φ(ra,rb) = 1
+2
+q j
+∑
+l=1
+|ral −rbl|,
+(10)
+where ral and rbl denote the l-th element of ra and rb, respectively.
+Calculating category dissimilarities using this definition for Φ is equivalent to the proposal (for
+categorical variables) by Ahmad and Dey (2007). However, as this relationship is not trivial and appears to
+be unknown, we present this here in some detail.
+5.2.1
+Ahmad and Dey’s categorical variable distance
+Ahmad and Dey (2007) argue that the dissimilarity between categories should be computed as a function of
+their distribution in the overall data set and in co-occurrence with other categories, rather than in isolation.
+The idea is to take into account co-occurrences of categories when constructing distances. The way they do
+this, is by considering all combinations of categories of one variable, and selecting the partitioning (that is,
+a combination of categories) for which the sum of proportions in the two complementary partitions for the
+two categories is maximal. Following Ahmad and Dey (2007), we can define the dissimilarity between
+categories a and b of variable i, with respect to the distribution over the categories of variable j, as
+δ(a,b) = max
+ωj (P(ωj|a)+P( ¯ωj|b)−1),
+(11)
+where ωj and its complement ¯ωj define a binary partition with respect to the categories of variable j, and
+P(ωj|a) denotes the proportion of observations with the category a of variable i, corresponding to the set
+of categories of variable j as defined by ωj. Note that the term −1 is only introduced to fix the upper
+limit of the dissimilarities at 1. The number of binary partitions for variable j, excluding the partitions
+containing all or no categories, equals 2q j −2, where qj gives number of categories of variable j, and hence
+this number grows exponentially when the number of categories for a variable increases. Ahmad and Dey
+(2007) propose an algorithm to calculate their distances. The order of their algorithm is O(Q∗2n+Q∗2 ¯q3),
+where Q∗ gives the total number of categories, n is the number of observations and ¯q denotes the average
+number of categories per variable. However, as we show below, and in more detail in Appendix A, for
+distances between categorical variables, the distance of Ahmad and Dey (2007) is equivalent to the total
+variation distance, and calculations using Equation 10 are much more efficient.
+9
+
+5.2.2
+Equivalence of Ahmad and Dey’s distance and the total variation distance between profiles
+When going from δ(a,b) to δ(b,a), the optimal partition ωj, that is, the combination of categories that
+maximizes the sum, is simply flipped (i.e., the complement is taken), hence Ahmad and Dey’s distance is
+symmetric:
+δ(a,b) = max
+ω (P(ω|a)+P( ¯ω|b)−1) = max
+ω (P(ω|b)+P( ¯ω|a)−1) = δ(b,a),
+where, for convenience, we dropped the subscripts j for the ω’s.
+As P( ¯ω|b) = 1−P(ω|b) and P( ¯ω|a) = 1−P(ω|a), we have
+δ(a,b) = max
+ω (P(ω|a)−P(ω|b))
+= max
+ω (P(ω|b)−P(ω|a))
+= max
+ω |P(ω|a)−P(ω|b)|.
+(12)
+Equation (12) shows that Ahmad and Dey’s distance is equal to finding the maximum difference between
+all combinations of observed proportions. This implies that we can express this distance as the supremum
+norm of a vector of differences between probabilities. The total variation distance as defined in (10) can
+also be defined as the largest difference between probabilities from two probability distributions that can be
+assigned to the same event. Therefore, the Ahmad and Dey (2007) distance for categorical variables is
+equivalent to the total variation distance. For the sake of completeness, we provide a complete proof of the
+equivalence in Appendix A.
+5.3
+Kullback-Leibler divergence between profiles
+Kullback-Leibler divergence (KL, Kullback and Leibler 1951; Kullback 1959) is an entropy-based measure
+of dissimilarity between probability distributions. Le and Ho (2005) define category dissimilarities for
+the categories of variable i by taking the sum of KL-divergences between the (empirical) conditional
+probability distributions over all other variables. Using similar notation as before, we can implement this
+divergence by setting all weights wij equal to one and by defining Φ as
+Φ(ra,rb) =
+q j
+∑
+l=1
+�
+ral log
+�ral
+rbl
+�
++rbl log
+�rbl
+ral
+��
+,
+where log() is the binary logarithm and ral and rbl denote, as before, l-th element of ra and rb, respectively.
+It is important to note that KL is not symmetric. Hence, distance calculations using ∆∆∆KL may result in
+non-symmetric distances.
+5.4
+χ2-distance between profiles
+A distance for categorical data, that has a strong link to the data visualization technique correspondence
+analysis, is the chi-squared distance. There exist several forms and implementations of the chi-square
+distance that differ with respect to the chosen standardization. That, is, chi-squared distance considers the
+squared differences between proportions divided by the expected proportions. For an n× p contingency
+matrix F j = Z′
+iZ j, the squared χ2-distance between rows a and b can be defined as
+s
+p
+∑
+l=1
+1
+f•l
+� fal
+fa•
+− fbl
+fb•
+�2
+,
+(13)
+where s is the sum of all elements of F and • denotes the summation in the appropriate dimension (rows or
+columns) of the matrix (see, e.g., Gifi 1990, p.266). In our notation, we can implement the chi-squared
+distances as category dissimilarities by defining
+Φ(ra,rb) =
+qj
+∑
+l=1
+1
+pl
+(ral −rbl)2,
+10
+
+where we dropped the constant s, pl corresponds to the l-th element of the j-th block of Pd and, as before,
+ral and rbl denote the l-th element of ra and rb, respectively.
+6
+Supervised association-based distances
+In a supervised setting, where we want to assign observations to classes (i.e., categories) for one variable,
+say y, based on observations on categorical variables xj where j = 1,...,Q, we can define a supervised
+variant of association-based categorical variable distances. That is, we can define category dissimilarities
+that take into account the association between variables y and x. Next, we can make predictions using either
+the K-nearest neighbors or a distance-based clustering method, where we fix the number of clusters to the
+number of classes and do a post-hoc comparison of clusters and classes. That is, we match the clusters to
+the true classes and assign labels accordingly.
+To define supervised association-based distances we create an n × c indicator matrix Zy, where c
+corresponds to the number of classes of y. If we add, to the right, this indicator matrix to Z and insert this
+supplemented Z into Equations (1) through (5), we can calculate category dissimilarities using Equations
+(8) and (9). Note that in this new setting, we have Q+1 variables and consequently Q+1 association-based
+category dissimilarity matrices ∆∆∆i. However, the (Q+1)-th diagonal block gives the category dissimilarities
+between the categories of the y variable. In a supervised setting, a category (class) of y is to be predicted
+based on data from the other Q variables. The category dissimilarities for y should therefore not be used.
+This is easily achieved by simply ignoring these in the overall block diagonal category dissimilarity matrix
+∆∆∆. That is, we construct ∆∆∆ by collecting only the first Q category dissimilarity matrices on its diagonal.
+As before, our framework allows for great flexibility in how to incorporate the information of variable y.
+In particular, the dissimilarity functions Φij and the weights wi j are pair-specific. One could, as suggested in
+Section 5, set all weights wij equal to 1, so that the category dissimilarities take into account all associations.
+We refer to this choice as “full supervised” dissimilarity.
+Alternatively, in a supervised setting, one may choose to have the category dissimilarities depend only
+on the association with the variable y. This corresponds to the choice wi j = 1 for j = Q+1 and 0 for all
+other pairs. In this case, category dissimilarities may better discriminate with respect to the classes of y.
+We refer to this choice as “supervised” dissimilarity. Note that both supervised variants require a choice of
+association-based dissimilarity functions Φi j, for all pairs of variables.
+7
+Aggregation and dissimilarity tuning
+Our general framework introduced in Section 3 allows for great flexibility in the implementation of
+distances between categorical variables. In particular, in the previous sections, we introduced a selection of
+category dissimilarity measures. However, there are many more options. For example, all 46 functions
+available in the R package philentropy, described in Drost (2018), can be used for association-based
+functions. Moreover, as is clear from Definition (9, all separate category dissimilarity measures can be
+combined and aggregated according to the researcher’s preferences. How to exactly determine which
+category dissimilarity and aggregation strategy is the most appropriate is non-trivial, and this choice may
+depend on the properties of the data and the research objectives.
+In several distance-based methods, for example cluster analysis and multidimensional scaling, a clear
+measure of fit is not available, as the methods tend to be primarily exploratory. That is, the goal is to find
+and interpret patterns in the data. As the interpretability of a solution is not easily quantified, validation is
+typically not trivial. If, however, a measure of fit can be calculated, we can use this to select an aggregation
+and category-dissimilarity definition. That is, we can apply several aggregation and category dissimilarity
+definitions, and compare the fit for each of them by considering the selected measure.
+In a supervised classification setting, where we have a data set for which the true classes are known,
+we can assess the fit by comparing true classes with “predicted” classes. A choice for aggregation and
+category dissimilarity definitions, can then be made based on the discrepancy between these. Therefore,
+the aggregation state (that is, the weights wi j) and the category dissimilarity function (that is, Φi j) can be
+11
+
+treated as tuning parameters. Note, however, that Equation 9 allows many combinations and some choices
+need to be made to restrict the total search space.
+8
+Applications
+To illustrate how our general framework can be used in practice, we consider distance-based methods for
+supervised and unsupervised learning. In a supervised setting, a distance-based approach is K-nearest
+neighbors averaging, which can be used in regression and classification problems: each new observation is
+labeled according to a set of K close training points (neighbors). In an unsupervised setting, distance-based
+cluster analysis aims to assign observations to groups (clusters) for which the within-cluster distances are
+small, whereas the distances between clusters are large.
+As a general setup, we consider nine different labeled data sets (see Table 1), all available via the UCI
+Machine Learning repository2. Each data set is split into five folds, for cross-validation. On the training
+folds, a block diagonal matrix ∆∆∆ of pair-wise category dissimilarities is calculated for each of the reviewed
+dissimilarity measures (see Table 2). The test fold is used for performance assessment of the considered
+methods. The performance metric depends on the considered method:
+• Accuracy of the nearest neighbors classifier: proportion of the test observations correctly classified
+(Metz 1978);
+• Adjusted Rand Index (ARI, Hubert and Arabie 1985) comparing the cluster allocation of the test
+observations to the true cluster allocation (the labels).
+The procedure is iterated until each fold is used as test. The whole cross-validation process is repeated
+10 times, for different random splits.
+Table 1: Data set information
+Dataset
+n
+p
+# clusters
+australian
+690
+8
+2
+balance
+625
+4
+3
+cars
+1728
+6
+4
+lympho
+148
+18
+4
+soybean (large)
+307
+35
+19
+tae
+151
+5
+3
+tictac
+958
+9
+2
+vote
+435
+16
+2
+wbcd
+699
+9
+2
+Table 2: Category dissimilarity measures considered in the experiments
+Independent
+Association-based
+SM (Sec. 4.1)
+TVD (Sec. 5.2)
+Eskin (Sec. 4.2)
+KL (Sec. 5.3)
+Lin (Sec. 4.3)
+KL (Sec. 5.3)
+IOF (Sec. 4.4)
+Supervised TVD, Supervised TVD-full (Sec. 6)
+OF (Sec. 4.5)
+Goodall 3 and 4 (Sec. 4.6)
+VE, VM (Sec. 4.7)
+2https://archive.ics.uci.edu/ml/index.php
+12
+
+8.1
+K-nearest neighbors of categorical data
+The KNN classification of the test observations is based on the calculation of the distance between
+each test observation and the training observations, as described in Section 3.1. Let ∆∆∆train denote the
+category dissimilarity matrix, where the subscript train indicates that if the category dissimilarities are data
+dependent, only observations of the training set were used. Furthermore, Ztest and Ztrain are the indicator
+matrices of the test and training observations, respectively. The distances of interest are in the columns of
+Ztrain∆∆∆trainZ′
+test
+and the nearest neighbors for the j-th test observation are the K smallest values in the j-th column.
+Since the lower the number of considered neighbors, the higher the flexibility of the classifier, K is a
+hyper-parameter. We tune this hyper-parameter using the repeated cross-validation validation procedure
+described in Section 8. In particular, we consider values for K ∈ {1,3,5,9,15,21} and, for each data
+set/category dissimilarity combination, the value of K is chosen that minimizes the cross-validation estimate
+of the classifier’s test accuracy.
+Figure 1 presents the accuracy assessment of the tuned KNN classifier for each considered data set and
+for each considered category dissimilarity definition. In each panel, the position of each point corresponds
+to the accuracy obtained using the indicated category dissimilarity definition; the size of each point is
+proportional to the tuned value of the hyper-parameter K. The lines centered at each point span twice the
+standard deviation of the accuracy over the 10 replicates. For each data set, the distances are reported
+in descending order, highlighting the best performing ones. For some data sets, e.g., australian, vote
+and wbcd, the accuracy is high almost irrespective to the chosen distance, with no variability over the 10
+cross-validation replicates. For smaller data sets, such as lympho and tae, there is more variability over the
+10 cross-validation replicates, as expected.
+tictac (n: 958, p: 9, cl: 2)
+vote (n: 435, p: 16, cl: 2)
+wbcd (n: 699, p: 9, cl: 2)
+lympho (n: 148, p: 18, cl: 4)
+soybeanlarge (n: 307, p: 35, cl: 19)
+tae (n: 151, p: 5, cl: 3)
+australian (n: 690, p: 8, cl: 2)
+balance (n: 625, p: 4, cl: 3)
+cars (n: 1728, p: 6, cl: 4)
+0.25
+0.50
+0.75
+1.00
+0.25
+0.50
+0.75
+1.00
+0.25
+0.50
+0.75
+1.00
+Of
+Lin
+Iof
+Var_mutability
+Var_entropy
+Gifi_chi2
+Kullback−Leibler
+Supervised_full
+Tot_var_dist
+Supervised
+Goodall_3
+Goodall_4
+Eskin
+Matching
+Iof
+Tot_var_dist
+Goodall_4
+Kullback−Leibler
+Eskin
+Matching
+Supervised_full
+Var_entropy
+Goodall_3
+Gifi_chi2
+Var_mutability
+Supervised
+Of
+Lin
+Of
+Lin
+Goodall_4
+Matching
+Eskin
+Var_mutability
+Gifi_chi2
+Var_entropy
+Goodall_3
+Kullback−Leibler
+Iof
+Supervised
+Tot_var_dist
+Supervised_full
+Kullback−Leibler
+Gifi_chi2
+Var_entropy
+Iof
+Var_mutability
+Eskin
+Matching
+Goodall_4
+Goodall_3
+Tot_var_dist
+Lin
+Of
+Supervised_full
+Supervised
+Goodall_4
+Lin
+Of
+Var_mutability
+Var_entropy
+Iof
+Eskin
+Matching
+Gifi_chi2
+Kullback−Leibler
+Goodall_3
+Tot_var_dist
+Supervised_full
+Supervised
+Lin
+Var_mutability
+Var_entropy
+Goodall_3
+Of
+Eskin
+Matching
+Iof
+Supervised_full
+Tot_var_dist
+Gifi_chi2
+Goodall_4
+Kullback−Leibler
+Supervised
+Lin
+Of
+Goodall_4
+Goodall_3
+Eskin
+Gifi_chi2
+Supervised_full
+Tot_var_dist
+Var_entropy
+Var_mutability
+Iof
+Supervised
+Matching
+Kullback−Leibler
+Lin
+Of
+Eskin
+Goodall_4
+Matching
+Var_mutability
+Kullback−Leibler
+Var_entropy
+Goodall_3
+Iof
+Gifi_chi2
+Supervised_full
+Tot_var_dist
+Supervised
+Lin
+Of
+Supervised
+Eskin
+Matching
+Var_mutability
+Var_entropy
+Goodall_4
+Goodall_3
+Gifi_chi2
+Kullback−Leibler
+Iof
+Supervised_full
+Tot_var_dist
+accuracy
+distances
+distance measure
+Eskin
+Gifi_chi2
+Goodall_3
+Goodall_4
+Iof
+Kullback−Leibler
+Lin
+Matching
+Of
+Supervised
+Supervised_full
+Tot_var_dist
+Var_entropy
+Var_mutability
+5−fold CV
+KNN
+Figure 1: KNN classification accuracy for each considered distance measure and data set
+8.2
+Partitioning around medoids
+Partitioning around medoids (PAM) is an iterative clustering procedure that takes as input a matrix of
+pair-wise distances between a set of observations. Within a cluster, a medoid corresponds to the median
+13
+
+observation, just like a centroid in K-means corresponds to the mean. The starting set of the K medoids
+is random, and each observation is assigned to the closest medoid; given the allocation of the obtained
+clusters, the medoids are updated accordingly. The procedure stops iterating when there are no changes in
+the set of medoids. Although we are working in an unsupervised context, the performance of PAM on each
+data set/category dissimilarity combination can be assessed via cross-validation by calculating distances
+based on the supervised association dissimilarities; this also allows for consistency with the KNN-based
+application.
+In particular, for each data set and each category dissimilarity definition, a medoids set is obtained by
+applying PAM to the training data. That is,
+D = Ztrain∆∆∆trainZ′
+train
+where for ∆∆∆train we consider all category dissimilarity matrices described in Sections 4 and 5 and reported
+in Table 2. Next, we compute the test observation-to-medoid distance matrix as follows,
+Zmedoid∆∆∆trainZ′
+test,
+and assign each test observation to the cluster corresponding to the nearest medoid.
+The results are reported in Figure 2. We observe that, with the exception of the cars data set, for which
+performance of all methods is poor, the supervised association-based distance generally performs well.
+In general, the KNN and PAM results lead to the following conclusions.
+• Data sets for which classification accuracy is high in the supervised setting also have higher ARI
+values in the unsupervised setting.
+• The choice of category dissimilarity does not appear to impact classification accuracy when the
+overall performance of the method is very poor (for example, for cars) or very good (e.g., for wbcd).
+• In an unsupervised setting, association-based measures seem to provide an edge: in wbcd, five out of
+the six measures with an ARI value above 0.75 are association-based.
+Note that extended results and the R code to reproduce them are available online 3.
+9
+Conclusion
+In this paper, we propose a general framework for implementing distances between categorical variables in
+a flexible, efficient and transparent manner. In detail, we show that both independent and association-based
+distances can be incorporated in our framework. The latter can therefore be used to implement several
+existing measures, as well as to easily introduce new highly customizable ones.
+Our proposal is valuable from a theoretical perspective because it allows assessing the differences
+between dissimilarity measures by simplifying the wide variety of notation and definitions used in the
+literature, and therefore making their implementation much more transparent. From an applied perspective,
+since the proposed framework is not method- or application-specific, it can allow the definition of problem-
+specific dissimilarity measures. With respect to this, it is important to outline that in a supervised context
+our proposal can be used to define measures that consider associations with a response variable.
+For the independent category dissimilarities, the definitions described in Section 4 (with the exception
+of the ordered category dissimilarities) are implemented in the nomclust R package (Šulc and ˇRezanková
+2015). However, association-based measures are not implemented in this package. In the catdist
+package4 the independent as well as the association-based measures presented in this paper are implemented.
+3https://alfonsoiodicede.github.io/blogposts_archive/distances_experiment_superv_
+unsuperv.html
+4Available on GitHub at https://github.com/alfonsoIodiceDE/catdist_package and will soon be re-
+leased on CRAN.
+14
+
+tictac (n: 958, p: 9, cl: 2)
+vote (n: 435, p: 16, cl: 2)
+wbcd (n: 699, p: 9, cl: 2)
+lympho (n: 148, p: 18, cl: 4)
+soybeanlarge (n: 307, p: 35, cl: 19)
+tae (n: 151, p: 5, cl: 3)
+australian (n: 690, p: 8, cl: 2)
+balance (n: 625, p: 4, cl: 3)
+cars (n: 1728, p: 6, cl: 4)
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+0.00
+0.25
+0.50
+0.75
+Goodall_3
+Var_mutability
+Eskin
+Var_entropy
+Lin
+Matching
+Of
+Goodall_4
+Kullback−Leibler
+Iof
+Supervised_full
+Gifi_chi2
+Supervised
+Tot_var_dist
+Lin
+Tot_var_dist
+Of
+Supervised_full
+Var_entropy
+Matching
+Gifi_chi2
+Goodall_3
+Kullback−Leibler
+Var_mutability
+Goodall_4
+Iof
+Eskin
+Supervised
+Of
+Lin
+Goodall_4
+Eskin
+Matching
+Var_entropy
+Var_mutability
+Goodall_3
+Kullback−Leibler
+Gifi_chi2
+Iof
+Tot_var_dist
+Supervised_full
+Supervised
+Lin
+Of
+Tot_var_dist
+Supervised_full
+Eskin
+Matching
+Goodall_4
+Goodall_3
+Var_entropy
+Var_mutability
+Iof
+Gifi_chi2
+Kullback−Leibler
+Supervised
+Goodall_4
+Lin
+Of
+Var_mutability
+Var_entropy
+Iof
+Eskin
+Matching
+Goodall_3
+Kullback−Leibler
+Gifi_chi2
+Supervised_full
+Tot_var_dist
+Supervised
+Lin
+Eskin
+Matching
+Var_entropy
+Var_mutability
+Of
+Iof
+Gifi_chi2
+Goodall_4
+Tot_var_dist
+Supervised_full
+Goodall_3
+Kullback−Leibler
+Supervised
+Gifi_chi2
+Kullback−Leibler
+Of
+Lin
+Goodall_4
+Matching
+Goodall_3
+Supervised_full
+Iof
+Eskin
+Var_entropy
+Var_mutability
+Tot_var_dist
+Supervised
+Goodall_4
+Of
+Eskin
+Lin
+Kullback−Leibler
+Iof
+Goodall_3
+Matching
+Var_mutability
+Tot_var_dist
+Supervised_full
+Gifi_chi2
+Var_entropy
+Supervised
+Lin
+Of
+Var_entropy
+Var_mutability
+Goodall_3
+Supervised_full
+Iof
+Tot_var_dist
+Gifi_chi2
+Kullback−Leibler
+Eskin
+Matching
+Goodall_4
+Supervised
+test ARI
+distances
+distance measure
+Eskin
+Gifi_chi2
+Goodall_3
+Goodall_4
+Iof
+Kullback−Leibler
+Lin
+Matching
+Of
+Supervised
+Supervised_full
+Tot_var_dist
+Var_entropy
+Var_mutability
+k−medoids
+PAM clustering
+Figure 2: ARI results on K-medoids
+To illustrate the importance of selecting the “best” or the “most appropriate” distance for the problem
+at hand, we used our framework in both supervised (via KNN) and unsupervised (via PAM) contexts.
+Applications on real-world data sets revealed that choosing a specific measure will not affect neither classifi-
+cation accuracy nor clustering performance, respectively, when the variables have no discriminatory power,
+there is no strong cluster structure in the data, or KNN/ PAM are not appropriate methods for the problem
+at hand. Similarly, there are cases where all measures perform equally well. Putting extreme scenarios
+aside, choosing the “most appropriate” measure can lead to a classification / clustering improvement and
+association-based measures outperformed, in many cases, independent category measures. A further, more
+structured study, using synthetic data as well as a larger collection of empirical data sets, is needed to
+appraise this claim. Such a study is beyond the scope of this paper.
+By using the general framework proposed in this paper, new or customized distances can easily be
+implemented. In fact, the supervised total variation distances introduced in Section 6, for example, are “new”
+measures. However, rather than introducing and appraising new measures, it might be more interesting to
+consider a more systematic comparison of the strengths and weaknesses of different dissimilarity measures
+for categorical variables that are already available in the literature.
+Appendix A
+Here we prove that the definition of category dissimilarities using total variance, as in Equation (10), is
+equivalent to the definition of category dissimilarities proposed in Ahmad and Dey (2007). Recall that, as
+explained in Section 5.2.2, Ahmad and Dey (2007) define the dissimilarity between categories a and b of
+variable i, with respect to the distribution over the categories of variable j, as
+δ(a,b) = max
+ωj (P(ωj|a)+P( ¯ωj|b)−1),
+(14)
+where ωj and its complement ¯ωj define a binary partition with respect to the categories of variable j, and
+P(ωj|a) denotes the proportion of observations with the category a of variable i, corresponding to the set
+15
+
+of categories of variable j as defined by ωj. In Section 5.2.2, we showed that Equation (14) is equivalent to
+δ(a,b) = max
+ωj
+��P(ωj|a)−P(ω j|b)
+��.
+(15)
+Let Kq j be a design matrix that defines all, except the empty and complete, binary partitions for the qj
+categories of variable j. Hence, Kq j is a qj ×q⋆
+j matrix of zeros and ones, where q⋆
+j = ∑
+qj−1
+l=1
+�qj
+l
+�
+= 2q j −2.
+We can re-express Equation (15) as
+δ(a,b) =
+���K′
+qj (ra −rb)
+���
+∞ =
+���K′
+q jdab
+���
+∞ ,
+where dab = (ra −rb) and ∥x∥∞ denotes the supremum norm of vector x, that is, the maximum element of
+x in absolute value.
+The number of columns of Kq j and hence the size of the vector from which we need to take the norm,
+grows exponentially with the number of categories. For example, for qj = 4, q⋆
+j = 14 but for qj = 8 we
+have q⋆
+j = 254. When considering binary partitions, only half of these combinations are needed. Still,
+when the number of categories of the categorical variables is not too small, considering all combinations
+becomes computationally expensive.
+A more efficient way to calculate the distances between categories a and b can be obtained using the
+following relationship
+���K′
+qjdab
+���
+∞ = 1
+2 ∥dab∥1 ,
+(16)
+where ∥x∥1 denotes the L1 norm of vector x, that is
+∥x∥1 = ∑
+i
+|xi|.
+To see that Equation (16) holds, note that the maximum, in absolute value, for the combinations of
+elements of dab is obtained by selecting the combination consisting of elements that have the same sign.
+Furthermore, as the sum of elements of dab equals zero, that is:
+dab1q j = (ra −rb)1qj = 0,
+it immediately follows that the sum of all positive elements equals the sum of all negative values. Therefore,
+���K′
+q jdab
+���
+∞ = ∑
+l:dl>0
+|dl| = ∑
+l:dl<0
+|dl|,
+where dl denotes the l-th element of dab. Finally, as
+∥dab∥1 = ∑
+l
+|dl| = ∑
+l:dl>0
+|dl|+ ∑
+l:dl<0
+|dl|
+the equivalence in Equation (16) immediately follows, and we can express Ahmad and Dey’s distance
+between categories a and b with respect to the categories of variable j, as
+δ(a,b) = 1
+2 ∥dab∥1 = 1
+2 ∥dab∥1 = 1
+2
+qj
+∑
+l=1
+|ral −rbl|.
+(17)
+Comparing Equations 10 and 17 shows that Ahmad and Dey’s distance is equivalent to the total variation
+distance introduced in Section 5.2.
+Funding
+The authors received no financial support for the research, authorship, and/or publication of this article.
+16
+
+Conflict of interest
+The authors declare that they have no conflict of interest.
+Data availability
+Extended results and the code to reproduce them are available online at https://alfonsoiodicede.
+github.io/blogposts_archive/distances_experiment_superv_unsuperv.html. The
+data used in this study were downloaded from the UCI repository (Dua and Graff 2017) and are also avail-
+able in the catdist package available on GitHub at https://github.com/alfonsoIodiceDE/
+catdist_package.
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+18
+

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+Fair Recommendation by Geometric Interpretation and Analysis of
+Matrix Factorization
+Hao Wang
+haow85@live.com
+Ratidar.com
+Beijing, China
+ABSTRACT
+Matrix factorization-based recommender system is in effect an angle preserving dimensionality reduction technique.
+Since the frequency of items follows power-law distribution, most vectors in the original dimension of user feature
+vectors and item feature vectors lie on the same hyperplane. However, it is very difficult to reconstruct the embeddings
+in the original dimension analytically, so we reformulate the original angle preserving dimensionality reduction problem
+into a distance preserving dimensionality reduction problem. We show that the geometric shape of input data of
+recommender system in its original higher dimension are distributed on co-centric circles with interesting properties, and
+design a paraboloid-based matrix factorization named ParaMat to solve the recommendation problem. In the experiment
+section, we compare our algorithm with 8 other algorithms and prove our new method is the most fair algorithm
+compared with modern day recommender systems such as ZeroMat and DotMat Hybrid.
+Keywords: matrix factorization, ParaMat, Linear Factorization, geometric interpretation, recommender system
+1. INTRODUCTION
+Today, every big internet company is into recommender systems. Recommender systems can boost traffic volume and
+increase sales revenues by a large margin (30%-40% for all sales on Amazon) while saving a huge amount of marketing
+budget. Although recommender systems suffered from a major setback in the first half of 2010’s when most companies
+agreed unanimously that recommender system could not serve as a stand-alone product, in a short period after the
+rampant spreading of pessimistic opinions, companies such as TikTok and Kuai Shou emerged as major players on
+global market as stand-alone recommender system products.
+Researchers invented recommender system a couple of decades ago, and the trickling stream of the AI technology
+becomes torrents today. The basic idea behind recommender system is to use big data analytical mechanisms to analyze
+historic information of users and items to recommend new items to users based on computed preference scores. For
+example, if we know Alice loves reading Su Tung-Po , The Three Kingdoms and many other Chinese Classics, we could
+recommend other Chinese Classics which she had not read for her. This procedure is much more effective to help Alice
+find her new books than search engines with which Alice needs to find what she likes by herself. The example we just
+illustrated is the primitive form of a recommender system technology named content-based recommendation. It is just
+one of the sub-fields of many recommender system research areas.
+Since the year of 2016, prestigious research venues such as RecSys [1][2] has witnessed the rise of deep neural network
+models. With more and more companies and schools taking up the course, the technical models of recommender systems
+are becoming more and more complex and individualistic. Since deep neural networks can represent any function, it
+enables researchers to choose the best model from a much larger candidate pool than in the age of shallow models.
+However, due to the number of choices in parameter tweaking, public knowledge of deep neural network becomes more
+and more individualistic – usually only a small hand of researchers know the reasons and principles behind the neural
+models that they produce.
+
+Although earlier models in the age of shallow models seem out-of-dated , they are still widely used in various companies.
+One of the major reasons of their longevity is their simplicity and interpretability. Due to the same reasons, we choose
+matrix factorization [3] as our main research target in this paper. We provide a geometric interpretation and analysis of
+the classic matrix factorization algorithm, and based on our observations and analysis, we propose a new algorithm
+named ParaMat that is much more interpretable , and at the same time very accurate when compared with other models.
+2. RELATED WORK
+Recommender system remains a popular research field irrespective of the up-and-down in the investment in the field. As
+one of the most successful recommendation paradigms, matrix factorization is provided with a probabilistic framework
+as the foundation in 2007 [4]. The probabilistic framework is named Probabilistic Matrix Factorization. Based on
+modification of this framework, ZeroMat [5] and DotMat [6] are proposed to solve the cold-start and sparsity problem
+without input data. The algorithms could predict user preferences fairly accurately with no historic information. The
+social implications of the 2 algorithms are astonishing - human cultures are locked into a predictable state only after a
+couple of years of evolution.
+Matrix factorization can also be used to solve the context-aware recommendation (CARS) problem [7] [8]. In 2021, a
+new CARS solution named MatMat [9] is introduced. Instead of scalar fitting, MatMat uses matrix factorization by
+matrix fitting to incorporate contextual information into the system. A practical example of MatMat named MovieMat
+[10] which solves CARS problem for movie recommendation is proposed in the same year. The algorithm incorporates
+no more than 6 contextual information fields in the algorithm and achieves better results than classic models.
+Fairness is a hot research topic in recent years. Google comes up with an algorithm named Focused Learning [11] in the
+year of 2017 which penalizes the matrix factorization loss function with a fairness metric. More researchers spend time
+and energy on fair Learning to Rank approaches [12] [13] [14] and publish extensively at top conferences such as SIGIR
+and KDD. However, due to its simplicity, matrix factorization still remains a good benchmark for fairness ideas. Zipf
+Matrix Factorization [15] is introduced in 2021 with the introduction of a fairness metric named Degree of Matthew
+Effect as a side product. MatRec [16] and KL-Mat [17] are also examples of fair recommender algorithms based on
+matrix factorization framework. In 2022, Wang [18] proposed a set of fairness metrics using extreme value theory.
+3. GEOMETRIC ANALYSIS OF MATRIX FACTORIZATION
+One of the most popular definition of the loss function of matrix factorization is as follows :
+L =
+i=1
+n
+j=1
+m
+Ri, j − Ui
+T ∙ Vj
+2
+�
+�
+(1)
+In this formula, ������, ��� represents the rating value that the i-th user gives on the j-th item. ������ is the user feature vector, and
+������ is the item feature vector. If ������ and ������ are not normalized in the real world applications, we usually encounter gradient
+explosion problems, so a more practical loss function of matrix factorization is actually defined as follows :
+L =
+i=1
+n
+j=1
+m
+Ri, j
+Rmax
+−
+Ui
+T ∙ Vj
+||Ui|| × ||Vj||
+2
+�
+�
+(2)
+
+Since
+Ui
+T∙Vj
+||Ui||×||Vj|| is actually the cosine between vectors Ui and Vj , we are actually looking for vectors in higher dimensions
+whose cosine values of angles between each pair are defined by
+Ri, j
+Rmax .
+As in most real world data sets,
+Ri, j
+Rmax follows power law distribution. To make the framework simpler, we assume the
+ratios follow Zipf Distribution. Namely, the frequency of occurrences of
+Ri, j
+Rmax is proportional to their own values. This
+brings about a very interesting geometric property of the input data set : The majority values of
+Ri, j
+Rmax are 1. This means to
+minimize L in Formula (2), most of ������ and ������ should be co-linear.
+A natural question arises for us : What does the vector space of ������ and ������ look like ? Let’s define the number of co-linear
+������ and ������ pairs as M, then
+���������,���
+������������ of the vector pairs have cosine value
+Ri, j
+Rmax . We find it extremely difficult to come up
+with a visualization of such space by analytical methods without intervention of computers.
+To examine the impact of the co-linearity property of the majority of ������ and ������ , we propose an algorithm as follows :
+We define the loss function of the new algorithm (which we name Linear Factorization) below :
+L =
+i=1
+n
+j=1
+m
+Ri, j
+Rmax
+−
+Ui
+T ∙ Vj
+||Ui|| × ||Vj||
+2
+�
+�
+Subject to :
+���=1
+���
+������ ∙ ���1,��� = 0
+�
+. . .
+���=1
+���
+������ ∙ ������,��� = 0
+�
+���������
+���=1
+���
+������ ∙ ���1,��� = 0
+�
+. . .
+���=1
+���
+������ ∙ ������,��� = 0
+�
+(3)
+We tested the algorithm on MovieLens 1 Million Dataset [19] (Fig. 1) , and discovered that the technical accuracy of our
+proposed algorithm can achieve competitive
+results with other algorithms. By this observation, we safely draw the
+conclusion that the co-linearity property of user and item feature vector space plays a vital role in the technical accuracy
+of matrix factorization algorithms. We also find out that Linear Factorization is the most fair recommender system
+among the 9 algorithms. We will provide bibliographic information for the algorithms in this experiment in the
+Experiment Section.
+
+However, due to the difficulty of designing a vector space functional that caters to all the geometric angle preserving
+vectors, we take a different route to solve the problem. Instead of considering
+Ri, j
+Rmax as consine angles between vectors, we
+consider the value as the distance from a vector to the origin. By doing this, the input user item rating values (by Zipf
+Law) becomes equidistantly distributed sample points on co-centric circles.
+The reason behind this geometric property is because the number of equidistantly distributed points on co-centric circles
+is proportional to the radius length. If we define the radii by
+Ri, j
+Rmax , the points of our newly designed geometry just
+become compliant with Zipf’s Law.
+To propose our new algorithm named ParaMat, we elevate our 2-D geometry into 3-D space by the following formula :
+z =
+Ri, j
+Rmax
+xk
+2 + yk
+2
+(4)
+We define the x and y as the products of user feature vector and item feature vector :
+xk = Ui
+T ∙ Vj
+yk = Wi
+T ∙ Pj
+(5)
+The loss function for ParaMat is defined as follows :
+L =
+i=1
+n
+j=1
+m
+Ri, j
+Rmax
+− d
+xk , yk ,
+Ri, j
+Rmax
+xk
+2 + yk
+2
+, 0 , 0 , 0
+2
+�
+�
+(6)
+Fig. 1 Linear Factorization comparison experiments on MovieLens 1 Million Dataset (MAE and Degree of
+Matthew Effect)
+
+2.0
+1.8
+ZeroMat
+1.6
+RandomPlacement
+ClassicMatrixFactorization
+DotMat
+1.4
+DotMat Hybrid
+ZipfPlacement
+PoissonMat
+1.2
+PoissonMatHybrid
+LinearFactorization
+1.0
+0.8
+0.2
+0.4
+0.6
+0.8
+1.0
+GradientLearningStep
+1e-6-0.032
+-0.034
+Effect
+ZeroMat
+RandomPlacement
+-0.036
+ofMatthew
+ClassicMatrixFactorization
+DotMat
+-0.038
+DotMatHybrid
+ZipfPlacement
+-0.040
+PoissonMat
+Degr
+PoissonMatHybrid
+LinearFactorization
+-0.042
+-0.044
+0.2
+0.4
+0.6
+0.8
+1.0
+GradientLearningStep
+1e-6We plug in the values of x and y by dot products of user feature vectors and item feature vectors, and substitute
+Ri, j
+Rmax by
+dot products of user feature vectors and item feature vectors produced by the classic matrix factorization. We obtain the
+following formula :
+L =
+i=1
+n
+j=1
+m
+Ri, j
+Rmax
+−
+Ui
+T ∙ Vj
+2 + Wi
+T ∙ Pj
+2 +
+Ri, j
+Rmax
+2
+Ui
+T ∙ Vj
+2 + Wi
+T ∙ Pj
+2
+2
+�
+�
+(7)
+We choose to optimize Formula (7) using Stochastic Gradient Descent (SGD) algorithm, after which we compute the
+unknown user item rating values using the following formula :
+Ri, j = Rmax
+Ui
+T ∙ Vj
+2 + Wi
+T ∙ Pj
+2 +
+Ri, j
+Rmax
+2
+Ui
+T ∙ Vj
+2 + Wi
+T ∙ Pj
+2
+(8)
+In the Experiment section, we show that ParaMat is the best algorithm when evaluated by fairness metric. After
+geometric analysis and derivation, we have obtained an effective solution for fairness problem in recommender systems.
+4. EXPERIMENT
+To evaluate the performance of ParaMat, we compare the algorithm with 8 other algorithms on both the MovieLens 1
+Million Dataset [19] and LDOS-CoMoDa [20] dataset. Among the algorithms, Random Placement means recommend
+uniformly randomly; Zipf Placement means recommend according to Power Law distribution; ZeroMat [5] and DotMat
+[6] are introduced in the Related Work section; PoissonMat [21] is an algorithm to appear at an international conference
+in 2022.
+Fig.2 illustrates the experimental results of comparison among 9 algorithms on the MovieLens 1 Million Dataset.
+ParaMat achieves the 5th place by technical accuracy metric MAE, but wins the 1st place by fairness metric Degree of
+Matthew Effect. By observation, ParaMat is a quite effective recommender system evaluated by fairness metric. Since
+the error curves are pretty cluttered, we also use Takens Embedding to visualize the curves in 2D (displayed in 3D grid
+for better visibility).
+
+1.8
+1.6
+ZeroMat
+RandomPlacement
+Classic Matrix Factorization
+MAE
+1.4
+DotMat Hybrid
+ZipfPlacement
+PoissonMat
+1.2
+PoissonMat Hybrid
+ParaMat
+1.0 
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Gradient Leaming Step
+le-50.0020
+ofMatthewEffect
+0.0025
+ZeroMat
+RandomPlacement
+ClassicMatrixFactorization
+0.0030
+DotMat
+DotMat Hybrid
+ZipfPlacement
+0.0035
+PoissonMat
+PoissonMat Hybrid
+ParaMat
+-0.0040
+1
+2
+E
+4
+5
+6
+7
+8
+Gradient Learning Step
+le-5Fig.3 demonstrates the experimental results of comparison among 9 algorithms on the LDOS-CoMoDa Dataset. ParaMat
+achieves the 4th place by technical accuracy metric MAE, but once again wins the 1st place by fairness metric Degree of
+Matthew Effect. By observation, ParaMat is the most fair recommender system. The result is more clearly demonstrated
+in 2-D via Takens Embedding.
+Fig. 2 ParaMat comparison on MovieLens 1 Million Dataset (MAE and Degree of Matthew Effect).
+The first row shows the MAE and DME curves of different algorithms. The bottom row visualizes the
+curves respectively using Takens Embedding.
+Fig. 3 ParaMat comparison on LDOS-CoMoDa Dataset (MAE and Degree of
+Matthew Effect). The first row shows the MAE and DME curves of different
+algorithms. The bottom row visualizes the curves respectively using Takens
+Embedding.
+
+ZeroMat
+RandomPlacement
+ClassicMatrixFactorization
+DotMat
+DotMat Hybrid
+Zipf Placement
+0.04
+PoissonMat
+PoissonMatHybrid
+0.02
+ParaMat
+0.00
+0.02
+0.04
+1.8
+1.6
+1.4
+1.0
+1.2
+1.2
+1.4
+1.6
+1.0ZeroMat
+Random Placement
+Classic Matrix Factorization
+DotMat
+DotMatHybrid
+ZipfPlacement
+PoissonMat
+0.04
+PoissonMat Hybrid
+.
+ParaMat
+0.02
+0.00
+0.02
+0.04
+0.0020
+-0.0025
+0.0040
+0.0030
+-0.0035
+0.0030
+-0.0035
+0.0025
+0.00402.0
+1.8
+ZeroMat
+1.6
+Random Placement
+Classic MatrixFactorization
+DotMat
+DotMat Hybrid
+Zipf Placement
+PoissonMat
+1.2
+PoissonMat Hybrid
+ParaMat
+1.0
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Gradient Learning Step
+le-50.032
+0.034
+Ettect
+ZeroMat
+RandomPlacement
+0.036
+Classic Matrix Factorization
+DotMat
+0.038
+DotMatHybrid
+ZipfPlacement
+PoissonMat
+-0.040
+PoissonMat Hybrid
+ParaMat
+0.042
+0.044
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Gradient Leaming Step
+le~5ZeroMat
+RandomPlacement
+Classic Matrix Factorization
+DotMat
+DotMat Hybrid
+ZipfPlacement
+PoissonMat
+3.04
+PoissonMat Hybrid
+0.02
+ParaMat
+0.00
+0.02
+0.04
+2.0
+1.8
+1.6
+0.8
+1.0
+1.4
+1.2
+1.2
+1.4
+1.6
+1.0
+1.8
+0.8ZeroMat
+Random Placement
+Classic Matrix Factorization
+DotMat
+DotMatHybrid
+ZipfPlacement
+0.04
+PoissonMat
+PoissonMatHybrid
+0.02
+ParaMat
+0.00
+0.02
+0.04
+0.032
+0.034
+0.036
+0.038
+0.040
+0.042
+0.0445. CONCLUSION
+In this paper, we propose geometry-powered fair recommender system algorithms named Linear Factorization and
+ParaMat. Linear Factorization assumes the user feature vectors and item feature vectors of matrix factorization lay on the
+same hyperplane due to the observation that user/item feature vectors are mostly co-linear. ParaMat also relies on the
+geometric observation that user item rating values mostly lie on co-centric circles equidistantly.
+Both Linear Factorization and ParaMat are not the best performing algorithm on technical accuracy metrics such as
+MAE, but they are the most fair algorithms among the 9 algorithms in our Experiment section. By analyzing the
+geometric space of the input data structure, we’ve come up with two effective fair recommendation algorithms.
+In future work, we would like to explore the geometric space of input data structures to other classic algorithms such as
+learning to rank and factorization machines. We are also very interested in geometric interpretation of deep learning
+models.
+REFERENCES
+[1] H. Guo, R. Tang, et. al., “DeepFM: A Factorization-Machine based Neural Network for CTR Prediction”,
+IJCAI, 2017
+[2] G. Zhou, X. Zhu, C. Song, et.al, “Deep Interest Network for Click-Through Rate Prediction”, KDD, 2018
+[3] R. Mehta; K. Rana, “A Review on Matrix Factorization Techniques in Recommender Systems”, CSCITA,
+2017
+[4] R.Salakhutdinov, A.Mnih, “Probabilistic Matrix Factorization”, Proceedings of the 20th International
+Conference on Neural Information Processing Systems, 2007
+[5] H. Wang, “ZeroMat: Solving Cold-start Problem of Recommender System with No Input Data”, IEEE 4th
+International Conference on Information Systems and Computer Aided Education (ICISCAE), 2021
+[6] H. Wang, “DotMat: Solving Cold-start Problem and Alleviating Sparsity Problem for Recommender Systems”,
+IEEE 5th International Conference on Electronics Technology (ICET 2022)
+[7] E. Coviello, K. Ellis, et. al. “A Scalable Model for Online Contextual Music Recommendations”, CARS
+Workshop, 2021
+[8] Y. Chen, J. Li, “Modeling Dynamic Attributes for Next Basket Recommendation”, CARS Workshop, 2021
+[9] H.Wang, “MatMat: Matrix Factorization by Matrix Fitting”, IEEE 4th International Conference on Information
+Systems and Computer Aided Education (ICISCAE), 2021
+[10]H.Wang, "MovieMat: Context-aware Movie Recommendation with Matrix Factorization by Matrix Fitting", 7th
+International Conference on Computer and Communications (ICCC), 2021
+[11]A. Beutel, Ed. Chi, et. al. “Beyond Globally Optimal: Focused Learning for Improved Recommendations”,
+WWW, 2017
+[12]H.Yadav, Z.Du, T.Joachims. “Fair Learning-to-Rank from Implicit Feedback.”, SIGIR, 2020
+[13]M.Morik, A.Singh, J.Hong, T.Joachims. “Controlling Fairness and Bias in Dynamic Learning-to-Rank”, SIGIR,
+2020
+[14]M.Zehlike, C.Castillo. “ Reducing Disparate Exposure in Ranking: A Learning to Rank Approach.”, SIGIR,
+2020
+[15]H. Wang, “Zipf Matrix Factorization : Matrix Factorization with Matthew Effect Reduction”, ICAIBD, 2021
+[16]H. Wang, B. Ruan, “MatRec: Matrix Factorization for Highly Skewed Dataset”, ICBDT, 2020
+[17]H. Wang, “KL-Mat: Fair Recommender System via Information Geometry”, icWCSN, 2022
+[18]H. Wang, “Fairness Metrics for Recommender Systems”, icWCSN, 2022
+[19]T. Bertin-Mahieux, B. Whitman, P. Lamere, The Million Song Dataset, ISMIR, 2011
+[20]ODIĆ, Ante, TKALČIČ, Marko, TASIČ, Jurij F., KOŠIR, Andrej: Predicting and Detecting the Relevant
+Contextual Information in a Movie-Recommender System, Interacting with Computers, Volume 25, Issue 1,
+2013
+[21]H. Wang, “PoissonMat: Remodeling Matrix Factorization using Poisson Distribution and Solving the Cold Start
+Problem without Input Data”, to appear, MLISE, 2022
+

DtE2T4oBgHgl3EQfSQej/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf,len=275
+page_content='Fair Recommendation by Geometric Interpretation and Analysis of  Matrix Factorization  Hao Wang  haow85@live.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='com  Ratidar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='com  Beijing, China  ABSTRACT  Matrix factorization-based recommender system is in effect an angle preserving dimensionality reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Since the frequency of items follows power-law distribution, most vectors in the original dimension of user feature  vectors and item feature vectors lie on the same hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' However, it is very difficult to reconstruct the embeddings  in the original dimension analytically, so we reformulate the original angle preserving dimensionality reduction problem  into a distance preserving dimensionality reduction problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We show that the geometric shape of input data of  recommender system in its original higher dimension are distributed on co-centric circles with interesting properties, and  design a paraboloid-based matrix factorization named ParaMat to solve the recommendation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' In the experiment  section, we compare our algorithm with 8 other algorithms and prove our new method is the most fair algorithm  compared with modern day recommender systems such as ZeroMat and DotMat Hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Keywords: matrix factorization, ParaMat, Linear Factorization, geometric interpretation, recommender system  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' INTRODUCTION  Today, every big internet company is into recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Recommender systems can boost traffic volume and  increase sales revenues by a large margin (30%-40% for all sales on Amazon) while saving a huge amount of marketing  budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Although recommender systems suffered from a major setback in the first half of 2010’s when most companies  agreed unanimously that recommender system could not serve as a stand-alone product, in a short period after the  rampant spreading of pessimistic opinions, companies such as TikTok and Kuai Shou emerged as major players on  global market as stand-alone recommender system products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Researchers invented recommender system a couple of decades ago, and the trickling stream of the AI technology  becomes torrents today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The basic idea behind recommender system is to use big data analytical mechanisms to analyze  historic information of users and items to recommend new items to users based on computed preference scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' For  example, if we know Alice loves reading Su Tung-Po , The Three Kingdoms and many other Chinese Classics, we could  recommend other Chinese Classics which she had not read for her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' This procedure is much more effective to help Alice  find her new books than search engines with which Alice needs to find what she likes by herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The example we just  illustrated is the primitive form of a recommender system technology named content-based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' It is just  one of the sub-fields of many recommender system research areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Since the year of 2016, prestigious research venues such as RecSys [1][2] has witnessed the rise of deep neural network  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' With more and more companies and schools taking up the course, the technical models of recommender systems  are becoming more and more complex and individualistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Since deep neural networks can represent any function, it  enables researchers to choose the best model from a much larger candidate pool than in the age of shallow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  However, due to the number of choices in parameter tweaking, public knowledge of deep neural network becomes more  and more individualistic – usually only a small hand of researchers know the reasons and principles behind the neural  models that they produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Although earlier models in the age of shallow models seem out-of-dated , they are still widely used in various companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  One of the major reasons of their longevity is their simplicity and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Due to the same reasons, we choose  matrix factorization [3] as our main research target in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We provide a geometric interpretation and analysis of  the classic matrix factorization algorithm, and based on our observations and analysis, we propose a new algorithm  named ParaMat that is much more interpretable , and at the same time very accurate when compared with other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' RELATED WORK  Recommender system remains a popular research field irrespective of the up-and-down in the investment in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' As  one of the most successful recommendation paradigms, matrix factorization is provided with a probabilistic framework  as the foundation in 2007 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The probabilistic framework is named Probabilistic Matrix Factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Based on  modification of this framework, ZeroMat [5] and DotMat [6] are proposed to solve the cold-start and sparsity problem  without input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The algorithms could predict user preferences fairly accurately with no historic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The  social implications of the 2 algorithms are astonishing - human cultures are locked into a predictable state only after a  couple of years of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Matrix factorization can also be used to solve the context-aware recommendation (CARS) problem [7] [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' In 2021, a  new CARS solution named MatMat [9] is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Instead of scalar fitting, MatMat uses matrix factorization by  matrix fitting to incorporate contextual information into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' A practical example of MatMat named MovieMat  [10] which solves CARS problem for movie recommendation is proposed in the same year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The algorithm incorporates  no more than 6 contextual information fields in the algorithm and achieves better results than classic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Fairness is a hot research topic in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Google comes up with an algorithm named Focused Learning [11] in the  year of 2017 which penalizes the matrix factorization loss function with a fairness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' More researchers spend time  and energy on fair Learning to Rank approaches [12] [13] [14] and publish extensively at top conferences such as SIGIR  and KDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' However, due to its simplicity, matrix factorization still remains a good benchmark for fairness ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Zipf  Matrix Factorization [15] is introduced in 2021 with the introduction of a fairness metric named Degree of Matthew  Effect as a side product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' MatRec [16] and KL-Mat [17] are also examples of fair recommender algorithms based on  matrix factorization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' In 2022, Wang [18] proposed a set of fairness metrics using extreme value theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' GEOMETRIC ANALYSIS OF MATRIX FACTORIZATION  One of the most popular definition of the loss function of matrix factorization is as follows :  L =  i=1  n  j=1  m  Ri, j − Ui  T ∙ Vj  2  �  �  (1)  In this formula, ������, ��� represents the rating value that the i-th user gives on the j-th item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' ������ is the user feature vector, and  ������ is the item feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' If ������ and ������ are not normalized in the real world applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' we usually encounter gradient  explosion problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' so a more practical loss function of matrix factorization is actually defined as follows :  L =  i=1  n  j=1  m  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j  Rmax  −  Ui  T ∙ Vj  ||Ui|| × ||Vj||  2  �  �  (2)  Since  Ui  T∙Vj  ||Ui||×||Vj|| is actually the cosine between vectors Ui and Vj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' we are actually looking for vectors in higher dimensions  whose cosine values of angles between each pair are defined by  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j  Rmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  As in most real world data sets,  Ri, j  Rmax follows power law distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' To make the framework simpler, we assume the  ratios follow Zipf Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Namely, the frequency of occurrences of  Ri, j  Rmax is proportional to their own values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' This  brings about a very interesting geometric property of the input data set : The majority values of  Ri, j  Rmax are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' This means to  minimize L in Formula (2), most of ������ and ������ should be co-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  A natural question arises for us : What does the vector space of ������ and ������ look like ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Let’s define the number of co-linear  ������ and ������ pairs as M, then  ���×������,���  ������������ of the vector pairs have cosine value  Ri, j  Rmax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We find it extremely difficult to come up  with a visualization of such space by analytical methods without intervention of computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  To examine the impact of the co-linearity property of the majority of ������ and ������ , we propose an algorithm as follows :  We define the loss function of the new algorithm (which we name Linear Factorization) below :  L =  i=1  n  j=1  m  Ri, j  Rmax  −  Ui  T ∙ Vj  ||Ui|| × ||Vj||  2  �  �  Subject to :  ���=1  ���  ������ ∙ ���1,��� = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  ���=1  ���  ������ ∙ ������,��� = 0  �  ���������  ���=1  ���  ������ ∙ ���1,��� = 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  ���=1  ���  ������ ∙ ������,��� = 0  �  (3)  We tested the algorithm on MovieLens 1 Million Dataset [19] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' 1) , and discovered that the technical accuracy of our  proposed algorithm can achieve competitive  results with other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' By this observation, we safely draw the  conclusion that the co-linearity property of user and item feature vector space plays a vital role in the technical accuracy  of matrix factorization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We also find out that Linear Factorization is the most fair recommender system  among the 9 algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We will provide bibliographic information for the algorithms in this experiment in the  Experiment Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  However, due to the difficulty of designing a vector space functional that caters to all the geometric angle preserving  vectors, we take a different route to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Instead of considering  Ri, j  Rmax as consine angles between vectors, we  consider the value as the distance from a vector to the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' By doing this, the input user item rating values (by Zipf  Law) becomes equidistantly distributed sample points on co-centric circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  The reason behind this geometric property is because the number of equidistantly distributed points on co-centric circles  is proportional to the radius length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' If we define the radii by  Ri, j  Rmax , the points of our newly designed geometry just  become compliant with Zipf’s Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  To propose our new algorithm named ParaMat, we elevate our 2-D geometry into 3-D space by the following formula :  z =  Ri, j  Rmax  xk  2 + yk  2  (4)  We define the x and y as the products of user feature vector and item feature vector :  xk = Ui  T ∙ Vj  yk = Wi  T ∙ Pj  (5)  The loss function for ParaMat is defined as follows :  L =  i=1  n  j=1  m  Ri, j  Rmax  − d  xk , yk ,  Ri, j  Rmax  xk  2 + yk  2  , 0 , 0 , 0  2  �  �  (6)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' 1 Linear Factorization comparison experiments on MovieLens 1 Million Dataset (MAE and Degree of  Matthew Effect)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  ZeroMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  RandomPlacement  ClassicMatrixFactorization  DotMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  DotMat Hybrid  ZipfPlacement  PoissonMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  PoissonMatHybrid  LinearFactorization  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  GradientLearningStep  1e-6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='032  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='034  Effect  ZeroMat  RandomPlacement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='036  ofMatthew  ClassicMatrixFactorization  DotMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='038  DotMatHybrid  ZipfPlacement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='040  PoissonMat  Degr  PoissonMatHybrid  LinearFactorization  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='042  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  GradientLearningStep  1e-6We plug in the values of x and y by dot products of user feature vectors and item feature vectors, and substitute  Ri, j  Rmax by  dot products of user feature vectors and item feature vectors produced by the classic matrix factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We obtain the  following formula :  L =  i=1  n  j=1  m  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j  Rmax  −  Ui  T ∙ Vj  2 + Wi  T ∙ Pj  2 +  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j  Rmax  2  Ui  T ∙ Vj  2 + Wi  T ∙ Pj  2  2  �  �  (7)  We choose to optimize Formula (7) using Stochastic Gradient Descent (SGD) algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' after which we compute the  unknown user item rating values using the following formula :  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j = Rmax  Ui  T ∙ Vj  2 + Wi  T ∙ Pj  2 +  Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' j  Rmax  2  Ui  T ∙ Vj  2 + Wi  T ∙ Pj  2  (8)  In the Experiment section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' we show that ParaMat is the best algorithm when evaluated by fairness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' After  geometric analysis and derivation, we have obtained an effective solution for fairness problem in recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' EXPERIMENT  To evaluate the performance of ParaMat, we compare the algorithm with 8 other algorithms on both the MovieLens 1  Million Dataset [19] and LDOS-CoMoDa [20] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Among the algorithms, Random Placement means recommend  uniformly randomly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Zipf Placement means recommend according to Power Law distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' ZeroMat [5] and DotMat  [6] are introduced in the Related Work section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' PoissonMat [21] is an algorithm to appear at an international conference  in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2 illustrates the experimental results of comparison among 9 algorithms on the MovieLens 1 Million Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  ParaMat achieves the 5th place by technical accuracy metric MAE, but wins the 1st place by fairness metric Degree of  Matthew Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' By observation, ParaMat is a quite effective recommender system evaluated by fairness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Since  the error curves are pretty cluttered, we also use Takens Embedding to visualize the curves in 2D (displayed in 3D grid  for better visibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  ZeroMat  RandomPlacement  Classic Matrix Factorization  MAE  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  DotMat Hybrid  ZipfPlacement  PoissonMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  PoissonMat Hybrid  ParaMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  0  1  2  3  4  5  6  7  8  Gradient Leaming Step  le-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0020  ofMatthewEffect  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0025  ZeroMat  RandomPlacement  ClassicMatrixFactorization  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0030  DotMat  DotMat Hybrid  ZipfPlacement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0035  PoissonMat  PoissonMat Hybrid  ParaMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0040  1  2  E  4  5  6  7  8  Gradient Learning Step  le-5Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='3 demonstrates the experimental results of comparison among 9 algorithms on the LDOS-CoMoDa Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' ParaMat  achieves the 4th place by technical accuracy metric MAE, but once again wins the 1st place by fairness metric Degree of  Matthew Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' By observation, ParaMat is the most fair recommender system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The result is more clearly demonstrated  in 2-D via Takens Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' 2 ParaMat comparison on MovieLens 1 Million Dataset (MAE and Degree of Matthew Effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  The first row shows the MAE and DME curves of different algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The bottom row visualizes the  curves respectively using Takens Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' 3 ParaMat comparison on LDOS-CoMoDa Dataset (MAE and Degree of  Matthew Effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The first row shows the MAE and DME curves of different  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' The bottom row visualizes the curves respectively using Takens  Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  ZeroMat  RandomPlacement  ClassicMatrixFactorization  DotMat  DotMat Hybrid  Zipf Placement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='04  PoissonMat  PoissonMatHybrid  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='02  ParaMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0ZeroMat  Random Placement  Classic Matrix Factorization  DotMat  DotMatHybrid  ZipfPlacement  PoissonMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='04  PoissonMat Hybrid  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  ParaMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content='8  ZeroMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='6  Random Placement  Classic MatrixFactorization  DotMat  DotMat Hybrid  Zipf Placement  PoissonMat  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content='038  DotMatHybrid  ZipfPlacement  PoissonMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content='8ZeroMat  Random Placement  Classic Matrix Factorization  DotMat  DotMatHybrid  ZipfPlacement  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='04  PoissonMat  PoissonMatHybrid  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='02  ParaMat  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='042  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='0445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' CONCLUSION  In this paper, we propose geometry-powered fair recommender system algorithms named Linear Factorization and  ParaMat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Linear Factorization assumes the user feature vectors and item feature vectors of matrix factorization lay on the  same hyperplane due to the observation that user/item feature vectors are mostly co-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' ParaMat also relies on the  geometric observation that user item rating values mostly lie on co-centric circles equidistantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  Both Linear Factorization and ParaMat are not the best performing algorithm on technical accuracy metrics such as  MAE, but they are the most fair algorithms among the 9 algorithms in our Experiment section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' By analyzing the  geometric space of the input data structure, we’ve come up with two effective fair recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content='  In future work, we would like to explore the geometric space of input data structures to other classic algorithms such as  learning to rank and factorization machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' We are also very interested in geometric interpretation of deep learning  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content='Castillo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
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+page_content=' Wang, “KL-Mat: Fair Recommender System via Information Geometry”, icWCSN, 2022  [18]H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Wang, “Fairness Metrics for Recommender Systems”, icWCSN, 2022  [19]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Bertin-Mahieux, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Whitman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Lamere, The Million Song Dataset, ISMIR, 2011  [20]ODIĆ, Ante, TKALČIČ, Marko, TASIČ, Jurij F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=', KOŠIR, Andrej: Predicting and Detecting the Relevant  Contextual Information in a Movie-Recommender System, Interacting with Computers, Volume 25, Issue 1,  2013  [21]H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}
+page_content=' Wang, “PoissonMat: Remodeling Matrix Factorization using Poisson Distribution and Solving the Cold Start  Problem without Input Data”, to appear, MLISE, 2022' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE2T4oBgHgl3EQfSQej/content/2301.03791v1.pdf'}

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+INFERENCE / Vol. 7, No. 3
+1 / 7
+A Truncated Manuscript
+Pierre Schapira
+Récoltes et Semailles I, II. Réflexions et témoignage  
+sur un passé de mathématicien 
+by Alexander Grothendieck 
+Editions Gallimard, 29,50 €.
+S
+trictly speaking, this essay is not solely a review 
+of Alexander Grothendieck’s Récoltes et Semailles 
+(Reaping and Sowing).1 Although the book as a 
+whole, as well as Grothendieck’s work and life, will be dis-
+cussed here, a good part of the essay is devoted to refuting 
+a thesis Grothendieck developed throughout his original 
+text. We have an important ally in this respect: the author 
+himself. In a series of important additions that have not 
+been incorporated in this new version, Grothendieck goes 
+back entirely on some of his assertions.
+T
+he figure of Grothendieck dominated much of 
+mathematics during the second half of the twen-
+tieth century. If his work is essentially concerned 
+with algebraic geometry, his vision and methods have 
+spread far beyond—to algebraic topology, representation 
+theory, complex geometry, symplectic geometry, algebraic 
+analysis and even, more recently, computational geometry. 
+In short, all linear mathematics, as opposed to dynamical 
+systems, probabilities, or purely geometric geometry, such 
+as Riemannian or Hamiltonian geometry. It was under 
+his influence that the language of derived categories and 
+sheaf theory were established in these fields. It was also 
+Grothendieck who had the intuition and then formulated 
+the main lines of the theory of derived categories. He left 
+it to his student Jean-Louis Verdier to write out all the 
+details for his thesis, in which Verdier clarified the key 
+notion of a triangulated category.2 But it was Grothend-
+ieck who situated sheaf theory and the six operations in 
+the theory of derived categories, to which we will return  
+later.
+Between 1950 and 1970, functions—possibly gen-
+eralized—were studied on real or complex manifolds, 
+and especially on Euclidean spaces, using the Fourier 
+transform. But on a complex manifold, when we refer 
+to functions we really mean holomorphic functions, and 
+these present a serious difculty. That is, they do not 
+exist, at least not globally on a compact manifold such as 
+the projective line—apart, of course, from the constants. 
+Global knowledge, therefore, does not provide any infor-
+mation, unlike the real diferentiable case, and one must 
+work locally. There is a rather extraordinary tool for this 
+purpose: sheaf theory, which was invented by Jean Leray 
+while he was a prisoner of war in Germany between 1941 
+and 1945.3 Leray’s text was somewhat incomprehensible 
+but was later clarified by Henri Cartan and Jean-Pierre 
+Serre, resulting in Roger Godement’s famous book, 
+Théorie des Faisceaux (Theory of Sheaves).4 Cartan and 
+Serre used this tool in their respective studies of holomor-
+phic functions in dimension ≥ 1, after the seminal work of 
+Kiyoshi Oka, giving rise to Cartan’s Theorems A and B and 
+Serre’s influential paper “Faisceaux algébriques cohérents” 
+(Coherent algebraic sheaves).5
+It is in this context that Grothendieck approached 
+algebraic geometry around 1955, providing a solid basis 
+for sheaf cohomology in a foundational paper published 
+by the Tōhoku Mathematical Journal.6 In this paper, one 
+already encounters—implicitly—the main difculty of cat-
+egory theory, namely the problem of universes, a problem 
+later solved by Grothendieck in SGA4. This was done in the 
+manner of another Alexander cutting the Gordian knot:7 
+Grothendieck poses the axiom that every set belongs to 
+a universe. This problem of universes, also known as the 
+inaccessible cardinals, outside of which category theory 
+cannot develop, is probably the reason why Bourbaki 
+gave up on categories and why Grothendieck then left the 
+group. Ralf Krömer has written an excellent article on this 
+issue.8
+During the 1940s and 1950s, there were two concep-
+tual revolutions whose importance was not immediately 
+understood: sheaf theory, as mentioned above, and cat-
+egory theory, the latter due to Samuel Eilenberg and 
+Saunders Mac Lane.9 Moreover, the categorical point of 
+view is part of a vast movement of ideas that embraced 
+the structuralist approach of Claude Lévi-Strauss and 
+the linguistics of Noam Chomsky. Instead of considering 
+sets endowed with certain structures, category theory 
+focuses on the relations that can exist between objects. A 
+category C is thus a family of objects—as a set is a family 
+of elements—but given two objects X and Y, there exists 
+a priori a set called HomC (X, Y) representing the mor-
+Published by Inference-Review.com  Vol 7, issue 3, Dec 2022.
+
+2 / 7
+BOOK REVIEWS
+phisms from X to Y, these data being, of course, subject to 
+a certain number of natural axioms—composition of mor-
+phisms, identity morphisms, etc. A new step then consists 
+in looking at morphisms between categories, which are 
+called functors. Certain key notions then emerge—such as 
+those of adjoint functors, final or initial objects, limits and 
+colimits—giving a precise and unifying meaning to many 
+ideas that run through mathematics.
+There is a family of categories that plays a central role: 
+these are the additive categories and, among them, the 
+abelian categories, which are modeled on the category of 
+modules over a ring. But if the vector spaces over a field are 
+replaced by the modules over a ring, the classical tensor 
+product and internal Hom functors are no longer exact, i.e., 
+they do not transform exact sequences into exact sequenc-
+es—a subspace does not always admit a supplementary. 
+It is thus necessary to consider the derived functors. We 
+then enter the domain of homological algebra, a natural 
+generalization of linear algebra. Here, the reference book 
+was initially that of Cartan and Eilenberg,10 before it was 
+dethroned by Grothendieck’s Tōhoku paper. But comput-
+ing the derived functor of the composition of two functors 
+requires the use of Leray’s spectral sequences, giving rise 
+to often inextricable calculations. This is where derived 
+categories show their power: in this language, everything 
+is remarkably simple.
+What are the six operations? With ordinary functions, we 
+have three natural operations, besides addition: the prod-
+uct and, associated with an application f : X → Y between 
+real manifolds, the integration which sends—modulo some 
+technical details—functions on X to functions on Y and the 
+composition by f which sends functions on Y to functions 
+on X. In sheaf theory, the tensor product ⊗
+L  is the analogue 
+of the product, the proper direct image Rf! is the analogue 
+of the integration, and the inverse image f –1 is the analogue 
+of the composition by f. But the tensor product has a right 
+adjoint, Rhom, the functor f –1 has a right adjoint, the direct 
+image Rf*, and the functor Rf! has a right adjoint, f !.
+The functor f !, which exists only in the derived frame-
+work, unlike the other five, was discovered by Grothendieck 
+in the context of étale cohomology and was subsequently 
+constructed by Verdier for locally compact spaces. As Gro-
+thendieck had seen, f ! provides a broad generalization of 
+Poincaré duality and this functor now plays a crucial role. 
+But since locally compact spaces appear more often than 
+the étale topology, it is the name Poincaré–Verdier, if not 
+just Verdier alone, that remains associated with duality. 
+This attribution is largely unfair and left Grothendieck 
+feeling somewhat bitter—rightly so.11
+One might think that such an abstract framework 
+dispenses with explicit computations, but this is a mis-
+conception: put simply, the computations are no longer 
+the same. If the direct image functor does not allow for 
+integrals to be computed explicitly, the formalism of the 
+six operations nevertheless gives rise to sophisticated 
+numerical results, such as the computation of dimensions 
+of cohomology spaces. The Riemann–Roch–Hirzebruch–
+Grothendieck theorem is a beautiful illustration.
+In the same vein, one of Grothendieck’s fundamen-
+tal discoveries was to develop sheaf theory on categories 
+and thus, in particular, on spaces that no longer have any 
+points. What do the sheaves require to exist? Namely, the 
+data of open sets and their inclusions, and the notion of a 
+covering. There is nothing to prevent the objects of a cat-
+egory from playing the role of the open sets, the category 
+is then called a pre-site, and it remains to define axiomat-
+ically what the coverings are in order to obtain a site—i.e., 
+a category with a Grothendieck topology. This natural 
+generalization of the usual topological spaces proves to be 
+extremely fruitful and analysts would, in fact, do well to 
+draw inspiration from it. On a real manifold, there are far 
+too many pathological open sets and too many coverings 
+if one is interested in what happens at the edge of an open 
+set.
+One then arrives at topos theory—topoi for the scholars. 
+The underlying idea, which in a particular situation goes 
+back to Israel Gelfand, is that a space—in this case, a site—
+can be reconstructed from the category of sheaves on that 
+site. A topos is then a category equivalent to a category 
+of sheaves. The category of sets, for example, is nothing 
+other than the topos associated with a point. But even if 
+topos theory has been used in a new proof of Paul Cohen’s 
+result on the independence of the continuum hypothesis, 
+its applications in mathematics are still uncertain. 
+This presentation of a selection of Grothendieck’s fun-
+damental ideas is far from complete and only reflects the 
+particular interests of the reviewer. In R&S, Grothendieck 
+lists what he considers the 12 key ideas of his work and 
+also provides a list of his students.12
+One should, of course, also mention his first works in 
+functional analysis, dating from around 1955, and scheme 
+theory, which revolutionized algebraic geometry, as well 
+as the intuition of motives, a partially conjectural theory 
+that was later developed by Pierre Deligne, Vladimir 
+Voevodsky, Joseph Ayoub, and many others. One should 
+also mention the fundamental text “A la poursuite des 
+champs” in which Grothendieck lays the foundations for 
+∞-categories and homotopical algebra.13 Indeed, if trian-
+gulated categories are an incredibly simple and efcient 
+tool, they have a defect which seriously limits their use, for 
+example, in gluing problems. This defect is linked to the 
+fact that a certain morphism is unique up to isomorphism, 
+but this isomorphism is not unique! The new theory of 
+∞-categories, to which the names Jacob Lurie, Graeme 
+Segal, Bertrand Töen, and a few others must be associated, 
+is in the process of completely supplanting the classical 
+theory of derived categories, although it is, for the time 
+being, not easily accessible, to say the least.
+P
+ierre cartier has written a remarkable article on 
+Grothendieck’s life and it is pointless to paraphrase 
+it here.14 There are also excellent articles by Allyn 
+
+INFERENCE / Vol. 7, No. 3
+3 / 7
+Jackson and Winfried Scharlau, as well as all the links 
+on the Grothendieck Circle website managed by Leila 
+Schneps.15
+Nonetheless, a few words on this subject are helpful. 
+Grothendieck’s father, Sascha Schapiro, was a Russian 
+anarchist who took part in the aborted revolution of 1905. 
+He then served ten years in the prisons of Czar Nicholas 
+II before being released following the revolution of 1917. 
+Despite initially being feted as a hero, Schapiro was soon 
+declared an enemy of the people. He later fought along-
+side the Republicans during the Spanish Civil war before 
+becoming a traveling photographer in France. In 1939, 
+Schapiro was interned at the Camp Vernet in the French 
+Pyrenees. He was handed over to the Nazis by the Vichy 
+police in 1942 and disappeared into Auschwitz.
+Grothendieck’s mother, Hanka, was an extreme-left 
+militant in Germany during the 1920s who emigrated to 
+France when Adolf Hitler came to power.16 Her son did not 
+join her until 1938, at the age of 10, after having lived in 
+hiding on a farm in Germany. Grothendieck spent part of 
+the war in Le Chambon-sur-Lignon at the famous Collège 
+Cévenol that saved so many Jewish children.
+Grothendieck’s mathematical life began in Nancy 
+during the 1950s, where Jean Dieudonné and Laurent 
+Schwartz took him under their wing. After his initial work 
+in functional analysis, which still remains fundamental, he 
+turned to algebraic geometry with great success, a story 
+that is now well-known.
+Grothendieck was one of the first two professors 
+appointed to the Institut des Hautes Études Scientifiques 
+(IHES) in 1959, where he obtained most of his results and 
+published, with the help of Jean Dieudonné, the other 
+professor at the IHES, the famous EGA (Éléments de 
+géométrie algébrique). He directed the seminar on alge-
+braic geometry, which resulted in a publication of more 
+than 5,000 pages coauthored with some of his students, 
+known as SGA (Séminaire de Géométrie Algébrique du Bois 
+Marie). Grothendieck was awarded the Fields Medal at 
+the International Congress of Mathematicians in 1966, 
+but did not travel to Moscow to receive it. In 1988, he won 
+the prestigious and well-funded Crafoord Prize, which he  
+refused.
+Grothendieck left the IHES in 1970 after discovering 
+that the institute benefited from military funding and 
+launched his own ecological crusade, first through the 
+journal Survivre, and then later Survivre et vivre. But Gro-
+thendieck not only left the IHES, he also left the world of 
+mathematics and, in particular, his students. He returned 
+to mathematics in 1983, but in a very diferent style, with 
+his publications “Esquisse d’un programme” (Sketch of a 
+Programme) and “À la poursuite des champs” (Pursuing 
+Stacks).17 After a year at the Collège de France, he was 
+appointed professor in Montpellier, where he worked 
+until his retirement in 1988. He spent his final years living 
+in the countryside in almost total seclusion, until his death 
+in 2014 at the age of 86.
+A
+s we have seen, Grothendieck is the author of a 
+considerable body of mathematical work. But he 
+is also the author of significant literary works. 
+Among them is R&S, which was published by Gallimard in 
+January 2022 after having been widely distributed on the 
+internet since Grothendieck first wrote the text in 1986. 
+Amounting to more than 1,900 pages, the book deals with 
+many subjects: the author’s journey as a mathematician, 
+his passions, his illusions and disillusions, the process of 
+creation, and a thousand other topics. It also includes long 
+passages on Yin and Yang, feminine and masculine ways 
+of doing mathematics, the mother, the father and child, 
+dreams, and so on. A large part of the text is devoted to 
+a revelation he is said to have experienced in 1976 and 
+a long period of meditation that followed. It is a kind of 
+self-analysis tinged, it has to be said, with a certain degree 
+of paranoia. A recurring theme is the sense of betrayal he 
+felt toward his former students, which is manifested in his 
+work being ignored and forgotten. The words “funeral,” 
+“deceased,” “hearse,” “massacre,” and “gravedigger,” and so 
+on, quickly become omnipresent after their appearance in 
+the table of contents. More generally, the book denounces 
+a loss of ethics among the entire mathematical community.
+Grothendieck explains to the reader that mathematics 
+“was better before”—that is, prior to 1960—as if the older 
+generation was irreproachable! In fact,  on the contrary, 
+it can be said that mathematicians have become much 
+more honest since the 1990s. The source of this miracle 
+has a name: arXiv. It is now becoming ever more difcult 
+to appropriate the ideas of others, although, of course, it 
+is still possible to some degree. The institution of math-
+ematics itself has also been greatly improved, or at least 
+has been greatly transformed. The system of mandarins 
+that dominated French mathematics until the 1970s, from 
+which Grothendieck did not experience any difculties 
+and about whom he does not say a word, has practically 
+disappeared.
+Grothendieck, who is very self-critical throughout the 
+text, sometimes ponders whether he might have been arro-
+gant or even contemptuous of those around him during his 
+heyday in the 1960s and 1970s. Despite these concerns, it 
+is clear that he cares little about ingratiating himself with 
+his readers. Instead, he ofers a book of more than 1,900 
+pages, while in response to a question about the IHES 
+library in its early days, he remarks: “We don’t read books, 
+we write them!”18 R&S contains many contradictions that 
+are only partly corrected by a series of Notes—some of 
+which, despite being of particular importance, are not 
+included in this new edition. Addressing these contradic-
+tions properly would undoubtedly have required the text 
+to be completely rewritten.
+Grothendieck is not paralyzed by any sense of false 
+modesty:
+The thing that struck me is that I do not remember having 
+known, even from the allusions of friends or colleagues 
+
+4 / 7
+BOOK REVIEWS
+who are better versed in history, of a mathematician apart 
+from myself who contributed such a multiplicity of inno-
+vative ideas, not more or less disjointed from one another, 
+but as part of a vast unifying vision (as was the case for 
+Newton and for Einstein in physics and cosmology, and for 
+Darwin and for Pasteur in biology).19
+Elsewhere, he writes: “It would seem that, as a servant of 
+a vast unifying vision born in me, I am ‘one of a kind’ in the 
+history of mathematics from its origin to the present day.”20
+Although the writing style is not lacking in inspiration, 
+it is nonetheless uneven and sometimes—deliberately—
+familiar. Grothendieck is not le duc de Saint-Simon.
+The following analysis will focus only on the content con-
+cerning mathematics and the world of mathematicians.
+In the text, Grothendieck complains at length that his 
+ideas have been plundered by his former students with-
+out reference to their master or that they have simply been 
+erased and forgotten. These assertions are not always 
+supported by solid arguments or precise references. But, 
+above all, it is the nature of discoveries to be trivialized 
+and their author forgotten, and all the more so when the 
+underlying ideas are often, in hindsight, obvious.
+Grothendieck’s reproaches are addressed to all his 
+pupils, and particularly to Deligne—whose name is almost 
+always preceded by the words “my friend,” insinuating 
+“my former friend”—and to Verdier. It is quite possible to 
+imagine that Deligne was only lightly involved with Gro-
+thendieck’s authorship of the motives or that the “Verdier 
+duality” already mentioned could just as well be called the 
+“Grothendieck duality.” But, otherwise, everyone knows 
+that it was Grothendieck who invented schemes, motives, 
+Grothendieck topologies, topoi, and, above all, that he 
+imposed the functorial point of view via the six opera-
+tions and the derived categories. Everyone knows that it 
+is thanks to the machinery devised by Grothendieck that 
+Deligne was able to prove André Weil’s last conjecture.
+In support of his claims about the total loss of ethics 
+in the mathematical community from the 1970s onwards, 
+Grothendieck’s entire argument is based on the unique 
+testimony of one and only one mathematician who came 
+to see him several times at his home in the countryside.
+It is common practice in ethnology to rely on an infor-
+mant from the group being studied and who speaks the 
+language. The problem is that the informant may not 
+always be all that reliable and can, in fact, say anything. 
+Here it is an even worse situation, since the informant 
+declares himself to be the first person afected by the story 
+he is going to tell, namely the Riemann–Hilbert (R–H) 
+correspondence.
+T
+he informant was able to convince Grothend-
+ieck that he was, in a certain sense, his spiritual 
+son—“a continuator of my work.”21 Furthermore, 
+the informant persuaded Grothendieck that he had been 
+able to demonstrate the R–H correspondence—without 
+the slightest advice and in the face of indiference, if not 
+outright hostility, from all around him.22 And that he had 
+done so using the language of derived categories for the 
+first time in this field. Speaking of this informant, Gro-
+thendieck also writes that “his pioneering work since 1972 
+has been done in complete solitude.”23
+All of this is grossly untrue.
+The informant did his postgraduate thesis in 1974 under 
+my direction and on a subject that I had proposed to him. 
+He had largely benefited from a private talk given by 
+Masaki Kashiwara in 1975 when the informant was start-
+ing to prepare his first article, which makes no mention of 
+this decisive talk. He also benefited from a copy of Kashi-
+wara’s 1970 thesis24—written in Japanese, but there is no 
+lack of translators—and from Christian Houzel’s repeated 
+advice throughout his thèse d’état. As for the derived cat-
+egories, they appear on the first page of the foundational 
+article by Mikio Sato, Takahiro Kawai, and Kashiwara 
+published in 1973.25
+The R–H correspondence is an “equivalence of cat-
+egories” formulated by Kashiwara in 1975 and was 
+demonstrated by the same author in 1980.26
+In passing, it is worth noting that interesting equiv-
+alences of categories build bridges between diferent a 
+priori unrelated fields of mathematics: here, the partial 
+diferential equations of analysis and the constructible 
+sheaves of algebraic topology. Another more recent and 
+very important equivalence is provided by Maxim Kontse-
+vich’s “mirror symmetry,” which links complex geometry 
+and symplectic geometry.
+Grothendieck’s entire statement about the role of his 
+protégé in the story of the R–H correspondence, scattered 
+and repeated throughout the 1,900 pages of his original 
+text, is therefore based on false testimonies. However, 
+with astonishing naïveté, our author takes everything 
+that his interlocutor tells him at face value and he is 
+quoted more than 200 times. Grothendieck goes on a cru-
+sade against Kashiwara, whom he goes so far as to call “a 
+ringleader [caïd in the original French] from across the 
+Pacific,”27 even though he had never communicated with 
+Kashiwara and had only a very fragmentary knowledge 
+of his work. By extension, it is the entire Sato school that 
+is labeled “ringleaders from across the Pacific.”28 With-
+out being exhaustive, which would be impossible in any 
+practical sense without copying the entire book, consider 
+the following quote from Note 458, which is not lacking 
+in unintentional irony: “the Sato school is said to have ini-
+tiated the method of surrounding itself with obscurity in 
+order to dominate.”29
+In 1981, Grothendieck reached the peak of his resent-
+ment with an event he referred to as “le Colloque Pervers” 
+(the Perverse Colloquium), a historically important con-
+ference to which his protégé was not invited. It was on this 
+occasion that Alexander Beilinson, Joseph Bernstein, and 
+
+INFERENCE / Vol. 7, No. 3
+5 / 7
+Deligne—not counting Ofer Gabber who refused to asso-
+ciate his name with it for obscure reasons—introduced 
+perverse sheaves.30 The idea of these sheaves—which are 
+not sheaves in the strict sense, but complexes of sheaves, 
+hence, perhaps, the adjective—arises naturally from the 
+R–H correspondence. Their definition already appears 
+implicitly in Kashiwara’s 1975 text. Grothendieck was 
+outraged that his protégé was completely ignored at this 
+colloquium when he should have been its star. If no ref-
+erence was made to the authorship of R–H at the event, 
+it was probably because the mathematical community 
+was aware of the controversies surrounding it and nobody 
+wanted to be involved. But if one reads what Grothendieck 
+writes in the additions to R&S, the notable absentee at this 
+colloquium was not, in fact, his informant, but Kashiwara! 
+In other words, all the pages and pages of indignation in 
+the original text are either entirely misplaced, or do not 
+defend the right people. There is no doubt in my mind that 
+Sato and his student Kashiwara were unjustly ignored, or 
+maybe even misunderstood, by the French school during 
+the 1980s, and by Bourbaki in particular, but that is another 
+story.31
+In the edition published by Gallimard, Grothendieck 
+cautiously goes back on his assertions concerning the 
+authorship of R–H and is willing to acknowledge that 
+Kashiwara could have played a role in it,32 perhaps even 
+a role equivalent to that of his informant. Finally, at the 
+end of Part III, he ofers “my most sincere apologies” to 
+Kashiwara.33 But 1,500 pages later, none of these admis-
+sions preclude Grothendieck from insulting Kashiwara 
+once again.
+I
+n 1986, I was aware of this part of Grothendieck’s 
+text—concerning the ringleaders, or caïds, as he 
+referred to them, from the other side of the Pacific—
+and I wrote to him about it on January 16. An important 
+correspondence followed that continued for several 
+months until around the end of March. Supported by the 
+testimony of Christian Houzel, I believe that I successfully 
+convinced Grothendieck that his version of R–H was com-
+pletely false.
+In a series of additions, adding about twenty pages 
+to the initial text of R&S, some extracts from which are 
+presented below,34 Grothendieck went back completely 
+on what he had written earlier. He finally afrmed that 
+it was indeed Kashiwara who first formulated the R–H 
+correspondence in 1975 and that it was also Kashiwara 
+who gave the first outline of a proof in 1980. It is to Gro-
+thendieck’s credit that he admits his error of judgement, 
+but because it had been propagated throughout 1,900 
+pages, it was an error that was not easily rectified. Gro-
+thendieck chose to address the problem using additions 
+and footnotes, but unfortunately, apart from a flat apol-
+ogy,35 none of these amendments appear in the published  
+book.
+Consider, for example, the following additions.
+Grothendieck, writing on May 9, 1986:
+After the provisional distribution of Récoltes et semailles, 
+from October last year, I was contacted by Pierre Schapira, 
+and then by Christian Houzel, to point out some glaring 
+inaccuracies in the version of events presented in Récol-
+tes et semailles. The situation was clarified considerably 
+during correspondence with both of them, which con-
+tinued between January and March of this year. It now 
+appears to me that in the “[Zoghman] Mebkhout version” 
+(which was not lacking in internal consistency) the true, 
+the tendentious and the downright false are inextricably 
+mixed.
+Grothendieck, also dated May 15, 1986:
+In retrospect, I am convinced that Kashiwara cannot be 
+reproached for the slightest incorrectness in this case. In 
+his presentation, he gives a statement and a first sketch of 
+a proof of a theorem, which he had indeed been the first 
+to conjecture as early as 1975… Moreover, he has the cor-
+rection to specify, as early as page 2: “Let us note that the 
+theorem is also proved by Mebkhout by a diferent way.” 
+This was even “lending to the rich,” because the previ-
+ous month, in his note to the CRAS [Comptes Rendus de 
+l’Académie des Sciences] of 3 March 1980, Mebkhout had 
+expressed himself in the hypothetical form “we hope to 
+show that…,” and without making the slightest allusion to 
+it…
+A
+ll of this raises some questions about the edi-
+torial process that led to the publication of R&S 
+by Gallimard in January 2022. In the foreword, 
+which is dated January 1986,36 Grothendieck thanks Chris-
+tian Bourgois and Stéphane Deligeorges for including his 
+text in the Épistémè collection. What happened between 
+this date and the publication by Gallimard? And, above all, 
+why do Grothendieck’s additions—incorporated prior to 
+May 29, 1986—not appear in the final published version, 
+while the brief apology that does appear clearly proves 
+that the Gallimard edition includes other elements added 
+after January 1986?37
+In this review, I have focused on the mathematical sec-
+tions of the book and the passages that discuss the history 
+of the R–H correspondence. The latter is far from being 
+an anecdotal component in the book. Indeed, Grothend-
+ieck refers to it constantly—it is a leitmotif. Unfortunately, 
+and as he admits with great frankness, Grothendieck was 
+misled by an informant lacking in objectivity, to say the 
+absolute least. With a disarming and, in a certain sense, 
+admirable degree of naivety, he also admits that he never 
+imagined that the information he was being fed could be 
+biased or incomplete, let alone downright false. Between 
+1955 and 1970, Grothendieck lived in a world of pure ideas. 
+He was immersed in mathematics to an extent that is hard 
+to imagine. When he emerged from the noosphere into the 
+
+6 / 7
+BOOK REVIEWS
+real world—that is, the social world—one can only imag-
+ine the harsh shock of everyday life and how crushed he 
+felt by what he perceived as a loss of ethics in the world of 
+science. But why should this world be any diferent from 
+the rest of society? The rigor of science has never been 
+reflected in its practitioners—examples are legion.
+Science is a great devourer of men and characters.38
+Pierre Schapira is Professor Emeritus at University Pierre 
+et Marie Curie (University of Paris 6).
+1. 
+The author would like to thank Leila Schneps for her critical 
+and constructive advice.
+2. 
+Alexander Grothendieck, Récoltes et Semailles: I, II. Réflex-
+ions et témoignage sur un passé de mathématicien (Paris: 
+Gallimard, 2022), 439.
+3. 
+See the historical article by Christian Houzel “Les débuts de 
+la théorie des faisceaux,” in Masaki Kashiwara and Pierre 
+Schapira, Sheaves on Manifolds, Grundlehren der Mathema-
+tischen Wissenschaften, vol. 292 (Berlin: Springer-Verlag, 
+1990), doi:10.1007/978-3-662-02661-8.
+4. 
+Roger Godement, Théorie des faisceaux (Paris: Hermann, 
+1958).
+5. 
+Jean-Pierre Serre, “Faisceaux Algebriques Coherents,” 
+Annals of Mathematics, 2nd Series 61, no. 2. (1955): 197–278, 
+doi:10.2307/1969915 .
+6. 
+Alexander Grothendieck, “Sur quelques points d’algèbre 
+homologique,” Tōhoku Mathematical Journal 9, no. 3 (1957): 
+119–21, doi:10.2748/tmj/1178244774. This article is analyzed 
+in detail in Rick Jardine, “Tōhoku,” Inference 1, no. 3 (2015), 
+doi:10.37282/991819.15.13.
+7. 
+Mike Artin, Alexandre Grothendieck, and Jean-Louis 
+Verdier, Théorie des topos et cohomologie étale des schémas, 
+Lecture Notes in Mathematics, vols. 269, 270, 305 (Berlin: 
+Springer-Verlag, 1972–73).
+8. 
+Ralf Krömer, “La « machine de Grothendieck » se fonde-
+t-elle seulement sur des vocables métamathématiques ? 
+Bourbaki et les catégories au cours des années cinquante,” 
+Revue d’histoire des mathématiques 12 (2006): 119–62.
+9. 
+Samuel Eilenberg and Saunders Mac Lane, “Natural Iso-
+morphisms in Group Theory,” Proceedings of the National 
+Academy of Sciences 28 (1942): 537–43, doi:10.1073/
+pnas.28.12.537; Samuel Eilenberg and Saunders Mac Lane, 
+“General Theory of Natural Equivalences,” Transactions 
+of the American Mathematical Society 58 (1945): 231–94, 
+doi:10.1090/S0002-9947-1945-0013131-6.
+10. Henri Cartan and Samuel Eilenberg, Homological Algebra 
+(Princeton, NJ : Princeton University Press, 1956).
+11. Grothendieck, Récoltes et Semailles, 158.
+12. Grothendieck, Récoltes et Semailles, 42, 377.
+13. Alexander Grothendieck, “A la poursuite des champs,” 
+(1987).
+14. Pierre Cartier, “A Country Known Only by Name,” Infer-
+ence 1, no. 1 (2014), doi:10.37282/991819.14.7. For additional 
+details, see “Sascha Shapiro,” Wikipedia.
+15. Allyn Jackson, “Comme Appelé du Néant—As if Summoned 
+from the Void: The Life of Alexandre Grothendieck,” Notices 
+of the AMS 51, no. 4 and 51, no. 10 (2004): 1,038–54 and 1,196–
+212; Winfried Scharlau, “Who is Alexander Grothendiek?,” 
+Notices of the AMS 55, no. 8 (2008): 930–41; Leila Schneps, 
+“The Grothendieck Circle” (grothendieckcircle.org).
+16. The wording “extreme-left militant in Germany in the 
+1920s” does not really have the same meaning as its transpo-
+sition a century later.
+17. Grothendieck, “A la poursuite des champs.”
+18. Jackson, “Comme Appelé du Néant,” 1,050.
+19. Grothendieck, Récoltes et Semailles, 935.
+20. Grothendieck, Récoltes et Semailles, 94.
+21. Grothendieck, Récoltes et Semailles, 1,664.
+22. Grothendieck, Récoltes et Semailles, 413.
+23. Grothendieck, Récoltes et Semailles, 1,663.
+24. Masaki Kashiwara, “Algebraic Study of Systems of Partial 
+Diferential Equations,” Master’s thesis (Tokyo University, 
+1970), Mémoires de la Société Mathématique de France 63 
+(1995).
+25. Mikio Sato, Takahiro Kawai, and Masaki Kashiwara, 
+“Microfunctions 
+and 
+Pseudo-Diferential 
+Equations, 
+Hyperfunctions and Pseudo-Diferential Equations,” in 
+Proceedings of a Conference at Katata, 1971; Dedicated to the 
+Memory of André Martineau, Lecture Notes in Mathematics, 
+vol. 287 (Berlin: Springer-Verlag, 1973), 265–529.
+26. A few words on the R–H correspondence follow. The 
+modern formulation uses the theory of D-modules, a theory 
+that was intuited by Sato in the 1960s, and fully imple-
+mented by Kashiwara in his thesis. (A related theory was 
+developed independently by Joseph Bernstein.) In everyday 
+language, a coherent D-module means a system of partial 
+diferential equations with holomorphic coefcients. Holo-
+nomic modules are the > 1 dimensional version of ordinary 
+diferential equations, and among them regular holonomic 
+modules generalize the classical notion of Fuchsian equa-
+tions. In 1975, Kashiwara showed that the functor Sol, 
+which associates the complex of its holomorphic solutions 
+to a holonomic module, takes its values in constructible 
+sheaves, the sheaves which behave locally as a direct sum 
+of constant sheaves along a stratification. In the same year 
+he conjectured that there exists a triangulated subcategory 
+of the holonomic modules, the regular holonomic modules, 
+on which the functor Sol induces an “equivalence of cate-
+gories.” Kashiwara proved his conjecture in 1980 and gave a 
+detailed account of the main steps in his proof at the École 
+Polytechnique seminar, which was published. His proof 
+uses Heisuke Hironaka’s singularity resolution theorem and 
+the precursor work of Deligne who treated the special case 
+of “meromorphic connections.”
+27. Grothendieck, Récoltes et Semailles, 1,656.
+28. Grothendieck, Récoltes et Semailles, 1,651.
+29. Grothendieck, Récoltes et Semailles, 1,650.
+
+INFERENCE / Vol. 7, No. 3
+7 / 7
+30. Alexander Beilinson, Joseph Bernstein, and Pierre Del-
+igne, “Faisceaux pervers, Analysis and topology on singular 
+spaces, I” (Luminy, 1981), Astérisque, no. 100 (Paris: Societé 
+Mathematique de France, Paris, 1982): 5–171.
+31. Pierre Schapira, “Mikio Sato, a Visionary of Mathematics,” 
+Notices of the AMS 54, no. 2 (2007); Pierre Schapira, “Fifty 
+Years of Mathematics with Masaki Kashiwara,” Proceed-
+ings of the International Congress of Mathematicians, Rio 
+de Janeiro, 2018, vol. 1, Plenary Lectures (Singapore: World 
+Scientific, 2018).
+32. Grothendieck, Récoltes et Semailles, 163.
+33. Grothendieck, Récoltes et Semailles, 164.
+34. These additions have just been posted on Leila Schneps’s 
+website, “The Grothendieck Circle” (grothendieckcircle.org).
+35. Grothendieck, Récoltes et Semailles, 163.
+36. Grothendieck, Récoltes et Semailles, 9–15.
+37. On the recently updated “Grothendieck Circle” (grothend-
+ieckcircle.org) website, it is written that Gallimard editions 
+will soon publish a new version of R&S augmented with 
+these Grothendieck additions.
+38. Adapted freely from a famous quotation by Leon Trotsky.
+DOI: 10.37282/991819.22.61
+Sorbonne Université, Cnrs IMJ-PRG
+pierre.schapira@imj-prg.fr
+http://webusers.imj-prg.fr/~pierre.schapira
+

GdE1T4oBgHgl3EQfFAO9/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf,len=349
+page_content='INFERENCE / Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3  1 / 7  A Truncated Manuscript  Pierre Schapira  Récoltes et Semailles I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Réflexions et témoignage  sur un passé de mathématicien  by Alexander Grothendieck  Editions Gallimard, 29,50 €.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  S  trictly speaking, this essay is not solely a review  of Alexander Grothendieck’s Récoltes et Semailles  (Reaping and Sowing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='1 Although the book as a  whole, as well as Grothendieck’s work and life, will be dis-  cussed here, a good part of the essay is devoted to refuting  a thesis Grothendieck developed throughout his original  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' We have an important ally in this respect: the author  himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In a series of important additions that have not  been incorporated in this new version, Grothendieck goes  back entirely on some of his assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  T  he figure of Grothendieck dominated much of  mathematics during the second half of the twen-  tieth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' If his work is essentially concerned  with algebraic geometry, his vision and methods have  spread far beyond—to algebraic topology, representation  theory, complex geometry, symplectic geometry, algebraic  analysis and even, more recently, computational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In short, all linear mathematics, as opposed to dynamical  systems, probabilities, or purely geometric geometry, such  as Riemannian or Hamiltonian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It was under  his influence that the language of derived categories and  sheaf theory were established in these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It was also  Grothendieck who had the intuition and then formulated  the main lines of the theory of derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He left  it to his student Jean-Louis Verdier to write out all the  details for his thesis, in which Verdier clarified the key  notion of a triangulated category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='2 But it was Grothend-  ieck who situated sheaf theory and the six operations in  the theory of derived categories, to which we will return  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Between 1950 and 1970, functions—possibly gen-  eralized—were studied on real or complex manifolds,  and especially on Euclidean spaces, using the Fourier  transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But on a complex manifold, when we refer  to functions we really mean holomorphic functions, and  these present a serious difculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' That is, they do not  exist, at least not globally on a compact manifold such as  the projective line—apart, of course, from the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Global knowledge, therefore, does not provide any infor-  mation, unlike the real diferentiable case, and one must  work locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' There is a rather extraordinary tool for this  purpose: sheaf theory, which was invented by Jean Leray  while he was a prisoner of war in Germany between 1941  and 1945.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='3 Leray’s text was somewhat incomprehensible  but was later clarified by Henri Cartan and Jean-Pierre  Serre, resulting in Roger Godement’s famous book,  Théorie des Faisceaux (Theory of Sheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='4 Cartan and  Serre used this tool in their respective studies of holomor-  phic functions in dimension ≥ 1, after the seminal work of  Kiyoshi Oka, giving rise to Cartan’s Theorems A and B and  Serre’s influential paper “Faisceaux algébriques cohérents”  (Coherent algebraic sheaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='5  It is in this context that Grothendieck approached  algebraic geometry around 1955, providing a solid basis  for sheaf cohomology in a foundational paper published  by the Tōhoku Mathematical Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='6 In this paper, one  already encounters—implicitly—the main difculty of cat-  egory theory, namely the problem of universes, a problem  later solved by Grothendieck in SGA4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This was done in the  manner of another Alexander cutting the Gordian knot:7  Grothendieck poses the axiom that every set belongs to  a universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This problem of universes, also known as the  inaccessible cardinals, outside of which category theory  cannot develop, is probably the reason why Bourbaki  gave up on categories and why Grothendieck then left the  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Ralf Krömer has written an excellent article on this  issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='8  During the 1940s and 1950s, there were two concep-  tual revolutions whose importance was not immediately  understood: sheaf theory, as mentioned above, and cat-  egory theory, the latter due to Samuel Eilenberg and  Saunders Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='9 Moreover, the categorical point of  view is part of a vast movement of ideas that embraced  the structuralist approach of Claude Lévi-Strauss and  the linguistics of Noam Chomsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Instead of considering  sets endowed with certain structures, category theory  focuses on the relations that can exist between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A  category C is thus a family of objects—as a set is a family  of elements—but given two objects X and Y, there exists  a priori a set called HomC (X, Y) representing the mor-  Published by Inference-Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='com  Vol 7, issue 3, Dec 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  2 / 7  BOOK REVIEWS  phisms from X to Y, these data being, of course, subject to  a certain number of natural axioms—composition of mor-  phisms, identity morphisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A new step then consists  in looking at morphisms between categories, which are  called functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Certain key notions then emerge—such as  those of adjoint functors, final or initial objects, limits and  colimits—giving a precise and unifying meaning to many  ideas that run through mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  There is a family of categories that plays a central role:  these are the additive categories and, among them, the  abelian categories, which are modeled on the category of  modules over a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But if the vector spaces over a field are  replaced by the modules over a ring, the classical tensor  product and internal Hom functors are no longer exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=',  they do not transform exact sequences into exact sequenc-  es—a subspace does not always admit a supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  It is thus necessary to consider the derived functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' We  then enter the domain of homological algebra, a natural  generalization of linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Here, the reference book  was initially that of Cartan and Eilenberg,10 before it was  dethroned by Grothendieck’s Tōhoku paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But comput-  ing the derived functor of the composition of two functors  requires the use of Leray’s spectral sequences, giving rise  to often inextricable calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This is where derived  categories show their power: in this language, everything  is remarkably simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  What are the six operations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' With ordinary functions, we  have three natural operations, besides addition: the prod-  uct and, associated with an application f : X → Y between  real manifolds, the integration which sends—modulo some  technical details—functions on X to functions on Y and the  composition by f which sends functions on Y to functions  on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In sheaf theory, the tensor product ⊗  L  is the analogue  of the product, the proper direct image Rf!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' is the analogue  of the integration, and the inverse image f –1 is the analogue  of the composition by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But the tensor product has a right  adjoint, Rhom, the functor f –1 has a right adjoint, the direct  image Rf*, and the functor Rf!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' has a right adjoint, f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='.  The functor f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=', which exists only in the derived frame-  work, unlike the other five, was discovered by Grothendieck  in the context of étale cohomology and was subsequently  constructed by Verdier for locally compact spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' As Gro-  thendieck had seen, f !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' provides a broad generalization of  Poincaré duality and this functor now plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  But since locally compact spaces appear more often than  the étale topology, it is the name Poincaré–Verdier, if not  just Verdier alone, that remains associated with duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  This attribution is largely unfair and left Grothendieck  feeling somewhat bitter—rightly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='11  One might think that such an abstract framework  dispenses with explicit computations, but this is a mis-  conception: put simply, the computations are no longer  the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' If the direct image functor does not allow for  integrals to be computed explicitly, the formalism of the  six operations nevertheless gives rise to sophisticated  numerical results, such as the computation of dimensions  of cohomology spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The Riemann–Roch–Hirzebruch–  Grothendieck theorem is a beautiful illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In the same vein, one of Grothendieck’s fundamen-  tal discoveries was to develop sheaf theory on categories  and thus, in particular, on spaces that no longer have any  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' What do the sheaves require to exist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Namely, the  data of open sets and their inclusions, and the notion of a  covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' There is nothing to prevent the objects of a cat-  egory from playing the role of the open sets, the category  is then called a pre-site, and it remains to define axiomat-  ically what the coverings are in order to obtain a site—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=',  a category with a Grothendieck topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This natural  generalization of the usual topological spaces proves to be  extremely fruitful and analysts would, in fact, do well to  draw inspiration from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' On a real manifold, there are far  too many pathological open sets and too many coverings  if one is interested in what happens at the edge of an open  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  One then arrives at topos theory—topoi for the scholars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  The underlying idea, which in a particular situation goes  back to Israel Gelfand, is that a space—in this case, a site—  can be reconstructed from the category of sheaves on that  site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A topos is then a category equivalent to a category  of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The category of sets, for example, is nothing  other than the topos associated with a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But even if  topos theory has been used in a new proof of Paul Cohen’s  result on the independence of the continuum hypothesis,  its applications in mathematics are still uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  This presentation of a selection of Grothendieck’s fun-  damental ideas is far from complete and only reflects the  particular interests of the reviewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In R&S, Grothendieck  lists what he considers the 12 key ideas of his work and  also provides a list of his students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='12  One should, of course, also mention his first works in  functional analysis, dating from around 1955, and scheme  theory, which revolutionized algebraic geometry, as well  as the intuition of motives, a partially conjectural theory  that was later developed by Pierre Deligne, Vladimir  Voevodsky, Joseph Ayoub, and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' One should  also mention the fundamental text “A la poursuite des  champs” in which Grothendieck lays the foundations for  ∞-categories and homotopical algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='13 Indeed, if trian-  gulated categories are an incredibly simple and efcient  tool, they have a defect which seriously limits their use, for  example, in gluing problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This defect is linked to the  fact that a certain morphism is unique up to isomorphism,  but this isomorphism is not unique!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The new theory of  ∞-categories, to which the names Jacob Lurie, Graeme  Segal, Bertrand Töen, and a few others must be associated,  is in the process of completely supplanting the classical  theory of derived categories, although it is, for the time  being, not easily accessible, to say the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  P  ierre cartier has written a remarkable article on  Grothendieck’s life and it is pointless to paraphrase  it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='14 There are also excellent articles by Allyn  INFERENCE / Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3  3 / 7  Jackson and Winfried Scharlau, as well as all the links  on the Grothendieck Circle website managed by Leila  Schneps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='15  Nonetheless, a few words on this subject are helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck’s father, Sascha Schapiro, was a Russian  anarchist who took part in the aborted revolution of 1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  He then served ten years in the prisons of Czar Nicholas  II before being released following the revolution of 1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Despite initially being feted as a hero, Schapiro was soon  declared an enemy of the people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He later fought along-  side the Republicans during the Spanish Civil war before  becoming a traveling photographer in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In 1939,  Schapiro was interned at the Camp Vernet in the French  Pyrenees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He was handed over to the Nazis by the Vichy  police in 1942 and disappeared into Auschwitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck’s mother, Hanka, was an extreme-left  militant in Germany during the 1920s who emigrated to  France when Adolf Hitler came to power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='16 Her son did not  join her until 1938, at the age of 10, after having lived in  hiding on a farm in Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck spent part of  the war in Le Chambon-sur-Lignon at the famous Collège  Cévenol that saved so many Jewish children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck’s mathematical life began in Nancy  during the 1950s, where Jean Dieudonné and Laurent  Schwartz took him under their wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' After his initial work  in functional analysis, which still remains fundamental, he  turned to algebraic geometry with great success, a story  that is now well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck was one of the first two professors  appointed to the Institut des Hautes Études Scientifiques  (IHES) in 1959, where he obtained most of his results and  published, with the help of Jean Dieudonné, the other  professor at the IHES, the famous EGA (Éléments de  géométrie algébrique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He directed the seminar on alge-  braic geometry, which resulted in a publication of more  than 5,000 pages coauthored with some of his students,  known as SGA (Séminaire de Géométrie Algébrique du Bois  Marie).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck was awarded the Fields Medal at  the International Congress of Mathematicians in 1966,  but did not travel to Moscow to receive it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In 1988, he won  the prestigious and well-funded Crafoord Prize, which he  refused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck left the IHES in 1970 after discovering  that the institute benefited from military funding and  launched his own ecological crusade, first through the  journal Survivre, and then later Survivre et vivre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But Gro-  thendieck not only left the IHES, he also left the world of  mathematics and, in particular, his students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He returned  to mathematics in 1983, but in a very diferent style, with  his publications “Esquisse d’un programme” (Sketch of a  Programme) and “À la poursuite des champs” (Pursuing  Stacks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='17 After a year at the Collège de France, he was  appointed professor in Montpellier, where he worked  until his retirement in 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He spent his final years living  in the countryside in almost total seclusion, until his death  in 2014 at the age of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  A  s we have seen, Grothendieck is the author of a  considerable body of mathematical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But he  is also the author of significant literary works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Among them is R&S, which was published by Gallimard in  January 2022 after having been widely distributed on the  internet since Grothendieck first wrote the text in 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Amounting to more than 1,900 pages, the book deals with  many subjects: the author’s journey as a mathematician,  his passions, his illusions and disillusions, the process of  creation, and a thousand other topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It also includes long  passages on Yin and Yang, feminine and masculine ways  of doing mathematics, the mother, the father and child,  dreams, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A large part of the text is devoted to  a revelation he is said to have experienced in 1976 and  a long period of meditation that followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It is a kind of  self-analysis tinged, it has to be said, with a certain degree  of paranoia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A recurring theme is the sense of betrayal he  felt toward his former students, which is manifested in his  work being ignored and forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The words “funeral,”  “deceased,” “hearse,” “massacre,” and “gravedigger,” and so  on, quickly become omnipresent after their appearance in  the table of contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' More generally, the book denounces  a loss of ethics among the entire mathematical community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck explains to the reader that mathematics  “was better before”—that is, prior to 1960—as if the older  generation was irreproachable!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In fact,  on the contrary,  it can be said that mathematicians have become much  more honest since the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The source of this miracle  has a name: arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It is now becoming ever more difcult  to appropriate the ideas of others, although, of course, it  is still possible to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The institution of math-  ematics itself has also been greatly improved, or at least  has been greatly transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The system of mandarins  that dominated French mathematics until the 1970s, from  which Grothendieck did not experience any difculties  and about whom he does not say a word, has practically  disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck, who is very self-critical throughout the  text, sometimes ponders whether he might have been arro-  gant or even contemptuous of those around him during his  heyday in the 1960s and 1970s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Despite these concerns, it  is clear that he cares little about ingratiating himself with  his readers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Instead, he ofers a book of more than 1,900  pages, while in response to a question about the IHES  library in its early days, he remarks: “We don’t read books,  we write them!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='18 R&S contains many contradictions that  are only partly corrected by a series of Notes—some of  which, despite being of particular importance, are not  included in this new edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Addressing these contradic-  tions properly would undoubtedly have required the text  to be completely rewritten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck is not paralyzed by any sense of false  modesty:  The thing that struck me is that I do not remember having  known,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' even from the allusions of friends or colleagues  4 / 7  BOOK REVIEWS  who are better versed in history,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' of a mathematician apart  from myself who contributed such a multiplicity of inno-  vative ideas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' not more or less disjointed from one another,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  but as part of a vast unifying vision (as was the case for  Newton and for Einstein in physics and cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' and for  Darwin and for Pasteur in biology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='19  Elsewhere, he writes: “It would seem that, as a servant of  a vast unifying vision born in me, I am ‘one of a kind’ in the  history of mathematics from its origin to the present day.”20  Although the writing style is not lacking in inspiration,  it is nonetheless uneven and sometimes—deliberately—  familiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck is not le duc de Saint-Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  The following analysis will focus only on the content con-  cerning mathematics and the world of mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In the text, Grothendieck complains at length that his  ideas have been plundered by his former students with-  out reference to their master or that they have simply been  erased and forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' These assertions are not always  supported by solid arguments or precise references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But,  above all, it is the nature of discoveries to be trivialized  and their author forgotten, and all the more so when the  underlying ideas are often, in hindsight, obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck’s reproaches are addressed to all his  pupils, and particularly to Deligne—whose name is almost  always preceded by the words “my friend,” insinuating  “my former friend”—and to Verdier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It is quite possible to  imagine that Deligne was only lightly involved with Gro-  thendieck’s authorship of the motives or that the “Verdier  duality” already mentioned could just as well be called the  “Grothendieck duality.” But, otherwise, everyone knows  that it was Grothendieck who invented schemes, motives,  Grothendieck topologies, topoi, and, above all, that he  imposed the functorial point of view via the six opera-  tions and the derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Everyone knows that it  is thanks to the machinery devised by Grothendieck that  Deligne was able to prove André Weil’s last conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In support of his claims about the total loss of ethics  in the mathematical community from the 1970s onwards,  Grothendieck’s entire argument is based on the unique  testimony of one and only one mathematician who came  to see him several times at his home in the countryside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  It is common practice in ethnology to rely on an infor-  mant from the group being studied and who speaks the  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The problem is that the informant may not  always be all that reliable and can, in fact, say anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Here it is an even worse situation, since the informant  declares himself to be the first person afected by the story  he is going to tell, namely the Riemann–Hilbert (R–H)  correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  T  he informant was able to convince Grothend-  ieck that he was, in a certain sense, his spiritual  son—“a continuator of my work.”21 Furthermore,  the informant persuaded Grothendieck that he had been  able to demonstrate the R–H correspondence—without  the slightest advice and in the face of indiference, if not  outright hostility, from all around him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='22 And that he had  done so using the language of derived categories for the  first time in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Speaking of this informant, Gro-  thendieck also writes that “his pioneering work since 1972  has been done in complete solitude.”23  All of this is grossly untrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  The informant did his postgraduate thesis in 1974 under  my direction and on a subject that I had proposed to him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  He had largely benefited from a private talk given by  Masaki Kashiwara in 1975 when the informant was start-  ing to prepare his first article, which makes no mention of  this decisive talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He also benefited from a copy of Kashi-  wara’s 1970 thesis24—written in Japanese, but there is no  lack of translators—and from Christian Houzel’s repeated  advice throughout his thèse d’état.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' As for the derived cat-  egories, they appear on the first page of the foundational  article by Mikio Sato, Takahiro Kawai, and Kashiwara  published in 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='25  The R–H correspondence is an “equivalence of cat-  egories” formulated by Kashiwara in 1975 and was  demonstrated by the same author in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='26  In passing, it is worth noting that interesting equiv-  alences of categories build bridges between diferent a  priori unrelated fields of mathematics: here, the partial  diferential equations of analysis and the constructible  sheaves of algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Another more recent and  very important equivalence is provided by Maxim Kontse-  vich’s “mirror symmetry,” which links complex geometry  and symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck’s entire statement about the role of his  protégé in the story of the R–H correspondence, scattered  and repeated throughout the 1,900 pages of his original  text, is therefore based on false testimonies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' However,  with astonishing naïveté, our author takes everything  that his interlocutor tells him at face value and he is  quoted more than 200 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck goes on a cru-  sade against Kashiwara, whom he goes so far as to call “a  ringleader [caïd in the original French] from across the  Pacific,”27 even though he had never communicated with  Kashiwara and had only a very fragmentary knowledge  of his work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' By extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' it is the entire Sato school that  is labeled “ringleaders from across the Pacific.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='28 With-  out being exhaustive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' which would be impossible in any  practical sense without copying the entire book,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' consider  the following quote from Note 458,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' which is not lacking  in unintentional irony: “the Sato school is said to have ini-  tiated the method of surrounding itself with obscurity in  order to dominate.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='29  In 1981,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck reached the peak of his resent-  ment with an event he referred to as “le Colloque Pervers”  (the Perverse Colloquium),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' a historically important con-  ference to which his protégé was not invited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It was on this  occasion that Alexander Beilinson, Joseph Bernstein, and  INFERENCE / Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3  5 / 7  Deligne—not counting Ofer Gabber who refused to asso-  ciate his name with it for obscure reasons—introduced  perverse sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='30 The idea of these sheaves—which are  not sheaves in the strict sense, but complexes of sheaves,  hence, perhaps, the adjective—arises naturally from the  R–H correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Their definition already appears  implicitly in Kashiwara’s 1975 text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck was  outraged that his protégé was completely ignored at this  colloquium when he should have been its star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' If no ref-  erence was made to the authorship of R–H at the event,  it was probably because the mathematical community  was aware of the controversies surrounding it and nobody  wanted to be involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But if one reads what Grothendieck  writes in the additions to R&S, the notable absentee at this  colloquium was not, in fact, his informant, but Kashiwara!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In other words, all the pages and pages of indignation in  the original text are either entirely misplaced, or do not  defend the right people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' There is no doubt in my mind that  Sato and his student Kashiwara were unjustly ignored, or  maybe even misunderstood, by the French school during  the 1980s, and by Bourbaki in particular, but that is another  story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='31  In the edition published by Gallimard, Grothendieck  cautiously goes back on his assertions concerning the  authorship of R–H and is willing to acknowledge that  Kashiwara could have played a role in it,32 perhaps even  a role equivalent to that of his informant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Finally, at the  end of Part III, he ofers “my most sincere apologies” to  Kashiwara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='33 But 1,500 pages later, none of these admis-  sions preclude Grothendieck from insulting Kashiwara  once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  I  n 1986, I was aware of this part of Grothendieck’s  text—concerning the ringleaders, or caïds, as he  referred to them, from the other side of the Pacific—  and I wrote to him about it on January 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' An important  correspondence followed that continued for several  months until around the end of March.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Supported by the  testimony of Christian Houzel, I believe that I successfully  convinced Grothendieck that his version of R–H was com-  pletely false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  In a series of additions, adding about twenty pages  to the initial text of R&S, some extracts from which are  presented below,34 Grothendieck went back completely  on what he had written earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' He finally afrmed that  it was indeed Kashiwara who first formulated the R–H  correspondence in 1975 and that it was also Kashiwara  who gave the first outline of a proof in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It is to Gro-  thendieck’s credit that he admits his error of judgement,  but because it had been propagated throughout 1,900  pages, it was an error that was not easily rectified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Gro-  thendieck chose to address the problem using additions  and footnotes, but unfortunately, apart from a flat apol-  ogy,35 none of these amendments appear in the published  book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Consider, for example, the following additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck, writing on May 9, 1986:  After the provisional distribution of Récoltes et semailles,  from October last year, I was contacted by Pierre Schapira,  and then by Christian Houzel, to point out some glaring  inaccuracies in the version of events presented in Récol-  tes et semailles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The situation was clarified considerably  during correspondence with both of them, which con-  tinued between January and March of this year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' It now  appears to me that in the “[Zoghman] Mebkhout version”  (which was not lacking in internal consistency) the true,  the tendentious and the downright false are inextricably  mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Grothendieck, also dated May 15, 1986:  In retrospect, I am convinced that Kashiwara cannot be  reproached for the slightest incorrectness in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In  his presentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' he gives a statement and a first sketch of  a proof of a theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' which he had indeed been the first  to conjecture as early as 1975… Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' he has the cor-  rection to specify,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' as early as page 2: “Let us note that the  theorem is also proved by Mebkhout by a diferent way.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  This was even “lending to the rich,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' because the previ-  ous month,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' in his note to the CRAS [Comptes Rendus de  l’Académie des Sciences] of 3 March 1980,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Mebkhout had  expressed himself in the hypothetical form “we hope to  show that…,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' and without making the slightest allusion to  it…  A  ll of this raises some questions about the edi-  torial process that led to the publication of R&S  by Gallimard in January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In the foreword,  which is dated January 1986,36 Grothendieck thanks Chris-  tian Bourgois and Stéphane Deligeorges for including his  text in the Épistémè collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' What happened between  this date and the publication by Gallimard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' And, above all,  why do Grothendieck’s additions—incorporated prior to  May 29, 1986—not appear in the final published version,  while the brief apology that does appear clearly proves  that the Gallimard edition includes other elements added  after January 1986?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='37  In this review, I have focused on the mathematical sec-  tions of the book and the passages that discuss the history  of the R–H correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The latter is far from being  an anecdotal component in the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Indeed, Grothend-  ieck refers to it constantly—it is a leitmotif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Unfortunately,  and as he admits with great frankness, Grothendieck was  misled by an informant lacking in objectivity, to say the  absolute least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' With a disarming and, in a certain sense,  admirable degree of naivety, he also admits that he never  imagined that the information he was being fed could be  biased or incomplete, let alone downright false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Between  1955 and 1970, Grothendieck lived in a world of pure ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  He was immersed in mathematics to an extent that is hard  to imagine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' When he emerged from the noosphere into the  6 / 7  BOOK REVIEWS  real world—that is, the social world—one can only imag-  ine the harsh shock of everyday life and how crushed he  felt by what he perceived as a loss of ethics in the world of  science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' But why should this world be any diferent from  the rest of society?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The rigor of science has never been  reflected in its practitioners—examples are legion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Science is a great devourer of men and characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='38  Pierre Schapira is Professor Emeritus at University Pierre  et Marie Curie (University of Paris 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  The author would like to thank Leila Schneps for her critical  and constructive advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Alexander Grothendieck, Récoltes et Semailles: I, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Réflex-  ions et témoignage sur un passé de mathématicien (Paris:  Gallimard, 2022), 439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  See the historical article by Christian Houzel “Les débuts de  la théorie des faisceaux,” in Masaki Kashiwara and Pierre  Schapira, Sheaves on Manifolds, Grundlehren der Mathema-  tischen Wissenschaften, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 292 (Berlin: Springer-Verlag,  1990), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='1007/978-3-662-02661-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Roger Godement, Théorie des faisceaux (Paris: Hermann,  1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Jean-Pierre Serre, “Faisceaux Algebriques Coherents,”  Annals of Mathematics, 2nd Series 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' (1955): 197–278,  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='2307/1969915 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Alexander Grothendieck, “Sur quelques points d’algèbre  homologique,” Tōhoku Mathematical Journal 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3 (1957):  119–21, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='2748/tmj/1178244774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' This article is analyzed  in detail in Rick Jardine, “Tōhoku,” Inference 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3 (2015),  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='37282/991819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Mike Artin, Alexandre Grothendieck, and Jean-Louis  Verdier, Théorie des topos et cohomologie étale des schémas,  Lecture Notes in Mathematics, vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 269, 270, 305 (Berlin:  Springer-Verlag, 1972–73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Ralf Krömer, “La « machine de Grothendieck » se fonde-  t-elle seulement sur des vocables métamathématiques ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Bourbaki et les catégories au cours des années cinquante,”  Revue d’histoire des mathématiques 12 (2006): 119–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  Samuel Eilenberg and Saunders Mac Lane, “Natural Iso-  morphisms in Group Theory,” Proceedings of the National  Academy of Sciences 28 (1942): 537–43, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='1073/  pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='537;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Samuel Eilenberg and Saunders Mac Lane,  “General Theory of Natural Equivalences,” Transactions  of the American Mathematical Society 58 (1945): 231–94,  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='1090/S0002-9947-1945-0013131-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Henri Cartan and Samuel Eilenberg, Homological Algebra  (Princeton, NJ : Princeton University Press, 1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 42, 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Alexander Grothendieck, “A la poursuite des champs,”  (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Pierre Cartier, “A Country Known Only by Name,” Infer-  ence 1, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 1 (2014), doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='37282/991819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' For additional  details, see “Sascha Shapiro,” Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Allyn Jackson, “Comme Appelé du Néant—As if Summoned  from the Void: The Life of Alexandre Grothendieck,” Notices  of the AMS 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 4 and 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 10 (2004): 1,038–54 and 1,196–  212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Winfried Scharlau, “Who is Alexander Grothendiek?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=',”  Notices of the AMS 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 8 (2008): 930–41;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Leila Schneps,  “The Grothendieck Circle” (grothendieckcircle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The wording “extreme-left militant in Germany in the  1920s” does not really have the same meaning as its transpo-  sition a century later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, “A la poursuite des champs.”  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Jackson, “Comme Appelé du Néant,” 1,050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 1,664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 1,663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Masaki Kashiwara, “Algebraic Study of Systems of Partial  Diferential Equations,” Master’s thesis (Tokyo University,  1970), Mémoires de la Société Mathématique de France 63  (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Mikio Sato, Takahiro Kawai, and Masaki Kashiwara,  “Microfunctions  and  Pseudo-Diferential  Equations,  Hyperfunctions and Pseudo-Diferential Equations,” in  Proceedings of a Conference at Katata, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Dedicated to the  Memory of André Martineau, Lecture Notes in Mathematics,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 287 (Berlin: Springer-Verlag, 1973), 265–529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' A few words on the R–H correspondence follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' The  modern formulation uses the theory of D-modules, a theory  that was intuited by Sato in the 1960s, and fully imple-  mented by Kashiwara in his thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' (A related theory was  developed independently by Joseph Bernstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=') In everyday  language, a coherent D-module means a system of partial  diferential equations with holomorphic coefcients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Holo-  nomic modules are the > 1 dimensional version of ordinary  diferential equations, and among them regular holonomic  modules generalize the classical notion of Fuchsian equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In 1975, Kashiwara showed that the functor Sol,  which associates the complex of its holomorphic solutions  to a holonomic module, takes its values in constructible  sheaves, the sheaves which behave locally as a direct sum  of constant sheaves along a stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' In the same year  he conjectured that there exists a triangulated subcategory  of the holonomic modules, the regular holonomic modules,  on which the functor Sol induces an “equivalence of cate-  gories.” Kashiwara proved his conjecture in 1980 and gave a  detailed account of the main steps in his proof at the École  Polytechnique seminar, which was published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' His proof  uses Heisuke Hironaka’s singularity resolution theorem and  the precursor work of Deligne who treated the special case  of “meromorphic connections.”  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 1,656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 1,651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 1,650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  INFERENCE / Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 7, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 3  7 / 7  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Alexander Beilinson, Joseph Bernstein, and Pierre Del-  igne, “Faisceaux pervers, Analysis and topology on singular  spaces, I” (Luminy, 1981), Astérisque, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 100 (Paris: Societé  Mathematique de France, Paris, 1982): 5–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Pierre Schapira, “Mikio Sato, a Visionary of Mathematics,”  Notices of the AMS 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 2 (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Pierre Schapira, “Fifty  Years of Mathematics with Masaki Kashiwara,” Proceed-  ings of the International Congress of Mathematicians, Rio  de Janeiro, 2018, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' 1, Plenary Lectures (Singapore: World  Scientific, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' These additions have just been posted on Leila Schneps’s  website, “The Grothendieck Circle” (grothendieckcircle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Grothendieck, Récoltes et Semailles, 9–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' On the recently updated “Grothendieck Circle” (grothend-  ieckcircle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='org) website, it is written that Gallimard editions  will soon publish a new version of R&S augmented with  these Grothendieck additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content=' Adapted freely from a famous quotation by Leon Trotsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='  DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='37282/991819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='61  Sorbonne Université, Cnrs IMJ-PRG  pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='schapira@imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='fr  http://webusers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='imj-prg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='fr/~pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}
+page_content='schapira' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE1T4oBgHgl3EQfFAO9/content/2301.02898v1.pdf'}

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+Quantum algorithm for finding minimum values in a Quantum Random Access Memory
+Anton S. Albino,1, ∗ Lucas Q. Galv˜ao,1, † Ethan Hansen,2, ‡ Mauro Q. Nooblath Neto,1, § and Clebson Cruz3, ¶
+1Latin American Quantum Computing Center, SENAI CIMATEC, Salvador, Brazil.
+2Zapata Computing, Canada
+3Grupo de Informa¸c˜ao Quˆantica, Centro de Ciˆencias Exatas e das Tecnologias,
+Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos. Rua Bertioga,
+892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil.
+Finding the minimum value in an unordered database is a common and fundamental task in computer science.
+However, the optimal classical deterministic algorithm can find the minimum value with a time complexity that
+grows linearly with the number of elements in the database. In this paper, we present the proposal of a quantum
+algorithm for finding the minimum value of a database, which is quadratically faster than its best classical
+analogs. We assume a Quantum Random Access Memory (QRAM) that stores values from a database and
+perform an iterative search based on an oracle whose role is to limit the searched values by controlling the states
+of the most significant qubits. A complexity analysis was performed in order to demonstrate the advantage of this
+quantum algorithm over its classical counterparts. Furthermore, we demonstrate how the proposed algorithm
+would be used in an unsupervised machine learning task through a quantum version of the K-means algorithm.
+Keywords: Quantum RAM, Minimum search, Grover’s Algorithm
+I.
+INTRODUCTION
+Random Access Memory (RAM) is a versatile, short-term
+memory used in computing for storing and retrieving infor-
+mation via bits [1]. Similarly, the concept of Quantum RAM
+(QRAM) emerges with the same goal but employing qubits
+to apply a superposition of states to achieve faster results for
+computational applications, whether quantum or classical [2–
+11]. Several works discuss the potential of its applications
+to optimize the execution of quantum algorithms, including
+quantum searching on a classical database [10, 12–15], col-
+lision finding [12, 16–18], and algorithms for solving linear
+systems [19–22], for instance.
+These results have attracted the attention of the scien-
+tific community in the past few years, leading to the de-
+velopment of QRAM architectures that demonstrate the po-
+tential for producing efficient results in quantum computing
+[3, 4, 6, 7, 9, 23–25]. Some models, such as Fanout quantum
+RAM [3, 7, 9] and Bucket-Brigade quantum RAM [3, 23, 25],
+illustrate potential future implementations of a QRAM in
+practical scenarios. In addition, recent experiments have re-
+vealed a designed architecture for hybrid quantum computers
+that use superconducting qubits and spin-qubit memory crys-
+tals capable, in theory, of implementing a QRAM in real sys-
+tems [11].
+Furthermore, other efforts have been made to construct
+quantum algorithms that are able to optimally access a QRAM
+in the process of searching for certain values stored in its cells
+[10]. In general, problems based on searching use the famous
+Grover’s algorithm to search quantum states in an unstruc-
+tured list [16, 26–29]. On the other hand, a well-known exam-
+ple of determining the minimal value in a list is the so-called
+∗ anton.albino@fieb.org.br
+† lucas.g5847@ufob.edu.br
+‡ 1ethanhansen@proton.me
+§ mauro.neto@fbter.org.br
+¶ clebson.cruz@ufob.edu.br
+D¨urr-Hoyer minimum finding algorithm [30], which employs
+Grover’s Algorithm as a fundamental subroutine to find the
+greatest or smallest entry in a list [31].
+In this scenario, based on the core concept of Durr-Hoyer’s
+algorithm, we apply Grover’s Algorithm as a subroutine to de-
+velop a quantum algorithm for identifying the smallest value
+in a classical data set stored in a QRAM. The proposed Quan-
+tum Minimum Search (QMS) algorithm is based on the itera-
+tive change of the oracle function, which limits the searched
+values by controlling the states of the most significant qubits.
+First, we present the description of the QMS algorithm, de-
+scribing the concept of QRAM and approaching an example
+to find the minimum in a list of four real values using the
+proposed algorithm. In sequence, we analyze the complex-
+ity of the QMS algorithm compared with classical algorithms.
+The results show that, whereas the complexity of the classi-
+cal algorithm grows linearly with the number of elements in
+the database, O(N), since the classical algorithms go through
+all the N items in the list, the presented QMS algorithm has
+a complexity of O(
+�
+N
+t ), with t being the number of marked
+states. Finally, we present an application of the proposed al-
+gorithm in the K-means problem of determining the optimal
+location of K-centroids in order to minimize the sum of all
+distances between the points and their respective centroids.
+II.
+QUANTUM MINIMUM SEARCH (QMS) ALGORITHM
+The search problem is a ubiquitous subject of discussion
+in classical computer science [26]. The problem consists of
+identifying the index of the database item (x) that fulfills some
+predetermined search criterion x = y, where y is the sought
+element, given an unstructured database with N elements. In
+this context, it is possible to prepare the so-called response
+function (R(x)) that translates database entries to True if the
+entry x matches the search criterion (x = y) or False if x �
+y. This is possible by using the so-called Oracle subroutine,
+which queries the database until the desired item is located.
+arXiv:2301.05122v1  [quant-ph]  12 Jan 2023
+
+2
+Consequently, the bigger the requested element’s position in
+the list, the greater the number of queries required to locate it.
+Therefore, the complexity of this task is exactly proportional
+to the number of items on the list [26, 27]. On average, N
+2
+queries are required, and the complexity of the classical search
+problem is thus defined as being of order O (N) [29].
+The renowned quantum search algorithm developed by
+Grover searches unstructured datasets and comprises an appli-
+cation that demonstrates the advantages of quantum comput-
+ing over classical analogs [26]. The introduction of the quan-
+tum superposition concept enables the algorithm the ability to
+map all the database items simultaneously, which allows for a
+reduction in the total number of queries, which gives an im-
+provement in the efficiency of the search process [27]. In this
+regard, Grover’s Algorithm presents a complexity that grows
+in order of O(
+√
+N), being quadratically faster than its classical
+counterpart [29]. Therefore, many algorithms use Grover’s
+method as a subroutine in order to optimize some quantum
+processes. A famous example is the so-called D¨urr-Hoyer’s
+Algorithm for finding a minimum value in an unstructured
+database [32]. Based on the Grovers algorithm, the achieve
+a quadratic speed-up in the minimum search problem, which
+complexity can be expressed as O(
+�
+N
+t ), where t is the num-
+ber of marked states [30].
+In this context, this study investigates the quantum mini-
+mum search (QMS) problem, proposing a quantum algorithm
+quadratically faster than any classical analogs for finding min-
+imal values in a quantum random access memory (QRAM).
+We proposed the algorithm’s description considering a data
+set represented by the vector ⃗y, which can be rewritten in the
+computational bases as quantum states. The problem is to find
+the minimum value of the list, ymin, using Grover’s Algorithm
+as a subroutine based on Durr-Hoyer’s approach.
+A.
+Quantum Random Access Memory (QRAM)
+In order to use Grover’s Algorithm to find a minimum in a
+classical data set, one proposal is to use a QRAM, typically
+meaning a large classical memory, which can be queried in
+a quantum superposition. It can be built using an equivalent
+quantum circuit in which classical data is stored in a quan-
+tum register in binary form. It can be done by creating two
+quantum registers (with n and m qubits, respectively) whose
+initialization should be
+|ψ0⟩ =
+1
+√
+2n
+2n−1
+�
+x=0
+|x⟩ ⊗ |0⟩⊗m ,
+(1)
+which can be implemented by the operation H⊗n ⊗ I⊗m that
+creates an equal superposition on the first register and keeps
+the second register in the state |0⟩⊗m. The quantum RAM is
+implemented in the second register by applying an operator
+UX given by multicontrolled-NOT operations. The goal is to
+store the classical values ⃗y = {y0, y1, y2, ..., yk} into quantum
+states in order to obtain
+|ψ1⟩ =
+1
+√
+2n
+2n−1
+�
+x=0
+|x⟩ ⊗ |yx⟩.
+(2)
+Thus, a quantum RAM can store 2n data values. In this
+scenario, we need to choose the number m of qubits used in
+the second register. Since we are searching for the minimum
+value of the whole dataset, a random index can serve as our
+first iteration. Thus, it is possible to specify the number m
+such that the number associated with such an index can be
+expressed on a computational binary basis.
+B.
+Finding the minimum in a QRAM
+In order to perform the task of finding a smallest value
+stored in a QRAM, the adopted strategy is by performing a
+search analyzing the most (or less, if we want the maximum
+value) significant bits from a single measurement. The full
+quantum circuit can be seen in Fig. 1, where the special sub-
+routine responsible for searching according to most significant
+qubits is the iterative phase flip, given by an operator P.
+Figure 1. Full quantum circuit for minimum search. UX is the rep-
+resentation of a QRAM; the operator P is changed iteratively by
+analysing the most significant qubits in the last measurement; and
+W is the diffuser operator. It is important to emphasize that the last
+qubit has its state initialized in |−⟩.
+The key idea of the algorithm is in the dynamics of the P op-
+erator. The additional register (qubits further down in Fig. 1)
+is used to represent the storage of classical values in QRAM
+and is also where the search is done. It is known that if the
+most significant qubits have bits in the 0 state, it means that,
+in the decimal base, this number is smaller than if the most
+significant qubits were in the 1 state. Based on this logic, the
+P operator can be constructed through multicontrolled-NOT
+having qubits either with control at 0 or with control at 1.
+Therefore, the algorithm that governs the dynamics of P is
+described on box Algorithm 1.
+
+H
+EE
+M
+人
+X
+Ux
+Xn
+P3
+Algorithm 1 Finding the minimum in a QRAM
+• Input A classical database ⃗y
+1. Take a random value yi = f(xi), whose binary represen-
+tation demands m bits.
+2. Initialize a quantum computer in the state |ψ0⟩
+=
+1
+√
+2n
+�2n
+x=0 |x⟩|0⟩⊗m|−⟩.
+3. Store the classical values in the QRAM in order to get
+the state |ψ1⟩ =
+1
+√
+2n
+�2n
+x=0 |x⟩|yx⟩|−⟩
+4. Apply the oracle operator P in order to guarantee that
+the most signicant qubit is 0, that is, the marked states
+is less than yi.
+5. Apply the diffuser operator, W, to amplify the marked
+states.
+6. Perform a measurement in the computational basis to
+obtain yi+1 < yi.
+7. If all qubits have analyzed:
+– end if
+else:
+– repeat steps
+• return yi
+Thus, Grover’s Algorithm can be used iteratively in or-
+der to amplify states (index) that correspond to smaller val-
+ues than the last one, quadratically faster than their clas-
+sical counterparts.
+For instance, supposing the following
+dataset ⃗y = {5, 4, 12, 10, 8}, the list entries can be repre-
+sented in the computational basis (with four qubits) as ⃗y =
+{|0101⟩, |0100⟩, |1100⟩, |1010⟩, |1000⟩}.
+Figure 2. Implementation of a quantum RAM, given by the operator
+UX, as a quantum circuit. Each gray block stores a classical value
+from the database.
+If the first guess is (purely classical), for instance, 10 →
+1010 it is very unlikely that this number is the lowest. This
+can be confirmed by looking for the number whose most sig-
+nificant qubit is |0⟩.
+Figure 3. Quantum operator P for searching all states whose the most
+significant qubit is in the state |0⟩.
+Thus, a Grover iteration with this oracle mark all states
+whose the most significant qubit is in state |0⟩. A diagram rep-
+resentation of the state before the measurement can be seen in
+Fig. 4.
+Figure 4. A diagramatic representation of the quantum state probabil-
+ities. The states |x0⟩⟩ = |000⟩ and |x1⟩ = |001⟩ are amplified beacause
+f(x0) = |0101⟩ and f(x1) = |0100⟩ whose the most significant qubits
+are |0⟩ for both.
+After that, by performing Grover’s search in that most sig-
+nificant qubit, the states |0100⟩ and |0101⟩ will be had equal
+probability to be measured. If we get the state |0101⟩ after the
+measurement, the next step is to search for values whose two
+first binary digits are |00⟩.
+Figure 5. Quantum operator P for searching all states whose the two
+most significant qubits are in the state |00⟩.
+If there are one or more with which it is satisfied, a number
+less than |0101⟩ will be measured (yi > yi+1), if not, a num-
+ber greater will likely be measured (yi < yi+1), because none
+rotation is performed in the initial state.
+In the case where yi < yi+1, the process shows that the min-
+imum is |0101⟩ or a less number whose the first two more
+significant qubits are also |01⟩, so it is necessary to search for
+values whose third most significant qubits are |010⟩. In this
+particular case, the only remaining task is to verify if |0100⟩
+is in the QRAM since it is the smaller possible number whose
+three most significant qubits are in the state |010⟩.
+
+UxX
+XX
+X
+X
+X4
+Figure 6. Quantum operator P for checking if the value |4⟩ ≡ |0100⟩
+is in the QRAM.
+In this example, the state |0100⟩ will be measured with ap-
+proximately 100% of probability (See Fig. 7). The process is
+iteratively done with the rest of the qubits in order to find the
+minimum ymin = |0100⟩ surely.
+Figure 7.
+Final distribution for the last iteration.
+In this case,
+f(001) = ymin = |0100⟩.
+The task of finding a minimum in a vector can be performed
+using an optimal algorithm, with time complexity O(|⃗y|). The
+quantum algorithm proposed in this work solves the same
+problem on a quantum computer by performing O(c
+�
+|⃗y|
+t )
+queries in Grover’s oracle, where c is a constant whose value
+is a number of digits in a binary representation of the initial
+value and t is the number of marked states. According to com-
+plexity theory, a constant doesn’t affect time complexity, then
+it is valid to rewrite the time complexity of this algorithm as
+O(
+�
+|⃗y|
+t ).
+C.
+Complexity analysis
+In order to analyze and compare the time complexity be-
+tween classical and quantum algorithms for different scenar-
+ios, we take two classical algorithms with different complexi-
+ties. For the proposed quantum algorithm, the same was done,
+but using an increase in complexity by increasing the number
+of bits in the database values (See Fig. 8). We know that
+classical and quantum algorithms have complexities O(ccN)
+and O(cq
+√
+N), respectively. The constants cc and cq indicate,
+respectively, the constant inherent complexity factor of each
+classical algorithm and the number of bits of the quantum al-
+gorithm’s initial guess, as explained in the procedure.
+Figure 8. Complexity analysis among algorithms. The shade be-
+tween lines represents the complexity range among classical and
+quantum algorithms. The upper and lower bounds of the classical
+algorithms (blue) have time complexities O( 3
+2 N − 2) and O(N − 1).
+For the case of the quantum algorithm (orange), the upper and lower
+limits were drawn for the cases where cq = 14 and cq = 6, respec-
+tively.
+III.
+K-MEANS CLUSTERING
+In order to demonstrate the application of minimum search
+in important computer science tasks, we implemented the
+clustering algorithm called K-means, well known in statistics
+and unsupervised machine learning. Given a set of points in
+Euclidean space, the algorithm aims to determine the optimal
+location of K-centroids in order to minimize the sum of all dis-
+tances between the points and their respective centroids. The
+objective function can be given by
+f(x, y) =
+K
+�
+j=0
+|S |
+�
+i=0
+∥p( j)
+i
+− ci∥2
+(3)
+where pi is an observed point in Euclidean space, |S | is the
+total number of observed points, ci is the position of a cen-
+troid and K is the predetermined number of centroids. Fig. 9
+shows a distribution of points in the Cartesian plane and the
+randomly initialized centroids before starting the optimization
+process. Although a simplified example, this one can be use-
+ful to visually demonstrate how the algorithm works.
+
+X
+X
+X
+X
+X
+XClassical
+810
+Quantum
+: cormplexity
+0
+400
+O
+0 -
+0
+100
+210
+300
+400
+500
+N5
+Figure 9. Distribution of |S | = 12 points (red) in two-dimensional
+Euclidean space. In this example, four centroids (K = 4), represented
+by stars (black), were randomly initialized.
+We assume that in the quantum version of K-means, the
+distances between each observed point, pi, and all centroids,
+S = {c0, c1, c2, c3}, are stored in QRAM in constant time, that
+is, O(1). Clusters are formed at each iteration by the proxim-
+ity between each point and its closest centroid. The average
+between the coordinates of each new cluster is calculated and
+becomes the new centroid. This process is carried out until a
+certain stopping criterion is satisfied. Fig. 10 shows the best
+clustering found by the algorithm.
+Figure 10. Optimal solution found by the quantum version of K-
+means. Each of the four clusters found by the algorithm is being
+represented by a color.
+Note that the only difference between this procedure to its
+classical analog is that the distances between each point and
+all centroids are stored in a QRAM, and the QMS is used to
+find the smallest one. Although we are using QMS for a spe-
+cific example, it can be useful for a huge amount of computa-
+tional tasks, such as unsupervised machine learning problems.
+IV.
+CONCLUSIONS
+Classical computing can be outperformed by quantum com-
+puting in a wide range of problems, from those with low to
+those with high levels of computational complexity. The clas-
+sical minimum search problem is characterized by linear com-
+plexity and is connected to an extensive variety of applications
+in the domain of computer science. In this scenario, this work
+proposed a quantum algorithm for finding the minimum value
+of a database that is quadratically faster than its best classical
+analogs. The algorithm is based on D¨urr-Hoyer’s approach for
+finding a minimum value in an unstructured list through the
+use of Grover’s algorithm as a subroutine applied to a QRAM
+that stores values from a defined database. Although it is not
+considered a complex task, our results show that the suggested
+QMS algorithm has the potential to significantly reduce the
+execution time of minimum search algorithms for cases where
+the database is very large. Moreover, an examination of the
+complexity of the studied problem was performed in order to
+highlight the advantages of this quantum algorithm over its
+classical analogs. Furthermore, we show how the suggested
+approach can be used in an unsupervised machine learning
+task by performing a quantum adaptation of the K-means al-
+gorithm. In conclusion, our results demonstrate that it is possi-
+ble to search for minimums in a classical database by utilizing
+information stored in a QRAM, which represents a significant
+contribution to the development of fault-tolerant quantum al-
+gorithms.
+ACKNOWLEDGMENTS
+We thank the Latin American Quantum Computing Cen-
+ter and the High-Performance Computing Center, both from
+SENAI CIMATEC for supporting this research. We also thank
+the Quantum Open Source Foundation (QOSF) mentoring
+program, whose developed project originated this article.
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+

IdE4T4oBgHgl3EQfhA2z/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf,len=374
+page_content='Quantum algorithm for finding minimum values in a Quantum Random Access Memory  Anton S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Albino,1, ∗ Lucas Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Galv˜ao,1, † Ethan Hansen,2, ‡ Mauro Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Nooblath Neto,1, § and Clebson Cruz3, ¶  1Latin American Quantum Computing Center, SENAI CIMATEC, Salvador, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  2Zapata Computing, Canada  3Grupo de Informa¸c˜ao Quˆantica, Centro de Ciˆencias Exatas e das Tecnologias,  Universidade Federal do Oeste da Bahia - Campus Reitor Edgard Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Rua Bertioga,  892, Morada Nobre I, 47810-059 Barreiras, Bahia, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Finding the minimum value in an unordered database is a common and fundamental task in computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  However, the optimal classical deterministic algorithm can find the minimum value with a time complexity that  grows linearly with the number of elements in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this paper, we present the proposal of a quantum  algorithm for finding the minimum value of a database, which is quadratically faster than its best classical  analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' We assume a Quantum Random Access Memory (QRAM) that stores values from a database and  perform an iterative search based on an oracle whose role is to limit the searched values by controlling the states  of the most significant qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' A complexity analysis was performed in order to demonstrate the advantage of this  quantum algorithm over its classical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Furthermore, we demonstrate how the proposed algorithm  would be used in an unsupervised machine learning task through a quantum version of the K-means algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Keywords: Quantum RAM, Minimum search, Grover’s Algorithm  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  INTRODUCTION  Random Access Memory (RAM) is a versatile, short-term  memory used in computing for storing and retrieving infor-  mation via bits [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Similarly, the concept of Quantum RAM  (QRAM) emerges with the same goal but employing qubits  to apply a superposition of states to achieve faster results for  computational applications, whether quantum or classical [2–  11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Several works discuss the potential of its applications  to optimize the execution of quantum algorithms, including  quantum searching on a classical database [10, 12–15], col-  lision finding [12, 16–18], and algorithms for solving linear  systems [19–22], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  These results have attracted the attention of the scien-  tific community in the past few years, leading to the de-  velopment of QRAM architectures that demonstrate the po-  tential for producing efficient results in quantum computing  [3, 4, 6, 7, 9, 23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Some models, such as Fanout quantum  RAM [3, 7, 9] and Bucket-Brigade quantum RAM [3, 23, 25],  illustrate potential future implementations of a QRAM in  practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In addition, recent experiments have re-  vealed a designed architecture for hybrid quantum computers  that use superconducting qubits and spin-qubit memory crys-  tals capable, in theory, of implementing a QRAM in real sys-  tems [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Furthermore, other efforts have been made to construct  quantum algorithms that are able to optimally access a QRAM  in the process of searching for certain values stored in its cells  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In general, problems based on searching use the famous  Grover’s algorithm to search quantum states in an unstruc-  tured list [16, 26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' On the other hand, a well-known exam-  ple of determining the minimal value in a list is the so-called  ∗ anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='albino@fieb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='br  † lucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='g5847@ufob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='br  ‡ 1ethanhansen@proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='me  § mauro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='neto@fbter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='br  ¶ clebson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='cruz@ufob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='br  D¨urr-Hoyer minimum finding algorithm [30], which employs  Grover’s Algorithm as a fundamental subroutine to find the  greatest or smallest entry in a list [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  In this scenario, based on the core concept of Durr-Hoyer’s  algorithm, we apply Grover’s Algorithm as a subroutine to de-  velop a quantum algorithm for identifying the smallest value  in a classical data set stored in a QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The proposed Quan-  tum Minimum Search (QMS) algorithm is based on the itera-  tive change of the oracle function, which limits the searched  values by controlling the states of the most significant qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  First, we present the description of the QMS algorithm, de-  scribing the concept of QRAM and approaching an example  to find the minimum in a list of four real values using the  proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In sequence, we analyze the complex-  ity of the QMS algorithm compared with classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  The results show that, whereas the complexity of the classi-  cal algorithm grows linearly with the number of elements in  the database, O(N), since the classical algorithms go through  all the N items in the list, the presented QMS algorithm has  a complexity of O(  �  N  t ), with t being the number of marked  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Finally, we present an application of the proposed al-  gorithm in the K-means problem of determining the optimal  location of K-centroids in order to minimize the sum of all  distances between the points and their respective centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  QUANTUM MINIMUM SEARCH (QMS) ALGORITHM  The search problem is a ubiquitous subject of discussion  in classical computer science [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The problem consists of  identifying the index of the database item (x) that fulfills some  predetermined search criterion x = y, where y is the sought  element, given an unstructured database with N elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In  this context, it is possible to prepare the so-called response  function (R(x)) that translates database entries to True if the  entry x matches the search criterion (x = y) or False if x �  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' This is possible by using the so-called Oracle subroutine,  which queries the database until the desired item is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='05122v1  [quant-ph]  12 Jan 2023  2  Consequently, the bigger the requested element’s position in  the list, the greater the number of queries required to locate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Therefore, the complexity of this task is exactly proportional  to the number of items on the list [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' On average, N  2  queries are required, and the complexity of the classical search  problem is thus defined as being of order O (N) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  The renowned quantum search algorithm developed by  Grover searches unstructured datasets and comprises an appli-  cation that demonstrates the advantages of quantum comput-  ing over classical analogs [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The introduction of the quan-  tum superposition concept enables the algorithm the ability to  map all the database items simultaneously, which allows for a  reduction in the total number of queries, which gives an im-  provement in the efficiency of the search process [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this  regard, Grover’s Algorithm presents a complexity that grows  in order of O(  √  N), being quadratically faster than its classical  counterpart [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Therefore, many algorithms use Grover’s  method as a subroutine in order to optimize some quantum  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' A famous example is the so-called D¨urr-Hoyer’s  Algorithm for finding a minimum value in an unstructured  database [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Based on the Grovers algorithm, the achieve  a quadratic speed-up in the minimum search problem, which  complexity can be expressed as O(  �  N  t ), where t is the num-  ber of marked states [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  In this context, this study investigates the quantum mini-  mum search (QMS) problem, proposing a quantum algorithm  quadratically faster than any classical analogs for finding min-  imal values in a quantum random access memory (QRAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  We proposed the algorithm’s description considering a data  set represented by the vector ⃗y, which can be rewritten in the  computational bases as quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The problem is to find  the minimum value of the list, ymin, using Grover’s Algorithm  as a subroutine based on Durr-Hoyer’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Quantum Random Access Memory (QRAM)  In order to use Grover’s Algorithm to find a minimum in a  classical data set, one proposal is to use a QRAM, typically  meaning a large classical memory, which can be queried in  a quantum superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' It can be built using an equivalent  quantum circuit in which classical data is stored in a quan-  tum register in binary form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' It can be done by creating two  quantum registers (with n and m qubits, respectively) whose  initialization should be  |ψ0⟩ =  1  √  2n  2n−1  �  x=0  |x⟩ ⊗ |0⟩⊗m ,  (1)  which can be implemented by the operation H⊗n ⊗ I⊗m that  creates an equal superposition on the first register and keeps  the second register in the state |0⟩⊗m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The quantum RAM is  implemented in the second register by applying an operator  UX given by multicontrolled-NOT operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The goal is to  store the classical values ⃗y = {y0, y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=', yk} into quantum  states in order to obtain  |ψ1⟩ =  1  √  2n  2n−1  �  x=0  |x⟩ ⊗ |yx⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  (2)  Thus, a quantum RAM can store 2n data values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this  scenario, we need to choose the number m of qubits used in  the second register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Since we are searching for the minimum  value of the whole dataset, a random index can serve as our  first iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Thus, it is possible to specify the number m  such that the number associated with such an index can be  expressed on a computational binary basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Finding the minimum in a QRAM  In order to perform the task of finding a smallest value  stored in a QRAM, the adopted strategy is by performing a  search analyzing the most (or less, if we want the maximum  value) significant bits from a single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The full  quantum circuit can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 1, where the special sub-  routine responsible for searching according to most significant  qubits is the iterative phase flip, given by an operator P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Full quantum circuit for minimum search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' UX is the rep-  resentation of a QRAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' the operator P is changed iteratively by  analysing the most significant qubits in the last measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' and  W is the diffuser operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' It is important to emphasize that the last  qubit has its state initialized in |−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  The key idea of the algorithm is in the dynamics of the P op-  erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The additional register (qubits further down in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 1)  is used to represent the storage of classical values in QRAM  and is also where the search is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' It is known that if the  most significant qubits have bits in the 0 state, it means that,  in the decimal base, this number is smaller than if the most  significant qubits were in the 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Based on this logic, the  P operator can be constructed through multicontrolled-NOT  having qubits either with control at 0 or with control at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Therefore, the algorithm that governs the dynamics of P is  described on box Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  H  EE  M  人  X  Ux  Xn  P3  Algorithm 1 Finding the minimum in a QRAM  Input A classical database ⃗y  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Take a random value yi = f(xi), whose binary represen-  tation demands m bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Initialize a quantum computer in the state |ψ0⟩  =  1  √  2n  �2n  x=0 |x⟩|0⟩⊗m|−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Store the classical values in the QRAM in order to get  the state |ψ1⟩ =  1  √  2n  �2n  x=0 |x⟩|yx⟩|−⟩  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Apply the oracle operator P in order to guarantee that  the most signicant qubit is 0, that is, the marked states  is less than yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Apply the diffuser operator, W, to amplify the marked  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Perform a measurement in the computational basis to  obtain yi+1 < yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' If all qubits have analyzed:  – end if  else:  – repeat steps  return yi  Thus, Grover’s Algorithm can be used iteratively in or-  der to amplify states (index) that correspond to smaller val-  ues than the last one, quadratically faster than their clas-  sical counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  For instance, supposing the following  dataset ⃗y = {5, 4, 12, 10, 8}, the list entries can be repre-  sented in the computational basis (with four qubits) as ⃗y =  {|0101⟩, |0100⟩, |1100⟩, |1010⟩, |1000⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Implementation of a quantum RAM, given by the operator  UX, as a quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Each gray block stores a classical value  from the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  If the first guess is (purely classical), for instance, 10 →  1010 it is very unlikely that this number is the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' This  can be confirmed by looking for the number whose most sig-  nificant qubit is |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Quantum operator P for searching all states whose the most  significant qubit is in the state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Thus, a Grover iteration with this oracle mark all states  whose the most significant qubit is in state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' A diagram rep-  resentation of the state before the measurement can be seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' A diagramatic representation of the quantum state probabil-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The states |x0⟩⟩ = |000⟩ and |x1⟩ = |001⟩ are amplified beacause  f(x0) = |0101⟩ and f(x1) = |0100⟩ whose the most significant qubits  are |0⟩ for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  After that, by performing Grover’s search in that most sig-  nificant qubit, the states |0100⟩ and |0101⟩ will be had equal  probability to be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' If we get the state |0101⟩ after the  measurement, the next step is to search for values whose two  first binary digits are |00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Quantum operator P for searching all states whose the two  most significant qubits are in the state |00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  If there are one or more with which it is satisfied, a number  less than |0101⟩ will be measured (yi > yi+1), if not, a num-  ber greater will likely be measured (yi < yi+1), because none  rotation is performed in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  In the case where yi < yi+1, the process shows that the min-  imum is |0101⟩ or a less number whose the first two more  significant qubits are also |01⟩, so it is necessary to search for  values whose third most significant qubits are |010⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this  particular case, the only remaining task is to verify if |0100⟩  is in the QRAM since it is the smaller possible number whose  three most significant qubits are in the state |010⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  UxX  XX  X  X  X4  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Quantum operator P for checking if the value |4⟩ ≡ |0100⟩  is in the QRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  In this example, the state |0100⟩ will be measured with ap-  proximately 100% of probability (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The process is  iteratively done with the rest of the qubits in order to find the  minimum ymin = |0100⟩ surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Final distribution for the last iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  In this case,  f(001) = ymin = |0100⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  The task of finding a minimum in a vector can be performed  using an optimal algorithm, with time complexity O(|⃗y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The  quantum algorithm proposed in this work solves the same  problem on a quantum computer by performing O(c  �  |⃗y|  t )  queries in Grover’s oracle, where c is a constant whose value  is a number of digits in a binary representation of the initial  value and t is the number of marked states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' According to com-  plexity theory, a constant doesn’t affect time complexity, then  it is valid to rewrite the time complexity of this algorithm as  O(  �  |⃗y|  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Complexity analysis  In order to analyze and compare the time complexity be-  tween classical and quantum algorithms for different scenar-  ios, we take two classical algorithms with different complexi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' For the proposed quantum algorithm, the same was done,  but using an increase in complexity by increasing the number  of bits in the database values (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' We know that  classical and quantum algorithms have complexities O(ccN)  and O(cq  √  N), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The constants cc and cq indicate,  respectively, the constant inherent complexity factor of each  classical algorithm and the number of bits of the quantum al-  gorithm’s initial guess, as explained in the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Complexity analysis among algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The shade be-  tween lines represents the complexity range among classical and  quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The upper and lower bounds of the classical  algorithms (blue) have time complexities O( 3  2 N − 2) and O(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  For the case of the quantum algorithm (orange), the upper and lower  limits were drawn for the cases where cq = 14 and cq = 6, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  K-MEANS CLUSTERING  In order to demonstrate the application of minimum search  in important computer science tasks, we implemented the  clustering algorithm called K-means, well known in statistics  and unsupervised machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Given a set of points in  Euclidean space, the algorithm aims to determine the optimal  location of K-centroids in order to minimize the sum of all dis-  tances between the points and their respective centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The  objective function can be given by  f(x, y) =  K  �  j=0  |S |  �  i=0  ∥p( j)  i  − ci∥2  (3)  where pi is an observed point in Euclidean space, |S | is the  total number of observed points, ci is the position of a cen-  troid and K is the predetermined number of centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 9  shows a distribution of points in the Cartesian plane and the  randomly initialized centroids before starting the optimization  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Although a simplified example, this one can be use-  ful to visually demonstrate how the algorithm works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  X  X  X  X  X  XClassical  810  Quantum  : cormplexity  0  400  O  0 -  0  100  210  300  400  500  N5  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Distribution of |S | = 12 points (red) in two-dimensional  Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this example, four centroids (K = 4), represented  by stars (black), were randomly initialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  We assume that in the quantum version of K-means, the  distances between each observed point, pi, and all centroids,  S = {c0, c1, c2, c3}, are stored in QRAM in constant time, that  is, O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Clusters are formed at each iteration by the proxim-  ity between each point and its closest centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The average  between the coordinates of each new cluster is calculated and  becomes the new centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' This process is carried out until a  certain stopping criterion is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' 10 shows the best  clustering found by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Optimal solution found by the quantum version of K-  means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Each of the four clusters found by the algorithm is being  represented by a color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  Note that the only difference between this procedure to its  classical analog is that the distances between each point and  all centroids are stored in a QRAM, and the QMS is used to  find the smallest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Although we are using QMS for a spe-  cific example, it can be useful for a huge amount of computa-  tional tasks, such as unsupervised machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  CONCLUSIONS  Classical computing can be outperformed by quantum com-  puting in a wide range of problems, from those with low to  those with high levels of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The clas-  sical minimum search problem is characterized by linear com-  plexity and is connected to an extensive variety of applications  in the domain of computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In this scenario, this work  proposed a quantum algorithm for finding the minimum value  of a database that is quadratically faster than its best classical  analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' The algorithm is based on D¨urr-Hoyer’s approach for  finding a minimum value in an unstructured list through the  use of Grover’s algorithm as a subroutine applied to a QRAM  that stores values from a defined database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Although it is not  considered a complex task, our results show that the suggested  QMS algorithm has the potential to significantly reduce the  execution time of minimum search algorithms for cases where  the database is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Moreover, an examination of the  complexity of the studied problem was performed in order to  highlight the advantages of this quantum algorithm over its  classical analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' Furthermore, we show how the suggested  approach can be used in an unsupervised machine learning  task by performing a quantum adaptation of the K-means al-  gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' In conclusion, our results demonstrate that it is possi-  ble to search for minimums in a classical database by utilizing  information stored in a QRAM, which represents a significant  contribution to the development of fault-tolerant quantum al-  gorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We thank the Latin American Quantum Computing Cen-  ter and the High-Performance Computing Center, both from  SENAI CIMATEC for supporting this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
+page_content=' We also thank  the Quantum Open Source Foundation (QOSF) mentoring  program, whose developed project originated this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE4T4oBgHgl3EQfhA2z/content/2301.05122v1.pdf'}
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MtAyT4oBgHgl3EQfs_ll/content/tmp_files/2301.00586v1.pdf.txt

@@ -0,0 +1,1576 @@
+arXiv:2301.00586v1  [math.FA]  2 Jan 2023
+Indeterminate Jacobi operators
+Christian Berg and Ryszard Szwarc
+January 3, 2023
+Abstract
+We consider the Jacobi operator (T, D(T)) associated with an in-
+determinate Hamburger moment problem, i.e., the operator in ℓ2 de-
+fined as the closure of the Jacobi matrix acting on the subspace of
+complex sequences with only finitely many non-zero terms. It is well-
+known that it is symmmetric with deficiency indices (1, 1).
+For a
+complex number z let pz, qz denote the square summable sequences
+(pn(z)) and (qn(z)) corresponding to the orthonormal polynomials pn
+and polynomials qn of the second kind. We determine whether linear
+combinations of pu, pv, qu, qv for u, v ∈ C belong to D(T) or to the
+domain of the self-adjoint extensions of T in ℓ2. The results depend
+on the four Nevanlinna functions of two variables associated with the
+moment problem. We also show that D(T) is the common range of
+an explicitly constructed family of bounded operators on ℓ2.
+Mathematics Subject Classification: Primary 47B25, 47B36, 44A60
+Keywords. Jacobi matrices and operators, indeterminate moment prob-
+lems.
+1
+Introduction
+We shall consider the Jacobi matrix J associated with a moment sequence
+s = (sn)n≥0 of the form
+sn =
+�
+xn dµ(x),
+n = 0, 1, . . . ,
+(1)
+1
+
+where µ is a positive measure on R with infinite support and moments of
+every order. It is a tridiagonal matrix of the form
+J =
+
+
+
+
+
+b0
+a0
+0
+. . .
+a0
+b1
+a1
+. . .
+0
+a1
+b2
+. . .
+...
+...
+...
+...
+
+
+
+
+ ,
+(2)
+where an > 0, bn ∈ R, n ≥ 0 are given by the three term recurrence relation
+xpn(x) = anpn+1(x) + bnpn(x) + an−1pn−1(x), n ≥ 0,
+a−1 := 0.
+Here (pn)n≥0 is the sequence of orthonormal polynomials associated with µ,
+hence satisfying
+�
+pn(x)pm(x) dµ(x) = δn,m,
+and pn is a real polynomial of degree n with positive leading coefficient. In
+this paper we follow the terminology of [17]. Basic results about the classical
+moment problem can also be found in [1] and [16]. Recent results about
+indeterminate moment problems can be found in [5], [6] [7], [8].
+It is clear that the proportional measures λµ, λ > 0 lead to the same
+Jacobi matrix J, and the well-known Theorem of Favard (see [17, Theorem
+5.14]) states that any matrix of the form (2) with an > 0, bn ∈ R comes
+from a unique moment sequence (sn) as above, normalized such that s0 = 1.
+In the following we shall always assume that this normalization holds, and
+consequently the solutions µ of (1) are probability measures and p0 = 1.
+The Jacobi matrix acts as a symmetric operator in the Hilbert space ℓ2 of
+square summable complex sequences. Its domain F consists of the complex
+sequences (cn)n≥0 with only finitely many non-zero terms, and the action is
+multiplication of the matrix J by c ∈ F considered as a column, i.e.,
+(Jc)n := an−1cn−1 + bncn + ancn+1,
+n ≥ 0.
+(3)
+Denoting (en)n≥0 the standard orthonormal basis of ℓ2, we have
+F = span{en|n ≥ 0}.
+Definition 1.1. The Jacobi operator associated with J is by definition the
+closure (T, D(T)) of the symmetric operator (J, F).
+2
+
+It is a classical fact that (T, D(T)) is a closed symmetric operator, and its
+deficiency indices are either (0, 0) or (1, 1). These cases occur precisely if the
+moment sequence (1) is determinate or indeterminate, i.e., there is exactly
+one or several solutions µ satisfying (1).
+By definition D(T) consists of those c ∈ ℓ2 for which there exists a se-
+quence (c(k)) ∈ F such that limk→∞ c(k) = c and (Jc(k)) is a convergent
+sequence in ℓ2. For such c we have Tc = limk→∞ Jc(k), and this limit is
+independent of the choice of approximating sequence (c(k)).
+Clearly, D(T) is closed under complex conjugation and
+Tc = Tc,
+c ∈ D(T).
+The purpose of the present paper is to study the Jacobi operator (T, D(T))
+as well as its self-adjoint extensions (Tt, D(Tt)), t ∈ R∗ := R ∪ {∞} in the
+indeterminate case. We shall in particular give some families of sequences
+c ∈ ℓ2 which belong to D(T), see Theorem 1.2–Theorem 1.4.
+Section 2 is devoted to the proof of Theorem 1.2 after a presentation of
+the deficiency spaces of (T, D(T)). The self-adjoint extensions of (T, D(T))
+as well as their corresponding N-extremal solutions to (1), cf.
+(26), are
+introduced in Section 3.
+In Theorem 3.2, Theorem 3.4 and Theorem 3.7 we describe vectors be-
+longing to D(Tt) \ D(T). Like the results in Theorem 1.2–Theorem 1.4, they
+depend on the Nevanlinna functions of two variables defined in (6), (7), (8),
+(9).
+In Section 4 we construct for each z0 ∈ C a bounded operator Ξz0 in ℓ2
+with range D(T). The restriction of Ξz0 to (T − z0I)(D(T)) is a bijection
+onto D(T) equal to (T − z0I)−1, see Theorem 4.3. It is based on a study of
+the function space E defined in (28), and known to be a de Branges space of
+entire functions by [10, Theorem 23]. We prove in particular Theorem 4.2,
+showing that E is stable under the formation of difference quotients.
+Various technical results about the Nevanlinna functions are given in
+Section 5.
+After this summary of the content of the present paper, we recall that
+the adjoint operator (T ∗, D(T ∗)) is the maximal operator associated with J,
+cf. [17, Proposition 6.5]. In fact, the matrix product of J and any column
+vector c makes sense, cf. (3), and D(T ∗) consists of those c ∈ ℓ2 for which
+the product Jc belongs to ℓ2. For c ∈ D(T ∗) we have T ∗c = Jc.
+In the determinate case with a unique solution µ of (1), the Jacobi opera-
+tor is self-adjoint and (pn) is an orthonormal basis of L2(µ). The self-adjoint
+3
+
+operator of multiplication Mµ in L2(µ) given by
+D(Mµ) = {f ∈ L2(µ) | xf(x) ∈ L2(µ)},
+Mµf(x) = xf(x)
+is unitarily equivalent with (T, D(T)) via the unitary operator U : ℓ2 → L2(µ)
+given by U(en) = pn, n ≥ 0. We shall not study the determinate case in this
+paper, but concentrate on the indeterminate case, where it is known that the
+set of solutions µ to (1) is an infinite convex set V . The polynomials of the
+second kind (qn) are given as
+qn(z) =
+� pn(z) − pn(x)
+z − x
+dµ(x),
+z ∈ C,
+where µ ∈ V is arbitrary.
+We define and recall
+pz := (pn(z)), qz := (qn(z)) ∈ ℓ2,
+z ∈ C,
+(4)
+where we have followed the terminology of [17]. It is known that ||pz|| and
+||qz|| are positive continuous functions on C. It is therefore possible for c ∈ ℓ2
+to define entire functions Fc, Gc as
+Fc(z) =
+∞
+�
+n=0
+cnpn(z),
+Gc(z) =
+∞
+�
+n=0
+cnqn(z),
+z ∈ C.
+(5)
+We also have the following four entire functions of two complex variables,
+called the Nevanlinna functions of the indeterminate moment problem:
+A(u, v)
+=
+(u − v)
+∞
+�
+k=0
+qk(u)qk(v)
+(6)
+B(u, v)
+=
+−1 + (u − v)
+∞
+�
+k=0
+pk(u)qk(v)
+(7)
+C(u, v)
+=
+1 + (u − v)
+∞
+�
+k=0
+qk(u)pk(v)
+(8)
+D(u, v)
+=
+(u − v)
+∞
+�
+k=0
+pk(u)pk(v),
+(9)
+see Section 7.1 in [17]. The two-variable functions were introduced in [11]
+in a slightly different form, which was subsequently used in [3],[14].
+An
+4
+
+approximation to the two-variable functions was already considered in [1, p.
+123]. If the functions of [11] are marked with a ∗, we have
+A∗(u, v)
+=
+−A(u, v), B∗(u, v) = −C(u, v),
+C∗(u, v)
+=
+−B(u, v), D∗(u, v) = −D(u, v).
+In the following we need several formulas about these functions, see Theo-
+rem 5.1 and Corollary 5.2 in the Appendix, but at this point we just recall
+that
+A(u, v)D(u, v) − B(u, v)C(u, v) = 1,
+u, v ∈ C.
+(10)
+We define entire functions of one variable by setting the second variable to
+0, i.e.,
+A(u) = A(u, 0), B(u) = B(u, 0), C(u) = C(u, 0), D(u) = D(u, 0),
+(11)
+and by specialization of (10) we get
+A(u)D(u) − B(u)C(u) = 1,
+u ∈ C.
+(12)
+By Section 6.5 in [17] we have
+pz, qz ∈ D(T ∗),
+T ∗pz = zpz, T ∗qz = e0 + zqz,
+z ∈ C.
+(13)
+Our first main result is the following:
+Theorem 1.2. For all z ∈ C we have pz, qz /∈ D(T).
+Let u, v ∈ C be given.
+(i) There exists α ∈ C such that pu+αpv ∈ D(T) if and only if D(u, v) = 0.
+In the affirmative case α is uniquely determined as α = B(u, v).
+(ii) There exists β ∈ C such that qu+βqv ∈ D(T) if and only if A(u, v) = 0.
+In the affirmative case β is uniquely determined as β = −C(u, v).
+(iii) There exists γ ∈ C such that pu+γqv ∈ D(T) if and only if B(u, v) = 0.
+In the affirmative case γ is uniquely determined as γ = −D(u, v). In
+particular pu + γqu /∈ D(T) for all u, γ ∈ C.
+The proof will be given in Section 2.
+We shall next give results about the zero-sets of the entire functions
+A, . . . , D of two variables.
+5
+
+Theorem 1.3. Let F(u, v) denote any of the four functions A, B, C, D on
+C2. For v ∈ C define
+Z(F)v := {u ∈ C | F(u, v) = 0}.
+(14)
+Then Z(F)v is countably infinite. If v ∈ R then Z(F)v ⊂ R, and if v is in
+either the upper or lower half-plane, then Z(F)v belongs to the same half-
+plane.
+As a follow up on the two previous theorems we have the following:
+Theorem 1.4. Let v ∈ R be given and consider the set of real zeros Z(F)v
+from Theorem 1.3.
+(i) Let u ∈ Z(D)v be such that u < v and such that ]u, v[∩Z(D)v = ∅.
+Then B(u, v) > 0, where pu + B(u, v)pv ∈ D(T) according to Theo-
+rem 1.2.
+(ii) Let u ∈ Z(A)v be such that u < v and such that ]u, v[∩Z(A)v = ∅. Then
+C(u, v) < 0, where qu − C(u, v)qv ∈ D(T) according to Theorem 1.2.
+The proofs of Theorem 1.3 and Theorem 1.4 will be given in Section 5.
+2
+Preliminaires and proof of Theorem 1.2
+Fix z0 ∈ C in the open upper half-plane and consider the deficiency spaces
+∆+(z0)
+=
+ker(T ∗ − z0I) = Cpz0
+∆−(z0)
+=
+ker(T ∗ − z0I) = Cpz0,
+cf. (13).
+We know from [2, section 80] that
+D(T ∗) = D(T) ⊕ ∆+(z0) ⊕ ∆−(z0),
+(15)
+and the sum is direct as indicated by the ⊕ signs.
+Proposition 2.1. For any λ ∈ C we have the decomposition from (15)
+pλ = sλ + s+
+λ pz0 + s−
+λ pz0,
+(16)
+6
+
+where sλ ∈ D(T) and
+s+
+λ =
+D(λ, z0)
+2iIm (z0)||pz0||2,
+s−
+�� = −
+D(λ, z0)
+2iIm (z0)||pz0||2.
+(17)
+Similarly, we have the decomposition
+qλ = rλ + r+
+λ pz0 + r−
+λ pz0,
+(18)
+where rλ ∈ D(T) and
+r+
+λ =
+C(λ, z0)
+2iIm (z0)||pz0||2,
+r−
+λ = −
+C(λ, z0)
+2iIm (z0)||pz0||2.
+(19)
+Proof. Applying the operator T ∗ − z0I to the Equation (16) gives
+(T ∗ − z0I)pλ = (T − z0I)sλ + s+
+λ (z0 − z0)pz0,
+which is the splitting of the left-hand side according to the orthogonal de-
+composition
+ℓ2 = (T − z0I)(D(T)) ⊕ ∆+(z0).
+(20)
+Therefore, s+
+λ (z0 − z0)pz0 is the orthogonal projection of (T ∗ − z0I)pλ onto
+∆+(z0), and hence
+2iIm (z0)s+
+λ pz0 = ⟨(T ∗ − z0I)pλ, pz0⟩
+pz0
+||pz0||2,
+which gives the first formula in (17). The second formula is obtained similarly
+by applying the operator (T ∗−z0I) to the Equation (16). Notice that ||pz0|| =
+||pz0||.
+Applying the operator T ∗ − z0I to the Equation (18) gives
+(T ∗ − z0I)qλ = (T − z0I)rλ + r+
+λ (z0 − z0)pz0,
+which is the splitting of the left-hand side according to the orthogonal de-
+composition (20). Therefore, r+
+λ (z0 − z0)pz0 is the orthogonal projection of
+(T ∗ − z0I)qλ onto ∆+(z0), and hence
+2iIm (z0)r+
+λ pz0 = ⟨(T ∗ − z0I)qλ, pz0⟩
+pz0
+||pz0||2,
+which gives the first formula in (19) because T ∗(qλ) = λqλ + e0 by (13). The
+second formula is obtained similarly by applying the operator (T ∗ − z0I) to
+the Equation (18).
+7
+
+Corollary 2.2. For all λ ∈ C we have pλ, qλ /∈ D(T).
+Proof. For λ = z0 we get from (16) and (17)
+sz0 = 0,
+s+
+z0 = 1,
+s−
+z0 = 0,
+showing that pz0 /∈ D(T). The case λ = z0 follows because D(T) is closed
+under complex conjugation. Since z0 in the upper half-plane is arbitrary, the
+assertion about pλ follows for λ ∈ C \ R.
+For λ ∈ R we note that s−
+λ = s+
+λ ̸= 0 because z �→ D(λ, z) has only real
+zeros, cf. [4, Theorem 3] or Theorem 1.3.
+For λ = z0 we get from (19) that
+r−
+z0 =
+−1
+2iIm (z0)||pz0||2 ̸= 0,
+showing that qz0 /∈ D(T) and hence also qz0 /∈ D(T). Since z0 in the upper
+half-plane is arbitrary, the assertion about qλ follows for λ ∈ C \ R.
+For λ ∈ R we note that r−
+λ = r+
+λ , and by (57) we have
+C(λ, z0) = D(z0)[A(λ) − ρC(λ)] ̸= 0,
+ρ := B(z0)/D(z0).
+because D has only real zeros. Furthermore, Im ρ > 0 because B/D is a Pick
+function, cf. Proposition 5.8, so also the second factor is non-zero.
+Remark 2.3. Concerning Corollary 2.2, it is clear that pλ /∈ D(T) for λ /∈ R
+because otherwise pλ would be an eigenvector for T with eigenvalue λ, and
+as T is symmetric, the eigenvalues are real.
+A small modification yields also that qλ /∈ D(T) for λ /∈ R.
+In fact,
+otherwise by symmetry of T
+⟨Tqλ, qλ⟩ = ⟨qλ, Tqλ⟩.
+(21)
+The left-hand side of (21) equals ⟨λqλ+e0, qλ⟩ = λ||qλ||2 because ⟨e0, qλ⟩ = 0.
+Similarly, the right-hand side of (21) equals λ||qλ||2, and finally λ must
+be real. We show later that (T, D(T)) has no eigenvalues at all, cf. (37).
+Proof of Theorem 1.2.
+Corollary 2.2 proves the first assertion, and from this assertion it is clear
+that there exists at most one number α satisfying pu + αpv ∈ D(T), and
+similarly with β, γ.
+8
+
+Let us now prove assertion (i) of the theorem.
+By (16) we get
+pu + αpv = (su + αsv) + (s+
+u + αs+
+v )pz0 + (s−
+u + αs−
+v )pz0,
+so pu + αpv ∈ D(T) if and only if
+s+
+u + αs+
+v = s−
+u + αs−
+v = 0,
+which by (17) is equivalent to
+D(u, z0) + αD(v, z0) = D(u, z0) + αD(v, z0) = 0.
+(22)
+The determinant of this linear system is
+D := D(u, z0)D(v, z0) − D(u, z0)D(v, z0)
+and using Lemma 5.7 with
+x =
+�B(u)
+D(u)
+�
+, y =
+�B(z0)
+D(z0)
+�
+, z = y, w =
+�B(v)
+D(v)
+�
+,
+we get from Corollary 5.2 and (58)
+D =
+����
+B(u)
+B(v)
+D(u)
+D(v)
+����
+����
+B(z0)
+B(z0)
+D(z0)
+D(z0)
+���� = D(u, v)D(z0, z0).
+However, D(z0, z0) = −2Im (z0)||pz0||2 ̸= 0 so D = 0 iff D(u, v) = 0. There-
+fore, if α is a solution to (22) we have D(u, v) = 0.
+Suppose next that
+D(u, v) = 0. To see that (22) has a solution α, we notice that D(v, z0) and
+D(v, z0) cannot both be zero. In fact, defining ρ := B(z0)/D(z0) we have
+Im (ρ) > 0 because B/D is a Pick function, cf. Proposition 5.8, and by (58)
+D(v, z0) = 0 iff B(v) = ρD(v) while D(v, z0) = 0 iff B(v) = ρD(v), so both
+equations cannot hold. Here we use that B(v) = D(v) = 0 is impossible
+because of (12).
+If D(v, z0) ̸= 0, then α := −D(u, z0)/D(v, z0) satisfies (22) because D =
+0. Similarly α := −D(u, z0)/D(v, z0) satisfies (22) if D(v, z0) ̸= 0.
+Furthermore, in the case D(v, z0) ̸= 0 we get using (54) and D(u, v) = 0
+that
+D(u, z0) = D(u, v)C(v, z0) − B(u, v)D(v, z0) = −B(u, v)D(v, z0),
+9
+
+so finally α = B(u, v). The case D(v, z0) ̸= 0 is similar.
+Proof of (ii):
+By (18) we get
+qu + βqv = (ru + βrv) + (r+
+u + βr+
+v )pz0 + (r−
+u + βr−
+v )pz0,
+so qu + βqv ∈ D(T) if and only if
+r+
+u + βr+
+v = r−
+u + βr−
+v = 0,
+which by (19) is equivalent to
+C(u, z0) + βC(v, z0) = C(u, z0) + βC(v, z0) = 0.
+(23)
+The determinant of this linear system is
+D1 := C(u, z0)C(v, z0) − C(u, z0)C(v, z0)
+and using Lemma 5.7 with
+x =
+�A(u)
+C(u)
+�
+, y =
+�B(z0)
+D(z0)
+�
+, z = y, w =
+�A(v)
+C(v)
+�
+we get from Corollary 5.2 combined with (55), (57), (58)
+D1 =
+����
+A(u)
+A(v)
+C(u)
+C(v)
+����
+����
+B(z0)
+B(z0)
+D(z0)
+D(z0)
+���� = A(u, v)D(z0, z0).
+As in case (i) we see that D1 = 0 iff A(u, v) = 0. Therefore, if β is a solution
+to (23), we have A(u, v) = 0. Suppose next that A(u, v) = 0. To see that
+(23) has a solution β, we notice as in (i) that C(v, z0) and C(v, z0) cannot
+both be zero. For this we use that A/C is a Pick function by Proposition 5.8.
+If C(v, z0) ̸= 0, then β := −C(u, z0)/C(v, z0) satisfies (23). Similarly
+β := −C(u, z0)/C(v, z0) satisfies (23) if C(v, z0) ̸= 0.
+Furthermore, in the case C(v, z0) ̸= 0 we get using (53) and A(u, v) = 0
+that
+C(u, z0) = C(u, v)C(v, z0) − A(u, v)D(v, z0) = C(u, v)C(v, z0),
+so finally β = −C(u, v). The case C(v, z0) ̸= 0 is similar.
+10
+
+Proof of (iii): By (16) and (18) we get
+pu + γqv = (su + γrv) + (s+
+u + γr+
+v )pz0 + (s−
+u + γr−
+v )pz0,
+so pu + γqv ∈ D(T) if and only if
+s+
+u + γr+
+v = s−
+u + γr−
+v = 0,
+which by (17) and (19) is equivalent to
+D(u, z0) + γC(v, z0) = D(u, z0) + γC(v, z0) = 0.
+(24)
+The determinant of this linear system is
+D2 := D(u, z0)C(v, z0) − D(u, z0)C(v, z0)
+and using Lemma 5.7 with
+x =
+�
+B(u)
+D(u)
+�
+, y =
+�
+B(z0)
+D(z0)
+�
+, z = y, w =
+�
+A(v)
+C(v)
+�
+we get from Corollary 5.2 combined with (56), (57), (58)
+D2 =
+����
+B(u)
+A(v)
+D(u)
+C(v)
+����
+����
+B(z0)
+B(z0)
+D(z0)
+D(z0)
+���� = B(u, v)D(z0, z0).
+As in case (i) we see that D2 = 0 iff B(u, v) = 0. Therefore, if γ is a solution
+to (24), we have B(u, v) = 0. Suppose next that B(u, v) = 0. We see like in
+(ii) that if C(v, z0) ̸= 0, then γ := −D(u, z0)/C(v, z0) satisfies (24), and if
+C(v, z0) ̸= 0, then γ := −D(u, z0)/C(v, z0) satisfies (24).
+We finally see that in both cases γ = −D(u, v) because of (54).
+□
+Remark 2.4. The case (ii) can be deduced from case (i) by using the obser-
+vation that the polynomials (qn+1(x)/q1(x))n≥0 are the orthonormal polyno-
+mials associated with the truncated Jacobi matrix J(1) obtained from J by
+removing the first row and column. See [1, p. 28]. We have
+J(1)(Sc) = S(Jc) − a0⟨c, e0⟩,
+c ∈ F,
+where S is the bounded shift operator in ℓ2 given by (Sc)n := cn+1, n ≥ 0.
+If we let (T (1), D(T (1))) denote the Jacobi operator associated with J(1),
+one can prove that
+v ∈ D(T (1)) ⇐⇒ v = Su, u ∈ D(T)
+and
+T (1)(Su) = S(Tu) − a0⟨u, e0⟩,
+u ∈ D(T).
+11
+
+3
+Self-adjoint extensions of the Jacobi oper-
+ator
+As before (sn) is an indeterminate moment sequence with s0 = 1.
+The
+corresponding Jacobi operator (T, D(T)) has deficiency indices (1, 1) and
+the self-adjoint extensions in ℓ2 can be parametrized as the operators Tt, t ∈
+R∗ = R ∪ {∞} with domain
+D(Tt) = D(T) ⊕ C(q0 + tp0) for t ∈ R,
+D(T∞) = D(T) ⊕ Cp0
+(25)
+and defined by the restriction of T ∗ to the domain, cf. [17, Theorem 6.23].
+We recall that p0, q0 are defined in (4).
+The purpose of this section is to give some results about the domains
+D(Tt) of the self-adjoint operators Tt, t ∈ R∗.
+For t ∈ R∗ we define the solutions to the moment problem
+µt(·) := ⟨Et(·)e0, e0⟩,
+(26)
+where Et(·) is the spectral measure of the self-adjoint operator Tt.
+The measures µt, t ∈ R∗ are precisely those measures µ ∈ V for which
+the polynomials C[x] are dense in L2(µ) according to a famous theorem of
+M. Riesz, cf. [15], and they are called N-extremal in [1] and von Neumann
+solutions in [16]. They form a closed subset of ext(V ), the set of extreme
+points of the convex set V . However, ext(V ) is known to be a dense subset
+of V . They are characterized by the formula
+� dµt(x)
+x − z = − A(z) + tC(z)
+B(z) + tD(z),
+z ∈ C \ R, t ∈ R∗,
+(27)
+where A, . . . , D are the entire functions given in (11), cf. [17, Theorem 7.6].
+Recall that (12) holds.
+We summarize some of the properties of µt, which can be found in [1] and
+[17].
+Proposition 3.1.
+(i) The solution µt is a discrete measure with support
+equal to the countable zero set Λt of the entire function B(z) + tD(z),
+with the convention that Λ∞ is the zero set of D. In particular Λt ⊂ R
+for t ∈ R∗.
+12
+
+(ii) The support of two different N-extremal solutions are disjoint, and each
+point x0 ∈ R belongs to the support of a unique N-extremal measure µt,
+where t ∈ R∗ is given as t = −B(x0)/D(x0) if D(x0) ̸= 0 and t = ∞ if
+D(x0) = 0.
+Let us consider the vector space
+E := {Fc(z) =
+∞
+�
+n=0
+cnpn(z) | c ∈ ℓ2}
+(28)
+of entire functions, cf. (5). It is a Hilbert space under the norm
+||Fc||2 =
+∞
+�
+n=0
+|cn|2 =
+�
+|Fc(x)|2 dµ(x),
+where µ ∈ V can be arbitrary. It is a reproducing kernel Hilbert space of
+functions with the reproducing kernel
+K(u, v) :=
+∞
+�
+n=0
+pn(u)pn(v),
+u, v ∈ C,
+in the sense that
+�
+K(u, x)Fc(x) dµ(x) = Fc(u),
+µ ∈ V, u ∈ C.
+Note that (pn) is an orthonormal basis of E, and the mapping c �→ Fc is a
+unitary operator of the Hilbert space ℓ2 onto E.
+For each N-extremal measure µt the mapping c �→ Fc|supp(µt) is a unitary
+operator of ℓ2 onto L2(µt). The inverse mapping is given by
+f �→
+�
+⟨f, pn⟩L2(µt)
+�
+n≥0 ,
+f ∈ L2(µt),
+and
+f(x) =
+∞
+�
+n=0
+⟨f, pn⟩L2(µt)pn(x),
+x ∈ supp(µt).
+(29)
+The series in (29) converges to f in L2(µt) and converges also locally uni-
+formly for x ∈ C, but f is apriori only defined on supp(µt), so the equality
+holds pointwise for x ∈ supp(µt). The series represents a holomorphic exten-
+sion of f to all of C.
+13
+
+The self-adjoint operator Tt from (25) is unitarily equivalent with the
+multiplication operator Mµt on L2(µt) given by
+Mµtf(x) = xf(x),
+f ∈ L2(µt), x ∈ supp(µt).
+Theorem 3.2. Let µt be an N-extremal measure and let λ ∈ C \ supp µt.
+Then
+wµt(λ)pλ + qλ ∈ D(Tt),
+(30)
+where
+wµt(λ) :=
+�
+1
+x − λ dµt(x) = −C(λ, x)
+D(λ, x),
+x ∈ supp µt.
+(31)
+In particular, the ratio C(λ, x)/D(λ, x) does not depend on x ∈ supp µt.
+Proof. Since λ /∈ supp µt the functions (x−λ)−1 and x(x−λ)−1 are bounded
+on supp µt and in particular they belong to L2(µt). Thus (x−λ)−1 ∈ D(Mµt)
+and we find
+� pn(x)
+x − λ dµt(x) = wµt(λ)pn(λ) + qn(λ),
+where wµt(λ) is given by the first equality of (31). Moreover, by (29)
+(x − λ)−1 =
+∞
+�
+n=0
+[wµt(λ)pn(λ) + qn(λ)]pn(x),
+x ∈ supp µt.
+(32)
+In view of the unitary equivalence of Tt and Mµt we get (30). Multiplying
+(32) sidewise by x − λ gives
+1 = wµt(λ)(x − λ)
+∞
+�
+n=0
+pn(λ)pn(x) + (x − λ)
+∞
+�
+n=0
+qn(λ)pn(x),
+x ∈ supp(µt).
+By (8) and (9) we therefore get
+wµt(λ)D(λ, x) + C(λ, x) = 0,
+x ∈ supp µt.
+Assume D(λ, x) = 0. Then C(λ, x) = 0, but this gives a contradiction to
+(10). Hence D(λ, x) ̸= 0 and
+wµt(λ) = −C(λ, x)
+D(λ, x),
+x ∈ supp µt,
+which gives the second part of (31).
+14
+
+Remark 3.3. Using the formulas (57) and (58) in the last expression in (31),
+we get formula (27) with t = −B(x)/D(x) independent of x ∈ supp(µt) if
+D(x) ̸= 0, and t = ∞ if D(x) = 0.
+Note also that wµt(λ)pλ + qλ /∈ D(T) by Theorem 1.2 (iii) because
+B(λ, λ) = −1.
+We know from Theorem 1.2 that pλ, qλ /∈ D(T) for every λ ∈ C. We shall
+now clarify when pλ, qλ belong to the domain of the self-adjoint extension Tt
+associated with the N-extremal measure µt.
+Theorem 3.4. For λ ∈ C and t ∈ R∗ we have
+pλ ∈ D(Tt) ⇐⇒ D(λ, x) = 0 ∀x ∈ supp(µt) ⇐⇒ λ ∈ supp(µt).
+(33)
+qλ ∈ D(Tt) ⇐⇒ C(λ, x) = 0 ∀x ∈ supp(µt).
+(34)
+Proof. We define the entire functions gλ, hλ ∈ L2(µt) by
+gλ(x) :=
+∞
+�
+k=0
+pk(λ)pk(x),
+hλ(x) :=
+∞
+�
+k=0
+qk(λ)pk(x),
+x ∈ C.
+If pλ ∈ D(Tt) then (Tt − λI)pλ = 0 by (13), because Tt is a restriction
+of T ∗. Furthermore, by the unitary equivalence between Tt and the multi-
+plication operator Mµt on L2(µt) we have xgλ(x) ∈ L2(µt) and D(λ, x) =
+(λ − x)gλ(x) = 0 in L2(µt). By discreteness of µt we get D(λ, x) = 0 for all
+x ∈ supp(µt). The last equivalence of (33) follows from Remark 5.10. On
+the other hand, it is easy to see that the last two equivalent conditions of
+(33) imply pλ ∈ D(Tt), because the zero-function x �→ (λ − x)gλ(x) as well
+as λgλ(x) are in L2(µt), hence also xgλ(x) ∈ L2(µt). Therefore gλ ∈ D(Mµt)
+and finally pλ ∈ D(Tt).
+If qλ ∈ D(Tt) then (Tt − λI)qλ = e0 by (13), because Tt is a restriction
+of T ∗. This shows that (x − λ)hλ(x) = 1 in L2(µt) and hence C(λ, x) = 0
+for all x ∈ supp(µt). On the other hand, if C(λ, x) = 0 for x ∈ supp(µt), we
+conclude that xhλ(x) ∈ L2(µt), hence qλ ∈ D(Tt). This establishes (34).
+We know from Proposition 3.1 that supp(µt) is the the zero set of the
+entire function B(z) + tD(z) understood as D(z) if t = ∞. Using this we
+get the following Corollary about pλ. We get a similar result about qλ from
+(57).
+15
+
+Corollary 3.5. For t ∈ R and t = ∞ we have
+pλ ∈ D(Tt)
+⇐⇒
+B(λ) + tD(λ) = 0,
+qλ ∈ D(Tt)
+⇐⇒
+A(λ) + tC(λ) = 0.
+pλ ∈ D(T∞)
+⇐⇒
+D(λ) = 0,
+qλ ∈ D(T∞)
+⇐⇒
+C(λ) = 0.
+In particular pλ and qλ only belong to D(Tt) if λ ∈ R, and for λ ∈ R they
+belong to a unique D(Tt). Furthermore, they never belong to the same domain
+D(Tt).
+Remark 3.6. Since D(T) ⊂ D(Tt) for all t ∈ R∗, it is clear that Corollary 3.5
+implies that pλ, qλ /∈ D(T) as stated in Corollary 2.2.
+We also have a kind of converse to Theorem 3.2.
+Theorem 3.7. Assume that λ, τ ∈ C are such that τpλ + qλ ∈ D(Tt) for
+some t ∈ R∗. Then λ /∈ supp(µt) and τ = wµt(λ) given by (31).
+Proof. Assume that λ ∈ supp(µt). By Theorem 3.4 we know that pλ ∈ D(Tt)
+and hence qλ ∈ D(Tt), contradicting Corollary 3.5.
+Having established λ /∈ supp(µt), we get by Theorem 3.2 that (τ −
+wµt(λ))pλ ∈ D(Tt), but since pλ /∈ D(Tt), we get τ − wµt(λ) = 0.
+Theorem 3.8. Let t ∈ R∗ and λ ∈ C \ supp(µt) be given. Then there exists
+a unique pair (s, c) ∈ D(T) × C depending on t, λ such that
+wµt(λ)pλ + qλ =
+�
+s + c(q0 + tp0),
+t ∈ R
+s + cp0,
+t = ∞.
+We have
+c =
+� −1/(B(λ) + tD(λ)),
+t ∈ R
+−1/D(λ),
+t = ∞,
+and s is given by inserting the value of c.
+Proof. Recall that λ ∈ C \ supp(µt) if and only if B(λ) + tD(λ) ̸= 0 when
+t ∈ R, and that λ ∈ C \ supp(µ∞) if and only if D(λ) ̸= 0. The existence
+and uniqueness of (s, c) follow from Theorem 3.2 and formula (25).
+In case t ∈ R we have
+wµt(λ)pλ + qλ − c(q0 + tp0) ∈ D(T),
+16
+
+and fixing z0 in the open upper half-plane we have by Proposition 2.1
+wµt(λ)s+
+λ + r+
+λ − c(r+
+0 + ts+
+0 ) = wµt(λ)s−
+λ + r−
+λ − c(r−
+0 + ts−
+0 ) = 0,
+or equivalently by (17) and (19)
+wµt(λ)D(λ, z0) + C(λ, z0) − c(C(0, z0) + tD(0, z0)) = 0,
+(35)
+and
+wµt(λ)D(λ, z0) + C(λ, z0) − c(C(0, z0) + tD(0, z0)) = 0.
+(36)
+From (36) and (31) we get for x ∈ supp(µt)
+c
+=
+−wµt(λ)D(λ, z0) + C(λ, z0)
+B(z0) + tD(z0)
+=
+C(λ, x)D(λ, z0) − D(λ, x)C(λ, z0)
+(B(z0) + tD(z0))D(λ, x)
+=
+−D(z0, λ)C(λ, x) − B(z0, λ)D(λ, x)
+(B(z0) + tD(z0))D(λ, x)
+=
+−
+D(z0, x)
+(B(z0) + tD(z0))D(λ, x) = −
+1
+B(λ) + tD(λ).
+Here we have first used (54) and next used (58) twice. Finally we recall that
+t = −B(x)/D(x) for x ∈ supp(µt), cf. Proposition 3.1.
+Note that (35) leads to the same expression for c.
+The case t = ∞ is treated in the same way.
+4
+Parametrizations of the domain of the Ja-
+cobi operator
+The Jacobi operator (T, D(T)) in the indeterminate case is regular in the
+sense of [13, p. 20], i.e., for any z ∈ C there exists d(z) > 0 such that
+||(T − zI)c|| ≥ d(z)||c||,
+c ∈ D(T).
+(37)
+For z ∈ C \ R this is true with d(z) = |Im (z)|, and for z ∈ R let t0 ∈ R∗ be
+such that z ∈ supp(µt0). For t ∈ R∗ \ {t0} the distance
+dt(z) := min{|z − x| | x ∈ supp(µt)} > 0,
+17
+
+can be used in (37), since we have
+||(T − zI)c||2 =
+�
+|(x − z)Fc(x)|2 dµt(x) ≥ dt(z)2||c||2,
+where Fc is given in (5).
+We have the orthogonal decomposition in closed subspaces
+ℓ2 = (T − zI)(D(T)) ⊕ Cpz,
+z ∈ C.
+(38)
+The operator (T, D(T)) has no eigenvalues, has empty continuous spec-
+trum, and the spectrum σ(T) = C is equal to the residual spectrum, cf. [18,
+p.209].
+For z0 ∈ C we have the orthogonal expansion
+pn(z) − pn(z0)
+z − z0
+=
+n−1
+�
+k=0
+an,k(z0)pk(z),
+z ∈ C
+(39)
+of the polynomial (pn(z) − pn(z0)/(z − z0) of degree n − 1, and it is easy to
+see that
+an,k(z0) =
+� pn(x) − pn(z0)
+x − z0
+pk(x) dµ(x) = qn(z0)pk(z0) − pn(z0)qk(z0), (40)
+where µ ∈ V is an arbitrary solution to (1), cf. [1, p. 18].
+Lemma 4.1. The coefficients an,k(z0) from (40) satisfy
+|an,k(z0)|2 ≤
+�
+|pn(z0)|2 + |qn(z0)|2� �
+|pk(z0)|2 + |qk(z0)|2�
+.
+(41)
+Therefore
+∞
+�
+n=k+1
+|an,k(z0)|2 ≤ (||pz0||2 + ||qz0||2)(|pk(z0)|2 + |qk(z0)|2).
+(42)
+Furthermore,
+∞
+�
+n=0
+����
+pn(z) − pn(z0)
+z − z0
+����
+2
+≤ ||pz||2 �
+||pz0||2 + ||qz0||2�2 .
+(43)
+In particular, for z → z0
+∞
+�
+n=0
+|p′
+n(z0)|2 ≤ ||pz0||2 �
+||pz0||2 + ||qz0||2�2 .
+18
+
+Proof. Formula (41) is a consequence of the Cauchy-Schwarz inequality.
+From (39) and (41) we get
+����
+pn(z) − pn(z0)
+z − z0
+����
+2
+≤
+n−1
+�
+k=0
+|an,k(z0)|2
+n−1
+�
+k=0
+|pk(z)|2
+≤
+||pz||2
+n−1
+�
+k=0
+�
+|pn(z0)|2 + |qn(z0)|2� �
+|pk(z0)|2 + |qk(z0)|2�
+,
+and finally
+∞
+�
+n=0
+����
+pn(z) − pn(z0)
+z − z0
+����
+2
+≤
+||pz||2
+∞
+�
+k=0
+�
+�
+|pk(z0)|2 + |qk(z0)|2�
+∞
+�
+n=k+1
+�
+|pn(z0)|2 + |qn(z0)|2�
+�
+≤
+||pz||2 �
+||pz0||2 + pz0||2�2 ,
+which yields (43).
+We shall now show that the Hilbert space E = {Fc(z)} defined in (28) is
+stable under difference quotients:
+Theorem 4.2. For c ∈ ℓ2 and z0 ∈ C there exists ξ(c, z0) ∈ ℓ2 such that
+Fc(z) − Fc(z0)
+z − z0
+= Fξ(c,z0)(z) ∈ E,
+(44)
+and the coordinates of ξ(c, z0) are defined by
+ξk(c, z0) =
+∞
+�
+n=k+1
+cnan,k(z0),
+k ≥ 0.
+(45)
+Furthermore,
+||ξ(c, z0)|| ≤ ||c||
+�
+||pz0||2 + ||qz0||2�
+.
+(46)
+Proof. The series in (45) is absolutely convergent being the product of two
+ℓ2 sequences. Furthermore, by the Cauchy-Schwarz inequality and (41) we
+get
+|ξk(c, z0)|2 ≤ ||c||2(||pz0||2 + ||qz0||2)
+�
+|pk(z0)|2 + |qk(z0)|2�
+,
+19
+
+and therefore (ξk(c, z0)) ∈ ℓ2 and (46) holds.
+We next find
+Fc(z) − Fc(z0)
+z − z0
+=
+∞
+�
+n=0
+cn
+pn(z) − pn(z0)
+z − z0
+,
+z ̸= z0.
+Inserting the expression (39) on the right-hand side, we get for z ̸= z0
+Fc(z) − Fc(z0)
+z − z0
+=
+∞
+�
+n=1
+cn
+n−1
+�
+k=0
+an,k(z0)pk(z)
+=
+∞
+�
+k=0
+pk(z)
+∞
+�
+n=k+1
+cnan,k(z0)
+=
+∞
+�
+k=0
+ξk(c, z0)pk(z),
+where the rearrangement is possible due to absolute convergence:
+����
+Fc(z) − Fc(z0)
+z − z0
+����
+≤
+∞
+�
+n=1
+|cn|
+n−1
+�
+k=0
+|an,k(z0)| |pk(z)|
+=
+∞
+�
+k=0
+|pk(z)|
+∞
+�
+n=k+1
+|cn| |an,k(z0)|
+≤
+||c||
+∞
+�
+k=0
+|pk(z)|
+�
+∞
+�
+n=k+1
+|an,k(z0)|2
+�1/2
+≤
+||c||
+∞
+�
+k=0
+|pk(z)|
+��
+||pz0||2 + ||qz0||2� �
+|pk(z0)|2 + |qk(z0)|2��1/2
+≤
+||c||
+�
+||pz0||2 + ||qz0||2�1/2 ||pz||
+� ∞
+�
+k=0
+�
+|pk(z0)|2 + |qk(z0)|2�
+�1/2
+=
+||c|| ||pz||
+�
+||pz0||2 + ||qz0||2�
+,
+where we have used (42).
+It is now clear that the entire functions z �→ (Fc(z) − Fc(z0))/(z − z0),
+with value F ′
+c(z0) for z = z0, and Fξ(c,z0)(z) agree.
+20
+
+Theorem 4.3. Let Ξz0 denote the bounded operator in ℓ2 defined by
+Ξz0(c) := ξ(c, z0),
+z0 ∈ C, c ∈ ℓ2,
+(47)
+where ξ(c, z0) is defined in Theorem 4.2.
+We have Ξz0(ℓ2) = D(T) and
+ker(Ξz0) = Ce0 for each z0 ∈ C.
+Furthermore, for z0 ∈ C
+(T − z0I)Ξz0(c) + Fc(z0)e0 = c,
+c ∈ ℓ2.
+(48)
+The restriction of Ξz0 to (T − z0I)(D(T)) is a bijection onto D(T) equal to
+(T − z0I)−1.
+Proof. Let U : ℓ2 → E denote the unitary mapping given by U(c) = Fc.
+Then
+U(Jc)(z) = z
+∞
+�
+k=0
+ckpk(z),
+c ∈ F, z ∈ C,
+i.e., U is the intertwining operator between J and the densely defined oper-
+ator of multiplication with z on C[z] ⊂ E. Therefore
+U(Tc)(z) = zFc(z),
+c ∈ D(T), z ∈ C.
+(49)
+For c ∈ ℓ2 and z0 ∈ C we have
+c − Fc(z0)e0 ⊥ pz0,
+so by (38) c − Fc(z0)e0 belongs to (T − z0I)(D(T)). Therefore, there exists
+a unique vector v ∈ D(T) such that
+c − Fc(z0)e0 = (T − z0I)(v),
+(50)
+and applying U to (50) we get by (49)
+Fc(z) − Fc(z0) = (z − z0)Fv(z),
+z ∈ C.
+Now (44) shows that Fv(z) = Fξ(c,z0)(z) for z ̸= z0, hence for all z, and finally
+v = ξ(c, z0), showing that Ξz0(c) ∈ D(T). Inserting v = ξ(c, z0) in (50) yields
+(48).
+For v ∈ D(T) we define c = (T − z0I)(v). Then Fc(z0) = 0 as c ⊥ pz0 by
+(38), and then (48) gives (T − z0I)Ξz0(c) = c. By injectivity of T − z0I we
+get v = Ξz0(c) = (T − z0I)−1(c).
+It is easy to see that ξ(e0, z0) = 0, hence Ce0 ⊆ ker(Ξz0), and from (48)
+the converse inclusion follows.
+21
+
+Remark 4.4. The operator Ξz0 defined in (47) is seen to satisfy
+Ξz0(en) =
+n−1
+�
+k=0
+an,k(z0)ek,
+n ≥ 1,
+and it follows easily that Ξz0(F) = F.
+Moreover, since (Ce0)⊥ = {c ∈ ℓ2 | c0 = 0}, we have the following
+parametrizations of D(T)
+D(T) = {Ξz0(c) | c ∈ ℓ2, c0 = 0},
+z0 ∈ C.
+5
+Appendix
+We need the following result about the Nevanlinna functions defined in the
+Introduction.
+Theorem 5.1. For u, v, w ∈ C we have
+A(u, v)
+=
+C(u, w)A(w, v) − A(u, w)B(w, v)
+(51)
+B(u, v)
+=
+D(u, w)A(w, v) − B(u, w)B(w, v)
+(52)
+C(u, v)
+=
+C(u, w)C(w, v) − A(u, w)D(w, v)
+(53)
+D(u, v)
+=
+D(u, w)C(w, v) − B(u, w)D(w, v).
+(54)
+From the obvious relations
+A(u, v) = −A(v, u), B(u, v) = −C(v, u), D(u, v) = −D(v, u)
+and putting w = 0 in the formulas of Theorem 5.1, we get the following
+formulas in terms of the one variable functions (11):
+Corollary 5.2. For u, v ∈ C we have
+A(u, v)
+=
+����
+A(u)
+A(v)
+C(u)
+C(v)
+����
+(55)
+B(u, v)
+=
+����
+B(u)
+A(v)
+D(u)
+C(v)
+����
+(56)
+C(u, v)
+=
+����
+A(u)
+B(v)
+C(u)
+D(v)
+����
+(57)
+D(u, v)
+=
+����
+B(u)
+B(v)
+D(u)
+D(v)
+���� .
+(58)
+22
+
+We have not been able to find the formulas of Theorem 5.1 in the litera-
+ture, so we indicate a proof. The formulas of Corollary 5.2 expressing the two
+variable functions in terms of the one variable functions were, as far as we
+know, first given in [11] and included in [17, exercise 7.8 (3)]. (Unfortunately
+there is a misprint in the exercise: B and C are interchanged.)
+We begin by introducing polynomial approximations to the Nevanlinna
+functions.
+Proposition 5.3. [17, Proposition 5.24] For u, v ∈ C and n ≥ 0 we have
+An(u, v)
+:=
+(u − v)
+n
+�
+k=0
+qk(u)qk(v) = an
+����
+qn+1(u)
+qn+1(v)
+qn(u)
+qn(v)
+����
+Bn(u, v)
+:=
+−1 + (u − v)
+n
+�
+k=0
+pk(u)qk(v) = an
+����
+pn+1(u)
+qn+1(v)
+pn(u)
+qn(v)
+����
+Cn(u, v)
+:=
+1 + (u − v)
+n
+�
+k=0
+qk(u)pk(v) = an
+����
+qn+1(u)
+pn+1(v)
+qn(u)
+pn(v)
+����
+Dn(u, v)
+:=
+(u − v)
+n
+�
+k=0
+pk(u)pk(v) = an
+����
+pn+1(u)
+pn+1(v)
+pn(u)
+pn(v)
+���� .
+It is important to notice that
+����
+An(u, v)
+Bn(u, v)
+Cn(u, v)
+Dn(u, v)
+���� = 1 for (u, v) ∈ C2,
+cf. [17, Equation(5.57)].
+For later use we introduce the transfer matrix with determinant 1
+hn(u, v) =
+�
+Cn(u, v)
+An(u, v)
+−Dn(u, v)
+−Bn(u, v)
+�
+,
+u, v ∈ C, n ≥ 0.
+(59)
+The name transfer matrix is motivated by
+Proposition 5.4. For u, v ∈ C, n ≥ 0 we have
+�
+pn(u)
+qn(u)
+pn+1(u)
+qn+1(u)
+�
+hn(u, v) =
+�
+pn(v)
+qn(v)
+pn+1(v)
+qn+1(v)
+�
+(60)
+23
+
+Proof. The four formulas of Proposition 5.3 can be expressed as the matrix
+equation
+hn(u, v) = an
+�
+qn+1(u)
+−qn(u)
+−pn+1(u)
+pn(u)
+� �
+pn(v)
+qn(v)
+pn+1(v)
+qn+1(v)
+�
+.
+However, by [17, Equation (5.52)]
+�
+qn+1(u)
+−qn(u)
+−pn+1(u)
+pn(u)
+�−1
+= an
+�
+pn(u)
+qn(u)
+pn+1(u)
+qn+1(u)
+�
+,
+and (60) follows.
+By the uniqueness of a matrix hn(u, v) satisfying (60) we get:
+Corollary 5.5. For u, v, w ∈ C, n ≥ 0 we have
+hn(u, w)hn(w, v) = hn(u, v),
+hn(u, v) = hn(v, u)−1.
+Proof of Theorem 5.1. Letting n tend to infinity in (59) we obtain the
+entire matrix function
+h(u, v) =
+�
+C(u, v)
+A(u, v)
+−D(u, v)
+−B(u, v)
+�
+,
+u, v ∈ C,
+with determinant 1 satisfying h(u, w)h(w, v) = h(u, v), which is equivalent
+to the formulas (51),(52),(53) and (54) of the theorem.
+□
+Remark 5.6. The M¨obius transformation M(u, v) : C∗ → C∗ defined by
+M(u, v)(z) :=
+C(u, v)z + A(u, v)
+−D(u, v)z − B(u, v),
+z ∈ C∗,
+maps the Weyl circle Kv onto the Weyl circle Ku, where Ku = R∗ if u ∈ R.
+For u ∈ C \ R the Weyl circle is defined in [17, Section 7.3].
+The following Lemma unifies some calculations:
+24
+
+Lemma 5.7. For vectors x, y, z, w ∈ C2 we have the following determinant
+equation
+����
+|x y|
+|x z|
+|w y|
+|w z|
+���� = |x w||y z|,
+where
+|x y| =
+����
+x1
+y1
+x2
+y2
+���� etc.
+Proof. First method: Direct computation.
+Second method: Define
+M(x, y, z, w) = |x y||z w| + |x z||w y| + |x w||y z|,
+where it should be noticed that y, z, w appear in its cyclic permutations.
+Clearly M is a 4-linear form on C2, and it is alternating, i.e., is zero, if
+any two arguments agree. An alternating 4-linear form on a vector space of
+dimension ≤ 3 is identically zero, hence M ≡ 0.
+In various proofs we need that certain functions are Pick function, i.e.,
+holomorphic functions in the cut plane C\R with certain properties, see [12].
+Proposition 5.8. The meromorphic functions B/D and A/C are Pick func-
+tions, i.e., they map the upper (resp. lower) open half-plane into itself.
+Proof. The result about B/D is in [3, Proposition 1.3]. The result about
+A/C can be deduced from the previous result by considering the indeter-
+minate moment problem corresponding to the truncated Jacobi matrix J(1)
+considered in Remark 2.4. There are simple relations between the Nevan-
+linna functions �A, . . . , �D of the truncated problem and those of the original
+moment problem, see [14]:
+A(z) = a−2
+0 �D(z),
+C(z) = −b0a−2
+0 �D(z) − �B(z),
+z ∈ C.
+Therefore −C/A = b0+a2
+0( �B/ �D), which shows that −C/A is a Pick function,
+and so is A/C.
+By a famous Theorem of M. Riesz each of the functions Fc defined in (5)
+are of minimal exponential type meaning that for each ε > 0 there exists a
+constant Cε > 0 such that
+|Fc(z)| ≤ Cεeε|z|,
+z ∈ C.
+25
+
+This follows from the Cauchy-Schwarz inequality because the norm ||pz|| sat-
+isfies the same inequality by [1, Theorem 2.4.3]. Using that the polynomials
+qn+1/q1 are the orthonormal polynomials for the indeterminate truncated Ja-
+cobi matrix J(1), cf. Remark 2.4, we also get that the functions Gc from (5)
+are of minimal exponential type.
+We next recall an important property of these functions in case they are
+not polynomials.
+Proposition 5.9. For each c ∈ ℓ2\F the functions Fc, Gc are transcendental
+and have a countably infinite set of zeros.
+In particular, for each v ∈ C the functions of the variable u, A(u, v), B(u, v),
+C(u, v), D(u, v) have a countably infinite set of zeros.
+Proof. An entire transcendental function f of minimal exponential type has
+a countably infinite set of zeros. In fact, the order ρ of f is either strictly
+less than 1 or equal to 1, and in the latter case the type of f is zero. In the
+first case the result follows from the Hadamard factorization Theorem, cf.
+[9, p.22]. In the second case the result follows from a Theorem of Lindel¨of,
+see [9, Theorem 2.20.3].
+For v ∈ C and F being one of the functions A, . . . , D, we see that u �→
+F(u, v) is an entire transcendental function of minimal exponential type.
+Proof of Theorem 1.3: Case 1: Let us first consider the case of D with
+Z(D)v = {u ∈ C | D(u, v) = 0} for given v ∈ C, cf. (14).
+If v ∈ R then Z(D)v ⊂ R by [4, Theorem 3], and furthermore Z(D)v
+equals the support of the unique N-extremal measure which contains v in
+the support.
+If v ∈ C \ R, then D(v) ̸= 0, and using (58) we get
+D(u, v) = D(v)[B(u) − ρD(u)], ρ := B(v)/D(v),
+so u ∈ Z(D)v iff B(u) = ρD(u). For such u we must have D(u) ̸= 0 for
+otherwise B(u) = D(u) = 0 contradicting (12). This gives u ∈ Z(D)v iff
+B(u)/D(u) = ρ. Using that B/D is a Pick function by Proposition 5.8, we
+see that u, v belong to the same half-plane.
+Case 2: We consider Z(B)v and use (56), viz.
+B(u, v) = B(u)C(v) − D(u)A(v).
+26
+
+Let first v ∈ R. If C(v) = 0 then A(v) ̸= 0 by (12), so B(u, v) = 0 iff
+D(u) = 0, hence Z(B)v ⊂ R. If C(v) ̸= 0 then
+B(u, v) = C(v)[B(u) − τD(u)], τ := A(v)/C(v) ∈ R,
+so Z(B)v is the zero set of B − τD, hence real by Proposition 3.1.
+Let next v ∈ C \ R. Then C(v) ̸= 0, so u ∈ Z(B)v iff B(u) = τD(u) with
+τ as above, but this is only possible if D(u) ̸= 0 and hence B(u)/D(u) = τ.
+Using that both A/C and B/D are Pick functions, cf. Proposition 5.8, we
+see that u, v belong to the same half-plane.
+Case 3: We consider Z(C)v and use (57), viz.
+C(u, v) = A(u)D(v) − C(u)B(v).
+By considering the cases v ∈ R and v ∈ C\R separately and factor out D(v)
+in case it is non-zero, we may proceed as in case 2.
+Finally, the case of Z(A)v follows from the case 1 by considering the
+truncated case as in Remark 2.4. □
+Remark 5.10. As noticed in the proof above one has for v ∈ R:
+D(u, v) = 0 ⇐⇒ u ∈ supp(µ),
+where µ is the N-extremal measure such that v ∈ supp(µ).
+Compare also with Remark 3.3.
+Proof of Proposition 1.4:
+Case (i): From the proof of Theorem 1.2 (i) we know that B(u, v) =
+−D(u, z)/D(v, z) for all z ∈ C \ R since D(v, z) ̸= 0 for these z. By assump-
+tion D(x, v) ̸= 0 for u < x < v, so by continuity B(u, v) = −D(u, x)/D(v, x)
+for these x. We next observe that
+B(u, v)v − x
+x − u = v − x
+D(v, x)
+D(u, x)
+u − x ̸= 0,
+u < x < v,
+(61)
+and
+lim
+x→u+
+D(u, x)
+u − x = ||pu||2,
+lim
+x→v−
+D(v, x)
+v − x = ||pv||2,
+so the function in (61) is positive for u < x < v, hence B(u, v) > 0.
+Case (ii): This case is reduced to case (i) for the truncated Jacobi matrix
+from Remark 2.4. If �A, . . . , �D denote the Nevalinna functions of two variables
+27
+
+for the truncated case, the following formulas can be found in [14]. (The
+reader is warned that this reference follows the normalization of [11].)
+�B(u, v) = (v − b0)A(u, v) − C(u, v),
+�D(u, v) = a2
+0A(u, v),
+and since we assume that A(u, v) = 0, we have −C(u, v) = �B(u, v) > 0.
+□
+References
+[1] N. I. Akhiezer, The Classical Moment Problem and Some Related Ques-
+tions in Analysis. English translation, Oliver and Boyd, Edinburgh,
+1965.
+[2] N. I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space.
+Two volumes bound as one. Dover 1993.
+[3] C. Berg, Indeterminate moment problems and the theory of entire func-
+tions, J. Comput. Appl. Math. 65 (1995), 27–55.
+[4] C. Berg and J. P. R. Christensen, Density questions in the classical
+theory of moments, Ann. Inst. Fourier 31, no. 3 (1981), 99–114.
+[5] C. Berg and R. Szwarc, The Smallest Eigenvalue of Hankel Matrices,
+Constr. Approx. 34 (2011), 107–133.
+[6] C. Berg and R. Szwarc, Inverse of infinite Hankel moment matrices,
+SIGMA 14 (2018), 109, 48 pages.
+[7] C. Berg and R. Szwarc, Closable Hankel Operators and Moment Prob-
+lems, Integr. Equ. Oper. Theory 92(1) (2020), 1–9.
+[8] C. Berg and R. Szwarc, Self-adjoint operators associated with Hankel
+moment matrices, Journal of Functional Analysis, 283 (2022), 109674.
+[9] R. P. Boas, Entire Functions. Acdemic Press Inc., Publishers, New York
+1954.
+[10] L. de Branges, Hilbert Spaces of Entire Functions. Prentice-Hall, Inc.
+Englewood Cliffs, N. J. 1968.
+28
+
+[11] H. Buchwalter and G. Cassier, La param´etrisation de Nevanlinna dans
+le probl`eme des moments de Hamburger, Expo Math. 2 (1984), 155–178.
+[12] W. F. Donoghue, Jr., Monotone Matrix Functions and Analytic Contin-
+uation. Springer-Verlag, Berlin, Heidelberg, New York, 1974.
+[13] M. L. Gorbachuk and V. I. Gorbachuk, M. G. Krein’s Lectures on Entire
+Operators. Birkh¨auser Verlag, Basel, Boston, Berlin, 1997.
+[14] H. L. Pedersen, The Nevanlinna matrix of entire functions associated
+with a shifted indeterminate Hamburger moment problem, Math. Scand.
+74 (1994), 152–160.
+[15] M. Riesz, Sur le probl`eme des moments et le th´eor`eme de Parseval cor-
+respondant, Acta Litt. Ac. Sci. Szeged 1 (1923), 209–225.
+[16] B. Simon, The classical moment problem as a self-adjoint finite difference
+operator, Adv. Math. 137 (1998), 82–203.
+[17] K. Schm¨udgen, The Moment Problem, Graduate Texts in Mathematics
+Vol. 277. Springer International Publishing AG 2017.
+[18] K. Yosida, Functional Analysis, Third Edition, Springer Verlag, Berlin,
+Heidelberg, New York, 1971.
+Christian Berg
+Department of Mathematical Sciences, University of Copenhagen
+Universitetsparken 5, DK-2100 Copenhagen, Denmark
+e-mail: berg@math.ku.dk
+Ryszard Szwarc
+Institute of Mathematics, University of Wroc�law
+pl. Grunwaldzki 2/4, 50-384 Wroc�law, Poland
+e-mail: szwarc2@gmail.com
+29
+

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+arXiv:2301.11830v1  [astro-ph.IM]  27 Jan 2023
+Detection of Ultra High Energy Cosmic Rays and Neutrinos with
+Lunar Orbital Radio Telescope
+Linjie Chen,1,2∗ Marc Klein Wolt,3 Amin Aminaei,4 Stijn Buitink,5 Heino Falcke3
+1National Space Science Center, Chinese Academy of Sciences, Beijing, 100190, China
+2National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100101, China
+3Astronomical Institute, Radboud University Nijmegen, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
+4UC Davis, Dept. of Physics and Astronomy, Physics Bldg, 1 Shields Ave, Davis, USA
+5Astrophysical Institute, Vrije Universiteit Brussel, Pleinlaan 2, Brussels, 1050, Belgium
+∗Corresponding author; E-mail: ljchen@nao.cas.cn.
+Jan. 25, 2023
+Particle cascades induced by ultra-high-energy (UHE) cosmic rays and neutrinos impacting on the lu-
+nar regolith usually radiate Cherenkov radio emissions due to the presence of excess negative charge,
+which is known as Askaryan effect. Several experiments have been carried out to detect the Cherenkov
+radio emissions in the lunar regolith. To prepare for future lunar Ultra-Long Wavelength (ULW, fre-
+quencies below 30 MHz) radio astronomy missions, we study the detection of the Cherenkov radio
+emissions with the ULW radio telescope that are operating at the lunar orbit. We have carried out in-
+strument modelling and analytic calculations for the analysis of aperture, flux and event rate, and the
+analyses show the detectability of the Cherenkov radiation. Based on the properties of the Cherenkov
+radiation, we have demonstrated that the cosmic ray and neutrino events could be reconstructed with
+the three ULW vector antennas onboard the lunar satellites via measurements of the Askaryan radio
+pulse intensity, polarizations, etc. The results obtained by this study would be useful for future lunar
+radio explorer mission, where the detections of UHE cosmic rays and neutrinos could be successfully
+attempted.
+1
+Introduction
+Studies on the behaviour of ultra-high energy (UHE) particles and their source of origins have been a major topic of
+interest in modern high energy astrophysics and particle physics. However, the origin of ultra-high-energy (> 1018
+eV) cosmic rays (UHECR) still remains unknown to date. It is mainly because of their deflected trajectories caused
+by cosmic magnetic fields. Above a threshold energy of ∼ 1019.6 eV, cosmic rays usually interact with cosmic
+microwave background to produce pions and lose a substantial amount of energy within a distance of tens of mega
+parsec (Mpc). This is well known as Greisen-Zatsepin-Kuzmin (GZK) effect ([28], [54]). The few cosmic rays
+that observed above this threshold energy must originate from the distant sources. In order to determine the source
+origin of such UHE cosmic rays, the detection of UHE neutrinos, those produced together with the cosmic rays
+1
+
+or during the GZK interaction, has usually been employed as one of the method. Since neutrinos are chargeless
+and weakly interacting particles, the propagating UHE neutrinos (UHECν) usually remain unaffected over large
+cosmic distances, and therefore, their arrival directions carry direct information of their source of origins.
+The chief problem of both UHECR and UHECν event detection come from their extreme rarity. The flux at
+GZK energy is around one particle per square kilometer per century, which calls for huge detectors (thousands km2)
+to collect a significant amount of events at ultra-high energies. For instance, even the Pierre Auger Observatory
+(PAO) with a collecting area of 3000 km2 can only detect of order 30 particles above the GZK energy per year [37].
+However, for the detection of particles above 1021 eV, the detectors are needed to have a thousand-fold increase in
+their collecting area. Thus, space-based detection systems are, therefore, appeared to be a feasible approach that is
+currently being attempted and implemented.
+The Moon acts as a huge natural detector with large collecting area and is first proposed to be a target to detect
+cosmic ray and neutrino showers in [20]. The detection is based on the Askaryan effect, which is pointed out by
+G.A. Askaryan from the Lebedev Physical Institute in 1961 [5]. When high-energy particle interaction occurs in
+a denser medium like ice, rock salt, lunar regolith, or the atmosphere, it attracts electrons from the surrounding
+medium and transfer them into the shower disk. With the annihilation of showering positrons in flight, a net
+excess of electrons are produced. The excess electrons rapidly appear and disappear to produce a radio pulse that
+could be observed by a sensitive telescope. Using the above technique several experiments have been proposed
+and performed to detect high-energy cosmic particles, such as Parkes ([32], [33]), GLUE ([26],[25]), LUNASKA
+[38],[39], NuMoon [13], RESUN [36], LaLuna [52], LOFAR [50], FAST [37], all of which employ the terrestrial
+telescopes to carry out the observations. In the literatures ([31], [53], [4]), the scheme of a lunar orbital satellite
+system was presented and analyzed to detect the UHE cosmic rays and neutrinos.
+The ultra-long wavelength (ULW) (≥ 10 m) range, the last unexplored region of the electromagnetic (EM)
+spectrum, is presently one of the new promising areas of radio astronomy. Strengthening the science cases for
+extending the radio astronomy into the unexplored ULW window has been significantly increasing. Due to the
+reflection of the ULW signals by the Earth’s ionosphere and the presence of the strong terrestrial anthropogenic
+radio frequency interference (RFI), the far side of Moon can be an ideal site for carying out the ULW radio
+observations. To date, various Moon-based telescopes have been made and launched for exploring the ULW
+region, such as OLFAR [22], FARSIDE [47], DARE [14], LRX [44], DSL ([12], [18]), NCLE [11], LFRS [41],
+etc. The lunar ULW radio telescopes provide ample opportunities to detect the Cherenkov radiation usually caused
+by the UHNCR or UHECν on the Moon.
+In this paper we present the radio detection of the UHNCR or UHECν with the lunar orbital ULW radio
+antennas. The paper is organized as follows. In Section 2 we describe the model for the detection of the lunar
+Cherenkov emission. The aperture and flux limit for the lunar UHECR and UHEC events are estimated and
+analyzed in Section 3. In Section 4 we discuss the characteristic of the UHNCR or UHECν with the multi-satellite
+detections. Finally, summary and conclusions are provided in Section 5.
+2
+
+2
+Model for radio emission detection
+When energetic particles interact in the denser medium like lunar regolith, intense coherent radio pulses are pro-
+duced in the frequency range from MHz to GHz. The radio emissions that escaped from the lunar surface could
+be detected by a Moon-based or terrestrial radio telescope as shown in Figure 1, which strongly depends on the
+sensitivity of the telescope. Though the terrestrial radio telescope have a huge collecting area and the best sensitiv-
+ity, the intensity of the lunar Askaryan radio emission is weak due to the long-distance attenuation. A lunar radio
+telescope may provide a reasonable detection capability for the UHECR and UHECν events since it is thousand
+times closer to the lunar surface than to the ground-based telescopes. For the detection of the UHE particles, the
+effective aperture is another vital factor which is directly dependent on the physical lunar surface illuminated by
+the antenna. The physical surface area is determined by the antenna beamwidth and antenna altitude from the lunar
+surface, and the altitude affects the detected electric field of the Askaryan radio emission. In order to study the
+lunar UHECR and UHECν detections, the electromagnetic properties of Cherenkov radiation have been modelled
+including the system parameters of the radio detector.
+Figure 1: Schematic view of lunar antenna detections of Askaryan radio emissions produced by the UHECR or
+UHECν in the lunar soil.
+2.1
+Lunar Cherenkov radio emission
+According to numerous Monte Carlo simulations, the Cherenkov radio emissions from the UHE hadronic showers
+have been parameterized for different dielectric media ([55], [7], [8], [25]). In the literatures ([31], [53], [23], [9], [4])
+the radio detection of lunar Askaryan pulse caused by the UHECR and UHECν interacting with Moon have been
+addressed. For the Cherenkov radiation, we utilized the formula given in Ref. ([39], [23], [9]) to calculate the
+electric field strength ε0 (µV m−1 MHz−1) at the Cherenkov angle θc = cos−1(1/n).
+ε0(E, R, ν) = 8.45 × 106 1
+R( Es
+E0
+)(
+ν
+GHz)(
+1
+1 + (ν/ν0)1.23 )
+(1)
+3
+
+R
+0.
+RM
+0
+Os
+v or CRwhere, R is the distance from the source point to the detector, Es is the shower energy in eV. E0 = 1020eV, ν0 =
+2.32GHz ([39], [9]). The angular distribution of the radio emissions around the Cherenkov angle θc is crucial for
+the detection of lunar UHNCR and UHECν events. Due to the loss of coherence, the radio emission decreases in
+the direction away from θc. [48] discussed the angular spread of the Cherenkov radiation around the Cherenkov
+angle. Through comparing several different formulae of intensity distribution, Scholten concluded that the gaussian
+parametrization formula is more accurate since it agrees with both the Monte Carlo simulations at high frequencies
+and the analytic results at low frequencies. In the present study, we use the formula given in [48] for angular spread
+of the Cherenkov radiation as follows,
+Iθ = sin θ
+sin θc
+e− Z2
+2 ,
+Z = cos θ − cos θc
+∆c sin θc
+(2)
+The spreading of the radiation intensity is determined by ∆c = 0.0754[L(E0)/(νL(Es)] (in radians), which is
+inversely proportional to the shower length and the radiation frequency. Here L(E is the shower length determined
+by the energy, L(E) = 12.7 + 2/3 log(E/E0) [48].
+In the detection analysis of lunar UHE particle events, a point-source is treated for the coherent Cherenkov
+radiation. When the Cherenkov radio emissions escape from the lunar surface, it will be reflected and transmitted
+at the interface. According the Fresnel law, the transmission coefficient for parallel polarization can be expressed
+as
+ˆt∥(θs) =
+2 cosr
+n cos r + cos i
+(3)
+where θs is the polar angle which is uniquely related to the angle r and i. r is the angle of refraction relative to the
+normal (outside the Moon) as the rays pass through the lunar surface into free space, i is the angle of incidence as
+shown in Figure 1. r and i have the relation as sin(i) = sin(r)/n, n is the refraction index of lunar regolith, and
+chosen to be 1.73. The detected radiation from a lunar regolith shower, with energy E, at a frequency ν and an
+angle θ, can be expressed as ε = ε0(E, R, ν) · ˆt∥(θs) · I(θ). Here the absorption of the radio waves before exiting
+the lunar surface has not been discussed, and it will be taken into account in the aperture calculations.
+2.2
+Detection with lunar ULW radio antenna
+The radio emission from the hadronic cascades induced by the UHE cosmic rays or neutrinos in the lunar regolith
+covers a broad frequency band from MHz to GHz, and it peaks in the GHz regime [39]. Therefore, most of the past
+lunar radio experiments to UHECR or UHECν operate at GHz frequency band ([32], [26], [38], [36]). With the
+decreasing frequency the peak intensity of the radiation decreases, however, the increasing angular spread allows
+the radio emission detected by the lunar radio telescope to escape from the lunar surface within a broader solid
+angle, which makes it increasingly efficient to detect the Cherenkov radiation at low frequencies. Furthermore, the
+surface roughness does not play an important role in the detection analysis at lower frequencies since a consider-
+able fraction of the radiation will penetrate the surface. According to these factors, some experiments running at
+low frequencies(∼ 150 MHz) for the lunar particle detection have been carried out or planned with the Westerbork
+Synthesis Radio Telescope (WSRT), NuMoon [13], the LOw Frequency ARray (LOFAR), and the Square Kilo-
+meter Array (SKA-low). Although the ULW band ≤ 30MHz is beyond the optimum frequency window where
+4
+
+the wavelength is comparable with the shower length [48], and the bandwidth is also limited at ULW band, the
+increased particle energy could well compensate the loss of electric field strength at lower frequency. The net effect
+is that the detection probability could increase reasonably at adequately high shower energies.
+In the present study, we discuss the UHECR and UHECν detection with lunar orbital ULW radio telescope.
+For the space ULW radio observations, a tripole antenna has been selected as the receiving element in most of the
+radio missions as DSL ([12], [18]), NCLE [11], LFRS [41], and OLFAR [15], since it has specific advantages of
+measuring the three-dimension electric field, estimating the directions of arrival (DOA) of incident signals with
+single antenna ([17], [16]), and protecting the desired signals from interference signals [19]. In the lunar ULW
+mission, the Dark Ages is undoubtedly one of the greatly interesting sciences, it requires the observation of the
+entire 1 − 30 MHz band, which could be extend up to 50 MHz for cross-referencing with the terrestrial radio
+facilities such as LOFAR. Therefore, the tripole antenna of the lunar ULW radio telescope is supposed to operate
+at the frequency range up to 40 or 50 MHz ([41], [12]).
+For a radio telescope, its sensitivity can be determined by the system equivalent flux density (SEFD) F =
+kTsys/Aeff in Jansky (10−26 Wm−2Hz−1), . Here k is the Boltzmann’s constant, Tsys the system temperature
+and Aeff the effective collecting area. However, a electric field strength (V/m/Hz) of a coherent radio pulse is
+usually utilized to characterize the Cherenkov radiation. In order to analyze the sensitivity of a radio telescope for
+the Askaryan pulse detection, it is required to express the telescope sensitivity in terms of the electric field strength.
+If a radio telescope has a flat system noise spectrum over the observed bandwidth, the equivalent root mean square
+(RMS) spectral electric field [9] can be written as follows,
+εrms = Erms
+∆ν ,
+Erms =
+�
+kTsysZ0∆ν
+Aeff
+(4)
+where Erms is the RMS electric field strength over the bandwidth ∆ν in one polarization, and Z0 is the impedance
+of free space. Then the sensitivity of an antenna to detect a coherent radio pulse (the minimum detectable electric
+field) can be defined as εmin = Nσεrms, where Nσ is the minimum number of standard deviations needed to reject
+statistical noise pulses.
+At the ULW band, the galactic synchrotron emission dominates the sky radio foreground, its spectrum varies
+greatly with the frequencies, the brightness temperature rises from about 104 K at 30 MHz to about 2.6 × 107 K at
+around 1 MHz; Furthermore, the antenna effective collecting area changes significantly with the frequencies too.
+Therefore, the Equation (4) cannot be applied directly to the electric field calculation for a ULW radio antenna.
+In order to truly characterize the equivalent spectral electric field of the telescope, the RMS electric field strength
+Erms should be re-computed by integrating the SEFD over the frequency. According to the studies in Ref. [16],
+the sky noise temperature is absolutely dominant in the total system temperature for the ULW radio antennas of
+different lengths, then Erms can be re-written as,
+Erms =
+�
+kZ0
+� νH
+νL
+Tsky(ν)
+Aeff(ν) dν
+(5)
+Where νL and νH are the lower and upper limit of the frequency respectively. Using a sky temperature model
+[40], the sensitivity of a ULW radio antenna longer than 5 meters [16] to detect the Askaryan radio pulse has been
+calculated with a nominal value Nσ = 3 for different center frequencies and bandwidths, as shown in Table 1. Note
+5
+
+Table 1: Sensitivity parameters of radio UHECR and UHECν observations, Frequencies and bandwidths are in
+MHz.
+Simulations
+νmin
+νmax
+∆ν
+ν0
+εmin(uV/m/MHz)
+Case I
+10
+30
+20
+20
+2.06
+Case II
+15
+25
+10
+20
+2.84
+Case III
+10
+40
+30
+25
+1.61
+Case IV
+15
+35
+20
+25
+1.93
+Case V
+20
+30
+10
+25
+2.67
+Case VI
+10
+50
+40
+30
+1.34
+Case VII
+15
+45
+30
+30
+1.53
+Case VIII
+20
+40
+20
+30
+1.83
+Case IX
+25
+35
+10
+30
+2.55
+that the antenna length has less influence on the sensitivity since the sky noise temperature is absolutely dominant.
+In order to improve the detection probability of lunar UHECR and UHECν events, we use the central frequency
+of 30 MHz and the bandwidth of 40 MHz to obtain the best sensitivity in our analysis.
+3
+Detection analysis for lunar UHECR and UHECν events
+We used the method presented in Ref. [31] to analyze the detection of lunar UHECR and UHECν events. In order
+to decide the cascade event rates for the UHECR and UHECν, one way is in term of the aperture size, which is
+multiplied by the isotropic cosmic ray or neutrino flux in the given energy bin to obtain the detection rate. We
+calculate the analytic aperture by integrating the angular aperture for specified energy E over the angles (θs, ϕs) at
+the lunar surface S, and over the angles (θn, ϕn) of the cascade axis relative to the normal to this surface, as shown
+in Figure 1.
+For a volume element near the lunar surface (z ≃ 0) with the polar angle θs, (see Figure 1) radio emissions
+from it received by the detector has a given refraction angle r, sin(r) = sin(θs)(RM + h)/R(θs). Here R(θs) =
+[R2
+M + (RM + h)2 − 2RM(RM + h) cos θs]1/2, is the distance between the surface element and the detector.
+The maximum value of the polar angle θmax
+s
+= arccos[RM/(RM + h)] is associated with the refraction angle
+r −→ π/2.
+At low frequencies, the lunar surface roughness will not have a major influence on the detection sensitivity, and
+thus can be neglected [48], which could be another advantage of UHE particle detection with a lunar ULW radio
+antenna. In our analysis, the Moon is assumed to be an approximately smooth sphere.
+3.1
+UHE cosmic rays
+For the UHE cosmic rays, they cannot penetrate through the lunar regolith, and cosmic ray events occur in the lunar
+regolith near the surface. Therefore, only the upper hemisphere (cosθn > 0) contributes to the aperture calculation
+for the cosmic ray cascades. The total aperture ACR(E) is defined by Equation (6) and (7) as the integral of the
+angular aperture ∆ΩCR(E, θs) over the surface S [31].
+ACR(E) = A0
+�
+∆ΩCR(E, θs)d cos θs
+(6)
+6
+
+∆ΩCR =
+�
+cos θnΘ[ε(E, θs, θ) − εrms] × Θ(cos θn)dϕnd cos θn
+(7)
+where, A0 = 2πR2
+M is the physical area of lunar hemisphere, the unit step function Θ(ε − εrms) only selects the
+cascade events that produce radio signals above the threshold value εmin. The radio attenuation length of lunar
+soil depends on the fraction of its composition. For the lunar regolith we use the value ℓ = 60λ (λ, the wavelength
+in the vacuum) given in the literatures ([39], [9]), it is a quite large values at low frequencies. Thus the attenuation
+of radio wave prior to their escape from the lunar surface can be ignored since the cosmic ray cascades exist only
+few meters deep.
+Figure 2: Askaryan radio emissions initiated by UHE cosmic rays propagate between the lunar soil and lunar
+orbital radio antenna. The lunar soil is modelled with two layers, the regolith and the sub-regolith.
+The lunar subsurface soil consists of several different strata according to the detection of Lunar Penetrating
+Radar. Therefore, the reflected radio waves by the strata interfaces that can be detected by the lunar orbital radio
+antenna should also be taken into account besides the direct radio waves emitted by the hadronic shower, as shown
+in Figure 2. Due to the very small contribution, the secondary reflections at the interfaces are neglected. In order to
+simplify the analysis, the lunar soil structure is modelled as only two layers, lunar regolith and lunar sub-regolith.
+The regolith is assumed to be 12 meter deep [45], and the refractive index of lunar regolith (n = 1.73) and sub-
+regolith (n = 2.5) ([39], [9]) have been used in the aperture calculation.
+Figure 3: Total aperture of UHE cosmic rays for a lunar orbital ULW radio antenna at different altitudes of 50 km,
+100 km, 300 km, and 500 km with a sensitivity of 1.34 uV/m/MHz.
+Using analytic integrations of the angular aperture over the Moon surface, the total aperture are computed
+7
+
+107
+106
+...................
+10
+50 km
+103
+100 km
+300 km
+500 km
+10
+1018
+1020
+1021
+1022
+1022
+E. [eV]To the antennafor the lunar orbital ULW radio antenna with different altitudes. As shown in Figure 3, with the higher altitude
+of lunar ULW radio antenna, the physical aperture illuminated by the antenna increases, however, the detected
+Cherenkov radiation on the antenna becomes weaker due to the distance-decay effect. As a result, the minimum
+(threshold) cascade energy at which the radio emission intensity on the antenna is equal to the threshold intensity
+εmin increases when the altitude rises. For all the altitudes, the aperture reaches around the peak above the energy
+of ∼ 1022 eV.
+Based on the model-independent estimate [46], for the zero detected events, the upper limit for differential (in
+energy) flux J(E) of primary particles in the observation time T, can be obtained by the following relation,
+dJ(E)
+dE
+≤
+Sup
+EA(E)T
+(8)
+Here the Poisson factor Sup = 2.3 for a limit with 90% confidence level.
+Figure 4: UHE cosmic ray flux of a lunar orbital ULW radio antenna at different altitudes of 50 km, 100 km, 300
+km, and 500 km for one-year observation. The results are compared with the flux limits of both SKA Low for a
+1000-hour observation between 100 MHz and 350 MHz (reproduced from Bray et al., 2014) and LOFAR for a
+90-day observation between 125 MHz and 175 MHz (reproduced from Singh et al., 2009).
+For one-year observation with the lunar orbital ULW radio antenna, the flux limits are calculated for different
+antenna altitudes. As shown in Figure 4, only the cosmic ray with the energy above a threshold can be detected by
+the lunar orbital antenna, it is 4 × 1019 eV for the antenna at the 50 km high orbit, and 4 × 1020 eV for the 500 km
+high orbit. The results in Figure 4 also show that the flux limit of single lunar orbital antenna will be competitive
+with the estimated limits of large terrestrial radio telescopes such as LOFAR [51] and SKA-Low [10]. Note that
+the flux limit of SKA-Low is calculated according to the SKA array for Phase I.
+For an isotropic cosmic ray or neutrino flux J(E), during the observation time T the number of detected events
+in the given energy interval between E1 and E2 can be obtained by N =
+� E2
+E1 T J(E)A(E)dE, where A(E) is the
+total aperture as a function of energy.
+Using the flux parametrization of the Auger Collaboration in Ref. [1], the expected event rates of UHE cosmic
+rays are calculated in the energy range between 1018 and 1021, as shown in Figure 5, for the lunar ULW radio
+antenna with different altitudes. It can be seen that the total detected cosmic ray events decreases for the antenna
+8
+
+108
+50 km
+100 km
+109
+500 km
+LOFAR Core
+SKA LOw
+10
+23.
+Flux
+11
+10°12
+101g
+10
+1021
+10
+1023
+E. [eV]Figure 5: The number of detected (per year) cosmic ray events for the energy E ≥ 1018 eV with the flux
+parametrization of the Auger Collaboration [1]. Top, the total number detected events as a function of lunar
+antenna orbital altitude; Bottom, the detected events per energy interval for the lunar antenna of 50 km-high orbit
+(left) and 200 km-high orbit (right).
+with higher altitudes. The detected event number reaches to about 400 for the 10 km-high orbit, and it drops rapidly
+from tens at the altitude of 200 km to zeros at the attitude above 500 km. The simulation results of detected cosmic
+ray events per energy interval show that the events mostly lie in the higher energy range with the increasing of
+orbital altitude.
+3.2
+UHE neutrinos
+The cascades initiated by the UHE neutrinos are different from that produced by the cosmic rays. Due to the long
+attenuation length for neutrinos in lunar regolith, most of the UHE neutrinos induce cascades very deep inside the
+Moon. Therefore, the attenuation of radio wave propagation prior to their escape from the lunar surface can not be
+neglected, and the contribution of both the upper and lower lunar hemisphere have to be taken into account while
+calculating the total aperture. Considering the neutrino absorption on the path L(z, θn), we integrate the angular
+aperture over z and obtain the contributions of the upper and lower hemisphere respectively [31] as follows,
+Aν(E) = A0
+�
+∆Ων(E, θs)d cos θs
+(9)
+∆Ω+
+ν =
+�
+2ℓ cos(i)
+LνN(Eν) ln(
+ε
+εmin
+) × Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+(10)
+∆Ω−
+ν =
+�
+2ℓ cos(i)
+LνN(Eν) ln(
+ε
+εmin
+) × exp[−2RM| cos θn|
+LνN(Eν)
+] × Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+(11)
+here, it is assumed that only 20% of the energy of the original neutrino goes into the hadronic particle cascade. ℓ is
+the radio wave attenuation length. Since the showers produced by the UHE neutrinos occur in the lunar soil over a
+9
+
+10
+Number of the detected events per year
+10
+10
+10
+10
+10°
+101
+102
+103
+Altitude of lunar UL radio antenna, km
+Events per energv interval for 50 km orbital altitude
+Events per energy interval for 200 km orbital altitude
+Number of detected events per year
+40
+detected events peryear
+20
+35
+30
+15
+25
+20
+10
+15
+10
+of
+5
+L
+0
+18
+0
+18.5
+19
+19.5
+20
+20.5
+21
+18
+18.5
+19
+19.5
+20
+20.5
+21
+E. [log(eV]
+E, [log(eV)]large range of depths from the surface, we use the value ℓ = 29λ of lunar sub-regolith ([39], [9]) for the aperture
+calculation. The neutrino-nucleon interaction length was LνN(km) = 122km(E0/Eν)1/3, where E0 = 1020eV
+([23], [9]). For the upper hemisphere cos θn > 0, we assume l/LνN ≪ 1 [31], which is valid for most of cases.
+Figure 6: Direct and reflected radio emissions initiated by the UHE neutrinos propagate between the lunar soil and
+lunar orbital radio antenna. The lunar soil is modelled with two layers, the regolith and the sub-regolith.
+For the UHE neutrinos, the multi-strata structure of the lunar subsurface soil has also been taken into account in
+the aperture calculation. Since the cascade showers generated by the UHE neutrinos mostly occur deep inside the
+lunar soil, that is in the lunar sub-regolith, we consider the refractions at both the lunar surface and the subsurface
+strata interface, as shown in Figure 6. Here, the reflected signals of the Askaryan radio emissions at the lunar
+surface is ignored to reduce the complexity of aperture analysis.
+Figure 7: Total aperture of the UHE neutrinos for a lunar orbital ULW radio antenna at different altitudes of 50
+km, 100 km, 300 km, and 500 km with sensitivity of 1.34 uV/m/MHz.
+Similar to the cosmic rays, the total aperture of the UHE neutrinos are calculated using analytic methods for
+the lunar orbital ULW radio antenna at different altitudes. As shown in Figure 7, the aperture increases with the
+rising of antenna attitude. However, the increase becomes slower above the attitude of 300 km. Being different
+from the cosmic rays, the aperture of the UHE neutrinos increase over the entire energy range from 1020 eV to
+1024 eV without reaching peaks for all the four altitudes. For 50 km distance, the energy of detectable events begin
+at 2 × 1020 eV. With the increasing of orbital attitudes, only higher energetic neutrino events become detectable,
+for 500 km attitude the threshold energy of detectable events increases to ∼ 2 × 1021 eV.
+10
+
+107
+JS
+103
+50 km
+100 km
+101
+300 km
+500 km
+101
+1020
+1021
+1022
+1032
+1024
+E. [ev]To the antennaFigure 8: The UHE neutrino flux of a lunar orbital ULW radio antenna at different altitudes of 50 km, 100 km,
+300 km, and 500 km for one-year observation. The results are compared with the flux limits of the SKA Low for
+a 1000-hour observation between 100 and 350 MHz (reproduced from Bray et al., 2014), LOFAR for a 90-day
+observation between 125 MHz and 175 MHz (reproduced from Singh et al., 2009). The blue chain-dotted lines
+show the predicted fluxes from the GZK [21], and topological defects (TD) [49]. The green dashed lines show the
+flux limits set by the past experiments of ANITA-II [24], IceCube [3] and Auger [2].
+Figure 8 shows the flux limits for one-year observation with the lunar orbital antennas of different altitudes.
+The flux limits of the SKA-Low for a 1000-hour observation, the LOFAR for a 90-day observation, the past
+experiments of ANITA-II [24], IceCube [3] and Auger [2] are also plotted for comparison, as well as the predicted
+fluxes from the models of the GZK [21] and topological defects (TD) [49]. The results indicate that the lunar
+orbital observations in one year will set the comparable flux limits with the estimated fluxes of the SKA-Low and
+the LOFAR core (even lower than the energy of ∼ 3 × 1022 eV), which signifies the higher detection efficiency of
+the lunar ULW radio antenna for the UHE neutrinos.
+Figure 9: The total number of detected (per year) neutrino events with the flux limits set by the ANITA-II experi-
+ment, and the predicted fluxes for the GZK neutrinos [21] and topological defects (TD) [21], as a function of the
+lunar antenna altitude in the energy E ≥ 1018 eV.
+Based on the predicted fluxes from the models of the GZK neutrinos, topological defects (TD), the neutrino
+11
+
+104
+10°
+GZK
+10g
+TD
+ANITA-II
+10
+10"4
+100
+10
+102
+10°
+Altitude of lunar ULW radio antenna, km10
+50 km
+100 km
+ANITA-II
+300 km
+2010 :
+500 km
+10*5
+. :
+LOFAR Core
+ceoube
+SKALOW
+20:13
+10°
+Auger
+xn
+109
+.: . .
+GZK
+TD
+10-11
+:i.
+1018
+1019
+1020
+1031
+10-2
+1023
+1024
+E. [eV]flux limit set by the experiment of ANITA-II, the expected event rates per year are estimated and presented in
+Figure 9 as a function of attitudes for the lunar orbital ULW radio antenna. It is evident from Figure 9 that the
+GZK neutrinos almost cannot be detected for all the altitudes. In order to increase the detection sensitivity of the
+GZK neutrinos, the aperture must be improved by increasing the effective receiving area of the radio telescope,
+for instance, employing a radio array. For the TD neutrino detection, the event rate depends on the altitude, and
+reaches the peak of about 70 per year at the attitude of about 100 km, which could be the optimum attitude for
+a lunar orbital ULW antenna. With the neutrino flux limit of ANITA-II, the expected neutrino events increase
+exponentially at the altitude range from 1 km to 1000 km, and thousands of events per year could be detected at
+most by the lunar ULW antenna.
+4
+Characteristic analysis of UHE cosmic rays and neutrinos
+For the detection of the UHE cosmic rays and neutrinos, the ultimate goal is to reconstruct the events to obtain all
+the parameters characterizing the cosmic rays or neutrinos, including the source, the primary particle energy, etc.
+In the literatures ([29], [30]), a random search method was employed to determine the UHECR event parameters
+for the LORD experiment under the assumption that the cosmic ray flux is known. Due to the lack of direct infor-
+mation, the reconstructed parameters have large uncertainties even for the detection with two satellites. Compared
+with the LORD experiment, the future lunar ULW radio missions like DSL ([12], [18]), OLFAR [22] have more
+advantages in the UHE particle detection. They consist of many antennas onboard micro-satellites to form a large
+radio array, which make it feasible to detect the UHECR and UHECν with multi satellites simultaneously. Fur-
+thermore, in lunar ULW radio observations, the tripole antenna can measure the three-dimension electric fields,
+which make it possible to measure the three-dimension polarization of the Askaryan radio emission.
+Figure 10: Scheme of Askaryan pulse detections in the lunar soil with three antennas onboard micro-satellites.
+In the detection of the UHECR and UHECν, if the Askaryan radio pulses can be observed by multi antennas on-
+12
+
+Rs1
+V or CR
+0board lunar satellites, the time delays can be measured between the radio pulses sensed by different antennas, and
+the particle cascade location (P) on the lunar surface can be solely determined with the time delays among at least
+three non-linearly distributed antennas, as shown in Figure 10. We know the Askaryan radio emission is highly
+linearly polarized, which is always in the same plane of the Poynting vector and the shower axis ([37], [34], [25]).
+Therefore, using the three-dimension polarizations of the electric fields (E1, E2, E3) measured by the tripole an-
+tennas onboard three orbital satellites, the show axis can be determined by the intersection line of the three planes
+of E1 −Rs1, E2−Rs2, E3−Rs3 shown in Figure 10, which means the original direction of the UHECR or UHECν
+is known now. For a linear ULW radio array like DSL, it is necessary to note that the cascade location deduced by
+the time delays could have two candidates symmetric with respect to the satellite orbit plane. However, since only
+the right location can account with the fact that the shower axis, the Poynting vectors and the polarizations of the
+Askaryan radio emissions are in the same plane, the measured polarizations of the coherent radio pulses on three
+tripole antennas will definitely exclude one of the candidate locations except for the case that the shower axis is
+parallel to the satellite orbit plane.
+Once the shower axis and the location of the UHE particle cascade are confirmed, the distance Rs between the
+particle cascade and the radio detector, the three angles θn, r, i shown in Figure 1 will also be known according
+to the simple geometrical relationship. Using the formulae (1),(2) and (3), the particle energy E can be calculated
+with the intensities of electric fields induced on the lunar orbital antennas. The results of the three satellites will be
+cross-checked with each other to improve the accuracy and reliability.
+5
+Summary and conclusions
+In the present study, using the lunar Askaryan technique, the detection of the UHE cosmic rays and neutrinos
+has been analytically analyzed for the future lunar ULW radio missions. Our results indicate that the single lunar
+ULW radio antenna could detect the cosmic ray and neutrino flux as low as that observed by the present or future
+experiments in the energy range E > 1020eV. With the known flux of the UHE cosmic rays, the simulated
+detectable events per year has found to increase when the antenna altitude is lowered, and it reaches the peak at
+the altitude of ∼ 10 km. For the UHE neutrinos, during the observations in one year, it is shown that the single
+lunar ULW radio antenna could detect ∼ 70 topological defect neutrinos at an optimal altitude of 100 km, while
+the GZK neutrinos could not be detected.
+Investigating the properties of the lunar Cherenkov radiation, it has been demonstrated that the UHECR or
+UHECν events could be reconstructed directly using the radio observations with at least three tripole antennas
+onboard the lunar satellites that makes it feasible to detect the UHECR and UHECν with the lunar ULW radio
+mission. The method could also be used to build a dedicated lunar radio telescope for the detections of the UHE
+cosmic ray and the UHE neutrinos.
+Acknowledgements
+This work is funded by the National Natural Science Foundation of China (NSFC) under No.11941003,11790305,
+11573043, the CE-4 mission of the Chinese Lunar Exploration Program: the Netherlands-China Low Frequency
+13
+
+Explorer (NCLE), and the Chinese Academy of Science Strategic Priority Research Program XDA15020200. The
+authors would like to thank Dr. C.W. James from the University of Adelaide, Australia, and Prof. Dr. Xuelei
+Chen from National Astronomical Observatories, Chinese Academy of Sciences for all the useful discussions. In
+addition the authors thank the referees for their useful comments and suggestions.
+Appendix
+A. Aperture calculation of UHE cosmic rays and neutrinos
+Based on the analytic methods used in [31], the modifications have been made to the aperture analysis for both
+cosmic rays and neutrinos in this paper. The differences include the calculations of electric field intensity and
+angular spread, the attenuation length and neutrino-nucleon interaction length of lunar regolith, the model of
+antenna sensitivity, etc., which have been addressed in this paper.
+For the detection of UHE cosmic rays, the total aperture is calculated by integrating the angular aperture over
+the upper hemisphere surface, where
+∆ΩCR =
+� 1
+0
+cos θnd cos θn
+� π
+−π
+Θ[ε(E, θs, θ) − εrms]dϕn
+(A.1)
+In the angular coordinate (θ, ϕ) with a polar axis aligned with the radiation direction (see Figure 1), it can be
+derived as,
+cos θn = sin θ sin i cosϕ − cos θ cos i
+(A.2)
+dΩ = −d cosθndϕn = −d cosθdϕ
+(A.3)
+Then the angular aperture can be re-written as,
+∆ΩCR = 2
+� cos θmin
+cos θmax
+d cos θ ×
+� ϕm
+0
+(sin θ sin i cos ϕ − cos θ cos i)dϕ
+(A.4)
+here cos ϕm = (cos θ cos i)/(sin θ sin i). θmin and θmax are the minimum and maximum angels of incidence
+within which the detected radiation intensity ε is stronger than εmin. The angular apertures calculated as functions
+of cos θs are shown in Figure 11 .
+For the detection of UHE neutrinos, the neutrino absorption on the path up to the shower production point has
+to be taken into account in the calculation of angular aperture, as well as the absorption of radio wave in the lunar
+regolith.
+∆Ων =
+�
+dz
+LνN
+�
+Θ{ε(E, θs, θ) × exp[−z/(2ℓ cosi)] − εrms} × exp[−L(z, θn)/LνN(Eν)]
+(A.5)
+here, L(z, θn) ≈ z/ cosθn for the lunar upper hemisphere cos θn > 0, and L(z, θn) ≈ 2RM | cos θn | for the lunar
+lower hemisphere cos θn < 0. By integrating over the cascade depth z, we obtain, respectively, the contributions
+of the upper and lower hemispheres to the neutrino angular aperture.
+∆Ω+
+ν ≃
+� zmax
+0
+1
+LνN
+exp(−
+z
+LνN cos θn
+)dz ×
+�
+Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+≃
+�
+cos θn{1 − exp[
+2ℓ cosi
+LνN cos θn
+ln(
+ε
+εmin
+)]} × Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+(A.6)
+14
+
+Figure 11: Angular aperture ∆ΩCR for the cosmic ray detection with a lunar orbital ULW radio antenna at different
+altitudes of 50km, 100km and 200km, and for the initial cosmic ray energy W = 1021eV .
+∆Ω−
+ν ≃
+� zmax
+0
+1
+LνN
+exp(−2RM cos θn
+LνN
+)dz ×
+�
+Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+≃
+� 2ℓ cosi
+LνN
+ln(
+ε
+εmin
+) exp(−2RM cos θn
+LνN
+) × Θ[ε(E, θs, θ) − εmin]dϕnd cos θn
+(A.7)
+where, zmax = 2ℓ cosi ln(ε/εmin). For the ultra-high energy neutrino, when ℓ/LνN ≪ 1, formula A.6 will be
+simplified to the formula (10). The angular aperture is then calculated numerically.
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+

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+DP-SIPS: A simpler, more scalable mechanism for differentially
+private partition selection
+Marika Swanberg∗
+Boston University
+Department of Computer Science
+marikas@bu.edu
+Damien Desfontaines
+Tumult Labs
+damien@desfontain.es
+Samuel Haney
+Tumult Labs
+sam.haney@tmlt.io
+ABSTRACT
+Partition selection, or set union, is an important primitive in differ-
+entially private mechanism design: in a database where each user
+contributes a list of items, the goal is to publish as many of these
+items as possible under differential privacy.
+In this work, we present a novel mechanism for differentially
+private partition selection. This mechanism, which we call DP-SIPS,
+is very simple: it consists of iterating the naive algorithm over
+the data set multiple times, removing the released partitions from
+the data set while increasing the privacy budget at each step. This
+approach preserves the scalability benefits of the naive mechanism,
+yet its utility compares favorably to more complex approaches
+developed in prior work.
+Along the way, this work also gives an alternate definition of
+approximate zero-concentrated DP, and reports some empirical
+observations on the utility of other partition selection mechanisms.
+KEYWORDS
+Differential privacy
+1
+INTRODUCTION
+Given a set of users, where each is associated to certain items, how
+can we publish as many of these items as possible in a differentially
+private manner? This problem, simple in its formulation, is an im-
+portant primitive for building differentially private systems: it is
+central to privately releasing the result of group-by aggregations
+where the group-by keys are a priori unknown, or impractical to
+enumerate. This is useful in applications like vocabulary extrac-
+tion [10], the private release of search queries [12], or in building
+general-purpose differentially private tooling [1, 6, 18]. In these
+last two use cases, scalability is a primary concern: the input of
+this operation might be too large to fit in the memory of a single
+machine, and the computation must be parallelized across multiple
+machines to avoid unacceptably long running times.
+Differentially private partition selection mechanisms have been
+proposed as early as 2009 [12], but recent work has shed a new light
+on this problem, and proposed alternative approaches that bring
+significant utility gains [5, 10]. To obtain these utility improvements,
+these newer mechanisms use a greedy approach: each user considers
+what items have been contributed by previous users so far, and
+“chooses” which items to contribute according to a policy, chosen
+carefully to maintain a sensitivity bound. Doing so, however, limits
+the scalability of the mechanism obtained in this way: each user
+chooses their contribution based on the contribution of all previous
+∗Work done while at Tumult Labs
+users, so the data has to be processed one user after another, and
+the overall algorithm cannot be parallelized.
+This raises a natural question: can we achieve the utility benefits
+of policy-based approaches, while preserving the scalability of more
+naive approaches? In this work, we show that both benefits can be
+combined. We introduce a new approach, which we call DP-SIPS,
+short for scalable, iterative partition selection. DP-SIPS relies a simple
+idea: rather than having to process the data of one user at a time,
+we run the naive, massively-parallelizable algorithm multiple times,
+splitting the privacy budget between each step. Surprisingly, this
+mechanism turns out to have a similar utility to greedy approaches,
+but scales horizontally: increasing the number of cluster nodes
+significantly reduces runtime.
+The rest of this paper is organized as follows:
+• In Section 2, we formally define the problem and the building
+blocks we use for DP-SIPS and its privacy accounting.
+• In Section 3, we give an alternate definition of approxi-
+mate zCDP and prove that it satisfies composition, post-
+processing, and other useful properties.
+• In Section 4, we detail existing approaches to differentially
+private partition selection.
+• In Section 5, we introduce our algorithm and the proof of its
+privacy guarantees.
+• In Section 6, we report on the experimental evaluation of
+DP-SIPS.
+• Finally, in Section 7, we discuss our results, report on some
+unsuccessful approaches that we tried, and outline directions
+for future work.
+2
+PRELIMINARIES
+A data set 𝑥 = (𝑊1, . . . ,𝑊𝑁 ) contains a set of user lists 𝑊𝑖 ∈ U∗.
+We refer to elements in U as items or partitions, and define partition
+selection (also called key selection or set union) as follows:
+Definition 1 (Private Partition Selection Problem). Given
+a (possibly unbounded) universe U of items and a data set 𝑥 =
+(𝑊1, . . . ,𝑊𝑁 ) of user lists 𝑊𝑖 ∈ U∗, an algorithm M solves the
+private partition selection problem if it is differentially private, and
+M outputs a set 𝑆 ⊆ ∪𝑖𝑊𝑖.
+We begin by presenting the standard notion of differential pri-
+vacy. Two data sets 𝑥,𝑥 ′ are neighbors if they differ on one user’s list:
+𝑥 = 𝑥 ′ ∪𝑊𝑖∗. Informally, differential privacy requires that an algo-
+rithm’s output is distributed similarly on every pair of neighboring
+data sets.
+Definition 2 (Differential Privacy [7, 8]). A randomized
+algorithm M : U∗ → Y is (𝜀,𝛿)-differentially private if for every
+arXiv:2301.01998v1  [cs.CR]  5 Jan 2023
+
+Marika Swanberg, Damien Desfontaines, and Samuel Haney
+pair of neighboring datasets 𝑥,𝑥 ′ ∈ U∗ and for all subsets 𝑌 ⊆ Y,
+Pr[M(𝑥) ∈ 𝑌] ≤ 𝑒𝜀 · Pr[M(𝑥 ′) ∈ 𝑌] + 𝛿.
+A common variant of differential privacy, called zCDP, is use-
+ful for analyzing algorithms that sample noise from a Gaussian
+distribution (as ours will). The definition of zCDP uses the Rényi
+Divergence:
+Definition 3 (Rényi Divergence). Fix two probability distribu-
+tions 𝑃 and 𝑄 over a discrete domain 𝑆. Given a positive 𝛼 ≠ 1, Rényi
+divergence of order 𝛼 of distributions 𝑃 and 𝑄 is
+𝐷𝛼 (𝑃||𝑄) =
+1
+1 − 𝛼 log
+�∑︁
+𝑥 ∈𝑆
+𝑃(𝑥)𝛼𝑄(𝑥)1−𝛼
+�
+.
+Definition 4 (𝜌-zCDP [3]). A randomized mechanism M : X∗ →
+Y satisfies 𝜌-zCDP if, for all 𝑥,𝑥 ′ ∈ X∗ differing on a single entry,
+𝐷𝛼 (𝑀(𝑥)||𝑀(𝑥 ′)) ≤ 𝜌 · 𝛼
+∀𝛼 ∈ (1, ∞).
+(1)
+We will relax this definition to a definition approximate zCDP
+in the following section. A common primitive in building private
+algorithms, the Gaussian Mechanism, satisfies 𝜌-zCDP.
+Definition 5 (Gaussian Distribution). The Gaussian distribu-
+tion with parameter 𝜎 and mean 0, denoted N (0, 𝜎2) is defined for
+all ℓ ∈ R and has probability density
+ℎ(ℓ) =
+1
+𝜎
+√
+2𝜋
+𝑒− ℓ2
+2𝜎2 .
+Definition 6 (ℓ2-Sensitivity). Let 𝑓 : U𝑛 → R𝑑 be a function.
+Its ℓ2-sensitivity is
+Δ𝑓 =
+max
+𝑥,𝑥′∈U
+𝑥,𝑥′𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠
+∥𝑓 (𝑥) − 𝑓 (𝑥 ′)∥2.
+Definition 7 (Gaussian Mechanism). Let 𝑓 : U𝑛 → R𝑑 be a
+function with ℓ2-sensitivity Δ𝑓 . Then the Gaussian mechanism is the
+algorithm
+M𝑓 (𝑥) = 𝑓 (𝑥) + (𝑍1, . . . ,𝑍𝑑),
+where 𝑍𝑖 ∼ N
+�
+0,
+Δ2
+𝑓
+2𝜌 · I
+�
+. Algorithm M𝑓 satisfies 𝜌-zCDP.
+3
+ALTERNATE DEFINITION OF
+APPROXIMATE ZCDP
+In this section, we give an alternate definition of the definition
+of approximate zCDP given in [3]. While Bun et. al. write that
+they intended for their definition to be a generalization of (𝜖,𝛿)-
+approximate differential privacy, the definition seems more similar
+to what is often called probabilistic differential privacy [13], although
+this may not be the interpretation the authors intended 1. As shown
+in [13], (an interpretation of) this definition is not closed under
+post processing, as the authors of [3] claim. To resolve this issue,
+we give a new definition of approximate zCDP.
+1From an exchange with the authors of [3], we believe the intended definition of
+approximate zCDP is more similar to the definition that we propose in this paper,
+depending on the interpretation of an “event” in their definition. However, the inter-
+pretation used in [13] that leads to a breakdown in closure under postprocesssing is
+likely to be a common one. For that reason, we give an alternate definition in this
+paper.
+Definition 8 (Approximate zCDP). A randomized mechanism
+𝑀 : X∗ → Y is 𝛿-approximately 𝜌-zCDP if, for all 𝑥,𝑥 ′ ∈ X∗
+differing on a single entry, letting 𝑃 and 𝑄 denote the distributions of
+𝑀(𝑥) and 𝑀(𝑥 ′) respectively, there exists a distribution 𝑃 ′ over the
+output domain of 𝑀 such that
+𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿, and
+𝐷𝛼 (𝑃 ′||𝑄) ≤ 𝜌 · 𝛼,
+for all 𝛼 ∈ (1, ∞). Here, 𝑇𝑉𝐷 is the total variation distance, which is
+defined formally as
+𝑇𝑉𝐷(𝑃,𝑄) = sup
+𝑆
+|𝑃(𝑆) − 𝑄(𝑆)|.
+Intuitively, this definition says that the mechanism should be
+"close" to one that satisfies pure zCDP, in terms of the distribution
+on each output.
+Note that we can define approximate differential privacy in sim-
+ilar way. That is, [9] shows the follow equivalence:
+Lemma 9 ([9] (Lemma 3.17)). A randomized mechanism 𝑀 :
+X∗ → Y is (𝜖,𝛿)-approximately DP if and only if, for all 𝑥,𝑥 ′ ∈ X∗
+differing on a single entry, letting 𝑃 and 𝑄 denote the distributions of
+𝑀(𝑥) and 𝑀(𝑥 ′) respectively, there exists a distribution 𝑃 ′ over the
+output domain of 𝑀 such that
+𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿, and
+𝑃 ′(𝑆) ≤ exp(𝜖)𝑄(𝑆)
+for all sets 𝑆 of possible outputs of 𝑀.
+We can use Definition 8 in place of Definition 15 from [3] and
+most results from [3] hold as is, except that our conversion from ap-
+proximate zCDP to approximate DP is slightly worse (see Lemma 12),
+and our composition bound is slightly worse (see Lemma 10). In
+the rest of this section, we state some facts about approximate
+zCDP that we will use in the rest of the paper. Proofs appear in
+Appendix C.
+First, we give composition and postprocessing lemmas.
+Lemma 10 (Composition). Suppose M : X𝑁 → Y and M′ :
+X𝑁 → Z satisfy 𝛿-approximate 𝜌-zCDP and 𝛿′-approximate 𝜌′-
+zCDP, respectively. Then the composition M′′(𝑥) = (M(𝑥), M′(𝑥))
+satisfies (𝛿 + 𝛿′)-approximate (𝜌 + 𝜌′)-zCDP.
+Lemma 11 (Post-Processing). Suppose M : X𝑁 → Y satisfies
+𝛿-approximate 𝜌-zCDP and let 𝑓 : Y → Z be a function. Then,
+𝑓 ◦ M satisfies 𝛿-approximate 𝜌-zCDP.
+Next, we can convert approximate zCDP to approximate DP
+using the following lemma.
+Lemma 12. Suppose we can show that every mechanism that sat-
+isfies 𝜌-zCDP must satisfy 𝜖∗(𝜌),𝛿∗(𝜌)-approximate DP. That is,
+(𝜖∗,𝛿∗) is a function converting a (pure) zCDP guarantee to an approx-
+imate DP guarantee. Suppose M : X𝑁 → Y satisfies 𝛿-approximate
+𝜌-zCDP. Then, M satisfies (𝜖∗(𝜌),𝛿 + 𝛿∗(𝜌))-DP.
+Lemma 12 differs from the conversion lemma in [3] in two ways.
+First, we are only able to show a slightly worse bound, 𝛿 + 𝛿∗(𝜌)
+rather than 𝛿 + (1 − 𝛿)𝛿∗(𝜌). Second, we change the presentation
+of the lemma so we can use any conversion function from zCDP
+to approximate DP. Specifically, [4] give the following conversion,
+improving on the conversion given in [3]:
+
+DP-SIPS: A simpler, more scalable mechanism for differentially private partition selection
+Lemma 13 ([4], Proposition 7). Suppose M : X𝑁 → Y satisfies
+𝜌-zCDP. Then M satisfies (𝜖,𝛿)-approximate DP for any 𝜖 > 0 and
+𝛿 =
+inf
+𝛼 ∈(1,∞)
+exp((𝛼 − 1)(𝛼 · 𝜌 − 𝜖))
+𝛼 − 1
+�
+1 − 1
+𝛼
+�𝛼
+.
+(2)
+Combining Lemma 12 and Lemma 13 gives us the following.
+Corollary 14. Suppose M : X𝑁 → Y satisfies 𝛿-approximate
+𝜌-zCDP. Then M satisfies (𝜖,𝛿 + 𝛿′)-approximate DP for any 𝜖 > 0
+and
+𝛿′ =
+inf
+𝛼 ∈(1,∞)
+exp((𝛼 − 1)(𝛼 · 𝜌 − 𝜖))
+𝛼 − 1
+�
+1 − 1
+𝛼
+�𝛼
+.
+(3)
+Finally, we define a notion of probabilistic zCDP that is inspired
+by, but not equivalent to, Definition 8.1 in [3]. We show that proba-
+bilistic zCDP implies approximate DP (though the converse is not
+true). This will be useful, e.g., in the proof of Theorem 17 since the
+probabilistic definition can be more natural to use.
+Definition 15 (Probabilistic zCDP). A randomized mechanism
+𝑀 : X∗ → Y is 𝛿-probabilistically 𝜌-zCDP if, for all 𝑥,𝑥 ′ ∈ X∗
+differing on a single entry, there exists an event 𝐸 ⊆ Y such that, for
+all 𝛼 ∈ (1, ∞),
+𝐷𝛼 (𝑀(𝑥)|𝐸||𝑀(𝑥 ′)) ≤ 𝜌 · 𝛼.
+and Pr[𝑀(𝑥) ∈ 𝐸] ≥ 1 − 𝛿. Here 𝑀(𝑥)|𝐸 denotes the output distri-
+bution of 𝑀(𝑥) conditioned on 𝐸.
+We next claim that Definition 15 implies Definition 8 (an analagous
+fact is true about probabilistic DP and approximate DP).
+Lemma 16. Any mechanism satisfying 𝛿-probabilistic 𝜌-zCDP also
+satisfies 𝛿-approximate 𝜌-zCDP.
+4
+PRIOR APPROACHES
+In this section we discuss three existing algorithms for differentially
+private partition selection. We begin with the naive algorithm,
+called Weighted Gaussian, in Section 4.1. In Section 4.2, we present
+Policy Gaussian [10] and Greedy updates Without sampling [5].
+The algorithms all have three main steps: first, they compute a
+weighted histogram, which is simply a mapping from an item𝑢 ∈ U
+to a weight 𝐻 [𝑢] ∈ R; second, they add calibrated noise to each item
+in the histogram; and lastly, they release items that are above some
+appropriately-chosen threshold 𝑇. The primary difference between
+the algorithms is in how they compute the weighted histogram.
+This high-level algorithm for private partition selection is described
+in Algorithm 1, which can be composed with different weighted
+histogram algorithms.
+4.1
+Baseline: Weighted Gaussian
+The Weighted Gaussian algorithm (Algorithm 2) is one of the sim-
+plest algorithms for private partition selection. To build a weighted
+histogram, the algorithm first pre-processes the data set by remov-
+ing any duplicates within a user’s set and truncating each user’s set
+to have at most Δ0 items. Then, the users compute a histogram with
+bounded ℓ2-sensitivity as follows: each user 𝑖 updates the weight
+𝐻 [𝑢] for each of the items 𝑢 in their set 𝑊𝑖 with the following rule:
+𝐻 [𝑢] ← 𝐻 [𝑢] +
+1
+√︁
+|𝑊𝑖 |
+.
+Algorithm 1 High-Level Partition Selection Algorithm
+Input: Data set of user partitions 𝑥 = (𝑊1, . . . ,𝑊𝑁 )
+Weighted Histogram Algorithm Weighted_Hist
+Threshold 𝑇
+Noise distribution D
+Output: Partitions 𝑆 ⊆ ∪𝑖𝑊𝑖
+1: Initialize empty set 𝑆 ← {}
+2: 𝐻 ← Weighted_Hist(𝑥)
+⊲ Compute weighted histogram
+3: for 𝑢 ∈ 𝑆𝑢𝑝𝑝(𝐻) do
+4:
+𝑍𝑢 ∼ D
+5:
+ˆ𝐻 [𝑢] ← 𝐻 [𝑢] + 𝑍𝑢
+6:
+if ˆ𝐻 [𝑢] ≥ 𝑇 then
+7:
+𝑆 ← 𝑆 ∪ {𝑢}
+return 𝑆
+The resulting weighted histogram has an ℓ2-sensitivity of 1, so it
+can be composed with the high-level algorithm using calibrated
+Gaussian noise and an appropriate threshold to ensure that the over-
+all algorithm satisfies 𝛿-approximate 𝜌-zCDP. In the statement of
+Algorithm 2, Φ(·) is used to denote the cumulative density function
+of the standard Gaussian distribution and Φ−1(·) is its inverse.
+Algorithm 2 Weighted Gaussian
+Input: Data set 𝑥 = (𝑊1, . . . ,𝑊𝑁 )
+Privacy parameters (𝜌,𝛿)
+Maximum per-user contribution Δ0
+Output: Partitions 𝑆 ⊆ ∪𝑖𝑊𝑖
+1: Initialize empty histogram 𝐻 ← {}
+2: Initialize empty set 𝑆 ← {}
+3: for 𝑖 = 1, . . . , 𝑁 do
+4:
+𝑊𝑖 ← get rid of duplicate items from 𝑊𝑖
+5:
+𝑊𝑖 ← uniformly sample at most Δ0 items from 𝑊𝑖
+6:
+for 𝑢 ∈ 𝑊𝑖 do
+7:
+𝐻 [𝑢] ← 𝐻 [𝑢] +
+1
+√
+|𝑊𝑖 |
+8: 𝜎 ←
+1
+√2𝜌
+9: 𝑇 ← max𝑘 ∈[Δ0]
+�
+1
+√
+𝑘 + 𝜎 · Φ−1 �
+(1 − 𝛿)1/𝑘��
+10: for 𝑢 ∈ 𝑆𝑢𝑝𝑝(𝐻) do
+11:
+ˆ𝐻 [𝑢] ← 𝐻 [𝑢] + N (0, 1
+2𝜌 )
+12:
+if ˆ𝐻 [𝑢] ≥ 𝑇 then
+13:
+𝑆 ← 𝑆 ∪ {𝑢}
+return 𝑆
+Theorem 17. Fix any 𝜌 > 0, any 𝛿 ∈ (0, 1), and any Δ0 ∈ N. The
+Weighted Gaussian algorithm (Algorithm 2) satisfies 𝛿-approximate
+𝜌-zCDP.
+See Appendix A for the proof of Theorem 17.
+The Weighted Gaussian algorithm benefits from being highly
+scalable. In particular, it lends itself well to parallel computation
+across several computers within a cluster, because the weights on
+each histogram item can be computed in parallel as well as the noise
+addition and thresholding steps (see Figure 4). Thus, Algorithm 2 is
+the standard approach for doing partition selection on data sets that
+are too large to fit in a single machine’s memory. Unfortunately,
+
+Marika Swanberg, Damien Desfontaines, and Samuel Haney
+Weighted Gaussian suffers from poor accuracy compared to the
+greedy approaches we discuss next.
+4.2
+Greedy Approaches
+One problem with the Weighted Gaussian algorithm is users waste
+their sensitivity budget on histogram items that are already well
+above the threshold. Most real-world data has highly skewed item
+frequencies, but Weighted Gaussian increments all items in a user’s
+set by the same amount.
+The two greedy algorithms we discuss next solve this problem by
+iterating through the users one-by-one and using an update policy
+and the current state of the histogram to decide how to allocate
+weight across the items in their set. That is, each user’s update de-
+pends on previous users’ updates. For example, in both algorithms,
+users do not contribute to items in 𝐻 that have already reached 𝑇 ∗,
+the threshold 𝑇 plus some positive buffer; items that have reached
+the buffered threshold are very likely to be returned after noise
+is added and thus do not need more weight. These adaptive up-
+date rules are carefully chosen so that the overall algorithm has
+bounded global sensitivity. As we will discuss in later sections, the
+main downside of these algorithms is their sequential nature. By
+design, they require iterating over the entire data set, which may
+be prohibitively slow for industrial data sets.
+4.2.1
+Policy Gaussian [10]. As with the Weighted Gaussian algo-
+rithm, the data set is pre-processed by removing duplicate items
+from each user’s set and truncating the user sets to some fixed
+maximum size Δ0. Then, the algorithm iterates sequentially over
+the users and, for each item 𝑢 in user 𝑖’s set𝑊𝑖, the user increments
+𝐻 [𝑢] by a weight that is proportional to 𝑇 ∗ − 𝐻 [𝑢], where 𝑇 ∗ is
+equal to 𝑇 plus some positive buffer. Essentially, items that are fur-
+ther from the buffered threshold get more weight added to them, and
+those that have already reached the buffered threshold get none. The
+weight that a user adds to each item is normalized so a single user’s
+update to 𝐻 has an ℓ2-norm of at most 1.
+Gopi et al. prove that the global ℓ2-sensitivity of the entire Policy
+Gaussian algorithm is bounded by 1, so applying the high-level al-
+gorithm (Algorithm 1) with appropriately-scaled Gaussian noise to
+each item and thresholding are sufficient for satisfying differential
+privacy.
+4.2.2
+Greedy updates Without sampling (GW) [5]. Carvalho et
+al. observe that, rather than removing duplicate items in a user’s
+set, one can use this frequency information to decide where to
+allocate sensitivity budget. GW iterates over the users, and each
+user computes 𝑢∗: the most frequent item in their list such that
+𝐻 [𝑢∗] is below the buffered threshold 𝑇 ∗. Then, the user incre-
+ments 𝐻 [𝑢∗] by min(1,𝑇 ∗ − 𝐻 [𝑢∗],𝑏𝑢𝑑𝑔𝑒𝑡) where 𝑏𝑢𝑑𝑔𝑒𝑡 is the
+user’s remaining ℓ1 budget. This process is repeated until the user’s
+initial budget of 1 is consumed or the user has no items left that are
+below 𝑇 ∗2. Carvalho et al. prove that their GW algorithm for build-
+ing a weighted histogram has a global ℓ1-sensitivity bound of 1, so
+running the high-level algorithm (Algorithm 1) with Laplace noise
+and thresholding is sufficient for satisfying differential privacy.
+2As the algorithm is presented in [5], it does not always terminate. In particular, they
+do not consider the case when a user has budget left but all of its items have reached
+𝑇 ∗, but adding this condition luckily does not affect the privacy proof.
+One notable feature of their algorithm is that it can use item fre-
+quency information from a public data set to increase the accuracy
+of the frequency estimates. We do not use this feature when com-
+paring it to other algorithms, since our goal is to create a general-
+purpose algorithm that could be incorporated into a privacy frame-
+work without requiring a data analyst to input a public data set.
+5
+DP-SIPS
+In this section we present DP-SIPS: Differentially Private Scalable,
+Iterative Partition Selection, detailed in Algorithm 3. The basic
+structure is quite simple: it runs Weighted Gaussian on the data set
+multiple times with increasing privacy budget (and corresponding
+decreasing thresholds); on each iteration, the partitions returned
+by Weighted Gaussian are removed from each of the users’ sets.
+Because Weighted Gaussian updates the histogram uniformly
+across a user’s items, when returned partitions are removed from
+the user’s set, they can allocate more weight to their remaining
+items (see Figure 1). The first iteration returns the very popular
+items using only a tiny fraction of the overall privacy budget, and
+subsequent iterations yield less frequent items. The action of the
+algorithm is twofold: in each iteration, the threshold is lowered at
+the same time as users allocate more weight to each of the items that
+remain in their sets (since the user sets get smaller when previously-
+returned items are removed). DP-SIPS is parallelizable because
+the steps within each iteration are parallelizable, and only a few
+iterations are required to achieve good accuracy (see Figure 4).
+Furthermore, the user sets are re-truncated on each iteration
+after the previously returned partitions are removed. For data sets
+that are both skewed in the item frequencies and in the sizes of
+users’ sets, this allows additional items to be included on each
+iteration.
+Algorithm 3 DP-SIPS: Scalable, Iterative Partition Selection
+Input: Data set of user partitions 𝑥 = (𝑊1, . . . ,𝑊𝑁 )
+Maximum per-user contribution Δ0
+Privacy parameters 𝜌,𝛿
+Number of iterations 𝐼
+Privacy budget allocation factor 𝑟
+Output: Subset 𝑆 ⊆ ∪𝑖𝑊𝑖
+1: 𝑆 ← {}
+2: for 𝑖 = 0, . . . , 𝐼 − 1 do
+3:
+(𝜌𝑖,𝛿𝑖) ←
+�
+𝜌 · 𝑟𝐼−𝑖−1 · 1−𝑟
+1−𝑟𝐼 ,
+𝛿 · 𝑟𝐼−𝑖−1 · 1−𝑟
+1−𝑟𝐼
+�
+4:
+𝑃𝑖 ← Weighted_Gauss (𝑥 = (𝑊1, . . . ,𝑊𝑁 ), 𝜌𝑖,𝛿𝑖, Δ0)
+5:
+for 𝑗 ∈ 𝑁 do
+6:
+𝑊𝑗 ← 𝑊𝑗 \ 𝑃𝑖
+⊲ Remove already-found paritions
+7:
+𝑆 ← 𝑆 ∪ 𝑃𝑖
+return 𝑆
+Theorem 18. For any 𝜌 > 0 and any 𝛿 ∈ (0, 1], Algorithm 3
+satisfies 𝛿-approximate 𝜌-zCDP.
+Proof. By Theorem 17, each call to Weighted_Gauss satisfies𝛿𝑖-
+approximate 𝜌𝑖-zCDP. Applying composition and postprocessing
+(Lemmas 10 and 11), Algorithm 3 satisfies
+��𝐼−1
+𝑖=0 𝛿𝑖
+�
+-approximate
+��𝐼−1
+𝑖=0 𝜌𝑖
+�
+-zCDP.
+
+DP-SIPS: A simpler, more scalable mechanism for differentially private partition selection
+A B C D E F G H I J K
+Weighted Gaussian
+DP-SIPS Iteration 1
+DP-SIPS Iteration 2
+DP-SIPS Iteration 3
+A B C D E F G H I J K
+A B C D E F G H I J K
+A B C D E F G H I J K
+Previously returned:
+A, B
+Previously returned:
+A, B, C, D, E
+A B C D E F G H I J K
+Weighted Gaussian
+DP-SIPS Iteration 1
+DP-SIPS Iteration 2
+DP-SIPS Iteration 3
+A B C D E F G H I J K
+A B C D E F G H I J K
+A B C D E F G H I J K
+Previously returned:
+A, B
+Previously returned:
+A, B, C, D, E
+Figure 1: Depiction of Weighted Gaussian noisy histogram (left) compared to intermediate SIPS noisy histograms (right three
+diagrams) on a skewed data set. Solid blue bars represent partitions that will be returned, and yellow outlines represent the
+weight of each item on the previous iteration. Although the threshold for Weighted Gaussian is lower than each of the SIPS
+thresholds, SIPS benefits from less-frequent items getting increased weight as returned partitions are removed from user sets.
+Now, we will solve for these summations using the closed-form
+formula for geometric sums.
+𝐼−1
+∑︁
+𝑖=0
+𝜌𝑖 =
+𝐼−1
+∑︁
+𝑖=0
+𝜌 · 𝑟𝐼−𝑖−1 · 1 − 𝑟
+1 − 𝑟𝐼 = 𝜌 · 1 − 𝑟
+1 − 𝑟𝐼 ·
+𝐼−1
+∑︁
+𝑖=0
+𝑟𝐼−𝑖−1
+= 𝜌 · 1 − 𝑟
+1 − 𝑟𝐼 ·
+𝐼−1
+∑︁
+𝑗=0
+𝑟 𝑗 = 𝜌 · 1 − ����
+1 − 𝑟𝐼 · 1 − 𝑟𝐼
+1 − 𝑟 = 𝜌.
+An identical calculation holds for �𝐼−1
+𝑖=0 𝛿𝑖. Thus, Algorithm 3 satis-
+fies 𝛿-approximate 𝜌-zCDP.
+□
+6
+EXPERIMENTAL RESULTS
+We use data sets of several different sizes from varied text domains
+to validate the empirical performance of the DP-SIPS algorithm. As
+with prior works, we focus on the problem of vocabulary discovery
+under the constraint of user-level privacy. We begin by describing
+the data sets and computing clusters for our experiments in Sec-
+tions 6.1 and 6.2. Then, in Section 6.3 we discuss how the empirical
+accuracy of DP-SIPS compares to that of existing algorithms, and in
+Section 6.4 we discuss the results from our scalability experiments.
+6.1
+Data sets
+We use six publicly-available data sets to study the accuracy of
+DP-SIPS against existing partition selection algorithms, and we
+use the largest of the six for scalability experiments on Amazon
+Elastic Map Reduce clusters. Reddit [15] is a data set of text posts
+collected from r/AskReddit which appeared as a benchmark in
+[5, 10]. We also use four data sets that appeared in [5]: Twitter [16],
+comprising customer support tweets to and from large corporations;
+Finance [2], financial headlines for stocks; IMDb [14], a set of movie
+reviews scraped from IMDb; Wikipedia [17], a set of wikipedia
+abstracts (where we treat each abstract as a separate user set).
+For the scalability experiments, we use Amazon [11], a publicly-
+available text data set from Kaggle of 4 million Amazon product
+reviews. Using the same methods as [5, 10], we preprocess all data
+sets using tokenization with nltk.word_tokenize, removing URLs
+and symbols, and lower casing all words.
+6.2
+Cluster settings
+For the two scalability experiments described in Tables 5 and 6, we
+implement all of the algorithms in PySpark to take advantage of
+parallelism in the algorithms. We then run the algorithms on the
+Amazon dataset on Amazon Elastic Map Reduce (EMR) clusters. In
+the first experiment, we use clusters with one master node of type
+m5a.2xlarge and varying numbers of core nodes of type m5a.2xlarge.
+All of the privacy parameters and hyperparameters are identical to
+those listed in Table 1.
+For the second experiment (Table 6), we fix a cluster with one
+master node and two core nodes and vary the machine size of the
+nodes. We modify the PySpark session settings to proportionally
+increase the number of cores and memory allocation for both dri-
+ver and executor nodes to reflect the available resources of the
+machines.
+In both benchmark experiments, we measure the amount of time
+required to run the algorithm (after the dataset has already been
+read in to PySpark) and return the number of partitions that were
+discovered, in order to ensure PySpark’s lazy evaluation executed
+the algorithm during the timing phase.
+6.3
+Accuracy results
+Because the goal of partition selection is to privately output as
+many partitions as possible, we measure accuracy as the number
+of partitions released. We test the accuracy of the four algorithms
+on the data sets described in Section 6.1: Weighted Gaussian (Al-
+gorithm 2), SIPS (our Algorithm 3), Policy Gaussian (DPSU) from
+[10], and Greedy updates Without sampling (GW) from [5]. Table 1
+shows the following trends for SIPS:
+• In general, the accuracy of our algorithm (SIPS) is on-par
+with or better than that of DPSU on the 6 data sets tested
+• The accuracy of SIPS is consistently approximately double
+that of Weighted Gaussian.
+The accuracy of GW is higher than that of DPSU on 4 of the 6
+data sets; however, surprisingly on the IMDb and Wikipedia data
+sets its accuracy is below that of Weighted Gaussian. To further
+investigate this phenomenon, we modify these two data sets in
+
+Marika Swanberg, Damien Desfontaines, and Samuel Haney
+addition to Finance to remove repeated items within each user’s
+set (see the data sets marked as “deduplicated” in Table 1). Without
+any item frequency information, the algorithm adds all of its weight
+to a randomly-selected item, so one would expect the performance
+on all three data sets to be worse than Weighted Gaussian. To the
+contrary, GW’s accuracy on the deduplicated Finance data set is
+still significantly higher than the others, and GW’s accuracy on the
+deduplicated IMDb and Wikipedia is again worse than Weighted
+Gaussian.
+One possible explanation is the relatively small vocabulary size
+compared to the number of users and the small user sets in the
+Finance data set, since stock headlines are often short and for-
+mulaic. We do not attempt to resolve this discrepancy in GW’s
+accuracy; however, we present these results as a caution when se-
+lecting a partition selection algorithm. Note also that the accuracy
+of the other three algorithms is unaffected by deduplication since
+they all perform this preprocessing step to the data sets.
+6.3.1
+Comparing zCDP to DP. We implement our algorithm to sat-
+isfy approximate zCDP to take advantage of its simpler composition
+properties over standard DP. In [10], the Policy Gaussian algorithm
+satisfies (𝜀,𝛿)-DP, but the threshold and Gaussian noise can easily
+be recalibrated to satisfy approximate zCDP instead. We do so in
+this work to facilitate the comparison to our algorithm. Unfortu-
+nately, the GW algorithm from [5] has a bounded ℓ1-sensitivity
+and uses Laplace noise, so it is not easily converted to satisfy ap-
+proximate zCDP without a significant loss in accuracy. Instead, we
+use Corollary 14 with 𝛼 = 10 to choose (𝜀,𝛿′) parameters implied
+by the settings of 𝜌 and 𝛿 used on the other algorithms. This con-
+version from approximate zCDP to approximate DP is not tight
+(meaning the given (𝜀,𝛿) may be higher than the true privacy guar-
+antee given by the approximate zCDP parameters). Because of the
+looseness of the conversion, the GW algorithm’s accuracy may be
+slightly inflated when compared to the other algorithms.
+6.3.2
+Selecting hyperparameters. Our algorithm has several hyper-
+parameters (aside from the privacy parameters 𝜌 and 𝛿) that need
+to be set by the data analyst: Δ0 the maximum number of items per
+user, 𝑟 the ratio between the privacy budget for iteration 𝑖 + 1 and
+𝑖, and 𝐼 the number of iterations in the algorithm. One option is
+to divide the privacy budget and try several hyperparameter set-
+tings and select the setting with the highest accuracy; however, this
+wastes a lot of privacy budget. We find that the accuracy of DP-SIPS
+is largely invariant to reasonable settings of the hyperparameters,
+and we provide general rules of thumb for selecting them.
+Figures 2 and 3 display the accuracy of our algorithm on five data
+sets with varying 𝑟 and Δ0, respectively. For the given setting of 𝛿
+and 𝜌, the accuracy of DP-SIPS is largely unaffected by different
+choices of 𝑟 and Δ0 across 5 data sets, and Table 2 shows that the
+accuracy of DP-SIPS only slightly increases with the number of
+iterations 𝐼 on the Reddit data set (for the given settings of the
+other parameters).
+While it is impossible to test the accuracy of every possible
+combination of parameters, these results suggest that DP-SIPS is
+generally insensitive to its hyperparameters. We recommend using
+the following settings: 𝑟 ∈ [0.2, 0.4], 𝐼 ≥ 3, and Δ0 set to an overes-
+timate of the true maximum number of per-user contributions. If
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Ratio Parameter r
+0
+5000
+10000
+15000
+20000
+25000
+30000
+Number of Partitions Returned
+Finance
+Twitter
+Wikipedia
+Reddit
+IMDB
+Figure 2: Number of partitions returned versus the choice
+of the privacy ratio parameter, 𝑟. All other hyperparameters
+are identical to those listed in Table 1. For all 5 data sets, the
+accuracy is maximized for 𝑟 ∈ [0.1, 0.4].
+5
+50
+100
+150
+200
+250
+300
+350
+400
+Maximum Number of Contributions per User
+0
+5000
+10000
+15000
+20000
+25000
+30000
+Number of Partitions Returned
+Finance
+Twitter
+Wikipedia
+Reddit
+IMDB
+Figure 3: Number of partitions returned for varying per-user
+maximum contribution bounds, Δ0. All other hyperparame-
+ters are identical to those listed in Table 1. On all 5 data sets,
+increasing Δ0 past 100 has almost no affect on accuracy, with
+maximum accuracy for Δ0 ∈ [5, 150]. We note that Finance is
+unusual as each financial headline is quite short.
+the true maximum number of contributions per user is public, Δ0
+should be set to that value.
+6.4
+Scalability results
+We benchmark the algorithms on the Amazon data set consisting
+of product reviews from 4 million users. Figure 5 shows the re-
+sults from the first experiment, in which we measure the runtimes
+of the algorithms on Amazon EMR clusters with a single master
+node and varying numbers of core nodes. All nodes are of type
+m5a.2xlarge, which has 8 processor cores and 32 GB of memory.
+Figure 5 illustrates the following runtime trends:
+
+DP-SIPS: A simpler, more scalable mechanism for differentially private partition selection
+Data set
+Wt. Gauss
+SIPS
+DPSU
+GW
+Reddit
+6,160
+11,392
+11,186
+11,984
+Twitter
+12,632
+23,649
+23,576
+27,184
+Finance
+17,350
+27,559
+29,005
+37,503
+IMDb
+3,728
+7,759
+5,845
+3,133
+Wikipedia
+11,340
+21,037
+18,129
+11,251
+Amazon
+67,522
+144,805
+143,997
+185,563
+Finance (deduplicated)
+-
+-
+-
+37,563
+IMDb (deduplicated)
+-
+-
+-
+3,005
+Wikipedia (deduplicated)
+-
+-
+-
+9,802
+Table 1: Number of partitions returned by Wt. Gauss (Algorithm 2), SIPS (Algorithm 3), DPSU (Policy Gaussian from [10], and
+GW (from [5]) on six data sets, and additionally we run GW on three data sets where duplicates within user lists have been
+removed. Note that the other three algorithms already deduplicate user sets. For SIPS we use 3 iterations and 𝑟 = 1/3. For Wt.
+Gaussian, SIPS, and DPSU, the privacy budget is set to 𝜌 = 0.1,𝛿 = 10−5, and the user contributions are truncated to Δ0 = 100.
+For GW, 𝜀 = 1.7 and 𝛿 = 8.1142 × 10−5.
+Start
+End Iteration
+End
+Start
+End
+Start
+End
+Figure 4: Parallelism diagram for Weighted Gaussian (left), SIPS (middle), and greedy algorithms (right). Arrows represent
+computational steps while boxes represent the data set. The steps of Weighted Gaussian can run in parallel on parts of the
+data set while greedy algorithms must run sequentially over the data set. Each iteration of SIPS can run in parallel, but the
+iterations must be done sequentially.
+Iterations 𝐼
+Partitions Returned
+1
+5,464
+2
+10,041
+3
+11,126
+4
+11,182
+5
+11,541
+6
+11,061
+8
+11,585
+10
+11,637
+Table 2: Accuracy as a function of the number of iterations
+on Reddit for Algorithm 3 with parameters 𝜌 = 0.1,𝛿 = 10−5,
+𝑟 = 1/3, and Δ0 = 50.
+• The runtimes of Weighted Gaussian and SIPS both decrease
+as the number of core nodes increases.
+• The runtimes of DPSU and GW remains approximately the
+same even as the number of core nodes increases.
+These results confirm the intuition that Weighted Gaussian and
+SIPS are both parallelizable and thus scale well with increased
+cluster sizes, even on large data sets. Additionally, it confirms our
+observation that since DPSU and GW need to iterate sequentially
+over each user, they do not scale well with larger cluster sizes as
+this iteration is the main bottleneck.
+Next, we run a slightly different scalability experiment: instead
+of increasing the number of core nodes, we increase the sizes of the
+core nodes. We use a single master node and two core nodes, and
+
+Marika Swanberg, Damien Desfontaines, and Samuel Haney
+2
+4
+8
+16
+24
+Number of Core Machines
+0
+100
+200
+300
+400
+500
+600
+Runtime (sec)
+DPSU
+GW
+SIPS
+Weighted Gaussian
+Figure 5: Algorithm runtimes on Amazon dataset with increas-
+ing number of m5a.2xlarge cores in the cluster. The algo-
+rithms were run with the same hyperparameters as those
+listed in Table 1.
+m5a.xlarge
+m5a.2xlarge
+m5a.4xlarge
+m5a.8xlarge
+Machine Sizes in Cluster
+0
+100
+200
+300
+400
+500
+600
+700
+Runtime (sec)
+DPSU
+GW
+SIPS
+Weighted Gaussian
+Figure 6: Algorithm runtimes on Amazon dataset with increas-
+ing sizes of nodes in a cluster with 1 master node and 2 core
+nodes. The algorithms were run with the same hyperparam-
+eters as those listed in Table 1.
+increase the sizes of all three nodes. Table 6 shows similar trends
+as the previous experiment: Weighted Gauss and SIPS scale with
+increased node sizes while DPSU and GW do not.
+Comparing Figures 5 and 6, we see that for DP-SIPS, the com-
+munication overhead between machines is small. Specifically we
+can compare SIPS’s performance on 8 cores in Figure 5, a cluster
+with 1 master node and 8 core nodes each of type m5a.2xlarge (8
+CPU cores, 32 GB memory) to m5a.8xlarge (32 CPU cores, 128 GB
+memory) in Figure 6, a cluster with 1 master node and 2 core nodes.
+In both, the core nodes have in total 8 · 8 = 2 · 32 CPU cores and
+8 · 32 = 2 · 128 GB of memory among them. The runtimes of SIPS
+in the first case is 58.41 seconds while in the second case, it is 66.28
+seconds, which suggests that the communication overhead is small.
+In fact, SIPS runs faster on a cluster of 9 smaller machines than a
+cluster of 3 larger machines. This discrepancy could be the result
+of any number of factors in the machine specifications.
+While the Amazon data set is not large enough that GW and DPSU
+are infeasible to use on it, the experiments suggest that SIPS could
+better handle industrial scale data sets with hundreds of millions
+of users (rather than just 4 million) where the greedy algorithms
+truly would be infeasible to use–all at little to no loss in accuracy.
+7
+DISCUSSION
+Differentially private partition selection (or set union) is a fun-
+damental problem for many private data analysis tasks. Prior ap-
+proaches to this problem either suffer from poor accuracy or are
+slow on large data sets because they are designed to sequentially
+iterate over the users. We present a simple algorithm for differ-
+entially private partition selection that achieves accuracy that is
+comparable to DPSU and runtimes that scale well with increased
+computational resources.
+7.1
+Unsuccessful Attempts
+One natural question we explored is: can we boost the accuracy of
+DPSU using our method of iterating several times and removing
+partitions from the data set that were previously returned? We
+found that empirically this did not boost the accuracy on the data
+sets that we tested. Intuitively, this makes sense because the per-
+user Gaussian update policy in DPSU prevents users from adding
+weight to items that have already reached the buffered threshold.
+Such items will very likely be released after the noise addition and
+thresholding steps. The DPSU approach achieves the same goal as
+iterating, albeit with more precision due to the greedy nature of
+the Policy Gaussian per-user updates.
+7.2
+Future work
+This work raises several natural questions. From a practical per-
+spective, one may wonder whether there is a scalable algorithm
+with higher accuracy than DP-SIPS. Another interesting line of
+inquiry could consider theoretical guarantees on the number of
+partitions released, possibly under some distributional assumptions
+on the data. We believe these lines of inquiry would give important
+insights into a fundamental problem in private data analysis.
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+movie
+reviews.
+https://www.kaggle.com/datasets/lakshmi25npathi/imdb-dataset-of-50k-
+movie-reviews.
+[15] Judy Hanwen Shen. Ask reddit. https://github.com/heyyjudes/differentially-
+private-set-union/tree/ea7b39285dace35cc9e9029692802759f3e1c8e8/data.
+[16] Thought Vector and Stuart Axelbrooke.
+Customer support on twitter.
+https://www.kaggle.com/datasets/thoughtvector/customer-support-on-twitter.
+[17] Mark
+Wijkhuizen.
+Simple/normal
+wikipedia
+abstracts
+v1.
+https://www.kaggle.com/datasets/markwijkhuizen/simplenormal-wikipedia-
+abstracts-v1.
+[18] Royce J Wilson, Celia Yuxin Zhang, William Lam, Damien Desfontaines, Daniel
+Simmons-Marengo, and Bryant Gipson. Differentially private SQL with bounded
+user contribution. Proceedings on Privacy Enhancing Technologies, 2:230–250,
+2020.
+A
+PROOF OF THEOREM 17
+Proof of Theorem 17. Since the algorithm is simply comput-
+ing a stable histogram, the proof follows from standard arguments.
+We will denote Algorithm 2 by M. Fix any 𝜌 > 0, any 𝛿 ∈ (0, 1),
+and any Δ0 ∈ N. Fix two neighboring data sets 𝑥 = (𝑊1, . . . ,𝑊𝑁 )
+and 𝑥 ′, where 𝑥 ′ contains one extra user set𝑊 ∗. Let 𝐸 be the event
+that M(𝑥 ′) ⊆ ∪𝑖 ∈[𝑁 ]𝑊𝑖; that is, the partitions returned for data set
+𝑥 ′ are in the support of M(𝑥). We will first argue that, conditioned
+on event 𝐸, the output distributions of M(𝑥) and M(𝑥 ′) satisfy
+pure zCDP (Definition 4). Then we will argue that, because of the
+thresholding step, event 𝐸 occurs with probability at least 1 − 𝛿.
+If we condition both M(𝑥) and M(𝑥 ′) on event 𝐸, then by the
+Gaussian mechanism (Definition 7) and post-processing (Lemma 10),
+Algorithm 2 satisfies pure zCDP (Definition 4).
+Now, it remains to show that Pr[𝐸] ≥ 1 − 𝛿. First, note that each
+user contributes to most Δ0 items in the histogram, and further
+note that event 𝐸𝑐 occurs when at least one item from 𝑊 ∗ that is
+not in 𝑥 is released. Let 𝑊 ′ denote the set of items in 𝑊 ∗ that do
+not appear in 𝑥. Then,
+Pr[𝐸] = Pr[∀𝑢 ∈ 𝑊 ′,𝑢 ∉ M(𝑥)]
+= Pr[∩𝑢∈𝑊 ′ ˆ𝐻 [𝑢] ≤ 𝑇]
+= Pr[∩𝑢∈𝑊 ′𝐻 [𝑢] + 𝑍𝑢 ≤ 𝑇]
+For 𝑍𝑢 ∼ N (0, 1/2𝜌)
+=
+�
+𝑢∈𝑊 ′
+Pr[𝐻 [𝑢] + 𝑍𝑢 ≤ 𝑇]
+By independence of 𝑍𝑢’s
+=
+�
+𝑢∈𝑊 ′
+Pr
+�
+1
+√︁
+|𝑊 ′|
++ 𝑍𝑢 ≤ 𝑇
+�
+=
+�
+Pr
+�
+1
+√︁
+|𝑊 ′|
++ 𝑍𝑢 ≤ 𝑇
+�� |𝑊 ′|
+Since 𝑍𝑢’s are i.i.d.
+≥ min
+𝑘 ∈[Δ0]
+��
+Pr
+� 1
+√
+𝑘
++ 𝑍𝑢 ≤ 𝑇
+��𝑘�
+Since |𝑊 ′| ≤ Δ0
+≥ 1 − 𝛿
+By definition of 𝑇 .
+□
+B
+AMAZON EMR CLUSTER SPECIFICATIONS
+Table 3 shows the number of CPU cores and amount of memory
+available to each type of machine on Amazon EMR that we use in
+our experiments.
+Machine Size
+Cores
+Memory (GB)
+m5a.xlarge
+4
+16
+m5a.2xlarge
+8
+32
+m5a.4xlarge
+16
+64
+m5a.8xlarge
+32
+128
+Table 3: Number of cores and memory per EMR machine of
+each size.
+C
+OMITTED PROOFS FOR SECTION 3
+In this section, we give the proofs omitted from Section 3
+Proof of Lemma 10. Suppose M : X∗ → Y and M′ : X∗ →
+Z satisfy 𝛿-approximate 𝜌-zCDP and 𝛿′-approximate 𝜌′-zCDP,
+respectively. Fix any neighboring data sets 𝑥,𝑥 ′, and let 𝑃1, 𝑄1,
+𝑃2, 𝑄2 denote the distributions of M(𝑥), M(𝑥 ′), M′(𝑥), M′(𝑥 ′)
+respectively. Then there exist distributions 𝑃 ′
+1 and 𝑃 ′
+2 such that
+𝑇𝑉𝐷(𝑃1, 𝑃 ′
+1) ≤ 𝛿,𝑇𝑉𝐷(𝑃2, 𝑃 ′
+2) ≤ 𝛿 and
+𝐷𝛼 (𝑃 ′
+1||𝑄1) ≤ 𝜌 · 𝛼, 𝐷𝛼 (𝑃 ′
+2||𝑄2) ≤ 𝜌 · 𝛼,
+for all 𝛼 ∈ (1, ∞). The output distribution of M′′(𝑥) is the joint
+distribution (𝑃1,𝑄1), and similarly (𝑃2,𝑄2) for M′′(𝑥 ′). Note that,
+(1) The joint distribution (𝑃 ′
+𝑖 ,𝑄′
+𝑖 ) satisfies𝑇𝑉𝐷((𝑃 ′
+𝑖 ,𝑄′
+𝑖 ), (𝑃𝑖,𝑄𝑖)) =
+𝑇𝑉𝐷(𝑃𝑖, 𝑃 ′
+𝑖 ) +𝑇𝑉𝐷(𝑄′
+𝑖,𝑄𝑖) ≤ 𝛿 + 𝛿′ since the random out-
+puts of the two mechanisms are independent.
+(2) 𝐷𝛼 ((𝑃 ′
+1, 𝑃 ′
+2)||(𝑄1,𝑄2)) ≤ (𝜌 + 𝜌′) · 𝛼 by the same argument
+as composition for pure zCDP.
+Therefore, M′′(𝑥) satisfies (𝛿 +𝛿′)-approximate (𝜌+𝜌′)-zCDP.
+□
+Proof of Lemma 11. Suppose M : X𝑁 → Y satisfies𝛿-approximate
+𝜌-zCDP and let 𝑓 : Y → Z be a function. Let 𝑃 and 𝑄 denote the
+
+Marika Swanberg, Damien Desfontaines, and Samuel Haney
+distributions of M(𝑥) and M(𝑥 ′) respectively. Let 𝑃 ′ be the dis-
+tribution promised by the definition of approximate zCDP such
+that
+𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿, and
+𝐷𝛼 (𝑃 ′||𝑄) ≤ 𝜌 · 𝛼
+for all 𝛼 ∈ (1, ∞). Let 𝑓 (𝑃) denote the distribution over Z obtained
+by setting 𝑓 (𝑃)(𝑆) = 𝑃(𝑓 −1(𝑆)) for all 𝑆 ⊆ Z, where 𝑓 −1(𝑆) is the
+preimage of 𝑆 in Y. Then the following are true:
+(1) 𝐷𝛼 (𝑓 (𝑃 ′)||𝑓 (𝑄)) ≤ 𝜌 · 𝛼
+(2) 𝑇𝑉𝐷(𝑓 (𝑃 ′), 𝑓 (𝑃)) ≤ 𝛿
+(1) follows from the fact that zCDP is closed under postprocessing.
+For (2), note that |𝑓 (𝑃)(𝑆) − 𝑓 (𝑃 ′)(𝑆)| = |𝑃(𝑓 −1(𝑆)) − 𝑃 ′(𝑓 −1(𝑆))|
+for all 𝑆 ⊆ Z. Therefore, 𝑇𝑉𝐷(𝑓 (𝑃 ′), 𝑓 (𝑃)) ≤ 𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿.
+□
+Proof of Lemma 12. Suppose M : X𝑁 → Y satisfies𝛿-approximate
+𝜌-zCDP and suppose every mechanism that satisfies 𝜌-zCDP must
+satisfy 𝜖∗(𝜌),𝛿∗(𝜌)-approximate DP. Let 𝑃 and 𝑄 denote the dis-
+tributions of 𝑀(𝑥) and 𝑀(𝑥 ′) respectively. By the definition of
+approximate zCDP, there exists 𝑃 ′ such that
+𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿, and
+𝐷𝛼 (𝑃 ′||𝑄) ≤ 𝜌 · 𝛼
+for all 𝛼 ∈ (1, ∞). Next, by assumption in the lemma, distributions
+𝑃 ′ and 𝑄 satisfy
+𝑃 ′(𝑆) ≤ exp(𝜖)𝑄(𝑆) + 𝛿.
+Then by Lemma 9, we have the following:
+𝑇𝑉𝐷(𝑃 ′′, 𝑃 ′) ≤ 𝛿∗(𝜌), and
+𝑃 ′′(𝑆) ≤ exp(𝜖)𝑄(𝑆).
+Then,
+𝑇𝑉𝐷(𝑃 ′′, 𝑃) ≤ 𝑇𝑉𝐷(𝑃 ′′, 𝑃 ′) +𝑇𝑉𝐷(𝑃 ′, 𝑃)
+= 𝛿 + 𝛿∗(𝜌).
+Applying Lemma 9 again completes the proof.
+□
+Proof of Lemma 16. Suppose M : X𝑁 → Y satisfies (𝜖,𝛿)-
+approximate differential privacy. Let 𝑃 and 𝑄 denote the distribu-
+tions of M(𝑥) and M(𝑥 ′) respectively. Let 𝑃 ′ denote 𝑀(𝑥)|𝐸. We
+claim that 𝑇𝑉𝐷(𝑃, 𝑃 ′) ≤ 𝛿. Note that for any 𝑠 ∈ ¬𝐸, 0 = 𝑃 ′(𝑠) ≤
+𝑃(𝑠) and for any 𝑠 ∈ 𝐸, 𝑃(𝑠) ≤ 𝑃 ′(𝑠). Therefore
+sup
+𝑆
+|𝑃(𝑆) − 𝑃 ′(𝑆)| = |𝑃(𝐸) − 𝑃 ′(𝐸)|
+≤ 𝛿.
+□
+Marika Swanberg, Damien Desfontaines, Samuel Haney„
+

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@@ -0,0 +1,1310 @@
+Dynamic Grained Encoder for Vision Transformers
+Lin Song1∗
+Songyang Zhang2,4,5∗
+Songtao Liu3
+Zeming Li3
+Xuming He2
+Hongbin Sun1†
+Jian Sun3
+Nanning Zheng1
+1 College of Artificial Intelligence, Xi’an Jiaotong University
+2 ShanghaiTech University
+3 Megvii Inc. (Face++)
+4University of Chinese Academy of Sciences
+5Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences
+stevengrove@stu.xjtu.edu.cn, sy.zhangbuaa@gmail.com,
+liusongtao@megvii.com, hexm@shanghaitech.edu.cn,
+{hsun, nnzheng}@mail.xjtu.edu.cn, {lizeming, sunjian}@megvii.com
+Abstract
+Transformers, the de-facto standard for language modeling, have been recently
+applied for vision tasks. This paper introduces sparse queries for vision trans-
+formers to exploit the intrinsic spatial redundancy of natural images and save
+computational costs. Specifically, we propose a Dynamic Grained Encoder for
+vision transformers, which can adaptively assign a suitable number of queries to
+each spatial region. Thus it achieves a fine-grained representation in discriminative
+regions while keeping high efficiency. Besides, the dynamic grained encoder is
+compatible with most vision transformer frameworks. Without bells and whistles,
+our encoder allows the state-of-the-art vision transformers to reduce computa-
+tional complexity by 40%-60% while maintaining comparable performance on
+image classification. Extensive experiments on object detection and segmenta-
+tion further demonstrate the generalizability of our approach. Code is available
+at https://github.com/StevenGrove/vtpack.
+1
+Introduction
+Following the evolution of network architectures in natural language processing (NLP), Vision
+Transformers [1–5] have recently attracted increasing research attention and demonstrated promising
+results on several vision tasks, such as image classification, object detection, and other pixel-level
+tasks. Vision transformers are notable for modeling long-range dependencies and introducing less
+inductive bias, considered to be a solid alternative to CNNs for vision tasks.
+One of the eminent obstacles for vision transformers is the high computational cost. Vision tasks
+typically require high-resolution image features to obtain detail and structure representation, which
+is critical for pixel-level tasks [6–10]. However, since the encoders in vision transformers need to
+establish pairwise relationships, high-resolution features could impose unacceptable computational
+and memory costs. Therefore, similar to the efficient transformers [11–13] in NLP, many variants [2–
+4] of vision transformers are proposed to perform sparse self-attentions with dense queries and sparse
+key-value pairs based on fixed pattern or heuristic rules.
+In this paper, we notice that different from natural language, natural images involve much spatial
+redundancy, especially in flat or low-texture regions [14–18]. This could enable the image features to
+have a low resolution in some regions while maintaining similar representational capabilities.
+To verify the spatial redundancy in vision transformers, we give an empirical analysis for DeiT [19]
+on ImageNet [20] classification dataset (the details refer to Sec. 3.1). It demonstrates the existence
+∗Equal contribution. This work was done in Megvii Research.
+†Corresponding author.
+35th Conference on Neural Information Processing Systems (NeurIPS 2021).
+arXiv:2301.03831v1  [cs.CV]  10 Jan 2023
+
+Dynamic
+Grained
+Router
+Vanilla Encoder Block
+Pooling
+Unpooling
+Dynamic Grained Encoder
+Reshape to 2D 
+Feature
+Mixed-Grained Patches
+Flatten
+x
+p
+q
+,k v
+y
+z
+Sparse Tokens
+z
+y
+ˆy
+Figure 1: The overall diagram of the proposed dynamic grained encoder. x is the input sequence,
+and y is the output sequence. The dynamic grained router automatically split a 2D feature into
+mixed-grained patches with a different number of tokens in a patch. Each patch is then flattened as a
+sparse query by an average pooling operator. The vanilla encoder block can be a standard transformer
+encoder or other efficient variants. Besides, the dash lines are only used in the training phase.
+of spatial redundancy in queries, and the complexity can be dramatically reduced by downsampling
+some highly redundant regions while maintaining comparable performance. These properties allow
+the queries to use mixed granularity to achieve a balance between effectiveness and efficiency, i.e.,
+more tokens in more discriminative regions while fewer tokens in less informative regions. However,
+the distribution of spatial redundancy varies greatly among different input images, making it difficult
+for a static method to handle complex and variable features.
+We thus attempt to explore a new perspective: introducing dynamic network mechanism into vision
+transformers to reduce the spatial redundancy of image features. As shown in Fig. 1, we propose
+a Dynamic Grained Encoder (DGE) to replace the vanilla encoder in vision transformers. It could
+assign a suitable number of queries for each region by using a dynamic grained router, e.g., the
+foreground regions of the cat head in Fig. 1 are assigned more queries than the background regions.
+Concretely, a reshaped 2D feature is first divided into regions using a fixed window. For each
+region, the number of patches is decided by a data-dependent routing process, and each patch is
+average pooled to obtain a 1D token. All the tokens are then concatenated into a sequence as the
+queries. Since our method focuses on the sparsity of queries, it is compatible with many efficient
+transformer encoders [2,3,11–13], making our approach available as a generic plugin in most vision
+transformers [1–3,19,21]. Furthermore, the output of the encoder is restored to the input resolution
+by an un-pooling operation and compensates for detailed information with the input feature.
+To demonstrate the effectiveness, we conduct extensive experiments on three typical vision transform-
+ers, i.e., DeiT [19], PVT [3] and DPVT, where DPVT is a new framework based on the deformable
+attention [2]. In the image classification task, our dynamic grained encoder allows these models to
+reduce computational complexity by 40%-60% while maintaining comparable performance. On the
+other hand, with lower computational complexity, the accuracy can be improved by up to 4.4% on
+ImageNet val set. In addition, the experiments on object detection and segmentation show the strong
+robustness and generalization of our method.
+2
+Related Work
+2.1
+Vision Transformer
+Recently, Vision Transformers, inspired by the significant success of transformer [22] achieved in the
+NLP field, have received more attention in the vision community. ViT [1], which converts the image
+into a sequence and applies the transformer encoder structure directly on it for image classification,
+has pioneered this direction in visual recognition. To tackle the issue of training efficiency and data
+efficiency, DeiT [19] introduces several training strategies to enable learning the vision transformer
+on ImageNet. PVT [3] further develops a feature pyramid based on the transformer structure and
+makes it applicable for the various downstream vision tasks. Swin [21] introduces the local window
+idea to improve the efficiency of the transformer structure. Our work mainly focuses on reducing the
+spatial redundancy and improving the model efficiency in a data-dependent manner, which is rarely
+explored in previous works and complementary with various vision transformer structures.
+2
+
+0
+20
+40
+60
+80
+100
+0.8~1.00.6~0.80.4~0.60.2~0.40.0~0.2
+Proportion (%)
+Correlation coefficient
+(a) Distribution
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+0.8
+0.6
+0.4
+0.2
+0
+0
+20
+40
+60
+80
+100
+Complexity Ratio
+Correlation coefficient threshold
+Top1 Accuracy (%)
+Complexity
+Accuracy
+(b) Accuracy v.s. Complexity
+Layer index of DeiT-S
+1 2 3 4 5 6 7 8 9 101112
+Correlation coefficient
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+(c) Layer
+Figure 2: Spatial redundancy statistics of the vanilla encoders in DeiT-S [19]. The correlation
+coefficient is used to measure the similarity of queries in a local region. Higher the correlation
+corresponds to more spatial redundancy. (a) indicates that most queries are highly redundant in a local
+region. (b) reflects that reducing the queries with high redundancy has little impact on performance.
+(c) means that the redundancy varies greatly in some layers.
+2.2
+Efficient Transformer
+To improve the efficiency of transformers, prior works mainly concentrate on reducing the quadratic
+computation of self-attention. These works can be roughly summarized as three types: learnable/fixed
+pattern based methods, low-rank/kernel based methods and memory based methods. Some recent
+approaches [12, 23–26] try to reduce the complexity of the self-attention mechanism by using a
+heuristic method to generate fixed or learnable patterns. Other efforts [11, 13, 27, 28] focus on
+utilizing the low-rank property of the attention matrix or introducing kernels to avoid computing the
+attention matrix explicitly. Moreover, some works [29–31] also explore the memory mechanism to
+improve efficiency. However, previous attempts mainly concentrate on the NLP tasks. Different from
+the language sequence, which has a highly abstract representation of information, natural images
+typically have much spatial redundancy. It makes the vision transformers require expensive costs for
+downstream vision tasks, especially the dense-prediction tasks, e.g., object detection, segmentation.
+Our work tries to utilize this intrinsic property of natural images to achieve redundancy reduction in a
+data-dependent manner.
+2.3
+Dynamic Network
+Dynamic networks [32] are proposed to adaptively change the network architecture and parame-
+ters according to input, which have been widely explored in computer vision and natural language
+processing tasks. Most of the dynamic networks focus on coarse-grained strategy by dropping
+blocks [33–36], pruning channels [37, 38] or adjusting layer-level scales [39, 40]. For instance,
+MSDNet [34] proposes an early existing mechanism to achieve efficient inference for image classifi-
+cation. Switch Transformer [41] uses the Mixture of Experts (MoE) model [42] to select different
+parameters for each input sample. DRNet [39] attempts to perform adaptive scale transformation
+in a feature pyramid network for semantic segmentation. The closest works to ours are probably
+the Dynamic Convolution [43] and the Dynamic Head [10], which use a learnable mask to skip
+specific spatial locations. However, they are only applicable to the CNN-based networks, and the
+skipping-location strategy could result in significant performance degradation for vision transformers
+(refer to Sec. 4.1.2). Different from them, our method adapts the region-level granularity to the input
+feature for the vision transformers, which is more general and flexible.
+3
+Method
+3.1
+Empirical Analyses on Spatial Redundancy
+To investigate the spatial redundancy of vision transformer on image data, we conduct a series
+of experiments on the ImageNet [20] val set with a pre-trained DeiT-S [19] model. Our main
+purpose is to explore the relationship among the granularity of queries, computational complexity,
+and classification performance. Specifically, for each encoder layer in DeiT-S, we reshape its input
+3
+
+queries (excluding the extra embedding) as a 2D feature map and split it into 2 × 2 non-overlap
+patches. For each patch, we calculate its average token, and measure the similarity of each token in
+the patch with the average token by using the Pearson Correlation Coefficient (PCC) metric.
+Then we have three valuable observations. (1) Queries share similar patterns in a local region. From
+the correlation coefficient histogram plotted in Fig. 2(a), most of the correlation coefficients are
+greater than 0.8, which indicates the queries typically have a strong correlation in a local region. (2)
+Large potential of reducing spatial redundancy. Furthermore, in each patch, we replace the tokens
+with the average token when their correlation coefficient is above a given threshold. As shown in
+Fig. 2(b), we illustrate the accuracy/complexity curve varying correlation thresholds. When the
+threshold is 0.9, the complexity decreases by 27%, but the top-1 accuracy decreases by only 0.3%.
+This evidence demonstrates the potential of reducing the spatial redundancy on vision transformers.
+(3) Static strategy is sub-optimal. As shown in Fig. 2(c), some encoders have large variance of
+correlation coefficients among different images. Thus, using data-independent methods to reduce
+spatial redundancy is sub-optimal, which may lead to considerable performance degradation. These
+observations motivate us to explore a data-dependent manner to reduce spatial redundancy.
+3.2
+Dynamic Grained Encoder
+3.2.1
+Overall Architecture
+In this paper, we propose a new encoder block for vision transformers, called Dynamic Grained
+Encoder (DGE). As shown in Fig. 1, the proposed encoder consists of two main modules, i.e.,
+dynamic grained router and vanilla encoder block. Specifically, the dynamic grained router adaptively
+generates mixed-grained patches for a 2D feature. The vanilla encoder block can be a standard
+encoder block [22] or other efficient variants [2, 9, 11–13, 44], which is made up of a multi-head
+attention and a feed-forward network. If there are extra tokens in the input sequence, such as class
+embedding in ViT [1], we handle them separately with the vanilla encoder. For ease of presentation,
+the rest of this section only considers the input sequence without extra tokens.
+Given an input sequence x ∈ R(H×W )×C for the dynamic grained encoder, (H, W) denotes the
+resolution of the feature, C is the number of channels. To compatible with most vanilla encoders,
+we only generate sparse queries q ∈ RN×C by the dynamic grained router, where N indicates the
+number of queries. Then the sparse queries as well as dense keys k and values v are transformed by
+a vanilla encoder. It is worth mentioning that keys and values can be sparse in the vanilla encoder to
+improve efficiency further. The output sequence of the vanilla encoder is restored to a 2D feature
+with the original resolution by using an un-pooling operation. Furthermore, to enhance the details of
+the output feature and alleviate the vanishing gradient problem, we add a residual connection [45] to
+fuse the input sequence.
+3.2.2
+Dynamic Grained Router
+To achieve dynamic grained patches in space, we first partition the 2D feature, denoting as z, into
+multiple regions, which can perform in regular or irregular ways. Although the irregular ways,
+e.g., superpixels [46] and segmentation [47], may lead to better performance, it is very unfriendly
+to memory access and inducing inefficiency. Therefore, as shown in Fig. 3, we adopt a S × S
+non-overlap window3 to split image features into multiple regular regions. Furthermore, we define a
+set of candidate granularities Φ = {φ1, φ2, ..., φK} to represent the optional patch size in a region,
+where K is the number of candidate granularities. The granularity denotes the side length of a patch,
+e.g., φ = 8 corresponds to an 8 × 8 patch. Since each patch is pooled into one query in the encoder,
+larger granularity indicates fewer queries and less computation. For convenience, we set the region
+size with the maximum granularity, i.e., S = max(Φ), in the experiments.
+Inference. For a region i ∈ {1, 2, ..., ⌈ H
+S ⌉·⌈ W
+S ⌉}, we use a gating network to select a granularity from
+the set of candidate granularities. Concretely, we reduce the region feature zi into a representative
+token by using the average pooling operation and linearly project it to the gating logits:
+h(zi) = 1
+S2
+S2
+�
+j=1
+zi,jW + b,
+(1)
+3Bottom-right padding is adopted on the feature if needed.
+4
+
+Multi-Grained
+Patch Split
+Gumbel Noise
+ArgMax
+MLP
+Pooling
+SoftMax
+Routing
+Region Split
+z
+iz
+2
+i =
+ip
+p
+Output Patches
+Dynamic Routing
+for Each Region
+Dynamic Routing for a Region
+Dynamic Grained Router
+iz
+z
+Figure 3: The diagram of the dynamic grained router in a DGE. As shown in the left part, a 2D
+feature is split into multiple regions. For each region, as shown in the right part, we generate multiple
+groups of patches with different granularities and select a specific group by the gating network. The
+Gumbel noise is added to achieve end-to-end training. Besides, the modules in dash lines are only
+used in the training phase.
+where W ∈ RC×K and b ∈ R1×K indicate the weight and bias, respectively. The gating logits is
+used to decide the granularity for the region by calculating the gating indices:
+θi = arg max
+k
+(h(zi)k) ∈ {1, 2, ..., K}.
+(2)
+As shown in Fig. 3, we split the region feature into multiple groups of patches1 with K granularities.
+We then choose a group of specific granularity according to the gating indices. We denote the selected
+group as z′i ∈ RNi×φ2
+θi×C, where Ni = ⌈ S
+φθi ⌉ · ⌈ S
+φθi ⌉ is the number of patches in the group.
+As shown in Fig. 1, to construct a sequence as queries, we use the spatial mean vector of each patch as
+the representative token by a pooling operation and concatenate all the tokens for the vanilla encoder:
+ˆyi = VanillaEncoder(qi, k, v), where qi =
+1
+φ2
+θi
+φ2
+θi
+�
+j=1
+z′
+i,j ∈ RNi×C.
+(3)
+Compared with the previous encoders [2,11–13], the number of queries is reduced to 1/φ2
+θi of the
+original, the efficiency of the encoder can be improved, and the acceleration is more significant when
+selected granularity θi is larger.
+Training. To enable the end-to-end training for the gating network, motivated by [43,48–50], we
+replace the determined decisions in Eq. 2 with a stochastic sampling process during the training phase.
+Specifically, given a categorical distribution with unnormalized log probabilities, a discrete gating
+index can be yielded with noise samples gj drawn from a standard Gumbel distribution:
+θi = arg max
+k
+(h(zi)k + gk), where gk ∼ Gumbel(0, 1).
+(4)
+Furthermore, since the Eq. 4 is a hard decision process, it is not straightforward to train the gating
+logits. To enable the back-propagation, we adopt the Gumbel-Softmax technique [51] to give a
+continuous and differentiable approximation by replacing the argmax with a softmax operation. The
+soft gating score for a region is then selected by the gating index:
+pi =
+exp((h(xi)θi + gθi)/τ)
+�K
+k exp((h(xi)k + gk)/τ)
+∈ [0, 1],
+(5)
+where a fixed temperature τ = 1 is used in our experiments for convenience. Similar with [43,52],
+we further use a straight-through estimator for the gradients of gating logits, which are obtained
+through the soft gating score pi during the backward pass:
+y′
+i =
+�
+ˆy
+forward
+pi · ˆy
+backward
+(6)
+The above stochastic process is only adopted in the training phase. Our method requires no random
+sampling and exponential functions during inference, guaranteeing high efficiency in practice.
+5
+
+(a) gating indices of different images
+(b) gating indices of different stages
+Figure 4: Visualization of predicted gating indices of PVT-S+DGE on ImageNet val set. The
+candidate granularity set is Φ = {1, 2, 4}, which are shown in red, green and blue respectively.
+Higher granularity corresponds to less computational complexity. Our dynamic encoder tends to
+assign more queries to the representative foreground regions than the background regions, thus
+significantly reducing the computational cost. The left and right parts of Fig.4(a) come from stage
+1 and stage 2 of PVT, respectively. From left to right, the heatmaps of each instance in Fig.4(b)
+correspond to stage 1, stage 2, and stage 3, respectively.
+3.2.3
+Budget Constraint
+In the absence of a budget constraint, our encoder typically prefers to assign more queries to
+each region to achieve high performance. To obtain a better balance between effectiveness and
+efficiency, we define a computational budget denoted as γ ∈ [0, 1], which corresponds to the desired
+computational complexity ratio relative to the vanilla encoder without dynamic grained.
+Given a vision transformer with L dynamic grained encoders, we can calculate the used computational
+complexity ratio of the transformer by:
+β =
+�L
+l Clψl
+�L
+l ClHlW l , where ψl =
+� �
+i φ2
+θi
+forward
+�
+i pl
+i · φ2
+θi
+backward
+(7)
+The Cl indicates the computational complexity required to compute a query in an encoder layer. The
+ψl corresponds to the number of queries, adopting a straight-through estimator to enable end-to-end
+training. This strategy ensures an accurate complexity estimation when computing the training loss.
+Moreover, we use the Euclidean distance for the budget loss to narrow the computational complexity
+to a predetermined bound:
+L = Ltask + λLbudget, where Lbudget = (β − γ)2.
+(8)
+The hyper-parameter λ balances losses among different tasks, making the gradients have the same
+order of magnitude. Besides, for batched image inputs, β is averaged along the batch dimension to
+estimate the average load of the network.
+4
+Experiment
+In this section, we apply our encoder to the state-of-the-art vision transformers and conduct exten-
+sive experiments on image classification, object detection, and segmentation. To demonstrate the
+generalization of our method, we conduct experiments on three Vision Transformer frameworks,
+i.e., DeiT [19], PVT [3] and DPVT. Where DPVT is a new framework we proposed, which is
+based on the architecture of PVT [3] but using the deformable attention [2] as the vanilla encoder.
+Different from the dense self-attention process in DeiT, PVT and DPVT utilize sparse key-value
+pairs in position-insensitive and position-sensitive ways, respectively. These three frameworks could
+represent the vanilla encoder used by most vision transformers.
+6
+
+767
+69
+71
+73
+75
+77
+79
+81
+83
+0
+2
+4
+6
+8
+10
+12
+14
+Top1 Accuracy (%)
+FLOPs (G)
+PVT
+PVT+DGE
+DeiT-Ti
+ResNet-18
+ResNet-101
+DeiT-S
+Ti
+S
+M
+L
+Ti
+S
+L
+M
+MSDNet
+(a) Model size (γ = 0.5)
+70
+72
+74
+76
+78
+80
+82
+1
+2
+3
+4
+5
+6
+7
+Top1 Accuracy (%)
+FLOPs (G)
+DPVT-S+DGE
+PVT-S+DGE
+DeiT-S+DGE
+DeiT-Ti
+PVT-Ti
+PVT-S
+DeiT-S
+DPVT-Ti
+DPVT-S
+(b) Budget
+Layer index of DeiT-S+DGE
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11 12
+Complexity Ratio
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+(c) Layer (γ = 0.5)
+41
+41.5
+42
+42.5
+43
+43.5
+100
+200
+300
+400
+500
+600
+700
+mIoU (%)
+FLOPs (G)
+PVT
+PVT+DGE
+768
+640
+512
+512
+640
+768
+(d) Resolution (γ = 0.5)
+Figure 5: Visualization of accuracy and computational complexity of different configurations. (a), (b)
+and (c) are evaluated on ImageNet val set. The PVT and PVT+DGE in (a) is scaled by model size,
+i.e., "tiny", "small" "medium" and "large". (b) indicates the performance of our method with different
+budget constraints. (c) reflects the distribution of computational complexity in different encoder
+layers of the DeiT-S+DGE. (d) is evaluated on ADE-20K val set with varying image resolutions.
+4.1
+Image Classification on ImageNet
+4.1.1
+Implementation Detail
+All the experiments for image classification are based on ImageNet [20] classification dataset. We use
+256×2564 as the input image resolution for training and evaluation. For a fair comparison, we follow
+the training settings in DeiT and PVT. Specifically, the random-size cropping, random horizontal
+flipping [53] and mixup [54] are used for data augmentation. We use the AdamW [55] optimizer with
+the weight decay of 0.05 and the momentum of 0.9. The learning rate is initially set to 0.001 and
+decreases according to the cosine schedule [56]. All the models are trained for 300 epochs with 128
+images per batch. The label-smoothing regularization is used in the training phase. Besides, for the
+dynamic grained encoders, λ is set to 1.0 and Φ is set to {1, 2, 4} by default. During the training
+phase, we use four compute nodes with 32 Nvidia Tesla V100 GPUs. For instance, we spend about
+1.2 days training the PVT-S with DGE model for 300 epochs. For the runtime evaluation, we measure
+the frameworks on both Intel Xeon Gold 6130 CPU and Nvidia Tesla V100 GPU to demonstrate the
+efficiency of our dynamic networks.
+4To achieve efficient region splitting, we choose 256 × 256 instead of 224 × 224 as it is divisible by more
+optional granularities. We re-train all involved vision transformers in this work for a fair comparison.
+7
+
+Table 1: Performance of dynamic grained encoder with different configurations on ImageNet val set.
+The budget for DGE is set to 0.5. "Region" means using region-wise routing instead of layer-wise
+routing in the encoder.
+Framework
+Dynamic
+Region
+Φ
+Top1 Acc
+Top5 Acc
+FLOPs
+#Param
+PVT-S
+
+-
+-
+80.2
+95.2
+6.2G
+28.2M
+PVT-S+DGE
+
+
+1, 2, 4
+79.1
+94.5
+3.4G
++12.1K
+
+0, 1
+78.8
+94.4
+3.5G
++8.1K
+1, 2
+80.0
+95.0
+3.5G
++8.1K
+1, 2, 4
+80.2
+95.0
+3.5G
++12.1K
+1, 2, 4, 8
+79.9
+95.0
+3.4G
++16.1K
+4.1.2
+Ablation Study
+Where are Fine-Grained Queries Assigned?
+To reveal the undergoing properties of our dynamic
+grained encoder, we illustrate the predicted gating indices θ on ImageNet val set, which is shown in
+Fig. 4. Without additional supervision other than classification, our dynamic network can generate
+instance-aware masks with rich details. It allows the encoder to assign more queries on the foreground
+regions with discriminative features than background regions. This ensures that the network can
+consume less computational cost while maintaining fine-grained representation. In addition, as
+presented in Fig. 4(b), the predicted gating indices have similar patterns among different stages in the
+PVT. It demonstrates the effectiveness for a pyramid network, which is crucial for applying to the
+downstream tasks.
+Dynamic vs Static
+To demonstrate the superiority of the dynamic mechanism, we give a compar-
+ison on the PVT framework with different model sizes in Fig. 5(a). For convenience, we fix the
+budget constraint γ at 0.5. Our dynamic grained encoder can reduce the computational complexity
+by half while maintaining comparable performance. On the other hand, with similar computational
+complexity, our method can improve the static transformers by up to 4.4%. The results demonstrate
+the effectiveness of our method even on the efficient vision transformers. In addition, as shown in
+Fig. 5(c), we calculate the complexity ratio of each layer in DeiT-S with DGE, where the complexity
+of the network in the middle layers varies significantly due to the dynamic mechanism. Interestingly,
+the deeper layer has lower average computational complexity, which means the deeper layer tends
+to assign fewer queries. Thus, DeiT is turned into a dynamic feature pyramid structure, which is
+consistent with the observation in CNNs.
+Budget Constraint and Candidate Granularity Set
+As illustrated in Fig. 5(b), we give a com-
+parison of varying the budget constraints γ, which is selected from {0.25, 0.5, 0.75, 1.0} respectively.
+The redundancy in space allows the network to achieve comparable performance with much less
+computational cost even on the efficient transformers, e.g., PVT and DPVT. Our encoder achieves the
+optimal balance between effectiveness and efficiency when the budget is about half. Therefore, we set
+the budget constraint to 0.5 for other experiments by default. In addition, we report the performance
+of PVT-S with DGE with different candidate granularity set Φ in Tab. 1. When Φ = {0, 1}, the
+gating indices degenerate into a learnable binary mask similar to dynamic convolutions [10,43], but
+this strategy results in significant performance degradation. There is no significant difference in
+performance between other granularity settings. The performance is highest when Φ = {1, 2, 4},
+which becomes our default setting.
+Region-wise Routing vs Layer-wise Routing
+The Fig. 4 clearly demonstrates that DGE can
+perform dynamic granularity in space to adapt to different object structures. Nevertheless, most
+previous dynamic networks are based on layer-wise routing [32]. To demonstrate the advantages of
+our method, we set the region size S × S to the input feature size so that DGE can be degraded from
+region-wise routing to layer-wise routing. As shown in Tab. 1, region-wise gating achieves 1.1%
+absolute gains over layer-wise gating with similar complexity, which agrees well with the empirical
+analysis in Sec.3.1.
+8
+
+Table 2: Performance of dynamic grained encoder on COCO val set. All experiments are conducted
+with 1x schedule [57]. Time and FLOPs are measured on an 800 × 1280 image. "C" and "G" indicate
+the backbone latency on CPU (Xeon 6130) and GPU (Tesla V100). All the budget for DGE is 0.5.
+Backbone
+Size
+#Param
+Latency
+FLOPS
+Mask R-CNN(1x)
+(M)
+C(ms)
+G(ms)
+(G)
+APb
+AP50
+b
+AP75
+b
+APm
+AP50
+m
+AP75
+m
+ResNet
+50
+44.2
+-
+-
+189
+38.0
+59.6
+41.4
+34.4
+55.1
+36.7
+PVT
+Small
+44.3
+880
+33
+251
+40.4
+62.9
+43.8
+37.8
+60.1
+40.3
+PVT+DGE
+Small
+44.3
+440
+26
+185
+40.1
+62.6
+43.2
+37.5
+59.7
+40.0
+DPVT
+Small
+37.7
+1090
+50
+186
+44.0
+65.9
+48.2
+40.3
+62.9
+43.4
+DPVT+DGE
+Small
+37.7
+720
+34
+147
+43.8
+65.7
+47.7
+40.0
+62.6
+43.2
+ResNet
+101
+63.2
+-
+-
+263
+40.4
+61.1
+44.2
+36.4
+57.7
+38.8
+ResNeXt
+101(32x4)
+62.8
+-
+-
+354
+41.9
+62.5
+45.9
+37.5
+59.4
+40.2
+PVT
+Medium
+63.9
+1260
+73
+339
+42.0
+64.4
+45.6
+39.0
+61.6
+42.1
+PVT+DGE
+Medium
+63.9
+620
+40
+228
+41.7
+64.1
+45.0
+38.3
+62.0
+40.6
+DPVT
+Medium
+49.9
+1800
+75
+236
+46.4
+68.0
+51.1
+42.0
+65.2
+45.2
+DPVT+DGE
+Medium
+49.9
+1240
+50
+169
+45.8
+67.2
+50.0
+41.4
+64.5
+44.6
+Table 3: Performance of different backbones for semantic
+segmentation on ADE-20K val set.The inference time
+(backbone) is measured for a 512 × 2048 input image.
+"C" and "G" indicate the latency on CPU and GPU.
+Backbone
+#Param
+FLOPs
+mIoU
+Latency
+(M)
+(G)
+(%)
+C(ms)
+G(ms)
+PVT-S
+28.2
+226
+41.8
+1350
+65
+PVT-S+DGE
+28.2
+155
+41.7
+720
+42
+PVT-M
+48.0
+316
+44.0
+1910
+100
+PVT-M+DGE
+48.0
+202
+43.9
+1100
+64
+DPVT-S
+21.7
+157
+44.4
+1470
+55
+DPVT-S+DGE
+21.7
+121
+44.4
+860
+32
+DPVT-M
+34.3
+209
+46.8
+1990
+110
+DPVT-M+DGE
+34.3
+148
+46.1
+1260
+50
+Table 4: Comparisons with state-of-the-art
+vision transformers on ADE-20K val set.
+FLOPs is tested on 512×2048 resolution.
+Backbone
+#Param
+FLOPs
+mIoU
+(M)
+(G)
+(%)
+ResNet-50 [45]
+28.5
+184
+36.7
+PVT-S [3]
+28.2
+226
+41.8
+Swin-Ti [21]
+31.9
+187
+41.5
+Twins-S [60]
+28.3
+174
+43.2
+DPVT-S+DGE
+21.7
+121
+44.4
+ResNet-101 [45]
+47.5
+262
+38.8
+PVT-M [3]
+48.0
+316
+44.0
+Swin-S [21]
+53.2
+280
+44.9
+Twins-B [60]
+60.4
+318
+45.3
+DPVT-M+DGE
+34.3
+148
+46.1
+4.2
+Experiments for Downstream Tasks
+4.2.1
+Object Detection/Instance Segmentation on COCO
+We apply our models for object detection and instance segmentation on the COCO dataset [58]. We
+resize the images so that the shorter side is 768 pixels. All experiments are conducted on 8 GPUs with
+2 images per GPU (effective minibatch size 16) for 90K iterations. The learning rate is initialized to
+1e-4, which is decreased by 10 at the 60K and 80K iteration. Following the settings in PVT [3], we
+report the performance with 1x training schedule [57,59].
+The results are reported in Tab. 2. When equipped with DGE, the PVT-S achieves comparable
+performance at 40.1% APbox with a significant complexity reduction (185G vs 251G) and inference
+speed up by 22%. Even with larger models or different vanilla encoders, our method is still effective
+and efficient. In addition, the proposed vision transformer variant, i.e., DPVT, is also competitive in
+terms of parameter, computational cost and performance. Moreover, DPVT-M+DGE achieves 45.8
+APbox with 169G FLOPs, even efficient than the ResNet-50 backbone.
+4.2.2
+Semantic Segmentation on ADE-20K
+We further evaluate our models as the backbones for Semantic-FPN [61] on ADE-20K [62] dataset.
+All the experiments are based on MM-Segmentation toolkit [63]. In the training phase, we follow
+the settings in PVT [3] and set the learning rate to 1e-4, which gradually decreases to 0 by the poly
+strategy [64]. The images are cropped to 512 × 512 and augmented with random scaling (from 0.5 to
+2.0) and flipping. All models are trained in 80k iterations with a batch size of 32.
+9
+
+We conduct several ablation studies by introducing the DGE block into PVT [3] and our proposed
+DPVT. As shown in Tab. 3, with our dynamic grained encoder, DPVT+DGE and PVT+DGE both
+achieve competitive performance with a significant computation cost reduction by about 30% FLOPs.
+On the other hand, PVT-M+DGE achieves 2.1% mIoU absolute gains over PVT-S but with less
+computational complexity. As illustrated in Fig. 5(d), this phenomenon also occurs for different
+image sizes on the same framework, e.g., our method has up to 1.2% mIoU absolute gains against
+the baseline with similar computational complexity. In addition, as shown in Tab. 4, our DPVT
+models with DGE are superior to the state-of-the-art vision transformers in terms of parameters,
+computational complexity and performance. These results well demonstrate the generalization ability
+and robustness of our method.
+5
+Conclusion
+In this paper, we analyze the spatial redundancy in vision transformers and propose a dynamic grained
+encoder to speed up inference. Our encoder can adaptively yield a suitable number of queries for
+different regions to reduce spatial redundancy while maintaining comparable performance. Besides,
+our encoder is compatible with many efficient transformers and can be trained in an end-to-end
+manner. The extensive experiments demonstrate the effectiveness and generalization of our method.
+In general, this paper explores a new perspective, i.e., leveraging the intrinsic properties of natural
+images with the dynamic network mechanism to achieve efficient vision transformers. We hope that
+our dynamic grained encoder can provide insights into future works and beyond.
+Acknowledgments and Disclosure of Funding
+This research was supported by National Key R&D Program of China (No. 2017YFA0700800),
+National Natural Science Foundation of China (No. 61790563 and 61774125), Shanghai Science and
+Technology Program (No. 21010502700).
+A
+Additional Experiments
+A.1
+Quantitative Analysis on Dynamic Grained Router
+We follow the weakly supervised segmentation [64] to show how well the dense query region captures
+the foreground region. The metric in [64] is used to measure the gating scores in each DGE layer.
+Specifically, we set the candidate granularities Φ to {1, 2}, so that the finer-grained gating scores are
+taken as a soft-segmentation of the image. We adopt the evaluation protocol in [64] to report the
+quantitative segmentation results. As shown in Tab. 5 and Tab. 6, our gating scores have significant
+superiority even over the weakly supervised method, i.e., GradCAM. These results demonstrate that
+the DGE could guide the transformer to focus on the foreground regions, which is consistent with the
+visualization.
+Table 5: The quantitative analysis on DeiT-S with DGE (γ = 0.5).
+Metric
+Random
+GradCAM [64]
+Layer 1
+Layer 4
+Layer 8
+Accuracy
+50.0
+64.4
+55.4
+56.3
+67.6
+mAP
+50.0
+71.6
+63.5
+60.7
+78.8
+mIoU
+31.9
+40.8
+36.4
+37.7
+48.2
+Table 6: The quantitative analysis on PVT-S with DGE (γ = 0.5).
+Metric
+Random
+Layer 1
+Layer 6
+Layer 11
+Layer 16
+Accuracy
+50.0
+55.4
+49.1
+67.8
+65.5
+mAP
+50.0
+68.0
+45.2
+71.3
+79.4
+mIoU
+31.9
+34.5
+32.5
+50.2
+46.6
+10
+
+A.2
+Runtime Analysis on GPUs
+The efficiency of our DGE modules on GPUs mainly relies on the throughput of sparse matrix
+multiplication, which is dependent on hardware architecture and code optimization. To demonstrate
+the potential of our method for parallel devices, we implement an optimized CUDA kernel with
+multiple streams for batched sparse matrix multiplication. With this kernel, we report the runtime
+comparison of different backbones for multiple downstream tasks on a Tesla V100 GPU. The results
+are reported in Tab. 7 and Tab. 8, where the latency indicates the runtime of backbone.
+Table 7: Runtime comparison of MaskRCNN (1x) framework on COCO val set (γ = 0.5).
+Backbone
+APb
+APm
+FLOPs
+Latency (CPU)
+Latency (GPU)
+PVT-S
+40.4
+37.8
+251G
+0.88s
+33ms
+PVT-S+DGE
+40.1
+37.5
+185G
+0.44s
+26ms
+DPVT-S
+44.0
+40.3
+186G
+1.09s
+50ms
+DPVT-S+DGE
+43.8
+40.0
+147G
+0.72s
+34ms
+PVT-M
+42.0
+39.0
+339G
+1.26s
+73ms
+PVT-M+DGE
+41.7
+38.3
+228G
+0.62s
+40ms
+DPVT-M
+46.4
+42.0
+236G
+1.80s
+75ms
+DPVT-M+DGE
+45.8
+41.4
+169G
+1.24s
+50ms
+Table 8: Runtime comparison of Semantic-FPN framework on ADE20K val set (γ = 0.5).
+Backbone
+mIoU
+FLOPs
+Latency (CPU)
+Latency (GPU)
+PVT-S
+41.8
+226G
+1.35s
+65ms
+PVT-S+DGE
+41.7
+155G
+0.72s
+42ms
+DPVT-S
+44.4
+157G
+1.47s
+55ms
+DPVT-S+DGE
+44.4
+121G
+0.86s
+32ms
+PVT-M
+44.0
+316G
+1.91s
+100ms
+PVT-M+DGE
+43.9
+202G
+1.10s
+64ms
+DPVT-M
+46.8
+209G
+1.99s
+110ms
+DPVT-M+DGE
+46.1
+148G
+1.26s
+50ms
+A.3
+Implementation Details for Complexity Computation
+We report the FLOPs following the conventional protocol of dynamic networks [32]. Specifically,
+we split the entire network into static and dynamic parts. The complexity of the static part, i.e., the
+modules without dynamic mechanism including the gating networks in DGE, is computed in the
+standard way [1,3,19]. For the complexity of the dynamic part, i.e., the dynamic modules in DGE,
+we accumulate the complexity associate with each enabled query according to the gating indices.
+References
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+A Stochastic Proximal Polyak Step Size
+Fabian Schaipp
+schaippf@ma.tum.de
+Department of Mathematics
+Technical University of Munich
+Robert M. Gower
+rgower@flatironinstitute.org
+Center for Computational Mathematics
+Flatiron Institute, New York
+Michael Ulbrich
+mulbrich@ma.tum.de
+Department of Mathematics
+Technical University of Munich
+Abstract
+Recently, the stochastic Polyak step size (SPS) has emerged as a competitive adap-
+tive step size scheme for stochastic gradient descent. Here we develop ProxSPS, a
+proximal variant of SPS that can handle regularization terms. Developing a prox-
+imal variant of SPS is particularly important, since SPS requires a lower bound of
+the objective function to work well. When the objective function is the sum of a
+loss and a regularizer, available estimates of a lower bound of the sum can be loose.
+In contrast, ProxSPS only requires a lower bound for the loss which is often readily
+available.
+As a consequence, we show that ProxSPS is easier to tune and more
+stable in the presence of regularization. Furthermore for image classification tasks,
+ProxSPS performs as well as AdamW with little to no tuning, and results in a network
+with smaller weight parameters. We also provide an extensive convergence analysis
+for ProxSPS that includes the non-smooth, smooth, weakly convex and strongly
+convex setting.
+1
+Introduction
+Consider problems of the form
+min
+x∈Rn f(x),
+f(x) := EP [f(x; S)] =
+�
+S
+f(x; s)dP(s),
+(1)
+where S is a sample space (or sample set). Formally, we can see S as a random variable mapping
+to S and P(s) as the associated probability measure.
+Let us assume that for each s ∈ S, the
+function f(·; s) : Rn → R is locally Lipschitz and hence possesses the Clarke subdifferential ∂f(·; s)
+(Clarke, 1983). Problems of form (1) arise in machine learning tasks where S is the space of available
+data points (Bottou et al., 2018). An efficient method for such problems is stochastic (sub)gradient
+descent (Robbins & Monro, 1951; Bottou, 2010; Davis & Drusvyatskiy, 2019), given by
+xk+1 = xk − αkgk,
+gk ∈ ∂f(xk; Sk),
+where Sk ∼ P.
+(SGD)
+1
+arXiv:2301.04935v1  [math.OC]  12 Jan 2023
+
+Moreover, we will also consider the composite problem
+min
+x∈Rn ψ(x),
+ψ(x) := f(x) + ϕ(x),
+(2)
+where ϕ : Rn → R ∪ {∞} is a proper, closed, and convex regularization function.
+In practical
+situations, the expectation in the objective function f is typically approximated by a sample average
+over N ∈ N data points. We formalize this special case with
+S = {s1, . . . , sN}, P(si) = 1
+N , fi := f(·; si)
+i = 1, . . . , N.
+(ER)
+In this case, problem (1) becomes the empirical risk minimization problem
+min
+x∈Rn
+1
+N
+N
+�
+i=1
+fi(x).
+1.1
+Background and Contributions
+Polyak step size. For minimizing a convex, possibly non-differentiable function f, Polyak (1987,
+Chapter 5.3) proposed
+xk+1 = xk − αkgk,
+αk = f(xk) − min f
+∥gk∥2
+,
+gk ∈ ∂f(xk) \ {0}.
+This particular choice of αk, requiring the knowledge of min f, has been subsequently called the
+Polyak step size for the subgradient method. Recently, Berrada et al. (2019); Loizou et al. (2021);
+Orvieto et al. (2022) adapted the Polyak step size to the stochastic setting: consider the (ER) case
+and assume that each fi is differentiable and that a lower bound C(si) ≤ infx fi(x) is known for all
+i ∈ [N]. The method proposed by (Loizou et al., 2021) is
+xk+1 = xk − min
+�
+γb, fik(xk) − C(sik)
+c∥∇fik(xk)∥2
+�
+∇fik(xk),
+(SPSmax)
+with hyper-parameters c, γb > 0 and where in each iteration ik is drawn from {1, . . . , N} uniformly at
+random. It is important to note that the initial work (Loizou et al., 2021) used C(si) = inf fi; later,
+Orvieto et al. (2022) established theory for (SPSmax) for the more general case of C(si) ≤ infx fi(x)
+and allowing for mini-batching. Other works analyzed the Polyak step size in the convex, smooth
+setting (Hazan & Kakade, 2019) and in the convex, smooth and stochastic setting (Prazeres &
+Oberman, 2021). Further, the stochastic Polyak step size is closely related to stochastic model-
+based proximal point (Asi & Duchi, 2019) as well as stochastic bundle methods (Paren et al., 2022).
+Contribution. We propose a proximal version of the stochastic Polyak step size, called ProxSPS,
+which explicitly handles regularization functions. Our proposal is based crucially on the fact that
+the stochastic Polyak step size can be motivated with stochastic proximal point for a truncated linear
+model of the objective function (we explain this in detail in Section 3.1). Our method has closed-form
+updates for squared ℓ2-regularization. We provide theoretical guarantees for ProxSPS for any closed,
+proper, and convex regularization function (including indicator functions for constraints). Our main
+results, Theorem 7 and Theorem 8, also give new insights for SPSmax, in particular showing exact
+convergence for convex and non-convex settings.
+2
+
+Lower bounds and regularization. Methods such as SPSmax need to estimate a lower bound C(s)
+for each loss function f(·; s). Though infx f(x; s) can be precomputed in some restricted settings,
+in practice the lower bound C(s) = 0 is used for non-negative loss functions.1 The tightness of the
+choice C(s) is further reflected in the constant σ2 := min f −EP [C(S)], which affects the convergence
+guarantees of SPSmax (Orvieto et al., 2022).
+Contribution. For regularized problems (2) and if ϕ is differentiable, the current proposal of SPSmax
+would add ϕ to every loss function f(·; s).
+In this case, for non-negative regularization terms,
+such as the squared ℓ2-norm, the lower bound C(s) = 0 is always loose. Indeed, if ϕ ≥ 0, then
+infx∈Rn(f(x; s)+ϕ(x)) ≥ infx∈Rn f(x; s) and this inequality is strict in most practical scenarios. For
+our proposed method ProxSPS, we now need only estimate a lower bound for the loss f(x; s) and
+not for the composite function f(x; s) + ϕ(x). Further, ProxSPS decouples the adaptive step size for
+the gradient of the loss from the regularization (we explain this in detail in Section 4.1 and Fig. 1).
+Proximal and adaptive methods.
+The question on how to handle regularization terms has
+also been posed for other families of adaptive methods. For Adam (Kingma & Ba, 2015) with ℓ2-
+regularization it has been observed that it generalizes worse and is harder to tune than AdamW
+(Loshchilov & Hutter, 2019) which uses weight decay. Further, AdamW can be seen as an approx-
+imation to a proximal version of Adam (Zhuang et al., 2022).2 On the other hand, Loizou et al.
+(2021) showed that – without regularization – default hyperparameter settings for SPSmax give very
+encouraging results on matrix factorization and image classification tasks. This is promising since
+it suggests that SPSmax is an adaptive method, and can work well across varied tasks without the
+need for extensive hyperparameter tuning.
+Contribution.
+We show that by handling ℓ2-regularization using a proximal step, our resulting
+ProxSPS is less sensitive to hyperparameter choice as compared to SPSmax. This becomes apparent
+in matrix factorization problems, where ProxSPS converges for a wide range of regularization pa-
+rameters and learning rates, while SPSmax is more sensitive to these settings. We also show similar
+results for image classification over the CIFAR10 dataset when using a ResNet56 and ResNet110
+model, where we also compare our method to AdamW.
+The remainder of our paper is organized as follows: we will first recall how the stochastic Polyak
+step size, in the case of ϕ = 0, can be derived using the model-based approach of (Asi & Duchi,
+2019; Davis & Drusvyatskiy, 2019) and how this is connected to SPSmax. We then derive ProxSPS
+based on the connection to model-based methods, and present our theoretical results, based on the
+proof techniques in (Davis & Drusvyatskiy, 2019).
+2
+Preliminaries
+2.1
+Notation
+Throughout, we will write E instead of EP . For any random variable X(s), we denote E[X(S)] :=
+�
+S X(s)dP(s). We denote (·)+ := max{·, 0}. We write ˜O when we drop logarithmic terms in the
+O-notation, e.g. ˜O( 1
+K ) = O( ln(1+K)
+K
+).
+1See for instance https://github.com/IssamLaradji/sps.
+2For SGD treating ℓ2-regularization as a part of the loss can be seen to be equivalent to its proximal version (cf.
+Appendix C).
+3
+
+2.2
+General assumptions
+Throughout the article, we assume the following:
+Assumption 1. It is possible to generate infinitely many i.i.d. realizations S1, S2, . . . from S.
+Assumption 2. For every s ∈ S there exists C(s) satisfying C(s) ≤ infx f(x; s).
+In many machine learning applications, non-negative loss functions are used and thus we can satisfy
+the second assumption choosing C(s) = 0 for all s ∈ S.
+2.3
+Convex analysis
+Let g : Rn → R be convex and α > 0. The proximal operator is given by
+proxαg(x) := arg min
+y
+g(y) + 1
+2α∥y − x∥2.
+Further, the Moreau envelope is defined by envα
+g (x) := miny g(y) +
+1
+2α∥y − x∥2, and its derivative
+is ∇envα
+g (x) = 1
+α(x − proxαg(x)) (Drusvyatskiy & Paquette, 2019, Lem. 2.1). Moreover, due to the
+optimality conditions of the proximal operator, if g ∈ C1 then
+ˆx = proxαg(x) =⇒ ∥∇g(ˆx)∥ = α−1∥x − ˆx∥ = ∥∇envα
+g (x)∥.
+(3)
+Davis & Drusvyatskiy (2019) showed how to use the Moreau envelope as a measure of stationarity:
+if ∥∇envα
+g (x)∥ is small, then x is close to ˆx and ˆx is an almost stationary point of g. Formally, the
+gradient of the Moreau envelope can be related to the gradient mapping (cf. (Drusvyatskiy & Lewis,
+2018, Thm. 3.5) and Lemma 11).
+We say that a function g : Rn → R is L-smooth if its gradient is L–Lipschitz, that is
+∥∇g(x) − ∇g(y)∥ ≤ L∥x − y∥,
+∀x, y ∈ Rn.
+(4)
+If g is L-smooth, then
+g(y) ≤ g(x) + ⟨∇g(x), y − x⟩ + L
+2 ∥y − x∥2
+for all x, y, ∈ Rn.
+A function g : Rn → R is ρ–weakly convex for ρ ≥ 0 if g + ρ
+2∥ · ∥2 is convex. Any L–smooth function
+is weakly convex with parameter less than or equal to L (Drusvyatskiy & Paquette, 2019, Lem. 4.2).
+The above results on the proximal operator and Moreau envelope can immediately be extended to
+g being ρ–weakly convex if α ∈ (0, ρ−1), since then g + ρ
+2∥ · ∥2 is convex.
+If we assume that each f(·; s) is ρs-weakly convex for ρs ≥ 0, then applying (Bertsekas, 1973, Lem.
+2.1) to the convex function f(·; s) + ρs
+2 ∥ · ∥2 yields that f + ρ
+2∥ · ∥2 is convex and thus f is ρ–weakly
+convex for ρ := E[ρS]. In particular, f is convex if each f(·; s) is assumed to be convex. For a weakly
+convex function g, we denote with ∂g the regular subdifferential (cf. (Davis & Drusvyatskiy, 2019,
+section 2.2) and (Rockafellar & Wets, 1998, Def. 8.3)).
+3
+The unregularized case
+For this section, consider problems of form (1), i.e. no regularization term ϕ is added to the loss f.
+4
+
+3.1
+A model-based view point
+Many classical methods for solving (1) can be summarized by model-based stochastic proximal point:
+in each iteration, a model fx(·; s) is constructed approximating f(·; s) locally around x. With Sk ∼ P
+being drawn at random, this yields the update
+xk+1 = arg min
+y∈Rn fxk(y; Sk) +
+1
+2αk
+∥y − xk∥2.
+(5)
+The theoretical foundation for this family of methods has been established by Asi & Duchi (2019)
+and Davis & Drusvyatskiy (2019). They give the following three models as examples:
+(i) Linear: fx(y; s) := f(x; s) + ⟨g, y − x⟩ with g ∈ ∂f(x; s).
+(ii) Full: fx(y; s) := f(y; s).
+(iii) Truncated: fx(y; s) := max{f(x; s) + ⟨g, y − x⟩, infz∈Rn f(z; s)} where g ∈ ∂f(x; s).
+It is easy to see that update (5) for the linear model is equal to (SGD) while the full model results in
+the stochastic proximal point method. For the truncated model, (5) results in the update
+xk+1 = xk − min
+�
+αk, f(xk; Sk) − infz∈Rn f(z; Sk)
+∥gk∥2
+�
+gk,
+gk ∈ ∂f(xk, Sk).
+(6)
+More generally, one can replace the term infx∈Rn f(x; Sk) with an arbitrary lower bound of f(·; Sk)
+(cf. Lemma 10). The model-based stochastic proximal point method for the truncated model is
+given in Algorithm 1. The connection between the truncated model and the method depicted in
+(6) is not a new insight and has been pointed out in several works (including (Asi & Duchi, 2019;
+Loizou et al., 2021) and (Berrada et al., 2019, Prop. 1)). For simplicity, we refer to Algorithm 1 as
+SPS throughout this article. However, it should be pointed out that this acronym (and variations of
+it) have been used for stochastic Polyak-type methods in slightly different ways (Loizou et al., 2021;
+Gower et al., 2021).
+Algorithm 1 SPS
+Require: x0 ∈ Rn, step sizes αk > 0.
+for k = 0, 1, 2, . . . , K − 1 do
+1. Sample Sk and set Ck := C(Sk).
+2. Choose gk ∈ ∂f(xk; Sk). If gk = 0, set xk+1 = xk. Otherwise, set
+xk+1 = xk − γkgk,
+γk = min
+�
+αk, f(xk; Sk) − Ck
+∥gk∥2
+�
+.
+(7)
+return xK
+For instance consider again the SPSmax method
+xk+1 = xk − min
+�
+γb, fik(xk) − C(sik)
+c∥∇fik(xk)∥2
+�
+∇fik(xk),
+(SPSmax)
+5
+
+where c, γb > 0. Clearly, for c = 1 and αk = γb, update (7) is identical to SPSmax. With this in mind,
+we can interpret the hyperparameter γb in SPSmax simply as a step size for the model-based stochastic
+proximal point step. For the parameter c on the other hand, the model-based approach motivates the
+choice c = 1, which potentially reduces the amount of hyperparameter tuning. However, according
+to (Loizou et al., 2021), c = 1/2 gives the best rate of convergence in the strongly convex case.
+4
+The regularized case
+Now we consider regularized problems of the form (2), i.e.
+min
+x∈Rn ψ(x),
+ψ(x) = f(x) + ϕ(x),
+where ϕ : Rn → R ∪ {∞} is a proper, closed, λ-strongly convex function for λ ≥ 0 (we allow
+λ = 0). For s ∈ S, denote by ψx(·; s) a stochastic model of the objective ψ at x. We aim to analyze
+algorithms with the update
+xk+1 = arg min
+x∈Rn ψxk(x; Sk) +
+1
+2αk
+∥x − xk∥2,
+(8)
+where Sk ∼ P and αk > 0. Naively, if we know a lower bound ˜C(s) of f(·; s) + ϕ(·), the truncated
+model could be constructed for the function f(x; s) + ϕ(x), resulting in
+ψx(y; s) = max{f(x; s) + ϕ(x) + ⟨g + u, y − x⟩, ˜C(s)},
+g ∈ ∂f(x; s),
+u ∈ ∂ϕ(x).
+(9)
+In fact, Asi & Duchi (2019) and Loizou et al. (2021) work in the setting of unregularized problems
+and hence their approaches would handle regularization in this way. What we propose instead, is to
+only truncate a linearization of the loss f(x; s), yielding the model
+ψx(y; s) = fx(y; s) + ϕ(y),
+fx(y; s) = max{f(x; s) + ⟨g, y − x⟩, C(s)},
+g ∈ ∂f(x; s).
+(10)
+Solving (8) with the model in (10) results in
+xk+1 = arg min
+y∈Rn
+max{f(xk; Sk) + ⟨gk, y − xk⟩, C(Sk)} + ϕ(y) +
+1
+2αk
+∥y − xk∥2.
+(11)
+The resulting model-based stochastic proximal point method is given in Algorithm 2 3. Lemma 12
+shows that, if proxϕ is known, update (11) can be computed by minimizing a strongly convex function
+over a compact one-dimensional interval. The relation to the proximal operator of ϕ motivates the
+name ProxSPS. Further, the ProxSPS update (11) has a closed form solution when ϕ is the squared
+ℓ2-norm, as we detail in the next section.
+Algorithm 2 ProxSPS
+Require: x0 ∈ Rn, step sizes αk > 0.
+for k = 0, 1, 2, . . . , K − 1 do
+1. Sample Sk and set Ck := C(Sk).
+2. Choose gk ∈ ∂f(xk; Sk).
+Update xk+1 according to (11).
+return xK
+3For ϕ = 0, Algorithm 2 is identical to Algorithm 1.
+6
+
+−6
+−4
+−2
+0
+2
+4
+6
+−1
+0
+1
+2
+3
+4
+5
+6
+x0
+x⋆
+ˆx1
+x1
+f(x; s) = ln(1 + exp(−0.5 · x)), α = 10.0, λ = 0.1
+f(·; s) + ϕ
+ProxSPS model
+SPS model
+ProxSPS objective
+SPS objective
+(a) Regularized logistic loss.
+−2
+0
+2
+−3
+−2
+−1
+0
+1
+2
+3
+ProxSPS
+−2
+0
+2
+−3
+−2
+−1
+0
+1
+2
+3
+SPS
+(b) Regularized squared loss with αk = 1, λ = 1.
+Figure 1: a) SPS refers to model (9) whereas ProxSPS refers to (10). We plot the corresponding
+model ψx0(y; s) and the objective function of (8). x1 (resp. ˆx1) denotes the new iterate for ProxSPS
+(resp. SPS), x⋆ is the minimizer of f(·; s) + ϕ. b) Streamlines of the vector field V (xk) := xk+1 − xk,
+for f(x) = ∥Ax − b∥2 and for the deterministic update, i.e. f(x; s) = f(x). ProxSPS refers to update
+(14) and SPS refers to (13). The circle marks the minimizer of f(x) + λ
+2 ∥x∥2.
+4.1
+The special case of ℓ2-regularization
+When ϕ(x) = λ
+2 ∥x∥2 for some λ > 0, ProxSPS (11) has a closed form solution as we show next
+in Lemma 1.
+For this lemma, recall that the proximal operator of ϕ(x) =
+λ
+2 ∥x∥2 is given by
+proxαϕ(x) =
+1
+1+αλx for all α > 0, x ∈ Rn.
+Lemma 1. Let ϕ(x) = λ
+2 ∥x∥2 and let g ∈ ∂f(x; s) and C(s) ≤ infz∈Rn f(z; s) hold for all s ∈ S.
+For ψx(y; s) = fx(y; s) + ϕ(y) with fx(y; s) = max{f(x; s) + ⟨g, y − x⟩, C(s)} consider the update
+xk+1 = arg min
+x∈Rn
+ψxk(x; Sk) +
+1
+2αk
+∥x − xk∥2.
+Denote Ck := C(Sk) and let gk ∈ ∂f(xk; Sk). Define
+τ +
+k :=
+�
+�
+�
+0
+if gk = 0,
+min
+�
+αk,
+�
+(1+αkλ)(f(xk;Sk)−Ck)−αkλ⟨gk,xk⟩
+∥gk∥2
+�
++
+�
+else.
+Update (11) is given by
+xk+1 =
+1
+1 + αkλ
+�
+xk − τ +
+k gk
+�
+= proxαkϕ(xk − τ +
+k gk).
+(12)
+See Lemma 9 in the appendix for an extended version of the above lemma and its proof.
+The
+update (12) can be naturally decomposed into two steps, one stochastic gradient step with an
+adaptive stepsize, that is ¯xk+1 = xk −τ +
+k gk followed by a proximal step xk+1 = proxαkϕ(¯xk+1). This
+decoupling into two steps, makes it easier to interpret the effect of each step, with τ +
+k adjusting for
+the scale/curvature and the following proximal step shrinking the resulting parameters. There is no
+clear separation of tasks if we apply the SPS method to the regularized problem, as we see next.
+7
+
+Algorithm 3 ProxSPS for ϕ = λ
+2 ∥ · ∥2
+Require: x0 ∈ Rn, step sizes αk > 0.
+for k = 0, 1, 2, . . . , K − 1 do
+1. Sample Sk and set Ck := C(Sk).
+2. Choose gk ∈ ∂f(xk; Sk). If gk = 0, set xk+1 =
+1
+1+αkλxk. Otherwise, set
+xk+1 =
+1
+1 + αkλ
+�
+xk − min
+�
+αk,
+�(1 + αkλ)(f(xk; Sk) − Ck) − αkλ⟨gk, xk⟩
+∥gk∥2
+�
++
+�
+gk
+�
+.
+return xK
+4.2
+Comparing the model of SPS and ProxSPS
+For simplicity, assume again the discrete sample space setting (ER) with differentiable loss functions
+fi and let ϕ = λ
+2 ∥ · ∥2. Clearly, the composite problem (2) can be transformed to an instance of (1)
+by setting ℓi(x) := fi(x) + λ
+2 ∥x∥2 and solving minx ℓ(x) with ℓ(x) :=
+1
+N
+�N
+i=1 ℓi(x). Assume that a
+lower bound ℓi ≤ infx ℓi(x) is known. In this case (9) becomes
+ψx(y; si) = max
+�
+fi(x) + λ
+2 ∥x∥2 + ⟨∇fi(x) + λx, y − x⟩, ℓi
+�
+.
+Due to Lemma 10, if ∇fik(xk) + λxk ̸= 0, the update (8) is given by
+xk+1 = xk − min
+�
+αk, fik(xk) + λ
+2 ∥xk∥2 − ℓik
+∥∇fik(xk) + λxk∥2
+�
+(∇fik(xk) + λxk).
+(13)
+We refer to this method, which is using model (9), as SPS. On the other hand, using model (10) and
+if ∇fik(xk) ̸= 0, the update of ProxSPS (12) is
+xk+1 =
+1
+1+αkλ
+�
+xk − min
+�
+αk,
+� (1+αkλ)(fik (xk)−C(sik ))−αkλ⟨∇fik (xk),xk⟩
+∥∇fik (xk)∥2
+�
++
+�
+∇fik(xk)
+�
+.
+(14)
+In Fig. 1a, we illustrate the two models (9) (denoted by SPS) and (10) (denoted by ProxSPS) for the
+logistic loss with squared ℓ2-regularization. We can see that the ProxSPS model is a much better
+approximation of the (stochastic) objective function as it still captures the quadratic behaviour of ϕ.
+Furthermore, as noted in the previous section, ProxSPS decouples the step size of the gradient and
+of the shrinkage, and hence the update direction depends on αk. In contrast, the update direction of
+SPS does not depend on αk, and the regularization effect is intertwined with the adaptive step size.
+Another way to see that the model (10) on which ProxSPS is based on is a more accurate model
+as compared to the SPS model (9), is that the resulting vector field of ProxSPS takes a more direct
+route to the minimum, as illustrated in Fig. 1b.
+Update (14) needs to compute the term ⟨∇fik(xk), xk⟩ while (13) needs to evaluate ∥xk∥2. Other
+than that, the computational costs are roughly identical. For (14), a lower bound ℓi is required. For
+non-negative loss functions, in practice both ℓi and C(si) are often set to zero, in which case (10)
+will be a more accurate model as compared to (9). 4
+4For single element sampling, inf ℓi can sometimes be precomputed (e.g. regularized logistic regression, see (Loizou
+et al., 2021, Appendix D)). But even in this restricted setting it is not clear how to estimate inf ℓi when using
+mini-batching.
+8
+
+4.3
+Convergence analysis
+For the convergence analysis of Algorithm 2, we can work with the following assumption on ϕ.
+Assumption 3. ϕ : Rn → R ∪ {∞} is a proper, closed, λ-strongly convex function with λ ≥ 0.
+Throughout this section we consider model (10), i.e. for g ∈ ∂f(x; s), let
+ψx(y; s) = fx(y; s) + ϕ(y),
+fx(y; s) = max{f(x; s) + ⟨g, y − x⟩, C(s)}.
+Let us first state a lemma on important properties of the truncated model:
+Lemma 2. Consider fx(y; s) = max{f(x; s) + ⟨g, y − x⟩, C(s)}, where g ∈ ∂f(x; s) is arbitrary and
+C(s) ≤ infz∈Rn f(z; s). Then, it holds:
+(i) The mapping y �→ fx(y; s) is convex.
+(ii) For all x ∈ Rn, it holds fx(x; s) = f(x; s). If f(·; s) is ρs–weakly convex for all s ∈ S, then
+fx(y; s) ≤ f(y; s) + ρs
+2 ∥y − x∥2
+for all x, y ∈ Rn.
+Proof.
+(i) The maximum over a constant and linear term is convex.
+(ii) Recall that C(s) ≤ f(y; s) for all y ∈ Rn. Therefore, fx(x; s) = max{C(s), f(x; s)} = f(x; s).
+From weak convexity of f(·; s) it follows f(x; s)+⟨g, y−x⟩ ≤ f(y; s)+ ρs
+2 ∥y−x∥2 and therefore
+fx(y; s) ≤ max{C(s), f(y; s) + ρs
+2 ∥y − x∥2} = f(y; s) + ρs
+2 ∥y − x∥2
+for all y ∈ Rn.
+4.3.1
+Globally bounded subgradients
+In this section, we show that the results for stochastic model-based proximal point methods in
+Davis & Drusvyatskiy (2019) can be immediately applied to our specific model – even though this
+model has not been explicitly analyzed in their article. This, however, requires assuming that the
+subgradients are bounded.
+Proposition 3. Let Assumption 3 hold and assume that there is an open, convex set U containing
+dom ϕ. Let f(·; s) be ρs–weakly convex for all s ∈ S and let ρ = E[ρS]. Assume that there exists
+Gs ∈ R+ for all s ∈ S, such that G :=
+�
+E[G2
+S] < ∞ and
+∥g(x; s)∥ ≤ Gs
+∀g(x; s) ∈ ∂f(x; s), ∀x ∈ U.
+(15)
+Then, ψx(y; s) satisfies (Davis & Drusvyatskiy, 2019, Assum. B).
+Proof. We verify properties (B1)–(B4) of (Davis & Drusvyatskiy, 2019, Assum. B), for r = ϕ and
+fx(·, ξ) = fx(·; s): (B1) is identical to Assumption 1. (B2) holds with τ = ρ due to our assumptions
+on ϕ, and Lemma 2, (ii) and the definition of f, i.e. f(x) = E[f(x; S)]. Due to Lemma 2, (i) and
+convexity of ϕ, the mapping fx(·; s)+ϕ(·) is convex which shows (B3) for η = 0 . Now, for arbitrary
+g ∈ ∂f(x; s) and x, y ∈ U, we have
+fx(x; s) − fx(y; s) ≤ f(x; s) − f(x; s) − ⟨g, y − x⟩ ≤ ∥g∥∥y − x∥ ≤ Gs∥y − x∥.
+Hence, (B4) holds for L = G and L(ξ) = Gs.
+9
+
+Corollary 4 (Weakly convex case). Let the assumptions of Proposition 3 hold with ρs > 0 for all
+s ∈ S. Let ρ = E[ρS] < ∞ and let ∆ ≥ env1/(2ρ)
+ψ
+(x0) − min ψ. Let {xk}k=0,...,K be generated by
+Algorithm 2 for constant step sizes αk =
+�
+2ρ +
+�
+4ρG2K
+∆
+�−1
+. Then, it holds
+E∥∇env1/(2ρ)
+ψ
+(xK
+∼)∥2 ≤ 8ρ∆
+K
++ 16G
+�
+ρ∆
+K ,
+where xK
+∼ is uniformly drawn from {x0, . . . , xK−1}.
+Proof. The claim follows from Proposition 3 and (Davis & Drusvyatskiy, 2019, Thm. 4.3), (4.16)
+setting η = 0, ¯ρ = 2ρ, T = K − 1 and βt = α−1
+k .
+Corollary 5 ((Strongly) convex case). Let the assumptions of Proposition 3 hold with ρs = 0 for
+all s ∈ S. Let λ > 0 and x⋆ = arg minx ψ(x). Let {xk}k=0,...,K be generated by Algorithm 2 for step
+sizes αk =
+2
+λ(k+1). Then, it holds
+E
+�
+ψ
+�
+2
+(K+1)(K+2)−2
+K
+�
+k=1
+(k + 1)xk�
+− ψ(x⋆)
+�
+≤
+λ
+(K + 1)2 ∥x0 − x⋆∥2 +
+8G2
+λ(K + 1).
+Proof. As ρs = 0 and hence ρ = 0, from the proof of Proposition 3 we have that (Davis & Drusvy-
+atskiy, 2019, Assum. B) is satisfied with τ = 0 (in the notation of (Davis & Drusvyatskiy, 2019)).
+Moreover, by Lemma 2, (i) and λ–strong convexity of ϕ, we have λ–strong convexity of ψx(·; s).
+The claim follows from Proposition 3 and (Davis & Drusvyatskiy, 2019, Thm. 4.5) setting µ = λ,
+T = K − 1 and βt = α−1
+k .
+4.3.2
+Lipschitz smoothness
+Assumption (15), i.e. having globally bounded subgradients, is strong: it implies Lipschitz continuity
+of f (cf. (Davis & Drusvyatskiy, 2019, Lem. 4.1)) and simple functions such as the squared loss do
+not satisfy this.
+Therefore, we provide additional guarantees for the smooth case, without the
+assumption of globally bounded gradients.
+The following result, similar to (Davis & Drusvyatskiy, 2019, Lem. 4.2), is the basic inequality for
+the subsequent convergence analysis.
+Lemma 6. Let Assumption 3 hold. Let xk+1 be given by (11) and ψxk be given in (10). For every
+x ∈ Rn it holds
+(1 + αkλ)∥xk+1 − x∥2 ≤ ∥xk − x∥2 − ∥xk+1 − xk∥2 + 2αk
+�
+ψxk(x; Sk) − ψxk(xk+1; Sk)
+�
+.
+(16)
+Moreover, it holds
+ψxk(xk+1; Sk) ≥ f(xk; Sk) + ⟨gk, xk+1 − xk⟩ + ϕ(xk+1).
+(17)
+Proof. The objective of (11) is given by Ψk(y) := ψxk(y; Sk) +
+1
+2αk ∥y − xk∥2. Using Lemma 2, (i)
+and λ-strong convexity of ϕ, Ψk(y) is (λ +
+1
+αk )–strongly convex. As xk+1 is the minimizer of Ψk(y),
+for all x ∈ Rn we have
+Ψk(x) ≥ Ψk(xk+1) + 1 + αkλ
+2αk
+∥xk+1 − x∥2 ⇐⇒
+(1 + αkλ)∥xk+1 − x∥2 ≤ ∥xk − x∥2 − ∥xk+1 − xk∥2 + 2αk
+�
+ψxk(x; Sk) − ψxk(xk+1; Sk)
+�
+.
+10
+
+Moreover, by definition of fx(y; s) in (10) we have
+ψxk(xk+1; Sk) = fxk(xk+1; Sk) + ϕ(xk+1) ≥ f(xk; Sk) + ⟨gk, xk+1 − xk⟩ + ϕ(xk+1).
+We will work in the setting of differentiable loss functions with bounded gradient noise.
+Assumption 4. The mapping f(·; s) is differentiable for all s ∈ S and there exists β ≥ 0 such that
+E∥∇f(x; S) − ∇f(x)∥2 ≤ β
+for all x ∈ Rn.
+(18)
+The assumption of bounded gradient noise (18) (in the differentiable setting) is indeed a weaker
+assumption than (15) since E[∇f(x; S)] = ∇f(x) and
+E∥∇f(x; S) − ∇f(x)∥2 ≤ β ⇐⇒ E∥∇f(x; S)∥2 ≤ ∥∇f(x)∥2 + β.
+Remark 1. Assumption 4 (and the subsequent theorems) could be adapted to the case where f(·; s)
+is weakly convex but non-differentiable: for fixed x ∈ Rn, due to (Bertsekas, 1973, Prop. 2.2) and
+(Davis & Drusvyatskiy, 2019, Lem. 2.1) it holds
+E[∂f(x; S)] = E
+�
+∂
+�
+f(x; S) + ρS
+2 ∥x∥2�
+− ρSx
+�
+= ∂f(x) + ρx − E[ρSx] = ∂f(x),
+where we used ρ = E[ρS]. Hence, for gs ∈ ∂f(x; s) we have E[gS] ∈ ∂f(x) and (18) is replaced by
+E∥gS − E[gS]∥2 ≤ β
+for all x ∈ Rn.
+However, as we will still require that f is Lipschitz-smooth, we present our results for the differen-
+tiable setting.
+The proof of the subsequent theorems can be found in Appendix A.2 and Appendix A.3.
+Theorem 7. Let Assumption 3 and Assumption 4 hold. Let f(·; s) be convex for all s ∈ S and
+let f be L–smooth (4). Let x⋆ = arg minx∈Rn ψ(x) and let θ > 1. Let {xk}k=0,...,K be generated by
+Algorithm 2 for step sizes αk > 0 such that
+αk ≤ 1 − 1/θ
+L
+.
+(19)
+Then, it holds
+(1 + αkλ)E∥xk+1 − x⋆∥2 ≤ E∥xk − x⋆∥2 + 2αkE[ψ(x⋆) − ψ(xk+1)] + θβα2
+k.
+(20)
+Moreover, we have:
+a) If λ > 0 and αk =
+1
+λ(k+k0) with k0 ≥ 1 large enough such that (19) is fulfilled, then
+E
+�
+ψ
+�
+1
+K
+K−1
+�
+k=0
+xk+1�
+− ψ(x⋆)
+�
+≤ λk0
+2K ∥x0 − x⋆∥2 + θβ(1 + ln K)
+2λK
+.
+(21)
+11
+
+b) If λ = 0 and αk =
+α
+√k+1 with α ≤ 1−1/θ
+L
+, then
+E
+�
+ψ
+�
+1
+�K−1
+k=0 αk
+K−1
+�
+k=0
+αkxk+1�
+− ψ(x⋆)
+�
+≤
+∥x0 − x⋆∥2
+4α(
+√
+K + 1 − 1) + θβα(1 + ln K)
+4(
+√
+K + 1 − 1).
+(22)
+c) If f is µ–strongly convex with µ ≥ 0,5 and αk = α fulfilling (19), then
+E∥xK − x⋆∥2 ≤ (1 + α(µ + 2λ))−K∥x0 − x⋆∥2 +
+θβα
+µ + 2λ.
+(23)
+Remark 2. If λ > 0, for the decaying step sizes in item a) we get a rate of ˜O( 1
+K ) if λ > 0.
+In the strongly convex case in item c), for constant step sizes, we get a linear convergence upto a
+neighborhood of the solution. Note that the constant on the right-hand side of (23) can be forced to be
+small using a small α. Further, the rate (23) has a 2λ term, instead of λ. This slight improvement
+in the rate occurs because we do not linearize ϕ in the ProxSPS model.
+Theorem 8. Let Assumption 3 and Assumption 4 hold. Let f(·; s) be ρs–weakly convex for all
+s ∈ S and let ρ := E[ρS] < ∞. Let f be L–smooth6 and assume that inf ψ > −∞. Let {xk}k≥0 be
+generated by Algorithm 2. For θ > 1, under the condition
+η ∈
+�
+(0,
+1
+ρ−λ)
+if ρ > λ
+(0, ∞)
+else
+,
+αk ≤ 1 − θ−1
+L + η−1 ,
+(24)
+it holds
+K−1
+�
+k=0
+αkE∥∇envη
+ψ(xk)∥2 ≤
+2(envη
+ψ(x0) − inf ψ)
+1 − η(ρ − λ)
++
+βθ
+η(1 − η(ρ − λ))
+K−1
+�
+k=0
+α2
+k.
+(25)
+Moreover, for the choice αk =
+α
+√k+1 and with α ≤ 1−θ−1
+L+η−1 , we have
+min
+k=0,...,K−1 E∥∇envη
+ψ(xk)∥2 ≤
+envη
+ψ(x0) − inf ψ
+α(1 − η(ρ − λ))(
+√
+K + 1 − 1) +
+βθ
+2η(1 − η(ρ − λ))
+α(1 + ln K)
+(
+√
+K + 1 − 1).
+If instead we choose αk =
+α
+√
+K and with α ≤
+√
+K 1−θ−1
+L+η−1 , we have
+E∥∇envη
+ψ(xK
+∼)∥2 ≤
+2(envη
+ψ(x0) − inf ψ)
+α(1 − η(ρ − λ))
+√
+K
++
+βθ
+η(1 − η(ρ − λ))
+α
+√
+K
+,
+where xK
+∼ is uniformly drawn from {x0, . . . , xK−1}.
+4.3.3
+Comparison to existing theory
+Recalling that Algorithm 1 is equivalent to SPSmax with c = 1 and γb = αk, we can apply Theorem 7
+and Theorem 8 for the unregularized case where ϕ = 0 and hence obtain new theory for (SPSmax). We
+start by summarizing the main theoretical results for SPSmax given in (Loizou et al., 2021; Orvieto
+et al., 2022): in the (ER) setting, recall the interpolation constant σ2 = E[f(x⋆; S) − C(S)] =
+5Note that as f(·; s) is convex, so is f, and that we allow µ = 0 here.
+6As f is ρ–weakly convex, this implies ρ ≤ L.
+12
+
+1
+N
+�N
+i=1 fi(x⋆) − C(si).
+If fi is Li-smooth and convex, (Orvieto et al., 2022, Thm. 3.1) proves
+convergence to a neighborhood of the solution, i.e. the iterates {xk} of SPSmax satisfy
+E[f(¯xK) − f(x⋆)] ≤ ∥x0 − x⋆∥2
+αK
++ 2γbσ2
+α
+,
+(26)
+where ¯xK :=
+1
+K
+�K−1
+k=0 xk, α := min{
+1
+2cLmax , γb}, and Lmax := maxi∈[N] Li.7 For the nonconvex
+case, if fi is Li-smooth and under suitable assumptions on the gradient noise, (Loizou et al., 2021,
+Thm. 3.8) states that, for constants c1 and c2, we have
+min
+k=1,...,K E∥∇f(xk)∥2 ≤
+1
+c1K + c2.
+(27)
+The main advantage of these results is that γb can be held constant; furthermore in the convex
+setting (26), the choice of γb requires no knowledge of the smoothness constants Li. For both results
+however, we can not directly conclude that the right-hand side goes to zero as K → ∞ as there is
+an additional constant. Choosing γb sufficiently small does not immediately solve this as c1, α and
+c2 all go to zero as γb goes to zero.
+Our results complement this by showing exact convergence for the (weakly) convex convex case, i.e.
+without constants on the right-hand side. This comes at the cost of an upper bound on the step
+sizes αk which depends on the smoothness constant L. For exact convergence, it is important to
+use decreasing step sizes αk: Theorem 8 shows that the gradient of the Moreau envelope converges
+to zero at the rate O(1/
+√
+K) for the choice of αk =
+α
+√
+K .8 Another minor difference to (Loizou
+et al., 2021) is that we do not need to assume Lipschitz-smoothness for all f(·; s) and work instead
+with the (more general) assumption of weak convexity. However, we still need to assume Lipschitz
+smoothness of f.
+Another variant of SPSmax, named DecSPS, has been proposed in (Orvieto et al., 2022): for unreg-
+ularized problems (1) it is given by
+xk+1 = xk − ˆγkgk,
+ˆγk = 1
+ck
+min
+�f(xk; Sk) − Ck
+∥gk∥2
+, ck−1ˆγk−1
+�
+(DecSPS)
+where {ck}k≥0 is an increasing sequence. In the (ER) setting, if all fi are Lipschitz-smooth and
+strongly convex, DecSPS converges with a rate of O(
+1
+√
+K ), without knowledge of the smoothness or
+convexity constants (cf. (Orvieto et al., 2022, Thm. 5.5)). However, under these assumptions, the
+objective f is strongly convex and the optimal rate is O( 1
+K ), which we achieve up to a logarithmic
+factor in Theorem 7, (21). Moreover, for DecSPS no guarantees are given for nonconvex problems.
+5
+Numerical experiments
+Throughout we denote Algorithm 1 with SPS and Algorithm 3 with ProxSPS. For all experiments
+we use PyTorch (Paszke et al., 2019).
+5.1
+General parameter setting
+For SPS and ProxSPS we always use C(s) = 0 for all s ∈ S. For αk, we use the following schedules:
+7The theorem also handles the mini-batch case but, for simplicity, we state the result for sampling a single ik in
+each iteration.
+8Notice that αk then depends on the total number of iterations K and hence one would need to fix K before
+starting the method.
+13
+
+• constant: set αk = α0 for all k and some α0 > 0.
+• sqrt: set αk = α0
+√j for all iterations k during epoch j.
+As we consider problems with ℓ2-regularization, for SPS we handle the regularization term by incor-
+porating it into all individual loss functions, as depicted in (13). With ϕ = λ
+2 ∥ · ∥2 for λ ≥ 0, we
+denote by ζk the adaptive step size term of the following algorithms:
+• for SPS we have ζk := f(xk;Sk)+ λ
+2 ∥xk∥2
+∥gk+λxk∥2
+(cf. (13) with ℓik = 0 ),
+• for ProxSPS we have ζk :=
+�
+(1+αkλ)f(xk;Sk)−αkλ⟨gk,xk⟩
+∥gk∥2
+�
++ and thus τ +
+k = min{αk, ζk} (cf.
+Lemma 1 with C(Sk) = 0).
+5.2
+Regularized matrix factorization
+Problem description: For A ∈ Rq×p, consider the problem
+min
+W1∈Rr×p,W2Rq×r Ey∼N(0,I)∥W2W1y − Ay∥2 =
+min
+W1∈Rr×p,W2Rq×r ∥W2W1 − A∥2
+F .
+For the above problem, SPSmax has shown superior performance than other methods in the numerical
+experiments of (Loizou et al., 2021). The problem can can be turned into a (nonconvex) empirical
+risk minimization problem by drawing N samples {y(1), . . . , y(N)}. Denote b(i) := Ay(i). Adding
+squared norm regularization with λ ≥ 0 (cf. (Srebro et al., 2004)), we obtain the problem
+min
+W1∈Rr×p,W2Rq×r
+1
+N
+N
+�
+i=1
+∥W2W1y(i) − b(i)∥2 + λ
+2
+�
+∥W1∥2
+F + ∥W2∥2
+F
+�
+.
+(28)
+This is a problem of form (2), setting x = (W1, W2), using a finite sample space S = {s1, . . . , sN},
+f(x; si) = ∥W2W1y(i) − Ay(i)∥2, and ϕ = λ
+2 ∥ · ∥2
+F . Clearly, zero is a lower bound of f(·; si) for all
+i ∈ [N].
+We investigate ProxSPS for problems of form (28) on synthetic data. For details on the experimental
+procedure, we refer to Appendix D.1.
+Discussion: We discuss the results for the setting matrix-fac1 in Table 1 in the appendix. We
+first fix λ = 0.001 and consider the three methods SPS, ProxSPS and SGD. Fig. 2 shows the objective
+function over 50 epochs, for both step size schedules sqrt and constant, and several initial values
+α0. For the constant schedule, we observe that ProxSPS converges quickly for all initial values
+while SPS is unstable. Note that for SGD we need to pick much smaller values here in order to avoid
+divergence (SGD diverges for the largest value of α0). Hence, SPS for large α0 is unstable, while
+for small α0 we can expect similar performance to SGD (as γk is capped by αk = α0). However, in
+the regime of small α0, convergence will be very slow. Hence, one of the main advantages of SPS,
+namely that its step size can be chosen constant and moderately large (compared to SGD), is not
+observed here. ProxSPS fixes this by admitting a large range of initial step sizes, which results in
+fast convergence, and therefore is more robust than SGD with respect to the tuning of α0.
+For the sqrt schedule, we observe in Fig. 2 that SPS can be stabilized by reducing the values of αk
+over the course of the iterations. However, for large α0 we still see instability in the early iterations,
+whereas ProxSPS does not show this behaviour. We again observe that ProxSPS is less sensitive with
+14
+
+0
+10
+20
+30
+40
+Epoch
+10−5
+10−4
+10−3
+10−2
+10−1
+ψ(xk) − mink ψ(xk)
+constant
+0
+10
+20
+30
+40
+Epoch
+10−5
+10−4
+10−3
+10−2
+10−1 sqrt
+α0
+2.0
+1.62
+1.25
+0.88
+0.5
+2.0
+1.62
+1.25
+0.88
+0.5
+0.7
+0.56
+0.41
+0.27
+0.12
+prox-sps
+sps
+sgd
+Figure 2: Objective function for the Matrix Factorization problem (28), with constant (left) and
+sqrt (right) step size schedule and several choices of initial values. Here mink ψ(xk) is the best
+objective function value found over all methods and all iterations.
+0
+10
+20
+30
+40
+Epoch
+10−4
+10−2
+100
+102
+104
+Validation Error
+constant
+0
+10
+20
+30
+40
+Epoch
+10−4
+10−3
+10−2
+10−1
+100
+sqrt
+α0
+2.0
+1.62
+1.25
+0.88
+0.5
+2.0
+1.62
+1.25
+0.88
+0.5
+0.7
+0.56
+0.41
+0.27
+0.12
+prox-sps
+sps
+sgd
+Figure 3: Validation error for the Matrix Factorization problem (28), with constant (left) and sqrt
+(right) step size schedule and several choices of initial values.
+respect to the choice of α0 as compared to SGD. The empirical results also confirm our theoretical
+statement, showing exact convergence if αk is decaying in the order of 1/
+√
+k. From Fig. 3, we can
+make similar observations for the validation error, defined as
+1
+Nval
+�Nval
+i=1 ∥W2W1y(i) − b(i)
+val∥2, where
+b(i)
+val are the Nval = N measurements from the validation set (cf. Appendix D.1 for details).
+We now consider different values for λ and only consider the sqrt schedule, as we have seen that
+for constant step sizes, SPS would not work for large step sizes and be almost identical to SGD for
+small step sizes. Fig. 4 shows the objective function and validation error. Again, we can observe
+that SPS is unstable for large initial values α0 for all λ ≥ 10−4. On the other hand, ProxSPS has a
+good performance for a wide range of α0 ∈ [1, 10] if λ is not too large. Indeed, ProxSPS convergence
+only starts to deteriorate when both α0 and λ are very large. For α0 = 1, the two methods give
+almost identical results. Finally, in Fig. 5a we plot the validation error as a function of λ (taking
+the median over the last ten epochs). The plot shows that the best validation error is obtained for
+λ = 10−4 and for large α0. With SPS the validation error is higher, in particular for large α0 and
+15
+
+0
+20
+40
+Epoch
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Objective ψ(xk)
+λ = 1e − 05
+0
+20
+40
+Epoch
+0.002
+0.004
+0.006
+0.008
+0.010
+λ = 0.0001
+0
+20
+40
+Epoch
+0.01
+0.02
+0.03
+0.04
+λ = 0.001
+0
+20
+40
+Epoch
+0.100
+0.125
+0.150
+0.175
+0.200
+0.225
+λ = 0.01
+0
+20
+40
+Epoch
+1.0
+1.5
+2.0
+2.5
+λ = 0.1
+prox-sps, sqrt, α0=10.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=10.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=1.0
+0
+20
+40
+Epoch
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+0.0025
+Validation Error
+λ = 1e − 05
+0
+20
+40
+Epoch
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+0.0025
+0.0030
+λ = 0.0001
+0
+20
+40
+Epoch
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+λ = 0.001
+0
+20
+40
+Epoch
+0.00
+0.02
+0.04
+0.06
+0.08
+λ = 0.01
+0
+20
+40
+Epoch
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+λ = 0.1
+prox-sps, sqrt, α0=10.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=10.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=1.0
+Figure 4: Objective function value and validation error over the course of optimization. For the
+validation error, we plot a rolling median over five epochs in order to avoid clutter.
+10−5
+10−4
+10−3
+10−2
+10−1
+λ
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Validation Error
+prox-sps, sqrt, α0=1.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=10.0
+sps, sqrt, α0=1.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=10.0
+(a) Validation error
+10−5
+10−4
+10−3
+10−2
+10−1
+λ
+4.00
+4.25
+4.50
+4.75
+5.00
+5.25
+5.50
+∥xk∥
+prox-sps, sqrt, α0=1.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=10.0
+sps, sqrt, α0=1.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=10.0
+(b) Model norm
+�
+∥W1∥2 + ∥W2∥2
+Figure 5: Validation error and model norm as a function of the regularization parameter λ. Shaded
+area is one standard deviation (computed over ten independent runs). For all values, we take the
+median over epochs [40, 50].
+λ. Finally, we plot the actual step sizes for both methods in Fig. 6. We observe that the adaptive
+step size ζk (Definition at end of Section 5.1) is typically larger and has more variance for SPS than
+ProxSPS, in particular for large λ. This increased variance might explain why SPS is unstable when
+α0 is large: the actual step size is the minimum between αk and ζk and hence both terms being
+large could lead to instability. On the other hand, if α0 = 1, the plot confirms that SPS and ProxSPS
+are almost identical methods as ζk > αk for most iterations. We plot the results for the setting
+matrix-fac2 of Table 1 in Appendix D.2.
+16
+
+Figure 6: Adaptive step size selection for SPS and ProxSPS. We plot ζk (see definition in Section 5.1)
+as dots for each iteration as well as their median over each epoch. For this plot, we use the results
+of only one of the ten runs.
+5.3
+Deep networks for image classification
+We train a ResNet56 and ResNet110 model (He et al., 2016) on the CIFAR10 dataset.
+We use
+the data loading and preprocessing procedure and network implementation from https://github.
+com/akamaster/pytorch_resnet_cifar10. We do not use batch normalization. The loss function
+is the cross-entropy loss of the true image class with respect to the predicted class probabilities,
+being the output of the ResNet56 network. We add λ
+2 ∥x∥2 as regularization term, where x consists
+of all learnable parameters of the model. The CIFAR10 dataset consist of 60,000 images, each of
+size 32 × 32, from ten different classes. We use the PyTorch split into 50,000 training and 10,000
+test examples and use a batch size of 128. For AdamW, we set the weight decay parameter to λ
+and set all other hyperparameters to its default. We use the AdamW-implementation from https:
+//github.com/zhenxun-zhuang/AdamW-Scale-free as it does not – in contrast to the Pytorch
+implementation – multiply the weight decay parameter with the learning rate, which leads to better
+comparability to SPS and ProxSPS for identical values of λ. For SPS and ProxSPS we use the sqrt-
+schedule and α0 = 1. We run each method repeatedly using (the same) three different seeds for the
+dataset shuffling.
+Discussion: For Resnet56, from the bottom plot in Fig. 7, we observe that both SPS and ProxSPS
+work well with ProxSPS leading to smaller weights. For λ = 5e−4, the progress of ProxSPS stagnates
+after roughly 25 epochs. This can be explained by looking at the adaptive step size term ζk in Fig. 9a:
+as it decays over time we have τ +
+k = ζk ≪ αk. Since every iteration of ProxSPS shrinks the weights
+17
+
+prox-sps, αo = 1.0, 入= le - 05
+prox-sps, αo = 5.0, 入= le - 05
+prox-sps, αo = 10.0, 入 = 1e - 05
+sps, αo = 1.0, 入= le - 05
+sps, αo = 5.0, 入 = le - 05
+sps, αo = 10.0, 入= 1e - 05
+103
+median(Sk)
+median(Sk)
+nedian(Sk)
+median(Sk)
+median(Sk)
+median(Sk)
+101
+10-3
+0.0001
+0.0001
+0.0001
+prox-sps, αo
+10.0.
+0.0001
+5.0.
+0.0001
+0.0. 
+0.0001
+.0. 入
+sps,α
+sps, αo
+sps, αo
+103
+median(Ck)
+median(Sk)
+nedian(Sk)
+nedian(Sh)
+median(Sk)
+nedian(Sk)
+101
+10-3
+prox-sps, αo = 1.0, 入 = 0.001
+prox-sps, αo = 5.0, 入 = 0.001
+10.0.)
+= 0.001
+sps, αo = 5.0, 入= 0.001
+sps, αo = 10.0, 入 = 0.001
+103
+nedian(Sk)
+median(Sk)
+median
+101
+10-1
+10-3
+prox-sps, αo = 1.0, 入= 0.01
+prox-sps, αo =
+= 5.0, 入= 0.01
+prox-sps, αo =
+sps, αo = 1.0, 入 = 0.0j
+sps, αo = 5.0, 入 = 0.01
+sps, αo = 10.0, 入 = 0.01
+103
+ median(Sk)
+median(Sk)
+edian(Sk)
+median(Sk)
+n(Sk)
+cep size
+101
+10-3
+prox-sps, αo
+1.0，入
+0.1
+5.0，)
+0.1
+prox-sps,
+10.0,入
+0.1
+.0,
+0.1
+5.0.
+sps,
+0.
+103
+median(Sk)
+median(Sk)
+ median(Sk)
+median(Sk)
+median(
+median(Sk)
+ size
+101
+10-3
+20
+0
+40 
+20
+40
+0
+20
+40
+0
+20
+40
+0
+20
+40
+20
+40
+0
+0
+Epoch
+Epoch
+Epoch
+Epoch
+Epoch
+Epoch0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Validation Accuracy
+λ = 5e − 06
+0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+λ = 5e − 05
+0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+λ = 0.0005
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=1.0
+adamw, constant, α0=0.001
+0
+25
+50
+75
+100
+Epoch
+80
+100
+120
+∥xk∥
+λ = 5e − 06
+0
+25
+50
+75
+100
+Epoch
+50
+60
+70
+80
+λ = 5e − 05
+0
+25
+50
+75
+100
+Epoch
+10
+20
+30
+40
+50
+60
+λ = 0.0005
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=1.0
+adamw, constant, α0=0.001
+Figure 7: ResNet56: (Top): Validation accuracy and model norm for three values of the regular-
+ization parameter λ. Validation accuracy is defined as the ratio of correctly labeled images on the
+validation set (i.e. Top-1 accuracy), plotted as five-epoch running median. (Bottom): With ∥xk∥
+we denote the norm of all learnable parameters at the k-th iteration. Shaded area is two standard
+deviations over three independent runs.
+by a factor
+1
+1+αkλ, this leads to a bias towards zero. This suggests that we should choose αk roughly
+of the order of ζk, for example by using the values of ζk from the previous epoch.
+For the larger model Resnet110 however, SPS does not make progress for a long time because the
+adaptive step size is very small (see Fig. 8 and Fig. 9b). ProxSPS does not share this issue and
+performs well after a few initial epochs. For larger values of λ, the training is also considerably
+faster than for AdamW. Generally, we observe that ProxSPS (and SPS for Resnet56) performs well in
+comparison to AdamW. This is achieved without extensive hyperparameter tuning (in particular this
+suggests that setting c = 1 in SPSmax leads to good results and reduces tuning effort).
+6
+Conclusion
+We proposed and analyzed ProxSPS, a proximal version of the stochastic Polyak step size.
+We
+arrived at ProxSPS by using the framework of stochastic model-based proximal point methods. We
+then used this framework to argue that the resulting model of ProxSPS is a better approximation
+as compared to the model used by SPS when using regularization. Our theoretical results cover
+a wide range of optimization problems, including convex and nonconvex settings. We performed
+a series of experiments comparing ProxSPS, SPS, SGD and AdamW when using ℓ2-regularization. In
+particular we found that SPS can be very hard to tune when using ℓ2-regularization, and in contrast,
+ProxSPS performs well for a wide choice of step sizes and regularization parameters. Finally, for our
+18
+
+0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Validation Accuracy
+λ = 5e − 06
+0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+λ = 5e − 05
+0
+25
+50
+75
+100
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+λ = 0.0005
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=1.0
+adamw, constant, α0=0.001
+0
+25
+50
+75
+100
+Epoch
+90
+95
+100
+105
+∥xk∥
+λ = 5e − 06
+0
+25
+50
+75
+100
+Epoch
+40
+50
+60
+70
+80
+90
+λ = 5e − 05
+0
+25
+50
+75
+100
+Epoch
+20
+40
+60
+80
+λ = 0.0005
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=1.0
+adamw, constant, α0=0.001
+Figure 8: ResNet110: Validation accuracy as five-epoch running median (top) and model norm
+(bottom) for three values of λ. Shaded area is two standard deviations over three independent runs.
+experiments on image classification, we found that ProxSPS is competitive to AdamW, whereas SPS
+can fail for larger models. At the same time ProxSPS produces smaller weights in the trained neural
+network. Having small weights may help reduce the memory footprint of the resulting network, and
+even suggests which weights can be pruned.
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+Figure 9: Adaptive step sizes for SPS and ProxSPS. See definition of ζk in Section 5.1. For this plot,
+we use the results of only one of the three runs.
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+
+prox-sps, αo = 1.0, 入 = 5e  06
+sps, α0 = 1.0, 入 = 5e - 06
+αk
+αk
+median(Sk)
+median(Sk)
+Step size
+101
+10-
+0
+10-3
+prox-sps, αo = 1.0, 入 = 5e - 05
+sps, α0 = 1.0, 入 = 5e - 05
+103
+αk
+αk
+median(Sk)
+median(Sk)
+Step size
+101
+10
+000
+00000
+10-
+pr0x-sps, αo = 1.0, 入 = 0.0005
+sps, α0 = 1.0, 入 = 0.0005
+103
+αk
+αk
+median(Sk)
+median(
+Ck
+Step size
+101
+10-1
+10-3
+0
+50
+100
+0
+50
+100
+Epoch
+Epochprox-sps, αo = 1.0, 入 = 5e 一 06
+sps, αo = 1.0, 入 = 5e  06
+102
+αk
+αk
+median(Sk)
+median(Sk)
+size
+10
+Step
+10
+10-10
+prox-sps, αo = 1.0, 入 = 5e - 05
+sps,α0 = 1.0, 入 = 5e - 05
+102
+αk
+αk
+median(Sk)
+medi
+ size
+10
+Step
+10
+prox-sps, αo = 1.0, 入 = 0.0005
+sps, αo = 1.0, 入 = 0.0005
+102
+αk
+αk
+median(Sk)
+ size
+10-6
+10-10
+0
+50
+100
+0
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+through proximal methods and scale-freeness. January 2022.
+21
+
+Contents
+1
+Introduction
+1
+1.1
+Background and Contributions
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+2
+Preliminaries
+3
+2.1
+Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+Convex analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3
+The unregularized case
+4
+3.1
+A model-based view point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+4
+The regularized case
+6
+4.1
+The special case of ℓ2-regularization
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+4.2
+Comparing the model of SPS and ProxSPS
+. . . . . . . . . . . . . . . . . . . . . . .
+8
+4.3
+Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4.3.1
+Globally bounded subgradients . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4.3.2
+Lipschitz smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.3.3
+Comparison to existing theory
+. . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5
+Numerical experiments
+13
+5.1
+General parameter setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+5.2
+Regularized matrix factorization
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5.3
+Deep networks for image classification . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+6
+Conclusion
+18
+A Missing Proofs
+23
+A.1 Proofs of model-based update formula . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+A.2 Proof of Theorem 7
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+A.3 Proof of Theorem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+B Auxiliary Lemmas
+28
+C Model equivalence for SGD
+30
+22
+
+D Additional information on numerical experiments
+30
+D.1 Matrix Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+D.2 Plots for matrix-fac2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+A
+Missing Proofs
+A.1
+Proofs of model-based update formula
+Lemma 9. For λ ≥ 0, let ϕ(x) = λ
+2 ∥x∥2 and let g ∈ ∂f(x; s) and C(s) ≤ infz∈Rn f(z; s) hold for
+all s ∈ S. For
+ψx(y; s) = fx(y; s) + ϕ(y),
+fx(y; s) = max{f(x; s) + ⟨g, y − x⟩, C(s)},
+consider the update
+xk+1 = arg min
+x∈Rn
+ψxk(x; Sk) +
+1
+2αk
+∥x − xk∥2.
+(29)
+Denote Ck := C(Sk) and let gk ∈ ∂f(xk; Sk). Define
+τ +
+k :=
+�
+�
+�
+0
+if gk = 0,
+min
+�
+αk,
+�
+(1+αkλ)(f(xk;Sk)−Ck)−αkλ⟨gk,xk⟩
+∥gk∥2
+�
++
+�
+else.
+Then, we have
+xk+1 =
+1
+1 + αkλxk −
+τ +
+k
+1 + αkλgk =
+1
+1 + αkλ
+�
+xk − τ +
+k gk
+�
+= proxαkϕ(xk − τ +
+k gk).
+(30)
+Define τk := 0 if gk = 0 and τk := min
+�
+αk, (1+αkλ)(f(xk;Sk)−Ck)−αkλ⟨gk,xk⟩
+∥gk∥2
+�
+else. Then, it holds
+τk ≤ τ +
+k and
+ψxk(xk+1; Sk) = f(xk; Sk) −
+αkλ
+1+αkλ⟨gk, xk⟩ −
+τk
+1+αkλ∥gk∥2 + ϕ(xk+1).
+(31)
+Proof. Note that max{f(xk; Sk) + ⟨gk, y − xk⟩, Ck} is convex as a function of y. The update is
+therefore unique. First, if gk = 0, then clearly xk+1 = proxαkϕ(xk) =
+1
+1+αkλxk and (31) holds
+true. Now, let gk ̸= 0. The solution of (29) is either in {y|f(xk; Sk) + ⟨gk, y − xk⟩ < Ck}, or in
+{y|f(xk; Sk) + ⟨gk, y − xk⟩ > Ck} or in {y|f(xk; Sk) + ⟨gk, y − xk⟩ = Ck}. We therefore solve three
+problems:
+(P1) Solve
+y+ = arg min
+y
+Ck + λ
+2 ∥y∥2 +
+1
+2αk
+∥y − xk∥2.
+Clearly, the solution is y+ =
+1
+1+αkλxk. This y+ solves (29) if f(xk; Sk)+⟨gk, y+ −xk⟩ < Ck.
+(P2) Solve
+y+ = arg min
+y
+f(xk; Sk) + ⟨gk, y − xk⟩ + λ
+2 ∥y∥2 +
+1
+2αk
+∥y − xk∥2.
+The optimality condition is 0 = αkgk + αkλy+ + y+ − xk.
+Thus, the solution is y+ =
+1
+1+αkλ(xk − αkgk). This y+ solves (29) if f(xk; Sk) + ⟨gk, y+ − xk⟩ > Ck.
+23
+
+(P3) Solve
+y+ = arg min
+y
+λ
+2 ∥y∥2 +
+1
+2αk
+∥y − xk∥2,
+s.t. f(xk; Sk) + ⟨gk, y − xk⟩ = Ck.
+The KKT conditions are given by
+αkλy + y − xk + µgk = 0,
+f(xk; Sk) + ⟨gk, y − xk⟩ = Ck.
+Taking the inner product of the first equation with gk, we get
+(1 + αkλ)⟨gk, y⟩ − ⟨gk, xk⟩ + µ∥gk∥2 = 0.
+From the second KKT condition we have ⟨gk, y⟩ = Ck − f(xk; Sk) + ⟨gk, xk⟩, hence
+(1 + αkλ)
+�
+Ck − f(xk; Sk) + ⟨gk, xk⟩
+�
+− ⟨gk, xk⟩ + µ∥gk∥2 = 0.
+Solving for µ gives µ = (1+αkλ)(f(xk;Sk)−Ck)−αkλ⟨gk,xk⟩
+∥gk∥2
+. From the first KKT condition, we
+obtain
+y+ =
+1
+1 + αkλ
+�
+xk − µgk
+�
+=
+1
+1 + αkλ
+�
+xk − (1 + αkλ)(f(xk; Sk) − Ck) − αkλ⟨gk, xk⟩
+∥gk∥2
+gk
+�
+.
+This y+ solves (29) if neither (P1) nor (P2) provided a solution.
+For all three cases, the solution takes the form y+ =
+1
+1+αkλ[xk −tgk] =: y(t). As ∥gk∥2 > 0, the term
+f(xk; Sk) + ⟨gk, y(t) − xk⟩ is strictly monotonically decreasing in t. We know f(xk; Sk) + ⟨gk, y(t) −
+xk⟩ = Ck for t = µ (from (P3)). Hence, f(xk; Sk) + ⟨gk, y(t) − xk⟩ < Ck (> Ck) if and only if
+t > µ (t < µ).
+We conclude:
+• If f(xk; Sk) + ⟨gk, y(0) − xk⟩ < Ck, then the solution to (P1) is the solution to (29). This
+condition is equivalent to µ < 0.
+• If f(xk; Sk) + ⟨gk, y(αk) − xk⟩ > Ck, then the solution to (P2) is the solution to (29). This
+condition is equivalent to αk < µ.
+• If neither 0 > µ nor αk < µ hold, i.e. if µ ∈ [0, αk], then the solution to (29) comes from
+(P3) and hence is given by y(µ).
+Altogether, we get that xk+1 =
+1
+1+αkλ[xk − τ +
+k gk] with τ +
+k = min{αk, (µ)+}.
+Now, we prove (31). Note that if gk ̸= 0, then τk = min{αk, µ} with µ defined as in (P3). In the
+case of (P1), we have ψxk(xk+1; Sk) = Ck + ϕ(xk+1). Moreover, it holds µ < 0 and as αk > 0 we
+have τk = µ. Plugging τk = µ into the right hand-side of (31), we obtain Ck + ϕ(xk+1).
+In the case of (P2) or (P3), we have Ck ≤ f(xk; Sk) + ⟨gk, xk+1 − xk⟩. Due to f(xk; Sk) + ⟨gk, y(t) −
+xk⟩ = f(xk; Sk) −
+1
+1+αkλ⟨gk, xk⟩ +
+t
+1+αkλ∥gk∥2, we obtain (31) as xk+1 = y(αk) and µ > αk in the
+case of (P2) and xk+1 = y(µ) and µ ≤ αk in the case of (P3).
+24
+
+Lemma 10. Consider the model fx(y; s) := max{f(x; s) + ⟨g, y − x⟩, C(s)} where g ∈ ∂f(x; s) and
+C(s) ≤ infz∈Rn f(z; s) holds for all s ∈ S. Then, update (5) is given as
+xk+1 = xk − γkgk,
+γk =
+�
+0
+if gk = 0,
+min
+�
+αk, f(xk;Sk)−C(Sk)
+∥gk∥2
+�
+else.
+where gk ∈ ∂f(xk; Sk). Moreover, it holds
+fxk(xk+1; Sk) = max{C(Sk), f(xk; Sk) − αk∥gk∥2},
+(32)
+and therefore fxk(xk+1; Sk) = f(xk; Sk) − γk∥gk∥2.
+Proof. We apply Lemma 9 with λ = 0. As f(xk; SK) ≥ C(Sk), we have that τ +
+k = τk = γk.
+A.2
+Proof of Theorem 7
+Proof of Theorem 7. In the proof, we will denote gk = ∇f(xk; Sk). We apply Lemma 6, (16) with
+x = x⋆. Due to Lemma 2 (ii) and convexity of f(·; s) it holds
+ψxk(x⋆; Sk) ≤ f(x⋆; Sk) + ϕ(x⋆).
+Together with (17), we have
+(1 + αkλ)∥xk+1 − x⋆∥2 ≤ ∥xk − x⋆∥2 − ∥xk+1 − xk∥2 + 2αk[ϕ(x⋆) − ϕ(xk+1)]
++ 2αk
+�
+f(x⋆; Sk) − f(xk; Sk) − ⟨gk, xk+1 − xk⟩
+�
+.
+(33)
+Smoothness of f yields
+−f(xk) ≤ −f(xk+1) + ⟨∇f(xk), xk+1 − xk⟩ + L
+2 ∥xk+1 − xk∥2.
+Consequently,
+− ⟨gk, xk+1 − xk⟩ = f(xk) − f(xk) − ⟨gk, xk+1 − xk⟩
+≤ f(xk) − f(xk+1) + ⟨∇f(xk) − gk, xk+1 − xk⟩ + L
+2 ∥xk+1 − xk∥2
+≤ f(xk) − f(xk+1) + θαk
+2 ∥∇f(xk) − gk∥2 +
+1
+2θαk
+∥xk+1 − xk∥2 + L
+2 ∥xk+1 − xk∥2.
+for any θ > 0, where we used Young’s inequality in the last step. Plugging into (33) gives
+(1 + αkλ)∥xk+1 − x⋆∥2 ≤ ∥xk − x⋆∥2 +
+�
+αkL + 1
+θ − 1
+�
+∥xk+1 − xk∥2 + 2αk[ϕ(x⋆) − ϕ(xk+1)]
++ 2αk
+�
+f(x⋆; Sk) − f(xk; Sk) + f(xk) − f(xk+1)
+�
++ θα2
+k∥∇f(xk) − gk∥2.
+Applying conditional expectation, we have E[f(x⋆; Sk)|Fk] = f(x⋆) and
+E[−f(xk; Sk) + f(xk)|Fk] = 0,
+E[∥∇f(xk) − gk∥2|Fk] ≤ β.
+Moreover, by assumption, αkL + 1
+θ − 1 ≤ 0. Altogether, applying total expectation yields
+(1 + αkλ)E∥xk+1 − x⋆∥2 ≤ E∥xk − x⋆∥2 + 2αkE[ψ(x⋆) − ψ(xk+1)] + θβα2
+k
+25
+
+which proves (20).
+Proof of a): let αk =
+1
+λ(k+k0). Denote ∆k := E∥xk − x⋆∥2. Rearranging and summing (20), we
+have
+K−1
+�
+k=0
+E[ψ(xk+1) − ψ(x⋆)] ≤
+K−1
+�
+k=0
+�
+1
+2αk ∆k − 1+αkλ
+2αk ∆k+1 + θβαk
+2
+�
+.
+Plugging in αk, we have 1+αkλ
+2αk
+= λ(k+k0)
+2
++ λ
+2 and thus
+K−1
+�
+k=0
+E[ψ(xk+1) − ψ(x⋆)] ≤
+K−1
+�
+k=0
+�
+λ(k+k0)
+2
+∆k − λ(k+1+k0)
+2
+∆k+1
+�
++ θβ
+2
+K−1
+�
+k=0
+1
+λ(k+k0).
+Dividing by K and using convexity of ψ9, we have
+E
+�
+ψ
+�
+1
+K
+K−1
+�
+k=0
+xk+1�
+− ψ(x⋆)
+�
+≤ λk0
+2K ∥x0 − x⋆∥2 +
+θβ
+2λK
+K−1
+�
+k=0
+1
+k+k0 .
+Finally, as k0 ≥ 1, we estimate �K−1
+k=0
+1
+k+k0 ≤ �K−1
+k=0
+1
+k+1 ≤ 1 + ln K by Lemma 13 and obtain (21).
+Proof of b): Similar to the proof above, we rearrange and sum (20) from k = 0, . . . , K − 1, and
+obtain
+K−1
+�
+k=0
+αkE[ψ(xk+1) − ψ(x⋆)] ≤ ∥x0 − x⋆∥2
+2
++ θβ �K−1
+k=0 α2
+k
+2
+.
+We divide by �K−1
+k=0 αk and use convexity of ψ in order to obtain the left-hand side of (22). Moreover,
+by Lemma 13 we have
+K−1
+�
+k=0
+αk ≥ 2α(
+√
+K + 1 − 1),
+K−1
+�
+k=0
+α2
+k ≤ α2(1 + ln K).
+Plugging in the above estimates, gives
+E
+�
+ψ
+�
+1
+�K−1
+k=0 αk
+K−1
+�
+k=0
+αkxk+1�
+− ψ(x⋆)
+�
+≤
+∥x0 − x⋆∥2
+4α(
+√
+K + 1 − 1) + θβα(1 + ln K)
+4(
+√
+K + 1 − 1).
+Proof of c): If f is µ–strongly–convex, then ψ is (λ + µ)–strongly convex and
+ψ(x⋆) − ψ(xk+1) ≤ − µ+λ
+2 ∥xk+1 − x⋆∥2.
+From (20), with αk = α, we get
+(1 + α(µ + 2λ))E∥xk+1 − x⋆∥2 ≤ E∥xk − x⋆∥2 + θβα2.
+Doing a recursion of the above from k = 0, . . . , K − 1 gives
+E∥xK − x⋆∥2 ≤ (1 + α(µ + 2λ))−K∥x0 − x⋆∥2 + θβα2
+K
+�
+k=1
+(1 + α(µ + 2λ))−k
+9By assumption f is convex and therefore ψ is convex.
+26
+
+Using the geometric series, �K
+k=1(1 + α(µ + 2λ))−k ≤ 1+α(µ+2λ)
+α(µ+2λ)
+− 1 =
+1
+α(µ+2λ), and thus
+E∥xK − x⋆∥2 ≤ (1 + α(µ + 2λ))−K∥x0 − x⋆∥2 +
+θβα
+µ + 2λ.
+A.3
+Proof of Theorem 8
+Proof of Theorem 8. In the proof, we will denote gk = ∇f(xk; Sk). By assumption f is ρ-weakly
+convex and hence ψ is (ρ − λ)-weakly convex if ρ > λ and convex if ρ ≤ λ. Hence, ˆxk := proxηψ(xk)
+is well-defined for η < 1/(ρ − λ) if ρ > λ and for any η > 0 else. Note that ˆxk is Fk–measurable.
+We apply Lemma 6, (16) with x = ˆxk. Due to Lemma 2 (ii) it holds
+ψxk(ˆxk; Sk) = fxk(ˆxk; Sk) + ϕ(ˆxk) ≤ f(ˆxk; Sk) +
+ρSk
+2 ∥ˆxk − xk∥2 + ϕ(ˆxk).
+Together with (17), this gives
+(1 + αkλ)∥xk+1 − ˆxk∥2 ≤(1 + αkρSk)∥xk − ˆxk∥2 − ∥xk+1 − xk∥2
++ 2αk
+�
+ϕ(ˆxk) − ϕ(xk+1) + f(ˆxk; Sk) − f(xk; Sk) − ⟨gk, xk+1 − xk⟩
+�
+Analogous to the proof of Theorem 7, due to Lipschitz smoothness, for all θ > 0 we have
+−f(xk; Sk) − ⟨gk, xk+1 − xk⟩ ≤ −f(xk; Sk) + f(xk)
+− f(xk+1) + θαk
+2 ∥∇f(xk) − gk∥2 +
+�
+1
+2θαk + L
+2
+�
+∥xk+1 − xk∥2.
+Plugging in gives
+(1 + αkλ)∥xk+1 − ˆxk∥2 ≤ (1 + αkρSk)∥xk − ˆxk∥2 + 2αk
+�
+ϕ(ˆxk) − ϕ(xk+1)
+�
++ 2αk
+�
+f(ˆxk; Sk) − f(xk; Sk) + f(xk) − f(xk+1) + θαk
+2 ∥∇f(xk) − gk∥2�
++
+� 1
+θ + αkL − 1
+�
+∥xk+1 − xk∥2.
+It holds E[f(ˆxk; Sk) − f(xk; Sk)|Fk] = f(ˆxk) − f(xk) and E[ψ(ˆxk)|Fk] = ψ(ˆxk). By Assumption 4,
+we have E[∥gk − ∇f(xk)∥2|Fk] ≤ β. Altogether, taking conditional expectation yields
+(1 + αkλ)E[∥xk+1 − ˆxk∥2|Fk] ≤ (1 + αkρ)∥xk − ˆxk∥2 + 2αkE
+�
+ψ(ˆxk) − ψ(xk+1)|Fk
+�
++ α2
+kθβ +
+� 1
+θ + αkL − 1
+�
+E[∥xk+1 − xk∥2|Fk].
+Next, the definition of the proximal operator implies that almost surely
+ψ(ˆxk) +
+1
+2η∥ˆxk − xk∥2 ≤ ψ(xk+1) +
+1
+2η∥xk+1 − xk∥2,
+and hence
+E
+�
+ψ(ˆxk) − ψ(xk+1)|Fk
+�
+≤ E
+� 1
+2η∥xk+1 − xk∥2 −
+1
+2η∥ˆxk − xk∥2|Fk
+�
+.
+Altogether, we have
+(1 + αkλ)E[∥xk+1 − ˆxk∥2|Fk] ≤ (1 + αk(ρ − η−1))∥xk − ˆxk∥2
++ α2
+kθβ +
+� 1
+θ + αkL + αkη−1 − 1
+�
+E[∥xk+1 − xk∥2|Fk].
+27
+
+From assumption (24), we can drop the last term. Now, we aim for a recursion in envη
+ψ. Using that
+1 + αk(ρ − η−1)
+1 + αkλ
+= 1 + αkλ − αkλ + αk(ρ − η−1)
+1 + αkλ
+= 1 + αk(ρ − η−1 − λ)
+1 + αkλ
+≤ 1 + αk(ρ − η−1 − λ),
+we get
+E[envη
+ψ(xk+1)|Fk] ≤ E[ψ(ˆxk) + 1
+2η ∥xk+1 − ˆxk∥2|Fk]
+≤ ψ(ˆxk) + 1
+2η ∥xk − ˆxk∥2
+�
+��
+�
+=envη
+ψ(xk)
++ 1
+2η
+�
+αk(ρ − η−1 − λ)
+�
+∥xk − ˆxk∥2 + α2
+k
+2η θβ.
+Now using ∥xk − ˆxk∥ = η∥∇envη
+ψ(xk)∥ we conclude
+E[envη
+ψ(xk+1)|Fk] ≤ envη
+ψ(xk) + η
+2
+�
+αk(ρ − η−1 − λ)
+�
+∥∇envη
+ψ(xk)∥2 + α2
+k
+2η θβ.
+Due to (24), we have η−1 + λ − ρ > 0. Taking expectation and unfolding the recursion by summing
+over k = 0, . . . , K − 1, we get
+K−1
+�
+k=0
+αk
+2 (1 − η(ρ − λ))E∥∇envη
+ψ(xk)∥2 ≤ envη
+ψ(x0) − E[envη
+ψ(xK)] +
+K−1
+�
+k=0
+α2
+k
+2η θβ.
+Now using that envη
+ψ(xK) ≥ inf ψ almost surely, we finally get
+K−1
+�
+k=0
+αkE∥∇envη
+ψ(xk)∥2 ≤
+2(envη
+ψ(x0) − inf ψ)
+1 − η(ρ − λ)
++
+βθ
+η(1 − η(ρ − λ))
+K−1
+�
+k=0
+α2
+k,
+(34)
+which proves (25). Now choose αk =
+α
+√k+1 and divide (34) by �K−1
+k=0 αk. Using Lemma 13 for
+�K−1
+k=0 αk and �K−1
+k=0 α2
+k, we have
+min
+k=0,...,K−1 E∥∇envη
+ψ(xk)∥2 ≤
+envη
+ψ(x0) − inf ψ
+α(1 − η(ρ − λ))(
+√
+K + 1 − 1) +
+βθ
+2η(1 − η(ρ − λ))
+α(1 + ln K)
+(
+√
+K + 1 − 1).
+Choosing αk =
+α
+√
+K instead, we can identify the left-hand-side of (34) as α
+√
+KE∥∇envη
+ψ(xK
+∼)∥2.
+Dividing by α
+√
+K and using �K−1
+k=0 α2
+k = α2, we obtain
+E∥∇envη
+ψ(xK
+∼)∥2 ≤
+2(envη
+ψ(x0) − inf ψ)
+α(1 − η(ρ − λ))
+√
+K
++
+βθ
+η(1 − η(ρ − λ))
+α
+√
+K
+.
+B
+Auxiliary Lemmas
+Lemma 11. Let ϕ be convex and f be L-smooth. For η > 0, define Gη(x) := η−1�
+x − proxηϕ(x −
+η∇f(x))
+�
+. For any η > 0 and x ∈ Rn, it holds
+1−Lη
+η
+∥Gη(x)∥ ≤ ∥∇envη
+ψ(x)∥ ≤ 1+Lη
+η
+∥Gη(x)∥.
+28
+
+Proof. This follows from the fact that ∥∇envη
+ψ(x)∥ = η−1∥x − proxηψ(x)∥ and applying (Drusvy-
+atskiy & Lewis, 2018, Thm. 3.5) with t = η, G = ϕ, Φ = ψ, and β = L.
+Lemma 12. Let c ∈ R, a, x0 ∈ Rn and β > 0 and let ϕ : Rn → R ∪ {∞} be proper, closed, convex.
+The solution to
+y+ = arg min
+y∈Rn
+�
+c + ⟨a, y⟩
+�
++ + ϕ(y) + 1
+2β ∥y − x0∥2
+(35)
+is given by
+y+ =
+�
+�
+�
+�
+�
+proxβϕ(x0 − βa),
+if c + ⟨a, proxβϕ(x0 − βa)⟩ > 0,
+proxβϕ(x0),
+if c + ⟨a, proxβϕ(x0)⟩ < 0,
+proxβϕ(x0 − βua)
+else, for u ∈ [0, 1] such that c + ⟨a, proxβϕ(x0 − βua)⟩ = 0.
+(36)
+Remark 3. The first two conditions can not hold simultaneously due to uniqueness of the solution.
+If neither of the conditions of the first two cases are satisfied, we have to find the root of u �→
+c + ⟨a, proxβϕ(x0 − βua)⟩ for u ∈ [0, 1]. Due to strong convexity of the objective in (35), we know
+that there exists a root and hence y+ can be found efficiently with bisection.
+Proof. The objective of (35) is strongly convex and hence there exists a unique solution. Due to
+(Beck, 2017, Thm. 3.63), y is the solution to (35) if and only if it satisfies first-order optimality, i.e.
+∃u ∈ ∂(·)+(c + ⟨a, y⟩) : 0 ∈ ua + ∂ϕ(y) + 1
+β (y − x0).
+(37)
+Now, as y = proxβϕ(z) ⇐⇒ 0 ∈ ∂ϕ(y) + 1
+β (y − z), it holds
+(37) ⇐⇒ ∃u ∈ ∂(·)+(c + ⟨a, y⟩) : 0 ∈ ∂ϕ(y) + 1
+β (y − (x0 − βua))
+⇐⇒ ∃u ∈ ∂(·)+(c + ⟨a, y⟩) : y = proxβϕ(x0 − βua).
+We distinguish three cases:
+1. Let ¯y := proxβϕ(x0 − βa) and suppose that c + ⟨a, ¯y⟩ > 0. Then ∂(·)+(c + ⟨a, ¯y⟩) = {1} and
+hence ¯y satisfies (37) with u = 1. Hence, y+ = ¯y.
+2. Let ¯y := proxβϕ(x0) and suppose that c+⟨a, ¯y⟩ < 0. Then ∂(·)+(c+⟨a, ¯y⟩) = {0} and hence
+¯y satisfies (37) with u = 0. Hence, y+ = ¯y.
+3. If neither the condition of the first nor of the second case of (36) are satisfied, then, as (37)
+is a necessary condition for the solution y+, it must hold c+⟨a, y+⟩ = 0. Hence, there exists
+a u ∈ ∂(·)+(c + ⟨a, y+⟩) = [0, 1] such that
+c + ⟨a, proxβϕ(x0 − uβa)⟩ = 0.
+Lemma 13. For any K ≥ 1 it holds
+K−1
+�
+k=0
+1
+k+1 = 1 +
+K−1
+�
+k=1
+1
+k+1 ≤ 1 +
+� K−1
+0
+1
+s+1ds = 1 + ln K,
+K−1
+�
+k=0
+1
+√k+1 ≥
+� K
+0
+1
+√s+1ds = 2
+√
+K + 1 − 2.
+29
+
+C
+Model equivalence for SGD
+In the unregularized case, the SGD update
+xk+1 = xk − αkgk,
+gk ∈ ∂f(xk; Sk),
+can be seen as solving (5) with the model
+fx(y; s) = f(x; s) + ⟨g, y − x⟩,
+g ∈ ∂f(x; s).
+Now, consider again the regularized problem (2) with ϕ(x) = λ
+2 ∥x∥2 and update (8) .
+On the one hand, the model ψx(y; s) = f(x; s) + ϕ(x) + ⟨g + λx, y − x⟩ with g ∈ ∂f(x; s) yields
+xk+1 = xk − αk(gk + λxk) = (1 − αkλ)xk − αkgk.
+(38)
+On the other hand, the model ψx(y; s) = f(x; s) + ⟨g, y − x⟩ + ϕ(y) with g ∈ ∂f(x; s) results in
+xk+1 = proxαkϕ(xk − αkgk) =
+1
+1 + αkλ
+�
+xk − αkgk
+�
+= (1 −
+αk
+1 + αkλλ)xk −
+αk
+1 + αkλgk.
+(39)
+Running (38) with step sizes αk = βk is equivalent to running (39) with step sizes
+αk
+1+αkλ = βk ⇐⇒
+αk =
+βk
+1−βkλ. In this sense, standard SGD can be seen to be equivalent to proximal SGD for ℓ2–
+regularized problems.
+D
+Additional information on numerical experiments
+D.1
+Matrix Factorization
+Synthetic data generation: We adapted the experimental setting of the deep matrix factorization
+experiments in (Loizou et al., 2021), but extending with regularization. We generate data in the
+following way: first sample B ∈ Rq×p with uniform entries in the interval [0, 1]. Then choose κ ∈ R
+(which will be our targeted inverse condition number) and compute A = DB where D is a diagonal
+matrix with entries from 1 to κ (equidistant on a logarithmic scale)10.
+In order to investigate
+the impact of regularization, we generate a noise matrix E with uniform entries in [−ε, ε] and set
+˜A := A⊙(1+E). We then sample y(i) ∼ N(0, I) and compute the targets b(i) = ˜Ay(i). A validation
+set of identical size is created by the same mechanism, but computing its targets, denoted by b(i)
+val,
+via the original matrix A instead of ˜A. The validation set contains Nval = N samples.
+Name
+p
+q
+N
+κ
+r
+ε
+matrix-fac1
+6
+10
+1000
+1e-5
+4
+0
+matrix-fac2
+6
+10
+1000
+1e-5
+10
+0.05
+Table 1: Matrix factorization synthetic datasets.
+Model and general setup: Problem (28) can be interpreted as a two-layer neural network without
+activation functions. We train the network using the squared distance of the model output and b(i)
+(averaged over a mini-batch) as the loss function. We run 50 epochs for different methods, step size
+10Note that (Loizou et al., 2021) uses entries from 1 to κ on a linear scale which, in our experiments, did not result
+in large condition numbers even if κ is very small.
+30
+
+schedules and values of λ. For each different instance, we do ten independent runs: each run has
+the identical training set and initialization of W1 and W2, but different shuffling of the training set
+and different samples y(i) for the validation set. In order to allow a fair comparison, all methods
+have identical train and validation sets across all runs. All metrics are averaged over the ten runs.
+We always use a batch size of 20.
+D.2
+Plots for matrix-fac2
+In this section, we plot additional results for Matrix Factorization, namely for the setting
+matrix-fac2 of Table 1. The results are qualitatively very similar to the setting matrix-fac1.
+0
+10
+20
+30
+40
+Epoch
+10−5
+10−4
+10−3
+10−2
+10−1
+ψ(xk) − mink ψ(xk)
+constant
+0
+10
+20
+30
+40
+Epoch
+10−5
+10−4
+10−3
+10−2
+10−1 sqrt
+α0
+2.0
+1.62
+1.25
+0.88
+0.5
+2.0
+1.62
+1.25
+0.88
+0.5
+0.7
+0.56
+0.41
+0.27
+0.12
+prox-sps
+sps
+sgd
+Figure 10: Objective function for the Matrix Factorization problem (28), with constant (left) and
+sqrt (right) step size schedule and several choices of initial values. Here mink ψ(xk) is the best
+objective function value found over all methods and all iterations.
+0
+10
+20
+30
+40
+Epoch
+10−2
+10−1
+100
+Validation Error
+constant
+0
+10
+20
+30
+40
+Epoch
+10−2
+10−1
+100
+sqrt
+α0
+2.0
+1.62
+1.25
+0.88
+0.5
+2.0
+1.62
+1.25
+0.88
+0.5
+0.7
+0.56
+0.41
+0.27
+0.12
+prox-sps
+sps
+sgd
+Figure 11: Validation error for the Matrix Factorization problem (28), with constant (left) and
+sqrt (right) step size schedule and several choices of initial values.
+31
+
+0
+20
+40
+Epoch
+0.00025
+0.00050
+0.00075
+0.00100
+0.00125
+0.00150
+Objective ψ(xk)
+λ = 1e − 05
+0
+20
+40
+Epoch
+0.0010
+0.0015
+0.0020
+0.0025
+0.0030
+0.0035
+λ = 0.0001
+0
+20
+40
+Epoch
+0.010
+0.015
+0.020
+0.025
+λ = 0.001
+0
+20
+40
+Epoch
+0.10
+0.15
+0.20
+0.25
+λ = 0.01
+0
+20
+40
+Epoch
+1.0
+1.5
+2.0
+2.5
+λ = 0.1
+prox-sps, sqrt, α0=10.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=10.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=1.0
+0
+20
+40
+Epoch
+0.0063
+0.0064
+0.0065
+0.0066
+Validation Error
+λ = 1e − 05
+0
+20
+40
+Epoch
+0.0065
+0.0070
+0.0075
+0.0080
+λ = 0.0001
+0
+20
+40
+Epoch
+0.006
+0.008
+0.010
+0.012
+0.014
+0.016
+0.018
+λ = 0.001
+0
+20
+40
+Epoch
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+λ = 0.01
+0
+20
+40
+Epoch
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+λ = 0.1
+prox-sps, sqrt, α0=10.0
+prox-sps, sqrt, α0=5.0
+prox-sps, sqrt, α0=1.0
+sps, sqrt, α0=10.0
+sps, sqrt, α0=5.0
+sps, sqrt, α0=1.0
+Figure 12: Objective function value and validation error over the course of optimization. For the
+validation error, we plot a rolling median over five epochs in order to avoid clutter.
+32
+

T9E2T4oBgHgl3EQftQh4/content/tmp_files/2301.04068v1.pdf.txt

@@ -0,0 +1,5168 @@
+BLOBBED TOPOLOGICAL RECURSION FROM EXTENDED
+LOOP EQUATIONS
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Abstract. We consider the N × N Hermitian matrix model with measure
+dµE,λ(M) = 1
+Z exp(− λN
+4 tr(M 4))dµE,0(M), where dµE,0 is the Gaußian mea-
+sure with covariance ⟨MklMmn⟩ =
+δknδlm
+N(Ek+El) for given E1, ..., EN > 0. It was
+previously understood that this setting gives rise to two ramified coverings x, y
+of the Riemann sphere strongly tied by y(z) = −x(−z) and a family ω(g)
+n
+of
+meromorphic differentials conjectured to obey blobbed topological recursion
+due to Borot and Shadrin. We develop a new approach to this problem via
+a system of six meromorphic functions which satisfy extended loop equations.
+Two of these functions are symmetric in the preimages of x and can be deter-
+mined from their consistency relations. An expansion at ∞ gives global linear
+and quadratic loop equations for the ω(g)
+n . These global equations provide the
+ω(g)
+n
+not only in the vicinity of the ramification points of x but also in the
+vicinity of all other poles located at opposite diagonals zi + zj = 0 and at
+zi = 0. We deduce a recursion kernel representation valid at least for g ≤ 1.
+1. Introduction
+1.1. Historical comments. This paper is the fifth in a series [GHW19, SW22,
+BHW22, HW21] which investigates and solves the quartic Kontsevich model. See
+[BGHW22] for a review. Back in 1991, Kontsevich [Kon92] constructed his classi-
+cal matrix model to prove Witten’s conjecture [Wit91] that the generating series
+of intersection numbers on the moduli space Mg,n of stable complex curves is a tau
+function of the integrable KdV hierarchy. It is formulated as an N ×N-Hermitian
+matrix model with covariance ⟨MklMmn⟩ =
+δknδlm
+N(Ek+El) for Ek > 0, deformed by a
+cubic potential
+iN
+6 Tr(M 3).
+Explicit results were computed for the correlation
+function in the classical Kontsevich model for instance in [MS91, EO07, Eyn16]
+and more recently for higher spectral dimension with smooth covariance renor-
+malised by quantum field theoretical techniques in [GSW17, GSW18, GHW23].
+The quartic Kontsevich model has the same covariance ⟨MklMmn⟩ =
+δknδlm
+N(Ek+El) for
+Ek > 0, but deformed by a quartic potential λN
+4 Tr(M 4). It is different from the
+generalised Kontsevich model [BCEGF21] (see for more details [BHW21, § 2.1]).
+2020 Mathematics Subject Classification. 05A15, 14H70, 14N10, 30F30, 32A20.
+Key words and phrases. (Blobbed) topological recursion, matrix models, exactly solvable
+models, enumerative geometry, Dyson-Schwinger equations.
+1
+arXiv:2301.04068v1  [math-ph]  10 Jan 2023
+
+2
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+The Hermitian 1-matrix models with trivial covariance but arbitrary polyno-
+mial deformation were understood properly in [Eyn05]. The solution of the more
+advanced Hermitian 2-matrix model [CEO06] shared similar structures to the
+Hermitian 1-matrix model and the Kontsevich model, which gave rise to a univer-
+sal definition of the theory of topological recursion (TR) [EO07] for general initial
+data (Σ, x, y, B), the so-called spectral curve. We assume in this article to have
+Σ ⋍ ˆC and two coverings x, y : Σ → Σ with simple distinct ramification points.
+These two coverings build in TR a meromorphic 1-form ω(0)TR
+1
+= y dx on Σ;
+ω(0)TR
+2
+= B is the symmetric meromorphic bilinear diffferential on Σ2 with double
+pole on the diagonal and no residue. From the initial data (Σ, x, y, B), Eynard and
+Orantin defined recursively in the negative Euler characteristic −χ = 2g + n − 2
+a family of symmetric meromorphic differentials ω(g)TR
+n
+on Σn with poles just at
+the ramification points of x for −χ > 0.
+The important insight coming from the Hermitian 2-matrix model was the cor-
+rect local interpretation around a ramification point βi through the deck trans-
+formation σi, giving a local map between two branches ramified at βi.
+The proof that the 2-matrix model satisfies TR has used a completely new
+technique due to the fact that the number of ramification points can be arbitrarily
+large depending on the deformation (the potential) of the model. The starting
+point was to look at two different classes of mixed correlators, where one of
+them is rational in x(z) rather than z [CEO06].
+We denote these functions
+by H(g),TR
+n+1
+(y(w); z; I) and P (g),TR
+n+1
+(y(w); x(z); I)1, where I = {u1, ..., un}. Here,
+H(g),TR is rational in {y(w), z} ∪ I and P (g),TR in {y(w), x(z)} ∪ I as indicated
+by the arguments.
+The two functions H(g),TR
+and P (g),TR
+together with W (g),TR
+n
+defined
+by du1...dunW (g),TR
+n+1
+(z; u1, ..., un)dx(z) := ω(g)TR
+n+1 (z, u1, ..., un) satisfy a Dyson-
+Schwinger equation (DSE)
+(y(w) − y(z))H(g),TR
+n+1
+(y(w); z; I) + P (g),TR
+n+1
+(y(w); x(z); I)
+(1.1)
+= −
+�
+g1+g2=g
+I1⊎I2=I
+(g2,I2)̸=(0,∅)
+H(g1),TR
+|I1|+1 (y(w); z; I1)W (g2),TR
+|I2|+1 (z; I2) − ∂x(z′)H(g−1),TR
+n+2
+(y(w); z; z′, I)
+��
+z′=z .
+Exactly the same DSE for some H(g),TR and P (g),TR (but with different x, y)
+appears in several other examples which are governed by TR (see for instance
+[BCEGF21, BH22]).
+The important observation is that the solution of P (g),TR has a representation
+in terms of symmetric functions of W (g′),TR
+n′
+, which are symmetric in all preimages
+of x in the variable z, i.e. symmetric in (ˆzk)k=0,1,...,d with x(z) = x(ˆzk) and ˆz0 = z.
+Furthermore, the function H(g),TR was chosen such that its asymptotic expansion
+1In [CEO06], we identify U (g) → �
+ui∈I ∂x(ui)H(g),T R and E(g) → �
+ui∈I ∂x(ui)P (g),T R
+
+BTR FROM EXTENDED LOOP EQUATIONS
+3
+in y(w) has as leading order W (g),TR, i.e.
+H(g),TR
+n+1
+(y(w); z; I) =
+1
+y(w)W (g),TR
+n+1
+(z; I) + O(y(w)−2) ,
+plus additional terms for (g, n) ∈ {(0, 0), (0, 1)}. From this one can show that
+the asymptotic expansion of the solution of P (g),TR recovers the so-called linear
+and quadratic loop equations [BEO15]), which are describing the local behaviour
+of W (g),TR around the ramification points.
+Finally, from the linear and qua-
+dratic loop equations, the well-known recursion formula for the W (g),TR
+n+1
+(z; I) or
+ω(g),TR
+n+1
+(z, I) can be deduced. Vica versa, for any spectral curve in TR, rˆole and
+formulae for H(g),TR, P (g),TR are completely determined. Actually, to prove that
+some example is governed by TR, the starting point is in general the equation
+(1.1).
+A family W (g),TR
+n+1
+(z; I) satisfies topological recursion if and only if it satisfies
+the linear and quadratic loop equations and all of its poles are for 2g + n −
+1 > 0 at the ramifications points of x. It is very natural to assume the linear
+and quadratic loop equations but to relax the assumption on the pole structure.
+This gave birth to a generalisation of TR by blobbed topological recursion (BTR)
+[BS17]. The motivation of formulating BTR came from the enumeration of stuffed
+maps [Bor14], which is a natural generalisation of the Hermitian 1-matrix model
+through a deformation (the potential) of higher topological structure. BTR is
+built not just by the initial data (Σ, x, y, B), but enriched by additional blobs
+φg′,n′ associated with a topology contributing if 2g + n − 2 ≤ 2g′ + n′ − 2. The
+meromorphic functions W (g)
+n+1(z; I) = PzW (g)
+n+1(z; I)+ HzW (g)
+n+1(z; I) decompose in
+BTR into a part PzW (g)
+n+1 with poles at the ramification points of x and a part
+HzW (g)
+n+1 with poles elsewhere. The decomposition is constructed by orthogonal
+projectors Pz, Hz. Due to the linear and quadratic loop equations, it is rather
+easy to show that the polar part PzW (g)
+n+1(z; I) is still determined by the TR
+recursion formula. However, the part HzW (g)
+n+1(z; I) depends highly on the model
+under consideration and therefore on the enriched initial data φg,n.
+1.2. Main result. Consider the N × N Hermitian matrix model with measure
+dµE,λ(M) =
+1
+Z exp(− λN
+4 tr(M 4))dµE,0(M) where dµE,0 is the Gaußian measure
+with covariance ⟨MklMmn⟩ =
+δknδlm
+N(Ek+El) for given E1, ..., EN > 0. Let e1, ..., ed be
+the pairwise distinct values in (E1, ..., EN) and r1, ..., rd their multiplicities. In
+previous work [SW22, BHW22] we showed that this setting (called the quartic
+Kontsevich model) is characterised by a generic ramified covering x : ˆC → ˆC of
+degree d+1 with simple poles (one located at ∞) and simple ramification points.
+The key observation was that the other ramified covering y of the spectral curve
+is simply given by
+y(z) = −x(−z).
+(1.2)
+
+4
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+This strong x, y-entanglement has exceptional consequences. We show in this
+paper that the Dyson-Schwinger equations established in [BHW22] give rise to a
+system of six functions which come in triples
+(P (g)
+n+1(x(w), x(z); I), H(g)
+n+1(x(w); z; I), U (g)
+n+1(w, z; I))
+and
+(Q(g)
+n+1(x(w), x(z); I), M (g)
+n+1(x(w); z; I), V (g)
+n+1(w, z; I)).
+Compared with the two functions (P (g),TR
+n+1
+(y(w), x(z); I), H(g),TR(y(w); z; I)) in
+TR, the rˆole of x, y is partly flipped. Partial fraction decompositions given in Def-
+inition 2.2 and the equations themselves given in Proposition 2.3 are intervowen.
+By extending the known techniques from the literature (applied for instance in
+[BCEGF21, BH22]), we show how our system of equations can be solved and
+how the Taylor expansion about
+1
+x(w) = 0 gives rise to globally defined linear and
+quadratic loop equations. They indicate poles at the ramification points z = βi of
+x, at the opposite diagonal z +ui = 0 and (for g ≥ 1) at z = 0. From the detailed
+structure a formula to compute ω(g)
+n
+recursively in decreasing Euler characteristic
+is deduced:
+Theorem 1.1. Let I = {u1, ..., un} and x be a generic ramified cover of ˆC. Let
+x, y be related via (1.2) and ω(0)
+2 (z, w) =
+dzdw
+(z−w)2 +
+dzdw
+(z+w)2.
+If the six function
+(P (g), H(g), U (g)) and (Q(g), M (g), V (g)) satisfy the interwoven DSEs of Proposi-
+tion 2.3 and the partial fraction decomposition of Definition 2.2, then the so-
+lution for ω(g)
+n+1(z, u1, ...un) = λ2g+n−1du1 · · · dunW (g)
+n+1(z; u1, ..., un)dx(z), where
+H(g)
+n+1(x(v); z; I) = −
+λW (g)
+n+1(z;I)
+x(v)
++ O(x(v)−2), is recursively computed for all I with
+2g + |I| ≥ 2 and for g < 2 via the recursion kernel representation
+ω(g)
+|I|+1(z, I) =
+�
+βi
+Res
+q→βi Ki(z; q)
+� �
+I1⊎I2=I
+g1+g2=g
+(gi,Ii)̸=(0,∅)
+ω(g1)
+|I1|+1(q, I1)ω(g2)
+|I2|+1(q, I2) + ω(g−1)
+|I|+2 (q, q, I)
+�
++
+|I|
+�
+j=1
+duj
+�
+Res
+q→−uj Kuj(z; q)
+� �
+I1⊎I2=I
+g1+g2=g
+(gi,Ii)̸=(0,∅)
+d−1
+uj (ω(g1)
+|I1|+1(q, I1)ω(g2)
+|I2|+1(q, I2))
++ d−1
+uj ω(g−1)
+|I|+2 (q, q, I) + (dx(q))2
+6
+∂2
+∂(x(q))2
+�ω(g−1)
+|I|+1 (q, I))
+dx(q)dx(uj)
+���
++ Res
+q→0 K0(z; q)
+� �
+I1⊎I2=I
+g1+g2=g
+(gi,Ii)̸=(0,∅)
+ω(g1)
+|I1|+1(q, I1)ω(g2)
+|I2|+1(q, I2) + ω(g−1)
+|I|+2 (q, q, I)
++ (dx(q))2
+2
+∂
+∂x(q)
+�d−1
+q′ ω(g−1)
+|I|+2 (q, q′, I)
+dx(q)
+���
+q′=q
+��
+,
+(1.3)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+5
+where βi are the ramification points of x, ω(0)
+2 (q, q) should be replaced by
+limq′→q(ω(0)
+2 (q, q′) −
+1
+(x(q)−x(q′))2) and the recursion kernels are given by Ki(z; q) =
+−
+( dz
+z−q −
+dz
+z−σi(q) )
+2(y(q)−y(σi(q)))dx(q), Ku(z; q) = −
+( dz
+z−q − dz
+z+u )
+2(y(q)+x(u))dx(q) and K0(z; q) = −
+( dz
+z−q − dz
+z )
+2(y(q)+x(q))dx(q).
+We do not see any obstacle to extend the result to all g; only the combinatorics
+becomes extremely involved and Taylor expansions of P (g)
+n+1(x(w), x(z); I) up to
+an order which increases as 4g become necessary. Already in the proof up to
+genus g ≤ 1 the employment of a loop insertion operator as an abbreviation
+for particular rational functions of (P (g), H(g), U (g)) and (Q(g), M (g), V (g)) was
+essential to master the combinatorics.
+A few remarks:
+• We establish the linear and quadratic loop equations globally on ˆC and
+not only in small neighbourhoods of the ramification points as required
+in the general formulation [BS17]. Therefore, we not only get a recursion
+formula for the ‘polar’ part Pzω(g)
+n+1(z, I) but for the entire ω(g)
+n+1(z, I).
+• In [HW21] we solved the genus-0 sector under the much weaker assumption
+that x, y are related as y(z) = −x(ιz) for some holomorphic involution ι
+on ˆC. We noticed that although one can solve (Q(0)
+1 , M (0)
+1 , V (0)
+1
+) also for
+y(z) = −x(ιz) along the lines of [SW22], the resulting expressions are of
+completely different type to which our tools do not apply. Nevertheless we
+would like to remark that the involution identity discovered in [HW21] was
+vastly extended in [Hoc22b, Hoc22a, ABDB+22] to a general approach to
+the x-y symmetry in topological recursion.
+• Considering in TR a more general version of (1.1) including intermediate
+correlators W (g),TR
+n,m
+(see [Hoc22b] for details), one can show that these
+correlatators can be derived via BTR with an astonishing similarity to
+the formula (1.3) except for the pole at the origin [Hoc23].
+The paper is organised as follows. In sec. 2 we derive from previous results
+a system of equations for (P (g)
+n , H(g)
+n , U (g)
+n ) and (Q(g)
+n , M (g)
+n , V (g)
+n ) and introduce
+the loop insertion operator. Sec. 3 solves P (0)
+|I|+1(x(v), x(z); I) by exploiting its
+symmetry in the preimages of x and its residues at x(v) = x(ui). The outcome
+gives the linear and quadratic loop equations for W (0)
+n . By essentially the same
+methods (but including poles at x(v) = x(0)) we identify Q(0)
+|I|+1(x(v), x(z); I) in
+sec. 3.3 and, with much larger effort, in particular in view of poles at x(v) = x(z),
+the functions P (1)
+|I|+1(x(v), x(z); I) in secs. 4.1 and 4.2. Again, this allows us to
+derive the global linear and quadratic loop equations for W (1)
+n
+from which we get
+in sec. 5 the recursion formula stated in Theorem 1.1.
+
+6
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Acknowledgements. AH was supported through the Walter-Benjamin fellow-
+ship2. RW was supported3 by the Cluster of Excellence Mathematics M¨unster
+and the CRC 1442 Geometry: Deformations and Rigidity.
+2. Setup
+2.1. Summary of prevous results. In [BHW22] we have shown that a
+quartic analogue of the Kontsevich matrix model gives rise to, and is com-
+pletely determined by, a coupled system of Dyson-Schwinger equations for
+three families of meromorphic functions Ω(g)
+n (z1, ..., zn), T (g)(u1, ..., un∥z, w|) and
+T (g)(u1, ..., un∥z|w|) on the Riemann sphere ˆC = C ∪ {∞}. Of particular interest
+are the Ω(g)
+n
+which give rise to meromorphic differentials
+ω(g)
+n (z1, ..., zn) = Ω(g)
+n (z1, ..., zn)
+n
+�
+j=1
+dx(zj) ,
+where
+(2.1a)
+x(z) = z − λ
+N
+d
+�
+k=1
+ϱk
+z + εk
+.
+(2.1b)
+The ramified cover4 x : ˆC → ˆC forms with its reflection y(z) = −x(−z) and
+ω(0)
+2 (z1, z2) =
+dz1 dz2
+(z1−z2)2 + dz1 dz2
+(z1+z2)2 a spectral curve in the spirit of topological recursion
+[EO07]. The parameters (λ, N, d, {ϱk, εk}) are defined by the initial data of the
+quartic Kontsevich model: It is a matrix model for N × N-Hermitian matrices
+M with covariance ⟨MklMmn⟩ =
+δknδlm
+N(Ek+El) for Ek > 0, deformed by a quartic
+potential λN
+4 Tr(M 4). If (e1, ..., ed) are the pairwise different values in (Ek), which
+arise with multiplicities r1, ..., rd, then (εk, ϱk) are determined by x(εk) = ek and
+ϱkx′(εk) = rk with limλ→0 εk = ek and limλ→0 ϱk = rk [GHW19, SW22].
+The system of Dyson-Schwinger equations for Ω(g)
+n
+and the two variants of T (g)
+in [BHW22] is most conveniently expressed in terms of meromorphic functions
+(W (g)
+n , U (g)
+n , V (g)
+n ) where multiple derivatives
+∂
+∂x(ui) are taken out:
+Ω(g)
+n+1(z, u1, ..., un) =: ∂nW (g)
+n+1(z; u1, ..., un)
+∂x(u1) · · · ∂x(un)
+,
+(2.2)
+T (g)
+n+1(u1, ..., un∥z, w|) =: ∂nU (g)
+n+1(z, w; u1, ..., un)
+∂x(u1) · · · ∂x(un)
+,
+2“Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
+Project-ID 465029630.”
+3“Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
+Project-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044
+390685587, Mathematics M¨unster: Dynamics – Geometry – Structure.”
+4The ramified cover x was called R in [SW22, BHW22].
+
+BTR FROM EXTENDED LOOP EQUATIONS
+7
+T (g)
+n+1(u1, ..., un∥z|w|) =: ∂nV (g)
+n+1(z, w; u1, ..., un)
+∂x(u1) · · · ∂x(un)
+.
+In terms of (U, V, W) and with I := {u1, ..., un} and |I| = n, the system reads:
+(a) DSE for U (g)
+n+1:
+(x(w) + y(z))U (g)
+|I|+1(z, w; I) + λ
+N
+d
+�
+k=1
+rkU (g)
+|I|+1(εk, w; I)
+x(z) − x(εk)
+(2.3)
+= δ|I|,0δg,0 + λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(z; I1)U (g2)
+|2|+1(z, w; I2) − λ
+|I|
+�
+j=1
+U (g)
+|I| (ui, w; I \ uj)
+x(uj) − x(z)
+− λ
+∂
+∂x(s)U (g−1)
+|I|+2 (z, w; I ∪ {s})
+���
+s=z − λ
+V (g−1)
+|I|+1 (z, w; I) − V (g−1)
+|I|+1 (w, w; I)
+x(w) − x(z)
+.
+(b) DSE for V (g)
+n+1:
+(x(z) + y(z))V (g)
+|I|+1(z, w; I) + λ
+N
+d
+�
+k=1
+rk
+V (g)
+|I|+1(εk, w; I)
+x(z) − x(εk)
+(2.4)
+= −λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(z; I1)V (g2)
+|I2|+1(z, w; I2) − λ
+|I|
+�
+j=1
+V (g)
+|I| (ui, w; I\ui)
+x(ui) − x(z)
+− λ
+∂
+∂x(s)V (g−1)
+|I|+2 (z, w; I ∪ {s})
+���
+s=z − λ
+U (g)
+|I|+1(z, w; I) − U (g)
+|I|+1(w, w; I)
+x(w) − x(z)
+.
+(c) Connecting equation for W (g)
+n+1:
+W (g)
+|I|+1(z; I) =
+δ|I|,1δg,0
+x(z) − x(u1) + δ|I|,0δg,0
+λ
+�
+x(z) + λ
+N
+d
+�
+k=1
+rk
+x(εk) − x(z)
+�
+(2.5)
++ 1
+N
+d
+�
+l=1
+rlU (g)
+|I|+1(z, εl; I) −
+|I|
+�
+j=1
+U (g)
+|I| (z, ui; I\ui) + V (g−1)
+|I|+1 (z, z; I) .
+The connecting equation (c) has been extended to include the consistency equa-
+tion W (0)
+1 (z; ∅) :=
+1
+λy(z) for the ramified cover x, see [SW22].
+This consis-
+tency equation turned the originally non-linear equation [GW09, GW14] for
+U (0)
+1 (z, w) ≡ U (0)
+1 (z, w; ∅) into the linear equation (2.3) for I = ∅; its solution
+
+8
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+is [GHW19, SW22]
+U (0)
+1 (z, w) =
+1
+x(z) + y(w)
+�d
+k=1(x(w) + y(ˆzk))
+�d
+k=1(x(w) − x(εk))
+(2.6)
+where {z = ˆz0, ˆz1, ..., ˆzd} := x−1(x(z)) is the set of preimages of x. Straight-
+forward manipulations show the symmetry U (0)
+1 (z, w) = U (0)
+1 (w, z). The basic
+equation for V (0)
+1
+(z, w) ≡ V (0)
+1
+(z, w; ∅) has been solved in [SW22]:
+V (0)
+1
+(z, w)
+(2.7)
+=
+λ
+(x(w) − x(z))2
+�
+U (0)
+1 (z, w) − (x(w) + x(z) − 2x(0)) Æ(x(z)) Æ(x(w))
+(x(w) + y(w))(x(z) + y(z))
+�
+,
+where Æ(x(z)) := �d
+k=1
+x(z)−x(αk)
+x(z)−x(εk) and z ∈ {0, ±α1, ..., ±αd} are the solutions of
+x(z) + y(z) = 0. The diagonal function is also expressed in terms of Æ(x(z))
+[SW22]:
+U (0)
+1 (z, z) = 2(x(z) − x(0))(Æ(x(z)))2
+(x(z) + y(z))2
+.
+(2.8)
+Remark 2.1. The equations (2.3) and (2.4) were derived in [BHW22] from
+identities for the two-point functions ⟨MabMba⟩ =
+�
+HN MabMba dµE,λ(M) and
+⟨MaaMbb⟩ =
+�
+HN MaaMbb dµE,λ(M), which are symmetric in a, b. The deriva-
+tion made a choice in the order of a, b. Repeating all steps in the other order one
+would get an equation
+(x(z) + y(w))U (g)
+|I|+1(z, w; I) + λ
+N
+d
+�
+k=1
+rkU (g)
+|I|+1(z, εk; I)
+x(w) − x(εk)
+(2.3’)
+= δ|I|,0δg,0 + λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(w; I1)U (g2)
+|2|+1(z, w; I2) − λ
+|I|
+�
+j=1
+U (g)
+|I| (z, ui; I \ uj)
+x(uj) − x(w)
+− λ
+∂
+∂x(s)U (g−1)
+|I|+2 (z, w; I ∪ {s})
+���
+s=w − λ
+V (g−1)
+|I|+1 (z, w; I) − V (g−1)
+|I|+1 (z, z; I)
+x(z) − x(w)
+and similarly for V (g)
+|I|+1(z, w; I).
+Exchanging w
+↔
+z in (2.3’) and com-
+paring with
+(2.3) shows that the pairs (U (g)
+|I|+1(z, w; I), V (g)
+|I|+1(z, w; I)) and
+(U (g)
+|I|+1(w, z; I), V (g)
+|I|+1(w, z; I)) satisfy the same system of equations.
+Since the
+solution is unique according to the construction below, we necessarily have sym-
+metry U (g)
+|I|+1(z, w; I) = U (g)
+|I|+1(w, z; I) and V (g)
+|I|+1(z, w; I) = V (g)
+|I|+1(w, z; I)) in the
+first two arguments.
+
+BTR FROM EXTENDED LOOP EQUATIONS
+9
+The solution of the system (2.3)+(2.4)+(2.5) in [BHW22] for (g, n)
+=
+{(0, 2), (0, 3), (0, 4), (1, 1)} provided strong support for the conjecture that the
+meromorphic differentials ω(g)
+n
+obey blobbed topological recursion, a systematic
+extension of TR due to Borot and Shadrin [BS17].
+In [HW21] we succeeded
+in solving the genus g = 0 sector in larger generality. The result of [HW21],
+restricted to y(z) = −x(−z), is covered by Thm. 1.1 for g = 0.
+2.2. Auxiliary functions. Our proof of Theorem 1.1 follows a very different
+strategy than the one for genus g = 0 given in [HW21].
+The key idea is to
+rewrite the Dyson-Schwinger equations (2.3) and (2.4) into equations for auxiliary
+functions which in one or two arguments are symmetric in the preimages of x
+(given in (2.1b)).
+For genus g ≥ 1 it is essential that y(z) = −x(−z).
+We
+separate the arguments in the functions below by a comma if the function is
+symmetric when exchanging the arguments, otherwise by a semicolon.
+Definition 2.2. For I = {u1, ..., un} the following combinations of the functions
+U, V are introduced:
+H(g)
+|I|+1(x(v); z; I)
+(2.9a)
+:= δg,0δI,∅ − λ
+N
+d
+�
+l=1
+rlU (g)
+|I|+1(z, εl; I)
+x(v) − x(εl)
++ λ
+|I|
+�
+j=1
+U (g)
+|I| (z, uj; I\uj)
+x(v) − x(uj)
+− λ
+V (g−1)
+|I|+1 (z, z; I)
+x(v) − x(z)
+,
+P (g)
+|I|+1(x(v), x(z); I)
+(2.9b)
+:=
+λδ|I|,1δg,0
+x(v) − x(u1) + δ|I|,0δg,0
+�
+x(v) + x(z) − λ
+N
+d
+�
+k=1
+rk
+x(v) − x(εk)
+�
+− λ
+N
+d
+�
+k=1
+rkH(g)
+|I|+1(x(v); εk; I)
+x(z) − x(εk)
++ λ
+|I|
+�
+j=1
+H(g)
+|I| (x(v); uj; I \ uj)
+x(z) − x(uj)
+,
+M (g)
+|I|+1(x(v); z; I)
+(2.9c)
+:= − λ
+N
+d
+�
+l=1
+rlV (g)
+|I|+1(z, εl; I)
+x(v) − x(εl)
++ λ
+|I|
+�
+j=1
+V (g)
+|I| (z, uj; I\uj)
+x(v) − x(uj)
+− λ
+U (g)
+|I|+1(z, z; I)
+x(v) − x(z) ,
+Q(g)
+|I|+1(x(v), x(z); I)
+(2.9d)
+:= − λ
+N
+d
+�
+k=1
+rk
+M (g)
+|I|+1(x(v); εk; I)
+x(z) − x(εk)
++ λ
+|I|
+�
+j=1
+M (g)
+|I|+1(x(v); uj; I\uj)
+x(z) − x(uj)
+.
+Proposition 2.3. The Dyson-Schwinger equations (2.3), (2.4) and (2.5) imply:
+H(g)
+|I|+1(x(v); z; I)
+(2.10a)
+
+10
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+= (x(z) + y(v))U (g)
+|I|+1(v, z; I) + λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(v; I1)U (g2)
+|I2|+1(v, z; I2)
++ λ
+∂
+∂x(s)U (g−1)
+|I|+2 (v, z; I ∪ s)
+���
+s=v + λ
+V (g−1)
+|I|+1 (v, z; I)
+x(z) − x(v)
+,
+P (g)
+|I|+1(x(v), x(z); I)
+(2.10b)
+= (x(v) + y(z))H(g)
+|I|+1(x(v); z; I) + λ
+�
+I1⊎I2=I
+g1+g1=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(z; I1)H(g)
+|I2|+1(x(v); z; I2)
++ λ
+∂
+∂x(s)H(g−1)
+|I|+2 (x(v); z; I ∪ s)
+���
+s=z + λ
+M (g−1)
+|I|+1 (x(v); z; I)
+x(v) − x(z)
+,
+M (g)
+|I|+1(x(v); z; I)
+(2.10c)
+= (x(v) + y(v))V (g)
+|I|+1(v, z; I) + λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(v; I1)V (g2)
+|I2|+1(v, z; I2)
++ λ
+∂
+∂x(s)V (g−1)
+|I|+2 (v, z; I ∪ s)
+���
+s=v + λ
+U (g)
+|I|+1(v, z; I)
+x(z) − x(v) ,
+Q(g)
+|I|+1(x(v), x(z); I)
+(2.10d)
+= (x(z) + y(z))M (g)
+|I|+1(x(v); z; I) + λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+W (g1)
+|I1|+1(z; I1)M (g2)
+|I2|+1(x(v); z; I2)
++ λ
+∂
+∂x(s)M (g−1)
+|I|+2 (x(v); z; I ∪ s)
+���
+s=z + λ
+H(g)
+|I|+1(x(v); z; I)
+x(v) − x(z)
+.
+Proof. Equations (2.10a) and (2.10c) are an obvious rewriting of (2.3) and (2.4),
+respectively. For the proof of (2.10b) one starts from (2.10a) for z �→ εk, multi-
+plied by multiplies by
+λ
+N
+rk
+x(z)−x(εk) and summed over k. A rearrangement taking
+(2.5) into acount gives the assertion. Similarly for (2.10d).
+□
+Inserting U (0)
+1 (v, z) given by (2.6) into (2.10a) and the result into (2.10b) gives
+H(0)
+1 (x(v); z) := H(0)
+1 (x(v); z; ∅) =
+d
+�
+k=1
+x(v) + y(ˆzk)
+x(v) − x(εk) ,
+(2.11a)
+P (0)
+1 (x(v), x(z)) := P (0)
+1 (x(v); x(z); ∅) =
+�d
+k=0(x(v) + y(ˆzk))
+�d
+k=1(x(v) − x(εk))
+.
+(2.11b)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+11
+Proposition 2.4. The functions P (g)
+n
+and Q(g)
+n
+are symmetric in their first two
+arguments.
+Proof. Expressing all H(g′)
+n′
+and M (g′)
+n′
+in (2.10b) and (2.10d) in terms of U (g′′)
+n′′
+and
+V (g′′)
+n′′
+via (2.10a) and (2.10c) gives a manifestly symmetric expression when taking
+the symmetries U (g)
+|I|+1(z, v; I) = U (g)
+|I|+1(v, z; I) and V (g)
+|I|+1(z, v; I) = V (g)
+|I|+1(v, z; I)
+according to Remark 2.1 into account.
+□
+The Dyson-Schwinger equations (2.10a), (2.10b), (2.10c) and (2.10d) can be
+disentangled into two separate systems:
+Proposition 2.5. Let (Ch)h∈N be the sequence of Catalan numbers, defined for
+instance by C0 = 1 and Ch+1 = �h
+l=0 ClCh−l. Then the linear combinations
+ˆU (g)
+|I|+1(v, z; I)
+(2.12a)
+:= U (g)
+|I|+1(v, z; I) +
+g−1
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hV (g−1−h)
+|I|+1
+(v, z; I) ,
+ˆH(g)
+|I|+1(x(v); z; I)
+(2.12b)
+:= H(g)
+|I|+1(x(v); z; I) +
+g−1
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hM (g−1−h)
+|I|+1
+(x(v); z; I) ,
+ˆP (g)
+|I|+1(x(v), x(z); I)
+(2.12c)
+:= P (g)
+|I|+1(x(v), x(z); I) +
+g−1
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hQ(g−1−h)
+|I|+1
+(x(v), x(z); I) ,
+ˆV (g)
+|I|+1(z, v; I)
+(2.12d)
+:= V (g)
+|I|+1(v, z; I) −
+g
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hU (g−h)
+|I|+1 (v, z; I) ,
+ˆ
+M (g)
+|I|+1(x(v); z; I)
+(2.12e)
+:= M (g)
+|I|+1(x(v); z; I) −
+g
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hH(g−h)
+|I|+1 (x(v); z; I) ,
+ˆQ(g)
+|I|+1(x(v), x(z); I)
+(2.12f)
+:= Q(g)
+|I|+1(x(v), x(z); I) −
+g
+�
+h=0
+(−1)hChλ1+2h
+(x(v) − x(z))2+4hP (g−h)
+|I|+1 (x(v), x(z); I)
+satisfy
+ˆH(g)
+I|+1(x(v); z; I)
+(2.13a)
+
+12
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+= (x(z) + y(v)) ˆU (g)
+|I|+1(v, z; I) + λ
+∂ ˆU (g−1)
+|I|+2 (v, z; I ∪ s)
+∂x(s)
+���
+s=v
++ λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+�
+W (g1)
+|I1|+1(v; I1) − (−1)g1λ2g1−1δI1,∅Cg1−1
+(x(z) − x(v))4g1−1
+�
+ˆU (g2)
+|I2|+1(v, z; I2) ,
+ˆP (g)
+I|+1(x(v), x(z); I)
+(2.13b)
+= (x(v) + y(z)) ˆH(g)
+|I|+1(x(v); z; I) + λ
+∂ ˆH(g−1)
+|I|+2 (x(v); z; I ∪ s)
+∂x(s)
+���
+s=z
++ λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+�
+W (g1)
+|I1|+1(z; I1) − (−1)g1λ2g1−1δI1,∅Cg1−1
+(x(v) − x(z))4g1−1
+�
+ˆH(g2)
+|I2|+1(x(v); z; I2) ,
+ˆ
+M (g)
+I|+1(x(v); z; I)
+(2.13c)
+= (x(v) + y(v))ˆV (g)
+|I|+1(v, z; I) + λ
+∂ ˆV (g−1)
+|I|+2 (v, z; I ∪ s)
+∂x(s)
+���
+s=z
++ λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+�
+W (g1)
+|I1|+1(v; I1) − (−1)g1λ2g1−1δI1,∅Cg1−1
+(x(v) − x(z))4g1−1
+�
+ˆV (g2)
+|I2|+1(v, z; I2) ,
+ˆQ(g)
+I|+1(x(v), x(z); I)
+(2.13d)
+= (x(z) + y(z)) ˆ
+M (g)
+|I|+1(x(v); z; I) + λ
+∂ ˆ
+M (g−1)
+|I|+2 (x(v); z; I ∪ s)
+∂x(s)
+���
+s=z
++ λ
+�
+I1⊎I2=I
+g1+g2=g
+(g1,I1)̸=(0,∅)
+�
+W (g1)
+|I1|+1(z; I1) − (−1)g1λ2g1−1δI1,∅Cg1−1
+(x(v) − x(z))4g1−1
+�
+ˆ
+M (g2)
+|I2|+1(x(v); z; I2) .
+Proof. Lengthy but straightforward. The Segner recursion Ch+1 = �h
+l=0 ClCh−l
+is a key step.
+□
+In the following cases it is safe to omit the hat: ˆU (0)
+n+1 = U (0)
+n+1, ˆH(0)
+n+1 = H(0)
+n+1,
+ˆP (0)
+n+1 = P (0)
+n+1.
+2.3. The loop insertion operator. Let ˜R be the ring of Q-polynomials in the
+variables
+�
+x, y, W (g)
+n+1, ˆU (g)
+n+1, ˆH(g)
+n+1, ˆP (g)
+n+1, ˆV (g)
+n+1, ˆ
+M (g)
+n+1, ˆQ(g)
+n+1
+�
+,
+(2.14a)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+13
+in x( . )-derivatives of them (e.g.
+∂2y(z)
+∂(x(z))2,
+∂2W (0)
+n+1(z;u1,...,un)
+∂x(z)∂x(u1)
+,
+∂4 ˆP (g)
+n+1(x(z),x(w);u1,...,un)
+∂(x(z))2∂x(w)∂x(un)
+)
+as well as reciprocals
+�
+1
+x(∗) − x(⋆),
+1
+x(∗) + y(⋆),
+1
+U (0)
+1
+,
+1
+H(0)
+1
+,
+1
+P (0)
+1
+,
+1
+ˆV (0)
+1
+,
+1
+ˆ
+M (0)
+1
+,
+1
+ˆQ(0)
+1
+�
+.
+(2.14b)
+We let I be the ideal in ˜R generated by
+�
+(x(v) + y(z)) ·
+1
+(x(v) + y(z)) − 1,
+U (0)
+1 (v, z) ·
+1
+U (0)
+1 (v, z)
+− 1,
+H(0)
+1 (x(v); z) ·
+1
+H(0)
+1 (x(v); z)
+− 1,
+P (0)
+1 (x(v), x(z)) ·
+1
+P (0)
+1 (x(v), x(z))
+− 1
+�
+and R = ˜R/I be the quotient. The variables (2.14) belong to the field of mero-
+morphic functions on several copies of ˆC. For our purpose they are considered
+as independent variables. We also consider ˆP (g)
+n+1(x(v), x(z); I) as independent of
+ˆP (g)
+n+1(x(v′), x(z′); I′) whenever x(v) ̸= x(v′) or x(z) ̸= x(z′) or I ̸= I′. Simi-
+larly for ˆQ(g)
+n+1. We consider ˆH(g)
+n+1(x(v); z; I) as independent of ˆH(g)
+n+1(x(v′); z′; I′)
+whenever x(v) ̸= x(v′) or z ̸= z′ or I ̸= I′. Similarly for ˆ
+M (g)
+n+1. We consider
+ˆU (g)
+n+1(v, z; I) as independent of ˆH(g)
+n+1(v′; z′; I′) whenever v ̸= v′ or z ̸= z′ or
+I ̸= I′. Similarly for ˆV (g)
+n+1.
+The Dyson-Schwinger equations (2.13) are polynomial equations fi = 0 for fi ∈
+R. In addition we have relations between residues which follow from (2.9) and
+(2.12) as well as from the condition [(x(v))−1]H(g)
+|I|+1(x(v); z; I) = −λW (g)
+n+1(z; I).
+The most decisive tool in our construction is a loop insertion operator.
+Definition 2.6. For u ∈ ˆC, the loop insertion operator Du : R → R is defined
+on the variables (2.14a) as
+Du(x(z)) = 0,
+Duy(z) = λW (0)
+2 (z; u),
+DuW (g)
+|I|+1(z; I) = W (g)
+|I|+2(z; I ∪ u),
+Du ˆU (g)
+|I|+1(v, z; I) = ˆU (g)
+|I|+2(v, z; I ∪ u),
+Du ˆV (g)
+|I|+1(v, z; I) = ˆV (g)
+|I|+2(v, z; I ∪ u),
+Du ˆH(g)
+|I|+1(x(v); z; I) = ˆH(g)
+|I|+2(x(v); z; I ∪ u),
+Du ˆ
+M (g)
+|I|+1(x(v); z; I) = ˆ
+M (g)
+|I|+2(x(v); z; I ∪ u),
+Du ˆP (g)
+|I|+1(x(v), x(z); I) = ˆP (g)
+|I|+2(x(v), x(z); I ∪ u),
+Du ˆQ(g)
+|I|+1(x(v), x(z); I) = ˆQ(g)
+|I|+2(x(v), x(z); I ∪ u)
+and extended to R by linearity, Leibniz rule, the requirement Du : I → I
+and commutation with any x( . )-derivative. For J = {u1, ..., un} we let DJ :=
+Du1 · · · Dun : R → R. Moreover, D∅ is defined as the identity operator.
+
+14
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+The condition Du : I → I just means that the loop insertion operator of the
+reciprocals in (2.14b) is given by the usual rules of calculus, e.g.
+I ∋Du
+�
+(x(v) + y(z)) ·
+1
+(x(v) + y(z))
+�
+− Du1
+= (x(v) + y(z))
+� λW (0)
+2 (z; u)
+(x(v) + y(z))2 + Du
+�
+1
+x(v) + y(z)
+��
+,
+i.e. Du
+�
+1
+x(v)+y(z)
+�
+= − λW (0)
+2
+(z;u)
+(x(v)+y(z))2 + I. We can also apply DI to logarithms
+DI log H(0)
+1 (x(v); z) =
+|I|
+�
+l=1
+(−1)l−1
+l
+�
+I1⊎...⊎Il=I
+I1,...Il̸=∅
+l�
+j=1
+H(0)
+|Ij|+1(x(v); z; Ij)
+H(0)
+1 (x(v); z)
+,
+(2.15)
+DI log P (0)
+1 (x(v), x(z)) =
+|I|
+�
+l=1
+(−1)l−1
+l
+�
+I1⊎...⊎Il=I
+I1,...Il̸=∅
+l�
+j=1
+P (0)
+|Ij|+1(x(v), x(z); Ij)
+P (0)
+1 (x(v), x(z))
+and similarly for DI log ˆ
+M (0)
+1 (x(x); z) and DI log ˆQ(0)
+1 (x(v), x(z)). In the same
+way one has
+DI log(x(v) + y(z)) =
+|I|
+�
+l=1
+(−1)l−1
+l
+�
+I1⊎...⊎Il=I
+I1,...Il̸=∅
+l�
+j=1
+λW (0)
+|Ij|+1(z; Ij)
+x(v) + y(z)
+.
+(2.16)
+The loop insertion operator is compatible with the Dyson-Schwinger equations
+(2.13).
+3. Solution for genus g = 0
+3.1. Auxiliary functions for genus g = 0. Starting point is the following
+observation:
+Lemma 3.1. For any I = {u1, ..., un} one has
+DI log P (0)
+1 (x(v), x(z)) = DI log(x(v) + y(z)) + DI log H(0)
+1 (x(v); z) ,
+(3.1)
+where the three terms are short-hand notations for (2.15) and (2.16).
+Proof. This is a combinatorial rewriting of (2.10b), which reads
+P (0)
+|I|+1(x(v), x(z); I)
+P (0)
+1 (x(v), x(z))
+=
+H(0)
+|I|+1(x(v); z; I)
+H(0)
+1 (x(v); z)
++
+λW (0)
+|I|+1(z; I)
+x(v) + y(z)
++
+�
+I′⊎I′′
+I′,I′′̸=∅
+λW (0)
+|I′|+1(z; I′)
+x(v) + y(z)
+H(0)
+|I′′|+1(x(v); z; I′′)
+H(0)
+1 (x(v); z)
+.
+(3.2)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+15
+We arrange products of these expressions into DI log P (0)
+1 (x(v), x(z)) in the second
+line of (2.15). A contribution with h ≥ 1 factors of
+H(0)
+|Ii|+1(x(v);z;Ii)
+H(0)
+1
+(x(v);z)
+and w ≥ 1
+factors of
+λW (0)
+|Ij|+1(z;Ij)
+x(v)+y(z)
+arises via the binomial theorem in
+(w+h−k)!
+(w−k)!(h−k)k! different
+ways if k times a factor of the second line of (3.2) occurs. The prefactor of such
+a term in DI log P (0)
+1 (x(v), x(z)) is (−1)w+h−k−1
+(w+h−k)
+. The sum over k equals
+min(w,h)
+�
+k=0
+(−1)k (w + h − k − 1)!
+(w − k)!(h − k)k! = 0 ,
+which follows e.g. from [Gou10, vol. 4, eq. (10.20)]
+n
+�
+k=0
+(−1)k
+�n
+k
+��x − k
+j
+�
+=
+�x − n
+j − n
+�
+when setting n = h, x = w + h − 1 and j = h − 1. Hence, all contributions with
+at least one factor
+H(0)
+|Ii|+1(x(v);z;Ii)
+H(0)
+1
+(x(v);z)
+and at least one factor
+λW (0)
+|Ij|+1(z;Ij)
+x(v)+y(z)
+cancel, and
+the assertion follows.
+□
+In complete analogy to Lemma 3.1 one establishes
+DI log H(0)
+1 (x(v); z) = DI log(x(z) + y(v)) + DI log U (0)
+1 (v, z) ,
+(3.3a)
+DI log ˆ
+M (0)
+1 (x(v), x(z)) = DI log(x(v) + y(v)) + DI log ˆV (0)
+1
+(v, z) ,
+(3.3b)
+DI log ˆQ(0)
+1 (x(v), x(z)) = DI log(x(z) + y(z)) + DI log ˆ
+M (0)
+1 (x(v); z) .
+(3.3c)
+Lemma 3.2.
+DI
+U (0)
+1 (v, z)
+H(0)
+1 (x(v); z)
+≡
+|I|
+�
+l=0
+�
+I0⊎...⊎Il=I\uj
+I1,...Il̸=∅
+U (0)
+|I0|+1(v, z; I0)
+H(0)
+1 (x(v); z)
+(−1)l
+l�
+i=1
+H(0)
+|Ii|+1(x(v); z; Ii)
+H(0)
+1 (x(v); z)
+= DI
+1
+x(z) + y(v) .
+(3.4)
+Proof. Clear for I = ∅. For I ̸= ∅, start from the Dyson-Schwinger equation
+(2.10a)
+U (0)
+|I0|+1(v, z; I)
+H(0)
+1 (x(v); z)
+=
+H(0)
+|I|+1(x(v), z; I)
+(x(z) + y(v))H(0)
+1 (x(v); z)
++
+(−λ)W (0)
+|I|+1(v; I)
+(x(z) + y(v))2
++
+�
+I′⊎I′′=I
+I′,I′′̸=∅
+(−λ)W (0)
+|I′|+1(v; I′)
+(x(z) + y(v))
+U (0)
+|I′′|+1(v, z; I′′)
+H(0)
+1 (x(v); z)
+,
+
+16
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+which iterates to
+U (0)
+|I0|+1(v, z; I)
+H(0)
+1 (x(v); z)
+=
+H(0)
+|I|+1(x(v), z; I)
+(x(z) + y(v))H(0)
+1 (x(v); z)
++
+1
+(x(z) + y(v))
+|I|
+�
+l=1
+�
+I1⊎...⊎Il=I
+I1,...,Il̸=∅
+l�
+i=1
+(−λ)W (0)
+|Ii|+1(v; Ii)
+(x(z) + y(v))
++
+|I|
+�
+l=1
+�
+I0⊎I1⊎...⊎Il=I
+I0,I1,...,Il̸=∅
+H(0)
+|I0|+1(x(v), z; I0)
+(x(z) + y(v))H(0)
+1 (x(v); z)
+l�
+i=1
+(−λ)W (0)
+|Ii|+1(v; Ii)
+(x(z) + y(v))
+.
+The last line is iteratively removed by
+|I|−1
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I
+I0,I1,...,Il̸=∅
+U (0)
+|I0|+1(v, z; I0)
+H(0)
+1 (x(v); z)
+l�
+i=1
+H(0)
+|Ii|+1(x(v), z; Ii)
+H(0)
+1 (x(v); z)
+=
+I
+�
+l=1
+(−1)l−1
+(x(z) + y(v))
+�
+I1⊎...⊎Il=I
+I1,...,Il̸=∅
+l�
+i=1
+H(0)
+|Ii|+1(x(v), z; Ii)
+H(0)
+1 (x(v); z)
+(*)
++
+1
+(x(z) + y(v))
+|I|
+�
+l=1
+�
+I1⊎...⊎Il=I
+I1,...,Il̸=∅
+l�
+i=1
+(−λ)W (0)
+|Ii|+1(v; Ii)
+(x(z) + y(v))
+.
+(**)
+Taking
+1
+x(z)+y(v) =
+U(0)
+1
+(v,z)
+H(0)
+1
+(x(v);z) into account, the line (*) corresponds to the case
+I0 = ∅ of the lhs. The line (**) equals DI
+1
+x(z)+y(v) so that the assertion follows.
+□
+The previous lemmas allow us to solve the genus g = 0 case:
+Proposition 3.3. The solution of the system (2.10a)+(2.10b) with (2.9a)+(2.9b)
+is
+DI log H(0)
+1 (x(v); z) =
+d
+�
+k=1
+DI log(x(v) + y(ˆzk)) + F (0)
+|I|+1(x(v); x(z); I) , (3.5)
+DI log P (0)
+1 (x(v), x(z)) =
+d
+�
+k=0
+DI log(x(v) + y(ˆzk)) + F (0)
+|I|+1(x(v); x(z); I)
+(3.6)
+[note that the sum starts with k = 1 in (3.5) but k = 0 in (3.5)] where
+F (0)
+|I|+1(x(v); x(z); I) =
+|I|
+�
+j=1
+DI\uj
+λ
+(x(v) − x(uj))(x(z) + y(uj)) .
+(3.7)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+17
+Proof. Both sides of (3.1) can only be a function of x(z) if (3.5) and (3.6) hold for
+some rational function F (0)
+|I|+1(x(v); x(z); I). Since
+H(0)
+|I|+1(x(v);z;I)
+H(0)
+1
+(x(v);z)
+is holomorphic at
+x(v)+y(z) = 0 by (2.9a), the function F (0)
+|I|+1 cannot have poles at x(v)+y(ˆzj) = 0.
+Recall from (2.9b) and (2.9a) that the only poles of x(v) �→
+P (0)
+|I|+1(x(v),x(z);I)
+P (0)
+1
+(x(v),x(z))
+are
+at x(v) = x(uj) for uj ∈ I and at the z with P (0)
+1 (x(v), x(z)) = 0. The latter are
+already given by the first line of (3.6) so that the only poles of F (0)
+|I|+1 are located
+at x(v) = x(uj) for every uj ∈ I. These poles are simple and arise via (2.9a):
+lim
+x(v)→x(uj)(x(v) − x(uj))H(0)
+|I|+1(x(v); z; I) = λU (0)
+|I| (uj, z; I \ uj) ,
+if uj ∈ I. This gives for the corresponding limit of (2.15)
+lim
+x(v)→x(uj)(x(v) − x(uj))DI log H(0)
+1 (x(v); z))
+= λ
+|I|−1
+�
+l=0
+�
+I0⊎...⊎Il=I\uj
+I1,...Il̸=∅
+U (0)
+|I0|+1(uj, z; I0)
+H(0)
+1 (x(uj); z)
+(−1)l
+l�
+i=1
+H(0)
+|Ii|+1(x(uj); z; Ii)
+H(0)
+1 (x(uj); z)
+= DI\uj
+λ
+x(z) + y(uj)
+(3.8)
+where Lemma 3.2 has been used. This finishes the proof.
+□
+3.2. Linear and quadratic loop equations for genus g = 0. From (2.9b),
+(2.9a) and the connecting equation (2.5) we read off
+P (0)
+1 (x(v), x(z)) = x(v) + x(z) − λ
+N
+d
+�
+k=1
+rk
+x(z) − x(εk) + O((x(v))−1) ,
+H(0)
+1 (x(v); z) = 1 +
+1
+x(v)
+�
+x(z) − y(z) − λ
+N
+d
+�
+k=1
+rk
+x(z) − x(εk)
+�
++ O((x(v))−2) ,
+P (0)
+2 (x(v), x(z); u1) =
+λ
+x(z) − x(u1) +
+λ
+x(v)
+� λ
+N
+d
+�
+k=1
+rkW (0)
+2 (εk; u1)
+x(z) − x(εk)
++
+x(z) − y(u1) − λ
+N
+�d
+k=1
+rk
+x(z)−x(εk)
+(x(z) − x(u1))
+�
++ O((x(v))−2)
+and then for |I| ≥ 2
+P (0)
+|I|+1(x(v), x(z); I) =
+λ
+x(v)
+� λ
+N
+d
+�
+k=1
+rkW (0)
+|I|+1(εk; I)
+x(z) − x(εk) −
+|I|
+�
+j=1
+λW (0)
+|I| (uj; I \ uj)
+x(z) − x(uj)
+
+18
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
++
+λδ|I|,2
+(x(z) − x(u1))(x(z) − x(u2))
+�
++ O((x(v))−2) .
+These expansions combine for I ̸= ∅ to
+DI log P (0)
+1 (x(v), x(z))
+(3.9)
+=
+λ
+(x(v))2
+� λ
+N
+d
+�
+k=1
+rkW (0)
+|I|+1(εk; I)
+x(z) − x(εk) − λ(1 − δ|I|,1)
+|I|
+�
+j=1
+W (0)
+|I|+1(uj; I \ uj)
+x(z) − x(uj)
+�
++
+λδ|I|,1
+(x(z) − x(u1))
+� 1
+x(v) − y(u1)
+(x(v))2
+�
++ O((x(v))−3) .
+Comparison with (3.6) and (3.7) yields the following main result:
+Proposition 3.4. The functions W (0)
+|I|+1 satisfy for I ̸= ∅ the linear loop equations
+d
+�
+k=0
+W (0)
+|I|+1(ˆzk; I) =
+δ|I|,1
+(x(z) − x(u1)) −
+|I|
+�
+j=1
+DI\uj
+�
+1
+x(z) + y(uj)
+�
+(3.10)
+and the quadratic loop equations
+−
+d
+�
+k=0
+y(ˆzk)W (0)
+|I|+1(ˆzk; I)
+(3.11)
+= λ
+2
+�
+I1⊎I2=I
+I1,I2̸=∅
+d
+�
+k=0
+W (0)
+|I1|+1(ˆzk; I1)W (0)
+|I2|+1(ˆzk; I2) −
+|I|
+�
+j=1
+x(uj)DI\uj
+�
+1
+x(z) + y(uj)
+�
++ λ
+N
+d
+�
+k=1
+rkW (0)
+|I|+1(εk; I)
+x(z) − x(εk) − λ(1 − δ|I|,1)
+|I|
+�
+j=1
+W (0)
+|I|+1(uj; I \ uj)
+x(z) − x(uj)
+−
+δ|I|,1y(u1)
+x(z) − x(u1) .
+3.3. Computation of ˆQ(0). Combining (2.7) with (2.12d) and (2.13c)+(2.13d)
+gives
+ˆQ(0)
+1 (x(v), x(z)) = −λ(x(v) + x(z) − 2x(0)) Æ(x(v)) Æ(x(z))
+(x(v) − x(z))2
+(3.12)
+= −λ(x(v) + x(z) − 2x(0))
+2(x(v) − x(z))2
+�
+P (0)
+1 (x(v), x(v))P (0)
+1 (x(z), x(z))
+(x(v) − x(0))(x(z) − x(0))
+,
+where (2.8) and (2.10a)+(2.10b) have been used.
+We take the logarithm of this equation and apply the loop insertion operator.
+The na¨ıve expectation is actually true if a symbolic expression D0
+Ix(0) is correctly
+identified:
+Proposition 3.5. For I ̸= ∅ one has
+DI log ˆQ(0)
+1 (x(v), x(z))
+(3.13)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+19
+= 1
+2DI log P (0)
+1 (x(v), x(v)) + 1
+2DI log P (0)
+1 (x(z), x(z))
+− 1
+2
+|I|
+�
+l=1
+(−1)l−1
+l
+�
+2l+1
+(x(v) + x(z) − 2x(0))l −
+1
+(x(v) − x(0))l −
+1
+(x(z) − x(0))l
+�
+×
+�
+I1⊎...⊎Il=I
+I1,..,Il̸=∅
+l�
+i=1
+D0
+Iix(0) ,
+where D0
+Iix(0) are symbolic expressions uniquely determined by the condition that
+DI log ˆQ(0)
+1 (x(v), x(z)) is holomorphic at v = 0.
+We provide parts of the proof as separate results. Let v ∈ {0, ±α1, ..., ±αd} be
+the set of zeros of x(v) + y(v) = 0.
+Lemma 3.6. U (0)
+|I|+1(z, z; I) is holomorphic at z = αk.
+Proof. Set v = z in (3.3a) and insert Proposition (3.3):
+DI log U (0)
+1 (z, z) =
+d
+�
+l=1
+DI log(x(ˆzl) + y(ˆzl)) − DI log(x(z) + y(z))
++
+|I|
+�
+j=1
+DI\uj
+λ
+(x(z) − x(uj))(x(z) + y(uj)) .
+By definition we have 0 = x(αk)+y(αk) = x(αk)−x(−αk) where the key identity
+(1.2) is used. This means that �
+αk
+1 = −αk is one of the preimages of x(αk). Again
+with (1.2) we conclude x(ˆz1) + y(ˆz1) = −(x(z) + y(z)) for z = αk. This means
+that log(−x(ˆz1) − y(ˆz1)) − log(x(z) + y(z)) is holomorphic in a neighbourhood of
+z = αk and leads to a holomorphic DI log(x(ˆz1)+y(ˆz1))−DI log(x(z)+y(z)).
+□
+The zeros ˆQ(0)
+1 (x(v), x(αk)) = 0 and P (0)
+1 (x(αk), x(αk)) = 0 produce (higher)
+poles in DI log ˆQ(0)
+1 (x(v), x(z)) and DI log P (0)
+1 (x(z), x(z)) at x(z) = x(αk). But
+these cancel exactly:
+Lemma 3.7. DI log ˆQ(0)
+1 (x(v), x(z)) − 1
+2DI log P (0)
+1 (x(z), x(z)) is holomorphic at
+x(z) = x(αk).
+Proof. Equations (3.1)+(3.3a) for v �→ z and (3.3a)+(3.3c) combine to
+DI log ˆQ(0)
+1 (x(v), x(z)) − 1
+2DI log P (0)
+1 (x(z), x(z))
+= DI log ˆV (0)
+1
+(v, z) − 1
+2DI log U (0)
+1 (z, z) + DI log(x(v) + y(v))
+
+20
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+≡
+|I|
+�
+l=1
+(−1)l−1
+l
+�
+I1⊎...⊎Il=I
+I1,...,Il̸=∅
+�
+l�
+i=1
+ˆV (0)
+|Ii|+1(v, z; Il)
+ˆV (0)
+1
+(v, z)
+− 1
+2
+l�
+i=1
+U (0)
+|Ii|+1(z, z; Il)
+U (0)
+1 (z, z)
++
+l�
+i=1
+W (0)
+|Ii|+1(v; Il)
+x(v) + y(v)
+�
+.
+Here ˆV (0)
+1
+(v, αk) ̸= 0 and U (0)
+1 (αk, αk) ̸= 0 from the explicit formulae. Every func-
+tion V (0)
+|I|+1(v, z; I) and thus also ˆV (0)
+|I|+1(v, z; I) is holomorphic at z = αk from the
+matrix model construction. In fact, this holomorphicity is the key assumption for
+the recursive solution [BHW22, Prop. E.4] which we reproduce by our algebraic
+method. The assertion follows with Lemma 3.6.
+□
+By (2.9b) and (2.9d), Q(0)
+|I|+1(x(v), x(z); I) and P (0)
+|I|+1(x(v), x(z); I) have simple
+poles at x(z) = x(uj) which give rise to simple poles of DI log ˆQ(0)
+1 (x(v), x(z))
+and 1
+2DI log P (0)
+1 (x(v), x(v)) at x(v) = x(uj). But they cancel:
+Lemma 3.8. DI log ˆQ(0)
+1 (x(v), x(z)) − 1
+2DI log P (0)
+1 (x(v), x(v)) is holomorphic at
+every x(v) = x(uj) with uj ∈ I.
+Proof. From Proposition 3.3 at z = v we get
+lim
+v→uj(x(v) − x(uj))DI log P (0)
+1 (x(v), x(v)) = DI\uj
+2λ
+x(uj) + y(uj) .
+Indeed, this term with numerator λ instead of 2λ arises from (3.7) at z = v. A
+second copy arises from a unique factor W (0)
+2 (ˆvk; uj) in the first line of (3.6); its
+residue is
+Res
+x(v)→x(uj) DI\uj
+d
+�
+k=0
+λW (0)
+2 (ˆvk; uj)dx(v)
+x(v) + y(ˆvk)
+=
+Res
+x(v)→x(uj) DI\uj
+d
+�
+k=0
+λW (0)
+2 (ˆvk; uj)dx(v)
+x(v) + y(uj)
+= DI\uj
+λ
+x(uj) + y(uj)
+by (3.10). Next, from (2.12f) and (2.9b)+(2.9d), all at v ↔ z, we get
+lim
+v→uj(x(v) − x(uj)) ˆQ(0)
+|I|+1(x(z), x(v); I)
+= λM (0)
+|I| (x(z); uj; I \ uj) −
+λ2H(0)
+|I| (x(z); uj; I \ uj)
+(x(uj) − x(z))2
+≡ λ ˆ
+M (0)
+|I| (x(z); uj; I \ uj) .
+Therefore, the residue of DI log ˆQ(0)
+1
+is
+lim
+v→uj(x(v) − x(uj))DI log ˆQ(0)
+1 (x(v), x(z))
+
+BTR FROM EXTENDED LOOP EQUATIONS
+21
+=
+|I|−1
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I\uj
+I1,...,Il̸=∅
+ˆ
+M (0)
+|I0|+1(x(z); uj; I0)
+ˆQ(0)
+1 (x(z), x(uj))
+l�
+i=1
+ˆQ(0)
+|Ii|+1(x(z); uj; Ii)
+ˆQ(0)
+1 (x(z), x(uj))
+.
+In complete analogy to the proof of Lemma 3.2 we have
+lim
+v→uj(x(v) − x(uj))DI log ˆQ(0)
+1 (x(v), x(z)) = DI\uj
+λ
+x(uj) + y(uj) ,
+which finishes the proof.
+□
+Proof of Proposition 3.5. Observe that (2.12f) implies limv→z
+ˆQ(0)
+1 (x(v),x(z);I)
+ˆQ(0)
+1 (x(v),x(z))
+=
+P (0)
+1
+(x(z),x(z);I)
+P (0)
+1
+(x(z),x(z)) so that DI log ˆQ(0)
+1 (x(v), x(z)) is holomorphic at v = z for I ̸= ∅.
+Together with Lemma 3.7 and Lemma 3.8, the only remaining candidates for poles
+of DI log ˆQ(0)
+1 (x(v), x(z))− 1
+2DI log P (0)
+1 (x(v), x(v))− 1
+2DI log P (0)
+1 (x(z), x(z)) are:
+• The zeros x(v)
+=
+x(0) and x(z)
+=
+x(0) of P (0)
+1 (x(v), x(v)) and
+P (0)
+1 (x(z), x(z)), respectively, which produce higher poles
+DI log P (0)
+1 (x(v), x(v)) =
+|I|
+�
+l=1
+fl(I)
+(x(v) − x(0))l + O((x(v) − x(0))0)
+and similarly for DI log P (0)
+1 (x(z), x(z)).
+Neither ˆQ(0)
+1 (x(v), x(z)) has a
+zero at x(v) = x(0) or x(z) = x(0) nor ˆQ(0)
+|I|+1(x(v), x(z); I) has a pole there
+so that there is no compensation. The coefficients fl(I) only depend on the
+uj ∈ I but not on v, z; they are the same for DI log P (0)
+1 (x(v), x(v)) and
+DI log P (0)
+1 (x(z), x(z)) because of the symmetry of DI log ˆQ(0)
+1 (x(v), x(z))
+in v ↔ z. We can trade the fl(I) for the symbolic expressions D0
+I′x(0).
+• The zeros x(v) = 2x(0) − x(z) of ˆQ(0)
+1 (x(v), x(z)), which produce higher
+poles
+DI log ˆQ(0)
+1 (x(v), x(z)) =
+|I|
+�
+l=1
+˜fl(I)
+(x(v) + x(z) − 2x(0))l
++ O((x(v) + x(z) − 2x(0))0) .
+Neither P (0)
+1 (x(v), x(v)) has a zero at x(v)
+=
+2x(0) − x(z) nor
+P (0)
+|I|+1(x(v), x(z); I) has a pole there so that there is no compensation.
+Viewed as function x(v) �→ DI log ˆQ(0)
+1 (x(v), x(z)), the coefficients ˜fl(I)
+are independent of x(v) and then by symmetry independent of x(z). The
+requirement limz→v
+ˆQ(0)
+1 (x(v),x(z);I)
+ˆQ(0)
+1 (x(v),x(z))
+=
+P (0)
+1
+(x(v),x(v);I)
+P (0)
+1
+(x(v),x(v))
+then fixes the relative
+factor between fl(I) and ˜fl(I) to be −2l+1 as given in (3.13).
+This completes the proof.
+□
+
+22
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Example 3.9. We have
+lim
+v→0(x(v) − x(0))P (0)
+1 (x(v), x(v); u)
+P (0)
+1 (x(v), x(v))
+= λ
+2W (0)
+2 (0; u)
+from Proposition 3.3. This identifies D0
+ux(0) = − λ
+2W (0)
+2 (0; u).
+4. Solution for g = 1
+4.1. The case I = ∅. We consider the relation between the ‘genus inser-
+tions’ into log P (0)
+1 (x(v), x(z)) and log H(0)
+1 (x(v); z). Equation (2.13b), divided
+by P (0)
+1 (x(v), x(z)), reads
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+−
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+(4.1)
+=
+λ
+x(v) + y(z)
+�∂Dw log H(0)
+1 (x(v); z)
+∂x(w)
+���
+w=z + W (1)
+1 (z) +
+λ
+(x(v) − x(z))3
+�
+=
+λW (1)
+1 (z)
+x(v) + y(z) +
+λ2
+(x(v) − x(z))3(x(v) + y(z))
++
+d
+�
+k=1
+λ2Ω(0)
+2 (z, ˆzk)
+2(x(v) + y(z))(x(v) + y(ˆzk)) +
+d
+�
+j=1
+λ2Ω(0)
+2 (ˆzj, z)
+2(x(v) + y(ˆzj))(x(v) + y(z))
++
+λ2
+(x(v) + y(z))
+∂
+∂x(w)
+1
+(x(v) − x(w))(x(z) + y(w))
+���
+w=z ,
+where ∂Dw log H(0)
+1
+(x(v);ˆzk)
+∂x(w)
+��
+w=z was provided by Proposition 3.3. The differentiation
+leads to Ω(0)
+2
+(see (2.2)) which we have symmetrised in both arguments.
+We
+understand ∂y(w)
+∂x(w) ≡ y′(w)
+x′(w).
+In order for
+ˆP (1)
+1
+(x(v),x(z))
+P (0)
+1
+(x(v),x(z)) to be a rational function of x(z), we need that
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+=
+d
+�
+k=0
+λW (1)
+1 (ˆzk)
+x(v) + y(ˆzk) + λ2
+2
+d
+�
+k,j=0
+k̸=j
+Ω(0)
+2 (ˆzj, ˆzk)
+(x(v) + y(ˆzj))(x(v) + y(ˆzk))
++
+d
+�
+k=0
+λ2
+(x(v) + y(ˆzk))
+∂
+∂x(w)
+1
+(x(v) − x(w))(x(z) + y(w))
+���
+w=ˆzk
++
+d
+�
+k=0
+λ2
+(x(v) − x(z))3(x(v) + y(ˆzk)) + F (1)
+1 (x(v); x(z))
+(4.2)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+23
+for some rational function F (1)
+1 , and that almost the same formula holds for
+ˆH(1)
+1
+(x(v);z)
+H(0)
+1
+(x(v);z), only with preimage sum �d
+k=1 instead of �d
+k=0. By (2.12b), (2.9a)
+and (2.9c), the function
+ˆH(1)
+1
+(x(v);z)
+H(0)
+1
+(x(v);z) is holomorphic at x(v) + y(z) = 0 so that F (1)
+1
+must be holomorphic at x(v) + y(ˆzk) = 0. It follows from (2.12c), (2.9b) and
+(2.9d) that the only other poles of x(v) �→
+ˆP (1)
+1
+(x(v),x(z))
+P (0)
+1
+(x(v),x(z)) are at x(v) = x(z) of
+order at most 2; more precisely
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+=
+λ
+(x(v) − x(z))2
+Q(0)
+1 (x(z), x(z))
+P (0)
+1 (x(z), x(z))
+(4.3)
++
+λ
+(x(v) − x(z))
+∂
+∂x(w)
+Q(0)
+1 (x(w), x(z))
+P (0)
+1 (x(w), x(z))
+���
+w=z + regular,
+where ‘regular’ means O((x(v) − x(z))0). To achieve this we necessarily need
+F (1)
+1 (x(v); x(z)) =
+3
+�
+a=1
+F (1)a
+1
+(x(z))
+(x(v) − x(z))a ,
+F (1)3
+1
+(x(z)) = −
+d
+�
+k=0
+λ2
+x(z) + y(ˆzk) ,
+F (1)2
+1
+(x(z)) = λQ(0)
+1 (x(z), x(z))
+P (0)
+1 (x(z), x(z))
+,
+F (1)1
+1
+(x(z)) =
+∂
+∂x(w)
+λQ(0)
+1 (x(w), x(z))
+P (0)
+1 (x(w), x(z))
+���
+w=z − 1
+2
+∂
+∂x(w)
+d
+�
+k=0
+λ2
+(x(z) + y(w))2
+���
+w=ˆzk.
+From (3.12) and (2.12f) we obtain a representation
+λQ(0)
+1 (x(w), x(z))
+P (0)
+1 (x(w), x(z))
+=
+λ2
+(x(w) − x(z))2
+�
+1 −
+�
+1 +
+(x(z) − x(w))2
+4(x(w) − x(0))(x(z) − x(0))
+× exp
+�1
+2 log P (0)
+1 (x(w), x(w))
+P (0)
+1 (x(z), x(z))
+− log P (0)
+1 (x(w), x(z))
+P (0)
+1 (x(z), x(z))
+��
+= −
+λ2
+8(x(z) − x(0))2 − λ2
+2
+∂2 log P (0)
+1 (x(w), x(z))
+∂x(w)∂x(z)
+���
+w=z
++ (x(w) − x(z))
+�
+λ2
+8(x(z) − x(0))3 − λ2
+2
+∂3 log P (0)
+1 (x(w), x(z))
+∂(x(w))2∂x(z)
+���
+w=z
+�
++ O((x(w) − x(z))2) .
+Inserting the resulting Taylor expansion into F (1)1
+1
+(x(z)) and F (1)2
+1
+(x(z)) leads to:
+
+24
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Proposition 4.1.
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+(4.4)
+=
+d
+�
+k=0
+λW (1)
+1 (ˆzk)
+x(v) + y(ˆzk) + λ2
+2
+d
+�
+k,j=0
+k̸=j
+Ω(0)
+2 (ˆzj, ˆzk)
+(x(v) + y(ˆzj))(x(v) + y(ˆzk))
+−
+λ2
+(x(v) − x(z))3
+�∂ log P (0)
+1 (x(v), x(w))
+∂x(w)
+− ∂ log P (0)
+1 (x(z), x(w))
+∂x(w)
+����
+w=z
++
+λ2
+2(x(v) − x(z))2
+∂2 log P (0)
+1 (x(z), x(w))
+∂x(z)∂x(w)
+���
+w=z
+−
+λ2
+8(x(v) − x(z))2(x(z) − x(0))2 +
+λ2
+8(x(v) − x(z))(x(z) − x(0))3 .
+4.2. The case I ̸= ∅. Our goal is to determine
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+=
+|I|
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I
+I1,...,Il̸=∅
+ˆP (1)
+|I0|+1(x(v), x(z); I0)
+P (0)
+1 (x(v), x(z))
+l�
+i=1
+P (0)
+|Ii|+1(x(v), x(z); Ii)
+P (0)
+1 (x(v), x(z))
+in parallel with DI
+ˆH(1)
+1
+(x(v);z)
+H(0)
+1
+(x(v);z). The Dyson-Schwinger equations (2.13b) for g ≤ 1
+imply:
+Lemma 4.2.
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+− DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+(4.5)
+=
+�
+I1⊎I2=I
+DI1
+�
+λ
+x(v) + y(z)
+��∂DI2∪w log H(0)
+1 (x(v); z)
+∂x(w)
+���
+w=z
++ W (1)
+|I2|+1(z; I2) +
+λδ|I2|,∅
+(x(v) − x(z))3
+�
+.
+Proof. By induction in |I|. The case |I| = 0 is (2.13b) for g = 1. Otherwise
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+− DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+=
+ˆP (1)
+|I|+1(x(v), x(z); I)
+P (0)
+1 (x(v), x(z))
+−
+�
+I1⊎I2=I
+I2̸=∅
+DI1
+� ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+�
+·
+P (0)
+|I2|+1(x(v), x(z); I2)
+P (0)
+1 (x(v), x(z))
+
+BTR FROM EXTENDED LOOP EQUATIONS
+25
+−
+ˆH(1)
+|I|+1(x(v); z; I)
+H(0)
+1 (x(v); z)
++
+�
+I1⊎I2=I
+I2̸=∅
+DI1
+� ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+�
+·
+H(0)
+|I2|+1(x(v); z; I2)
+H(0)
+1 (x(v); z)
+=
+�
+I1⊎I2=I
+I1̸=∅
+λW (0)
+|I1|+1(z; I1)
+x(v) + y(z)
+ˆH(1)
+|I2|+1(x(v); z; I2)
+H(0)
+1 (x(v); z)
+(*)
++
+�
+I1⊎I2=I
+λ(W (1)
+|I1|+1(z; I1) +
+λδ|I1|,0
+(x(v)−x(z))3)
+x(v) + y(z)
+H(0)
+|I2|+1(x(v); z; I2)
+H(0)
+1 (x(v); z)
+(‡)
++
+λ
+x(v) + y(z)
+∂
+∂x(w)
+H(0)
+|I|+2(x(v); z; I ∪ w)
+H(0)
+1 (x(v); z)
+���
+w=z
+(§)
+−
+�
+I1⊎I2⊎I3=I
+I3̸=∅
+�∂DI2∪w log H(0)
+1 (x(v); z)
+∂x(w)
+���
+w=z + W (1)
+|I1|+1(z; I1) +
+λδ|I1|,0
+(x(v) − x(z))3
+�
+× DI2
+�
+λ
+x(v) + y(z)
+�
+·
+P (0)
+|I3|+1(x(v), x(z); I3)
+P (0)
+1 (x(v), x(z))
+(†)
+−
+�
+I1⊎I2⊎I3=I
+I2̸=∅
+DI1
+� ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+�
+·
+λW (0)
+|I2|+1(z; I2)
+x(v) + y(z)
+H(0)
+|I3|+1(x(v); z; I3)
+H(0)
+1 (x(v); z)
+,
+(**)
+where the induction hypothesis was used. The lines (*) and (**) cancel when
+distinguishing I3 = ∅ and I3 ̸= ∅. Take (3.2), multiplied by
+λ
+x(v)+y(z):
+λP (0)
+|˜I|+1(x(v), x(z); ˜I)
+(x(v) + y(z))P (0)
+1 (x(v), x(z))
+=
+λH(0)
+|˜I|+1(x(v); z; ˜I)
+(x(v) + y(z))H(0)
+1 (x(v); z)
++
+�
+˜I′⊎˜I′′=˜I
+˜I′̸=∅
+λW (0)
+|˜I′|+1(z; ˜I′)
+x(v) + y(z)
+λH(0)
+|˜I′′|+1(x(v); z; ˜I′′)
+(x(v) + y(z))H(0)
+1 (x(v); z)
+.
+For ˜I′′ ̸= ∅, take this equation for ˜I �→ ˜I′′, multiply by
+(−λ)W (0)
+|˜I′|+1(z;˜I′)
+x(v)+y(z)
+and sum
+over ˜I′ ⊎ ˜I′′ = ˜I with ˜I′ ̸= ∅. Repeat until all products of
+(−λ)W (0)
+|Ii|+1(z;Ii)
+x(v)+y(z)
+with
+λH(0)
+|I0|+1(x(v);z;I0)
+(x(v)+y(z))H(0)
+1
+(x(v);z) are removed. The result is
+�
+I2⊎I3=I′
+I3̸=∅
+DI2
+�
+λ
+x(v) + y(z)
+�P (0)
+|I3|+1(x(v), x(z); I3)
+P (0)
+1 (x(v), x(z))
+(4.6)
+
+26
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+=
+λ
+x(v) + y(z)
+H(0)
+|I′|+1(x(v); z; I′)
+H(0)
+1 (x(v); z)
++
+�
+I2⊎I3=I′
+I3̸=∅
+λW (0)
+|I3|+1(z; I3)
+x(v) + y(z) DI2
+λ
+x(v) + y(z) ,
+which we use in (†). Multiplied with (W (1)
+|I1|+1(z; I1) +
+λδ|I1|,0
+(x(v)−x(z))3), this cancels all
+terms with I2 ̸= ∅ in (‡) and completes the case I2 = ∅ in (‡) to the last line of
+the assertion (4.5). Similarly, expanding
+DI∪w log H(0)
+1 (x(v); z)
+=
+|I|
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I
+I1,...,Il̸=∅
+H(0)
+|I0|+2(x(v); z; I0 ∪ w)
+H(0)
+1 (x(v); z)
+l�
+i=1
+H(0)
+|Ii|+1(x(v); z; Ii)
+H(0)
+1 (x(v); z)
+we get for the product with the first term on the rhs of (4.6)
+�
+I2∪I′=I
+I′̸=∅
+DI∪w log H(0)
+1 (x(v); z)
+λ
+x(v) + y(z)
+H(0)
+|I′|+1(x(v); z; I′)
+H(0)
+1 (x(v); z)
+=
+λ
+x(v) + y(z)
+�H(0)
+|I|+2(x(v); z; I ∪ w)
+H(0)
+1 (x(v); z)
+− DI∪w log H(0)
+1 (x(v); z)
+�
+.
+The first term on the rhs cancels the line (§), and the second term completes the
+final term in (4.6) to the second line of the assertion (4.5).
+□
+We conclude from Proposition 3.3:
+∂
+∂x(w)DI∪w log H(0)
+1 (x(v); z)
+���
+w=z
+(4.7)
+=
+d
+�
+k=1
+|I|
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I
+I1,...Il̸=∅
+λDI0Ω(0)
+2 (ˆzk, z)
+x(v) + y(ˆzk)
+l�
+j=1
+λW (0)
+|Ij|+1(ˆzk; Ij)
+x(v) + y(ˆzk)
+−
+|I|
+�
+j=1
+DI\uj
+λ2Ω(0)
+2 (z, uj)
+(x(v) − x(uj))(x(z) + y(uj))2 + DI
+λ
+(x(v) − x(z))2(x(z) + y(z))
++
+∂
+∂x(w)DI
+λ
+(x(v) − x(z))(x(z) + y(w))
+���
+w=z .
+We insert (4.7) into (4.5) and get with the derivation property of DI:
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+− DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+= DI
+λW (1)
+1 (z)
+x(v) + y(z) +
+d
+�
+k=1
+DI
+λ2Ω(0)
+2 (ˆzk, z)
+(x(v) + y(ˆzk))(x(v) + y(z))
+
+BTR FROM EXTENDED LOOP EQUATIONS
+27
+−
+|I|
+�
+j=1
+λ3
+(x(v) − x(uj))DI\uj
+Ω(0)
+2 (z, uj)
+(x(v) + y(z))(x(z) + y(uj))2
++
+λ2
+(x(v) − x(z))3DI
+1
+(x(z) + y(z))
++
+λ2
+(x(v) − x(z))
+∂
+∂x(w)DI
+1
+(x(v) + y(z))(x(z) + y(w))
+���
+w=z .
+As before, in order for DI
+ˆP (1)
+1
+(x(v),x(z))
+P (0)
+1
+(x(v),x(z)) to be a rational function of x(z), we need
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+= K(1)
+0,|I|+1(x(v); z; I) + F (1)
+|I|+1(x(v); x(z); I) ,
+(4.8a)
+DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+= K(1)
+1,|I|+1(x(v); z; I) + F (1)
+|I|+1(x(v); x(z); I) ,
+(4.8b)
+where (note the difference in the lower subscript A between (4.8a) and (4.8b))
+K(1)
+A,|I|+1(x(v); z; I)
+(4.9)
+:=
+d
+�
+k=A
+DI
+λW (1)
+1 (ˆzk)
+x(v) + y(ˆzk) + λ2
+2
+d
+�
+j,k=A
+j̸=k
+DI
+Ω(0)
+2 (ˆzj, ˆzk)
+(x(v) + y(ˆzj))(x(v) + y(ˆzk))
+−
+|I|
+�
+j=1
+λ3
+(x(v) − x(uj))
+d
+�
+k=A
+DI\uj
+Ω(0)
+2 (ˆzk, uj)
+(x(v) + y(ˆzk))(x(z) + y(uj))2
++
+λ2
+(x(v) − x(z))3DI
+d
+�
+k=A
+1
+(x(z) + y(ˆzk))
++
+λ2
+(x(v) − x(z))
+d
+�
+k=A
+∂
+∂x(w)DI
+1
+(x(v) + y(ˆzk))(x(z) + y(w))
+���
+w=ˆzk
+and F (1)
+|I|+1(x(v); x(z); I) is some rational function (the same in (4.8a) and (4.8b))
+in both x(v) and x(z). In fact, also K(1)
+0,|I|+1(x(v); z; I) is rational in both x(v)
+and x(z). Since DI
+ˆH(1)
+1
+(x(v),x(z))
+H(0)
+1
+(x(v),x(z)) is holomorphic at x(v) + y(z) = 0, the rational
+function x(v) �→ F (1)
+|I|+1 can only have poles at x(v) = x(z) and at x(v) = x(uj)
+with uj ∈ I. With the behavior resulting from (2.9a) and (2.12b),
+ˆH(g)
+|I|+1(x(v); z; I)
+H(0)
+1 (x(v); z)
+=
+|I|
+�
+j=1
+λ
+(x(v) − x(uj))
+ˆU (g)(z, uj; I \ uj)
+H(0)
+1 (x(uj); z)
++ O((x(v) − x(uj))0)
+(4.10)
+
+28
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+and (4.7) we see that F (1)
+|I|+1 has first-order poles at x(v) = x(uj). We determine
+them in Proposition 4.5 below.
+Near x(v) = x(z) we have in (4.9)
+K(1)
+0,|I|+1(x(v); z; I)
+(4.11)
+=
+λ2
+(x(v) − x(z))
+d
+�
+k=0
+∂
+∂x(w)DI
+1
+(x(z) + y(ˆzk))(x(z) + y(w))
+���
+w=ˆzk
++
+λ2
+(x(v) − x(z))3DI
+d
+�
+k=0
+1
+(x(z) + y(ˆzk)) + O((x(v) − x(z))0)
+= −
+λ2
+(x(v) − x(z))
+1
+2
+∂3
+∂(x(z))2∂x(w)
+d
+�
+k=0
+DI log(x(z) + y( ˆwk))
+���
+w=z
++
+λ2
+(x(v) − x(z))3DI
+d
+�
+k=0
+1
+(x(z) + y(ˆzk)) + O((x(v) − x(z))0)
+= −
+λ2
+(x(v) − x(z))
+�1
+2
+∂3(DI log P (0)
+1 (x(z), x(w)))
+∂(x(z))2∂x(w)
+���
+w=z
++
+|I|
+�
+j=1
+DI\uj
+λ
+(x(z) − x(uj))3(x(z) + y(uj))2
+�
++
+λ2
+(x(v) − x(z))3DI
+d
+�
+k=0
+1
+(x(z) + y(ˆzk)) + O((x(v) − x(z))0) .
+In the last step we have used Proposition 3.3. On the other hand, from (2.12c)
+we get
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+=
+λ
+(x(v) − x(z))2DI
+Q(0)
+1 (x(z), x(z))
+P (0)
+1 (x(z), x(z))
+(4.12)
++
+λ
+(x(v) − x(z))
+∂
+∂x(w)
+�
+DI
+Q(0)
+1 (x(w), x(z))
+P (0)
+1 (x(w), x(z))
+�
+w=z
++ O((x(v) − x(z))0) ,
+where
+DI
+Q(0)
+1 (x(z), x(z))
+P (0)
+1 (x(z), x(z))
+:=
+|I|
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I
+I1,...,Il̸=∅
+Q(0)
+|I0|+1(x(w), x(z); I0)
+P (0)
+1 (x(w), x(z))
+l�
+i=1
+P (0)
+|Ii|+1(x(w), x(z); Ii)
+P (0)
+1 (x(w), x(z))
+.
+
+BTR FROM EXTENDED LOOP EQUATIONS
+29
+These properties lead to an expansion
+F (1)
+|I|+1(x(v); x(z); I) =
+3
+�
+a=1
+F (1)a
+|I|+1(x(z); I)
+((x(v) − x(z))a +
+|I|
+�
+j=1
+ˆF (1)j
+|I| (x(z); I \ uj)
+x(v) − x(uj)
+(4.13)
+where (4.11) and (4.12) combine to
+F (1)3
+|I|+1(x(z); I) = −
+d
+�
+k=0
+DI
+λ2
+x(z) + y(ˆzk)
+(4.14a)
+F (1)2
+|I|+1(x(z); I) = λDI
+Q(0)
+1 (x(z), x(z))
+P (0)
+1 (x(z), x(z))
+(4.14b)
+F (1)1
+|I|+1(x(z); I) = λ
+∂
+�
+DI
+Q(0)
+1 (x(z), x(w))
+P (0)
+1 (x(z), x(w))
+�
+∂x(w)
++ λ2
+2
+∂3(DI log P (0)
+1 (x(z), x(w)))
+∂(x(z))2∂x(w)
++
+|I|
+�
+j=1
+DI\uj
+λ3
+(x(z) − x(uj))3(x(z) + y(uj))2
+���
+w=z .
+(4.14c)
+It remains to determine the functions ˆF (1)j
+|I| (x(z); I \ uj). We will need two
+lemmas:
+Lemma 4.3.
+ˆU (1)
+|I|+1(v, z; I)
+=
+�
+I1⊎I2=I
+DI1
+1
+(x(z) + y(v)) ·
+�
+ˆH(1)
+|I2|+1(x(v); z; I2) − λ
+∂U (0)
+|I|+2(v, z; I ∪ s)
+∂x(s)
+���
+s=v
+− λ
+�
+I′
+2⊎I′′
+2 =I2
+�
+W (1)
+|I′
+2|+1(v; I′
+2) +
+λδI′
+2,∅
+(x(z) − x(v))3
+�
+U (0)
+|I′′
+2 |+1(v, z; I′′
+2 )
+�
+.
+Proof. Resolve (2.13a) at g = 1 for the first term ˆU (1)(v, z; I), which becomes
+− �
+I1⊎I2=I,I1̸=∅
+λW (0)
+|I1|+1(v;I1)
+x(z)+y(v)
+ˆU (1)(v, z; I2) plus other terms. Iterate this procedure
+for every ˆU (1)(v, z; I2) and so on. The resulting products of
+λW (0)
+|Ii|+1(v;Ii)
+x(z)+y(v)
+can be
+collected to DI1
+1
+(x(z)+y(v)).
+□
+Lemma 4.4.
+DI
+U (0)
+2 (v, z; s)
+H(0)
+1 (x(v); z)
+= −DI
+λ
+�
+W (0)
+2 (v; s) −
+1
+x(v)−x(s)
+�
+(x(z) + y(v))2
+(4.15)
+
+30
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
++
+d
+�
+k=1
+DI
+λW (0)
+|I0|+2(ˆzk; s)
+(x(v) + y(ˆzk))(x(z) + y(v))
+−
+|I|
+�
+i=1
+DI\ui
+λ2W (0)
+2 (ui; s)
+(x(v) − x(ui))(x(z) + y(ui))2(x(z) + y(v))
+−
+�
+I⊎I′′=I
+DI′
+λ
+x(z) + y(v)DI′′
+1
+(x(z)+y(v)) −
+1
+(x(z)+y(s))
+(x(v) − x(s))
+.
+Proof. With (3.4) one has
+DI
+U (0)
+2 (v, z; s)
+H(0)
+1 (x(v); z)
+= DI∪s
+U (0)
+2 (v, z)
+H(0)
+1 (x(v); z)
++ DI
+�
+U (0)
+2 (v, z)
+H(0)
+1 (x(v); z)
+Ds log H(0)
+1 (x(v); z)
+�
+= DI∪s
+1
+x(z)+y(v) +
+�
+I⊎I′′=I
+DI′
+1
+x(z)+y(v)DI′′∪s log H(0)
+1 (x(v); z).
+The first term equals DI∪s
+1
+x(z)+y(v) = −DI
+λW (0)
+2
+(v;s)
+(x(z)+y(v))2 and gives partly the first
+line of (4.15). The other part cancels with a term in the last line of (4.15); we
+will need this combination. The last term is known from (3.5) and (3.7):
+DI′′∪s log H(0)
+1 (x(v); z)
+(4.16)
+=
+d
+�
+k=1
+|I′′|+1
+�
+l=1
+(−1)l−1
+l
+�
+I1⊎...⊎Il=I′′∪s
+I1,...Il̸=∅
+l�
+i=1
+λW (0)
+|Ii|+1(ˆzk; Ii)
+x(v) + y(ˆzk)
++
+|I′′|
+�
+i=1
+DI′′\uiDs
+λ
+(x(v)−x(ui))(x(z)+y(ui)) + DI′′
+λ
+(x(v)−x(s))(x(z)+y(s)) .
+The middle line of (4.16) equals �d
+k=1 DI′′ λW (0)
+2
+(ˆzk;s)
+x(v)+y(ˆzk) and gives with the deriva-
+tion property of the DI the second line of (4.15). In the last line of (4.16) we
+have Ds
+λ
+(x(v)−x(ui))(x(z)+y(ui)) = −
+λ2W (0)
+2
+(ui;s)
+(x(v)−x(ui))(x(z)+y(ui))2 which gives the third line of
+(4.15). The final term of (4.16) gives the missing part of the last line of (4.15).
+□
+Proposition 4.5. One has
+ˆF (1)j(x(z); I \ uj)
+(4.17)
+= DI\uj
+�λ3Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 +
+λ3
+2
+∂2
+∂(x(uj))2
+1
+(x(z)+y(uj)) − λ2�
+W (1)
+1 (uj) +
+λ
+(x(z)−x(uj))3
+�
+(x(z) + y(uj))2
+�
++
+|I|
+�
+i=1
+i̸=j
+DI\{ui,uj}
+λ4Ω(0)
+2 (ui, uj)
+(x(uj) − x(ui))(x(z) + y(ui))2(x(z) + y(uj))2 ,
+
+BTR FROM EXTENDED LOOP EQUATIONS
+31
+where Ω(0)reg(u, u) := lims→u
+�
+Ω(0)
+2 (u, s) −
+1
+(x(u)−x(s))2
+�
+and DIΩ(0)reg
+2
+(u, u) =
+∂
+∂x(s)W (0)
+|I|+2(u; I ∪ s)
+��
+s=u for I ̸= ∅.
+Proof. The residue of (4.13) times dx(v) at x(v) = x(uj) is with (4.8b) and (4.9)
+given by
+ˆF (1)j
+|I| (x(z); I \ uj) = λ3
+d
+�
+k=1
+DI\uj
+Ω(0)
+2 (ˆzk, uj)
+(x(uj) + y(ˆzk))(x(z) + y(uj))2
+(4.18)
++ lim
+v→uj(x(v) − x(uj))DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+.
+The limit in the last line follows from (4.10) and the derivation property of DI:
+lim
+v→uj(x(v) − x(uj))DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+= λ
+� |I|−1
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I\uj
+I1,...,Il̸=∅
+ˆU (1)
+|I0|+1(z, uj; I0)
+H(0)
+1 (x(uj); z)
+l�
+i=1
+H(0)
+|Ii|+1(x(uj); z; Ii)
+H(0)
+1 (x(uj); z)
+−
+|I|−1
+�
+l=0
+(−1)l(l+1)
+�
+I−1⊎I0⊎I1⊎...⊎Il=I\uj
+I1,...,Il̸=∅
+U (0)
+|I−1|+1(z, uj; I−1)
+H(0)
+1 (x(uj); z)
+ˆH(1)
+|I0|+1(x(uj); z; I0)
+H(0)
+1 (x(uj); z)
+×
+l�
+i=1
+H(0)
+|Ii|+1(x(uj), x(z); Ii)
+H(0)
+1 (x(uj); z)
+�
+≡ λDI\uj
+�
+ˆU (1)
+1 (uj, z)
+H(0)
+1 (x(uj); z)
+−
+U (0)
+1 (uj, z)
+H(0)
+1 (x(uj); z)
+ˆH(1)
+1 (x(uj); z)
+H(0)
+1 (x(uj); z)
+�
+= λDI\uj
+ˆU (1)
+1 (uj, z)
+H(0)
+1 (x(uj); z)
+− λ
+�
+I1⊎I2=I\uj
+DI1
+U (0)
+1 (uj, z)
+H(0)
+1 (x(uj); z)
+DI2
+ˆH(1)
+1 (x(uj); z)
+H(0)
+1 (x(uj); z)
+.
+We insert (3.4) and expand the other operations DI\uj and DI2:
+lim
+v→uj(x(v) − x(uj))DI
+ˆH(1)
+1 (x(v); z)
+H(0)
+1 (x(v); z)
+(4.19)
+= λ
+|I|−1
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I\uj
+I1,...,Il̸=∅
+� ˆU (1)
+|I0|+1(uj, z; I0)
+H(0)
+1 (x(uj); z)
+
+32
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+−
+�
+I′
+0⊎I′′
+0 =I0
+DI′
+0
+1
+x(z) + y(uj)
+ˆH(1)
+|I′′
+0 |+1(x(uj); z; I′′
+0 )
+H(0)
+1 (x(uj); z)
+�
+l�
+i=1
+H(0)
+|Ii|+1(x(uj); z; Ii)
+H(0)
+1 (x(uj); z)
+= −λ2
+|I|−1
+�
+l=0
+(−1)l
+�
+I0⊎I1⊎...⊎Il=I\uj
+I1,...,Il̸=∅
+l�
+i=1
+H(0)
+|Ii|+1(x(uj); z; Ii)
+H(0)
+1 (x(uj); z)
+×
+�
+�
+I′
+0⊎I′′
+0 =I0
+DI′
+0
+1
+x(z) + y(uj) ·
+∂
+∂x(s)
+U (0)
+|I′′
+0 |+2(uj, z; I′′
+0 ∪ s)
+H(0)
+1 (x(uj); z)
+���
+s=uj
++
+�
+I′
+0⊎I′′
+0 ⊎I′′′
+0 =I0
+DI′
+0
+1
+x(z) + y(uj)
+�
+W (1)
+|I′′
+0 |+1(uj; I′′
+0 ) +
+λδI′′
+0 ,∅
+(x(z)−x(uj))3
+�U (0)
+|I′′′
+0 |+1(uj, z; I′′′
+0 )
+H(0)
+1 (x(uj); z)
+�
+= −λ2
+�
+I′⊎I′′=I\uj
+DI′
+1
+x(z) + y(uj) ·
+∂
+∂x(s)DI′′ U (0)
+2 (uj, z; s)
+H(0)
+1 (x(uj); z)
+���
+s=uj
+− DI\uj
+λ2�
+W (1)
+1 (uj) +
+λ
+(x(z)−x(uj))3
+�
+(x(z) + y(uj))2
+.
+Here Lemma 4.3 was used to get the second equality and (3.4) together with
+derivation property of DI to get the last equality.
+Next, in Lemma 4.4 we set v �→ uj and differentiate with respect to x(s) at
+s = uj. With the definition of Ω(0)reg
+2
+(uj, uj) given in the Proposition we get
+∂
+∂x(s)DI
+U (0)
+2 (uj, z; s)
+H(0)
+1 (x(uj); z)
+���
+s=uj
+= −DI
+λΩ(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))2 +
+d
+�
+k=1
+DI
+λΩ(0)
+2 (ˆzk, uj)
+(x(uj) + y(ˆzk))(x(z) + y(uj))
+−
+|I|
+�
+i=1
+DI\ui
+λ2Ω(0)
+2 (ui, uj)
+(x(uj) − x(ui))(x(z) + y(ui))2)(x(z) + y(uj))
+−
+�
+I⊎I′′=I
+DI′
+λ
+2(x(z) + y(uj))
+∂2
+∂(x(uj))2DI′′
+1
+(x(z) + y(uj)) .
+This result is inserted into (4.19) and then into (4.18). The sum over preimages
+cancels, the remainder simplifies to the assertion (4.17).
+□
+We proceed with the terms F (1)a
+|I|+1 in (4.14) which contribute to ˆP 1
+|I|+1. For them
+we need DI
+Q(0)
+1 (x(v),x(z))
+P (0)
+1
+(x(v),x(z)) which in the case I ̸= ∅ coincide with DI
+ˆQ(0)
+1 (x(v),x(z))
+P (0)
+1
+(x(v),x(z)). The
+function ˆQ(0)
+1 (x(v), x(z)) was given in (3.12). By Proposition 3.5, the action of
+
+BTR FROM EXTENDED LOOP EQUATIONS
+33
+DI on functions of log ˆQ(0)
+1 (x(v), x(z)) is the same as the combined action of DI
+on P (0)
+1 (x(z), x(z)) and P (0)
+1 (x(v), x(v)) and symbolic expressions D0x(0) as given
+in Proposition 3.5. Understanding D[0]
+I
+as DI when acting on P and D0
+I when
+acting on x(0), this means
+DI
+Q(0)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+= −
+λ
+(x(v) − x(z))2D[0]
+I
+��
+1 +
+(x(v) − x(z))2
+4(x(v) − x(0))(x(z) − x(0))
+× exp
+�1
+2 log P (0)
+1 (x(v), x(v)) + 1
+2 log P (0)
+1 (x(z), x(z)) − log P (0)
+1 (x(v), x(z))
+��
+.
+We need this expression at and near the diagonal x(v) = x(z). Taylor expansion
+gives
+DI
+Q(0)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+(4.20)
+= −D0
+I
+λ
+8(x(z) − x(0))2 − λ
+2
+∂2�
+DI log P (0)
+1 (x(w), x(z))
+�
+∂x(w)∂x(z)
+���
+w=z
++ (x(v) − x(z))
+�
+D0
+I
+λ
+8(x(z) − x(0))3 − λ
+2
+∂3�
+DI log P (0)
+1 (x(w), x(z))
+�
+∂x(w)∂(x(z))2
+���
+w=z
+�
++ O((x(v) − x(z))2) .
+We insert (4.20) into (4.14) and the result together with Proposition 4.5 into
+(4.13) to get the part F (1)
+|I|+1(x(v); x(z); I) of DI
+ˆP (1)
+1
+(x(v),x(z))
+P (0)
+1
+(x(v),x(z)) in (4.8a). The other
+part is K0,|I|+1(x(v); z; I) given in (4.9), which for A = 0 is a function of x(z). To
+simplify the total expression we write the last line of (4.9) for A = 0 as
+λ2
+(x(v) − x(z))
+d
+�
+k=0
+∂
+∂x(w)DI
+1
+(x(v) + y(ˆzk))(x(z) + y(w))
+���
+w=ˆzk
+= −
+λ2
+(x(v) − x(z))
+d
+�
+k=0
+DI
+y′(ˆzk)
+x′(ˆzk)(x(v) + y(ˆzk))(x(z) + y(ˆzk))2
+= −
+λ2
+(x(v) − x(z))3
+d
+�
+k=0
+DI
+�
+y′(ˆzk)
+x′(ˆzk)(x(v) + y(ˆzk)) −
+y′(ˆzk)
+x′(ˆzk)(x(z) + y(ˆzk))2
+�
+−
+λ2
+(x(v) − x(z))2
+d
+�
+k=0
+DI
+y′(ˆzk)
+x′(ˆzk)(x(z) + y(ˆzk))2
+
+34
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+= −
+λ2
+(x(v) − x(z))3
+∂
+∂x(w)
+d
+�
+k=0
+DI
+�
+log(x(v) + y( ˆwk)) − log(x(z) + y( ˆwk))
+�
+w=z
++
+λ2
+(x(v) − x(z))2
+∂2
+∂x(z)∂x(w)
+d
+�
+k=0
+DI log(x(z) + y( ˆwk))
+���
+w=z
+= −
+λ2
+(x(v) − x(z))3
+∂
+∂x(w)
+�
+DI log P (0)
+1 (x(v), x(w)) − DI log P (0)
+1 (x(z), x(w))
+�
+w=z
++
+λ2
+(x(v) − x(z))2
+∂2
+∂x(z)∂x(w)DI log P (0)
+1 (x(z), x(w))
+���
+w=z
+−
+|I|
+�
+j=1
+DI\uj
+�
+λ3
+(x(v) − x(z))(x(v) − x(uj))(x(z) − x(uj))2(x(z) + y(uj))2
+�
+,
+where Proposition 3.3 has been used. Putting everything together, we arrive (for
+I ̸= ∅) at
+Theorem 4.6. The auxiliary functions ˆP (1)
+|I|+1 of genus g = 1 are determined by
+DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+(4.21)
+=
+d
+�
+k=0
+DI
+λW (1)
+1 (ˆzk)
+x(v) + y(ˆzk) + λ2
+2
+d
+�
+j,k=0
+j̸=k
+DI
+Ω(0)
+2 (ˆzj, ˆzk)
+(x(v) + y(ˆzj))(x(v) + y(ˆzk))
+−
+|I|
+�
+j=1
+1
+(x(v) − x(uj))
+d
+�
+k=0
+DI\uj
+λ3Ω(0)
+2 (ˆzk, uj)
+(x(v) + y(ˆzk))(x(z) + y(uj))2
+−
+λ2
+(x(v) − x(z))3
+∂
+∂x(w)
+�
+DI log P (0)
+1 (x(v), x(w)) − DI log P (0)
+1 (x(z), x(w))
+�
+w=z
++
+λ2
+(x(v) − x(z))2
+�
+− D0
+I
+1
+8(x(z) − x(0))2 + 1
+2
+∂2(DI log P (0)
+1 (x(w), x(z)))
+∂x(w)∂x(z)
+���
+w=z
+�
++
+1
+x(v) − x(z)D0
+I
+λ2
+8(x(z) − x(0))3
++
+|I|
+�
+j=1
+1
+(x(v) − x(uj))DI\uj
+�λ3Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 −
+λ2W (1)
+1 (uj)
+(x(z) + y(uj))2
++
+λ3
+2(x(z) + y(uj))2
+∂2
+∂(x(uj))2
+1
+(x(z) + y(uj))
+�
+
+BTR FROM EXTENDED LOOP EQUATIONS
+35
++
+|I|
+�
+i,j=1
+i<j
+DI\{ui,uj}
+λ4Ω(0)
+2 (ui, uj)
+(x(v) − x(uj))(x(v) − x(ui))(x(z) + y(ui))2(x(z) + y(uj))2 .
+The Theorem also holds for I = ∅ where it specifies to Proposition 4.1.
+4.3. Loop equations for genus g = 1. From (4.21) we extract
+[(x(v))−1]DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+= λ
+d
+�
+k=0
+W (1)
+|I|+1(ˆzk; I) + D0
+I
+λ2
+8(x(z) − x(0))3
+(4.22)
++
+|I|
+�
+j=1
+DI\uj
+�λ3Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 −
+λ2W (1)
+1 (uj)
+(x(z) + y(uj))2
+−
+λ3
+2(x(z) + y(uj))2
+∂2
+∂(x(uj))2
+1
+(x(z) + y(uj))
+�
+and
+[(x(v))−2]DI
+ˆP (1)
+1 (x(v), x(z))
+P (0)
+1 (x(v), x(z))
+(4.23)
+= −λ
+d
+�
+k=0
+y(ˆzk)W (1)
+|I|+1(ˆzk; I) − λ2
+�
+I1⊎I2=I
+I2̸=∅
+d
+�
+k=0
+W (1)
+|I|+1(ˆzk; I1)W (0)
+|I|+1(ˆzk; I2)
+−
+|I|
+�
+j=1
+d
+�
+k=0
+DI\uj
+λ3Ω(0)
+2 (ˆzk, uj)
+(x(z) + y(uj))2
+(*)
++ λ2
+2
+d
+�
+j,k=0
+j̸=k
+DIΩ(0)
+2 (ˆzj, ˆzk) + λ2
+2
+∂2(DI log P (0)
+1 (x(w), x(z)))
+∂x(w)∂x(z)
+���
+w=z
+(†)
+− D0
+I
+λ2
+8(x(z) − x(0))2 + x(z)D0
+I
+λ2
+8(x(z) − x(0))3
++
+|I|
+�
+j=1
+x(uj)DI\uj
+�λ3Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 −
+λ2W (1)
+1 (uj)
+(x(z) + y(uj))2
++
+λ3
+2(x(z) + y(uj))2
+∂2
+∂(x(uj))2
+1
+(x(z) + y(uj))
+�
++ 1
+2
+|I|
+�
+i,j=1
+i̸=j
+DI\{ui,uj}
+λ4Ω(0)
+2 (ui, uj)
+(x(z) + y(ui))2(x(z) + y(uj))2 .
+(**)
+The combination of terms in the lines (*), (†) and (**) simplifies considerably:
+
+36
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Lemma 4.7.
+−
+|I|
+�
+j=1
+d
+�
+k=0
+DI\uj
+λ3Ω(0)
+2 (ˆzk, uj)
+(x(z) + y(uj))2 + λ2
+2
+∂2(DI log P (0)
+1 (x(w), x(z)))
+∂x(w)∂x(z)
+���
+w=z
++ λ2
+2
+d
+�
+j,k=0
+j̸=k
+DIΩ(0)
+2 (ˆzj, ˆzk) + 1
+2
+|I|
+�
+i,j=1
+i̸=j
+DI\{ui,uj}
+λ4Ω(0)
+2 (ui, uj)
+(x(z) + y(ui))2(x(z) + y(uj))2
+= −λ2
+2
+d
+�
+k=0
+DIΩ(0)reg
+2
+(ˆzk, ˆzk) + λ3
+6
+|I|
+�
+j=1
+∂
+∂x(uj)
+�
+DI\uj
+1
+(x(z) + y(uj))3
+�
+.
+(4.24)
+Proof. We start from (3.6) for v �→ z �→ w and differentiate with respect to x(z).
+In the second step we take (3.10) for I �→ I ∪ ˆwl into account:
+∂(DI log P (0)
+1 (x(z), x(w)))
+∂x(z)
+(4.25)
+=
+d
+�
+l=0
+DI
+1
+x(z) + y( ˆwl) −
+|I|
+�
+j=1
+DI\uj
+λ
+(x(z) − x(uj))2(x(w) + y(uj))
+= −DI
+d
+�
+k,l=0
+�
+W (0)
+2 (ˆzk; ˆwl) −
+δk,l
+(x(ˆzk) − x( ˆwl))
+�
+−
+|I|
+�
+j=1
+DI\uj
+�
+λ
+(x(z) − x(uj))2(x(w) + y(uj)) +
+d
+�
+l=0
+D ˆwl
+1
+x(z) + y(uj)
+�
+.
+We have �d
+l=0 D ˆwl
+1
+x(z)+y(uj) = − �d
+l=0
+λW (0)
+2
+(uj; ˆwl)
+(x(z)+y(uj))2.
+When differentiating with
+respect to x(w) at w = z, this term is the first one in (4.24), but with relative
+factor −2. Moreover, in the limit w → z the contributions with l ̸= k in the third
+line of (4.25) cancel the first term of the second line of (4.24), whereas for k = l
+the regularised Ω(0)reg
+2
+(ˆzk, ˆzk) appears. We thus have
+−
+|I|
+�
+j=1
+d
+�
+k=0
+DI\uj
+λ3Ω(0)
+2 (ˆzk, uj)
+(x(z) + y(uj))2 + λ2
+2
+∂2(DI log P (0)
+1 (x(w), x(z)))
+∂x(w)∂x(z)
+���
+w=z
++ λ2
+2
+d
+�
+j,k=0
+j̸=k
+DIΩ(0)
+2 (ˆzj, ˆzk)
+= −λ2
+2
+d
+�
+k=0
+DIΩ(0)reg
+2
+(ˆzk, ˆzk)
+
+BTR FROM EXTENDED LOOP EQUATIONS
+37
++ λ2
+2
+|I|
+�
+j=1
+�
+I1⊎I2=I\uj
+DI1
+λ
+(x(z) + y(uj))2DI2
+�
+δI2,∅
+(x(z) − x(uj))2 −
+d
+�
+l=0
+Ω(0)
+2 (uj, ˆzl)
+�
+.
+In the DI2 operation we use the symmetry under uj ↔ ˆzl to take a x(uj) derivative
+out. Using (3.10) we then get
+DI2
+�
+δI2,∅
+(x(z) − x(uj)2) −
+d
+�
+l=0
+Ω(0)
+2 (uj, ˆzl)
+�
+=
+∂
+∂x(uj)
+�
+δI2,∅
+(x(z) − x(uj)) −
+d
+�
+l=0
+W (0)
+2 (zl; I2 ∪ uj)
+�
+=
+∂
+∂x(uj)
+�
+DI2
+1
+x(z) + y(uj) +
+|I2|
+�
+i=1
+DI2\uiDuj
+1
+x(z) + y(ui)
+�
+=
+∂
+∂x(uj)DI2
+1
+x(z) + y(uj) −
+|I2|
+�
+i=1
+DI2\ui
+λΩ(0)
+2 (ui, uj)
+(x(z) + y(ui))2 .
+Inserted back, the last term leads to a double sum over pairs i ̸= j and a DI\{ui,uj}
+operation, which exactly cancels the second term of the second line of (4.24). An
+obvious rearrangement of �
+I1⊎I2=I\uj DI1
+1
+(x(z)+y(uj))2
+∂
+∂x(uj)DI2
+1
+x(z)+y(uj) confirms
+the assertion.
+□
+On the other hand, comparing (2.9a) with (2.5) we get as leading coefficient
+[(x(v))−1]H(g)
+|I|+1(x(v); q; I) = −λW (g)
+|I|+1(q; I) for any g ≥ 1.
+Then (2.9b) and
+(2.12c) show
+[(x(v))−2]
+ˆP (g)
+|I|+1(x(v), x(z); I)
+P (0)
+1 (x(v), x(z))
+(4.26)
+= λ
+N
+d
+�
+l=1
+λW (g)
+|I|+1(εl; I)
+x(z) − x(εl) − λ2
+|I|
+�
+j=1
+W (g)
+|I| (uj; I \ uj)
+x(z) − x(uj)
+for g ≥ 1
+as leading coefficient. Comparison with (4.22), (4.23) and Lemma 4.7 gives:
+Proposition 4.8. The genus-1 meromorphic functions W (1)
+|I|+1(z; I) satisfy the
+linear loop equation
+d
+�
+k=0
+W (1)
+|I|+1(ˆzk; I) = −D0
+I
+λ
+8(x(z) − x(0))3
+(4.27)
+−
+|I|
+�
+j=1
+DI\uj
+�λ2Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 −
+λW (1)
+1 (uj)
+(x(z) + y(uj))2
+
+38
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+−
+λ2
+2(x(z) + y(uj))2
+∂2
+∂(x(uj))2
+1
+(x(z) + y(uj))
+�
+and the quadratic loop equation
+−
+d
+�
+k=0
+y(ˆzk)W (1)
+|I|+1(ˆzk; I)
+(4.28)
+= λ
+�
+I1⊎I2=I
+I2̸=∅
+d
+�
+k=0
+W (1)
+|I|+1(ˆzk; I1)W (0)
+|I|+1(ˆzk; I2) + λ
+2
+d
+�
+k=0
+DIΩ(0)reg
+2
+(ˆzk, ˆzk)
+(*)
+− λ2
+6
+|I|
+�
+j=1
+∂
+∂x(uj)
+�
+DI\uj
+1
+(x(z) + y(uj))3
+�
+(†)
++ D0
+I
+λ
+8(x(z) − x(0))2 − x(z)D0
+I
+λ
+8(x(z) − x(0))3
+(§)
+−
+|I|
+�
+j=1
+x(uj)DI\uj
+�λ2Ω(0)reg
+2
+(uj, uj)
+(x(z) + y(uj))3 −
+λW (1)
+1 (uj)
+(x(z) + y(uj))2
+(‡)
+−
+λ2
+2(x(z) + y(uj))2
+∂2
+∂(x(uj))2
+1
+(x(z) + y(uj))
+�
+(‡)
++ λ
+N
+d
+�
+l=1
+W (1)
+|I|+1(εl; I)
+x(z) − x(εl) − λ
+|I|
+�
+j=1
+W (1)
+|I| (uj; I \ uj)
+x(z) − x(uj)
+.
+5. The recursion formula
+We learn from the loop equations (3.11)+(3.10) for g = 0, the loop equations
+(4.28)+(4.27) for g = 1 and the identity y(u) = −x(−u), see (1.2):
+• The only poles of z �→ x′(z)W (g)
+|I|+1(z; I) for 2g + |I| > 1 are located at
+the ramification points z = βi of x, at the opposite diagonals z = −ui for
+ui ∈ I and (for g ≥ 1) at z = 0. The pole at z = ui in the last line of
+(4.28) is due to the particular case I2 = {ui} in the first term of the line
+(*) of (4.28); it is not a pole of x′(z)W (1)
+|I|+1(z; I). Similarly, the pole at
+z = εk in the last line of (4.28) is due to the pole of y(z) at z = εk in the
+first line of (4.28). The same discussion applies to (3.11).
+Moving the integration contour to the complementary poles, we get
+x′(z)W (g)
+|I|+1(z; I)dz = Res
+q→z
+x′(q)W (g)
+|I|+1(q; I)dqdz
+q − z
+(5.1)
+=
+2d
+�
+i=1
+Res
+q→βi
+dz
+z − qx′(q)W (g)
+|I|+1(q; I)dq
+
+BTR FROM EXTENDED LOOP EQUATIONS
+39
++
+|I|
+�
+i=1
+Res
+q→−ui
+dz
+z − qx′(q)W (g)
+|I|+1(q; I)dq + Res
+q→0
+dz
+z − qx′(q)W (g)
+|I|+1(q; I)dq .
+• The linear loop equations (3.10) and (4.27) imply that the poles of z �→
+x′(z)W (g)
+|I|+1(I; z)dz at z = −ui and z = 0 have vanishing residue,
+Res
+z→ui x′(z)W (g)
+|I|+1(I; z)dz = 0 ,
+Res
+z→0 x′(z)W (g)
+|I|+1(I; z)dz = 0
+(since the integrands are total differentials in z). Renaming z �→ q, we
+thus have
+0 = −
+|I|
+�
+i=1
+Res
+q→−ui
+dz
+z + ui
+x′(q)W (g)
+|I|+1(I; q)dq − Res
+q→0
+dz
+z x′(q)W (g)
+|I|+1(I; q)dq ,
+which we can safely add to (5.1).
+• Let σi(z) be the local Galois conjugate near βi, i.e. the unique preim-
+age ˆzji = σi(z) ∈ x−1(x(z)) with limz→βi σi(z) = βi and σi(z) ̸≡ z.
+A residue at βi is invariant under Galois involution, Resq→βi f(q)dq =
+Resq→βi f(σi(q))dσi(q), so that
+Res
+q→βi
+dz
+z − qx′(q)W (g)
+|I|+1(q; I)dq
+= 1
+2 Res
+q→βi
+� dz
+z − qx′(q)W (g)
+|I|+1(q; I)dq +
+dz
+z − σi(q)x′(σi(q))W (1)
+|I|+1(σi(q); I)dσi(q)
+�
+= 1
+2 Res
+q→βi
+� dz
+z − q −
+dz
+z − σi(q)
+�
+x′(q)W (g)
+|I|+1(q; I)dq
++ 1
+2 Res
+q→βi
+dz
+z − σi(q)
+�
+x′(q)W (g)
+|I|+1(q; I)dq + x′(σi(q))W (g)
+|I|+1(σi(q); I)dσi(q)
+�
+.
+The last line vanishes because x′(q)dq = x′(σi)dσi(q) and W (g)
+|I|+1(q; I) +
+W (g)
+|I|+1(σi(q); I) is holomorphic at q = βi as consequence of the linear loop
+equations (3.10) and (4.27).
+In summary,
+x′(z)W (g)
+|I|+1(z; I)dz =
+2d
+�
+i=1
+Res
+q→βi
+�1
+2
+� dz
+z − q −
+dz
+z − σi(q)
+�
+x′(q)W (g)
+|I|+1(q; I)dq
+�
++
+|I|
+�
+i=1
+Res
+q→−ui
+�� dz
+z − q −
+dz
+z + ui
+�
+x′(q)W (g)
+|I|+1(q; I)dq
+�
++ Res
+q→0
+�� dz
+z − q − dz
+z
+�
+x′(q)W (g)
+|I|+1(q; I)dq
+�
+,
+(5.2)
+where the last line vanishes identically for g = 0.
+We can eventually complete the
+
+40
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+Proof of Theorem 1.1 for g ≤ 1. We identify the three contributions on the rhs
+of (5.2).
+(a) (poles at z = βi) For g = 1, only the lhs and the line (*) of the rhs of
+(4.28) have poles at z = βi; more precisely only the principal preimage
+k = 0 and the Galois preimage k = ki ∈ {1, ..., d} with ˆzki = σi(z). All
+terms with other k and all other lines in (4.28) are holomorphic at z = βi.
+Similarly, only the lhs and the first term of the rhs of (4.28) have poles at
+z = βi; more precisely only the principal preimage k = 0 and the Galois
+preimage k = ki ∈ {1, ..., d} with ˆzki = σi(z). The lhs of the quadratic
+loop equations for z �→ q, multiplied by x′(q)dq = x′(σi)dσi(q), is written
+as
+−
+d
+�
+k=0
+y(ˆqk)W (g)
+|I|+1(ˆqk; I)x′(q)dq
+=
+�
+− y(q)W (g)
+|I|+1(q; I) − y(σi(q))W (g)
+|I|+1(σi(q); I)
+�
+x′(q)dq + O((q − βi)1)dq
+= −(y(q) − y(σi(q)))W (1)
+|I|+1(q; I)x′(q)dq
+− y(σi(q))(W (g)
+|I|+1(q; I) + W (g)
+|I|+1(σi(q); I))x′(q)dq + O((q − βi)1)dq
+(**)
+= −(y(q) − y(σi(q)))W (g)
+|I|+1(q; I)x′(q)dq + O((q − βi)1)dq ,
+where we used that (3.10) and (4.27) make the whole line (**) of or-
+der O((q − βi)1)dq.
+Thus, the quadratic loop equations multiplied by
+x′(q)dq
+−(y(q)−y(σi(q))) (which is of order O((q − βi)0)dq), takes the form
+W (g)
+|I|+1(q; I)x′(q)dq
+(5.3)
+= −
+λx′(q)dq
+(y(q) − y(σi(q)))
+�
+q′∈{q,σi(q)}
+�1
+2
+�
+I1⊎I2=I
+g1+g1=g
+(gi,Ii)̸=(0,∅)
+W (g1)
+|I1|+1(q′; I1)W (g2)
+|I2|+1(q′; I2)
++ 1
+2DIΩ(g−1)reg
+2
+(q′, q′)
+�
++ O((q − βi)0)dq .
+When inserting this into (5.2), both cases q′ = q and q′ = σi(q)
+give the same contribution since the residue at βi is invariant un-
+der local Galois involution.
+When translating to ω(g)
+n+1(z, u1, ..., un) =
+λ2g+n−1du1 · · · dunW (g)
+n+1(z; u1, ..., un)dx(z) the first line of (1.3) with recur-
+sion kernel Ki(z; q) results.
+(b) (poles at z = −uj) For g = 1, these are present on the lhs and the line (*)
+of the rhs of (4.28), there only in the principal preimage k = 0, and in the
+lines (†) and (‡) of (4.28), there only in the j-summands. The line (†), by
+
+BTR FROM EXTENDED LOOP EQUATIONS
+41
+the linear loop equation (3.10) at genus g = 0, takes the form
+(4.28)† = −λ2
+12
+∂3
+∂x(uj)∂(x(z))2
+�
+DI\uj
+1
+(x(z) + y(uj))
+�
+= λ2
+12
+∂3W (0)
+|I|+1(z; I)
+∂x(uj)∂(x(z))2 + O((z + uj)0)
+= λ2
+12
+∂2DI\ujΩ(0)
+2 (z, uj)
+∂(x(z))2
++ O((z + uj)0) .
+By the linear loop equation (4.27), the j-summand of the lines (‡) can be
+written as
+(4.28)‡ = x(u)W (1)
+|I|+1(z; I) + O((z + uj)0) .
+Similarly, only the first two lines of (3.11) have poles at z = −uj, again
+only the principal preimages k = 0 and the j-summand of the last term
+of the second line of (3.11). The latter is with (3.10) also of the form
+x(u)W (0)
+|I|+1(z; I) + O((z + uj)0) that we noticed for g = 1. We bring these
+terms x(u)W (g)
+I|+1(z; I) to the lhs of the quadratic loop equations, rename
+z �→ q and multiply by
+x′(q)dq
+−(y(q)+x(uj)) to get
+W (g)
+|I|+1(q; I)x′(q)dq
+(5.4)
+=
+λx′(q)dq
+−(y(q) + x(uj))
+�1
+2
+�
+I1⊎I2=I
+g1+g1=g
+(gi,Ii)̸=(0,∅)
+W (g1)
+|I1|+1(q; I1)W (g2)
+|I2|+1(q; I2) + 1
+2DIΩ(g−1)reg
+2
+(q, q)
++ λ
+12
+∂2DI\ujΩ(g−1)
+2
+(q, uj)
+∂(x(q))2
+�
++ O((q + uj)−1)dq .
+Note that the undetermined residue poses no problem since it is
+projected away in (5.2).
+When translating to ω(g)
+n+1(z, u1, ..., un)
+=
+λ2g+n−1du1 · · · dunW (g)
+n+1(z; u1, ..., un)dx(z), the second and third lines of
+(1.3) with recursion kernel Kuj(z; q) result. Here one has take into ac-
+count that the differential duj does not commute with Kuj(z; q).
+This
+makes it necessary to keep the primitive d−1
+uj inside the residue.
+(c) (pole at z = 0) There is no such pole for g = 0. For g = 1, this pole is
+present on the lhs and the line (*) of the rhs of (4.28), there only in the
+principal preimage k = 0, and in both terms of the line (§). We write the
+first term as
+D0
+I
+λ
+8(x(z) − x(0))2 = −λ
+8
+∂2(D0
+I log(x(z) − x(0)))
+∂(x(z))2
+
+42
+ALEXANDER HOCK AND RAIMAR WULKENHAAR
+= −λ
+8
+∂2(DI log P (0)
+1 (x(z), x(z)))
+∂(x(z))2
++ O(z0)
+by Proposition 3.5. Next, from (4.25) we know
+∂(DI log P (0)
+1 (x(z), x(z)))
+∂(x(z))
+= lim
+w→z 2∂(DI log P (0)
+1 (x(z), x(w)))
+∂(x(z))
+���
+w=z
+= −2DIW (0)reg
+2
+(z; z) + O(z0) .
+By the linear loop equation (4.27), the second term in the line (§) of (4.28)
+can be written as
+−x(z)D0
+I
+λ
+8(x(z) − x(0))3 = x(z)W (1)
+|I|+1(z; I) + O(z0) .
+We bring this term to the lhs, rename z �→ q and multiply by
+x′(q)dq
+−(y(q)+x(q))
+to get with the previous considerations
+W (1)
+|I|+1(q; I)x′(q)dq
+(5.5)
+=
+λx′(q)dq
+−(y(q) + x(q))
+� �
+I1⊎I2=I
+I2̸=∅
+W (1)
+|I1|+1(q; I1)W (0)
+|I2|+1(q; I2) + 1
+2DIΩ(0)reg
+2
+(q, q)
++ 1
+4
+∂(DIW (0)reg
+2
+(q; q))
+∂x(q)
+�
++ O(q−1) .
+Again the undetermined residue poses no problem since it is pro-
+jected away in (5.2).
+When translating to ω(g)
+n+1(z, u1, ..., un)
+=
+λ2g+n−1du1 · · · dunW (g)
+n+1(z; u1, ..., un)dx(z), the last two lines of (1.3) with
+recursion kernel K0(z; q) result.
+□
+6. Outlook
+We continued the work of [GHW19, SW22, BHW22, HW21] and pushed
+the proof of the conjecture [BHW22, Conj. 6.1] that the quartic Kontsevich
+model obeys blobbed topological recursion [BS17] to genus g ≤ 1. The method
+that we developed in this paper is general and powerful enough to achieve the
+proof to any g. The Dyson-Schwinger equations (2.13b) provide the difference
+DIDg log P (0)
+1 (x(v), x(z)) − DIDg log H(0)
+1 (x(v); z), where DI is the loop inser-
+tion operator of Definition 2.6 and Dg a ‘genus insertion’ still to make precise.
+Then DIDg log P (0)
+1 (x(v), x(z)) is this difference symmetrised in all preimages ˆzk
+plus a function of (x(v), x(z); I) with simple poles at x(v) = x(ui) and poles at
+x(v) = x(z) up to order 4g + 3. The principal part of the corresponding Laurent
+series is uniquely determined by (2.9a), (2.9b), (2.12b) and (2.12c) in terms of
+Taylor expansions of Q(h)
+1 (x(v), x(z); I) and P (h)
+1 (x(v), x(z); I) for h < g at the
+
+BTR FROM EXTENDED LOOP EQUATIONS
+43
+diagonal x(v) = x(z). The required functions Q are determined before via a sim-
+ilar analysis of DIDg−1 log ˆQ(0)
+1 (x(v), x(z)) − DIDg−1 log ˆ
+M (0)
+1 (x(v); z). After all,
+there is no principal obstacle to produce the solution DIDg log P (0)
+1 (x(v), x(z))
+from which by expansion about x(v) = ∞ we get global linear and quadratic
+loop equations for W (g)
+|I|+1. This shows that the quartic Kontsevich model satisfies
+blobbed TR. Working out the details will be challenging, however, as the for-
+mulae become increasingly lengthy with larger g and Taylor expansions to order
+4g + 1 are necessary.
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+

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+Scaling transition for singular linear random fields on Z2:
+spectral approach
+Donatas Surgailis
+January 6, 2023
+Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius, Lithuania
+Abstract
+We study partial sums limits of linear random fields X on Z2 with spectral density f(x) tending to ∞, 0
+or to both (along different subsequences) as x → (0, 0). The above behaviors are termed (spectrum) long-
+range dependence, negative dependence, and long-range negative dependence, respectively, and assume an
+anisotropic power-law form of f(x) near the origin. The partial sums are taken over rectangles whose sides
+increase as λ and λγ, for any fixed γ > 0. We prove that for above X the partial sums or scaling limits
+exist for any γ > 0 and exhibit a scaling transition at some γ = γ0 > 0; moreover, the ‘unbalanced’ scaling
+limits (γ ̸= γ0) are Fractional Brownian Sheet with Hurst parameters taking values from [0, 1]. The paper
+extends [24, 19, 28] to the above spectrum dependence conditions and/or more general values of Hurst
+parameters.
+Keywords: Linear random field, Gaussian random field, spectral density, long-range dependence, negative
+dependence, long-range negative dependence, hyperbolic dependence, self-similarity, anisotropic scaling,
+scaling transition, fractional Brownian sheet,
+1
+Introduction
+A stationary random field (RF) X = {X(t); t ∈ Zd} with finite second moment and covariance r(t) =
+Cov(X(0), X(t)) is said (i) long-range dependent (LRD) if �
+t∈Zd |r(t)| = ∞;
+(ii) short-range depen-
+dent (SRD) if �
+t∈Zd |r(t)| < ∞,
+�
+t∈Zd r(t) ̸= 0, and (iii) negatively dependent (ND) if �
+t∈Zd |r(t)| <
+∞, �
+t∈Zd r(t) = 0 (for RF indexed by continuous argument, the above concepts are analogously defined
+with sums replaced by integrals over Rd). The above classification, albeit not completely satisfactory, is
+important in limit theorems, see [13, 16, 28]. Related but not equivalent definitions of LRD, SRD and ND
+are given in terms of spectrum or moving-average coefficients of RF. In the sequel we refer to the above co-
+variance or moving-average characterizations as spatial domain properties, to be distinguished from frequency
+domain (spectrum) characterizations used in the literature. A very general form of such frequency domain
+concepts assumes the existence of spectral density f(u), u ∈ Πd := [−π, π]d which is bounded outside of the
+1
+arXiv:2301.02015v1  [math.PR]  5 Jan 2023
+
+origin and such that (i’) limu→0 f(u) = ∞ (spectrum LRD), (ii’) limu→0 f(u) > 0 (spectrum SRD), and
+(iii’) limu→0 f(u) = 0 (spectrum ND). Moreover, in cases (i’) and (iii’) it is often additionally assumed that
+spectral density varies regularly as u → 0 [13, 22, 25]. It is clear that in dimensions d ≥ 2 a function can
+increase regularly to ∞ along one direction and vanish along another one. Such a behavior is not reflected in
+(i’)-(iii’), calling for a new concept (iv’) 0 = lim infu→0 f(u) < lim supu→0 f(u) = ∞ which we term spectrum
+long-range negative dependence (spectrum LRND). Properties (i’), (iii’) and (iv’) can be jointly dubbed as
+singular behaviors, the singularity limited to the origin and including both infinite and/or zero limits.
+The present paper studies anisotropic partial sums limits for a class of stationary linear RFs X on Z2 with
+spectrum LRD/ND/LRND properties. The partial sums
+Sλ,γ(x) :=
+�
+t∈K[λx1,λγx2]
+X(t),
+x = (x1, x2) ∈ R2
++
+(1.1)
+are taken over rectangles K[λx1,λγx2] := {t ∈ Z2 : 1 ≤ t1 ≤ λx1, 1 ≤ t2 ≤ λγx2} ⊂ Z2 whose sides increase as
+O(λ) and O(λγ) as λ → ∞, for a fixed γ > 0. Our main object are anisotropic scaling limits
+d−1
+λ,γSλ,γ(x)
+fdd
+−→ Vγ(x),
+x ∈ R2
++,
+λ → ∞,
+(1.2)
+where dλ,γ → ∞ is a normalization. It was proved in [24, 23, 18, 19, 4, 20, 28] and other works that for
+a large class of RFs nontrivial scaling limits Vγ(x) in (1.2) exist for any γ > 0. Moreover, the limit family
+{Vγ; γ > 0} may exhibit a trichotomy called scaling transition at some (nonrandom) point γ = γ0 > 0
+meaning that Vγ = V± (γ ≶ γ0) do not depend on γ and are different; see Definition 1. For linear RFs, these
+works can be summarized as saying that a large class of RFs on Z2 parade two types of the limit behavior
+in (1.2): either (I) Vγ is a standard Brownian Sheet B1/2,1/2 for any γ > 0, or (II) scaling transition at some
+γ0 > 0 exists such that all unbalanced limits Vγ, γ ̸= γ0 are Fractional Brownian Sheet (FBS) BH1,H2 with
+Hurst parameters (H1, H2) = (H+
+1 , H+
+2 ) (γ > γ0), = (H−
+1 , H−
+2 ) (γ < γ0), (H+
+1 , H+
+2 ) ̸= (H−
+1 , H−
+2 ). Behavior
+(I) is rather standard under SRD conditions, see [6, 7, 23] and the references therein. Behavior (II) is more
+interesting and was established under LRD and ND conditions in [24, 28]. However, these results exclude
+LRND singularity and some particular (boundary) values of parameters.
+The present paper extends [24, 19, 28] and some other work on type (II) behavior for linear RFs X under
+spectrum LRD, ND, and LRND conditions. It appears that the spectral approach used in this work is more
+efficient and simple compared to ‘time-domain’ approach in [19, 28] and other papers; moreover, it can be
+used to obtain the covariance structure of the scaling limits in a non-Gaussian context as well. It is assumed
+that X have a moving-average (MA) representation
+X(t) =
+�
+s∈Z2
+a(t − s)ε(s),
+t ∈ Z2,
+(1.3)
+with standardized i.i.d.
+innovations {ε(s); s ∈ Z2}, Eε(s) = 0, Eε(s)2 = 1 and deterministic coefficients
+a(t), t ∈ Z2. The latter coefficients satisfy the necessary condition �
+t∈Z2 a(t)2 < ∞ for the convergence of
+(1.3) but all other conditions for various behaviors in (1.2) in this paper refer to the spectral density of X
+2
+
+Figure 1. Level graphs of spectral density under LRD, ND, LRND and ‘hyperbolic dependence’ conditions. a) f(x) =
+ρ−1
+2 (x), υ1 = 0.8, υ2 = 1.2; b) f(x) = ρ(x), υ1 = 0.5, υ2 = 1; c) f(x) = ρ−1(x)|x2/ρ(x)1/υ2|0.5, υ1 = 0.5, υ2 = 1; d)
+f(x) = |x1|−0.2|x2|−.5
+written as the squared Fourier transform f(x) = (2π)2|�a(x)|2, �a(x) := (2π)−2 �
+t∈Z2 eit·xa(t), x ∈ Π2 of
+MA coeffiecients. For identification of the covariance structure of the limit Gaussian RF in (1.2), the basic
+assumption is that f(x) behaves like a power function
+ρ(x) := |x1|υ1 + |x2|υ2,
+x = (x1, x2) ∈ R2
+(1.4)
+at the origin, where υi > 0, i = 1, 2 are given parameters, or, more precisely,
+f(x) ∼ L(x)/ρ(x)
+(‘spectrum LRD’)
+and
+f(x) ∼ L(x)ρ(x)
+(‘spectrum ND’),
+(1.5)
+where L(x) is an ‘angular function’ satisfying some boundedness conditions. Clearly, the first relation in
+(1.5) implies spectrum LRD provided L(x) is bounded from below, while the second relation relation in (1.5)
+3
+
+a)b)c)d)implies spectrum ND provided L(x) is bounded from above. The above choice of ρ(x) as the ‘radial function’
+is rather flexible: we can use other forms of (1.4), e.g. with
+ρp(x) := (|x1|pυ1 + |x2|pυ2)1/p,
+p > 0,
+x = (x1, x2) ∈ R2
+(1.6)
+instead of (1.4) as well, since the change of ρ(x) to ρp(x) amounts to a change of the angular function which
+does not affect the limit results. Spectrum LRND may arise if these boundedness conditions on L(x) are
+violated, e.g. when L(x) decreases as |xi|µi with exponent µi > 0 when xi → 0 in the first relation in (1.5).
+We show that in this case the limits in (1.2) may exist, together with a scaling transition and the limit
+Gaussian RF depends on parameters υi, µi, i = 1, 2. We also consider the case when the spectral density
+asymptotically factorizes into a product of power functions |xi|−υi, i = 1, 2. This form of singularity (studied
+in [24, 6] and termed hyperbolic dependence in the present paper) allows for spectrum LRD, ND, or LRND
+depending on the sign of υi, but leads to very different limit results (the absence of scaling transition). Fig. 1
+illustrates different types of singular spectral densities studied in this paper. We also expect that our results
+can be extended to RFs on d-dimensional lattice, d ≥ 3 arbitrary although the description of the limit RFs
+is more complicated, see [4, 27].
+The rest of the paper is organized as follows. Sec. 2 provides rigorous assumptions on f(x) and some
+preliminary facts used in the subsequent text. Sections 3-5 contain the main Theorems 1, 2, 3 and 4, under
+respective LRD, ND, LRND and hyperbolic dependence premises.
+Notation. In this paper,
+fdd
+−→ ,
+fdd
+= , and
+fdd
+̸= , denote respectively the weak convergence, equality, and
+inequality, of finite dimensional distributions. C stands for a generic positive constant which may assume
+different values at various locations and whose precise value has no importance. R+ := (0, ∞), R2
+0 := R2 \
+{(0, 0)}, Π := [−π, π], |x| := |x1| + |x2|, x · y := x1y1 + x2y2, x = (x1, x2), y = (y1, y2), 1 := (1, 1), 0 := (0, 0).
+I(A) denotes the indicator function of a set A.
+2
+Preliminaries and assumptions
+Definition 1. [24, 23, 19] We say that a stationary RF X = {X(t); t ∈ Z2} exhibits scaling transition if
+non-trivial limits in (1.2) exist for any γ > 0 and there exists a γ0 > 0 such that
+Vγ
+fdd
+= V+ (γ > γ0),
+Vγ
+fdd
+= V− (0 < γ < γ0),
+V+
+fdd
+̸= aV− (∀ a > 0).
+(2.1)
+In such a case, RFs V± will be called the unbalanced limits, and RF V0 the well-balanced limit of X.
+A weaker version of Definition 1 in [7] requires that Vγ is independent of γ for γ > γ0 and γ < γ0 in a small
+neighborhood of γ0. There exist RFs which exhibit scaling transition in the above weaker sense at several
+(possibly, infinite) number of points γ > 0 [18, 7]. However, for RFs discussed in this paper Definition 1
+suffices.
+Recall that the covariance function of stationary zero-mean RF X = {X(t); t ∈ Z2} is the Fourier transform
+of its spectral density, viz., r(t) := EX(0)X(t) =
+�
+Π2 eit·uf(u)du, t ∈ Z2. Whence, the covariance function
+4
+
+of the normalized partial sums RF in (1.1) writes as
+Rλ,γ(x, y)
+:=
+d−2
+λ,γESλ,γ(x)Sλ,γ(y) = d−2
+λ,γ
+�
+Π2
+2
+�
+i=1
+D[λγixi](ui)D[λγiyi](ui)f(u)du,
+x, y ∈ R2
++, (2.2)
+where (γ1, γ2) := (1, γ) and Dn(u) := �n
+t=1 eitu = (eiu − ei(n+1)u)/(1 − eiu), u ∈ Π (the real-valued function
+Dn(u)e−i(n+1)u/2 is the Dirichlet kernel).
+Proposition 1. Let X be a linear RF in (1.3) with spectral density f and Sλ,γ be the partial sum RF in
+(1.1), γ > 0. Assume that
+Rλ,γ(x, y) → Rγ(x, y),
+λ → ∞
+(∀ x, y ∈ R2
++)
+(2.3)
+and
+|gλ|∞ := sup
+u∈Z2
+���
+�
+t∈K[λx1,λγx2]
+a(t − u)
+��� = o(dλ,γ),
+λ → ∞.
+(2.4)
+Then (1.2) holds, where Vγ is a Gaussian RF on R2
++ with zero mean EVγ(x) = 0 and covariance EVγ(x)Vγ(y) =
+Rγ(x, y), x, y ∈ R2
++.
+Proof. The finite-dimensional convergence in (1.2) is equivalent to one-dimensional convergence Sλ,γ :=
+d−1
+λ,γ
+�m
+j=1 θj Sλ,γ(xj)
+d
+−→ �m
+j=1 θjVγ(xj) =: Vγ for any θj ∈ R, xj ∈ R2
++, j = 1, · · · , m, m ≥ 1. From (2.3)
+we have E(Sλ,γ)2 → �m
+j,j′=1 θjθj′Rγ(xj, xj′) =: Rγ ≥ 0. Let Rγ > 0 then Sλ,γ = d−1
+λ,γ
+�
+u∈Z2 gλ(u)ε(u) is a
+normalized weighted linear form in i.i.d. r.v.s with weights gλ(u) := �m
+j=1 θj
+�
+t∈K[λx1j,λγx2j] a(t − u), xj =
+(x1j, x2j). Relation (2.4) implies the Lindeberg condition, viz., for any τ > 0
+�
+u∈Z2
+g2
+λ(t)E[ε(u)2I(|gλ(u)ε(u)| > τdλ,γ)] = o(d2
+λ,γ),
+λ → ∞
+(2.5)
+since Eε(u)2I(|gλ(u)ε(u)| > τdλ,γ) ≤ Eε(u)2I
+�
+|ε(u)| > τdλ,γ/|gλ|∞
+�
+→ 0 in view of (2.4) and Eε(0)2 < ∞.
+Thus, Sλ,γ
+d
+−→ N(0, Rγ), in other words, the distribution of (d−1
+λ,γSλ,γ(xj); j = 1, · · · , m) tends to a Gaussian
+distribution on Rm with mean zero and covariance matrix (Rγ(xj, xj′))j,j=1,··· ,m, proving the proposition. □
+Let υi > 0, i = 1, 2, Υ :=
+1
+υ1 + 1
+υ2 and ρ(x), ρp(x) be as in (1.4), (1.6). We note the elementary inequality:
+for any p > 0 there exist constants 0 < C1 ≤ C2 < ∞ such that
+C−ρ(x) ≤ ρp(x) ≤ C+ρ(x),
+(∀ x ∈ R2).
+(2.6)
+We also use the fact [19] that for any δ > 0, w > 0
+�
+|x|<δ
+ρ(x)−wdx < ∞ ⇐⇒ w < Υ,
+�
+|x|>δ
+ρ(x)−wdx < ∞ ⇐⇒ w > Υ.
+(2.7)
+Following [20], we say that a measurable function L : R2
+0 → R is generalized invariant if the function
+x �→ L(λ1/υ1x1, λ1/υ2x2) does not depend on λ > 0. Every generalized invariant function L can be represented
+as
+L(x) = ˜L(x1/ρ(x)1/υ1, x2/ρ(x)1/υ2),
+x ∈ R2
+0,
+(2.8)
+5
+
+where ˜L is the restriction of L to S1 := {x ∈ R2
+0 : ρ(x) = 1}.
+Assumption (F)LRD The spectral density f satisfies
+f(x) = ρ(x)−1L(x)
+�
+1 + o(1)
+�
+,
+|x| → 0,
+(2.9)
+where ρ(x) as in (1.4), Υ > 1 and L(x) = L(−x), x ∈ R2
+0 is a strictly positive continuous generalized invariant
+function. Moreover, f is bounded on {x ∈ Π2 : |x| > δ}, for any δ > 0.
+Assumption (F)LRND,i
+(i = 1, 2) The spectral density f satisfies Assumption (F)LRD except that strict
+positivity of L(x) is replaced by
+˜L(x) = ℓi|xi|µi(1 + o(1))
+(xi → 0, x ∈ S1)
+(2.10)
+for some µi > 0, ℓi > 0.
+Assumption (F)ND The spectral density f(x), x ∈ Π2 is a bounded continuous function such that
+f(x) = ρ(x)L(x)
+�
+1 + o(1)
+�
+,
+|x| → 0,
+where
+(2.11)
+where ρ(x) as in (1.4), and L(x), x ∈ R2
+0 is as in (2.9).
+With generalized homogeneous functions ρ(x)−1L(x), ρ(x)L(x) in (2.9), (2.11) we associate Gaussian RFs
+V0,1/ρ, V0,ρ by
+V0,1/ρ(x) :=
+�
+R2
+�2
+j=1
+1−eiujxj
+iuj
+�
+L(u)/ρ(u) Z(du),
+V0,ρ(x) :=
+�
+R2
+�2
+j=1
+1−eiujxj
+iuj
+�
+L(u)ρ(u) Z(du),(2.12)
+x ∈ R2
++, where {Z(du)} is a complex-valued Gaussian noise with zero mean and E|Z(du)|2 = du. The
+existence of V0,1/ρ, V0,ρ will be established in the following sections.
+Definition 2. Let V = {V (x); x ∈ R2
++} be a RF and γ > 0, H ≥ 0, Hi ≥ 0, i = 1, 2. We say that
+(i) V is (γ, H)-self-similar (SS) if
+{V (λx1, λγx2); x ∈ R2
++} fdd
+= {λHV (x); x ∈ R2
++},
+∀λ > 0.
+(2.13)
+(ii) V is (H1, H2)-multi-self-similar (MSS) is
+{V (λ1x1, λ2x2); x ∈ R2
++} fdd
+= {λH1
+1 λH2
+2 V (x); x ∈ R2
++},
+∀λ1, λ2 > 0.
+(2.14)
+(γ, H)-SS property is a particular case of the operator self-similarity property introduced in [2] and corre-
+sponding to scaling x → λEx with diagonal matrix E = diag(1, γ). Particularly, (1, H)-SS property coincides
+with the usual H-SS property for RFs on R2
++ [25]. (H1, H2)-MSS property was introduced in [12]. It implies
+(γ, H)-SS property with any γ > 0 and H = H1 + γH2. Under mild additional assumptions, scaling limits
+V X
+γ
+in (1.2) are (γ, H)-SS RFs and the normalization dλ,γ is regularly varying at infinity with index H [23].
+6
+
+Definition 3. (Standard) Fractional Brownian Sheet (FBS) BH1,H2 = {BH1,H2(x); x ∈ R2
++} with (H1, H2) ∈
+[0, 1]2 is defined as a Gaussian process with zero-mean and covariance function EBH1,H2(x)BH1,H2(y) =
+�2
+i=1 rHi(xi, yi), x, y ∈ R2
++, where for x, y ∈ R+,
+rH(x, y)
+:=
+1
+2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+x2H + y2H − |x − y|2H,
+0 < H ≤ 1,
+2,
+H = 0, x = y,
+1,
+H = 0, x ̸= y.
+(2.15)
+Usually, FBS is defined for (H1, H2) ∈ (0, 1]2 or even (H1, H2) ∈ (0, 1)2, see [1]. Extension of FBS to
+H1 ∧ H2 = 0 was introduced in [28]. (2.15) implies that the restriction of FBS BH1,H2 to horizontal/vertical
+line agrees with fractional Brownian motion (FBM) BH = {BH(x); x ∈ R+} with covariance function rH(x, y)
+and the corresponding Hurst parameter H = Hi ∈ [0, 1], i = 1, 2. Note r0(x, y) = limH↓0 rH(x, y). FBM B0
+is 0-SS and an extremely singular (non-measurable) process, see [25, pp.256–257]. It can be represented as
+B0
+fdd
+= { 1
+√
+2(W(x)−W(0)); x ∈ R+}, where W(x), x ∈ [0, ∞), is (uncountable) family of independent N(0, 1)
+r.v.s. The above B0 is different from the ’regularized’ FBM with H = 0 defined in [10, p.2985]. FBS BH1,H2
+with H1 ∧ H2 = 0 and their α-stable extensions appeared in limit theorems for RFs [28, 21]. We also recall
+spectral representation of FBS with (H1, H2) ∈ (0, 1)2:
+BH1,H2(x)
+=
+κ−1
+�
+R2
+2
+�
+j=1
+1 − eixjuj
+iuj|uj|Hj− 1
+2
+Z(du)
+(2.16)
+where
+κ2
+:=
+�
+R2
+2
+�
+j=1
+|1 − eiuj|2
+|uj|2Hj+1 du =
+2
+�
+j=1
+π
+HjΓ(2Hj) sin(Hjπ)
+(2.17)
+and Z(du) is the same Gaussian white noise as in (2.12); see [1]. For H1 ∨ H2 = 1, FBS is a line RF of the
+form B1,H2(x) = x1BH2(x2) (0 < H2 < 1), BH1,1(x) = x2BH1(x1) (0 < H1 < 1), B1,1(x) = x1x2Z, where BH
+is FBM and Z ∼ N(0, 1). For H1 ∧ H2 = 0, FBS allow a construction of finite-dimensional distributions via
+independent FBM or uncountable family of independent Gaussian variables [28].
+3
+Long-range dependence
+Define
+H+
+1
+:=
+1
+2(1 + (υ1 ∧ 1)),
+H+
+2 := 1
+2(1 + υ2 −
+υ2
+υ1 ∨ 1),
+(3.1)
+H−
+1
+:=
+1
+2(1 + υ1 −
+υ1
+υ2 ∨ 1),
+H−
+2 := 1
+2(1 + (υ2 ∧ 1)),
+Note H+
+1 = 1 for υ1 ≥ 1, H+
+2 = 1/2 for υ1 ≤ 1. Analogous relations are satisfied by H−
+i , i = 1, 2.
+Theorem 1. Let X in (1.1) be a stationary linear RF on Z2 in (1.3) with spectral density f satisfying
+Assumption (F)LRD. Then:
+7
+
+• The scaling limits in (1.2) exist for any γ > 0 and satisfy (2.1) with γ0 = υ1
+υ2 and the unbalanced limits
+are given by
+V+
+:=
+κ+
+�
+�
+�
+�
+�
+BH+
+1 ,1/2
+υ1 ≤ 1,
+B1,H+
+2 ,
+υ1 ≥ 1,
+V− := κ−
+�
+�
+�
+�
+�
+B1/2,H−
+2
+υ2 ≤ 1,
+BH−
+1 ,1,
+υ2 ≥ 1,
+(3.2)
+where H±
+i , i = 1, 2 as in (3.1) and the constants κ± > 0 in (3.2) are defined in (3.13), (3.14) (or can
+be determined from these expressions by symmetry).
+• The well-balanced limit V0 = V0,1/ρ is given in (2.12).
+• The normalization in (1.2) is given by dλ,γ := λH(γ)(log+ λ)1/2 if γ > γ0 and υ1 = 1 or γ < γ0 and
+υ2 = 1, and dλ,γ := λH(γ) in the remaining cases, where H(γ) is the weighted linear combination with
+weights (1, γ) of the Hurst indices of FBM on the r.h.s. (3.2), viz.,
+H(γ) =
+�
+�
+�
+�
+�
+H+
+1 + γH+
+2 ,
+γ ≥ γ0,
+H−
+1 + γH−
+2 ,
+γ ≤ γ0.
+(3.3)
+• The RF X exhibits scaling transition at γ0.
+Proof. The simple bound
+�� �
+t∈K[λx1,λγx2] a(t − u)
+�� ≤ Cλ(1+γ)/2� �
+t∈Z2 a(t)2�1/2 ≤ Cλ(1+γ)/2 reduces the
+criterion (2.4) to H(γ) > (1 + γ)/2. This follows from (3.3) since H±
+i
+≥ 1/2, i = 1, 2. Whence, for (1.2) it
+suffices to prove the convergence of covariance functions in (2.3). The latter relation is essentially proved in
+[24] except for the cases υ1 = 1 or υ2 = 1. The subsequent proof is limited to these cases. Note that the
+definitions in (3.2), (3.3) given by different expressions agree for υi = 1, i = 1, 2 and/or γ = γ0. By symmetry,
+it suffices to discuss the case υ1 = 1 only and consider the limits in (2.3) for γ > γ0, γ < γ0, and γ = γ0
+separately. W.l.g. λ > 1.
+Case γ > γ0 = 1/υ2 (υ1 = 1). In this case, V+ = κ+B1,1/2 and d2
+λ,γ = λ2+γ log λ, according to (3.2). In the
+integral in (2.2), change the variables: u1 → u1/λγυ2, u2 → u2/λγ and note that
+λ−1D[λx1](u1/λγυ2) → x1,
+λ−γD[λγx2](u2/λγ) → 1 − eix2u2
+−iu2
+,
+(3.4)
+fλ(u) := λ−γυ2f(u1/λγυ2, u2/λγ) → L(u)/ρ(u) =: f0(u)
+point-wise for any u ∈ R2
+0 as λ → ∞, due to γ > γ0. Moreover, (2.9) implies
+|fλ(u) − f0(u)| ≤
+1
+|u1| + |u2|υ2 δ
+� u1
+λγυ2 , u2
+λγ
+�
+,
+(3.5)
+where δ(u) is a bounded function tending to 0 as u → 0.
+Thus, Rλ,γ(x, y) = �2
+i=1 R(i)
+λ,γ(x, y), where
+R(i)
+λ,γ(x, y) :=
+1
+log λ
+�
+λγυ2Π×λγΠ G(i)
+λ (u)du, i = 1, 2 with
+G(1)
+λ (u)
+:=
+D[λx1](u1/λγυ2)D[λy1](u1/λγυ2)
+λ2
+· D[λγx2](u2/λγ)D[λγy2](u2/λγ)
+λ2γ
+· f0(u)
+(3.6)
+→
+x1y1 · (1 − eix2u2)(1 − e−iy2u2)
+|u2|2
+· f0(u) =: G(u),
+8
+
+and G(2)
+λ (u) analogously defined with f0(u) replaced by fλ(u) − f0(u).
+Let us prove R(2)
+λ,γ(x, y) → 0. From elementary bound |Dn(x)| ≤ Cn/(1 + n|x|), x ∈ Π and (3.5) we get
+|R(2)
+λ,γ(x, y)|
+≤
+C
+log λ
+�
+λγυ2Π×λγΠ
+du
+(1 + u2
+2)(|u1| + |u2|υ2)δ
+� u1
+λγυ2 , u2
+λγ
+�
+=
+C
+log λ
+�
+R
+du2
+1 + u2
+2
+�
+|u1|≤λγυ2/|u2|υ2
+du1
+|u1| + 1δ
+�u1|u2|υ2
+λγυ2
+, u2
+λγ
+�
+.
+(3.7)
+Split the integration region on the r.h.s.
+of (3.7) into three parts corresponding to {|u2|
+λγ
+≥ ϵ}, {|u2|
+λγ
+<
+ϵ, |u1|( |u2|
+λγ )υ2 < ϵ}, and {|u2|
+λγ < ϵ, |u1|( |u2|
+λγ )υ2 ≥ ϵ}, T1, T2 and T3, say, so that |R(2)
+λ,γ(x, y)| ≤ C(log λ)−1 �3
+i=1
+�
+Ti . . .
+=: (log λ)−1 �3
+i=1 Ri. The inner integral in (3.7) on T1 is bounded by C log(1/ϵ)υ2) implying R1 → 0 (λ →
+∞, ∀ϵ > 0). Next, by the property δ(u) → 0, for any δ′ > 0 there exists ϵ > 0 s.t. R2 ≤ δ′(log λ)−1 �
+R(1 +
+u2
+2)−1 log(λγ/|u2|)υ2du2 ≤ Cδ′ vanishes with ϵ → 0 uniformly in λ ≥ 2.
+Finally, for any ϵ > 0 fixed,
+R3 ≤ C(log λ)−1 �
+R(1 + u2
+2)−1�
+log(λγ/|u2|)υ2 − log ϵ(λγ/|u2|)υ2�
+= O(1/ log λ) → 0 as λ → ∞, thus ending
+the proof of R(2)
+λ,γ(x, y) → 0.
+Next, we prove the limits
+lim
+λ→∞ R(1)
+λ,γ(x, y) = lim
+λ→∞
+˜Rλ,γ(x, y) = κ2
++E[B1,1/2(x)B1,1/2(y)] = κ2
++x1y1(x2 ∧ y2)
+(3.8)
+where
+˜Rλ,γ(x, y)
+:=
+1
+log λ
+�
+λγυ2Π×λγΠ
+G(u)du
+=
+x1y1
+�
+|u2|≤λγπ
+(1 − eix2u2)(1 − e−iy2u2)
+|u2|2
+du2 ×
+1
+log λ
+�
+|u1|≤λγυ2π
+f0(u)du1.
+(3.9)
+Consider the second limit in (3.8).
+By definition, f0(u) = L
+�
+u1
+|u1|+|u2|υ2 ,
+u2
+(|u1|+|u2|υ2)1/υ2
+�
+(|u1| + |u2|υ2)−1
+behaves as L(1, 0)|u1|−1 when |u1| → ∞, indeed, for any u2 ̸= 0
+|u2|υ2(|u1| + 1)f0(u1|u2|υ2, u2) = L
+� u1 + 1
+|u1| + 1, sgn(u2)
+|u1| + 1
+�
+→ L(1, 0),
+|u1| → ∞
+(3.10)
+by the conditions on L(·) in the theorem. Then, the last term on the r.h.s. of (3.9) can be rewritten as
+the sum of two terms: the first term L(1,0)
+log λ
+�
+|u1|≤(λγ/|u2|)υ2π(|u1| + 1)−1du1 → 2L(1, 0)γυ2 (∀u2 ̸= 0), and the
+second term
+1
+log λ
+�
+|u1|≤(λγ/|u2|)υ2π
+du1
+|u1| + 1
+�
+L
+� u1 + 1
+|u1| + 1, sgn(u2)
+|u1| + 1
+�
+− L(1, 0)
+�
+→ 0
+(∀ u2 ̸= 0)
+(3.11)
+in view of (3.10) and boundedness of L(·). The proof of the second limit in (3.8) follows from (3.9)-(3.11)
+and the DCT using the bound (log λ)−1�� �
+|u1|≤λγυ2π g+(u)du1
+�� ≤ C(log λ)−1 �
+|u1|≤λγυ2π(|u1| + |u2|υ2)−1du1 ≤
+C(1 + | log(|u2|)|) and the integrability of (1 + | log(|u2|)|)(1 + u2
+2)−1 on R.
+It remains to prove that R(1)
+λ,γ(x, y) − ˜Rλ,γ(x, y) → 0.
+W.l.g., take x = y = 1 and let R(1)
+λ,γ(1, 1) −
+˜Rλ,γ(1, 1) =: R′
+λ,γ. Then
+|R′
+λ,γ|
+≤
+1
+log λ
+�
+λγυ2Π×λγΠ
+���
+|D[λ](u1/λγυ2)D[λγ](u2/λγ)|2
+λ2+2γ
+− |1 − eiu2|2
+u2
+2
+���
+du
+|u1| + |u2|υ2
+=
+1
+log λ
+�
+λγυ2Π×λγΠ
+δλ(u)du
+(1 + u2
+2)(|u1| + |u2|υ2) = o(1)
+(3.12)
+9
+
+since δλ(u) → 0 uniformly in u in the last integral, see (4.6), and the r.h.s. of (3.12) with δλ(u) replaced by
+1 is bounded. This ends the proof of (3.8) with
+κ2
++ := 2L(1, 0)γυ2
+�
+R
+|(1 − eiu)/u|2du = 4πL(1, 0)γυ2
+(3.13)
+and completes the proof of (2.3) when γ > γ0.
+Case γ < γ0 = 1/υ2 (υ1 = 1). There are three subcases: (a) υ2 < 1, (b) υ2 > 1, and (c) υ2 = 1. Accordingly,
+V− = κ−B1/2,H−
+2 in subcase (a), V− = κ−BH−
+1 ,1 in subcase (b), and V− = κ−B1/2,1 in subcase (c). Subcases
+(a) and (b) do not involve logarithmic normalization and are essentially treated in [24, Thm.3.1]. Subcase
+(c) is symmetric to the above case γ > γX
+0 , υ1 = 1 with (x1, υ1) and (x2, υ2) exchanged, by noting that the
+latter discussion applies to any υ2 > 0 including υ2 = 1.
+Case γ = γ0 = 1/υ2 (υ1 = 1). In this case, V0 = V0,1/ρ is given in (2.12) and the result follows from [24] with
+small changes, thus ending the proof of Theorem 1.
+□
+Remark 1. The squared asymptotic constant in (3.2) for υ1 ̸= 1 takes the form
+κ2
++
+=
+�
+�
+�
+�
+�
+L(1, 0)
+�
+R2
+�2
+j=1
+��(1 − eiuj)/uj
+��2|u1|−υ1du,
+υ1 < 1,
+�
+R2
+��(1 − eiu2)/u2
+��2L(u)ρ(u)−1du,
+υ1 > 1,
+(3.14)
+(κ2
+− is defined symmetrically with u1, u2, υ1, υ2 exchanged by u2, u1, υ2, υ1). For υ1 < 1 the integral in (3.14)
+can be explicitly evaluated as κ2
++ = L(1, 0)(2π)2/Γ(2 + υ1) sin((1 + υ1)π/2), see (2.17).
+4
+Negative dependence
+This sec. describes anisotropic scaling limits in (1.2) of linear RFs satisfying Assumption (F)ND. Define
+H+
+1 := 1
+2(1 − (υ1 ∧ 1)),
+H−
+2 := 1
+2(1 − (υ2 ∧ 1)),
+γ0 := υ1 ∧ 1
+υ2 ∧ 1.
+(4.1)
+Note H+
+1 , H−
+2 ∈ [0, 1/2) and H+
+1 = 0 (respectively, H−
+2 = 0) is equivalent to υ1 ≥ 1 (respectively, υ2 ≥ 1).
+We also set H+
+2 = H−
+1 := 1/2.
+Theorem 2. Let X in (1.1) be a stationary linear RF on Z2 in (1.3) with spectral density f satisfying
+Assumption (F)ND. Then:
+• The scaling limits in (1.2) exist for any γ > 0 and satisfy (2.1) with γ0, H±
+i in (4.1) and the unbalanced
+limits given by
+V+
+:=
+κ+BH+
+1 ,1/2,
+V− := κ−B1/2,H−
+2 .
+(4.2)
+The asymptotic constants κ± > 0 are written in (4.9), (4.14), (4.19), (4.21), and (??) (or can be
+determined from these expressions by symmetry).
+10
+
+• The well-balanced limit is given by
+V0
+:=
+�
+�
+�
+�
+�
+V0,ρ,
+H+
+1 ∧ H−
+2 > 0,
+κ+BH+
+1 ,1/2 + κ−B1/2,H−
+2 ,
+H+
+1 ∧ H−
+2 = 0,
+(4.3)
+where BH+
+1 ,1/2 and B1/2,H−
+2 are independent and V0,ρ is defined in (2.12).
+• The normalization in (1.2) is given by dλ,γ := λH(γ)(log+ λ)1/2 in the cases γ ≥ γ0, H+
+1 = 0 or γ ≤
+γ0, H−
+2 = 0, and dλ,γ := λH(γ) in the remaining cases, with
+H(γ)
+:=
+�
+�
+�
+�
+�
+H+
+1 + (γ/2),
+γ ≥ γ0,
+(1/2) + γH−
+2 ,
+γ ≤ γ0.
+(4.4)
+• The RF X exhibits scaling transition at γ0.
+Proof.
+Similarly as in the proof of Theorem 1 we check the asymptotic gaussianity criterion in (2.4),
+reducing the proof to the convergence in (2.3) of covariance functions. Relation f(u) = (2π)2|�a(u)|2 and
+the assumptions on f imply that |�a(u)| ≤ C is bounded.
+Whence, with (γ1, γ2) := (1, γ), we get that
+sups∈Z2
+�� �
+t∈K[λx1,λγx2] a(t−s)
+�� ≤
+�
+Π2
+�2
+i=1
+��D[λγixi](ui)
+�� |�a(u)|du ≤ C
+�
+Π2
+�2
+i=1
+��D[λγixi](ui)
+�� du ≤ C(log λ)2
+so that (2.4) holds since H(γ) in (4.4) satisfy H(γ) > 0.
+The subsequent proof of (2.3) is split into parts I and II according to whether υi ̸= 1, i = 1, 2, or υi =
+1 (∃i = 1, 2). In turn, each part is split into several cases and subcases which are treated separately. Part II
+involves the logarithmic factor in the normalization and is more delicate.
+Part I. Case υ1 ∨ υ2 < 1, γ0 = υ1/υ2.
+Then H+
+1 = (1 − υ1)/2 ∈ (0, 1/2), H−
+2 = (1 − υ2)/2 ∈ (0, 1/2). By
+change of variables: u1 → u1/λ, u2 → u2/λγ we rewrite (2.2) as Rλ,γ(x, y) =
+�
+λΠ×λγΠ Gλ(u)du where
+Gλ(u)
+:=
+D[λx1](u1/λ)D[λy1](u1/λ)
+λ2
+· D[λγx2](u2/λγ)D[λγy2](u2/λγ)
+λ2γ
+· fλ(u),
+(4.5)
+fλ(u)
+:=
+f(u1/λ, u2/λγ)/λ2H(γ)−1−γ.
+By (2.9) and the definition of H(γ) in (4.4),
+λ−1D[λx1](u1/λ) → 1 − eix1u1
+−iu1
+,
+λ−γD[λγx2](u2/λγ) → 1 − eix2u2
+−iu2
+,
+(4.6)
+fλ(u) → gγ(u) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+f0(u),
+γ = γ0,
+f0(u1, 0),
+γ > γ0,
+f0(0, u2),
+γ < γ0,
+point-wise for any u ∈ R2
+0, where f0(u) := L(u)ρ(u), see (2.11). Particularly, f0(u1, 0) = |u1|−υ1L(1, 0),
+f0(0, u2) = |u2|−υ2L(0, 1), u = (u1, u2) ∈ R2
+0 since L(1, 0) = L(−1, 0), L(0, 1) = L(0, −1).
+We also see
+11
+
+from (2.16), (2.12) that the covariance function of the limit RF with appropriately chosen κ± writes as
+EVγ(x)Vγ(y) =
+�
+R2 G(u)du, where
+G(u) :=
+2
+�
+j=1
+�1 − eixjuj
+iuj
+��1 − e−iyjuj
+−iuj
+�
+gγ(u)
+(4.7)
+is the product of the corresponding limit functions in (4.6) and Gλ(u) → G(u), u ∈ R2
+0 according to (4.6).
+The proof of
+�
+R2 Gλ(u)du →
+�
+R2 G(u)du in all three cases γ > γ0, γ < γ0, and γ = γ0 now follows from the
+dominating convergence theorem (DCT) using the bound
+|λ−1D[λx]( u
+λ)| ≤ Cx(1 + |([λx]/λ)u|) ≤ C/(1 + |u|),
+|u| < λπ,
+for any fixed x ∈ R, x ̸= 0, implying
+|Gλ(u)| ≤ C �2
+i=1(1 + u2
+i )−1 ×
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρ(u),
+γ = γ0,
+ρ(u1, 0),
+γ > γ0,
+ρ(0, u2),
+γ < γ0.
+(4.8)
+Hence, the r.h.s. of (4.8) denoted by ¯G(u) is integrable:
+�
+R2 ¯G(u)du < ∞. The asymptotic constants κ± in
+(4.2) are determined by
+�
+R2 |G(u)|2du = 1/κ2
++(γ > γ0), = 1/κ2
+−(γ < γ0) for x = y = 1 in (4.7) and take a
+similar form as in (3.14); particularly,
+κ2
++
+:=
+L(1, 0)
+�
+R2 |u1|υ1
+2
+�
+j=1
+|(1 − eiuj)/uj|2du.
+(4.9)
+Case υ1 < 1 < υ2, γ0 = υ1.
+Subcase γ > υ1. We prove (2.3) with the same V+ = κ+BH+
+1 ,1/2 as in the previous case using a modified
+argument, as follows. Split f(u) = f1(u) + ˜f1(u), where f1(u) = f(u1, 0), ˜f1(u) := f(u) − f(u1, 0) and
+accordingly, Rλ,γ(x, y) = R1(x, y) + ˜R1(x, y). The convergence R1(x, y) → κ2
++EBH+
+1 ,1/2(x)BH+
+1 ,1/2(y) for
+any γ > 0 follows as in the case υ1 ∨ υ2 < 1 above. Whence, it suffices to prove ˜R1(x, x) → 0, or
+Jλ :=
+�
+Π2 | ˜f1(u)| |D[λx1](u1)D[λγx2](u2)|2du
+=
+o(λ1−υ1+γ).
+(4.10)
+We have | ˜f1(u)| ≤ C �3
+k=1 |h1,k(u)|, where h1,1(u) := ρ(u) − ρ(u1, 0) = |u2|υ2, h1,2(u) := ρ(u1, 0)(L(u) −
+L(u1, 0)) and h1,3(u) = ρ(u)δ(u), where δ(u) is a bounded function tending to 0 as |u| → 0. Accordingly,
+Jλ ≤ C �3
+k=1 Jλ,k where Jλ,k is defined as in (4.10) with | ˜f1(u)| replaced by |h1,k(u)|. Here,
+Jλ,1
+≤
+�
+Π2 |u2|υ2 |1 − ei[λx1]u1|2
+|u1|2|u2|2
+du ≤ λ
+�
+R×Π
+|1 − eiu1x1|2
+|u1|2
+|u2|υ2−2du ≤ Cλ = o(λ1−υ1+γ)
+(4.11)
+since γ > υ1. Next,
+Jλ,2
+≤
+�
+Π
+|u1|υ1 |1 − ei[λx1]u1|2
+|u1|2
+du1 ×
+�
+Π
+|L(u1, u2) − L(u1, 0)||1 − ei[λγx2]u2|2
+|u2|2
+du2
+(4.12)
+≤
+Cλ1−υ1+γ
+�
+R
+|u1|υ1 |1 − eiu1x1|2
+|u1|2
+du1 ×
+�
+R
+��L
+�u1
+λ , u2
+λγ
+�
+− L
+�u1
+λ , 0
+���|1 − eiu2x2|2
+|u2|2
+du2
+=
+o(λ1−υ1+γ)
+12
+
+by the DCT and the fact that
+��L
+� u1
+λ , u2
+λγ
+�
+− L
+� u1
+λ , 0
+��� is bounded and tends to 0 for any u ∈ R2
+0 by the
+continuity of ˜L in (2.8) and the fact that γ > υ1 > υ1/υ2. A similar argument also entails Jλ,3 = o(λ1−υ1+γ),
+proving (4.10) and the convergence (1.2) in the subcase γ > υ1.
+Subcase γ < υ1. We prove (2.3) with V− = κ−B1/2,0, d2
+λ,γ = λ and κ− in (4.14). Split f(u) = f2(u) +
+˜f2(u), ˜f2(u) := f(u) − f(0, u2) and accordingly, Rλ,γ(x, y) = R2(x, y) + ˜R2(x, y). It suffices to prove
+R2(x, y) → κ2
+−(x1 ∧ y1) × 1
+2(1 + I(x2 = y2))
+and
+˜R2(x, x) → 0.
+(4.13)
+To show the first relation in (4.13), note that
+R2(x, y)
+=
+1
+λ
+�
+Π
+D[λx1](u)D[λy1](u)du ×
+�
+Π
+f(0, v)D[λγx2](v)D[λγy2](v)dv =: J1 × J2,
+where
+J1
+→
+�
+R
+(1 − eix1u)(1 − e−iy1u)
+|u|2
+du = (x1 ∧ y1)
+�
+R
+|1 − eiu|2
+|u|2
+du
+follows by the DCT, and
+J2
+=
+�
+Π
+f(0, v)
+|1 − eiv|2
+�
+1 − ei[λγx2]v − e−i[λγy2]v + ei([λγx2]−[λγy2])v�
+dv
+→
+�
+Π
+f(0, v)
+|1 − eiv|2 dv ×
+�
+�
+�
+�
+�
+2,
+x2 = y2,
+1,
+x2 ̸= y2,
+by the Lebesgue-Riemann theorem [26, Thm.1.2] and the integrability of f(0, v)|1 − eiv|−2. Whence, the
+asymptotic constant in (4.13) equals
+κ2
+− := 2
+�
+R
+|(1 − eiu)/u|2du ×
+�
+Π
+f(0, v)|1 − eiv|−2dv.
+(4.14)
+To show the second relation in (4.13), similarly as in the subcase γ > υ1 above write | ˜f2(u)| ≤ C �3
+k=1 |h2,k(u)|
+and ˜R2(x, x) ≤ C �3
+k=1 ˜R3,k(x, x) accordingly, where h2,1(u) := ρ(u) − ρ(0, u2) = |u1|υ1, h2,2(u) :=
+ρ(0, u2)(L(u) − L(0, u2)) and h2,3(u) := ρ(u)δ(u), where δ(u) is a bounded function tending to 0 as |u| → 0.
+Then ˜R2,3(x, x) = o(1),
+| ˜R2,1(x, x)|
+≤
+Cλ−1
+�
+Π2 |u1|υ1 |1 − ei[λx1]u1|2|1 − ei[λγx2]u2|2
+|1 − eiu1|2|1 − eiu2|2
+du
+≤
+Cλγ−υ1
+�
+R
+|u1|υ1−2|1 − eix1u1|2du1 ×
+�
+R
+|u2|−2|1 − eix2u2|2du2
+=
+O(λγ−υ1) = o(1)
+(4.15)
+and
+| ˜R2,2(x, x)|
+≤
+C
+�
+Π
+|u2|υ2−2du2
+�
+R
+��L
+�u1
+λ , u2) − L(0, u2)
+��|(1 − eix1u1)/u1|2du1, = o(1),
+(4.16)
+proving (4.13).
+13
+
+Subcase γ = υ1. We prove (2.3) with V0 = κ+BH+
+1 ,1/2 + κ−B1/2,0 a sum of independent limits in subcases
+γ > υ1 and γ < υ1. Split f(u) = f1(u)+f2(u)+ ˜f12(u), ˜f12(u) := f(u)−f(u1, 0)−f(0, u2) and, accordingly,
+Rλ,γ(x, y) = R1(x, y) + R2(x, y) + ˜R12(x, y). Note H+
+1 + (υ1/2) = 1/2. The convergences R1(x, y) →
+κ2
++EBH+
+1 ,1/2(x, y)BH+
+1 ,1/2(x, y) and R2(x, y) → κ2
+−EB1/2,0(x, y)B1/2,0(x, y) were proved in subcases γ >
+γ0, υi < 1, i = 1, 2 and γ < γ0, υ1 < 1 < υ2, respectively. It remains to show
+˜R12(x, x) → 0.
+(4.17)
+Write f(u) = f0(u)T(u), where f0(u) = ρ(u)L(u) and T(u) := f(u)
+f0(u), u ∈ Π2 are a continuous functions, see
+Assumption (F)ND. Then, ˜f12(u) = �2
+i=1 ˜f12,i(u), where
+˜f12,1(u) := |u1|υ1( �T(u) − �T(u1, 0)),
+˜f12,2(u) := |u2|υ2( �T(u) − �T(0, u2)),
+�T(u) := L(u)T(u).
+Accordingly, ˜R12(x, x) = �2
+i=1 ˜R12,i(x, x). Then
+˜R12,1(x, x)
+≤
+�
+R2 |u1|υ1δλ,1(u)
+2
+�
+j=1
+|(1 − eiujxj)/uj|2du,
+(4.18)
+where δλ,1(u) :=
+�� �T
+� u1
+λ , u2
+λυ1
+�
+− �T
+� u1
+λ , 0
+���I(u ∈ λΠ × λυ1Π) is bounded uniformly in λ > 1 and the integral on
+the r.h.s. with δλ,1(u) replaced by 1 converges due to υ1 < 1. Also note that δλ,1(u) → 0 as T(u), u ∈ Π2
+is continuous and L
+� u1
+λ , u2
+λυ1
+�
+− L
+� u1
+λ , 0
+�
+→ 0 due to υ2 > 1. This proves (4.17) for ˜R12,1(x, x) instead of
+˜R12(x, x). The proof of ˜R12,2(x, x) → 0 is similar to (4.16), with
+��L
+� u1
+λ , u2) − L(0, u2)
+�� there replaced by
+δλ,2(u) :=
+�� �T
+� u1
+λ , u2) − �T(0, u2)
+�� → 0 in view of the above mentioned properties of L(u) and T(u).
+Case υi > 1, i = 1, 2, γ0 = 1.
+By symmetry, it suffices to consider the case γ ≤ 1. Split f(u) = f1(u) +
+f2(u) + ˜f12(u), Rλ,γ(x, y) = R1(x, y) + R2(x, y) + ˜R12(x, y) as in the previous case. Then R2(x, y) →
+κ2
+−EB1/2,0(x)B1/2,0(y) as in (4.13) while R1(x, x) ≤ Cλγ−1 �
+Π |u1|υ1−2du1 ×
+�
+R |(1 − eiu2x2)/u2|2du2 =
+O(λγ−1) = o(1) is negligible when γ < 1; for γ = 1 we have R1(x, y) → κ2
++EB0,1/2(x)B0,1/2(y) analogously
+to (4.13) with
+κ2
++ := 2
+�
+R
+|(1 − eiu)/u|2du ×
+�
+Π
+f(v, 0)|1 − eiv|−2dv.
+(4.19)
+The proof of ˜R12(x, x) → 0 for γ ≤ 1 follows as in (4.16). This ends the proof of Part I.
+Part II. Case υ1 = 1, υ2 < 1, γ0 = 1/υ2.
+Let first γ > 1/υ2, d2
+λ,γ = λγ log+ λ. Split f(u) = f1(u) + ˜f1(u)
+and Rλ,γ(x, y) = R1(x, y) + ˜R1(x, y) as in the case υ1 < 1 < υ2 above. Then
+R1(x, y)
+∼
+1
+log λ
+�
+λΠ
+(1 − eix1u1)(1 − e−iy1u1)
+|u1|2
+λf(u1/λ, 0)du1
+×
+�
+λγΠ
+(1 − eix2u2)(1 − e−iy2u2)
+|u2|2
+du2 =: J1 × J2,
+(4.20)
+where J2 → κ2(x2 ∧y2), κ2 :=
+�
+R |(1−eiu)/u|2du and we need to show the limit of J1. We have λf(u1/λ, 0) =
+λL(1, 0)|u1/λ|(1+δ(u1/λ)) = L(1, 0)|u1|(1+δ(u1/λ)) where δ(u) is a bounded function tending to 0 as u → 0.
+14
+
+Therefore, J1 = J′
+1 + J′′
+1 , where
+J′
+1 := 2L(1, 0)
+log λ Re
+� λπ
+0
+(1 − eix1u)(1 − e−iy1u)u−1du → 2L(1, 0)
+�
+�
+�
+�
+�
+2,
+x1 = y1,
+1,
+x1 ̸= y1
+as limλ→∞
+� λ
+1 eixuu−1du exists for any x ̸= 0, and |J′′
+1 | ≤ C(log λ)−1‘
+� λπ
+0 (u ∧ 1)2u−1δ(u/λ)du → 0 fol-
+lows as in (3.7) by splitting the last integral over sets u/λ < ϵ and ϵ ≤ u/λ ≤ π. Whence, R1(x, y) →
+κ2
++EB0,1/2(x)B0,1/2(y), where
+κ2
++ = 4L(1, 0)κ2 = 4L(1, 0)
+�
+R
+|(1 − eiu)/u|2du.
+(4.21)
+To show that ˜R1(x, x) is negligible, use the decomposition of ˜f1(u) following (4.10) so that | ˜R1(x, x)| ≤
+C �3
+k=1 ˜R1,k(x, x) where ˜R1,k(x, x) = Jλ,k/ log λ and Jλ,k are as in the aforementioned proof. Then ˜R1,1(x, x) =
+o(1) as in (4.11) while ˜R1,2(x, x) ≤ C
+�
+R(1 + u2
+2)−1δλ(u2)du2, c.f. (4.12), with
+δλ(u2) :=
+1
+log λ
+�
+|u1|≤λπ
+(1 + |u1|)−1��L
+�u1
+λ , u2
+λγ
+�
+− L
+�u1
+λ , 0
+���du1
+bounded and tending to 0 as λ → ∞ for any fixed u2 ̸= 0 by continuity of ˜L due to γυ2 > 1. Finally,
+˜R1,3(x, x) ≤ C(log λ)−1 �
+λΠ(1 + |u1|)−1du1
+�
+R δ(u1/λ, u2/λγ)(1 + u2
+2)−1du2 + o(1) = o(1) follows by splitting
+the last integral over |u1| < ϵλ and |u1| > ϵλ, using
+� πλ
+ϵλ u−1
+1 du1 = log(1/ϵ) < ∞ for any small ϵ > 0.
+Next, let γ < 1/υ2, d2
+λ,γ = λ2H(γ), 2H(γ) = 1 + γ(1 − υ2).
+Note λγυ2f(u1/λ, u2/λγ) → f0(0, u2) =
+L(0, 1)|u2|υ2. Then Rλ,γ(x, y) → κ2
+−EB1/2,H−
+2 (x)B1/2,H−
+2 (y) follows as in the case υ1 ∧ υ2 < 1, γ < γ0, with
+κ2
+− = L(0, 1)
+�
+R2 |u2|υ2 �2
+j=1 |(1 − eiuj)/uj|2du, c.f. (4.9).
+Let γ = 1/υ2, then d2
+λ,γ = λ2H(γ) log λ, 2H(γ) =
+1
+υ2 . We have
+Rλ,γ(x, y)
+∼
+1
+log λ
+�
+λΠ×λ1/υ2Π
+2
+�
+i=1
+(1 − eixiui)(1 − e−iyiui)
+|ui|2
+λf(u1/λ, u2/λ1/υ2)du
+(4.22)
+where λf(u1/λ, u2/λ1/υ2) = f0(u)(1+δ(u1/λ, u2/λ1/υ2)) = �3
+k=1 fk(u), f1(u) := L(u)|u1|, f2(u) := L(u)|u2|υ2,
+f3(u) := f0(u)δ(u1/λ, u2/λ1/υ2), and limu→0 δ(u) = 0. Accordingly, Rλ,γ(x, y) ∼ �3
+k=1 Rk(x, y); we will
+show that R1(x, y) is the main term and Rk(x, y) → 0, k = 2, 3. Indeed,
+R1(x, y)
+=
+1
+log λ
+� λπ
+−λπ
+(1 − eix1u)(1 − e−iy1u)
+|u|
+hλ(u)du,
+where hλ(u) :=
+�
+|w|≤λ1/υ2π(1 − eix2w)(1 − e−iy2w)|w|−2L(u, w)dw → h(u) (λ → ∞) and where
+h(u)
+:=
+�
+R
+(1 − eix2w)(1 − e−iy2w)
+|w|2
+L(u, w)dw
+(4.23)
+→
+L(1, 0)
+�
+R
+(1 − eix2w)(1 − e−iy2w)
+|w|2
+dw = L(1, 0)κ2(x2 ∧ y2)
+as |u| → ∞, see (2.16), (2.17) for the last equality. Whence, the convergence
+R′
+1(x, y) :=
+1
+log λ
+� λπ
+−λπ
+(1 − eix1u)(1 − e−iy1u)
+|u|
+h(u)du → κ2
++EB0,1/2(x)B0,1/2(y)
+15
+
+with κ2
++ in (4.21) follows as in (4.20) and the same limit for R1(x, y) requires few changes. Next, |R2(x, x)| ≤
+C(log λ)−1 �
+R2(1+u2
+1)−1(1+u2
+2)−1|u2|υ2du = O(1/ log λ). Finally, |R3(x, x)| ≤ C(log λ)−1� � λπ
+0 (1+u)−1 �
+R(1+
+w2)−1δ(u/λ, w/λ1/υ2)dw + O(1)
+�
+→ 0 follows as in the case γ > 1/υ2.
+Case υ1 = 1, υ2 > 1, γ0 = 1. Let γ ≥ 1, d2
+λ,γ = λγ log+ λ. Then
+Rλ,γ(x, y)
+∼
+1
+log λ
+�
+λΠ×λγΠ
+2
+�
+i=1
+(1 − eixiui)(1 − e−iyiui)
+|ui|2
+λf(u1/λ, u2/λγ)du
+where λf(u1/λ, u2/λγ) → f0(u1, 0) = L(1, 0)|u| due to γυ2 > 1. Then Rλ,γ(x, y) → κ2
++EB0,1/2(x)B0,1/2(y)
+as in (4.20) with κ2
++ in (4.21). Next, let γ < 1. Then dλ,γ = λ1/2 and
+Rλ,γ(x, y)
+∼
+� πλ
+−πλ
+(1 − eix1u)(1 − e−iy1u)
+|u|2
+du ×
+�
+Π
+(1 − eiλγx2w)(1 − e−iλγy2w)
+|1 − eiv|2
+f(u/λ, w)dw
+∼
+κ2
+− EB1/2,0(x)B1/2,0(y)
+as in (4.13) with κ2
+− in (4.14).
+Case υ1 = υ2 = γ0 = 1.
+The convergence in (2.3) for γ ̸= 1 leading to limits V+ = κ+B0,1/2 and V− =
+κ−B1/2,0 by symmetry follow as in the case υ1 = 1, υ2 > 1, with with κ2
++ in (4.21) and κ2
+− = 4L(0, 1)κ2. Let
+us prove that for γ = 1 (2.3) tends to the sum of the latter limits leading to
+V0
+=
+κ+B0,1/2 + κ−B1/2,0,
+with
+d2
+λ,γ = λ log+ λ,
+where B0,1/2 and B1/2,0 are mutually independent. Proceeding as in (4.22) we see that fλ(u) := λf(u1/λ, u2/λ)
+→ f0(u) = L(u)(|u1| + |u2|) and Rλ,γ(x, y) behaves asymptotically as the sum of two terms
+Rk(x, y)
+:=
+1
+log λ
+�
+(λΠ)2 L(u1, u2)|uk|
+2
+�
+j=1
+(1 − eixjuj)(1 − e−iyjuj)
+|uj|2
+du,
+k = 1, 2
+which tend to κ2
++EB0,1/2(x)B0,1/2(y) and κ2
+−EB1/2,0(x)B1/2,0(y), respectively. We also need to check that
+the term R3(x, y) corresponding to fλ(u)−f0(u) is negligible, viz., |R3(x, y)| ≤ C(log λ)−1 �
+(λΠ)2(|u1|+|u2|)
+δ(u1/λ, u2/λ) �2
+j=1(1 + |uj|2)−1du → 0. We omit these details being similar to (4.22). This ends the proof
+of Part II and Theorem 2, too.
+□
+Remark 2. Following the terminology in time series [13], the asymptotic constants κ2
+± may be dubbed
+‘long-range variances’. It is notable that the only case when κ2
+± depend on the spectral density outside of
+the origin are (4.14), (4.19) corresponding to H±
+i
+= 0 or spectrum ND under ‘edge effects’, see Remark 3.
+Obviously, these expressions for κ2
+± make sense for continuous f (the continuity can be relaxed) but not for
+an arbitrary integrable or bounded f. On the other hand, continuity of f is a consequence of summability of
+covariance function hence occurs under covariance SRD and ND.
+5
+Long-range negative and hyperbolic dependence
+Recall the asymptotic form of the spectral density under Assumption (F)LRND,2:
+f(u) ∼ f0(u) =
+L(u)
+|u1|υ1 + |u2|υ2
+|u| → 0,
+(5.1)
+16
+
+where L(u) = L(−u), u ∈ R2
++ is a continuous generalized invariant function such that
+L(u) ∼ ℓ
+�
+|u2|
+ρ(u)1/υ2
+�µ
+(u2 → 0)
+(5.2)
+for some µ ∈ (0, 1), ℓ > 0. Define
+H+
+1
+:=
+1
+2
+�
+1 +
+�
+(υ1 + µυ1
+υ2
+) ∧ 1
+��
+,
+H+
+2 := 1
+2
+�
+1 −
+�
+µ ∧ (υ2
+υ1
+− υ2)
+��
+,
+(5.3)
+H−
+1
+:=
+1
+2
+�
+1 + υ1 −
+υ1
+υ2 ∨ 1
+�
+,
+H−
+2 := 1
+2
+�
+1 + (υ2 ∧ 1)
+�
+Note H−
+i , i = 1, 2 in (5.3) are the same as in Theorem 1, (3.1), whereas the expressions for H+
+i , i = 1, 2 in (5.3)
+and (3.1) agree if and only if µ = 0. Also note that H±
+1 , H−
+2 ∈ [1/2, 1] while H+
+2 = 1
+2(1+υ2 − υ2
+υ1 ) ∈ [1/2, 1] for
+υ1 ≥ 1; for υ1 ≤ 1 we have H+
+2 ∈ (0, 1/2]. Finally, H(γ) in (5.5) is a continuous function of γ, υi, i = 1, 2, µ,
+its value H(γ0) = 1
+2(1 + υ1 + υ1
+υ2 ) at γ = γ0 := υ1
+υ2 being the same as in Theorem 1 independently of µ.
+The following theorem excludes some particular cases of parameters µ, υi, i = 1, 2 which may require
+extra logarithmic normalizing factor. It also leaves open the question about the scaling limits when both
+Assumptions (F)LRND,1 and (F)LRND,2 are satisfied.
+Theorem 3. Let X in (1.1) be a stationary linear RF on Z2 in (1.3) with spectral density f satisfying
+Assumption (F)LRND,2. In addition, let υ1 ̸= 1, υ1 + υ2
+υ1 ̸= 1. Then:
+• The scaling limits in (1.2) exist for any γ > 0 and satisfy (2.1) with γ0 = υ1
+υ2 and the unbalanced limits
+are given by
+V+
+:=
+κ+BH+
+1 ,H+
+2 ,
+V− := κ−BH−
+1 ,H−
+2 ,
+(5.4)
+where H±
+i , i = 1, 2 as in (5.3). The asymptotic constant κ+ is defined in (5.6), (5.7), whereas κ− is the
+same as in Theorem 1.
+• The well-balanced limit V0 := V0,1/ρ is given in (2.12).
+• The normalization in (1.2) is given by dλ,γ := λH(γ) with
+H(γ)
+:=
+�
+�
+�
+�
+�
+H+
+1 + γH+
+2 ,
+γ ≥ γ0,
+H−
+1 + γH−
+2 ,
+γ ≤ γ0.
+(5.5)
+• The RF X exhibits scaling transition at γ0 = υ1
+υ2 .
+Proof. Since (5.5) satisfy H(γ) > 1/2 (∀ γ > 0), the Lindeberg criterion in (2.4) is satisfied as in Theorem
+1. For γ ≤ γ0 the results of Theorem 3 and their proof completely agree with those of Theorem 1. The
+subsequent proof on (2.3) is limited to γ > γ0 and split into three cases as follows.
+Case υ1 + υ1
+υ2 < 1. According to (5.1)-(5.2) for γ > υ1/υ2 we have that
+f(u1/λ, u2/λγ)
+∼
+λυ1
+|u1|υ1 × ℓ
+���
+u2
+λγ
+| u1
+λ |υ1/υ2
+���
+µ
+= ℓλ2H(γ)−1−γ
+2
+�
+j=1
+|uj|1−2H+
+j
+17
+
+point-wise in u = (u1, u2) ∈ R2
+0. Then, the limit in (2.3) with Vγ = V+ = κ+BH+
+1 ,H+
+2 follows similarly as in
+Theorem 1 with
+κ2
++
+=
+ℓ
+2
+�
+j=1
+�
+R
+|(1 − eiu)/u|2|u|1−2H+
+j du.
+(5.6)
+Case υ1 < 1 < υ1 + υ1
+υ2 . Then (2H+
+1 , 2H+
+2 ) = (1+υ1+ µυ1
+υ2 , 1−µ) if µ < υ2
+υ1 −υ2, = (2, 1+υ2− υ2
+υ1 ) if µ > υ2
+υ1 −υ2.
+The proof of the limit V+ = κ+BH+
+1 ,H+
+2 for µ < υ2
+υ1 − υ2 is the same as in the case υ1 + υ1
+υ2 < 1 above, and is
+omitted. Next, let µ > υ2
+υ1 − υ2. Due to γ > υ1/υ2 we see that
+λ−γυ2f
+�
+u1
+λγυ2/υ1 , u2
+λγ
+�
+→ f0(u),
+λ−1D[λx]
+�
+u1
+λγυ2/υ1
+�
+→ x,
+λ−γD[λγx]
+�u2
+λγ
+�
+→ 1 − eixu2
+−iu2
+point-wise for any ui ̸= 0, i = 1, 2, x > 0. We also have
+�
+R g+(u1, u2)du1 = |u2|1−2H+
+2 �
+R f0(u, 1)du where the
+last integral converges due to (5.2) and µ > υ2
+υ1 − υ2. Then, V+ = κ+B1,H+
+2 follows as in [24, Thm.3.1]. The
+asymptotic constant κ+ in both cases of µ is given by
+κ2
++
+=
+�
+�
+�
+�
+�
+ℓ �2
+j=1
+�
+R |(1 − eiu)/u|2|u|1−2H+
+j du,
+µ < υ2
+υ1 − υ2,
+�
+R
+��(1 − eiv)/v
+��2|v|1−2H+
+2 dv ×
+�
+R g+(u, 1)du,
+µ > υ2
+υ1 − υ2.
+(5.7)
+Case υ1 > 1. In this case, the results do not depend on µ and completely agree with those in Theorem 1,
+including the proof. This ends the proof of Theorem 3.
+□
+Next, we formalize the meaning of hyperbolic dependence mentioned in the Introduction. Let
+f(u) = f0,hyp(u)(1 + o(1)),
+|u| → 0,
+where
+f0,hyp(u) := L(u)
+2
+�
+i=1
+|ui|−υi,
+u ∈ Π2,
+(5.8)
+|υi| < 1, i = 1, 2 and
+L(u) := ˜L
+�
+u1
+(|u1||υ1| + |u2||υ2|)1/|υ1| ,
+u2
+(|u1||υ1| + |u2||υ2|)1/|υ2|
+�
+(5.9)
+is a generalized invariant function corresponding to generalized homogeneous function ρ(u) = |u1||υ1| +
+|u2||υ2|, u = (u1, u2) ∈ R2
+0. Let
+Hi := 1
+2(1 + υi),
+i = 1, 2.
+(5.10)
+The class in (5.8) includes (separately) fractionally integrated spectral densities �2
+i=1 |1 − eiui|−υi which play
+an important role in spatial statistics [5, 14, 17]. If L(u) is separated from 0 and ∞, the spectral density in
+(5.8) satisfies (spectrum) LRD, ND or LRND properties depending on the sign of υi, i = 1, 2 except that it
+explodes/vanishes on the coordinate axis ui = 0 when υi ̸= 0 as well and represents a different class from
+those discussed in Theorems 1-3 Introduce a Gaussian RF
+V0,hyp(x)
+:=
+�
+R2
+2
+�
+j=1
+1 − eiujxj
+iuj
+�
+f0,hyp(u)Z(du),
+x ∈ R2
++,
+(5.11)
+where Z(du) is the same white noise as in (2.12). In the case when L(u) = ℓ > 0 is a constant function, the
+RF V0,hyp is a multiple of FBS: V0,hyp = κℓBH1,H2 with Hi, i = 1, 2 given in (5.10) and κ > 0 as in (2.17).
+18
+
+Theorem 4. be a stationary linear RF on Z2 in (1.3) with spectral density f in (5.8)-(5.9), where υi ∈
+(−1, 1), i = 1, 2 and ˜L(x) is a strictly positive continuous function. Then:
+• The scaling limits in (1.2) exist for any γ > 0 and satisfy
+Vγ =
+�
+�
+�
+�
+�
+κγBH1,H2,
+γ ̸= |υ1|
+|υ2|,
+V0,hyp,
+γ = |υ1|
+|υ2|,
+(5.12)
+where Hi, i = 1, 2 as in (5.10) and
+κ2
+γ :=
+2
+�
+j=1
+�
+R
+|(1 − eiu)/u|2|u|1−2Hjdu ×
+�
+�
+�
+�
+�
+L(1, 0),
+γ > |υ1|
+|υ2|,
+L(0, 1),
+γ < |υ1|
+|υ2|.
+(5.13)
+• The normalization in (1.2) is given by dλ,γ := λH(γ), H(γ) := H1 + γH2.
+• The RF X does not exhibit scaling transition.
+We omit the proof of Theorem 4 since it resembles [24] and the previous proofs. The gaussianity criterion
+(2.4) holds for sign(υ1) = sign(υ2) as in Theorems 1 and 2; for υ1υ2 < 0 from (5.8) we get |�a(u)| ≤
+C �2
+i=1 |ui|−υi/2 and (2.4) follows similarly.
+The absence of scaling transition at γ =
+|υ1|
+|υ2| is clear from
+(5.12)-(5.13) and the fact that L(1, 0) > 0, L(0, 1) > 0 according to the positivity assumption on L (the last
+conclusion may fail if L(1, 0) and/or L(0, 1) vanish).
+We end the paper with two remarks on possible extensions of this work.
+Remark 3. More general scaling schemes and ‘edge effects’. In a more abstract setting, a RF is often indexed
+by test functions, through which scaling operations are defined; see [8, 9, 11]. Accordingly, one can study
+anisotropic scaling limits of integrals
+Sλ,γ(φ) :=
+�
+R2 φ(λ−Γt)X(⌈t⌉)dt
+(5.14)
+involving X in (1.3) extended to R2; ⌈t⌉ := (⌈t1⌉, ⌈t2⌉), t = (t1, t2) ∈ R2, and a re-scaled function φ : R2 → R,
+λ−Γt := (λ−1t1, λ−γt2), Γ := diag(1, γ), for a given γ > 0 and each φ from a class Φ of (test) functions. The
+sum in (1.1) corresponds to (5.14) with indicator function φ(t) = I(t ∈]0, x]). For (5.14), the scaling limit
+d−1
+λ,γ(Sλ,γ(φ) − ESλ,γ(φ))
+d
+−→ Vγ(φ) is a RF indexed by φ ∈ Φ. Isotropic (γ = 1) scaling limits in spirit of
+(5.14) for different classes of RF on Rd were studied in [9, 3, 15] and other papers. Extending Theorems 1-4
+to scaling limits of integral functionals in (5.14) for suitable class Φ of test functions which include indicator
+functions φ(t) = I(t ∈ A) of general bounded sets A ⊂ R2 is an interesting problem. Of particular interest
+is the case of ND RF X, where ‘edge effects’ can be expected following [16, 28], leading to scaling limits Vγ
+‘living’ on the boundary of A or the discontinuity set of φ.
+Remark 4. Incongruous scaling and dependence axis. For linear LRD RF X in (1.3) [20] define the de-
+pendence axis of X as the direction on the plane along which a(t) decay at the smallest rate. In spectral
+19
+
+terms, we may define the dependence axis as a direction along which the spectral density f(x) grows at the
+fastest rate when |x| → 0. Under Assumption (F)LRD with υ1 ̸= υ2 such dependence axis coincides with
+one of the coordinate axes. The scaling in (1.2) involves rectangles with sides parallel to the coordinate axis.
+Using the terminology in [20] we may say that the scaling in (1.2) and in Theorem 1 is congruous with the
+dependence axis of X. [20] showed that incongruous scaling of linear LRD RF in (1.3) may dramatically
+change the scaling limits in (1.2) and the scaling transition point γ0. We expect that the spectrum approach
+in our paper may lead to a comprehensive treatment of incongruous scaling limits under various dependence
+assumptions.
+Acknowledgements
+The author thanks Vytaut˙e Pilipauskait˙e for useful comments and Remigijus Lapinskas for help with Figure
+1 graphs.
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+21
+

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+Folded Optimization for End-to-End Model-Based Learning
+James Kotary , My H. Dinh and Ferdinando Fioretto
+Syracuse University
+{jkotary, mydinh, ffiorett}@syr.edu
+Abstract
+The integration of constrained optimization models
+as components in deep networks has led to promis-
+ing advances in both these domains. A primary
+challenge in this setting is backpropagation through
+the optimization mapping, which typically lacks
+a closed form. A common approach is unrolling,
+which relies on automatic differentiation through
+the operations of an iterative solver. While flexi-
+ble and general, unrolling can encounter accuracy
+and efficiency issues in practice. These issues can
+be avoided by differentiating the optimization map-
+ping analytically, but current frameworks impose
+rigid requirements on the optimization problem’s
+form. This paper provides theoretical insights into
+the backpropagation of unrolled optimizers, which
+lead to a system for generating equivalent but effi-
+ciently solvable analytical models. Additionally, it
+proposes a unifying view of unrolling and analyti-
+cal differentiation through constrained optimization
+mappings. Experiments over various structured pre-
+diction and decision-focused learning tasks illustrate
+the potential of the approach both computationally
+and in terms of enhanced expressiveness.
+1
+Introduction
+The integration of optimization mappings in deep networks
+has shown to be an effective framework for enforcing certain
+types of structures in deep models. Here, an optimization prob-
+lem is viewed as a mapping from some undefined parameters
+to a resulting optimal solution. Outputs of the mapping are
+guaranteed to obey the problem’s constraints, which can be
+either predefined or learned [Kotary et al., 2021].
+These approaches are known to increase accuracy and effi-
+ciency on specialized learning tasks by imparting task-specific
+structural knowledge. For example, they have been used
+to design efficient multi-label classifiers based on sparsity-
+enforcing mechanisms [Martins and Astudillo, 2016], learn-
+ing to rank tasks by exploiting simple optimization models
+[Adams and Zemel, 2011], and even enhance decision-focused
+pipelines by the integration of predictive components with op-
+erational decision models [Wilder et al., 2019a].
+While, in general, constrained optimization mappings can
+be used as components in neural networks in a similar man-
+ner to linear layers or activation functions [Amos and Kolter,
+2017], networks with optimization layers require end-to-end
+training by stochastic gradient descent, which in turn requires
+differentiation of the optimization mappings for backpropaga-
+tion of gradients.
+This poses unique challenges, partly due to their lack of
+a closed form, and modern approaches typically follow one
+of two strategies. In unrolling, an optimization algorithm is
+executed entirely on the computational graph, and backpropa-
+gated by automatic differentiation from optimal solutions to
+the underlying problem parameters. The approach is adaptable
+to many problem classes, but has been shown to suffer from
+time and space inefficiency, as well as vanishing gradients
+[Monga et al., 2021]. Analytical differentiation is a second
+strategy that circumvents those issues by forming analytical
+models for the derivatives of an optimization mapping and
+solving them exactly. However, current frameworks have rigid
+requirements on the form of the optimization problems, re-
+lying on transformations to canonical convex cone programs
+before applying a standardized procedure for their solution and
+differentiation, based on cone programming [Agrawal et al.,
+2019a]. This system precludes the use of specialized solvers
+that are best-suited to handle various optimization problems,
+and inherently restricts itself only to convex problems.1
+Contributions.
+To address these limitations, this paper pro-
+poses a novel analysis of unrolled optimization, which results
+in closed-form models for the backpropagation of an unrolled
+optimizer. Theoretically, the result is significant because it
+establishes an equivalence between unrolling and analytical
+differentiation, and allows for convergence of the backward
+pass to be analyzed in unrolling. Practically, it allows for
+the forward and backward passes of unrolled optimization
+to be disentangled and solved separately, using blackbox im-
+plementations of specialized algorithms. More specifically,
+the paper makes the following novel contributions: (1) A
+theoretical analysis of unrolling that leads to an efficiently
+solvable closed-form model, whose solution is equivalent to
+the backward pass of an unrolled optimizer. (2) Building on
+this analysis, it proposes a system for generating analytically
+1A discussion of related work on differentiable optimization and
+decision-focused learning is provided in Appendix A.
+arXiv:2301.12047v1  [cs.LG]  28 Jan 2023
+
+differentiable optimizers from unrolled implementations, ac-
+companied by a Python library called fold-opt to facilitate
+automation. (3) Its effectiveness is evaluated on a diverse set
+of end-to-end optimization and learning tasks, demonstrat-
+ing efficiency and modeling advantages compared to current
+differentiable optimization approaches. Importantly, we re-
+port the first demonstration of decision-focused learning with
+nonconvex decision models.
+2
+Setting and Goals
+In this paper, the goal is to differentiate mappings that are
+defined as the solution to an optimization problem. Consider
+the parameterized problem (1) which defines an optimization
+mapping from a vector of parameters c ∈ Rp to its associated
+optimal solution x⋆(c) ∈ Rn:
+x⋆(c) = argmin
+x
+f(x, c)
+(1a)
+subject to: g(x, c) ≤ 0,
+(1b)
+h(x, c) = 0,
+(1c)
+in which f is the objective function, and g and h are vector-
+valued functions capturing the inequality and equality con-
+straints of the problem, respectively. It is assumed throughout
+that for any c, the associated optimal solution x⋆(c) can be
+found by conventional solution methods, within some toler-
+ance in solver error. This coincides with the “forward pass” of
+the mapping in a neural network. The primary challenge is
+to compute its backward pass, which amounts to finding the
+Jacobian matrix of partial derivatives ∂x⋆(c)
+∂c
+.
+The parameters c can be thought of as a prediction from
+previous layers of a neural network, or as learnable parameters
+of the problem (1), analogous to the weights of a linear layer,
+or as some combination of both. In any case, ∂x⋆(c)
+∂c
+is required
+to backpropagate gradients from a downstream loss function
+to the underlying parameters of the machine learning model.
+Backpropagation.
+Backpropagation is the calculation of
+gradients from a downstream loss function L to the param-
+eters of a learning model. The primary goal of this paper
+is backpropagation through the function x⋆(c), as defined
+in Problem (1). The Jacobian matrix of the vector-valued
+function x⋆(c) : Rp → Rn is a matrix ∂x⋆
+∂c in Rn×p, whose
+elements at (i, j) are the partial derivatives ∂x⋆
+i (c)
+∂cj . Backpropa-
+gation through x⋆(c) amounts to computing ∂L
+∂c given ∂x⋆(c)
+∂c
+,
+and can be performed by finding the Jacobian-vector product
+∂L
+∂c = ∂L
+∂x⋆ · ∂x⋆(c)
+∂c
+.
+(2)
+In deep learning, backpropagation is typically accomplished
+through automatic differentiation (AD), which propagates gra-
+dients from the loss function through the arithmetic operations
+and low-level functions of a deep model by repeatedly apply-
+ing the multivariate chain rule. This requires a record of all
+the operations performed during the forward pass and their
+dependencies, known as the computational graph.
+z
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+x?(c)
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+L(x?)
+<latexit sha1_base64="VNMivXQp7mx3+oNvqOGvP/v5aNA=">AC63icfVHNbtNAEN6Yv2L+UpC4cLGIkApCkdeJ03CrBAcOIrUtJViE63Xk3TV3bW1u4ZGi5+CG+KExAlegQfh
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+unrolling
+solver
+features
+DNN 
+prediction
+unfolding
+fixed-point folding
+all operations on the computation graph
+segments of the computation graph 
+replaced with precomputed optimization 
+steps and derivatives 
+blackbox forward pass and
+backward pass, repeated
+Figure 1: Compared to unrolling, unfolding requires fewer oper-
+ations on the computational graph by replacing inner loops with
+Jacobian-vector products. Fixed-point folding models the unfolding
+analytically, allowing for blackbox optimization implementations.
+3
+From Unrolling to Unfolding
+By unrolling, which performs each arithmetic operation of an
+optimization algorithm on the computational graph, optimiza-
+tions can be backpropagated without explicitly representing
+their Jacobians. However, these graphs can become very large
+after many iterations. The strategy taken in this paper for back-
+propagation through the function x⋆(c), as in unrolling, is to
+backpropagate through the steps of an optimizer which solves
+Problem (1). However, we begin by proposing a variation
+on unrolling in which the optimizer steps are grouped into
+subroutines that can be differentiated analytically.
+This paper considers iterative optimization algorithms,
+which refine an initial starting point x0 by repeated appli-
+cation of an update routine, which we view as a function.
+For optimization variables x ∈ Rn, the update function is a
+vector-valued function U : Rn → Rn:
+xk+1(c) = U(xk(c), c).
+(U)
+and the iteration (U) converges if xk(c) → x⋆(c) as k → ∞.
+Evaluating the update function U may also require the use
+of optimization algorithms that are difficult to differentiate.
+In such cases, a full unrolling of (U) would involve unrolling
+each inner algorithm used in the evaluation of U. However, if
+the Jacobians of U can be modeled analytically, it is possible to
+avoid unrolling each evaluation of U by modeling its backward
+pass with a Jacobian-vector product instead (see Equation (2)).
+Then, only the outer iterations (U) need to be unrolled.
+This type of partial unrolling, which allows for backpropa-
+gating large segments of computation at a time by leveraging
+analytically differentiated subroutines, is referred to as unfold-
+ing. It is made possible by the fact that U is often easier to
+differentiate than the overall optimization mapping x⋆(c).
+There are two main advantages of unfolding over unrolling:
+(1) Analytical differentiation of U allows for removal of the
+inner loops of unrolling, greatly reducing the total number
+of unrolled operations as depicted in Figure 1, which shows
+forward pass steps in red with their corresponding backward
+passes in blue. More importantly, (2) unfolded optimizers can
+be converted to analytically differentiated ones by the method
+proposed in the next sections. Thus, unfolding will be an in-
+termediate step in a system for converting unrolled optimizers
+to analytically differentiated optimization mappings.
+
+C
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+blackbox 
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+x?(c) + ↵rf(x?(c), c)
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+Figure 2: Unfolding Projected Gradient Descent at x⋆ consists of
+alternating gradient step S with projection PC. Each function’s
+forward and backward pass are in blue and red, respectively.
+Definition 1 (Unfolding). An unfolded differentiable optimiza-
+tion of the form (U) is one in which the backpropagation of U
+at each step does not require unrolling an iterative algorithm.
+It is worth noting that when U has a closed-form formula
+and is easy to differentiate, its derivative rule may be natively
+implemented in an automatic differentiation environment (e.g.
+PyTorch). In such cases, there is no distinction between un-
+rolling and unfolding the iteration (U). Therefore, a method for
+converting an unfolding of (U) into analytical differentiation
+(proposed in Section 5) can also be applied to fully unrolled
+optimizers by applying it successively from the innermost to
+the outermost loop of the unrolling. Alternatively, when U
+consists of a nontrivial optimization with known derivatives,
+such as a Quadratic Program, it can be differentiated directly
+to produce an unfolding of (U), as exemplified next.
+Inner Optimizations
+When U is nontrivial to evaluate, it can be viewed as the
+following composition of functions:
+U(xk(c), c) := ¯U (O(S(xk(c), c)), xk(c), c) ,
+(O)
+wherein the primary difficulty in differentiating U is posed
+by the inner optimization sub-routine O : Rn → Rn. Prior
+to O, a setup step S is commonly performed, viewed as a
+differentiable closed-form function, such as a gradient descent
+step. If O can be differentiated analytically, then so can U,
+by applying a chain rule through Equation (O). This can be
+easily automated using automatic differentiation tools like
+PyTorch. This implementation style aligns with the goal of
+the paper, which is to convert unrolled optimizers to ones with
+analytically modeled Jacobians by replacing unrolled loops
+with equivalent closed-form models.
+The next three examples illustrate the unfolding concept,
+highlighting the roles of U, O and S. They will be used to
+construct analytically differentiable optimization mappings
+for a variety of learning tasks in Section 6. The role of these
+components is also depicted in Figure 2 which illustrates one
+iteration of unfolding projected gradient descent.
+Projected gradient descent
+Given a problem
+min
+x∈C f(x)
+(3)
+where f is differentiable and C is a convex set, Projected
+Gradient Descent (PGD) follows the update function
+xk+1 = PC(xk − αk∇f(xk)),
+(4)
+where O = PC is the Euclidean projection onto C, and
+S(x) = x−α∇f(x) is a gradient descent step w.r.t. the objec-
+tive function. The operation ∇f(x) is typically in closed-form
+and easy to differentiate; the projection is the main difficulty
+and depends on the set C. Many simple C have differen-
+tiable closed form projections to facilitate unfolding of (4) (see
+[Beck, 2017]). Further, when C is linear, PC is a quadratic
+programming (QP) problem for which a differentiable solver
+qpth is available from Amos and Kolter [2017].
+Proximal gradient descent
+More generally, to solve prob-
+lems of the form
+min
+x
+f(x) + g(x)
+(5)
+where f is differentiable and g is a closed convex function,
+proximal gradient descent follows the update function
+xk+1 = Proxαkg (xk − αk∇f(xk)) .
+(6)
+Here O is the proximal operator, defined as
+Proxg(x) = argmin
+y
+�
+g(y) + 1
+2∥y − x∥2
+�
+,
+(7)
+and its difficulty depends on g. Many simple proximal opera-
+tors can be represented in closed form and have simple deriva-
+tives. For example, when g(x) = λ∥x∥1, then Proxg = Tλ(x)
+is the soft thresholding operator, whose closed-form formula
+and derivative are given in Appendix C.
+Sequential quadratic programming
+A broadly applicable
+method for solving continuous optimization problems is Se-
+quential Quadratic Programming (SQP). SQP solves the over-
+all optimization Problem (1) by approximating it at each step
+by a QP problem, whose objective is a second-order approx-
+imation of the problem’s Lagrangian function, subject to a
+linearization of its constraints. SQP is well-suited for com-
+posing unfolded optimizers, as it can solve a broad class of
+convex and nonconvex problems and can readily be unfolded
+by implementing its QP step (shown in Appendix C) with the
+differentiable qpth solver [Amos and Kolter, 2017].
+Alternating
+Direction
+Method
+of
+Multipliers
+The
+ADMM-based QP solver of Boyd et al. [2011] is detailed
+in Appendix C. It can be unfolded easily, as its inner
+optimization step is a simpler equality-constrained quadratic
+program, whose solution as linear system of equations has
+a by AD natively in PyTorch. Once converted to analytical
+differentiation (see Section 5), it can be used in place of qpth
+in the PGD and SQP examples above to compose analytical
+backward passes.
+The examples presented above illustrate a common pattern
+in the construction of general-purpose optimization algorithms
+(U), which is to build them from simpler models that can be
+solved by more basic optimizers, such as O. This paper is
+motivated by a similar principle for constructing differentiable
+optimizers. When the inner optimizations of an update func-
+tion U can be differentiated analytically, the only iteration
+that relies on backpropagation by automatic differentiation
+is the outermost one (U). This conversion from unrolling to
+unfolding facilitates the next step: the conversion from unfold-
+ing to an efficient and fully analytical differentiation. This
+approach, called folded optimization, is the main contribution
+of the paper and will be developed next.
+
+asas
+acaPc
+as0
+10
+20
+30
+40
+50
+60
+70
+Unfolded PGD Iteration
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative L1 Error
+Fwd. Pass: x0 =
+Fwd. Pass: x0 = x
+Bwd. Pass: x0 =
+Bwd. Pass: x0 = x
+Figure 3: Forward and backward pass error in unfolding PGD
+4
+Unfolding at a Fixed Point
+In the previous section, unfolded optimization was introduced
+as a variation of unrolling in which the inner iterative subrou-
+tines are differentiated analytically, while the outer iteration of
+(U) is still “unrolled”. Next, it will be demonstrated that this
+outer unrolling can be replaced with a simple linear system
+of equations, which model the Jacobians of the overall opti-
+mization mapping x⋆(c). Additionally, it will be shown that
+unfolding (U) is computationally equivalent to solving this
+linear system using a well-known iterative method. By model-
+ing the unfolding process in closed form, it will be possible to
+solve the resulting linear model using improved methods.
+Optimization procedures of the form (U) generally require
+a starting point x0, which is often chosen arbitrarily, since
+forward-pass convergence xk → x⋆ is guaranteed regardless
+of starting point. It is natural to then ask how the choice of x0
+affects the convergence of the backward pass.
+Definition 2. Suppose that an unfolded iteration (U) produces
+a convergent sequence of solution iterates limk→∞xk = x⋆
+in its forward pass. Then convergence of the backward pass is
+limk→∞
+∂xk
+∂c (c) = ∂x⋆
+∂c (c).
+(8)
+Effect of the starting point on backpropagation.
+Con-
+sider the optimization mapping (20) which maps feature em-
+beddings to smooth top-k class predictions, and will be used
+to learn multilabel classification later in Section 6. A loss
+function L targets ground-truth top-k indicators, and the result
+of the backward pass is the gradient ∂L
+∂c . To evaluate back-
+ward pass convergence in unfolding of this optimization, we
+measure the relative L1 errors of the forward and backward
+passes w.r.t. the equivalent result at full convergence. We con-
+sider two starting points: the precomputed optimal solution
+xa
+0 = x⋆, and a uniform random vector xb
+0 = η ∼ U(0, 1).
+The former case is illustrated in Figure 2, in which xk remains
+stationary at each step.
+Figure 3 reports the errors of the forward and backward pass
+at each iteration of the unfolded PGD under these two starting
+points. The figure shows that when starting the unfolding
+from the precomputed optimal solution xa
+0 the forward pass
+error remains within error tolerance to zero. This is because
+x⋆(c) = U(x⋆(c), c) is a fixed point of (U). Interestingly
+though, the backward pass also converges, but at a slightly
+faster rate than when starting from a random vector xb
+0.
+Next, we show that this phenomenon holds in general: when
+an unfolded optimizer is iterated at a precomputed optimal
+solution, its backward pass converges. The proof elucidates
+a connection between unrolling and analytical differentiation
+of the fixed-point conditions of the iteration (U). First, some
+practical remarks on unfolding at the precomputed optimum.
+Fixed-Point Unfolding: Forward Pass
+Notice that using a precomputed optimum for unfolding is not
+the most efficient method, as it requires solving the optimiza-
+tion problem to completion before unfolding, which requires
+its own additional solver iterations (U). However, since the
+optimal solution x⋆ is a fixed point of the iteration (U), all
+iterations will result in xk = x⋆. Furthermore, since x⋆ is
+already known and U(x⋆) = x⋆, there is no need to evaluate
+the forward pass of U at each unfolded iteration, including any
+associated inner optimization problems.
+This means that the forward optimization step of the map-
+ping c → x⋆(c) can be implemented as a blackbox software,
+using any solver algorithm, without the need for it to be the
+same as the algorithm used for backpropagation. This enables
+the use of highly optimized solvers, such as Gurobi, and is
+a major advantage over other solvers such as cvxpy, which
+requires conversion of any problem to a convex cone program
+before solving it with a specialized operator-splitting solver,
+rendering it inefficient for many optimization problems.
+Fixed-Point Unfolding: Backward Pass
+Notice also that the backward pass of a fixed point unfolding
+of (U) does need to be computed, as seen in Figure 3, to
+produce gradients for backpropagation. However, since the
+xk are fixed at each iteration, so is the associated Jacobian
+operator ∂U(xk)
+∂xk
+= ∂U(x⋆)
+∂x⋆ , used to model the backward pass
+of the update function U. Therefore the calculation of this
+Jacobian need be performed only once. What then remains
+is only to iterate this backward pass until convergence to an
+accurate gradient of x⋆(c); we will show that this is equivalent
+to the algorithm (LFPI) of the following Lemma, which we
+call Linear Fixed-Point Iteration.
+Lemma 1 (Quarteroni et al. [2010]). Let B ∈ Rn×n and
+b ∈ Rn. For any z0 ∈ Rn, the iteration
+zk+1 = Bzk + b
+(LFPI)
+converges to the solution z of the linear system z = Bz + b
+whenever B is nonsingular and has spectral radius ρ(B) < 1.
+Furthermore, the asymptotic convergence rate for zk → z is
+− log ρ(B).
+(9)
+LFPI is a foundational iterative linear system solver, and can be
+applied to any linear system Ax=b by rearranging z=Bz+b
+and identifying A=I−B. To proceed, we derive an analytical
+model for the desired gradients ∂x⋆
+∂c (c). Consider the fixed-
+point conditions of the iteration (U), assuming xk → x⋆:
+x⋆(c) = U(x⋆(c), c)
+(FP)
+Differentiating (FP) with respect to c,
+∂x⋆
+∂c (c) = ∂U
+∂x⋆ (x⋆(c), c)
+�
+��
+�
+Φ
+·∂x⋆
+∂c (c) + ∂U
+∂c (x⋆(c), c)
+�
+��
+�
+Ψ
+, (10)
+
+by the chain rule and recognizing the implicit and explicit
+dependence of U on the independent parameters c. Equation
+(10) will be called the differential fixed-point conditions. Rear-
+ranging (10), the desired ∂x⋆
+∂c (c) can be found in terms of Φ
+and Ψ as defined above, to yield the system (DFP) below.
+The results discussed next operate under the mild assump-
+tions that x⋆:Rn →Rn is differentiable in an open set C, and
+Equation (FP) holds for c ∈ C. Additionally U is assumed
+differentiable on an open set containing the point (x⋆(c), c).
+Lemma 2. When I is the identity operator and Φ nonsingular,
+(I − Φ)∂x⋆
+∂c = Ψ.
+(DFP)
+The result follows from the Implicit Function theorem
+[Munkres, 2018], and implies that the Jacobian ∂x⋆
+∂c can be
+found as the solution to a linear system once the prerequisite
+partial derivatives of U are known. Note that any algorithm
+which solves Problem (1) can be used to create fixed-point
+conditions for backpropagation, and does not have to be the
+same as the blackbox algorithm that solves the forward pass
+mapping c → x⋆(c). In this spirit, the technique differs from
+early proposals of [Amos and Kolter, 2017] and [Wilder et
+al., 2019b], which apply implicit differentiation of a quadratic
+program’s KKT conditions of optimality or the fixed-point
+conditions of a K-means clustering algorithm, respectively.
+5
+Folded Optimization
+We are now ready to discuss the central result of the paper.
+Informally, it states that the backward pass of an iterative
+solver (U), unfolded at a precomputed optimal solution x⋆(c),
+is equivalent to solving the linear equations (DFP) using linear
+fixed-point iteration, as outlined in Lemma 1.
+This has significant implications for unrolling optimization.
+It shows that unrolling is computationally equivalent to solving
+closed-form equations using a specific algorithm and does not
+require automatic differentiation. It also provides new insights
+into the convergence properties of the backward pass, which
+also apply to situations where the starting point, x0, is not
+precomputed. Finally, it highlights that the backpropagation
+of unrolled optimization is generally suboptimal in terms of
+efficiency, since more efficient algorithms can be used to solve
+(DFP) in place of its inherent LFPI implementation.
+The following results hold under the mild assumptions that
+the parameterized optimization mapping x⋆ converges for
+certain parameters c through a sequence of iterates xk(c) →
+x⋆(c) using algorithm (U), and that Φ is nonsingular with a
+spectral radius ρ(Φ) < 1.
+Theorem 1. The backward pass of an unfolding of algorithm
+(U), starting at the point xk = x⋆, is equivalent to linear fixed-
+point iteration on the linear system (DFP), and will converge
+to its unique solution at an asymptotic rate of
+− log ρ(Φ).
+(11)
+Proof. Since U converges given any parameters c ∈ C, Equa-
+tion (FP) holds for any c ∈ C. Together with the assumption
+the U is differentiable on a neighborhood of (x⋆(c), c),
+(I − Φ)∂x⋆
+∂c = Ψ
+(12)
+holds by Lemma 2. When (U) is unfolded, its backpropagation
+rule can be derived by differentiating its update rule:
+∂
+∂c [ xk+1(c) ] = ∂
+∂c [ U(xk(c), c) ]
+(13a)
+∂xk+1
+∂c
+(c) = ∂U
+∂xk
+∂xk
+∂c + ∂U
+∂c ,
+(13b)
+where all terms on the right-hand side are evaluated at c and
+xk(c). Note that in the base case k = 0, since in general x0 is
+arbitrary and does not depend on c, ∂x0
+∂c = 0 and
+∂x1
+∂c (c) = ∂U
+∂c (x0, c).
+(14)
+This holds also when x0 =x⋆ w.r.t. backpropagation of (U),
+since x⋆ is precomputed outside the computational graph of
+its unfolding. Now since x⋆ is a fixed point of (U),
+xk(c) = x⋆(c)
+∀k ≥ 0,
+(15)
+which implies
+∂U
+∂xk
+(xk(c), c) = ∂U
+∂x⋆ (x⋆(c), c) = Φ,
+∀k ≥ 0
+(16a)
+∂U
+∂c (xk(c), c) = ∂U
+∂c (x⋆(c), c) = Ψ,
+∀k ≥ 0.
+(16b)
+Letting Jk := ∂xk
+∂c (c), the rule (13b) for unfolding at a fixed
+point x⋆ becomes, along with initial conditions (14),
+J0 = Ψ
+(17a)
+Jk+1 = ΦJk + Ψ.
+(17b)
+The result then hold by direct application of Lemma 1 to (17),
+recognizing zk = Jk , B = Φ and z0 = b = Ψ.
+The following is a direct result from the proof of Theorem 1.
+Corollary 1. Backpropagation of the fixed-point unfolding
+consists of the following rule:
+J0 = Ψ
+(18a)
+Jk+1 = ΦJk + Ψ,
+(18b)
+where Jk := ∂xk
+∂c (c).
+The proof illustrates that in the LFPI applied through fixed-
+point unfolding, the initial iterate is J0 = Ψ. However, con-
+vergence to the true Jacobian is guaranteed regardless of the
+initial iterate.
+Figure 4 shows a simplified computational graph of unfold-
+ing the iteration (U) at a precomputed fixed point x⋆. Forward
+pass operations are shown in blue arrows, and consist of re-
+peated application of the update function U. Its first input,
+x⋆(c), is produced by the previous call to U while the second
+input c is always at the base of the graph. The corresponding
+backward passes are shown in red, as viewed through the Ja-
+cobians ∂U
+∂x and ∂U
+∂c , which equal Φ and Ψ at each iteration
+since xk = x⋆ ∀k. Comparing to Equation (13b), notice that
+∂U
+∂c contributes to the chain rule additively while ∂U
+∂x does so
+multiplicatively. This causes the resulting multivariate chain
+rule to take the linear fixed-point iteration form of Lemma 1.
+Theorem 1 specifically applies to the case where the ini-
+tial iterate is the precomputed optimal solution, x0 = x⋆,
+
+Figure 4: Computational graph for unfolding three iterations of (U)
+at a precomputed optimal solution x⋆
+however, it also has implications for the general case where
+x0 is arbitrary. If the forward pass converges, meaning that
+xk → x⋆ as k → ∞, this case becomes identical to the one
+proved in Theorem 1 and a similar asymptotic convergence
+result applies. If xk → x⋆ and Φ is a nonsingular operator
+with ρ(Φ) < 1, the following result holds.
+Corollary 2. When the parametric Problem (1) can be solved
+by an iterative algorithm of the form (U) and the forward pass
+of the unfolded algorithm converges, the backward pass con-
+verges at an asymptotic rate that is bounded by − log ρ(Φ).
+The result above helps explain why the forward and back-
+ward pass in the experiment of Section 4 converge at different
+rates. If the forward pass converges faster, the overall conver-
+gence rate of an unfolding is limited by that of the backward
+pass. Even if the initial point x0 is close to the optimal solution
+x⋆, the backward pass still needs to converge at its own rate.
+Therefore, warm-starting the LFPI applied in the backward
+pass from previous steps can help accelerate training.
+More generally, these results are applicable to backward
+pass convergence of any unrolled optimization by applying
+them at the level of each iterative procedure that is unrolled.
+Fixed-Point Folding
+Section 4 showed that unfolding from a precomputed optimal
+solution yields the same gradients as when unfolding from
+an arbitrary point. However, this method is less efficient as
+it includes unnecessary forward-pass optimization steps at
+the optimum. Section 5 proved that the backward pass of
+this fixed-point unfolding is equivalent to directly solving the
+differential fixed-point conditions (DFP) using LFPI.
+To improve efficiency and building on these findings, we
+propose to replace unfolding at the fixed-point x⋆ with the an-
+alytical solution of (DFP). This technique leads to fixed-point
+folding, a system for converting any unrolled implementation
+of an optimization method (U) into a folded optimizer that
+eliminates unrolling entirely.
+It is important to note that as per Definition 1, the innermost
+optimization loop of a full unrolling can be considered an
+unfolding and can be differentiated through fixed-point folding.
+Subsequently, the next innermost loop can now be considered
+unfolded and the same process can be applied until all unrolled
+optimization loops are replaced with their analytical models.
+This procedure is illustrated schematically in Figure 1, which
+depicts fixed-point folding, where the gray arrows symbolize a
+blackbox forward pass and the long red arrows illustrate that a
+static backward pass iterates at the fixed point. The procedure
+is also exemplified by f-PGDb (introduced in Section 6), which
+applies successive fixed-point folding through ADMM and
+PGD to compose an analytical backward pass model.
+Stepsize Selection
+Many optimization algorithms rely on
+a parameter such as a stepsize, which may be constant or
+change according to some rule at each iteration. Since folded
+optimization depends on a model of some algorithm at its
+fixed point to compute gradients, it also requires a stepsize
+to be chosen. In general, this stepsize need not be chosen
+so that the forward pass optimization converges; in all the
+example algorithms of this paper, the fixed point remains
+stationary even for large stepsizes. Instead, the stepsize should
+be chosen according to its effect on the spectral radius ρ(Φ)
+(see Theorem 1). For example, in the case of folded PGD,
+Φ = ∂
+∂xPC(x⋆ − α∇f(x⋆)),
+(19)
+which depends explicitly on the constant stepsize α. In prac-
+tice, it is observed that larger α lead to convergence in less
+LFPI iterations during fixed-point folding. However when α
+becomes too large, the resulting gradients explode. A large
+range of α tend to result in backward-pass convergence to
+the same gradients, so that careful stepsize selection is typi-
+cally not required. For the purpose of optimizing efficiency,
+Φ could be analyzed to determine its optimal α, but such an
+analysis is not pursued within the scope of this paper.
+6
+Experiments
+This section evaluates folded optimization on four end-to-
+end optimization and learning tasks. It is primarily evaluated
+against cvxpy, which is the preeminent general-purpose dif-
+ferentiable optimization solver. Two crucial limitations of
+cvxpy are its efficiency and expressiveness. This is due to its
+reliance on transforming general optimization programs to con-
+vex cone programs, before applying a standardized operator-
+splitting cone program solver and differentiation scheme (see
+Appendix A). This precludes the incorporation of problem-
+specific solvers in the forward pass and limits its use to convex
+problems only. One major advantage of fold-opt is the
+modularity of its forward optimization pass, which can apply
+any black-box algorithm to produce x⋆(c). In each experiment
+below, this is used to demonstrate a different advantage.
+A summary of results is provided below for each study, and
+a more complete specification is provided in Appendix D.
+Implementation details.
+All the folded optimizers used in
+this section were produced using the accompanying Python
+library fold-opt, which supplies routines for constructing
+and solving the system (DFP), and integrating the resulting
+Jacobian-vector products into the computational graph of Py-
+Torch for backpropagation. When the linear system (DFP)
+is represented explicitly, it can be solved by a user-input
+blackbox linear solver, as can the forward-pass optimization
+solver, as mentioned in Section 4. Implementation details of
+fold-opt can be found in Appendix B.
+The experiments test four folded optimizer: (1) f-PGDa
+applies to optimization mappings with linear constraints, and is
+based on folding projected gradient descent steps, where each
+
+auau
+ac=亚0
+20
+40
+60
+80
+Training Iteration
+0
+1
+2
+3
+4
+5
+Avg. Regret: Test Set
+Model
+Two-Stage
+Integrated
+0
+20
+40
+60
+80
+100
+Training Epoch
+16
+18
+20
+22
+24
+26
+28
+30
+Mean Square Error: Test
+7
+9
+11
+13
+15
+17
+19
+0
+10
+20
+30
+40
+50
+Avg Regret: Test
+0.00
+0.02
+0.04
+0.06
+0.08
+Training Epoch
+Deg. Nonlinearity
+1
+2
+3
+Model
+cvxpy
+SQP
+(a)
+(b)
+(c)
+Figure 5: Bilinear decision focus (a), Enhanced Denoising with f-FDPG (b), and Portfolio optimization (c).
+inner projection is a QP solved by the differentiable QP solver
+qpth [Amos and Kolter, 2017]. (2) f-PGDb is a variation
+on the former, in which the inner QP step is differentiated by
+fixed-point folding of the ADMM solver detailed in Appendix
+C. (3) f-SQP applies to optimization with nonlinear constraints
+and uses folded SQP with the inner QP differentiated by qpth.
+(4) f-FDPG comes from fixed-point folding of the Fast Dual
+Proximal Gradient Descent (FDPG) shown in Appendix C.
+The inner Prox is a soft thresholding operator, whose simple
+closed form is differentiated by AD in PyTorch.
+Enabling nonconvex optimization mappings.
+The first ex-
+periment showcases the ability of folded optimization to be
+applied in nonconvex decision-focused settings. In this experi-
+ment, we predict the coefficients of a bilinear program
+x⋆(c, d) = argmax
+0≤x,y≤1
+cT x + xT Qy + dT y
+s.t.
+�
+x = p,
+�
+y = q,
+in which two separable linear programs are confounded by
+a nonconvex quadratic objective term Q. Costs c and d are
+predicted by a 5-layer network, while p and q are constants.
+Such programs have numerous industrial applications such
+as optimal mixing and pooling in gas refining [Audet et al.,
+2004]. Here we focus on the difficulty posed by the problem’s
+form and propose a task to evaluate f-PGDb in learning with
+nonconvex mappings. Feature and cost data are generated by
+the process described in Appendix D, along with 15 distinct
+Q for a collection of nonconvex problems.
+It is known that PGD converges to local optima in non-
+convex problems [Attouch et al., 2013], and this folded im-
+plementation uses the Gurobi nonconvex QP solver to find
+a global optimum. Since no known general framework can
+accommodate nonconvex optimization mappings in end-to-
+end models, we benchmark against the two-stage approach,
+in which the costs c, and d are targeted to ground-truth costs
+by MSE loss and the optimization problem is solved as a sep-
+arate component from the learning task (see Appendix E for
+additional details). The integrated f-PGDb model minimizes
+solution regret (i.e., suboptimality) directly. [Elmachtoub and
+Grigas, 2021]. Notice in Figure 5(a) how f-PGDb achieves
+much lower regret for each of the 15 nonconvex objectives.
+Improving performance with specialized solvers.
+This
+experiment illustrates the efficiency benefit of incorporating
+problem-specific solvers. The optimization models a denoiser
+x⋆(D) = argmin
+x
+1
+2∥x − d∥2 + λ∥Dx∥1,
+which seeks to recover the true signal x⋆ from a noisy input
+d and is often best handled by variants of Dual Proximal
+Gradient Descent. Classically, D is a differencing matrix so
+that ∥Dx∥1 represents total variation. Here we initialize D to
+this classic case and learn a better D by targeting a set of true
+signals with MSE loss and adding Gaussian noise to generate
+their corresponding noisy inputs. Figure 5(b) shows test MSE
+throughout training due to f-FDPG for various choice of λ.
+Appendix F shows comparable results from the framework of
+Amos and Kolter [2017], which converts the problem to a QP
+form (see Appendix C) in order to differentiate the mapping
+analytically with qpth. Small differences in these results
+likely stem from solver error tolerance in the two methods.
+However, f-FDPG computes x⋆(D) up to 40 times faster.
+Measuring the accuracy of backpropagation.
+Since gra-
+dient errors accumulate at each training step, we ask how
+precise are the operations performed by fold-opt in the
+backward pass. This experiment compares the backpropaga-
+tion of both f-PGDa and f-SQP with that of cvxpy, by using
+the forward pass of cvxpy in each model as a control factor.
+This experiment, adapted from Berrada et al. [2018], imple-
+ments a smooth top-5 classification model on noisy CIFAR-
+100. The optimization below maps image feature embeddings
+c from DenseNet 40-40 [Huang et al., 2017], to smoothed top-
+k binary class indicators (see Appendix D for more details):
+x⋆(c)=argmax
+0≤x≤1
+cT x +
+�
+i
+xi log xi s.t.
+�
+x = k (20)
+Appendix F shows that all three models have indistinguish-
+able classification accuracy throughout training, indicating
+the backward pass of both fold-opt models is precise and
+agrees with a known benchmark even after 30 epochs of train-
+ing on 45k samples. On the other hand, the more sensitive test
+set shows marginal accuracy divergence after a few epochs.
+Improving accuracy with specialized solvers.
+Having es-
+tablished the equivalence in performance of the backward pass
+across these models, the final experiment describes a situation
+in which cvxpy makes non negligible errors in the forward
+pass of a problem with nonlinear constraints:
+x⋆(c) = argmax
+0≤x
+cT x s.t. xT Vx ≤ γ,
+�
+x = 1.
+(21)
+
+This model describes a risk-constrained portfolio optimiza-
+tion where V is a covariance matrix, and the predicted cost
+coefficients c represent assets prices [Elmachtoub and Gri-
+gas, 2021]. A 5-layer ReLU network is used to predict future
+prices c from exogenous feature data, and trained to minimize
+regret (the difference in profit between optimal portfolios un-
+der predicted and ground-truth prices) by integrating Problem
+(21). The folded f-SQP layer used for this problem employs
+Gurobi QCQP solver in its forward pass. This again highlights
+the ability of fold-opt to accommodate a highly optimized
+blackbox solver. Figure 5(c) shows test set regret throughout
+training, three synthetically generated datasets of different
+nonlinearity degrees. Notice the accuracy improvements of
+fold-opt over cvxpy. Such dramatic differences can be
+explained by non-negligible errors made in cvxpy’s forward
+pass optimization on some problem instances, which occurs
+regardless of error tolerance settings (please see Appendix
+D for details). In contrast, Gurobi agrees to machine preci-
+sion with a custom SQP solver, and solves about 50% faster
+than cvxpy. This shows the importance of highly accurate
+optimization solvers for accurate end-to-end training.
+7
+Conclusions
+This paper introduced folded optimization, a framework for
+generating analytically differentiable optimization solvers
+from unrolled implementations. Theoretically, folded opti-
+mization was justified by a novel analysis of unrolling at a
+precomputed optimal solution, which showed that its back-
+ward pass is equivalent to solution of a solver’s differential
+fixed-point conditions, specifically by fixed-point iteration on
+the resulting linear system. This allowed for the convergence
+analysis of the backward pass of unrolling, and evidence that
+the backpropagation of unrolling can be improved by using
+superior linear system solvers. The paper showed that folded
+optimization offers substantial advantages over existing differ-
+entiable optimization frameworks, including modularization
+of the forward and backward passes and the ability to handle
+nonconvex optimization.
+Acknowledgments
+This research is partially supported by NSF grant 2007164 and
+NSF CAREER Award 2143706. Fioretto is also supported by
+a Google Research Scholar Award and an Amazon Research
+Award. Its views and conclusions are those of the authors only.
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+
+A
+Related Work
+This section categorizes end-to-end optimization and learn-
+ing approaches into those based on unrolling, and analytical
+differentiation. Since this paper focuses on converting un-
+rolled implementations into analytical ones, each category is
+reviewed first below.
+Unrolling optimization algorithms.
+Automatic Differenti-
+ation (AD) is the primary method of backpropagating gradients
+in modern deep models for training with stochastic gradient
+descent. Modern machine learning frameworks such as Py-
+Torch have natively implemented differentiation rules for a
+variety of functions that are commonly used in deep models,
+as well as interfaces to define custom differentiation rules for
+new functions [Paszke et al., 2017]. As a mainstay of deep
+learning, AD is also a natural tool for backpropagating through
+constrained optimization mappings. Unrolling refers to the
+execution of an optimization algorithm, entirely on the compu-
+tational graph, for backpropagation by AD from the resulting
+optimal solution to its input parameters. Such approaches are
+general and apply to a broad range of optimization models.
+They can be performed simply by implementing a solution
+algorithm within an AD framework, without the need for an-
+alytical modeling of an optimization mapping’s derivatives
+[Domke, 2012]. However, unrolling over many iterations has
+been shown to encounter issues of time and memory ineffi-
+ciency due to the size of its computational graph [Amos and
+Kolter, 2017]. Further issues encountered in unrolling, such
+as vanishing and exploding gradients, are reminiscent of recur-
+rent neural networks [Monga et al., 2021]. On the other hand,
+unrolling may offer some unique practical advantages, like the
+ability to learn optimization parameters such as stepsizes to
+accelerate the solution of each optimization during training
+[Shlezinger et al., 2022].
+Analytical differentiation of optimization models.
+Differ-
+entiation through constrained argmin problems in the con-
+text of machine learning was discussed as early as Gould et
+al. [2016], who proposed first to implicitly differentiate the
+argmin of a smooth, unconstrained convex function by its first-
+order optimality conditions, defined when the gradient of the
+objective function equals zero. This technique is then extended
+to find approximate derivatives for constrained problems, by
+applying it to their unconstrained log-barrier approximations.
+Subsequent approaches applied implicit differentiation to the
+KKT optimality conditions of constrained problems directly
+[Amos and Kolter, 2017; Amos et al., 2019], but only on spe-
+cial problem classes such as Quadratic Programs. Konishi
+and Fukunaga [2021] extend the method of Amos and Kolter
+[2017], by modeling second-order derivatives of the optimiza-
+tion for training with gradient boosting methods. Donti et al.
+[2017] uses the differentiable quadratic programming solver
+of [Amos and Kolter, 2017] to approximately differentiate gen-
+eral convex programs through quadratic surrogate problems.
+Other problem-specific approaches to analytical differentia-
+tion models include ones for sorting and ranking [Blondel et
+al., 2020], linear programming [Mandi and Guns, 2020], and
+convex cone programming [Agrawal et al., 2019b].
+The first general-purpose differentiable optimization solver
+was proposed in Agrawal et al. [2019a], which leverages
+the fact that any convex program can be converted to a con-
+vex cone program [Nemirovski, 2007]. The equivalent cone
+program is subsequently solved and differentiated follow-
+ing Agrawal et al. [2019b], which implicitly differentiates
+a zero-residual condition representing optimality [Busseti et
+al., 2019]. A differentiable solver library cvxpy is based on
+this approach, which converts convex programs to convex cone
+programs by way of their graph implementations as described
+in Grant and Boyd [2008]. The main advantage of the sys-
+tem is that it applies to any convex program and has a simple
+symbolic interface. A major disadvantage is its restriction to
+solving problems only in a standard convex cone form with
+an ADMM-based conic programming solver, which performs
+poorly on some problem classes, as seen in Section 6.
+A related line of work concerns end-to-end learning with dis-
+crete optimization problems, which includes linear programs,
+mixed-integer programs and constraint programs. These prob-
+lem classes often define discontinuous mappings with respect
+to their input parameters, making their true gradients unhelpful
+as descent directions in optimization. Accurate end-to-end
+training can be achieved by smoothing the optimization map-
+pings, to produce approximations which yield more useful
+gradients. A common approach is to augment the objective
+function with smooth regularizing terms such as euclidean
+norm or entropy functions [Wilder et al., 2019a; Ferber et al.,
+2020; Mandi and Guns, 2020]. Others show that similar effects
+can be produced by applying random noise to the objective
+[Berthet et al., 2020; Paulus et al., 2020], or through finite dif-
+ference approximations [Poganˇci´c et al., 2019; Sekhar Sahoo
+et al., 2022]. This enables end-to-end learning with discrete
+structures such as constrained ranking policies [Kotary et al.,
+2022], shortest paths in graphs [Elmachtoub and Grigas, 2021],
+and various decision models [Wilder et al., 2019a].
+B
+Implementation Details
+The purpose of the fold-opt library is to facilitate the con-
+version of unrolled optimization code into analytically mod-
+eled differentiable optimization. The following steps can be
+used to replace any loop of a functioning unrolled implementa-
+tion. To convert a full unrolling with one or more nested loops,
+the process is applied from the innermost to the outermost
+loop. (1) After executing a blackbox optimization, initialize
+the precomputed optimal solution x⋆ onto the computational
+graph of PyTorch. (2) Execute a single step of the unrolled
+loop’s update function to get x⋆⋆(c) = U(x⋆, c) and save its
+computational graph; in principle, the forward execution can
+be overridden with its known result x⋆. (3) Backpropagate
+an identity matrix from x⋆⋆(c) to x⋆ and from x⋆⋆(c) to c to
+get Φ and Ψ, respectively (see Section 5). (4) Solve equation
+(DFP) and apply the Jacobian-vector product to backpropa-
+gate incoming gradients. The library contains functions to
+automate each of the above steps.
+C
+Optimization Models
+Soft Thresholding Operator
+The soft thresholding opera-
+tor defined below arises in the solution of denoising problems
+proximal gradient descent variants as the proximal operator to
+
+the ∥ · ∥1 norm:
+Tλ(x) = [|x| − λe]+ · sgn(x)
+Fast Dual Proximal Gradient Descent
+The following is an
+FDPG implementation from Beck [2017], specialized to solve
+the denoising problem
+x⋆(D) = argmin
+x
+1
+2∥x − d∥2 + λ∥Dx∥1,
+of Section 6. Letting uk be the primal solution iterates, with
+t0 = 1 and arbitrary w0 = y0:
+uk = DT wk + d
+(22a)
+yk+1 = wk − 1
+4Duk + 1
+4T4λ(Duk − 4wk)
+(22b)
+tk+1 = 1 +
+�
+1 + 4t2
+k
+2
+(22c)
+wk+1 = yk+1 +
+�tk − 1
+tk+1
+�
+(yk+1 − yk)
+(22d)
+Quadratic Programming by ADMM
+A Quadratic Pro-
+gram is an optimization problem with convex quadratic objec-
+tive and linear constraints. The following ADMM scheme of
+Boyd et al. [2011] solves any quadratic programming problem
+of the standard form:
+argmax
+x
+1
+2xT Qx + pT x
+(23a)
+s.t. Ax = b
+(23b)
+x ≥ 0
+(23c)
+by declaring the operator splitting
+argmax
+x
+f(x) + g(z)
+(24a)
+s.t. x = z
+(24b)
+with f(x) = 1
+2xT Qx + pT x, dom(f) = {x : Ax = b},
+g(x) = δ(x ≥ 0) and where δ is the indicator function.
+This results in the following ADMM iterates:
+1. Solve
+�
+P + ρI
+AT
+A
+0
+� �
+xk+1
+ννν
+�
+=
+�
+−q + ρ(zk − uk)
+b
+�
+2. zk+1 = (xk+1 + uk)+
+3. uk+1 = uk + xk+1 − zk+1
+Where (1) represents the KKT conditions for equality-
+constrained minimization of f, (2) is projection onto the posi-
+tive orthant, and (3) is the dual variable update.
+Sequential Quadratic Programming
+For an optimization
+mapping defined by Problem (1) where f, g and h are contin-
+uously differentiable, define the operator T as:
+T (x,λλλ) = argmin
+d
+∇f(x)T d + dT ∇2L(x,λλλ)d
+(25a)
+s.t. h(x) + ∇h(x)T d = 0
+(25b)
+g(x) + ∇g(x)T d ≤ 0
+(25c)
+where dependence of each function on parameters c is hidden.
+The function L is a Lagrangian function of Problem (1). Then
+given initial estimates of the primal and dual solution (x0, λ0),
+sequential quadratic programming is defined by
+(d,µµµ) = T (xk,λλλk)
+(26a)
+xk+1 = xk + αkd
+(26b)
+λλλk+1 = αk(µµµ − λλλk)
+(26c)
+Here, the inner optimization O = T as in Section 3.
+Denoising Problem - Quadratic Programming form
+The
+following quadratic program is equivalent to the unconstrained
+denoising problem of Section 6:
+x⋆(D) = argmin
+x,t
+1
+2∥x − d∥2 + λ−→1 t
+(27a)
+s.t.
+Dx ≤ t
+(27b)
+− t ≤ Dx
+(27c)
+D
+Experimental Details
+Additional details for each experiment of Section 6 are de-
+scribed in their respective subsections below. Note that in all
+cases, the machine learning models compared in Section 6 use
+identical settings within each study, with the exception of the
+optimization components being compared.
+D.1
+Nonconvex Bilinear Programming
+Data generation
+Data is generated as follows for the non-
+convex bilinear programming experiments. Input data con-
+sists of 1000 points ∈ R10 sampled uniformly in the interval
+[−2, 2]. To produce targets, inputs are fed into a randomly ini-
+tialized 2-layer neural network with tanh activation, and gone
+through a nonlinear function x cos 2x+ 5
+2 log
+x
+x+2 +x2 sin 4x
+to increase the nonlinearity of the mapping between inputs
+and targets. Train and test sets are split 90/10.
+Settings
+A 5-layer NN with ReLU activation trained to pre-
+dict cost c and d. We train model with Adam optimizer on
+learning rate of 10−2 and batch size 32 for 5 epochs.
+Nonconvex objective coefficients Q are pre-generated ran-
+domly with 15 different seeds. Constraint parameters are
+chosen arbitrarily as p = 1 and q = 2. The average solving
+time in Gurobi is 0.8333s, and depends per instance on the
+predicted parameters c and d. However the average time tends
+to be dominated by a minority of samples which take up to
+∼ 3 min. This issue is mitigated by imposing a time limit
+in solving each instance. While the correct gradient is not
+guaranteed under early stopping, the overwhelming majority
+of samples are fully optimized under the time limit, mitigating
+any adverse effect on training. Differences in training curves
+under 10s and 120s timeouts are negligible due to this effect;
+the results reported use the 120s timeout.
+D.2
+Enhanced Denoising
+Data generation
+The data generation follows Amos and
+Kolter [2017], in which 10000 random 1D signals of length
+100 are generated and treated as targets. Noisy input data is
+generated by adding random perturbations to each element of
+each signal, drawn from independent standard-normal distri-
+butions. A 90/10 train/test split is applied to the data.
+
+Settings
+A learning rate of 10−3 and batch size 32 are used
+in each training run. Each denoising model is initialized to the
+classical total variation denoiser by setting the learned matrix
+of parameters D ∈ R99×100 to the differencing operator, for
+which Di,i = 1 and Di,i+1 = −1 ∀i with all other values 0.
+D.3
+Multilabel Classification
+Dataset
+We follow the experimental settings and implemen-
+tation provided by Berrada et al. [2018]. Each model is eval-
+uated on the noisy top-5 CIFAR100 task. CIFAR-100 labels
+are organized into 20 “coarse” classes, each consisting of 5
+“fine” labels. With some probability, random noise is added
+to each label by resampling from the set of “fine” labels. The
+50k data samples are given a 90/10 training/testing split.
+Settings
+The DenseNet 40-40 architecture is trained by SGD
+optimizer with learning rate 10−1 and batch size 64 for 30
+epochs to minimize a cross-entropy loss function.
+D.4
+Portfolio Optimization
+Data Generation
+The data generation follows exactly the
+prescription of Appendix D in Elmachtoub and Grigas [2021].
+Uniform random feature data are mapped through a random
+nonlinear function to create synthetic price data for training
+and evaluation. A random matrix is used as a linear mapping,
+to which nonlinearity is introduced by exponentiation of its
+elements to a chosen degree. The studies in Section 6 use
+degrees 1, 2 and 3.
+Settings
+A five-layer ReLU network is trained to predict
+asset prices c ∈ R20 using Adam optimizer with learning rate
+10−2 and batch size 32.
+E
+Decision-Focused Learning
+For unfamiliar readers, this section provides background on the
+decision-focused learning setting, also known as predict-and-
+optimize, which characterizes the first and last experiments
+of Section 6 on bilinear programming and portfolio optimiza-
+tion. In this paper, those terms refer to settings in which an
+optimization mapping
+x⋆(c) = argmin
+x
+f(x, c)
+(28a)
+subject to: g(x) ≤ 0,
+(28b)
+h(x) = 0,
+(28c)
+represents a decision model and is parameterized by the vector
+c, but only in its objective function. The goal of the supervised
+learning task is to predict ˆc from feature data such that the
+resulting x⋆(ˆc) optimizes the objective under ground-truth
+parameters ¯c, which is f(x⋆(ˆc), ¯c). This is equivalent to
+minimizing the regret loss function:
+regret(ˆc, ¯c) = f(x⋆(ˆc), ¯c) − f(x⋆(¯c), ¯c),
+(29)
+which measures the suboptimality, under ground-truth objec-
+tive data, of decisions x⋆(ˆc) resulting from prediction ˆc.
+When x⋆ and f are differentiable, the prediction model for
+ˆc can be trained to minimize regret directly in an integrated
+predict-and-optimize model. Since the task amounts to pre-
+dicting ˆc under ground-truth ¯c, a two-stage approach is also
+0
+20
+40
+60
+80
+100
+Training Epoch
+16
+18
+20
+22
+24
+26
+28
+30
+Mean Square Error: Test
+7
+9
+11
+13
+15
+17
+19
+(a) f-FDPG
+0
+20
+40
+60
+80
+100
+Training Epoch
+16
+18
+20
+22
+24
+26
+28
+30
+Mean Square Error: Test
+7
+9
+11
+13
+15
+17
+19
+(b) qpth
+Figure 6: Enhanced Denoiser Test Loss
+available which does not require backpropagation through x⋆.
+In the two-stage approach, the loss function MSE(ˆc, ¯c) is used
+to directly target ground-truth parameters, but the final test
+criteria is still measured by regret. Since the integrated ap-
+proach minimizes regret directly, it generally outperforms the
+two-stage in this setting.
+0
+10
+20
+30
+Training Epoch
+0.2
+0.4
+0.6
+0.8
+Top-1 Acc: Train Set
+0
+20
+Training Epoch
+0.2
+0.4
+0.6
+0.8
+Top-k Acc: Train Set
+Model
+cvxpy
+PGD
+SQP
+0
+10
+20
+30
+Training Epoch
+0.2
+0.4
+0.6
+0.8
+Top-1 Acc: Test Set
+0
+20
+Training Epoch
+0.2
+0.4
+0.6
+0.8
+Top-k Acc: Test Set
+Model
+cvxpy
+PGD
+SQP
+Figure 7: Multilabel Classification Accuracy
+F
+Additional Figures
+Enhanced denoising experiment.
+Figure 6 shows test loss
+curves, for a variety of λ, in learning enhanced denoisers
+with the chosen baseline method qpth. As per the original
+experiment of Amos and Kolter [2017], the implementation is
+facilitated by conversion to the quadratic programming form
+of model (27). The results from f-FDPG are again shown
+alongside for comparison. Small differences between the
+results stem from the slightly different solutions found by
+their respective solvers at each training iteration, due to their
+differently-defined error tolerance thresholds.
+Multilabel classification experiment.
+Figure 7 shows Top-
+1 and Top-k accuracy on both train and test sets where k = 5.
+Accuracy curves are indistinguishable on the training set even
+after 30 epochs. On the test set, generalization error manifests
+slightly differently for each model in the first few epochs.
+

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@@ -0,0 +1,1328 @@
+{Semi}-Structured Object Sequence Encoders
+Rudra Murthy V1 , Riyaz Bhat1 , Chulaka Gunasekara1 , Hui Wan1 , Tejas Indulal
+Dhamecha2∗ , Danish Contractor1 , Marina Danilevsky1
+1IBM Research, India
+2Microsoft, Bangalore
+rmurthyv@in.ibm.com, {Riyaz.Bhat,chulaka.gunasekara,Danish.Contractor}@ibm.com, {hwan,
+mdanile}@us.ibm.com, tdhamecha@microsoft.com
+Abstract
+In this paper we explore the task of modeling (semi)
+structured object sequences; in particular we fo-
+cus our attention on the problem of developing a
+structure-aware input representation for such se-
+quences. In such sequences, we assume that each
+structured object is represented by a set of key-
+value pairs which encode the attributes of the struc-
+tured object. Given a universe of keys, a sequence
+of structured objects can then be viewed as an
+evolution of the values for each key, over time.
+We encode and construct a sequential representa-
+tion using the values for a particular key (Tempo-
+ral Value Modeling - TVM) and then self-attend
+over the set of key-conditioned value sequences to
+a create a representation of the structured object se-
+quence (Key Aggregation - KA). We pre-train and
+fine-tune the two components independently and
+present an innovative training schedule that inter-
+leaves the training of both modules with shared at-
+tention heads. We find that this iterative two part-
+training results in better performance than a unified
+network with hierarchical encoding as well as over,
+other methods that use a record-view representation
+of the sequence [de Souza Pereira Moreira et al.,
+2021] or a simple flattened representation of the se-
+quence. We conduct experiments using real-world
+data to demonstrate the advantage of interleaving
+TVM-KA on multiple tasks and detailed ablation
+studies motivating our modeling choices. We find
+that our approach performs better than flattening se-
+quence objects and also allows us to operate on sig-
+nificantly larger sequences than existing methods.
+1
+Introduction
+Recent work on employing self-supervision during pre-
+training has led to the development of a number of
+transformer-based large language models (LLM). In partic-
+ular, such LLMs are pre-trained with self-supervised objec-
+tives based on different forms of input masking [Devlin et al.,
+∗Work done while employed at IBM Research, India
+2019; Yang et al., 2019], auto-regressive pre-training [Rad-
+ford et al., 2019; Brown et al., 2020; Workshop et al., 2022],
+or both [Beltagy et al., 2020a; Lewis et al., 2020], and then
+applied to downstream tasks after fine-tuning [Devlin et al.,
+2019; Yang et al., 2019; Lewis et al., 2020], zero-shot and in-
+context learning with prompting[Brown et al., 2020]. These
+methods have also been applied jointly to images and text
+[Ramesh et al., 2021; Rombach et al., 2022]. One of the
+key elements in such models is the choice of representation
+of the input tokens to the transformer; in the case of natu-
+ral language byte-pair encodings are often used to represent
+sub-words and sequences of text are modeled then using these
+encodings.
+In this paper, we explore the task of modeling (semi) struc-
+tured object sequences; in particular we focus our attention
+on the problem of developing a structure-aware input repre-
+sentation for such sequences. In such sequences, we assume
+that each structured object is represented by a set of key-value
+pairs which encode the attributes of the structured object (Fig-
+ure 1(a)). Given a universe of keys, a sequence of structured
+objects can then be viewed as a column-evolution of the val-
+ues for each key, over time. A trivial method of representing
+such sequences would be to flatten each structured object and
+view them as individual words for tokenization in natural lan-
+guage.1 However, this causes the sequence length to become
+extremely large (thousands of tokens). For instance, in our
+study of structured objects from user-interaction sessions on
+software from a large cloud-based service provider, we found
+these objects could contain, on average 25 fields with values
+that include timestamps, identifiers, log messages, etc (aver-
+age 7 words). A session length of 15 minutes results in 105
+such session objects, amounting to nearly 18000 words which
+would further increase the sequence length after sub-words
+are created.
+Contributions: To overcome these challenges, we use a
+modular, two-part hierarchical encoding strategy. First, we
+encode and construct a sequential representation using the
+values for a particular key (Temporal Value Modeling). This
+may be achieved using any encoder.2 We then self-attend over
+the over the set of key-conditioned value sequences to a create
+a representation of the structured object sequence (Key Ag-
+1with markers to indicate boundaries for each structured objects
+2We use BERT in our experiments
+arXiv:2301.01015v1  [cs.CV]  3 Jan 2023
+
+Figure 1: {Semi}-Structured Object Sequences: (a) Generic representation consisting of multiple key-value pairs at time steps t0 . . . tn. (b)
+Sequences of events triggered by the use of a graphical user interface. (c) Viewing a text paragraph as a sequence of sentences. (d) Viewing
+a conversation as a sequence of role annotated turns.
+gregation). We develop a novel sequential transfer learning
+training paradigm to facilitate information sharing between
+two independently trained sub-networks. In contrast to tradi-
+tional approaches based on parameter sharing [Baxter, 1997;
+Duong et al., 2015; Zhang et al., 2022], fine tuning of pre-
+trained models [Devlin et al., 2019], and the use of adapter
+layers [Pfeiffer et al., 2020], we share complete attention
+heads between two networks. First we pre-train the TVM
+network with shared attention heads in place and then use the
+frozen representations from this network to initialize the KA
+network which has its own untrained attention heads as well
+the shared attention heads from the TVM network as part of
+its parameters. We utilize a training schedule that interleaves
+the training of both modules to iteratively train the TVM and
+KA modules. We find that this iterative two part-training re-
+sults in better performance than a unified network with hier-
+archical encoding (with no attention-head sharing) as well as
+over, other methods that use a record-view representation of
+the sequence [de Souza Pereira Moreira et al., 2021].
+We present experiments using real-world data to demon-
+strate the advantage of TVM-KA on multiple tasks and de-
+tailed ablation studies motivating our modeling choices. We
+find that our approach significantly outperforms baseline
+methods for encoding semi-structured object sequences.
+In summary, our paper makes the following contributions:
+(i) We present a two part encoder, TVM-KA that mod-
+els structured sequence objects (ii) It easily accommodates
+the modeling of unstructured natural language text as well
+as modeling semi-structured natural language data (iii) We
+present detailed experiments to demonstrate the strengths of
+our work and include ablation studies motivating our model-
+ing choices.
+2
+Modeling
+Let Ji = {ki,1 : vi,1, ki,2 : vi,2, . . . , ki,n : vi,n} de-
+note a structured object Ji, containing n key-value pairs
+< kj, vj > . A sequence of structured objects is denoted
+as J = [J1, J2, J3, . . . , JN] that is a (temporal) sequence of
+structured objects corresponding to the N time steps. The
+goal of our modeling is to learn a representation of a se-
+quence of structured objects J ; and subsequently, learn f :
+Embd(J ) → {1, 2, . . . , C} for an end-task, such as a C-way
+classification task.
+We develop a modular two-part modeling strategy to rep-
+resent a sequence of structured objects.
+1. Our first module called the Temporal Value Modeler
+(TVM), is used to learn a combined representation (re-
+ferred to as the key-representations) for the different val-
+ues that each key takes in the sequence.
+2. The second module, called the Key-Aggregator (KA)
+uses the key-representations corresponding to each key,
+to create an overall representation for J .
+Temporal Value Modeling : Let k be a key from the univer-
+sal set of all the keys K in the sequence. The sequence of the
+values corresponding to the key is represented as V(k) and a
+
+t1
+>
+Key l: <value>,
+Key l: <value>,
+Key l: <value>,
+Key l: <value>,
+Key_2: <value>,
+Key_2: <value>,
+Key 2: null,
+Key 2: <value>,
+(a)
+Key_3: <value>
+Key_3: <value>
+Key_3: <value>
+Key 3: null
+event:add text,
+event:add node,
+event:delete node,
+color:white,
+color:blue,
+color:null,
+Key_l: <value>,
+text: "Node 1"
+text:null
+text:null
+Key_2: <value>,
+(b)
+node id: 2301,
+node_id: 2301
+node_id: 2301
+Key_3: null
+time_stamp:
+time stamp:
+time stamp:
+<value>
+t1
+text: As of 2022,
+text: New Delhi
+text: The river
+it has
+(c)
+is the capital of
+Yamuna flows
+population of 21
+text: .
+India.
+through the city.
+million.
+ti
+6
+utterance: Sure,
+utterance:Hi,
+I can help with
+utterance: I'm
+my system is
+you that. Which
+on my linux
+(d)
+very slow.
+machine are you
+machine.
+utterance: ..
+role: user
+on?
+role: user
+time_stamp...
+role: agent
+time_stamp...
+time_stamp: ...TextEncoder
+Key-Aggregator
+Cross Entropy 
+Loss
+TextEncoder
+MLM Loss
+(Frozen)
+TextEncoder
+Key-Aggregator
+Cross Entropy 
+Loss
+Pre-training
+Fine-Tuning
+Key-Aggregator
+MLM Loss
+Key-Aggregator
+End-to-EndNetwork Training
+Two-Part Network Training
+Layer 1
+Layer 12
+Layer 2
+Layer 1
+Layer 12
+Layer 2
+Word+Pos
+Embd
+attention 
+head 
+sharing
+TextEncoder
+KeyAggregator
+Text Encoder
+S(k1)=[CLS] k1 [VAL_SEP] V(k1)1 [VAL_SEP] V(k1)2[VAL_SEP]…
+S(k2)=[CLS] k2 [VAL_SEP] V(k2)1 [VAL_SEP] V(k2)2[VAL_SEP]…
+…
+S(kN)=[CLS] kN [VAL_SEP] V(kN)1 [VAL_SEP] V(kN)2[VAL_SEP]…
+S(k)
+(sequence of values of 
+a key)
+KR(k)
+(representation 
+of 
+a key)
+[S(k1), S(k2)... S(kN)]
+[KR(k1), KR(k2)... KR(kN)] 
+(representations of all keys)
+Embd(J)
+(representation of 
+structured object 
+sequence)
+Figure 2: (left) Proposed two-part training in contrast to end-to-end training and (right) Attention Head Sharing
+value sequence S(k) from it as the following:
+V (k)
+=
+[vi,j | ki,j = k, ∀i ∈ {1, 2, . . . , N}]
+(1)
+S(k)
+=
+[CLS] k[VAL_SEP]V (k)1 [VAL_SEP]
+V (k)2 [VAL_SEP] . . . V (k)≤N
+(2)
+where [VAL_SEP] and [CLS] are special tokens.
+Note that this formulation allows us to accommodate the
+modeling of natural language text as in Figure 1 (c). Specifi-
+cally, in Eq. 2, if N = 1 such, V (k) = [v1,i]. V (k) reduces
+to modeling the tokens using the TextEncoder. For illustra-
+tion, if the TextEncoder is based on BERT [Devlin et al.,
+2019], Eq. 3 reduces to the encoding scheme typically em-
+ployed in BERT where the [VAL_SEP] corresponds to an
+end of sentence marker.
+With any choice of a transformer-based [Vaswani et al.,
+2017] language encoder, an embedding for S(k), termed as
+the key-representation (KR), can be obtained:
+KR(k) = TextEncoder(S(k))[0]
+(3)
+TextEncoder gives us a dim x L dimension tensor where
+dim is the output embedding dimension size and L corre-
+sponds to the tokenized sequence length for S(k). We use
+output embedding representation at the first position as the
+key-representation.
+Key-Aggregation: Once we create key-representations we
+need to utilize them for an end-task. We encode the key-
+representations KR(k), k ∈ K using another encoder. The
+choice of encoder used can be task specific as we make no
+assumptions about the encoder.
+Embd(J ) = KeyAggregator ({KR(k) | k ∈ K})
+(4)
+Column-evolution vs. Record-view Representation: In-
+stead of the column-evolution representation used by the
+TVM one could construct a record-level view to model the
+sequence [de Souza Pereira Moreira et al., 2021]. Specifi-
+cally, instead of modeling the evolution of columns in a semi-
+structured object sequence using V (k) and S(k) for each key,
+k such models view the sequence as a series of Ji. However,
+the record-view representation forces the network to com-
+press information present in multiple keys and values of a
+record (Ji) which can create an information bottleneck for
+the downstream task. We compare and contrast the benefit
+of these alternative views in the experiments (Section 3) and
+discussion section (Section 5) of this paper.
+Challenges of Scalable Training: The training of the hierar-
+chical two-part network for
+first obtaining the key-representations (Eq. 3) followed by
+obtaining structured object sequence representation (Eq. 4),
+could be done end-to-end where the network parameters are
+directly trained for the downstream task as illustrated in Fig-
+ure 2. However, due to challenges of scale, it is infeasible
+to train the network end-to-end. Specifically, the sequence
+length (N) can be very large3, and the size of the universe of
+the keys present in the structured object sequence can also be
+very large. 4
+Therefore, the end-to-end model architecture operating
+over a batch of J sequences would exceed the memory of
+most commodity GPUs.
+By a conservative estimate, for
+n = 11 and N = 512, a typical 120M parameter model,
+would exceed 40gb RAM limit with batch-size of 2.
+Interleaved Task Training: To address this challenge, we
+use a sequential task training paradigm where we interleave
+the training of the TVM and KA components.
+Note that
+3We experiment with object sequences with 3000 time steps.
+4Real-world data can have hundreds of keys in each structured
+object.
+
+our training paradigm is different from traditional training
+schedules for sequential task training where one network is
+fully trained before the next module or from fine-tuning ap-
+proaches where a part of the network maybe initialized with
+a pre-trained model and additional layers of the network ini-
+tialized randomly and then updated for an end-task.
+The interleaved training procedure is described in Algo-
+rithm 1. In our work we use the Masked language modeling
+(MLM) objective [Devlin et al., 2019] to train the TVM com-
+ponent and an end-task specific objective for training the VA.
+Algorithm 1 Interleaved training
+1: Initialize Temporal Value Modeler Mv, Key Aggregator Mk
+parameters randomly.
+2: Prepare the dataset Dv consisting of value Sequences S(k) as
+per Eq. 2
+3: for i = 1,2,. . . ,p do
+4:
+▷ TVM training
+5:
+Update TVM Mv model parameters with MLM objective on
+Dv.
+6:
+Prepare the dataset Dk consisting of key-representations
+KR(k) as per Eq. 3
+7:
+▷ KA training
+8:
+Update Key Aggregator Mk model parameters with cross-
+entropy loss for downstream task.
+9: end for
+Although, the interleaved training allows the modules to be
+trained independently, it makes the overall training harder as
+the two networks are disjoint. To overcome this limitation,
+we propose a novel model parameter sharing strategy as de-
+scribed below.
+Sharing Attention Heads: The TVM network creates a rep-
+resentation for each key by attending on the values that occur
+in the sequence for each of them. The KA network then uses
+these representations to learn the end-task. However, if the
+KA network could influence how these representations are
+created for each key, it could perhaps help improve the per-
+formance of the KA on the end-task. We therefore, introduce
+hard-parameter sharing between the TVM and KA compo-
+nents. Instead of sharing specific layers, we share some of the
+attention heads between the two networks. We hypothesize
+that by sharing a few attention heads between the two net-
+works, the KA will be able to increasingly rely on the shared
+attention heads because as training progresses and updates the
+parameters used in these heads, it will have an effect of ad-
+justing the key-representations from the TVM in a way that
+could help improve overall end-task performance.
+Formally, let the TVM and KA networks use h = p + q
+heads in multi-head attention of the transformer-based net-
+work. Then at any given layer in the network, their multihead
+attention layers are defined respectively as
+TE MultiHead(Q, K, V )
+=
+Concat(heads0, . . . , headsp,
+headt1, . . . , headtq)W O
+t
+KA MultiHead(Q, K, V )
+=
+Concat(heads0, . . . , headsp,
+headk1, . . . , headkq)W O
+k
+where headi
+=
+softmax
+�
+QW Q
+i
+�
+KW K
+i
+�T
+√dk
+�
+V W V
+i
+where s0, . . . , sp denotes the p shared attention heads,
+t1, . . . , tq denotes TextEncoder specific attention heads
+in the TVM, and k1, . . . , kq denotes KA specific attention
+heads, W Q
+i , W K
+i , are W V
+i projection matrices for query, key,
+and value for the i attention head. W O
+t and W O
+k are the out-
+put projection matrices for a layer of TextEncoder and the
+KA. We summarize our parameter sharing approach in Fig.
+2.
+The use of interleaved training as outlined in Algorithm
+1 prevents the problem of catastrophic forgetting [French,
+1999; McCloskey and Cohen, 1989; McClelland et al., 1995;
+Kumaran et al., 2016; Ratcliff, 1990] when the KA is trained.
+Further, it is possible that when the TVM is trained for the
+first time it may rely heavily on the heads that are shared.
+Thus, any change to the representation from these heads
+could lead to poorer key-representations and attention shar-
+ing in that case would be counter-productive. To overcome
+this problem, we apply DropHead [Zhou et al., 2020] on the
+shared attention heads in TVM and pre-train the model before
+beginning the interleaving schedule.
+Choice of Encoders: An advantage of the modular two-part
+encoder is that we can easily plug-in existing text encoder for
+TVM. We use BERT [Devlin et al., 2019] in our experimen-
+tation.
+For Key Aggregation we use the same model as the TVM
+TextEncoder but we do not use positional embeddings since
+we encode a set (and not a sequence) of keys.
+3
+Experiments
+Our experiments are designed to answer the following ques-
+tions: (i) How helpful is the TVM-KA architecture over
+baseline strategies that involve flattening semi-structured se-
+quence objects?
+(ii) How important is the use of shared-
+attention heads for fine-tuning of the Key Aggregator? (iii)
+How does the model compare to record-view representations
+[de Souza Pereira Moreira et al., 2021].
+3.1
+Data
+We choose two application/cloud logs datasets and one e-
+commerce purchase history dataset. The first application logs
+dataset, referred to as the ‘Cloud Service Logs’, is an in-
+ternal dataset consisting of interaction traces typically used
+for product usage analysis. We also use publicly available
+LogHub [He et al., 2020] dataset that consists of system log
+messages from Hadoop distributed file system and a publicly
+available e-commerce dataset consisting of product purchase
+information [stanley et al., 2017].
+Cloud Service Logs Data: Application event traces from
+a large cloud provider
+In the Cloud Service Logs (CSL) dataset, application event
+traces are logged in the cloud provider website. Each user
+is assigned a unique identifier. Event types range from lo-
+gin, browsing, account creation/maintenance/update, UI nav-
+igation, search, service creation/deletion, app interactions,
+
+Dataset
+Train
+Dev
+Test
+# classes
+Task
+Metric
+Cloud Service Logs
+12,833
+1,605
+1,604
+3
+Milestone Prediction
+Micro-F1 & Macro- F1
+LogHub
+402,542
+57,506
+115,012
+2
+Anomaly Detection
+F1
+Instacart
+780,003
+97,501
+97,500
+3,212
+Next Product Prediction
+Recall@k
+Table 1: Dataset Statistics
+among several others. We have about ?? unique event types.
+Each event has an associated payload which provides context
+around the event. For example, if a user performed a search,
+the payload captures the search query and the page where the
+search was performed. If a user interacted with a service, the
+payload captures the service ID and action performed on the
+service, among other information.
+Our raw data is a snapshot of application event traces span-
+ning 3-months comprising of about 450M events.
+Using
+these, we build our user sessions. A user session is essen-
+tially a temporal sequence of event traces for that user.
+We constructed user sessions for 100k users. The applica-
+tion events corresponding to 1) plan upgrade, and 2) opening
+chat bot (to seek help) are considered as milestone events.
+These milestone events are chosen as they represent revenue
+generation and user experience, respectively. The case of no
+milestone event occurring is treated as third class.
+From the traces, the temporal sequences of 300 events are
+extracted to predict if a milestone (or no-milestone) event will
+occur in next 50 time steps.
+Instacart eCommerce Data
+The publicly available Instacart dataset5 contains 3 million
+grocery purchase orders of nearly 200, 000 users of the appli-
+cation. Each order consists of multiple products and each
+structured object associated with a product contains meta-
+data such as the day of the week, the product category, de-
+partment, aisle, etc. We reprocess this dataset to create se-
+quences of product purchases and evaluate models on task
+of next product prediction. We create variable length train-
+ing instances from each user’s order history by sampling be-
+tween 50 to 200 previous product purchases for a certain tar-
+get product. Additionally, we only sample a training instance
+if the target product has been ordered at-least 50 times across
+users.
+As per our task formulation, we predict the product name
+given the sequence of product orders6, which is effectively
+a classification task over a universe of 3212 products. Ex-
+isting work on this dataset has focused on a simpler binary
+prediction task where models are asked to predict whether a
+particular item is likely to be purchased again.7
+LogHub Data
+HDFS-1 from LogHub [He et al., 2020] is utilized for log
+anomaly detection task. Originally, the dataset consists of log
+5https://tech.instacart.com/3-million-instacart-orders-open-
+sourced-d40d29ead6f2
+6We use the complete structured object.
+7https://www.kaggle.com/competitions/instacart-market-basket-
+analysis/leaderboard
+lines. We utilize Drain [He et al., 2017] log parser, that iden-
+tifies 48 log templates. In a semi-manual fashion, we assign
+key names to the value slots of the templates. Thus, each log
+line is converted to a structured object. Each structured ob-
+ject contains about 10 key-value pairs. A block consists of
+a sequence of such structured objects. The binary task is to
+predict if a block is anomalous or not.
+3.2
+Encoders
+Baselines: We experiment with BERT [Devlin et al., 2019]
+encoders as baseline.
+We flatten each key-value pair in a
+structured object and encode them with special markers in-
+dicating boundaries for objects and timesteps. We fine-tune
+the pre-trained encoders for each evaluation task and report
+their performance. In addition, instead of using a pre-trained
+encoder for BERT we also use the equivalent untrained model
+(with the same number of parameters) which is directly opti-
+mized for the evaluation tasks.
+Encoders for TVM-KA: One of the advantages of the TVM-
+KA architecture is agnostic to the choice of encoder. We use
+the same encoder architectures used in our baselines to fa-
+cilitate a direct comparison in performance. Recall that the
+TVM module and KA module share attention heads to facili-
+tate sharing of information between them.
+To pre-train the TVM, we mask 15
+3.3
+Hyperparameters and Training schedule
+Due to the compute limitations,
+we estimate hyper-
+parameters with Cloud Service Logs datasets and use them
+for other two datasets.
+We tune for the learning rates
+in {1e−4, 3e−4, 5e−4, 1e−5, 3e−5, 5e−5, 1e−6, 3e−6, 5e−6},
+and number of shared heads p in {2, 4, 6, 8}. The drophead
+mechanism is only activated during TVM training, with drop-
+head probability set to 0.2.
+For the TVM training in the first iteration, we vary learning
+rates and observe model convergence (in terms of train and
+dev loss) after a fixed 100K steps. The best learning rate for
+TVM is identified from this exercise. A similar approach is
+used for identifying the best learning rate for the KA training
+stage too, albeit for a smaller number of update steps. In
+the first iteration, TVM training is done for 1 epoch. Then
+we proceed with the interleaving step. We alternate between
+TVM training and KA training with their number of training
+steps in 2:1 proportion; specifically, for cloud service logs
+dataset, the number of TVM training steps is 50K and the
+number of KA training steps is around 100K.
+3.4
+Results
+In this section, we report the results from our experiments.
+
+Comments
+Cloud Service Logs
+Instacart
+Loghub
+Macro F1
+Micro F1
+Recall @10
+F1
+Flattened Encoding
+Pre-Trained
+74.77
+80.31
+16.4
+61.63
+Random
+50.22
+72.38
+9.6
+53.62
+TVM-KA
+No Interleaving
+73.22
+84.14
+17.5
+98.64
+Interleaving
+79.60
+86.49
+22.5
+99.32
+Table 2: Comparison of TVM-KA model with the baseline approaches on various datasets. All our transformer encoders have the same
+architecture and number of parameters as bert-base-uncased. In our proposed approach, both the TVM and KA have the same architecture
+and number of parameters as bert-base-uncased. The results from our approach are statistically significant compared to the baseline approach
+Model
+Configuration
+# Parameters
+Cloud Service Logs
+Loghub
+Joint Modeling
+No TVM Pre-Training
+209.42M
+69.61
+53.62
+TVM Pre-Training
+209.42M
+77.68
+70.72
+Record View
+104.41M
+77.33
+99.51
+Flattened Encoding
+bert-base-uncased
+104.41M
+74.77
+61.63
+bert-large-uncased
+319.62M
+76.59
+75.86
+Flattened Encoding
+allenai/longformer-base-4096
+141.77M
+73.71
+98.54
+google/bigbird-roberta-base
+122.13M
+70.69
+99.31
+TVM-KA
+With Interleaved Training
+199.28M
+79.60
+99.32
+Table 3: Ablation Experiments
+Comparison with Flattened encoding: Table 2 compares
+the performance of our approach with the baseline models.
+Our approach obtains statistically significant results com-
+pared to the baseline approaches on all three datasets. Loghub
+data suffers the most from information compression due to
+the stripping of sequences to 512 tokens. For Cloud Service
+Logs we report both Macro and Micro F1-Score due to class
+imbalance.
+For the baseline approach, we could either train an encoder
+with randomly initialized weights or an existing pre-trained
+model. We observe that training (fine-tuning) a pre-trained
+encoder results in the best performance compared to training
+an encoder with randomly initialized weights for the down-
+stream task.
+Importance of interleaved training: As seen from Table 2,
+our interleaving approach outperforms the model trained with
+no interleaving emphasizing that interleaving is a crucial step
+in our approach.
+3.5
+Ablation Study for Modeling choices
+Joint Modeling vs Interleaved: We observe that joint mod-
+eling performs poorly compared to our interleaving approach.
+In the joint modeling approach, the weights of the Temporal
+Value Modeler (TVM) are randomly initialized. We could
+train only the weights of the TVM using Masked Language
+Modeling (MLM) objective [Devlin et al., 2019] followed by
+joint training.
+We observe that joint modeling largely benefits from ini-
+tializing the TVM weights. When we train (pre-train) TVM
+using MLM objective, we are inducing a prior on the TVM
+which is not the case when we jointly train TVM-KA with
+randomly initialized weights.
+However, the performance of the joint model is poor com-
+pared to the interleaved training. By interleaving and sharing
+attention heads between TVM and KA, we are inducing task
+bias which benefits the model for the end task.
+Comparison with record-view We additionally compare our
+approach with an alternate view of the data, namely, the
+record view. Given a temporal sequence of JSON objects, we
+obtain a representation for the entire JSON. The sequence of
+the JSON representation is now passed through a transformer
+encoder followed by a sequence classifier module.
+We observe that our model obtains comparable perfor-
+mance when compared with the record view model.
+Effect of parameter size/model capacity To rule out the
+possibility of model capacity being the bottleneck for the
+baseline approach, we fine-tune bert-large-uncased model
+which has 3x the parameter of bert-base-uncased model. The
+results are presented in Table 3.
+We observe significant improvements on both Cloud Ser-
+vice Logs and Loghub datasets when we use bert-large-
+uncased over bert-base-uncased model with the flattened en-
+coding. However, the results are poor compared to our TVM-
+KA model. Thereby, demonstrating that the sequence length
+limit largely impacts the model performance with flattened
+encoding which is absent in our TVM-KA approach.
+
+Figure 3: Performance of Flattened Encoding models and out TVM-KA on Cloud Service. We divide the test set into four bins based on the
+sequence length
+Figure 4: Effect of sharing different attention heads between TVM and KA
+Comparison with Long Sequence Models: In the previous
+ablation, we observed that the sequence length limit is re-
+sponsible for existing pre-trained models performing poorly
+with the flattened encoding. We now use pre-trained models
+
+1.0
+bert-random
+bert-pretrained
+bert-large-pretrained
+longformer
+0.8
+TVM KA
+e
+0.6
+-Scor
+F1
+acro 
+M
+0.4
+0.2
+0.0
+: 1024
+< len(x)
+Λ=
+512
+Seguence
+Lenath100
+Cloud Service Logs
+Instacart (right)
+20
+80
+15
+60
+Recall@10
+F1
+acro
+Ma(
+10
+40
+20 
+5
+0
+0
+0
+2
+4
+6
+8
+Number of Shared Attention Headsnamely Longformer [Beltagy et al., 2020b], BigBird [Zaheer
+et al., 2020] which can handle longer sequences of up to 4096
+tokens (as opposed to 512 tokens of BERT models) with the
+flattened encoding. The results are presented in Table 3.
+We observe that ability to handle longer sequence length
+benefits the Loghub dataset. The performance of these mod-
+els is comparable with our TVM-KA approach. On the con-
+trary, we do not observe any improvements over the bert-
+base-uncased models for Cloud Service Logs.
+Additionally, we report the performance of the above mod-
+els on sequences from each test set having varying sequence
+lengths. Figure 3 demonstrates the performance vs sequence
+length comparison. We observe that our TVM-KA approach
+consistently outperforms flattened encoding baselines on all
+sequence length bins. Surprisingly, pre-trained longformer
+gives comparable performance with bert-base-uncased and
+bert-large-uncased models.
+Number of Shared Attention Heads We use BERT encoder
+[Devlin et al., 2019] to model both TVM and KA compo-
+nents in our model. This allows us to share attention heads
+between the TVM and KA components. bert-base-uncased
+has 12 attention heads at each encoder layer. We experiment
+with sharing 0, 2, 4, 6, 8 attention heads between TVM and
+KA components.
+Figure 4 presents the performance of our TVM-KA ap-
+proach with different numbers of shared attention heads be-
+tween TVM and KA components. In general, we observe that
+sharing of 4 and 6 attention heads helps the most. While not
+sharing any attention heads or sharing more results in poor
+performance.
+4
+Related Work
+Our work relates to multiple areas in existing literature. First,
+modeling multiple sets of key-value entries has similarities to
+modeling the rows and columns in a table. Second, the pre-
+dictive tasks we apply our models are related to multi-variate
+regression and spatio-temporal modeling tasks. Finally, our
+two stage TVM-KA architecture is a form of hierarchical en-
+coding, forms of which have been used for multiple tasks in
+the past – from tabular data modeling to encoding text.
+Modeling Tabular and Timeseries Data: In our work we
+explicitly model the temporal sequences for values for each
+key that is encountered in the sequences. In this sense the na-
+ture of the data we are modeling is related to modeling tabular
+data where each row is a time-step and the columns indicate
+different values for a particular field. However, even though
+at the surface the modeling of data appears related, the actual
+tasks and models developed for tasks on tabular data cannot
+be applied to semi-structured sequence objects. This is be-
+cause the work on modeling textual tabular data often focuses
+on retrieving information from cells
+[Zayats et al., 2021;
+Wang et al., 2021; Iida et al., 2021], multi-hop reasoning
+across information in different cells across parts of the table
+[Chen et al., 2021; Chen et al., 2020; Zhao et al., 2022], com-
+bining information present in tables and unstructured text for
+information seeking tasks [Li et al., 2021; Zhu et al., 2021;
+Zayats et al., 2021; Chen et al., 2020], etc. In addition, work
+on modeling time-series data present in the form of tables has
+been focused on numerical data [Zhou et al., 2021; Zerveas
+et al., 2021] with purpose-built task specific architectures that
+cannot be easily adapted for other tasks [Wu et al., 2021;
+Padhi et al., 2021].
+Modeling Temporal Graph Sequences and Recommender
+Systems: Finally our work is also related to a rich body
+of work related to modeling temporal graphs and recom-
+mender systems [Xu et al., 2021b; Grigsby et al., 2021;
+de Souza Pereira Moreira et al., 2021].
+Temporal Graph
+evolution problems involve constructing representations to
+enable tasks such as link prediction [Sankar et al., 2020;
+Xu et al., 2021a], item recommendation in user sessions
+[Hsu and te Li, 2021], answering queries on graphs and se-
+quences [Saxena et al., 2022], classifying graph instances in
+a sequence,[Xu et al., 2021a; Xu et al., 2021b] etc. How-
+ever, we find that such approaches either do not scale for long
+sequences for the tasks we experiment with or are likely to
+suffer from an information bottleneck resulting in lower per-
+formance as compared to TVM-KA.
+Fundamentally, we find that such approaches also limit our
+ability to model text sequences as in Figure 1(c), because such
+a model would create a sequence representation over graph-
+computed sentence representations (each sentence would be
+graph and the tokens within that sentence a node). As a result,
+sentence tokens would not directly interact across times-step
+boundaries. Similarly in the case of inputs with features such
+as in Figure 1(d) it would limit the ability to model such ut-
+terances across time-step boundaries since a representation
+would first be created using fields and meta-data within each
+turn (time-step) and then aggregated as a sequence. We dis-
+cuss more about how temporal graphs sequences and our ap-
+proach to model structured object sequences differ in Section
+5.
+5
+Discussion
+The choice of a column-evolution representation enables us
+to encode larger objects as well as long sequences. Our ex-
+periments demonstrate that by using the two-part TVM-KA
+architecture we are able to incorporate information about the
+downstream task as a form of inductive bias for the value
+modeler network that provides key-representations. We hy-
+pothesize it is this task-specific bias that helps our model per-
+form better than joint-modeling (when computationally fea-
+sible).
+However,
+the column-evolution representation of se-
+quences does not allow it to support tasks as sequence tagging
+of the structured objects. In addition, it also does not allow
+it to model graph sequences effectively as it does not use a
+global view of a structured objects. Thus, it may not be able
+learn patterns across fields at different time steps. For such
+tasks, a record-view representation is more helpful. The both
+representations have their strengths and weaknesses and the
+choice of using a column-view representation vs a a record-
+view representation should be made keeping the downstream
+task in mind. To the best of our knowledge we are the first
+to demonstrate the use of a column-view representation for
+structured object sequence encoding at scale.
+
+6
+Conclusion
+In this paper, we have presented a two-part encoder to model
+structured sequence objects. We additionally present a novel
+interleaving scheme to train our two-part encoder. We also
+induce task bias into the model by sharing attention heads
+between Temporal Value Modeler and Key Aggregator com-
+ponents. Our proposed approach outperforms baseline ap-
+proaches which flatten structured sequence objects and gives
+comparable performance to record-view approaches.
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+

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+Extremal problems for the eccentricity matrices of complements of
+trees
+Iswar Mahato ∗
+M. Rajesh Kannan†
+January 5, 2023
+Abstract
+The eccentricity matrix of a connected graph G, denoted by E(G), is obtained from the
+distance matrix of G by keeping the largest nonzero entries in each row and each column, and
+leaving zeros in the remaining ones. The E-eigenvalues of G are the eigenvalues of E(G), in which
+the largest one is the E-spectral radius of G. The E-energy of G is the sum of the absolute values
+of all E-eigenvalues of G. In this article, we study some of the extremal problems for eccentricity
+matrices of complements of trees and characterize the extremal graphs. First, we determine the
+unique tree whose complement has minimum (respectively, maximum) E-spectral radius among
+the complements of trees. Then, we prove that the E-eigenvalues of the complement of a tree are
+symmetric about the origin. As a consequence of these results, we characterize the trees whose
+complement has minimum (respectively, maximum) least E-eigenvalues among the complements
+of trees. Finally, we discuss the extremal problems for the second largest E-eigenvalue and the
+E-energy of complements of trees and characterize the extremal graphs. As an application, we
+obtain a Nordhaus-Gaddum type lower bounds for the second largest E-eigenvalue and E-energy
+of a tree and its complement.
+AMS Subject Classification (2010): 05C50, 05C35.
+Keywords. Complements of trees, Eccentricity matrix, E-spectral radius, Second largest E-
+eigenvalue, Least E-eigenvalue, E-energy.
+1
+Introduction
+Throughout the paper, we consider finite, simple, and connected graphs. Let G = (V (G), E(G))
+be a graph with the vertex set V (G) = {v1, v2, . . . , vn} and the edge set E(G) = {e1, e2, . . . , em}.
+The number of vertices in G is the order of G. If two vertices u and v in G are adjacent, then we
+write u ∼ v, otherwise u ≁ v. The adjacency matrix A(G) of G is the n × n matrix with its rows
+and columns indexed by the vertices of G, and the entries are defined as
+A(G)uv =
+�
+1
+if u ∼ v,
+0
+otherwise.
+∗Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India.
+Email:
+iswarmahato02@gmail.com, iswarmahato02@iitkgp.ac.in
+†Department of Mathematics, Indian Institute of Technology Hyderabad, Hyderabad 502285, India. Email: ra-
+jeshkannan1.m@gmail.com, rajeshkannan@math.iith.ac.in
+1
+arXiv:2301.01708v1  [math.CO]  4 Jan 2023
+
+Let λ1(G) ≥ λ2(G) ≥ . . . ≥ λn(G) be the eigenvalues of the adjacency matrix A(G) of G. The
+largest eigenvalue of A(G) is the spectral radius of G. The energy (or the A-energy) of G is defined
+as EA(G) = �n
+i=1
+��λi(G)
+��. The distance between the vertices u and v in G, denoted by dG(u, v), is
+the length of a shortest path between them in G, and define dG(u, u) = 0 for all u ∈ V (G). The
+distance matrix D(G) of G is the n×n matrix whose rows and columns are indexed by the vertices
+of G and the (u, v)-th entry is equal to dG(u, v). Let NG(v) denote the collection of vertices that
+are adjacent to the vertex v in G, and NG(v) is called the neighborhood of v in G. The eccentricity
+eG(v) of a vertex v ∈ V (G) is the maximum distance from v to all other vertices of G.
+The
+maximum eccentricity of all vertices of G is the diameter of G, which is denoted by diam(G). The
+diametrical path is a path whose length is equal to the diameter of G.
+The eccentricity matrix of a graph G on n vertices, denoted by E(G), is the n × n matrix whose
+rows and columns are indexed by the vertices of G, and the entries are defined as
+E(G)uv =
+�
+dG(u, v)
+if dG(u, v) = min{eG(u), eG(v)},
+0
+otherwise.
+The eccentricity matrix E(G) of a graph G is a real symmetric matrix, and hence all of its eigenvalues
+are real. The eigenvalues of E(G) are the E-eigenvalues of G, in which the largest one is the E-
+spectral radius of G. The set of all E-eigenvalues of G is the E-spectrum of G. If ξ1 > ξ2 > . . . > ξk
+are the distinct E-eigenvalues of G, then we write the E-spectrum of G as
+Specε(G) =
+�
+ξ1
+ξ2
+. . .
+ξk
+m1
+m2
+. . .
+mk
+�
+,
+where mi is the multiplicity of ξi for i = 1, 2, . . . , k.
+Spectral extremal problems are one of the interesting problems in spectral graph theory. Re-
+cently, the extremal problems for eccentricity matrices of graphs have gained significant importance
+and attracted the attention of researchers. In [23], Wei et al. considered the extremal problem for
+E-spectral radius of trees and determined the trees with minimum E-spectral radius among all trees
+on n vertices. Also, they characterized the trees with minimum E-spectral radius among the trees
+with a given diameter. In [14], the authors studied the minimal problem for E-spectral radius of
+graphs with a given diameter and characterized the extreme graphs. Moreover, they identified the
+unique bipartite graph with minimum E-spectral radius. Wang et al. [19] characterized the graphs
+with minimum and second minimum E-spectral radius as well as the graphs with maximum least
+and second least E-eigenvalues. Recently, He and Lu [4] considered the maximal problem for the
+E-spectral radius of trees with the fixed odd diameter and determined the extremal trees. Wei et
+al. [25] characterized the trees with second minimum E-spectral radius and identified the trees with
+the small matching number having the minimum E-spectral radius. Very Recently, Wei and Li [24]
+studied the relationship between the majorization and E-spectral radius of complete multipartite
+graphs and determined the extremal complete multipartite graphs with minimum and maximum
+E-spectral radius. Mahato and Kannan [16] considered the extremal problem for the second largest
+E-eigenvalue of trees and determined the unique tree with minimum second largest E-eigenvalue
+among all trees on n vertices other than the star. For more advances on the eccentricity matrices
+of graphs, we refer to [7, 8, 13, 15, 17, 18, 21, 22].
+The eccentricity energy (or the E-energy) of a graph G is defined [20] as
+EE(G) =
+n
+�
+i=1
+|ξi| ,
+2
+
+where ξ1, ξ2, . . . , ξn are the E-eigenvalues of G. Recently, many researchers focused on the eccen-
+tricity energy of graphs. In [20], Wang et al. studied the E-energy of graphs and obtained some
+bounds for the E-energy of graphs and determined the corresponding extremal graphs. Lei et al.
+[8] obtained an upper bound for the E-energy of graphs and characterized the extremal graphs.
+Very recently, Mahato and Kannan [16] studied the minimization problem for the E-energy of trees
+and characterized the trees with minimum E-energy among all trees on n vertices. For more details
+about the E-energy of graphs, we refer to [14, 15, 18].
+Motivated by the above-mentioned works, in this article, we study the extremal problems for
+eccentricity matrices of complements of trees and characterize the extremal graphs for the E-spectral
+radius, second largest E-eigenvalue, least E-eigenvalue and E-energy among the complements of
+trees. The extremal problems for the complements of trees with respect to the other graph matrices
+have been studied in [2, 9, 10, 11]. For a tree T, let T c be the complement of T. Throughout the
+paper, we always assume that T c is connected; hence, T is not isomorphic to the star graph. Let
+T c
+n denote the complements of all trees on n vertices. Note that if T is a tree with diam(T) > 3,
+then E(T c) = 2A(T). If diam(T) = 3, then E(T c) ≥ 2A(T) entrywise. Since A(T) is irreducible,
+therefore E(T c) is also irreducible.
+The article is organized as follows: In section 2, we collect needed notations and some prelim-
+inary results. In section 3, we characterize the extremal graphs with the minimum and maximum
+E-spectral radius among the complements of trees. As a consequence, we determine the unique
+graphs with the minimum and maximum least E-eigenvalues among the complements of trees. We
+discuss the extremal problems for the second largest E-eigenvalue and the E-energy of complements
+of trees in section 4 and section 5, respectively.
+2
+Preliminaries
+In this section, we introduce some notations and collect some preliminary results, which will be
+used in the subsequent sections. First, we define the quotient matrix and equitable partition.
+Definition 2.1 (Equitable partition [1]). Let A be a real symmetric matrix whose rows and columns
+are indexed by X = {1, 2, . . . , n}. Let Π = {X1, X2, . . . , Xm} be a partition of X. The characteristic
+matrix C is the n×m matrix whose j-th column is the characteristic vector of Xj (j = 1, 2, . . . , m).
+Let A be partitioned according to Π as
+A =
+�
+�����
+A11
+A12
+. . .
+A1m
+A21
+A22
+. . .
+A2m
+...
+. . .
+...
+...
+Am1
+Am2
+. . .
+Amm
+�
+�����
+,
+where Aij denotes the submatrix (block) of A formed by rows in Xi and the columns in Xj. If qij
+denotes the average row sum of Aij, then the matrix Q = (qij) is called the quotient matrix of A.
+If the row sum of each block Aij is a constant, then the partition Π is called equitable partition.
+In the following theorem, we state a well-known result about the spectrum of a quotient matrix
+corresponding to an equitable partition.
+Theorem 2.1 ([1]). Let Q be a quotient matrix of any square matrix A corresponding to an equitable
+partition. Then the spectrum of A contains the spectrum of Q.
+3
+
+Figure 1: The tree T a,b
+n,3, where a + b = n − 4.
+Figure 2: The tree Da,b
+n,d, where a + b = n − d − 1 and d ≥ 4.
+Let Kn, Pn, and K1,n−1 denote the complete graph, the path, and the star on n vertices,
+respectively. For d = 3, let T a,b
+n,d be the tree obtained from P4 = v0v1v2v3 by attaching a pendant
+vertices to v1 and b pendent vertices to v2, where a + b = n − 4 and b ≥ a ≥ 0. For d ≥ 4, let Da,b
+n,d
+be the tree obtained from Pd+1 = v0v1v2 . . . vd by attaching a pendant vertices to v1 and b pendent
+vertices to vd−1, where a + b = n − d − 1 and b ≥ a ≥ 0. The trees T a,b
+n,3 and Da,b
+n,d are depicted in
+Figure 1 and Figure 2, respectively. For a real number x, let ⌊x⌋ denote the greatest integer less
+than or equal to x, and ⌈x⌉ denote the least integer greater than or equal to x. In the following
+lemma, we compute the E-spectrum of the complement of the tree T a,b
+n,3, where a + b = n − 4 and
+b ≥ a ≥ 0.
+Lemma 2.1. For b ≥ a ≥ 0 with a + b = n − 4, the E-spectrum of (T a,b
+n,3)c is given by
+SpecE
+�
+(T a,b
+n,3)c�
+=
+�
+�
+�
+−
+�
+4n+1±√
+(4n+1)2−64(a+1)(b+1)
+2
+0
+�
+4n+1±√
+(4n+1)2−64(a+1)(b+1)
+2
+1
+n − 4
+1
+�
+�
+� .
+Proof. Let T a,b
+n,3 be the tree obtained from P4 = v0v1v2v3 by attaching a pendant vertices u1, . . . , ua
+to v1 and b pendent vertices w1, . . . , wb to v2, where a + b = n − 4 and b ≥ a ≥ 0. Then the
+4
+
+Vo
+V1
+V2
+V3
+bVo
+V1
+V2
+Vd-2
+Vd-1
+Vd
+a
+beccentricity matrix of (T a,b
+n,3)c is given by
+E
+�
+(T a,b
+n,3)c�
+=
+u1
+. . .
+ua
+v0
+v1
+v2
+v3
+w1
+. . .
+wb
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+u1
+0
+. . .
+0
+0
+2
+0
+0
+0
+. . .
+0
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+ua
+0
+. . .
+0
+0
+2
+0
+0
+0
+. . .
+0
+v0
+0
+. . .
+0
+0
+2
+0
+0
+0
+. . .
+0
+v1
+2
+. . .
+2
+2
+0
+3
+0
+0
+. . .
+0
+v2
+0
+. . .
+0
+0
+3
+0
+2
+2
+. . .
+2
+v3
+0
+. . .
+0
+0
+0
+2
+0
+0
+. . .
+0
+w1
+0
+. . .
+0
+0
+0
+2
+0
+0
+. . .
+0
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+wb
+0
+. . .
+0
+0
+0
+2
+0
+0
+. . .
+0
+.
+It is easy to see that the rank of E
+�
+(T a,b
+n,3)c�
+is 4. Therefore, 0 is an eigenvalue of E
+�
+(T a,b
+n,3)c�
+with
+multiplicity n − 4.
+If U = {u1, . . . , ua, v0} and W = {v3, w1, . . . , wb}, then Π1 = U ∪{v1}∪{v2}∪W is an equitable
+partition of E
+�
+(T a,b
+n,3)c�
+with the quotient matrix
+Q1 =
+�
+�
+�
+�
+�
+0
+2
+0
+0
+2(a + 1)
+0
+3
+0
+0
+3
+0
+2(b + 1)
+0
+0
+2
+0
+�
+�
+�
+�
+� .
+By a direct calculation, the eigenvalues of Q1 are
+±
+�
+4a + 4b + 17 +
+�
+(4a + 4b + 17)2 − 64(a + 1)(b + 1)
+2
+and
+±
+�
+4a + 4b + 17 −
+�
+(4a + 4b + 17)2 − 64(a + 1)(b + 1)
+2
+.
+Now, the proof follows from Theorem 2.1 by substituting a + b = n − 4.
+In the following theorem, we find the adjacency energy of the path graph on n vertices. We
+include proof of this result for the sake of completeness.
+Theorem 2.2. The energy of a path Pn on n vertices is given by
+EA(Pn) =
+�
+�
+�
+�
+�
+2
+�
+cot
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is odd,
+2
+�
+csc
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is even.
+Proof. We know that the eigenvalues of Pn are 2 cos πk
+n+1, k = 1, 2, . . . , n. By using the formula
+cos x + cos 2x + . . . + cos nx = sin
+� nx
+2
+�
+csc
+� x
+2
+�
+cos
+� (n+1)x
+2
+�
+(for a proof of this identity, we refer to
+5
+
+[6]), we have
+EA(Pn) =
+�
+�
+�
+�
+�
+4
+�
+cos
+�
+π
+n+1
+�
++ cos
+� 2π
+n+1
+�
++ . . . + cos
+� (n−1)π
+2(n+1)
+��
+if n is odd,
+4
+�
+cos
+�
+π
+n+1
+�
++ cos
+� 2π
+n+1
+�
++ . . . + cos
+�
+nπ
+2(n+1)
+��
+if n is even;
+=
+�
+�
+�
+4 sin
+� (n−1)π
+4(n+1)
+�
+csc
+�
+π
+2(n+1)
+�
+cos
+� π
+4
+�
+if n is odd,
+4 sin
+�
+nπ
+4(n+1)
+�
+csc
+� (n+2)π
+4(n+1)
+�
+cos
+� π
+4
+�
+if n is even;
+=
+�
+�
+�
+�
+�
+2
+�
+cos
+�
+π
+2(n+1)
+�
+− sin
+�
+π
+2(n+1)
+��
+csc
+�
+π
+2(n+1)
+�
+if n is odd,
+2
+�
+sin
+� π
+2
+�
+− sin
+�
+π
+2(n+1)
+��
+csc
+�
+π
+2(n+1)
+�
+if n is even;
+=
+�
+�
+�
+�
+�
+2
+�
+cot
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is odd,
+2
+�
+csc
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is even.
+Now, we collect some results related to the spectral radius, second-largest eigenvalue, and energy
+for the adjacency matrices of trees.
+Theorem 2.3 ([12]). Let T be a tree on n vertices. Then λ1(T) ≥ 2 cos
+π
+n+1 with equality if and
+only if T ∼= Pn.
+Theorem 2.4 ([5, Theorem 2]). Let T be a tree on n ≥ 4 vertices and T ≇ K1,n−1.
+Then
+λ1(T) ≤
+�
+n−1+
+√
+n2−6n+13
+2
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Theorem 2.5 ([26, Theorem 2]). Let T be a tree of order n ≥ 4 and T ≇ K1,n−1, T 0,n−4
+n,3
+. Then
+λ2(T) ≥ 1.
+Theorem 2.6 ([5, Theorem 10]). Let T be a tree with n ≥ 4 vertices.
+1. If T ≇ T s−1,s−1
+n,5
+, then λ2(T) ≤
+�
+n−3
+2 . The equality holds if and only if n = 2s + 3 and
+T ∼= T s−1,s−1
+n,4
+, T s−2,s−1
+n,5
+, T s−2,s−2
+n,6
+.
+2. If T ∼= T s−1,s−1
+n,5
+, then λ2(T) = x >
+�
+n−3
+2 , where x is the positive root of the equation
+x3 + x2 − (s + 1) − s = 0.
+Lemma 2.2 ([3, Proposition 4]). Let T be a tree on n ≥ 2 vertices such that T is not isomorphic
+to K1,n−1, T 0,n−4
+n,3
+, T 1,n−3
+n,3
+and T 0,n−5
+n,4
+. Then
+EA(T) > EA(T 0,n−5
+n,4
+) > EA(T 1,n−3
+n,3
+) > EA(T 0,n−4
+n,3
+) > EA(K1,n−1).
+Lemma 2.3 ([3, Proposition 3]). Let T be a tree on n ≥ 2 vertices such that T ≇ K1,n−1, Pn. Then
+EA(K1,n−1) < EA(T) < EA(Pn).
+6
+
+Next, we state a result about the minimum second largest E-eigenvalue of trees.
+Theorem 2.7 ([16]). Let T be a tree on n ≥ 4 vertices other than the star. Then
+ξ2(T) ≥
+�
+13n − 35 −
+√
+169n2 − 974n + 1417
+2
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+The following theorem is about the characterization of trees with minimum E-energy.
+Theorem 2.8 ([16]). Let T be a tree on n ≥ 5 vertices. Then
+EE(T) ≥ 2
+�
+13n − 35 + 8
+√
+n − 3
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+3
+Extremal problems for E-spectral radius and least E-eigenvalue
+In this section, we characterize the graphs with the minimum and maximum E-spectral radius
+among the complements of all trees on n vertices. As a consequence, we determine the graphs for
+which the least E-eigenvalues attain the minimum and maximum value among T c
+n , where T c
+n denote
+the complements of all trees on n vertices. First, we give an ordering of the complements of trees
+with diameter 3 according to their E-spectral radius.
+Theorem 3.1. Let T be a tree with diameter 3, that is, T ∼= T a,b
+n,3 with a + b = n − 4, b ≥ a ≥ 0.
+Then the complements of T a,b
+n,3 can be ordered with respect to their E-spectral radius as follows:
+ξ1
+�
+(T 0,n−4
+n,3
+)c�
+> ξ1
+�
+(T 1,n−3
+n,3
+)c�
+> . . . > ξ1
+��
+T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+�c�
+.
+Proof. By Lemma 2.1 , we have
+ξ1
+�
+(T a,b
+n,3)c�
+=
+�
+4n + 1 +
+�
+(4n + 1)2 − 64(a + 1)(b + 1)
+2
+=
+�
+4n + 1 +
+�
+(4n + 1)2 − 64(n − 3 + a(n − 4 − a))
+2
+(since, a + b = n − 4).
+Now, consider the function f(x) = 4n + 1 +
+�
+(4n + 1)2 − 64(n − 3 + x(n − 4 − x)), where
+0 ≤ x ≤ ⌊n−4
+2 ⌋. Therefore,
+f′(x) =
+−32(n − 4 − 2x)
+�
+(4n + 1)2 − 64(n − 3 + x(n − 4 − x))
+≤ 0.
+Hence, f(x) is an decreasing function of x, where 0 ≤ x ≤ ⌊ n−4
+2 ⌋. Therefore, ξ1
+�
+(T a,b
+n,3)c�
+=
+�
+f(a)
+2
+is an decreasing function for 0 ≤ a ≤ ⌊ n−4
+2 ⌋. This completes the proof.
+7
+
+As a consequence of the above result, we get the following corollaries.
+Corollary 3.1. Let T be a tree on n ≥ 4 vertices with diam(T) = 3. Then
+ξ1(T c) ≤
+�
+4n + 1 +
+�
+(4n + 1)2 − 64(n − 3)
+2
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. The proof follows from Lemma 2.1 and Theorem 3.1.
+Corollary 3.2. If T is a tree with diameter 3, then
+ξ1(T c) ≥
+�
+�
+�
+�
+�
+�
+4n+1+√72n−63
+2
+if n is even,
+�
+4n+1+√72n−47
+2
+if n is odd.
+The equality holds if and only if T ∼= T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+.
+Proof. If T ∼= T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+, by Lemma 2.1 it follows that
+ξ1(T c) =
+�
+�
+�
+�4n + 1 +
+�
+(4n + 1)2 − 64
+�
+⌈ n−2
+2 ⌉⌊ n−2
+2 ⌋
+�
+2
+=
+�
+�
+�
+�
+�
+�
+4n+1+√72n−63
+2
+if n is even,
+�
+4n+1+√72n−47
+2
+if n is odd.
+If T ≇ T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+, then, by Theorem 3.1, it follows that ξ1(T c) > ξ1
+��
+T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+�c�
+. This
+completes the proof.
+In the following theorem, we characterize the trees whose complement has the minimum E-
+spectral radius among the complements of all trees on n vertices. Note that if T is a tree with
+diam(T) > 3, then E(T c) = 2A(T). If diam(T) = 3, then E(T c) ≥ 2A(T) entrywise.
+Theorem 3.2. Let T be a tree of order n ≥ 4. Then
+ξ1(T c) ≥ ξ1(P c
+n)
+with equality if and only if T ∼= Pn.
+Proof. Since P4 is the only tree on 4 vertices with connected complement, we assume that n ≥ 5.
+For n ≥ 5, we have ξ1(P c
+n) = 2λ1(Pn) = 4 cos
+π
+n+1 < 4.
+If T is a tree with diameter 3, by Corollary 3.2 it follows that
+ξ1(T c) ≥
+�
+�
+�
+�
+�
+�
+4n+1+√72n−63
+2
+if n is even,
+�
+4n+1+√72n−47
+2
+if n is odd.
+It is easy to check that
+�
+4n+1+√72n−47
+2
+>
+�
+4n+1+√72n−63
+2
+> 4. Therefore, ξ1(T c) ≥ ξ1(P c
+n).
+If T is a tree with diam(T) ≥ 4, then the proof follows from Theorem 3.2.
+8
+
+Now, we determine the unique tree whose complement has maximum E-spectral radius in T c
+n .
+Theorem 3.3. Let T be a tree of order n ≥ 4. Then
+ξ1(T c) ≤ ξ1
+�
+(T 0,n−4
+n,3
+)c�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. If T is a tree with diameter 3, then the proof follows from Corollary 3.1. If T is a tree with
+diam(T) ≥ 4, then ξ1(T c) = 2λ1(T) and the proof follows from Theorem 2.4.
+Now, we consider the extremal problem for the least E-eigenvalue of complements of trees and
+characterize the extremal graphs. First, we show that the E-eigenvalues of the complements of trees
+are symmetric about the origin.
+Theorem 3.4. Let T be a tree of order n ≥ 4. Then the eigenvalues of E(T c) are symmetric about
+the origin, that is, if λ is an eigenvalue of E(T c) with multiplicity k, then −λ is also an eigenvalue
+of E(T c) with multiplicity k.
+Proof. If T is a tree with diameter 3, then the proof follows from Lemma 2.1. Let T be a tree with
+diam(T) ≥ 4. Then E(T c) = 2A(T). Since every tree is a bipartite graph, the eigenvalues of A(T)
+are symmetric about the origin, and hence the E-eigenvalues of T c are also symmetric about the
+origin.
+Let ξn(T c) denote the least E-eigenvalue of E(T c). In the following theorems, we characterize
+the trees whose complements have minimum and maximum least E-eigenvalue in T c
+n , respectively.
+Theorem 3.5. Let T be a tree of order n ≥ 4. Then
+ξn(T c) ≥ ξn
+�
+(T 0,n−4
+n,3
+)c�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. The proof follows from Theorem 3.3 and Theorem 3.4.
+Theorem 3.6. Let T be a tree of order n ≥ 4. Then
+ξn(T c) ≤ ξn(P c
+n)
+with equality if and only if T ∼= Pn.
+Proof. The proof follows from Theorem 3.2 and Theorem 3.4.
+4
+Extremal problems for the second largest E-eigenvalue
+In this section, we study the extremal problems for the second largest E-eigenvalue of complements
+of trees and characterize the extremal graphs. First, we give an ordering of the complements of
+trees with diameter 3 according to their second-largest E-eigenvalues.
+9
+
+Theorem 4.1. Let T be a tree with diameter 3, that is, T ∼= T a,b
+n,3 with a+b = n−4, b ≥ a ≥ 0. Then
+the complements of the trees T a,b
+n,3 can be ordered with respect to their second largest E-eigenvalues
+as follows:
+ξ2
+�
+(T 0,n−4
+n,3
+)c�
+< ξ2
+�
+(T 1,n−3
+n,3
+)c�
+< . . . < ξ2
+��
+T
+⌊ n−2
+2 ⌋,⌈ n−2
+2 ⌉
+n,3
+�c�
+.
+Proof. By Lemma 2.1, it follows that
+ξ2
+�
+(T a,b
+n,3)c�
+=
+�
+4n + 1 −
+�
+(4n + 1)2 − 64(a + 1)(b + 1)
+2
+=
+�
+4n + 1 −
+�
+(4n + 1)2 − 64(n − 3 + a(n − 4 − a))
+2
+(since, a + b = n − 4).
+Now, consider the function f(x) = 4n + 1 −
+�
+(4n + 1)2 − 64(n − 3 + x(n − 4 − x)), where
+0 ≤ x ≤ ⌊n−4
+2 ⌋. Therefore,
+f′(x) =
+32(n − 4 − 2x)
+�
+(4n + 1)2 − 64(n − 3 + x(n − 4 − x))
+≥ 0.
+Hence, f(x) is an increasing function of x for 0 ≤ x ≤ ⌊n−4
+2 ⌋. Therefore, ξ2
+�
+(T a,b
+n,3)c�
+=
+�
+f(a)
+2
+is an
+increasing function for 0 ≤ a ≤ ⌊n−4
+2 ⌋. This completes the proof.
+As a consequence of the above result, we get the following corollaries.
+Corollary 4.1. If T is a tree with diameter 3, then
+ξ2(T c) ≥ 2
+�
+4n + 1 −
+�
+(4n + 1)2 − 64(n − 3)
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. The proof follows from Lemma 2.1 and Theorem 4.1.
+Corollary 4.2. If T is a tree with diameter 3, then
+ξ2(T c) ≤
+�
+�
+�
+�
+�
+�
+4n+1−√72n−63
+2
+if n is even,
+�
+4n+1−√72n−47
+2
+if n is odd.
+The equality holds if and only if T ∼= T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+.
+Proof. If T ∼= T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+, by Lemma 2.1 it follows that
+ξ2(T c) =
+�
+�
+�
+�4n + 1 −
+�
+(4n + 1)2 − 64
+�
+⌈ n−2
+2 ⌉⌊ n−2
+2 ⌋
+�
+2
+=
+�
+�
+�
+�
+�
+�
+4n+1−√72n−63
+2
+if n is even,
+�
+4n+1−√72n−47
+2
+if n is odd.
+10
+
+If T ≇ T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+, by Theorem 4.1 it follows that ξ2(T c) < ξ2
+��
+T
+⌊ n−2
+2 ⌋,⌈ n−2
+2 ⌉
+n,3
+�c�
+. This completes
+the proof.
+In the following theorem, we determine the unique tree whose complement has minimum second
+largest E-eigenvalue in T c
+n .
+Theorem 4.2. Let T be a tree of order n ≥ 4. Then
+ξ2(T c) ≥ ξ2
+�
+(T 0,n−4
+n,3
+)c�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. If T is a tree on n ≥ 4 vertices with diameter 3, then the proof follows from Corollary 4.1.
+Let T be a tree with diam(T) ≥ 4. Then E(T c) = 2A(T), and hence by Theorem 2.5, we
+have ξ2(T c) ≥ 2. Note that
+�
+(4n − 7)2 + 144 > 4n − 7 and hence (4n + 1 −
+�
+(4n − 7)2 + 144) <
+8.
+Therefore, ξ2
+�
+(T 0,n−4
+n,3
+)c�
+=
+�
+4n+1−√
+(4n−7)2+144
+2
+< 2.
+Thus, ξ2(T c) > ξ2
+�
+(T 0,n−4
+n,3
+)c�
+.
+This
+completes the proof.
+Next, we characterize the maximal graphs for the second largest E-eigenvalue of complements
+of trees.
+Theorem 4.3. Let T be a tree with n ≥ 4 vertices.
+1. If T ≇ T s−1,s−1
+n,5
+, then ξ2(T) ≤
+�
+2(n − 3). The equality holds if and only if n = 2s + 3 and
+T ∼= T s−1,s−1
+n,4
+, T s−2,s−1
+n,5
+, T s−2,s−2
+n,6
+.
+2. If T ∼= T s−1,s−1
+n,5
+, then ξ2(T) = 2x >
+�
+2(n − 3), where x is the positive root of the equation
+x3 + x2 − (s + 1) − s = 0.
+Proof. If T is a tree on n ≥ 4 vertices with diameter 3, by Corollary 4.2 it follows that
+ξ2(T c) ≤
+�
+�
+�
+�
+�
+�
+4n+1−√72n−63
+2
+if n is even,
+�
+4n+1−√72n−47
+2
+if n is odd.
+It is easy to check that
+�
+4n+1−√72n−47
+2
+<
+�
+4n+1−√72n−63
+2
+<
+�
+2(n − 3) for n ≥ 4. Therefore,
+ξ2(T c) <
+�
+2(n − 3).
+Let T be a tree with diam(T) ≥ 4.
+Then ξ2(T c) = 2λ2(A(T)) and the proof follows from
+Theorem 2.6.
+Now, we give a Nordhaus-Gaddum type lower bound for the second largest E-eigenvalue of a
+tree and its complement, which directly follows from Theorem 2.7 and Theorem 4.2.
+Theorem 4.4. Let T be a tree of order n ≥ 4. Then
+ξ2(T) + ξ2(T c) ≥
+��
+13n − 35 −
+√
+169n2 − 974n + 1417
+2
++
+�
+4n + 1 −
+√
+16n2 − 56n + 193
+2
+�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+11
+
+5
+Extremal problems for E-energy
+In this section, we characterize the complements of trees with the minimum and maximum E-
+energy among the complements of all trees on n vertices, respectively.
+To begin with, in the
+following lemma, we give an ordering of the complements of trees with diameter 3 according to
+their E-energy.
+Theorem 5.1. Let T be a tree with diameter 3, that is, T ∼= T a,b
+n,3 with a + b = n − 4, b ≥ a ≥ 0.
+Then the complements of T a,b
+n,3 can be ordered with respect to their E-energy as follows:
+EE
+�
+(T 0,n−4
+n,3
+)c�
+< EE
+�
+(T 1,n−3
+n,3
+)c�
+< . . . < EE
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c�
+.
+Proof. By Lemma 2.1, it follows that EE
+�
+(T a,b
+n,3)c�
+= 2
+�
+ξ1
+�
+(T a,b
+n,3)c�
++ ξ2
+�
+(T a,b
+n,3)c��
+, where
+ξ1
+�
+(T a,b
+n,3)c�
+=
+�
+4n + 1 +
+�
+(4n + 1)2 − 64(a + 1)(b + 1)
+2
+and
+ξ2
+�
+(T a,b
+n,3)c�
+=
+�
+4n + 1 −
+�
+(4n + 1)2 − 64(a + 1)(b + 1)
+2
+.
+Therefore,
+EE
+�
+(T a,b
+n,3)c�
+= 2
+�
+4n + 1 + 8
+�
+(a + 1)(b + 1)
+= 2
+�
+4n + 1 + 8
+�
+n − 3 + a(n − 4 − a)
+(since, a + b = n − 4).
+Now, consider the function f(x) = 4n + 1 + 8
+�
+n − 3 + x(n − 4 − x), where 0 ≤ x ≤ ⌊ n−4
+2 ⌋.
+Therefore,
+f′(x) =
+8(n − 4 − 2x)
+4n + 1 + 8
+�
+n − 3 + x(n − 4 − x)
+≥ 0.
+Hence, f(x) is an increasing function of x, where 0 ≤ x ≤ ⌊n−4
+2 ⌋. Therefore, EE
+�
+(T a,b
+n,3)c�
+= 2
+�
+f(a)
+is an increasing function for 0 ≤ a ≤ ⌊n−4
+2 ⌋. This completes the proof.
+As a consequence of the above result, we get the following corollaries.
+Corollary 5.1. If T is a tree with diameter 3, then
+EE(T c) ≥ 2
+�
+4n + 1 + 8
+√
+n − 3
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. By Lemma 2.1, it follows that
+ξ1
+�
+(T 0,n−4
+n,3
+)c�
+=
+�
+4n + 1 +
+�
+(4n + 1)2 − 64(n − 3)
+2
+,
+and
+ξ2
+�
+(T 0,n−4
+n,3
+)c�
+=
+�
+4n + 1 −
+�
+(4n + 1)2 − 64(n − 3)
+2
+.
+12
+
+Thus,
+EE
+�
+(T 0,n−4
+n,3
+)c�
+= 2
+�
+ξ1
+�
+(T 0,n−4
+n,3
+)c�
++ ξ2
+�
+(T 0,n−4
+n,3
+)c��
+= 2
+�
+4n + 1 + 8
+√
+n − 3.
+Now, the proof follows from Theorem 5.1.
+Corollary 5.2. If T is a tree with diameter 3, then
+EE(T c) ≤
+�
+�
+�
+2√8n − 7
+if n is even,
+2
+�
+4n + 1 + 4
+√
+n2 − 4n + 3
+if n is odd.
+The equality holds if and only if T ∼= T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+.
+Proof. It follows from Lemma 2.1 that
+ξ1
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c�
+=
+�
+�
+�
+�4n + 1 +
+�
+(4n + 1)2 − 64
+�
+⌈ n−2
+2 ⌉⌊ n−2
+2 ⌋
+�
+2
+=
+�
+�
+�
+�
+�
+�
+4n+1+√72n−63
+2
+if n is even,
+�
+4n+1+√72n−47
+2
+if n is odd.
+and
+ξ2
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c�
+=
+�
+�
+�
+�4n + 1 −
+�
+(4n + 1)2 − 64
+�
+⌈ n−2
+2 ⌉⌊ n−2
+2 ⌋
+�
+2
+=
+�
+�
+�
+�
+�
+�
+4n+1−√72n−63
+2
+if n is even,
+�
+4n+1−√72n−47
+2
+if n is odd.
+Therefore,
+EE
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c�
+= 2
+�
+ξ1
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c�
++ ξ2
+�
+(T
+⌊ n−4
+2 ⌋,⌈ n−4
+2 ⌉
+n,3
+)c��
+=
+�
+�
+�
+2√8n − 7
+if n is even,
+2
+�
+4n + 1 + 4
+√
+n2 − 4n + 3
+if n is odd.
+Now, the proof follows from Theorem 5.1.
+Next, we find the E-energy of (T 0,n−5
+n,4
+)c by computing the E-spectrum of (T 0,n−5
+n,4
+)c. This will
+be used in the proof of Theorem 5.2.
+Lemma 5.1. For n ≥ 5, the E-energy of EE
+�
+(T 0,n−5
+n,4
+)c�
+is given by
+EE
+�
+(T 0,n−5
+n,4
+)c�
+= 2
+�
+4(n − 1) + 8
+√
+2n − 7.
+13
+
+Figure 3: The tree T 0,n−5
+n,4
+.
+Proof. The eccentricity matrix of (T 0,n−5
+n,4
+)c is given by
+E
+�
+(T 0,n−5
+n,4
+)c�
+=
+v0
+v1
+v2
+v3
+v4
+w1
+. . .
+wn−5
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+v0
+0
+2
+0
+0
+0
+0
+. . .
+0
+v1
+2
+0
+2
+0
+0
+0
+. . .
+0
+v2
+0
+2
+0
+2
+0
+0
+. . .
+0
+v3
+0
+0
+2
+0
+2
+2
+. . .
+2
+v4
+0
+0
+0
+2
+0
+0
+. . .
+0
+w1
+0
+0
+0
+2
+0
+0
+. . .
+0
+...
+...
+...
+...
+...
+...
+...
+...
+...
+wn−5
+0
+0
+0
+2
+0
+0
+. . .
+0
+.
+It is easy to see that the rank of E
+�
+(T 0,n−5
+n,4
+)c�
+is 4, and hence 0 is an eigenvalue of E
+�
+(T 0,n−5
+n,4
+)c�
+with multiplicity n − 4.
+If W = {v4, w1, . . . , wn−5}, then Π2 = {v0} ∪ {v1} ∪ {v2} ∪ {v3} ∪ U is an equitable partition of
+E
+�
+(T 0,n−5
+n,4
+)c�
+with the quotient matrix
+Q2 =
+�
+�
+�
+�
+�
+�
+�
+0
+2
+0
+0
+0
+2
+0
+2
+0
+0
+0
+2
+0
+2
+0
+0
+0
+2
+0
+2(n − 4)
+0
+0
+0
+2
+0
+�
+�
+�
+�
+�
+�
+�
+.
+Now, the eigenvalues of Q2 are
+−
+�
+2n − 2 ± 2
+�
+n2 − 10n + 29,
+0,
+and
+�
+2n − 2 ± 2
+�
+n2 − 10n + 29.
+Therefore, by Theorem 2.1, we have
+SpecE
+�
+(T 0,n−5
+n,4
+)c�
+=
+�
+−
+�
+2n − 2 ± 2
+√
+n2 − 10n + 29
+0
+�
+2n − 2 ± 2
+√
+n2 − 10n + 29
+1
+n − 4
+1
+�
+.
+Hence, EE
+�
+(T 0,n−5
+n,4
+)c�
+= 2
+��
+2n − 2 + 2
+√
+n2 − 10n + 29 +
+�
+2n − 2 − 2
+√
+n2 − 10n + 29
+�
+.
+If
+α =
+�
+2n − 2 + 2
+√
+n2 − 10n + 29 and β =
+�
+2n − 2 − 2
+√
+n2 − 10n + 29, then α2 +β2 = 4n−4 and
+14
+
+Vo
+V1
+V2
+V3
+V4
+W1
+Wn-52αβ = 8√2n − 7. Therefore,
+EE
+�
+(T 0,n−5
+n,4
+)c�
+= 2(α + β)
+= 2
+��
+α2 + β2 + 2αβ
+�
+= 2
+��
+4n − 4 + 8
+√
+2n − 7
+�
+.
+Theorem 5.2. Let T be a tree of order n ≥ 4. Then
+EE(T c) ≥ EE
+�
+(T 0,n−4
+n,3
+)c�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
+Proof. For n = 4, T 0,0
+n,3 is the only tree with a connected complement. For n = 5 and 6, it is easy
+to see that EE(T c) > EE
+�
+(T 0,n−4
+n,3
+)c�
+(see Appendix). So, let us assume that T is a tree on n ≥ 7
+vertices.
+If T is a tree with diam(T) = 3, then the proof follows from Corollary 5.1. If T is a tree with
+diam(T) ≥ 4, then, by Lemma 2.2, it follows that
+EE(T c) = 2EA(T) ≥ 2EA(T 0,n−5
+n,4
+) = EE
+�
+(T 0,n−5
+n,4
+)c�
+.
+Again, by Lemma 5.1, we have EE
+�
+(T 0,n−5
+n,4
+)c�
+= 2
+�
+4(n − 1) + 8√2n − 7. Note that 4(n − 1) +
+8√2n − 7 > 4n + 1 + 8√n − 3 for n ≥ 7 and hence EE
+�
+(T 0,n−5
+n,4
+)c�
+> EE
+�
+(T 0,n−4
+n,3
+)c�
+for n ≥ 7. This
+completes the proof.
+In the following theorem, we characterize the complements of trees with maximum E-energy.
+Theorem 5.3. Let T be a tree of order n ≥ 4. Then
+EE(T c) ≤ EE(P c
+n)
+with equality if and only if T ∼= Pn.
+Proof. For n = 4, P4 is the only tree with a connected complement. For n = 5 and 6, it is easy to
+see that EE(T c) ≤ EE(P c
+n) with equality if and only if T ∼= Pn (see, Appendix). So, let us assume
+that T is a tree on n ≥ 7 vertices. For n ≥ 7, it follows from Theorem 2.2 that
+EE(P c
+n) = 2EA(Pn) =
+�
+�
+�
+�
+�
+4
+�
+cot
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is odd,
+4
+�
+csc
+�
+π
+2(n+1)
+�
+− 1
+�
+if n is even;
+≥
+�
+cot
+�
+π
+2(n + 1)
+�
+− 1
+�
+.
+We know that cot x >
+� 1
+x +
+1
+x−π
+�
+for 0 < x < π
+2 . Since 0 <
+π
+2(n+1) < π
+2 , therefore
+EE(P c
+n) ≥ 4
+�
+cot
+�
+π
+2(n + 1)
+�
+− 1
+�
+> 4
+�4n(n + 1)
+(2n + 1)π − 1
+�
+.
+(1)
+15
+
+Let T be a tree with diam(T) = 3. Therefore, by Corollary 5.2 it follows that
+EE(T c) ≤
+�
+�
+�
+2√8n − 7
+if n is even,
+2
+�
+4n + 1 + 4
+√
+n2 − 4n + 3
+if n is odd;
+≤ 2
+√
+8n − 7.
+By using (1) one can check that
+EE(P c
+n) > 4
+�4n(n + 1)
+(2n + 1)π − 1
+�
+> 2
+√
+8n − 7 ≥ EE(T c)
+for n ≥ 7.
+Thus, for any tree T with diameter 3, EE(P c
+n) > EE(T c).
+Let T be a tree with diam(T) ≥ 4. Therefore, E(T c) = 2A(T) and hence EE(T c) = 2EA(T).
+Now, by Lemma 2.3 it follows that
+EE(T c) = 2EA(T) ≤ 2EA(Pn) = EE(P c
+n)
+and the equality holds if and only if T ∼= Pn. This completes the proof.
+Finally, we obtain a Nordhaus-Gaddum type lower bound for the E-energy of a tree and its
+complement, which directly follows from Theorem 2.8 and Theorem 5.2.
+Theorem 5.4. Let T be a tree of order n ≥ 4. Then
+EE(T) + EE(T c) ≥ 2
+��
+13n − 35 + 8
+√
+n − 3 +
+�
+4n + 1 + 8
+√
+n − 3
+�
+with equality if and only if T ∼= T 0,n−4
+n,3
+.
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+Appendix
+Figure 4: List of trees on 5 and 6 vertices with connected complements.
+Trees(T)
+EE(T c)
+Trees(T)
+EE(T c)
+T1
+≈ 10.4528
+T5
+≈ 13.798
+T2
+≈ 10.9284
+T6
+≈ 12.3108
+T3
+≈ 11.6372
+T7
+≈ 13.9756
+T4
+= 12
+Table 1: E-energy of complements of the trees T1-T7.
+18
+
+Ti
+T2
+T3
+T4
+Ts
+T7