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+arXiv:2301.00507v1  [math.DG]  2 Jan 2023
+On Geodesics of Sprays and Projective Completeness
+Guojun Yang
+Abstract
+Geodesics, which play an important role in spray-Finsler geometry, are integral
+curves of a spray vector field on a manifold. Some comparison theorems and rigidity
+issues are established on the completeness of geodesics of a spray or a Finsler metric. In
+this paper, projectively flat sprays with weak Ricci constant (eps. constant curvature)
+are classified at the level of geodesics. Further, a geodesic method is introduced to
+determine an n-dimensional spray based on a family of curves with 2(n−1) free constant
+parameters as geodesics. Finally, it shows that a spray is projectively complete under
+certain condition satisfied by the domain of geodesic parameter of all geodesics.
+Keywords: Spray, Geodesic, Completeness, Path Space, Finsler Metric
+MR(2000) subject classification:
+53B40, 53C60
+1
+Introduction
+Spray geometry studies the properties of sprays on a manifold, and it is closely related to
+Finsler geometry. Every Finsler metric induces a natural spray but there are a lot of sprays
+which are not Finsler-metrizable (not be induced by any Finsler metric) ([3, 5, 10]). So
+a popular topic is to investigate whether a given spray is metrizable or not, and what’s
+more important is to give necessary and sufficient conditions for certain class of sprays to
+be metrizable ([2, 11, 12]). It is also important to investigate the properties of some special
+classes of sprays, for example, (locally) projectively flat sprays, Berwald sprays, sprays of
+scalar (resp. isotropic, constant) curvature, Hamel (resp. Funk) sprays ([2, 6, 11, 12]).
+A spray G on a manifold M defines a special vector filed on a conical region C of
+T M \ {0}, and it naturally defines its integral curves and the projections of the integral
+curves onto the manifold M are called geodesics. Geodesics play an important role in the
+studies of comparison theorems and rigidity issues on spray or Finsler manifolds. In [8], Z.
+Shen studies two pointwise projectively related Einstein Finsler metrics and determine the
+metrics along geodesics. In [10], the present author obtains a comparison theorem on the
+Ricci curvatures of a spay and a Finsler metric which are pointwise projectively related and
+the corresponding projective factor is estimated. In [1], R. Bryant proves that a geodesically
+reversible Finlser metric on S2 with positive constant flag curvature is a Riemann metric.
+In [7], C. Robles classifies geodesics of Randers metrics of constant flag curvature. In [4],
+L. Huang and X. Mo obtain the relation between the geodesics of two Finsler metrics F
+and ˜F, where ˜F is defined by the navigation data (F, V ) with V being a homothetic vector
+field of F. In this paper, we study projectively flat sprays with weak Ricci constant, the
+construction of sprays from a geodesic method and the projective completeness of sprays.
+In [11], it introduces sprays of constant curvature and a spray G of constant curvature
+is weakly Ricci constant (the Ricci curvature is constant along any geodesic of G). For
+two pointwise projectively related sprays, they have same geodesics as point sets and their
+geodesic parameters are closely related by the projective factor. Starting from this fact, we
+can determine a projectively flat spray with weak Ricci constant at geodesic level.
+1
+
+We consider a projectively flat spray manifold (G, M), that is,
+Gi = �Gi + Pyi,
+(1)
+where �G is a locally Minkowski spray on M. We have the following theorem.
+Theorem 1.1 If the spray G in (1) is weakly Ricci constant Ric;0 = 0 or of constant
+curvature, then along any geodesic x = x(s) of G, P(s) := P(x(s), x′(s)) is given by one of
+the following cases:
+P(s) =
+1
+s + κ,
+P(s) = −c · tan(cs + κ),
+P(s) = −c(1 − κe2cs)
+1 + κe2cs ,
+(2)
+where c, κ are constant. Further, if G is complete, then P(s) is given by
+P(s) = −c(1 − κe2cs)
+1 + κe2cs .
+(3)
+In Theorem 1.1, we can further give the relation between the geodesic parameters of G
+and �G by (2) (see Proposition 3.1, Corollary 3.3).
+The family of geodesics of an n-dimensional spray considered as point sets or paths is
+dependent on 2(n − 1) free constant parameters. A path space is a family of curves satis-
+fying certain conditions (Definition 4.1). We can freely give many interesting path spaces,
+especially in dimension two. Starting from a path space, we can construct its corresponding
+spray.
+Theorem 1.2 In an n-dimensional path space G, all paths in a local coordinate system (xi)
+can be parameterized under a variable t with 2(n−1) free constant parameters u, v as follows:
+x = x(t) = σ(t; u, v),
+(u, v ∈ Rn−1).
+(4)
+Further, the parametric equation (4) induces a spray G whose geodesics are given by (4)
+with t as its geodesic parameter, and if a new variable s = s(t) = s(t; u, v) is given with
+s′(t) > 0, then it gives a spray ¯G ∈ Proj(G) with s as its geodesic parameter.
+If a family of curves can be parameterized in the form (4), then with an auxiliary pa-
+rameter c > 0 multiplied by t in (4), we can obtain the corresponding spray by eliminating
+the parameters u, v, c, t. We give some examples to show how to solve the sprays from given
+path spaces (see Examples 4.5-4.8).
+In the study of rigidity issues on a Finsler or spray manifold, it is important to assume
+that the (Finsler) spray in consideration be (positively/negatively) complete. A given spray
+is not necessarily (positively/negatively) complete. So a natural problem is whether a spray
+can be projectively (positively/negatively) complete or not. We solve this problem under
+certain conditions in the following result (Theorem 1.3).
+Theorem 1.3 Let G be a spray on a manifold M with its each geodesic x = x(t) being
+defined on the maximal interval I given by one case of the following
+I = (a, b),
+or (a, +∞),
+or (−∞, b),
+(5)
+where a = a(u, v) < 0, b = b(u, v) > 0 with u = x(0), v = x′(0) are C∞ functions on a
+conical region C of T M \ {0}. Then G is projectively (positively/negatively) complete on C.
+In Theorem 1.3, usually we can also put u, v as that in (4) (see Example 5.5). If (5)
+is not satisfied, it is uncertain that G is projectively complete (cf.
+Example 5.5)).
+We
+give Examples 5.2-5.5 as an application of Theorem 1.3. A Finsler metric is not necessarily
+projectively (positively/negatively) complete, namely, if G in Theorem 1.3 is a Finsler spray,
+the corresponding spray projective to G may not be a Finsler spray.
+2
+
+2
+Geodesic parameters in projective relations
+A spray on M, in our consideration, is a smooth vector field G on a conical region C of
+T M \ {0} (an important case is C = T M \ {0}) expressed in a local coordinate system
+(xi, yi) in T M as follows
+G = yi ∂
+∂xi − 2Gi ∂
+∂yi ,
+where Gi are local homogeneous functions satisfying Gi(x, λy) = λ2Gi(x, y) for λ > 0. If
+C = T M \ {0}, G is called regular; otherwise, it is called singular.
+The integral curves of G projected onto M are the geodesics of G. Let x = x(s) be a
+geodesic of G. Then it satisfies the following ODE:
+d2xi
+ds2 + 2Gi(x, dx
+ds ) = 0,
+where s is called a geodesic parameter of the geodesic x = x(s). Reparameterizing a geodesic
+x = x(s) by a general parameter t with ds/dt > 0, we have
+d2xi
+dt2 + 2Gi(x, dx
+dt ) = γ(t)dxi
+dt ,
+(6)
+where γ(t) is given by
+γ(t) = d2s
+dt2
+�ds
+dt = − d2t
+ds2
+�
+( dt
+ds)2.
+(7)
+Let G, ¯G be two sprays pointwise projectively related by ¯Gi = Gi + Pyi. Let x = x(t)
+be a geodesic of G or ¯G as a point set for a general parameter t. Then along the geodesic
+x = x(t), it follows from (6) and (7) that
+2P(t) = ¯s′′(t)
+¯s′(t) − s′′(t)
+s′(t) ,
+�
+P(t) := P(x(t), x′(t))
+�
+,
+(8)
+where s, ¯s are the geodesic parameters of the curve x = x(t) in G, ¯G respectively.
+In
+particular, along a geodesic x = x(s) of G, it follows from (8) that
+2P(s) = ¯s′′(s)
+¯s′(s) ,
+�
+P(s) := P(x(s), x′(s))
+�
+,
+(9)
+If we express the geodesic x = x(s) of G as the geodesic x = x(¯s) of ¯G, by (9), we have
+2P(¯s) = 2P(x(s), x′(s))ds
+d¯s =
+¯s′′(s)
+�
+¯s′(s)
+�2 ,
+�
+P(¯s) := P(x(¯s), x′(¯s))
+�
+.
+(10)
+So if P(s) or P(¯s) is known, the relation ¯s = ¯s(s) can be obtained from (9) or (10).
+Example 2.1 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn. Define a
+projectively flat spray G by
+Gi = Pyi,
+P := cF,
+where c is a constant. Any geodesic x = x(t) (as a point set) of G is given by
+x = x(t) = vt + u,
+�
+−
+1
+F(u, −v) < t <
+1
+F(u, v)
+�
+,
+3
+
+where u, v ∈ Rn are constant vectors. We have
+F(vt + u, v) =
+F(u, v)
+1 − tF(u, v).
+(11)
+Let s be a geodesic parameter of G. Then by (9) and (11) we have
+s′′(t)
+s′(t) = 2cF(vt + u, v) =
+2cF(u, v)
+1 − tF(u, v),
+(12)
+integration of which with s(0) = 0 gives
+s = s(t) =
+�
+κ ln
+�
+1 − tF(u, v)
+�
+,
+(c = 1
+2),
+κ
+�
+1 −
+�
+1 − tF(u, v)
+�1−2c�
+,
+(c ̸= 1
+2),
+(13)
+where κ is a constant with κ < 0 for c ≥ 1/2, and κ > 0 for c < 1/2. Thus the spray
+is positively complete for c ≥ 1/2, and any geodesic is defined on a finite open interval for
+c < 1/2. Besides, the spray G is (locally) metrizable if and only if c = 0, 1, 1/2 (see [10]).
+Example 2.2 In Example 2.1, if the spray G is given by
+Gi(y) := Pyi,
+P := c
+�
+F(y) − F(−y)
+�
+,
+then by (11) and
+F(vt + u, −v) =
+F(u, −v)
+1 + tF(u, −v).
+(14)
+it follows from (9) that
+s′′(t)
+s′(t) =
+2cF(u, v)
+1 − tF(u, v) −
+2cF(u, −v)
+1 + tF(u, −v),
+integration of which with s(0) = 0 gives
+s = s(t) = κ
+� t
+0
+��
+1 − tF(u, v)
+��
+1 + tF(u, −v)
+��−2c
+dt,
+(15)
+where κ > 0 is constant. From (15), it is clear to conclude that G is complete if c ≥ 1/2; s
+is bounded in a finite open interval if c < 1/2.
+3
+Projective flat sprays with weak Ricci constant
+For a spray G, the Riemann curvature tensor Ri
+k is defined by
+Ri
+k := 2∂kGi − yj(∂j ˙∂kGi) + 2Gj( ˙∂j ˙∂kGi) − ( ˙∂jGi)( ˙∂kGj),
+where we define ∂k := ∂/∂xk, ˙∂k := ∂/∂yk. The trace of Ri
+k is called the Ricci curvature,
+Ric := Ri
+i.
+For a spray tensor T = Tidxi as an example, the horizontal and vertical
+derivatives of T with respect to Berwald connection are given by
+Ti;j = δjTi − TrGr
+ij,
+Ti.j = ˙∂jTi,
+(δi := ∂i − Gr
+i ˙∂r, Gk
+ir := ˙∂r ˙∂iGk)).
+4
+
+A spay is called weakly Ricci constant if Ric;0 := Ric;ryr = 0. A spray G is said to be of
+constant curvature if Ri
+k is given by Ri
+k = Rδi
+k − τkyi with ([11])
+τi;k = 0 ( ⇔ R = τk = 0, or R;i = 0(R ̸= 0)).
+By definition, it is clear that a spray of constant curvature is weakly Ricci constant. For
+two pointwise projectively related sprays G, ¯G with ¯Gi = Gi + Pyi, their Ricci curvatures
+Ric, ¯Ric are related by
+¯Ric = Ric − (n − 1)(P;0 − P 2).
+(16)
+We consider a projectively flat spray manifold (G, M) given by (1), that is,
+Gi = �Gi + Pyi,
+where �G is a locally Minkowski spray on M ( �G has local straight lines as geodesics). If
+G is weakly Ricci constant, then we can determine the projective factor P along geodesics,
+which is shown in Theorem 1.1.
+Proof of Theorem 1.1 : By (16) and �Gi = Gi − Pyi, the Ricci curvature Ric of G is
+given by
+Ric = −(n − 1)(P 2 + P;0).
+Therefore, Ric;0 = 0 is equivalent to P;0;0 + 2PP;0 = 0. Then along a geodesic x = x(s) of
+G, we have
+P ′′(s) + 2P(s)P ′(s) = 0.
+whose solution is given by one of the three cases in (2). Further, if G is complete, it is clear
+that (3) follows from (2).
+Q.E.D.
+If the spray G in (1) is weakly Ricci constant Ric;0 = 0, then applying (2) and (10), we
+obtain the following proposition.
+Proposition 3.1 Let the spray G in (1) be weakly Ricci constant (esp. of constant curva-
+ture). For any geodesic σ, let s and t be the geodesic parameters of σ with respect to G and
+�G respectively. Then s = s(t) is given by one of the following cases:
+s = at, (a > 0);
+s = b ln(1 + at), (ab > 0);
+(17)
+s =
+bt
+1 + at, (a ̸= 0, b > 0);
+s = c
+�
+arctan(at + b) − arctan b
+�
+, (ac > 0);
+(18)
+s = c ln 1 + bt
+1 + at,
+�
+(b − a)c > 0, ab ̸= 0
+�
+,
+(19)
+where a, b, c are constant, and in (19), it further requires s′(t) > 0 (see Remark 3.2).
+Proof : By (10) we need to solve the following ODE with initial conditions:
+s′′(t)
+s′(t) = 2P(s)s′(t),
+(s(0) = 0, s′(t) > 0),
+integration of which gives
+s′(t) = ae2 � P (s)ds,
+�
+e−2 � P (s)dsds = at + b,
+(20)
+5
+
+where a, b are two constants. Now P(s) is given by (2) from Theorem 1.1, and thus we can
+obtain s = s(t) by plugging P(s) into (20).
+If P(s) = 0, then (20) gives s = at+b. Since s(0) = 0, s′(t) > 0, we obtain s = at (a > 0),
+which gives the first formula in (17).
+If P(s) = c ̸= 0 is constant, then (20) gives the second formula in (17) with ab > 0.
+If P(s) is given by the first formula in (2), then (20) gives
+s = −κ +
+1
+at + b,
+which can be rewritten as the form of the first formula in (18) by s(0) = 0, s′(t) > 0.
+If P(s) is given by the second formula in (2) (c ̸= 0), then (20) gives
+s = −κ − arctan(at + b)
+c
+,
+which can be rewritten as the second formula in (18) by s(0) = 0, s′(t) > 0.
+If P(s) is given by the third formula in (2) (cκ ̸= 0), then (20) gives
+s = 1
+2c ln
+�
+1
+at + b − 1
+κ
+�
+which can be rewritten as the formula in (19) by s(0) = 0, s′(t) > 0.
+Q.E.D.
+In (19), by s′(t) > 0, we have further restriction on the constant parameters a, b, c, which
+is shown in the following remark.
+Remark 3.2 In (19), let t be defined on the maximal interval (κ1, κ2) with κ1 < 0 < κ2. It
+is easy to conclude the following cases from s′(t) > 0:
+a > 0, b > 0 :
+�
+t ∈ (κ1, κ2) ⊂ (− 1
+a, +∞),
+(b < a)
+t ∈ (κ1, κ2) ⊂ (− 1
+b, +∞), (b > a),
+a < 0, b < 0 :
+�
+t ∈ (κ1, κ2) ⊂ (−∞, − 1
+a),
+(b > a)
+t ∈ (κ1, κ2) ⊂ (−∞, − 1
+b), (b < a),
+a > 0, b < 0 : t ∈ (κ1, κ2) ⊂ (−1
+a, −1
+b),
+a < 0, b > 0 : t ∈ (κ1, κ2) ⊂ (−1
+b , −1
+a).
+By Proposition 3.1 and Remark 3.2, we directly obtain the following corollary.
+Corollary 3.3 If the spray G in Proposition 3.1 (P ̸= 0) is complete, then s = s(t) is given
+by one of the following two cases:
+s = b ln(1 + at), (ab > 0),
+(21)
+s = c ln 1 + bt
+1 + at,
+�
+(b − a)c > 0, ab < 0
+�
+,
+(22)
+where in (21) and (22), we respectively have
+t ∈ (−∞, −1
+a) if a < 0, and t ∈ (−1
+a, +∞) if a > 0;
+t ∈ (−1
+a, −1
+b) if a > 0, b < 0, and t ∈ (−1
+b , −1
+a) if a < 0, b > 0.
+6
+
+Now in the following, we give some projectively flat sprays to verify the above results on
+the projective factors and the geodesic parameters.
+Example 3.4 Consider the spray G in Example 2.1. A direct computation shows that G
+is weakly Ricci constant or of constant curvature if and only if c = 0, 1, 1/2. Let x = x(t) =
+vt + u be a geodesic (as a point set) of G, and the geodesic parameter s in G is given by
+(13). Then it follows from (10) and (13) that P = cF is given by
+P(s) =
+�
+− 1
+2κ,
+(c = 1
+2)
+c
+2c−1 ·
+1
+s−κ,
+(c ̸= 1
+2).
+(23)
+It is clear from (23) that P(s) is in one of the forms in (2) if and only if c = 0, 1, 1/2.
+Meanwhile, s = s(t) is given in Proposition 3.1 if and only if c = 1, 1/2, 1, and in this case,
+s = s(t) is in the respective forms shown in (17) and the first formula in (18).
+Example 3.5 Let G be the spray in Example 2.2 with c = 1/2. G is complete and it is of
+constant curvature. Then it follows from (15) that
+s = κ ln 1 + tF(u, −v)
+1 − tF(u, v) ,
+(24)
+where κ > 0 is a constant. In this case, s = s(t) is in the form (22), and it is easy to verify
+that P(s) is in the form of the third formula in (2) by plugging (24) into (10).
+Example 3.6 Let G be a spray on Rn defined by
+Gi := Pyi,
+P := − ⟨x, y⟩
+1 + |x|2 .
+G is metrizable and it is of constant curvature. Let x = x(t) = vt + u be a geodesic (as a
+point set) of G, and by (9), the geodesic parameter s of G satisfies
+s′′(t)
+s′(t) = −
+2(|v|2t + ⟨u, v⟩)
+1 + |u|2 + 2⟨u, v⟩t + |v|2t2 .
+Solving the ODE, we obtain
+s = s(t) = κ1 + κ2 arctan
+|v|2t + ⟨u, v⟩
+�
+(1 + |u|2)|v|2 − ⟨u, v⟩2 ,
+where κ1, κ2 are constant. It is clear that s = s(t) is in the form of the second formula in
+(18) if s(0) = 0, and P(s) is in the form of the second formula in (2) by plugging the above
+s = s(t) into (10).
+4
+Construction of sprays from geodesics
+Given a family of curves G on a manifold, if G can constitute the geodesics of a spray G
+on M, how can we solve G (at least locally)? A spray induces a (local) semispray and two
+pointwise projectively related sprays induce a same semispray (cf. [9]). A semispray can
+also be considered as a special parameterized family of curves, which forms a path space.
+In this section, we will start from a path space and introduce some ways to construct
+sprays based on a path space and its parameterization. We call it the geodesic method of
+construction of sprays.
+Similarly to a spray, a path space G on a manifold M is usually defined on a conical
+region C of T M \ {0} (see Definition 4.1), and G is called singular if C ̸= T M \ {0}.
+7
+
+Definition 4.1 Let G be a family of C∞ parameterized curves (called paths) on an n-
+dimensional manifold M. G or (M, G) is called an n-dimensional path space if on a conical
+region C of T M it satisfies
+(i) for y ∈ Cx, there is a curve σ : (−ǫ, ǫ) → M in G with σ′(0) = y;
+(ii) for any σ, τ in G with σ′(0) = τ ′(0), σ and τ coincide in a small intervals of 0;
+(iii) if a curve σ is in G, then for any constants λ > 0 and to, the curve η is also in G,
+where η is defined by η(t) := σ(λt + to).
+An equivalent version of Definition 4.1 in regular case is refereed to [9] (P52).
+Example 4.2 Consider a set G of a family of curves x = x(s) on R2 in the form
+x(s) = σ(s; xo, yo),
+�
+x(0) = xo = (a, b),
+x′(0) = yo = (u, v)
+�
+,
+σ(s; xo, yo) := (a, b) + (u, v)s − (0, 1)
+�1
+3u3s3 + au2s2�
+,
+where a, b, u, v are arbitrary parameters. It can be directly verified that G is a path space on
+R2, since Definition 4.1 (i) (ii) automatically hold, and Definition 4.1 (iii) follows from
+σ(λs + so; xo, yo) = σ(s; ˆxo, ˆyo),
+where we define
+xo = (a, b),
+yo = (u, v),
+ˆxo = (ˆa,ˆb),
+ˆyo = (ˆu, ˆv),
+ˆa := a + uso,
+ˆb := b + vso − 1
+3u2(3a + uso)(so)2,
+ˆu := λu,
+ˆv := λv − λu2so(2a + uso).
+For a path space G, we have different ways to parameterize the paths in G under a para-
+metric variable and some constant parameters (see Theorem 1.2 and Lemma 4.3). Example
+4.2 satisfies (25) and (26) in the following Lemma 4.3 with
+f(s; xo, yo) := −(0, 1)
+�1
+3u3s3 + au2s2�
+,
+xo = (a, b), yo = (u, v).
+Lemma 4.3 An n-dimensional path space (M, G) is locally expressed as the following family
+of curves x = x(s) with arbitrary constant parameters xo, yo ∈ Rn:
+x(s) = σ(s; xo, yo) = xo + yos + f(s; xo, yo),
+(25)
+where f is a smooth function satisfying f(0; xo, yo) = f ′(0; xo, yo) = 0 and
+f(s; ˆxo, ˆyo) = f(λs + so; xo, yo) − f(so; xo, yo) − λf ′(so; xo, yo)s,
+(26)
+�
+ˆxo := xo + yoso + f(so; xo, yo),
+ˆyo := λyo + λf ′(so; xo, yo)
+�
+.
+It is clear that the collection of geodesics of a spray naturally forms a path space. Shen
+proves the converse in the following lemma ([9]). We also give the proof for convenience.
+Lemma 4.4 A path space G induces a spray G with the set of geodesics of G being G.
+Proof : Let (G, M) be a path space on a conical region C. For a given y ∈ Cx, there is a
+curve σ : (−ǫ, ǫ) → M in G with σ(0) = x, σ′(0) = y by Definition 4.1 (i). Define
+Gi(y) := −1
+2
+d2σ
+ds2 (0),
+8
+
+which is independent of the choice of σ by Definition 4.1 (ii). We are going to verify that G
+is a spray. For any constant λ > 0, let η(s) := σ(λs) ∈ G (see Definition 4.1 (iii)). Then we
+have
+Gi(λy) = −1
+2
+d2η
+ds2 (0) = −1
+2λ2 d2σ
+ds2 (0) = λ2Gi(y),
+which implies that Gi is positively homogeneous of degree two. Further, for any η : (a, b) →
+M in G and any fixed t ∈ (a, b), define γ(s) := η(s + t). Then we have
+η′(t) = γ′(0),
+η′′(t) = γ′′(0).
+So by the definition of Gi, we get
+Gi�
+η′(t)
+�
+= Gi�
+γ′(0)
+�
+= −1
+2
+d2γi
+ds2 (0) = −1
+2
+d2ηi
+ds2 (t),
+which implies that η satisfies the following ODE:
+d2ηi
+ds2 + 2Gi�dη
+ds
+�
+= 0.
+Therefore, G is a spray, and the set of geodesics of G coincides with G.
+Q.E.D.
+In Lemma 4.4, by different choices of the parametric variables, it (locally) induces a
+projective class Proj(G) of G, each of which is projective to G.
+For a given path space G, it induces a spray G by Lemma 4.4. Then G defines a semispray
+ˆG (see [9]: P37) and the geodesics of G and ˆG are closely related (see [9]: Lemma 3.1.1).
+Therefore, in G, any path can be locally expressed as
+xa = xa(x1; u, v),
+(u, v ∈ Rn−1, 2 ≤ a ≤ n),
+where u, v are free constant parameters. So all paths in an n-dimensional path space depend
+only on 2(n−1) free constant parameters, where the Jaccobi determinant is not zero, namely,
+det
+�∂xa/∂u
+∂xa/∂v
+∂ya/∂u
+∂ya/∂v
+�
+̸= 0,
+�
+ya := dxa
+dx1
+�
+.
+Then we obtain Theorem 1.2 for the construction of sprays based on the parametric equations
+of path spaces.
+If we write (4) in the form
+x(t) = σ(λt + µ, u, v),
+(27)
+where λ, µ are constant numbers, then under the 2n constant parameters λ, µ, u, v, this
+family of curves satisfies Definition 4.1 (i)(ii)(iii).
+For instance, the 2-dimensional path
+space in Example 4.2 can be written as the following family of curves
+x(s) = τ(λs + µ; bo, vo) = (0, bo) + (1, vo)(λs + µ) − 1
+3(0, 1)(λs + µ)3,
+�
+λ := u, µ := a, vo := v
+u + a2, bo = b − avo + 1
+3a3�
+.
+By Theorem 1.2, if the set A of a family of curves on an n-dimensional manifold defines
+a path space, then A just depends on 2(n − 1) free constant parameters. For example, in
+Rn, all circles with fixed radius cannot define a path space when n ≥ 3, because in this case,
+the circles depend on more than 2(n − 1) free constant parameters.
+9
+
+Now we introduce a method of constructing a spray G determined by a path space
+considered as the geodesics of G, which is similar to Okubo’s method for the construction of
+a Finlser metric from a hypersurface as its indicatrix. We can start from a family of curves
+given by (25) or (4) to determine a corresponding spray.
+Method (I): For a family of curves given by (25) satisfying (26), actually we can reduce
+one constant parameter since (25) can be written as (if y1
+o ̸= 0)
+x(t) = σ(t; xo, ¯yo) = xo + ¯yot + f(t; xo, ¯yo),
+�
+t := y1
+os, ¯ya
+o := ya
+o/y1
+o, ¯yo := (¯ya
+o)
+�
+.
+Let a path space be determined by (25) and we put
+x = xo + yos + f(s; xo, yo),
+y(= dx
+ds ) = yo + f ′(s; xo, yo)),
+(28)
+Gi := −1
+2
+d2x
+ds2 = −1
+2f ′′(s; xo, yo)).
+(29)
+Then we obtain a spray G from (29) by eliminating xo, yo, s in (29) from (28), where s is a
+geodesic parameter of the spray G.
+Method (II): Suppose that a family of curves are given by the parametric equation (4) with
+2(n − 1) free constant parameters u, v. This case is more convenient to construct sprays.
+With an auxiliary parameter c > 0, we put
+x = σ(cs; u, v),
+y = dx
+ds = cdσ
+dˆs (cs; u, v),
+ˆs := cs.
+(30)
+Theoretically, we can express c, s, u, v as functions of x, y from (30). Then plugging them
+into the following
+Gi := −1
+2
+d2xi
+ds2 = c2 d2σi
+dˆs2 (cs; u, v),
+(31)
+we obtain a spray G given by (31), where s is a geodesic parameter of the spray G.
+Now in the following Examples 4.5-4.8, we use Method (I) or Method (II) to show how
+we construct sprays from given path spaces by eliminating the corresponding parameters.
+Example 4.5 Consider a set G of a family of curves on R3:
+x(s) = (a, b, c) + (u, v, w)s − (0, 1, 0)h(s),
+�
+h(s) := −1
+3(u3 + w3)s3 − (au2 + cw2)s2�
+,
+where a, b, c, u, v, w are constant parameters. G is a path space. By (28) we get
+x1 = a + us,
+x3 = c + ws,
+y1 = u,
+y3 = w.
+(32)
+By (29), the induced spray G is given by
+G1 = −1
+2
+d2x1
+ds2 = 0,
+G3 = −1
+2
+d2x3
+ds2 = 0,
+G2 = −1
+2
+d2x2
+ds2 = (u3 + w3)s + (au2 + cw2)
+= (u3 + w3)s +
+�
+(x1 − us)u2 + (x3 − ws)w2� �
+by (32)
+�
+= x1u2 + x3w2 = x1(y1)2 + x3(y3)2 �
+by (32)
+�
+.
+10
+
+G has zero Riemann curvature and so it is metrizable (a Finsler spray) ([11]).
+Example 4.6 Let G be the set of all circles with fixed radius r on R2. We parameterize G
+by
+x1(s) = a + r cos s,
+x2(s) = b + r sin s,
+where a, b are arbitrary constant parameters. G depends on just two free constant parameters.
+By Theorem 1.2, G defines a spray G on R2 with s as a geodesic parameter of G. We show
+the spray as follows. With an auxiliary parameter c > 0, it follows from (30) that
+x1 = a + r cos cs,
+x2 = b + r sin cs,
+y1 = −cr sin cs,
+y2 = cr cos cs.
+(33)
+Then plugging the latter two formulas of (33) into (31) yields a spray G given by
+G1 = −c2r cos cs = −1
+r y2�
+(y1)2 + (y2)2,
+G2 = −c2r sin cs = 1
+r y1�
+(y1)2 + (y2)2.
+This circle spray first appears in [9] (P49), and even locally it is not metrizable ([11]).
+Example 4.7 Consider a family of semicircles G on the positive semi-plane R2
++ with center
+on x1-axis and arbitrary radius. Note that G is singular at the direction parallel to x2-axis.
+We can parameterize G by
+x1 = a + b coss,
+x2 = b sin s,
+(x2 > 0, b ≥ 0),
+where a, b are arbitrary constant parameters. G depends on just two free constant parameters.
+By Theorem 1.2, G defines a spray G on R2
++ with s as a geodesic parameter of G. With an
+auxiliary parameter c > 0, by (30) we get
+x1 = a + b cos cs,
+x2 = b sin cs,
+y1 = −bc sincs,
+y2 = bc coscs.
+(34)
+Then similarly, by the elimination of the parameters a, b, c, s in (31) from (34), the spray G
+with s being a geodesic parameter is given by
+G1 = −y1y2
+2x2 ,
+G2 = (y1)2
+2x2 .
+(35)
+The spray G is regular on R2
++ (any straight lines parallel to x2-axis are geodesics of G). G
+is of isotropic curvature, and locally it is not metrizable by the method in [2, 11].
+Example 4.8 Let Bn be the unit ball in Rn and G be all circle arcs in Bn which are
+perpendicular to the boundary Sn−1 = ∂Bn. Let s be the arc-length parameter of a circle arc
+induced by the Euclidean metric. What is the spray G induced by G with s being a geodesic
+parameter of G (see Example 4.1.4 in [9])? We will show that G is given by
+Gi = ⟨x, y⟩yi − |y|2xi
+1 − |x|2
+,
+(36)
+which is not metrizable by [11]. Now for arbitrarily given p, q ∈ Sn−1, there is a circle arc
+γ in G, in which γ is perpendicular to Sn−1 at p, q. Let C be the circle with γ ⊂ C. The
+center and radius of C are respectively given by
+τ(p + q),
+|p − τ(p + q)|,
+(τ := (1 + pq)−1),
+11
+
+where pq is the Euclidean inner product of p, q. Then γ is parameterized by the equation
+x(s) = x(s; p, q) = [p − τ(p + q)] cos s + |p − τ(p + q)|p sin s + τ(p + q).
+(37)
+Since there are just 2(n − 1) free constant parameters in (37), the family of curves in the
+form (37) define a path space by Theorem 1.2. Now based on (30) and (31), we can give
+the spray G from (37) with s being a geodesic parameter of G. With an auxiliary parameter
+c > 0, by (30) we put
+x = [p − τ(p + q)] cos cs + |p − τ(p + q)|p sin cs + τ(p + q),
+(38)
+y = −c[p − τ(p + q)] sin cs + c|p − τ(p + q)|p cos cs.
+(39)
+By (31) we have
+2Gi := c2�
+[p − τ(p + q)]i cos cs + |p − τ(p + q)|pi sin cs
+�
+.
+(40)
+By a direct lengthy computation, we can eliminate the parameters p, q, c, s in (40) from (38)
+and (39) (the details are omitted). Finally, the spray G is given by (36).
+5
+Projectively complete sprays
+For a given spray G, if we know the general solutions of all geodesics of G, then under another
+parameter as a geodesic parameter, we can determine a corresponding spray projectively
+related to G. Now suppose that the general solutions of geodesics of G are locally given by
+x = σ(t) = σ(t; u, v),
+�
+u, v ∈ Rn−1�
+,
+(41)
+where t is a geodesic parameter of G and u, v are free constant parameters. Sometimes, it is
+also convenient to put u = σ(0), v = σ′(0) for the elimination of parameters. Make a change
+of the variables from t to s with
+t = t(s) = t(s; u, v),
+dt
+ds > 0.
+(42)
+With an auxiliary parameter c > 0, we put
+x = σ(t(cs); u, v),
+y = dx
+ds = cdσ
+dt
+dt
+ds,
+(43)
+where dt/ds, as a function of s, takes the value at cs. Further, we have
+d2xi
+ds2 = c2 d2σi
+dt2
+� dt
+ds
+�2 + c2 dσi
+dt
+d2t
+ds2
+= −2Gi(x, dσ
+dt )c2� dt
+ds
+�2 + c2 dσi
+dt
+d
+dt
+� dt
+ds
+� dt
+ds
+= −2Gi(x, y) + c d
+dt
+� dt
+ds
+�
+yi.
+(44)
+Expressing c, t in terms of x, y from (43), and then plugging c, t into (44), we obtain a spray
+¯G given by
+¯Gi = Gi − 1
+2
+d
+dt
+� dt
+ds
+�
+cyi = Gi + Pyi,
+(45)
+�
+P = P(x, y) := −1
+2
+d
+dt
+� dt
+ds
+�
+c
+�
+,
+with s being a geodesic parameter of ¯G.
+12
+
+Lemma 5.1 Suppose that the general solutions of geodesics of a spray G are given by (41).
+Let s be another parameter related to t by (42). Then a spray ¯G projective to G with s being
+its geodesic parameter is given by (45), where c, t are determined by (43).
+Under certain condition, a spray can be projectively (positively/negatively) complete,
+which is shown in Theorem 1.3. Now we give the proof of Theorem 1.3.
+Proof of Theorem 1.3 : Let G be a spray on a manifold M. For an arbitrary geodesic
+x = x(t), suppose that t belongs to the maximal interval I given by (5).
+If I = (a, +∞) or I = (−∞, b), we respectively make a change of the variables from t to
+s by
+s = ln(1 − t
+a),
+or
+s = − ln(1 − t
+b),
+(46)
+either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞).
+If I = (a, b), make a change by (46) and then we respectively have
+s ∈
+�
+− ∞, ln(1 − b
+a)
+�
+,
+or
+s ∈
+�
+− ln(1 − a
+b ), +∞).
+If I = (a, b), make a change of the variables from t to s by
+s = ln 1 − t/a
+1 − t/b ,
+or
+s = tan
+�
+π
+b − a(t − a + b
+2
+)
+�
++ tan
+�b + a
+b − a
+π
+2
+�
+,
+(47)
+either of which gives s(0) = 0, s′(t) > 0 and the maximal interval of s with s ∈ (−∞, +∞).
+Therefore, by the change (46) or (47), we obtain a (positively/negatively) complete spray
+which is projective to G. This completes the proof.
+Q.E.D.
+As an application of Theorem 1.3, we give the following Examples 5.2-5.4 to show the
+construction of the (positively/negatively) complete sprays projective to given sprays.
+Example 5.2 Let F be the Funk metric on a strongly convex domain Ω ⊂ Rn.
+The
+Minkowski spray G = 0 on Ω has its geodesics given by
+x(t) = vt + u,
+�
+−
+1
+F(u, −v) < t <
+1
+F(u, v)
+�
+,
+where u, v ∈ Rn are arbitrary constant vectors. By (46), put t = t(s) as
+s = − ln
+�
+1 − tF(u, v)
+�
+.
+(48)
+With s being a geodesic parameter, we obtain a projectively flat and positively complete spray
+¯G, which will be shown to be the Finsler spray induced by F, namely,
+¯Gi = 1
+2Fyi.
+(49)
+Actually, it follows from (2) and (11) that
+dt
+ds =
+1
+F(u, v) − t =
+1
+F(vt + u, v).
+(50)
+Then (43) gives
+x = vt + u,
+y = cv dt
+ds.
+(51)
+13
+
+It is clear from (51) and (50) that
+F(x, y) = cF(vt + u, v) dt
+ds = c.
+(52)
+Therefore, by (45), (50) and (52), the spray G is given by (49).
+Example 5.3 In Example 5.2, by (47), put t = t(s) as
+s = ln 1 + tF(u, −v)
+1 − tF(u, v) .
+(53)
+With s being a geodesic parameter, we obtain a projectively flat and complete spray ¯G, which
+will be shown to be the Finsler spray induced by the Klein metric ¯F(x, y) :=
+�
+F(x, y) +
+F(x, −y)
+�
+/2, namely,
+¯Gi(x, y) = 1
+2
+�
+F(x, y) − F(x, −y)
+�
+yi.
+(54)
+Firstly, by (53), we get
+dt
+ds =
+�
+1 − tF(u, v)
+��
+1 + tF(u, −v)
+�
+F(u, v) + F(u, −v)
+,
+(55)
+d
+dt
+� dt
+ds
+�
+= F(u, −v) − F(u, v) − 2tF(u, v)F(u, −v)
+F(u, v) + F(u, −v)
+.
+(56)
+Secondly, (43) gives
+x = vt + u,
+y = cv dt
+ds,
+from which we have
+F(x, y) = F(vt + u, v)c dt
+ds,
+F(x, −y) = F(vt + u, −v)c dt
+ds.
+(57)
+Plugging (11), (14) and (55) into (57), we obtain
+c = F(x, y) + F(x, −y),
+t = F(x, y)F(u, −v) − F(x, −y)F(u, v)
+F(u, v)F(u, −v)[F(x, y) + F(x, −y)].
+(58)
+Finally, by (58) and (56), it follows from (45) that the spray ¯G is given by (54).
+Example 5.4 For the spray G in Example 5.2, we will introduce a different way from that
+in Example 5.3 to make G be complete, which is actually to use (46) to make complete the
+Finsler spray induced by the Funk metric F. For a geodesic x(t) = vt + u of G, put
+s = ln
+�
+1 − ln(1 − tF(u, v))
+a
+�
+,
+a := ln
+�
+1 + F(u, v)
+F(u, −v)
+�
+.
+In a similar way to that for the computation in Example 5.3, we obtain a projectively flat
+and complete spray ¯G given as follows:
+¯Gi(y) = Gi
+F (y) + 1
+2
+F(y)
+ln
+F (−y)
+F (y)+F (−y)
+yi,
+�
+Gi
+F (y) := 1
+2F(y)yi�
+.
+¯G is of scalar curvature and actually we can verify that ¯G is not metrizable by using the
+method in [2].
+14
+
+Example 5.5 For the family of semicircles G on R2
++ as shown in Example 4.7, we can
+parameterize them in the following form
+x1 = u − v sin t,
+x2 = v cos t,
+(x2 > 0, v ≥ 0),
+(59)
+where u, v are arbitrary constant parameters. We have shown in Example 4.7 that the spray
+G determined by G is given by (35), that is,
+G1 = −y1y2
+2x2 ,
+G2 = (y1)2
+2x2 .
+(60)
+We can make G be projectively complete on the conical region C with the direction (0, 1)
+being deleted from T R2
++ \ {0}. Since −π/2 < t < π/2 for any u, v in (59), by (47), we let
+s = tan t.
+Then by (45), we get a complete spray ¯G projective to G with the projective factor P being
+given by
+P = c
+�
+− 1
+2
+d
+dt
+� dt
+ds
+��
+t=cs = c
+�
+cos t sin t
+�
+t=cs =
+c2s
+1 + c2s2 .
+(61)
+Now it follows from (43) that
+x1 = u −
+vcs
+√
+1 + c2s2 ,
+x2 =
+v
+√
+1 + c2s2 ,
+y1 =
+−vc
+(1 + c2s2)3/2 ,
+y2 =
+−bc2s
+(1 + c2s2)3/2 ,
+from which we get
+s = −
+x2y2
+(y1)2 + (y2)2 ,
+c = −(y1)2 + (y2)2
+x2y1
+.
+Plugging s, c in the above into (61) yields P = −y2/x2. Thus the spray ¯G is given by
+¯G1 = G1 + Py1 = −3y1y2
+2x2 ,
+¯G2 = G2 + Py2 = (y1)2 − 2(y2)2
+2x2
+.
+¯G is complete on the conical region C but not complete in the direction (0, 1). We don’t know
+whether the spray G in (60) can be projectively complete or not on T R2 \ {0}. Besides, ¯G
+is of isotropic curvature, and locally it is not metrizable by the method in [2, 11].
+References
+[1] R. Bryant, Geodesically reversible Finlser 2-spheres of constant curvature, Inspired by
+S. S. Chern, 95-111, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, 2006.
+[2] I. Bucataru and Z. Muzsnay, Finsler metrizable isotropic sprays and Hilbert’s Fourth
+Problem, J. Aust. Math. Soc. 97 (2014), 27-47.
+[3] S. G. Elgendi and Z. Muzsnay, Metrizability of holonomy invariant projective defor-
+mation of sprays, Cana. Math. Bulletin, 2020.
+15
+
+[4] L.Huang and X. Mo, On geodesics of Finsler metrics via navigation problem, P. Am.
+Math. Soc., 139 (8) (2011), 3015-3024.
+[5] Y. Li, X. Mo and Y. Yu, Inverse problem of sprays with scalar curvature, Intern. J.
+Math. 30(6), 2019.
+[6] B. Li and Z. Shen, Sprays of isotropic curvature, Intern. J. math. 2019.
+[7] C. Robles, Geodesics in Randers spaces of constant curvature, Trans. Amer. Math.
+Soc. 359 (2007), 1633-1651.
+[8] Z. Shen: On projectively related Einstein metrics in Riemann-Finsler geometry, Math.
+Ann., 320(2001), 625-647.
+[9] Z. Shen: Differential geometry of spray and Finsler spaces, Kluwer Academic Publish-
+ers, Dordrecht, 2001.
+[10] G. Yang, Some classes of sprays in projective spray geometry, Diff. Geom. Appl., 29
+(2011), 606-614.
+[11] G. Yang, On sprays of scalar curvature and metrizability, J. Geom. Anal., 2022 (in
+press).
+[12] G. Yang, Sprays on Hamel-Funk functions model, preprint.
+Guojun Yang
+Department of Mathematics
+Sichuan University
+Chengdu 610064, P. R. China
+yangguojun@scu.edu.cn
+16
+

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+A Multi-Source Information Learning Framework
+for Airbnb Price Prediction
+Lu Jiang1, Yuanhan Li1, Na Luo1, Jianan Wang2,∗, Qiao Ning3,∗
+1Information Science and Technology, Northeast Normal University, Changchun
+2College of Physics, Northeast Normal University, Changchun
+3Information Science and Technology, Dalian Maritime University, Dalian
+{jiangl761, liyh447, luon110, wangjn}@nenu.edu.cn, ningq669@dlmu.edu.cn
+Corresponding author*
+Abstract—With the development of technology and sharing
+economy, Airbnb as a famous short-term rental platform, has
+become the first choice for many young people to select. The
+issue of Airbnb’s pricing has always been a problem worth
+studying. While the previous studies achieve promising results,
+there are exists deficiencies to solve. Such as, (1) the feature
+attributes of rental are not rich enough; (2) the research on
+rental text information is not deep enough; (3) there are few
+studies on predicting the rental price combined with the point of
+interest(POI) around the house. To address the above challenges,
+we proposes a multi-source information embedding(MSIE) model
+to predict the rental price of Airbnb. Specifically, we first selects
+the statistical feature to embed the original rental data. Secondly,
+we generates the word feature vector and emotional score
+combination of three different text information to form the text
+feature embedding. Thirdly, we uses the points of interest(POI)
+around the rental house information generates a variety of spatial
+network graphs, and learns the embedding of the network to
+obtain the spatial feature embedding. Finally, this paper combines
+the three modules into multi source rental representations, and
+uses the constructed fully connected neural network to predict
+the price. The analysis of the experimental results shows the
+effectiveness of our proposed model.
+I. INTRODUCTION
+Accommodation sharing systems are being introduced to
+more and more cities recently, and therefore they have gener-
+ated huge amounts of data. Airbnb is an online marketplace
+for sharing home and experience which is suffering from the
+chaotic pricing problem. Tenants need to know the reasonable
+price of this rental house to prevent being deceived. The
+homeowner needs to customize a reasonable price for their
+short-term rental house to attract more customers. Therefore,
+airbnb price prediction plays a key role in accommodation
+sharing systems. However, rapid increase in the number
+of tenants and homeowners makes traditional manual-based
+methods [1] time-consuming and inefficient. Computational
+methods have received more attention for accurate airbnb price
+prediction [2].
+Computational methods for price prediction can be mainly
+divided into two categories: (1) feature-based methods [3], and
+(2) deep learning methods [4, 5]. In feature-based methods,
+various types of features extraction strategies are utilized to
+extract price correlated features for tenants and homeowners.
+Feature-based methods transform price prediction into a ma-
+chine learning methods, such as support vector machine(SVM)
+and random forest. For instance, in order to distinguish from
+the traditional method of formulating prices, Li et al. selects
+rough set (RS) and SVM algorithms to establish a new math-
+ematical model of pricing on the basis of hedonic price [6].
+PR Kalehbastiet al. proposed a price prediction model us-
+ing machine learning, deep learning, and natural language
+processing techniques to embed the features of the rentals,
+owner characteristics, and the customer reviews [7]. Deep
+learning methods, which use multi-layer neural network to
+map the correlation between input features and output results.
+For instance, Chen et al. applied auto regressive integrated
+moving average model to generate the baseline while LSTM
+networks to build prediction model [8].
+However, the research of airbnb price prediction based on
+feature-based methods consider the single feature in most
+cases. With the development of representation learning [9, 10],
+the spatial embedding [11, 12] have received more attention.
+There has been work to model the statistical features, text
+feature and spatial features related to housing prices, but there
+is no unified framework to integrate the above features. Based
+on the above disadvantages, we proposes a prediction model
+based on multi-source information embedding to study the
+Airbnb price problem. The major contributions are summa-
+rized below.
+• Firstly, in order to obtain the best feature set, this paper
+selects the features of the house itself to obtain statistical
+information features.
+• Secondly, the text information in this paper is divided into
+three categories, and the house description and landlord
+introduction are converted into feature matrix. The tenant
+reviews are then converted into sentiment scores about
+each house.
+• Then, we uses different types of point-of-interest (POI)
+data and houses to form various spatial network graphs
+and learns their network embeddings to obtain spatial
+information features.
+• Finally, we combines these three types of feature embed-
+dings are combined into multi-resource housing features
+as input, and the neural network constructed in this paper
+is used for price prediction. The effectiveness of our
+model is demonstrated with two real data.
+arXiv:2301.01222v1  [cs.LG]  1 Jan 2023
+
+II. PRELIMINARY
+We first introduce some key definitions and the problem
+definition. Then, we present the overview of the proposed
+method.
+A. Definitions and Problem Statement
+Definition 1. Statistic Feature The statistics feature con-
+structed by our model is S =(s1, s2, ..., sn), where si
+is the preprocessed listing features includes ’host since’,
+‘host is superhost’, ’verification’, etc.
+Definition 2. Text Feature There are three types of text
+features: listing description, host introduction, and tenant
+review. We convert listing description and host introduction
+into feature vector L =(l1, l2, ..., ln) and H =(h1, h2, ..., hn),
+and transform the tenant review to sentiment score R
+=(r1, r2, ..., rn). Thus, we define the text features as T =
+(L, H, R).
+Definition 3. Spatial Feature We first combine each rental
+house and the POI with in 1,000m around it into a spatial
+network G = (V, E, W). Then, we learn network embedding
+through SDNE [13], and get the spatial feature matrix P
+=(p1, p2, ..., pn).
+Definition 4. Problem Statement In this paper, we study the
+problem of airbnb price prediction. We formulate the problem
+as a multi-source feature embedding task. Formally, we aim
+to find a mapping function f : (S, T, P) → V that takes
+the statistic feature S, text feature T, spatial feature P as
+input, and outputs a unified vectorized representations V , for
+predicting the specific listing price.
+B. Framework Overview
+Figure 1 shows an overall framework for the multi-source
+feature embedding. Specifically, we embed the original data
+from three aspects. (1) For the statistical feature embedding,
+we uses Lasso CV to select the feature set with the rental
+house feature. (2) For the text feature embedding, we divides
+the text feature into three categories, include house descrip-
+tion, landlord introduction and tenant comments. Through the
+negative sampling CBOW model, the house description and
+landlord introduction are converted into word feature vectors,
+and the Bayesian model based on naive Bayesian principle is
+used to convert tenant comments into emotional scores, we
+combine them as the text feature. (3) For the spatial feature
+embedding, we collects different types of POIs, and combines
+the POI of each house and the surrounding area within 1,000m
+into a spatial network. Through the SDNE model to learn the
+spatial feature. (4) Three different features are combined into
+a multi-source feature and input into the neural network to
+obtain the final rental price.
+III. MULTI-SOURCE INFORMATION LEARNING
+In this section, we introduce the core architecture of our
+framework as follows: (1) statistic feature embedding; (2) text
+feature embedding; (3) spatial feature embedding.
+A. Statistics Feature Embedding
+Each
+house’s
+statistic
+feature
+is
+represented
+by
+a
+245-
+dimensional
+vector
+which
+describes
+the
+listing
+of
+a
+house,
+including
+listing id,
+host id,
+host since,
+host response rate, host is superhost, host has profile pic,
+host identity verified,
+bathrooms,
+bedrooms,
+latitude,
+longitude, accommodates, security deposit, guests included,
+verification, etc.
+We use the Lasso CV to do the feature set selection. The
+loss function is defined as follows:
+obj = 1
+2
+n
+�
+i=1
+�
+yi − wT xi
+�2 + α
+m
+�
+j=1
+|wi|
+(1)
+where n is the number of houses, m is the number of
+parameters, α is the regularization coefficient, α �m
+j=1 |wi| is
+the L1 regularization term, yi is the rental price, xi is the
+statistical features of rental housing, w is the coefficient matrix
+of rental housing features, xi = si. Statistical feature matrix
+S =(s1, s2, ..., sn). Lasso CV can compress the coefficients of
+unimportant features to 0, realizing the purpose of feature se-
+lection, and ultimately leaving the important statistical feature
+set that this paper wants.
+B. Text Feature Embedding
+We extract three types of text data from the original data,
+there are listing description, host introduction and tenant re-
+view. Listing description mainly about introducing the location
+of the house, surrounding environment, indoor layout and
+housing regulations, etc. Host introduction mainly introduces
+the age, height, occupation, hobbies and personality of the
+host, and tenant review expresses the tenants’ feelings about
+housing rentals and the evaluation of the host’s attitudes. Since
+tenant review contain emotional value, we use two different
+methods to model text feature. We first use CBOW [14]
+model to embed the text feature of listing description, host
+introduction. We selects the Wikipedia Chinese thesaurus after
+preprocessing as the training corpus W, the objective function
+is defined as follows:
+L =
+�
+c∈W
+�
+�
+�log
+�
+σ
+�
+xT
+c θc��
++
+�
+u∈NEG(c)
+log
+�
+σ
+�
+−xT
+c θu��
+�
+�
+�
+(2)
+Then the above objective function is optimized by using the
+random gradient rise method to obtain:
+L(c, u) = Lc(u) log
+�
+σ
+�
+xT
+c θu��
++[1 − Lc(u)] log
+�
+1 − σ
+�
+xT
+c θu��
+(3)
+Then calculate the gradient of L(c, u) to obtain:
+v(˜c) := v(˜c) + η
+�
+u∈{c}∪NEG(c)
+∂L(c, u)
+∂xc
+(4)
+
+Geospatial Information Embedding
+Text  Information Embedding
+listing information
+host introduction
+tenant review
+feature vector
+feature vector
+sentiment score
+Original
+Listing
+Data
+preprocess
+Feature 
+Selection
+Statistics Information Embedding
+Real 
+Airbnb 
+Listing 
+Data
+Multi
+Source
+Listing
+Feature
+listing
+POI
+Predict
+Price
+Fig. 1: Framework Overview.
+In this paper, we set the dimension of the word vector as
+100. l and h represent the embedding of listing description,
+host introduction, respectively.
+l = 1
+Z
+Z
+�
+i=1
+v (˜cl)
+(5)
+h = 1
+z
+z
+�
+i=1
+v (˜ch)
+(6)
+Therefore, we get the text information feature matrix of the
+listing description and host introduction: L =(l1, l2, ..., ln) and
+H =(h1, h2, ..., hn).
+For the tenant review embedding, since it contains strong
+emotional expression, in order to reflect whether the tenants’
+evaluation of the house is positive or negative, we uses the
+naive Bayes method to generate the corresponding emotional
+score r ∈ [0, 1] for each house, where 0 represents the negative
+and 1 represents the positive. Specifically, the probability that
+a tenant review text belongs to the positive class can be
+expressed as:
+P (pos | c1, . . . cd) = P (c1, . . . cd | pos) P(pos)
+P (c1, . . . cd)
+(7)
+After simplifying the above formula, we can obtain:
+P (pos | c1, . . . cd) =
+1
+1 + γ
+(8)
+In this work, a text represents a tenant’s review, and a tenant
+has many reviews, so the emotional score of a tenant can be
+expressed as:
+r = 1
+q
+q
+�
+i=1
+P (pos | c1, . . . cd)
+(9)
+where q represents the number of reviews on a rental,
+d represents the total number of words in a review, and
+P (pos | c1, . . . cd) represents the probability that the review
+belongs to the category of positive emotions. Therefore, the
+emotional score vector of tenant reviews can be expressed as
+R =(r1, r2, ..., rn).
+C. Spatial Feature Embedding
+We proposes a method to learn spatial embedding. First, POI
+is divided into 8 different types, and the rented houses and the
+surrounding different type POIs form a spatial network. Then
+learn the network embedding of these spatial graphs through
+SDNE model. This method can accurately capture the spatial
+features related to important POIs such as scenic spots and
+railway stations.
+We uses Euclidean distance to calculate the weight W
+between house and poi as follows:
+W = R · arccos( dis ) · π/180
+(10)
+where dis = sin(LatA) sin(LatB) cos(LonA − LonB) +
+cos(LatA) cos(LatB), the two types of nodes, A and B,
+
+Net-
+work1
+Learning Network Embedding
+Net-
+work2
+Net-
+work3
+Net-
+workkInput
+layer
+Hidden
+layer
+Output
+layeirepresent the rental houses and POI respectively. LonA and
+LonB are their longitudes, LatA and LatB are their latitudes,
+and R is the average radius of the earth, taking the value of
+6371.004km.
+In SDNE model, the encoder is from xi to y(k)
+i
+, the decoder
+is from y(k)
+i
+to �xi, y(k)
+i
+is the node embedding of vi, in this
+paper,y(k)
+i
+= pi. The formula of encoder is:
+y(k)
+i
+= σ
+�
+W (k)y(k−1)
+i
++ b(k)�
+(11)
+Therefore, the spatial embedding can be expressed as P
+=(p1, p2, ..., pn). After we get the statistic embedding, the text
+embedding and the spatial embedding. We combine the above
+features into a multi-source feature M = (S, T, P), and use
+the fully connected neural network to predict the rental price.
+We take the multi-source feature matrix M =(m1, m2, ..., mn)
+as the input of the neural network, then obtain as follows:
+y = wT m + b,
+A = σ(y)
+(12)
+where y is the actual rental price, m is the multi-source
+feature, w is the parameter matrix, and b is the offset term.
+A is the activation function, we uses ReLU function as the
+activation function. At last, the output layer uses a neuron to
+output and get the predicted price �yi.
+IV. EXPERIMENT
+In this section, we first introduce two real dataset and
+evaluation metrics. Then, we design experiments to answer
+the following three questions:
+• Q1. How is the performance of our proposed MSIE in
+the airbnb price prediction task?
+• Q2. How do the feature combination affect the price
+prediction performance?
+• Q3. What is the key influence on the airbnb price?
+A. Dataset
+We collect the dataset from an open online airbnb website.
+Table I shows the statistics of our two real airbnb datasets
+from two cities: Beijing and Shanghai after preprocess.
+TABLE I: Statistics of the data
+City
+# Houses
+# Reviews
+Time Period
+Beijing
+10779
+191876
+01/2017-06/2019
+Shanghai
+8638
+159069
+01/2020-07/2021
+Besides, we also collect the POI of the Beijing and Shanghai
+in Table II. We divide it into 8 categories, include, Education,
+Entertainment, Food, Beverage Shopping, Tourist, Transporta-
+tion, Medical Service, and Public Service.
+B. Evaluation Metrics
+We evaluate the model performances in terms of the fol-
+lowing metrics.
+TABLE II: POI Categories
+Number
+POI Category Name
+#Beijing
+#Shanghai
+1
+Education
+8711
+2635
+2
+Entertainment
+6501
+2607
+3
+Food
+5744
+6301
+4
+Beverage Shopping
+6601
+5632
+5
+Tourist
+6713
+4176
+6
+Transportation
+4322
+1753
+7
+Medical Service
+3660
+2862
+8
+Public Service
+5976
+3699
+(1) Mean Absolute Error(MAE) represents the average of
+the absolute value of the error between the predicted value
+and the true value.
+MAE = 1
+n
+n
+�
+i=1
+|ˆy − y|
+(13)
+(2) Mean Squared Error(MSE) is a measure of the close-
+ness of the predicted value relative to the actual value.
+MSE = 1
+n
+n
+�
+i=1
+(ˆy − y)2
+(14)
+(3) Root Mean Squared Error(RMSE) is defined as fol-
+lows:
+RMSE =
+�
+�
+�
+� 1
+n
+n
+�
+i=1
+(ˆy − y)2
+(15)
+where ˆy is the predicted price from the regression and y is the
+actual price. The lower the RMSE, the better the method.
+(4) The coefficient of determination(R2) convert the pre-
+dicted results into accuracy, the results are between [0,1].
+R2 = 1 −
+�n
+i=1 (ˆyi − yi)2
+�n
+i=1 (¯yl − yi)2
+(16)
+The higher the value of R2 , the more accurate the estima-
+tion method.
+C. Baseline Algorithms
+To prove the effectiveness of our model, we compare our
+method with the following algorithms.
+(1)Extreme Gradient Boosting XGBOOST [15] is an
+improvement on the boosting algorithm based on Gradient
+Boosting Decision Tree to make it faster and more efficient.
+(2)Random Forest RF [16] is an algorithm that integrates
+multiple trees through the idea of integrated learning. Its basic
+unit is the decision tree.
+(3)Support Vector Regression SVR [17] is an algorithm
+that applies support vector machine to regression problems.
+(4)TAPE TAPE [18] analyzed the relationship between
+the description of each rental and the price, and added the
+geographical factor component to recommend a reasonable
+price for each new rental of the landlord.
+
+MAE
+0.0
+0.1
+0.2
+0.3
+0.4
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(a) MAE
+MSE
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(b) MSE
+RMSE
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(c) RMSE
+R2
+0.0
+0.2
+0.4
+0.6
+0.8
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(d) R2
+Fig. 2: Overall comparison on Beijing dataset.
+MAE
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(a) MAE
+MSE
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(b) MSE
+RMSE
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(c) RMSE
+R2
+0.0
+0.2
+0.4
+0.6
+0.8
+XGB
+RF
+SVR
+TAPE
+GSNE
+MSIE
+(d) R2
+Fig. 3: Overall comparison on Shanghai dataset.
+(5)GSNE GSNE [19] is a geospatial embedding framework,
+which can accurately capture the geospatial neighborhood re-
+lationship between houses and surrounding POIs. Essentially,
+it is to learn the low dimensional Gaussian embedding on
+the geospatial network node, and can be combined with the
+regression method, which has a certain effect on house price
+prediction.
+Besides, our proposed model has three variants of the
+feature set combination: (1) MSIE-S, where the model utilizes
+the statistic feature; (2) MSIE-ST, where the model utilizes
+the statistic feature and text feature; (3) MSIE-STP, where
+the model utilizes the statistic feature, text feature and spatial
+feature. We evaluate these three variants with our model.
+In the experiment, we split the dataset into two nonover-
+lapping sets: for all records, the earliest 80% of records
+are the training set and the remaining 20% are testing set.
+We implement the model by Pytorch and run the code on
+Windows10, Inter(R) Core(TM) i7-7700HQ @2.80GHZ and
+memory size 8G.
+D. Overall Performances
+We present the results for “MAE”, “MSE”, “RMSE” and
+“R2”, compared with baseline algorithms. Figure 2 and Fig-
+ure 3 show that our proposed method ”MSIE” outperform the
+baselines over both the Beijing and Shanghai dataset. The
+lower value of ”MAE”, ”MSE”, ”RMSE”, and the higher
+value of ”R2”, means the performance better. In all cases, we
+observe an improvement with respect to baseline algorithms,
+especially on “MSE” and “R2”. One interesting observation is
+that the traditional machine learning method(such as, ”XGB”,
+”RF” and ”SVR”) performs better than ”TAPE”. We analysis
+(a) Beijing
+(b) Shanghai
+Fig. 4: The loss curve on two datasets.
+the reason is that our algorithm feature engineering proposed
+in this paper is well done and has universality.
+Besides, we uses the fully connected neural network con-
+structed as the prediction model. In order to prevent over
+fitting, we sets 128 neurons in the input layer, uses 2 hidden
+layers and set Epoch=120, the size of Batch as 256. Figure 4
+show the loss curves of neural network training on the two
+datasets respectively.
+E. Robustness Check
+We evaluate the feature embedding contribution on mod-
+eling representations. To set the control group, we de-
+velop a variant of the proposed ”MSIE”, namely ”MSIE-S”,
+”MSIE-ST”, ”MSIE-STP”. ”MSIE-S”, ”MSIE-ST”, ”MSIE-
+STP” takes the different combination of feature set as the
+input, while other component of remains the same. Table III
+and Table IV show the comparison results. We can observe
+
+LossCurve
+1.0
+Training loss
+0.8
+0.6
+loss
+0.4
+0.2
+0.0
+Fo
+20
+40
+60
+80
+100
+120
+epochLoss Curve
+1.0
+Training loss
+0.8
+0.6
+loss
+0.4 -
+0.2 
+0.0
+0
+20
+40
+60
+80
+100
+120
+epochTABLE III: The feature combination on Beijing dataset.
+Feature set
+MAE
+MSE
+RMSE
+R2
+MSIE-S
+0.3652
+0.2341
+0.4839
+0.5545
+MSIE-ST
+0.2941
+0.1688
+0.4109
+0.6786
+MSIE-STP
+0.2905
+0.1669
+0.4086
+0.6824
+TABLE IV: The feature combination on Shanghai dataset.
+Feature set
+MAE
+MSE
+RMSE
+R2
+MSIE-S
+0.4003
+0.2852
+0.5340
+0.5824
+MSIE-ST
+0.3512
+0.2371
+0.4869
+0.6527
+MSIE-STP
+0.3310
+0.2065
+0.4544
+0.6977
+that the performance of ”MSIE-STP” outperforms ”MSIE-
+S” and ”MSIE-ST” in terms of the four metrics over both
+two datasets. The results validate that the integration of text
+feature and spatial feature indeed enhances the modeling of
+price prediction.
+F. Analysis of Key Influence
+In order to analysis the key feature-influence of price, ac-
+cording to the previous studies, we use three feature selection
+method, include manual selection, P-value [20] and Lasso
+CV [21]. We use R2 as an indicator to analyze, and the results
+are shown in Figure 5. The best result is to use Lasso CV to
+select features from the original data.
+(a) Beijing
+(b) Shanghai
+Fig. 5: The R2 with different feature selection method.
+Then, we selects 20 features with the highest correlation for
+rental price prediction according to P-value, and uses Lasso
+CV to select features to obtain statistical information as input
+for prediction. Figure 6 show the 10 features with the highest
+correlation with rental prices in the two datasets.
+V. RELATED WORK
+In this work, we propose a model to predict the price of
+listings on Airbnb. Some studies used sentiment analysis to
+study the problem of Airbnb. Martinez R D et al. studied
+the relationship between Airbnb host’s listing description and
+occupancy rates by sentiment analysis [22]. Zhang et al.
+proposed a text analytics framework to study the relationship
+among self-description, trust perception and purchase behavior
+(a) Beijing
+(b) Shanghai
+Fig. 6: The feature most relevant to price.
+on Airbnb. They used text mining to extracted sentiment
+intensity and use regression method to identify the impact
+of linguistic and semantic features on trust perception [23].
+Kalehbast P R et al. used sentiment analysis and machine
+learning to predict the price of listings on Airbnb [24].
+Most researches have proposed many works to study the
+price by using the listing information. Wang et al. studied
+the relationship between a price and its determinants by
+various listing information (e.g., host identity verified, accom-
+modates, wireless Internet, amenities and services, and free
+parking) [25]. P.Choudhary et al. analyzed Airbnb listings in
+San Francisco to better understand how different attributes
+(e.g., bedrooms, location, and listing type) can be used to
+accurately predict the price of a new listing, which is optimal
+in terms of the host’s profitability yet affordable to their
+guests [26]. Shen et al. analyzed the relationship between
+the description of each listing and its price, and proposed a
+text-based model to recommend a reasonable price for newly
+added listings [27]. Tang et al. labeled text information for
+nine handpicked classes, extracted image-related features, and
+finally used all features to predict a listing’s neighborhood and
+its price [28].
+VI. CONCLUSION
+In this paper, we proposes a prediction model based on
+multi-source information embedding to study the Airbnb price
+problem. Specifically, in order to obtain the best feature set,
+we first selects the features of the house itself to obtain
+statistical information features. Secondly, the text information
+in this paper is divided into three categories, and the house
+description and landlord introduction are converted into feature
+matrix. The tenant reviews are then converted into sentiment
+scores about each house. Then, we uses different types of
+point-of-interest (POI) data and houses to form various spatial
+network graphs and learns their network embeddings to obtain
+spatial information features. Finally, we combines these three
+types of feature embeddings are combined into multi-resource
+housing features as input, and the neural network constructed
+in this paper is used for price prediction. The effectiveness
+of our model is demonstrated with two real data. For future
+work, we plan to combine some heuristic methods [29, 30] to
+further improve performance.
+
+0.8
+0.7 
+0.6
+0.5
+0.4 
+0.3 
+0.2
+0.1 -
+0.0
+Manual selection
+P-value
+Lasso CV0.8
+0.7 
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1 -
+0.0
+Manual selection
+p-value
+Lasso CV1.0
+price
+1.00
+0.63
+0.56
+0.56
+0.44
+0.41
+0.30
+0.24
+0.22
+0.21
+accommodates
+0.63
+1.00
+0.82
+0.82
+0.63
+0.19
+0.34
+0.26
+0.22
+0.8
+bedrooms
+0.56
+0.82
+1.00
+0.28
+0.72
+0.64
+0.14
+0.28
+0.23
+0.23
+Entirehome/apt
+0.56
+0.38
+0.28
+1.00
+0.21
+0.09
+0.35
+0.18
+0.04
+-0.01
+0.6
+beds
+0.44
+0.82
+0.72
+0.21
+1.00
+09:0
+0.10
+0.27
+0.23
+0.20
+bathrooms
+0.41
+0.63
+0.64
+0.09
+0.60
+1.00
+0.06
+0.15
+0.25
+0.26
+0.4
+TV
+0.30
+0.19
+0.14
+0.35
+0.10
+0.06
+1.00
+800
+0.08
+0.06
+guests_included
+0.24
+0.28
+0.18
+0.27
+0.15
+0.08
+1.00
+0.04
+0.02
+0.2
+Suitable_for_events
+0.22
+0.26
+0.23
+0.04
+0.23
+0.25
+0.08
+0.04
+1.00
+0.24
+latitude
+0.21
+0.22
+0.23
+-0.01
+0.20
+0.26
+0.06
+0.02
+0.24
+1.00
+0.0
+price
+commodates
+bedrooms
+ire home/apt
+beds
+bathrooms
+2
+sts_included
+_for_events
+latitude1.00
+price
+1.00
+0.72
+0.66
+0.61
+0.42
+0.34
+0.20
+0.20
+0.20
+0.19
+accommodates
+0.72
+1.00
+0.87
+0.85
+0.49
+0.28
+0.18
+0.23
+0.22
+0.23
+0.75
+bedrooms
+0.66
+0.87
+1.00
+0.89
+0.50
+0.23
+0.17
+0.22
+0.20
+0.21
+beds
+0.61
+0.85
+0.89
+1.00
+0.47
+0.18
+0.18
+0.22
+0.21
+0.21
+0.50
+Entire villa
+0.42
+0.49
+0.50
+0.47
+1.00
+0.18
+0.22
+0.22
+0.17
+0.22
+Entirehome/apt
+0.34
+0.28
+0.23
+0.18
+0.18
+1.00
+-0.23
+-0.11
+-0.05
+-0.17
+0.25
+_Backyard
+0.20
+0.18
+0.17
+0.18
+0.22
+-0.23
+1.00
+0.41
+0.27
+0.49
+_Barbecue_utensils
+0.20
+0.23
+0.22
+0.22
+0.22
+-0.11
+0.41
+1.00
+0.48
+0.62
+0.00
+Board_games
+0.20
+0.22
+0.20
+0.21
+0.17
+-0.05
+0.27
+0.48
+1.00
+0.33
+_BBQ_grill
+0.19
+0.23
+0.21
+0.21
+0.22
+-0.17
+0.49
+0.62
+1.00
+price
+commodates
+bedrooms
+speq
+Entire villa
+ire home/apt
+cue_utensils
+oardACKNOWLEDGMENTS
+This work is supported by the Natural Science Research
+Foundation of Jilin Province of China under Grant No.
+YDZJ202201ZYTS415, the Fundamental Research Funds for
+the Central Universities 2412019ZD013, NSFC (under Grant
+Nos.61976050 and 61972384).
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+

49FIT4oBgHgl3EQf7St_/content/tmp_files/2301.11397v1.pdf.txt

@@ -0,0 +1,1532 @@
+IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,  MANUSCRIPT ID 
+1 
+ 
+Automating Knowledge-Driven Model 
+Recommendation: Methodology, Evaluation, 
+and Key Challenges 
+Adam A. Butchy, Cheryl A. Telmer, and Natasa Miskov-Zivanov 
+Abstract—There is significant interest in using existing repositories of biological entities, relationships, and models to automate 
+biological model assembly and extension. Current methods aggregate human-curated biological information into executable, 
+simulatable models, but these models do not resemble human curated models and do not recapitulate experimental results. Here, 
+we outline the process of automated model assembly and extension, while demonstrating it on both synthetic models and human-
+curated models of biological signaling networks. We begin with an iterative, greedy, and combinatoric approach to automated 
+assembly and demonstrate the key difficulties inherent to contextless assembly. We publicly release the software used in this 
+paper to enable further exploration of this problem. 
+Index Terms— Automatic Model Creation; Biological Networks; Extending Biological Networks; Model Construction; Network 
+Reconstruction.  
+——————————   u   —————————— 
+1 INTRODUCTION
+omputational approaches to modeling large complex 
+systems standardize the representation of knowledge, 
+while simulation of computational models illuminates the 
+dynamics of systems, allowing for discoveries and theoret-
+ical advances [1]. Due to the complexity and redundancy 
+of biological systems, computational models are difficult 
+and laborious to create and update. There are two main ap-
+proaches to modeling these systems, bottom-up and top-
+down [2]. In a bottom-up approach, known molecular in-
+teractions are assembled into a model to help explain the 
+system’s behavior and predict how the system will re-
+spond to new stimuli or inputs. This method has been used 
+extensively by biologists, biochemists, and molecular biol-
+ogists to manually create models based on the interactions 
+within cells involved in signaling that are supported by sci-
+entific literature. In a top-down approach, experimental 
+data—usually collected with high-throughput methods—
+is used to infer correlations between element behavior and 
+determine causal relationships. Top-down approaches em-
+ploy many different methods such as Bayesian Inference 
+[3], ANOVA calculations [4], and Fuzzy Logic [5]. In both 
+the mechanistic bottom-up approach and the data-driven 
+top-down approach, the model is used to predict the be-
+havior of individual elements in the network [6, 7]. Re-
+cently, there has been a push to integrate the two methods, 
+using experimental data to inform the bottom-up ap-
+proach, and incorporating prior knowledge into the top-
+down approach to reduce the number of potential models 
+[8-12]. Despite these hybrid approaches, this problem re-
+mains a combinatoric one, with large, complex systems be-
+ing prohibitively difficult to investigate and model manu-
+ally. 
+It is a direct result of these factors that system and com-
+putational biologists have endeavored to automate the 
+process of model creation and extension. To automatically 
+create models, information can be extracted from litera-
+ture, queried from databases, or taken from existing path-
+ways and models. Public databases such as Reactome [13], 
+MetaCyc [14], OmniPath [15], and STRING [16] offer easy 
+access to millions of interactions. Additionally, there exist 
+a number of model databases with published models that 
+are publicly available such as The Nature Pathway Interac-
+tion Database [17], WikiPathways [18], BioModels [19], the 
+Cell Collective [20], and KEGG pathways [21]. These data-
+bases contain highly targeted, curated published and un-
+published models which are created for specific biological 
+context and may not be generalizable to explain other phe-
+nomena. When new interactions are discovered, and de-
+scribed in a scientific publication, state-of-the-art machine 
+reading engines such as REACH [22], TRIPS [23], and 
+EVEX [24] can extract them, together with other relevant 
+information. These automated readers are able to extract 
+tens of thousands of biological entity interactions from 
+hundreds of papers in a few hours, and produce a ma-
+chine-readable, structured output [22]. Despite this abun-
+dance of available interactions, there is still no efficient 
+way to assemble them into accurate models that correctly 
+reflect the system under investigation and the same biolog-
+ical context and recapitulate the observed experimental be-
+havior. 
+Recently, a few tools, such as Path2Models [25] and IN-
+DRA [26, 27], have been created to help modelers collect 
+biological interactions, assemble a model, and perform 
+xxxx-xxxx/0x/$xx.00 © 200x IEEE        Published by the IEEE Computer Society 
+———————————————— 
+• A.A. Butchy is with the Department of Bioengineering, University of Pitts-
+burgh, Pittsburgh, PA 15213. E-mail: adam.butchy@pitt.edu. 
+• C.A. Telmer is with the Department of Biological Sciences, Carnegie 
+Mellon University, Pittsburgh, PA 15213. E-mail: ctelmer@cmu.edu. 
+• N. Miskov-Zivanov is with the Departments of Electrical and Computer 
+Engineering, Bioengineering, and Computational Biology, University of 
+Pittsburgh, Pittsburgh, PA 15213. E-mail: nmzivanov@pitt.edu. 
+ 
+C 
+
+2 
+IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,  MANUSCRIPT ID 
+ 
+simulations. These tools assemble quantitative and quali-
+tative models using available pathway information; how-
+ever, the quality of the assembled models is dependent 
+upon the modeling approach, and the granularity of the 
+information they are given. These techniques rely on accu-
+rate information, and their performance suffers when the 
+interaction information is incomplete, from a different bio-
+logical context, or erroneous. Other methods have been 
+proposed to automatically expand, test, and select the best 
+model, with respect to a given performance metric. These 
+approaches integrate stochastic model simulations with 
+statistical model checking only [28], or also incorporating 
+Markov clustering [29], or genetic algorithm [30], and 
+therefore have different strengths and weaknesses. The 
+Markov clustering approach to model extension is well 
+suited for the combinatorial explosion in the number of 
+possible model extensions while the genetic algorithm ap-
+proach is overwhelmed by large number of extensions. 
+Markov clustering prioritizes strongly connected compo-
+nents at the expense of interactions involving nodes of low 
+degree. The genetic algorithm explores the effect of single 
+extensions distributed throughout the network. 
+In this work, we examine the complexities inherent to 
+automatic model assembly and extension. We use two 
+novel algorithms, Breadth First Addition (BFA) and Depth 
+First Addition (DFA), which utilize the same principles as 
+the breadth-first search and depth-first search algorithms 
+in network studies [31] to illustrate the key limitations of 
+iterative model assembly and extension. In contrast to pre-
+vious work [28-30], these methods not only represent a 
+new approach to bottom-up model assembly but are also 
+used to demonstrate the existence of key biological prop-
+erties which hinder automated modeling of biological sys-
+tems. We demonstrate these properties using both syn-
+thetic networks, Erdös-Rényi random networks (ER) [32] 
+and Barabási-Albert scale-free networks (BA) [33], as well 
+as two published expert curated and validated models, a T 
+cell large granular lymphocyte (TLGL) leukemia model 
+[34], and a model of  naïve Tcell differentiation (Tcell) [35]. 
+By using different network structures, we are able to more 
+comprehensively explore automated model assembly and 
+identify the main difficulties with the BFA and DFA ap-
+proaches. 
+2 METHODS 
+2.1 Discrete Models and Simulations 
+The underlying structure of models that we study here is a 
+network 𝐺(𝑉, 𝐸), where 𝑉 is a set of nodes (model ele-
+ments), and 𝐸 is a set of directed edges (regulatory influ-
+ences between elements). A few toy examples of such net-
+works are shown in Figure 1 (A). Model elements usually 
+represent proteins, genes, chemicals, or biological pro-
+cesses. For each model element 𝑣! ∈ 𝑉 (𝑖 = 1. . 𝑁, where 
+𝑁 = |𝑉|), we define an update rule 𝑣! = 𝑓"!(𝑣#, 𝑣$, … , 𝑣%), 
+which can either be a constant (input nodes in network 𝐺) 
+or it can depend on a subset of elements from 𝑉. In the lat-
+ter case, for each element 𝑣! this subset is often referred to 
+as an influence set for 𝑣! and it consists of its positive (acti-
+vating)	 and negative (inhibiting) regulators. Positive 
+regulators of 𝑣! comprise set 𝑉&'(
+!  and are represented with 
+regular arrowheads in Figure 1 (A). Negative regulators of 
+𝑣! comprise set 𝑉)*+
+!  and are represented with blunt arrow-
+heads in Figure 1 (A). 
+The high throughput retrieval of interaction infor-
+mation from literature typically only includes knowledge 
+of the sign of influence (positive or negative) and rarely ad-
+ditional information about relationships between regula-
+tors. In such cases, logic functions and elements with two 
+levels, 0 (low) and 1 (high), have been found most suitable. 
+To broaden the application beyond just Boolean functions 
+to other cases where interactions were enriched either 
+through manual curation or more specific information re-
+trieval, we will assume that each element 𝑣! can have 𝐿! 
+number of discrete levels. While the choice of function 
+does not affect the main algorithms described in Section 
+2.2, in order to simulate models, and closely approximate 
+different functions, including Boolean, we adopted the 
+common approach that computes a (weighted) sum of reg-
+ulator values to determine element update values. The 
+general form of this function is: 
+𝑔"! = 𝑓"!(𝑣#, ����$, … , 𝑣%) = ∑
+𝑤,𝑣,
+""∈.#$%
+!
+− ∑
+𝑤/𝑣/
+"&∈.'()
+!
+ 
+(1) 
+The weighting factors 𝑤, and 𝑤/ can be used to account for 
+different influence strengths for regulators. To remain 
+within boundaries of the allowed levels for element 𝑣! (0..	
+𝐿! − 1), the function 𝑔"! is then used to determine a suitable 
+increment/decrement for 𝑣!, 𝛿"! = 𝑓(𝑔"!), such that:  
+𝑣!,)*12 = 8
+0
+𝑣! + 𝛿"! ≤ 0
+𝑣! + 𝛿"!
+0 < 𝑣! + 𝛿"! < 𝐿! − 1
+𝐿! − 1
+𝑣! + 𝛿"! ≥ 𝐿! − 1
+ 
+(2) 
+Together, the set of model elements 𝑉, element influences 
+forming the set 𝐸, and the set of element update rules 𝐹, 
+comprise an Executable Model, ℳ(𝑉, 𝐸, 𝐹), a model that in-
+cludes all the necessary information for simulation and dy-
+namic analysis.  
+We use the Discrete, Stochastic, Heterogeneous simula-
+tor (DiSH) [36] which allows for simulations of discrete 
+models with various types of update functions, and has 
+several different simulation schemes, that can be either de-
+terministic or stochastic. For the analysis we conducted 
+here, we used the USB-RSQ simulation scheme in DiSH 
+(uniform, step-based, random-order, sequential update 
+scheme, described in detail in [36]) . It has been shown pre-
+viously [36, 37] that, by taking into account the random-
+ness in timing of signaling events, the USB-RSQ simulation 
+scheme is able to recapitulate the network dynamics within 
+cells. DiSH simulates the models starting from an initial 
+state 𝒒ℳ,4 = A𝑠"*,4, 𝑠"+,4, …	, 𝑠",,4C (assigned before simula-
+tions), where 𝑠"!,4 denotes the state value of element 𝑣! at 
+time point 𝑡 = 0, and for a pre-defined number of time 
+steps, 𝑇 (e.g., when the steady state is reached). Each such 
+simulation run, 𝑟, yields for every model element 𝑣! ∈ 𝑉, a 
+trajectory of values, 𝒔"!
+5 =	A𝑠"!,#
+5 , 𝑠"!,$
+5 , …	𝑠"!,6
+5
+C, where 𝑠"!,2
+5  is 
+the state value of element 𝑣! at time point 𝑡 (𝑡 = 1, . . , 𝑇) 
+within run 𝑟. Due to the randomness of the update scheme, 
+element trajectories may vary across multiple runs that 
+
+BUTCHY ET AL.:  TITLE 
+3 
+ 
+start with the same initial state. Therefore, for the same 
+time step 𝑡, following the approach from [36], we compute 
+the mean of values 𝑠"!,2
+5  across different runs, to obtain av-
+erage trajectories for all elements. More formally, we com-
+pute an average element trajectory of element 𝑣! as: 
+𝒔H"! = 1
+𝑅 J 𝒔"!
+5
+7
+58#
+= 1
+𝑅 JA𝑠"!,#
+5 , 𝑠"!,$
+5 , …	𝑠"!,6
+5
+C
+7
+58#
+ 
+= A𝑠̅"!,#, 𝑠̅"!,$, …	𝑠̅"!,6C 
+ 
+(3) 
+where 𝑅 is the overall number of conducted simulation 
+runs. For example, in Figure 1 (B), we illustrate simulation 
+trajectories for elements of the toy models in Figure 1 (A). 
+We denote average model state for model ℳ(𝑉, 𝐸, 𝐹) at time 
+step 𝑡 as a vector of average element states at time step 𝑡:  
+ 
+𝒒ℳ,2
+9"+ = A𝑠̅"*,2, 𝑠̅"+,2, …	, 𝑠̅",,2C 
+(4) 
+We define model behavior resulting from a specific initial 
+model state 𝒒ℳ,4 = (S#, S$, …	, S%) as: 
+ 
+𝑸ℳ = A𝒒ℳ,4, 𝒒ℳ,#
+9"+, …	, 𝒒ℳ,6
+9"+C  
+(5) 
+2.2 Extension method inputs 
+We define here inputs used by extension methods and by 
+our evaluation methodology: Baseline Model, Golden 
+Model, and Candidate Knowledge.  
+Existing models of a system of interest are often lever-
+aged and contextualized for a specific purpose. The Base-
+line Model is the existing, high confidence model before 
+updating with extensions. As a special case, we can also 
+assume that the Baseline Model is an empty network with 
+no nodes or edges. The Golden Model is assumed to 
+contain all relevant knowledge about the system, including 
+accurate element relationships and update functions. The 
+Candidate Knowledge is a set of directed edges, including 
+their source and target nodes, which are candidates for ad-
+dition to the Baseline Model. 
+Given the Golden Model knowledge, through simula-
+tions, for different initial states representing different con-
+ditions and scenarios, we can obtain Golden Model behav-
+ior, 𝑸:;, as in Equation 5. 𝑸:; represents the true expected 
+behavior of the system being modeled. As part of 𝑸:;, we 
+also obtain the average Golden Model state at the final sim-
+ulation time step 𝑇 (e.g., steady state), 𝒒:;,6
+9"+ .   
+The above definition of Golden Model is important for 
+the rest of our discussion since Golden Model is used as an 
+input to our evaluation methodology. However, in real sce-
+narios, the Golden Model is usually not known in advance. 
+Instead, the goal of model assembly and extension algo-
+rithms is to discover the Golden Model, while only the real 
+system behavior, i.e., measured state values for system 
+components, may be available. The system state data can 
+be used to form the target behavior 𝑸N. Ideally, the Golden 
+Model behavior is identical to the target behavior, ���:; =
+𝑸N. The target state at time 𝑇 is part of the target behavior 
+and is denoted as 𝒒O6.  
+As will be detailed in the following sub-sections, exten-
+sion algorithms start with the Baseline Model for which 
+𝑸<; ≠ 𝑸N. Next, they add selected edges from the Candi-
+date Knowledge to create new models, called Candidate 
+Models, which are then iteratively updated and simulated 
+to obtain 𝑸=; in each iteration, and to ultimately find a 
+model that most closely reproduces the target behavior 𝑸N. 
+Figure 1. A toy example illustrating directed cyclic network models explored in this work and the flow of the proposed meth-
+odology for evaluating extension algorithms. (A) (top) An example Golden Model used in evaluation; (middle) Example input 
+graphs, Candidate Knowledge, and Baseline Model, used in extension methods ([28-30] and this work); (bottom) An example 
+Candidate Model recommended by extension methods. (B) Average element trajectories obtained from stochastic simulation 
+for the three example models (Golden, Baseline, and Candidate). (C) An example iterative procedure that uses the Total Model 
+Error (TME) metric to evaluate each intermediate Candidate Model.  
+ 
+
+A
+Golden Model
+B
+c
+Iterative Extension*
+Golden Model Behavior
+A
+B
+a
+C
+D
+e
+E
+TME
+F
+Edge Removal
+Candidate
+Baseline Model
+Baseline Model Behavior
+Knowledge
+F
+0
+A
+B
+B
+1
+2
+B
+D
+D
+(Baseline)
+E
+E
+Iteration
+B
+A
+C
+F
+c
+Where:
+a = TME of Baseline Model
+Model Extension
+b = TME of Candidate Model 3
+c = TME of Candidate Model 1
+Candidate Model
+Candidate Model Behavior
+d = TME of Candidate Model 2
+A
+e = TME of Candidate Model 4
+E
+B
+f = TME of Candidate Model 5
+D
+→ = Path of minimizing TME
+C
+→ = Explored TME paths
+F4 
+IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,  MANUSCRIPT ID 
+ 
+2.3 Model evaluation metric 
+Given two models, ℳ# and ℳ$, if they have the same ele-
+ment sets, 𝑉ℳ* ≡ 𝑉ℳ+ ≡ 𝑉 (𝑁 = |𝑉|), and if we simulate 
+them starting from the same initial state, 𝒒ℳ*,4 = 𝒒ℳ+,4 =
+A𝑠"*,4, 𝑠"+,4, …	, 𝑠",,4C, to obtain their behaviors, 𝑸ℳ* and 
+𝑸ℳ+, respectively, we can compute the difference between 
+the two model behaviors, Δ2A𝑸ℳ*, 𝑸ℳ+C at any simulation 
+time step 𝑡 as:    
+ 
+Δ2A𝑸ℳ*, 𝑸ℳ+C = ∑
+S𝑠̅"!,2
+ℳ* − 𝑠̅"!,2
+ℳ+S
+%
+!8#
+ 
+(6) 
+In other words, Δ2 finds the absolute difference between an 
+element’s average state in time step 𝑡, in model ℳ# (𝑠̅"!,2
+ℳ*) 
+and in model ℳ$ (𝑠̅"!,2
+ℳ+) and sums these differences across 
+all model elements.  
+From (6), we derive the Total Model Error (TME) metric, 
+as Δ6, when 𝑡 = 𝑇, between a Candidate Model behavior 
+𝑸=; and known target behavior 𝑸N: 
+ 
+TMEA𝑸=;, 𝑸NC = Δ6A𝑸=;, 𝑸NC = ∑
+S𝑠̅"!,6
+=; − 𝑠̂"!,6S
+%
+!8#
+ (7) 
+Or, in the case when a Golden Model is used: 
+ TME(𝑸=;, 𝑸:;) = Δ6(𝑸=;, 𝑸:;) = ∑
+S𝑠̅>!,6
+=; − 𝑠̅>!,6
+:;S
+%
+!8#
+ (8) 
+Besides the above defined Δ2, other types of functions 
+could be used to compute the difference between two mod-
+els, such as the squared error, or more statistic-based eval-
+uation methods like the Chi-squared test to compare the 
+Figure 2. The Breadth and Depth First Addition (BFA and DFA, respectively) algorithms. Top: The pseudocode for the two 
+algorithms. Bottom: An example illustrating the Candidate Knowledge and Baseline Model inputs and steps for BFA and DFA 
+algorithms: (A, D) The inputs to the BFA and DFA algorithms. (B) In the BFA extension process, the Baseline Model is extended 
+with single interactions from Candidate Knowledge and the TME is calculated for each Candidate Model. The Candidate Model 
+with the lowest TME is selected and becomes the Baseline Model for the next iteration. (E) In the DFA extension process, the 
+Baseline Model is extended with a single interaction from Candidate Knowledge and the TME is calculated to determine if the 
+Candidate Model has a lower TME than the Baseline Model. As soon as the TME decreases, that edge of Candidate Knowledge 
+is incorporated into the Candidate Model, and it becomes the Baseline Model for the next iteration. (C, F) For both algorithms, 
+the process is repeated with the remaining Candidate Knowledge until all edges are added back, the TME reaches zero, or there 
+are no edges that reduce the TME below its current lowest value. 
+  
+ 
+ 
+
+Algorithm: Breadth First Addition (BFA)
+Algorithm: Depth First Addition (DFA)
+Input: baseline model (MBM), list of edges (ENEw), TME of the baseline model
+Input: baseline model, list of edges, expected performance of the golden
+(TMEBM), expected performance of the golden model (QGM)
+model, current TME
+Output: extended baseline model that minimizes the TME
+Output: extended baseline model that minimizes the TME
+1: while (TMEBM!= 0) and (ENEw != [)
+1: 
+EADDED = FALSE
+2:
+Initialize scores = []
+2:
+for edge in ENEw:
+3:
+for edge in ENEw:
+3:
+4:
+McM = a candidate model is created by adding the edge to the MBM
+4:
+ simulate McM
+5:
+simulate McM
+5:
+TME(QGM,QcM ) use the TME function to compare
+6:
+TME(QGM,QcM ) use the TME function to compare the
+6:
+the candidate model to the expected performance
+7:
+7:
+candidate model to the expected performance of the golden model
+of the golden model
+8:
+8:
+Append TMEcM to the scores list
+if TMEcM < TMEBM:
+9:
+9:
+end for
+McM = a candidate model is created by
+10:
+10:
+find index = min(scores)
+adding the edge to the MBM
+11:
+11:
+TMEcM = scores(index)
+MBM = McM
+12:
+12:
+if TMEcM < TMEBM:
+ENEw.delete(edge)
+13:
+13:
+ McM = a candidate model is created by adding
+14:
+14:
+the edge to the MBM
+EADDED = TRUE
+15:
+15:
+MBM = McM
+exit for loop
+16:
+16:
+17:
+ENEw.delete(index)
+17:
+end if
+18:
+TMEBM = TMEcM
+18:
+end for
+19:
+else
+19:
+if (EADDED =- FALSE)
+20:
+return MBM
+20:
+return MBM
+21:
+end if
+21:
+end if
+22: end while
+22:
+: end while
+23: return MBM
+23: return MBMCandidate
+Baseline Model
+Candidate
+Baseline Model
+A
+Knowledge
+TME = 5.0
+Knowledge
+TME = 5.0
+BFA
+DFA
+B
+D
+Inputs
+Inputs
+2)
+2) 
+3)
+3) 
+ B
+E
+First
+First
+Extension
+Extension
+Round
+Round
+Extension 1
+Extension 2
+Extension 3
+Extension 1
+TME = 3.0
+TME = 2.0
+TME = 6.0
+TME = 3.0
+c
+:
+Second
+Second
+Extension
+Extension
+Round
+Round
+Extension 2&1
+Extension 2&3
+Extension 1&2
+Extension 1&3
+TME = 0.0
+TME = 4.0
+TME = 0.0
+TME = 4.0BUTCHY ET AL.:  TITLE 
+5 
+ 
+distribution of model states at time step 𝑡. We use the ab-
+solute difference of the model’s end state (𝑡 = 𝑇) for a few 
+reasons: it would not exaggerate the effect of large differ-
+ences (as would be observed in the squared error); it is less 
+computationally expensive than the Chi-squared test; and 
+it more accurately matches how computational biologists 
+compare computational model simulations against sparse 
+biological measurements, where the full time-course of the 
+model elements is often unknown. 
+2.4 Methodology for evaluating model extension 
+In this work, we are interested in evaluating automated 
+model extension, that is, the limitations of automatically 
+extending the Baseline Model with behavior 𝑸<; to 
+achieve the target or Golden Model behavior 𝑸:;. There-
+fore, in our studies we assume that the Golden Model is 
+known, and to obtain Baseline Models we use the proce-
+dure illustrated in Figure 1 (A) and described as follows.  
+For a given Golden Model, we create multiple Baseline 
+Models by removing edges from the Golden Model, in order 
+to disrupt its behavior and to determine whether the ex-
+tension algorithms are able to recover the Golden Model 
+from a range of Baseline Models. The removed edges form 
+the Candidate Knowledge sets (Figure 1 (A)). The extension 
+algorithms are given the Baseline Model and the Candi-
+date Knowledge and tasked with extending the Baseline 
+Model using edges from the Candidate Knowledge, to cre-
+ate Candidate Models (Figure 1 (A)) and reproduce the 
+Golden Model behavior.  
+Using the DiSH simulator, we simulate the Golden, 
+Baseline, and Candidate Models to observe how elements 
+of each model behave over time, and to obtain model be-
+haviors 𝑸:;, 𝑸<;, 𝑸=;, respectively (Figure 1 (B)). The goal 
+of this procedure is to find Candidate Model(s) with be-
+havior similar to the Golden Model behavior. By tracking 
+TME (Equation 8) across consecutive extension iterations, 
+we can add Candidate Knowledge to the Baseline Model 
+to form new Candidate Models and determine whether 
+these new models perform more closely to the Golden 
+Model (Equation 8). If the TME decreases, the Candidate 
+Model is considered an improvement to the Baseline 
+Model. If the TME increases, the Candidate Model is 
+considered worse than the model from previous iteration, 
+and the Candidate Knowledge incorporated is removed 
+from the model. At each iteration, all Candidate 
+Knowledge is added one interaction at a time and the TME 
+is calculated. Candidate Knowledge with the largest de-
+crease of TME is incorporated.  
+2.5 The Breadth First and Depth First Algorithms 
+In this analysis, we employ two algorithms to illustrate two 
+different philosophies in automated assembly and exten-
+sion; namely (i) incorporating the least amount of infor-
+mation necessary into the model that best improves the 
+model and (ii) incorporating the most amount of infor-
+mation into the model as long as it relates to and improves 
+the model. These algorithms are called the: (i) Breadth First 
+Addition (BFA) algorithm that compares all potential ad-
+ditions against each other to only add the best supported 
+information at any one time, and the (ii) Depth First Addi-
+tion (DFA) algorithm that incorporates any new infor-
+mation that improves the model. The pseudocode for the 
+two algorithms is shown in Figure 2 (top) and we depict 
+example demonstrations for both algorithms in Figure 2 
+(bottom). 
+The Breadth First Addition (BFA) algorithm starts by 
+evaluating the contribution of each new edge to decreasing 
+TME, that is, it simulates the model that consists of the 
+original Baseline Model and a selected new edge, and then 
+computes TME of that extended model according to Eq 8. 
+Next, it permanently incorporates the new edge that leads 
+to the largest decrease in the original TME, and then it re-
+peats the steps with this new extended model, i.e., similar 
+to what was done with the original model, it evaluates ad-
+dition of the remaining edges to this new model by com-
+puting their TME values. This process is repeated until at 
+least one of the following conditions is satisfied: (i) the ex-
+tended model matches the expected end values of the 
+Golden Model; (ii) there are no more edges to evaluate; (iii) 
+no edge can be added to the Baseline Model without in-
+creasing TME. The pseudocode and the toy example for the 
+BFA algorithm are shown in Error! Reference source not 
+found. (left).  
+The Depth First Addition (DFA) algorithm, similar to 
+Figure 3. Network structure illustration, standard graph attributes, and node degree distribution histograms for different net-
+work types: Erdos-Renyi random networks, Barabasi-Albert scale-free networks, and two human-curated published biological 
+networks, TLGL and Tcell. 
+ 
+
+Erdos-Renyi
+Barabasi-Albert
+Published
+Published
+Network Type
+Network
+Network
+TLGL Model
+Tcell Model
+Network Structure
+Number of Nodes
+48.4 ± 1.4
+50.0 ± 0.0
+87
+80
+Number of Edges
+85.5 ± 8.9
+96.0 ± 0.0
+171
+122
+Model Density
+0.04 ± 0.00
+0.04 ± 0.00
+0.024
+0.019
+Model Average Degree
+3.53 ± 0.31
+3.84 ± 0.00
+4.07
+3.05
+Undirected Model Clustering
+0.06 ± 0.03
+0.20 ± 0.06
+0.28
+0.0
+Undirected Model Diameter
+6.9 ± 0.7
+5.0 ± 0.4
+4.0
+12.0
+Number of Models Used
+50
+50
+1
+1
+Node Degree Distribution
+0.0
+89 104
+0.0
+89 10+6 
+IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,  MANUSCRIPT ID 
+ 
+the BFA algorithm, starts with evaluation of edges by com-
+puting their contribution to decreasing TME of the Base-
+line Model. Different from BFA, as soon as it finds an edge 
+which leads to a TME lower than the current TME, it adds 
+that edge to the Baseline Model. These steps are then re-
+peated using the new extended model and the remaining 
+edges. Same as for the BFA algorithm, the DFA algorithm 
+stops when at least one of the three conditions above, (i)-
+(iii) is satisfied. The pseudocode and the toy example for 
+the DFA algorithm are shown in Error! Reference source n
+ot found. (right). 
+3 RESULTS 
+We describe here our experimental setup, including the set 
+of benchmarks that we created (Sections 3.1 and 3.2), and 
+we follow with a discussion of the outcomes of our study 
+(Sections 3.3-3.5).  
+3.1 Benchmarks: Synthetic and Curated Models 
+In this analysis, we explore how the BFA and DFA algo-
+rithms affect automated assembly and extension of two 
+types of synthetic networks and two manually curated 
+published biological signaling pathway networks.  
+The Erdos-Renyi (ER) network type is considered a ran-
+dom graph and does not share many similarities to biolog-
+ical networks. The Barabasi-Albert (BA) network type is a 
+scale-free network that has many shared characteristics 
+with biological networks (most notably their node-degree 
+distribution). Since we generated the ER and BA networks 
+in a random manner, we created 50 models for each net-
+work type. We employed the python package, NetworkX 
+[38] to create all synthetic networks.  
+The last two networks we used in our studies are the 
+human-curated biological model of T cell large granular 
+lymphocyte (TLGL) leukemia [34] and the biological 
+model of naïve T cell differentiation (Tcell) [35]. The TLGL 
+model has been used previously [39, 40] to perform struc-
+tural and dynamic analysis in order to identify potential 
+therapeutic targets, while the Tcell model was created to 
+explore the control circuitry of naïve T cell differentiation 
+[41][42]. 
+In Figure 3, we show example networks illustrating dif-
+ferent structure of these models. We also list several de-
+scriptive statistics for networks to demonstrate the similar-
+ities and differences between these network types. Model 
+Density is the fraction of edges present over all possible 
+edges between nodes. Model Average Degree is the sum of 
+each node’s degree across all model nodes (with degree be-
+ing the number of edges that are incident to the node), di-
+vided by the number of nodes in the graph. Undirected 
+Model Clustering [43] is a measure of the degree to which 
+nodes in a graph tend to cluster together in groups of local 
+triangles. Undirected Model Diameter is the maximum dis-
+tance from any node in the network to any other node. In 
+the last row in Figure 3, we provide histograms of the Node 
+Degree Distribution metric. In the case of ER and BA net-
+works, the histograms show average values for 50 gener-
+ated models. 
+3.2 Experimental Setup 
+For the purposes of the evaluation discussed here, we 
+assume that each model element 𝑣! ∈ 𝑉 (𝑖 = 1. . ��, where 
+𝑁 = |𝑉|), can be in one of the three states, OFF (value 0), 
+LOW activity (value 1), and HIGH activity (value 2). This 
+assumption makes the synthetic networks comparable to 
+the published biological models. We randomly initialized 
+the synthetic networks (as they are not based on human-
+curated or biological knowledge) while we initialized the 
+Tcell [35] and TLGL [34] models based on the values listed 
+in their corresponding publications. As nodes and edges 
+are added back into the model, we assume that the initial 
+state value of each model element 𝑣!, is 𝑠"!,#
+5
+= 1. For each 
+created model, we conducted 𝑅 = 100 simulation runs. For 
+synthetic models, we simulated ER and BA models each 
+with T = 2,500 time steps, while we simulated human cu-
+rated models—TLGL and Tcell— for T = 5,000 time steps. 
+The simulation length was governed by how long each net-
+work type required to reach a steady state.   
+3.3 Network structure and baseline information 
+complicate model assembly 
+For each Golden Model, we used five different removal 
+probabilities 𝑝5*?'"9@ ∈ [0.10, 0.25, 0.50, 0.75, 1.00] to ran-
+domly select edges for removal from the Golden Model. 
+Edges that were removed formed the Candidate 
+Knowledge and the remaining edges formed the Baseline 
+Model. When 𝑝5*?'"9@ = 1.00, the Baseline Model is empty 
+(no edges) and both the BFA and DFA algorithms will at-
+tempt to reassemble the biological networks with only 
+Candidate Knowledge. In all conducted studies (𝑝5*?'"9@ ∈
+[0.10, 0.25, 0.50, 0.75, 1.00]), both the BFA and DFA algo-
+rithms were given the exact same Baseline Models and 
+Candidate Knowledge and tasked to reconstruct the 
+Golden Model. The recall—or ratio of edges returned to the 
+Baseline Model out of all removed edges—is shown in Fig-
+ure 4 for each network type (Erdos-Renyi - blue, Barabasi-
+Figure 4. Recall distributions for all explored scenarios, for 
+each network type (Erdos-Renyi - blue, Barabasi-Albert - red, 
+TLGL - green, Tcell – purple) and at different edge removal 
+probability (𝑝5*?'"9@ ∈ [0.10, 0.25, 0.50, 0.75, 1.00]). (A) BFA 
+algorithm results and (B) DFA algorithm results. 
+ 
+ 
+
+口
+ER
+口
+BA
+TLGL
+Tcell
+A
+1.0-
+0.8
+recall
+0.6
+0.4
+0.2
+0.0
+10
+25
+50
+75
+100
+Premoval
+B
+1.0-
+0.8
+recall
+0.6
+0.4
+0.2
+美
+0.0
+10
+25
+50
+75
+100
+PremovalBUTCHY ET AL.:  TITLE 
+7 
+ 
+Albert - red, TLGL - green, Tcell - purple) and each algo-
+rithm (BFA – part A, DFA – part B).  
+In general, network type drastically affects recall rates, 
+and for the most part, each network’s recall trends down 
+with higher 𝑝5*?'"9@. This makes intuitive sense as the 
+more edges that are removed from each network, the more 
+information there is to add back, and therefore the recall 
+has a larger denominator (i.e., the size of the Candidate 
+Knowledge set). Even with many missing edges, both BFA 
+and DFA can still converge on local minima as long as each 
+edge reduces TME. Both ER and Tcell network types corre-
+spond to higher rates of recall than in BA and TLGL. As 
+both BFA and DFA add edges back based on each edge’s 
+effect on TME, this points to ER and Tcell networks having 
+more edges which tangibly reduce TME. BA networks are 
+noted for their hub and spoke structure, with a small num-
+ber of highly connected nodes, and a large number of 
+sparsely connected nodes. These networks are known for 
+their redundancy, with the removal of an edge often com-
+pensated for by the rest of the network, the behavior that 
+is observed in our results (Figure 4). 
+3.4 Model performance is difficult to encapsulate 
+into one metric to optimize 
+We also examined the relationship between the selected 
+𝑝5*?'"9@ and TME. We expected the TME to be proportional 
+to the amount of the information removed from the model 
+(i.e., the number of edges in the Candidate Knowledge). To 
+explore the effect of network structure on automated as-
+sembly and extension, we evaluated the starting TME of 
+each Baseline Model. For each Baseline Model of each net-
+work type, we calculated the actual percentage of edges re-
+moved based on the 𝑝5*?'"9@. This percentage was termed 
+the “Percent Removed”. For each network type, we plotted 
+the Percent Removed from the Golden Model and the TME 
+before extension started. Next, starting with a Golden 
+Model of each network type, we removed every combina-
+tion of two edges and calculated the TME of the resultant 
+Baseline Models. The results of these two analyses are 
+shown in Figure 5. 
+We observed from our analysis that TME is not propor-
+tional to missing information and that the contribution of 
+different edges to the model’s TME can vary. At higher lev-
+els of Percent Removed, the relationship to TME is not lin-
+ear. This points to the fact that even with only a few edges 
+missing, a model can have quite high TME. We found that 
+while TME does generally increase with more information 
+removed, this increase is not directly proportional or con-
+sistent with information removed. TME functions as a sim-
+plified error function that approximates the Baseline Mod-
+els deviation from Golden Model behavior but does not 
+completely reflect how much information is missing from 
+the Baseline Model or indicate how much information the 
+algorithm must add back.  
+Additionally, network type has a large influence on the 
+TME response to missing information. Networks like the 
+Barabasi-Albert networks appear more robust to infor-
+mation removal, with no single edge resulting in large 
+changes in TME. This same behavior is not observed in the 
+Erdos-Renyi or human curated models where only a few 
+edges can strongly affect TME. Indeed, returning to Figure 
+4, it appears that BA networks are some of the hardest to 
+assemble and extend with automated methods relying on 
+error evaluation, due to each edge only contributing a little 
+to TME. A more comprehensive error function would re-
+quire more information about the Golden Model’s network 
+Figure 5. (A) Percent Removed plotted against TME for the ER, BA, TLGL, and Tcell network types. (B) The effect of the removal 
+of pairs of edges from the network, first edge index indicated by the x-axis value, second edge index indicated by the y-axis 
+value. The TME values are represented with shades of blue, from the minimum observed (i.e., no error, TME=0, shown in white) 
+to the maximum observed (TME=50, shown in blue). Solid blue lines show the importance of particular edges to model perfor-
+mance and TME. 
+ 
+
+ER
+BA
+TLGL
+Tcell
+60
+607
+60
+601
+A
+40
+40J
+40
+40
+.
+！
+.
+:
+20
+20}
+20
+20
+:
+0
+.
+.
+of
+0+
+01
+0
+50
+100
+0
+50
+100
+0
+50
+100
+0
+50
+100
+Percent Removed
+Percent Removed
+Percent Removed
+Percent Removed
+82
+72-
+50
+B
+92
+162
+72-
+82
+142-
+62
+0 62
+Second Edge Removed
+ Second Edge Removed
+72-
+40
+122
+52
+ 62-
+ 52-
+42
+30
+82
+42-
+ 32-
+ 62
+32
+20
+22
+22
+42
+12
+A2
+22
+12-
+10
+2
+2-4
+21
+2122232425262
+7282
+122232425262728292
+.、
+22
+42
+62
+82102122142162
+2
+12
+22
+62
+72
+Do
+First Edge Removed
+First Edge Removed
+First Edge Removed
+First Edge Removed8 
+IEEE TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS,  MANUSCRIPT ID 
+ 
+structure and dynamics; however, this proves elusive as 
+the more information about the Golden Model there is, the 
+easier this problem becomes.  
+3.5 Initialization values play a small but important 
+role in network assembly 
+Finally, when adding Candidate Knowledge back into 
+the model, if a new node is introduced into the Baseline 
+Model, there is no information surrounding how it should 
+be initialized. In Figure 6 we show the effects of different 
+initialization 
+assumptions 
+when 
+adding 
+Candidate 
+Knowledge back into the model. Each network type was 
+extended with BFA and DFA algorithms using one of five 
+different initialization schemes: initializing new model el-
+ements with a fixed value (0, 1 or 2), initializing the model 
+with the correct initialization used in the Golden Model, 
+and randomly assigning an initial value. In general, initial-
+ization does not play a large role in automated assembly or 
+extension. In Figure 6, there appears to be little difference 
+between initialization types for the ER and BA network 
+types, and the TLGL model. Although the human curated 
+models (TLGL and Tcell) do diverge slightly from this 
+trend, this is much more prominent for the Tcell model, 
+which is an outlier, with automated model assembly and 
+extension suffering due to the focused nature of the model. 
+This is not to say that initialization is a problem that can be 
+disregarded in model assembly, rather it is to be considered 
+after the correct structure of the model has been identified. 
+This is particularly true in the case of logical models where 
+initialization can impact downstream model elements de-
+pending upon nature of the logic functions that are used. 
+For example, initializing to 0 a model element involved in 
+many “AND” operations will affect downstream model el-
+ements. As described in Section 2.1, we used summation 
+functions in this analysis. This choice likely made the role 
+of initialization less important, as the inclusion of a new 
+edge (and thus, a new regulator for some element in the 
+model) would not impact the effect of other regulators in 
+such a substantial way as would be present with logic 
+update functions. 
+Taken together, the discussion in Sections 3.3-3.5 and 
+Figures 4-6 demonstrate the key difficulties to automated 
+model assembly and extension. Several methods exist 
+which create such automated pipelines but do not focus on 
+how they incorporate biological information into executa-
+ble models [27, 44, 45]. To date, only a few methods have 
+been proposed to automatically assemble and extend mod-
+els, while also evaluating the available information and its 
+impact on the created executable model [28-30]. Still, even 
+these methods do not fully assess the structural and dy-
+namic impacts of adding new biological information to an 
+executable model, and therefore do not address the com-
+plexities to this problem. 
+4 CONCLUSION 
+In this paper, we have presented an automated assembly 
+and extension pipeline to depict the types and magnitudes 
+of the problems facing computational and system biolo-
+gists as they work to solve automated model assembly. 
+Through the largest assembly and extension analysis of 
+synthetic and human-curated models to date, we have 
+characterized the complexities of the automated model as-
+sembly problem. Our findings demonstrate that iterative 
+model assembly, devoid of context, lacking starting struc-
+tural information in the form of a baseline model, and 
+without robust dynamic information describing the golden 
+model’s behavior, is intractable.  More often, model assem-
+bly creates models which perform similarly in dynamics, 
+but do not represent the full information of a full “Golden” 
+network.  
+In this paper, we have demonstrated that particular fo-
+cus must be paid to a model’s structure and baseline infor-
+mation, as these can complicate model assembly. In pick-
+ing a metric to optimize during model assembly, we have 
+illustrated that a single metric more often serves to sim-
+plify the golden model, rather than recapitulate it. Lastly, 
+we have shown that initializing model elements only play 
+Figure 6. The ten networks for each network type were disassembled and then reassembled (either through BFA or DFA) under 
+different initialization schemes. In “0” new model elements are initialized with a starting value of 0. Similarly, “1” and “2” 
+follow similar schemes. “Golden” initializes the model element as it would be seen in the Golden Model, while “Random” 
+randomly initializes the model element. In each assembly, the number of edges added back were recorded and used to calculate 
+each assembly method’s recall. 
+ 
+
+ER
+BA
+TLGL
+Tcell
+1.0-
+1.0-
+1.0
+1.0-
+0.8-
+0.8-
+0.8-
+0.8-
+50%
+0.4-
+0.4-
+TT
+IT
+0.2-
+0.2-
+0.2-
+0.2 
+0.0
+0.0
+0.0 
+0.0.
+BFA
+DFA
+BFA
+DFA
+BFA
+DFA
+BFA
+DFA
+1.0-
+1.0
+1.0-
+1.0
+0.8-
+0.8-
+0.8-
+0.8-
+ 0.6
+ 0.6-
+100%
+& 0.4-
+P 0.4
+0.2-
+1
+0.2-
+0.2-
+0.2 
+一
+0.0
+0.0
+0.0
+DFA
+0.0
+BFA
+DFA
+BFA
+BFA
+DFA
+BFA
+DFA
+0
+1
+□2
+ Golden
+RandomBUTCHY ET AL.:  TITLE 
+9 
+ 
+a small role in network assembly.   
+In future work, we plan to further investigate the effect 
+of network type, additional parametrization of update 
+functions (e.g., timing effects), methods to determine initial 
+state for simulations, and other error functions on the qual-
+ity of recommended Candidate Models. We will also ex-
+plore the effect of erroneous Candidate Knowledge on ex-
+tension methods.  
+ACKNOWLEDGMENT 
+NMZ is the corresponding author. This work was funded 
+in part by DARPA award W911NF-17-1-0135. The authors 
+would like to thank Kai-Wen Liang for his instrumental 
+work in the implementation of the BFA algorithm. 
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+ 
+Adam A. Butchy. Adam is a PhD candi-
+date in the Bioengineering Department 
+at the University of Pittsburgh. Adam 
+completed a B.S. in Chemical Engineer-
+ing and a B.S. in Biochemistry at Villa-
+nova University. He is working on Dis-
+crete Modeling of Macrophage Activa-
+tion and its role in the cancer microenvi-
+ronment and lung. 
+ 
+Cheryl A. Telmer Dr. Telmer is a Re-
+search Biologist at Carnegie Mellon Uni-
+versity. Cheryl and Natasa began work-
+ing together as iGEM advisors in 2013 
+and have expanded their collaborations 
+through the DARPA Big Mechanism and 
+World Modelers programs. Biologists 
+are constantly trying new tools that have 
+the potential to improve our understand-
+ing of complex systems, and the standardized representation and 
+computational modeling approaches being developed by the Melody 
+Lab are a great contribution. 
+ 
+Natasa Miskov-Zivanov Dr. Miskov-Zi-
+vanov is an Assistant Professor of Elec-
+trical and Computer Engineering, Bioen-
+gineering, and Computational and Sys-
+tems Biology at the University of Pitts-
+burgh. She received a B.Sc. degree in 
+electrical engineering and computer sci-
+ence from University of Novi Sad, Serbia 
+and M.Sc. and Ph.D. degrees in electri-
+cal and computer engineering from Carnegie Mellon University. Be-
+fore joining University of Pittsburgh as a faculty, she spent several 
+years as a postdoctoral researcher in Computational and Systems Bi-
+ology at the University of Pittsburgh, and as research scientist and 
+instructor in Computer Science and in Electrical and Computer Engi-
+neering at Carnegie Mellon University. Dr. Miskov-Zivanov’s research 
+interests include hybrid, knowledge-driven and data-driven, model 
+recommendation and reasoning for complex systems with applica-
+tions in systems and synthetic biology.  
+ 
+

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+arXiv:2301.04413v1  [cs.IR]  11 Jan 2023
+CoSPLADE: Contextualizing SPLADE for
+Conversational Information Retrieval
+Nam Le Hai1[0000−0002−9020−8790], Thomas Gerald2, Thibault Formal1,3,
+Jian-Yun Nie4, Benjamin Piwowarski1[0000−0001−6792−3262], and Laure
+Soulier1,2[0000−0001−9827−7400]
+1 Sorbonne Université, CNRS, ISIR, F-75005 Paris, France
+first.last @sorbonne-universite.fr
+2 Université Paris-Saclay, CNRS, LISN, 91405 Orsay France first.last @lisn.fr
+3 Naver Labs Europe, Meylan, France first.last @naverlabs.com
+4 University of Montreal, Montreal, Canada nie@iro.umontreal.ca
+Abstract. Conversational search is a difficult task as it aims at retriev-
+ing documents based not only on the current user query but also on the
+full conversation history. Most of the previous methods have focused on
+a multi-stage ranking approach relying on query reformulation, a criti-
+cal intermediate step that might lead to a sub-optimal retrieval. Other
+approaches have tried to use a fully neural IR first-stage, but are ei-
+ther zero-shot or rely on full learning-to-rank based on a dataset with
+pseudo-labels. In this work, leveraging the CANARD dataset, we propose
+an innovative lightweight learning technique to train a first-stage ranker
+based on SPLADE. By relying on SPLADE sparse representations, we
+show that, when combined with a second-stage ranker based on T5Mono,
+the results are competitive on the TREC CAsT 2020 and 2021 tracks.
+Keywords: information retrieval · conversational search · first-stage
+ranking.
+1
+Introduction
+With the introduction of conversational assistants like Siri, Alexa or Cortana,
+conversational Information Retrieval, a variant of adhoc IR, has emerged as
+an important research domain [4,6]. In conversational IR, a search is conducted
+within a session, and the user’s information need is expressed through a sequence
+of queries, similarly to natural conversations – thus introducing complex inter-
+dependencies between queries and responses.
+Not surprisingly, neural IR models have been shown to perform the best on
+conversational IR [5,7]. Most prior works rely on a Historical Query Expansion
+step [34], i.e. a query expansion mechanism that takes into account all past
+queries and their associated answers. Such query expansion model is learned on
+the CANARD dataset [8], which is composed of a series of questions and their
+associated answers, together with a disambiguated query, referred to as gold
+query in this paper. However, relying on a reformulation step is computationally
+
+2
+Le Hai et al.
+costly and might be sub-optimal as underlined in [13,16]. Krasakis et al. [13]
+proposed to use ColBERT [12] in a zero-shot manner, replacing the query by the
+sequence of queries, without any training of the model. Lin et al. [16] proposed
+to learn a dense contextualized representation of the query history, optimizing
+a learning-to-rank loss over a dataset composed of weak labels. This makes the
+training process complex (labels are not reliable) and long.
+In this work, we follow this direction of research but propose a much lighter
+training process for the first-stage ranker, where we focus on queries and do not
+make use of any passage – and thus of a learning-to-rank training. It moreover
+sidesteps the problem of having to derive weak labels from the CANARD dataset.
+Given this strong supervision, we can consider more context – i.e. we use the
+answers provided by the system the user is interacting with, which allows to
+better contextualize the query, as shown in our experiments. The training loss we
+propose leverages the sparse representation of queries and documents provided
+by the SPLADE model [9]. In a nutshell, we require that the representation of
+the query matches that of the disambiguated query (i.e. the gold query). Our
+first-stage ranker achieves high performances, especially on recall – the most
+important measure in a multi-stage approach, comparable to the best systems
+in TREC CAsT [7], but also on precision-oriented measures – which shows the
+potential of our methodology.
+Finally, to perform well, the second-stage ranker (i.e. re-ranker) needs to
+consider the conversation as well, which might require a set of heuristics to select
+some content and/or query reformulation such as those used in [18]. Leveraging
+the fact that our first-stage ranker outputs weights over the (BERT) vocabulary,
+we propose a simple mechanism that provides a conversational context to the
+re-ranker in the form of keywords selected by SPLADE.
+In summary, our contributions are the following:
+1. We propose a new loss to optimize a first-stage ranker resulting in a lightweight
+training strategy and state-of-the-art results in terms of recall;
+2. We show that, when combined with a second-stage ranker based on a context
+derived from the SPLADE query representation of the first stage, we obtain
+results on par with the best approaches in TREC CAsT 2020 and 2021.
+2
+Related Works
+The first edition [5] of the TREC Conversational Assistance Track (CAsT) was
+implemented in 2019, providing a new challenge on Conversational Search. The
+principle is the following: a user queries the system with questions in natural
+language, and each time gets a response from the system. The challenge differs
+from classical search systems as involving previous utterances (either queries
+or answers) is key to better comprehending the user intent. In conversational
+IR, and in TREC CAsT [6,5,7] in particular, the sheer size of the document
+collection implies to design an efficient (and effective) search system.
+Conversational IR is closely related to conversational Question-Answering
+[25,27,26] in the sense that they both include interaction turns in natural lan-
+guage. However, the objective is intrinsically different. While the topic or the
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+3
+context (i.e., the passage containing answers) is known in conversational QA,
+conversational IR aims to search among a huge collection of documents with po-
+tentially more exploratory topics. With this in mind, in the following, we focus
+on the literature review of conversational IR.
+We can distinguish two lines of work in conversational search. The first one
+[29,30,32,3] focuses on a Contextual Query Reformulation (CQR) to produce
+a (plain or bag-of-words) query, representing ideally the information need free
+of context, which is fed into a search model. One strategy of CQR consists in
+selecting keywords from previous utterances by relying on a graph weighted by
+either word2vec similarity [29], term-based importance using BM25 [19], or clas-
+sification models [30]. Other approaches [14,19,18,33,28] leverage the potential
+of generative language models (e.g., GPT2 or T5) to rewrite the query. Such
+approaches are particularly effective, reaching top performances in the TREC
+CAsT 2020 edition [5]. Query reformulation models also differ in the selected
+evidence sources. Models either focus on the early stage of the conversation [1],
+on a set of the queries filtered either heuristically [2] or by a classification model
+[21], or on both previous queries and documents [31]. Finally, to avoid the prob-
+lem of generating a single query, [14,20] have proposed to use different generated
+queries and aggregate the returned documents.
+The reformulation step is however a bottleneck since there is no guarantee
+that the “gold query” is optimal and thus generalizes well [16,13]. Moreover,
+generating text is time-consuming. To avoid these problems, the second line of
+work aims to directly integrate the conversation history into the retrieval model,
+bypassing the query reformulation step. As far as we know, only a few studies
+followed this path in conversational search. Qu et al [24] compute a query rep-
+resentation using the k last queries in the dialogue [15]. Similarly Lin et al. [16]
+average contextualized tokens embeddings over the whole query history. The
+representation is learned by optimizing a learning-to-rank loss over a collection
+with weak labels, which requires much care to ensure good generalization. Fi-
+nally, Krasakis et al. [13] use a more lexical neural model, i.e. ColBERT [12],
+to encode the query with its context – but they do not finetune it at all. In
+this work, we go further by using a sparse model SPLADE [9], using a novel
+loss tailored to such sparse representations, and by using a lightweight train-
+ing procedure that does not rely on passages, but only on a dataset containing
+reformulated queries.
+3
+Model
+In TREC CAsT [5,7], each retrieval session contains around 10 turns of ex-
+change. Each turn corresponds to a query and its associated canonical answer5
+is provided as context for future queries. Let us now introduce some notations
+that we use to describe our model. For each turn n ≤ N, where N is the last
+turn of the conversation, we denote by qn and an respectively the corresponding
+query and its response. Finally, the context of a query qn at turn n corresponds
+5 Selected by the organizer as the most relevant answer of a baseline system.
+
+4
+Le Hai et al.
+to all the previous queries and answers, i.e. q1, a1, q2, a2, ..., qn−1, an−1. The
+main objective of the TREC CAsT challenges is to retrieve, for each query qn
+and its context, the relevant passages.
+In the next sections, we present our first-stage ranker and second-stage re-
+ranker, along with their training procedure, both based, directly or indirectly,
+on the SPLADE (v2) model described in [9]. SPLADE has shown results on
+par with dense approaches on in-domain collections while exhibiting stronger
+abilities to generalize in a zero-shot setting [9]. It outputs a sparse representation
+of a document or a query in the BERT vocabulary, which is key to our model
+during training and inference. The SPLADE model we use includes a contextual
+encoding function, followed by some aggregation steps: ReLU, log saturation,
+and max pooling over each token in the text. The output of SPLADE is a sparse
+vector with only positive or zero components in the BERT vocabulary space R|V |.
+In this work, we use several sets of parameters for the same SPLADE architecture
+and distinguish each version by its parameters θ, and the corresponding model
+by SPLADE(. . . ; θ).
+3.1
+First stage
+The original SPLADE model [9] scores a document using the dot product be-
+tween the sparse representation of a document ( ˆd) and of a query (ˆq):
+s(ˆq, ˆd) = ˆq · ˆd
+(1)
+In our work, like in [16], we suppose that the document representation has
+been sufficiently well-tuned on the standard ad-hoc IR task. The document
+embedding ˆd is thus obtained using the pre-trained SPLADE model, i.e. ˆd =
+SPLADE([CLS] d; θSP LADE) where θSP LADE are the original SPLADE param-
+eters obtained from HuggingFace6. These parameters are not fine-tuned during
+the training process. We can thus use standard indices built from the original
+SPLADE document representations to retrieve efficiently the top-k documents.
+In the following, we present how to contextualize the query representation using
+the conversation history. Then, we detail the training loss aiming at reducing
+the gap between the representation of the gold query and the contextualized
+representation.
+Query representation. Like state-of-the-art approaches for first-stage conversa-
+tional ranking [16,13], we contextualize the query with the previous ones. Going
+further, we propose to include the answers in the query representation process,
+which is easier to do thanks to our lightweight training.
+To leverage both contexts, we use a simple model where the contextual query
+representation at turn n, denoted by ˆqn,k, is the combination of two representa-
+tions, ˆqqueries
+n
+which encodes the current query in the context of all the previous
+6 The weights can be found at https://huggingface.co/naver/splade-cocondenser-ensembledistil
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+5
+queries, and ˆqanswers
+n,k
+which encodes the current query in the context of k the
+past answers7. Formally, the contextualized query representation ˆqn,k is:
+ˆqn,k = ˆqqueries
+n
++ ˆqanswers
+n,k
+(2)
+where we use two versions of SPLADE parameterized by θqueries for the full
+query history and θanswers,k for the answers. These parameters are learned by
+optimizing the loss defined in Eq. (8).
+Following [16], we define ˆqqueries
+n
+to be the query representation produced by
+encoding the concatenation of the current query and all the previous ones:
+ˆqqueries
+n
+= SPLADE([CLS] qn [SEP] q1 [SEP] . . . [SEP] qn−1; θqueries)
+(3)
+using a set of specific parameters θqueries.
+To take into account the answers that the user had access to, we need to
+include them in the representation. Following prior work [2], we can consider a
+various number of answers k, and in particular, we can either choose k = 1 (the
+last answer) or k = n−1 (all the previous answers). Formally, the representation
+ˆqanswers
+n,k
+is computed as:
+ˆqanswers
+n,k
+= 1
+k
+n−1
+�
+i=n−k
+SPLADE(qn [SEP] ai; θanswers,k)
+(4)
+Training Based on the above, training aims at obtaining a good representation
+ˆqn for the last issued query qn, i.e. to contextualize qn using the previous queries
+and answers. To do so, we can leverage the gold query q∗
+n, that is, a (hopefully)
+contextualized and unambiguous query. We can compute the representation ˆq∗
+n
+of this query by using the original SPLADE model, i.e.
+ˆq∗
+n = SPLADE(q∗
+n; θSP LADE)
+(5)
+For example, for a query "How old is he?" the matching gold query could be
+"How old is Obama?". The representation of the latter given by SPLADE would
+be as follows:
+[(”Obama”, 1.5), (”Barack”, 1.2), (”age”, 1.2), (”old”, 1.0), (”president”, 0.8), ...]
+where the terms “Obama” and “Barack” clearly appear alongside other words
+related to the current query (“old” and the semantically related “age”).
+We can now define the goal of the training, which is to reduce the difference
+between the gold query representation ˆq∗
+n and the representation ˆqn,k computed
+by our model. An obvious choice of a loss function is to match the predicted
+and gold representations using cosine loss (since the ranking is invariant when
+scaling the query). However, as shown in the result section, we experimentally
+7 In the experiments, we also explore an alternative model where answers and queries
+are considered at once.
+
+6
+Le Hai et al.
+found better results with a modified MSE loss, whose first component is the
+standard MSE loss:
+LossMSE(ˆqn,k, ˆq∗
+n) = MSE(ˆqn,k, ˆq∗
+n)
+(6)
+In our experiments, we observed that models trained with the direct MSE do
+not capture well words from the context, especially for words from the answers.
+The reason is that the manually reformulated gold query usually only contains a
+few additional words from the previous turns that are directly implied by the last
+query. Other potentially useful words from the answers may not be included. This
+is a conservative expansion strategy which may not be the best example to follow
+by an automatic query rewriting process. We thus added an asymmetric MSE,
+designed to encourage term expansion from past answers, but avoid introducing
+noise by restricting the terms to those present in the gold query q∗
+n. Formally,
+our asymmetric loss is:
+Lossasym(ˆqanswers
+n,k
+, ˆq∗
+n) =
+�
+max(ˆq∗
+n − ˆqanswers
+n,k
+, 0)
+�2
+(7)
+where the maximum is component-wise. This loss thus pushes the answer-biased
+representation ˆqanswers
+n,k
+to include tokens from the gold answer. Contrarily to
+MSE, it does not impose (directly) an upper bound on the components of the
+ˆqanswers
+n,k
+representation – this is done indirectly through the final loss function
+described below.
+The final loss we optimize is a simple linear combination of the losses defined
+above, and only relies on computing two query representations:
+Loss(ˆqn,k, ˆq∗
+n) = LossMSE(ˆqn,k, ˆq∗
+n) + Lossasym(ˆqanswers
+n,k
+, ˆq∗
+n)
+(8)
+There is an interplay between the two components of the global loss. More pre-
+cisely, Lossasym pushes the ˆqanswers
+n,k
+representation to match the golden query
+representation ˆq∗
+n if it can, and LossMSE pushes the queries-biased representa-
+tion ˆqn,k to compensate if not. It thus puts a strong focus on extracting infor-
+mation from past answers, which is shown to be beneficial in our experiments.
+Implementation details. For the first-stage, we initialize both encoders (one en-
+coding the queries, and the other encoding the previous answer) with pre-trained
+weights from SPLADE model for adhoc retrieval. We use the ADAM optimizer
+with train batch size 16, learning rate 2e-5 for the first encoder and 3e-5 for the
+second. We fine-tune for only 1 epoch over the CANARD dataset.
+3.2
+Reranking
+We perform reranking using a T5Mono [22] approach, where we enrich the raw
+query qn with keywords identified by the first-stage ranker. Our motivation is
+that these words should capture the information needed to contextualize the raw
+query. The enriched query q+
+n for conversational turn n is as follows:
+q+
+n = qn. Context : q1 q2 . . . qn−1. Keywords : w1, w2, ..., wK
+(9)
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+7
+where the wi are the top-K most important words that we select by leveraging
+the first-stage ranker as follows. First, to reduce noise, we only consider words
+that appear either in any query qi or in the associated answers ai (for i ≤ n−1).
+Second, we order words by using the maximum SPLADE weight over tokens
+that compose the word.8
+We denote the T5 model fine-tuned for this input as T 5+. As in the original
+paper [22], the relevance score of a document d for the query qn is the proba-
+bility of generating the token “true” given a prompt pt(q+
+n , d) = “Query: q+
+n .
+Document: d. Relevant:”:
+score(q+
+n , d; θ) =
+pT 5(true|pt(q+
+n , d); θ)
+pT 5(true|pt(q+
+n , d); θ) + pT 5(false|pt(q+
+n , d); θ)
+(10)
+where θ are the parameters of the T5Mono model.
+Differently to the first stage training, we fine-tune the ranker by aligning the
+scores of the documents, and not the weight of a query (which is obviously not
+possible with the T5 model). Here the “gold” score of a document is computed us-
+ing the original T5Mono with the gold query q∗
+n. The T5 model is initialized with
+weights made public by the original authors9, denoted as θT 5. More precisely,
+we finetune the pre-trained T5Mono model using the MSE-Margin loss [11]. The
+loss function for the re-ranker (at conversation turn n, given documents d1 and
+d2) is calculated as follows:
+LR =
+��
+s(q+
+n , d1; θT 5+) − s(q+
+n , d2; θT 5+)
+�
+− (s(q∗
+n, d1; θT 5) − s(q∗
+n, d2; θT 5))
+�2
+We optimize the θT 5+ parameters by keeping the original θT 5 to evaluate the
+score of gold queries.
+Implementation details. We initialize θT 5+ as θT 5, and fine-tune for 3 epochs,
+with a batch size of 8 and a learning rate 1e-4. We sample pairs (d1, d2) using the
+first-stage top-1000 documents: d1 is sampled among the top-3, and d2 among
+the remaining 997 to push the model to focus on important differences in scores.
+4
+Experimental Protocol
+We designed the evaluation protocol to satisfy two main evaluation objectives:
+(i) Evaluating separately the effectiveness of the first-stage and the second-stage
+ranking components of our CoSPLADE model; (ii) Comparing the effectiveness
+of our CoSPLADE model with TREC CAsT 2020 and 2021 participants.
+8 To improve coherence, we chose to make keywords follow their order of appearance
+in the context, but did not vary this experimental setting.
+9 We used the Huggingface checkpoint https://huggingface.co/castorini/monot5-base-msmarco
+
+8
+Le Hai et al.
+4.1
+Datasets
+To train our model, we used the CANARD corpus10, a conversational dataset fo-
+cusing on context-based query rewriting. More specifically, the CANARD dataset
+is a list of conversation histories, each being composed of a series of queries, short
+answers (human written) and reformulated queries (contextualised). The train-
+ing, development, and test sets include respectively 31.538, 3.418, and 5.571
+contextual and reformulated queries.
+To evaluate our model, we used the TREC CAsT 2020 and 2021 datasets
+which include respectively 25 and 26 information needs (topics) and a document
+collection composed of the MS MARCO dataset, an updated dump of Wikipedia
+from the KILT benchmark, and the Washington Post V4 collection. For each
+topic, a conversation is available, alternating questions and responses (manually
+selected passages from the collection, aka canonical answers). For each question
+(216 and 239 in total), the dataset provides its manually rewritten form as well
+as a set of about 20 relevant documents. We use the former to define an upper-
+bound baseline (Splade_GoldQuery).
+4.2
+Metrics and baselines
+We used the official evaluation metrics considered in the TREC CAsT 2020 and
+2021, namely nDCG@3, MRR, Recall@X, MAP@X, nDCG@X, where the cut-off
+is set to 1000 for the CAsT 2020 and 500 for the CAsT 2021. For each metric,
+we calculate the mean and variance of performance across the different queries
+in the dataset. With this in mind, we present below the different baselines and
+scenarios used to evaluate each component of our model.
+First-stage ranking scenarios. To evaluate the effectiveness of our first-stage
+ranking model (Section 3.1), we compare our approach CoSPLADE, based on
+the query representation of Eq. (2) with different variants (the document en-
+coder is set to the original SPLADE encoder throughout our experiments):
+SPLADE_rawQuery (lower bound): SPLADE [10] using only the original
+and ambiguous user queries qn; SPLADE_goldQuery (kind of upper bound):
+SPLADE using the manually rewritten query q∗
+n; CQE [16], a state-of-the-art
+dense contextualized query representation learned using learning-to-rank on a
+dataset with pseudo-labels.
+To model answers when representing the query using ˆqanswers
+n,k
+, we used two
+historical ranges (“All” with k = n−1 answers and “Last” where we use only the
+last one, i.e. k = 1) and three types of answer inputs: Answer in which answers
+are the canonical answers; Answer-Short in which sentences are filtered as
+in the best performing TREC CAsT approach [18]. This allows for consistent
+input length, at the expense of losing information; Answer-Long As answers
+from CANARD are short (a few sentences extracted from Wikipedia – contrarily
+to CAsT ones), we expand them to reduce the discrepancy between training and
+10 https://sites.google.com/view/qanta/projects/canard
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+9
+inference. For each sentence, we find the Wikipedia passage it appears in (if it
+exists in ORConvQA [23]), and sample a short snippet of 3 adjacent sentences
+from it.
+Finally, we also conducted ablation studies (on the best of the above vari-
+ants) by modifying either the way to use the historical context or the training
+loss: flatContext a one-encoder version of our SPLADE approach in which we
+concatenate all information of the context to apply SPLADE to obtain a single
+representation of the query (instead of two representations ˆqqueries
+n
+and ˆqanswers
+n,k
+as in Equations 2 and 3) trained using a MSE loss function (Eq. 6) since there is
+no more two representations. MSE the version of our SPLADE approach trained
+with the MSE loss (Eq. 6) instead of the proposed one (Eq. 8); cosine the ver-
+sion of our SPLADE approach trained with a cosine loss instead of the proposed
+loss (Eq. 8). The cosine loss is interesting because it is invariant to the scaling
+factor that preserves the document ordering (Eq.1).
+Second-stage ranking scenarios. We consider different scenario for our second-
+stage ranking model: T5Mono_RawQuery the T5Mono ranking model [22]
+applied on raw queries; T5Mono_GoldQuery the T5Mono ranking model
+applied on gold queries; T5Mono_CQR the T5Mono ranking model applied
+on query reformulation generated with a pre-trained T5 (using the CANARD
+dataset); CoSPLADE_[context]_[number] : different versions of our second-
+stage ranking model input (Eq. 9), varying 1) the number K of keywords identi-
+fied as relevant by the first-stage ranker: 5, 10, 20, and 2) the presence or absence
+of the past queries within the reformulation.
+TREC participant baselines. For each evaluation campaign (2020 and 2021),
+we also compare our model with the best, the median and the lowest TREC
+CAsT participants presented in the two overviews [5,7], where participant are
+ranked according to the nDCG@3 metric.
+5
+Results
+5.1
+First-stage ranking effectiveness
+In this section, we focus on the first-stage ranking component of our CoSPLADE
+model. To do so, we experiment different scenarios aiming at evaluating the
+impact of the designed loss (Eq. 8) and the modeling/utility of evidence sources
+(Equations 3 and 4). Results of these different baselines and scenarios on the
+TREC CAsT 2021 dataset are provided in Table 1. Similar trends are observed
+on CAsT 2020, but are not reported due to space limit.
+In general, one can see that all variants of our approach (CoSPLADE_*
+models) outperform the scenario applying the initial version of SPLADE on raw
+and, more importantly, gold queries. This is very encouraging since this latter
+scenario might be considered an oracle, i.e. the query is manually disambiguated.
+Finally, we improve the results over CQE [16] for all the metrics – showing that
+
+10
+Le Hai et al.
+Recall@500 MAP@500
+MRR
+nDCG@500 nDCG@3
+Baselines
+SPLADE_rawQuery
+30.8±2.7
+5.5±0.9
+21.3±2.9
+17.8±1.8
+13.1±2.1
+SPLADE_goldQuery
+68.8±2.0
+16.1±1.2
+55.5±3.3
+42.8±1.7
+38.3±2.8
+CQE [17] from [7]
+79.1
+28.9
+60.3
+55.7
+43.8
+Effect of answer processing: CoSPLADE_. . .
+AllAnswers
+79.5±2.2
+28.8±1.7
+61.7±3.1
+55.3±2.0
+46.5±2.9
+AllAnswers-short
+72.8±2.6
+25.7±1.9
+54.4±3.3
+49.5±2.3
+40.1±3.0
+AllAnswers-long
+80.4±2.1
+29.3±1.8
+62.0±3.2
+55.6±2.1
+48.9±3.0
+LastAnswer
+83.4±2.0
+31.2±1.8
+61.8±3.1
+58.1±2.0
+47.4±3.0
+LastAnswer-short
+79.2±2.2
+28.1±1.8
+61.4±3.3
+54.3±2.1
+46.4±3.0
+LastAnswer-long
+85.2±1.8
+32.0±1.7 64.3±03.0 59.4±1.9
+48.6±3.0
+CoSPLADE_LastAnswer-long variants
+flatContext
+77.0±2.0
+26.0±2.0
+55.0±3.0
+52.0±2.0
+42.0±3.0
+MSE loss
+70.9±2.4
+21.6±1.7
+48.7±3.4
+45.2±2.3
+39.6±3.1
+cosine loss
+70.4±2.5
+22.6±1.7
+52.5±3.3
+46.9±2.2
+39.0±3.0
+Table 1. Effectiveness of different scenarios of our first-stage ranking model on the
+TREC CAsT 2021.
+our simple learning mechanism, combined with SPLADE, allows for achieving
+SOTA performance.
+Leveraging queries and answers history better contextualizes the current query.
+The results of the flatContext scenario w.r.t. to the SPLADE_goldQuery allows
+comparing the impact of evidence sources related to the conversation since they
+both use the same architecture (SPLADE). We can observe that it obtains better
+results than SPLADE_goldQuery (e.g., 77 vs. 68.8 for the Recall@500 metric),
+highlighting the usefulness of context to better understand the information need.
+More detailed answers perform better. Since answers are more verbose than ques-
+tions, including them is more complex, and we need to study the different pos-
+sibilities (CoSPLADE_AllAnswers* and CoSPLADE_LastAnswer*). One can
+see that: 1) trimming answers (*-short) into a few keywords is less effective than
+considering canonical answers, but 2) it might be somehow effective when com-
+bined with the associated Wikipedia passage (*-long). Moreover, it seems more
+effective to consider only the last answer rather than the whole set of answers
+in the conversation history11. Taking all together, these observations highlight
+the importance of the way to incorporate information from answers into the
+reformulation process.
+Dual query representation with asymmetric loss leverages sparse query represen-
+tations. The results of the flatContext scenario show that considering at once
+past queries and answers perform better (compared to the MSE loss scenario
+which is directly comparable). However, if we separate the representations and
+11 This might be due to the simple way to use past answers, i.e. Eq. 4, but all the other
+variations we tried did not perform better
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+11
+Recall@500 MAP@500
+MRR
+nDCG@500 nDCG@3
+Baselines
+T5Mono_RawQuery
+78.4±2.3
+21.0±1.8
+39.6±3.2
+45.9±2.1
+28.4±3.0
+T5Mono_GoldQuery
+86.1±1.7
+44.1±1.9
+78.7±2.7
+68.5±1.8
+64.6±2.8
+T5Mono_CQR
+80.4±2.2
+30.0±1.9
+58.2±3.4
+55.3±2.1
+44.6±3.2
+coSPLADE-based second stage variants
+CoSPLADE_NoContext_5
+84.3±1.8
+31.7±2.0
+61.6±3.3
+58.1±2.0
+45.9±3.1
+CoSPLADE_NoContext_10
+83.1±1.9
+32.0±1.7
+66.0±3.1
+59.1±1.9
+49.8±2.9
+CoSPLADE_NoContext_20
+84.8±1.7
+33.4±1.8
+66.0±3.0
+60.4±1.8
+49.6±2.9
+CoSPLADE_Context_5
+85.0±1.7
+35.0±1.8
+68.4±3.0
+61.7±1.9
+51.5±02.9
+CoSPLADE_Context_10
+84.8±1.7
+36.5±1.9 67.8±3.1
+63.0±1.9
+53.3±3.1
+CoSPLADE_Context_20
+84.9±1.7
+35.5±1.8 69.8±3.0
+62.2±1.9
+54.4±2.9
+Table 2. Effectiveness of different scenarios of our second-stage ranking model on
+TREC CAsT 2021.
+use an asymmetric loss function, the conclusion changes. Moreover, the compar-
+ison of our best scenario CoSPLADE_LastAnswer-long with a similar scenario
+trained by simply using a MSE or a cosine losses reveals the effectiveness of our
+asymmetric MSE (Equation 7). Remember that this asymmetric loss encourages
+the consideration of previous answers in the query encoding. This reinforces our
+intuition that the conversation context, and particularly verbose answers, is im-
+portant for the conversational search task. It also reveals that the context should
+be included at different levels in the architecture (input and loss).
+5.2
+Second-stage ranking effectiveness
+In this section, we rely on the CoSPLADE_LastAnswer-long model as a first
+stage ranker, and evaluate different variants of the second-stage ranking method
+relying on the T5Mono model. For fair comparison, we also mention results ob-
+tained by a T5Mono ranking model applied on raw and gold queries, as well as
+query reformulated using a T5 generative model. Results on the TREC CAsT
+2021 dataset are presented in Table 2.
+The analysis of the CoSPLADE model variants allows to highlight different
+observations regarding the usability of the context and the number of keywords
+added to the query. First, adding the previous questions to the current query
+in the prompt (i.e., “Context”) seems to improve the query understanding and,
+therefore, positively impacts the retrieval effectiveness. For instance, when 5
+keywords are added, the context allows reaching 51.5% for the nDCG@3 against
+45.9% without context. Second, the effectiveness metrics tend to increase with
+the number of additional keywords, particularly for scenarios without context,
+which is sensible. This trend is less noticeable for the scenarios with context since
+the best metrics are alternatively obtained by the scenario adding either 10 or
+20 keywords. It is worth noting however that adding 10 or 20 keywords is more
+valuable than adding only 5 (e.g. 54.4% vs. 51.5% for the nDCG@3 metric). It
+thus seems that 1) keywords help to reformulate the initial information need,
+
+12
+Le Hai et al.
+2) but they can lead to saturation when they are both numerous and combined
+with other information.
+By comparing the best model scenarios with the more basic scenarios apply-
+ing the T5Mono second-stage ranker on raw and gold queries, we can observe that
+our method allows improving the retrieval effectiveness regarding initial queries
+but is not sufficient for reaching the performance of T5Mono_GoldQuery. How-
+ever, results obtained when applying T5Mono on queries reformulated by T5
+highlight that the contextualization of an initial query is a difficult task. Indeed,
+the T5Mono_CQR scenario is less effective than the T5Mono_GoldQuery one
+with between 6 and 20 points of difference depending on the metrics.
+Moreover, it is interesting to note that the SPLADE model applied on raw
+and gold queries (first-stage ranking in Table 1) obtains lower results than the
+T5Mono model on the same data (second-stage ranking in Table 2). It can be ex-
+plained by the purpose of those two architectures which are different: SPLADE is
+a sparse model focusing on query/document expansion while T5Mono is partic-
+ularly devoted to increase precision. However, it is worth noting that combining
+SPLADE and T5Mono as first and second-stage rankers reaches the highest ef-
+fectiveness results in our experimental evaluation. This shows the effectiveness
+of CoSPLADE to both contextualize queries and effectively rank documents.
+5.3
+Effectiveness compared to TREC CAsT participants
+We finally compare our approach with TREC CAsT participants for the 2020
+and 2021 evaluation campaigns. For both years, we can see that we obtain effec-
+tiveness metrics that are very close or higher than the ones reached by the best
+participants. Indeed, CoSPLADE surpasses the best TREC participant for the
+2020 evaluation campaign regarding Recall@1000 and nDCG@1000. For 2021,
+our model obtains better results than the best one for the MRR and nDCG@3
+metrics. For both years, the best participant is the h2oloo team [18,7] where
+they use query reformulation techniques, either using AllenAI or T5. Our re-
+sults suggest that our approach focusing on a sparse first-stage ranking model
+allows combining the benefit of query expansion and document ranking in a sin-
+gle model that eventually helps the final reranking step. In other words, simply
+rewriting the query without performing a joint learning document ranking can
+hinder the overall performance of the search task.
+6
+Conclusion
+In this paper, we have shown how a sparse retrieval neural IR model, namely
+SPLADE [9], could be leveraged together with a lightweight learning process
+to obtain a state-of-the-art first-stage ranker. We further showed that this first-
+stage ranker could be used to provide context to the second-stage ranker, leading
+to results comparable with the best-performing systems. Future work may ex-
+plore strategies to better capture the information from the context or to explicitly
+treat user feedback present in the evaluation dataset.
+
+CoSPLADE: Contextualizing SPLADE for Conversational IR
+13
+TREC CAsT 2020
+Recall@1000 MAP@1000
+MRR
+nDCG@1000 nDCG@3
+TREC Participant (best)
+63.3
+30.2
+59.3
+52.6
+45.8
+TREC Participant (median)
+52.1
+15.1
+42.2
+36.4
+30.4
+TREC Participant (low)
+27.9
+1.0
+5.9
+11.1
+2.2
+CoSPLADE
+82.4±2.0
+26.9±1.5
+58.1±2.9
+54.2±1.8
+44.0±2.7
+TREC CAsT 2021
+Recall@500
+MAP@500
+MRR
+nDCG@500 nDCG@3
+TREC Participants 1 (best)
+85.0
+37.6
+67.9
+63.6
+52.6
+TREC Participants 2 (median)
+36.4
+17.6
+53.4
+33.6
+37.7
+TREC Participants 3 (low)
+58.9
+7.6
+27.0
+31.4
+15.4
+CoSPLADE
+84.9±1.7
+35.5±1.8
+69.8±3
+62.2±1.9
+54.4±2.9
+Table 3. TREC CAsT 2020 and 2021 performances regarding participants
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+

5tE2T4oBgHgl3EQfkQe0/content/tmp_files/2301.03977v1.pdf.txt

@@ -0,0 +1,1182 @@
+Service Differentiation and Fair Sharing
+in Distributed Quantum Computing
+Claudio Cicconettia,∗, Marco Contia, Andrea Passarellaa
+aIIT, National Research Council, Pisa, Italy
+Abstract
+In the future, quantum computers will become widespread and a network of
+quantum repeaters will provide them with end-to-end entanglement of remote
+quantum bits. As a result, a pervasive quantum computation infrastructure will
+emerge, which will unlock several novel applications, including distributed quan-
+tum computing, that is the pooling of resources on multiple computation nodes
+to address problem instances that are unattainable by any individual quantum
+computer. In this paper, we first investigate the issue of service differentiation
+in this new environment. Then, we define the problem of how to select which
+computation nodes should participate in each pool, so as to achieve a fair share
+of the quantum network resources available. The analysis is performed via an
+open source simulator and the results are fully and readily available.
+Keywords:
+Distributed Quantum Computing, Quantum Internet, Quantum
+Routing
+1. Introduction
+Quantum Computing (QC) exploits the properties of matter at very small
+scale to solve some problems much faster than a classical counterpart. Even
+though QC has been theorized 40 years ago [1], only recently the technology
+evolution and a spur of investments have made it possible to obtain practical
+∗Corresponding author
+Email addresses: c.cicconetti@iit.cnr.it (Claudio Cicconetti), m.conti@iit.cnr.it
+(Marco Conti), a.passarella@iit.cnr.it (Andrea Passarella)
+Preprint submitted to Elsevier
+January 11, 2023
+arXiv:2301.03977v1  [quant-ph]  10 Jan 2023
+
+results and speculate about approaching mass deployments [2]. QC is being al-
+ready used in the chemical and pharmaceutical industry, while new applications
+are being progressively unlocked in material science, Machine Learning (ML)
+and engineering optimization, production and logistics, post-quantum security
+[3]. Essentially, the computational advantage of QC stems from the proper-
+ties of superposition and entanglement of the qubits (i.e., the “quantum bits”):
+(i) superposition, which means that a qubit can be in a combination of multiple
+states at the same time; and (ii) entanglement, which is a property exhibited by
+a set of qubits that maintain their correlation even separated in space or time.
+We can expect that the computational power of a single QC will remain
+relatively limited in the near future, due to scalability issues in maintaining
+a very stable and controlled environment to cope with the flimsy nature of
+qubits. On the other hand, the realization of the Quantum Internet is progress-
+ing steadily [4], with the long-term goal to enable the entanglement of qubits that
+reside in QCs across geographical distances. With the diffusion of QCs and their
+gradual interconnection via quantum networks, a pervasive infrastructure will
+therefore materialize, with the potential to combine opportunistically resources
+from multiple QCs for the execution of specialized algorithms in a distributed
+fashion. A general framework for such distributed quantum computing has been
+proposed in [5], where the authors propose practical examples, e.g., a quantum
+version of the k-means clustering, which is used in unsupervised ML.
+A preliminary analysis of the allocation of resources among multiple quantum
+computers based on the characteristics of the underlying quantum network has
+been presented in [6]. In the same work, we have also proposed a practical solu-
+tion inspired by a well-known algorithm in classical data networks, i.e., Deficit
+Round Robin (DRR) [7], which we have evaluated through simulations. We
+have found that some fundamental properties of quantum networks immensely
+impact on the provisioning of resources, which calls for new research in this
+area. This is especially manifest when considering networks of first-generation
+(1G) quantum repeaters [8], which do not have error-correction capabilities and
+are expected to be next in line for the industrialization and mass deployment
+2
+
+in the following years [9].
+The contribution of this paper is twofold.
+1. We evaluate the performance of the resource allocation algorithm proposed
+in [6] with differentiated services coexisting within the same quantum net-
+work. Furthermore, we do so by comparing the performance with two alter-
+native algorithms, inspired by equivalents in classical problems with similar
+settings. This extends and completes the preliminary analysis in our previous
+work.
+2. We define a new problem related to fair share of resources in a quantum
+network among multiple applications wishing to perform distributed QC:
+how to best choose the peers among those available? After introducing a
+mathematical formulation of the problem, we propose a greedy approxima-
+tion algorithm, which is then evaluated thoroughly and compared to two
+alternatives.
+All the experiments in the paper are carried out via simulations, which are
+fully reproducible and publicly available on GitHub, including the simulation
+software source code, the scripts to run the analysis, and the artifacts and plots.
+The rest of this paper is structured as follows. We summarize the system
+model assumptions and findings in [6] in Sec. 2. We then review the related work
+on routing in quantum network in Sec. 3. The main contributions are reported
+in Sec. 4, where we study the service differentiation, and in Sec. 5, where we
+tackle the problem of fair sharing of resources. Sec. 6 concludes the paper and
+identifies the most important open research directions in this context.
+2. System Model
+In this section, we describe in short the quantum network abstract model
+adopted in the paper (Sec. 2.1), the resource allocation algorithm proposed in [6]
+(Sec. 2.2), and the simulation methodology and tool (Sec. 2.3). For more details,
+we refer the reader to [6], in particular sections II and IV, and references within.
+3
+
+end-to-end
+entanglement 
+path
+Figure 1: Quantum network model. End-to-end entanglement can be established between two
+nodes s and d for which there is a path in G(V, E), where the intermediate nodes perform
+entanglement swapping. F is the fidelity with which the local link EPR pairs are generated; q
+is the measurement success probability, which affects the entanglement swapping procedure;
+Cij is the capacity of edge eij, in EPR-pairs/s.
+2.1. Quantum network model
+The quantum network model is illustrated in Fig. 1 as a graph G(V, E),
+where nodes represent quantum devices (repeaters or computers), and edges
+represent direct quantum communication links between them [10]. We assume
+that maximally entangled EPR (Einstein–Podolsky–Rosen) pairs, e.g., |Φ+⟩,
+are generated periodically at each link, and they have initial fidelity equal to
+¯F ∈ [0.5, 1]. The fidelity is a measure of how close a given quantum state is to
+a reference state, with 1 meaning that they are identical. Every edge eij ∈ E
+has a given capacity Cij, which is the rate of generation of EPR-pairs, which
+in turn depends on the physical characteristics of the quantum network devices
+and links.
+Quantum networks will offer the capability to produce entangled
+EPR-pairs between remote nodes, i.e., nodes that are interconnected through
+intermediate hops, which will perform the procedure of entanglement swapping
+to this purpose.
+Such a procedure is stochastic in nature, in 1G quantum
+repeaters, and we assume that it succeeds with probability q [11]. An end-to-
+end EPR-pair can only be used in a meaningful manner if all the entanglement
+4
+
+swaps along the path have succeeded, which leads to the following formula to
+compute the maximum net rate that can be used by a quantum application
+consuming resources along a path p = {(s, v1), . . . , (vN, d)}:
+r(p) ≤ mineij∈p{Cij}
+q|p|−1
+,
+(1)
+where |p| is the length of the path p, in number of edges. Furthermore, en-
+tanglement swapping reduces the fidelity of the end-to-end entangled EPR-pair
+according to the following formula [12]:
+F(p) = 1
+4 + 3
+4
+�4 ¯F − 1
+3
+�|p|
+.
+(2)
+We can say that r(p) and F(p) define the effective rate at which two end-points
+can transfer EPR-pairs, which is the logical equivalent of throughput in classical
+networks.
+In our previous work [6] we have classified the quantum applications in two
+categories:
+– Flows. They are characterized by the need for two specific nodes to exchange
+a constant flow of EPR pairs in a point-to-point manner for the whole dura-
+tion of a session. If the rate of EPR pairs falls below the requested amount,
+then the application Quality of Service (QoS) degrades. Examples of such ap-
+plications are: clock synchronization and Quantum Key Distribution (QKD).
+– Apps. In this category we find distributed quantum computing applications,
+each characterized by a given quantum computer (host) running an algorithm
+that pools the resources of a number of other quantum computers (peers or
+workers). There is no required EPR-pair rate, but the application wishes
+to consume as many EPR-pairs as possible to complete the execution faster.
+Such kind of service is equivalent to best-effort or elastic traffic in a classical
+data network.
+Since the network operator might provide a differentiated
+service, we foresee that each app is also assigned a weight (ρ), which is a
+relative indication of how much throughput (in EPR-pair/s) it should be
+given in the long-term compared to another app with a different weight.
+5
+
+For both categories, we foresee the minimum fidelity (F min) can be also a
+user requirement. Since in this work we focus on distributed quantum computing
+applications, whose traffic is better modeled by apps than flows, in the rest of
+the paper we do not consider further the latter.
+2.2. QDRR resource allocation algorithm
+We report below a recap of the apps’ resource allocation algorithm in [6],
+which in the following will be called Quantum DRR (QDRR) as it was inspired
+by the well-known DRR algorithm [7]. The basic idea of QDRR is to provide all
+applications with a fair chance to be allocated a fraction of the quantum network
+resources. This is enforced by visiting the applications in a round robin: at each
+visit, the application can be allocated capacity across multiple paths towards
+its peers, up until a given amount that is proportional to the application’s
+weight. Shortest paths are always preferred to longer ones, because they are
+more efficient. A more detailed explanation follows.
+The algorithm has two system parameters, which are set based on our pre-
+vious results: k = 4 and the round size φ = 10 EPR-pairs/s. For a set of apps
+i ∈ A, each defined by a host node {hi} and a set of candidate peers Wi, QDRR
+consists of the following steps:
+0. ∀i ∈ A, ∀j ∈ Wi: find k shortest paths from hi to wij and add them to
+Pi; at the end, each Pi contains up to k paths for each possible peer of i.
+Initialize the active list of apps L with the identifiers of all the apps A. Copy
+the graph G(V, E) into a temporary copy G′.
+1. If L = ∅ terminate. Otherwise, let a be the next application to be visited in
+L in round robin order.
+2. Set the residual capacity that can be used by the current app a in this round
+to δa = φ
+ρa
+�
+i∈A ρi , i.e., a fraction of the round size φ proportional to its
+priority.
+3. Select the shortest path p ∈ Pa. The shortest path is the one that requires
+the least amount of resources among those available for the current app,
+6
+
+according to Eq. (1), and gives the maximum fidelity, according to Eq. (2).
+If p is not feasible anymore because it contains edges that have been removed
+from G′ discard it and move to the next shortest path. If Pa = ∅ remove a
+from the active list L and continue from Step 1.
+4. Determine the gross rate to be assigned to the current application a along
+path p at this round as R = min {δa, mine∈p Ce} ≥ 0. The corresponding net
+rate will be r = R · q|p|−1, as per Eq. (1).
+5. Remove R from the capacity of all the edges along the path p. Remove from
+G′ all the vanishing edges.
+6. Update δa ← δa − R. If δa = 0 restart from Step 1, otherwise continue from
+Step 3.
+2.3. Simulation methodology and tool
+We conclude the section by describing the methodology and tool adopted
+for the performance evaluation in Sec. 4 and Sec. 5.
+Like in [13], we use a Poisson Point Process (PPP) to generate the position
+of an average of µ nodes in a flat square grid with edge size 60 km; a link is
+added between two nodes with probability plink = 0.5 if their Euclidean distance
+is smaller than a threshold τ. The capacity of each link is drawn from a r.v.
+uniformly distributed between 1 Bell pair/s and 400 Bell pairs/s, as in [10]. The
+initial fidelity of Bell pairs is ¯F = 0.95, which is widely used in the literature,
+and the entanglement swapping success probability is q = 0.5, which is the
+best value that can be obtained with linear optics components [11]. Based on
+previous results in [6] we have selected two representative topologies:
+– dense: µ = 100, τ = 20 km;
+– sparse: µ = 50, τ = 15 km.
+The following metrics are used to evaluate the performance:
+– The net rate of the apps, i.e., the number of EPR-pairs that the end-points
+can consume in the unit of time, which is a direct operational measure from
+the point of view of the end users;
+7
+
+– The max-min fairness, which is the difference between the top and lowest net
+rates assigned to the apps.
+– The fidelity, weighted for each app on the net rate assigned to the correspond-
+ing peer, which impacts on the accuracy and convergence of the distributed
+QC applications.
+– The inter-class unfairness index, for a class of apps with R priority weights
+ρj, provided that each app is allocated net rate ri, defined as:
+�
+�
+�
+�
+R
+�
+i=2
+� ri
+r1
+− ρi
+ρ1
+�2
+(3)
+This measures the distance of the net allocations with respect to an ideal case
+where the proportions between rates is exactly the same as the proportions
+between priorities.
+We used a Monte Carlo approach: for any combination of the parameters
+under study, we simulated 6,000 drops with randomly generated networks and
+random workload. Statistical significance has been verified for all the metrics
+in the experiments performed, but we seldom include error bars in plots for
+better readability. The simulation tool used is a custom simulator, developed
+in C++ and using the Boost Graph Library, available as open source under a
+MIT license on GitHub:
+https://github.com/ccicconetti/quantum-routing/
+For full reproducibility, the repository also includes the scripts to run the
+experiments, as well as the artifacts obtained and the Gnuplot files to produce
+the plots: see tag v1.5, experiments labeled 004 and 005.
+3. Related Work
+The literature on quantum networking and distributed QC is not vast: even
+though the basic ingredients have been known since a long time ago —consider
+8
+
+for instance the seminal paper by Bouwmeester et al.
+on quantum telepor-
+tation [14] published on Nature in 1997— only recently there have been in-
+vestments in an order of magnitude sufficient for technology to take off. This
+revamped interest has triggered new research activities in this area, briefly re-
+viewed below.
+In general terms, the problem of quantum routing is formulated as follows:
+given a network of quantum nodes (repeaters or computers) and a set of traffic
+flows identified by their sources, destinations, and application requirements (e.g.,
+the minimum fidelity), find the “best” paths that fulfill the constraints. Some
+works have studied the problem by reusing the findings in the area of routing
+in classical networks.
+Van Meter et al.
+proposed a quantum version of the
+famous Dijkstra’s shortest path algorithm, which was shown to give very good
+performance with an appropriate selection of the routing metric that considers
+the specific properties of quantum networks [15]. More recently, Caleffi et al.
+have proposed a slightly less efficient variation of Dijkstra’s algorithm that can
+work with non-isotonic routing metrics, which they have advocated to provide
+superior performance in selected use cases [16].
+Dijkstra’s algorithm is also
+the subject of [17], where the authors lay some mathematical foundations that
+allow them to derive upper bounds of performance in specific network topologies,
+including grid and ring.
+A different direction is explored by Pant et al., who studied the distribution
+of routing information to the nodes [18]; for this they propose a time-slotted
+approach: in the first part of the slot every repeater tries to create a local entan-
+glement with all its neighbors, then in the second part the paths are established
+as instructed by a centralized authority. One interesting aspect of the paper
+is that multiple paths are selected for the same (source, destination) to maxi-
+mize the rate of end-to-end EPR-pairs. We have also adopted this time-slotted
+model in [19], where we have investigated the issue of “scheduling” of traffic
+flows, i.e., determining the order in which to assign paths to pending requests,
+in case the network resources are not sufficient to serve them all. This prob-
+lem is called “distribution” in [13], where the authors formulate it as an Integer
+9
+
+Linear Programming (ILP), for which they derive closed formula performance
+bounds in the case of a homogeneous chain of quantum repeaters. The issue
+is also addressed in [10], where the authors have proposed to split the overall
+quantum routing problem in two to reduce the computational complexity: first,
+they determine the rates achievable by the traffic flows under the given network
+constraints using an approach based on multi-commodity flow optimization,
+then they map these rates to paths. The paper adopts a network model using
+probabilistic entanglement swapping, which we reuse in this work (described in
+Sec. 2).
+An important reference for our study is [20], where the authors study the
+allocation strategy of traffic flows for which the paths have been pre-determined:
+they do so by borrowing the fairness concept from data networks and re-using
+traditional algorithms from the relevant literature. In our paper, we also bor-
+row from the same literature, though we apply the concepts to a different
+class of applications, as it will be clear in the next section.
+As a matter of
+fact, all the scientific works cited above have focused on point-to-point traffic
+flows, while in this paper we focus on a different type of traffic that is more
+suitable to model distributed QC, with distinguishing features that do not al-
+low the reuse of state-of-the-art solutions. Rather, we claim that any existing
+routing/allocation/scheduling solutions should work in parallel to our proposed
+scheme to provide an effective resource allocation to each of the two traffic
+classes.
+In addition to mere routing aspects, system-wide studies have also been
+published. We mention [21], which is a compendium of several previous studies
+from the same authors that illustrates an overall architecture of the Quantum
+Internet, also including application, protocol, and deployment aspects at a high
+level. On the other hand, other works have focused on specific components,
+which are complementary to the research activity presented, e.g., [22] on con-
+gestion control in transport protocols and [23] on the link layer, with a focus on
+hardware and physical-layer considerations.
+Furthermore, some research groups have been working to define the basic
+10
+
+principles of distributed QC. Parekh et al. have defined an elegant frame-
+work for the parallel execution of a broad class of quantum algorithms on mul-
+tiple nodes [5], both using remote entanglement and with Local Operations and
+Classical Communication (LOCC) only, also studying in depth three classes
+of algorithms: variational quantum eigensolver, low-depth quantum amplitude
+estimation, and quantum k-means clustering. In [24] the authors address the
+problem of the efficient compilation of circuits for distributed QC by considering
+that some gate operations will be executed remotely, hence with much different
+latency and reliability than on-chip operations. The research of Dahlber et al.
+moved in the same direction and went as far as defining a set of low-level in-
+structions (called NetQASM) for distributed QC systems seamlessly supporting
+local and remote gates [25]. These works confirm that there is a growing interest
+in distributed QC, which is a motivation for our work.
+On another line of research, solutions have been proposed to trade capacity
+for fidelity, by using purification (or distillation) techniques [26].
+In brief,
+they consist in entangling multiple pairs of qubits with low fidelity and then
+collapsing them into a single one with high fidelity. We do not consider network-
+level purification in this work to remain consistent with the positioning of our
+contribution within the realm of 1G-repeater quantum networks. End-to-end
+purification is also possible, that is the operation is performed by quantum
+computers after the qubits have been entangled all along the path(s). This is
+studied, e.g., in [27], where the authors propose a quantum routing algorithm
+that maximizes the rate of EPR-pairs, while deciding not only the paths but
+also the purification patterns. These works complement our contributions since
+they operate on constant-rate point-to-point flows only and they do not take into
+account network provisioning issues.
+Finally, in line with the vast majority of prior works, we only consider bi-
+partite entanglement, i.e., made of two qubits, each situated in a quantum
+computer.
+While there are some promising theoretical studies on repeater-
+assisted multi-partite entanglement, i.e., involving more than two qubits (e.g.,
+[28]), the research in that area is in still its infancy. One noteworthy contribution
+11
+
+is [29], where the authors propose to adopt n-fusion of bi-partite entanglements
+to create higher level entanglements between n > 2 quantum computers. An
+appealing property that they demonstrate, under some assumptions, is that the
+entanglement rate between nodes remains constant with increasing distance,
+in number of hops. Ways to exploit this phenomenal quality are still under
+study. Multi-partite entanglement in a quantum network is generated starting
+from elementary bi-partite entanglements, which is the subject of this work.
+4. Service Differentiation
+In this section we extend the study in [6] by analyzing the QDRR algorithm
+along two directions which have remained so far uninvestigated: service differ-
+entiation by assigning apps different ρ values (Sec. 4.1) and apps with different
+fidelity thresholds F min (Sec. 4.2). In all the simulations in this section the
+peers are selected as follows: for each app i on node v we draw at random be-
+tween 2 and 4 candidate nodes by sampling in a uniform manner from the set
+of all nodes that are reachable from v in 2–7 hops. For benchmarking purposes,
+QDRR is compared to two baseline algorithms: random and best-fit. The Step 0
+in Sec. 2.2 is the same for all the algorithms, that is for each app we find k = 4
+shortest paths to reach any peer. All the algorithms then loop through all the
+possible paths for all candidates for each app until there are no more feasible
+paths to be assigned, but they differ in how they do so: QDRR is descriped by
+Steps 1–6 in Sec. 2.2 (and in far more details in Sec. IV-C in [6]), while:
+– Random: at each iteration one app with remaining paths is chosen at random
+and assigned its shortest path among any of its peers, which is allocated
+the maximum rate along the path. Random is representative of allocation
+algorithms that provide fair access to the quantum network resources in a
+per-app manner.
+– Best-fit: at each iteration, select the app with the shortest path to reach
+one of its peers and allocate the maximum rate along that path. Best-fit is
+representative of allocation algorithms that strive to maximize the efficiency,
+12
+
+that is the ratio between net entanglement rate and the quantum network
+capacity allocated.
+ 0
+ 10
+ 20
+ 30
+ 40
+ 50
+ 60
+ 70
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Iterations (x1000)
+Load (#apps)
+Dense|Random
+Dense|BestFit
+Dense|QDRR
+Sparse|Random
+Sparse|BestFit
+Sparse|QDRR
+Figure 2: Number of iterations (expect Step 0 in Sec. 2.2) with random, best-fit, QDRR, in a
+dense vs. sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.
+Unlike QDRR, both random and best-fit can be considered greedy algo-
+rithms, since they never backtrack to a previously selected combination of
+(app, peer, path), which is always allocated as much throughput as possi-
+ble. Therefore, their worst-case computational complexity (expect Step 0) is
+O (k|A| log |A|E[Wi]): k is the number of shortest paths selected in Step 0; |A|
+is the number of apps; log |A| takes into account the random selection or the ex-
+traction from an sorted data structure, respectively for the random and best-fit
+resource allocation algorithms; and, E[Wi] is the average number of peers per
+app. The complexity of QDRR is discussed in [6] and it depends on the choice
+of φ. To give an idea of the relative average complexity between QDRR and
+random/best-fit, we report their number of iterations in the simulations dis-
+cussed in Sec. 4.1 below in Fig. 2. As can be seen, in a sparse scenario the time
+complexity of QDRR is only marginally higher than that of greedy algorithms
+random and best-fit, but it becomes clearly higher in a dense scenario. If this is
+an issue, the value of φ can always be tuned to reduce the number of iterations,
+trading off fairness for speed.
+4.1. Different traffic priorities
+In a first batch of results we increase the load of the network from 10 to
+1000 apps. For every app, its priority weight ρ is drawn randomly in {1, 2, 4},
+13
+
+ 0
+ 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Residual/total capacity
+Load (#apps)
+Dense|Random
+Dense|BestFit
+Dense|QDRR
+Sparse|Random
+Sparse|BestFit
+Sparse|QDRR
+Figure 3: Ratio between the residual capacity and the total capacity with random, best-fit,
+QDRR, in a dense vs. sparse topology, when increasing the number of apps with ρ ∈ {1, 2, 4}.
+while F min is the same for all and equal to 0.7. As shown in Fig. 3, for all
+the allocation algorithms and topologies the relative residual capacity decreases
+with a sub-linear trend as the load increases, which is due to the exponential
+relation between the net rate, in EPR-pairs/s, and the number of hops as per
+Eq. (1).
+The utilization is higher in a sparse topology, while the difference
+between the allocation algorithms is negligible. In the following we report only
+the results in a dense topology, due to limited space; the complete results can
+be retrieved from the public GitHub repo above.
+ 120
+ 140
+ 160
+ 180
+ 200
+ 220
+ 240
+ 260
+ 280
+ 300
+ 320
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Max-min fairness (EPR pairs/s)
+Load (#apps)
+Random (class avg)
+Best-fit (class avg)
+QDRR (ρ = 1)
+QDRR (ρ = 2)
+QDRR (ρ = 4)
+Figure 4: Max-min fairness with random, best-fit, QDRR, in a dense topology, when increasing
+the number of apps with ρ ∈ {1, 2, 4}.
+We begin by showing the max-min fairness in Fig. 4. Since random and
+best-fit do not differentiate based on the ρ values, for them we show an aggre-
+gate average, while we keep separate curves for QDRR. With very low loads, the
+max-min fairness is good, i.e., low, for all allocation algorithms, because there is
+14
+
+little contention on resources. However, as the load increases, the max-min fair-
+ness increases steeply and the behavior is significantly affected by the allocation
+algorithm: with random the curve reaches a peak, which then slowly decreases
+towards high loads; best-fit performs worst, as expected, by continuing to in-
+crease, even though only slightly after the initial spur; for all traffic categories,
+QDRR provides increasingly better fairness as the load increases. The latter can
+be explained as follows. When there are few apps, it is likely that there are not
+many shared nodes/paths, hence QDRR does not really have a chance to dis-
+tribute the resources proportional to the apps’ weights; on the other hand, with
+more apps, it is increasingly easier for QDRR to enforce priorities by regulating
+the resources in common.
+ 1
+ 1.2
+ 1.4
+ 1.6
+ 1.8
+ 2
+ 2.2
+ 2.4
+ 2.6
+ 2.8
+ 3
+ 3.2
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Inter-class unfairness
+Load (#apps)
+Random
+Best-fit
+QDRR
+Figure 5: Inter-class unfairness index with random, best-fit, QDRR, in a dense topology, when
+increasing the number of apps with ρ ∈ {1, 2, 4}.
+To better show the service differentiating behavior of QDRR, we show the
+inter-class unfairness index, as defined in Eq. (3), in Fig. 5. It is clear that
+QDRR is the only allocation algorithm providing the apps with a clear service
+differentiation, which improves as the load increases for the same reason above.
+Finally, in Fig. 6 we show the net rate/app, in EPR-pairs/s: even though
+it slowly decreases for all the resource allocation algorithms, we can see that
+random and best-fit achieve better rates. Therefore, QDRR is effective in dif-
+ferentiating service across apps with different priority weights, but this incurs a
+cost, in terms of net rate.
+15
+
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Net rate/app (EPR pairs/s)
+Load (#apps)
+Random (class avg)
+Best-fit (class avg)
+QDRR (ρ = 1)
+QDRR (ρ = 2)
+QDRR (ρ = 4)
+Figure 6: Net rate/app with random, best-fit, QDRR, in a dense topology, when increasing
+the number of apps with ρ ∈ {1, 2, 4}.
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Max-min fairness (EPR pairs/s)
+Load (#apps)
+Dense|Random
+Dense|Best-fit
+Dense|QDRR
+Sparse|Random
+Sparse|Best-fit
+Sparse|QDRR
+Figure 7: Max-min fairness with random, best-fit, QDRR, in dense vs. sparse topologies, when
+increasing the number of apps with F min ∈ {0.7, 0.8, 0.9}.
+16
+
+4.2. Different fidelity thresholds
+We now report the results obtained in a new batch, which follows the same
+direction as above, but we set ρ = 1 for all apps and draw randomly the mini-
+mum fidelity F min from {0.7, 0.8, 0.9}, instead. We show the max-min fairness
+in Fig. 7, for both topologies. Like in Sec. 4.1, QDRR achieves significantly
+better performance than random and best-fit, the latter performing worst.
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 700
+ 800
+ 900
+ 1000
+Net rate/app (EPR pairs/s)
+Load (#apps)
+Dense|Random
+Dense|Best-fit
+Dense|QDRR
+Sparse|Random
+Sparse|Best-fit
+Sparse|QDRR
+Figure 8: Net rate/app with random, best-fit, QDRR, in dense vs. sparse topologies, when
+increasing the number of apps with F min ∈ {0.7, 0.8, 0.9}.
+However, as can be seen in Fig. 8, also with a mix of fidelity thresholds, the
+net rate/app of QDRR is slightly less than that with both greedy allocation
+strategies, which confirms the trade-off already identified in the previous batch
+of experiments. It is worth noting that such a trade-off is well-known also in
+different contexts: for instance, in cellular systems, greedy scheduling algorithms
+that prioritize user terminals with good channel conditions (often known as “max
+C/I”) are bound to provide a higher cell throughput at the cost of an inferior
+fairness compared to milder strategies such as Proportional Fair [30].
+In conclusion, like for heterogeneous weights, QDRR can provide service dif-
+ferentiation to apps with different fidelity thresholds, but the net rate achievable
+is slightly reduced compared to alternatives that do not differentiate.
+5. Fair Sharing
+In this section we address a problem that is preliminary and complementary
+to that defined in our previous work [6] and investigated in Sec. 4 with differen-
+17
+
+tiated service. So far we have assumed that all nodes are equal, and each host
+is wishing to cooperate with a given set of workers, without elaborating further
+on how such a set is selected; for performance evaluation purposes, such a set
+was selected randomly from candidates depending only on the distance as per
+the specific scenario simulated. In the following, instead, we address specifically
+this issue: indeed, how does one decide which are the possible workers of a host
+node?
+end users
+data centers
+Figure 9: Example of quantum network with three end users, labeled from u1 to u3, wishing to
+host distributed quantum computing algorithms with data center nodes d1 or d2, represented
+with multiple co-located circles to indicate that they are expected to be more powerful than end
+users and, possibly, they might have a more complex internal structure that is not elaborated
+further in this paper; the other nodes in G(V, E) participate to the end-to-end entanglement of
+qubits as intermediate hops. Like in Sec. 2.1/Fig. 1, the network is characterized by capacity
+Cij of the link between nodes i and j, initial generation fidelity F, and entanglement swapping
+success probability q.
+We adapt our system model to the new landscape by specializing the role of
+nodes. As illustrated in the example in Fig. 9, we assume that nodes can be of
+three types: (i) end users (ui): they act as the “home QCs” of customers wish-
+ing to run quantum algorithms on them, also exploiting quantum computation
+resources offered by other nodes through distributed quantum computing; a cus-
+tomer operates the end user via a classical computer for, e.g., input preparation,
+18
+
+circuit compilation, and quantum network resource reservations; (ii) data centers
+(dj): these are QCs that can provide customers with extra quantum computa-
+tion capacity to be added to their respective end users for the purpose of solving
+bigger instances of their problems via distributed quantum computing, exploit-
+ing end-to-end entanglement of qubits through an underlying quantum network;
+(iii) intermediate nodes: quantum repeaters who do not consume or offer QC
+capacity but contribute to the quantum network by performing entanglement
+swapping between links as instructed by the resource allocation algorithm (e.g.,
+QDRR). A node can play any combination of the three roles above. We assume
+that end user ui may perform distributed quantum computing with any combi-
+nation of data centers {dj}, following commercial agreements that are out of the
+scope of this work. All other quantum network assumptions in Sec. 2.1 remain
+the same. In Sec. 5.1 we formulate the problem in mathematical terms and
+propose a solution, which is then evaluated via simulation in Sec. 5.2, compared
+to two alternatives.
+5.1. Quantum Workers’ Assignment Problem
+In its most general formulation, the problem of how to best assign each host
+a set of workers depends on several factors that may be not known or under the
+control of the quantum network operator. They include, for instance: the quan-
+tum algorithms to be run, the different characteristics of the end user and data
+center QCs, the schedule of the task execution, not to mention administrative
+factors (contracts, partnerships, billing issues) and technical constraints (do the
+QCs need to have the same hardware or software?). Since both quantum net-
+working and distributed quantum computing are in their infancy, we consider
+unrealistic to address the problem under such general settings. Rather, we focus
+on aspects that are captured by our network model (in Sec. 2.1) with the goal
+of providing an initial understanding of the problem, to be used as a stepping
+stone by future studies as the technologies involved become more mature. Our
+formulation, in natural language, is the following:
+Quantum Workers’ Assignment Problem (QWAP): find the sets of workers, se-
+19
+
+lected from the data center nodes, to be assigned to each host, from the end
+user nodes, so that the overall profit of the hosts is maximum while balancing
+the load of data centers.
+The problem can be formulated in a formal manner once we define the no-
+tions of “profit” and “load balancing”. Based on the prior works in the literature
+(see Sec. 3), we consider the net entanglement rate in Eq. (1) as the profit, i.e.,
+between two possible data centers d1 and d2 considered as candidate workers
+for end user u, we prefer the one that potentially achieves the highest net en-
+tanglement rate. On the other hand, we introduce load balancing as follows.
+First, we define the system parameter W as the target number of workers per
+end user. Then, we force each data center to be assigned as worker to at most
+B end users, where B is determined as the minimum value that allows this con-
+straint to be provided with given W, Nu end users, and Nd data centers. This
+way, we force an even load across data centers. Note that this formulation can
+be trivially extended to the case where data centers have different capabilities
+by introducing appropriate weights, which we do not consider in this work to
+keep the notation succinct.
+Assuming without loss of generality, again to simplify notation, that all end
+users have one and only one request to run distributed quantum computing, the
+QWAP can be expressed as an optimization problem with objective function:
+max
+Nu
+�
+u=1
+Nd
+�
+d=1
+πudxud
+(4)
+such that:
+20
+
+Nu
+�
+u=1
+xud ≤ B
+d = 1, . . . , Nd
+(5)
+Nd
+�
+d=1
+xud ≤ W
+u = 1, . . . , Nu
+(6)
+xud ∈ {0, 1}
+(7)
+B =
+�Nu · W
+Nd
+�
+(8)
+where the profit πud between end user u and data center d is defined by:
+πud =
+�
+�
+�
+�
+�
+0
+if ∀p ∈ Pu,d : F(p) ≤ F min
+u
+rud¯p
+where ¯p = arg maxp
+�
+rudp|F(p) ≥ F min
+u
+� ,
+(9)
+where F(p) is the fidelity of the end-to-end entanglement along the path p,
+according to Eq. (2), F min
+u
+is the minimum fidelity requested by end user u,
+Pu,d is the set of all paths between u and d in G(V, E), and the net rate rudp
+between end user u and data center d along path p is as follows:
+rudp = max(i,j)∈p {Cij}
+q|p|−1
+.
+(10)
+The output assignment integer variables, as per Eq. (8), are the xud, with
+the constraints as follows: Eq. (5) ensures that no data center is assigned more
+than its fair share of B users, where B is computed via Eq. (8); Eq. (6) ensures
+that no end user is assigned more than W workers. The profit πud, defined
+through Eqs. (9–10), corresponds to the maximum net rate of EPR-pairs/s that
+can assigned to end user u along any path towards data center d that fulfils its
+minimum fidelity requirement. Before proceeding, we state two key observations
+about the QWAP.
+Observation#1.
+In a general graph G(V, E), the number of paths between
+two nodes can be exponential with the size of the graph. For instance, in a
+complete graph this number is ⌊(V − 2)!e⌋. Therefore, the preparation of the
+problem input in Eq. (9) might be very computation-intensive in practice. In
+21
+
+the evaluation below, we adopt a reasonable approximation: rather than finding
+all the paths Pud between nodes u and d, we use a reduced set P¯k
+ud that consists
+of the ¯k shortest paths (¯k = 10 in the simulations in Sec. 5.2). The rationale is
+that long paths have a small chance of being selected as ¯p in the second branch
+of Eq. (9), because the net rate decreases exponentially with the path length as
+per Eq. (10).
+Observation#2. When W = 1, then Eqs. (4–10) above can be trivially trans-
+formed into an assignment problem, whose optimum solution can be found ef-
+ficiently.
+With W > 1, however, the QWAP becomes a “multiple knapsack
+problem”, which is NP-hard, but for which several efficient heuristics are well-
+known in the operations research literature (e.g., [31]).
+Based on the two observations above, we propose our load balancing al-
+gorithm to solve the QWAP, which we implemented in our simulator and eval-
+uated in the next section. The idea of the load balancing algorithm is to find
+the optimal allocation for the first worker of each end user using the Hungar-
+ian algorithm [32], which finds the best (exact) solution in assignment problem
+instances; then, it progresses by considering one more worker at a time, until
+W, each time invoking the Hungarian algorithm again on the data centers with
+residual slots. The algorithm is greedy because it never backtracks prior deci-
+sions and it always terminates after a fixed number of iterations. More formally,
+the load balancing algorithm consists of the following three steps:
+1. Prepare the problem input, in particular the profits πud, by finding up to ¯k
+shortest paths between u and d in G(E, V ) using Yen’s algorithm [33].
+2. Determine B based on Nu, Nd, and W using Eq. (8).
+3. Arrange the output of Step 1 in a profit matrix where the rows are the end
+users and there are B columns for each data center, i.e., the matrix size is
+Nu × BNd. Then run W iterations of the Hungarian algorithm. After each
+iteration, set πud ← 0 in all columns where a data center has been assigned
+(avoids that the constraint Eq. (5) is violated) and in all cells that refer to the
+same pair u, d that has been assigned (avoids that a user is assigned multiple
+22
+
+times the same data center).
+5.2. Evaluation
+Yen's algorithm using Dijkstra with Fibonacci heap
+preparation
+execution
+for each end user
+and data center
+number of shortest
+paths to search
+Hungarian algorithm
+number of iterations
+Dijkstra with Fibonacci heap
+for each end user
+and worker
+for each end user and worker
+random selection
+of data center
+a)
+b)
+c)
+Figure 10: Worst case time complexity of a) the load balancing algorithm to solve the QWAP
+in Sec. 5.1 vs. the comparison algorithms b) random and c) shortest path.
+In this section we evaluate the load balancing algorithm defined in Sec. 5.1
+using the simulator described in Sec. 2.3. End users and data centers are selected
+randomly from the set of nodes V , with the following composition of the nodes:
+10% end users, 10% data centers, 80% intermediate nodes.
+As comparison
+algorithms, we define:
+– Random, which assigns each end user W data centers at random, thus it
+maximizes fairness.
+– Shortest path, which assigns each end user the W data centers that are closest
+in G(E, V ), in number of hops, thus it maximizes the net rate.
+After the assignment, the resources are allocated using QDRR.
+We report in Fig. 10 the worst case time complexity of the three algorithms,
+where we assume that the shortest path is computed using Dijkstra’s algorithm
+with the help of a Fibonacci heap to keep edges sorted [34]. As can be seen, the
+23
+
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+ 450
+ 500
+ 1
+ 2
+ 3
+ 4
+ 5
+Net rate/app (EPR-pairs/s)
+W
+Random
+Shortest path
+Load balancing
+ 0
+ 0.5
+ 1
+ 1.5
+ 2
+ 2.5
+ 3
+ 3.5
+ 4
+ 4.5
+ 1
+ 2
+ 3
+ 4
+ 5
+Max-min num users per data center
+W
+Random
+Shortest path
+Load balancing
+Figure 11: Net rate/app (top) and max-min number of users per data center (bottom) with
+random, shortest path, and load balancing, in a dense topology, with 10 apps, when increasing
+W from 1 to 5.
+load balancing algorithm is far more complex than random and shortest-path,
+in both the preparation and the execution phases; random, in particular, does
+not even depend on the graph size. However, in the following we will see that
+the added complexity brings benefits that can be of potential interest to the
+future quantum network operators.
+In Fig. 11 (top) we show the net rate/app with Nu = 10 and W increasing
+from 1 to 5, in a dense topology. As can be seen, the random algorithm per-
+forms poorly, because it does not consider at all the network topology. On the
+other hand, shortest path and load balancing perform similarly, with the latter
+exhibiting a higher net rate with small values of W. In Fig. 11 (bottom) we
+plot a measure of the spread of resources, as the max-min number of users per
+data center. Load balancing performs consistently and significantly better than
+both random and shortest path, with the latter exhibiting the highest unfair-
+ness. We note that our conclusions are limited to the settings of the scenarios
+24
+
+simulated; in particular, different topologies or link capacity distributions can
+lead to situations where the gap between load balancing and either random or
+shortest path is reduced significantly. However, a crucial advantage of our pro-
+posed solution, compared to its alternatives under test, is that it can adapt to
+different settings, thus it can perform well even when the scenario is not known
+or changes dynamically.
+ 0.82
+ 0.84
+ 0.86
+ 0.88
+ 0.9
+ 0.92
+ 0.94
+ 0.96
+load balancing
+random
+shortest-path
+load balancing
+random
+shortest-path
+Fidelity
+Dense|10 apps
+Dense|20 apps
+Sparse|10 apps
+Sparse|20 apps
+Figure 12:
+Fidelity with random, shortest path, and load balancing, in dense vs. sparse
+topologies, with 10 vs. 20 apps and W = 1.
+The fidelity is shown in Fig. 12, with Nu = {10, 20} and in dense vs. sparse
+topologies. The number of end users/apps, i.e., Nu, does not affect the perfor-
+mance in a noticeable manner. On the other hand, the fidelity is generally lower
+in sparse topologies, as expected. Random performs worse, while load balanc-
+ing and shortest path give similar results, with the former performing slightly
+worse only in the sparse case. This can be explained by following the same line
+of reasoning for the net rate/app above.
+To conclude the analysis, we modify the mix of nodes.
+In one batch of
+simulations we increase the ratio of end users from 10% (like in the results so
+far) to 50%; in another one we do the same for data centers. Results are shown
+only for load balancing in Fig. 13, in terms of the net rate, with Nu = 15 and
+W = 3. As the ratio of data centers increases (green curves), the net rate/app
+increases almost linearly, as well. On the other hand, increasing the number
+of users (blue curves), only provides a sub-linear performance improvement.
+This is because the former case corresponds to increasing the physical resources
+25
+
+ 50
+ 100
+ 150
+ 200
+ 250
+ 300
+ 350
+ 400
+ 450
+ 500
+ 550
+ 600
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+Net rate per app (EPR-pairs/s)
+Ratio of (end users | data centers)
+[end users]|Dense
+[end users]|Sparse
+[data centers]|Dense
+[data centers]|Sparse
+Figure 13: Net rate/app with load balancing, in dense vs. sparse topologies, with 15 apps and
+W = 3, when increasing the fraction of nodes as either end users or data centers.
+provided to the users, while the latter only to a more uniform distribution of the
+same resources. The conclusions are the same for dense and spare topologies,
+though the net rate for the latter is significantly lower.
+In conclusion, assigning data centers to end users through the load balancing
+algorithm, which provides an approximate solution of the QWAP, achieves an
+even distribution of resources, better than both random and shortest path as-
+signment, without compromising on the net rate and fidelity of the end-to-end
+entanglement paths.
+6. Conclusions
+In this paper we have studied two open issues in quantum networking for
+distributed quantum computing. First, we have assessed through simulation
+the effectiveness of the QDRR resource allocation algorithm [6] in handling sce-
+narios where applications have different priority weights or minimum fidelity
+requirements. Second, we have defined a novel problem involving the selection
+of workers for a set of nodes hosting computation, called the Quantum Work-
+ers’ Assignment Problem (QWAP), which we have modeled as an optimization
+problem and for which we have proposed a heuristic called “load balancing”.
+The latter has been evaluated in comparison to alternatives seeking to maxi-
+mize only fairness and the net rate of end-to-end entanglement, respectively, and
+the results have shown that load balancing achieves an excellent compromise in
+26
+
+terms of the two metrics. The source code and simulation scripts are publicly
+available to the community.
+Further open research areas are: the use of purification to increase fidelity
+at the expense of capacity; modeling distributed QC applications to understand
+their characteristic time scales and requirements; integration with link layer pro-
+tocols; incorporation in the simulation of more realistic models for the quantum
+channel and repeaters.
+Acknowledgment
+Work co-funded by EU, PON Ricerca e Innovazione 2014–2020 FESR/FSC
+Project ARS01_00734 QUANCOM, and European High-Performance Comput-
+ing Joint Undertaking (JU) under grant agreement No 101018180 HPCQS. The
+paper reflects only the authors’ view and the funding agencies are not respon-
+sible for any use that may be made of its content.
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+

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+arXiv:2301.03332v1  [math.AP]  9 Jan 2023
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN
+INEQUALITY ON THE H-TYPE GROUP
+QIAOHUA YANG
+Abstract. We determine the optimal constant in the L2 Folland-Stein in-
+equality on the H-type group, which partially confirms the conjecture given
+by Garofalo and Vassilev (Duke Math. J., 2001). The proof is inspired by the
+work of Frank and Lieb (Ann. of Math., 2012) and Hang and Wang.
+1. Introduction
+Let G be a stratified, simply connected nilpotent Lie group (in short a Carnot
+group) of step r. Denote by g the Lie algebra of G. It is known that g = �r
+i=1 Vi
+satisfying (see e.g. [10])
+[V1, Vj] = Vj+1, 1 ≤ j ≤ r − 1; [V1, Vr] = {0}.
+As a simply connected nilpotent group, G is differential with RN, N = �r
+i=1 dim Vi,
+via the exponential map exp : g → G. There is a natural family of nonisotropic
+dilations δλ : g → g for λ > 0 and we define it as follows:
+δλ(X1 + · · · + Xr) = λX1 + · · · + λrXr, Xj ∈ Vj, 1 ≤ j ≤ r.
+The homogeneous dimension of G, associated with δλ, is Q = �r
+j=1 j dim Vj. Via
+the exponential map exp : g → G, we define the group of dilations on G as follows:
+δλ(g) = exp ◦δλ ◦ exp−1(g), g ∈ G.
+Set nj = dim Vj, 1 ≤ j ≤ r. Let {X1, · · · , Xn1} be a basis of V1 and denote
+by ∇G = (X1, · · · , Xn1) the horizontal gradient of G. The sub-Laplacian on G is
+∆G = �n1
+i=1 X2
+i . The Sobolev space W 1,p
+0
+(G) is the closure of C∞
+0 (G) with respect
+to the norm
+∥u∥W 1,p
+0
+(G) =
+��
+G
+|∇Gu|pdg
+� 1
+2
+,
+where dg is the Haar measure on G. We remark that the Haar measure on G,
+induced by the exponential mapping from the Lebesgue measure on g = RN, coin-
+cides the Lebesgue measure on RN. The Folland-Stein inequality on G reads that
+there exits some constant C > 0 such that for each u ∈ W 1,p
+0
+(G) (see [8, 9]),
+��
+G
+|u|
+pQ
+Q−p dg
+� Q−p
+pQ
+≤ C
+��
+G
+|∇Gu|pdg
+� 1
+p
+, 1 < p < Q.
+(1.1)
+2000 Mathematics Subject Classification. Primary: 43A80; 46E35; 22E25.
+Key words and phrases. Folland-Stein inequality; Heisenberg group; H-type group; best
+constant.
+The
+work
+was
+partially
+supported
+by
+the
+National
+Natural
+Science
+Foundation
+of
+China(No.11201346).
+1
+
+2
+QIAOHUA YANG
+For the existence and regularity of minimizers of the Folland-Stein inequality (1.1),
+we refer to [24].
+The Heisenberg group is the simplest example of Carnot group of step 2. We
+denote it by Hn = (Cn × R, ◦). The group law on Hn is given by
+(z, t) ◦ (z′, t′) = (z + z′, t + t′ + 2Imz · z′),
+where z · z′ = �n
+j=1 zj¯z′
+j. The homogeneous norm on Hn is given by
+|(z, t)| = (|z|4 + t2)
+1
+4 .
+In a series of papers [18, 19, 20], Jerison and Lee, among other results, determined
+the explicit computation of the extremal functions in (1.1) in the case p = 2 and
+G = Hn. In fact, the extremal functions are, up to group translations and dilations,
+c((1 + |z|2)2 + t2)− Q−2
+4 , c ∈ R.
+Such inequalities play an important role in the study of CR Yamabe problems.
+Later, in a celebrated paper [11], Frank and Lieb established sharp Hardy-Littlewood-
+Sobolev inequalities on Hn. We state the result as follows:
+Theorem 1.1 (Frank-Lieb). Let 0 < λ < Q and p =
+2Q
+Q−λ. Then for any f, g ∈
+Lp(Hn),
+�����
+� �
+Hn×Hn
+f(z, t)g(z′, t′)
+|(z, t)−1 ◦ (z′, t′)|λ dzdtdz′dt′
+����� ≤
+� πn+1
+2n−1n!
+�λ/Q n!Γ( Q−λ
+2 )
+Γ2( Q−2λ
+4
+)
+∥f∥p∥g∥p,
+with equality if and only if, up to group translations and dilations,
+f = c((1 + |z|2)2 + t2)− 2Q−λ
+4
+, g = c′((1 + |z|2)2 + t2)− 2Q−λ
+4
+for some c, c′ ∈ C.
+In particular, choosing λ = Q−2 in Theorem 1.1 yields the Jerison and Lee’s in-
+equality. Using the method in [11], Frank and Lieb [12] also gave a new, rearrangement-
+free proof of sharp Hardy-Littlewood-Sobolev inequalities on Rn. Recently, Hang
+and Wang [15] present a shorter proof of the Frank-Lieb inequality, in which they
+bypasses the subtle proof of existence and the Hersch-type argument via subcritical
+approximation.
+Some of the results of Theorem 1.1 have been generalized to the cases of quater-
+nionic Heisenberg group and octonionic Heisenberg group (see [4, 5, 16, 17]). We
+note that Heisenberg group, quaternionic Heisenberg group and octonionic Heisen-
+berg group are known as the groups of Iwasawa type, i.e., the nilpotent component
+in the Iwasawa decomposition of simple groups of rank one (see e.g. [6]).
+The aim of this paper is to look for the optimal constant of (1.1) when p = 2 and
+G is a group of Heisenberg type (in short a H-type group). Recall that a H-type
+group G is a Carnot group of step two with the following properties (see Kaplan
+[21]): the Lie algebra g of G is endowed with an inner product ⟨, ⟩ such that, if z
+is the center of g, then [z⊥, z⊥] = z and moreover, for every fixed z ∈ z, the map
+Jz : z⊥ → z⊥ defined by
+⟨Jz(v), ω⟩ = ⟨z, [v, ω]⟩, ∀ω ∈ z���
+(1.2)
+is an orthogonal map whenever ⟨z, z⟩ = 1. It is known (see [6]) that a H-type group
+G is the group of Iwasawa type if and only if its Lie algebra satisfies the following
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+3
+J2-condition: for any v ∈ z⊥ and z, z′ ∈ z such that ⟨z, z′⟩ = 0, there exists z′′ ∈ z
+such that
+JzJz′v = Jz′′v.
+Therefore, most of H-type groups are not groups of Iwasawa type.
+Set m = dim z⊥ and n = dim z. Since G has step two, we can fix on G a system
+of coordinates (x, t) such that the group law on G has the form (see [2])
+(x, t) ◦ (x′, t′) =
+�
+xi + x′
+i, i = 1, 2, · · · , m
+tj + t′
+j + 1
+2⟨x, U (j)x′⟩, j = 1, 2, · · · , n
+�
+(1.3)
+for suitable skew-symmetric matrices U (j)’s. Nextly, we set
+U(ξ) =
+��
+1 + |x|2
+4
+�2
++ |t|2
+�− Q−2
+4
+, ξ = (x, t) ∈ G;
+(1.4)
+Uλ,η(ξ) =λ
+Q−2
+2 U(δλ(η−1 ◦ ξ)),
+η ∈ G.
+(1.5)
+It has been shown that [m(Q − 2)]
+Q−2
+4 Uλ,η(ξ) satisfies the Yamabe-type equation
+(see [13, 14])
+∆G[m(Q − 2)]
+Q−2
+4 Uλ,η + {[m(Q − 2)]
+Q−2
+4 Uλ,η}
+Q+2
+Q−2 = 0,
+or equivalently,
+∆GUλ,η + m(Q − 2)U
+Q+2
+Q−2
+λ,η
+= 0.
+(1.6)
+In the paper [13], Garofalo and Vassilev gave the following conjecture:
+Conjecture (Garofalo-Vassilev). In a H-type group G, the functions [m(Q −
+2)]
+Q−2
+4 Uλ,η(ξ) are the only nontrivial entire solutions to
+�
+∆Gu + u
+Q+2
+Q−2 = 0,
+u ∈ W 1,2
+0
+(G), u ≥ 0.
+If the conjecture is true, then one can obtain the optimal constant of L2 Folland-
+Stein inequality on H-type groups. In this paper we shall use the method given by
+Frank and Lieb [11, 12] and Hang and Wang [15] to determine the optimal constant,
+instead of proving the conjecture directly. To this end, we have
+Theorem 1.2. It holds that
+�
+G
+|∇Gu|2dxdt ≥ Sm,n
+��
+G
+|u|
+2Q
+Q−2 dxdt
+� Q−2
+Q
+, u ∈ W 1,2
+0
+(G),
+(1.7)
+where
+Sm,n = 4− 2n
+Q m(Q − 2)π
+m+n
+Q
+� Γ( m+n
+2
+)
+Γ(m + n)
+�1/Q
+.
+The inequality is sharp and an extremal function is
+U(x, t) =
+��
+1 + |x|2
+4
+�2
++ |t|2
+�− Q−2
+4
+.
+
+4
+QIAOHUA YANG
+By Theorem 1.2, it is easy to see that the functions cUλ,η(ξ)(c ∈ R) are also
+extremal functions of inequality (1.7).
+As an application of Theorem 1.2, we study the eigenvalues of
+−∆Gv = µU
+4
+Q−2
+λ,η v, v ∈ W 1,2
+0
+(G).
+(1.8)
+We note that the eigenvalues of (1.8) play an important role in the study of stability
+for the Folland-Stein inequality (see [1, 3, 7] for the case of Euclidean space). In
+Lemma 3.2 we show that the embedding map W 1,2
+0
+(G) ֒→ L2(G, U(x, t)
+4
+Q−2 dxdt) is
+compact. So the spectrum of (1.8) is discrete. Furthermore, we have the following
+theorem:
+Theorem 1.3. Let µi, i = 1, 2, · · · be the eigenvalues of (1.8) given in increasing
+order. Then
+(1) µ1 = m(Q − 2) is simple with eigenfunction Uλ,η.
+(2) µ2 = m(Q + 2) and {∂λUλ,η, ∇ηUλ,η} are eigenfunctions.
+Furthermore, the eigenvalues do not depend on λ and η.
+Remark 1.4. It seems that µ2 has multiplicity m + n + 1 with corresponding
+eigenspace spanned by {∂λUλ,η, ∇ηUλ,η}.
+However, we fail to prove it.
+Once
+it has been proven, it would provide a generalization of the results of Bianchi and
+Egnell ([1], Lemma A1) to the setting of H-type groups.
+2. preliminaries on H-type groups
+In the rest of paper, we let G be a H-type group with group law given by (1.3).
+The nonisotropic dilations δλ on G is
+δλ(x, t) = (λx, λ2t).
+For (x, t) ∈ G, the homogeneous norm of (x, t) is
+ρ(x, t) =
+�|x|4
+16 + |t|2
+� 1
+4
+.
+With this norm ρ, we can define the ball centered at origin with radius R
+BR(0) = {(x, t) ∈ G : ρ(x, t) < R}
+and the unit sphere Σ = ∂B1(0) = {(x, t) ∈ G : ρ(x, t) = 1}.
+Given any (x, t) ∈ G with ρ(x, t) ̸= 0, we set x∗ =
+x
+ρ(x,t) and t∗ =
+t
+ρ(x,t)2 . The
+polar coordinates on G associated with ρ are the following (see [10]):
+�
+G
+f(x, t)dxdt =
+� ∞
+0
+�
+Σ
+f(ρx∗, ρ2t∗)ρQ−1dσdρ, f ∈ L1(G).
+The following theorem was proved in [2], Theorem A.2.:
+Theorem 2.1. G is a H-type group if and only if G is (isomorphic to) Rm+n
+with the group law in (1.3) and the matrices U (1), U (2), · · · , U (n) have the following
+properties:
+(1) U (j) is a m × m skew symmetric and orthogonal matrix, for every j =
+1, 2, · · · , n;
+(2) U (i)U (j) + U (j)U (i) = 0 for every i, j ∈ {1, 2, · · · , n} with i ̸= j.
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+5
+The vector field in the Lie algebra g that agrees at the origin with
+∂
+∂xj (j =
+1, · · · , m) is given by
+Xj =
+∂
+∂xj
++ 1
+2
+n
+�
+k=1
+� m
+�
+i=1
+U (k)
+i,j xi
+�
+∂
+∂tk
+and g is spanned by the left-invariant vector fields X1, · · · , Xm, T1 =
+∂
+∂t1 , · · · , Tn =
+∂
+∂tn . Furthermore (see [2], Page 200, (A.4) ),
+[Xi, Xj] =
+n
+�
+r=1
+U (r)
+i,j Tr, i, j ∈ {1, 2, · · · , n}.
+(2.1)
+The exponential map exp : g → G is
+exp : g → Rm+n,
+m
+�
+i=1
+xiXi +
+n
+�
+j=1
+tjTj �→ (x, t).
+We note that by exponential mapping, the group law (1.3) is nothing but the
+Baker-Campbell-Hausdorff formula (see [2], the proof of Theorem A.2)
+exp X ◦ exp Y = exp(X + Y + 1
+2[X, Y ]), X, Y ∈ g.
+Using (2.1), we have that for t = (t1, · · · , tn) = t1T1 + · · · + tnTn and x =
+(x1, · · · , xm) = x1X1 + · · · + xmXm, the map Jt, defined by (1.2), is (see also
+[2], Page 201)
+Jtx =
+n
+�
+r=1
+m
+�
+i=1
+trxiJTr(Xi) =
+n
+�
+r=1
+m
+�
+i=1
+trxi
+
+
+m
+�
+j=1
+U (r)
+i,j Xj
+
+
+=
+m
+�
+j=1
+� n
+�
+r=1
+m
+�
+i=1
+trxiU (r)
+i,j
+�
+Xj.
+Since Jt is an orthogonal map whenever |t| = 1, we obtain
+|Jtx|2 = |t|2|x|2 =
+m
+�
+j=1
+� n
+�
+r=1
+m
+�
+i=1
+trxiU (r)
+i,j
+�2
+.
+(2.2)
+The horizontal gradient on G is ∇G = (X1, · · · , Xm). The sub-Laplacian on G
+is given by (see [2], Remark A.6.)
+∆G =
+m
+�
+j=1
+X2
+j =
+m
+�
+j=1
+�
+∂
+∂xj
++ 1
+2
+n
+�
+k=1
+� m
+�
+i=1
+U (k)
+i,j xi
+�
+∂
+∂tk
+�2
+= ∆x + 1
+4|x|2∆t +
+n
+�
+k=1
+⟨x, U (k)∇x⟩ ∂
+∂tk
+,
+where
+∆x =
+m
+�
+j=1
+� ∂
+∂xj
+�2
+, ∆t =
+n
+�
+k=1
+� ∂
+∂tk
+�2
+.
+We remark that ∆G is homogeneous of degree two with respect to δλ.
+By using (1.6), we have the following Hardy inequality (see [22], Corollary 1.4
+for Hardy inequality of fractional powers of the sublaplacian on G).
+
+6
+QIAOHUA YANG
+Lemma 2.2. It holds that, for u ∈ W 1,2
+0
+(G),
+�
+G
+|∇Gu|2dxdt ≥ m(Q − 2)
+�
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt,
+with equality if and only if u = cU(x, t), where c ∈ R and U(x, t) is given by (1.4).
+Proof. We have, through integration by parts,
+0 ≤
+�
+G
+U 2|∇G(U(x, t)−1u)|2dxdt
+=
+�
+G
+���∇Gu − u
+U ∇GU
+���
+2
+dxdt
+=
+�
+G
+|∇Gu|2dxdt +
+�
+G
+|∇GU|2
+U 2
+u2dxdt −
+�
+G
+1
+U ⟨∇Gu2, ∇GU⟩dxdt
+=
+�
+G
+|∇Gu|2dxdt +
+�
+G
+u2 1
+U ∆GUdxdt
+=
+�
+G
+|∇Gu|2dxdt − m(Q − 2)
+�
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt.
+(2.3)
+To get the last equality, we use (1.6). The desired result follows.
+□
+Set η = (y1, · · · , ym, w1, · · · , wn) ∈ G. By (1.6), we have
+∆G
+∂Uλ,η
+∂yj
++ m(Q + 2)U
+4
+Q−2
+λ,η
+∂Uλ,η
+∂yj
+= 0,
+j = 1, , · · · , m,
+∆G
+∂Uλ,η
+∂wr
++ m(Q + 2)U
+4
+Q−2
+λ,η
+∂Uλ,η
+∂wr
+= 0,
+r = 1, , · · · , n,
+∆G
+∂Uλ,η
+∂λ
++ m(Q + 2)U
+4
+Q−2
+λ,η
+∂Uλ,η
+∂λ
+= 0.
+(2.4)
+Furthermore, we have the following lemma:
+Lemma 2.3. It holds that
+m
+�
+j=1
+����
+∂Uλ,η
+∂yj
+|λ=1,η=0
+����
+2
++
+n
+�
+r=1
+����
+∂Uλ,η
+∂wr
+|λ=1,η=0
+����
+2
++ 1
+4
+����
+∂Uλ,η
+∂λ |λ=1,η=0
+����
+2
+= (Q − 2)2
+16
+U(ξ)2.
+Proof. It is easy to see η−1 = −η. Therefore, by (1.3) and (1.5), we have
+Uλ,η(x, t) =λ
+Q−2
+2
+
+
+�
+1 + λ2
+4
+m
+�
+i=1
+(xi − yi)2
+�2
++ λ4
+n
+�
+r=1
+�
+tr − wr − ⟨y, U (r)x⟩
+2
+�2
+
+− Q−2
+4
+.
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+7
+We compute
+∂Uλ,η
+∂yj
+|λ=1,η=0 = − Q − 2
+4
+U(ξ)
+Q+2
+Q−2
+�
+2
+�
+1 + |x|2
+4
+�
+·
+�
+−xj
+2
+�
++
+n
+�
+r=1
+tr
+�
+−
+m
+�
+i=1
+U (r)
+j,i xi
+��
+=Q − 2
+4
+U(ξ)
+Q+2
+Q−2
+��
+1 + |x|2
+4
+�
+xj +
+n
+�
+r=1
+m
+�
+i=1
+trxiU (r)
+j,i
+�
+, j = 1, · · · , m;
+∂Uλ,η
+∂wr
+|λ=1,η=0 = − Q − 2
+4
+U(ξ)
+Q+2
+Q−2 (−2tr)
+=Q − 2
+2
+U(ξ)
+Q+2
+Q−2 tr,
+r = 1, · · · , n;
+∂Uλ,η
+∂λ |λ=1,η=0 =Q − 2
+2
+U(ξ) − Q − 2
+4
+U(ξ)
+Q+2
+Q−2
+�
+2
+�
+1 + |x|2
+4
+�
+· |x|2
+2
++ 4|t|2
+�
+= − Q − 2
+2
+U(ξ)
+Q+2
+Q−2
+�
+−1 + |x|4
+16 + |t|2
+�
+.
+Since each U (j)(1 ≤ j ≤ n) is a m × m skew symmetric matrix, we have, by using
+(2.2),
+m
+�
+j=1
+����
+∂Uλ,η
+∂yj
+|λ=1,η=0
+����
+2
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+m
+�
+j=1
+��
+1 + |x|2
+4
+�
+xj −
+n
+�
+r=1
+m
+�
+i=1
+trxiU (r)
+i,j
+�2
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+��
+1 + |x|2
+4
+�2
+|x|2 + |t|2|x|2−
+2
+�
+1 + |x|2
+4
+�
+n
+�
+r=1
+tr
+
+
+m
+�
+i=1
+m
+�
+j=1
+U (r)
+i,j xixj
+
+
+
+
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+��
+1 + |x|2
+4
+�2
+|x|2 + |t|2|x|2
+�
+.
+To get the last equality, we use the fact
+m
+�
+i=1
+m
+�
+j=1
+U (r)
+i,j xixj = 0
+
+8
+QIAOHUA YANG
+since U (r)(1 ≤ r ≤ n) is a m × m skew symmetric matrix. Therefore, we have
+m
+�
+j=1
+����
+∂Uλ,η
+∂yj
+|λ=1,η=0
+����
+2
++
+n
+�
+r=1
+����
+∂Uλ,η
+∂wr
+|λ=1,η=0
+����
+2
++ 1
+4
+����
+∂Uλ,η
+∂λ |λ=1,η=0
+����
+2
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+��
+1 + |x|2
+4
+�2
+|x|2 + |t|2|x|2
+�
++ (Q − 2)2
+4
+U(ξ)2 Q+2
+Q−2 |t|2+
+(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+�
+−1 + |x|4
+16 + |t|2
+�2
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+��
+1 + |x|2
+4
+�2
+|x|2 + |t|2|x|2 + 4|t|2 +
+�
+−1 + |x|4
+16 + |t|2
+�2�
+=(Q − 2)2
+16
+U(ξ)2 Q+2
+Q−2
+��
+1 + |x|2
+4
+�2
++ |t|2
+�2
+=(Q − 2)2
+16
+U(ξ)2.
+To get the third equality, we use the fact
+�
+−1 + |x|4
+16 + |t|2
+�2
+=
+��
+1 + |x|2
+4
+�2
++ |t|2 − 2
+�
+1 + |x|2
+4
+��2
+=
+��
+1 + |x|2
+4
+�2
++ |t|2
+�2
++ 4
+�
+1 + |x|2
+4
+�2
+− 4
+�
+1 + |x|2
+4
+� ��
+1 + |x|2
+4
+�2
++ |t|2
+�
+=
+��
+1 + |x|2
+4
+�2
++ |t|2
+�2
+−
+�
+1 + |x|2
+4
+�2
+|x|2 − |t|2|x|2 − 4|t|2.
+This completes the proof of Lemma 2.3.
+□
+For simplicity, we set
+ωj =
+4
+Q − 2U(ξ)−1 ∂Uλ,η
+∂yj
+|λ=1,η=0, j = 1, · · · , m;
+ωj+r =
+4
+Q − 2U(ξ)−1 ∂Uλ,η
+∂wr
+|λ=1,η=0, r = 1, · · · , n;
+ωm+n+1 =
+2
+Q − 2U(ξ)−1 ∂Uλ,η
+∂λ |λ=1,η=0.
+(2.5)
+By Lemma 2.3 and (2.4), we have
+m+n+1
+�
+j=1
+ω2
+j =1;
+(2.6)
+∆G(U(ξ)ωj) + m(Q + 2)U(ξ)
+Q+2
+Q−2 ωj =0, 1 ≤ j ≤ m + n + 1.
+(2.7)
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+9
+3. Proof of Theorem 1.2 and 1.3
+In this section, we shall prove Theorem 1.2 and 1.3. The proof depends on a
+scheme of subcritical approximation due to Hang and Wang [15]. We first establish
+the following subcritical Sobolev inequality on G.
+Lemma 3.1. Let 2 ≤ p <
+2Q
+Q−2. There exists C > 0 such that for each u ∈ W 1,2
+0
+(G),
+�
+G
+|∇Gu|2dxdt ≥ C
+��
+G
+|u|pU(x, t)
+2Q
+Q−2 −pdxdt
+� 2
+p
+.
+Proof. By H¨older’s inequality, we have
+�
+G
+|u|pU(x, t)
+2Q
+Q−2 −pdxdt
+=
+�
+G
+�
+|u|U(x, t)
+2
+Q−2
+�Q− Q−2
+2
+p
+|u|
+Q
+2 (p−2)dxdt
+≤
+��
+G
+|u|2U(x, t)
+4
+Q−2 dxdt
+� 2Q−(Q−2)p
+4
+��
+G
+|u|
+2Q
+Q−2 dxdt
+� (Q−2)(p−2)
+4
+=
+��
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt
+� 2Q−(Q−2)p
+4
+��
+G
+|u|
+2Q
+Q−2 dxdt
+� (Q−2)(p−2)
+4
+≤C
+�
+G
+|∇Gu|2dxdt,
+where C is a positive constant independent of u. To get the last inequality above,
+we use Folland-Stein inequality (1.1) and Lemma 2.2. This completes the proof of
+Lemma 3.1.
+□
+By Lemma 3.1, we have
+W 1,2
+0
+(G) ֒→ Lp(G, U(x, t)
+2Q
+Q−2 −pdxdt).
+Furthermore, the embedding map is compact.
+Lemma 3.2. Let 2 ≤ p <
+2Q
+Q−2. The embedding map W 1,2
+0
+(G) ֒→ Lp(G, U(x, t)
+2Q
+Q−2 −pdxdt)
+is compact.
+Proof. The proof is similar to that given by Schneider (see [23], section 2.2).
+Let φ : G → [0, 1] be a cut-off function that is equal to one in B1(0) and zero
+outside of B2(0). Consider the operator
+IR : W 1,2
+0
+(G) ֒→ Lp(G, U(x, t)
+2Q
+Q−2 −pdxdt)
+
+10
+QIAOHUA YANG
+defined by IR(u) = u(x, t)φ
+� x
+R,
+t
+R2
+�
+. Since the imbedding map W 1,2
+0
+(B2(0)) ֒→
+Lp(B2(0)) is compact, so is IR. Moreover, by H¨older’s inequality,
+�
+G
+|u − IR(u)|pU(x, t)
+2Q
+Q−2 −pdxdt
+≤
+�
+G\BR(0)
+|u|pU(x, t)
+2Q
+Q−2 −pdxdt
+≤
+��
+G\BR(0)
+|u|
+2Q
+Q−2 dxdt
+� Q−2
+2Q p ��
+G\BR(0)
+U(x, t)
+2Q
+Q−2 dxdt
+�1− Q−2
+2Q p
+≤C
+��
+G
+|∇Gu|2dxdt
+� p
+2 ��
+G\BR(0)
+U(x, t)
+2Q
+Q−2 dxdt
+�1− Q−2
+2Q p
+.
+To get the last inequality above, we use the Folland-Stein inequality (1.1). By polar
+coordinates, we have
+�
+G\BR(0)
+U(x, t)
+2Q
+Q−2 dxdt =
+�
+G\BR(0)
+��
+1 + |x|2
+4
+�2
++ |t|2
+�− Q
+2
+dxdt
+≤
+�
+G\BR(0)
+1
+( |x|2
+4 + |t|2)
+Q
+2 dxdt
+=
+� ∞
+R
+�
+Σ
+1
+ρ2Q ρQ−1dρdσ
+=|Σ|
+1
+QRQ → 0, R → ∞,
+where |Σ| is the volume of Σ. Therefore, the embedding map
+W 1,2
+0
+(G) ֒→ Lp(G, U(x, t)
+2Q
+Q−2 −pdxdt)
+is a limit of compact operators and thus it is compact. The proof of Lemma 3.2 is
+thereby completed.
+□
+By Lemma 3.2, the minimization problem
+Λp = inf
+��
+G
+|∇Gu|2dxdt :
+�
+G
+|u|pU(x, t)
+2Q
+Q−2 −pdxdt = 1
+�
+, 2 ≤ p <
+2Q
+Q − 2,
+(3.1)
+has a positive solution u.
+We shall show that such u satisfies a moment zero
+condition. The main result is the following lemma:
+Lemma 3.3. Let 2 ≤ p <
+2Q
+Q−2 and u be a positive solution of (3.1). Then we have
+�
+G
+upU(x, t)
+2Q
+Q−2 −pωidxdt = 0, i = 1, 2, · · · , m + n + 1,
+(3.2)
+where ωi(1 ≤ i ≤ m + n + 1) is given by (2.5).
+Proof. For simplicity, we set
+Fp(u) =
+�
+G |∇Gu|2dxdt
+��
+G |u|pU(x, t)
+2Q
+Q−2 −pdxdt
+� 2
+p
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+11
+and
+uλ−1,η−1(ξ) =λ− Q−2
+2 u(δλ−1(η ◦ ξ)), λ > 0, η = (y1, · · · , ym, w1, · · · , wn) ∈ G.
+A simple calculation shows
+�
+G
+|∇Guλ−1,η−1|2dxdt =
+�
+G
+|∇Gu|2dxdt;
+�
+G
+up
+λ−1,η−1U(x, t)
+2Q
+Q−2 −pdxdt =
+�
+G
+upUλ,η(x, t)
+2Q
+Q−2 −pdxdt,
+where Uλ,η is given by (1.5). Therefore,
+Fp(uλ−1,η−1(ξ)) =
+�
+G |∇Gu|2dxdt
+��
+G |u|pUλ,η(x, t)
+2Q
+Q−2 −pdxdt
+� 2
+p .
+(3.3)
+Since u is a positive solution of (3.1), we have
+∂
+∂yj
+Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, j = 1, · · · , m;
+∂
+∂wr
+Fp(uλ−1,η−1(ξ))|λ=1,η=0 =0, r = 1, · · · , n;
+���
+∂λFp(uλ−1,η−1(ξ))|λ=1,η=0 =0.
+(3.4)
+Combining (3.3) and (3.4) yields (3.2). This completes the proof of Lemma 3.3.
+□
+Remark 3.4. We remark that Lemma 3.3 is also valid for u > 0 satisfying the
+Yamabe-type equation
+∆Gu + ΛpupU(x, t)
+2Q
+Q−2 −p = 0.
+The proof is same and we omit it (see [15], Corollary 1 for the case of CR sphere).
+Lemma 3.5. It holds that, for any u ∈ W 1,2
+0
+(G),
+m+n+1
+�
+i=1
+�
+G
+|∇G(uωi)|2dxdt =
+�
+G
+|∇Gu|2dxdt + 4m
+�
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt.
+
+12
+QIAOHUA YANG
+Proof. Let u = U(ξ)v. We compute, through integration by parts,
+m+n+1
+�
+i=1
+�
+G
+|∇G(uωi)|2dxdt =
+m+n+1
+�
+i=1
+�
+G
+|∇G(vUωi)|2dxdt
+=
+m+n+1
+�
+i=1
+�
+G
+|Uωi∇Gv + v∇G(Uωi)|2dxdt
+=
+m+n+1
+�
+i=1
+��
+G
+|∇Gv|2U 2ω2
+i dxdt +
+�
+G
+|∇G(Uωi)|2v2dxdt+
+1
+2
+�
+G
+⟨∇G(Uωi)2, ∇Gv2⟩dxdt
+�
+=
+m+n+1
+�
+i=1
+��
+G
+|∇Gv|2U 2ω2
+i dxdt −
+�
+G
+v2Uωi∆G(Uωi)dxdt
+�
+=
+�
+G
+|∇Gv|2U 2dxdt + m(Q + 2)
+�
+G
+v2U
+2Q
+Q−2 dxdt.
+(3.5)
+To get the last equality, we use (2.7). On the other hand, by (2.3), we have
+�
+G
+|∇Gv|2U 2dxdt =
+�
+G
+���∇G
+u
+U
+���
+2
+U 2dxdt
+=
+�
+G
+|∇Gu|2dxdt − m(Q − 2)
+�
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt.
+(3.6)
+Substituting (3.6) into (3.5), we obtain
+m+n+1
+�
+i=1
+�
+G
+|∇G(uωi)|2dxdt =
+�
+G
+|∇Gu|2dxdt + 4m
+�
+G
+u2
+(1 + |x|2
+4 )2 + |t|2 dxdt.
+This completes the proof of Lemma 3.5.
+□
+Now we can give the proof of Theorem 1.2. The idea is due to Frank and Lieb
+[11, 12] and Hang and Wang [15].
+Proof of Theorem 1.2. Let 2 ≤ p <
+2Q
+Q−2 and up be a positive solution of
+(3.1). The 2nd variation of the functional Fp around up shows that
+�
+G
+|∇Gf|2dxdt
+�
+G
+up
+pU(x, t)
+2Q
+Q−2 −pdxdt−
+(p − 1)
+�
+G
+|∇Gup|2dxdt
+�
+G
+up−2
+p
+Uλ,η(x, t)
+2Q
+Q−2 −pf 2dxdt ≥ 0
+for any f with
+�
+G
+up
+pUλ,η(x, t)
+2Q
+Q−2 −pfdxdt = 0.
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+13
+By Lemma 3.3, we may choose f = upωi, i = 1, 2, · · · , m + n + 1. Summing the
+corresponding inequalities for all such f’s yields, in view of (2.6) and Lemma 3.5,
+0 ≤
+m+n+1
+�
+i=1
+�
+G
+|∇G(upωi)|2dxdt − (p − 1)
+�
+G
+|∇Gup|2dxdt
+=4m
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt − (p − 2)
+�
+G
+|∇Gup|2dxdt,
+i.e.
+(p − 2)
+��
+G
+|∇Gup|2dxdt − m(Q − 2)
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+�
+≤m(Q − 2)
+� 2Q
+Q − 2 − p
+� �
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+≤m(Q − 2)
+� 2Q
+Q − 2 − p
+� ��
+G
+up
+pU(x, t)
+2Q
+Q−2 −pdxdt
+� 2
+p ��
+G
+U(x, t)
+2Q
+Q−2 dxdt
+�1− 2
+p
+=m(Q − 2)
+� 2Q
+Q − 2 − p
+� ��
+G
+U(x, t)
+2Q
+Q−2 dxdt
+�1− 2
+p
+→ 0, p ր
+2Q
+Q − 2.
+To get the last equality, we use the fact
+�
+G up
+pU(x, t)
+2Q
+Q−2 −pdxdt = 1. Therefore, by
+Lemma 2.2, we obtain
+�
+G
+|∇Gup|2dxdt − m(Q − 2)
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt → 0, p ր
+2Q
+Q − 2,
+or equivalently,
+�
+G
+|∇G(U −1up)|2U 2dxdt → 0, p ր
+2Q
+Q − 2.
+So we can choose a sequence {pk : k = 1, 2, · · ·} such that pk ր
+2Q
+Q−2 and upk
+converges to a nonzero function c0U (for reader’s convenience, we prove it in Lemma
+3.6). Thus c0U is an extremal function of
+Λ = inf
+��
+G
+|∇Gu|2dxdt :
+�
+G
+|u|
+2Q
+Q−2 dxdt = 1
+�
+.
+The value Sm,n has been calculated in [13], Theorem 1.6. The proof of Theorem
+1.2 is thereby completed.
+Lemma 3.6. Let up(2 ≤ p <
+2Q
+Q−2) be a positive solution of (3.1). If
+�
+G
+|∇Gup|2dxdt − m(Q − 2)
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt → 0, p ր
+2Q
+Q − 2,
+then there exists c0 > 0 and a sequence {pk : k = 1, 2, · · ·} such that pk ր
+2Q
+Q−2 and
+�
+G
+|∇G(upk − c0U)|2dxdt → 0, k → ∞.
+
+14
+QIAOHUA YANG
+Proof. By Lemma 2.2, µ1 = m(Q − 2) is simple with eigenfunction U of (1.8) with
+λ = 1 and η = 0. Decompose up as
+up = λpU + vp
+(3.7)
+with
+λp =
+�
+G U
+Q+2
+Q−2 updxdt
+�
+G U
+2Q
+Q−2 dxdt
+> 0.
+Then vp ⊥ U, i.e.
+�
+G
+U
+4
+Q−2 · Uvpdx =
+�
+G
+U
+Q+2
+Q−2 vpdxdt = 0,
+�
+G
+⟨∇GU, ∇Gvp⟩dx = 0.
+(3.8)
+Therefore, we have
+�
+G
+|∇Gvp|2dxdt ≥ µ2
+�
+G
+U
+4
+Q−2 · v2
+pdx = µ2
+�
+G
+v2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt,
+(3.9)
+where µ2 is the second eigenvalue of (1.8) with with λ = 1 and η = 0. We compute,
+by using (3.8) and (3.9),
+�
+G
+|∇Gup|2dxdt − m(Q − 2)
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+=
+�
+G
+�
+λ2
+p|∇GU|2 + |∇Gvp|2�
+dxdt − µ1
+�
+G
+λ2
+pU 2 + v2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+=
+�
+G
+|∇Gvp|2dxdt − µ1
+�
+G
+v2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+=µ1
+µ2
+��
+G
+|∇Gvp|2dxdt − µ2
+�
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+�
++
+µ2 − µ1
+µ2
+�
+G
+|∇Gvp|2dxdt
+≥µ2 − µ1
+µ2
+�
+G
+|∇Gvp|2dxdt.
+Therefore,
+�
+G
+|∇Gvp|2dxdt
+≤
+µ2
+µ2 − µ1
+��
+G
+|∇Gup|2dxdt − m(Q − 2)
+��
+G
+u2
+p
+(1 + |x|2
+4 )2 + |t|2 dxdt
+�
+→ 0, p ր
+2Q
+Q − 2.
+(3.10)
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+15
+On the other hand, by (3.7), Minkowski’s inequalities, H¨older’s inequality and
+(1.1), we have
+λp
+��
+G
+U(x, t)
+2Q
+Q−2 dxdt
+� 1
+p
+=
+��
+G
+(up − vp)pU(x, t)
+2Q
+Q−2 −pdxdt
+� 1
+p
+≤
+��
+G
+up
+pU(x, t)
+2Q
+Q−2 −pdxdt
+� 1
+p
++
+��
+G
+|vp|pU(x, t)
+2Q
+Q−2 −pdxdt
+� 1
+p
+=1 +
+��
+G
+|vp|pU(x, t)
+2Q
+Q−2 −pdxdt
+� 1
+p
+≤1 +
+��
+G
+|vp|
+2Q
+Q−2 dxdt
+� Q−2
+2Q ��
+G
+U
+2Q
+Q−2 dxdt
+� 1
+p − Q−2
+2Q
+≤1 + C
+��
+G
+|∇Gvp|2dxdt
+� 1
+2 ��
+G
+U
+2Q
+Q−2 dxdt
+� 1
+p − Q−2
+2Q
+.
+(3.11)
+Substituting (3.10) into (3.11), we obtain
+lim sup
+pր 2Q
+Q−2
+λp ≤
+��
+G
+U(x, t)
+2Q
+Q−2 dxdt
+�− Q−2
+2Q
+.
+Therefore, there exists c0 ≥ 0 and a sequence {pk : k = 1, 2, · · · } such that pk ր
+2Q
+Q−2 and
+λpk → c0, k → ∞.
+(3.12)
+We claim that
+�
+G
+|∇G(upk − c0U)|2dxdt → 0, k → ∞.
+(3.13)
+In fact, by using (3.10) and (3.12), we obtain
+�
+G
+|∇G(upk − c0U)|2dxdt
+=
+�
+G
+|∇G(vpk + (λpk − c0)U)|2dxdt
+=
+�
+G
+|∇Gvpk|2dxdt + (λpk − c0)2
+�
+G
+|∇GU|2dxdt → 0, k → ∞.
+This proves the claim.
+Finally, we show that c0 > 0. In fact, if c0 = 0, then by (3.13),
+�
+G
+|∇Gupk|2dxdt → 0, k → ∞.
+
+16
+QIAOHUA YANG
+On the other hand, by H¨older’s inequality and (1.1), we obtain
+1 =
+��
+G
+upk
+pkU(x, t)
+2Q
+Q−2 −pkdxdt
+� 1
+pk
+≤
+��
+G
+|upk|
+2Q
+Q−2 dxdt
+� Q−2
+2Q ��
+G
+U
+2Q
+Q−2 dxdt
+� 1
+pk − Q−2
+2Q
+≤C
+��
+G
+|∇Gupk|2dxdt
+� 1
+2 ��
+G
+U
+2Q
+Q−2 dxdt
+� 1
+pk − Q−2
+2Q
+→ 0, k → ∞,
+which is a contradiction. So c0 > 0. The proof of Lemma 3.6 is thereby completed.
+□
+Finally, we give the proof of Theorem 1.3.
+Proof of Theorem 1.3 A simple scaling argument shows that the eigenvalues
+do not depend on λ and η. So we may assume λ = 1 and η = 0.
+From Lemma 2.2 we know that µ1 = m(Q − 2) is simple with eigenfunction U.
+Nextly, we show µ2 ≥ m(Q + 2). Let V ̸= 0 be a eigenfunction of µ2. Then
+(3.14)
+µ2 =
+�
+G |∇GV |2dxdt
+�
+G U
+4
+Q−2 V 2dxdt
+.
+Furthermore, since V ⊥ U, we have
+(3.15)
+�
+G
+⟨∇GU, ∇GV ⟩dxdt = 0,
+�
+G
+U
+4
+Q−2 · UV dxdt =
+�
+G
+U
+Q+2
+Q−2 V dxdt = 0.
+Set
+Φ(ǫ) =
+�
+G |∇G(U + ǫV )|2dxdt
+��
+G |U + ǫV |
+2Q
+Q−2 dxdt
+� Q−2
+Q , ǫ ∈ R.
+By Theorem 1.2, U is an extremal function of Folland-Stein inequality (1.7). So we
+have Φ′(0) = 0 and Φ′′(0) ≥ 0. We compute
+Φ′(ǫ) =2
+�
+G⟨∇G(U + ǫV ), ∇GV ⟩dxdt
+��
+G |U + ǫV |
+2Q
+Q−2 dxdt
+� Q−2
+Q
+−
+2
+�
+G |∇G(U + ǫV )|2dxdt
+��
+G |U + ǫV |
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+|U + ǫV |
+4
+Q−2 (U + ǫV )V dxdt
+=Φ1(ǫ) − Φ2(ǫ),
+where
+Φ1(ǫ) =2
+�
+G⟨∇G(U + ǫV ), ∇GV ⟩dxdt
+��
+G |U + ǫV |
+2Q
+Q−2 dxdt
+� Q−2
+Q
+;
+Φ2(ǫ) =2
+�
+G |∇G(U + ǫV )|2dxdt
+��
+G |U + ǫV |
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+|U + ǫV |
+4
+Q−2 (U + ǫV )V dxdt.
+
+THE OPTIMAL CONSTANT IN THE L2 FOLLAND-STEIN INEQUALITY
+17
+By using (3.15), we have
+Φ′
+1(0) =2
+�
+G |∇GV |2dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� Q−2
+Q
+− 4
+�
+G⟨∇GU, ∇GV ⟩dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+U
+Q+2
+Q−2 V dxdt
+=2
+�
+G |∇GV |2dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� Q−2
+Q ;
+Φ′
+2(0) =4
+�
+G⟨∇GU, ∇GV ⟩dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+U
+Q+2
+Q−2 V dxdt
+− 8(Q − 1)
+Q − 2
+�
+G |∇GU|2dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� 3Q−2
+Q
+��
+G
+U
+4
+Q−2 V 2dxdt
+�2
++ 2(Q + 2)
+Q − 2
+�
+G |∇GU|2dxdt
+��
+G U
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+U
+4
+Q−2 V 2dxdt
+=2(Q + 2)
+Q − 2
+�
+G |∇GU|2dxdt
+��
+G U
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+U
+4
+Q−2 V 2dxdt.
+Therefore,
+0 ≤ Φ′′(0) =Φ′
+1(0) − Φ′
+2(0)
+=2
+�
+G |∇GV |2dxdt
+��
+G |U|
+2Q
+Q−2 dxdt
+� Q−2
+Q
+− 2(Q + 2)
+Q − 2
+�
+G |∇GU|2dxdt
+��
+G U
+2Q
+Q−2 dxdt
+� 2Q−2
+Q
+�
+G
+U
+4
+Q−2 V 2dxdt,
+i.e.
+�
+G |∇GV |2dxdt
+�
+G |U|
+4
+Q−2 V 2dxdt
+≥ Q + 2
+Q − 2
+�
+G |∇GU|2dxdt
+�
+G |U|
+2Q
+Q−2 dxdt
+.
+(3.16)
+Combing (3.14) and (3.16) yields
+µ2 ≥ Q + 2
+Q − 2
+�
+G |∇GU|2dxdt
+�
+G |U|
+2Q
+Q−2 dxdt
+= Q + 2
+Q − 2µ1 = m(Q + 2).
+On the other hand, by (2.4), {∂λUλ,η|λ=1,η=0, ∇ηUλ,η|λ=1,η=0} are eigenfunctions
+of m(Q + 2). So µ2 = m(Q + 2). This completes the proof of Theorem 1.3.
+References
+[1] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.
+[2] A. Bonfiglioli, F. Uguzzoni, Nonlinear Liouville theorems for some critical problems on H-type
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+[3] H. Brezis, E. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal., 62(1985),
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+Heisenberg group, Calc. Var. 55, Article number: 11 (2016).
+
+18
+QIAOHUA YANG
+[6] M. Cowling, A. H. Dooley, A. Kor´anyi, F. Ricci, H-type groups and Iwasawa decompositions,
+Adv. Math., 87 (1991), 1-41.
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+School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s
+Republic of China
+Email address: qhyang.math@whu.edu.cn
+

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+Measuring Power with a Saturated Photodiode
+Shiekh Zia Uddin1,*
+1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
+*suddin@mit.edu
+ABSTRACT
+Accurate measurement of optical power is pivotal in many applications and scientific research. However, traditional power
+meters are unable to measure power levels beyond a certain saturation point, limiting their usefulness in high-power applications.
+In this technical note, I discuss how optical power can be measured using a saturated photodiode. I demonstrate that by
+monitoring both the dc photocurrent and ac noise, it is possible to accurately measure power levels beyond its saturation point.
+Keywords: Power meter, Photodiode, Saturation, Noise.
+Introduction
+Optical power measurement is a critical aspect of many applications. It is the conventional wisdom that a saturated photodiode
+can not be used to measure power. Saturation power of photodiodes can be pushed to higher levels by applying a reverse bias
+voltage, however there is a limit to the amount of bias voltage due to the reverse breakdown which can be catastrophic to the
+diode. This limitation can be problematic in high-power applications, where it is important to be able to accurately measure
+power levels at high speed. In this technical note, I discuss a method for measuring optical power using a saturated photodiode.
+I demonstrate that the photocurrent noise decreases with power beyond saturation which can be used to accurately measure
+power at levels beyond its saturation point. This information might be useful in photon noise measurements.
+Background
+If a photoevent generated at t = 0 produces an electric pulse h(t), of area e, in the external circuit. A photoevent generated
+at time t1 then produces a displaced pulse, h(t −t1). Dividing the time axis into incremental time intervals ∆t so that the
+probability p that a photoevent occurs within an interval is P = ηΦ∆t. The electric current i at time t is written as
+i(t) = ∑
+l
+Xlh(t −l∆t),
+(1)
+where Xl assumes the value 1 with probability p, and 0 with probability 1− p. The variables Xl are independent. The mean
+value of Xl is E[Xl] = 0×(1− p)+1× p = p. Its mean-square value is E[X2
+l ] = 02 ×(1− p)+12 × p = p. The mean of the
+product XlXk is p2 if l ̸= k, and p if l = k. The mean and mean-square values of i(t) are now determined via
+i = E[i(t)] = ∑
+l
+ph(t −l∆t),
+(2)
+E[i2(t)] = ∑
+l ∑
+k
+E[XlXk]h(t −l∆t)h(t −k∆t)
+(3)
+= ∑∑
+l̸=k
+p2h(t −l∆t)h(t −k∆t)+∑
+l
+ph2(t −l∆t)
+(4)
+Substituting p = ηΦ∆t, and taking the limit ∆t → 0 so that the summations become integrals, previous equations yield,
+respectively,
+E[i(t)] = ηΦ
+�
+h(t)dt,
+(5)
+E[i2(t)] =
+�
+ηΦ
+�
+h(t)dt
+�2
++ηΦ
+�
+h2(t)dt
+(6)
+The limits of the integration is zero to infinity. It follows that
+σ2
+i = E[i2]−E[i]2 = ηΦ
+�
+h2(t)dt
+(7)
+arXiv:2301.02658v1  [physics.ins-det]  7 Jan 2023
+
+Definition of the bandwidth B as
+B = 1
+2e2
+� ∞
+0 h2(t)dt =
+� ∞
+0 h2(t)dt
+2(
+� ∞
+0 h(t)dt)2 ,
+(8)
+can be readily verified by noting that the Fourier transform of h(t) is its transfer function H(v). The area under h(t) is simply
+H(0) = e. In accordance with Parseval’s theorem, the area under h2(t) is equal to the area under the symmetric function |H(v)|2,
+so that
+B =
+� ∞
+0
+����
+H(v)
+H(0)
+����
+2
+dv
+(9)
+The quantity B is therefore the power-equivalent spectral width of the function H(v) (i.e., the bandwidth of the device/circuit
+combination). As an example, if H(v) = 1 for −Vc < v < Vc and 0 elsewhere, we get B = Vc. Using this definition of bandwidth,
+we get back our familiar expression for the noise in photocurrent
+σ2
+i = 2eE[i]B.
+(10)
+So far this is a standard derivation of the shot noise1. Note that this expression hinges on the assumption that the area under h(t)
+is simply e, basically one photoevent cause one electrons worth of charge to flow as current. However in saturation its definitely
+not the case and the photodiode response becomes a function of intensity. In the first order approximation, one can make the
+assumption that h(t) = f(Φ)g(t), where f(Φ) is a power dependent function that is 1 at low power and decreases at high power
+and area under g(t) is e. Then we get
+E[i] = ηΦ f(Φ)
+�
+g(t)dt
+(11)
+σ2
+i = ηΦ f 2(Φ)
+�
+g2(t)dt,
+(12)
+which gives us the key insight that the average value of current and noise power scales differently with incident optical power.
+If we take a ratio
+E[i]2
+σ2
+i
+= η2Φ2 f 2(Φ)e2
+ηΦ f 2(Φ)2e2B ∝ Φ
+(13)
+we find that the signal to noise ratio (SNR) in principle is proportional to incident power despite the nonlinearity in the response.
+Even if this proportionality does not hold exactly, with proper calibration therefore it should be possible to measure the power
+with a saturated photodiode by measuring both the average photocurrent and the photocurrent noise.
+Results
+Figure 1A shows a schematic diagram of a standard photodiode driving circuit. The photodiode can be modelled as a current
+source in parallel with a diode2. A reverse bias voltage is applied to set the operating point, but there is a limit to how much
+voltage can be applied defined by junction breakdown3. Such a circuit can be simulated using conventional electrical circuit
+theory4 (MATLAB code below), the resulting operating current with realistic circuit parameters is shown in Fig. 1B. We can see
+that the photodiode saturates at some power and the saturation knee increases with reverse bias voltage. The highest measurable
+power is determined by the highest reverse bias voltage that can be applied across a junction. Below saturation the current is
+linear with incident power, which is expected.
+Experimental data of a reverse biased photodiode is shown in Fig. 2 where reverse bias is seen to increase the saturation
+power. Before saturation the voltage is proportional to power. After saturation no measurement of power is possible, which is
+the current paradigm.
+We can now attempt to measure power beyond saturation. In Fig. 3A we show the photovoltage and the photocurrent noise
+as a function of incident power. Photocurrent noise is measured around 1.5 MHz with a spectrum analyzer (10 kHz resolution
+bandwidth, 1 MHz span, with preamplifier on and no attenuation, electronic noise floor is −165 dBm/Hz). Below saturation
+photovoltage and noise increases simultaneously. As the photovoltage is saturated, the photocurrent noise suddenly decreases.
+This qualitatively follows our theoretical voltage and noise shown in Fig. 1. In Fig. 3B we show the photovoltage and SNR
+calculated from the experimental data. Beyond saturation SNR changes with incident power, which can be used to measure
+power after proper calibration. Such behaviour also holds at other noise frequencies as long as they are lower than the badngap
+of the photodiode and away from 1/ f noise.
+.
+2/4
+
+Figure 1. Schematic diagram of a photodiode driving circuit and simulated operating current I0.
+Figure 2. Output voltage across the 25Ω load resistance from a Thorlabs FDGA055 InGaAs photodiode at different reverse
+bias excited by a 1070 nm CW laser. It has a 0.95 A/W responsivity, 2.5 ns rise time and 0.5 mm active area diameter.
+Figure 3. (A) Output voltage and noise from a Excelitas C30641GH6 InGaAs photodiode at 30 V reverse bias excited by a
+1550 nm femtosecond pulsed laser. It saturates around 25 mW. (B) Even though photovoltage has saturated, the SNR shows
+response beyond the saturation power.
+3/4
+
+A
+B
+100
+C
+-140
+Current
+Photon
+Vr (V)
+ (mA)
+80
+Noise (dBm/Hz)
+0
+g Current (
+10
+60
+VR (V)
+Load
+20
+-160
+0
+R
+30
+40
+10
+Photocurrent
+20
+Ip
+Operating
+30
+20
+Bias
+0
+VR
+-180
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+100
+Incident Power (mW)
+Incident Power (mW)1000
+Voltage (mV)
+100
+Vr (V)
+0
+5
+10
+15
+18
+20
+25
+30
+SL
+10
+1
+10
+100
+Incident Power (mWPhotovoltage (mV) 
+B
+30
+1.0
+-135
+Photovoltage (mV)
+25
+10
+Noise (dBm/Hz)
+0.8
+20
+(norm.)
+-145
+50
+0.6
+I
+0.4
+SNR
+-155
+工
+5
+0.2
+-165
+0.1
+0.0
+0
+10
+20
+30
+40
+50
+0
+10
+20
+30
+40
+50
+Power (mW)
+Power (mw)Discussion
+Even when a photodiode is saturated, the information about the photon flux intensity are not completely lost and in a way
+encoded in the photocurrent noise, which can be practically used to measure power at high speed after proper calibration.
+Codes
+1
+%% Matlab code to solve for photocurrent and noise in a circuit
+2
+clc;clear all;close all;
+3
+P=linspace(0,100,1000)*1e-3; % Incident power in W
+4
+Responsivity=1;
+5
+Ip=P*Responsivity; % Expected photocurrent
+6
+R=500;
+7
+V=[linspace(-50,0,1e5),linspace(0,.7,1e5)];
+8
+VR=30; % Reverse Bias Voltage
+9
+Iop=zeros(size(P)); % Operating Photocurrent
+10
+11
+for indx=1:length(P)
+12
+I1=-Ip(indx)+0.1e-9*exp(V/.0259);
+13
+I2=-(V+VR)/R;
+14
+[¬,pos]=min(abs(I1-I2));
+15
+Iop(indx)=-I2(pos);
+16
+end
+17
+18
+figure(1);subplot(121), plot(P/1e-3,Iop); % Operating Current vs Power
+19
+f=Iop./P; %nonlinear response function
+20
+S=10*log10(2*1.6e-19*R*1*(P/1e-3).*(f.^2)); % Noise power vs optical power
+21
+subplot(122), plot(P/1e-3,S);
+References
+1. Saleh, B. E. & Teich, M. C. Fundamentals of Photonics (John Wiley & Sons, 2019).
+2. Bhattacharya, P. Semiconductor Optoelectronic Devices (Prentice-Hall, Inc., 1997).
+3. Neamen, D. A. Semiconductor Physics and Devices: Basic Principles (McGraw-hill, 2003).
+4. Sedra, A. S., Smith, K. C., Carusone, T. C. & Gaudet, V. Microelectronic Circuits, vol. 4 (Oxford University Press New
+York, 2004).
+5. Thorlabs FDGA05. https://www.thorlabs.com/thorproduct.cfm?partnumber=FDGA05 (2022). Accessed: 2022-12-10.
+6. Excelitas C30641GH. https://www.excelitas.com/product/c30641gh-ingaas-pin-1mm-18 (2022). Accessed: 2022-12-10.
+Acknowledgements
+The author acknowledges Nicholas Rivera, Jamison Sloan, Yannick Salamin, chatGPT for their discussions. All equipment
+used in the experiments are properties of MIT.
+4/4
+

C9E0T4oBgHgl3EQfyQLe/content/tmp_files/load_file.txt

@@ -0,0 +1,153 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf,len=152
+page_content='Measuring Power with a Saturated Photodiode  Shiekh Zia Uddin1,*  1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  suddin@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='edu  ABSTRACT  Accurate measurement of optical power is pivotal in many applications and scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' However, traditional power  meters are unable to measure power levels beyond a certain saturation point, limiting their usefulness in high-power applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  In this technical note, I discuss how optical power can be measured using a saturated photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' I demonstrate that by  monitoring both the dc photocurrent and ac noise, it is possible to accurately measure power levels beyond its saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Keywords: Power meter, Photodiode, Saturation, Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Introduction  Optical power measurement is a critical aspect of many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' It is the conventional wisdom that a saturated photodiode  can not be used to measure power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Saturation power of photodiodes can be pushed to higher levels by applying a reverse bias  voltage, however there is a limit to the amount of bias voltage due to the reverse breakdown which can be catastrophic to the  diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' This limitation can be problematic in high-power applications, where it is important to be able to accurately measure  power levels at high speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' In this technical note, I discuss a method for measuring optical power using a saturated photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  I demonstrate that the photocurrent noise decreases with power beyond saturation which can be used to accurately measure  power at levels beyond its saturation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' This information might be useful in photon noise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Background  If a photoevent generated at t = 0 produces an electric pulse h(t), of area e, in the external circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' A photoevent generated  at time t1 then produces a displaced pulse, h(t −t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Dividing the time axis into incremental time intervals ∆t so that the  probability p that a photoevent occurs within an interval is P = ηΦ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The electric current i at time t is written as  i(t) = ∑  l  Xlh(t −l∆t),  (1)  where Xl assumes the value 1 with probability p, and 0 with probability 1− p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The variables Xl are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The mean  value of Xl is E[Xl] = 0×(1− p)+1× p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Its mean-square value is E[X2  l ] = 02 ×(1− p)+12 × p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The mean of the  product XlXk is p2 if l ̸= k, and p if l = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The mean and mean-square values of i(t) are now determined via  i = E[i(t)] = ∑  l  ph(t −l∆t),  (2)  E[i2(t)] = ∑  l ∑  k  E[XlXk]h(t −l∆t)h(t −k∆t)  (3)  = ∑∑  l̸=k  p2h(t −l∆t)h(t −k∆t)+∑  l  ph2(t −l∆t)  (4)  Substituting p = ηΦ∆t, and taking the limit ∆t → 0 so that the summations become integrals, previous equations yield,  respectively,  E[i(t)] = ηΦ  �  h(t)dt,  (5)  E[i2(t)] =  �  ηΦ  �  h(t)dt  �2  +ηΦ  �  h2(t)dt  (6)  The limits of the integration is zero to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' It follows that  σ2  i = E[i2]−E[i]2 = ηΦ  �  h2(t)dt  (7)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='02658v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='ins-det]  7 Jan 2023  Definition of the bandwidth B as  B = 1  2e2  � ∞  0 h2(t)dt =  � ∞  0 h2(t)dt  2(  � ∞  0 h(t)dt)2 ,  (8)  can be readily verified by noting that the Fourier transform of h(t) is its transfer function H(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The area under h(t) is simply  H(0) = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' In accordance with Parseval’s theorem, the area under h2(t) is equal to the area under the symmetric function |H(v)|2,  so that  B =  � ∞  0  ����  H(v)  H(0)  ����  2  dv  (9)  The quantity B is therefore the power-equivalent spectral width of the function H(v) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=', the bandwidth of the device/circuit  combination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' As an example, if H(v) = 1 for −Vc < v < Vc and 0 elsewhere, we get B = Vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Using this definition of bandwidth,  we get back our familiar expression for the noise in photocurrent  σ2  i = 2eE[i]B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  (10)  So far this is a standard derivation of the shot noise1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Note that this expression hinges on the assumption that the area under h(t)  is simply e, basically one photoevent cause one electrons worth of charge to flow as current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' However in saturation its definitely  not the case and the photodiode response becomes a function of intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' In the first order approximation, one can make the  assumption that h(t) = f(Φ)g(t), where f(Φ) is a power dependent function that is 1 at low power and decreases at high power  and area under g(t) is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Then we get  E[i] = ηΦ f(Φ)  �  g(t)dt  (11)  σ2  i = ηΦ f 2(Φ)  �  g2(t)dt,  (12)  which gives us the key insight that the average value of current and noise power scales differently with incident optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  If we take a ratio  E[i]2  σ2  i  = η2Φ2 f 2(Φ)e2  ηΦ f 2(Φ)2e2B ∝ Φ  (13)  we find that the signal to noise ratio (SNR) in principle is proportional to incident power despite the nonlinearity in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Even if this proportionality does not hold exactly, with proper calibration therefore it should be possible to measure the power  with a saturated photodiode by measuring both the average photocurrent and the photocurrent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Results  Figure 1A shows a schematic diagram of a standard photodiode driving circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The photodiode can be modelled as a current  source in parallel with a diode2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' A reverse bias voltage is applied to set the operating point, but there is a limit to how much  voltage can be applied defined by junction breakdown3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Such a circuit can be simulated using conventional electrical circuit  theory4 (MATLAB code below), the resulting operating current with realistic circuit parameters is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' We can see  that the photodiode saturates at some power and the saturation knee increases with reverse bias voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' The highest measurable  power is determined by the highest reverse bias voltage that can be applied across a junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Below saturation the current is  linear with incident power, which is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Experimental data of a reverse biased photodiode is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 2 where reverse bias is seen to increase the saturation  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Before saturation the voltage is proportional to power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' After saturation no measurement of power is possible, which is  the current paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  We can now attempt to measure power beyond saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 3A we show the photovoltage and the photocurrent noise  as a function of incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Photocurrent noise is measured around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='5 MHz with a spectrum analyzer (10 kHz resolution  bandwidth, 1 MHz span, with preamplifier on and no attenuation, electronic noise floor is −165 dBm/Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Below saturation  photovoltage and noise increases simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' As the photovoltage is saturated, the photocurrent noise suddenly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  This qualitatively follows our theoretical voltage and noise shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 3B we show the photovoltage and SNR  calculated from the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Beyond saturation SNR changes with incident power, which can be used to measure  power after proper calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Such behaviour also holds at other noise frequencies as long as they are lower than the badngap  of the photodiode and away from 1/ f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  2/4  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Schematic diagram of a photodiode driving circuit and simulated operating current I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Output voltage across the 25Ω load resistance from a Thorlabs FDGA055 InGaAs photodiode at different reverse  bias excited by a 1070 nm CW laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' It has a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='95 A/W responsivity, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='5 ns rise time and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='5 mm active area diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' (A) Output voltage and noise from a Excelitas C30641GH6 InGaAs photodiode at 30 V reverse bias excited by a  1550 nm femtosecond pulsed laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' It saturates around 25 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' (B) Even though photovoltage has saturated, the SNR shows  response beyond the saturation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  3/4  A  B  100  C  140  Current  Photon  Vr (V)  (mA)  80  Noise (dBm/Hz)  0  g Current (  10  60  VR (V)  Load  20  160  0  R  30  40  10  Photocurrent  20  Ip  Operating  30  20  Bias  0  VR  180  0  20  40  60  80  100  0  20  40  60  80  100  Incident Power (mW)  Incident Power (mW)1000  Voltage (mV)  100  Vr (V)  0  5  10  15  18  20  25  30  SL  10  1  10  100  Incident Power (mWPhotovoltage (mV)  B  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='0  135  Photovoltage (mV)  25  10  Noise (dBm/Hz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='8  20  (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=')  145  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='6  I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='4  SNR  155  工  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='2  165  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='0  0  10  20  30  40  50  0  10  20  30  40  50  Power (mW)  Power (mw)Discussion  Even when a photodiode is saturated, the information about the photon flux intensity are not completely lost and in a way  encoded in the photocurrent noise, which can be practically used to measure power at high speed after proper calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Codes  1  %% Matlab code to solve for photocurrent and noise in a circuit  2  clc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='clear all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='close all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  3  P=linspace(0,100,1000)*1e-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Incident power in W  4  Responsivity=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  5  Ip=P*Responsivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Expected photocurrent  6  R=500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  7  V=[linspace(-50,0,1e5),linspace(0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='7,1e5)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  8  VR=30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Reverse Bias Voltage  9  Iop=zeros(size(P));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Operating Photocurrent  10  11  for indx=1:length(P)  12  I1=-Ip(indx)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='1e-9*exp(V/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='0259);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  13  I2=-(V+VR)/R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  14  [¬,pos]=min(abs(I1-I2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  15  Iop(indx)=-I2(pos);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  16  end  17  18  figure(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='subplot(121), plot(P/1e-3,Iop);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Operating Current vs Power  19  f=Iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='/P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' %nonlinear response function  20  S=10*log10(2*1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='6e-19*R*1*(P/1e-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' *(f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='^2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' % Noise power vs optical power  21  subplot(122), plot(P/1e-3,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Saleh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' & Teich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Fundamentals of Photonics (John Wiley & Sons, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Bhattacharya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Semiconductor Optoelectronic Devices (Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Neamen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Semiconductor Physics and Devices: Basic Principles (McGraw-hill, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Sedra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=', Smith, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=', Carusone, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' & Gaudet, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Microelectronic Circuits, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' 4 (Oxford University Press New  York, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Thorlabs FDGA05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='thorlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='com/thorproduct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='partnumber=FDGA05 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Accessed: 2022-12-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Excelitas C30641GH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='excelitas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='com/product/c30641gh-ingaas-pin-1mm-18 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' Accessed: 2022-12-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  Acknowledgements  The author acknowledges Nicholas Rivera, Jamison Sloan, Yannick Salamin, chatGPT for their discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content=' All equipment  used in the experiments are properties of MIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}
+page_content='  4/4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E0T4oBgHgl3EQfyQLe/content/2301.02658v1.pdf'}

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+Robust knapsack ordering for a partially-informed
+newsvendor with budget constraint
+Guus Boonstra
+Retail Consulting Department, IG&H Consultants, guus.boonstra@igh.com
+Wouter J.E.C. van Eekelen
+Department of Econometrics and Operations Research, Tilburg University, w.j.e.c.vaneekelen@tilburguniversity.edu
+Johan S.H. van Leeuwaarden
+Department of Econometrics and Operations Research, Tilburg University, j.s.h.vanleeuwaarden@tilburguniversity.edu
+This paper studies the multi-item newsvendor problem with a constrained budget and information about
+demand limited to its range, mean and mean absolute deviation. We consider a minimax model that deter-
+mines order quantities by minimizing the expected overage and underage costs for the worst-case demand
+distributions. The resulting optimization problem turns out to be solvable by a method reminiscent of the
+greedy algorithm that solves the continuous knapsack problem, purchasing items in order of marginal value.
+This method has lower computational complexity compared to directly solving the model and leads to a
+simple policy that (i) sorts items based on their marginal effect on the total cost and (ii) determines order
+quantities according to this ranking until the budget is spent.
+Key words : distributionally robust optimization, multi-item newsvendor model, knapsack problem,
+minimax analysis, inventory management
+History : This paper was first submitted on March 8, 2022.
+1.
+Introduction
+The newsvendor model is one of the cornerstones of inventory management, introduced by
+Arrow et al. (1951) for finding the order quantity that minimizes expected costs in view
+of unknown demand and the trade-off between leftovers and lost sales. The newsvendor
+model finds many applications in e.g. perishable food, fashion and high-tech industries,
+particularly when the total time span of production and lead times exceeds the market
+lifetime of a product; see Nahmias (1982) and Fisher and Raman (1996).
+Manufacturers and retailers need to decide how to employ the available budget or re-
+sources when determining the optimal order quantities of different products. A budget
+constraint makes the problem multidimensional—as ordering more of one item leaves less
+budget for other items—and gives rise to a challenging optimization problem. Hadley and
+Whitin (1963) solve this problem with Lagrangian optimization. Abdel-Malek et al. (2004)
+and Lau and Lau (1996) provide alternative solution methods, Erlebacher (2000) estab-
+lishes closed-form solutions for special demand distributions and Nahmias and Schmidt
+1
+arXiv:2301.02662v1  [math.OC]  5 Jan 2023
+
+2
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+(1984) develop heuristic solutions. All these works are for the full information setting,
+where the demand distributions for all items are fully specified. In this paper we perform
+a distribution-free analysis of the multi-item newsvendor problem with budget constraint.
+This analysis does not rely on full specification of the demand distributions, but only re-
+quires for each item knowledge of the mean, mean absolute deviation (MAD) and range.
+Given this partial demand information, we obtain a robust ordering policy by employing
+distributionally robust optimization (DRO) methods.
+The newsvendor model in this paper seeks to minimize the expected costs as function
+of the order quantity. The cost function depends on the order quantity, but also on the
+demand, which is a random variable with some distribution. Given the demand distribu-
+tion, the single-item newsvendor model finds the optimal order quantity that minimizes
+the expected costs. In traditional approaches, the demand distribution is fully specified,
+so that the expected costs can be calculated, and the optimal order quantity can be deter-
+mined. A robust version of this problem assumes partial information, and only knows that
+the demand distribution belongs to some ambiguity set that contains all distributions that
+comply with this partial information. We adopt a minimax strategy that can be viewed as
+a game between the newsvendor and nature: the newsvendor first picks the order quantity
+after which nature chooses a demand distribution that maximizes the expected costs. The
+goal then becomes to solve this minimax problem.
+The way we solve this minimax problem in this paper fits in a much richer class of DRO
+approaches that first calculate worst-case model performance, over the set of distributions
+satisfying some partial information, and then optimize against these worst-case circum-
+stances. Such DRO techniques found applications in many domains including scheduling
+(Kong et al., 2013; Mak et al., 2014), portfolio optimization (Popescu, 2007; Delage and
+Ye, 2010), pricing (Elmachtoub et al., 2021; Chen et al., 2022; Kleer and van Leeuwaarden,
+2022), complex networks (van Leeuwaarden and Stegehuis, 2021), and inventory manage-
+ment (Scarf, 1958; Gallego, 1992; Perakis and Roels, 2008; Ben-Tal et al., 2013). A classic
+distributionally robust approach is due to Scarf (1958), who considered the single-item
+newsvendor problem with mean-variance demand information. Scarf was able to derive
+explicit expressions for the worst-case distribution, and solved the minimax problem to
+obtain the optimal order quantity. Whether a minimax problem is solvable depends on
+both the function to be optimized and the choice of ambiguity set. There are many ways
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+3
+to characterize a set of distributions. In DRO, one can define ambiguity by using distance-
+based metrics, such as total variation or Kullback-Leibler distance. Another popular class
+of ambiguity uses summary statistics. The ambiguity set studied in this paper contains all
+distributions with known mean and MAD. The maximization part of the minimax problem
+can then be viewed as a semi-infinite linear optimization problem with three constraints,
+and an infinite number of variables (all distributions in the ambiguity set). In fact, such
+minimax problems are related to generalized moment bound problems, for which general
+theory says there exists an extremal distribution solving the maximization part with at
+most a number of support points equal to the number of moment constraints (Rogosinski,
+1958). See Rahimian and Mehrotra (2019) for overviews of many more DRO applications
+and techniques.
+For the multi-item newsvendor model in this paper, we solve the multi-dimensional mini-
+max problem with a random vector that describes the demand for all items. Compared with
+tractable one-dimensional problems such as the single-item newsvendor model, applying
+DRO techniques to such problems with multiple random variables might present consider-
+able challenges in terms of computational complexity. For example, given information on
+the mean and covariance of the demands, the distributionally robust multi-item newsvendor
+is significantly harder to solve than its single-item counterpart (Hanasusanto et al., 2015).
+However, for the multi-item newsvendor model in conjunction with mean-MAD ambiguity,
+solving the minimax problem becomes tractable, and in fact has an elegant algorithmic
+solution. The key insight will prove to be that the worst-case demand distribution—the
+solution to the maximization part of the minimax problem—is identical for any order
+quantity. As a result, the minimax problem reduces to a known-distribution optimization
+problem. This known distribution is in fact, for each item, a unique three-point distribu-
+tion. In turn, the minimization problem with this known (discrete) distribution can be
+solved using a reduction to a knapsack problem.
+The main contributions of this paper are as follows:
+(i) Solution of minimax problem. We solve the minimax problem for mean-MAD ambi-
+guity and a budget constraint. We first show that the worst-case scenarios arise when
+item demands follow specific three-point distributions that comply with the partial
+demand information. We minimize the associated worst-case costs to obtain a robust
+
+4
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+ordering policy as the solution to a knapsack problem. As opposed to existing meth-
+ods for the newsvendor model under full demand information, the knapsack problem
+leads to an effective closed-form ordering policy, also for scenarios with many items.
+As such, the present paper further develops DRO theory that uses MAD information
+to formulate tractable minimax problems.
+(ii) Budget consistency. The robust ordering policy only depends on the minimal, mean
+and maximal demand for each item. Hence, the worst-case distributions are indepen-
+dent of all other model parameters, which makes the robust ordering policy ‘budget
+consistent’. When the budget is increased, the orders for the original budget remain
+unaltered, while only the additional budget is further divided over the items. Such
+budget consistency is useful because the optimization model needs to be solved only
+once. That is, for the initial budget value the decision maker can generate an ordered
+list of items as the solution to the knapsack problem, using only standard spreadsheet
+software, and this solution is valid for all budget levels. In contrast, most other exact
+and robust methods for the multi-item newsvendor model do not have this feature,
+which means that the decision maker has to recompute the optimal policy for each
+budget level.
+(iii) Performance of ordering policy. Through a range of numerical examples we demon-
+strate the performance of the knapsack ordering. We draw comparisons with full infor-
+mation settings and other robust approaches that require partial demand information
+by assessing the so-called expected value of additional information (EVAI). Overall,
+the performance of the robust policy only deviates a few percent from the optimal
+performance with full information availability. We also quantify the value of MAD
+information by comparing the performance with the situations when only the mean
+and range of demand is known, and show that MAD indeed provides crucial infor-
+mation for providing good performance. In addition, we construct an ordering policy
+that attains the optimal value of a matching minimin problem which, in conjunction
+with the optimal value of the minimax problem, yields tight performance guarantees.
+We next discuss some related literature on the newsvendor model. Gallego and Moon
+(1993) consider the multi-item newsvendor model with budget constraint when the mean
+and variance of demand is known. Gallego and Moon (1993) extend the ideas in Scarf
+(1958) to obtain an optimization problem that can be solved with Lagrange multiplier
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+5
+techniques, similar to the full information setting with a known distribution. In contrast,
+our minimax analysis with mean-MAD-range information yields a knapsack ordering pol-
+icy that generates a sorted list and prescribes to sort items successively according to that
+list, with order sizes equal to the minimal, mean or maximum demand. Other related
+works that consider the multi-item newsvendor model under partial information include
+Vairaktarakis (2000), who assumes only the support of demand is known, and Ardestani-
+Jaafari and Delage (2016) who assume knowledge of partial moments and rephrase the
+robust optimization problem as a tractable linear program. Natarajan et al. (2018) assume
+knowledge of mean, variance and semivariance, for which the newsvendor model is solvable
+in the single-item setting using a semi-infinite linear program, but largely intractable in
+the multi-item setting. Natarajan et al. (2018) therefore consider a relaxation that gives
+a semidefinite program (SDP) to find a lower bound (which is not tight). Hanasusanto
+et al. (2015) consider mean and covariance knowledge. They prove that the distributionally
+robust problem is NP-hard but admits a semidefinite programming formulation with an ex-
+ponential number of inequalities (that grows in the number of items). Xu et al. (2018) and
+Natarajan and Teo (2017) present more tractable bounds for mean-covariance information.
+In the present paper we assume only marginal information is available, since covariance
+information and other dependency structures are difficult to estimate, and fixing covari-
+ance information often leads to difficult optimization problems with non-intuitive solutions
+(policies). The knapsack ordering policy that we obtain in this paper deals with the worst-
+case demand distributions among all demand distributions with a given mean, MAD and
+range, not conditioning on a specific dependency structure. This approach makes the knap-
+sack ordering policy robust, but also suitable for scarce-data settings, as the mean, MAD
+and range are relatively easy to estimate.
+Section 2 introduces the single-item model and the multi-item model with budget, under
+the traditional assumption of full information about the demand distributions. In Section 3
+we present our main results for the distributionally robust setting with partial information.
+Section 4 presents a detailed numerical study that demonstrates the robust policies. We
+present conclusions and several directions for future work in Section 5. Supplementary
+material appears in the Electronic Companion (EC), including several proofs, additional
+numerical experiments, and model extensions.
+
+6
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+2.
+Classical newsvendor analysis
+We introduce the newsvendor model and several well-known results in Section 2.1 for the
+single-item setting, and in Section 2.2 for the multi-item setting with budget constraint.
+2.1.
+Classical single-item setting
+Consider an item with purchase price c and selling pricing p. The decision maker places
+an order of size q. The demand for items is assumed to be the random variable D with
+distribution function FD(·). Unsold items will be salvaged at the end of the period for
+salvage value s per item. The mark-up m > 0 represents the profit per sold item and
+satisfies p = c(1 + m) and the discount factor d > 0 captures the loss through s = (1 − d)c.
+The expected costs consist of two terms: opportunity costs of lost sales and overage costs
+in case of overstocking. This gives the cost function
+G(q,D) =
+�
+�
+�
+�
+�
+(p − c)(D − q)
+if q ⩽ D,
+(c − s)(q − D)
+if q > D.
+(1)
+The case q ⩽ D amounts to lost sales and q > D results in overstocking. The objective is to
+order the quantity q of items that minimizes the expected costs. Let E denote expectation,
+and define µ = E[D] and x+ = max(x,0). Write the expected costs as
+C(q) := E[G(q,D)] = (c−s)q+(p−s)E(D−q)+−(c−s)µ = c
+�
+d(q − µ) + (m + d)E(D − q)+�
+.
+(2)
+To keep notation simple (and without loss of generality) set c = 1. Then, the optimal order
+quantity
+q∗ = argmin
+q⩾0
+C(q) ≡ argmin
+q⩾0
+dq + (m + d)E(D − q)+,
+(3)
+is given by
+q∗ = inf
+�
+q : F(q) ⩾
+m
+m + d
+�
+.
+(4)
+A proof of (4) is provided in most standard textbooks on inventory management; see e.g.
+Hadley and Whitin (1963); Silver et al. (1998); Nahmias (2009).
+2.2.
+Multi-item setting
+Consider n different items and order qi units for item i for a given period where i = 1,...,n.
+For item i, the unit purchasing and selling price are ci and pi respectively. Possible leftovers
+will be salvaged at the end of the period for unit salvage value si. We define the model
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+7
+in terms of the mark-up mi > 0 and discount factor di > 0. The mark-up represents the
+profit per sold unit and the discount factor the loss, i.e. pi = ci(1 + mi) and si = (1 − di)ci.
+The random demand for item i in one period is represented by the nonnegative random
+variable Di, distributed according to Fi(·).
+As in the single-item setting, we minimize the expected costs. Define the multi-item cost
+function as
+G(q,D) :=
+n
+�
+i=1
+ci
+�
+di(qi − Di) + (mi + di)(Di − qi)+�
+.
+(5)
+We also introduce the budget constraint �n
+i=1 ciqi ⩽ B with B the available budget. The
+multi-item newsvendor model, with decision vector q = (q1,...,qn), is then given by
+min
+q
+C(q) := E[G(q,D)] =
+n
+�
+i=1
+ci
+�
+di(qi − µi) + (mi + di)E(Di − qi)+�
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+qi ⩾ 0,
+i = 1,...,n.
+(6)
+Its solution, referred to as the optimal ordering policy, will be denoted by q∗. In the
+single-item setting the purchase costs had no influence on the objective function, but in
+the multi-item setting the optimal order quantity is affected by ci. It is well known that
+model (3) is a convex optimization problem. In (6) we take the summation over n convex
+functions, which preserves convexity. Moreover, the constraints form a convex set, so that
+(6) is a convex optimization problem (Boyd and Vandenberghe, 2004).
+3.
+Proposed robust approach
+Section 3.1 presents the robust ordering policy for the single-item setting. This result serves
+as building block for the robust analysis of the multi-item setting in Section 3.2, which
+describes the optimal policy as the solution of a linear program (LP). In Section 3.3 we
+show that this LP can be viewed as a knapsack problem. All these results are based on a
+tight upper bound for the cost function. In Section 3.4 we derive a matching tight lower
+bound for the cost function.
+3.1.
+Distribution-free ordering policy for single item
+Let P denote a probability distribution, and write EP for E to emphasize that the expec-
+tation is taken with respect to the distribution P of D. The MAD for random demand D
+
+8
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+is defined as δ := EP|D − µ|, where µ is the expected value of D. Similar to the variance,
+the MAD is a measure of dispersion or variability. We mention several properties of MAD
+in EC.2. For the random variable D with mean µ, MAD δ, and (bounded) support [a,b],
+where 0 ⩽ a ⩽ b < ∞, the mean-MAD ambiguity set is defined as
+P(µ,δ) := {P|EP[D] = µ, EP|D − µ| = δ, supp(D) ⊆ [a,b]}.
+We thus assume that the ‘true’ distribution ˜P of the random demand D is contained in
+this ambiguity set, that is, ˜P ∈ P(µ,δ).
+To obtain the robust order quantity, we solve
+min
+q
+max
+P∈P(µ,δ) dq + (m + d)EP(D − q)+,
+for which we first consider maxP∈P(µ,δ) EP(D − q)+. To characterize this tight bound, we
+apply a general upper bound for convex functions of a random variable by Ben-Tal and
+Hochman (1972). To make this paper self-contained, we provide a proof of the following
+result in EC.1.
+Lemma 1. The extremal distribution that solves
+max
+P∈P(µ,δ) EP(D − q)+ is a three-point dis-
+tribution on the values a, µ and b that does not depend on q.
+From the proof of Lemma 1, it follows that the worst-case probability distribution of D,
+the extremal distribution that solves maxP∈P(µ,δ) EP(D − q)+, is a three-point distribution
+defined as
+P(D = x) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+δ
+2(µ − a),
+for x = a,
+1 −
+δ
+2(µ − a) −
+δ
+2(b − µ), for x = µ,
+δ
+2(b − µ),
+for x = b.
+(7)
+Applying this worst-case distribution, the robust order quantity follows from solving
+qU = argminq CU(q) with
+CU(q) := d(q − µ) + δ(m + d)
+2(µ − a) (a − q)+ + (m + d)
+�
+1 −
+δ
+2(µ − a) −
+δ
+2(b − µ)
+�
+(µ − q)+
++ δ(m + d)
+2(b − µ) (b − q)+.
+(8)
+To illustrate the mean-MAD bound and robust order quantity qU, consider an example in
+which D is distributed according to a beta distribution with both shape parameters set
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+9
+to 1. For a general beta distribution, a = 0 and b = 1. In Figure 1a, we have m = 1 and
+d = 0.8. This leads to qU = µ. In Figure 1b, the mark-up increases to m = 3. In this case
+the mean-MAD order quantity increases to qU = b. When computing this upper bound,
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Order quantity
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+Expected costs
+Beta
+Mean-MAD bound
+Mean-variance bound
+(a) m = 1
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Order quantity
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Expected costs
+Beta
+Mean-MAD bound
+Mean-variance bound
+(b) m = 3
+Figure 1
+Mean-MAD and mean-variance bounds and corresponding ordering policies. The upper curve corre-
+sponds to the mean-variance upper bound that follows from P(1/2,1/12). The middle curve depicts the
+mean-MAD upper bound. The ‘true’ cost function assumes that D follows a beta distribution with
+both shape parameters equal to 1 (the lower curve).
+observe that the mean-MAD bound touches the ‘true’ cost function in the points a,µ and b.
+This property actually holds in general. Clearly, for q = a or b, it holds that CU(q) = C(q).
+When q = µ, the cost function equals
+C(µ) = d(µ − µ) + (m + d)E(D − µ)+ = δ(m + d)
+2
+= CU(µ),
+since E(D − µ)+ = E|D − µ|/2.
+By analyzing (8) one can obtain an explicit ordering rule for qU. The objective func-
+tion of (8) is composed of piecewise linear functions. By exploiting this structure, we
+can construct an explicit ordering policy. For scalars α1,...,αm,ν1,...,νm ∈ R, f(x) =
+maxi=1,...,m{αix+νi} denotes a convex, piecewise linear function. The function CU(q) in (8)
+admits a representation of the form
+CU(q) = d(q − µ) + (m + d)E(D − q) = m(µ − q) =: f0(q),
+for q ∈ [0,a) and
+CU(q) = d(q − µ) + (m + d)
+�
+1 −
+δ
+2(µ − a) −
+δ
+2(b − µ)
+�
+(µ − q) + δ(m + d)
+2(b − µ) (b − q)
+= q(δ(m + d)
+2(µ − a) − m) + ν1 =: f1(q),
+
+10
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+for q ∈ [a,µ), where ν1 is some constant value. For q ∈ [a,µ), the mean-MAD objective
+function is defined by the linear function f1(q). For the interval q ∈ [µ,b], we obtain
+CU(q) = d(q − µ) + δ(m + d)
+2(b − µ) (b − q) = q
+�
+d − δ(m + d)
+2(b − µ)
+�
++ ν2 =: f2(q)
+for some constant ν2. The cost function is thus the pointwise maximum of the three linear
+functions f0(q), f1(q) and f2(q):
+CU(q) = max{f0(q), f1(q), f2(q)}.
+Since CU(q) = maxj=0,1,2{αjq + νj} is a convex function, it holds that α0 ⩽ α1 ⩽ α2. Since
+we assume that m > 0, we know that α0 < 0. Therefore, from the derivatives α1, α2 of
+CU(q), we can derive an explicit order quantity by examining for which linear piece the
+slope turns positive. This allows us to state Theorem 1.
+Theorem 1 (Mean-MAD order quantity). The
+robust
+order
+quantity
+qU
+∈
+argminq CU(q) is given by
+(a) If m <
+δd
+2(µ − a) − δ, then qU = a.
+(b) If
+δd
+2(µ − a) − δ < m < d(2(b − µ) − δ)
+δ
+, then qU = µ.
+(c) If d(2(b − µ) − δ)
+δ
+< m, then qU = b.
+(d) If m =
+δd
+2(µ − a) − δ and m = d(2(b − µ) − δ)
+δ
+, then qU ∈ [a,µ] and qU ∈ [µ,b], respec-
+tively.
+According to Theorem 1, the robust order quantity qU for mean-MAD-range information
+consists of three predictable values (minimal, mean, maximum demand) that do not depend
+on the mark-up m and discount factor d, whereas the conditions that dictate how much
+to order do depend on them (in addition to the demand mean, MAD and range).
+3.2.
+Multiple items and budget constraint
+A distribution-free analysis of the multi-item model requires a multivariate ambiguity set.
+As in the single-item case, the partial information is the mean µi, MAD δi and support
+supp(Di) = [ai,bi] for each random variable Di, i = 1,...,n. The mean-MAD ambiguity set
+is defined as
+P(µ,δ) := {P|EP (Di) = µi, EP |Di − µi| = δi, supp(Di) ⊆ [ai,bi], ∀i}.
+(9)
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+11
+We henceforth assume that the distribution of the vector of random variables D =
+(D1,...,Dn) belongs to this ambiguity set, i.e., P ∈ P(µ,δ). Since the objective function in
+(6) is separable, one can apply the single-item bound to each term E(Di − qi)+ in the
+summation individually. The following result, for the multi-item problem, is then a direct
+consequence of Lemma 1.
+Lemma 2. The extremal distribution that solves max
+P∈P(µ,δ) EP[G(q,D)] consists for each Di
+of a three-point distribution with values ξ(i)
+1 = ai, ξ(i)
+2 = µi, ξ(i)
+3 = bi and probabilities
+p(i)
+1 =
+δi
+2(µi − ai),
+p(i)
+2 = 1 −
+δi
+2(µi − ai) −
+δi
+2(bi − µi),
+p(i)
+3 =
+δi
+2(bi − µi).
+(10)
+For the multi-item newsvendor model based on mean-MAD ambiguity, we use Lemma 2
+to solve the maximization part of
+min
+q:�
+i ciqi⩽B,qi⩾0 max
+P∈P(µ,δ) EP
+�
+n
+�
+i=1
+cidi(qi − µi) + ci(mi + di)(Di − qi)+ �
+,
+(11)
+and obtain
+min
+q
+n
+�
+i=1
+ci
+�
+di(qi − µi) + (mi + di)
+�
+p(i)
+1 (ai − qi)+ + p(i)
+2 (µi − qi)+ + p(i)
+3 (bi − qi)+��
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+qi ⩾ 0,
+i = 1,...,n.
+(12)
+The objective function of (12) has a piecewise linear structure. Moreover, because of this
+result and since the constraints are linear, (12) can be cast as a linear program (LP). In
+particular, as explained below, the robust ordering policy qU can be found by solving
+min
+q
+n
+�
+i=1
+max
+j=0,1,2{αi,jqi + νi,j}
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+qi ⩾ 0,
+i = 1,...,n,
+(13)
+where
+αi,0 = −cimi,
+νi,0 = cimiµi,
+αi,1 = ci
+�δi(mi + di)
+2(µi − ai) − mi
+�
+,
+νi,1 = ci(mi + di)
+�
+µi −
+δiai
+2(µi − ai)
+�
+− cidiµi,
+αi,2 = ci
+�
+di − δi(mi + di)
+2(bi − µi)
+�
+,
+νi,2 = ciδi(mi + di)bi
+2(bi − µi)
+− cidiµi,
+for i = 1,...,n.
+
+12
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+Let fi,j(x) = αi,jx + νi,j for i = 1,...,n and j = 0,1,2. From the single-item case, we know
+that the objective, for each item i, can be written as maxj=0,1,2{fi,j(qi)} with αi,0 ⩽ αi,1 ⩽
+αi,2, and thus the objective functions of (12) and (13) are equal, which makes the two
+models equivalent. Since we know from linear programming theory that convex, piecewise
+linear objective functions can be written as linear constraints, problem (13) admits an LP
+representation (Boyd and Vandenberghe, 2004).
+3.3.
+Knapsack algorithm
+It turns out that problem (13) is intimately related to the continuous knapsack problem,
+thus making available efficient sorting-based algorithms to solve (13). We next describe an
+efficient algorithm that determines the robust ordering policy.
+Define the linear funtion fi,j for each item i, and let αi,j represent its derivative with
+respect to qi, for items i = 1,...,n and linear pieces j = 0,1,2. That is,
+dfi,j(qi)
+dqi
+= αi,j.
+For each item i, fi,0, fi,1 and fi,2 represent the marginal effect on the value of (13) when
+we increase qi to ai,µi and bi respectively. The parameter αi,j represents the slope of these
+linear functions and an order quantity is increased only when αi,j < 0, because otherwise
+it will not reduce the expected costs. We consecutively allocate budget to the item that
+causes the largest relative decrease in expected costs; that is, item k with the smallest
+negative derivative αk,i relative to its cost ck. Define the set of all items as N = {1,...,n}.
+Since only order quantities that decrease the expected costs are considered, define the
+ordered set:
+G := {(i,j) | αi,j < 0,i ∈ N,j ∈ {0,1,2}},
+(14)
+where the ordering is determined according to the value of αi,j/ci. For m = |G|, this
+ordering is represented by the sequence (i1,j1),...,(im,jm) for which it holds that
+αi1,j1/ci1 ⩽ ··· ⩽ αim,jm/cim. Here G contains tuples (i,j) for which i represents an item in
+the newsvendor model and j a linear piece of the piecewise function. As these functions
+are convex, the linear pieces appear for each item i in increasing order in the set G. We
+can now state the knapsack algorithm for the distribution-free multi-item newsvendor
+model.
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+13
+Algorithm 1 (Knapsack algorithm). For a budget level B ⩾ 0, the ordering policy
+qU is found by the following procedure:
+(i) Initialize by setting q = (0,...,0), and construct G. Continue to (ii).
+(ii) Select the first element (i,j) ∈ G. If the set G is empty, the optimal solution is qU = q.
+Otherwise, continue to (iii).
+(iii) If j = 0, set qi = ai. If j = 1, set qi = µi. If j = 2, set qi = bi. Continue to (iv).
+(iv) Determine whether the budget constraint �n
+i=1 ciqi ⩽ B is violated. If so, set qi such
+that ciqi = B − �
+k∈N|k̸=i ckqk, and the optimal solution is qU = q. Otherwise, remove
+element (i,j) from G and return to step (ii).
+This algorithm yields an optimal solution to (13), as asserted in the following theorem.
+Theorem 2 (Knapsack ordering policy). The robust ordering policy qU that solves
+the multi-item newsvendor model (13) is determined by Algorithm 1.
+Proof.
+To prove that this algorithm produces an optimal solution, we construct a con-
+tinuous knapsack problem that solves (13). In the following, (ik,jk) corresponds to the kth
+entry of the ordered sequence of items in G. Define the following auxiliary model:
+min
+x
+m
+�
+k=1
+pkxk
+s.t.
+m
+�
+k=1
+ckxk ⩽ B,
+0 ⩽ xk ⩽ uk
+∀k = 1,...,m,
+(15)
+where
+uk =
+�
+�
+�
+�
+�
+�
+�
+aik,
+for jk = 0
+µik − aik, for jk = 1
+bik − µik, for jk = 2
+and pk = αik,jk and ck = cik. From the order of the sequence, it follows that p1/c1 ⩽ ... ⩽
+pm/cm. Assume that (x∗
+1,...,x∗
+m) is an optimal solution to optimization problem (15).
+For i ∈ N, let qU
+i = �
+k=1,...,m|i=ik x∗
+k. Since αi,0 ⩽ αi,1 ⩽ αi,2, the pieces jk appear in G in
+increasing order for each item i. Thus, in an optimal solution, uik,jk will only be attained if
+its predecessor uik,jl is also attained. By construction, qU is feasible for (13). Moreover, the
+objective values of problems (13) and (15) only differ by a constant term, so both problems
+have the same optimal solution. For the continuous knapsack problem, a greedy allocation
+produces an optimal solution (see EC.3). Hence, qU = (qU
+1 ,...,qU
+n ) is optimal for (13).
+□
+
+14
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+Theorem 2 shows that there exists a ranking for the selection of items. Take an initial
+budget B = 0. If we increase the budget B by some small value, we first increase item i
+to ai for the item that has the highest mark-up mi. This makes sense intuitively because
+the product with the highest mark-up is most profitable and, since qi < ai, we have no
+risk of overstocking. We successively select the items with the greatest marginal benefit
+αi,j/ci, and increase the order quantity consecutively to either ai, µi or bi. This procedure
+continues until we have spent the entire budget, or reached the uncapacitated optimum.
+Items that are ordered in the beginning of this procedure have the largest impact on the
+decrease in costs for the multi-item newsvendor model.
+As the main complexity of the knapsack algorithm in Theorem 2 stems from sorting
+the set G, the greedy approach is of computational complexity O(nlog n). Moreover, the
+solution can be found in O(n) time by first identifying the critical element (is,js) that will
+violate the budget constraint, as proposed by Balas and Zemel (1980) for the continuous
+knapsack problem. One then compares each αi,j/ci with the ratio of the critical element to
+determine the optimal allocation of budget to the items. The optimal solution can also be
+found through the LP (13), which we solve with the simplex method. We remark that a
+single iteration of the simplex method takes O(n2) arithmetic operations (Ill´es and Terlaky,
+2002), which exceeds the time requirement of the knapsack algorithm.
+3.4.
+A matching lower bound
+The robust analysis so far was based on finding a tight upper bound on the cost function
+when we know the mean, MAD and range of the demand distributions. When additional
+information is available, we can also construct a matching lower bound. We include the
+skewness information βi = P(Di ⩾ µi) in the mean-MAD ambiguity set to obtain the tight
+lower bound. For the random variables D = (D1,...,Dn), define the ambiguity set as
+P(µ,δ,β) := {P|P ∈ P(µ,δ), P(Di ⩾ µi) = βi, i = 1,...,n}
+with P(µ,δ,β) ⊆ P(µ,δ). The proof of the following result is identical to that of Lemma 2,
+but now uses the tight lower bound for a convex function of random variables discussed in
+Ben-Tal and Hochman (1972). To make this paper self-contained, a proof for the univariate
+case is provided in EC.1. This is sufficient since the univariate result can be applied to
+each term of the summation in G(q,D) separately, as with Lemma 2.
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+15
+Lemma 3. The extremal distribution that solves
+min
+P∈P(µ,δ,β) EP[G(q,D)] consists for each
+Di of a two-point distribution with values µi + δi
+2βi, µi −
+δi
+2(1−βi) and probabilities βi, 1 − βi,
+respectively.
+Using this result, we obtain
+min
+q
+CL(q) :=
+n
+�
+i=1
+ci
+�
+di(qi − µi) + (mi + di)
+�
+βi(µi + δi
+2βi
+− qi)+ + (1 − βi)(µi −
+δi
+2(1 − βi) − qi)+
+��
+s.t
+n
+�
+i=1
+ciqi ⩽ B,
+qi ⩾ 0,
+for i = 1,...,n,
+(16)
+as a model to provide a lower bound for the multi-item newsvendor. As the objective
+function in problem (16) also consists of piecewise linear functions, there exists an LP
+representation and knapsack algorithm for (16) similar to the results for problem (12).
+We can now solve (13) and (16) to obtain tight performance intervals for the multi-item
+newsvendor model, using recent DRO results (see EC.4 and Postek et al., 2018). For all
+feasible ordering policies q and P ∈ P(µ,δ,β), it holds that
+C(q) ∈
+�
+CL(q),CU(q)
+�
+.
+In addition, for the optimal solutions to the newsvendor problem and its distributionally
+robust counterparts,
+C(q∗) ∈
+�
+CL(qL),CU(qU)
+�
+.
+One can find the tightest upper and lower bounds, based on mean-MAD ambiguity, for
+the multi-item newsvendor model by calculating the optimal solutions to models (12) and
+(16), respectively.
+4.
+Numerical examples of robust ordering
+We will now illustrate and visualize the robust ordering policies. To demonstrate the
+‘budget-consistency’ property, Section 4.1 applies the knapsack algorithm for a setting
+where the budget is increased. In Section 4.2 we contrast the performance of the knapsack
+policy for partial demand information against that of the optimal solution for the full
+information setting. Our code is made available in the form of an online supplement.
+
+16
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+4.1.
+Numerical illustration of the ‘budget-consistency’ property
+We illustrate the knapsack algorithm and the process of allocating budget to different order
+quantities for items in the newsvendor model. Consider n = 5 identically distributed items
+with support a = 10, b = 50 and mean µ = 30. From Figure 2, we can infer that item 1
+is the most profitable. Low budget levels are allocated to this item such that we obtain
+q1 = µ. Item number 3 is the last item to which the budget is allocated. Hence, it is the
+least profitable item. Table 1 displays the ordered set G. From this table, we can indeed
+infer that item 1 has the smallest value for αi,0/ci and therefore is increased first.
+0
+50
+100
+150
+200
+250
+300
+Budget
+0
+10
+20
+30
+40
+50
+Order quantity
+Item 1
+Item 2
+Item 3
+Item 4
+Item 5
+Figure 2
+Development of the order quantities when the budget increases according to the knapsack algorithm
+Table 1
+Table containing αi,j/ci and corresponding information of the ordered set G
+G
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+αi,j/ci
+-0.92 -0.75 -0.72 -0.49 -0.3 -0.15 -0.1 -0.08 -0.03 -0.01 0.14 0.42 0.45 0.7 0.7
+Function piece
+0
+1
+0
+1
+0
+0
+1
+2
+1
+0
+1
+2
+2
+2
+2
+Item
+1
+1
+2
+2
+4
+5
+4
+1
+5
+3
+3
+5
+2
+4
+3
+Figure 2 nicely illustrates that when the budget is increased, the orders for the original
+budget remain unaltered, while only the additional budget is further divided over the items.
+To further illustrate the ‘budget-consistency’ property, consider the multi-item newsvendor
+model for which n = 2, m2 = 2, the remaining cost parameters equal 1, and demand is
+identically distributed according to a symmetric triangle distribution supported on [10,50].
+In Figure 3 we plot the expected costs and order quantities for various budget levels.
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+17
+Figure 3a contains the allocation between both order quantities. For low budget values, one
+first increases the order quantity of item one, the most profitable item. Figure 3b shows the
+upper bound (12) and lower bound (16) that together lead to a tight performance interval
+for the expected costs.
+For the sake of comparison, we also show results for the partial demand information
+setting considered in Gallego and Moon (1993), assuming that the mean and variance of
+demands are known; see EC.5 for more details. The results of Gallego and Moon (1993) de-
+pend (non-trivially) on all model parameters, including the budget B. This lack of budget-
+consistency forces the decision maker to solve an optimization problem, see (EC.13), for
+each budget level separately, and explains the smooth curve in Figure 3a. In contrast,
+our knapsack algorithm generates a sorted ordering list that does not depend on B, and
+prescribes to sort items successively according to that list, with order sizes equal to the
+minimal, mean or maximum demand.
+0
+5
+10
+15
+20
+25
+30
+Order quantity of item 1
+0
+5
+10
+15
+20
+25
+30
+35
+Order quantity of item 2
+Optimal order quantity
+Mean-MAD policy
+Mean-variance policy
+(a) Ordering policy
+0
+10
+20
+30
+40
+50
+60
+Budget
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Expected costs
+Triangular
+Mean-MAD lower bound
+Mean-MAD upper bound
+Mean-variance bound
+(b) Newsvendor costs
+Figure 3
+Mean-variance and mean-MAD bounds and ordering policies for the newsvendor model. The mean-
+variance curves are obtained through solving (EC.13). The mean-MAD policy corresponds to the optimal
+solution of (12). The mean-MAD upper and lower bounds correspond to the extremal three- and two-
+point distributions, respectively. The ‘true’ cost function assumes that D follows a symmetric triangular
+distribution on [10,50].
+We emphasize that these results are not meant to numerically compare the mean-MAD
+and mean-variance policies, because the displayed differences merely express different ways
+of dealing with ambiguity. Indeed, it is hard to compare both policies as the respective
+ambiguity sets can contain vastly different distributions. For instance, a finite variance
+excludes distributions with an infinite second moment, while finite MAD does not. For
+
+18
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+our purposes, MAD and variance are equally adequate descriptors of dispersion, and both
+are easily calibrated on data using basic statistical estimators. The crucial difference in
+the DRO context of this paper is that MAD leads to a simple, budget-consistent ordering
+policy.
+4.2.
+Expected value of additional information
+We introduce as performance measure the expected value of additional information (EVAI),
+defined as
+EVAI(qU
+B) = C(qU
+B) − C(q∗
+B)
+C(q∗
+B)
+,
+where qU
+B is the robust ordering policy and q∗
+B is the optimal ordering policy when the
+joint demand distribution is known. We let B run from 0 to �n
+i=1 q∗
+i =: Bopt, and consider
+nine different demand distributions, listed in Table 2.
+Table 2
+Nine distributions used for multi-item performance analysis
+Case
+Case
+Case
+1
+Uniform[10,50]
+4
+Beta(1,3) on [0,50]
+7
+Triangular(10,50,18)
+2
+Uniform[10,100]
+5
+Beta(2,2) on [0,50]
+8
+Triangular(10,50,30)
+3
+Uniform[10,200]
+6
+Beta(3,1) on [0,50]
+9
+Triangular(10,50,42)
+We consider n = 25 items. For each item i, let ci = di = 1 and assume identically dis-
+tributed demand. For example, in Case 2 the demand Di for each item i follows the uniform
+distribution with parameters ai = 10 and bi = 100. Table 3 provides an overview for the
+mark-up, representing low, average and high margins.
+For the low margin regime, Figure 4 shows results for each of the nine cases, for both the
+robust ordering policy with mean-MAD-range information, and for the policy that uses
+the additional information βi = P(Di ⩾ µi). For the former, the worst performance over all
+nine cases has a maximum deviation of approximately 23% compared to the optimal order
+quantity q∗
+B. Overall, the performance of the robust policy only deviates a few percent from
+the optimal performance with full information availability. For the uniformly distributed
+cases (Cases 1-3), the performance decreases when the range increases. For beta distributed
+demand (Cases 4-6), right-tailed distributions perform worse than left-tailed distributions.
+This effect is also observed for the triangular distributions (Cases 7-9). The policy with
+additional information βi = P(Di ⩾ µi) performs somewhat better in most cases.
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+19
+Table 3
+Mark-up values for all 25 items in the newsvendor model
+Mark-up
+m1
+m2
+m3
+m4
+m5
+m6
+m7
+m8
+m9
+m10
+m11
+m12
+m13
+Low margin
+0.1
+0.14 0.18 0.21 0.25 0.29 0.33 0.36
+0.4
+0.44 0.48 0.51 0.55
+Average margin
+1
+1.13 1.25 1.38
+1.5
+1.63 1.75 1.88
+2
+2.13 2.25 2.38
+2.5
+High margin
+4
+4.21 4.42 4.63 4.83 5.04 5.25 5.46 5.67 5.88 6.08 6.29
+6.5
+Mark-up
+m14
+m15
+m16
+m17
+m18
+m19
+m20
+m21
+m22
+m23
+m24
+m25
+Low margin
+0.59 0.63 0.66
+0.7
+0.74 0.78 0.81 0.85 0.89 0.93 0.96
+1
+Average margin 2.63 2.75 2.88
+3
+3.13 3.25 3.38
+3.5
+3.63 3.75 3.88
+4
+High margin
+6.71 6.92 7.12 7.33 7.54 7.75 7.96 8.17 8.37 8.58 8.79
+9
+Figure 5 shows similar results for high margins. The EVAI for the robust policy remains
+mostly below 10% for lower budget levels, but starts increasing rapidly when the budget
+approaches Bopt (i.e., when approaching the unconstrained model). When the budget is less
+restrictive, additional distributional information provides substantial value. In particular,
+since the policy uses skewness information βi, it performs better (in expectation) for higher
+budget levels than the robust ordering policy. We present some more performance plots
+for the average margin setting and additional numerical experiments with mean-variance
+information in EC.6.
+We next quantify the value of MAD information by comparing the performance with
+the situations when only the mean and range of demand is known. For the low margin
+setting, Figure 6 shows the EVAI for the ordering policy with only mean-range informa-
+tion. Like the mean-MAD policy, this policy follows from a discrete distribution, in this
+case the extremal distribution on {a,b} with probabilities b−µ
+b−a and µ−a
+b−a that attains the
+Edmundson-Madansky bound (see Ben-Tal and Hochman, 1972). That is, instead of the
+worst-case three-point distribution, we take the expectation in (6) over this two-point dis-
+tribution and find the robust mean-range ordering policy using the resulting LP. The plots
+clearly demonstrate that knowledge on dispersion in terms of MAD improves performance
+considerably.
+
+20
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+0
+224
+449
+674
+0.00
+0.05
+0.09
+0.14
+Case 1: Uniform (10,50)
+Mean-MAD
+Mean-MAD-
+0
+401
+802
+1204
+0.00
+0.05
+0.09
+0.14
+Case 2: Uniform (10,100)
+0
+755
+1510 2265
+0.00
+0.05
+0.09
+0.14
+Case 3: Uniform (10,200)
+0
+70
+140
+211
+0.00
+0.06
+0.11
+0.17
+Case 4: Beta (1,3)
+0
+187
+374
+561
+0.00
+0.07
+0.13
+0.20
+Case 5: Beta (2,2)
+0
+312
+624
+937
+0.00
+0.06
+0.12
+0.18
+Case 6: Beta (3,1)
+0
+190
+380
+571
+0.00
+0.08
+0.15
+0.23
+Case 7: Triangular (10,50,18)
+0
+236
+472
+709
+0.00
+0.07
+0.13
+0.20
+Case 8: Triangular (10,50,30)
+0
+277
+554
+831
+0.00
+0.06
+0.11
+0.17
+Case 9: Triangular (10,50,42)
+Figure 4
+The results for the low margin setting. The x-axis corresponds to B and the y-axis to the EVAI.
+0
+370
+740
+1110
+0.00
+0.11
+0.23
+0.34
+Case 1: Uniform (10,50)
+Mean-MAD
+Mean-MAD-
+0
+729
+1458 2187
+0.00
+0.11
+0.23
+0.34
+Case 2: Uniform (10,100)
+0
+1446 2892 4339
+0.00
+0.12
+0.23
+0.35
+Case 3: Uniform (10,200)
+0
+201
+403
+605
+0.00
+0.21
+0.42
+0.63
+Case 4: Beta (1,3)
+0
+319
+638
+958
+0.00
+0.18
+0.37
+0.55
+Case 5: Beta (2,2)
+0
+396
+792
+1188
+0.00
+0.11
+0.23
+0.34
+Case 6: Beta (3,1)
+0
+306
+612
+918
+0.00
+0.18
+0.36
+0.54
+Case 7: Triangular (10,50,18)
+0
+329
+658
+987
+0.00
+0.19
+0.38
+0.57
+Case 8: Triangular (10,50,30)
+0
+361
+722
+1084
+0.00
+0.20
+0.41
+0.61
+Case 9: Triangular (10,50,42)
+Figure 5
+The results for the high margin setting. The x-axis corresponds to B and the y-axis to the EVAI.
+
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+21
+0
+224
+449
+674
+0.00
+0.26
+0.51
+0.77
+Case 1: Uniform (10,50)
+Mean-MAD
+Mean-MAD-
+E-M
+0
+401
+802
+1204
+0.00
+0.26
+0.51
+0.77
+Case 2: Uniform (10,100)
+0
+755
+1510 2265
+0.00
+0.26
+0.51
+0.77
+Case 3: Uniform (10,200)
+0
+70
+140
+211
+0.00
+0.16
+0.32
+0.48
+Case 4: Beta (1,3)
+0
+187
+374
+561
+0.00
+0.45
+0.90
+1.35
+Case 5: Beta (2,2)
+0
+312
+624
+937
+0.00
+1.03
+2.07
+3.10
+Case 6: Beta (3,1)
+0
+190
+380
+571
+0.00
+0.34
+0.69
+1.03
+Case 7: Triangular (10,50,18)
+0
+236
+472
+709
+0.00
+0.54
+1.09
+1.63
+Case 8: Triangular (10,50,30)
+0
+277
+554
+831
+0.00
+0.63
+1.26
+1.89
+Case 9: Triangular (10,50,42)
+Figure 6
+The results for the low margin setting. The x-axis corresponds to B and the y-axis to the EVAI. The
+E-M performance plot refers to the model with only mean information.
+5.
+Conclusions
+This paper establishes new ordering policies for the newsvendor with partial demand in-
+formation (mean, MAD and range) with a budget constraint. The ordering policies follow
+from a minimax approach, where we search for the order quantities with minimal costs
+for the maximal (worst-case) cost function restricted to demand distributions that comply
+with the partial information.
+The minimax analysis for the multi-item setting gives rise to a knapsack problem, and
+the solution of this knapsack problem in fact is the ordering policy. This policy prescribes
+to sort items based on their marginal effect on the total costs, reminiscent of the greedy
+algorithm that solves the continuous knapsack problem. The ordering policy only orders
+the minimum, mean or maximum demand for each item. Hence, the decision maker can
+rank the items based on their marginal effects, and then start ordering items according to
+this list until the budget is spent. The fact that the ranking list is easy to generate, and
+that the ‘order of ordering’ does not depend on the budget, makes the policy transparent
+and easy to implement. Existing approaches for full and partial (such as mean-variance)
+knowledge of the demand distribution lack this property of ‘budget-consistency’.
+
+22
+Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+The minimax approach provides robustness, with an ordering policy that protects against
+all distributions that comply with the partial information. This approach avoids the need
+to estimate the demand distribution, which can be a daunting process in practice and
+is prone to errors. However, the minimax approach comes at the risk of being overly
+conservative. Through extensive numerical experiments we compared the robust policies
+for partial demand settings with the policies for full demand settings, and observed that
+the proposed policies perform well.
+At the heart of our analysis lies the idea to set up the robust minimax analysis with
+MAD information. With MAD as dispersion measure we obtained a tractable optimization
+model, with a solution in terms of a robust ordering policy that satisfies the budget-
+consistency property. Using MAD to formulate solvable minimax problems can also be
+applied to other inventory models. We demonstrate this idea in EC.7 for three extended
+settings: the newsvendor with multiple contraints, the newsvendor with unreliable supply,
+and the risk-averse newsvendor. In all three cases, the minimax analysis leads to a tractable
+mathematical program, either a knapsack problem or a linear program.
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+25
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+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec1
+E-Companion to “Robust knapsack ordering for a
+partially-informed newsvendor with budget constraint”
+EC.1.
+Proofs
+Proof of Lemma 1.
+In their original work, Ben-Tal and Hochman (1972) prove this
+result for general convex functions by dividing the support into two intervals [a,µ] and [µ,b]
+and then applying the Edmundson-Madansky bound to both subintervals. The following
+proof uses semi-infinite programming duality and is taken from van Eekelen et al. (2022).
+Consider a general convex function f(x) (this includes (x − q)+ as a special case). For
+X ∼ P ∈ P(µ,δ), we solve
+max
+P(x)⩾0
+� b
+a
+f(x)dP(x)
+s.t.
+� b
+a
+dP(x) = 1,
+� b
+a
+xdP(x) = µ,
+� b
+a
+|x − µ|dP(x) = δ,
+(EC.1)
+Consider the dual of (EC.1),
+min
+λ0,λ1,λ2
+λ0 + λ1µ + λ2δ
+s.t.
+M(x) := λ0 + λ1x + λ2|x − µ| ⩾ f(x), ∀x ∈ [a,b].
+(EC.2)
+The function M(x) has a ‘kink’ at x = µ. Since the dual problem (EC.2) has three variables,
+the optimal M(x) touches f(x) at three points: x = a, µ and b. For this choice of M(x),
+λ0 = f(a) − λ1a − λ2(µ − a), λ1 = 1
+2
+�f(b) − f(µ)
+b − µ
++ f(µ) − f(a)
+µ − a
+�
+,
+λ2 = 1
+2
+�f(b) − f(µ)
+b − µ
+− f(µ) − f(a)
+µ − a
+�
+.
+Because the majorant is piecewise linear and convex, we can majorize every convex function
+f(x) by letting M(x) touch at the boundary points a,b and at the kink point x = µ.
+According to the complementary slackness property, these points constitute the support
+of the extremal distribution, and the optimal probabilities follow from solving the linear
+system resulting from the equations of (EC.1). This is a linear system of three unknown
+probabilities and three equations, with the solution
+pa =
+δ
+2(µ − a),
+pµ = 1 −
+δ
+2(µ − a) −
+δ
+2(b − µ),
+pb =
+δ
+2(b − µ).
+Finally, for these primal and dual solutions, we verify that the objective values of problems
+(EC.1) and (EC.2) agree, which confirms that strong duality holds.
+□
+
+ec2e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+Proof of Lemma 3.
+We prove this result for general convex f(x). For a random variable
+X with distribution P ∈ P(µ,d,β), the tight lower bound follows from
+max
+P(x)⩾0
+� b
+a
+f(x)dP(x)
+s.t.
+� b
+a
+dP(x) = 1,
+� b
+a
+xdP(x) = µ,
+� b
+a
+|x − µ|dP(x) = δ,
+� b
+a
+1{x⩾µ}dP(x) = β.
+(EC.3)
+Consider the dual of (EC.3),
+min
+λ0,λ1,λ2
+λ0 + λ1µ + λ2δ + λ3β
+s.t.
+M(x) := λ0 + λ1x + λ2|x − µ| + λ31{x⩾µ} ⩽ f(x), ∀x ∈ [a,b].
+(EC.4)
+Here M(x) has both a ‘kink’ and a jump discontinuity at x = µ. Let the function M(x)
+touch the epigraph of f(x) in two points on opposite sides of µ. If we insert this knowledge,
+the constraints in the dual problem reduce to two equality constraints. From the Karush-
+Kuhn-Tucker conditions, we deduce the optimal tangent points:
+x1 = µ + δ
+2β ,
+x2 = µ −
+δ
+2(1 − β),
+which correspond to υ1 and υ2. Substituting this solution and solving for λ0,λ1,λ2 and λ3
+gives
+λ0 = f(υ2) + (λ1 − λ2)δ
+2(1 − β) − λ1µ,
+λ3 = f(υ1) − f(υ2) +
+λ2δ
+(1 − β) − (λ2 + λ1)δ
+2β(1 − β) ,
+and hence the optimal value is given by βf(υ1) + (1 − β)f(υ2). To ensure the solution is
+dual feasible, we assign suitable values to the two free decision variables. That is, we let
+λ1 + λ2 and λ1 − λ2 equal the slope of f(x) at x = υ1 and υ2, respectively. The optimal
+probabilities of (EC.3) are obtained by solving the linear system resulting from (EC.3).
+□
+EC.2.
+Known properties of MAD
+We recall some well-known properties of the MAD; see e.g. Ben-Tal and Hochman (1985).
+Denote by σ2 the variance of the random variable X, whose distribution is known to belong
+to the set P(µ,δ). Then
+δ2
+4β(1 − β) ⩽ σ2 ⩽ δ(b − a)
+2
+.
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec3
+In particular, since
+δ2 ⩽ 4β(1 − β)σ2 ⩽ σ2,
+it holds that δ ⩽ σ. For a proof, we refer the reader to Ben-Tal and Hochman (1985). For
+the distributions used in the paper, explicit formulas for δ are available:
+• Uniform distribution on [a,b]:
+δ = 1
+4(b − a)
+• Beta distribution with parameters k,λ on support [a,b]:
+δ =
+2kkλλΓ(k + λ)
+(k + λ)k+λ+1Γ(k)Γ(λ)(b − a)
+• Triangular distribution on [a,b] with mode c:
+δ =
+�
+�
+�
+�
+�
+2(b+c−2a)3
+81(a−b)(a−c),
+for a + b < 2c,
+2(a+c−2b)3
+81(a−b)(b−c),
+for a + b > 2c
+• Normal distribution N(µ,σ2):
+δ =
+�
+2
+πσ
+• Gamma distribution with parameters λ and k (for which µ = k/λ):
+δ =
+2kk
+Γ(k)exp(k)
+1
+λ.
+The MAD is known to satisfy the bound
+0 ⩽ δ ⩽ 2(b − µ)(µ − a)
+b − a
+.
+(EC.5)
+Let β = P(X ⩾ µ). For example, in the case of continuous symmetric distribution of X we
+know that β = 0.5. This quantity is known to satisfy the bounds:
+δ
+2(b − µ) ⩽ β ⩽ 1 −
+δ
+2(µ − a).
+(EC.6)
+EC.3.
+The knapsack problem
+The knapsack problem (Kellerer et al., 2004) is an integer programming problem and can
+be formulated as
+max
+x
+�
+i=1
+pixi
+s.t.
+n
+�
+i=1
+cixi ⩽ B,
+xi ∈ {0,1},
+1 = 1,...,n.
+(EC.7)
+
+ec4e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+for decision variable x, budget B, price p > 0 and costs c. Assume B < �n
+i=1 ci. The contin-
+uous version is obtained by considering the linear relaxation, i.e., we replace the integrality
+constraints by 0 ⩽ xi ⩽ 1, i = 1,...,n. The so-called greedy choice algorithm produces an
+optimal solution for the continuous knapsack problem.
+We first renumber the items xi such that p1/c1 ⩾ ... ⩾ pn/cn. Hence, the first item causes
+the largest increase in value relative to its costs. We now iterate over x1,...,xn and in each
+iteration, set xi to its maximum capacity. When the budget constraint is violated, set
+xi = B −
+i−1
+�
+i=1
+cixi.
+This greedy choice algorithm produces the optimal solution to (EC.7). Below we will state
+its proof, which is an adaptation from the proof in Kellerer et al. (2004).
+Assume that without loss of generality that p1/c1 > ··· > pn/cn. If we would have pi/ci =
+pi+1/ci+1 for some i, then we are indifferent between those items and the proof below can
+be easily adapted to satisfy this. The greedy choice algorithm produces a solution such
+that, for some index j, we have 1 = x1 = ··· = xj−1 > xj ⩾ xj+1 = ··· = xn = 0. Suppose
+we would have a different feasible optimal solution y ̸= x. Since pi > 0 and �n
+i=1 ci > B, it
+must hold that �n
+i=1 ciyi = B as otherwise we could spend additional capital to increase
+the optimal value. Because p1/c1 ⩾ ... ⩾ pn/cn, there exists a smallest index k such that
+yk < 1 and let l be the smallest index such that k < l and yl > 0. This solution must exists,
+else we would have y = x. Now, we will increase the value of yk and decrease the value of
+yl. By choosing ϵ = min{ck(1 − yk),clyl} > 0 and increasing yk by ϵ/ck and decreasing yl
+by ϵ/cl, we maintain feasibility and preserve �n
+i=1 ciyi = B. The solution value changes by
+pkϵ/ck −plϵ/cl = ϵ(pk/ck − pl/cl) > 0. This contradicts the assumption that y is an optimal
+solution. Therefore, x is optimal which concludes the proof.
+EC.4.
+DRO results
+In Ben-Tal and Hochman (1972), the following result was proved (for a much larger class
+of functions f(y,X) than in our case):
+Proposition EC.1. If f(y,·) is convex,
+sup
+P∈P(µ,δ)
+EP[f(y,X)] = gU(y) =
+�
+κ∈{1,2,3}n
+n
+�
+i=1
+p(i)
+κi f(y,ξ(1)
+κ1 ,...,ξ(n)
+κn ),
+(EC.8)
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec5
+with p(i)
+κi ,ξ(i)
+κi defined as in Lemma 2. If f(y,·) is concave,
+sup
+P∈P(µ,δ,β)
+EP[f(y,X)] = gL(y) =
+�
+κ∈{1,2}n
+n
+�
+i=1
+ˆp(i)
+κi f(y,υ(1)
+κ1 ,...,υ(n)
+κn ),
+(EC.9)
+with υ(i)
+1 = µi + δi
+2βi, υ(i)
+2 = µi −
+δi
+2(1−βi) and ˆp(i)
+1 = βi, ˆp(i)
+2 = 1 − βi.
+Hence, gU(·) in (EC.8) inherits the convexity in y from f(·,X) and its functional form
+depends only on the form of f(·,X) (and similarly for gL(·)). The upper and lower bound
+give a closed interval for
+ValP(y) = EP[f(y,X)]
+∀P ∈ P(µ,δ,β).
+(EC.10)
+Corollary EC.1. If f(y,·) is convex for all y then ValP(y) ∈ [gL(y),gU(y)] ∀P ∈
+P(µ,δ,β). If f(y,·) is concave for all y then ValP(y) ∈ [gU(y),gL(y)] ∀P ∈ P(µ,δ,β).
+From Proposition EC.1 we see that the extremal distribution is independent of y. Hence,
+we can substitute the 3n terms. This leads to a convex function in y, and hence the
+minimization problem over y is tractable.
+EC.5.
+Robust analysis with mean-variance knowledge
+EC.5.1.
+Scarf’s result for single item
+Scarf (1958) introduced a distribution-free analysis for the single-item newsvendor model
+by assuming that the decision maker only knows the mean and variance of the demand.
+Define the ambiguity set containing all distributions with the same mean and variance as
+P(µ,σ) := {P|EP(D) = µ, EP(D2) = σ2 + µ2}.
+Scarf (1958) determined an upper bound on the cost function C(q) by finding the worst-
+case distribution in the ambiguity set. To find the order quantity that protects against the
+ambiguity in P(µ,σ), the following minimax optimization problem is solved:
+min
+q
+max
+P∈P(µ,σ) dq + (m + d)EP(D − q)+.
+Since
+max
+P∈P(µ,σ) EP(D − q)+ ⩽
+�
+σ2 + (µ − q)2 + (µ − q)
+2
+,
+
+ec6e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+this minimax optimization problem becomes minq maxP CS(q) with
+CS(q) := d(q − µ) + (m + d)
+�
+σ2 + (µ − q)2 + (µ − q)
+2
+.
+(EC.11)
+and solution
+qS := argmin
+q
+CS(q) = µ + σ
+2
+��m
+d −
+�
+d
+m
+�
+.
+(EC.12)
+The quantity qS is known as Scarf’s order quantity which prescribes to order more than
+the expected demand when m > d, and less than the expected demand when d < m.
+EC.5.2.
+Gallego and Moon
+When the model is based on mean-variance information, Gallego and Moon (1993) formu-
+late the problem as
+min
+q CS(q) :=
+n
+�
+i=1
+ci
+�
+�di(qi − µi) + (mi + di)
+�
+σ2
+i + (qi − µi)2 − (qi − µi)
+2
+�
+�
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+(EC.13)
+q ⩾ 0.
+The optimal solution to problem (EC.13) is referred to as qS. Applying Scarf’s bound
+for each item individually results in (EC.13). Similar to the full information setting with
+a known distribution, this optimization problem can be solved with Lagrange multiplier
+techniques.
+EC.6.
+Additional numerical experiments
+This section presents additional numerical results. Section EC.6.1 presents the performance
+plots for the average margin setting. We compare the mean-MAD and mean-variance
+ordering policies in Section EC.6.2.
+EC.6.1.
+More mean-MAD results
+Figure EC.2 depicts the results for the average profitability scenario. A quick glance re-
+veals that these plots exhibit a different impression than the low profitability scenario. We
+conclude that the mean-MAD EVAI remains below some bound for budget levels ranging
+from zero to two-thirds of the maximum budget. For all cases, this bound on the EVAI
+is around 10%.As the budget passes two-thirds of the maximum budget, the performance
+starts to decrease. However, the mean-MAD-β EVAI decreases when approaching the max-
+imal budget.
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec7
+10
+20
+30
+40
+50
+0.024
+0.025
+0.026
+Case 1: Uniform (10,50)
+10 25 40 55 70 85 100
+0.0105
+0.0110
+0.0115
+Case 2: Uniform (10,100)
+10 40 70 100130160190
+0.0050
+0.0052
+0.0054
+Case 3: Uniform (10,200)
+10
+20
+30
+40
+50
+0.00
+0.03
+0.06
+0.09
+0.12
+Case 4: Beta (1,3)
+10
+20
+30
+40
+50
+0.00
+0.03
+0.06
+0.09
+0.12
+Case 5: Beta (2,2)
+10
+20
+30
+40
+50
+0.00
+0.03
+0.06
+0.09
+0.12
+Case 6: Beta (3,1)
+10
+20
+30
+40
+50
+0.00
+0.02
+0.04
+Case 7: Triangular c = (18)
+10
+20
+30
+40
+50
+0.00
+0.02
+0.04
+Case 8: Triangular c = (30)
+10
+20
+30
+40
+50
+0.00
+0.02
+0.04
+Case 9: Triangular c = (42)
+Figure EC.1
+Nine probability density functions used for multi-item performance analysis
+0
+314
+628
+942
+0.00
+0.07
+0.13
+0.20
+Case 1: Uniform (10,50)
+Mean-MAD
+Mean-MAD-
+0
+602
+1205 1808
+0.00
+0.07
+0.13
+0.20
+Case 2: Uniform (10,100)
+0
+1180 2360 3540
+0.00
+0.07
+0.13
+0.20
+Case 3: Uniform (10,200)
+0
+137
+275
+413
+0.00
+0.04
+0.08
+0.12
+Case 4: Beta (1,3)
+0
+263
+527
+791
+0.00
+0.08
+0.15
+0.23
+Case 5: Beta (2,2)
+0
+367
+735
+1103
+0.00
+0.07
+0.13
+0.20
+Case 6: Beta (3,1)
+0
+252
+505
+758
+0.00
+0.05
+0.11
+0.16
+Case 7: Triangular (10,50,18)
+0
+287
+574
+861
+0.00
+0.07
+0.14
+0.21
+Case 8: Triangular (10,50,30)
+0
+330
+661
+992
+0.00
+0.11
+0.21
+0.32
+Case 9: Triangular (10,50,42)
+Figure EC.2
+The results for the average margin setting. The x-axis corresponds to B and the y-axis to the EVAI.
+EC.6.2.
+Mean-variance comparison
+We start the performance analysis for the low margin scenario. The x-axis refers to the
+budget level B, and the y-axis refers to the EVAI. In each plot, the blue line corresponds
+
+ec8e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+to the EVAI for the mean-MAD model and the orange line to the mean-variance EVAI.
+Figure EC.3 contains the performance plots for each of the nine cases we are considering.
+0
+224
+449
+674
+0.00
+0.05
+0.09
+0.14
+Case 1: Uniform (10,50)
+Mean-MAD
+Mean-variance
+0
+401
+802
+1204
+0.00
+0.05
+0.09
+0.14
+Case 2: Uniform (10,100)
+0
+755
+1510 2265
+0.00
+0.05
+0.09
+0.14
+Case 3: Uniform (10,200)
+0
+70
+140
+211
+0.00
+0.06
+0.11
+0.17
+Case 4: Beta (1,3)
+0
+187
+374
+561
+0.00
+0.07
+0.13
+0.20
+Case 5: Beta (2,2)
+0
+312
+624
+937
+0.00
+0.06
+0.12
+0.18
+Case 6: Beta (3,1)
+0
+190
+380
+571
+0.00
+0.08
+0.15
+0.23
+Case 7: Triangular (10,50,18)
+0
+236
+472
+709
+0.00
+0.07
+0.13
+0.20
+Case 8: Triangular (10,50,30)
+0
+277
+554
+831
+0.00
+0.06
+0.11
+0.17
+Case 9: Triangular (10,50,42)
+Figure EC.3
+The results for the low margin scenario. The x-axis corresponds to the budget level and the y-axis
+to the EVAI.
+In Figure EC.3 we compare the mean-MAD policy with the mean-variance ordering
+policy in terms of EVAI for the scenario with low margins and a total of nine ground-
+truth demand distributions. While both policies generally give low EVAIs, the EVAI of
+the mean-variance policy is typically lower. We stress that this does not mean that the
+mean-variance policy is better. Indeed, a fair numerical comparison is impossible, as the
+respective ambiguity sets can contain vastly different distributions. While a finite variance
+excludes distributions with infinite-second moment, MAD does not. In general, the worst-
+case scenarios or extremal distributions are ‘more extreme’ for MAD than for variance.
+This also offers a possible explanation for the slightly higher EVAI.
+EC.7.
+Extensions
+We now present a distribution-free analysis for three extensions of the multi-item newsven-
+dor model. Section EC.7.1 deals with multiple constraints, Section EC.7.2 considers uncer-
+tain supply, and Section EC.7.3 discusses the risk-averse newsvendor where the conditional
+value at risk (CVaR) is chosen as objective function.
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec9
+EC.7.1.
+Multiple constraints
+Lau and Lau (1996) consider the newsvendor problem with multiple constraints, and pro-
+pose a numerical solution procedure that computes the Lagrange multipliers as roots of a
+system of nonlinear equations. Perakis et al. (2020) also consider multiple capacity con-
+straints in a retail environment, and distinguish between warehouse capacity and inventory
+availability constraints. By exploiting Lagrangian duality the problem is decomposed into
+two subproblems, which are solved iteratively by binary search.
+We now argue that the distribution-free analysis developed in the present paper also
+carries over to the setting with multiple constraints, and takes the form
+min
+q
+n
+�
+i=1
+ci
+�
+di(qi − µi) + (mi + di)
+�
+p(i)
+1 (ai − qi)+ + p(i)
+2 (µi − qi)+ + p(i)
+3 (bi − qi)+��
+s.t.
+n
+�
+i=1
+ci,jqi ⩽ Bj
+j = 1,...,m
+qi ⩾ 0
+i = 1,...,n.
+(EC.14)
+By introducing dummy variables τ (i)
+k , we reformulate problem (EC.14) as
+min
+q,τ
+n
+�
+i=1
+ci
+�
+di(qi − µi) + (mi + di)
+�
+p(i)
+1 τ (i)
+1 + p(i)
+2 τ (i)
+2 + p(i)
+3 τ (i)
+3
+��
+s.t.
+n
+�
+i=1
+ci,jqi ⩽ Bj,
+j = 1,...,m,
+τ (i)
+k ⩾ ξ(i)
+k − qi,
+k = 1,2,3; i = 1,...,n,
+τ (i)
+k ⩾ 0,
+k = 1,2,3; i = 1,...,n,
+qi ⩾ 0,
+i = 1,...,n,
+(EC.15)
+which remains a tractable LP, solvable for large-scale problems with interior-point meth-
+ods. Moreover, by solving the dual problem of (EC.15), shadow prices of the m budget
+constraints can be computed that quantify marginal expected net benefit of allocating an
+additional unit of budget to Bj, j = 1,...m.
+EC.7.2.
+Supply and demand uncertainty
+The newsvendor might take different decisions when the delivery of an order for q units is
+not necessarily complete (uncertain supply). K¨aki et al. (2015) consider uncertain supply
+and uncertain demand, when supply and demand are independent or follow a particular
+
+ec10e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+copula-based dependency structure. In the mean-variance setting and under the indepen-
+dence assumption, Gallego and Moon (1993) solve the distribution-free newsvendor prob-
+lem with random yield, but assume the yield is a binomial random variable that depends
+on the order size q. That is, when an order for q units is made, each individual unit is
+received with some fixed probability, or is not delivered at all.
+As opposed to Gallego and Moon (1993), we do introduce an ambiguity set for the
+random supply. Consider the setting with multiplicative yield Zi, where the random supply
+is given by Zi·qi. Assume Zi has mean ˜µi, MAD ˜δi and support [˜ai,˜bi], where 0 ⩽ ˜ai ⩽ ˜bi ⩽ 1.
+The distribution of Zi then resides in P(˜µi,˜δi). The extremal three-point distribution for Zi
+has probabilities
+˜p(i)
+1 =
+˜δi
+2(˜µi − ˜ai),
+˜p(i)
+2 = 1 −
+˜δi
+2(˜µi − ˜ai) −
+˜δi
+2(˜bi − ˜µi)
+,
+˜p(i)
+3 =
+˜δi
+2(˜bi − ˜µi)
+,
+and is supported on ζ(i)
+1 = ˜ai, ζ(i)
+2 = ˜µi, ζ(i)
+3 = ˜bi, respectively. The multi-item newsvendor
+with supply ambiguity is equivalent to
+min
+q
+n
+�
+i=1
+max
+P∈Pi EP
+�
+ci
+�
+di(Zi · qi − Di) + (mi + di)(Di − Zi · qi)+��
+s.t.
+n
+�
+i=1
+ciqi ⩽ Bj,
+j = 1,...,m,
+qi ⩾ 0,
+i = 1,...,n,
+(EC.16)
+with Pi := P(µi,δi) × P(˜µi,˜δi). Since the newsvendor problem is jointly convex in the pairwise
+independent random variables Di and Zi, the distributions that maximize the objective
+function of (EC.16) are the extremal three-point distributions. Applying these worst-case
+distributions to (EC.16) results in
+min
+q
+n
+�
+i=1
+ci
+�
+di( ˜µiqi − µi) + (mi + di)
+�
+κ∈{1,2,3}2
+p(i)
+κ1 ˜p(i)
+κ2τ (i)
+κ
+�
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+τ (i)
+κ ⩾ ξ(i)
+κ1 − ζ(i)
+κ2 qi,
+κ ∈ {1,2,3}2; i = 1,...,n,
+τ (i)
+κ ⩾ 0,
+κ ∈ {1,2,3}2; i = 1,...,n,
+qi ⩾ 0,
+i = 1,...,n.
+(EC.17)
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec11
+To demonstrate the distribution-free newsvendor with uncertain supply, consider the
+one-dimensional case with random demand D with a uniform distribution on [20,80] and
+multiplicative yield Z uniformly distributed on [0.65,0.95]. Figure EC.4 depicts the tight
+lower and upper bounds that follow from optimizing over the ambiguity sets that contain
+the distributions of D and Z. As the extremal distributions are discrete, the objective
+function of (EC.17) admits a piecewise linear representation.
+0
+20
+40
+60
+80
+100
+120
+Order quantity
+15
+20
+25
+30
+35
+40
+45
+50
+Expected costs
+Uniform distributions
+Mean-MAD upper bound
+Mean-MAD lower bound
+Figure EC.4
+Tight bounds for the multi-item newsvendor with uncertain supply yield, where m = 1 and d =
+0.8. The upper piecewise linear function is obtained by evaluating E[D − Z · q], with D following
+the extremal distribution that lies in P(50,15,20,80) and Z the worst-case three-point distribution in
+P(0.8,0.075,0.65,0.95). The lower bound follows from the best-case two-point distributions. The middle
+curve depicts the ‘true’ costs, where D has a uniform distribution on [20,80], and Z is uniformly
+distributed on [0.65,0.95].
+Because problem (EC.16) can be written in terms of a piecewise linear function, the
+optimal solution follows from a knapsack algorithm similar to Theorem 2. Further, one can
+gain additional insights by explicitly deriving the optimal order quantities for the robust
+single-item model, as in Theorem 1. The problem is similar for additive yield, also resulting
+in a three-point distribution for the worst case. Other directions for future research include
+solving (EC.16) with multiple unreliable and non-identical suppliers (Dada et al., 2007)
+and the newsvendor problem with fixed ordering costs and supplier capacity restrictions
+(Merzifonluoglu and Feng, 2014).
+
+ec12e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+EC.7.3.
+Risk aversion
+We next consider a risk-averse decision maker, as in Chen et al. (2010), who makes decisions
+based on CVaR. The decision maker no longer optimizes the expected costs, but instead
+minimizes the average value of the costs exceeding the γth-quantile of the newsvendor’s
+cost distribution. For the cost function G(q,D), CVaR can be calculated by solving a
+convex minimization problem (Rockafellar and Uryasev, 2000):
+min
+θ∈R
+�
+θ +
+1
+1 − γ E(G(q,D) − θ)+
+�
+.
+Calculating CVaR requires full knowledge of the demand distribution. However, in practice,
+committing to a particular distribution might be problematic for the decision maker if
+there is not enough data available. Hence, we consider the partial information setting as
+in Zhu and Fukushima (2009); Delage and Ye (2010), and seek to solve
+min
+q:�
+i ciqi⩽B,qi⩾0 max
+P∈P(µ,δ) min
+θ∈R
+�
+θ +
+1
+1 − γ EP(G(q,D) − θ)+
+�
+.
+(EC.18)
+Let us first consider the single-item model. Because the objective function of (EC.18)
+is finite, P(µ,δ) is weakly compact as supp(D) is compact, and the objective function of
+(EC.18) is linear in P and convex in θ, we are allowed to interchange the maximization and
+minimization operators by virtue of the minimax theorem (Shapiro and Kleywegt, 2002).
+Since (G(q,D) − θ)+ is a convex function of the uncertain demand, the three-point distri-
+bution (10) also maximizes EP(G(q,D)−θ)+. When β = P(D ⩾ µ) is known, the two-point
+distribution in Lemma 3 attains the matching lower bound. For the multivariate problem,
+notice that (G(q,D) − θ)+ is again a convex function of the uncertain demand, where
+D ∼ P ∈ P(µ,δ). By Proposition EC.1 and the reasoning above, the risk-averse newsvendor
+
+e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendorec13
+admits the following LP representation:
+min
+q,τ,η,θ θ +
+1
+1 − γ
+�
+κ∈{1,2,3}n
+n
+�
+i=1
+p(i)
+κi ηκ
+s.t.
+n
+�
+i=1
+ciqi ⩽ B,
+ηκ ⩾
+�
+n
+�
+i=1
+ci
+�
+di(qi − ξ(i)
+κi ) + (mi + di)τ (i)
+κ
+��
+− θ,
+κ ∈ {1,2,3}n,
+ηκ ⩾ 0,
+κ ∈ {1,2,3}n,
+τ (i)
+κ ⩾ ξ(i)
+κi − qi,
+κ ∈ {1,2,3}n; i = 1,...,n,
+τ (i)
+κ ⩾ 0,
+κ ∈ {1,2,3}n; i = 1,...,n,
+qi ⩾ 0,
+i = 1,...,n.
+(EC.19)
+We show in Figure EC.5 the bounds for the single-item model with demand having
+support [10,50], µ = 30, δ = 20/3 and β = 1/2. Solving (EC.19) for γ = 0.75,0.95 and
+different order sizes yields the upper bounds. We solve an analogous problem, but with the
+expectation taken over the extremal two-point distribution, stated in Lemma 3, to obtain
+the tight lower bounds. As a point of reference, we also plot the exact values of the CVaR
+and expected costs when D follows a symmetric triangular distribution on [10,50].
+10
+15
+20
+25
+30
+35
+40
+CVaR99%
+6
+8
+10
+12
+14
+16
+18
+20
+C(q)
+q = 10
+q = 20
+q = 30
+q = 40
+q = 50
+Triangular
+Mean-MAD upper bound
+Mean-MAD lower bound
+(a) Expected costs and CVaR
+10
+15
+20
+25
+30
+35
+40
+45
+50
+Order quantity
+5
+10
+15
+20
+25
+30
+CVaR75%
+Triangular
+Mean-MAD upper bound
+Mean-MAD lower bound
+(b) Mean-MAD bounds for CVaR
+Figure EC.5
+An illustration of the tight mean-MAD bounds for the risk-averse newsvendor with CVAR as objec-
+tive criterion, where m = 1, d = 0.8 and γ = 0.75,0.99. The middle curve corresponds to the CVaR
+when D follows a symmetric triangular distribution on [10,50]. The upper and lower bounds follow
+from optimizing over the ambiguity sets that contain this distribution.
+Solving (EC.19) can be challenging since the objective function (G(q,D) − θ)+ is no
+longer separable, thus resulting in an exponential number of variables and constraints. To
+
+ec14e-companion to Boonstra, van Eekelen, and van Leeuwaarden: Robust knapsack ordering for a partially-informed newsvendor
+alleviate this computational difficulty, one might resort to sampling-based procedures such
+as sample average approximation (Shapiro et al., 2009).
+We also mention ambiguous chance constraints that can be conservatively approximated
+by CVaR (Nemirovski and Shapiro, 2007). In the risk-averse newsvendor setting, the deci-
+sion maker introduces an ambiguous chance constraint that restricts the probability of the
+costs exceeding a certain threshold t to be less than 1 − γ, considering all distributions in
+the ambiguity set. For the multi-item setting, this means ensuring
+P(G(q,D) > t) ⩽ 1 − γ,
+∀P ∈ P(µ,δ),
+which is implied by
+max
+P∈P(µ,δ) CVaRγ[G(q,D)] ⩽ t.
+In addition, the newsvendor might require a minimal probability that all customer orders
+will be completely covered by the inventory on hand, i.e., the type-1 service level (Silver
+et al., 1998). When several of these probabilistic constraints are interrelated, the decision
+maker should conservatively approximate joint chance constraints. For this one can again
+use CVaR; see Chen et al. (2010); Zymler et al. (2013); Roos and den Hertog (2020). Adding
+ambiguous chance constraints to the models developed in this paper is a worthwhile topic
+for further research.
+

DNFQT4oBgHgl3EQf_zdP/content/tmp_files/2301.13459v1.pdf.txt

@@ -0,0 +1,1163 @@
+Learning Generalized Hybrid Proximity Representation for
+Image Recognition
+1st Zhiyuan Li
+Department of Computer Science
+University of Cincinnati
+Cincinnati, OH, United States
+li3z3@mail.uc.edu,
+2nd Anca Ralescu
+Department of Computer Science
+University of Cincinnati
+Cincinnati, OH, United States
+ralescal@ucmail.uc.edu
+Abstract—Recently, deep metric learning techniques received
+attentions, as the learned distance representations are useful to
+capture the similarity relationship among samples and further
+improve the performance of various of supervised or unsuper-
+vised learning tasks. We propose a novel supervised metric
+learning method that can learn the distance metrics in both
+geometric and probabilistic space for image recognition. In
+contrast to the previous metric learning methods which usually
+focus on learning the distance metrics in Euclidean space, our
+proposed method is able to learn better distance representation
+in a hybrid approach. To achieve this, we proposed a Generalized
+Hybrid Metric Loss (GHM-Loss) to learn the general hybrid
+proximity features from the image data by controlling the trade-
+off between geometric proximity and probabilistic proximity.
+To evaluate the effectiveness of our method, we first provide
+theoretical derivations and proofs of the proposed loss function,
+then we perform extensive experiments on two public datasets
+to show the advantage of our method compared to other state-
+of-the-art metric learning methods.
+Index Terms—Deep metric learning, proximity, probability
+distribution, representation learning, image classification
+I. INTRODUCTION
+Metric learning takes input data to learn the similar and
+dissimilar features between samples. The learned distance
+metric provides a meaningful and robust representation to
+discriminate the proximity or distance between samples and
+can be further utilized for both supervised and unsupervised
+learning tasks [1]. Recently, deep learning-based metric learn-
+ing algorithms, i.e., deep metric learning, were widely applied
+in the computer vision area by developing either a novel
+network architecture or an intuitive and efficient loss function
+[2]–[4]. Some typical works, such as the Siamese network [5],
+Triplet network [2], SupCon [6], aim to formulate an instance
+discrimination task to learn a useful feature representation
+by optimizing the proximity function in the Euclidean space,
+i.e., geometric distance or Cosine proximity between the
+feature embeddings. In this paper, we seek to address the
+inadequacies of geometric proximity of recent state-of-the-
+art metric learning methods by reconsidering an alternative
+Copyright (c) 2022 IEEE. Personal use of this material is permitted.
+Permission from IEEE must be obtained for all other uses, in any current or
+future media, including reprinting/republishing this material for advertising or
+promotional purposes, creating new collective works, for resale or redistribu-
+tion to servers or lists, or reuse of any copyrighted component of this work
+in other works.
+approach in which learned distance metrics are not biased to
+only geometric proximity.
+Metric learning methods have shown an excellent classifi-
+cation performance in image recognition applications, due to
+their extraordinary ability to discriminate similar information
+between samples [1], [4], [7], [8]. Such metric learning tasks
+can be either supervised or self-supervised. In supervised
+metric learning, the model learns to pull together the samples
+from the same classes and push away the samples from dif-
+ferent classes [9]. Self-supervised metric learning, also named
+contrastive learning, requires a data augmentation step to cre-
+ate some pseudo-ground-truth from the data itself, where the
+augmentation from the same sample is included in “positive”
+pairs and the augmentation from different samples is included
+in “negative” pairs [10]. Similar to supervised metric learning,
+the self-supervised model learns similar representations from
+the positive pairs and should be different than the representa-
+tions of the negative pairs. Various types of metric/contrastive
+learning works have been developed for image pattern recogni-
+tion applications, including image classification [6], [11]–[13],
+image clustering [14], [15], image segmentation [16], [17],
+image reconstruction [18], [19], and object detection [20]. All
+these works used geometric proximity (e.g., Cosine similarity
+or Euclidean distance) as the proximity function in the training
+an objective loss to learn the geometric representation of the
+samples. However, the probability distribution of the samples
+should not be ignored.
+To overcome these limitations and boost the prediction per-
+formance of metric learning, we proposed a novel supervised
+metric learning method to learn the hybrid proximity that
+combines the proximity in both geometric and probabilistic
+space. To achieve it, we defined a supervised Generalized
+Hybrid Metric Loss (GHM-Loss) to better learn the distance
+representations in both geometric and probabilistic space. We
+noticed that even if the geometric distance is small, the
+probabilistic distance can be large when the sample variance
+is large (Figure 1). This observation reminds us to reconsider
+that the model may not sufficiently learn the distance features
+based only on the geometric distance between data points.
+Thus, enabling the model to partially learn the probabilistic
+distance controls the trade-off between two types of distance
+representation (geometric and probabilistic).
+arXiv:2301.13459v1  [cs.CV]  31 Jan 2023
+
+𝑋2
+𝑋3
+𝐷 𝜇𝑋1, 𝜇𝑋2 > 𝐷 𝜇𝑋2, 𝜇𝑋3
+𝐾𝐿 𝑃𝑋1||𝑃𝑋2 < 𝐾𝐿 𝑃𝑋2||𝑃𝑋3
+𝑋1
+Fig. 1.
+Geometric distance vs. probabilistic distance between probability
+distributions X1, X2 and X3. Without considering the variances, the geomet-
+ric distance between mean values of µX1, µX2, and µX3 cannot represent
+their probabilistic distance. D: Geometric distance; KL: Kullback–Leibler
+divergence.
+Our proposed GHM-Loss is formulated by a hybrid di-
+vergence underlying the geometric and probabilistic space,
+which is a generalized distance loss form for many defined
+metric learning methods, including Triplet [2], N-pairs [21],
+Max Margin [22], NTXent [13], SupCon [6], etc. We first
+theoretically showing the advantage of using the probabilistic
+distance in metric learning compared to the geometric-based
+distance, and we employed two public datasets to show the
+effectiveness of our method compared to other state-of-the-
+art metric/contrastive learning methods. We also investigated
+the superiority of the proposed GHM-Loss with other met-
+ric/contrastive learning loss functions. To sum up, our main
+findings and contributions to this work are as follows:
+1) We propose a novel supervised metric learning method
+for enhancing the performance of image recognition
+by defining a Generalized Hybrid Metric Loss (GHM-
+Loss). The proposed GHM-Loss is able to learn better
+distance representation that controls the trade-off be-
+tween the geometric-based and the probabilistic distance
+from feature embeddings.
+2) We define two proximity functions with certain proper-
+ties in geometric and probabilistic space, respectively,
+and provide proof for each property. Meanwhile, we
+theoretically show the advantage of the GHM-Loss by
+including the probabilistic proximity for learning the
+distance between distributions.
+3) Our approach is supported both by a theoretical discus-
+sion and by extensive experiments performed on two
+common image classification tasks to demonstrate the
+effectiveness of our method compared to other state-of-
+the-art metric learning methods.
+II. RELATED WORK
+In this section, we first discuss some state-of-the-art meth-
+ods of deep metric learning and some of its applications in
+the computer vision domain. We further review related works
+on contrastive learning.
+A. Metric Learning
+1) Traditional Metric Learning: The early stage of the ma-
+chine learning techniques requires a hand-crafted processing
+step, i.e., feature engineering, such as feature selection and
+feature extraction before training a machine learning model
+for supervised (e.g., classification) or unsupervised learning
+(e.g., clustering) tasks [7], [23], [24]. These methods, including
+linear projections, i.e., principal component analysis (PCA)
+[25], decomposition, i.e., non-negative matrix factorization
+(NMF) [26] to extract useful feature information and are not
+directly within the classification structure, resulting in a limited
+performance on the certain complex structure, such as high-
+dimensional data and non-linearity. Unlike traditional machine
+learning approaches, metric learning performs the learning
+process on the data to learn a distance feature representation
+by decreasing the distance between similar samples and in-
+creasing the distance between dissimilar ones in a embedding
+space. The learned distance features will have a high ability
+to discriminate the classes of the sample data. Usually, metric
+learning approaches apply linear transformation techniques
+to the input data and map it to a new feature space with
+a higher-class separation [27]. However, these methods lack
+the generalization capability and nonlinear knowledge of the
+attributes [28].
+2) Deep Metric Learning: Unlike traditional metric learn-
+ing methods, deep metric learning relies on training deep
+neural networks with activation functions that capture nonlin-
+ear properties [4], and it has dominated metric representation
+learning in the image recognition community [2], [3], [5], [6],
+[29]–[31]. For example, the Siamese network [5] used two
+identical convolutional neural networks (CNNs) to encode a
+pair of input samples and minimize the contrastive loss to learn
+the representative distance features. Similar to the Siamese
+network, Hoffer et al [2] proposed a Triplet network, including
+the anchor, positive (similar), and negative (dissimilar) sample,
+which learns the inequality that the positive sample stays closer
+to the anchor compared to the negative sample. Afterward,
+Wang et al [3] defined a new Angular loss to constrain
+the angle at the negative sample from the Triplet network.
+Later, Sohn [21] proposed an N-pair loss to address the
+slow convergence problem of the Triplet loss. More recently,
+Khosla et al [6] developed a supervised contrastive learning
+framework with the more general form of metric learning loss,
+i.e., SupCon, and showed the effectiveness of classification
+performance compared to the Triplet loss and the N-pair loss.
+These existing works have shown great promise for metric
+and feature representation learning in a variety of image
+classification tasks. Nevertheless, to the best of our knowledge,
+most previous studies are focused on learning the geometric-
+based metrics of the embedding space, while the probabilistic-
+based metrics are usually ignored. In this work, our method
+is able to learn the meaningful metrics of both geometric and
+probabilistic space.
+
+B. Contrastive Representation Learning
+The main purpose of deep metric learning and contrastive
+learning is to train a deep learning model to learn the distance
+feature representations in an embedding space. The main
+difference between these two methods is that contrastive learn-
+ing is closely related to the self-supervised learning domain,
+which contains a data augmentation step to generalize an
+arbitrary number of positive and negative sample pairs from
+each sample [32]. Given the stunning achievement of self-
+supervised representation learning, many contrastive learning
+methods have been developed for various computer vision
+tasks [33]–[36]. For example, Ye et al [37] proposed an
+embedding contrastive learning method with the Siamese
+network to learn the invariant features of embedding space.
+Chen et al [13] developed a famous contrastive learning
+framework, SimCLR, in which the model is pretrained to
+discriminate the positive pairs of data augmentation from the
+same source image, demonstrating superior performance in
+ImageNet classification. Similarly, He et al [12] proposed the
+Moco v1, to maximize the proximity between the positive
+pairs based on the monument network encoder. Additional
+studies that are similar to SimCLR and Moco, including
+BYOL [38], SimSam [39] Barlow Twins [40], etc., show the
+exceptional performance of learned feature representation for
+further supervised or unsupervised tasks.
+…
+𝑥1
+𝑥2
+𝑥𝑁−1
+𝑥𝑁
+𝐟1
+𝐟2
+𝐟𝑁−1
+𝐟𝑁
+…
+𝐟1
+𝐟2
+𝐟𝑁−1
+𝐟𝑁
+Pull
+Push
+Learning Hybrid Metrics
+Softmax
+Encoder F(∙; 𝜽)
+Classification
+Normal
+Normal
+…
+Fibrosis
+Glaucoma
+MLP
+Feature Extraction
+MLP
+Input
+𝐿2
+Fig. 2. The overview of our proposed framework (example of fundus disease
+diagnosis). We use a pretrained convolutional neural network (CNN) and a
+multi-layer perceptron (MLP) to encode each image to the embedded feature.
+Afterward, we propose a metric learning branch that is supervised with the
+proposed GHM-Loss which is trained together with the cross-entropy loss of
+a classification branch in a multi-task scheme.
+III. METHODOLOGY
+A. Overview
+Our proposed supervised metric learning framework is il-
+lustrated in Figure 2. We first denote a training image dataset
+D = {xi, yi}N
+i=1, where yi is the label of image xi, a set
+of indices of all the positive samples for a randomly selected
+image xi in a batch, U(i) = {j ∈ Θ|yj = yi, j ̸= i}, a set of
+indices of all negative samples for a randomly selected image
+xi in a batch, V (i) := {j ∈ Θ|yj ̸= yi, i ̸= j}. The problem
+formulation is to learn a network F(·; θ) that maps each input
+xi to a L2 normalized d-dimensional feature embedding fi,
+such as fi = F(xi; θ) ∈ Rd. To achieve this, we use a pre-
+trained CNN, i.e., ResNet18, followed by a MLP to produce N
+high-level feature vectors and perform two supervised learning
+branches. The first branch is a metric learning task, which aims
+to learn the robust metrics by pulling all the samples with
+indices U(i) and pushing away all the samples with indices
+V (i). Meanwhile, the embedded feature fi is connected to
+another MLP layer with a Softmax to generate the predicted
+probability for the class label yi and is supervised with cross-
+entropy loss to perform a classification task. Below, we will
+elaborate on the procedure of the each branch, including the
+definition of GHM-Loss and its advantage, and other network
+details.
+B. Generalized Hybird Metric Loss
+1) General Loss Form: To perform the metric learning
+branch, we propose a general form metric loss function,
+in which the network can learn the proximity information
+between the embedded features {f1, f2, . . . , fN} by optimizing
+the loss. Let S(·) denote the proximity function for two input
+vectors fi and fj. That is, for fi, fj ∈ Rd, S(fi, fj) : Rd → R1.
+Thus, the probability of xi, xu, u ∈ U(i) is being recognized
+as yi is defined by
+p(yi|xi, xu) =
+exp [S(fi, fu)]
+�
+j��Θ,j̸=i exp [S(fi, fj)]
+(1)
+On the other hand, the probability of xi, xv, v ∈ V (i) is not
+being recognized as yi is defined by
+p(yi|xi, xv) =
+exp [S(fi, fv)]
+�
+j∈Θ,j̸=i exp [S(fi, fj)]
+(2)
+Next, assume that all the probabilities of different images be-
+ing recognized as image xi are independent, let q(yi|xi, xv) =
+1−p(yi|xi, xv) thus, the objective likelihood function that we
+are interested is defined by
+ℓi =
+�
+u∈U(i)
+�
+v∈V (i)
+p(yi|xi, xu)q(yi|xi, xv)
+(3)
+Correspondingly, the negative log likelihood over all the data
+points indexed by Θ yields:
+L∗ = −
+�
+i∈Θ
+∥V (i)∥
+�
+u∈U(i)
+log p(yi|xi, xu)
+−
+�
+i∈Θ
+∥U(i)∥
+�
+v∈V (i)
+log q(yi|xi, xv)
+(4)
+where ∥U(i)∥ and ∥V (i)∥ denotes the size of the set U(i)
+and V (i), respectively.
+2) Geometric Proximity: We first consider the proximity
+in the geometric space. Given a pair vectors fi and fj, the
+proximity function Sg(fi, fj) satisfies the following properties:
+1 Sg(fi, fj) ∈ [0, 1];
+2 Sg(fi, fj) = Sg(fj, fi);
+3 ∀c ∈ [fi, fj], Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}.
+The proximity measures that satisfy the above properties
+include Cosine similarity.
+
+Proof. Using Cosine similarity as the proximity metrics, such
+that Sg(fi, fj) = fi · fj/∥fi∥∥fj∥ satisfies each property above.
+1. Obviously, Sg(fi, fj) ∈ [0, 1].
+2. Obviously, this property is true.
+3. Let c ∈ [fi, fj], we have |fi−c| ≤ |fi−fj|, |c−fj| ≤ |fi−
+fj|, which means that Sg(fi, c) ≥ Sg(a, b), Sg(c, fj) ≥
+S(a, b). Thus, Sg(fi, fj) ≤ min{Sg(fi, c), Sg(c, fj)}∀c ∈
+[fi, fj].
+3) Probabilistic Proximity: Instead of only using geometric
+proximity, which ignores the sampling probability distribution,
+we consider the probabilistic proximity to summarize the
+distribution of the embedded features {fi, f2, . . . , fN}. Given
+a pair vectors fi and fj, with size of |fi|, |fj|, the probabilistic
+proximity function satisfies the following properties:
+1. Sp(fi, fj) ∈ [0, 1];
+2. Sp(fi, fj) = Sp(fj, fi);
+3. Sp(fi, fj) = 0 if and only if fi = fj;
+4. Sp(fi, fj) ≤ Sp(fi, fc) + Sp(fc, fj) under the certain
+condition, in which |fc| = |fi| = |fj|.
+We use a Gaussian mixture model (GMM) to represent the
+empirical distribution fi, which is defined by
+p(fi) =
+�
+k∈K
+wkN(fi; µk, σ2
+k)
+(5)
+where wk is a latent variable followed by a categorical distri-
+bution, denoting the k-th component, and N is the Gaussian
+probability density function with parameters µk and σk, which
+is defined as
+N(fi; µk, σ2
+k) =
+1
+�
+2πσ2
+k
+exp
+�
+− 1
+2σ2
+k
+(fi − µk)2
+�
+(6)
+Using this model, the probabilistic distance between p(fi)
+and p(fj) is chosen with the symmetric divergence, i.e.,
+Jensen–Shannon (JS)-divergence. For simplicity, we use pi
+and pj to denotes the probability distributions of fi and fj,
+respectively. Therefore, the Sp(fi, fj) is denoted by
+Sp(fi, fj) = 1
+2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)]
+(7)
+where ¯pij = (pi + pj) /2, dKL(·) presents a function of the
+Kullback–Leibler (KL)-divergence.
+Proof. We prove that Sp(fi, fj) satisfies the properties of
+defined probabilistic proximity function.
+1. The range of the JS-divergence is within 0 and 1, thus
+Sp(fi, fj) ∈ [0, 1] is true.
+2. Obviously, based on Eq (7), it is easy to have Sp(fi, fj) =
+1
+2 [dKL(pi∥¯pij) + dKL(pj∥¯pij)] = Sp(fi, fj).
+3. Sp(fi, fj) ≥ 0, as a sum of nonnegative terms. To have
+Sp(fi, fj) = 0, each term of Sp(fi, fj) must be 0. Thus,
+Sp(fi, fj) = 0 if and only if dKL(pi∥¯pij) = dKL(pj∥¯pij).
+Since dKL(pi∥pj) = 0 if and only if pi = pj, thus,
+Sp(fi, fj) = 0 if and only if fi = fj.
+Now we prove property 4. Using the Shannon entropy,
+H(pi) = − �
+pi∈pi pi log pi, the explicit form of Sp(fi, fj)
+can be written as
+Sp(fi, fj) = H(¯pij) − 1
+2 [H(pi) + H(pj)]
+Assume that H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc), thus,
+Sp(fi, fc) + Sp(fc, fj) − Sp(fi, fj) can be rewritten as
+H(¯pic) − H(¯pc) + H(¯pcj) − H(¯pij) ≥ 0
+Thus, Sp(fi, fc) + Sp(fc, fj) ≥ Sp(fi, fj) is true if and only if
+H(¯pic) + H(¯pcj) ≥ H(¯pij) + H(¯pc).
+C. Learning Hybrid Proximity
+1) Generalized Hybrid Metric Loss: The learning objective
+loss function of the metric learning branch is the convex
+combination of geometric proximity loss and probabilistic
+proximity loss. As such, the objective is denoted by
+L∗
+GHM = λL∗
+g − (1 − λ)L∗
+p
+(8)
+where λ ∈ [0, 1] indicates the weighting factor to control the
+geometric proximity loss, L∗
+g, and the probabilistic proximity
+loss, L∗
+p. In this way, the network is able to capture both
+geometric and probabilistic information during the training
+process.
+2) Comparing With Geometric Proximity: To show the
+advantage of including the probabilistic proximity loss in the
+metric learning branch using the probabilistic view, we com-
+pare the geometric proximity and the probabilistic proximity
+between two probability distribution.
+Consider a KL-divergence between fi and fj. For simplicity,
+we use p(x) and q(x) to represent p(fi) and p(fj), respectively,
+and assume x ∼ N(µ, σ2). Thus, the expanded form of
+dKL(p∥q) for two Gaussians is denoted as
+dKL(p∥q) =
+�
+x
+p(x) log p(x) dx −
+�
+x
+p(x) log q(x) dx
+(9)
+In here, we derive the result using the fact of [41]. For the
+first term,
+�
+x p(x) log p(x) dx can be expanded as
+−
+�
+x
+p(x) log
+�
+2πσ2p dx −
+�
+x
+p(x)(x − µp)2
+2σ2p
+dx
+= − log
+�
+2πσ2p −
+1
+2σ2p
+�
+x
+p(x)(x − µp)2 dx
+(10)
+Next, we expand the quadratic form:
+− log
+�
+2πσ2p −
+1
+2σ2p
+�
+Ep(x2) − Ep(x)2�
+= − log
+�
+2πσ2p − 1
+2
+(11)
+Following the same derivation,
+�
+x p(x) log q(x) dx can be
+expand by
+�
+x
+p(x) log q(x) dx = − log
+�
+2πσ2q −
+�
+σ2
+q + (µp − µq)2�
+2σ2q
+(12)
+
+Assume σ2
+p = σ2
+q = c, c is a constant, based on Eq (10)-(12),
+the KL-divergence between p(x) and q(x) is given by
+dKL(p∥q) = −1
+2 − 1
+2c + 1
+2c (µp − µq)2
+(13)
+that is a linear function consisting of L2 distance between
+two mean values, showing that the probabilistic proximity also
+considers the variation of the sampling distribution, while the
+geometric proximity does not. This derivation also supports
+the phenomenon in Figure 1.
+D. Network Implementation Details
+As illustrated in Figure 2, the proposed framework consists
+of a feature extraction backbone and two supervised learning
+branches: one for metric learning and one for classification. We
+used a pretrained ResNet18 [42], following the same setting as
+the previous work [36]. We used max pooling on the attention
+map after the last layer of the residual block in ResNet18.
+Then, we flatted the output to a vector and sequentially connect
+it with a MLP layer, batch normalization, and ReLU to reduce
+the feature dimension to 128. Next, each fi 1) was connected
+with a L2 normalization layer, i.e., ∥fi∥ = 1 to calculate the
+hybrid proximity of the metric learning branch, and 2) connect
+to another MLP layer and Softmax for classification.
+The classification branch is to take the input batch {x}b
+i=1 to
+generate a prediction output. We optimized the cross-entropy
+loss, LCE, together with the metric learning loss, L∗
+GHM, in
+a multi-task learning scheme. Thus, we defined our total
+objective loss as the weighted combination of a metric learning
+branch and a classification branch. The learning objective loss
+is denoted by
+Ltotal = βL∗
+GHM + LCE
+(14)
+where β indicates the weighting factor to control the impor-
+tance of the GHM-Loss In our experiments, we set β = 1 and
+λ = 0.5, we also analyze the effects of both β and λ using
+a grid search. Each input image of a batch was randomly
+scaled within a factor range of [0.3, 1.0], and cropped into
+patches of size 224 x 224. We set the batch size b = 8 in
+the experiment and trained our framework using an Adam
+optimization, the learning rate and weight decay are set to
+0.0001. We train our network for 2000 epochs. The whole
+framework was implemented using python 3.8, Scikit-Learn
+0.24.1, Pytorch 1.9.1, and Cuda 11.1 with a NVIDIA GeForce
+GTX 1660 SUPER GPU.
+IV. DATA AND EXPERIMENTS
+A. Datasets
+To show the effectiveness of our method, same as Li
+et al [36], we perform two binary (normal and abnormal)
+classification tasks by diagnosing pathological myopia (PM)
+and age-related macular degeneration (AMD) on two public
+ophthalmic disease datasets of iChallenge-PM and iChallenge-
+AMD.
+1) iChallenge-PM: iChallenge-PM [43] contains 1200 an-
+notated retinal fundus images in which 50% are PM subjects.
+More details of the iChallenge-PM dataset can be found on
+the [43]. We perform a 10-fold cross-validation to evaluate our
+method.
+2) iChallenge-AMD: There is a total of 1200 color fundus
+images of the iChallenge-AMD dataset [44], in which 77% are
+non-AMD subjects and 23% are AMD subjects. It provides the
+disc boundaries and fovea locations, as well as the boundaries
+of kinds of lesions. More details of the iChallenge-AMD
+dataset can be found on [44]. Note, that we only used the
+training dataset (400 fundus images) since only the training
+dataset is released with annotations. We perform a 10-fold
+cross-validation to evaluate our method.
+B. Model Comparison Setting
+1) Evaluation Metrics: We used AUC, accuracy, precision,
+recall, and F1-score to assess the classification performance.
+AUC stands for Area Under the Receiver Operating Character-
+istic (ROC) curve. The definition of accuracy, precision, recall,
+and F1-score are denoted by:
+Accuracy = (TP + TN)/(TP + TN + FP + FN)
+Precision = TP/(TP + FP)
+Recall = TP/(TP + FN)
+F1 = 2 ∗ (Precision ∗ Recall)/(Precision + Recall)
+where TP, TN, FP, and FN indicate the true positive, true
+negative, false positive, and false negative, respectively.
+To provide the statistical analysis of our method, we con-
+ducted a non-parametric Wilcoxon test [45] with a α level of
+0.05. A p-value less than 0.05 is considered as statistical sig-
+nificant for all inference. All statistical tests in the experiments
+were performed using R-4.0.3 (RStudio, Boston, MA, USA).
+2) Competing State-of-the-Art Methods: To have a fair
+comparison, we trained all peer methods with the pretrained
+ResNet18 with the same hyperparameters, network architec-
+tures, and optimizer under the 10-fold cross-validation. Since
+our framework consists of metric learning and classification
+branches, we fix the classification branch and only modify
+the metric learning part when compared with other metric
+learning methods in the experiment. Our proposed method was
+compared with other deep metric learning methods, Siamese
+[5], Triplet [2], SupCon [6], N-pair [21], and InfoNCE [46].
+We run these metric learning methods with the code released
+on iChallenge-PM and iChallenge-AMD datasets. We also
+provided a supervised ‘Baseline’ method by modifying the
+output layer of the last fully connected layer of the ResNet18
+to 2 and trained with cross-entropy loss.
+C. Comparison on the iChallenge-PM Dataset
+We compared with other state-of-the-art methods on the
+iChallenge-PM Dataset. The results are shown in Table I.
+We found that each method can achieve over 95% prediction
+performance on all evaluation metrics, which indicates that the
+patterns of pathological myopia in color fundus images are
+
+obvious. We can see that N-pair [21] achieved a limited result
+and is due to this method requires large, annotated training
+data that may not be suitable for the color fundus images.
+Notably, our method significantly outperformed other peer
+metric learning methods with 99.08% (p<0.0001) on AUC
+and 99.01% (p<0.0001) on accuracy for PM diagnosis. These
+results further demonstrate the effectiveness of our method
+compared to other state-of-the-art metric learning methods.
+TABLE I
+MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS
+ON THE ICHALLENGE-PM DATASET (UNIT: %).
+AUC
+Accuracy
+Precision
+Recall
+F1
+Baseline
+96.01
+95.45
+94.51
+97.25
+95.34
+Siamese [5]
+97.45
+97.30
+96.15
+96.60
+96.58
+Triplet [2]
+97.95
+98.64
+97.49
+96.14
+97.21
+SupCon [6]
+98.06
+98.22
+98.36
+97.29
+97.64
+N-pair [21]
+95.36
+95.83
+96.41
+97.25
+96.12
+InfoNCE [46]
+98.11
+97.91
+96.83
+97.59
+97.36
+Ours
+99.08
+99.01
+98.08
+99.12
+98.40
+D. Comparison on the iChallenge-AMD Dataset
+We compared with other state-of-the-art methods on the
+iChallenge-AMD Dataset. As shown in Table II, we can
+see that our method achieved the best prediction performance
+among other competing metric learning methods. Compared
+to the second-best method, InfoNCE [46], our method signif-
+icantly improved the performance, i.e., 78.69% vs. 76.75%
+(p<0.0001) on AUC and 88.04 % vs. 86.51% (p<0.0001)
+on accuracy. Notably, our method also outperformed the
+supervised ‘Baseline’ method on all evaluation metrics. These
+results demonstrated the effectiveness of the proposed method.
+TABLE II
+MODEL COMPARISONS WITH OTHER DEEP METRIC LEARNING METHODS
+ON THE ICHALLENGE-AMD DATASET (UNIT: %).
+AUC
+Accuracy
+Precision
+Recall
+F1
+Baseline
+76.51
+84.16
+82.54
+76.18
+78.86
+Siamese [5]
+67.58
+82.45
+72.54
+68.26
+70.14
+Triplet [2]
+69.52
+84.29
+76.87
+72.48
+73.21
+SupCon [6]
+73.24
+85.64
+78.42
+74.15
+76.05
+N-pair [21]
+69.58
+83.41
+75.14
+70.54
+71.86
+InfoNCE [46]
+76.75
+86.51
+85.36
+72.35
+77.95
+Ours
+78.69
+88.04
+82.95
+75.28
+78.24
+E. Comparison with Transfer Learning Models
+To show the robustness of learned features of our method,
+we compared our method with the ImageNet pretrained mod-
+els, including VGG-19 [47], InceptionNet v1 [48], and Effi-
+cientNet B0 [49] on the iChallenge-AMD dataset. We modified
+the output channel of the last fully connected layer in each
+pretrained model to 2 and trained them with cross-entropy
+loss. To have a fair comparison, all the models were trained
+with the same number of epochs, learning rate, and weight
+decay term on a 10-fold cross validation. The results are shown
+in Table III. We can see that Efficient Net achieves the best
+prediction performance among the transfer learning models.
+Compared to Efficient Net, it is observed that our method can
+achieve a higher prediction performance with around 1.5%
+(p<0.0001) on AUC and 7% (p<0.0001) on accuracy. Note,
+we trained our method with only 400 color fundus images
+and performed better than ImageNet models, which were
+pretrained with more than 1 million natural images. With this
+observation, the results further show the practical value of our
+method.
+TABLE III
+MODEL COMPARISONS WITH IMAGENET TRANSFER LEARNING MODELS
+ON THE ICHALLENGE-AMD DATASET (UNIT: %).
+AUC
+Accuracy
+Precision
+Recall
+F1
+VGG-19 [47]
+74.14
+81.52
+76.54
+72.36
+73.89
+Inception v1 [48]
+76.32
+77.35
+78.39
+75.54
+76.28
+Efficient B0 [49]
+77.25
+81.52
+80.32
+79.25
+79.52
+Ours
+78.69
+88.04
+82.95
+75.28
+78.24
+F. Analytical Study
+TABLE IV
+THE IMPORTANCE OF THE GHM-LOSS IN THE METRIC LEARNING
+BRANCH ON THE ICHALLENGE-AMD DATASET (UNIT: %).
+AUC
+Accuracy
+Precision
+Recall
+F1
+β = 0.0
+75.41
+83.21
+80.54
+72.88
+76.15
+β = 0.5
+76.85
+85.42
+80.95
+74.54
+77.28
+β = 1.0
+78.69
+88.04
+82.95
+75.28
+78.24
+β = 2.0
+72.45
+79.41
+77.66
+70.23
+73.59
+1) Importance of the GHM-Loss: The proposed method
+consists of metric learning branch and classification branch in
+a multi-task scheme, in which we trained GHM-Loss together
+with the cross-entropy loss. In this section, we analyzed
+the importance of the GHM-Loss of our method on the
+iChallenge-AMD dataset. We first fix the λ = 0.5 in GHM-
+Loss and trained our framework with different β in Eq (14),
+where β is the importance of the metric learning branch.
+β = 0.0 denotes that the framework is only trained with
+the cross-entropy loss. As β increases, the more weight or
+importance of the GHM-Loss in the network training.
+The results are shown in Table IV. As we can see, when
+β = 0.0, the network only learns the classification branch
+and the result is 75.41% on AUC and 83.21% on accuracy.
+As β increases, we found that the prediction performance
+improves to the best when β reached 1 (e.g., 78.69% on AUC,
+88.04% on accuracy). However, when β continues increasing,
+the prediction performance starts to drop apparently from
+78.69% to 72.45% on AUC. The comparison shows that both
+the metric learning branch and classification branch equally
+contributed to our framework for PM diagnosis.
+2) Effects of Weighting Factors in the GHM-Loss: We
+analyzed the effects of the weighting factors, i.e., β, λ in
+the GHM-Loss on the iChallenge-AMD Dataset, in which
+β indicates the importance of the metric learning branch of
+our method and λ controls the weight size between geomet-
+ric proximity and probabilistic proximity of the GHM-Loss.
+Note, λ = 0.0 denotes that only probabilistic proximity was
+
+considered between fi and fj. As we can see in Figure 3,
+for each fix β, the classification performance increases to the
+best performance when λ is reached 0.5 and drops apparently
+as it continues to increase. These results demonstrate that 1)
+both metric learning and classification branches are useful of
+our method and 2) both geometric and probabilistic proximity
+should be captured between fi and fj in the training.
+Fig. 3.
+Classification performance comparison on the iChallenge-AMD
+Dataset with different weighting factors β and λ of the GHM-Loss. We use
+AUC to choose the optimal β and λ using a grid search.
+3) Visualization of the Feature Distribution: We visualized
+the feature embedding distribution, i.e., f1 (red line) and f2,
+after ResNet18 for a positive pair color fundus image on
+the iChallenge AMD dataset. The feature distributions are
+shown in Figure 4. Before optimization, we can see that the
+distribution of feature embeddings from a positive pair sample
+are independent without overlaps. However, the probabilistic
+distance of f1 (red line) and f2 is reduced and stays close
+to each other after optimizing the network. Since we use
+GMM to approximate the empirical distribution of each feature
+embedding, the probability parameters of µ and σ of all
+the images with the same label should be closed to each
+other, thus, resulting in the similar probability densities. This
+visualization also demonstrate that the proposed GHM-Loss
+can efficiently capture the probabilistic patterns during the
+training process.
+V. DISCUSSION
+Metric learning is an important technique in visual repre-
+sentation area by learning the distance metric, which can be
+further used to perform supervised and unsupervised learning
+tasks, such as image classification [6], [12], [13], image
+clustering [14], [50], and object detection [20], [51], etc. With
+the advances of deep learning techniques, deep metric learn-
+ing has been widely studied in the metric learning research
+community. Although promising results were obtained on
+previous works [2], [5], [6], [21], [46], these methods usually
+ignore the probability distribution of the feature embeddings
+during the training process, which may lead an inaccurate
+Before Optimization
+After Optimization
+Fig. 4. The feature distribution between a positive pair of f1 (red line) and
+f2 (blue line) of color fundus images during the training process on the
+iChallenge AMD dataset. We applied Gaussian mixture model (GMM) to
+approximate the empirically distribution of these features. The probabilistic
+proximity between f1 (red line) and f2 are reduced after optimization.
+prediction. In this work, we present a novel supervised metric
+learning method that consists of learning both geometric and
+probabilistic proximity for image recognition. We formulate a
+Generalized Hybrid Metric Loss (GHM-Loss) to better learn
+the distance representation, where geometric-based distance
+and probabilistic-based distance are learned. Our method is
+validated on two public ophthalmic disease datasets (e.g.,
+iChallenge-PM and iChallenge-AMD), in which our method
+can significantly outperform other state-of-the-art metric learn-
+ing methods. With a convex combination of the geometric
+proximity and probabilistic proximity, our method consistently
+achieves the best prediction performance than the individual
+proximity.
+Although our method outperforms other state-of-the-art
+metric learning methods, it comes with limitations. Our
+method is a supervised learning approach, which relies on
+a large number of annotated training data, and it is costly
+to obtain. In future, we will investigate the unsupervised
+metric learning approach or self-supervised learning approach
+to address the human effort issue on image recognition com-
+munities. The exploration of probabilistic unsupervised/self-
+supervised metric learning would be our future work.
+VI. CONCLUSION
+In this paper, we present a novel supervised metric learning
+method for image recognition. Our main idea is to learn a
+hybrid proximity that consists of both geometric-based metric
+and probabilistic-based metric. The geometric proximity of
+proposed GHM-Loss helps the model learn the similarity
+information under the Euclidean space and the probabilistic
+proximity proposed GHM-Loss learns the similarity under the
+empirical probability distribution. With extensive experiments,
+our method consistently achieves the excellent prediction
+performance compared with the other state-of-the-art metric
+learning methods, showing the effectiveness of learned dis-
+tance features of our method in image recognition.
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+

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+Disorder-induced finite center-of-mass momentum Cooper pairing and its consequences to the
+critical temperature and superconducting gap of overdoped cuprates
+Victor Velasco1 and Marcello B. Silva Neto1
+1Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, Brazil
+One of the most studied classes of unconventional high-temperature superconductors is the hole-doped
+cuprates, where special attention is given to those doped with extra interstitial oxygens. In this context, the
+formation of spatially inhomogeneous agglomerates of dopant oxygen atoms in the form of nanosized puddles
+is not only relevant, but also subject of intense recent experimental and theoretical surveys. Following these
+efforts, in this work we show the consequences of the presence of networks of oxygen puddles in the supercon-
+ducting state of overdoped cuprates. Starting from the inhomogeneous disordered background brought by the
+network of puddles, we show that an effective interaction between electrons can be mediated by the local vibra-
+tional degrees of freedom of each puddle, but the pairs arising from this interaction have a finite center-of-mass
+momentum p, thus breaking up the Cooper channel. Furthermore, we derive an analytical expression for the
+amplitude of the superconducting gap ∆k in terms of disorder and finite center-of-mass momentum and show
+that amplitude fluctuations are induced in the superconducting state by the presence of the puddles, where both
+the gap and the critical temperature are affect and reduced by disorder and finite momentum pairs. Finally, we
+discuss our findings in the context of networks of superconducting oxygen nano-puddles in cuprates.
+I.
+INTRODUCTION
+It is a well known fact within the Bardeen-Cooper-
+Schrieffer (BCS) theory of superconductivity that the two
+quasi-particles forming the bound states that constitute the
+superconductor, named Cooper pairs, have momentum k and
+−k, near the Fermi surface, with oposite spins ↑ and ↓, form-
+ing a singlet with zero center-of-mass momentum [1], in
+what is usually called the Cooper channel. However, the ex-
+istence of a finite-momentum superconducting ground state
+has recently been raised theoretically [2–7] and supported
+by several experiments in correlated quantum materials [8–
+11]. Moreover, the possibility of emergent finite-momentum
+pair states, in the form of pair density waves, in a variety
+of well-established superconducting compounds, for example
+transition-metal dichalcogenides and in cuprates [12], points
+to the importance of understanding the intrinsic characteris-
+tics of these states and its interplay with other common fea-
+tures present in these systems, such as disorder [13] and in the
+presence of magnetic fields [14].
+Although condensed matter models usually start from the
+notion of a perfect crystal, a plethora of notable effects are
+only accessible when this notion is no longer true. One fa-
+mous example is the problem of the high-Tc superconductiv-
+ity on cuprates, in which a region of d-wave pairing occurs in
+the form of a dome-shaped area and as a function of doping in
+its phase diagram. Here, doping, either intentional or acciden-
+tal, usually takes place, for example, via chemical substitution
+in La2−xSrxCuO4 [15], or via inclusion of interstitial dopant
+oxygen atoms (Oi) in Bi2Sr2CaCu2O8+δ [16], La2CuO4+y
+[17] or YBa2Cu3O6.5+y [18], which can be treated as point-
+like scattering centers as well as extended defects that intro-
+duce disorder and deviate the neighboring atoms from their
+crystallographic positions. This brings to light a fundamen-
+tal question regarding the context of the dome-shaped area
+of high temperature superconductivity in cuprates, on what
+mechanism is responsible for the reduction in Tc upon over-
+doping as well as to the subsequent disappearance of super-
+conductivity at a critical doping. Usually, this is ascribed to
+intrinsic effects, in which pairing correlations diminish with
+doping, due to screening of local Coulomb interactions [19],
+but some authours have also addressed the role of disorder in
+surpressing superconductivity [20–22]. Disorder, however, is
+usually incorporated as random on-site energies in Hubbard-
+like models that can lead to Anderson localization phenomena
+[23–25], thus it is important to extend this effects to include
+also the possiblity of severe structural disorder within finite
+regions of the crystal.
+One of the most significant results from the study of disor-
+der effects in superconductivity is the well known Anderson’s
+theorem, which states that both the transition temperature, Tc,
+and the isotropic gap, ∆0, of s−wave superconductors are in-
+sensitive to the presence of weak disorder at the mean-field
+level of BCS-like models [26–28]. One of the requirements
+of the theorem is that the density of states remains unchanged
+when compared to the pure metal case. As such, if the in-
+fluence of disorder is strong enough to deplete the density
+of states, the theorem no longer holds, and disorder dramat-
+ically affects superconductivity [29]. In this case of strong
+disorder and high concentration of impurity centers, the su-
+perconducting correlation length is comparable to the disor-
+der correlation length, and the mean-field equations can lead
+to self-organized granularity where fluctuations of the local
+order parameter are present [30]. This is likely to be the case
+for overdoped cuprate superconductors with high concentra-
+tion of interstitial oxygens that can lead to the formation of
+nanosized oxygen puddles, regions with agglomeration of Oi,
+that support superconductivity [17, 18, 31–33].
+The case of unconventional high temperature d−wave su-
+perconductivity in hole-doped cuprates is of experimental and
+theoretical relevance since its discovery [34]. Apart from sev-
+eral different physical characteristics, one of the main differ-
+ences between these materials and the conventional BCS su-
+perconductors is that the superconducting gap amplitude is
+not homogenous when the system undergoes the supercon-
+ducting transition. This is evidenced by scanning tunneling
+microscopy (STM) spectra in Bi2Sr2CaCu2O8+δ at different
+arXiv:2301.02892v1  [cond-mat.supr-con]  7 Jan 2023
+
+2
+doping levels, where the inhomogenous gap in the supercon-
+ducting regime is revealed to be represented by a variety of
+gap sizes and amplitues occuring in all samples as the con-
+centration of dopants is varied [16]. Most remarkably, there
+is a clear correlation between the position of Oi agglomer-
+ates and the amplitudes of the gaps, since regions with larger
+groups of dopants are observed to correspond to regions of
+larger gap amplitudes [16]. Paralelly, Oi dopants have been
+observed to self-organize into nanosize regions, or puddles,
+as mentioned above, via µXRS in HgBa2CuO4+δ [35], as
+well as in other cuprate compounds [36]. Remarkably, it has
+been observed that spatial variations in the self-organization
+of the nanosized Oi-rich puddles have a direct effect on su-
+perconductivity, through variations in the critical temperature
+[37]. Therefore, it is of paramount importance a deeper un-
+derstanding, from a theoretical perspective, of the role of the
+oxygen puddles in the physics of hole-doped cuprates.
+In this work, we aim to investigate the effects of how struc-
+tural disorder caused by the agglomeration of Oi in puddles
+is responsible for the appearence of finite (nonzero) center-of-
+mass (CM) momentum Cooper pairs in overdoped cuprates.
+This will be done by making use of a previously reported
+model proposed to describe how superconductivity rises in
+cuprates, on the underdoped side of the phase diagram, in
+terms of the phase synchronization of networks of nanosized
+superconducting puddles, rich in interstitial dopant oxygens
+[38]. Following, we extend the puddle model to derive analyt-
+ical expressions showing how the superconducting gap, and
+thus the critical temperature, are affected by the presence of
+Cooper pairs with finite CM momentum and structural dis-
+order.
+Finally, we show numerically that both Tc and ∆0
+decrease with increasing disorder, thus pointing to a simple
+physical mechanism to explain the closing of the supercon-
+ducting dome-shaped area of the phase diagram, as being due
+to the reduction of the available phase space for Cooper pair-
+ing due to the development of a nonzero, finite CM momen-
+tum Cooper pairs.
+This paper is divided as following: in Sec. II we explain
+the puddle model, which is the base for the calculations pre-
+sented in this work, and derive the effective interaction be-
+tween electrons and the network of puddles, giving rise to a
+finite CM momentum pair state. Following, in Sec. III we de-
+scribe the effects of structural disorder that the agglomeration
+of Oi within each puddle causes to the system. In Sec. IV
+we derive the the self-consistent equation for the amplitude of
+the superconducting gap in terms of disorder and finite CM
+momentum Cooper pairs. Then Sec. V is devoted to the nu-
+merical calculations. Finally, we discuss the implications of
+our results within the framework of networks of nano-sized
+puddles and summarize our findings in Sec. VI.
+II.
+INHOMOGENEOUS OXYGEN PUDDLES
+The oxygen rich nanopuddles have different elastic proper-
+ties than their surroundings, and can therefore be considered
+as elastic insertions in an otherwise homogeneous medium,
+with its own vibrational mode, forming a network of super-
+Figure 1. Pictorical view of the disordered background introduced by
+the network of puddles (blue) in the system. The network consists of
+puddles of different sizes, defined by the radius of each insertion.
+Electrons (black and red) scatter in each puddle and, in the super-
+conducting state, percolate within the network.
+conducting nanoscale puddles, as shown in Fig. 1, which is
+the starting point for the model that captured how supercon-
+ductivity may arise in cuprates due to the phase synchroniza-
+tion of each nanopuddle [38]. In terms of the Kuramoto model
+for sychronization of phase oscillators [39, 40], each nano-
+sized puddle is assigned to a phase, that in the underdoped
+regime evolves independently of the others, giving rise to lo-
+calized patches of superconductivity, as revaled by STM and
+other techniques. With increased concentration of Oi through
+doping, the superfluid density is responsible for the enhance-
+ment of the interactions between the puddles and, in terms
+of the Kuramoto model, to lock their phases in a synchronous
+way. Following a BCS-like procedure, the order parameter for
+synchronization is connected to the amplitude of the bulk su-
+perconductor gap, that is non zero only after the locking of the
+global phase in the synchronized phase. The synchronization
+and the large frequency of the global network of puddles is
+also responsible for large values of Tc in the optimally doped
+cuprates within the model.
+Inspired by these experimental and theoretical findings,
+we introduce a model Hamiltonian that captures the inter-
+action between electrons and localized vibrations that arise
+from the agglomeration of interstitial oxygens in one pud-
+dle. This interaction must be local, since each electron will
+only interact with the quantized vibration whenever it is in
+the region defined by the puddle (see Fig. 1). The minimal
+model that captures this physical situation can be divided in
+H = Hel + Hp + Hel−p, with
+Hel =
+�
+k,σ
+ξkc†
+k,σck,σ +
+�
+k,k′
+Tk,k′c†
+k′,σck,σ,
+where the first term represents a band of electrons with dis-
+persion ξk measured relative to the chemical potential, with
+creation c†
+k,σ and annihilation ck,σ fermionic operators. The
+second term represents the scattering of electrons in each in-
+
+3
+homogeneity described by the puddles, with strenght con-
+troled by the spin-preserving momentum transfer disorder ma-
+trix Tk,k′. The oxygen puddles are described by local phonon
+modes
+Hp =
+�
+q
+ℏωqa†
+qaq,
+with frequencies ωq and the creation (a†
+q) and annihilation
+(aq) bosonic operators, responsible for the description of the
+localized vibration of each puddle. Finally, the interaction
+term can be described as
+Hel−p =
+�
+r,R,σ
+g(r − R)c†
+r,σcr,σ
+�
+a†
+R + aR
+�
+,
+where r and R are the electron and puddle locations, respec-
+tively. The puddle is a finite size region in space, thus R de-
+fines the center of this region that can be modeled as a sphere.
+The interaction strenght g(r − R) is only relevant whenever
+the electron is in the region around the puddle, which can
+be modeled using a Gogny-type short range interaction that
+is dependent on the radius of the oxygen agglomeration re-
+gion [41]. After performing the transformation to momentum
+space, the interaction term is written as
+Hel−p =
+�
+k,k′,σ,q
+M(q, k − k′)c†
+k,σck′,σ
+�
+a†
+−q + aq
+�
+, (1)
+where
+M(q, k − k′) =
+�
+R
+g(k − k′) exp [i(q − [k − k′]) · R]
+is associated with the fact that the puddles are not present in all
+sites, rather they are inhomogeneously distributed around the
+system, thus the summation has to be retained only to these
+regions, which is relevant for the case of Bi2Sr2CaCu2O8+δ,
+since locations of dopant oxygens are observed to be consis-
+tent with the position inferred from local strain analysis of the
+incommensurate structure, as imaged by scanning transmis-
+sion electron microscopy (STEM) [42], which means that the
+crucial oxygen dopants are periodically distributed in corre-
+lation with local strain. However, not all strained regions are
+occupied with dopant oxygen atoms, that is the distribution of
+Oi is inhomogeneous, which justifies our approximation and
+is consistent with STM measurements [43]. In the limits of a
+clean or a totally doped system, this term can be treated ex-
+actly. The factor g(k − k′) is the Fourier transform of the in-
+teracting potential between the electrons and the puddles and
+controls the momentum transfer between the incoming and
+scattered electron.
+One can see from Eq. (1) that the presence of a finite den-
+sity of puddles spread around the systems give rise to a off-
+diagonal term associated with the momentum transfer k − k′
+that comes from the interacting potential. In the limit that the
+summation over M(q, k − k′) can be made exactly, one re-
+covers the usual definition of an electron-phonon interaction,
+where the momentum transfer is the momentum of the local
+phononic mode q, as in the Frohlich [44] and Holstein [45]
+models, for example. In order to explore the effects of this
+kind of interaction in the form of pairing, we introduce an
+unitary transformation H′ = e−SHeS, with an ansatz for the
+transformation matrix
+S =
+�
+k,k′,σ,q,Q
+M(q, k − k′)c†
+k,σck′,σ
+�
+xa†
+−q + yaq
+�
+,
+where x and y are factors determined a posteriori. After the
+transformation (see Appendix A for details), we end with an
+effective interaction written as
+Heff =
+�
+k,k′
+���
+p,p′
+V (k, k′)f(p, p′)c†
+k,↑c†
+p−k,↓cp′−k′,↓ck′,↑
+(2)
+with V (k, k′) = D(k, k′)|g(k − k′)|2 being the potential
+arising from the interaction between electrons and puddles,
+D(k, k′) the phononic propagator associated with the lo-
+cal phonon modes produced by the vibrating puddles and
+f(p, p′) = �
+R e−i(p−p′)·R the phase factor controlling mo-
+mentum transfer between the interacting electrons.
+In the
+regime where the phononic propagator is negative, given that
+ξk ≈ ξk′, we have an effective attractive interaction between
+the electrons mediated by the nanopuddles. Remarkably, this
+interaction leads to the formation of finite center-of-mass mo-
+mentum Cooper pairs represented by p and p′. Therefore,
+from the perspective of inhomogeneously distributed puddles
+bringing disorder to an otherwise clean medium, a bound state
+between two electrons can be formed with a finite center-of-
+mass momentum that is associated with the strenght of the
+interaction between the electrons forming the pair and the ag-
+glomeration of interstitial dopant oxygens in one nanopuddle.
+It is important to notice that the states arising from the ef-
+fective Hamiltonian in Eq. (2) are different from other pro-
+posed pair states with finite CM momentum, as for example
+the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, where fi-
+nite center-of-mass momentum Cooper pairs can be stabilized
+under a finite magnetic field via the Zeeman coupling [46, 47],
+and the recently proposed current driven FFLO state [48].
+Moreover, it has been shown that, even without the presence
+of a magnetic field or other external potentials, a finite CM
+momentum Cooper pair can be stable in a superconducting
+ground state as pointed in Ref. [7], but the authors do not ex-
+plore the effects that can give rise to this kind of state. Here
+we start from the fact that nanosized puddles are formed via
+doping and the responsible for the CM momentum of the pairs
+is disorder induced by the puddles in the system.
+Eventhough we are not considering any specific form for
+the interaction potential g(k − k′), it is important to com-
+ment that the only requirement is that it must be a finite size
+potential in real space, which means that is not a point-like
+disorder center that is scattering the electrons in the interac-
+tion term of Eq. (1), rather is a region in space defined by
+the agglomeration of oxygen interstitials. In this case, we can
+point to potentials like the Woods-Saxon potential [49] that is
+
+4
+Figure 2. Top: Structure factor for disordered media from Hose-
+mann’s paracrystalline theory [58], given by eq. (6) in the text. The
+pristine case corresponds to the ℓ → ∞ limit, where the structure
+factor is given by delta-peaks at reciprocal lattice vectors and mo-
+mentum is conserved (here ℓ is a measure of disorder and for this
+reason should be inversely related to the residual resistivity shift due
+to structural disorder, ℓ ∝ 1/δρ0). Bottom left:: − Bragg diffraction
+pattern for a structually disordered medium, showing Bragg peaks at
+the central region and Bragg rings at the outter region; Bottom right
+− plot of the structure factor as a function of momentum transfer,
+∆Q, showing well defined Bragg peaks, for small momentum trans-
+fer, at the reciprocal lattice vectors, G, while the Bragg peaks be-
+come ever broader, at larger momentum transfer, eventually merging
+into rings.
+used to describe the forces applied on protons and neutrons in
+the atomic nucleous or the Gogny-type interactions [50–52],
+which is another kind of nucleon-nucleon potential that has
+also found applications in astrophysics [53], as possible can-
+didates to describe the electron-puddle interaction. However,
+a precise and detailed description of such potential would re-
+quired more knowledege about the formation of the nanosized
+puddles and its effects on the crystal structure of the host ma-
+terial, which would affect the electronic degrees of freedom
+[54], but this is outside the scope of the present study.
+III.
+STRUCTURAL DISORDER
+Before we proceed to the characterization of the supercon-
+ducting state that arises from the effective Hamiltonian de-
+rived in the last section, it is important to briefly discuss which
+kind of disorder is giving the Cooper pairs a finite CM mo-
+mentum. In order to do that, we introduce concepts arising
+from the study of structural disorder, which is the kind of per-
+turbation that the agglomeration of Oi causes in the crystalline
+structure of different cuprate systems, as for example by tilt-
+ing the CuO6 octahedra in La2CuO4+δ [55] and by altering
+the distance between the apical oxygen and the planar copper
+atom in Bi2Sr2CaCu2O8+δ [56].
+Translational invariance is one of the most fundamental
+properties of pristine crystals.
+The concept of a Brillouin
+zone, that repeats itself by translations of reciprocal lattice
+vectors, allows us to organize electrons in energy bands,
+ϵn(k), labeled by a band index, n, and function of a quasi-
+momentum (wave-vector) quantum number, k, in terms of
+which periodic Bloch wave-functions, un,k(r), are defined.
+A perfect crystal is characterized by very intense and sharp
+peaks in the Fraunhofer diffraction pattern of Bragg scatter-
+ing experiments. The existence of such sharp peaks follows
+directly from Heisenberg’s uncertainty principle and their lo-
+cation is determined by the crystalline-lattice structure factor.
+Simply put, an extended Bloch wave with well defined mo-
+mentum state, k, that interacts with ions located at arbitrary
+positions, ri, of the crystal (infinite uncertainty ∆r → ∞),
+scatters into another extended Bloch wave with momentum
+state, k′, with zero uncertainty, ∆k → 0. The entire process
+carries a phase
+φ(k′ − k) =
+1
+√
+N
+�
+ri
+friei(k′−k)·ri,
+(3)
+where N is the number of lattice sites in the crystal and fri is
+an atomic form factor that gives the probability that an atom
+is located at a certain crystallographic position. The scattered
+intensity is proportional to |φ(k′−k)|2 and is thus determined
+by the lattice structure factor,
+S(k′ − k) = 1
+N
+�
+ri,rj
+frifrjei(k′−k)·(ri−rj).
+(4)
+For a pristine crystal all atoms are at their ideal locations,
+fri = frj = 1 and thus S(k′ − k) = �
+g δk′−k,g, where g is
+a reciprocal lattice vector. The Fraunhoffer diffraction pattern
+in this case thus corresponds to δ−like peaks as shown in Fig.
+2 and the kinematic constraint of quasi-momentum conserva-
+tion,
+k′ = k + g,
+(5)
+forms the basis for Bloch’s theorem. In the opposite limit of a
+random atom gas, however, an extended Bloch wave with well
+defined momentum state, k, that interacts with ions located at
+a particular, well defined position, ri, of the crystal (zero un-
+certainty ∆r → 0), scatters into another extended Bloch wave
+with momentum state, k′, with infinite uncertainty, ∆k → ∞.
+In this case, frifrj = δri,rj, and S(q) = 1. There are no
+kinematic constraints whatsoever relating k and k′ to g and
+the Fraunhoffer diffraction pattern in this case corresponds to
+an isotropic disc of even intensity, as shown in Fig. 2.
+Interpolating between the pristine and random limits de-
+scribed above by increasing disorder is pivotal to the descrip-
+tion of inherently inhomogeneous systems, such as the one
+of ramdom oxygen puddles described in the present work. If
+disorder is of the first type, namely weak disorder, all atoms
+deviate only slightly from their ideal positions in the crys-
+tal, independently of the deviations of their neighbors [57].
+This is the case of pointlike defects, thermal vibrations or
+micro-mechanical strains, and this kind of disorder preserves
+long range crystalline order. In this case the widths of the
+peaks in the Fraunhoffer diffraction pattern are not affected,
+
+Bragg diffraction patterns
+lα1/po
+measures the amount of distortions
+Smax(g)
+S℃(k'- k) = k'-k-q,0
+1＋l2(k'kqg)
+g0
+20
+Structure factor Sq(k'-k)
+15
+10
+5
+0
+0
+2
+4
+Reciprocal vector AQ=k-k-q (units of G)
+uncertainty in reciprocal lattice
+breakdown momentum conservation5
+and only their intensity is slightly reduced since for uncor-
+related Gaussian disorder, frifrj = D2 < 1, where D2
+is the Debye-Waller factor. The structure factor is given by
+S(k′ − k) = D2 �
+g δk′−k,g. If disorder of the second type,
+namely strong disorder, however, the atoms deviate signifi-
+cantly from their ideal positions in the crystal, and deviations
+amogst neighboring atoms are correlated. This is the case of
+extended defects, amorphous regions, molten materials, etc,
+and this type of disorder causes the loss of long range crys-
+talline order. In these paracrystalline structures, not only the
+intensity of the diffraction peaks will decrease but, most im-
+portantly, their widths will suffer from a nonlinear increase of
+their integral breadth, δg, for successive orders of Bragg re-
+flections. The complete paracrystalline theory was proposed
+by Hosemann [58]. Hosemann included fluctuations of vari-
+ance σ that introduce correlations between pairs of atoms,
+⟨frifrj⟩, that decrease with separation ultimately causing the
+peaks in the structure factor of the material to broaden the
+larger the reciprocal lattice. The result is a structure factor
+composed by a sum of Lorentzians [59]
+Sq(k′ − k) =
+�
+g
+Smax(g)
+1 + ℓ2
+hkl(q − k′ + k − g)2 ,
+(6)
+of amplitudes Smax(g) = 4/σ2g2 and breadths for Bragg
+reflections, |δg| ≡ 1/ℓhkl = σ2π2(h2 + k2 + l2)/a0, given in
+terms of the original lattice parameter a0 and the momentum
+transfer, q. Hosemann’s paracrystalline theory allows us then
+to interpolate continuously between pristine and random cases
+through the fluctuation parameter σ:
+• for σ → 0 we have ℓhkl → ∞, ∀h, k, l and we obtain
+Sq(k′ − k) = �
+g δq,k′−k+g, enforcing the kinematic
+constraint of momentum conservation, q = k′ − k + g,
+typical of pristine crystals [59];
+• for σ → ∞ we have ℓhkl → 0, ∀h, k, l and we end
+up with Sq(k′ − k) = Smax(0) → 1, isotropic, for
+arbitrary q, k, k′ and determined solely by the g = 0
+contribution, typical of infinite, aperiodic systems [59];
+• for 0 ≤ σ ≤ ∞ we have ∞ ≥ ℓhkl ≥ 0 and the
+structure factor, Sq(k′ −k), will be composed by sharp
+Bragg peaks at small g (large ℓhkl) and isotropic discs
+for larger g (small ℓhkl), as shown in Fig. 2, relax-
+ing the kinematic constraint of momentum conserva-
+tion, q ̸≈ k′ − k + g, typical of a paracrystal, liquids,
+strongly disordered or amorphous systems [59].
+IV.
+DISORDER AND GAP FLUCTUATIONS
+We now address how the superconducting state of the effec-
+tive interaction derived in Sec. II is affected by the structural
+disorder effects introduced in the previous section. We start
+from the effective Hamiltonian in Eq. (2) and, within a mean-
+field decoupling of the quartic term, write the equation for the
+superconducting gap as
+∆k = −
+�
+k′,p′
+Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩ ,
+(7)
+where we set p = 0, since we want to describe ampli-
+tude fluctuations for the superconducting gap in the Cooper
+channel.
+For the superconducting state formed by singlet
+pairs with finite CM momentum, the system can be repre-
+sented by the spin-independent imaginary time Green’s func-
+tion G(k, k′, τ) = −
+�
+Tτck,σ(τ)c†
+k′,σ(0)
+�
+and the anomolous
+pair propagators F(k, k′, τ)
+=
+⟨Tτck,σ(τ)ck′σ′(0)⟩ and
+F∗(k, k′, τ) =
+�
+Tτc†
+k,σ(τ)c†
+k′,σ′(0)
+�
+for σ ̸= σ′. Within
+Nambu’s formalism, we can write the decoupled effective
+Hamiltonian from Eq. (2) and the electronic components from
+Hel in matrix form and derive in first order perturbation theory
+the electronic Green’s function for an inhomogeneous system
+with disorder as
+G (k, k′, iωn) = G0 (k, k′, iωn)
++
+�
+p,p′
+G0 (k, p, iωn) Tp,p′σ3G (p′, k′, iωn) ,
+where G0 (k, k′, iωn) is the matrix form of the translation-
+ally invariant electronic Green’s function in frequency space,
+iωn are the fermionic Matsubara frequencies and σ3 is a Pauli
+matrix.
+The diagonal elements of this matrix are defined
+by the bare Green’s function in the superconducting state,
+G0(k, iωn), and its off-diagonal terms are represented by the
+anomalous propagators F0(k, iωn) which are written as
+G0 (k, iωn) =
+− (iωn + ξk)
+ω2n + ξ2
+k + |∆k|2 ,
+F0 (k, iωn) =
+∆k
+ω2n + ξ2
+k + |∆k|2 .
+In order to proceed, we shall take a couple of approximations:
+first we consider the case of overdoped cuprates, which puts
+the system in a high concentration of disorder, thus Tp,p′ =
+T f(p, p′), where disorder influences the momentum trans-
+fer controled by the phase factor f(p, p′) with strenght T .
+Second we assume that for a translationally invariant system
+the normal and anomalous Green’s functions can be rewrit-
+ten as G0(k, k′, iωn) = G0(k, iωn)δk,k′ and F0(k, k′, iωn) =
+F0(k, iωn)δ−k,k′. Following these couple of approximations,
+the first order pertubation theory expansion of the interacting
+Green’s function is simplified
+G (k, k′, iωn) = G0 (k, iωn) δk,k′
++ T fk,k′G0 (k, iωn) σ3G0 (k′, iωn) . (8)
+From the gap equation in Eq. (7) and from the definition of
+the anomalous propagator, we write
+
+6
+∆k = −
+�
+k′,p′
+Vk,k′f0,p′ ⟨cp′−k′↓ck′↑⟩
+= −
+�
+k′,p′
+Vk,k′f0,p′
+�
+1
+β
+�
+ωn
+F (p′ − k′, k′, iωn)
+�
+,(9)
+with β = 1/T being the inverse temperature (in units of
+kB = 1). By using the matrix form in Eq. (8), we get the form
+of the interacting anomalous propagator, where it is worth not-
+ing that the normal and anomalous propagators mix in the
+impurity scattering.
+Despite the anomalous Green’s func-
+tion being invariant for time reversal, the normal one is not,
+and since disorder produces the transformation F0(k, iωn) ↔
+G0(k, iωn) we clearly see this is a mechanism that breaks time
+reversal invariance. As a consequence, this mechanism breaks
+the Cooper pair that leaks into the normal metal surrounding
+the puddles.
+In order to understand the effects of disorder and finite CM
+momentum in the gap equation, we substitute the form of the
+anomalous propagator given by the matrix in Eq. (8) inside
+Eq. (9) to write the gap equation as ∆k = ∆BCS
+k
++ δ∆k,
+where
+∆BCS
+k
+= −
+�
+k′
+Vk,k′∆k′
+2Ek′
+tanh
+�βEk′
+2
+�
+,
+(10)
+is the BCS limit for the gap equation, arising from the first
+term in Eq. (8), with the bare anomolous propagators and
+Ek =
+�
+ξ2
+k + ∆2
+k. Then
+δ∆k = T
+�
+k′,p′
+Vk,k′f0,p′fp′,0
+1
+β
+�
+ωn
+{F0G0 + G0F0}(11)
+is the correction to the superconductor gap due to effects of
+disorder in the system. The factor [f0,p′fp′,0] can be treated
+within a mean over disorder in order to calculate the inter-
+ference factor as [f0,p′fp′,0] = |f0,p′|2 → S(p′), where
+S(0, p′) is the static structure factor.
+Thus, the correction
+to the gap equation can be written in terms of the structure
+factor and we see that fluctuations associated with small CM
+momentum p′ → 0 are absent, since the structure factor
+S(p′) → 0 and the gap equation is dominated by the BCS
+contribution. On the other hand, fluctuations associated with a
+finite center-of-mass momentum dominate over the BCS con-
+tribution when p′ ≫ 0 and S(p′) → 1. In a general manner,
+the structure factor can be written as a sum of Lorentzians with
+peaks in wave vectors of the reciprocal lattice, as discussed in
+Sec. III and shown in Fig. 2.
+Finally, we proceed by taking the Matsubara summations
+over the set of mixed Green’s functions as in Eq. (11) to arrive
+at the correction in terms of the disorder strenght T and the
+finite CM momentum of the Cooper pairs p′ as
+δ∆k = T
+�
+k′,p′
+Vk,k′S (p′) 1
+2
+� ∆k′,p′
+Ek′−p′
+ξk′
+Ek′ + ∆k′
+Ek′
+ξk′−p′
+Ek′−p′
+�
+×
+�
+�
+�
+Ek′−p′ tanh
+�
+βEk′
+2
+�
+− Ek′ tanh
+� βEk′−p′
+2
+�
+E2
+k′−p′ − E2
+k′
+�
+�
+� .
+(12)
+It is importance to notice the dependence of the correction
+on the structure factor S(p′) controlling momentum transfer.
+In the limit of small amount of disorder, the so called first-
+type disorder [57], as discussed in Sec. III, pointlike deffects
+does not affect the BCS gap, in accordance with Anderson’s
+Theorem, as we shall see in the next section. On the other
+hand, in the limit of high concentration of puddles, the system
+is in the limit of second-type disorder, associated with strain-
+induced lattice deformations, and both the amplitude of the
+superconducting gap and the critical temperature are affected.
+In order to proceed to the numerical analsysis, we perform
+an approximation for the structure factor based on the limits
+of disorder discussed above. For the first-type disorder, we
+choose S(p′) = δ0,p′, since no momentum transfer will be
+associated with pairs with finite CM momentum in the dilute
+limit. On the other hand, for the second-type disorder, we
+write S(p′) = 1, assuming a system with high concentration
+of puddles. These two limits for the disorder of the 1st and
+2nd types can be understood as a hard cutoff for the CM mo-
+mentum distribution within the structure factor and are made
+to simplify Eq. (12) to the following numerical analysis.
+V.
+NUMERICAL ANALYSIS
+In order to fully understand the effects of disorder and
+CM momentum of the Cooper pairs in the superconduct-
+ing gap amplitude we perfom a numerical integration of Eq.
+(12). We use the decomposition Vk,k′ = −V0η(k)η(k′) and
+∆k = ∆0η(k), where η(k) = cos kx − cos ky is a d−wave
+form factor, which gives the amplitude fluctuations of the or-
+der parameter with the same symmetry. When stated for com-
+parison, we shall also use Vk,k′ = −V0 and ∆k = ∆0 when
+considering a s−wave symmetry for the interaction and the
+gap. For the calculations in the square lattice, we consider a
+two-dimensional electronic dispersion with nearest- and next-
+nearest-neighbor hopping elements (t, t′) as
+ϵk = −2t (cos kx + cos ky) + 4t′ cos kx cos ky − µ, (13)
+where µ is the chemical potential that controls the electronic
+density. This type of electronic dispersion is general for 2D
+transport in strongly correlated systems and is suitable for the
+description of the conduction band associated with the CuO2
+planes of high-Tc cuprates.
+In the following calculations, all parameters are defined in
+units of 4t and we set µ/4t = −0.45, away from the half-
+filled case µ/4t = 0.0 (see Fig. 3), since the mean-field theory
+
+7
+Figure 3. Fermi surface structure used in calculations. Left: The 3D
+plot of Eq. (13) in the first Brillouin zone in yellow and the chemical
+potential cut defining the Fermi level in blue. Right: The Fermi level
+defined by the cut at µ/4t = −0.45. The vectors k, fixed in the
+direction (0, π), and k′, varying across the Fermi surface, are also
+shown.
+yields incorrect results for a two-dimensional lattice near half-
+filling [60] and we avoid particle-hole symmetry [61]. For
+this reason, we can take t′ = 0. We also set V0/4t = 1.0,
+in the limit where the mean-field theory is still valid. For the
+summations over p′, we define p′ = k − k′, where k, k′ are
+the momenta of the two paired electrons, which we set |k| =
+|k′| = kF as two momenta in the Fermi surface. The CM
+momenta are then defined by fixing k in the direction of the
+point (0, π) and by varying k′ across the Fermi surface, as
+shown in Fig. 3.
+We start by analyzing the zero temperature limit T = 0 of
+Eq. (12), where the hyperbolic tangents can be simplified. In
+Fig. 4 we show how the gap amplitude ∆0 is affect by disor-
+der T in the limit of disorder of the 1st type, S(p′) = δ0,p′,
+or weak concentration of puddles, and strong concentration,
+S(p′) = 1, in the limit of disorder of the 2nd kind. The
+gap amplitude is insensitive to disorder in the dilute limit for
+s−wave pairing, thus ∆0 = ∆BCS
+0
+and the BCS limit is re-
+covered, in accordance with Anderson’s theorem. However,
+in the opposite limit, the disorder strongly affects the ampli-
+tude of the gap for d−wave pairing, introducing fluctuations
+and decreasing its absolute value in about 50% in the strong
+disorder limit, when compared to the clean case.
+It is worth noting that the reduction is not linear as the
+strenght of disorder approaches the values of the fixed pairing
+potential, T → V0, where the pertubation theory still holds.
+This can be traced back to the fact that the gap equation is
+a self-consistent equation for the aboslute value of ∆0, even
+after the approximations considered. Thus we see that even
+in the zero temperature limit, disorder tends to destroy super-
+conductivity in a system with high concentration of oxygen
+interstititals, as in the overdoped cuprates.
+We also investigate the effects of specific finite CM momen-
+tum on the amplitude of the gap when T = 0. We choose a set
+of momenta {p} and substitute in Eq. (12) the corresponding
+structure factor, namely S(ps) = δp′,ps, where ps are the mo-
+menta in the set. All ps are multiples of kF of each direction
+considered, namely (0, π) and (π, π). In Fig. 5 we display the
+Figure 4. T = 0 limit for the amplitude fluctuatios of the supercon-
+ducting gap as a function of disorder strenght compared to the clean
+system. Dilute limit (red), for disorder of the 1st kind and s−wave
+symmetry, and high concentration of puddles for disorder of the 2nd
+kind and d−wave symmetry (blue). The black dashed line is a guide
+to the eye. Gap values are given in terms of ∆0 in the absence of
+disorder T = 0.
+evolution of the amplitude of the superconducting order pa-
+rameter ∆0 as a function of the CM momentum of the Cooper
+pairs p, for fixed disorder strenght T = 0.1. The supercon-
+ducting order parameter is modulated, with period determined
+by the distance between adjacents Fermi surfaces in each di-
+rection, being 3.75|kF| for (0, π) and 5.75|kF| for (π, π). Re-
+markably, this is in direct contact with the diffraction pattern
+displayed in Fig. 2. However, since we are considering a
+hard cutoff for the structure factor in terms of delta functions,
+the amplitude of the gap modulation is not altered by the dis-
+tance from the origin. We expect that by including a more
+realistic model for the structure factor, the amplitudes of the
+modulations will decay with p, with its effect stronger in the
+(π, π) direction, since larger reciprocal lattice vectors G im-
+ply a broader structure factor, thus diminishing the amplitude
+of the superconducting gap. Altogether, the interplay between
+disorder and finite center-of-mass momentum Cooper pairs is
+able to strongly affect the superconducting order parameter.
+Now we turn to the finite temperature case T ̸= 0 for the
+d−wave symmetric order parameter to understand how disor-
+der and CM momenta for the Cooper pairs affects the critical
+temperature Tc. In Fig. 6 we show the evolution of the su-
+perconducting gap with temperature, for different values of
+the disorder strenght T . It is clear that with increasing dis-
+order, not only ∆0(0) decreases, as pointed in the zero tem-
+perature limit, but we also evidence a decrease in the criti-
+cal temperature Tc, defined as the value of temperature that
+∆0(T, T ) → 0, with disorder, as shown in the inset. This
+means that pair breaking is induced by the scattering of the
+finite CM momentum Cooper pairs with the nanosized oxy-
+gen puddles of the system and by increasing disorder, Tc is
+significantly reduced.
+This pair breaking effect is due to the fact that the phase
+space required to pair formation is reduced when p increases
+in absolute value. In the small scattering momentum transfer
+
+2
+k
+03
+斤-2
+0
+-1
+0
+ky
+元
+元
+0
+2
+Kx
+元
+2
+-2
+-2
+-2
+0
+2
+kx1.0
+0.9
+△o(T)/△o(0)
+0.8
+0.7
+2nd d wave
+0.6
+lst s wave
+0.0
+0.2
+0.4
+0.6
+0.8
+T8
+Figure 5. Left: Extended Brillouin zones in the upper positive part of
+momentum space. The arrows indicate the distance between the cen-
+ters of each Fermi surface in terms of the Fermi vector |kF | of each
+direction considered. Right: The amplitude of the superconducting
+order parameter as a function of different CM momentum vectors
+|p|, in the directions (0, π) and (π, π). ∆0(p) is given in units of
+the gap at p = 0.
+sector, p < |kF|, the gap is almost unnafacted by the presence
+of disorder when comprared to the value when p = 0, since
+the shape of the Fermi surface intersection of the two paired
+electrons suffers little change. However, when p approaches
+the maximum absolute value of 2|kF| within the first Brillouin
+zone, the phase space for pair formation is greatly reduced and
+disorder induces pair breaking, captured by the reduction of
+the superconducting order paremeter. The modulation occurs
+for p > 2|kF|, since electrons from different Brillouin zones
+participate in the scattering and pairing process. Therefore,
+these results point to the combined effect of finite center-of-
+mass momentum pairs being scattered by structural disorder
+induced by the network of oxygen puddles as a mechanism
+for the reduction of the superconducting gap and the critical
+temperature in the overdoped regime.
+VI.
+CONCLUSION AND DISCUSSION
+In this work we presented an extension of the proposed
+model for the formation of networks of puddles and its ef-
+fects on the superconductivity in oxygen-doped cuprates [38].
+We show that the presence of puddles, in the overdoped side
+of the phase diagram, introduces strong disorder in the sys-
+tem that induces the formation of finite center-of-mass mo-
+mentum Cooper pairs. We derive an analytical expression for
+the amplitude fluctuations in the superconducting gap induced
+by the puddles, within a mean-field BCS-like approach, in
+terms of the disorder strenght T and the finite CM momenta
+p. We numerically solve this expression to show that even
+in the zero temperature limit the gap is strongly affected by
+disorder-induced CM Cooper pairs. In the limit of strong dis-
+order, the gap tends to close and, in the finite temperature case,
+Tc tracks the reduction of the superconducting gap, also being
+strongly affected by disorder. It is important to emphasize
+that we do not account the effect of longer-range Coulomb re-
+pulsion, restricting the application of our results to screened
+systems [68].
+Figure 6. Temperature dependence of the superconducting order pa-
+rameter for different values of disorder strenght (colored bar). Gap
+alues are given in terms of the clean case T = 0 and temperature in
+terms of T 0
+c also of the clean case. Inset: The critical temperature
+dependence normalized to the clean value as a function of disorder
+strenght. The black dashed line is a guide to the eye.
+The experimental observations of structural scale invari-
+ance of dopants detected by scanning micro-x-ray diffraction
+[36], the promotion of critical temperature [37], the agglom-
+eration of interstitial oxygens in regions of strong local strain
+in the crystal structure of cuprate superconductors [42, 43]
+and the proposed theoretical reports regarding the presence
+of networks of nanoscale superconducting islands in high-
+temperature superconductors [62–65] are in close connection
+with the results reported here. Eventhough we are showing
+that the superconducting state is depleted in the presence of
+strong disorder in the overdoped regime, it is clear from the
+above mentioned surveys that the importance of these net-
+works and its interplay with electronic degrees of freedom
+pass across the whole phase diagram of hole-doped cuprates.
+In Ref. [38], the present authors show how the complex
+networks formed by the oxygen puddles can transionate to a
+synchronized phase, controlled by the superfluid density, in a
+way that the concentration of dopant atoms controls the emer-
+gence of local superconductivity in the underdoped regime
+and how the systems evolves to a bulk superconductor as the
+concentration of dopants, thus puddles, increases as the sys-
+tems approaches the optimally doped regime. It is important
+to emphasize that within this framework, the state studied in
+this work is described by the bulk superconductor state in the
+synchronized phase of the network formed by the oxygen pud-
+dles (see Fig. 1), in the sense that we require the network of
+puddles to be fully synchronized in order to the band of elec-
+trons to interact with the global mode of vibration of the syn-
+chronized network. Our approach is based on a mean-field
+approximation for the complex network, therefore we point to
+the importance of describing different topologies for the orga-
+nization of the puddles and how this can affect not only the
+transition to the superconducting state [66], but also its pos-
+sible interplay with the superconducting fluctuations of pre-
+formed Cooper pairs observed in the pseudogap phase above
+Tc [67], in terms of local superconductivity.
+
+8
+1.00
+(0, πt)
+(π, )
+6
+0.98
+[ KF
+0.96
+4
+5
+K
+7
+.75/kFl
+0.94
+3
+2
+0.92
+0
+0.90
+-2
+0
+2
+4
+6
+8
+01234567891011
+kx
+Ip/ / / kFl1.0
+1.0
+0.8
+0.8
+.0.6
+0.6
+T)/△o(
+1.0
+Ao(T,
+0.4
+0.4
+.0.8
+0.2
+0.2
+0.6
+0.0
+0.2
+0.4
+0.6
+0.8
+T
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.09
+Appendix A: Unitary transformation
+In this Appendix section, we show the derivation of the
+effective Hamiltonian containing the pairing interaction be-
+tween two electrons forming a Cooper pair with finite center-
+of-mass momentum. The starting point is the full Hamiltonian
+written in momentum space H = Hel+Hp+Hel−p, which is
+the summation over the contributions of the electrons, puddles
+and electron-puddle interaction, respectively. Introducing an
+unitary transformation of the form H′ = e−SHeS, where S
+is the transformation matrix introduced in Sec. II, we can ex-
+pand the exponentials up to second order in powers of S to
+write the transformed Hamiltonian as
+H′ = H + [H, S] + 1
+2[[H, S], S],
+(A1)
+and by treating Hel−p as a perturbation, we can divide the full
+Hamiltonian as H = H0 + Hel−p, where H0 contains the
+kinetic terms of electrons and puddles, to write
+H′ = H0 + Hel−p + [H0, S] + [Hel−p, S] + 1
+2[[H0, S], S].
+Since the goal is to eliminate the interaction, the defining
+equation for the transformation matrix comes from the elim-
+ination of the first-order term [H0, S] + Hel−p = 0, from
+which we can extract the factors x and y for S. In this way,
+the transformed Hamiltonian can be written in terms of an ef-
+fective interaction that comes from recombining the terms in
+the commutators
+H′ = H0 + 1
+2[Hel−p, S],
+(A2)
+thus the problem is reduced to an effective system described
+by H = H0 + Heff, where Heff =
+1
+2 [Hel−p, S].
+By
+performing the calculation over the commutator [H0, S], the
+choice of x and y that eliminate the first-order term is given
+by
+xk,k′,q =
+1
+ξk′ − ξk − ωq
+,
+yk,k′,q =
+1
+ξk′ − ξk + ωq
+,
+and the transformation matrix S is fully defined. Then we
+proceed to the calculation of the effective Hamiltonian that
+comes from the commutator of the now defined matrix S and
+the electron-puddle interaction, which gives a combination of
+M(q, Q)M(−q, Q′), where Q = k − k′ and Q′ = k′′ − k′′′
+are two auxiliar variables that accomodate the variety of in-
+dices arising from the commutator. Recalling the definition of
+the factor M given in the main text, we see that
+M(q, Q)M(−q, Q′) =
+�
+R,R′
+g(Q)g(Q′)
+× ei(R−R′)·qe−i(Q·R+Q′·R′),
+which can be simplified by taking R = R′ since each R de-
+scribes the position of a nanosized puddle and we are assum-
+ing the dilute limit of oxygen puddles, as discussed in the
+main text, in accordance with STEM and STM measurements
+[42, 43]. In this way, the effective Hamiltonian is written as
+Heff =
+�
+k′,k′′′,q,Q,Q′
+V (q, Q, Q′)M(q, Q)M(−q, Q′)
+× c†
+k′′′+Q′c†
+k′+Qck′ck′′′,
+(A3)
+with V (q, Q, Q′) = ωq/[(ξk′′′ − ξk′′′+Q′)2 − ω2
+q]. Proceed-
+ing with the calculation, we note that within BCS theory, the
+effective Hamiltonian describes the interaction between elec-
+trons with opposite momenta k′ = −k′′′, with zero CM mo-
+mentum. However, in our case, the auxiliar variables Q and
+Q′ introduces a momentum transfer connected with a finite
+CM momentum for the pairs, for each fermionic operator in
+the effective Hamiltonian that comes from the commutator
+[Hel−p, S]. In this sense, we perform a change of variables
+introducing the finite CM momentum k′ + k′′′ = p, in a way
+that we can eliminate the dependence on the auxiliar variables.
+The new variables introduced are written as k = k′′′ + Q′ and
+−k + p′ = k′ + Q, where p and p′ are the CM momenta of
+the Cooper pairs. In the limit where the interaction g(k, k′) is
+independent of the CM momenta, we can decouple the effec-
+tive interaction and end up with the effective Hamiltonian
+Heff =
+�
+k,k′
+�
+p,p′
+V (k, k′)f(p, p′)c†
+k,↑c†
+p−k,↓cp′−k′,↓ck′,↑,
+with
+V (k, k′) =
+ω0
+(ξk′ − ξk)2 − ω2
+0
+|g(k − k′)|2
+f(p, p′) =
+�
+R
+e−i(p′−p)·R
+where we assume ωq = ω0, a dispersionless phonon mode for
+each puddle.
+
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+

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@@ -0,0 +1,645 @@
+arXiv:2301.02663v1  [math.GR]  6 Jan 2023
+ON THE CHARACTERIZATION OF ALTERNATING GROUPS BY CODEGREES
+MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG
+Abstract. Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. Define the
+codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)|
+χ(1)
+and denote by cod(G) := {cod(χ) | χ ∈ Irr(G)} the codegree
+set of G. Let An be an alternating group of degree n ≥ 5. In this paper, we show that An is determined up
+to isomorphism by cod(An).
+1. Introduction
+Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. For any χ ∈ Irr(G),
+define the codegree of χ as cod(χ) := |G:ker(χ)|
+χ(1)
+. Then define the codegree set of G as cod(G) := {cod(χ) |
+χ ∈ Irr(G)}. The concept of codegrees was originally considered in [8], where the codegree was defined as
+cod(χ) :=
+|G|
+χ(1), and it was later modified to its current definition by [22] so that cod(χ) is the same for
+G and G/N when N ≤ ker(χ). Several properties of codegrees have been studied, such as the relationship
+between the codegrees and element orders, codegrees of p-groups, and groups with few codegrees.
+The codegree set of a group is closely related to the character degree set of a group, which is defined as
+cd(G) := {χ(1) | χ ∈ Irr(G)}. The relationship between the character degree set and a group’s structure is
+an active area of research – many properties of a group’s structure are largely determined by its character
+degree set. In 1990, Bertram Huppert made the following conjecture about the relationship between a simple
+group H and a finite group G having equal character degree sets.
+Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that
+cd(H) = cd(G). Then G ∼= H × A, where A is an abelian group.
+Huppert’s conjecture has been verified for many cases such as the alternating groups, sporadic groups,
+and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie type with
+high rank. Recently, a similar conjecture related to codegrees has been posed.
+Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a
+finite group such that cod(H) = cod(G). Then G ∼= H.
+This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question
+20.79 [18]. It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23
+and PSL(3, 3) by [1, 3, 13]. The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [19] and
+2G2(q) in [14]. Recently, the authors verified the conjecture for all sporadic simple groups in [11].
+In this paper, we provide a general proof verifying this conjecture for all alternating groups of degree
+greater than or equal to 5. The methods used may be generalized to simple groups of Lie type, giving
+promising results for characterizing all simple groups by their codegree sets.
+Theorem 1.1. Let An be an alternating group of degree n ≥ 5 and G a finite group. If cod(G) = cod(An),
+then G ∼= An.
+Throughout the paper, we follow the notation used in Isaacs’ book [16] and the ATLAS of Finite Groups
+[9].
+2000 Mathematics Subject Classification. 20C15, 20D06.
+1
+
+2. Preliminary Results
+We first introduce some lemmas which will be used later.
+Lemma 2.1. [21, Lemma 4.2] Let S be a finite nonabelian simple group. Then there exists 1S ̸= χ ∈ Irr(S)
+that extends to Aut(S).
+Lemma 2.2. [17, Theorem 4.3.34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St
+where Si ∼= S is a nonabelian simple group for each i = 1, . . . , t. If χ ∈ Irr(S) extends to Aut(S), then
+χ × · · · × χ ∈ Irr(N) extends to G.
+Lemma 2.3. [13, Remark 2.6] Let G be a finite group and S a finite nonabelian simple group with cod(G) =
+cod(S). Then G is a perfect group.
+Lemma 2.4. [15] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G).
+Then |S| divides |G|.
+Lemma 2.5. Let G be a finite group with N ⊴ G. Then cod(G/N) ⊆ cod(G).
+Proof. From [16, Lemma 2.22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}.
+Take any ˆχ ∈ Irr(G/N). By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ)
+are equal. In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|. Thus |G/N : ker(ˆχ)| =
+|G|/|N|
+| ker(χ)|/|N| =
+|G|
+| ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G).
+□
+Lemma 2.6. Let G be a finite group with normal subgroups N and M such that N ≤ M. Then, cod(G/M) ⊆
+cod(G/N).
+Proof. By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N,
+and by Lemma 2.5, cod(G/M) ⊆ cod(G/N).
+□
+Lemma 2.7. Let S be a finite nonabelian simple group and G be a nontrivial finite group with cod(G) ⊆
+cod(S). Then, |S| < |G| · |Irr(G)|.
+Proof. We know that for each irreducible character χ ∈ Irr(S), χ(1)2 < |S|. Because S is simple, if χ is non-
+trivial, then ker(χ) = 1, so cod(χ) =
+|S|
+χ(1) >
+�
+|S|. Then, since cod(G) ⊆ cod(S), for each irreducible non-
+trivial character ψ ∈ Irr(G), cod(ψ) >
+�
+|S|. Thus, |G:ker(ψ)|
+ψ(1)
+>
+�
+|S| which implies that
+|G|
+|ker(ψ)|√
+|S| > ψ(1).
+So, ψ(1) <
+|G|
+√
+|S|.
+Then �
+ψ∈Irr(G) ψ(1)2 < | Irr(G)| |G|2
+|S| , and by character theorems, we’ll have |G| <
+| Irr(G)| |G|2
+|S| . Thus |S| < |G| · |Irr(G)|.
+□
+3. Main Results
+We start with some lemmas which limit the simple groups whose codegree set can be contained in the
+codegree set of an alternating group.
+Lemma 3.1. Let H be an alternating group of degree m ̸= n, where m, n ≥ 5. Then cod(H) ̸⊆ cod(An).
+Proof. Suppose cod(Am) ⊆ cod(An). Then, from Lemma 2.4, |Am| divides |An|, so m < n. Let ax denote
+the minimal non-trivial codegree of Ax.
+We show that an−1 < an so that cod(Am) ̸⊆ cod(An) follows
+immediately.
+We know that irreducible representations of the symmetric group Sn are in one-to-one correspondence
+with the partitions of n. Let λ be a partition of n and Vλ be the corresponding irreducible representation
+of Sn. We note that a partition of n can be visualized by a Young diagram and we let hλ(i, j) be the hook
+length of the (i, j)th square of the Young diagram corresponding to λ, i.e. the number of cells (a, b) of λ
+such that a = i and b ≥ j or b = j and a ≥ i. By the hook length formula,
+n!
+dim(Vλ) = � hλ(i, j) := Hλ.
+Let Uλ be an irreducible constituent of the restriction of Vλ to An, ResSn
+An Vλ. If λ is not self-conjugate
+(λ ̸= λ′), then ResSn
+An Vλ remains irreducible, so Uλ = ResSn
+An Vλ. In this case,
+n!
+dim(Uλ) = Hλ. If λ is self-
+conjugate, then the restriction of Vλ to An splits into two irreducible representations of the same dimension,
+so dim(Uλ) = 1
+2dim(Vλ). In this case,
+n!
+dim(Uλ) = 2Hλ.
+2
+
+Now, an = min{
+n!/2
+dim(Uλ) | Uλ ∈ Irr(An)} = 1
+2 min({Hλ | λ ̸= λ′} ∪ {2Hλ | λ = λ′}). We want to show that
+an−1 < an. First, assume that an = 1
+22Hλ for some λ = λ′. Then we can remove a square from λ to give a
+non-self-conjugate partition µ of n − 1. Since Hµ < Hλ < 2Hλ and an−1 ≤ 1
+2Hµ, we know an−1 < an.
+Now assume that an = 1
+2Hλ for some λ ̸= λ′. Then if n ≥ 3, we can remove a square from λ to obtain a
+non-self-conjugate partition µ of n − 1. Since Hµ < Hλ and an−1 ≤ 1
+2Hµ, an−1 < an. Thus, if m < n, then
+am < an, contradicting the assumption that cod(Am) ⊆ cod(An).
+□
+Lemma 3.2. Let H be a sporadic simple group or the Tits group. Then if n ≥ 5, cod(H) ̸⊆ cod(An).
+Proof. In search of a contradiction, let H be a sporadic simple group or the Tits group such that cod(H) ⊆
+cod(An). From Lemmas 2.7 and 2.4, we deduce a tight restriction on the order of H. Namely, |H| = |An|/k
+where 1 ≤ k < |Irr(H)| is an integer. Now, for each sporadic (or Tits) group H, we can computationally
+check (using Julia [6]) which alternating groups An satisfy both |H| divides |An| and |An|
+|H| < |Irr(H)|. We
+find only one possible exception: An = A10 and H = J2 where |A10|
+|J2| = 3 < 21 = |Irr(J2)|. In this case, we
+check that cod(J2) ̸⊆ cod(A10) using the ATLAS [9].
+□
+Lemma 3.3. Let H be a classical simple group of Lie type. Then cod(H) ̸⊆ cod(An) for all n ≥ 5.
+Proof. There are 6 families of classical simple groups of Lie type.
+These are PSL(m + 1, q), Ω(2m +
+1, q), PSp(2m, q), O+(2m, q), PSU(m + 1, q), and O−(2m, q).l We prove the lemma in each case. Let k(G)
+denote the number of conjugacy classes of G, we reproduce [12, Table 2] for reference.
+Table 1. Class Numbers for Classical Groups
+G
+k(G) ≤
+Comments
+SL(n, q)
+2.5qn−1
+SU(n, q)
+8.26qn−1
+Sp(2n, q)
+10.8qn
+q odd
+Sp(2n, q)
+15.2qn
+q even
+SO(2n + 1, q)
+7.1qn
+q odd
+Ω(2n + 1, q)
+7.3qn
+q odd
+SO±(2n, q)
+7.5qn
+q odd
+Ω±(2n, q)
+6.8qn
+q odd
+O±(2n, q)
+9.5qn
+q odd
+SO±(2n, q)
+14qn
+q even
+O±(2n, q)
+15qn
+q even
+(1) Let H = PSL(m + 1, q) where q = pk and m ≥ 1. From the order formula found in [7], qm(m+1)/2
+divides |PSL(m + 1, q)|. From Legendre’s formula, we know that for any prime p, |n!|p ≤ p
+n
+p−1 .
+If q = pk, then we have |n!|q ≤ q
+n
+k(p−1) and thus |An|q ≤ q
+n
+k(p−1) . By Lemma 2.4, |PSL(m + 1, q)|
+divides |An|, so qm(m+1)/2 divides |An|. Thus m(m+1)
+2
+≤
+n
+k(p−1), giving n ≥ m(m+1)k(p−1)
+2
+. Therefore,
+|An| ≥
+���A m(m+1)k(p−1)
+2
+���.
+Now, we note that k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) since PSL(m + 1, q) is a quotient of
+SL(m + 1, q). Then from Table 1, we have that |Irr(PSL(m + 1, q))| = k(PSL(m + 1, q)) ≤ k(SL(m +
+1, q)) ≤ 2.5qm.
+Applying Lemma 2.7 gives |An| < |PSL(m + 1, q)| · |Irr(PSL(m + 1, q))|. Hence
+|A m(m+1)k(p−1)
+2
+| < |PSL(m + 1, q)| · 2.5qm. Now we show that if we consider the left and right sides
+as functions of m with constants p and k, then asymptotically, the value of |A m(m+1)k(p−1)
+2
+| grows
+faster than that of |PSL(m + 1, q)| · 2.5qm. We know that the left function behaves asymptotically
+as (m2)!, and using the order formula for PSL(m + 1, q), we know that the right function behaves
+asymptotically as qf(m), where f(m) is a polynomial with degree 2. Thus the left function grows
+faster than the right function since x! >> cx for any constant c when x is large. Similarly, we can
+prove this result considering the two sides as functions of p and k.
+3
+
+Then, we search for the maximum possible value of m which satisfies the inequality given the
+smallest possible values of p and k, which are 2 and 1, respectively. We find that m ≤ 6 and, using
+a similar process for p and k, that p ≤ 17 and k ≤ 63. Now, we have limited our search to a finite
+number of groups which we can check in the same way as for the sporadic groups. From this, we
+find a small list of exceptions, listed in Table 2:
+Table 2. Exceptions satisfying |PSL(m + 1, q)| divides |An| and
+|An| < |PSL(m + 1, q)| · 2.5qm
+m
+q
+n
+1
+4
+5
+1
+4
+6
+1
+8
+7
+1
+9
+6
+1
+9
+7
+1
+5
+5
+1
+5
+6
+1
+7
+7
+2
+4
+8
+2
+4
+9
+3
+2
+8
+3
+2
+9
+Now, all of these exceptions can be found in the ATLAS, and it is routine to check that none of
+these groups satisfy cod(PSL(m + 1, q)) ⊆ cod(An) unless PSL(m + 1, q) ∼= An. Thus, if PSL(m +
+1, q) ̸∼= An, then cod(PSL(m + 1, q) ̸⊆ cod(An).
+(2) Let H = Ω(2m+1, q) where q = pk is odd and m ≥ 2. Note that when q = 2k is even, Ω(2m+1, q) ∼=
+PSp(2m, q), which we deal with in the next case. From [7], qm2 divides |Ω(2m + 1, q)|. Thus, using
+Table 1 similarly to above, |Am2k(p−1)| < |Ω(2m+ 1, q)|·7.3qm. As above, we computationally check
+that we get a contradiction if m > 2, p > 3, or k > 1, so m = 2, p = 3, and k = 1 is the only
+possibility. We get the list of exceptions listed in Table 3 after checking divisibility.
+Table 3. Exceptions satisfying |Ω(2m + 1, q)| divides |An| and
+|An| < |Ω(2m + 1, q)| · 7.3qm
+m
+q
+n
+2
+3
+9
+Again, we check the ATLAS and find that cod(Ω(5, 3)) ̸⊆ cod(A9).
+(3) Let H = PSp(2m, q) where q = pk and m ≥ 3. From [7], qm2 divides |PSp(2m, q)|. Since PSp(2m, q)
+is a quotient of Sp(2m, q), we have k(PSp(2m, q)) ≤ k(Sp(2m, q)). From Table 1, |Am2k(p−1)| <
+|PSp(2m, q)| · 15.2qm. We computationally check that we get a contradiction if m > 4, p > 2, or
+k > 2, so m = 3 or 4, p = 2, and k = 1 or 2 are the only possibilities. We get no exceptions after
+checking divisibility.
+(4) Let H = O+(2m, q) where q = pk and m ≥ 4. From [7], qm(m−1) divides |O+(2m, q)|. Using Table
+1, we have that |Am(m−1)k(p−1)| < |O+(2m, q)| · 15qm. As above, we computationally check that we
+get a contradiction if m > 4, p > 2, or k > 1 so m = 4, p = 2, and k = 1 is the only possibility, and
+we get no possible exceptions after checking divisibility.
+(5) Let H = PSU(m+1, q) where q = pk and m ≥ 2. From [7], qm(m+1)/2 divides |PSU(m+1, q)|. Since
+PSU(m + 1, q) is a quotient of SU(m + 1, q), we have k(PSU(m + 1, q)) ≤ k(SU(m + 1, q)). From
+Table 1, |A m(m+1)k(p−1)
+2
+| < |PSU(m + 1, q)| · 8.26qm. Again, we computationally check that we get a
+4
+
+contradiction if m > 6, p > 7, or k > 42 so m ≤ 6, p ≤ 7, and k ≤ 42 are the only possibilities. We
+get Table 4 after checking divisibility:
+Table 4. Exceptions satisfying |PSU(m + 1, q)| divides |An| and
+|An| < |PSU(m + 1, q)| · 8.26qm
+m
+q
+n
+2
+3
+9
+3
+2
+9
+We check the ATLAS to find that cod(PSU(3, 3)) ̸⊆ cod(A9), and we note that PSU(4, 2) ∼=
+Ω(5, 3), which we have already ruled out.
+(6) Let H = O−(2m, q) where q = pk and m ≥ 4. From [7], qm(m−1) divides |O−(2m, q)|. Thus, using
+Table 1 similarly to above, |Am(m−1)k(p−1)| < |O−(2m, q)| · 15qm. Again, we computationally check
+that we get a contradiction if m > 5, p > 3, or k > 3 so m ≤ 5, p ≤ 3, and k ≤ 3 are the only
+possibilities, and we get no possible exceptions after checking divisibility.
+□
+Lemma 3.4. Let H be an exceptional simple group of Lie type. Then if n ≥ 5, cod(H) ̸⊆ cod(An).
+Proof. There are 10 familes of exceptional simple groups of Lie type (other than the Tits group). These are
+E6(q), E7(q), E8(q), F4(q), G2(q),2 E6(q),3 D4(q),2 B2(q),2 F4(q), and 2G2(q). We prove the lemma in each
+case. First, we reproduce [12, Table 1] for reference.
+Table 5. Class Numbers for Exceptional Groups
+G
+k(G) ≤
+Comments
+2B2(q)
+q + 3
+q = 22m+1
+2G2(q)
+q + 8
+q = 32m+1
+G2(q)
+q2 + 2q + 9
+2F4(q)
+q2 + 4q + 17
+q = 22m+1
+3D4(q)
+q4 + q3 + q2 + q + 6
+F4(q)
+q4 + 2q3 + 7q2 + 15q + 31
+E6(q)
+q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60
+2E6(q)
+q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62
+E7(q)
+q7 + q6 + 2q5 + 7q4 + 17q3 + 35q2 + 71q + 103
+E8(q)
+q8 + q7 + 2q6 + 3q5 + 10q4 + 16q3 + 40q2 + 67q + 112
+(1) Let H ∼= E6(q) where q = pk. From the order formula found in [7], q36 divides |E6(q)|. From [5],
+we know that for any prime p, |n!|p ≤ p
+n
+p−1 . If q = pk, then we have |n!|q ≤ q
+n
+k(p−1) and thus
+|An|q ≤ q
+n
+k(p−1) where |An|p is the p-part of An. By Lemma 2.4, |E6(q)| divides |An| so q36 divides
+|An|. Thus 36 ≤
+n
+k(p−1) and n ≥ 36k(p − 1). Therefore, |An| ≥ |A36k(p−1)|.
+Now, we note from Table 5 that |Irr(E6(q))| = k(E6(q)) ≤ q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60.
+Applying Lemma 2.7 gives |An| < |E6(q)|·|Irr(E6(q))|. Hence, |A36k(p−1)| < |E6(q)|·(q6 +q5 +2q4 +
+2q3 + 15q2 + 21q + 60). As with the classical Lie type groups, we can computationally find an upper
+bound on p and k since the left side grows faster in terms of p and k than the right side. In this
+case, we find that no values of p and k satisfy the inequality, since substituting p = 2 and k = 1
+gives |A36| > |E6(2)| · (26 + 25 + 2 · 24 + 2 · 23 + 15 · 22 + 21 · 2 + 60). Thus, there are no possible
+values for q and n such that cod(E6(q)) ⊆ cod(An).
+(2) Let H ∼= E7(q) where q = pk. From [7], q63 divides |E7(q)|. From Table 5, |A63k(p−1)| < |E7(q)| ·
+(q7 +q7 +2q5 +7q4 +17q3 +35q2+71q +103). We computationally check that we get a contradiction
+for p = 2, k = 1, so there are no possible exceptions.
+5
+
+(3) Let H ∼= E8(q) where q = pk. From [7], q120 divides |E8(q)|. Thus, using Table 5 as above, we have
+|A120k(p−1)| < |E8(q)|·(q8+q7+2q6+3q5+10q4+16q3+40q2+67q+112). Now, we computationally
+check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.
+(4) Let H ∼= F4(q) where q = pk. From [7], q24 divides |F4(q)|. From Table 5, |A24k(p−1)| < |F4(q)|·(q4 +
+2q3 + 7q2 + 15q + 31). Again, we computationally check that we get a contradiction for p = 2, k = 1,
+so there are no possible exceptions.
+(5) Let H ∼= G2(q) where q = pk. From [7], q6 divides |G2(q)|. Thus, using Table 5 as above, |A6k(p−1)| <
+|G2(q)| · (q2 + 2q + 9). Now, we find that p = 2, k = 1 satisfies the inequality, but any other values
+of p and k do not. However, we note that G2(2) is not simple, so we instead consider its derived
+subgroup G2(2)′ (which still satisfies the above inequality). We check for exceptions where |G2(2)′|
+divides |An| and |An| < |G2(2)′| · (22 + 2 · 2 + 9), but there are none.
+(6) Let H ∼= 2E6(q) where q = pk.
+From [7], q36 divides |2E6(q)|.
+Using Table 5, |A36k(p−1)| <
+|2E6(q)| · (q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62). Again, we computationally check that we get a
+contradiction for p = 2, k = 1, so there are no possible exceptions.
+(7) Let H ∼= 3D4(q) where q = pk. From [7], q12 divides |3D4(q)|. Thus, using Table 5 similarly to
+above, |A12k(p−1)| < |3D4(q)| · (q4 + q3 + q2 + q + 6). Now, we find that p = 2, k = 1 satisfies the
+inequality, but any other values of p and k do not. As for the sporadic groups, we check for possible
+exceptions where |3D4(2)| divides |An| and |An| < |3D4(2)| · (24 + 23 + 22 + 2 + 2), but there are
+none.
+(8) Let H ∼= 2B2(q) where q = 22m+1 and m ≥ 1. From [7], q2 divides |2B2(q)|. From Table 5, we
+have that |A2(2m+1)| < |2B2(q)| · (q + 3). In this case, we computationally check that we get a
+contradiction if m > 4, so m must be less than 5. However, checking the divisibility condition, we
+get no exceptions.
+(9) Let H ∼= 2F4(q) where q = 22m+1 and m ≥ 1. From [7], q12 divides |2F4(q)|. Thus, using Table
+5 as above, |A12(2m+1)| < |2F4(q)| · (q2 + 4q + 17). Now, we computationally check that we get a
+contradiction for m = 1, so there are no exceptions
+(10) Let H ∼= 2G2(q) where q = 32m+1 and m ≥ 1.
+From [7], q3 divides |2G2(q)|.
+From Table 5,
+|A3(2m+1)·2| < |2G2(q)| · (q + 8). Again, we computationally check that we get a contradiction for
+m = 1, so there are no exceptions.
+□
+Theorem 3.5. Let G be a finite group such that cod(G) = cod(An) where n ≥ 5. Let N be a maximal
+subgroup of G. Then, G/N ∼= An.
+Proof. By Lemma 2.3, G is perfect. Thus G/N is a nonabelian simple group. By Lemma 2.6, we have
+cod(G/N) ⊆ cod(G) = cod(An). By Lemmas 3.1, 3.2, 3.3, and 3.4, G/N cannot be an alternating group
+of degree m ̸= n, a sporadic simple group or the Tits group, a classical simple group of Lie type, or an
+exceptional simple group of Lie type. Thus, G/N ∼= An.
+□
+Now we present the proof of Theorem 1.1.
+Proof. Let G be a minimal counterexample and N be a maximal normal subgroup of G. By Lemma 2.3, G
+is perfect, and by Theorem 3.5, G/N ∼= An. In particular, N ̸= 1 as G ̸∼= An.
+Step 1: N is a minimal normal subgroup of G.
+Suppose L is a non-trivial normal subgroup of G with L < N. Then by Lemma 2.6, we have cod(G/N) ⊆
+cod(G/L) ⊆ cod(G).
+However, cod(G/N) = cod(An) = cod(G) so equality must be obtained in each
+inclusion. Thus, cod(G/L) = cod(An) which implies that G/L ∼= An since G is a minimal counterexample.
+This is a contradiction since we also have G/N ∼= An, but L < N.
+Step 2: N is the only non-trivial, proper normal subgroup of G.
+Otherwise we assume M is another proper nontrivial normal subgroup of G. If N is included in M, then
+M = N or M = G since G/N is simple, a contradiction. Then N ∩ M = 1 and G = N × M. Since M is also
+a maximal normal subgroup of G, we have N ∼= M ∼= An. Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(M) such that
+cod(ψ1) = cod(ψ2) = max(cod(An)). Set χ = ψ1 · ψ2 ∈ Irr(G). Then cod(χ) = (max(cod(An)))2 /∈ cod(G),
+a contradiction.
+Step 3: For each non-trivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful.
+6
+
+We construct Irr(G/N) as the same as Lemma 2.5. Then it follows by the definition of Irr(G|N) that if
+χ ∈ Irr(G|N), N ̸≤ ker(χ). Thus since N is the unique nontrivial, proper, normal subgroup of G, ker(χ) = G
+or ker(χ) = 1. Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N).
+Step 4: N is an elementary abelian group.
+Suppose that N is not abelian. Since N is a minimal normal subgroup, by [10, Theorem 4.3A (iii)],
+N = Sn where S is a nonabelian simple group and n ∈ Z+.
+By Lemmas 2.1 and 2.2, there is a non-
+trivial character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G). Now, ker(ψ) = 1 by Step 3, so cod(ψ) =
+|G|/ψ(1) = |G/N| · |N|/χ(1). However, by assumption, we have that cod(G) = cod(An) = cod(G/N). Thus,
+cod(ψ) ∈ cod(G) = cod(G/N), so cod(ψ) = |G/N|/φ(1) for some φ ∈ Irr(G/N). Hence, |G/N| is divisible by
+cod(ψ) which contradicts the fact that cod(ψ) = |G/N| · |N|/χ(1), as χ(1) ̸= |N|. Thus N must be abelian.
+Now to show that N is elementary abelian, let a prime p divide |N|. Then N has a p-Sylow subgroup
+K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N. Thus,
+K is a normal subgroup of G, so K = N as N is minimal.
+Thus |N| = pn. Now, take the subgroup
+N p = {np | n ∈ N} of N, which is proper by Cauchy’s theorem. Since N p is characteristic in N, it must
+be normal in G, so N p is trivial by the uniqueness of N. Thus every element of N has order p, and N is
+elementary abelian.
+Step 5: CG(N) = N.
+First note that since N is normal, CG(N) ⊴ G. Additionally, since N is abelian by Step 4, N ≤ CG(N).
+By the maximality of N, we must have CG(N) = N or CG(N) = G. If CG(N) = N, we are done.
+If not, then CG(N) = G, so N must be in the center of G. Then since N is the unique minimal normal
+subgroup of G by Step 2, we must have that |N| is prime. If not, there always exists a proper non-trivial
+subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of N. Moreover,
+since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur multiplier of
+G/N [16, Corollary 11.20].
+Now, we note that it is well-known that for n > 7, the Schur multiplier of An is Z2, so G ∼= 2.An.
+From [20], 2.An always has a character degree of order 2⌊(n−2)/2⌋. Let χ be such an irreducible character of
+2.An with χ(1) = 2⌊(n−2)/2⌋. Recall that by Step 2, there is only one non-trivial proper normal subgroup of
+G ∼= 2.An. In particular N ∼= Z2 is the only non-trivial proper normal subgroup of G. Thus |ker(χ)| = 1
+or 2. Then we have cod(χ) = |2.An:ker(χ)|
+χ(1)
+. If |ker(χ)| = 1, then cod(χ) =
+n!
+2⌊(n−2)/2⌋ , and if |ker(χ)| = 2,
+then cod(χ) =
+n!/2
+2⌊(n−2)/2⌋ =
+n!
+2⌊n/2⌋ . In either case, for any prime p ̸= 2, | cod(χ)|p = |n!|p = |An|p. Since
+cod(G) = cod(An), we know that cod(χ) ∈ cod(An). Therefore, there is a character degree of An which is a
+power of 2.
+However, from [20], we know that for n > 7, An only has a character degree equal to a power of 2 when
+n = 2d + 1 for some positive integer d. In this case, 2d = n − 1 ∈ cd(An) so we need |An|
+n−1 =
+|2.An|
+2⌊(n−2)/2⌋ or
+|2.An|
+2⌊n/2⌋ . Hence,
+1
+n−1 =
+2
+2⌊(n−2)/2⌋ =
+1
+2⌊(n−2)/2⌋−1 or
+1
+2⌊n/2⌋−1 so n − 1 = 2⌊(n−2)/2⌋−1 or 2⌊n/2⌋−1. However, the
+only integer solution to either of these equations occurs when n = 9 and 9 − 1 = 8 = 23 = 2⌊9/2⌋−1. In this
+case, we check the ATLAS [9] to find that the codegree sets of A9 and 2.A9 do not have the same order.
+This is a contradiction, so CG(N) = N.
+Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents
+of λIG(λ), where IG(λ) is the inertia group of λ ∈ G. Then |IG(λ)|
+ϑ(1)
+∈ cod(G). Also, ϑ(1) divides |IG(λ)/N|,
+and |N| divides |G/N|. Lastly, IG(λ) < G, i.e. λ is not G-invariant.
+Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ). Let χ be an irreducible constituent of
+ϑG. By [16, Corollary 5.4], we know χ ∈ Irr(G), and by [16, Definition 5.1], we have χ(1) =
+|G|
+|IG(λ)| · ϑ(1).
+Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) =
+|G|
+χ(1) = |IG(λ)|
+ϑ(1) , so |IG(λ)|
+ϑ(1)
+∈ cod(G). Now,
+since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)|
+|N| , so |N| divides
+|IG(λ)|
+ϑ(1) .
+Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides
+|G/N|.
+Next, we want to show IG(λ) is a proper subgroup of G. To reach a contradiction, assume IG(λ) = G.
+Then ker(λ) ⊴ G. From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime
+7
+
+order. Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is
+abelian, a contradiction.
+Step 7: Final contradiction.
+From Step 4, N is an elementary abelian group of order pm for some prime p and integer m ≥ 1. By
+the Normalizer-Centralizer theorem, An ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and m > 1. Note that in
+general, Aut(N) = GL(m, p). By Step 6, |N| divides |G/N|, so we know that |N| = pm divides |An| and
+G/N ∼= An ≲ GL(m, p). We prove by contradiction that this cannot occur.
+First, we claim that if pm divides |An| and An ≲ (GL(m, p), then p must equal 2. To show this, we note
+that for p > 2, by [5], we have that if pm divides |An|, then m < n
+2 . However, Theorem 1.1 of [24] shows that
+if n > 6, the minimal faithful degree of a modular representation of An over a field of characteristic p is at
+least n − 2. Since embedding An as a subgroup of GL(m, p) is equivalent to giving a faithful representation
+of degree m over a field of characteristic p, we have that m ≥ n − 2. This is a contradiction since n
+2 > n − 2
+implies n < 4. Therefore, p = 2.
+Now, let p = 2.
+As above, from [5], we obtain |n!|2 ≤ 2n−1.
+Thus, if 2m divides |An|, then m ≤
+|An|2 ≤ 2n−2. Now, Theorem 1.1 of [23] shows that if n > 8, then the minimal faithful degree of a modular
+representation of An over a field of characteristic 2 is at least n − 2. Therefore, we must have m ≥ n − 2, so
+m = |An|2 = 2n−2 is the only option.
+Let λ ∈ Irr(N), ϑ ∈ Irr(IG(λ)|λ), and T := IG(λ). Then 1 < |G : T | < |N| = 2n−2 for |G : T | is
+the number of all conjugates of λ. By Step 5, we know that
+|T |
+ϑ(1) ∈ cod(G) and moreover that |N| divides
+|T |
+ϑ(1). Since |N|2 = |An|2 and cod(G) = cod(An), we know that
+��� |T |
+ϑ(1)
+���
+2 = |N|2. Thus
+��� |T/N|
+ϑ(1)
+���
+2 = 1 so the
+2-parts of |T/N| and ϑ(1) are equal. Thus for every ϑ ∈ Irr(T | λ), we have |ϑ(1)|2 = |T/N|2. However,
+|T/N| = �
+ϑ∈Irr(T |λ) ϑ(1)2. Hence, if |ϑ(1)|2 = 2k ≥ 2 for every ϑ ∈ Irr(T | λ), we would have |T/N|2 = 22k
+contradicting the fact that |ϑ(1)|2 = |T/N|2. Therefore, |T/N|2 = 1. Thus, since |G/N|2 ≥ |N|2 = 2n−2, we
+have |G : T |2 = |G/N : T/N|2 ≥ 2n−2, so |G : T | ≥ 2n−2 = |N|, which is a contradiction.
+We have one final exception to consider: n = 8, p = 2, and m = 4, 5 or 6. In this case, A8 ∼= GL(4, 2) and
+26 divides |A8|. Now, cod(A8) = {1, 26·32·5, 25·32·5, 24·32·7, 26·3·5, 24·32·5, 26·32, 26·7, 23·32·5, 32·5·7, 25·32}
+from [13]. We will look at each possibility for m in turn.
+First, let m = 4. Then we have G/N ∼= A8 ∼= GL(4, 2), N = (Z2)4 so G is an extension of GL(4, 2) by N.
+Suppose first that this extension is split and G is a semidirect product. This semidirect product is defined
+by a homomorphism φ : GL(4, 2) → Aut((Z2)4) ∼= GL(4, 2). However, since GL(4, 2) is simple, ker(φ) = 1 or
+GL(4, 2). In the first case, we have the trivial direct product, so there are at least two copies of GL(4, 2) as
+normal subgroups of G, which contradicts Step 2. In the second case, φ is some automorphism of GL(4, 2).
+Here, we can check using GAP that any such φ creates a semidirect product GL(4, 2) ⋊φ (Z2)4 which does
+not have the same codegree set as A8. Now, suppose that the extension is non-split. Then, [4] gives that
+there is a unique non-split extension 24.GL(4, 2). However, we find using GAP that it doesn’t have the same
+codegree set as A8.
+Second, let m = 5.
+As above, |G : T | < |N| = 25 and
+|T |
+ϑ(1) ∈ cod(G) such that 25 divides
+|T |
+ϑ(1).
+Further, | |T/N|
+ϑ(1) |2 ≤ 2 so |T/N|2 ≤ 4 and |G/N : T/N|2 ≥ 16. Thus, we have 16 divides |G/N : T/N| and
+|G/N : T/N| < 32. But we check the index of all subgroups of G/N ∼= A8 using GAP and find that none of
+them satisfy these two properties.
+Finally, let m = 6. Now, |N|2 = |A8|2. For this case the same argument as above for general An holds,
+and we reach a contradiction. Thus we find that every |N| = pm produces a contradiction, so N = 1 and
+G ∼= An.
+□
+4. Acknowledgements
+This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230-
+21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the
+supervision of Yang. The authors gratefully acknowledge the financial support of NSF and NSA, and also
+thank Texas State University for providing a great working environment and support. Yang was also partially
+supported by grants from the Simons Foundation (#499532, #918096, to YY). The authors would also like
+to thank Prof. Richard Stanley for his help.
+8
+
+References
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+[5] C. Bessenrodt, H. P. Tong-Viet, J. Zhang, Huppert’s conjecture for alternating groups. J. Algebra, 470 (2017), 353-378.
+[6] J. Bezanson, S. Karpinski, V. B. Shah, A. Edelman, Julia: A fast dynamic language for technical computing. ArXiv
+Preprint, ArXiv:1209.5145.
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+submitted.
+[12] J. Fulman and R. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with appli-
+cations to derangements. Trans. Amer. Math. Soc., 364 (2012), 3023-3070.
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+with few codegrees. Comm. Algebra, 50 (2022), 3932-3939.
+[14] H. Guan, X. Zhang, Y. Yang, Recognizing Ree groups
+2G2(q) using the codegree set. Bull. Aust. Math. Soc.,
+https://www.doi.org/10.1017/S0004972722001022.
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+[16] I. M. Isaacs, Character Theory of Finite Groups. New York Academic Press, 1976.
+[17] G. James and A. Kerber, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Company, 1981.
+[18] E. I. Khukrho and V. D. Mazurov, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 20. Russian Academy
+of Sciences, 2022.
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+[20] G. Malle and A.E. Zalesskii, Prime power degree representations of quasi-simple groups. Arch. Math., 77 (2001), 461-468.
+[21] A. Moret´o, Complex group algebra of finite groups: Brauer’s problem 1. Adv. Math., 208 (2007), 236-248.
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+154 (1977), 104-113.
+Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.dolorfino19@kzoo.edu
+Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.com
+Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.edu
+Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.edu
+Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.edu
+9
+

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+Mechanical feedback linearization of single-input
+mechanical control systems
+Marcin Nowicki1 and Witold Respondek2,3
+1Poznan University of Technology, Institute of Automatic Control
+and Robotics, Piotrowo 3a, 61-138 Pozna´n, Poland
+2Lodz University of Technology, Institute of Automatic Control, B.
+Stefanowskiego 18, 90-537 Lodz, Poland
+3INSA de Rouen Normandie, Laboratoire de Math´ematiques,
+76801 Saint-Etienne-du-Rouvray, France
+January 3, 2023
+Abstract
+We present a new type of feedback linearization that is tailored for me-
+chanical control systems. We call it a mechanical feedback linearization.
+Its basic feature is preservation of the mechanical structure of the system.
+For mechanical systems with a scalar control, we formulate necessary and
+sufficient conditions that are verifiable using differentiations and algebraic
+operations only. We illustrate our results with several examples.
+1
+Introduction
+An N-dimensional control-affine system with a scalar control
+˙z = F(z) + G(z)u,
+(Σ)
+where z ∈ Z, an open subset of RN, and u ∈ R, is said to be (locally) feedback
+linearizable (F-linearizable) if there exist a (local) diffeomorphism Φ : Z → RN
+and an invertible feedback of the form u = α(z) + β(z)˜u such that the control
+system (Σ), in the new coordinates ˜z = Φ(z) and with the new control ˜u, is a
+controllable linear system of the form ˙˜z = A˜z + b˜u. A geometric solution to the
+problem of feedback linearization (inspired by [1], and developed independently
+in [2] and [3]) provides powerful techniques for designing a closed-loop control
+system that have been used in numerous engineering applications.
+From a
+theoretical point of view, that result identifies a class of nonlinear systems that
+can be considered as linear ones in a well-chosen coordinates and with respect
+to a well-modified control.
+1
+arXiv:2301.00087v1  [math.OC]  31 Dec 2022
+
+In this paper, we state and study the following fundamental question: if a
+nonlinear control system (Σ) is mechanical and feedback linearizable, are those
+two structures compatible? That is, can we feedback linearize the system pre-
+serving its mechanical structure? For mechanical control systems, it is natural
+to consider mechanical feedback equivalence (in particular, to a linear form)
+under mechanical transformations (coordinates changes and feedback) that pre-
+serve the mechanical structure of the system. In our recent paper [4], we showed
+that even in the simplest underactuated case of 2 degrees of freedom, the struc-
+tures (linear and mechanical) may not conform trivially. In the present paper,
+we treat the single-input case in its full generality.
+There are several motivations for preserving the mechanical structure when
+feedback linearizing the system. First, our formulation of the problem of me-
+chanical linearization preserves configurations and velocities. We reckon that it
+is essential that new configurations (of the linearized system) are functions of the
+original configurations only, as well as new velocities are true physical velocities
+(in contrast to pseudo-velocities). Therefore, we do not lose the physical inter-
+pretation of the system. This could be useful, e.g. for mechanical systems with
+constraints on configurations, which are transformed into linear constraints on
+configurations. Second, the configuration trajectories are preserved too, which
+could be useful in e.g. the motion planning problem (the most natural way to
+state the problem for mechanical systems is to follow configuration trajectories).
+Third, it is worth mentioning that mechanical feedback linearizability guaran-
+tees the linearizing outputs to be functions of configurations only. This may
+be of constructional importance because one needs only configuration sensors,
+not those of velocities. The next argument is the fact that the resultant linear
+mechanical system allows us to employ dedicated techniques for mechanical sys-
+tems. An example of such technique is the natural frequency method of tuning
+a linear feedback. Finally, when applying mechanical feedback linearization, the
+physical interpretation of the external action (force, torque, etc.) is preserved
+but is lost for general feedback linearization.
+This work is a mechanical counterpart of the classical results on feedback
+linearization of control systems [1], [2], [3], see also monographs [6], [7]. Our
+intention is to formulate conditions for mechanical linearization (shortly, MF-
+linearization) in a possibly similar manner (e.g. using involutivity of certain
+distributions).
+For a geometric approach to mechanical control systems see [5], [8], [9], [10].
+For mathematical preliminaries concerning the Lie derivative, the Lie bracket,
+distributions, etc., see [6], [7]. For linearization of mechanical control systems
+along controlled trajectories see [11]. For mechanical state-space linearization of
+mechanical control systems see [12] and [13]. Compare also [14], for a pioneering
+work on (partial) feedback linearization of mechanical systems.
+Although the state-space of mechanical control system is the tangent bundle
+TQ of the configuration space Q, we formulate our conditions using objects on
+Q only. The key here is a geometric approach to mechanical systems [5] and
+considering the Euler-Lagrange equations as the geodesic equation under an
+influence of external forces.
+2
+
+The outline of the paper is as follows. In Section 2, we state the problem. In
+Section 3, we develop further the problem of mechanical feedback linearization
+and formulate the main result, separately, for mechanical systems with n ≥ 3
+in Theorem 1, and with n = 2 in Theorem 2.
+In Section 4, we provide an
+application of our results to MF-linearization of several mechanical systems.
+Section 6 contains technical results used in proofs that could be of independent
+interest.
+1.1
+Notation
+Throughout the Einstein summation convention is assumed, i.e. any expression
+containing a repeated index (upper and lower) implies the summation over that
+index up to n, e.g. ωiXi = �n
+i=1 ωiXi.
+AT
+transpose of a matrix (of a vector) A,
+In
+n × n identity matrix,
+Q
+configuration manifold,
+X(Q)
+the set of smooth vector fields on a manifold Q,
+TxQ
+tangent space at x ∈ Q,
+TQ = �
+x∈Q TxQ
+tangent bundle of Q,
+x = (x1, . . . , xn)
+a local coordinate system on Q,
+φ
+a diffeomorphism of Q, and Φ a diffeomorphism of TQ,
+Dφ = ∂φ
+∂x
+the Jacobian matrix of a diffeomorphism φ,
+∂˜xi
+∂xj := ∂φi
+∂xj
+the (i, j)-element of the Jacobian matrix Dφ,
+∂xj
+∂˜xi
+the (j, i)-element of the inverse of the Jacobian matrix Dφ,
+LXα
+Lie derivative of a function α defined as LXα = ∂α
+∂xi Xi,
+[X, Y ] = ∂Y
+∂x X − ∂X
+∂x Y = adXY
+Lie bracket of vector fields,
+∂
+∂xi
+the i-th unity vector field, and dxi the i-th unity covector field, in a co-
+ordinate system x = (x1, . . . , xn),
+Ei = span
+�
+adj
+eg, 0 ≤ j ≤ i
+�
+distribution on Q spanned by adj
+eg,
+∇
+covariant derivative, and ∇2 second covariant derivative,
+Γi
+jk
+Christoffel symbols of the second kind of ∇,
+2
+Problem statement
+Consider an n-dimensional configuration space Q (an open subset of Rn or, in
+general, an n-dimensional manifold) equipped with a symmetric affine connec-
+tion ∇. The operator of the affine connection ∇ allows to define intrinsically the
+acceleration as the covariant derivative ∇ ˙x(t) ˙x(t), see e.g. [5,8,17]. The covari-
+ant derivative ∇ : X(Q) × X(Q) → X(Q) of an arbitrary vector field Y = Y i ∂
+∂xi
+with respect to X = Xi ∂
+∂xi in coordinates reads
+∇XY =
+�∂Y i
+∂xj Xj + Γi
+jkXjY k
+� ∂
+∂xi .
+(1)
+3
+
+A mechanical control system (MS) is a 4-tuple (Q, ∇, g, e), where g and e are,
+respectively, controlled and uncontrolled vector fields on Q. A curve x(t) : I →
+Q, I ⊂ R, is a trajectory of (MS) if it satisfies the following equation
+∇ ˙x(t) ˙x(t) = e (x(t)) + g (x(t)) u,
+(2)
+which can be viewed as an equation that balances accelerations of the system,
+where the left-hand side represents geometric accelerations (i.e. accelerations
+caused by the geometry of the system) and the right-hand side represents ac-
+celerations caused by external actions on the system (controlled or not). Notice
+that (2) is a second-order differential equation on Q (indeed, using (1) we con-
+clude that ∇ ˙x ˙x depends on ¨x, see [5] for details) and can be rewritten as a
+system of first-order differential equations on TQ, which we also call a mechan-
+ical control system (MS):
+˙xi = yi
+˙yi = −Γi
+jk(x)yjyk + ei(x) + gi(x)u,
+(MS)
+for 1 ≤ i ≤ n, where (x, y) =
+�
+x1, . . . , xn, y1, . . . , yn�
+are local coordinates
+on the tangent bundle TQ of the configuration manifold Q, and Γi
+jk(x) are
+Christoffel symbols of the affine connection ∇ that correspond to the Cori-
+olis and centrifugal forces.
+The vector fields e(x) = (e1(x), . . . , en(x))T and
+g(x) = (g1(x), . . . , gn(x))T correspond to, respectively, uncontrolled and con-
+trolled actions on the system. Throughout all objects are assumed to be smooth
+and the word smooth means C∞-smooth.
+Our obvious inspirations are Lagrangian mechanical control systems without
+dissipative forces. For the correspondence between (MS) and the Lagrangian
+equations of dynamics see [5], [8], [9] and our recent papers [13], [16]. However,
+we will consider throughout a more general class of mechanical control systems
+allowing for any symmetric (not necessarily a metric) connection and any (not
+necessarily potential) vector field e(x).
+Consider the group of mechanical feedback transformations GMF generated
+by the following transformations:
+(i) changes of coordinates in TQ given by Φ : TQ → T ˜Q
+(x, y) �→ (˜x, ˜y) = Φ(x, y) =
+�
+φ(x), ∂φ
+∂x(x)y
+�
+,
+(3)
+called a mechanical diffeomorphism, where φ : Q → ˜Q is a diffeomorphism
+and ∂φ
+∂x its Jacobian matrix,
+(ii) mechanical feedback transformations, denoted (α, β, γ), of the form
+u = γjk(x)yjyk + α(x) + β(x)˜u,
+(4)
+where γjk, α, β are smooth functions on Q satisfying
+γjk = γkj, β(·) ̸= 0. The matrix γ = (γjk) represents a (0, 2)−tensor field.
+4
+
+Even if the diffeomorphism φ is possibly local on Q, the action of ∂φ
+∂x(x) is always
+global on fibers TxQ.
+Definition 1. The system (MS) is MF-linearizable if there exist mechanical
+feedback transformations (Φ, α, β, γ) ∈ GMF bringing (MS) into a linear con-
+trollable mechanical system of the form
+˙˜xi = ˜yi
+˙˜yi = Ei
+j ˜xj + bi˜u,
+(LMS)
+where (˜x, ˜y) are coordinates on TRn = Rn × Rn, the matrix E = (Ei
+j) is an
+n × n real-valued matrix, the vector field b = bi ∂
+∂˜xi is constant, and the pair
+(E, b) is controllable (see [15]).
+Represent (MS) as ˙z = F(z) + G(z)u, where z = (x, y) ∈ TQ, F = yi ∂
+∂xi +
+�
+−Γi
+jk(x)yjyk + ei(x)
+�
+∂
+∂yi , and G = gi(x) ∂
+∂yi . The problem that we formulate
+and solve in the paper is whether (MS) is MF-linearizable? That is, do there
+exist Φ = (˜x, ˜y) = (φ, ∂φ
+∂xy) and (α, β, γ) such that
+∂Φ
+∂z (z)
+�
+F + G(yT γy + α)
+�
+(z) =
+�
+˜y
+E˜x
+�
+,
+∂Φ
+∂z (z) (Gβ) (z) =
+�0
+b
+�
+?
+Note that MF-linearizability is stronger than the classical feedback lineariz-
+ability since, for the latter, Φ : TQ → R2n can be any diffeomorphism (need not
+be of mechanical form (3)) and yT γ(x)y +α(x) can be replaced by any function
+α(x, y) on TQ and β(x) by any invertible function β(x, y) on TQ.
+If we neglect the mechanical structure of ˙z = F(z) + G(z)u, and consider
+it as a general control system, we can ask if the system is F-linearizable. The
+well-known answer [2,3] asserts that, locally, this is the case if and only if the
+distributions Di = span
+�
+adj
+F G, 0 ≤ j ≤ i
+�
+are involutive and of constant rank
+for i = 0, ..., 2n − 1 and D2n−1 = TQ. The natural question arises whether, for
+F-linearizable (MS), the general feedback transformations (Φ(z), α(z), β(z)) are
+mechanical (i.e. of the form (3) and (4)) or whether they can be replaced by
+mechanical ones.
+Example 1:
+Consider the mechanical system
+˙x1 = y1
+˙x2 = y2
+˙y1 = −x2(y1)2 + x2
+˙y2 = u,
+(5)
+on R4. This system is locally F-linearizable. Indeed, the local diffeomorphism
+˜z = Φ(z), where z = (x1, x2, y1, y2), ˜z = (˜x1, ˜x2, ˜y1, ˜y2), given by
+˜x1 = x1
+˜x2 = x2 − x2(y1)2
+˜y1 = y1
+˜y2 =
+�
+(y1)2 − 1
+� �
+2(x2)2y1 − y2�
+,
+5
+
+together with the feedback u = 2(x2)3 +6(x2 −(x2)2)(y1)2 +
+˜u
+(y1)2−1, render the
+original system linear and controllable
+˙˜x1 = ˜y1
+˙˜x2 = ˜y2
+˙˜y1 = ˜x2
+˙˜y2 = ˜u.
+Therefore, the system is F-linearizable. Note, however, that neither the change
+of coordinates nor the feedback is mechanical (˜x2 depends on velocities, and
+the function β depends on velocities as well) so the mechanical structure is
+not preserved. Our question is whether this system can be linearized by other
+transformations that preserve the mechanical structure, i.e.
+can it be MF-
+linearized?
+The group of mechanical transformations GMF = {(Φ, α, β, γ)} preserves
+trajectories, that is, maps the trajectories of (MS) into those of its MF-equivalent
+system ( �
+MS). Indeed, if z (t, z0, u(t)) is a trajectory of (MS) (passing through
+z0 = (x0, y0) and corresponding to a control u(t)), then ˜z (t, ˜z0, ˜u(t)) = Φ (z (t, z0, u(t)))
+is a trajectory of ( �
+MS) passing through ˜z0 = Φ(z0) = (φ(x0), ∂φ
+∂x(x0)y0) and
+corresponding to ˜u(t), where u(t) = y(t)T γ (x(t)) y(t) + α (x(t)) + β (x(t)) ˜u(t).
+Moreover, via φ : Q → ˜Q, it establishes a correspondence between configuration
+trajectories in Q and ˜Q, i.e. ˜x (t, ˜z0, ˜u(t)) = φ (x(t, z0, u(t))), making the fol-
+lowing diagram commutative (notice, however, that π (z(t, z0, u)) = x(t, z0, u)
+depends on z0 = (x0, y0), i.e. an initial configuration x0 and initial velocity y0):
+z(t, z0, u)
+˜z(t, ˜z0, ˜u)
+x(t, z0, u)
+˜x(t, ˜z0, ˜u)
+(Φ,α,β,γ)
+π
+π
+(φ,α,β,γ)
+where π : TQ → Q, π(z) = π(x, y) = x, is the canonical projection which
+assigns to the pair (x, y) the point x at which the velocity y is attached.
+3
+Mechanical feedback linearization
+Our main result uses two basic ingredients: the covariant derivative of the con-
+nection ∇, see (1), and the involutivity of suitable distributions. We will also
+need the second covariant derivative of a vector field Z in the directions (X, Y ),
+which is a mapping
+∇2 : X(Q) × X(Q) × X(Q) → X(Q)
+∇2
+X,Y Z = ∇X∇Y Z − ∇∇XY Z.
+(6)
+For properties of the second covariant derivative see Lemma 1 in Appendix.
+In order to formulate the result, we associate with (MS) the following se-
+quence of nested distributions E0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Ei ⊂ . . . ⊂ TQ, where
+E0 = span {g} ,
+Ei = span
+�
+adj
+eg, 0 ≤ j ≤ i
+�
+.
+6
+
+Remark 1. To analyze the behavior of the distributions Ei under mechanical
+feedback transformations (α, β, γ) notice, first, that Ei are invariant under γ
+since γ does not act on them. If the distributions Ei are involutive, then they
+are invariant under feedback transformations of the form (α, β, 0), i.e. for γ = 0
+they remain unchanged if we replaced e and g by, respectively, e + gα and βg,
+cf. [6], [7].
+Now, we formulate our main result for MF-linearization.
+First, we state
+a theorem for (MS) with n ≥ 3 degrees of freedom. The remaining case of
+n = 2 degrees of freedom is treated in Theorem 2. For an explanation of that
+distinction, see the comment before Theorem 2 and Remark 3 for a comparison
+of both results.
+By a local MF-linearization around x0 ∈ Q we mean that it holds on
+�
+x∈O TxQ, where O is a neighborhood of x0; recall that all transformations
+are global on tangent spaces TxQ.
+Theorem 1. Assume n ≥ 3. A mechanical control system (MS) is, locally
+around x0, MF-linearizable to a controllable (LMS) if and only if
+(MF1) rank En−1 = n,
+(MF2) Ei is involutive and of constant rank, for 0 ≤ i ≤ n − 2,
+(MF3) ∇adieg g ∈ E0
+for 0 ≤ i ≤ n − 1,
+(MF4) ∇2
+adkeg,adj
+eg e ∈ E1
+for 0 ≤ k, j ≤ n − 1,
+Remark 2. Notice that (MF1)-(MF2) are the classical conditions (see [2,3,6,
+7]) that assure F-linearization of the system ˙x = e(x)+g(x)u on Q via ˜x = φ(x)
+and u = α(x)+β(x)˜u. The remaining two, (MF3)-(MF4), can be interpreted as
+compatibility conditions that guarantee vanishing the Christoffel symbols Γi
+jk in
+the linearizing coordinates ˜x = φ(x), except for those that can be compensated
+by feedback u = γjk(x)yjyk + ˜u.
+Proof. In the proof we will use two Lemmata 1 and 2, given in Appendix, that
+are of independent interest.
+Necessity. For (LMS), we have Γi
+jk = 0, e = Ex and g = b. It follows that
+adi
+eg = (−1)iEib and therefore, using the definitions of ∇, given by (1), and of
+∇2, given by (6), we calculate
+∇adiegadj
+eg = 0,
+∇2
+adkeg,adj
+ege = 0,
+(7)
+which implies that (MF1)-(MF4) hold for (LMS) (in particular, (MF1) holds
+because (LMS) is assumed controllable). To prove necessity of (MF1)-(MF4),
+we will show that they are MF-invariant.
+All conditions (MF1)-(MF4) are
+expressed in a geometrical way, therefore they are invariant under diffeomor-
+phisms. The conditions (MF1) and (MF2) are mechanical feedback invariant,
+see Remark 1. It remains to show that (MF3) and (MF4) are invariant under the
+7
+
+mechanical feedback u = γjk(x)yjyk+α(x)+β(x)˜u. For the closed-loop system,
+denoted by ”∼”, the Christoffel symbols ˜Γi
+jk of ˜∇, ˜e, and ˜g are, respectively,
+given by
+˜Γi
+jk = Γi
+jk − giγjk,
+˜e = e + gα,
+˜g = gβ.
+(8)
+For any X, Y ∈ X(Q), we have ˜∇XY = ∇XY − γ(X, Y )g = ∇XY
+mod E0,
+where γ(X, Y ) = γjkXjY k ∈ C∞(Q), therefore
+˜∇adi
+˜e˜g˜g = ∇adi
+˜e˜g˜g − γ(adi
+˜e˜g, ˜g)g = ∇adi
+˜e˜g˜g
+mod E0.
+By ∇X˜g = ∇X (gβ) = ∇Xg + (LXβ) g, it follows that instead of calculating
+∇adi
+˜e˜g˜g it is enough to calculate ∇adi
+˜e˜gg, since the second term (LXβ) g ∈ E0.
+For i=0, we have ∇˜gg = ∇(gβ)g = β∇gg ∈ E0. It is easy to show that for any
+1 ≤ j ≤ n − 1, we have
+adj
+˜e˜g = βadj
+eg + dj−1,
+(9)
+where dj−1 ∈ Ej−1. Assume ∇adl
+˜e˜gg ∈ E0, for 0 ≤ l ≤ i − 1. Then, by formula
+(9), ∇adi
+˜e˜gg = β∇adiegg + ∇di−1g ∈ E0, because the first term is in E0 by (MF3)
+and the second by the induction assumption. We have thus proved necessity of
+(MF3).
+To show necessity of (MF4), using Lemma 1, calculate
+˜∇2
+X,Y Z = ˜∇X ˜∇Y Z − ˜∇ ˜∇XY Z
+= ˜∇X (∇Y Z − γ(Y, Z)g) − ˜∇(∇XY −γ(X,Y )g)Z
+= ∇2
+X,Y Z − γ(Y, Z)∇Xg + γ(X, Y )∇gZ
+mod E0.
+(10)
+By the above formula, we get
+˜∇2
+adk
+˜e ˜g,adj
+˜e˜g˜e =∇2
+adk
+˜e ˜g,adj
+˜e˜g˜e − γ(adj
+˜e˜g, ˜e)∇adk
+˜e ˜gg
++ γ(adk
+˜e˜g, adj
+˜e˜g)∇g˜e
+mod E0.
+The second term, on the right hand side, is in E0 (by (MF3) and its invariance),
+while the third term is a function multiplying
+∇g˜e = ∇g (e + gα) = ∇ge + α∇gg + Lgα g ∈ E1,
+since for (LMS) we have ∇ge = −adeg = −Eb ∈ E1.
+The first term ∇2
+adk
+˜e ˜g,adj
+˜e˜g˜e is, by (9) and Lemma 1(i), a linear combination
+with smooth coefficients of ∇2
+adieg,adleg˜e, with 0 ≤ i ≤ k and 0 ≤ l ≤ j. Thus
+we calculate ∇2
+adi
+eg,adl
+eg˜e = ∇2
+adieg,adlege+∇2
+adieg,adleg(gα). The first term vanishes
+since (7) holds for (LMS). We calculate the second term using Lemma 1(iii),
+and we have ∇2
+adi
+eg,adl
+eg(gα) = α∇2
+adieg,adlegg + Ladi
+egα∇adlegg + Ladlegα∇adi
+egg +
+(∇2
+adi
+eg,adl
+egα)g ∈ E0 because the first three terms vanish, due to (7), and
+8
+
+the last one is in E0. Summarizing the above calculations, we conclude that
+˜∇2
+adk
+˜e ˜g,adj
+˜e˜g˜e ∈ E1 = ˜E1, which proves necessity of (MF4).
+Sufficiency. We will transform the system (MS), satisfying (MF1)-(MF4),
+into (LMS) in two steps. In the first step, we will normalize the vector fields
+e and g and show that condition (MF4) implies zeroing some of the Christoffel
+symbols Γi
+jk, which exhibit a triangular form in the normalizing coordinates. In
+the second step, we compensate the remaining Christoffel symbols.
+By conditions (MF1)-(MF2), there exists a function h satisfying Ladj
+egh = 0,
+for 0 ≤ j ≤ n − 2, and Ladn−1
+e
+gh ̸= 0, and thus (˜x, ˜y) = (φ(x), ∂φ
+∂x(x)y) is a
+local mechanical diffeomorphism, where φ(x) = (Ln−1
+e
+h, . . . , Leh, h)T that can
+be completed by a feedback transformation (α, β, 0) that map, respectively, βg
+into ˜g = (1, 0, . . . , 0)T , e + gα into ˜e = (0, ˜x1, . . . , ˜xn−1)T , and Γi
+jk into ˜Γi
+jk, see
+the classical results of feedback linearization [2], [6], [7]. Then, (Φ, α, β, γ) ∈
+GMF , where (˜x, ˜y) = Φ(x, y) =
+�
+φ(x), ∂φ
+∂x(x)y
+�
+with φ, α, β just defined and
+γjk = ˜Γ1
+jk(˜x), brings (MS) into (we drop ”tildas” for readability)
+˙x1 = y1
+˙xi = yi
+˙y1 = u
+˙yi = −Γi
+jkyjyk + xi−1,
+2 ≤ i ≤ n,
+(11)
+to which Lemma 2 applies.
+We will show that the Christoffel symbols Γi
+jk of (11) satisfy
+Γi
+kj = 0
+for 1 ≤ k ≤ n − 1, 1 ≤ j ≤ i ≤ n,
+Γi
+nj =
+�
+0
+for 1 ≤ j < i ≤ n
+λ(xn)
+for 2 ≤ j = i ≤ n.
+(12)
+For system (11), we have adk−1
+e
+g = (−1)k−1
+∂
+∂xk and, in particular, g =
+∂
+∂x1 .
+Calculate ∇adk−1
+e
+gg = (−1)k−1∇
+∂
+∂xk gi ∂
+∂xi = (−1)k−1∇
+∂
+∂xk
+∂
+∂x1 = (−1)k−1Γi
+k1
+∂
+∂xi .
+It follows that Γi
+k1 = Γi
+1k = 0, for 2 ≤ i ≤ n by (MF3), and for i = 1 by the
+above form.
+Rewrite (MF4) as ∇2
+adk−1
+e
+g,adj−1
+e
+ge = 0 mod E1, for 1 ≤ j, k ≤ n, and apply
+it successively for j = 1, . . . , n and for all 1 ≤ k ≤ n. For j = 1, first calculate
+∇ge = ∇
+∂
+∂x1 e =
+∂
+∂x2 + Γi
+1ses ∂
+∂xi =
+∂
+∂x2
+and then
+∇adk−1
+e
+g (∇ge) = (−1)k−1∇
+∂
+∂xk
+∂
+∂x2 = (−1)k−1Γi
+k2
+∂
+∂xi .
+On the other hand, ∇adk−1
+e
+gg = (−1)k−1∇
+∂
+∂xk
+∂
+∂x1 = (−1)k−1Γ1
+k1
+∂
+∂x1 = 0 and
+hence ∇∇adk−1
+e
+gge = 0. Thus, by (6),
+∇2
+adk−1
+e
+g,ge = ∇adk−1
+e
+g (∇ge) − ∇∇adk−1
+e
+gge =
+= (−1)k−1Γi
+k2
+∂
+∂xi = 0
+mod E1,
+9
+
+implying that Γi
+k2 = Γi
+2k = 0 for any 3 ≤ i ≤ n.
+For j = 2, calculate
+∇adege = −∇
+∂
+∂x2 e = − ∂
+∂x3 + Γi
+2ses ∂
+∂xi = − ∂
+∂x3 − d
+where d = d1(x) ∂
+∂x1 + d2(x) ∂
+∂x2 ∈ E1, and then
+∇adk−1
+e
+g (∇adege) = (−1)k∇
+∂
+∂xk
+� ∂
+∂x3 + d
+�
+=
+= (−1)k �
+Γi
+k3 + Γi
+k1d1 + Γi
+k2d2� ∂
+∂xi =
+= (−1)kΓi
+k3
+∂
+∂xi
+mod E1.
+On the other hand,
+∇adk−1
+e
+gadeg = (−1)k∇
+∂
+∂xk
+∂
+∂x2 = (−1)kΓi
+k2
+∂
+∂xi =
+= (−1)k
+�
+Γ1
+k2
+∂
+∂x1 + Γ2
+k2
+∂
+∂x2
+�
+and ∇∇adk−1
+e
+gadege = (−1)kΓ2
+k2
+∂
+∂x3
+mod E1. It follows that, modulo E1,
+∇2
+adk−1
+e
+g,adege = (−1)k
+� n
+�
+i=4
+Γi
+k3
+∂
+∂xi + (Γ3
+k3 − Γ2
+k2) ∂
+∂x3
+�
+,
+and, using (MF4), we conclude Γi
+k3 = Γi
+3k = 0 for any 4 ≤ i ≤ n and Γ3
+k3 = Γ2
+k2.
+Following the same line (with a more tedious calculation), one can prove the
+general induction step. Namely, assuming, for a fixed j,
+Γj
+kj = Γj−1
+kj−1
+Γi
+ks = Γi
+sk = 0
+s + 1 ≤ i ≤ n,
+1 ≤ s ≤ j,
+(13)
+one shows by calculating ∇2
+adk−1
+e
+g,adj−1
+e
+ge, with the help of (24) of Lemma 2,
+that
+Γj+1
+kj+1 = Γj
+kj
+Γi
+kj+1 = 0
+for j + 2 ≤ i ≤ n
+and thus, by the induction assumption and symmetry of the Christoffel symbols,
+Γi
+ks = Γi
+sk = 0
+s + 1 ≤ i ≤ n,
+1 ≤ s ≤ j + 1.
+(14)
+It follows that for each 1 ≤ k ≤ n the matrices consisting of Christoffel symbols
+(Γi
+kj), for 2 ≤ i, j ≤ n are upper triangular. By the induction argument, (13)
+holds for all 2 ≤ j ≤ n and implies, for any 1 ≤ k ≤ n − 1,
+Γ2
+k2 = . . . = Γn−1
+kn−1 = Γn
+kn = 0.
+10
+
+since Γn
+kn = Γn
+nk = 0 (as n > k). On the other hand, for k = n, (13) implies
+Γ2
+n2 = . . . = Γn−1
+nn−1 = Γn
+nn = λ(x)
+for a function λ(x).
+Therefore for each 1 ≤ k ≤ n the matrices (Γi
+kj), for
+2 ≤ i, j ≤ n, are strictly upper triangular, and the last one, for k = n, is upper
+triangular with all diagonal elements equal to each other, which we denote by
+λ(x). The matrices read
+�
+Γi
+kj
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+Γ2
+k3
+Γ2
+k4
+. . .
+Γ2
+kn−2
+Γ2
+kn−1
+Γ2
+kn
+0
+0
+Γ3
+k4
+. . .
+Γ3
+kn−2
+Γ3
+kn−1
+Γ3
+kn
+...
+0
+0
+0
+. . .
+0
+Γn−2
+kn−1
+Γn−2
+kn
+0
+0
+0
+. . .
+0
+0
+Γn−1
+kn
+0
+0
+0
+. . .
+0
+0
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+for 1 ≤ k ≤ n − 1, and
+�
+Γi
+nj
+�
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+λ
+Γ2
+n3
+Γ2
+n4
+. . .
+Γ2
+nn−2
+Γ2
+nn−1
+Γ2
+nn
+0
+λ
+Γ3
+n4
+. . .
+Γ3
+nn−2
+Γ3
+nn−1
+Γ3
+nn
+...
+0
+0
+0
+. . .
+λ
+Γn−2
+nn−1
+Γn−2
+nn
+0
+0
+0
+. . .
+0
+λ
+Γn−1
+nn
+0
+0
+0
+. . .
+0
+0
+λ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+and are thus of the desired triangular structure (12) and it remains to prove
+that λ = λ(xn). Note that in the above matrices we skip the first row Γ1
+kj and
+the first column Γi
+k1. This is due to the fact that Γ1
+kj = 0 (which can always be
+achieved by a suitable feedback transformation) and Γi
+k1 = 0 by (MF3).
+Notice
+that we have En−2 = span
+� ∂
+∂x1 , . . . ,
+∂
+∂xn−1
+�
+and thus applying (24) of Lemma 2,
+for j = n and any 1 ≤ k ≤ n, we conclude (set Γn
+kn+1 = 0)
+(−1)n+k−2∇2
+adk−1
+e
+g,adn−1
+e
+ge = ∇2
+∂
+∂xk ,
+∂
+∂xn e
+=
+�∂Γn
+ns
+∂xk es + Γn
+nk+1 + Γn
+kn+1 − Γn−1
+kn
++ (Γd
+nsΓn
+kd − Γd
+knΓn
+ds)es
+� ∂
+∂xn
+mod En−2
+=
+� ∂λ
+∂xk en + Γn
+nk+1 − Γn−1
+kn
+� ∂
+∂xn
+mod En−2,
+(15)
+since, due to the triangular structure (14), Γn
+ns = 0 except for s = n giving
+Γn
+nn = λ and, moreover, the equality Γd
+nsΓn
+kd−Γd
+knΓn
+ds = 0 holds. Indeed, in the
+latter, Γn
+kd = 0 except d = k = n giving Γn
+nsΓn
+nn − Γn
+nnΓn
+ns = 0 and Γn
+ds = 0
+except for d = s = n giving Γn
+nnΓn
+kn − Γn
+knΓn
+nn = 0.
+11
+
+For (15) we will apply (MF4) in three cases. First, if 1 ≤ k ≤ n − 2, then,
+modulo En−2, we have
+� ∂λ
+∂xk en + Γn
+nk+1 − Γn−1
+kn
+�
+∂
+∂xn =
+� ∂λ
+∂xk xn−1
+�
+∂
+∂xn = 0,
+since all Γn
+nk+1 = 0 and all Γn−1
+kn
+= 0 by (14) and k ≤ n − 2.
+Second, for
+k = n − 1, we have modulo En−2,
+�
+∂λ
+∂xn−1 en + Γn
+nn − Γn−1
+n−1n
+� ∂
+∂xn =
+�
+∂λ
+∂xn−1 en + λ − λ
+�
+∂
+∂xn
+=
+�
+∂λ
+∂xn−1 xn−1
+�
+∂
+∂xn = 0.
+Therefore
+∂λ
+∂xk = 0, for 1 ≤ k ≤ n − 1, implying that λ is a function of the last
+variable xn only, i.e. λ = λ(xn), which gives the system in the desired form (12)
+Third, for k = n, we have modulo En−2,
+� ∂λ
+∂xn en+Γn
+nn+1−Γn−1
+nn
+� ∂
+∂xn =
+� ∂λ
+∂xn xn−1− Γn−1
+nn
+� ∂
+∂xn = 0,
+implying that Γn−1
+nn
+= Leλ, since ∂λ(xn)
+∂xn xn−1 = Leλ.
+Now, transform system (11), satisfying (12), via the local mechanical diffeo-
+morphism Φ : TQ → T ¯Q
+¯x = φ(x)
+¯y = Dφ(x)y,
+where
+φ(x) =
+�
+Ln−1
+e
+h, . . . , Leh, h
+�T ,
+(16)
+with h(xn) =
+� xn
+0
+Λ(s2)ds2, where Λ(s2) = exp
+�� s2
+0 λ(s1)ds1
+�
+.
+Denote by ¯Γi
+jk, ¯e, ¯g the objects of the system expressed in coordinates ¯x =
+φ(x). Applying feedback ¯u = −¯Γ1
+jk¯yj ¯yk + Ln
+e h + uLgLn−1
+e
+h, the transformed
+system becomes
+˙¯x1 = ¯y1
+˙¯xi = ¯yi
+˙¯y1 = ¯u
+˙¯yi = −¯Γi
+jk¯yj ¯yk + ¯xi−1,
+2 ≤ i ≤ n,
+(17)
+whose vector fields are ¯e = ¯xi−1 ∂
+∂¯xi , where x0 = 0, and ¯g =
+∂
+∂¯x1 . Transformed
+system (17) is still of the form (11) and at the moment we ignore how Γi
+jk have
+been changed into ¯Γi
+jk. Below we will prove that all ¯Γi
+jk vanish. To this end,
+we first calculate explicitly the time-evolution of the pair (¯xn, ¯yn)
+˙¯xn = d
+dth(xn) = Λ(xn) ˙xn = Λ(xn)yn = ¯yn
+˙¯yn = d
+dt (Λ(xn)yn) = Λ(xn)λ(xn) ˙xnyn + Λ(xn) ˙yn
+= Λ(xn)λ(xn)ynyn + Λ(xn) ˙yn
+= Λ(xn)λ(xn)ynyn + Λ(xn)
+�
+−Γn
+nn(xn)ynyn + xn−1�
+= Λ(xn)xn−1 = ¯xn−1,
+12
+
+since ¯xn−1 = Leh = Λ(xn)xn−1. It follows that ¯Γn
+jk = 0, for all 1 ≤ k, j ≤ n.
+For transformed system (17), we rewrite (24) by adding ”bars” as
+∇2
+adk−1
+¯e
+¯g,adj−1
+¯e
+¯g¯e = (−1)j+k
+�∂¯Γi
+js
+∂¯xk ¯es + ¯Γi
+jk+1
++ ¯Γi
+kj+1 + (¯Γd
+js¯Γi
+kd − ¯Γd
+kj ¯Γi
+ds)¯es − ¯Γi−1
+kj
+� ∂
+∂¯xi
+(18)
+and by (MF4), we have
+∇2
+adk−1
+¯e
+¯g,adj−1
+¯e
+¯g¯e = (−1)j+k¯an
+kj(¯x) ∂
+∂¯xn = 0
+mod En−2,
+where ¯an
+kj(¯x) =
+∂¯Γn
+js
+∂¯xk ¯es + ¯Γn
+jk+1 + ¯Γn
+kj+1 + (¯Γd
+js¯Γn
+kd − ¯Γd
+kj ¯Γn
+ds)¯es − ¯Γn−1
+kj , which
+implies (since ¯Γn
+kj = 0, for 1 ≤ j, k ≤ n) that ¯an
+kj(¯x) = ¯Γn−1
+kj
+= 0. Now assume
+¯Γi
+kj = 0 for a certain 1 ≤ i ≤ n − 1 and any 1 ≤ j, k ≤ n. Then (18) and (MF4)
+imply ¯Γi−1
+kj
+= 0. Therefore we have proved that all Christoffel symbols of (17)
+vanish and thus the system is a linear controllable (LMS), since the vector field
+¯e = ¯xi−1 ∂
+∂¯xi is linear and ¯g =
+∂
+∂¯x1 is constant.
+The above theorem does not work for systems with 2 degrees of freedom,
+i.e. for n=2, as that case is too restrictive for involutivity, see Remark 3 below.
+Therefore we state the following theorem for MF-linearization of (MS) with 2
+degrees of freedom.
+Theorem 2. A mechanical system (MS) with 2 degrees of freedom is, locally
+around x0, MF-linearizable to a controllable linear (LMS) if and only if it
+satisfies in a neighborhood of x0
+(MF1)’ g and adeg are independent at x0,
+(MF3)’ ∇g g ∈ E0 and ∇adeg g ∈ E0,
+(MF5)’ ∇2
+g,adeg adeg − ∇2
+adeg,g adeg ∈ E0.
+Remark 3. If n = 2, then E0 is of rank 1, thus involutive and (MF2) is trivially
+satisfied, and so is (MF4) because E1 = TQ (cf. Theorem 1). Therefore (MF2)’
+and (MF4)’ are absent and replaced by (MF5)’ that guarantees that we can
+compensate the Christoffel symbols (as do (MF3)-(MF4) for n ≥ 3).
+Proof. Necessity. Note that (MF1)’ is equivalent to (MF1) and (MF3)’ is (MF3)
+of Theorem 1. Although Theorem 1 applies to n ≥ 3, the necessity part of its
+proof remains valid for any n ≥ 2 so it shows necessity of (MF1)’-(MF3)’.
+Therefore we need to show necessity of (MF5)’. For a controllable (LMS) we
+have Γi
+jk = 0, g = b and adeg = −Eb are independent, and
+∇adiegadj
+eg = 0, ∇2
+adj
+eg,adkegadi
+eg = 0,
+�
+adj
+eg, adk
+eg
+�
+= 0,
+(19)
+13
+
+for 0 ≤ i, j, k ≤ 1. We will use formula (10) to show that (MF5)’ is invariant
+under mechanical feedback. Denote ˜∇, ˜e, ˜g, γ as in (8). Then we calculate
+˜∇2
+˜g,ad˜e˜gad˜e˜g =∇2
+˜g,ad˜e˜gad˜e˜g − γ(ad˜e˜g, ad˜e˜g)∇˜g˜g
++ γ(˜g, ad˜e˜g)∇˜gad˜e˜g
+mod E0,
+˜∇2
+ad˜e˜g,˜gad˜e˜g =∇2
+ad˜e˜g,˜gad˜e˜g − γ(g, ad˜e˜g)∇ad˜e˜g˜g
++ γ(ad˜e˜g, ˜g)∇˜gad˜e˜g
+mod E0.
+The second terms of the right hand side of both equations are in E0 due to the
+feedback invariance of (MF3)’, while the third terms are equal since γ(X, Y ) =
+γ(Y, X) is symmetric. Therefore we conclude
+˜∇2
+˜g,ad˜e˜gad˜e˜g − ˜∇2
+ad˜e˜g,˜gad˜e˜g
+= ∇2
+˜g,ad˜e˜gad˜e˜g − ∇2
+ad˜e˜g,˜gad˜e˜g
+mod E0.
+Denoting ad˜e˜g = βadeg + d0g (see (9)) and by Lemma 1 (i), we have
+∇2
+˜g,ad˜e˜gad˜e˜g = ∇2
+βg,βadeg+d0gad˜e˜g
+= β2∇2
+g,adegad˜e˜g + βd0∇2
+g,gad˜e˜g
+∇2
+ad˜e˜g,˜gad˜e˜g = ∇2
+βadeg+d0g,βgad˜e˜g
+= β2∇2
+adeg,gad˜e˜g + βd0∇2
+g,gad˜e˜g,
+where the last terms on the right are equal, implying
+∇2
+˜g,ad˜e˜gad˜e˜g − ∇2
+ad˜e˜g,˜gad˜e˜g
+= β2 �
+∇2
+g,adegad˜e˜g − β2∇2
+adeg,gad˜e˜g
+�
+and it remains to prove that ∇2
+g,adegad˜e˜g − ∇2
+adeg,gad˜e˜g ∈ E0, which we show
+using Lemma 1(iii), where X, Y stand for either g or adeg. Denote ∇Xβ = LXβ
+and ∇2
+X,Y β = LXLY β − L∇XY β (see Lemma 1) and calculate
+∇2
+X,Y ad˜e˜g = ∇2
+X,Y
+�
+βadeg + d0g
+�
+= β∇2
+X,Y adeg
++ LXβ∇Y adeg + LY β∇Xadeg +
+�
+∇2
+X,Y β
+�
+adeg
++ d0∇2
+X,Y g + LXd0∇Y g + LY d0∇Xg +
+�
+∇2
+X,Y d0�
+g
+=
+�
+∇2
+X,Y β
+�
+adeg
+mod E0,
+since all ∇2
+X,Y X = 0 and ∇XY = 0 , see (19). Therefore we have
+∇2
+g,adegad˜e˜g − ∇2
+adeg,gad˜e˜g
+=
+�
+∇2
+g,adegβ − ∇2
+adeg,gβ
+�
+adeg
+mod E0.
+Finally, we calculate
+∇2
+g,adegβ − ∇2
+adeg,gβ = LgLadegβ − L∇gadegβ
+−
+�
+LadegLgβ − L∇adeggβ
+�
+= L[g,adeg]β = 0,
+14
+
+which shows necessity of (MF5)’.
+Sufficiency.
+By (MF1)’, rank E1 = 2, and E0 = span {g} is of constant
+rank 1 and thus always involutive, hence the system is, locally around x0 (since
+g(x0) ̸= 0), MF-equivalent to (cf. (11))
+˙x1 = y1
+˙x2 = y2
+˙y1 = u
+˙y2 = −Γ2
+jkyjyk + x2.
+We have g =
+∂
+∂x1 , adeg = − ∂
+∂x2 and now we calculate
+∇gg = Γ2
+11
+∂
+∂x2
+∇adegg = −Γ2
+12
+∂
+∂x2 ,
+which by (MF3)’ are in E0 = span
+� ∂
+∂x1
+�
+, implying Γ2
+11 = Γ2
+12 = Γ2
+21 = 0. It
+follows ∇gg = ∇adegg = ∇gadeg = 0, and ∇adegadeg = Γ2
+22
+∂
+∂x2 and thus
+∇2
+g,adeg adeg − ∇2
+adeg,g adeg = ∇g∇adegadeg
+− ∇∇gadegadeg − ∇adeg∇gadeg − ∇∇adeggadeg
+= ∇g∇adegadeg = ∇
+∂
+∂x1 Γ2
+22
+∂
+∂x2 = ∂Γ2
+22
+∂x1
+∂
+∂x2
+implying, by (MF5)’, ∂Γ2
+22
+∂x1 = 0, i.e. Γ2
+22(x2) = λ(x2).
+Now, we transform the system via the local mechanical diffeomorphism Φ :
+TQ → T ¯Q (compare to (16))
+¯x = φ(x)
+¯y = Dφ(x)y,
+where
+φ(x) = (Leh, h)T ,
+with h(x2) =
+� x2
+0
+Λ(s2)ds2 and Λ(s2) = exp
+�� s2
+0 λ(s1)ds1
+�
+.
+We calculate the evolution of the pair (¯x(t), ¯y(t)) of transformed coordinates,
+using
+d
+dth
+�
+x2(t)
+�
+= Λ
+�
+x2(t)
+�
+˙x2(t) and
+d
+dtΛ
+�
+x2(t)
+�
+= λ
+�
+x2(t)
+�
+Λ
+�
+x2(t)
+�
+˙x2(t);
+first we get
+˙¯x2 = d
+dth(x2) = Λ(x2)y2 = ¯y2
+˙¯y2 = Λ(x2)λ(x2)y2y2 + Λ(x2) ˙y2 = Λ(x2)λ(x2)y2y2
++ Λ(x2)
+�
+−λ(x2)y2y2 + x2�
+= Λ(x2)x1 = ¯x1
+and then
+˙¯x1 = Λ(x2)y1 + d
+dtΛ
+�
+x2(t)
+�
+x1y2 = ¯y1
+˙¯y1 = −¯Γ1
+jk¯yj ¯yk + L2
+eh + uLgLeh,
+where we denote by ¯Γ1
+jk the Christoffel symbols in the ˙¯y1-equation of the trans-
+formed system. Applying the feedback ¯u = −¯Γ1
+jk¯yj ¯yk + L2
+eh + uLgLeh, we get
+a controllable linear mechanical system in the canonical form ˙¯x1 = ¯y1, ˙¯y1 =
+¯u, ˙¯x2 = ¯y2, ˙¯y2 = ¯x1.
+15
+
+4
+Examples
+Example 1 (cont.): For system (5), we have g =
+∂
+∂x2 and adeg = − ∂
+∂x1 are in-
+dependent. We check MF-linearizability using Theorem 2. A simple calculation
+shows that ∇gg = ∇adegg = 0 ∈ E0, but ∇2
+g,adeg adeg−∇2
+adeg,g adeg =
+∂
+∂x1 /∈ E0,
+therefore the system is not MF-linearizable.
+Thus (5) is an example of a system that is F-linearizable but not MF-
+linearizable. For such systems the choice is: either to F-linearize for the price
+of loosing the mechanical structure or to keep the mechanical structure but to
+get rid of the linearization.
+Example 2: Consider the equation of dynamics of the Inertia Wheel Pen-
+dulum [18] with constant parameters m0, md, J2:
+˙x1 = y1,
+˙x2 = y2,
+˙y1 = e1 + g1u,
+˙y2 = e2 + g2u,
+e1 = m0
+md sin x1, e2 = − m0
+md sin x1, g1 = − 1
+md , g2 = md+J2
+J2md .
+We will verify whether the conditions of Theorem 2 are satisfied. First, we
+calculate adeg = ( m0
+m2
+d cos x1) ∂
+∂x1 − ( m0
+m2
+d cos x1) ∂
+∂x2 . It can be checked that g and
+adeg are independent for x1 ̸= ± π
+2 , which corresponds to the horizontal position
+of the pendulum, therefore (MF1)’ is satisfied everywhere except for x1 = ± π
+2 .
+Next, we verify condition (MF2)’ by calculating ∇gg = ∇adegg = 0 ∈ E0.
+Finally, a direct calculation shows
+∇2
+g,adeg adeg = ∇2
+adeg,g adeg =
+= (m2
+0
+m5
+d
+cos2 x1) ∂
+∂x1 − (m2
+0
+m5
+d
+cos2 x1) ∂
+∂x2 ,
+thus ∇2
+g,adeg adeg − ∇2
+adeg,g adeg = 0 ∈ E0 satisfies (MF5)’. The system is MF-
+linearizable. A linearizing function is h(x) = md+J2
+J2
+x1 + x2 (all others giving
+MF-linearization are of the form σ h(x), where σ ∈ R\ {0}). Due to the proof of
+Theorem 2, the linearizing diffeomorphism is (˜x, ˜y) = Φ(x, y) = (φ(x), Dφ(x)y)
+with φ(x) = (h, Leh)T . The system in new coordinates reads
+˙˜x1 = md + J2
+J2
+y1 + y2 = ˜y1
+˙˜y1 = md + J2
+J2
+�m0
+md
+sin x1 − 1
+md
+u
+�
+− m0
+md
+sin x1 + md + J2
+m2J2
+u
+= m0
+J2
+sin x1 = Leh = ˜x2
+(20)
+˙˜x1 = m0
+J2
+cos x1y1 = ˜y2
+˙˜y2 = −m0
+J2
+sin x1y1y1 +
+m2
+0
+2mdJ2
+sin(2x1) −
+m0
+mdJ2
+cos x1u = ˜u.
+Example 3: We will study MF-linearizability of the TORA3 system (see
+Figure 1), which is based on the TORA system (Translational Oscillator with
+16
+
+Figure 1: The TORA3 system
+Rotational Actuator) studied in the literature, e.g. [19] (however we add gravita-
+tional effects). It consists of a two dimensional spring-mass system, with masses
+m1, m2 and spring constants k1, k2, respectively. A pendulum of length l3, mass
+m3, and moment of inertia J3 is added to the second body. The displacements
+of the bodies are denoted by x1 and x2, respectively, and the angle of the pen-
+dulum by x3. The gravitational constant is a and u is a torque applied to the
+pendulum. The kinetic energy is
+T =1
+2m1( ˙x1)2 + 1
+2(m2 + m3)( ˙x2)2
++ 1
+2(J3 + m3l2
+3)( ˙x3)2 + m3l3 cos x3 ˙x2 ˙x3,
+and the mass matrix depends on the configurations. The potential energy is
+V = 1
+2k1(x1)2 + 1
+2k2(x2 − x1)2 − m3l3a cos x3. The equations of the dynamics
+read
+m1¨x1 + k1x1 − k2
+�
+x2 − x1�
+= 0
+(m2 + m3)¨x2 + m3l3 cos x3¨x3 − m3l3 sin x3( ˙x3)2
++k2
+�
+x2 − x1�
+= 0
+m3l3 cos x3¨x2 + (m3l2
+3 + J3)¨x3 + m3l3a sin x3 = u,
+which can be rewritten on TQ as
+˙x1 = y1
+˙y1 = η1
+˙x2 = y2
+˙y2 = −¯Γ2
+33y3y3 + η2 + τ 2u
+˙x3 = y3
+˙y3 = −¯Γ3
+33y3y3 + η3 + τ 3u
+(21)
+where ¯Γ2
+33 =
+−ν0 sin x3
+ν1+ν2 sin2 x3 , ¯Γ3
+33 = ν2 sin x3 cos x3
+ν1+ν2 sin2 x3 , η1 = − k1
+m1 x1 + k2
+m3
+�
+x2 − x1�
+, η2 =
+1
+2 ν2a sin 2x3−ν3(x2−x1)
+ν1+ν2 sin2 x3
+,
+η3 =
+ν4(x2−x1) cos x3−ν5 sin x3
+ν1+ν2 sin2 x3
+, τ 2 = −m3l3 cos x3
+ν1+ν2 sin2 x3 , τ 3 =
+m2+m3
+ν1+ν2 sin2 x3 , with constant
+parameters:
+ν0 = m3l3(m3l2
+3 + J3), ν1 = m2m3l2
+3 + J3(m2 + m3), ν2 = m2
+3l2
+3, ν3 =
+k2
+�
+m3l2
+3 + J3
+�
+, ν4 = m3l3k2 ν5 = m3l3a(m2 + m3).
+17
+
+--
+m1
+m2
+ki
+k2
+W
+--
+W
+aTo simplify calculations we apply to the system a preliminary mechanical
+feedback1 u =
+1
+τ 3
+�¯Γ3
+33y3y3 − η3 + ¯u
+�
+which yields
+˙x1 = y1
+˙x2 = y2
+˙x3 = y3
+˙y1 = −µ1x1 + µ2x2
+˙y2 = µ3 sin x3y3y3 + µ4(x1 − x2) − µ3 cos x3u
+˙y3 = ¯u,
+(22)
+with µ1 = k1+k2
+m1 , µ2 = k2
+m1 , µ3 =
+m3l3
+m2+m3 , µ4 =
+k2
+m2+m3 .
+Since conditions (MF1)-(MF4) of Theorem 1 are MF-invariant, we will check
+them for system (22). To summarize:
+Γ2
+33 = −µ3 sin x3,
+and
+Γi
+jk = 0
+otherwise,
+e =
+�
+−µ1x1 + µ2x2� ∂
+∂x1 + µ4
+�
+x1 − x2� ∂
+∂x2
+g = −µ3 cos x3 ∂
+∂x2 +
+∂
+∂x3 = g2 ∂
+∂x2 +
+∂
+∂x3 .
+We have (notice that calculations are performed on Q only)
+adeg =
+�
+µ2µ3 cos x3� ∂
+∂x1 −
+�
+µ3µ4 cos x3� ∂
+∂x2 ,
+ad2
+eg = µ3 cos x3
+�
+(µ1µ2 + µ2µ4) ∂
+∂x1 −
+�
+µ2µ4 + µ2
+4
+� ∂
+∂x2
+�
+,
+therefore rank E2 = 3 for x3 ̸= ± π
+2 , and (MF1) is satisfied. Now
+[g, adeg] = −
+�
+µ2µ3 sin x3� ∂
+∂x1 +
+�
+µ3µ4 sin x3� ∂
+∂x2 ∈ E1
+and (MF2) is satisfied. Then, for any vector field v = vi(x) ∂
+∂xi ,
+∇vg =
+�∂g2
+∂x3 + Γ2
+33
+�
+v3 ∂
+∂x2 = 0,
+thus (MC3) is satisfied if we replace v by, in particular, g, adeg, ad2
+eg. Finally,
+for (MF4), we calculate
+∇2
+g,ge =
+�
+µ2µ3 sin x3� ∂
+∂x1 −
+�
+µ3µ4 sin x3� ∂
+∂x2 ∈ E1,
+∇2
+adkeg,adj
+ege = 0
+otherwise,
+thus, the system is MF-linearizable. Now, choose h =
+µ4
+µ2 x1 + x2 + µ3 sin x3
+(whose differential dh annihilates g and adeg), thus we take a linearizing diffeo-
+morphism (˜x, ˜y) =
+�
+φ(x), ∂φ
+∂x(x)y
+�
+, with φ(x) =
+�
+h, Leh, L2
+eh
+�T . The linearized
+1This preliminary feedback is not necessary and it is possible to check the conditions and
+to linearize the system without it, since our method and conditions are feedback invariant.
+18
+
+system is in the form of (LMS) and reads
+˙˜x1 = µ4
+µ2
+y1 + y2 + µ3 cos x3y3 = ˜y1
+˙˜y1 = µ4
+µ2
+˙y1+ ˙y2+µ3(cos x3 ˙y3−sin x3y3y3)= µ4(µ2 − µ1)
+µ2
+x1 = ˜x2
+˙˜x2 = µ4(µ2 − µ1)
+µ2
+y1 = ˜y2
+˙˜y2 = µ4(µ2 − µ1)
+µ2
+˙y1 = µ4(µ2 − µ1)
+µ2
+�
+µ2x2 − µ1x1�
+= ˜x3
+˙˜x3 = µ1µ4(µ1 − µ2)
+µ2
+y1 + µ4(µ2 − µ1)y2 = ˜y3
+˙˜y3 = (µ2 − µ1)µ3µ4 sin x3y3y3 − (µ1 − µ2)(µ2
+1 + µ2µ4)µ4
+µ2
+x1
++ (µ1 − µ2)(µ1 + µ4)µ4x2 + (µ1 − µ2)µ3µ4 cos x3u = ˜u.
+5
+Conclusions
+In this paper, we consider MF-linearization of mechanical control systems (MS)
+with scalar control. We formulate the problem as a particular case of feedback
+linearization preserving the mechanical structure of (MS) so that the trans-
+formed system is both linear and mechanical. As we showed in [4] and confirmed
+in this paper, even in the simplest case, the class of MF-linearizable systems is
+substantially smaller than that of general F-linearizable ones. Therefore, a nat-
+ural question arises, namely to compare the conditions presented in this paper
+with those for F-linearization.
+The answer lies in the interplay between the
+distributions Ei = span
+�
+adj
+eg, 0 ≤ j ≤ i
+�
+and the ”usual” for F-linearization
+Di = span
+�
+adj
+F G, 0 ≤ j ≤ i
+�
+. We will address this problem in the future.
+6
+Appendix
+The following lemma can be proved by a direct calculation.
+Lemma 1. The second covariant derivative ∇2
+X,Y Z satisfies the following prop-
+erties:
+(i) linearity over C∞(Q) in X and Y :
+∇2
+(α1X1+α2X2),Y Z = α1∇2
+X1,Y Z + α2∇2
+X2,Y Z
+∇2
+X,(α1Y1+α2Y2)Z = α1∇2
+X,Y1Z + α2∇2
+X,Y1Z
+(ii) linearity over R in Z:
+∇2
+X,Y (a1Z1 + a2Z2) = a1∇2
+X,Y Z1 + a2∇2
+X,Y Z2
+19
+
+(iii) the product rule:
+∇2
+X,Y (βZ) =β∇2
+X,Y Z + LXβ∇Y Z
++ LY β∇XZ +
+�
+∇2
+X,Y β
+�
+Z,
+where ∇2
+X,Y β = LXLY β−L∇XY β ∈ C∞(Q), Xi, Yi, Zi ∈ X(Q), αi, β ∈ C∞(Q),
+and ai ∈ R.
+The following lemma is crucial for the proof of Theorem 1.
+Lemma 2. For the system
+˙x1 = y1
+˙xi = yi
+˙y1 = u
+˙yi = −Γi
+jkyjyk + xi−1,
+2 ≤ i ≤ n,
+(23)
+we have for any 1 ≤ k, j ≤ n,
+∇2
+adk−1
+e
+g,adj−1
+e
+ge = (−1)j+k
+�∂Γi
+js
+∂xk es + Γi
+jk+1 + Γi
+kj+1 − Γi−1
+kj
++ (Γd
+jsΓi
+kd − Γd
+kjΓi
+ds)es
+� ∂
+∂xi .
+(24)
+Proof. For system (23) we calculate ∇2
+adk−1
+e
+g,adj−1
+e
+ge = (−1)j+k∇2
+∂
+∂xk ,
+∂
+∂xj e =
+∇
+∂
+∂xk ∇
+∂
+∂xj e − ∇∇
+∂
+∂xk
+∂
+∂xj e,
+where ∇
+∂
+∂xj e =
+�
+∂ed
+∂xj + Γd
+jses�
+∂
+∂xd , and
+∇
+∂
+∂xk
+�
+∇
+∂
+∂xj e
+�
+= ∇
+∂
+∂xk
+� ∂ed
+∂xj
+�
+∂
+∂xd + ∇
+∂
+∂xk
+�
+Γd
+jses�
+∂
+∂xd
+= ∂ed
+∂xj ∇
+∂
+∂xk
+∂
+∂xd + L
+∂
+∂xk
+� ∂ed
+∂xj
+�
+∂
+∂xd +
+�
+Γd
+jses�
+∇
+∂
+∂xk
+∂
+∂xd
++
+�
+L
+∂
+∂xk
+�
+Γd
+js
+�
+es + L
+∂
+∂xk (es) Γd
+js
+�
+∂
+∂xd
+= ∂ed
+∂xj Γi
+kd
+∂
+∂xi + Γd
+jsesΓi
+kd
+∂
+∂xi +
+�
+∂Γi
+js
+∂xk es + ∂es
+∂xk Γi
+js
+�
+∂
+∂xi
+=
+�
+∂Γi
+js
+∂xk es + Γi
+jk+1 + Γi
+kj+1 + Γd
+jsΓi
+kdes
+�
+∂
+∂xi ,
+since
+∂ed
+∂xj = 1, if d = j + 1, and zero otherwise, and thus
+∂ed
+∂xj Γi
+kd = Γi
+kj+1
+(analogously for the other derivatives).
+Now, using ∇
+∂
+∂xk
+∂
+∂xj = Γd
+kj
+∂
+∂xd , we
+calculate
+∇∇
+∂
+∂xk
+∂
+∂xj e = ∇Γd
+kj
+∂
+∂xd e = Γd
+kj
+� ∂ei
+∂xd + Γi
+dses
+� ∂
+∂xi
+=
+�
+Γi−1
+kj
++ Γd
+kjΓi
+dses� ∂
+∂xi ,
+so we have
+20
+
+∇2
+∂
+∂xk ,
+∂
+∂xj e = ∇
+∂
+∂xk
+�
+∇
+∂
+∂xj e
+�
+− ∇∇
+∂
+∂xk
+∂
+∂xj e
+=
+�∂Γi
+js
+∂xk es + Γi
+jk+1 + Γi
+kj+1 − Γi−1
+kj
++ (Γd
+jsΓi
+kd − Γd
+kjΓi
+ds)es
+� ∂
+∂xi .
+which yields (24).
+References
+[1] R. W. Brockett, ”Feedback invariants for nonlinear systems”, in Proc. IFAC
+Congress, Helsinki, 1978.
+[2] B. Jakubczyk and W. Respondek, ”On linearization of control systems”,
+Bull. Acad. Polonaise Sci., Ser. Sci. Math., vol. 28, pp. 517-522, 1980.
+[3] L. R. Hunt and R. Su, ”Linear equivalents of nonlinear time varying sys-
+tems”, Proc. of the MTNS, pp. 119-123, Santa Monica, 1981.
+[4] M. Nowicki and W. Respondek, ”A classification of feedback linearizable
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+[5] F. Bullo and A.D. Lewis, Geometric Control of Mechanical Systems,
+Springer-Verlag, 2004.
+[6] H. Nijmeijer, A.J. van der Schaft, Nonlinear Dynamical Control Systems,
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+
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+22
+

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+Universality of neural dynamics on complex networks
+Vaiva Vasiliauskaite†∗ and Nino Antulov-Fantulin†
+Computational Social Science, ETH Z¨urich, 8092 Z¨urich, Switzerland
+(Dated: January 13, 2023)
+This paper discusses the capacity of graph neural networks to learn the functional form of ordinary
+differential equations that govern dynamics on complex networks. We propose necessary elements
+for such a problem, namely, inductive biases, a neural network architecture and a learning task. Sta-
+tistical learning theory suggests that generalisation power of neural networks relies on independence
+and identical distribution (i.i.d.) of training and testing data. Although this assumption together
+with an appropriate neural architecture and a learning mechanism is sufficient for accurate out-of-
+sample predictions of dynamics such as, e.g. mass-action kinetics, by studying the out-of-distribution
+generalisation in the case of diffusion dynamics, we find that the neural network model: (i) has a
+generalisation capacity that depends on the first moment of the initial value data distribution; (ii)
+learns the non-dissipative nature of dynamics implicitly; and (iii) the model’s accuracy resolution
+limit is of order O(1/√n) for a system of size n.
+Introduction
+Dynamics in a complex networked sys-
+tem is modelled as a set of n ordinary differential equa-
+tions (ODEs) that describe the rate of change of a quan-
+tity xi(t) for each node i and are coupled via adjacency
+matrix A ∈ Rn×n. A general form of these equations is
+˙xi = L(xi(t)) +
+�
+j
+AijQ(xi(t), xj(t))
+(1)
+= F(xi(t), x(t), A)
+where L describes self-interactions, Q is a function that
+models pairwise interactions between neighbours and �
+is an aggregation function.
+With appropriate choices
+of functions L, Q, � this definition is a general form
+for models of epidemic processes, biochemical dynamics,
+birth–death processes, gene regulatory dynamics [1], as
+well as dynamics that show chaotic behaviour [2].
+The initial value problem of a set of ODEs such as Eq.
+1 together with an initial condition x(t0), has a solution
+that satisfies
+x(t) = x(t0) +
+� t
+t0
+FFF(x(t′), A)dt′
+(2)
+and describes a set of trajectories of the dynamics, if the
+system was initialised at x(t0).
+Appropriately setup, a neural network ΨΨΨ(x;ωωω) has ca-
+pacity to approximate any continuous function F with
+compact support [3]. In practice, learning the weights is
+usually done via some variant of backpropagation algo-
+rithm [4].
+Notably, neural networks can also be used to approx-
+imate dynamical systems [5] and find solutions of initial
+and boundary value problems of differential equations [6].
+A dynamical system is that in which FFF describes the time
+dependence of x in an ambient space. Notably, if FFF is
+known, the description quality of the course of dynamics
+∗ vvasiliau@ethz.ch
+† Authors contributed equally to this work.
+is independent of a coordinate in the space. For exam-
+ple, Newton’s laws of motion describe the trajectory of a
+bouncing ball regardless of its longitudinal and latitudi-
+nal position. Recovering universal dynamical principles
+from empirical data has been shown to belong to NP-
+hard class [7].
+Despite, the hardness of problem, in recent years, dif-
+ferent classes of neural networks were used to learn dif-
+ferent parts of dynamics from empirical data, including
+graph neural networks [8] and their differential [9] coun-
+terparts [10]; reservoir computers [11, 12] as well as re-
+gression techniques [13, 14] or to learn control dynam-
+ics [15].
+Here we discuss architectural design choices and induc-
+tive biases that are crucial for a neural network model
+that approximates dynamics evolving on complex net-
+works. We then study the model’s generalisation capac-
+ity using simple models of deterministic dynamics [1].
+Lastly, we discuss our work in the context of learning
+principles that govern dynamics in complex system from
+perspective of generalization to unseen initial conditions.
+Inductive biases for dynamics on complex networks
+There are several important inductive biases and assump-
+tions worth noting about the complex network dynamics
+and its neural approximations.
+1.
+Network structure: There exists a known static
+network represented as an adjacency matrix A. There-
+fore it is reasonable to take a GNN [16] as the candidate
+for ΨΨΨ. A single-layer graph convolution network can be
+defined as
+ΨΨΨgnn(x) = (σ [ΦΦΦxW + b]) Wagg.
+(3)
+where x ∈ Rn×d is an input, ΦΦΦ ∈ Rn×n is a graph oper-
+ator (e.g. ΦΦΦ = ˜D− 1
+2 ˜A ˜D− 1
+2 [17]), W ∈ Rd×h, b ∈ Rn×1,
+Wagg ∈ Rh×d are trainable parameters and σ is a non-
+linear function. Different versions of GNN with respect
+to different expressive power for Weisfeiler-Lehman iso-
+morphism are described in [18].
+2.
+Self-Interaction:
+The model includes a self-
+interaction part that approximates L(·).
+arXiv:2301.04900v1  [cond-mat.stat-mech]  12 Jan 2023
+
+2
+3.
+Neighbour-Interaction:
+The model includes a
+neighbour interaction part that approximates Q(·, ·).
+Note that a single-layer GNN, such as a convolutional
+graph neural network has no mixed quadratic terms
+xixj and therefore does not simply satisfy such a con-
+dition.
+Although theoretically it should still be pos-
+sible to approximate nonlinear quadratic terms with a
+single layer neural network with an arbitrary width, in
+practice it can be challenging and require either a very
+large number of hidden neurons, or an exotic learning
+mechanism that goes beyond the standard gradient de-
+scend.
+Alternatively, one can improve expressivity of
+the model by increasing its depth, i.e. using multi-layer
+GNNs or message-passing neural networks [19] to rep-
+resent ΨΨΨ(x;ωωω).
+Here ωωω includes graph operator terms
+ΦΦΦk, k ∈ {1, 2, ..., K} where K is the depth of the neural
+network.
+4. Spatiotemporal locality: The dynamical process
+that follows Eq. 1 must be local, that is, the function
+Q(·, ·) encodes interactions between neighbours.
+How-
+ever, including terms ΦΦΦk in a multi-layer graph neural
+network allows for k-hop interactions via length k walks
+in a network at a timescale smaller than the infinitesimal
+dt thereby subdividing dt to k intervals and breaking an
+assumption of temporal locality.
+5. Aggregation of neighbour-interactions: The ag-
+gregation can itself be non-linear.
+6. Initial value condition: Initial values are preserved
+during training: x0: ΨΨΨ(x0) → x0. If the neural network
+straightforwardly approximates the RHS of Eq. 2, then
+enconding and decoding layers must be pseudo-inverses
+of each other, see App. A.
+7. Conservation/dissipation laws. If the system is
+closed, it does not exchange energy or mass with the en-
+vironment, therefore a conservation law holds, namely
+�
+i
+dxi(t)
+dt
+= C
+∀t.
+(4)
+A constraint on a neural network to satisfy conservation
+laws can be imposed via a regularisation term in the loss
+function,
+R(D) =
+1
+|D|
+�
+x∈D
+|FFF(x)1 − ΨΨΨ(x)1| ,
+that penalises the model weights which produce predic-
+tions which do not respect the conservation law Eq. 4.
+Here D is the dataset over which the loss is calculated.
+The strength of the regulariser term can be modulated
+by mutiplying R(D) with a non-negative real number λ.
+Architecture
+Given the inductive biases for dynamics
+on networks, we propose a neural network model of the
+following form:
+˙x = ψψψℓ(x) + ψψψ
+�
+(x)
+(5)
+ψψψ
+�
+(x) = vec−1�
+ψψψq3�
+vec
+�
+ΦΦΦ ⊙
+�
+ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� ���
+where ψψψ(x) is a single hidden layer neural network
+are given by (3).
+The mappings of local interaction
+are summarised in App. B. The design choices of Eq.
+5 comply with the inductive biases stated earlier.
+To
+this end, we performed vetorisation of input to the
+function ψψψ
+�
+[ψψψq3 (·)].
+This function can approximate
+any invariant poolings of a set [20] or a multiset [18].
+Notably, we also assumed that Q(·, ·) is factorisable.
+Since it can be approximated by Chebyshev polynomials,
+and, according to the strictly real fundamental theorem
+of algebra [21], it is possible to factorise polynomial
+function to two factors. Alternatively, one can use deep
+sets [20] as arguments to approximate Q(·, ·).
+In order to guarantee the local existence and unique-
+ness of the solution to the initial value problem, by Pi-
+card–Lindel¨of theorem the neural network ΨΨΨ needs to be
+Lipschitz continuous. To enforce Lipschitz continuity of
+ΨΨΨ, we will be using 1-Lipschitz activation functions such
+as ReLU, sigmoid, softmax, or tanh.
+Learning task
+We formulate two distinct statistical
+learning settings that relate to an increasing strength of
+generality in the approximation of a dynamical system.
+1.
+Regression task to approximate FFF by ΨΨΨ: An
+appropriate “proto data set” here is
+D = {(x(t)α, y(t)α)},
+s.t. x(t)α ∈ Rn, y(t)α ∈ Rn, x(0)α ∼ fx(0)(x), t = [0, T] ∈ R.
+our labels are defined as y(t)α = FFF(x((t))α), α denotes
+α-th initial condition x(0)α sampled from a predefined
+distribution fx(0)(x); all others points x(t)α are obtained
+following Eq. 2. Here the functional mapping that is be-
+ing learnt is ˆFFF : Rn → Rn and is obtained by minimising
+the loss L between the true labels y and the labels f(x)
+obtained by the current model:
+ˆFFF = arg
+min
+f:Rn→Rn
+E
+P(x,y) L(f(x), y).
+Here E is an expectation operator, P(x, y) is the data
+sampling distribution.
+At the moment, samples from the “proto data set” are
+not independent: those trajectories that were obtained
+from the same initial condition are non-i.i.d. Such sam-
+pling is compulsory for the Uniform Law of Large num-
+bers, that together with capacity control ensures general-
+isation from train to test set [22, 23]. To ensure statistical
+independence of samples, we create finite train and test
+sets of size m1, m2 by using a specific distribution P over
+a “proto data set��
+Dtrain ∪ Dtest ∼ P(x, y).
+Specifically,
+we randomly delegate (x(t)α, y(t)α) to
+either Dtrain or Dtest thereby ensuring an i.i.d. condition
+by dropping information on the initial conditions and
+time.
+
+3
+Dynamics
+L
+Q
+Ltrain
+reg
+Ltest
+reg
+≈reg
+Ltrain
+traj
+Ltest
+traj
+≈traj
+Heata
+–
+B(xj − xi) 2.03 ± 1.03
+2.14 ± 1.08
+✓
+1.39 ± 0.59 1.47 ± 0.63
+✓
+MAKb
+F − Bxb
+i
+Rxj
+0.41 ± 1.08
+0.44 ± 1.14
+✓
+1.48 ± 0.05 1.55 ± 0.04
+×
+PDc
+−Bxb
+i
+Rxa
+j
+4.68 ± 12.82 4.72 ± 12.89
+✓
+3.03 ± 0.03 3.04 ± 0.03
+✓
+MMd
+−Bxi
+R
+xh
+j
+1+xh
+j
+7.68 ± 5.36
+7.83 ± 5.47
+✓
+5.93 ± 0.12 5.94 ± 0.14
+✓
+SISe
+−Bxi
+(1 − xi)xj
+1.16 ± 3.62
+1.31 ± 4.07
+✓
+1.54 ± 0.01 1.64 ± 0.02
+×
+a B = 0.05.
+b B = 0.1, R = 1, f = 0.5.
+c B = 2, R = 0.3, a = 1.5, b = 3.
+d B = 4, R = 0.5, h = 3.
+e B = 5, R = 0.5.
+TABLE I. Generalisation of a neural network model Eq. 5 trained on dynamics from [1] in the regression task setting, and
+the trajectory learning setting. Reported loss values are multiplied by a factor 10−2. In columns denoted “≈” we indicate for
+which dynamics the train loss is approximately similar (“✓”) or different (“×”) from the test loss.
+2.
+Trajectory learning setting that approxi-
+mates x(t): here the train set contains m1 initial condi-
+tions x(0)α as inputs, while each label corresponds to tra-
+jectories yα = {x(t)α}, where t = 0, ∆t, 2∆t, ....k∆t = T
+that were realised from the initial condition x(0)α:
+Dtrain = {(x(0)α, yα)},
+s.t. x(0)α ∈ Rn, yα ∈ Rkn, x(0)α ∼ fx(0)(x), α ∈ [1, m1],
+yα = {x(0)α, x(∆t)α, ..., x(k∆t)α}
+and test set Dtest is constructed analogously from m2
+initial conditions that are sampled from the same distri-
+bution x(0)α ∼ fx(0)(x). The mapping learnt here is of
+the following form:
+ˆFFF : Rn → Rkn and is realised by
+computing an initial value problem Eq. 2 using a neural
+network ΨΨΨ in replacement of FFF.
+Experiments and Results
+We consider models with
+h′ = 6, h = 8, h′′ = 5, hd = 3, trained in 1000 epochs us-
+ing Adam optimiser with learning rate of 10−2 and weight
+decay 10−3. All activations are ReLU. Unless otherwise
+stated, the initial values in both the train set and the test
+set are sampled from B[a = 5, b = 5]. For numerical in-
+tegration, an explicit Runge-Kutta method of order 5(4)
+is used [24].
+The training loss function is the average L1 norm. For
+the regression task, the loss is
+Ltrain
+reg
+=
+1
+Nreg
+�
+x,y∈Dtrain
+�
+||f(x) − y||1 + λR(x)
+�
+,
+where Nreg = |Dtrain|(xmax − xmin). For the trajectory
+learning task, the loss is defined as:
+Ltrain
+traj =
+1
+Ntraj
+�
+x(0),y∈Dtrain
+T/∆t
+�
+k=0
+(6)
+�
+||x(k∆t) − ˆx(k∆t)||1 + λR(x(k∆t))
+�
+Here
+the
+normalisation
+constant
+is
+Ntraj
+=
+|Dtrain|nT(xmax − xmin)/∆t.
+λ = 0 and the regu-
+larisation terms are nil for the first part of the analysis.
+The training sets include samples from 103 trajectories,
+the testing sets – from 102 trajectories and the batch
+size is 10. The parameters for numerical integration are
+∆t = 0.01, T = 1.5. In all cases, a graph was sampled
+from Erd¨os-R´enyi ensemble with p = 0.5 and � = �
+j.
+Tab. I shows that the trained neural network model Eq.
+5 can well-approximate the true dynamics and generalise
+to unseen initial values well, provided fx(0)(x) is used for
+generating both, a training test and a sampling test.
+Generalisation
+Crucially, the universality of the neu-
+ral approximation exemplified in Tab. I is only at the low-
+est level that is attainable by putting strong constraints
+on a test set (that are in accordance with statistical learn-
+ing theory): the two sets must be statistically equivalent.
+If the distribution of initial values is irrelevant for the
+steady state solution, the neural model also inadvertently
+universally approximates the dynamical system.
+However, it seems reasonable to ask if a neural net-
+work can do better. In Tab. II we propose three tiers of
+universality of approximation FFF ≈ ΨΨΨ in terms of statis-
+tical properties of training and testing samples. In this
+context, the statistical learning theory concerns only the
+lowest level of generality.
+Level
+Relation between f and g
+Bottom fX0 ≡ gX0 or fX0,X∞ = fX0fX∞
+Mid
+fX0 ̸≡ gX0, sup fX0 = sup gX0
+Top
+fX0 ̸≡ gX0, sup fX0 ̸= sup gX0
+TABLE II. Generalisation levels for a neural approximation
+FFF ≈ ΨΨΨ is encompassed in model’s ability to extrapolate pre-
+dictions to data that was not used during training. Probabil-
+ity density functions related to training data are denoted by
+“f”; related to testing data are denoted by “g”.
+More sophisticated, mid and top level generalisations
+would enable a faithful prediction in cases where the con-
+straints on statistical properties of data are relaxed, for
+example, where fX0 is not the same as gX0.
+Diffusion
+A dynamical system whose faith and the
+course of action depend on the distribution of initial val-
+ues enables us to study the limits of generalisation of a
+
+4
+neural network. Diffusion equation on a graph is a good
+example due to its simplicity and known analytical solu-
+tion of the form
+x(t) =
+�
+i
+ai(0)e−Bλitvi,
+ai(0) = x(t)⊤vi,
+(7)
+where λi, vi are ith eigenvalue and eigenvector of the
+graph Laplacian and the steady state solution is given
+by
+lim
+t→∞ xi(t) = 1
+n
+�
+j
+xj(0)
+∀i.
+Perturbation of the initial value x(0) by δ ∼ fδ such
+that xδ
+i (0) = xi(0) + δ gives a difference in the steady
+state solutions of ⟨xδ
+i (0)⟩i − ⟨xi(0)⟩i = γ.
+Fig. 1 shows how the loss accumulates over the inte-
+gration time t for the neural network model ΨΨΨ for tra-
+jectories in the train and in the test sets. In addition,
+we consider a perturbation (NN,pert) where the initial
+value is sampled from a different distribution, namely,
+gX0(x) = B(6, 5), while the neural network was trained
+using fX0(x) = B(5, 5). This figure shows that the neu-
+ral network prediction is reasonable, under i.i.d. sampling
+condition for an initial condition in train and test set.
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+t
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+traj(t)
+NN,train
+NN,test
+NN,pert
+Numerical
+FIG. 1.
+Node average loss between the analytical solution
+and:
+1) the numerical solution (numerical), 2) the neu-
+ral network solution for a subset of initial conditions in the
+training set (NN,train) as well as a subset of a testing set
+(NN,test).
+The original x(0) ∼ B(5, 5), whereas the per-
+turbed (NN,pert) initial values x′(0) ∼ B(6, 5).
+The loss
+is computed for the trajectory learning task using Ntraj =
+100 trajectories in each case using an equation Ltraj(t) =
+1
+Ntraj
+�
+x(0),y∈D ||x(t)− ˆx(t)||1. The errors show one standard
+deviation.
+Fig. 2 follows the same analysis and shows that by
+varying the parameters of the beta distribution B(a, b),
+the loss in the steady state (averaged over the last 10
+steps of the simulation) is proportional to the differ-
+ence in expectation value of the beta-distribution used
+in training, and in testing to generate the initial values.
+All in all, these results show that the neural network ap-
+proximation of the differential form is exclusive to the
+statistical properties of the training set.
+Upsofar, conservation law (4) and the effect of the reg-
+ulariser were not considered. We study it in Fig. 3 for a
+small case with a graph composed of N = 2 nodes. This
+figure presents two key findings: Fig. 3a) clearly shows
+that ΨΨΨ is biased towards the training set; whereas in Fig.
+3b) it is clear that ΨΨΨ has the property of implicit dissipa-
+tive (conservation) regularization. Even in the case of no
+explicit regularization of the dissipative term, the neural
+network optimises towards a less dissipative regime. This
+is of particular importance, since some systems in Tab. I
+are non-dissipative and some are dissipative.
+2
+4
+6
+8
+a
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+train
+b = a
+b = 5
+FIG. 2.
+Generalisation of ΨΨΨ to unseen initial conditions. The
+neural network was trained using initial values sampled from
+x0 ∼ B(5, 5) until it achieved the loss train. Its prediction
+capacity was then tested on dynamics with initial conditions
+x0 ∼ B(a, b = a) (red circles) as well as x0 ∼ B(a, b = 5) (blue
+triangles). The dashed orange line is a function |0.5−a|/(a+
+5)).
+The loss is computed for the trajectory learning task
+using Ntraj = 100 trajectories in each case using (6), omitting
+the term xmax−xmin in the normalisation and considering the
+last 10 timesteps. The errors show one standard deviation
+across trajectories.
+Next, we turn our attention to analyse the out-of-
+sample loss for system of n coupled differential equations
+(coupling with Erd¨os-R´enyi model) and diffusion dynam-
+ics. Notably, the steady state solution is governed by the
+average value ⟨x0⟩, and since we have n nodes in our sys-
+tem this value has variance ∝ 1/√n. This implies that
+it is easier to accurately predict dynamics with a larger
+number of differential equations. In Fig. 4, we show that
+indeed, test loss is inversely proportional to the system’s
+size.
+Discussion
+In this paper, we proposed a variant of a
+Neural ODE model which implements a set of inductive
+biases suitable for complex dynamics on graphs and elic-
+its dynamical models in complex networked systems di-
+rectly from time series these systems produce. While we
+showed the presence of generalisation out-of-sample for
+a wide range of dynamical models, perhaps more impor-
+tantly such an exercise reflects on generalisation capacity
+only at the most trivial level. Multiple out-of-distribution
+
+5
+epoch=0
+epoch=1000
+epoch=2000
+a)
+b)
+𝜆 = 1
+𝜆 = 0
+Legend
+FIG. 3. Learning diffusion on a fully connected n = 2 network
+using the regression training paradigm and a conservation
+law regulariser. The training sample consists of datapoints
+obtained from trajectories generated using x0 ∼ [0.2, 0.7] +
+N(0, 0.1), the testing sample: x0 ∼ [0.3, 0.8] + N(0, 0.1). a)
+shows an example of a training process, namely by contrast-
+ing the true (continuous lines) and learnt (dotted) trajectories
+of an initial value problem as predicted after indicated train-
+ing epochs, using λ = 1.
+b) shows the loss and the value
+of the regulariser over training period in the case where the
+regulariser plays a part in training (λ = 1, same training as
+in a)), and when it does not (λ = 0). The results in b) are
+obtained from 10 independent runs.
+tests suggest that the neural network approximation is
+valid only for a specific probability distribution of initial
+values, which was also used to generate the training sam-
+ples. Furthermore, even if we kept the statistics intact,
+we observe that it is harder to achieve accurate predic-
+tions in small-size systems as opposed to large-scale ones,
+due to presence of fluctuations that scale as O(1/√n) for
+a system of size n.
+Appendix A: Encoding and decoding layers
+Preceding the differential model layer ΨΨΨ, one can en-
+code the input via ΨΨΨe : x ∈ Rn×d → x ∈ Rn×de [10],
+in which case, the state space is of n × de dimensions
+instead of n × d.
+To revert back to the original n × d
+space, a decoding function ΨΨΨd is used at the end. The
+embedding respects the initial values iff ΨΨΨe =
+�
+ΨΨΨd�−1. If
+the encoding and decoding are obtained via linear lay-
+ers without bias terms, they are represented by matri-
+ces We ∈ Rd×de and Wd ∈ Rde×d. So after a forward
+pass, the initial values are modified if WeWd ̸= I. This
+only holds if the two matrices are inverses to each other.
+Since these matrices are not square, one can use a Moore-
+Penrose inverse, which is a generalisation of the tradi-
+tional inverse. We want Wd to be a right inverse of We,
+0
+50
+100
+n
+0.00
+0.05
+0.10
+0.15
+FIG. 4. Test loss (computed for the last 10 time steps of the
+simulation) for a regression learning task at varied network
+sizes.
+The training and testing datasets are sampled from
+B(1, 1). Averages are evaluated using 1000 test samples; for
+training, 100 trajectories were used. The figure indicates that
+the larger the network, the smaller the average loss and the
+variance.
+defined as: Wd = W∗
+e(WeW∗
+e)−1 . Here W∗
+e denotes
+a Hermitian transpose of We, however in our case it is
+equivalent to a transpose, since We is defined over real
+numbers.
+Appendix B: Neural network mappings
+The mappings of functions that constitute the neural
+network model defined in Eq. 3 are defined as (here we
+consider input x ∈ Rn×1×d, a three-dimensional tensor,
+and tensor dimension is counted starting from 1):
+1. ψψψℓ : Rn×1×d → Rn×1×d, k = 3 mode product with
+W ∈ Rd×h′ i.e. Rn×1×d ×3 Rd×h′ ∈ Rn×1×h′ and
+C ∈ Rh′×d.
+2. ψψψq1,ψψψq2 : Rn×1×d → Rn×1×h, k = 3 mode product
+with W ∈ Rd×h: Rn×1×d ×3 Rd×h ∈ Rn×1×h and
+C = I.
+3. x⊤1 : Rn×1×h → Rh×n×1.
+4. x⊤2 : Rn×1×h → Rh×1×n.
+5.
+�
+ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2�
+:
+Rh×n×1 ×3 Rh×1×n
+∈
+Rh×n×n.
+6. ΦΦΦ ⊙
+�
+ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2�
+: Rn×n ⊙ Rh×n×1 ×3
+Rh×1×n ∈ Rh×n×n.
+Here an operator ⊙ denotes
+a standard “broadcasted” element-wise multiplica-
+tion.
+7. vec(·): Rh×n×n → Rn2h×1.
+8. ψψψq3 : Rn2h×1 → Rn2h×1, W ∈ R1×h′′ and C ∈
+Rh′′×1.
+9. vec−1(·): Rn2h×1 → Rn×nh.
+
+6
+10. ψψψ
+�
+= ψ(�(·)), where we use �(·) as invariant
+pooling layer Rn×nh → Rn×1 and then apply de-
+coding layer ψ that maps Rn×1 → Rn×d, with
+W ∈ R1×hd and C ∈ Rhd×d.
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+

FtE4T4oBgHgl3EQfHAzF/content/tmp_files/load_file.txt

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+page_content='Universality of neural dynamics on complex networks  Vaiva Vasiliauskaite†∗ and Nino Antulov-Fantulin†  Computational Social Science, ETH Z¨urich, 8092 Z¨urich, Switzerland  (Dated: January 13, 2023)  This paper discusses the capacity of graph neural networks to learn the functional form of ordinary  differential equations that govern dynamics on complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' We propose necessary elements  for such a problem, namely, inductive biases, a neural network architecture and a learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Sta-  tistical learning theory suggests that generalisation power of neural networks relies on independence  and identical distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=') of training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Although this assumption together  with an appropriate neural architecture and a learning mechanism is sufficient for accurate out-of-  sample predictions of dynamics such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' mass-action kinetics, by studying the out-of-distribution  generalisation in the case of diffusion dynamics, we find that the neural network model: (i) has a  generalisation capacity that depends on the first moment of the initial value data distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' (ii)  learns the non-dissipative nature of dynamics implicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' and (iii) the model’s accuracy resolution  limit is of order O(1/√n) for a system of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Introduction  Dynamics in a complex networked sys-  tem is modelled as a set of n ordinary differential equa-  tions (ODEs) that describe the rate of change of a quan-  tity xi(t) for each node i and are coupled via adjacency  matrix A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' A general form of these equations is  ˙xi = L(xi(t)) +  �  j  AijQ(xi(t), xj(t))  (1)  = F(xi(t), x(t), A)  where L describes self-interactions, Q is a function that  models pairwise interactions between neighbours and �  is an aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  With appropriate choices  of functions L, Q, � this definition is a general form  for models of epidemic processes, biochemical dynamics,  birth–death processes, gene regulatory dynamics [1], as  well as dynamics that show chaotic behaviour [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The initial value problem of a set of ODEs such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  1 together with an initial condition x(t0), has a solution  that satisfies  x(t) = x(t0) +  � t  t0  FFF(x(t′), A)dt′  (2)  and describes a set of trajectories of the dynamics, if the  system was initialised at x(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Appropriately setup, a neural network ΨΨΨ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='ωωω) has ca-  pacity to approximate any continuous function F with  compact support [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In practice, learning the weights is  usually done via some variant of backpropagation algo-  rithm [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Notably, neural networks can also be used to approx-  imate dynamical systems [5] and find solutions of initial  and boundary value problems of differential equations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  A dynamical system is that in which FFF describes the time  dependence of x in an ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Notably, if FFF is  known, the description quality of the course of dynamics  ∗ vvasiliau@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='ch  † Authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  is independent of a coordinate in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' For exam-  ple, Newton’s laws of motion describe the trajectory of a  bouncing ball regardless of its longitudinal and latitudi-  nal position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Recovering universal dynamical principles  from empirical data has been shown to belong to NP-  hard class [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Despite, the hardness of problem, in recent years, dif-  ferent classes of neural networks were used to learn dif-  ferent parts of dynamics from empirical data, including  graph neural networks [8] and their differential [9] coun-  terparts [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' reservoir computers [11, 12] as well as re-  gression techniques [13, 14] or to learn control dynam-  ics [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Here we discuss architectural design choices and induc-  tive biases that are crucial for a neural network model  that approximates dynamics evolving on complex net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' We then study the model’s generalisation capac-  ity using simple models of deterministic dynamics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Lastly, we discuss our work in the context of learning  principles that govern dynamics in complex system from  perspective of generalization to unseen initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Inductive biases for dynamics on complex networks  There are several important inductive biases and assump-  tions worth noting about the complex network dynamics  and its neural approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Network structure: There exists a known static  network represented as an adjacency matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' There-  fore it is reasonable to take a GNN [16] as the candidate  for ΨΨΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' A single-layer graph convolution network can be  defined as  ΨΨΨgnn(x) = (σ [ΦΦΦxW + b]) Wagg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  (3)  where x ∈ Rn×d is an input, ΦΦΦ ∈ Rn×n is a graph oper-  ator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' ΦΦΦ = ˜D− 1  2 ˜A ˜D− 1  2 [17]), W ∈ Rd×h, b ∈ Rn×1,  Wagg ∈ Rh×d are trainable parameters and σ is a non-  linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Different versions of GNN with respect  to different expressive power for Weisfeiler-Lehman iso-  morphism are described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Self-Interaction:  The model includes a self-  interaction part that approximates L(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='04900v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='stat-mech]  12 Jan 2023  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Neighbour-Interaction:  The model includes a  neighbour interaction part that approximates Q(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Note that a single-layer GNN, such as a convolutional  graph neural network has no mixed quadratic terms  xixj and therefore does not simply satisfy such a con-  dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Although theoretically it should still be pos-  sible to approximate nonlinear quadratic terms with a  single layer neural network with an arbitrary width, in  practice it can be challenging and require either a very  large number of hidden neurons, or an exotic learning  mechanism that goes beyond the standard gradient de-  scend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Alternatively, one can improve expressivity of  the model by increasing its depth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' using multi-layer  GNNs or message-passing neural networks [19] to rep-  resent ΨΨΨ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='ωωω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Here ωωω includes graph operator terms  ΦΦΦk, k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=', K} where K is the depth of the neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Spatiotemporal locality: The dynamical process  that follows Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 1 must be local, that is, the function  Q(·, ·) encodes interactions between neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  How-  ever, including terms ΦΦΦk in a multi-layer graph neural  network allows for k-hop interactions via length k walks  in a network at a timescale smaller than the infinitesimal  dt thereby subdividing dt to k intervals and breaking an  assumption of temporal locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Aggregation of neighbour-interactions: The ag-  gregation can itself be non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Initial value condition: Initial values are preserved  during training: x0: ΨΨΨ(x0) → x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' If the neural network  straightforwardly approximates the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 2, then  enconding and decoding layers must be pseudo-inverses  of each other, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Conservation/dissipation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' If the system is  closed, it does not exchange energy or mass with the en-  vironment, therefore a conservation law holds, namely  �  i  dxi(t)  dt  = C  ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  (4)  A constraint on a neural network to satisfy conservation  laws can be imposed via a regularisation term in the loss  function,  R(D) =  1  |D|  �  x∈D  |FFF(x)1 − ΨΨΨ(x)1| ,  that penalises the model weights which produce predic-  tions which do not respect the conservation law Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Here D is the dataset over which the loss is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The strength of the regulariser term can be modulated  by mutiplying R(D) with a non-negative real number λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Architecture  Given the inductive biases for dynamics  on networks, we propose a neural network model of the  following form:  ˙x = ψψψℓ(x) + ψψψ  �  (x)  (5)  ψψψ  �  (x) = vec−1�  ψψψq3�  vec  �  ΦΦΦ ⊙  �  ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2� ���  where ψψψ(x) is a single hidden layer neural network  are given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The mappings of local interaction  are summarised in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The design choices of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  5 comply with the inductive biases stated earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  To  this end, we performed vetorisation of input to the  function ψψψ  �  [ψψψq3 (·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  This function can approximate  any invariant poolings of a set [20] or a multiset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Notably, we also assumed that Q(·, ·) is factorisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Since it can be approximated by Chebyshev polynomials,  and, according to the strictly real fundamental theorem  of algebra [21], it is possible to factorise polynomial  function to two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Alternatively, one can use deep  sets [20] as arguments to approximate Q(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  In order to guarantee the local existence and unique-  ness of the solution to the initial value problem, by Pi-  card–Lindel¨of theorem the neural network ΨΨΨ needs to be  Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' To enforce Lipschitz continuity of  ΨΨΨ, we will be using 1-Lipschitz activation functions such  as ReLU, sigmoid, softmax, or tanh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Learning task  We formulate two distinct statistical  learning settings that relate to an increasing strength of  generality in the approximation of a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Regression task to approximate FFF by ΨΨΨ: An  appropriate “proto data set” here is  D = {(x(t)α, y(t)α)},  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' x(t)α ∈ Rn, y(t)α ∈ Rn, x(0)α ∼ fx(0)(x), t = [0, T] ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  our labels are defined as y(t)α = FFF(x((t))α), α denotes  α-th initial condition x(0)α sampled from a predefined  distribution fx(0)(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' all others points x(t)α are obtained  following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Here the functional mapping that is be-  ing learnt is ˆFFF : Rn → Rn and is obtained by minimising  the loss L between the true labels y and the labels f(x)  obtained by the current model:  ˆFFF = arg  min  f:Rn→Rn  E  P(x,y) L(f(x), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Here E is an expectation operator, P(x, y) is the data  sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  At the moment, samples from the “proto data set” are  not independent: those trajectories that were obtained  from the same initial condition are non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Such sam-  pling is compulsory for the Uniform Law of Large num-  bers, that together with capacity control ensures general-  isation from train to test set [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' To ensure statistical  independence of samples, we create finite train and test  sets of size m1, m2 by using a specific distribution P over  a “proto data set”  Dtrain ∪ Dtest ∼ P(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Specifically,  we randomly delegate (x(t)α, y(t)α) to  either Dtrain or Dtest thereby ensuring an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' condition  by dropping information on the initial conditions and  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  3  Dynamics  L  Q  Ltrain  reg  Ltest  reg  ≈reg  Ltrain  traj  Ltest  traj  ≈traj  Heata  –  B(xj − xi) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='08  ✓  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='63  ✓  MAKb  F − Bxb  i  Rxj  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='41 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='44 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='14  ✓  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='04  ×  PDc  −Bxb  i  Rxa  j  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='68 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='72 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='89  ✓  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03  ✓  MMd  −Bxi  R  xh  j  1+xh  j  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='68 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='36  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='83 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='47  ✓  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='14  ✓  SISe  −Bxi  (1 − xi)xj  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='16 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='62  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='31 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='07  ✓  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='02  ×  a B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  b B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='1, R = 1, f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  c B = 2, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='3, a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5, b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  d B = 4, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5, h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  e B = 5, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Generalisation of a neural network model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 5 trained on dynamics from [1] in the regression task setting, and  the trajectory learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Reported loss values are multiplied by a factor 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In columns denoted “≈” we indicate for  which dynamics the train loss is approximately similar (“✓”) or different (“×”) from the test loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Trajectory learning setting that approxi-  mates x(t): here the train set contains m1 initial condi-  tions x(0)α as inputs, while each label corresponds to tra-  jectories yα = {x(t)α}, where t = 0, ∆t, 2∆t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='.k∆t = T  that were realised from the initial condition x(0)α:  Dtrain = {(x(0)α, yα)},  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' x(0)α ∈ Rn, yα ∈ Rkn, x(0)α ∼ fx(0)(x), α ∈ [1, m1],  yα = {x(0)α, x(∆t)α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=', x(k∆t)α}  and test set Dtest is constructed analogously from m2  initial conditions that are sampled from the same distri-  bution x(0)α ∼ fx(0)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The mapping learnt here is of  the following form:  ˆFFF : Rn → Rkn and is realised by  computing an initial value problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 2 using a neural  network ΨΨΨ in replacement of FFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Experiments and Results  We consider models with  h′ = 6, h = 8, h′′ = 5, hd = 3, trained in 1000 epochs us-  ing Adam optimiser with learning rate of 10−2 and weight  decay 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' All activations are ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Unless otherwise  stated, the initial values in both the train set and the test  set are sampled from B[a = 5, b = 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' For numerical in-  tegration, an explicit Runge-Kutta method of order 5(4)  is used [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The training loss function is the average L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' For  the regression task, the loss is  Ltrain  reg  =  1  Nreg  �  x,y∈Dtrain  �  ||f(x) − y||1 + λR(x)  �  ,  where Nreg = |Dtrain|(xmax − xmin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' For the trajectory  learning task, the loss is defined as:  Ltrain  traj =  1  Ntraj  �  x(0),y∈Dtrain  T/∆t  �  k=0  (6)  �  ||x(k∆t) − ˆx(k∆t)||1 + λR(x(k∆t))  �  Here  the  normalisation  constant  is  Ntraj  =  |Dtrain|nT(xmax − xmin)/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  λ = 0 and the regu-  larisation terms are nil for the first part of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The training sets include samples from 103 trajectories,  the testing sets – from 102 trajectories and the batch  size is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The parameters for numerical integration are  ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='01, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In all cases, a graph was sampled  from Erd¨os-R´enyi ensemble with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5 and � = �  j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' I shows that the trained neural network model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  5 can well-approximate the true dynamics and generalise  to unseen initial values well, provided fx(0)(x) is used for  generating both, a training test and a sampling test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Generalisation  Crucially, the universality of the neu-  ral approximation exemplified in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' I is only at the low-  est level that is attainable by putting strong constraints  on a test set (that are in accordance with statistical learn-  ing theory): the two sets must be statistically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  If the distribution of initial values is irrelevant for the  steady state solution, the neural model also inadvertently  universally approximates the dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  However, it seems reasonable to ask if a neural net-  work can do better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' II we propose three tiers of  universality of approximation FFF ≈ ΨΨΨ in terms of statis-  tical properties of training and testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In this  context, the statistical learning theory concerns only the  lowest level of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Level  Relation between f and g  Bottom fX0 ≡ gX0 or fX0,X∞ = fX0fX∞  Mid  fX0 ̸≡ gX0, sup fX0 = sup gX0  Top  fX0 ̸≡ gX0, sup fX0 ̸= sup gX0  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Generalisation levels for a neural approximation  FFF ≈ ΨΨΨ is encompassed in model’s ability to extrapolate pre-  dictions to data that was not used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Probabil-  ity density functions related to training data are denoted by  “f”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' related to testing data are denoted by “g”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  More sophisticated, mid and top level generalisations  would enable a faithful prediction in cases where the con-  straints on statistical properties of data are relaxed, for  example, where fX0 is not the same as gX0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Diffusion  A dynamical system whose faith and the  course of action depend on the distribution of initial val-  ues enables us to study the limits of generalisation of a  4  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Diffusion equation on a graph is a good  example due to its simplicity and known analytical solu-  tion of the form  x(t) =  �  i  ai(0)e−Bλitvi,  ai(0) = x(t)⊤vi,  (7)  where λi, vi are ith eigenvalue and eigenvector of the  graph Laplacian and the steady state solution is given  by  lim  t→∞ xi(t) = 1  n  �  j  xj(0)  ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Perturbation of the initial value x(0) by δ ∼ fδ such  that xδ  i (0) = xi(0) + δ gives a difference in the steady  state solutions of ⟨xδ  i (0)⟩i − ⟨xi(0)⟩i = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 1 shows how the loss accumulates over the inte-  gration time t for the neural network model ΨΨΨ for tra-  jectories in the train and in the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In addition,  we consider a perturbation (NN,pert) where the initial  value is sampled from a different distribution, namely,  gX0(x) = B(6, 5), while the neural network was trained  using fX0(x) = B(5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' This figure shows that the neu-  ral network prediction is reasonable, under i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' sampling  condition for an initial condition in train and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='50  t  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='07  traj(t)  NN,train  NN,test  NN,pert  Numerical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Node average loss between the analytical solution  and:  1) the numerical solution (numerical), 2) the neu-  ral network solution for a subset of initial conditions in the  training set (NN,train) as well as a subset of a testing set  (NN,test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The original x(0) ∼ B(5, 5), whereas the per-  turbed (NN,pert) initial values x′(0) ∼ B(6, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The loss  is computed for the trajectory learning task using Ntraj =  100 trajectories in each case using an equation Ltraj(t) =  1  Ntraj  �  x(0),y∈D ||x(t)− ˆx(t)||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The errors show one standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 2 follows the same analysis and shows that by  varying the parameters of the beta distribution B(a, b),  the loss in the steady state (averaged over the last 10  steps of the simulation) is proportional to the differ-  ence in expectation value of the beta-distribution used  in training, and in testing to generate the initial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  All in all, these results show that the neural network ap-  proximation of the differential form is exclusive to the  statistical properties of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Upsofar, conservation law (4) and the effect of the reg-  ulariser were not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' We study it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 3 for a  small case with a graph composed of N = 2 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' This  figure presents two key findings: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 3a) clearly shows  that ΨΨΨ is biased towards the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' whereas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  3b) it is clear that ΨΨΨ has the property of implicit dissipa-  tive (conservation) regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Even in the case of no  explicit regularization of the dissipative term, the neural  network optimises towards a less dissipative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' This  is of particular importance, since some systems in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' I  are non-dissipative and some are dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  2  4  6  8  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='30  train  b = a  b = 5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Generalisation of ΨΨΨ to unseen initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The  neural network was trained using initial values sampled from  x0 ∼ B(5, 5) until it achieved the loss train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Its prediction  capacity was then tested on dynamics with initial conditions  x0 ∼ B(a, b = a) (red circles) as well as x0 ∼ B(a, b = 5) (blue  triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The dashed orange line is a function |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='5−a|/(a+  5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The loss is computed for the trajectory learning task  using Ntraj = 100 trajectories in each case using (6), omitting  the term xmax−xmin in the normalisation and considering the  last 10 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The errors show one standard deviation  across trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Next, we turn our attention to analyse the out-of-  sample loss for system of n coupled differential equations  (coupling with Erd¨os-R´enyi model) and diffusion dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Notably, the steady state solution is governed by the  average value ⟨x0⟩, and since we have n nodes in our sys-  tem this value has variance ∝ 1/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' This implies that  it is easier to accurately predict dynamics with a larger  number of differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 4, we show that  indeed, test loss is inversely proportional to the system’s  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Discussion  In this paper, we proposed a variant of a  Neural ODE model which implements a set of inductive  biases suitable for complex dynamics on graphs and elic-  its dynamical models in complex networked systems di-  rectly from time series these systems produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' While we  showed the presence of generalisation out-of-sample for  a wide range of dynamical models, perhaps more impor-  tantly such an exercise reflects on generalisation capacity  only at the most trivial level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Multiple out-of-distribution  5  epoch=0  epoch=1000  epoch=2000  a)  b)  𝜆 = 1  𝜆 = 0  Legend  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Learning diffusion on a fully connected n = 2 network  using the regression training paradigm and a conservation  law regulariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The training sample consists of datapoints  obtained from trajectories generated using x0 ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='7] +  N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='1), the testing sample: x0 ∼ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='8] + N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' a)  shows an example of a training process, namely by contrast-  ing the true (continuous lines) and learnt (dotted) trajectories  of an initial value problem as predicted after indicated train-  ing epochs, using λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  b) shows the loss and the value  of the regulariser over training period in the case where the  regulariser plays a part in training (λ = 1, same training as  in a)), and when it does not (λ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The results in b) are  obtained from 10 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  tests suggest that the neural network approximation is  valid only for a specific probability distribution of initial  values, which was also used to generate the training sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Furthermore, even if we kept the statistics intact,  we observe that it is harder to achieve accurate predic-  tions in small-size systems as opposed to large-scale ones,  due to presence of fluctuations that scale as O(1/√n) for  a system of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Appendix A: Encoding and decoding layers  Preceding the differential model layer ΨΨΨ, one can en-  code the input via ΨΨΨe : x ∈ Rn×d → x ∈ Rn×de [10],  in which case, the state space is of n × de dimensions  instead of n × d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  To revert back to the original n × d  space, a decoding function ΨΨΨd is used at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The  embedding respects the initial values iff ΨΨΨe =  �  ΨΨΨd�−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' If  the encoding and decoding are obtained via linear lay-  ers without bias terms, they are represented by matri-  ces We ∈ Rd×de and Wd ∈ Rde×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' So after a forward  pass, the initial values are modified if WeWd ̸= I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' This  only holds if the two matrices are inverses to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Since these matrices are not square, one can use a Moore-  Penrose inverse, which is a generalisation of the tradi-  tional inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' We want Wd to be a right inverse of We,  0  50  100  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='15  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Test loss (computed for the last 10 time steps of the  simulation) for a regression learning task at varied network  sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  The training and testing datasets are sampled from  B(1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Averages are evaluated using 1000 test samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' for  training, 100 trajectories were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' The figure indicates that  the larger the network, the smaller the average loss and the  variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  defined as: Wd = W∗  e(WeW∗  e)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Here W∗  e denotes  a Hermitian transpose of We, however in our case it is  equivalent to a transpose, since We is defined over real  numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Appendix B: Neural network mappings  The mappings of functions that constitute the neural  network model defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' 3 are defined as (here we  consider input x ∈ Rn×1×d, a three-dimensional tensor,  and tensor dimension is counted starting from 1):  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' ψψψℓ : Rn×1×d → Rn×1×d, k = 3 mode product with  W ∈ Rd×h′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' Rn×1×d ×3 Rd×h′ ∈ Rn×1×h′ and  C ∈ Rh′×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' ψψψq1,ψψψq2 : Rn×1×d → Rn×1×h, k = 3 mode product  with W ∈ Rd×h: Rn×1×d ×3 Rd×h ∈ Rn×1×h and  C = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' x⊤1 : Rn×1×h → Rh×n×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' x⊤2 : Rn×1×h → Rh×1×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  �  ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2�  :  Rh×n×1 ×3 Rh×1×n  ∈  Rh×n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' ΦΦΦ ⊙  �  ψψψq1(x)⊤1 ×k ψψψq2(x)⊤2�  : Rn×n ⊙ Rh×n×1 ×3  Rh×1×n ∈ Rh×n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  Here an operator ⊙ denotes  a standard “broadcasted” element-wise multiplica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' vec(·): Rh×n×n → Rn2h×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' ψψψq3 : Rn2h×1 → Rn2h×1, W ∈ R1×h′′ and C ∈  Rh′′×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
+page_content=' vec−1(·): Rn2h×1 → Rn×nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtE4T4oBgHgl3EQfHAzF/content/2301.04900v1.pdf'}
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+Transition between metastable equilibra:
+applications to binary-choice games
+A. Antonov(1)∗, A. Leonidov(1,2), and A. Semenov(1,3)
+(1) P.N. Lebedev Physical Institute, Moscow, Russia
+(2) Moscow Institute of Physics and Technology, Dolgoprudny, Russia
+(3) Higher School of Economics, Moscow, Russia
+Abstract
+Transitions between metastable equilibria in the low-temperature phase
+of dynamical Ising game with activity spillover are studied in the infinite
+time limit.
+It is shown that exponential enhancement due to activity
+spillover previously found for finite-time transitions in [1] is absent in the
+infinite time limit. Analytical description for infinite time trajectory is
+developed and compared with results of exact numerical analysis.
+∗antonov@lpi.ru
+1
+arXiv:2301.01285v1  [cond-mat.stat-mech]  3 Jan 2023
+
+1. Introduction
+Studies of noisy binary choice games are of special interest because of the
+existence of close parallels to statistical physics of spin systems, in particular to
+static and dynamic properties of phase transitions in them [2, 3, 4]. These par-
+allels are particularly intriguing because of the fundamentally different origins
+of equilibria in game theory and statistical physics: in game theory equilibration
+is a result of balancing individual interests while in statistical physics equilibra-
+tion is a search of a global minimum of free energy. For the noisy binary choice
+problem on complete graphs it is long known, see [2] and references therein,
+that for a special choice of noise game-theoretic equilibria are characterised by
+the same mean-field Curie-Weiss equation as that describing phase transitions
+in magnetics, see e.g. [4]. The properties of static and dynamic equilibria in
+noisy binary choice games were studied in [5, 6, 7] for arbitrary noise, and com-
+plete and random graph topologies. It was established in particular that static
+game-theoretic equilibria in noisy binary choice games on graphs correspond to
+the so-called quantal response/expectation equilibria [8].
+The dynamics of games can, however, be fundamentally different from con-
+ventional spin dynamics due to a variety of possible mechanisms. One of these
+is a possibility of activity spillover (self-excitation) that was intensively studied
+for so-called Hawkes processes [9] with applications to finance [10, 11], earth-
+quakes [12] and other subjects, see the recent review in [13]. A master equation
+formalism for such processes was developed in [14, 15]. The effects of an ac-
+tivity spillover different from the Hawkes self-excitation mechanism for a noisy
+binary choice game (Ising game) on complete graphs was studied in [1]. The
+main focus of [1] was in studying transitions between metastable equilibria in
+the low-temperature phase taking place at finite time. It was observed that
+activity spillover leads to an exponential acceleration of such transitions. The
+present paper complements the analysis of [1] by studying transitions between
+metastable equilibria in the limit of infinite time. The importance of studying
+this limit is, first, in establishing a link with a rich literature on Kramers rate
+[16] and, second, in that in this limit the exponential enhancement is absent
+and an analysis of pre-exponential contribution is necessary. In analysing this
+problem we develop an analytical description of the infinite-limit trajectory and
+suggest an analytical formula for the transition rate that is compared with the
+results of exact numerical simulations.
+2. Model
+We consider a dynamical noisy binary choice game of N agents on a complete
+graph topology. Each agent i has two possible strategies si = ±1 so the system
+is fully described by the vector st = (s1, . . . , sn)t at given time t. The temporal
+evolution of the strategies configuration st → st+δt within a small time interval
+δt is assumed to be driven by a strategy flip si → −si of some agent i with the
+flip probability
+Prob[si → −si|(t; t + δt)] = λi(t)δt γi(si → −si|s−i,t)
+(1)
+where λi(t) is an activity rate of the agent i, i.e. λi(t)δt is a time-dependent
+probability for an agent i to be active and have a possibility to change a strategy
+2
+
+within a time interval (t, t + δt) while γ(si → −si|s−i,t) is a probability, for an
+active agent i, of a strategy flip dependent of the current configuration s−i,t of
+strategies in the neighbourhood of this node. In what follows we shall assume
+a noisy best response (Ising-Glauber) flip rate1. For a complete graph topology
+at large N, it is the same for all agents
+γ(m(t)) = 1
+2 [1 − si tanh (βJm(t))]
+→
+γ±(m(t)) = 1
+2 [1 ± tanh(βJm(t))]
+(2)
+where β = 1/T is an inverse temperature, J is an Ising coupling constant,
+γ± = γ(∓s → ±s) and m(t) =
+1
+N
+�N
+i=1 si. For the compete graph topology
+activity rates {λi} are also the same for all agents, λi(t) = λ(t) for any i.
+The time-dependence of the activity rate λ(t) is due to the spillover effect
+driven by the past events of strategy flips assumed to be described by the Hawkes
+process [9] with an exponential kernel:
+λ(t) = λ0 + µ
+N
+�
+τk<t
+e−b(t−τk),
+(3)
+where {τk} are times at which strategy flip of one of agents took place. The
+spillover effect described by (3) can be termed realised activity spillover.
+The time-dependent state of the system is described by the probability dis-
+tribution P(m, λ; t) and the character of its evolution depends on parameters
+λ0, µ, b, βJ. The particular case of µ = 0 corresponds to a standard Poisson
+dynamics of the system with constant intensity λ(t) = λ0 so that the probability
+distribution describing it is reduced to P(m(t); t). To investigate the effects of
+realised activity spillover, in what follows we compare the properties of Hawkes
+and Poisson dynamic games.
+In the limit N → ∞, the probability density function P(m, λ; t) obeys the
+Fokker-Planck equation derived in [1]
+∂tP
+=
+∂i(fiP) + 1
+N ∂i∂j (gijP)
+fi
+=
+�
+λ [m − tanh(βJm)]
+−λ [1 − m tanh(βJm)] + b [λ − λ0]
+�
+gij
+=
+�λ [1 − m tanh(βJm)]
+−λ [m − tanh(βJm)]
+−λ [m − tanh(βJm)]
+λ [1 − m tanh(βJm)]
+�
+,
+(4)
+where summation over repeated indices is assumed. Here and in what follows
+the indices i and j represent coordinates m and λ, and the following rescaling
+was performed:
+λ → 2λ
+µ , λ0 → 2λ0
+µ , b → 2b
+µ , t → µt
+2 .
+(5)
+The Fokker-Planck equation (4) describes Brownian motion in an external
+vector field fi in the plane (λ, m) subject to noise effects described by the
+matrix gij and corresponds to a mean field game-type description of the dynamic
+Ising game under consideration2. We also note that the considered system is
+1This choice corresponds to the Gumbel noise in the individual agents utilities.
+2The standard description of a mean field game includes, in addition to a Fokker-Planck
+equation, additional equations describing optimal control, see e.g. [17].
+3
+
+symmetric with respect to m-axis, and the external field is non-gradient, i.e.
+∂λfm ̸= ∂mfλ.
+In such a parametrisation, the process of realised activity spillover is con-
+trolled by a single memory kernel parameter b. In the special case of Poisson
+game with µ = 0 the rescaling in (5) is of course not relevant and the corre-
+sponding one-dimensional dynamics along the m axis can be studied by simply
+taking λ(t) ≡ λ0.
+Depending on parameters of the system, the considered system can have
+different equilibrium configurations (λeq, meq) given by the zeros of vector field
+fi = 0 corresponding to stable fixed points. As we can see from Eq. (4), for
+both Hawkes and Poisson games, for any time-dependent activity rate in the
+Hawkes game λ(t), the m-equilibria are described [1] by the same Curie-Weiss
+equation as in [2, 3, 5]
+meq = tanh(βJmeq).
+(6)
+For high temperatures βJ < 1 the system has one equilibrium meq = 0, and
+for low temperatures βJ > 1 it has two symmetrical (meta)stable equlibria at
+meq = ±m0(β) as well as the unstable one at m = 0 serving as a separatrix
+separating the two stable ones.
+The λ - equilibria are more complicated and depend on both temperature
+βJ and self-excitation memory kernel parameter b.
+In the high-temperature phase βJ < 1 and b > 1, we have the equilibrium
+configuration of the form
+m = 0, λ = bλ0
+b − 1
+while for b < 1 we have a blow-up solution with λ → ∞ for m = 0.
+In the low-temperature phase βJ > 1, three following modes are possible:
+• Mode 1 “calm agents”: if b > 1, then we have two (meta)stable equilibrium
+configurations at
+m = ±m0(β), λ =
+bλ0
+b − 1 + m2
+0(β) = ˜λ(m0)
+as well as the unstable saddle one at
+m = 0, λ = bλ0
+b − 1
+• Mode 2 “excited agents”: if 1 − m2
+0(β) < b < 1, then we still have equilib-
+rium configurations at
+m = ±m0(β), λ = ˜λ(m0),
+but the saddle configuration is now absent:
+m = 0, λ → ∞
+• Mode 3 “physcho agents”: if b < 1 − m2
+0(β), then
+λ → ∞
+for all extrema of the m-axis.
+4
+
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+βJ
+b
+Mode 2
+Mode 3
+Single equilibrium
+Mode 1
+Figure 1. Phase diagram of all possible modes in the (b, βJ) plane for λ0 = 1.
+The red area (single equilibrium) here denotes the presence of only equilibrium
+along the m-axis. Blue, yellow and green areas correspond to Modes 1, 2 and
+3, respectively (see the description in the main text).
+5
+
+The phase diagram showing the above modes is given in Fig. 1.
+At the timescale of τλ ∼ 1/b, in Modes 1 and 2 the system relaxes to the ap-
+propriate temperature-dependent equilibrium while the Mode 3 does not corre-
+spond to any equilibrium. The dependence of such a relaxation on temperature
+βJ and Hawkes parameter b in the Modes 1 and 2 was studied in [1].
+At low temperatures βJ > 1, the equilibrium configurations for Modes 1 and
+2 are in fact metastable due to noise-induced transitions of the type m0(β) ↔
+−m0(β) taking place at large timescale τ ≫ τλ.
+The saddle we introduced
+for Mode 1 then has the following physical meaning: it is the point where the
+transition trajectory from one equilibrium to another at the infinite time limit
+crosses the separatrix m = 0. [18]
+To consider these transitions, here and in what follows we fix βJ = 1.5 to
+establish the mode with two metastable equilibria (Mode 1 or 2, see Fig. 1).
+For our convenience, in what follows we shall consider the transition −m0(β) →
+m0(β).
+3. Transition between metastable equilibria
+3.1. Long-time behaviour of probability density function
+The subject of our study is a comparison of the transition probability be-
+tween the states (m(ta), λ(ta)) and (m(tb), λ(tb)) within the time interval [ta, tb]
+for Hawkes and Poisson Ising games. In what follows we shall use a condensed
+notation xa,b = (m(ta,b), λ(ta,b)) and fix [ta, tb] = [0, τ] so that the transition
+probability between two metastable states is
+P(xb, t|xa, 0) ≡ P(xb, t)
+���
+x(0)=xa
+(7)
+where m(0) = −m0(β), λ(0) = ˜λ(m0) and m(τ) = m0(β), λ(τ) = ˜λ(m0). The
+transition probability (7) obeys [1] the Fokker-Planck equation (4).
+In the previous paper we have compared the probabilities of transition be-
+tween metastable equilibria in Hawkes and Poisson Ising games within a finite
+time interval [0, τ] and demonstrated an exponential acceleration of this tran-
+sition in the Hawkes case. The main goal of the present paper is to calculate
+this transition probability in the limit τ → ∞. To discuss this limit let us use,
+following [1], the analogy with classical mechanics. A formal justification for it
+can be found, e.g., in [19].
+As the diffusion coefficient in (4) is proportional to 1/N, in the limit of N →
+∞, for solving the Fokker-Planck equation we can use the WKB approximation.
+Introducing an analogue of action S(x, t) through P(x, t) ∝ e−NS(x,t), we get
+the following Hamilton-Jacobi equation for S:
+∂tS(x, t) = fi(x(t))∂S(x, t)
+∂xi
+− gij(x(t))∂S(x, t)
+∂xi
+∂S(x, t)
+∂xj
+.
+(8)
+On can also introduce an analogue of the Hamiltonian
+H(p, x; t) = −fi(x(t))pi(t) + gij(x(t))pi(t)pj(t),
+pi = ∂S
+∂xi
+(9)
+6
+
+The time evolution of the system is then given by the corresponding Hamilton
+equations
+˙xi(t) + fi(x(t))
+=
+2gij(x(t))pj(t)
+˙pi(t) − pj∂ifj(x(t))
+=
+−pj∂igjk(x(t))pk(t).
+(10)
+The system of Hamilton equations (10) has the first integral H(p, x) = E. As
+will be shown later, the value of E implicitly sets conditions on the transition
+time τ from one metastable equilibrium to another in the classical problem.
+The leading contribution to the transition probability has the form
+P(xi, xf; τ) ∝ e−NS
+(11)
+where the exponential factor S can be calculated by implementing the Mopertui
+principle [20]
+S = S0 − Eτ =
+�
+i
+� ∞
+0
+pi(t) ˙xi(t)dt − Eτ =
+�
+i
+�
+trajectory
+pidxi − Eτ.
+(12)
+The transition trajectory itself is determined by equations (10), the first
+integral H(p, x) = E and, obviously, should minimise the trajectory-depended
+term S0. Transition time is set by E via relation τ = ∂S0/∂E. [20]
+In [1] we considered transition probability from one metastable equilibrium
+to another in finite time (E ̸= 0) and found out that the probability exponen-
+tially increases due to activity spillover. In the present study we augment the
+results of [1] by considering introducing transition rates in the infinite time limit
+corresponding to E = 0.
+The system of differential equation (10) for E = 0 is solvable in quadratures.
+The corresponding solution for the transition trajectory can naturally be broken
+into two pieces.
+The first piece corresponding to transition from the initial equilibrium to
+separatrix −m0(β) → 0. The corresponding formulae read
+˙m(t)
+=
+λ(t)[m − tanh(βJm)],
+(13)
+˙λ(t)
+=
+λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0)
+−
+λ(t)[m − tanh(βJm)]2
+1 − m tanh(βJm)
+,
+(14)
+pm
+=
+m − tanh(βJm)
+1 − m tanh(βJm),
+(15)
+pλ
+=
+0.
+(16)
+The second piece corresponding to transition from the separatrix to another
+equilibrium 0 → m0(β) . The corresponding formulae read
+˙m(t)
+=
+−λ(t)[m − tanh(βJm)],
+(17)
+˙λ(t)
+=
+λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0),
+(18)
+pm
+=
+0,
+(19)
+pλ
+=
+0.
+(20)
+7
+
+We note that despite the symmetry with respect to m-axis, the transition tra-
+jectory is asymmetric as the external field is non-gradient.
+In accordance with the classification of modes introduced in Section 2, for
+different values of the parameter b the Hawkes transition trajectory does either
+pass through the saddle point where it has the discontinuity (Mode 1) or di-
+verges at the separatrix m = 0 (Mode 2). The trajectories for various values of
+parameter b are shown in Fig. 2.
+1
+2
+3
+4
+5
+6
+7
+−1
+0
+m0
+−m0
+1
+λ
+m
+b = 2.0
+b = 1.5
+b = 1.2
+b = 0.5
+Figure 2.
+Transition trajectories
+−m0(β) → m0(β) at the infinite time
+limit E = 0 given by Eqs. 13-16 (left half) and Eqs. 17-20 (right half) for
+b = 0.5, 1.2, 1.5, 2.0 at βJ = 1.5, λ0 = 1.
+The trajectories for Mode 1
+(b = 1.2, 1.5, 2.0) are defined and have a discontinuity at the saddle, and the
+trajectory for Mode 2 (b = 0.5) diverges at m = 0.
+From Eqs. (12),(13-20) it follows that in the infinite time limit for which
+E = 0, the exponential factor S for Poisson and Hawkes Ising games is the
+same is equal for Poisson and Hawkes games for all b:
+S =
+0
+�
+−m0(β)
+m − tanh(βJm)
+1 − m tanh(βJm)dm
+(21)
+Therefore, for understanding a possible difference between the Hawkes and
+Poisson Ising games in the infinite time limit, an analysis of pre-exponential
+factor of the transition rate is required.
+8
+
+3.2. Pre-exponential factor of the transition rate
+The calculation of the pre-exponential factor for the one-dimensional Pois-
+son game closely follows the original calculation by Kramers [16] and can be
+done analytically, see e.g. [21, 22]. A more general result for larger number
+of dimensions, including the case of non-potential fields, was obtained in [23].
+However, this result is not applicable in our the two-dimensional Hawkes game,
+since the transition trajectory in the non-gradient field has a discontinuity, see
+a related discussion in [24].
+When the trajectory is defined (Mode 1), we can use analogies with one-
+dimensional motion. In the Kramers’ problem for the potential with smooth
+barrier the pre-exponential factor of escape rate depends on second derivatives
+of the potential both for stationary attractor and a saddle.
+However, if the
+potential barrier is edge-shaped, the result depends only on the second derivative
+of the potential at stationary attractor.[25]. That leads us to an assumption that
+in the Hawkes game acceleration with respect to Poisson one is caused only by
+a corresponding change in the activity of agents in the equilibrium state, with
+the rest of motion having non-significant effect on the transition time. That
+means average transition times in the Hawkes and Poisson games with intensity
+˜λ(m0) are equal. Therefore a ratio of transition times in the original Hawkes
+and Poisson games can be written in the following form:
+ttr,P
+ttr,H
+≃
+b
+b − 1 + m2
+0(β).
+(22)
+To check the above-formulated assumption we have performed computer
+simulations of Hawkes and Poisson games as well as those of Langevin equations
+that correspond to Eq. 4. We have also checked that the transition time ratio
+does not depend on number of agents for N ≥ 20, i.e. when the number of
+agents is sufficiently large. A comparison of the results of these simulations
+with Eq. 22 is shown in Fig. 3.
+From Fig. 3 we see that the activity in the Hawkes game as compared to the
+Poisson one is indeed enhanced. A more detailed conclusion is that in the regime
+corresponding to Mode 1 the formula in Eq. (22) works well for the Mode 1 for
+both continuum and discrete cases, but in the regime corresponding to Mode
+2 it is, due to the presence of divergence the continuum generalisation of the
+game, not in agreement with the exact discrete formulation. Despite this, the
+shape of the transition trajectory still provides us a qualitatively correct insight
+into the behaviour of agents, see Fig. 4.
+Let us note that the decision process does significantly intensify around the
+separatrix, i.e. when are uncertain of which of the two (quasi)stable equilibria
+to choose. Once the decision is made, the agents calm down.
+4. Conclusions
+We have studied the self-excited Ising game on a complete graph. Inspite
+of its simplicity, it has rich dynamics exhibiting various types of behaviour.
+Competition of “calming down” and “activation” in the Hawkes self-excitation
+mechanism at different levels of noise results in three possible modes (phases).
+9
+
+1
+1.2
+1.4
+1.6
+1.8
+2
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+ttr,P/ttr,H
+b
+game
+langevin
+analytics
+Mode 1
+Mode 2
+Figure 3.
+Ratio of transition times in Hawkes and Poisson cases.
+Triangles
+show simulation results for games (discrete model), and squares show results
+for Langevin equations (continuum model). The line refers to the theoretical
+prediction given by Eq. (22). Dashed line b = 1 separates Mode 1 (blue area)
+from Mode 2 (yellow area).
+10
+
+2
+4
+6
+8
+10
+12
+14
+−1
+0
+1
+λ
+m
+Analytics
+Agent model
+Figure 4. Example of transition in Hawkes game (red line) for b = 0.5, λ0 =
+1, N = 20. As in the corresponding transition trajectory (blue line), the inten-
+sity of decision making process increases near m = 0.
+We expect that this competition might play an important role in other situ-
+ations, e.g. for non-exponential Hawkes kernels [26] or for more complicated
+graph topology.
+Another focus in this work was to investigate the probability of transition
+between metastable equilibria in the infinite time limit. This is a very challeng-
+ing task for a multi-dimensional case when the external field is non-gradient and
+has a discontinuity. Also, since in the relevant one-dimensional case (i.e. when
+the potential field only has the discontinuity) it is known that the dynamics for
+such fields is rather different that for smooth potential fields [27], it would be
+natural to assume a similar situation in the multi-dimensional case. However,
+based on the intuitive understanding of the considered model, we have presented
+an approach that allows us to reduce the problem to calculating the transition
+time in the corresponding one-dimensional model. The analytically calculated
+transition trajectory also gave us a qualitative insight into the behaviour of
+agents in the corresponding discrete system.
+As for further developments of the suggested approach, an interesting idea
+would be working out its generalisation for two- and multi-dimensional systems.
+Compared to other another existing approaches for treating the case of non-
+gradient external field (see e.g. [28, 29]), this newly introduced method could
+present a workable alternative due to its simplicity.
+11
+
+References
+[1] A. Antonov, A. Leonidov, and A. Semenov. Self-excited ising game. Physica
+A: Statistical Mechanics and its Applications, 561:125305, 2021.
+[2] Lawrence Blume and Steven Durlauf. Equilibrium concepts for social inter-
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+[3] Jean-Philippe Bouchaud. Crises and collective socio-economics phenomena:
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+2013.
+[4] Silvio Salinas. Introduction to statistical physics. Springer Science & Busi-
+ness Media, 2001.
+[5] Andrey Leonidov, Alexey Savvateev, and Andrew G Semenov.
+Quan-
+tal response equilibria in binary choice games on graphs. arXiv preprint
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+[6] A. Leonidov, A. Savvateev, and A. Semenov. Qre in the ising game. CEUR
+Workshop proceedings, 2020.
+[7] Andrey Leonidov, Alexey Savvateev, and Andrew G Semenov. Ising game
+on graphs. arXiv preprint arXiv:2108.00824, 2021.
+[8] Jacob K Goeree, Charles A Holt, and Thomas R Palfrey. Quantal response
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+[11] Stephen J. Hardiman, Nicolas Bercot, and Jean-Philippe Bouchaud. Criti-
+cal reflexivity in financial markets: a hawkes process analysis. The European
+Physical Journal B, 86:442, 2013.
+[12] Yosihiko Ogata. Statistical models for earthquake occurrences and residual
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+tion, 83(401):9–27, 1988.
+[13] Patrick J. Laub, Thomas Taimre, and Philip K. Pollett. Hawkes processes,
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+[14] Kiyoshi Kanazawa and Didier Sornette. Field master equation theory of
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+[15] Kiyoshi Kanazawa and Didier Sornette.
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+12
+
+[17] Jean-Michel Lasry and Pierre-Louis Lions. Mean field games. Japanese
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+[18] Haidong Feng, Kun Zhang, and Jin Wang. Non-equilibrium transition state
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+[19] V. P. Maslov and M. V. Fedoriuk. Semiclassical Approximation in Quantum
+Mechanics. Reidel, Dordrecht, 1981.
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+Mechanics. Vol. 1.
+Butterworth-
+Heinemann, 1976.
+[21] B. Caroli, C. Caroli, and B. Roulet.
+Diffusion in a bistable potential:
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+Journal of Statistical Physics, 26:83–
+111, 1981.
+[22] Sidney Coleman. Aspects of Symmetry: Selected Erice Lectures. Cambridge
+University Press, 1985.
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+transition rate formula to irreversible diffusion processes. Annales Henri
+Poincar´e, 17:3499–3532, 2016.
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+puting the quasi-potential. Journal of Scientific Computing, 75, 2018.
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+approach to kramers’ diffusion problem. SIAM Journal on Applied Math-
+ematics, 42(4):835–849, 1982.
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+Gould. Trades, Quotes and Prices: Financial Markets Under the Micro-
+scope. Sect. 9.3.4. Cambridge University Press, 2018.
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+tions, 136(1):124–146, 1986.
+[28] Nicholas Paskal and Maria Cameron. An efficient jet marcher for computing
+the quasipotential for 2d sdes, 2021.
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+and Krasimira Tsaneva-Atanasova.
+Quasipotentials for coupled escape problems and the gate-height bifurca-
+tion, 2022.
+13
+

KNAzT4oBgHgl3EQfVfxu/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf,len=355
+page_content='Transition between metastable equilibra:  applications to binary-choice games  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Antonov(1)∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Leonidov(1,2), and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Semenov(1,3)  (1) P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Lebedev Physical Institute, Moscow, Russia  (2) Moscow Institute of Physics and Technology, Dolgoprudny, Russia  (3) Higher School of Economics, Moscow, Russia  Abstract  Transitions between metastable equilibria in the low-temperature phase  of dynamical Ising game with activity spillover are studied in the infinite  time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  It is shown that exponential enhancement due to activity  spillover previously found for finite-time transitions in [1] is absent in the  infinite time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Analytical description for infinite time trajectory is  developed and compared with results of exact numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  ∗antonov@lpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='ru  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='01285v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='stat-mech]  3 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Introduction  Studies of noisy binary choice games are of special interest because of the  existence of close parallels to statistical physics of spin systems, in particular to  static and dynamic properties of phase transitions in them [2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' These par-  allels are particularly intriguing because of the fundamentally different origins  of equilibria in game theory and statistical physics: in game theory equilibration  is a result of balancing individual interests while in statistical physics equilibra-  tion is a search of a global minimum of free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' For the noisy binary choice  problem on complete graphs it is long known, see [2] and references therein,  that for a special choice of noise game-theoretic equilibria are characterised by  the same mean-field Curie-Weiss equation as that describing phase transitions  in magnetics, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The properties of static and dynamic equilibria in  noisy binary choice games were studied in [5, 6, 7] for arbitrary noise, and com-  plete and random graph topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' It was established in particular that static  game-theoretic equilibria in noisy binary choice games on graphs correspond to  the so-called quantal response/expectation equilibria [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The dynamics of games can, however, be fundamentally different from con-  ventional spin dynamics due to a variety of possible mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' One of these  is a possibility of activity spillover (self-excitation) that was intensively studied  for so-called Hawkes processes [9] with applications to finance [10, 11], earth-  quakes [12] and other subjects, see the recent review in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' A master equation  formalism for such processes was developed in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The effects of an ac-  tivity spillover different from the Hawkes self-excitation mechanism for a noisy  binary choice game (Ising game) on complete graphs was studied in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The  main focus of [1] was in studying transitions between metastable equilibria in  the low-temperature phase taking place at finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' It was observed that  activity spillover leads to an exponential acceleration of such transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The  present paper complements the analysis of [1] by studying transitions between  metastable equilibria in the limit of infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The importance of studying  this limit is, first, in establishing a link with a rich literature on Kramers rate  [16] and, second, in that in this limit the exponential enhancement is absent  and an analysis of pre-exponential contribution is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In analysing this  problem we develop an analytical description of the infinite-limit trajectory and  suggest an analytical formula for the transition rate that is compared with the  results of exact numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Model  We consider a dynamical noisy binary choice game of N agents on a complete  graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Each agent i has two possible strategies si = ±1 so the system  is fully described by the vector st = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' , sn)t at given time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The temporal  evolution of the strategies configuration st → st+δt within a small time interval  δt is assumed to be driven by a strategy flip si → −si of some agent i with the  flip probability  Prob[si → −si|(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' t + δt)] = λi(t)δt γi(si → −si|s−i,t)  (1)  where λi(t) is an activity rate of the agent i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' λi(t)δt is a time-dependent  probability for an agent i to be active and have a possibility to change a strategy  2  within a time interval (t, t + δt) while γ(si → −si|s−i,t) is a probability, for an  active agent i, of a strategy flip dependent of the current configuration s−i,t of  strategies in the neighbourhood of this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In what follows we shall assume  a noisy best response (Ising-Glauber) flip rate1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' For a complete graph topology  at large N, it is the same for all agents  γ(m(t)) = 1  2 [1 − si tanh (βJm(t))]  →  γ±(m(t)) = 1  2 [1 ± tanh(βJm(t))]  (2)  where β = 1/T is an inverse temperature, J is an Ising coupling constant,  γ± = γ(∓s → ±s) and m(t) =  1  N  �N  i=1 si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' For the compete graph topology  activity rates {λi} are also the same for all agents, λi(t) = λ(t) for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The time-dependence of the activity rate λ(t) is due to the spillover effect  driven by the past events of strategy flips assumed to be described by the Hawkes  process [9] with an exponential kernel:  λ(t) = λ0 + µ  N  �  τk<t  e−b(t−τk),  (3)  where {τk} are times at which strategy flip of one of agents took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The  spillover effect described by (3) can be termed realised activity spillover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The time-dependent state of the system is described by the probability dis-  tribution P(m, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' t) and the character of its evolution depends on parameters  λ0, µ, b, βJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The particular case of µ = 0 corresponds to a standard Poisson  dynamics of the system with constant intensity λ(t) = λ0 so that the probability  distribution describing it is reduced to P(m(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' To investigate the effects of  realised activity spillover, in what follows we compare the properties of Hawkes  and Poisson dynamic games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In the limit N → ∞, the probability density function P(m, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' t) obeys the  Fokker-Planck equation derived in [1]  ∂tP  =  ∂i(fiP) + 1  N ∂i∂j (gijP)  fi  =  �  λ [m − tanh(βJm)]  −λ [1 − m tanh(βJm)] + b [λ − λ0]  �  gij  =  �λ [1 − m tanh(βJm)]  −λ [m − tanh(βJm)]  −λ [m − tanh(βJm)]  λ [1 − m tanh(βJm)]  �  ,  (4)  where summation over repeated indices is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Here and in what follows  the indices i and j represent coordinates m and λ, and the following rescaling  was performed:  λ → 2λ  µ , λ0 → 2λ0  µ , b → 2b  µ , t → µt  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (5)  The Fokker-Planck equation (4) describes Brownian motion in an external  vector field fi in the plane (λ, m) subject to noise effects described by the  matrix gij and corresponds to a mean field game-type description of the dynamic  Ising game under consideration2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' We also note that the considered system is  1This choice corresponds to the Gumbel noise in the individual agents utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  2The standard description of a mean field game includes, in addition to a Fokker-Planck  equation, additional equations describing optimal control, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  3  symmetric with respect to m-axis, and the external field is non-gradient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  ∂λfm ̸= ∂mfλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In such a parametrisation, the process of realised activity spillover is con-  trolled by a single memory kernel parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In the special case of Poisson  game with µ = 0 the rescaling in (5) is of course not relevant and the corre-  sponding one-dimensional dynamics along the m axis can be studied by simply  taking λ(t) ≡ λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Depending on parameters of the system, the considered system can have  different equilibrium configurations (λeq, meq) given by the zeros of vector field  fi = 0 corresponding to stable fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' As we can see from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' (4), for  both Hawkes and Poisson games, for any time-dependent activity rate in the  Hawkes game λ(t), the m-equilibria are described [1] by the same Curie-Weiss  equation as in [2, 3, 5]  meq = tanh(βJmeq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (6)  For high temperatures βJ < 1 the system has one equilibrium meq = 0, and  for low temperatures βJ > 1 it has two symmetrical (meta)stable equlibria at  meq = ±m0(β) as well as the unstable one at m = 0 serving as a separatrix  separating the two stable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The λ - equilibria are more complicated and depend on both temperature  βJ and self-excitation memory kernel parameter b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In the high-temperature phase βJ < 1 and b > 1, we have the equilibrium  configuration of the form  m = 0, λ = bλ0  b − 1  while for b < 1 we have a blow-up solution with λ → ∞ for m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In the low-temperature phase βJ > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' three following modes are possible:  Mode 1 “calm agents”: if b > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' then we have two (meta)stable equilibrium  configurations at  m = ±m0(β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' λ =  bλ0  b − 1 + m2  0(β) = ˜λ(m0)  as well as the unstable saddle one at  m = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' λ = bλ0  b − 1  Mode 2 “excited agents”: if 1 − m2  0(β) < b < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' then we still have equilib-  rium configurations at  m = ±m0(β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' λ = ˜λ(m0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  but the saddle configuration is now absent:  m = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' λ → ∞  Mode 3 “physcho agents”: if b < 1 − m2  0(β),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' then  λ → ∞  for all extrema of the m-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2  βJ  b  Mode 2  Mode 3  Single equilibrium  Mode 1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Phase diagram of all possible modes in the (b, βJ) plane for λ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The red area (single equilibrium) here denotes the presence of only equilibrium  along the m-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Blue, yellow and green areas correspond to Modes 1, 2 and  3, respectively (see the description in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  5  The phase diagram showing the above modes is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  At the timescale of τλ ∼ 1/b, in Modes 1 and 2 the system relaxes to the ap-  propriate temperature-dependent equilibrium while the Mode 3 does not corre-  spond to any equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The dependence of such a relaxation on temperature  βJ and Hawkes parameter b in the Modes 1 and 2 was studied in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  At low temperatures βJ > 1, the equilibrium configurations for Modes 1 and  2 are in fact metastable due to noise-induced transitions of the type m0(β) ↔  −m0(β) taking place at large timescale τ ≫ τλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The saddle we introduced  for Mode 1 then has the following physical meaning: it is the point where the  transition trajectory from one equilibrium to another at the infinite time limit  crosses the separatrix m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [18]  To consider these transitions, here and in what follows we fix βJ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5 to  establish the mode with two metastable equilibria (Mode 1 or 2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  For our convenience, in what follows we shall consider the transition −m0(β) →  m0(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Transition between metastable equilibria  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Long-time behaviour of probability density function  The subject of our study is a comparison of the transition probability be-  tween the states (m(ta), λ(ta)) and (m(tb), λ(tb)) within the time interval [ta, tb]  for Hawkes and Poisson Ising games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In what follows we shall use a condensed  notation xa,b = (m(ta,b), λ(ta,b)) and fix [ta, tb] = [0, τ] so that the transition  probability between two metastable states is  P(xb, t|xa, 0) ≡ P(xb, t)  ���  x(0)=xa  (7)  where m(0) = −m0(β), λ(0) = ˜λ(m0) and m(τ) = m0(β), λ(τ) = ˜λ(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The  transition probability (7) obeys [1] the Fokker-Planck equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In the previous paper we have compared the probabilities of transition be-  tween metastable equilibria in Hawkes and Poisson Ising games within a finite  time interval [0, τ] and demonstrated an exponential acceleration of this tran-  sition in the Hawkes case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The main goal of the present paper is to calculate  this transition probability in the limit τ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' To discuss this limit let us use,  following [1], the analogy with classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' A formal justification for it  can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=', in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  As the diffusion coefficient in (4) is proportional to 1/N, in the limit of N →  ∞, for solving the Fokker-Planck equation we can use the WKB approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Introducing an analogue of action S(x, t) through P(x, t) ∝ e−NS(x,t), we get  the following Hamilton-Jacobi equation for S:  ∂tS(x, t) = fi(x(t))∂S(x, t)  ∂xi  − gij(x(t))∂S(x, t)  ∂xi  ∂S(x, t)  ∂xj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (8)  On can also introduce an analogue of the Hamiltonian  H(p, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' t) = −fi(x(t))pi(t) + gij(x(t))pi(t)pj(t),  pi = ∂S  ∂xi  (9)  6  The time evolution of the system is then given by the corresponding Hamilton  equations  ˙xi(t) + fi(x(t))  =  2gij(x(t))pj(t)  ˙pi(t) − pj∂ifj(x(t))  =  −pj∂igjk(x(t))pk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (10)  The system of Hamilton equations (10) has the first integral H(p, x) = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' As  will be shown later, the value of E implicitly sets conditions on the transition  time τ from one metastable equilibrium to another in the classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The leading contribution to the transition probability has the form  P(xi, xf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' τ) ∝ e−NS  (11)  where the exponential factor S can be calculated by implementing the Mopertui  principle [20]  S = S0 − Eτ =  �  i  � ∞  0  pi(t) ˙xi(t)dt − Eτ =  �  i  �  trajectory  pidxi − Eτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (12)  The transition trajectory itself is determined by equations (10), the first  integral H(p, x) = E and, obviously, should minimise the trajectory-depended  term S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Transition time is set by E via relation τ = ∂S0/∂E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [20]  In [1] we considered transition probability from one metastable equilibrium  to another in finite time (E ̸= 0) and found out that the probability exponen-  tially increases due to activity spillover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In the present study we augment the  results of [1] by considering introducing transition rates in the infinite time limit  corresponding to E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The system of differential equation (10) for E = 0 is solvable in quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The corresponding solution for the transition trajectory can naturally be broken  into two pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The first piece corresponding to transition from the initial equilibrium to  separatrix −m0(β) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The corresponding formulae read  ˙m(t)  =  λ(t)[m − tanh(βJm)],  (13)  ˙λ(t)  =  λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0)  −  λ(t)[m − tanh(βJm)]2  1 − m tanh(βJm)  ,  (14)  pm  =  m − tanh(βJm)  1 − m tanh(βJm),  (15)  pλ  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (16)  The second piece corresponding to transition from the separatrix to another  equilibrium 0 → m0(β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The corresponding formulae read  ˙m(t)  =  −λ(t)[m − tanh(βJm)],  (17)  ˙λ(t)  =  λ(t)[1 − m tanh(βJm)] − b(λ(t) − λ0),  (18)  pm  =  0,  (19)  pλ  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (20)  7  We note that despite the symmetry with respect to m-axis, the transition tra-  jectory is asymmetric as the external field is non-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  In accordance with the classification of modes introduced in Section 2, for  different values of the parameter b the Hawkes transition trajectory does either  pass through the saddle point where it has the discontinuity (Mode 1) or di-  verges at the separatrix m = 0 (Mode 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The trajectories for various values of  parameter b are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  1  2  3  4  5  6  7  −1  0  m0  −m0  1  λ  m  b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='0  b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2  b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Transition trajectories  −m0(β) → m0(β) at the infinite time  limit E = 0 given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 13-16 (left half) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 17-20 (right half) for  b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='0 at βJ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5, λ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  The trajectories for Mode 1  (b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='0) are defined and have a discontinuity at the saddle, and the  trajectory for Mode 2 (b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5) diverges at m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' (12),(13-20) it follows that in the infinite time limit for which  E = 0, the exponential factor S for Poisson and Hawkes Ising games is the  same is equal for Poisson and Hawkes games for all b:  S =  0  �  −m0(β)  m − tanh(βJm)  1 − m tanh(βJm)dm  (21)  Therefore, for understanding a possible difference between the Hawkes and  Poisson Ising games in the infinite time limit, an analysis of pre-exponential  factor of the transition rate is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Pre-exponential factor of the transition rate  The calculation of the pre-exponential factor for the one-dimensional Pois-  son game closely follows the original calculation by Kramers [16] and can be  done analytically, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' A more general result for larger number  of dimensions, including the case of non-potential fields, was obtained in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  However, this result is not applicable in our the two-dimensional Hawkes game,  since the transition trajectory in the non-gradient field has a discontinuity, see  a related discussion in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  When the trajectory is defined (Mode 1), we can use analogies with one-  dimensional motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' In the Kramers’ problem for the potential with smooth  barrier the pre-exponential factor of escape rate depends on second derivatives  of the potential both for stationary attractor and a saddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  However, if the  potential barrier is edge-shaped, the result depends only on the second derivative  of the potential at stationary attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' That leads us to an assumption that  in the Hawkes game acceleration with respect to Poisson one is caused only by  a corresponding change in the activity of agents in the equilibrium state, with  the rest of motion having non-significant effect on the transition time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' That  means average transition times in the Hawkes and Poisson games with intensity  ˜λ(m0) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Therefore a ratio of transition times in the original Hawkes  and Poisson games can be written in the following form:  ttr,P  ttr,H  ≃  b  b − 1 + m2  0(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  (22)  To check the above-formulated assumption we have performed computer  simulations of Hawkes and Poisson games as well as those of Langevin equations  that correspond to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' We have also checked that the transition time ratio  does not depend on number of agents for N ≥ 20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' when the number of  agents is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' A comparison of the results of these simulations  with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 22 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 3 we see that the activity in the Hawkes game as compared to the  Poisson one is indeed enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' A more detailed conclusion is that in the regime  corresponding to Mode 1 the formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' (22) works well for the Mode 1 for  both continuum and discrete cases, but in the regime corresponding to Mode  2 it is, due to the presence of divergence the continuum generalisation of the  game, not in agreement with the exact discrete formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Despite this, the  shape of the transition trajectory still provides us a qualitatively correct insight  into the behaviour of agents, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Let us note that the decision process does significantly intensify around the  separatrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' when are uncertain of which of the two (quasi)stable equilibria  to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Once the decision is made, the agents calm down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Conclusions  We have studied the self-excited Ising game on a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Inspite  of its simplicity, it has rich dynamics exhibiting various types of behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Competition of “calming down” and “activation” in the Hawkes self-excitation  mechanism at different levels of noise results in three possible modes (phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  9  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='8  2  ttr,P/ttr,H  b  game  langevin  analytics  Mode 1  Mode 2  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Ratio of transition times in Hawkes and Poisson cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Triangles  show simulation results for games (discrete model), and squares show results  for Langevin equations (continuum model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The line refers to the theoretical  prediction given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Dashed line b = 1 separates Mode 1 (blue area)  from Mode 2 (yellow area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  10  2  4  6  8  10  12  14  −1  0  1  λ  m  Analytics  Agent model  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Example of transition in Hawkes game (red line) for b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='5, λ0 =  1, N = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' As in the corresponding transition trajectory (blue line), the inten-  sity of decision making process increases near m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  We expect that this competition might play an important role in other situ-  ations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' for non-exponential Hawkes kernels [26] or for more complicated  graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Another focus in this work was to investigate the probability of transition  between metastable equilibria in the infinite time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' This is a very challeng-  ing task for a multi-dimensional case when the external field is non-gradient and  has a discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Also, since in the relevant one-dimensional case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' when  the potential field only has the discontinuity) it is known that the dynamics for  such fields is rather different that for smooth potential fields [27], it would be  natural to assume a similar situation in the multi-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' However,  based on the intuitive understanding of the considered model, we have presented  an approach that allows us to reduce the problem to calculating the transition  time in the corresponding one-dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' The analytically calculated  transition trajectory also gave us a qualitative insight into the behaviour of  agents in the corresponding discrete system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  As for further developments of the suggested approach, an interesting idea  would be working out its generalisation for two- and multi-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Compared to other another existing approaches for treating the case of non-  gradient external field (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' [28, 29]), this newly introduced method could  present a workable alternative due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
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+page_content=' Leonidov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Semenov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Self-excited ising game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
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+page_content=' Crises and collective socio-economics phenomena:  Simple models and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
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+page_content='  Quan-  tal response equilibria in binary choice games on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
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+page_content=' A singular perturbation  approach to kramers’ diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' SIAM Journal on Applied Math-  ematics, 42(4):835–849, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  [26] Jean-Philippe Bouchaud, Julius Bonart, Jonathan Donier, and Martin  Gould.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Trades, Quotes and Prices: Financial Markets Under the Micro-  scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Cambridge University Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Dekker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Kramers’ activation rate for a sharp edged potential barrier:  The double oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' Physica A: Statistical Mechanics and its Applica-  tions, 136(1):124–146, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  [28] Nicholas Paskal and Maria Cameron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content=' An efficient jet marcher for computing  the quasipotential for 2d sdes, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  [29] Peter Ashwin,  Jennifer Creaser,  and Krasimira Tsaneva-Atanasova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  Quasipotentials for coupled escape problems and the gate-height bifurca-  tion, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNAzT4oBgHgl3EQfVfxu/content/2301.01285v1.pdf'}

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+MNRAS 000, 1–9 (2021)
+Preprint 4 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Dissecting the active galactic nucleus in Circinus IV. MUSE NFM
+observations unveil a tuning-fork ionised outflow morphology
+D. Kakkad1,★ M. Stalevski2,3, M. Kishimoto4, S. Knežević2, D. Asmus5,6, F. P. A. Vogt7
+1Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+2Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia
+3Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281-S9, Gent, 9000, Belgium
+4Department of Astrophysics & Atmospheric Sciences, Faculty of Science, Kyoto Sangyo University, Kamigamo-motoyama, Kita-ku, Kyoto 603-8555, Japan
+5Department of Physics & Astronomy, University of Southampton, Southampton, SO17 1BJ, UK
+6Gymnasium Schwarzenbek, 21493 Schwarzenbek, Germany
+7Federal Office of Meteorology and Climatology - MeteoSwiss, Chemin de l’Aérologie 1, 1530 Payerne, Switzerland
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present the ionised gas outflow morphology in the Circinus galaxy using the Narrow Field Mode (NFM) of the MUSE
+instrument on board the Very Large Telescope (VLT). The NFM observations provide a spatial resolution of ∼0.1′′, corresponding
+to a physical scale of ∼2 pc, one of the highest spatial resolution achievable using ground-based AO-assisted observations in the
+optical wavelengths. The MUSE observations reveal a collimated clumpy outflow profile originating near the AGN location and
+extending up to 1.5′′ (∼30 pc) in the NW direction. The collimated structure then fragments into two filaments, giving the entire
+outflowing gas a “tuning-fork” morphology. These structures remain undetected in the lower spatial resolution MUSE Wide
+Field Mode data. We explain the origin of this tuning-fork structure to the interaction of the outflow with a dense clump in the
+interstellar medium (ISM) as the outflow propagates outward. The origin of the collimated structure itself could be from jet-ISM
+interactions on small scales. These observations also provide evidence to the origin of the ionised gas filaments previously
+observed in the Circinus galaxy out to kiloparsec scales. We find instantaneous and time-averaged mass outflow rates of 10−2
+M⊙ yr−1 and 10−4 M⊙ yr−1, respectively. Based on the star formation rate in the Circinus galaxy reported in the literature, the
+observed ionised outflows are not expected to regulate star formation within the ∼100 pc scales probed by the NFM data.
+Key words: galaxies:active – galaxies:individual – galaxies:ISM – galaxies: kinematics and dynamics – galaxies: nuclei –
+galaxies: Seyfert
+1 INTRODUCTION
+The so-called Unified Model (UM) of the active galactic nuclei
+(AGN) consists of a central black hole surrounded by an equato-
+rial torus-like structure, which is responsible for the angle-dependent
+obscuration of the accretion disk and in some cases, may include col-
+limated jets along the polar directions (e.g., Antonucci 1993; Urry
+& Padovani 1995; Netzer 2015). The equatorial torus has been be-
+lieved to dominate the infrared emission from the AGN (see Ramos
+Almeida & Ricci 2017, and the references therein). There have been
+intensive efforts in the literature, both from an observational as well
+as modelling perspective, to study the nature of this torus, such as
+its geometry (e.g., Hönig 2019; García-Burillo et al. 2021), what
+are the typical dust covering factors (e.g., Elitzur 2012; Stalevski
+et al. 2016; Toba et al. 2021) and whether the material is clumpy or
+smooth (e.g., Dullemond & van Bemmel 2005; Marin et al. 2015;
+García-González et al. 2017).
+High resolution mid-infrared observations over the past decade
+have now challenged these simplified torus model that have dusty
+★ E-mail: dkakkad@stsci.edu
+clumps only in the equatorial region (Nenkova et al. 2002). Several
+studies in the literature now show strong infrared emission along
+the polar direction on the scales of a few parsecs (e.g., Hönig et al.
+2012, 2013; López-Gonzaga et al. 2016; Hönig & Kishimoto 2017;
+Leftley et al. 2018) to hundreds of parsecs (e.g., Braatz et al. 1993;
+Bock et al. 2000; Asmus et al. 2016; Asmus 2019). As a result,
+the dust emission around the AGN is believed to consist of two
+components: an equatorial thin disk and a polar extended feature that
+could originate from the winds from the central engine (see Hönig
+2019, and the references therein).
+The polar wind is believed to have a multi-phase composition
+ranging from dust to ionised and molecular gas components. In fact,
+recent high spatial resolution observations of nearby AGN have tar-
+geted the molecular gas distribution around the torus using ALMA,
+revealing high velocity outflows in the molecular gas phase (e.g.,
+Gallimore et al. 2016; Combes et al. 2019; García-Burillo et al.
+2019; Lopez-Rodriguez et al. 2020). In order to get a holistic view
+of the multi-phase dusty gas flows around the torus, it is imperative
+to obtain the morphology and kinematics of the ionised gas on the
+same scales as the molecular gas and infrared emission. Furthermore,
+obtaining outflow morphology at such small spatial scales can also
+© 2021 The Authors
+arXiv:2301.00825v1  [astro-ph.GA]  2 Jan 2023
+
+2
+D. Kakkad et al.
+give clues into the outflow launching mechanism and connect them
+to the observed kiloparsec scales structure or outflows, whenever
+available in the literature. Thanks to the Narrow Field Mode (NFM)
+capabilities of the Multi Unit Spectroscopic Explorer (MUSE Bacon
+et al. 2010) at the Very Large Telescope (VLT), such high spatial
+resolution observations can now be performed with ground-based
+Integral Field Spectroscopic instruments operating at optical wave-
+lengths. The optical wavelengths provide access to bright emission
+lines such as the [O iii]𝜆5007 that trace ionised gas in the Narrow
+Line Region (NLR). Furthermore, emission lines such as H𝛼, H𝛽,
+[N ii]𝜆𝜆6549, 6585 and [S ii]𝜆𝜆6716, 6731 help derive dust extinc-
+tion maps and diagnostic diagrams that trace the source of ionisation
+across the field-of-view.
+The Circinus galaxy is the closest Seyfert 2 galaxy (∼4.2 Mpc
+away, z = 0.001 Freeman et al. 1977) and hosts an infrared-bright
+AGN (e.g., Jarrett et al. 2019). The polar axis of the AGN in the
+Circinus galaxy is seen almost edge-on and is therefore an ideal target
+to study the relative gas and dust structure in a typical obscured AGN.
+Recent mid-infrared (MIR) observations using the upgraded VISIR
+instrument at the Very Large Telescope (VLT) suggests the presence
+of dust in the form of a hollow cone at the edges of the ionised outflow
+(e.g., Stalevski et al. 2017). The polar elongation of the infrared
+emission has also been reported on parsec scales (e.g., Tristram
+et al. 2014; Stalevski et al. 2019) using Mid-Infrared Interferometric
+Instrument (MIDI) and the Multi AperTure mid-infrared Spectro-
+Scopic Experiment (MATISSE, e.g., Isbell et al. 2022) at the VLT.
+The presence of dust in the form of a hollow cone in the polar region is
+also confirmed by multi-band optical polarimetry with VLT/FORS2
+(Stalevski et al., submitted) The galaxy also hosts powerful outflows
+in the ionised gas phase, visible in the form of a one-sided ionisation
+cone extended up to ∼1 kiloparsec (e.g., Marconi et al. 1994; Veilleux
+& Bland-Hawthorn 1997; Mingozzi et al. 2019; Fonseca-Faria et al.
+2021; Kakkad et al. 2022). The Circinus galaxy hosts an obscured
+AGN with nuclear star formation that dominates the dust emission
+on scales of hundreds of parsec (e.g., Matt et al. 2000; Arévalo et al.
+2014).
+In this fourth paper in the series, we map the morphology and
+kinematics of ionised gas in the Circinus galaxy at ∼2 pc resolution
+using MUSE-NFM observations. We present a model of the ionisa-
+tion cone and the resulting outflowing gas structure. The observed
+ionised outflow morphology obtained from the NFM observations
+is compared with the larger scale outflows observed with the Wide
+Field Mode (WFM) of MUSE, to understand outflow propagation
+across the host galaxy. We locate the regions with high dust extinc-
+tion and compare this with the archival mid-infrared images. Lastly,
+the MUSE data is compared with other archival radio observations
+to infer the presence of jet-ISM interaction in the host galaxy.
+Throughout this paper, e adopt the following ΛCDM cosmological
+parameters: 𝐻0 = 70 km s−1, ΩM = 0.3 and ΩΛ = 0.7. All the maps
+use the following convention: North is up and East is to left.
+2 MUSE-NFM OBSERVATIONS & DATA REDUCTION
+The observations were performed using the Laser Tomographic
+Adaptive Optics (LTAO) assisted Narrow Field Mode of the MUSE
+instrument, on board Unit Telescope 4 of the VLT1. The observa-
+tions were carried out on the nights of 29 and 30 April 2019 with a
+DIMM seeing in the range 0.89–1.01′′. We observed the galaxy with
+1 ESO programme ID: 0103.B-0396(A)
+an optimised sequence: O-S-O-O-S-O (O = Object, S = Sky), where
+the Sky was obtained at an offset position ∼1.5 arcmin away outside
+the galaxy. We performed small dithering between the individual
+science exposures and rotated the field by 90 degrees on each subse-
+quent exposure to eliminate the impact of bad pixels and to average
+out the patterns of slicers and channels. The nucleus of the Circinus
+galaxy has an H-band magnitude of 13.4 and therefore, served as the
+Adaptive Optics (AO) reference for the Wavefront Sensor (WFS).
+The total on-source exposure time was ∼4000s.
+The raw data was reduced using the standard MUSE pipeline
+(e.g., Weilbacher et al. 2014, 2020). The pipeline performs bias
+correction, flat fielding, wavelength and astrometry calibration, sky
+subtraction and flux calibration. The final data cube consisted of a
+field-of-view of ∼7.5×7.5 arcsec2 centred on the nucleus (AGN) with
+a spatial sampling of 0.025′′. As the observations were performed
+in the Nominal mode, this provided a uniform wavelength coverage
+between 4800–9300 Å with a gap between 5780–6050 Å due to the
+presence of a notch filter that suppresses the Sodium laser light. The
+spectral PSF (also known as the Line Spread Function, LSF) is in
+the range 2.5–2.9 Å, with the best resolution obtained at the redder
+end of the spectra. This corresponds to a velocity resolution of ∼150
+km s−1 at the location of [O iii]𝜆5007 line, one of the emission lines
+that will be analysed in this paper. The LTAO-assisted observations
+resulted in a spatial PSF of ∼0.1′′, determined using one of the point
+sources in the field-of-view. This spatial resolution is one of the
+highest that can be achieved using ground-based IFS observations.
+At the redshift of the Circinus, this corresponds to a physical scale of
+∼2 pc, which means that the observations can potentially resolve the
+region near the AGN torus. With a field-of-view of 7.5′′×7.5′′, the
+NFM observations trace spatial scales up to ∼100 pc from the AGN
+location.
+3 ANALYSIS
+In order to derive the flux and velocity maps from the optical emis-
+sion lines, we first subtract the stellar continuum emission across the
+MUSE field-of-view. We perform the stellar continuum emission us-
+ing the LZIFU tool (e.g., Ho et al. 2016; Kreckel et al. 2018), which
+adopts the penalized pixel fitting routine (PPXF Cappellari & Em-
+sellem 2004; Cappellari 2017) to fit the stellar continuum using input
+stellar spectrum templates (from González Delgado et al. (2005)) or
+modelled simple stellar populations (SSPs) that are convolved with
+parametrised velocity distributions. In doing so, regions in the spectra
+with strong skylines and emission lines intrinsic to the host galaxy
+were masked. The key emission lines that were masked were H𝛽,
+[O iii]𝜆𝜆4959, 5007, [N ii]𝜆𝜆6549, 6585, H𝛼 and [S ii]𝜆𝜆6716, 6731
+([S ii] doublet hereafter). We also mask the notch filter region in the
+spectrum that is contaminated by the sodium doublet emission from
+the lasers. Being a Seyfert 2 galaxy, the Circinus galaxy does not
+display broad emission lines from the Broad Line Region (BLR).
+The resulting stellar continuum-subtracted data cubes were then
+used to analyse the morphology, kinematics and the ionisation mech-
+anism of the gas using strong emission lines in the optical spectra. For
+instance, we used the [O iii]𝜆5007 line ([O iii] hereafter) to trace the
+ionised gas in the Narrow Line Region (NLR), the Balmer lines H𝛼
+and H𝛽 lines are used to trace potential star formation and dust extinc-
+tion in the host galaxy (using Balmer decrement), the [N ii]𝜆𝜆6549,
+6585 lines are used in investigating the ionisation source in each
+pixel (AGN or star formation) and the [S ii] doublet are used to trace
+the ionised gas electron density. We model these emission lines with
+multiple Gaussian functions using the scipy.curve-fit package in
+MNRAS 000, 1–9 (2021)
+
+NFM observations of Circinus
+3
+Figure 1. The top left panel shows an RGB colour image of the Circinus galaxy, derived from the NFM data. The field-of-view of the RGB image is 7.5′′×7.5′′.
+The red hue indicated the H𝛼 emission, while the ionised gas cone is apparent in green. The white square at the edge of the ionisation cone shows the pixel from
+where the example spectra shown in this figure was obtained. The middle and right panels in the top row shows the example of a stellar continuum fit and the
+middle and right panels in the bottom row shows the emission line fitting results of H𝛽, [O iii]𝜆𝜆4959,5007, [N ii]𝜆𝜆6549,6585, H𝛼 and [S ii]𝜆𝜆6716,6731.
+In all the spectra, the grey curve shows the extracted data from the pixel location shown in the median image, the magenta curve shows the stellar continuum
+model, the green and blue curves show the individual Gaussian components used to model the emission lines and the red curve shows the overall model. The
+bottom left panel illustrates the definition of outflow and systemic flux used in this paper. Further details are given in Sect. 3.
+python (see Virtanen et al. 2020). Initially a single Gaussian is fitted
+to the emission line profile, and additional Gaussian functions were
+added if the 𝜒2 is minimised, and until the line fluxes are stable within
+∼10%. Circinus does not display BLR emission in its H𝛽 or H𝛼 pro-
+files and the maximum number of Gaussians required to model an
+emission line was two. We do not assign any physical significance to
+the individual Gaussian components as we follow a non-parametric
+approach to derive the properties associated with the systemic and
+outflow components. The non-parametric approach was chosen over
+a parametric model as it does not depend on the choice of the models
+used for the emission lines (e.g., Gaussian, Lorentzian or a power-
+law). In addition, we tied the line centroids of [O iii] line with that of
+H𝛽 and the [N ii] and [S ii] lines with that of H𝛼 based on the expected
+location of the respective atomic species. The emission line ratios
+[O iii]𝜆5007:[O iii]𝜆4959 and [N ii]𝜆6585:[N ii]𝜆6549 were set ap-
+proximately equal to 3:1 based on expectations from theory (e.g.,
+Osterbrock & Ferland 2006; Dimitrijević et al. 2007). Lastly, the
+FWHM of the individual Gaussian components of the [O iii] line
+was coupled with H𝛽 and the H𝛼 FWHM with that of [N ii].
+From the line fitting procedure, we are interested in the following
+parameters: The 10th percentile velocity: 𝑣10 (blue-shifted [O iii] ve-
+locity that contains 10% of the overall [O iii] flux), the width of the
+emission line: 𝑤80 (width containing 80% of the flux of the emission
+line, see Harrison et al. 2014; Kakkad et al. 2016; Wylezalek et al.
+2020) and the flux of the outflowing and systemic components of
+each of the emission lines. To determine the fluxes, we first define
+the zero velocity location in the emission line spectra, which is the
+location of the peak of the emission line. The flux within ±300 km
+s−1 on either side of this zero velocity location is considered to be
+the systemic component of the emission line. The choice of 300 km
+s−1 is based on the line width (FWHM) cut of ∼600 km s−1 which
+is often employed in the literature to distinguish between outflow-
+ing and non-outflowing gas (e.g., Kakkad et al. 2020). Using similar
+arguments, we use the flux outside of the ±300 km s−1 channels to
+define the flux associated with outflows. Using lower velocity cuts
+such as 250 or 200 km s−1 yield similar results, but due to the pos-
+sibility of contamination from the non-outflowing gas at the lower
+velocities, we make a conservative cut of 300 km s−1. Figure 1 shows
+an example of the analysis methods and the fitting results presented
+in this section.
+4 RESULTS & DISCUSSION
+In this section, we show the results of the analysis methods described
+in Sect. 3. The main aim of this section is to derive the ionised
+gas outflow morphology and kinematics close to the AGN torus and
+compare this with archival multi-wavelength data, including the low
+spatial resolution MUSE-WFM data.
+4.1 The parsec-scale ionised outflow in the Circinus galaxy
+Figure 2 shows the flux maps of the systemic (left panel) and outflow-
+ing component (middle panel) of the [O iii] emission line from the
+NFM data. The systemic [O iii] flux map traces the ionisation cone
+originating from the AGN location and extends towards the NW di-
+rection from the nucleus. The presence of the ionisation cone in the
+Circinus galaxy has previously also been reported in the literature, in-
+cluding MUSE Wide Field Mode (WFM) observations (e.g., Marconi
+et al. 1994; Fischer et al. 2013; Mingozzi et al. 2019; Fonseca-Faria
+MNRAS 000, 1–9 (2021)
+
+3500
+Data
+3500
+Stellar continuum
+3000
+3000
+erg/s/cm?/A)
+2500
+2500
+2000
+2000
+[e-17
+1500
+1500
+Flux
+1000
+1000
+500
+500
+0
+0
+4800
+4850
+4900
+4950
+5000
+5050
+6550
+6600
+6650
+6700
+6750
+Rest-frame wavelength[A]
+Rest-frame wavelength[A]
+2.5
+Outflow
+Data
+Systemic
+3.0
+Component 1
+Component2
+2.0
+erg/s/cm²/A)
+2.5
+Model
+2.0
+1.5
+1.5
+[e-17
+1.0
+1.0
+0.5
+0.5
+0.0
+0.0
+-300
+0
+300
+4800
+4850
+4900
+4950
+5000
+5050
+65506600
+665067006750
+v [km/s]
+Rest-frameWavelength[A]
+Rest-frameWavelength[A]4
+D. Kakkad et al.
+Figure 2. The left panel shows the systemic [O iii] flux (|𝑣 | < 300 km s−1) that traces the ionisation cone. The middle panel shows the outflowing component
+of the [O iii] flux (|𝑣 | > 300 km s−1). The outflow morphology suggests the gas propagation along a collimated structure before it fragments into two filaments,
+giving it a "tuning-fork" resemblance. The green contours in the middle panel shows the [O iii] outflow contours within the collimated structure to highlight
+that the outflowing structure itself shows the presence of clumps. Thanks to the 0.1′′ spatial resolution achieved with the NFM observations, such structures are
+barely visible in archival WFM data shown in the right panel. In all the panels, the blue "X" marks the AGN location and the black bar on the bottom left shows
+the 20 pc scale.
+Figure 3. The top left panel shows the stellar velocity map obtained from the stellar continuum fitting. The stellar velocity map shows a smooth rotation-like
+profile of the host galaxy. The [O iii] centroid velocity profile (top centre panel) approximately mimics the stellar velocity field, suggesting that the bulk of the
+ionised gas cone co-rotates with the host galaxy. The top right panel shows the residual map after subtracting the stellar velocity map from the [O iii] velocity
+map. We observe residuals at the locations of the "tuning-fork" structure (red contour), suggesting that it is a part of the non-rotation component. The positive
+residuals in the filament directed towards the West and the negative residuals in the filament towards North shows that the outflow itself is co-rotating with the
+ionised gas and the host galaxy. The bottom panels show the non-parametric velocities, 𝑣10 and 𝑤80, described in Sect. 3. Both these velocities confirm that
+the high velocity regions are along the collimated structure that fragments into two filaments ∼1.5′′ from the AGN location. Furthermore, the presence of this
+structure in the 𝑣10 map shows that the dominant component of the outflow is blue-shifted. The black "X" in all maps indicate the AGN location.
+et al. 2021; Kakkad et al. 2022). The systemic flux dominates the
+bulk of the ionised gas flux in the host galaxy by nearly two orders of
+magnitude, compared to the [O iii] outflow flux. The [O iii] outflow
+map (middle panel, Fig. 2), on the other hand, shows a collimated
+structure that originates close to the AGN location and extends to-
+wards the NW of the nucleus (same direction and approximately the
+same PA as the ionisation cone). The collimated structure itself is
+not uniform and shows multiple clumps. Such clumps have also been
+previously reported in extended radial ionised gas filaments of the
+Circinus galaxy (e.g., Veilleux & Bland-Hawthorn 1997). We note
+that the location of the first clump is not coincident with the AGN
+location, but ≈0.4′′ NW of the nucleus. In Section 5, we further dis-
+cuss the origin of these clumps and whether they could be produced
+by the shocks within the outflowing wind.
+Beyond ∼1.5′′ from the AGN location (∼30 pc) in the NW direc-
+tion, the collimated structure then fragments into two filaments, one
+towards the West and another towards North, which gives the overall
+outflow morphology a "tuning-fork" resemblance. The impact of the
+high resolution NFM observations is clear from these observations
+as such pc-scale filaments and fragmenting structures are not visible
+in the archival low resolution (∼0.5′′) MUSE WFM data, as shown
+in the right panel of Fig. 2.
+Figure 3 shows velocity maps of the stellar component and the
+ionised gas of the Circinus galaxy, derived from the NFM observa-
+tions. The top left panel in Fig. 3 shows the stellar velocity distribution
+MNRAS 000, 1–9 (2021)
+
+1e-17.
+10
+1e-19
+9
+3
+3
+8
+8
+2
+2
+arcsec]
+6
+1
+5
+0
+0
+4
+4
+-1
+m
+2
+2
+2
+2
+20 pc
+-3
+20 pc
+-3
+1
+[ol] systemic flux
+[olll] outflowflux
+C
+-2
+0
+2
+-2
+0
+2
+Ax [arcsec]
+△x [arcsec][Olll] outflow flux WFM
+1e-18
+6
+3 
+5
+2 -
+4
+y [arcsec]
+1 
+erg/s/cm2
+0 -
+X
+-11
+2
+-2 -
+1
+-3
+0
+-2
+0
+2
+△x [arcsec]100
+100
+30
+75
+75
+2
+2
+2
+20
+50
+50 
+Ay [arcsec]
+V[ol] [km s-1]
+10
+25
+[t-s 
+25
+V[OI] - V*
+0
+0
+[km
+0
+0
+0
+0
+-25
+-25
+-10
+-50
+-50 
+-2
+-2 
+-2 
+-75
+-75
+-20
+100
+-100
+-2
+30
+-2
+0
+2
+0
+2
+-2 
+0
+2
+Ax[arcsec]
+Ax[arcsec]
+Ax[arcsec][OI] V10
+[OI|] W80
+-50
+380
+2
+2
+[arcsec]
+-95
+320
+-1
+-1
+'s
+0
+0
+S
+km
+km
+-135
+260
+-2
+-2
+-180
+200
+-2
+0
+2
+-2
+0
+2
+Ax[arcsec]
+Ax[arcsec]NFM observations of Circinus
+5
+Figure 4. The map shows the mass outflow rate distribution for the ionised
+gas derived from the [O iii]𝜆5007 line in the MUSE-NFM observations of
+the Circinus galaxy. The mass outflow rates are higher in regions with higher
+outflow velocity.
+in the host galaxy obtained from the stellar continuum modelling. The
+velocity map shows a smooth gradient indicating a rotation-like pro-
+file, with the axis of rotation aligned approximately along the axis of
+the ionisation cone. The [O iii] centroid map, shown in the top centre
+panel of Fig. 3, mimics the stellar velocity map i.e., the ionised gas
+co-rotates with the host galaxy. The [O iii] centroid velocity profile is
+also consistent with previous MUSE-WFM results from the literature
+(e.g., Fonseca-Faria et al. 2021). On subtracting the stellar velocity
+component from the [O iii] centroid velocity, we see clear residuals
+at the locations of the "tuning-fork" structure, as shown in the top
+right panel of Fig. 3. This proves that the outflow flux shown in the
+middle panel in Fig. 2 is indeed part of the non-systemic component
+of the host galaxy. Furthermore, the positive and negative residuals
+in the West and North arms respectively in the residual map in the
+top right panel of Fig. 3 indicates that the fork structure itself is
+co-rotating with the host galaxy and the ionisation cone. The bottom
+panels in Fig. 3 show the [O iii] 𝑣10 and the 𝑤80 maps (left and right
+panels respectively). Both the 𝑣10 and 𝑤80 maps confirm the results
+seen in the outflow maps i.e., the high velocity regions are collimated
+up to ∼1.5′′ from the AGN location before they fragment into two
+filaments. The presence of the tuning-fork structure in the 𝑣10 map
+suggests that most of the observed outflow flux is dominated by the
+blue-shifted emission. We note that the stellar velocity map in the
+top left panel of Fig. 3 also shows a "V-shaped" structure at approx-
+imately the same location where the high velocity regions fragment
+into the two arms, suggesting that the material within the cone is both
+outflowing and co-rotating with the host galaxy.
+We also derived the ionised gas mass outflow rate using the
+[O iii] line adopting methods from the literature (e.g., Rupke et al.
+2005; Genzel et al. 2011; Veilleux et al. 2020; Kakkad et al. 2022).
+We report two kinds of outflow rate values: Instantaneous outflow
+rates ( �𝑀inst) is the sum of mass outflow rates calculated for each
+pixel, and time-averaged mass outflow rate (𝑀Tavg) calculated by
+taking averaged quantities over the whole outflowing region. These
+quantities can be computed using the following equations:
+�𝑀inst =
+∑︁
+pix
+𝑀out · 𝑣out
+Δ𝑅
+(1)
+�𝑀Tavg = 𝑀out· < 𝑣out >
+𝑅
+(2)
+In Equation 1, the mass of the outflowing ionised gas, 𝑀out and the
+velocity of the ionised gas, 𝑣out is computed for each pixel and Δ𝑅
+is the size of the pixel. In Equation 2, on the other hand, 𝑀out is the
+total outflowing gas mass computed from the outflowing [O iii] flux
+and 𝑣out is the average velocity over the outflowing region (∼300 km
+s−1). The parameter, 𝑅, in Eq. 2 is the distance of the outflow from
+the AGN location. As we are using spatially-resolved observations,
+we do not need to assume an outflow geometry or outflow density
+for the time-averaged quantity. The outflow density in both cases
+is obtained from the flux ratio of the outflowing components of
+[S ii]𝜆𝜆6716, 6731.
+The mass outflow rate map, representing the instantaneous mass
+outflow rates (Eq. 1), is shown in Fig. 4, where the mass outflow rate
+was calculated for each pixel. The advantage of using this method is
+that variation in the outflow parameters such as outflow density and
+velocity can be incorporated without the need for any assumptions
+on outflow models. We find the median outflow density across the
+field-of-view, calculated using the flux ratio of [S ii]𝜆𝜆6716, 6731
+to be ∼200 cm−3 (e.g., Sanders et al. 2016; Kaasinen et al. 2017;
+Kakkad et al. 2018). The total summed instantaneous outflow rate
+is 0.01 M⊙ yr−1 (an average of 3×10−7 M⊙ yr−1 per pixel where
+outflow is detected), which is two orders of magnitude less than the
+total instantaneous outflow rate value reported with MUSE-WFM
+observations (Kakkad et al. 2022). The time-averaged outflow rate
+computed using Eq. 2 is 10−4 M⊙ yr−1. The obscured star formation
+rate (SFR) in the Circinus galaxy is reported to be between 3–8 M⊙
+yr−1. The orders of magnitude difference between the SFR and the
+ionised outflow rate within a radius of ∼100 pc of the AGN location,
+therefore, suggests that the observed ionised outflow is not expected
+to shut down star formation in the host galaxy. However, this may not
+be true for kiloparsec-scale molecular outflows where the outflow
+rate in the molecular gas phase has been reported to be ∼0.35–12.3
+M⊙ yr−1 (see Zschaechner et al. 2016). The high molecular outflow
+rate in kiloparsec-scales can, therefore, regulate star formation. A
+multi-phase approach to high resolution gas kinematics, by tracing
+warm and cold molecular gas components, is required to robustly
+confirm whether these AGN outflows affect star formation within
+∼100 pc of the AGN.
+Lastly, using spatially resolved Baldwin, Phillips & Terlevich di-
+agrams (e.g., Baldwin et al. 1981; Veilleux & Osterbrock 1987), we
+infer that the dominant source of ionisation across the NFM field-of-
+view is the AGN and the ionisation by star formation is negligible or
+absent (Figure 5). The ionisation structure is consistent with previ-
+ous WFM results in the literature (e.g., Mingozzi et al. 2019; Kakkad
+et al. 2022). The ionisation by the AGN is observed for both systemic
+as well as outflowing components. Therefore, the current observa-
+tions also do not support a scenario where these outflows trigger star
+formation activity in the vicinity of the AGN.
+4.2 The dust-outflow connection in Circinus
+Previous mid-infrared observations of the Circinus galaxy estab-
+lished that a major fraction of dust emission is coming from the
+polar region, tentatively associated with dusty winds driven by radi-
+ation pressure (e.g., Stalevski et al. 2017; Venanzi et al. 2020). Even
+though far away from the central engine, dust and gas are expected to
+be coupled and co-spatial, and until recently the models of infrared
+emission ignored this polar dust component. The spatially-resolved
+optical spectra from the MUSE-NFM mode can be used to derive
+extinction maps from Balmer decrement (H𝛼/H𝛽) to confirm the
+presence of dust along the polar direction. Therefore, we derived the
+host galaxy extinction, 𝐴V, across the NFM field-of-view using the
+MNRAS 000, 1–9 (2021)
+
+OutflowRatemap
+1e-6
+3
+4
+2
+[arcsec]
+1
+3
+0
+2
+-1
+-2
+1
+-3
+0
+-2
+0
+2
+△x[arcsec]6
+D. Kakkad et al.
+Figure 5. The left panel shows the ionisation structure (AGN in red and composite in orange) in the field-of-view probed by the MUSE-NFM data. The right
+panel shows the location of each pixel in the classical [N ii] BPT diagram. The solid and dashed black curves are obtained from Kauffmann et al. (2003) and
+Kewley et al. (2001) and divide the plots between regions ionised by AGN, star formation and composite processes. The systemic flux of the emission lines was
+used while plotting this diagram. However, the results are similar if the outflowing components is used. The figure highlights that the gas is ionised primarily by
+the AGN.
+Figure 6. Dust extinction (𝐴V) map of the Circinus galaxy using the NFM
+observations. The background image shows the extinction map obtained from
+the systemic components ofH𝛼 and H𝛽, thecyan contoursshow theextinction
+from the outflowing components and the magenta contours show the location
+of high velocity ionised gas outflow. The dust extinction is dominant along
+the polar direction, consistent with previous mid-infrared observations in the
+literature.
+Balmer Decrement parameter. We assumed a Calzetti et al. (2000)
+dust attenuation law with 𝑅V = 4.05 and a fixed temperature of 10,000
+K, which is the typical electron temperature in the NLR. We note
+that the Circinus galaxy suffers from Galactic extinction of 𝐴V ∼2
+(see For et al. 2012). However, the [O iii] outflow morphology and
+the associated velocities and mass outflow rates will not change on
+correcting the Galactic extinction. The extinction map is shown in
+Fig. 6. The background map in Fig. 6 shows the extinction map ob-
+tained from the systemic components of the flux ratio, H𝛼/H𝛽. The
+cyan contours show the extinction (𝐴V > 1) obtained from the out-
+flowing component of H𝛼 and H𝛽, and the magenta contours show
+the location of the [O iii] ionised gas outflow from the middle panel
+in Figure 2.
+While the map in Fig. 6 shows potential dust distribution both along
+the disk (consistent with the results reported in Mingozzi et al. (2019)
+and Fonseca-Faria et al. (2021)) as well as the polar direction, the
+overall distribution is dominant along the polar direction. This result
+is consistent with the previous results with mid-infrared emission,
+which was also dominant along the polar direction (e.g., Jaffe et al.
+2004; Tristram et al. 2007, 2014; Asmus et al. 2016). This suggests
+that the dust along the polar direction might be a part of lower velocity
+gas (compared to the high velocity collimated outflow observed here)
+surrounding the ionised gas outflow. The extinction from the systemic
+components peaks at the location of the AGN and gradually falls off
+to 𝐴V = 0 at a distance of ≈3′′ i.e., ≈60 pc.
+The extinction obtained from the outflowing H𝛼 and H𝛽 compo-
+nents shows non-uniform clumps scarcely distributed along the polar
+direction. We attribute these clumps to be a part of the outflowing
+gas and dust. It is worth noting that one of these clumps is almost
+at the tip of the collimated component of the outflow, approximately
+where the outflow filament fragments into two arms. The observa-
+tion, therefore, might support a picture where the ionised gas outflow
+chooses the path of least resistance and therefore fragments into the
+two filaments, avoiding the region radially outward towards where
+the dust clump is present.
+5 DISCUSSION
+The results presented in Section 4 highlights the complex structures
+within an outflow that are revealed from high resolution observations
+in the vicinity of the AGN. The presence of an outflow in the form of
+an ionisation cone in the Circinus galaxy has been known for nearly
+three decades, and thanks to the current state-of-the-art instrumen-
+tation that we are now able to resolve parsec scale emission using
+optical spectra. The origin of the initial clumpy collimated structure
+observed in the NFM data could be due to a small-scale radio jet.
+The Circinus galaxy is known to host a radio jet, although its PA is
+aligned closer to the edge of the ionisation cone (e.g., Elmouttie et al.
+1998). However, these are lower spatial resolution radio imaging ob-
+servations and the regions close to the AGN torus is not resolved.
+Due to the precession and interaction with the surrounding medium,
+jets in AGN are known to bend and change directons at larger scales
+(e.g., as in the case of NGC 1068 Gallimore et al. 2004). Therefore,
+future work will target the regions closer to the AGN to search for
+the potential impact of small-scale jets that may be aligned along the
+MNRAS 000, 1–9 (2021)
+
+1.5
+3
+AGN
+1.0
+2
+0.5
+1
+Log [OI]/Hβ
+[arcsec]
+0
+0.0
+-0.5
+-2 
+-1.0
+Star
+-3 
+formation
+Composite
+-1.5
+-3
+-2
+0
+1
+2
+3
+0.8
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+Ax [arcsec]
+Log [NII]/Hα6
+3
+5
+2
+4
+[arcsec]
+7
+0
+m
+以
+-1
+2
+-2
+1
+[ooutflow
+-3
+0
+2
+Ax [arcsec]NFM observations of Circinus
+7
+Figure 7. The background image shows the [O iii]𝜆5007 flux map from the
+MUSE-WFM data, which shows an overall asymmetric conical morphology
+with two distinct filaments on hundreds of parsec scales. The red contours
+show the [O iii] outflow morphology obtained from the NFM observations
+in the middle panel of Fig. 2. Comparing the NFM contours with that of the
+WFM [O iii] flux map, it appears that the two filaments observed in the WFM
+data have their origins in parsec-scale outflow traced with the NFM.
+Figure 8. A cartoon model of the ionised outflow observed with the MUSE-
+NFM data in the Circinus galaxy. The outflow might be launched as a col-
+limated structure by the radio jet, which then fragments into two filaments
+probably due to the presence of a dense clump at the tip of the collimated
+structure. The dust is distributed along the ionisation cone, apparent from the
+extinction maps and also consistent with archival mid-infrared observations.
+axis of the ionisation cone. The relative orientations of the radio jet
+and the ionisation cone in the Circinus galaxy are depicted in Fig. 8.
+The outflow itself is not composed of uniformly distributed ionised
+gas, but shows the presence of clumps, confirming previous observa-
+tions from lower resolution data of other targets that the outflowing
+media are non-uniform in nature (e.g., Kakkad et al. 2018, 2022).
+The presence of clumps or knots along outflow/ionised gas filaments
+in the Circinus galaxy has also been previously reported in Veilleux
+& Bland-Hawthorn (1997) and these structures have been attributed
+to bow-shocked features that resemble Herbig Haro objects that in-
+teract strongly with the surrounding ISM. The observed morphology
+in the NFM data could be a smaller-scale version of the observations
+in larger scales.
+The collimated component of the outflow most likely fragments
+into two components because of the presence of an obstruction in
+the path of the outflow. In the NFM data, there is an indication of
+the presence of a dust clump via extinction maps obtained from the
+outflowing component of H𝛼 and H𝛽 emission. Infrared observations
+targeting primarily the dust emission could confirm this scenario.
+Filament structures are also observed in images that trace gas across
+larger scales (e.g., Marconi et al. 1994; Veilleux & Bland-Hawthorn
+1997) and it is probable that the NFM data presented in this paper
+reveals the origin of the kiloparsec-scale filaments. Fig. 7 shows
+the ionised gas morphology traced by the archival MUSE-WFM
+observations (background) and the emission from the outflowing
+[O iii] component from the MUSE-NFM data (red contours). The
+spatial distribution of the filaments observed in WFM is strongly
+suggestive that their origin is to be found in the fragmented arms of
+the tuning fork observed in the NFM observations.
+Fig. 8 shows an overall cartoon model of the ionised outflow in the
+Circinus galaxy, that shows the ionised outflow that fragments into
+two filaments, probably due to the presence of a dense clump at the
+tip of the collimated structure. The dust itself is distributed around the
+ionisation cone and it envelops the lower velocity ionised gas within
+the conical structure. We speculate that the outflow itself might be
+launched due to the radio jet, which cannot be robustly confirmed
+based on the available archival radio observations, as mentioned
+earlier.
+As the NFM observations have only recently targeted the nearby
+galaxies, it is unclear if such outflow structures and fragmentations
+within outflows are common. If this is indeed observed for majority
+of the galaxies, conventional outflow models that use ionisation cone
+morphology may need to be revised to account for the results from
+these high spatial resolution data.
+6 SUMMARY & CONCLUSIONS
+We presented the MUSE NFM observations of the Circinus galaxy at
+a spatial resolution of 0.1′′ (∼2 pc) that resolves the regions close to
+the AGN torus. We derived the properties of the ionised gas outflow
+using the [O iii]𝜆5007 emission line and the dust distribution using
+Balmer Decrement. We follow a non-parametric approach to analyse
+the emission lines, so the derived properties are independent of the
+fitting functions used. The main results of this work is summarized
+below:
+• The flux distribution of the systemic component of [O iii] emis-
+sion, defined by the velocity components within ±300 km s−1, shows
+a conical morphology, which is also observed in larger spatial scales
+up to hundreds of parsecs in the literature. Archival radio observa-
+tions show that the radio jet is aligned approximately with the edge
+of the ionisation cone. The flux distribution of the outflowing com-
+ponent of the [O iii] emission, on the other hand, shows a collimated
+structure up to ∼30 pc before fragmenting into two arms that overall
+mimics a “tuning-fork” shape. The outflowing structure itself is not
+smooth and shows clumps at several locations.
+• A comparison between the stellar kinematic map and the
+[O iii] centroid map suggests that the ionised gas (both the systemic
+as well as the outflowing component) co-rotates with the host galaxy.
+Both the non-parametric velocity distributions, 𝑣10 and 𝑤80 show
+similar structures as the outflow flux i.e., the tuning-fork shapes. Most
+of this outflow is blue-shifted, consistent with the outflow models of
+the Circinus galaxy reported in the literature.
+• We find a total instantaneous mass outflow rate of ∼0.01 M⊙
+yr−1 (3×10−7 on average per pixel of size 0.5 pc) and a time-average
+mass outflow rate of 10−4 M⊙ yr−1. This is much lower than the star
+formation within the galaxy and therefore, the ionised gas outflow
+is not expected to regular star formation within a radius of ∼100 pc
+from the AGN location.
+MNRAS 000, 1–9 (2021)
+
+Dust
+clumps
+Radio jet
+lonisationcone
+AGN20
+15
+[arcsec]
+10
+5
+Ay I
+0
+-5 
+-10
+-10
+-5
+0
+5
+10
+15
+20
+△x [arcsec]8
+D. Kakkad et al.
+• The extinction maps using the systemic components of the H𝛽
+and H𝛼 line show that the dust distribution is concentrated along the
+ionisation cone i.e., the polar direction, consistent with the archival
+mid-infrared observations. The extinction map obtained from the
+outflowing components of H𝛽 and H𝛼 shows a scarce distribution,
+with a clump approximately at the tip of the collimated part of the
+outflow. This dust clump might explain the fragmentation in the
+outflowing ionised gas.
+• We combine previous reported results gathered from the litera-
+ture and the observed outflows from the NFM data presented in this
+paper to present a model of the ionised gas outflow in the Circinus
+galaxy. We suggest that the observed outflow in the Circinus galaxy
+is composed of high velocity collimated gas that is enveloped by
+lower velocity dusty ionised gas. The presence of a dust clump at the
+tip of the collimated part of the outflow might be responsible for the
+fragmentation in the outflowing gas. The collimated outflow might
+be launched by a radio jet. Although the jet, observed in low spatial
+resolution radio observations, does not show a 1:1 alignment with
+the outflowing cone PA, they are known to bend and change at larger
+scales.
+While the MUSE-NFM only provides the kinematic information
+in the ionised gas phase, outflows have been known in exist also in
+the molecular gas phase in the Circinus galaxy. A multi-wavelength
+approach to trace gas in the other phases such as warm and cold
+molecular, at the same spatial resolution of ∼2 pc, will be key to
+obtaining a holistic view into the outflow-AGN connection in the
+Circinus galaxy. Furthermore, high spatial resolution radio obser-
+vations will verify if the observed collimated outflow is a result of
+jet-ISM interaction on small scales.
+ACKNOWLEDGEMENTS
+We would like to thank the anonymous referee for insightful com-
+ments and suggestions to improve this manuscript. The authors also
+thank T. Fischer for useful discussions. Based on observations from
+the ESO programme ID 0103.B-0396. M.S. and S.K. acknowledge
+support by the Science Fund of the Republic of Serbia, PROMIS
+6060916, BOWIE and by the Ministry of Education, Science and
+Technological Development of the Republic of Serbia through con-
+tract No. 451-03-9/2022-14/200002. D.A. acknowledges funding
+through the European Union’s Horizon 2020 and Innovation pro-
+gramme under the Marie Sklodowska-Curie grant agreement no.
+793499 (DUSTDEVILS).
+DATA AVAILABILITY
+The data presented in this paper is available with ESO archive under
+the programme ID: 0103.B-0396.
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+

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+Text to Point Cloud Localization with Relation-Enhanced Transformer
+Guangzhi Wang1, Hehe Fan2, Mohan Kankanhalli2
+1Institute of Data Science, National University of Singapore
+2School of Computing, National University of Singapore
+guangzhi.wang@u.nus.edu, hehe.fan@nus.edu.sg, mohan@comp.nus.edu.sg
+Abstract
+Automatically localizing a position based on a few natural
+language instructions is essential for future robots to commu-
+nicate and collaborate with humans. To approach this goal,
+we focus on the text-to-point-cloud cross-modal localization
+problem. Given a textual query, it aims to identify the de-
+scribed location from city-scale point clouds. The task in-
+volves two challenges. 1) In city-scale point clouds, similar
+ambient instances may exist in several locations. Searching
+each location in a huge point cloud with only instances as
+guidance may lead to less discriminative signals and incor-
+rect results. 2) In textual descriptions, the hints are provided
+separately. In this case, the relations among those hints are
+not explicitly described, leading to the difficulties of learn-
+ing relations. To overcome these two challenges, we propose
+a unified Relation-Enhanced Transformer (RET) to improve
+representation discriminability for both point cloud and nat-
+ural language queries. The core of the proposed RET is a
+novel Relation-enhanced Self-Attention (RSA) mechanism,
+which explicitly encodes instance (hint)-wise relations for the
+two modalities. Moreover, we propose a fine-grained cross-
+modal matching method to further refine the location predic-
+tions in a subsequent instance-hint matching stage. Experi-
+mental results on the KITTI360Pose dataset demonstrate that
+our approach surpasses the previous state-of-the-art method
+by large margins.
+Introduction
+Understanding natural language instructions in the 3D real
+world is a fundamental skill for future artificial intelligence
+assistants to collaborate with humans. In this paper, we fo-
+cus on the outdoor environment and study the task of natural
+language-based localization from city-scale point clouds. As
+shown in Figure 1, given a linguistic description of a posi-
+tion, which contains several hints, the goal of the task is to
+find out the target location from a large-scale point cloud.
+This task can effectively help mobile robots, such as self-
+driving cars and autonomous drones, cooperate with humans
+to coordinate actions and plan their trajectories. By under-
+standing the destination from natural language instructions,
+it reduces the human effort required for manual operation.
+However, this task is intrinsically challenging. Precise lo-
+calization requires both correct language interpretation and
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+Heading to a place:
+[hint1] east of a dark-green terrain.
+[hint2] south of a gray road.
+[hint3] west of a dark-green traffic sign.
+[hint4] south of a green terrain.
+Textual Query
+Localization
+Figure 1: Illustration of the text to point cloud localization
+task. Given a textual query, which usually contains several
+independent hints, the goal is to localize the point of interest
+in a huge city-scale point cloud.
+effective large-scale point cloud understanding. Considering
+the difficulties, an existing method (Kolmet et al. 2022) first
+divides a city-wide point cloud into several cells, and then
+solves this task in a Coarse-to-Fine manner.
+The goal of the ‘coarse’ stage is to find out the target
+cell that contains the queried location according to the given
+natural language descriptions. In this stage, the instances
+included in point cloud cells and those mentioned in lan-
+guage descriptions are mainly used for text-to-point-cloud
+retrieval based on their types, without considering their rela-
+tions. In the ‘fine’ stage, each object in the textual query is
+matched with an in-cell point cloud instance, whereby a tar-
+get location will be predicted from each hint. This pioneer-
+ing method sets up a significant starting point for tackling
+the challenging task. However, it fails to consider the intrin-
+sic relations in both stages, resulting in sub-optimal perfor-
+mance.
+For the coarse stage, because similar ambient instances
+may exist in several cells, performing retrieval based on only
+the cell-contained and query-related instance types without
+considering their relations may lead to low discriminabil-
+arXiv:2301.05372v1  [cs.CV]  13 Jan 2023
+
+ity for both cell and query representations, which inevitably
+leads to ambiguity. Based on those low-discriminability rep-
+resentations, it is difficult to find out the correct cell. In the
+fine stage, we observe that insufficient cross-modal collabo-
+ration leads to difficulties in location refinement. Given the
+retrieved cell, precise location prediction requires joint un-
+derstanding of both point clouds and textual queries. How-
+ever, in the previous method (Kolmet et al. 2022), the cross-
+modal collaboration is only performed from textual queries
+to point clouds in a single step, which results in optimization
+difficulty for multi-task learning.
+In this work, we aim to solve the aforementioned short-
+comings in both stages. For the coarse stage, we pro-
+pose to encode pairwise instance relations to improve rep-
+resentation discriminability for both modalities, which is
+achieved through a novel Relation-Enhanced Transformer
+(RET) architecture. In particular, the in-cell point cloud in-
+stance relations are modeled as their geometric displace-
+ments, while computed as the fusion of hint representations
+in the linguistic domain. These relations from two modali-
+ties are respectively incorporated into their representation in
+a unified manner, which is achieved through the proposed
+Relation-enhanced Self-Attention (RSA) mechanism. For
+the fine stage, we perform Cascaded Matching and Refine-
+ment (CMR) to enhance cross-modal collaboration. In par-
+ticular, different from (Kolmet et al. 2022) which achieves
+this objective in a single step, we perform description-
+instance matching and position refinement in two sequential
+steps. Such formulation allows us to minimize the optimiza-
+tion difficulty of multi-objective learning and noisy interme-
+diate results, thereby improving cross-modal collaboration.
+We validated the effectiveness of our method on the
+KITTI360Pose benchmark (Kolmet et al. 2022). Extensive
+experiments demonstrate that the proposed method can sur-
+pass the previous approach by a large margin, leading to new
+state-of-the-art results. Our contributions are three-fold:
+• We propose a novel Relation-Enhanced Transformer
+(RET) to improve representation discriminability for
+both point clouds and textual queries. The core com-
+ponent of RET is the Relation-enhanced Self-Attention
+(RSA) mechanism, which encodes instance (hint) rela-
+tions for the two modalities in a unified manner.
+• We propose to perform cross-modal instance matching
+and position refinement in two sequential steps. This for-
+mulation allows us to minimize the optimization diffi-
+culty of multi-task learning and the influence of noisy
+intermediate results, thereby improving cross-modal col-
+laboration for fine-grained location prediction.
+• We perform extensive experiments on the KITTI360Pose
+dataset (Kolmet et al. 2022). The results show that our
+approach can surpass previous method by a large margin,
+resulting in new state-of-the-art performance. Additional
+ablation studies further demonstrate the effectiveness of
+each component in the proposed method.
+Related Work
+Transformer and Attention Mechanism. Transformer and
+self-attention mechanism (Vaswani et al. 2017; Fan, Yang,
+and Kankanhalli 2021) has become increasingly popular in
+recent years. Although first proposed for natural language
+processing, with architectural adaptation, Transformer has
+been widely applied to many vision tasks including visual
+recognition (Dosovitskiy et al. 2020; Liu et al. 2021), object
+detection (Carion et al. 2020; Zhu et al. 2020) and seman-
+tic segmentation (Cheng, Schwing, and Kirillov 2021). Be-
+sides, the transformer-based architectures are also utilized to
+model cross-modal (e.g., vision and language) relations (Tan
+and Bansal 2019; Lu et al. 2019; Li et al. 2019; Zhang et al.
+2021; Li et al. 2022). In these architectures, the attention
+mechanism is widely employed to implicitly learn relations
+among the input tokens. Nevertheless, without explicit rela-
+tion encoding, the vanilla Transformer can only encode rela-
+tions implicitly with the help of positional encoding (Doso-
+vitskiy et al. 2020). To facilitate better relation modeling,
+some works modulate the attention computation process
+by explicitly incorporating element relations. For example,
+(Wu et al. 2021) modified the attention mechanism via uni-
+fied relative position bias to improve visual recognition. For
+object detection, spatial relations between bounding boxes
+are introduced to modulate the attention weights (Liu et al.
+2022; Gao et al. 2021). For dynamic point cloud analy-
+sis, displacement between points (Fan, Yang, and Kankan-
+halli 2022) is utilized for point-specific attention computa-
+tion. In this work, we propose to model relations for both
+point clouds and language queries by explicitly incorporat-
+ing intra-modality relations in a unified manner.
+Visual Localization. The task that is most related to ours is
+vision-based localization (Arandjelovic et al. 2016; Brach-
+mann et al. 2017; Hausler et al. 2021), which is to estimate a
+pose based on an image or image sequence. Existing meth-
+ods mostly solve this task in two stages (Sarlin et al. 2019;
+Sattler, Leibe, and Kobbelt 2016; Zhou et al. 2020). The first
+stage finds a subset of all images using image retrieval-based
+techniques (Arandjelovic et al. 2016; Hausler et al. 2021;
+Torii et al. 2015), while the second stage establishes pixel-
+wise correspondence between the query image and the re-
+trieved one to predict the precise pose. In this work, we also
+study the task of localization in a coarse-to-fine manner, but
+differ from visual localization in that: 1) we try to infer the
+location from city-wide point clouds instead of images. 2)
+we try to estimate the pose from textual query rather than
+images. Compared to visual localization, our task requires
+multi-modal understanding and is more challenging to solve.
+3D Language Grounding. As we humans live in a 3D
+world and communicate through natural language, recent
+work has begun to investigate the tasks on the cross-modal
+understanding of 3D vision and natural language. Among
+these tasks, the one that is most related to ours is 3D lan-
+guage grounding, which aims at localizing an object in
+point clouds from a given natural language query. For ex-
+ample, ScanRefer (Chen, Chang, and Nießner 2020) stud-
+ies 3D language grounding from real-life in-door scenes.
+ReferIt3D (Achlioptas et al. 2020) studies a related task un-
+der a simpler setting, which assumes the object instances
+are segmented in advance. InstanceRefer (Yuan et al. 2021)
+improves previous methods by adopting a 3D panoptic seg-
+mentation backbone, utilizing multi-level visual context. Re-
+
+east of a dark-green terrain.
+south of a gray road.
+south of a green terrain. 
+west of a dark-green traffic sign. 
+Split
+...
+Cells
+Textual Query  
+Hint Encoder
+north of a dark-green smallpole . 
+east of a green pole. 
+...
+Instance Encoder
+Hints  
+Instances 
+Relation-Enhanced
+Self-Attention
+Add & LayerNorm
+Feed Foward Network
+Add & LayerNorm
+Relation-Enhanced
+Self-Attention
+Add & LayerNorm
+Feed Foward Network
+Add & LayerNorm
+Instance-wise
+Relation  
+x
+x
+...
+(a) Hint-Instance Matching
+...
+...
+...
+...
+Feature
+Pooling
+(b) Offset Prediction
+Offsets
+Matching
+Coarse Stage
+Fine Stage
+Cross-modal
+Fusion
+Multi-Layer
+Perceptron
+Figure 2: Framework of the proposed method. The city-scale point cloud is first divided into individual cells. Then, in the
+coarse stage, the cells and the textual query are respectively encoded with the proposed Relation-Enhanced Transformer (RET),
+which are later used for query-cell matching. In the fine stage, each hint is matched with an in-cell instance. Then, cross-modal
+fusion dynamically aggregates hints and instance representations for offset prediction. The target location is predicted based on
+matching results and offset predictions.
+cently, graph structure (Feng et al. 2021) is also utilized to
+improve the representation learning qualities.
+Methodology
+Preliminaries
+Given a textual query, our goal is to identify the position it
+describes from a city-scale point cloud. To handle the large-
+scale point cloud, we divide each scene into a set of cubic
+cells of fixed size by a preset stride. Each cell C contains a
+set of p point cloud instances, which are encoded by Point-
+Net++ (Qi et al. 2017) into vector representations {pi}p
+i=1.
+Following (Kolmet et al. 2022), the textual query T is repre-
+sented as a set of hints {hj}h
+j=1, each encoding the direction
+relation between the target location and an instance.
+Inspired by the existing work (Kolmet et al. 2022), given
+the cell splits, we solve this task in a coarse-to-fine manner
+with two stages. The coarse stage is formulated as textual
+query based cell retrieval. The goal of this stage is to train
+a model that encodes C and T into a joint embedding space
+whereby matched query-cell pairs are close while those un-
+matched are pulled apart (Kiros, Salakhutdinov, and Zemel
+2014). In the fine stage, given a retrieved cell, we aim to
+refine the position prediction by utilizing fine-grained cross-
+modal information. In particular, we first match each hint
+in the query with an in-cell instance by formulating it as an
+optimal transport problem (Liu et al. 2020). After that, with
+the matching results, we predict the target location through
+a cross-modal fusion of point cloud instance and hint repre-
+sentations. Based on the fused representation, we predict the
+target location for each matched instance. Finally, we obtain
+the target location prediction based on a weighted combi-
+nation of the matching and location prediction results. The
+framework of our method is shown in Figure 2. In the fol-
+lowing of this section, we will explain the proposed method
+for coarse stage and fine stage. After that, our training and
+inference procedure will be detailed.
+Coarse Stage: Relation-Enhanced Transformer
+After the cell split, the goal of the coarse stage is to suc-
+cessfully retrieve the cell C given a textual query T . To ap-
+proach this objective, we need to encode C and T into a joint
+embedding space. An intuitive solution is to encode both
+C and T based on the instances they contained as is done
+in (Kolmet et al. 2022). However, with such representations,
+the low discriminability for cells and textual queries results
+in poor retrieval performance. We argue that this can be at-
+tributed to the following two reasons. On the one hand, the
+outdoor scenes are often of low diversity, whereby a group
+of mentioned instances can appear at multiple different lo-
+cations. Thus, simply describing a cell with its contained in-
+stances can result in less discriminative representations. On
+the other hand, the textual queries often contain limited clues
+compared to the point clouds, making this cross-modality re-
+trieval especially challenging. To this end, we propose to ex-
+plicitly encode instance-relations to provide more discrimi-
+native representations for both modalities.
+The Transformer (Vaswani et al. 2017) has been widely
+utilized for relation-based representation learning in vari-
+ous tasks (Hu et al. 2018; Liu et al. 2021; Fan, Yang, and
+Kankanhalli 2022). The key component of the Transformer
+is the Self-Attention (SA) operation:
+Attn(Q, K, V ) = Softmax(QKT /
+√
+d)V ,
+(1)
+
+Pooling
+Matmul
+Add
+Figure 3: Illustration of the proposed Relation-enhanced
+Self-Attention (RSA) mechanism. Pairwise relations are ex-
+plicitly encoded into the value computation process.
+where d is the representation dimension and Q, K, V
+∈
+RN×d are the query, key and value matrices by transform-
+ing in-cell instances (or hints for textual queries) with corre-
+sponding linear transformations:
+Q = W QX, K = W KX, V = W V X,
+(2)
+with W ∗ ∈ Rd×d are learnable matrices and X = P ∈
+Rp×d or H ∈ Rh×d represents stacked instances1.
+Despite its generality, the vanilla SA lacks explicit rela-
+tions in both modalities, thus is less informative to represent
+the cell and query. To this end, we propose a novel Relation-
+Enhanced Transformer (RET) to model explicit instance re-
+lations in both point clouds and textual descriptions. Our
+RET is a stack of multiple Transformer encoder layers, ex-
+cept that, in place of SA, we propose a Relation-enhanced
+Self-Attention (RSA) to explicitly incorporate relation in-
+formation into value computation. The computation process
+is shown as follows and illustrated in Figure 3.
+RSA(Q, K, V , R) = Softmax(QKT /
+√
+d)(V +Pool(R, 1)),
+(3)
+where R ∈ RN×N×d captures pairwise relations with
+Rij ∈ Rd representing the relation between the i-th and j-
+th instance (hint). Pool(R, 1) indicates pooling tensor R
+along dimension 1. In this way, our model can explicitly
+encode instance relations through this computation process,
+leading to more informative representations.
+The definition of relation varies flexibly with task objec-
+tive and input modality. For point cloud data, we take the
+geometric displacement of two instances as their relations,
+as direction is often mentioned in textual queries and thus
+informative for retrieval:2
+RV
+ij = W V (ci − cj),
+(4)
+where ci ∈ R3 represents the center coordinate of the i-th
+instance and W v ∈ Rd×3 transforms the displacement into
+1Note that the attention operation is often performed in different
+subspaces with multiple heads, which is omitted for simplicity.
+2We have also tried other features such as number of points
+and bounding boxes of instances but didn’t observe performance
+improvement.
+embedding space. For the linguistic description, we compute
+the hint relation as the concatenation of their embeddings:
+RL
+ij = W L[hi; hj],
+(5)
+where W L ∈ Rd×2d transforms the linguistic feature into
+representation space. With the computation of RSA, the
+instance-wise relations for different modalities can be uni-
+formly incorporated into query or cell representations
+Finally, the cell (description) representations Cm (Tm) are
+obtained via a pooling operation over all instances (hints)
+output from the RET for cross-modal retrieval.
+Fine Stage: Cascaded Matching and Refinement
+Following the coarse stage, we aim to refine the location pre-
+diction within the retrieved cell in the fine stage. Inspired
+by (Kolmet et al. 2022), we perform instance matching and
+location refinement to utilize the fine-grained visual and lin-
+guistic information, which involves the following two objec-
+tives: (1) For each hint, we find the in-cell instance it refers
+to via a matching process. (2) For each matched pair (i, j),
+a regressor predicts an offset ˆti ∈ R2 for each matched hint
+hj, which represents the offset from the instance center ci
+to the target location.3
+Previous method (Kolmet et al. 2022) achieves the two
+objectives within a single step. However, given the objec-
+tive of both hint-instance matching and offset prediction,
+the multi-task learning process introduces optimization dif-
+ficulty. Furthermore, in the early training steps, the matcher
+is only partially trained, which produces noisy matching re-
+sults. The regressor learns and makes predictions based on
+this noisy results, leading to unstable learning process and
+sub-optimal performance.
+To this end, we propose a Cascaded Matching and Refine-
+ment (CMR) strategy for the fine stage, where hint-instance
+matching and offset regression are sequentially performed.
+Specifically, following (Kolmet et al. 2022), we first train
+the SuperGlue (Sarlin et al. 2020) matcher for hint-instance
+matching, which is formulated as an optimal-transport prob-
+lem. Given the trained matcher, we obtain a set of hint-
+instance matching results {pi, hj, wi}h
+j=1, where wi repre-
+sents the confidence of the match. Then, to reduce the noise
+for regression, we predict the target location according to
+matched instances only.
+Precise location prediction requires proper understand-
+ing on both point cloud (what and where the referred in-
+stance is, e.g., dark-green terrain) and language de-
+scription (what is the relation between the matched instance
+and the target location, e.g., east of). For this, we pro-
+pose to facilitate cross-modal collaboration via the Cross-
+Attention (CA) mechanism, which is commonly used for
+cross-modality information fusion.
+CA(H, P ) = Attn(W QH, W KP , W V P ),
+(6)
+where H, P represent hints and instances, respectively, and
+W ∗ are learnable transformation matrices. Shortcut connec-
+tion and layer normalization (Ba, Kiros, and Hinton 2016)
+3For position prediction, we ignore the height information and
+considers 2D coordinates only.
+
+Table 1: Performance comparison on the KITTI360Pose.
+Method
+Localization Recall (ϵ < 5/10/15m) ↑
+Validation Set
+Test Set
+k = 1
+k = 5
+k = 10
+k = 1
+k = 5
+k = 10
+Text2Pos (Kolmet et al. 2022)
+0.14/0.25/0.31
+0.36/0.55/0.61
+0.48/0.68/0.74
+0.13/0.21/0.25
+0.33/0.48/0.52
+0.43/0.61/0.65
+RET (Ours)
+0.19/0.30/0.37
+0.44/0.62/0.67
+0.52/0.72/0.78
+0.16/0.25/0.29
+0.35/0.51/0.56
+0.46/0.65/0.71
+follows the cross-attention operation. With these operations,
+the hint representation hi is accordingly updated to ˜hi by
+dynamically fusing visual information. As such, the infor-
+mation in the two modalities are joint utilized with the help
+of cross-modal collaboration.
+Then, we predict the offset (the direction vector from in-
+stance center to target location) from the updated hint:
+ˆti = MLP(˜hj).
+(7)
+To utilize the matching results, the final prediction is ob-
+tained via a weighted combination of each hint’s prediction:
+ˆg =
+�
+i
+wi
+�
+m wm
+(ci + ˆti),
+(8)
+where wi ∈ [0, 1] is the confidence score of the match
+(pi, hj, wi) and is set to 0 for non-matched instances. To
+filter out noisy matches, we consider only matches with con-
+fidence score greater than 0.2.
+Training and Inference
+Training. For the coarse stage, we train the proposed RET
+for cross-modal retrieval with pairwise ranking loss (Kiros,
+Salakhutdinov, and Zemel 2014):
+Lcoarse =
+Nb
+�
+m=1
+Nb
+�
+n̸=m
+[α − ⟨Cm, Tm⟩ + ⟨Cm, Tn⟩]+
++
+Nb
+�
+m=1
+Nb
+�
+n̸=m
+[α − ⟨Tm, Cm⟩ + ⟨Tm, Cn⟩]+,
+(9)
+where Nb is the batch size, α is a hyper-parameter to con-
+trol the separation strength and ⟨·, ·⟩ represents inner product
+between vectors. This loss function encourages the represen-
+tation of matched description-cell pair to be by a margin α
+closer than those unmatched. For the fine stage, we employ
+the loss in (Sarlin et al. 2020) to train the matcher, while L2
+loss is applied to train the offset regressor.
+Inference. We first encode all cells and queries into a joint
+embedding space with the proposed Relation-Enhanced
+Transformer. Then, for each query representation, we re-
+trieve top-k cells with highest similarity. For each retrieved
+cell, we use the SuperGlue matcher trained in the fine stage
+to match each hint with an in-cell instance, which is fol-
+lowed by offset prediction based on the fused representa-
+tions. Finally, the position prediction is given by Eq. 8.
+Experiments
+Dataset and Implementation Details
+Dataset Details. We evaluate our method on the recently
+proposed KITTI360Pose dataset (Kolmet et al. 2022), which
+is built upon the KITTI360 dataset (Liao, Xie, and Geiger
+2021) with sampled locations and generated hints. It con-
+tains point clouds of a total of 9 scenes, covering 14,934
+positions with a total area of 15.51km2. We follow (Kol-
+met et al. 2022) to use five scenes for training, one for val-
+idation, and the remaining three for testing. We sample the
+cells of size 30m with a stride of 10m. For more details on
+the dataset preprocessing, please refer to our supplementary
+material.
+Implementation Details For the coarse stage, we trained
+the model with AdamW optimizer (Loshchilov and Hutter
+2018) with a learning rate of 2e-4. The models are trained
+for a total of 18 epochs while the learning rate is decayed
+by 10 at the 9-th epoch. The α is set to 0.35. For the fine
+stage, we first train the matcher with a learning rate of 5e-
+4 for a total of 16 epochs. Afterwards, we fix the matcher
+and train the regressor based on the matching results for 10
+epochs with a learning rate of 1e-4. The regressor is for-
+mulated as a 3 layer Multi-Layer Perceptron. Both of the
+two steps adopt an Adam (Kingma and Ba 2014) optimizer.
+The RET has 2 encoder layers for both point cloud part and
+linguistic part, each utilizing the Relation-enhanced Atten-
+tion (RSA) mechanism with 4 heads and hidden dimension
+2048. For the two stages, we encode each instance in the cell
+with PointNet++ (Qi et al. 2017) provided by Text2Pos (Kol-
+met et al. 2022) for a fair comparison. The hint representa-
+tions are obtained by concatenating learned word embed-
+dings. More details are provided in our appendix.4
+Comparison with the State-of-the-art
+We compared our method with Text2Pos (Kolmet et al.
+2022) on the KITTI360Pose dataset. Following (Kolmet
+et al. 2022), we report top-k (k = 1/5/10) recall rate of dif-
+ferent error ranges ϵ < 5/10/15m for comprehensive com-
+parison. The results are shown in Table 1. Text2Pos gives
+a recall of 0.14 when k = 1 and ϵ < 5m. In contrast,
+our method can significantly improve the recall rate to 0.19,
+which amounts to 35.7% relative improvement upon the
+baseline. Furthermore, when we relax the localization error
+constraints or increase k, consistent improvements upon the
+baseline can also be observed. For example, with ϵ < 5m,
+our method achieves top-5 recall rate of 0.44, which is 8%
+higher than previous state-of-the-art. Similar improvements
+can also be seen on the test set, showing our method is su-
+perior to the baseline method.
+Ablation Studies
+In this section, we perform ablation studies for both stages
+to investigate the effectiveness of each proposed component
+4Code available at: https://github.com/daoyuan98/text2pos-ret
+
+Table 2: Ablation study of the Relation-Enhanced Trans-
+former (RET) on KITTI360Pose validation set. ”wo X rela-
+tion” indicates replacing the proposed RSA with the vanilla
+Self-Attention in corresponding modality.
+Method
+k = 1 ↑
+k = 3 ↑
+k = 5 ↑
+w/o both relations
+0.11
+0.24
+0.32
+w/o linguistic relation
+0.14
+0.28
+0.37
+w/o visual relation
+0.16
+0.30
+0.40
+Full (Ours)
+0.18
+0.34
+0.44
+Table 3: The effects of #layers of RET and #heads of RSA.
+#Layers
+#Heads
+k = 1 ↑
+k = 3 ↑
+k = 5 ↑
+1
+4
+0.16
+0.31
+0.40
+1
+8
+0.16
+0.30
+0.40
+2
+2
+0.17
+0.32
+0.42
+2
+4
+0.18
+0.34
+0.44
+2
+8
+0.16
+0.31
+0.40
+3
+4
+0.16
+0.32
+0.39
+3
+8
+0.15
+0.29
+0.37
+in our method. The ablation studies for coarse stage and fine
+stage are provided separately for clear investigation.
+Coarse Stage. We study the importance of explicit relation
+incorporation in the coarse stage. Since the coarse stage is
+formulated as a retrieval task, we use top-1/3/5 recall rate as
+evaluation metric, whereby the cell that contains the ground
+truth location is defined as positive.
+Relation Incorporation. We first study the necessity of ex-
+plicit relation modeling for both point cloud and textual
+queries. The results are shown in Table 2. It can be observed
+that relation modeling contributes significantly to successful
+retrieval. In particular, without any relation incorporation,
+the top-5 recall rate is 0.32. With the explicit fusion of lin-
+guistic relation, we observe an increase of 0.05 recall rate
+under same condition. Besides, with the incorporation of vi-
+sual (point cloud instance) relations only, the top-5 recall
+rate can be improved by 0.08, indicating explicit relations
+in the point clouds play a more important role. Finally, with
+both relations, we achieve an improvement of 0.12 at top-5
+recall rate upon that without any relation, showing that both
+visual and linguistic relations are necessary and complemen-
+tary to improve the cell retrieval performance.
+RET Hyper-parameters. We also studied the importance of
+the hyper-parameters involved in RET, namely the number
+of layers of RET and the number of heads of RSA. The re-
+sults are shown in Table 3. It can be observed that, thanks to
+the strong relation modeling capacity of the proposed RET,
+we can obtain the best performance with 2 layers and 4 heads
+in the RSA. Decreasing and increasing the number of layers
+both lead to worse performance, which may be attributed to
+underfitting and overfitting, respectively.
+Fine Stage. The objective of the fine stage is to correctly
+match linguistic hints and point cloud instances and regress
+the target location. Thus, we study the performance of the
+matcher and regressor, respectively.
+Table 4: Comparison of training strategy and matcher per-
+formance on the KITTI360Pose dataset.
+Strategy
+Train
+Validation
+Precision ↑
+Recall ↑
+Precision ↑
+Recall ↑
+joint
+98.12
+98.16
+86.67
+87.59
+cascade(ours)
+98.89
+99.04
+92.18
+93.01
+Table 5: Ablation study on the regression error of the fine-
+stage on the KITTI360Pose dataset.
+Method
+Train Error ↓
+Validation Error ↓
+w/o cascade training
+10.24 (+1.72)
+10.01 (+0.86)
+w/o cross-attention
+9.57 (+1.05)
+9.56 (+0.41)
+w/o confidence weighting
+9.02 (+0.50)
+9.23 (+0.08)
+Ours
+8.52
+9.15
+Matcher. Following (Sarlin et al. 2020), we take precision
+and recall as the the evaluation metric of the matcher. With
+an identical matcher architecture, we investigate the impact
+of training strategy on the matcher performance. The results
+are shown in Table 4. It can be seen that compared with joint
+training (Kolmet et al. 2022), our cascaded training achieves
+not only high precision and recall in the training set, but
+also stronger generalization on the validation set. The re-
+sults demonstrate that the cascade training strategy is able to
+mitigate the multi-task optimization difficulty.
+Regressor. The regressor predicts the target location based
+on the the matching results. We study the effects of cas-
+caded training, cross-attention based cross-modal fusion and
+confidence weighting for final location prediction. We use
+regression error as evaluation metric and compare different
+versions on both KITTI360Pose training and validation set.
+The results are shown in Table. 5. Without cascaded training
+strategy, the regressor achieves an error of 10.24 and 10.01
+on the training and validation set, respectively, which is 1.72
+and 0.86 higher than that with cascaded training. This re-
+sult suggests that our cascaded training strategy also allevi-
+ates the optimization difficulty of the regressor, which was
+caused by the noisy intermediate results. Furthermore, with-
+out cross-attention mechanism, the regression error also in-
+creases by a considerable margin, showing that cross-modal
+collaboration is important for precise location prediction. Fi-
+nally, with confidence-based weighting, we can further re-
+duce the regression error on both the training and validation
+set, suggesting this information from the trained matcher can
+be further utilized to improve performance.
+Visualizations
+Embedding Space Visualization. We visualize the learned
+embedding space via T-SNE (Van der Maaten and Hin-
+ton 2008) in Figure 5. It can be observed that the base-
+line method Text2Pos (Kolmet et al. 2022) results in a less
+discriminative space, where positive cells are relatively far
+away from the query and sometimes separated across the
+embedding space. In contrast, our method draw positive cell
+and query representations closer in the embedding space, re-
+sulting in a more informative embedding space for retrieval.
+
+Ground Truth
+Top-1
+Top-2
+Top-3
+Ground Truth
+Top-1
+Top-2
+Top-3
+(a) 
+(b) 
+(c) 
+(e) 
+(d) 
+(f) 
+557.85
+10.00
+20.00
+0.00
+10.00
+50.99
+10.00
+0.0
+819.08
+10.00
+0.00
+64.03
+14.14
+211.90
+221.36
+455.41
+1150.00
+218.40
+Building
+Pole
+Traffic Light
+Traffic Sign
+Parking
+Sidewalk
+Vegetation
+Terrain
+Road
+Wall
+Garage
+Figure 4: Qualitative retrieval results on KITTI360Pose validation set. The red dot in the ground truth cell indicates the target
+location. In each retrieved cell, the number in the lower right indicates the center distance between this cell and the ground
+truth. Green box indicates positive cell which contains the target location, while red box indicates negative cells.
+Text2Pos
+Ours
+Textual Query
+Negative Cell
+Positive Cell
+Figure 5: T-SNE visualization of embedding space for the
+coarse stage. A cell is considered as positive if it contains
+the location described by the query. Compared with baseline
+method (Kolmet et al. 2022), our method can produce better
+representation where positive cells are closer to the target.
+Qualitative Cell Retrieval Results. We show some exam-
+ple text to point cloud retrieval results in Figure. 4. For a
+given query, we visualize the top-3 retrieved cells. A re-
+trieved cell is defined as positive if it contains the target lo-
+cation. It can be observed that, our method can retrieve the
+ground truth cell or those close in most cases. Sometimes,
+negative cells can also be retrieved, e.g., top-1 in (a) and
+top-3 in (e). It can be seen that these retrieved negative cells
+exhibit high semantic similarity with the ground truth cell,
+even though far away from it. We also show a failure case (f),
+where the retrieved cells are all negative. It can be seen that
+even though far away from the target location, all these neg-
+ative cells have instances similar to the ground truth. These
+observations suggest that outdoor scenes are indeed of low
+diversity, indicating that successful retrieval requires highly
+discriminative representations to disambiguate the cells.
+Conclusion
+In this work, we proposed a novel method for precise
+text-based localization from large-scale point clouds. Our
+method employs a coarse-to-fine principle and pipelines this
+process into two stages. For the coarse stage which is formu-
+lated as a textual query based cell retrieval task, we aim to
+improve representation discriminability for both point cloud
+and query representations. This is achieved through explicit
+modeling of instance relations and implemented via a newly
+proposed Relation-Enhanced Transformer (RET). The core
+of RET is a novel Relation-enhanced Self-Attention (RSA)
+mechanism, whereby the instance relations for the two
+modalities are explicitly incorporated into the value com-
+putation process in a unified manner. For the fine stage,
+our method performs description-instance matching and
+position refinement in a cascaded way, whereby cross-
+modal information collaboration is enhanced through the
+cross-attention mechanism. Extensive experiments on the
+KITTI360Pose dataset validated the effectiveness of the pro-
+posed method, which achieves new state-of-the-art perfor-
+mance. Additional ablation studies further corroborate the
+effectiveness of each component in the proposed method.
+Acknowledgement
+This research is supported by the National Research Foun-
+dation, Singapore under its Strategic Capability Research
+Centres Funding Initiative. Any opinions, findings and con-
+clusions or recommendations expressed in this material are
+those of the author(s) and do not reflect the views of National
+Research Foundation, Singapore.
+
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+arXiv:2301.11864v1  [physics.flu-dyn]  27 Jan 2023
+Under consideration for publication in J. Fluid Mech.
+1
+Gravity can lead to multiple peaks in the early stages
+of coffee ring formation
+M. R. M O O R E1
+AND A. W.
+W R A Y2
+1Department of Mathematics, School of Natural Sciences, University of Hull, Cottingham Road, Hull, HU6 7RX,
+UK
+2Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street,
+Glasgow G1 1XH, UK
+(Received ?; revised ?; accepted ?. - To be entered by editorial office)
+We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically-
+relevant assumptions that the contact line is pinned and the solutal P´eclet number, Pe is large, we identify
+two fundamental regimes that depend on the size of the Bond number, Bo. When Bo = O(1), the asymptotic
+structure of solute transport follows directly from the surface tension-dominated regime, whereby advection
+drives solute towards the contact line, only to be countered by local diffusive effects, leading to the for-
+mation of the famous “coffee ring”. For larger Bond numbers, we identify the distinguished limit in which
+Bo−1/2Pe2/3 = O(1), where the diffusive boundary layer is comparable to the surface tension boundary
+layer. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare
+our predictions to numerical simulations of the full model. Our analysis identifies the effect of gravity on
+the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass
+profile. Furthermore, we reveal that, for certain values of Bo, Pe and the evaporation time, a secondary
+peak may exist inside the classical coffee ring. We find that the onset of this secondary peak is linked to the
+change in behaviour of the critical point in the droplet centre. Both the onset and the peak characteristics
+are shown to be independent of Pe, but solutal diffusion may act to remove the secondary peak when the
+classical coffee ring becomes so large as to subsume it.
+Key words:
+1. Introduction
+The evaporation of sessile droplets has received significant attention in recent years, being the subject
+of several major reviews (Cazabat & Guena 2010; Lohse et al. 2015; Brutin & Starov 2018; Wilson &
+D’Ambrosio 2023) due to its ubiquity in theoretical, experimental and industrial settings. A particular
+phenomenon of interest is the so-called “coffee ring effect”, in which a solute in such an evaporating droplet
+ends up preferentially accumulated at the contact line (Deegan et al. 1997, 2000). This effect is very robust,
+occurring even when the solution is initially uniformly dispersed throughout the droplet, and even when the
+evaporative flux is not preferentially localised at the contact line (Boulogne et al. 2016).
+Motivated by typical physical parameters, models of such systems typically assume that the P´eclet number
+is sufficiently large that diffusive effects can be neglected, and so dynamics of the solute inside the droplet
+are governed purely by convection (Deegan et al. 1997; Wray et al. 2021). This unphysical assumption leads
+to a variety of undesirable side-effects, including singular accumulations of residue, and solute not being
+conserved (Deegan et al. 2000).
+A variety of attempts have been made to resolve this problem phenomenologically, including via the
+incorporation of jamming effects (Popov 2005; Kaplan & Mahadevan 2015). However, jamming effects only
+become significant close to the particle packing fraction, and the assumptions underpinning the model fail
+long before this point. In particular, the assumption that diffusive effects can be ignored breaks down in a
+diffusive boundary layer close to the contact line (Moore et al. 2021), as might be anticipated from the singular
+accumulation in the na¨ıve, convection-only model. This boundary layer and its growth and dynamics have
+been analysed and understood via matched asymptotics and careful numerics in situations where droplets
+are small, and thus exist at quasi-static equilibrium due to surface tension (Moore et al. 2022), but little is
+known for larger droplets where the effects of gravity are important.
+Investigations of larger droplets have a long history, dating back to numerical integration of the appropriate
+Laplace equations by Padday (1971) and Boucher & Evans (1975), with a variety of studies via asymptotics of
+
+2
+M. R. Moore & A. W. Wray
+their shape (Rienstra 1990; O’Brien 1991; Yariv 2022) and stability (Pozrikidis 2012) in the intervening time.
+The effect of gravity on droplets, and especially their internal flows, has experienced a recent resurgence of
+interest due to the experiments of Edwards et al. (2018), which showed that the dynamics of binary droplets
+can be sensitively dependent on droplet inclination (and hence gravity). This has since received extensive
+investigation both experimentally and numerically (Li et al. 2019; Pradhan & Panigrahi 2017).
+Notably, however, despite the original experiments of Deegan et al. (1997) involving large droplets, there
+have been relatively few investigations of particle transport inside them, with those available being principally
+experimental (Sandu & Fleaca 2011; Hampton et al. 2012; Devlin et al. 2016). This is perhaps because of the
+robustness of the coffee-stain effect: asymptotic and numerical investigations (Barash et al. 2009; Kolegov
+& Lobanov 2014) confirm the experimental results that the ring-stain is preserved unless additional physics
+are incorporated, such as continuous particle deposition (Devlin et al. 2016). However, this neglects the bulk
+of the story, including the dynamics of the residue over the course of the lifetime of the droplets: a critical
+omission in situations such as continuous particle deposition. We show in the present work that the dynamics
+are actually quite subtle and complex, and certainly merit detailed investigation.
+The structure of this paper is therefore as follows. In §2, we describe the equations governing the fluid
+flow and solute transport for the problem of a thin droplet evaporating under a diffusive flux, in particular
+highlighting the effect of gravity in the model. We nondimensionalise the model and introduce the three key
+dimensionless numbers in the model: the capillary, Bond and P´eclet numbers. In §3, we completely solve
+for the liquid flow in the limit in which the solute is dilute, so that the flow and solute transport problems
+decouple. We discuss pertinent features of the resulting fluid velocity and droplet shape, and in particular
+how these features vary with the Bond number.
+The bulk of the analysis in this paper concerns the influence of gravity on solute transport within the
+droplet, which we analyse in the physically-relevant large-P´eclet number limit in §4. We find that there are
+two distinct regimes depending on the relative sizes of the Bond and P´eclet numbers. In the first, where
+the Bond number is moderate, we extend the asymptotic analysis of Moore et al. (2021) to include the
+effect of gravity. However, when the Bond number is also large, a more complex asymptotic analysis is
+necessary, which is presented in detail in Appendix A. In each asymptotic regime, we derive predictions for
+the distribution of the solute mass within the droplet and compare the results to numerical simulations of
+the full advection-diffusion problem. In particular, while we find the expected ‘nascent coffee ring’ profile in
+the solute mass, for certain input parameters, we also find evidence of a novel phenomenon whereby a second
+peak may also develop in the mass profile inside the classical coffee ring.
+We analyse both of these peaks in detail in §5. In particular, for the classical coffee ring, we discuss
+the effect of gravity in each of the two asymptotic regimes discussed in §4 and Appendix A, while for the
+secondary peak, we investigate the key role gravity plays in its existence and how the secondary peak may
+also be subsumed in the classical coffee ring for certain values of the Bond and P´eclet numbers. Finally, in
+§6, we summarize our findings and discuss implications to various applications, as well as avenues for future
+study.
+2. Problem configuration
+We consider the configuration depicted in figure 1 in which an axisymmetric droplet of initial volume
+V ∗
+0 evaporates from a solid substrate. Here and hereafter, an asterisk denotes a dimensional variable. We
+let (r∗, θ, z∗) be cylindrical polar coordinates centred along the line of symmetry of the droplet with the
+substrate lying in the plane z∗ = 0: by axisymmetry, we shall assume that all the variables are independent
+of θ. The droplet contact line is thus circular and we assume that it is pinned throughout the drying process,
+which is observed in practice for a wide range of liquids for the majority of the drying time (Deegan et al.
+1997; Hu & Larson 2002; Kajiya et al. 2008; Howard et al. 2023). We let r∗ = R∗ be the radius of the contact
+line. Throughout this analysis, we shall assume that the droplet is thin, which reduces to the assumption
+that
+0 < δ = V ∗
+0
+R∗3 ≪ 1.
+(2.1)
+As we discuss presently, the thin-droplet assumption allows us to greatly simplify the flow and solute transport
+models; the assumption has been extensively-validated and has shown to be reasonable even for droplets that
+should realistically fall outside of this regime (Larsson & Kumar 2022).
+The droplet consists of a liquid of constant density and viscosity denoted by ρ∗ and µ∗, respectively. The
+droplet free surface is denoted by z∗ = h(r∗, t∗) and the air-water surface tension coefficient, σ∗ is assumed
+to be constant.
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+3
+z∗
+r∗
+2R∗
+z∗ = h∗(r∗, t∗)
+E∗(r∗)
+Figure 1: A side-on view of a solute-laden droplet evaporating under an evaporative flux E∗(r∗) from a solid
+substrate that lies in the plane z∗ = 0. The droplet is axisymmetric and the contact line is assumed to be
+pinned on the substrate at r∗ = R∗. The droplet free surface is denoted by h∗(r∗, t∗). The solute is assumed
+to be inert and sufficiently dilute that the flow of liquid in the droplet is decoupled from the solute transport.
+The liquid evaporates into the surrounding air and we assume that the evaporative process is quasi-steady,
+which is a reasonable assumption for a wide range of liquid-substrate configurations (Hu & Larson 2002).
+While there are a number of different viable evaporation models depending on the physical and chemical
+characteristics of the problem (Murisic & Kondic 2011), for the purposes of this analysis, we assume that
+the dominant process of vapour transport from the droplet surface is diffusion, so that the evaporative flux
+E∗(r∗) is given by
+E∗(r∗) = 2D∗(c∗
+s − c∗
+∞)
+π
+√
+R∗2 − r∗2 ,
+(2.2)
+where D∗ is the diffusion coefficient and c∗
+s, c∗
+∞ are the surface and ambient vapour concentrations, respec-
+tively (Deegan et al. 2000; Murisic & Kondic 2011).
+The droplet contains an inert solute of initially uniform concentration φ∗
+0. The solute is assumed to be
+sufficiently dilute that the flow and transport problems completely decouple. We shall discuss the validity of
+the dilute assumption further in §6.
+2.1. Flow model
+The droplet is assumed to be sufficiently thin and the evaporation-induced flow sufficiently slow that the
+flow is governed by the lubrication equations
+∂h∗
+∂t∗ + 1
+r∗
+∂
+∂r∗ (r∗h∗u∗) = − E∗
+ρ∗ ,
+(2.3)
+u∗ = − h∗2
+3µ∗
+∂p∗
+∂r∗ ,
+(2.4)
+p∗ = p∗
+atm − ρ∗g∗(z∗ − h∗) − σ∗ 1
+r∗
+∂
+∂r∗
+�
+r∗ ∂h∗
+∂r∗
+�
+,
+(2.5)
+for 0 < r∗ < R∗, t∗ > 0, where u∗(r∗, t∗) is the depth-averaged radial fluid velocity, p∗(r∗, z∗, t∗) is the liquid
+pressure and p∗
+atm denotes atmospheric pressure (Hocking 1983; Deegan et al. 2000; Oliver et al. 2015).
+Equations (2.3)–(2.5) must be solved subject to the symmetry conditions
+r∗h∗u∗ = ∂h∗
+∂r∗ = 0
+at
+r∗ = 0,
+(2.6a, b)
+and the fact that the free surface touches down at, and we require no-flux of liquid through, the pinned
+contact line, that is
+h∗ = r∗h∗u∗ = 0
+at
+r∗ = R∗.
+(2.7a, b)
+We close the problem by specifying the initial droplet profile, that is
+h∗(r∗, 0) = h∗
+0(r∗)
+for
+0 < r∗ < R∗.
+(2.8)
+It is worth noting at this stage that, while this initial condition is needed to fully specify the mathematical
+problem, in our analysis, we do not explicitly use the initial condition (2.8). In what follows, it is assumed
+
+4
+M. R. Moore & A. W. Wray
+that the rate of evaporation is sufficiently slow that the droplet quickly relaxes under capillary action to the
+quasi-steady profile found in §3 (see, for example, Lacey (1982); De Gennes (1985); Oliver et al. (2015)).
+Thus, we shall for simplicity assume that h0(r) is of the same functional form of the free surface we find in
+§3. While this assumption is reasonable for a wide range of applications, for extremely rapid evaporation (for
+example, laser-induced evaporation, Volkov & Strizhak (2019)), a more careful consideration of the evolution
+after deposition would be needed.
+Assuming the contact line is pinned, the volume of the droplet V ∗(t∗) is given by
+V ∗(t∗) = 2π
+� R∗
+0
+r∗h∗(r∗, t∗) dr∗,
+V ∗(0) = V ∗
+0 .
+(2.9)
+The total mass loss due to evaporation F ∗(t∗) is given by
+F ∗(t∗) = 2π
+� R∗
+0
+r∗E∗(r∗) dr∗ = 4D∗(c∗
+s − c∗
+∞)R∗.
+(2.10)
+Thus, conservation of mass in the liquid phase is
+dV ∗
+dt∗ = −F ∗
+ρ∗ = −4D∗(c∗
+s − c∗
+∞)R∗
+ρ∗
+(2.11)
+so that
+V ∗(t∗) = V ∗
+0 − 4D∗(c∗
+s − c∗
+∞)R∗t∗
+ρ∗
+.
+(2.12)
+In particular, the dryout time, that is the time when the drop has fully evaporated, is
+t∗
+f =
+ρ∗V ∗
+0
+4D∗(c∗s − c∗∞)R∗ .
+(2.13)
+2.2. Solute model
+The droplet is assumed to be sufficiently thin that the transport of the solute is governed by the depth-
+averaged advection-diffusion equation
+∂
+∂t∗ (h∗φ∗) + 1
+r∗
+∂
+∂r∗
+�
+r∗
+�
+h∗u∗φ∗ − D∗
+φh∗ ∂φ∗
+∂r∗
+��
+= 0
+(2.14)
+for 0 < r∗ < R∗, t∗ > 0, where φ∗(r∗, t∗) is the depth-averaged solute concentration and D∗
+φ is the solutal
+diffusion coefficient (Wray et al. 2014; Pham & Kumar 2017; Moore et al. 2021).
+While there is an acknowledged effect of the solute particles eventually being trapped at and transported
+along the free surface (Kang et al. 2016; D’Ambrosio 2022), this effect is less pronounced for thin droplets,
+where the capture tends to occur closer to the contact line due to the stronger outward radial flow. Thus, we
+shall neglect its effects here as our study concerns the interplay between gravity, surface tension and solute
+advection/diffusion. A more focused analysis on the final deposit profile would certainly need to account for
+such effects.
+Equation (2.14) must be solved subject to the symmetry condition
+∂φ∗
+∂r∗ = 0
+at
+r∗ = 0,
+(2.15)
+and the condition that there can be no flux of solute particles through the pinned contact line,
+r∗
+�
+h∗u∗φ∗ − D∗
+φh∗ ∂φ∗
+∂r∗
+�
+= 0
+at
+r∗ = R∗.
+(2.16)
+Finally, we impose the initially uniform distribution of solute throughout the droplet, so that
+φ∗(r∗, 0) = φ∗
+0
+for
+0 < r∗ < R∗.
+(2.17)
+2.3. Non-dimensionalization
+We assume that the fluid velocity is driven by evaporation and, for now, we retain both gravity and surface
+tension, so that the pertinent scalings are
+(r∗, z∗) = R∗(r, δz),
+u∗ = D∗(c∗
+s − c∗
+∞)
+δρ∗R∗
+u,
+t∗ = t∗
+ft,
+φ∗ = φ∗
+0φ,
+(h∗, h∗
+0) = δR∗(h, h0),
+p∗ = p∗
+atm + µ∗D∗(c∗
+s − c∗
+∞)
+δ3ρ∗R∗2
+p
+V ∗ = V ∗
+0 V.
+(2.18)
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+5
+Note, in particular, that the choice of timescale fixes the dimensionless dryout time to be t = 1.
+Upon substituting the scalings (2.18) into (2.3)–(2.5), we see that
+∂h
+∂t + 1
+4r
+∂
+∂r (rhu) = −
+1
+2π
+√
+1 − r2 ,
+(2.19)
+u = h2
+3Ca
+∂
+∂r
+�
+−Boh + 1
+r
+∂
+∂r
+�
+r∂h
+∂r
+��
+,
+(2.20)
+for 0 < r < 1, 0 < t < 1, where the Capillary and Bond numbers are defined by
+Ca = µ∗D∗(c∗
+s − c∗
+∞)
+δ4ρ∗R∗σ∗
+and
+Bo = ρ∗g∗R∗2
+σ∗
+,
+(2.21)
+respectively.
+Under scalings (2.18), the symmetry conditions (2.6) become,
+rhu = ∂h
+∂r = 0
+at
+r = 0,
+(2.22a, b)
+while the contact line conditions (2.7) are
+h = rhu = 0
+at
+r = 1.
+(2.23a, b)
+The initial condition (2.8) becomes
+h(r, 0) = h0(r)
+for
+0 < r < 1.
+(2.24)
+Finally, the dimensionless form of conservation of liquid volume conditions (2.9) and (2.12) is
+1 − t = 2π
+� 1
+0
+rh(r, t) dr.
+(2.25)
+After scaling, the solute transport equation (2.14) becomes
+∂
+∂t (hφ) + 1
+4r
+∂
+∂r
+�
+r
+�
+huφ − h
+Pe
+∂φ
+∂r
+��
+= 0
+(2.26)
+for 0 < r <, 0 < t < 1, where the solutal P´eclet number is
+Pe = D∗(c∗
+s − c∗
+∞)
+δρ∗D∗
+φ
+.
+(2.27)
+The symmetry (2.15) and boundary conditions (2.16) become
+∂φ
+∂r = 0
+at
+r = 0
+(2.28)
+and
+r
+�
+huφ − h
+Pe
+∂φ
+∂r
+�
+= 0
+at
+r = 1,
+(2.29)
+respectively. Finally, the initial condition (2.17) becomes
+φ(r, 0) = 1
+for
+0 < r < 1.
+(2.30)
+2.4. Integrated mass variable formulation
+The assumption that the solute is dilute decouples the flow and solute transport problems, so that we may
+solve for h and u from (2.19)–(2.25) independently of the solute concentration, φ. We shall discuss the
+resulting flow solution shortly in §3.
+First, however, we present a reformulation of the solute transport problem (2.26)–(2.30), which will greatly
+aid us in our asymptotic and numerical investigations. In this, we follow Moore et al. (2021, 2022) by
+introducing the integrated mass variable
+M(r, t) =
+� r
+0
+sh(s, t)φ(s, t) ds.
+(2.31)
+By integrating the advection-diffusion equation (2.26) from 0 to r and applying the no-flux condition (2.29),
+
+6
+M. R. Moore & A. W. Wray
+we find that
+∂M
+∂t +
+�u
+4 +
+1
+4Pe
+�1
+r + 1
+h
+∂h
+∂r
+�� ∂M
+∂r −
+1
+4Pe
+∂2M
+∂r2
+= 0
+for
+0 < r, t < 1.
+(2.32)
+This must be solved subject to the boundary conditions
+M(0, t) = 0,
+M(1, t) = 1
+2π
+for
+t > 0,
+(2.33a, b)
+where the latter condition dictates that mass is conserved along a radial ray, which replaces the no-flux
+condition (2.29). Finally, the initial condition (2.30) becomes
+M(r, 0) =
+� r
+0
+sh(s, 0) ds
+for
+0 < r < 1.
+(2.34)
+Finally, we note that, once we have determined the integrated mass variable from (2.32)–(2.34), the solute
+mass m = φh can then be retrieved from
+m = 1
+r
+∂M
+∂r .
+(2.35)
+3. Flow solution in the large-Ca limit
+We now suppose that surface tension dominates viscosity in the flow problem, that is Ca ≫ 1. Importantly,
+this means that the problems for the free surface profile and the flow velocity decouple, an assumption that
+is valid for a wide range of different liquids and evaporation models in practice (Moore et al. 2021, 2022).
+Unlike these previous studies, however, we shall retain gravity in (2.20) to investigate what role it plays in
+the formation of the nascent coffee ring.
+To this end, we neglect the left-hand side of (2.20), so that upon integrating and applying the symmetry
+condition (2.22), the contact line condition (2.23a) and the conservation of liquid volume condition (2.25),
+we deduce that
+h(r, t) = (1 − t)
+π
+I0(
+√
+Bo)
+I2(
+√
+Bo)
+�
+1 − I0(
+√
+Bo r)
+I0(
+√
+Bo)
+�
+,
+(3.1)
+where Iν(z) is the modified Bessel function of the first kind of order ν.
+With the free surface found, the velocity is determined immediately from (2.19) and the no-flux condition
+(2.23b) to be
+u(r, t) = 1
+rh
+�
+2
+π
+�
+1 − r2 + 4I0(
+√
+Bo)
+πI2(
+√
+Bo)
+�r2 − 1
+2
++
+1
+√
+BoI0(
+√
+Bo)
+(I1(
+√
+Bo) − rI1(
+√
+Bo r))
+��
+.
+(3.2)
+Notably, as in the surface tension-dominated regime where Bo → 0, time is separable in both the free
+surface and fluid velocity profiles, and so merely acts to scale the functional form. In particular, this means
+that the streamlines and pathlines coincide, which we shall exploit when considering the regime in which
+solutal diffusion is negligible in §5.2.
+We display the scaled forms of the free surface and fluid velocity for various values of the Bond number
+in figure 2a,b. For the droplet free surface profile, we see the expected transition from the spherical cap for
+Bo → 0 (Deegan et al. 2000) to the flat ‘pancake’ droplet for Bo → ∞ (Rienstra 1990). For each Bond
+number, the velocity is singular at the contact line — as expected for a diffusive evaporative flux (see, for
+example, Deegan et al. (2000)). We see that as the effect of gravity increases, the sharp increase in u occurs
+closer to the contact line, corresponding to the progressively smaller region in which surface tension effects
+are important.
+Finally, since this will be important in our discussions of the secondary peaks seen in the solute mass profile
+in §5.2, we show the divergence of the fluid velocity in figure 2c. For small Bond numbers, the divergence is
+monotonically increasing with r and, as with the velocity, singular at the contact line. However, for moderate
+and large Bond numbers ≳ 15, we see a clear change of behaviour, with a region of non-monotonic behaviour
+in the droplet interior. This behaviour is accentuated as Bo → ∞.
+For future reference, the asymptotic behaviours of the free surface and fluid velocity as r → 1− for
+Bo = O(1) are given by
+h = θc(t; Bo)(1 − r) + O((1 − r)2),
+(3.3)
+u =
+2χ
+θc(t; Bo)(1 − r)−1/2 + O
+�
+(1 − r)1/2�
+,
+(3.4)
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+7
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+0
+1
+2
+3
+4
+5
+0
+0.5
+1
+0
+1
+2
+3
+4
+5
+Figure 2: (a) The quasi-steady droplet free surface, (b) the fluid velocity, and (c) the divergence of the
+velocity displayed for Bo = 0.1 (black), 1 (dark purple), 10 (blue), 20 (cyan), 50 (green) and 100 (yellow).
+Notably, we see the transition from the spherical cap to the ‘pancake’ droplet profile as the effect of gravity
+increases. The divergence of the fluid velocity also shows a transition from a monotonic to a non-monotonic
+profile as the Bond number increases.
+where
+θc(t; Bo) = − lim
+r→1−
+∂h
+∂r = (1 − t)ψ(Bo),
+ψ(Bo) =
+√
+BoI1(
+√
+Bo)
+πI2(
+√
+Bo)
+(3.5)
+is the leading order contact angle in the thin droplet limit and
+χ =
+√
+2
+π
+(3.6)
+is the dimensionless coefficient of the inverse square root singularity at the contact line in the evaporative
+flux (2.2). Note that we have chosen this notation to highlight the similarities with the previous analysis of
+Moore et al. (2022), who consider a surface tension-dominated droplet of arbitary contact set.
+On the other hand, if we take 1 − r = O(1) and consider the large-Bo limit of (3.1), (3.2), we find that
+h = h0(t) + Bo−1/2h1(t) + O(Bo−1),
+(3.7)
+u = u0(r, t) + Bo−1/2u1(r, t) + O(Bo−1),
+(3.8)
+as Bo → ∞, where
+h0(t) = (1 − t)
+π
+,
+h1(t) = 2(1 − t)
+π
+,
+(3.9a, b)
+and
+u0(r, t) = 2
+√
+1 − r2
+r(1 − t) (1 −
+�
+1 − r2),
+u1(r, t) =
+4
+r(1 − t)(1 −
+�
+1 − r2).
+(3.10a, b)
+Notably, in the droplet bulk, the droplet free surface h is flat to all orders: the aforementioned characteristic
+of ‘pancake’ droplets associated with large Bond numbers (Rienstra 1990). These expansions break down
+close to the contact line where surface tension effects become important. We find that for 1 − r = Bo−1/2¯r,
+we have
+h = ¯h0(¯r, t) + Bo−1/2¯h1(¯r, t) + O(Bo−1),
+(3.11)
+u = Bo−1/4 �
+¯u0(¯r, t) + Bo−1/4¯u1(¯r, t) + Bo−1/2¯u2(¯r, t) + O(Bo−3/4)
+�
+(3.12)
+as Bo → ∞, where
+¯h0(¯r, t) = 1
+π (1 − t)(1 − e−¯r),
+(3.13)
+¯h1(¯r, t) = 2(1 − t)
+π
+�
+1 − e−¯r�
+− (1 − t)¯r
+2π
+e−¯r,
+(3.14)
+
+8
+M. R. Moore & A. W. Wray
+and
+¯u0(¯r, t) =
+2
+√
+2¯r
+(1 − t)(1 − e−¯r),
+(3.15)
+¯u1(¯r, t) =
+4
+(1 − t) −
+4¯r
+(1 − t)(1 − e−¯r),
+(3.16)
+¯u2(¯r, t) =
+3¯r3/2
+√
+2(1 − t)(1 − e−¯r) −
+4
+√
+2¯r
+(1 − t)(1 − e−¯r) +
+√
+2¯r3/2e−¯r
+(1 − t)(1 − e−¯r)2 .
+(3.17)
+We note here that as ¯r → 0, we retrieve the expect inverse square root singularity in the fluid velocity.
+4. Solute transport in the large-Pe limit
+Having fully determined the leading-order flow, we now seek to understand the transport of solute within
+the drop and to make predictions about the early-stages of coffee ring formation. We follow the analyses of
+Moore et al. (2021, 2022) by considering the physically-relevant regime in which Pe ≫ 1. In this regime, in
+the bulk of the droplet, advection dominates solutal diffusion, with the latter only being relevant close to
+the contact line.
+Previous studies of this problem have concentrated on surface tension-dominated drops (i.e. Bo → 0) and
+have shown how the competition between solutal advection and diffusion near the contact line leads to the
+early stages of coffee ring formation in drying droplets. In this analysis, we wish to investigate how this
+behaviour changes as we allow Bo to vary, which we pursue using a hybrid asymptotic-numerical approach.
+There are naturally several different asymptotic regimes depending on the relative sizes of Bo and Pe, but
+these broadly fall into two categories
+i) intermediate Bond number, Bo = O(1), where the asymptotic structure of the solute transport depends
+solely on the large P´eclet number;
+ii) large Bond number, Bo ≫ 1, where the asymptotic structure of the solute transport now depends on
+the relative sizes of Bo and Pe.
+In the first regime where Bo = O(1), Pe ≫ 1, the asymptotic structure of the flow is a natural extension of
+the surface tension-dominated case considered in Moore et al. (2021). In the droplet bulk where 1−r = O(1),
+solute advection dominates diffusion. However, close to the contact line, a balance between solute advection
+and diffusion occurs when
+rhuφ ∼ rh
+Pe
+∂φ
+∂r =⇒ 1 − r = O(Pe−2).
+(4.1)
+We discuss the asymptotic solution for this regime in §4.1.
+In the second regime, there are several different possibilities depending on the relative sizes of the boundary
+layer where surface tension enters the flow profile and the solutal diffusion boundary layer. The richest
+distinguished asymptotic limit is that in which these boundary layers are comparable. As detailed in §3, for
+large Bond number the free surface is flat in the bulk of the droplet, with the effect of surface tension restricted
+to a boundary layer at the contact line of size 1 − r = O(Bo−1/2), where h = O(1) and u = O(Bo−1/4).
+Turning to the solute transport equation (2.26), since h is order unity and u is square root bounded in this
+region, advection and diffusion are comparable when
+1 − r = O(Pe−2/3).
+(4.2)
+Hence, in the most general limit in which the size of the two boundary layers are comparable, we have
+α = Bo−1/2Pe2/3 = O(1).
+(4.3)
+The asymptotic analysis in this regime is somewhat more involved, so for brevity, we present the details in
+Appendix A.
+4.1. Asymptotic solution when Bo = O(1)
+In this section, we present the asymptotic solution of the solute transport problem as Pe → ∞ when
+Bo = O(1). The analysis herein is a natural extension of Moore et al. (2021). For the purposes of this section,
+we shall use the concentration form of the advection-diffusion equation (2.26)–(2.30) and, in particular, find
+the solution in terms of the solute mass m = φh, where h is given by (3.1).
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+9
+4.1.1. Outer region
+In the droplet bulk where 1−r = O(1), we seek a solution of the form m = m0(r, t)+O(Pe−1) as Pe → ∞.
+Substituting into (2.26), (2.30), we find that
+∂m0
+∂t
++ 1
+4r
+∂
+∂r (rm0u) = 0
+for
+0 < r < 1, t > 0
+(4.4)
+where u is given by (3.2), subject to m(r, 0) = h(r, 0). This is the usual advection equation, with solution
+given by
+m0(r, t) = h(R, 0)
+J(R, t) ,
+(4.5)
+where R is the initial location of the point that is at r at time t and J(R, t) is the Jacobian of the Eulerian-
+Lagrangian transformation, that satisfies Euler’s identity,
+D
+Dt(log J) = 1
+4r
+∂
+∂r(ru),
+J(R, 0) = 1,
+(4.6)
+where D/Dt is the convective derivative.
+A straightforward asymptotic analysis of (4.4) reveals that
+u∂m
+∂r ∼ m
+r
+∂
+∂r (ru)
+(4.7)
+as r → 1−, so that m0 = O(√1 − r) as r → 1−, and hence the concentration φ0 is square root singular. This
+sharp local concentration increase necessitates the inclusion of a diffusive boundary layer.
+4.1.2. Inner region
+Close to the contact line, we set
+r = 1 − Pe−2ˆr,
+h = Pe−2ˆh,
+u = Peˆu,
+m = Pe2 ˆm,
+(4.8)
+where the last scaling on the mass comes from global conservation of solute considerations (Moore et al.
+2021). We seek an asymptotic solution of the form ˆm = ˆm0(ˆr, t) + O(Pe−1) and find to leading order
+∂
+∂ˆr
+��
+2χ
+θc(t; Bo)
+√
+ˆr
+− 1
+ˆr
+�
+ˆm0 + ∂ ˆm0
+∂ˆr
+�
+= 0
+in
+ˆr > 0, t > 0
+(4.9)
+such that
+�
+2χ
+θc(t; Bo)
+√
+ˆr
+− 1
+ˆr
+�
+ˆm0 + ∂ ˆm0
+∂ˆr
+= 0
+for
+ˆr = 0.
+(4.10)
+It is straightforward to show that the solution to (4.9)–(4.10) is given by
+ˆm0(ˆr, t) = C(t; Bo)ˆrexp
+�
+−
+4χ
+θc(t; Bo)
+√
+ˆr
+�
+,
+(4.11)
+where, by pursuing a similar matching process to Moore et al. (2022), we find that the coefficient C(t; Bo)
+is given by
+C(t) =
+64χ4
+3θc(t; Bo)4 N(t; Bo),
+(4.12)
+where N(t; Bo) is the leading-order accumulated mass advected into the contact line region up to time t,
+viz.
+N(t; Bo) = 1
+4
+� t
+0
+m0(r, τ)u(r, τ) dτ.
+(4.13)
+It is worth noting that this solution follows directly from the Bo = 0 regime discussed in Moore et al. (2021,
+2022), with the alterations due to gravity entering into the accumulated mass flux into the contact line and
+the leading order contact angle. In particular, we note that in the limit Bo → 0, since ψ = 4/π + O(Bo), this
+yields the expected form found in the surface tension-dominated problem in Moore et al. (2022) (see §3.7.2
+therein). We display the accumulated mass flux and the local contact angle for a wide range of Bond numbers
+in figure 3. We see that as the influence of gravity increases, the acccumulated mass flux into the contact
+line at a fixed percentage of the evaporation time is reduced from the surface tension-dominated regime. On
+the other hand, the local contact angle increases, commensurate with the droplet profile transitioning from
+
+10
+M. R. Moore & A. W. Wray
+0
+0.5
+1
+0
+1
+2
+3
+4
+Figure 3: (a) The accumulated mass flux, N(t; Bo) as defined by (4.13) and (b) the leading order local
+contact angle θc(t; Bo) as defined by (3.5), for Bo = 10−2 (purple), Bo = 10−1 (purple) (dark blue), Bo = 1
+(light blue), Bo = 10 (green) and Bo = 102 (yellow).
+a spherical cap to a ‘pancake’ droplet. We note that this combined behaviour leads to C(t; Bo) decreasing
+as Bo increases. We discuss how these findings impact coffee ring formation in more detail in §5.1.1.
+4.1.3. Composite solution
+We may use van Dyke’s rule (Van Dyke 1964) to formulate a leading-order composite solution for the
+solute mass that is valid throughout the drop by combining the leading-order-outer solution (4.5) and the
+leading-order-inner solution (4.11), finding
+mcomp(r, t) = mouter(r, t) + Pe2m
+�
+Pe2(1 − r), t
+�
+.
+(4.14)
+4.2. Comparisons between the numerical and asymptotic results
+Our asymptotic predictions are compared to numerical simulations of the full advection-diffusion problem for
+the integrated mass variable given by (2.32)–(2.34). The integrated mass variable is chosen over the solute
+mass m or the concentration φ since it is better behaved close to the contact line. The numerical procedure
+requires careful consideration of the thin diffusive boundary layer and we follow a similar approach to that
+described for the surface tension-dominated problem by Moore et al. (2021). We give a summary of the
+methodologies in Appendix B.
+We begin by comparing the asymptotic predictions of the solute mass profiles to numerical solutions in
+the regime where Bo = O(1). In figure 4, we display asymptotic (dashed, red) and numerical (solid, blue)
+curves at 10% intervals of the total drying time for Pe = 102, Bo = 1 (a,b) and Pe = 102, Bo = 30 (c,d).
+In each figure, we see excellent agreement between the simulations of the full system and the leading-order
+composite solution (4.14). There is a clear formation of the expected coffee ring in the region near the contact
+line, where solutal diffusion and advection interact. We see that increasing the Bond number in this regime
+leads to a slight reduction of the size of the coffee ring.
+This behaviour is reminiscent of the Bo = 0 regime considered previously by Moore et al. (2021). However,
+in the later stages of the Pe = 102, Bo = 30 example, we see evidence of a qualitative difference in behaviour,
+with the formation of another peak in the mass profile in the droplet interior (see inset in figure 4(c)).
+Henceforth, we shall refer to the classical coffee ring as the primary peak and this new feature as the
+secondary peak. The presence of the secondary peak depends on the Bond number, as there is no secondary
+peak in any of the profiles when Bo = 1, but it also depends on the drying time, as the peak only develops
+in the later stages of evaporation when Bo = 30 (between 60 − 70% of the drying time). Noticeably, the
+secondary peak is significantly smaller in magnitude than the primary peak.
+For larger Bond numbers, we compare the numerical results to the asymptotic predictions in Appendix
+A. In figure 5, we display results for Pe = 102, Bo = 105 (α ≈ 0.07) (a,b) and Pe = 103 and Bo = 104
+(α = 1) (c,d). In each case, we display the composite profile for the solute mass given by (A 34). In each
+figure, we see that after an initial transient the asymptotic predictions and numerical results are again in
+excellent agreement. Moreover, we see further evidence of the existence of a secondary peak in the case
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+11
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+10 -6
+10 -4
+10 -2
+10 0
+10 -3
+10 -1
+10 1
+10 3
+10 5
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+10 -6
+10 -4
+10 -2
+10 0
+10 -3
+10 -1
+10 1
+10 3
+10 5
+0.4
+0.6
+0.1
+0.105
+Figure 4: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative
+flux for (a,b) Pe = 102 and Bo = 1 (c,d) Pe = 102 and Bo = 30. In each figure, the bold, black curve
+represents the initial mass profile, which corresponds to the droplet free surface profile (3.1). We also display
+plots at time intervals of 0.1 up to t = 0.9 in which solid, blue curves represent the results from the numerical
+solution of (2.32)–(2.34) and the dashed, red curves show the leading-order composite mass profile, given by
+(4.14). The right-hand figures display a close-up of the profiles near the contact line. In (c), the inset shows
+a close up of the mass profile in the droplet interior at t = 0.9 where we see a clear formation of a secondary
+peak.
+Pe = 103, Bo = 104 regimes, where the peak appears much earlier and is noticeably larger than that in
+the previous example (cf. figure 4c, where Pe = 102, Bo = 30). However, we also note again the strong
+dependence of the secondary peak on Bo and, possibly, Pe, as there is no evidence of such an interior peak
+when Pe = 102, Bo = 105.
+These findings prompt us to investigate this new feature more closely, alongside a discussion of how the
+characteristics of the primary peak — and hence the classical coffee ring — depend on the Bond number.
+5. Properties of the two peaks
+Given the excellent comparisons displayed in the previous section, we seek to use our asymptotic results to
+investigate properties of the nascent coffee ring and, in particular, the new feature of these moderate-to-large
+Bond number regimes: the secondary peak.
+5.1. Primary peak
+We shall begin by discussing the effect of the Bond number on the primary peak. As in previous studies
+of the surface tension-dominated regime, the formation of the primary peak is driven by the competing
+diffusive and advective solute fluxes (Moore et al. 2021, 2022) and is always present in the large-Pe regime.
+Furthermore, since all of the features of interest are well within the solutal diffusion boundary layer, we will
+
+12
+M. R. Moore & A. W. Wray
+Mass
+Mass
+Figure 5: Profiles of the solute mass when an axisymmetric droplet evaporates under a diffusive evaporative
+flux for (a,b) Pe = 102 and Bo = 105 (α ≈ 0.07) (c,d) Pe = 103 and Bo = 104 (α = 1). In each figure,
+the bold, black curve represents the initial mass profile (2.34). We also display plots at time intervals of 0.1
+up to t = 0.9 in which solid, blue curves represent the results from the numerical solution of (2.32)–(2.34)
+and the dashed, red curves show the composite mass profiles, given by (A 33) for the integrate mass variable
+and (A 34) for the solute mass, respectively. Note that in (c,d), we can clearly see the development of the
+secondary peak behind the primary peak.
+use the inner solution — as discussed in §4.1.2 in the Bo = O(1) regime and §A.2 in the large-Bo regime —
+to do this.
+5.1.1. Bo = O(1) regime
+When the Bond number is order unity, the analysis is a natural extension of that in Moore et al. (2021,
+2022). The local solute profile is dominated by the leading-order inner solution (4.11). Introducing the time-
+dependent P´eclet number
+Pet =
+Pe
+1 − t,
+(5.1)
+the nascent coffee ring profile may be seen to have the similarity form
+ˆm0(R, t)
+Pe2
+t N(t; Bo) =
+2χ
+3ψ(Bo)f
+�√
+R, 3,
+4χ
+ψ(Bo)
+�
+,
+R = Pe2
+t(1 − r)
+(5.2)
+where ψ and χ retain their definitions from (3.5) and (3.6) as the initial local contact angle and the coefficient
+of the evaporative flux singularity, respectively, and f(x, k, l) = lkxk−1e−lx/Γ(k) is the probability density
+function of a gamma distribution. It is this functional form which describes the characteristic narrow, sharp
+peak of the coffee ring.
+Since the definition of R only depends on the time-dependent P´eclet number, we can clearly illustrate the
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+13
+0
+20
+40
+60
+80
+100
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+Figure 6: The similarity profile (5.2) of the leading-order-inner solute mass profile for Bo = 10−2 (purple),
+Bo = 10−1 (purple) (dark blue), Bo = 1 (light blue), Bo = 10 (green) and Bo = 102 (yellow).
+effect of gravity by plotting the similarity profile (5.2) for a range of Bond numbers in figure 6. We see that,
+as the effect of gravity increases, the height of the primary peak decreases, and the peak moves further from
+the pinned contact line. Moreover, the shape of the primary peak tends towards a shallower, wider profile.
+Notably, this behaviour is driven purely by changes in ψ(Bo); as we saw in figure 3a, the accumulated mass
+flux into the contact line decreases with the Bond number, clearly this acts to accentuate this behaviour.
+We can expand upon these results by finding the leading order asymptotic prediction of the primary peak
+height and location, which are given by
+rpeak,I(t; Bo) = 1 − ψ(Bo)2
+4Pe2
+t χ2 ,
+mpeak,I(t; Bo) = 16Pe2
+tN(t; Bo)χ2
+3e2ψ(Bo)2
+,
+(5.3)
+respectively. Notably, while gravity only influences the location of the primary peak through the initial local
+contact angle, ψ(Bo), the height depends on gravity through both the contact angle and the accumulated
+mass flux, N(t; Bo). In particular, referring back to figure 3, this means that gravity has a stronger effect on
+the peak height than its location.
+We illustrate the veracity of these asymptotic predictions by comparing them to the corresponding numer-
+ical results for Pe = 102 and a range of Bond numbers in figure 7. As anticipated from the comparisons of
+the solute mass profiles, we see excellent agreement between the asymptotic predictions and the numerical
+results. In particular, in figure 7a, we note that as the influence of gravity increases (i.e. Bo increases), the
+coffee ring effect is inhibited: although a peak clearly still forms, it is lower for large Bond number at a similar
+stage of the drying process. This effect varies nonlinearly with time (cf. figure 3a). For example, considering
+the cases Bo = 1/2 and Bo = 30, after 50% of the drying time, the peak height is reduced by a factor of
+≈ 3.97, while at 60% of the drying time, the reduction is a factor of ≈ 3.85 and at 90% of the drying time,
+it is ≈ 3.63.
+Similarly, in figure 7b, we see that as the Bond number increases, the location of primary peak moves
+further from the contact line and that this significantly increases as the Bond number gets larger. For
+Bo = 1/10, 1/2, 1 the location is almost indistinguishable from the zero-Bond number solution — where
+Pe2
+t (1 − r) = 2 (Moore et al. (2022)) — but for Bo = 30, this has increased to ≈ 6.79.
+It is worth noting that in all this analysis, the P´eclet number simply acts to scale the above findings. For
+a larger P´eclet number, the height of the primary peak increases, while it is located closer to the contact
+line. This is precisely what is seen for the Bo = 0 regime (Moore et al. 2021).
+5.1.2. Large-Bo regime
+In the large-Bo regime, given the size of the primary peak, we anticipate that the leading-order-inner
+solution
+˜
+M0(˜r, t) as given by (A 16) should reasonably capture the features of the primary peak. However,
+
+14
+M. R. Moore & A. W. Wray
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+2
+4
+6
+8
+10
+Figure 7: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak,
+mpeak,I(t)/Pe2/3
+t
+and (b)) its location Pe2/3
+t
+(1 − rpeak,I(t)) in the Bo = O(1) regime as given by (5.3). For
+each curve, Pe = 102, while the Bond number varies according to Bo = 1/10 (dark purple), Bo = 1/2 (blue),
+Bo = 1 (green) and Bo = 30 (yellow).
+0
+0.2
+0.4
+0.6
+0.8
+1
+10-6
+10-4
+10-2
+100
+102
+104
+106
+0
+0.2
+0.4
+0.6
+0.8
+1
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+100
+Figure 8: Numerical (circles) and asymptotic predictions (solid lines) of (a)) the height of the primary peak,
+MI(t) = mpeak,I(t)/Pe2/3 and (b)) its location ηI(t) = Pe2/3(1−rpeak,I(t)) as given by (5.8)–(5.9). Results are
+presented for Pe = 102, Bo = 103 (α ≈ 0.68, yellow), Pe = 103, Bo = 104 (α = 1, green), Pe = 104, Bo = 105
+(α ≈ 1.47, blue) and Pe = 105, Bo = 105 (α ≈ 6.81, dark purple).
+unlike its moderate-Bo counterpart, there is no simple similarity form for the solution in this regime, so that
+we proceed more carefully.
+We denote the height and location of the primary peak by
+mpeak,I(t) = Pe2/3MI(t),
+rpeak,I(t) = 1 − Pe−2/3ηI(t),
+(5.4a, b)
+respectively. By (2.35), the location of the maximum ηI(t) satisfies
+0 = ∂2 ˜
+M0
+∂˜r2 (ηI(t), t).
+(5.5)
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+15
+Utilizing (A 13), we find that
+∂2 ˜
+M0
+∂˜r2 (ηI(t), t) = −
+�
+˜u0 − 1
+˜h0
+∂˜h0
+∂˜r
+�
+∂ ˜
+M0
+∂˜r
+�����
+(ηI(t),t)
+= 0.
+(5.6)
+Since ∂ ˜
+M0/∂˜r > 0 for ˜r > 0, we conclude
+˜u0(ηI(t), t) −
+1
+˜h0(ηI(t), t)
+∂˜h0
+∂˜r (ηI(t), t) = 0
+(5.7)
+so that
+ηI(t) = α
+2 W0
+�(1 − t)2
+4α3
+�
+,
+(5.8)
+where W0(x) is the Lambert-W function (i.e. the solution to wew = x).
+With ηI(t) in hand, the corresponding height of the ring at the peak is then given by
+MI(t) =
+� −B0(t)
+I(ηI(t), t)
+�
+=
+�
+N(t)
+I(ηI(t), t)
+�� ∞
+0
+1
+I(s, t) ds
+�−1�
+,
+(5.9)
+where I(r, t) is given by (A 15) and N(t) is the leading-order accumulated mass flux into the boundary layer
+(A 18). Note that, in this regime, N(t) is independent of α and, hence, the Bond number, but the function
+I(r, t) does change with α.
+In figure 8, we plot the asymptotic predictions of the location and height of the primary deposit peak
+against the simulation results for a range of different P´eclet and Bond numbers (and, correspondingly, α).
+There are several discernible features. After an initial transient, the location of the peak is captured extremely
+well by the asymptotic prediction (5.8) for each case presented. This initial transient is primarily due to the
+lack of a distinct peak at early stages of the drying process; a period of time is necessary for sufficient solute
+to be advected to the contact line. This process takes longer for smaller P´eclet numbers, i.e. when diffusion
+is relatively stronger. The height of the primary peak is captured quite well by the asymptotic prediction
+(5.9), particularly for larger P´eclet numbers and as time increases. It is worth noting that the error in the
+approximation of the height is O(Pe1/3), so for an improved estimation of the primary peak height, it would
+be necessary to consider the first two inner solutions ˜
+M0(˜r, t) and ˜
+M1(˜r, t). While this is possible, the results
+do not have a simple analytic form, so are not practical to work with. We also note that, as the droplet
+evaporates, the primary peak both increases in size and moves closer to the contact line, i.e. MI(t) increases
+and ηI(t) decreases as t increases.
+5.2. Secondary peak
+As evidenced by the solute mass profiles, the behaviour of the secondary peak — and indeed, even its presence
+— is more complex than that of the primary peak, which always forms in the large-Pe regime. We have seen,
+for example in figure 4 in the Bo = O(1) regime, that the presence of the peak varies with both Bo and
+drying time, while when Bo ≫ 1, we have also seen variation with Pe (and hence α), see for example figure
+5. This gives a clear indication that we need to treat this feature more carefully.
+To begin, we will consider whether or not the secondary peak is present. We shall first fix the P´eclet
+number and use the numerical results to produce a regime diagram in (Bo, t)-parameter space indicating
+whether one or two peaks are present in the solute mass profile. We note here that these are the only options
+that we have been able to find — we have found no instances of more than two peaks appearing.
+We show the results for Pe = 102 in figure 9a. In the figure, solute profiles with one peak — i.e. only the
+classical coffee ring — are denoted by blue circles, while solute profiles exhibiting two peaks are denoted
+by red circles. We see a strong nonlinear dependence on both Bond number and dryout time. In particular,
+there is a band of Bond numbers between around Bo ≈ 10 and Bo ≈ 30000 that may lead to secondary peak
+formation, although the existence of a peak also depends strongly on t for a fixed Bond number. We note
+that for Bo ≲ 10, there is only one peak for any t, in agreement with the classical Bo = 0 regime. Moreover,
+for very large Bond number Bo ≳ 30000, again we see that there is only one peak.
+We illustrate the effect of the P´eclet number by plotting the equivalent regime diagram for Pe = 103
+in figure 9b. Remarkably, the onset of the secondary peak appears to be unaffected by the increase of the
+P´eclet number, although the band of Bond numbers for which we see two peaks is significantly widened into
+larger Bo. Notably, however, the shape of the curve delineating between two peaks / one peak for large Bond
+number appears to be independent of Pe, only its location has shifted.
+
+16
+M. R. Moore & A. W. Wray
+10-3
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+0
+0.2
+0.4
+0.6
+0.8
+1
+One peak
+Two peaks
+10-3
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+0
+0.2
+0.4
+0.6
+0.8
+1
+One peak
+Two peaks
+Figure 9: (Bo, t)-regime diagram illustrating the presence of either one (blue circles) or two (red circles)
+peaks in the solute mass profile for (a) Pe = 102 and (b) Pe = 103. The data is extracted from the numerical
+simulations and demonstrates a clear band of Bond numbers for which two peaks may exist in the profile.
+In each figure, the black curve denotes the asymptotic prediction of when the centre of the droplet changes
+from a maximum to a minimum as given by (5.21).
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+17
+10-6
+10-3
+100
+10-2
+10-1
+100
+101
+102
+103
+10-6
+10-3
+100
+10-2
+10-1
+100
+101
+102
+103
+10-6
+10-3
+100
+10-2
+10-1
+100
+101
+102
+103
+Figure 10: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 20. The deposit profile is
+displayed on a doubly-logarithmic plot at 25% (a)), 35% (b)) and 75% (c)) of the drying time in order to
+catch the emergence of the secondary peak. In each of a) − c), the primary peak is indicated by a red circle,
+while the secondary peak is indicated by a black circle (when it exists).
+5.2.1. Onset of the secondary peak
+In this section, we seek to investigate some of the phenomena around the onset of the secondary peak in
+more detail. We saw that for a fixed P´eclet number, there was a distinct switch from one to two peaks for
+Bond number Bo ≈ 10 and that this switch appears to be independent of Pe. This suggests that secondary
+peak formation is not a result of the interplay between solutal advection and diffusion that drives the classical
+coffee ring.
+In order to investigate the reasons behind the presence of or lack of a secondary peak, in figure 10, we
+plot numerical results for the solute profiles in a droplet with Pe = 102, Bo = 20 at 25%, 35% and 75% of
+the drying time. In the figure, the primary and secondary peaks are indicated by the red and black circles,
+respectively. We clearly see in figure 10a that at 25% of the drying time there is only one peak, but by 35%
+of the drying time, the secondary peak has emerged close to the droplet centre. As the droplet evaporates
+further to 75% of the drying time the secondary peak has moved further towards the droplet contact line.
+This particular example gives us a strong indication that the secondary peak initially arises from the centre
+of the drop and, in particular, appears to be linked with a transition from the centre being a maximum in
+solute mass profile — as it is for the classical coffee ring of Deegan et al. (1997, 2000) — to a minimum.
+To investigate this postulate, we consider the behvaiour close to the droplet centre. To simplify things,
+since the initial emergence of the secondary peak appears to be independent of the P´eclet number, we neglect
+solutal diffusion completely, taking Pe = ∞, so that the solute mass m satisfies the first-order semi-linear
+equation
+∂m
+∂t + 1
+4r
+∂
+∂r (rmu) = 0,
+m(r, 0) = h(r, 0),
+(5.10)
+where, since the emergence appears to be rooted in the region where Bo ≈ 10, we consider the moderate
+Bond number regime and retain the full expressions for the droplet free surface h and fluid velocity u given
+by (3.1)–(3.2).
+We seek an asymptotic solution of (5.10) as r → 0. First, we note that for small arguments, the free surface
+and velocity have the following asymptotic expansions:
+h(r, t) ∼ (1 − t)
+�
+H0(Bo) + H1(Bo)r2 + o(r2)
+�
+,
+(5.11)
+u(r, t) ∼
+1
+(1 − t)
+�
+U0(Bo)r + U1(Bo)r3 + o(r3)
+�
+(5.12)
+
+18
+M. R. Moore & A. W. Wray
+as r → 0, where
+H0(Bo) = (I0(
+√
+Bo) − 1)
+πI2(
+√
+Bo)
+,
+(5.13)
+H1(Bo) = −
+Bo
+4πI2(
+√
+Bo)
+,
+(5.14)
+U0(Bo) = 2
+√
+Bo −
+√
+BoI0(
+√
+Bo) − 2I1(
+√
+Bo)
+√
+Bo(1 − I0(
+√
+Bo))
+,
+(5.15)
+U1(Bo) = −(Bo3/2 −
+√
+BoI0(
+√
+Bo) +
+√
+BoI0(
+√
+Bo)2 + 2I1(
+√
+Bo) − 2BoI1(
+√
+Bo) − 2I0(
+√
+Bo)I1(
+√
+Bo)
+4
+√
+Bo(1 − I0(
+√
+Bo))2
+.
+(5.16)
+Now, by the symmetry of the problem, the droplet centre must be a critical point, so we seek a solution
+of the form m = m0(t) + m1(t)r2 + o(r2) as r → 0. Upon substituting this ansatz and the above forms for h
+and u into (5.10), straightforward calculation yields
+m0(t) = H0(1 − t)U0/2,
+(5.17)
+m1(t) =
+�2U1H0
+U0
++ H1
+�
+(1 − t)U0 − 2U1H0
+U0
+(1 − t)U0/2.
+(5.18)
+Hence, given that initially the droplet has a maximum at its centre for any Bo, we deduce that the
+maximum becomes a minimum at the critical time tc such that
+m1(tc) = 0.
+(5.19)
+Since 2U1H0/U0 + H1 < 0, H0 > 0, U0 > 0 for all Bo, (5.19) only has solutions for Bo > Boc where
+U1(Boc) = 0
+=⇒
+Boc ≈ 15.21.
+(5.20)
+When Bo > Boc, we may solve (5.19) explicitly to find
+tc(Bo) = 1 −
+�
+2U1(Bo)H0(Bo)
+2U1(Bo)H0(Bo) + H1(Bo)U0(Bo)
+�2/U0(Bo)
+.
+(5.21)
+This critical curve in figure 9 is displayed as the solid black curve and we see that there is excellent
+agreement between this prediction and the transition from one to two peaks. But, what is causing the
+transition? Since the phenomenon is independent of the P´eclet number, it is purely a result of the droplet
+geometry and the evaporation-driven flow. In particular, we note that the critical Bond number Boc given by
+(5.20) is linked to the change in sign of U1, which is equivalent to requiring that (1 − t)∇ · (uer) is decreasing
+near r = 0. This correlates with the profiles of the divergence of u displayed in figure 2c, where we see this
+change in sign clearly as the Bond number increases.
+Notably, considering the curve displayed in figure 9, we see that for Bo close to Boc, the secondary peak only
+emerges very late in the dryout process, but as the Bond number increases, it appears almost instantaneously.
+Hence, from this analysis alone, we might expect there to always be two peaks for Bo > Boc, but clearly this
+is not the case. We now investigate why in more detail.
+5.2.2. Loss of the secondary peak
+Given its clear variation with each of t, Bo and Pe, it is perhaps unsurprising that it is more challenging to
+determine an analytical expression for the location of the right-hand boundary between two peaks and one
+peak in figure 9), and unfortunately we have been unable to do so. However, it is relatively straightforward
+to illustrate why the transition occurs by considering a specific example.
+In figure 11, we plot solute mass profiles for Pe = 102 and Bo = 103 at 5%, 20%, 50% and 90% of the
+drying time indicating the primary and secondary peaks by red and black circles where appropriate. Note
+that, for such a large Bond number, the critical time at which we would expect a secondary peak to be
+present may be found from (5.21) to be tc ≈ 2.8 × 10−10. We see in figure 11a that, indeed, after 5% of the
+drying time, the secondary peak has emerged and is visible close to the droplet centre — moreover, at this
+stage, the primary peak associated with the coffee ring has yet to fully develop (so that the ‘one peak’ at
+this stage in figure 9a is in fact the secondary peak!). However, by the time we reach 20% of the drying time,
+both peaks are clearly visible, with the primary peak now approximately 50% larger than the secondary
+peak.
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+19
+10-6
+10-4
+10-2
+100
+10-2
+10-1
+100
+10-6
+10-4
+10-2
+100
+10-2
+10-1
+100
+10-6
+10-4
+10-2
+100
+10-2
+10-1
+100
+101
+10-6
+10-4
+10-2
+100
+10-2
+100
+102
+Figure 11: Solute profiles for an evaporating droplet with Pe = 102 and Bo = 103 displayed on a doubly-
+logarithmic plot at 5% (a)), 20% (b)), 50% (c)) and 90% (d)) of the drying time. In each figure, the primary
+peak is indicated by a red circle, while the secondary peak is indicated by a black circle when either exists.
+Increasing time further, we see that the primary peak continues to grow rapidly so that, by 50% of the
+drying time, it is so large, that it has subsumed the secondary peak into its upstream tail. That is, the
+secondary peak is still present according to the Pe = ∞ theory, but due to the fact that Pe is actually finite
+and the corresponding presence of the classical coffee ring, we do not see the secondary peak.
+If we then increase t even further, we see that by 90% of the drying time, the secondary peak has reemerged
+from the lee of the primary peak. By this stage of the evaporation process, the primary peak has moved
+significantly closer to the contact line — here 1 − rpeak,I ≈ 1.4 × 10−4, while the secondary peak is located
+at 1 − r ≈ 4.8 × 10−2, so that it is sufficiently far behind the primary peak to be visible.
+Thus, the loss of the secondary peak appears to be intrinsically tied to both the location, size and shape of
+the primary peak. Given that this behaviour largely occurs in the regime in which Bo ≫ 1, these properties
+of the primary peak are given by (5.8), (5.9) and the derivative of (A 16), respectively. Clearly, therefore, the
+behaviour is strongly dependent on t, Bo and Pe (cf. figure 8, for example).
+5.2.3. Properties of the secondary peak
+Given its dependence on the various parameters of the model, discerning the properties of the secondary
+peak analytically is challenging, particularly in the Bo = O(1)-regime since, in this case, the peak tends to
+be situated in the droplet bulk, so that we are unable to use the simpler forms of the inner solution described
+in §4.1.2.
+Hence, we utilize the numerical results to track the height mpeak,II(t) and location rpeak,II(t) of the
+secondary peak when it exists and we display the results for several different values of Pe, Bo in figure
+12. In the figure, results for Pe = 102 and Pe = 103 are denoted by circles and squares, respectively. The
+Bond number is represented by the colour, with results for Bo = 20 (purple), 50 (dark blue), 100 (light
+blue) and 1000 (green). It is evident that for each of the Bond numbers represented, increasing the P´eclet
+number appears to have negligible effect on both the size and location of the secondary peak. However, both
+
+20
+M. R. Moore & A. W. Wray
+0
+0.5
+1
+0
+0.1
+0.2
+0.3
+0.4
+0
+0.5
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Figure 12: Numerical predictions of (a)) the height of the secondary peak, mpeak,II(t) and (b)) its location
+rpeak,II(t) for different values of Pe, Bo. The symbols denote different P´eclet numbers: Pe = 100 (circles),
+Pe = 1000 (squares); while the colours denote different Bond numbers: Bo = 20 (purple), Bo = 50 (dark
+blue), Bo = 100 (light blue), Bo = 1000 (green).
+properties do vary with the Bond number. In particular, as the Bond number increases, the secondary peak
+is situated closer to the contact line at the same stage of the drying process, and similarly, for a fixed Bond
+number, the peak gets closer to the contact line as the droplet evaporates. On the other hand, variations
+of the secondary peak height with Bo are much less trivial, although for all of the displayed results, we see
+that the height of the secondary peak decreases as the droplet evaporates. This is in stark contrast to the
+primary peak, which always grows as more solute is transported to the contact line.
+Thus, we conclude that the secondary peak is predominantly driven by the Bond number. Indeed, it is
+only for sufficiently large Bond numbers that we find a second peak at all, and the properties of that peak
+then depend strongly on the size of Bo. The only role played by the P´eclet number appears to be in the
+disappearance of the secondary peak when it is subsumed by the primary peak, which is typically orders of
+magnitude larger and always closer to the contact line.
+6. Summary and discussion
+In this paper, we have performed a detailed asymptotic and numerical analysis into the effect of gravity on
+the famous coffee ring phenomenon observed in solute-laden droplets. In the physically-relevant limit of small
+droplet capillary number, Ca ≪ 1, and large solutal P´eclet number, Pe ≫ 1, we identified two asymptotic
+regimes based on the size of the Bond number, Bo:
+i) a moderate Bond number regime, where Bo = O(1);
+ii) a large Bond number regime, Bo ≫ 1.
+In the first of these regimes, gravity acts to flatten the droplet profile from the spherical cap of the zero-
+gravity problem, while reducing the liquid velocity. Moreover, the asymptotic structure of the solute transport
+follows exactly that presented by Moore et al. (2021) for surface tension-dominated droplets, with advection
+dominating in the droplet bulk, while the competition between advection and diffusion in a boundary layer
+of width of O(Pe−2) near the pinned contact line drives the nascent coffee ring. Gravity acts to modify the
+surface tension-dominated solution both through the accumulated mass flux of solute into the contact line
+and a parameter dependent on the local contact angle. In particular, as the Bond number increases, the
+height of the nascent coffee ring is reduced — which is consistent with the reduced flow velocity as Bo is
+increased. Moreover, the peak is situated further from the contact line.
+To categorize the role of gravity more explicitly, we derived an approximate similarity profile, ˆm0, for the
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+21
+nascent coffee ring profile, given by
+ˆm0(R, t)
+Pe2
+t N(t; Bo) =
+2χ
+3ψ(Bo)f
+�√
+R, 3,
+4χ
+ψ(Bo)
+�
+,
+R = Pe2
+t(1 − r)
+(6.1)
+where Pet = Pe/(1−t) is the time-dependent P´eclet number, N(t; Bo) is the accumulated mass flux of solute
+at the contact line from the droplet bulk, χ is the coefficient of the inverse square root singularity in the
+evaporative flux at the contact line; ψ(Bo) is the leading order initial local contact angle; and f(x, k, l) =
+lkxk−1e−lx/Γ(k)! is the probability density function of a gamma distribution. Clearly, the Bond number acts
+to scale the coffee ring profile through the accumulated mass flux, while it acts to change the shape of the
+profile through the initial contact angle ψ(Bo).
+In the second regime, the Bond number is large, so that the droplet is approximately flat, with surface
+tension confined to a narrow region near the pinned contact line — a ‘pancake’ or ‘puddle’ droplet. Thus,
+the asymptotic analysis discussed above is no longer valid, since there are two competing boundary layers
+near the edge of the droplet — the diffusion boundary layer in the solute transport and the surface tension
+boundary layer in the droplet free surface profile (and, hence, the liquid velocity). We derived the resulting
+solute distribution in the most general regime in which the two boundary layers are comparable, which
+reduces to the assumption that α = Bo−1/2Pe2/3 = O(1). Under this assumption, diffusion and advection
+balance in a region of size Pe−2/3 near the contact line, noticeably larger than in the moderate gravity
+regime. This is a further indication of gravity acting to shift the coffee ring further from the contact line
+and, moreover, tends to cause shallower solute profiles in the boundary layer region.
+The asymptotic analysis in the large-Bond number regime is more challenging than that in the moderate
+Bond number regime and, in particular, the nascent coffee ring no longer collapses onto an approximate
+similarity profile. However, we were able to derive expressions for the location (5.8) and height (5.9) of the
+peak, demonstrating that it still strongly depends on the accumulated mass flux of solute into the contact
+line alongside the parameter α. In particular, increasing α leads to higher coffee rings that are located closer
+to the contact line.
+In each regime, we demonstrated that our asymptotic predictions were in excellent agreement with nu-
+merical simulations of the full advection-diffusion problem for the solute mass distribution.
+Alongside the anticipated nascent coffee ring driven by the competition between advection and diffusion
+of the solute, our asymptotic and numerical analysis also revealed a novel phenomenon: that the solute
+profile may have a secondary peak. The secondary peak was characterized by being situated upstream of and
+significantly smaller than the primary coffee ring. Moreover, the presence of this peak strongly depended on
+the Bond number, P´eclet number and evaporation time.
+Further investigation revealed that, for a fixed P´eclet number, there exists a band in (Bo, t)-space at which
+two peaks are present in the profile. We demonstrated that the onset of this band is independent of the P´eclet
+number and is caused by the critical point at the centre of the droplet changing in type from a maximum
+(as in the spherical cap droplet in the Bo = 0 regime) to a minimum. When the critical point at the droplet
+centre changes type, an internal maximum forms downstream of the centre and it is this that corresponds to
+the secondary peak. This behaviour only occurs above a critical Bond number, Boc ≈ 15.21, and then only
+after a given drying time, given by
+tc(Bo) = 1 −
+�
+2U1(Bo)H0(Bo)
+2U1(Bo)H0(Bo) + H1(Bo)U0(Bo)
+�2/U0(Bo)
+.
+(6.2)
+In particular, as Bo increases, tc decreases, so the secondary peak emerges earlier in the evaporative process.
+These predictions were shown to be in excellent agreement to the numerical results and, remarkably, are
+independent of the P´eclet number.
+However, the above analysis suggests that for all Bo > Boc and t > tc a secondary peak exists — something
+that we did not find in our analysis. The reason for this discrepancy was shown to be due to the presence
+of the primary peak. In particular, as time increases, the secondary peak is located further from the droplet
+centre so that it may get subsumed in the tail of the primary peak. For a fixed Bond number, this possibility
+was shown to depend strongly on both the P´eclet number and the evaporation time; this is due to the fact
+that the size of the primary peak increases with both t and Pe, while the size of the secondary peak only
+varies with t.
+Beyond this subsuming effect, however, we were able to demonstrate that the P´eclet number plays negligible
+role in the size and location of the secondary peak for a range of Bond numbers, suggesting that this feature
+may be reliably controlled simply by altering Bo.
+In previous studies of coffee ring formation (e.g. Deegan et al. (2000); Popov (2005); Moore et al. (2021),
+
+22
+M. R. Moore & A. W. Wray
+gravity has frequently been neglected under the assumption of small Bond number, which is a reasonable
+assumption for sufficiently small droplets. However, given that the Bond number may be increased in an
+experimental or industrial setting by steadily increasing the droplet radius, the influence of gravity may be
+of fundamental interest in applications that utilize droplet drying to adaptively control the shape of the
+residual deposit, such as colloidal patterning (Harris et al. 2007; Choi et al. 2010) and fabrication techniques
+using inkjet printing (Layani et al. 2009). Our analysis thus plays a dual role in the field. First, we have
+presented the first formal categorization of the role of gravity in the early-stages of coffee ring formation and
+given a quantitative prediction of the resulting features of the solute profile. Second, we have found a novel
+phenomenon — the secondary peak — which may also be exploited in such processes, particularly when the
+size of the primary peak can be carefully controlled. This is particularly relevant given that the secondary
+peak emerges at a relatively moderate critical Bond number.
+There are, naturally, limitations to our analysis. Throughout, we have assumed that the contact line
+remains pinned as the droplet evaporates. This has been shown to be a reasonable assumption for many
+configurations (see, for example, the experiments in Deegan et al. (2000); Kajiya et al. (2008); Howard et al.
+(2023)) and may further be enhanced by solute aggregation near the edge of the droplet (Orejon et al.
+(2011); Weon & Je (2013); Larson (2014)). However, at late stages of the evaporation, the contact line may
+depin and become mobile, moving inwards towards the droplet centre. The contact line may then become
+pinned at a new location and the process may repeat. This behaviour is known as ‘stick-slip’ evaporation
+and represents an important class within the field that is beyond the scope of the present study, but may
+represent an interesting future direction in terms of the effect of gravity, particularly with the presence of
+the secondary peak and its associated increased solute mass, which may promote re-pinning.
+Another effect that we have neglected in the present analysis is the possibility of solute becoming trapped
+at the free surface of the droplet. If this occurs, the solute is then transported to the contact line along
+the free surface, and has been suggested as an alternative mechanism for coffee ring formation (Kang et al.
+(2016)). This behaviour has been demonstrated to occur for a wide variety of droplets but is more pronounced
+for droplets with large contact angles Kang et al. (2016) or when vertical diffusion happens over a longer
+timescale than evaporation D’Ambrosio (2022). Since we deal with the opposite case of a thin drop with
+fast vertical diffusion (i.e. so that the solute concentration may be assumed to be uniform across the droplet
+to leading order), we have not considered this phenomenon here. It would be interesting to see how such
+behaviour impacted the solute profile in the current case, although it should be noted that the aforementioned
+studies neglect gravity entirely.
+A further aspect that would form the basis of an exciting future study surrounds the assumption that
+the solute is dilute in the droplet. Naturally, the build up of the solute in the coffee ring means that the
+concentration rapidly approaches levels where finite particle size effects can no longer be ignored. This has
+been analysed in detail for surface tension-dominated droplets in Moore et al. (2021, 2022) and a similar
+analysis would follow here with the inclusion of gravity. One possible aspect that would differentiate droplets
+where gravity is included is in the vicinity of the secondary peak. It is an interesting open question as to
+whether the dilute assumption may also break down in the vicinity of the secondary peak as well as the
+primary. Once finite particle size effects become important, there are a number of different approaches that
+can be followed to continue the analysis, such as a sharp transition between a dilute and jammed region
+(Popov (2005)), using a viscosity and solute diffusivity that vary with concentration (Kaplan & Mahadevan
+(2015)) or through more complicated two-phase suspension models (see, for example, Guazzelli & Pouliquen
+(2018)).
+Our analysis has concentrated on a diffusive evaporative model, while there are many situations where
+other evaporative models may be appropriate. Examples include water evaporating on glass, which may more
+appropriately be modelled using a kinetic evaporative model (Murisic & Kondic (2011)), droplets evaporating
+above a bath of the same liquid, where the evaporation is effectively constant (Boulogne et al. (2016)) and
+situations where a mask is used to control the evaporative flux so that it is stronger towards the centre
+(Vodolazskaya et al. (2017)). The analysis herein could readily be extended to such situations, although we
+note that for evaporative fluxes with different — including non-singular — behaviour close to the contact
+line, the size of the boundary layer regions near the contact line in which solutal diffusion and surface tension
+are relevant will change accordingly, as for surface tension-dominated droplets (Moore et al. 2021).
+A future direction of interest would be to extend the analysis herein to non-axisymmetric droplets. Such
+droplets occur widely in applications, particularly in printing OLED/AMOLED screens (see, for example,
+Mai & Richerzhagen (2007); Huo et al. (2020)). It is well-known that droplet geometry plays a strong role in
+the behaviour of the evaporative flux (S´aenz et al. (2017); Wray & Moore (2023)) and the transient and final
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+23
+deposit profiles (Freed-Brown (2015); S´aenz et al. (2017); Moore et al. (2022)). It would be of significant
+theoretical and practical interest to explore the behaviour of the secondary peak in such problems as well.
+Finally, we note that another context in which gravity may play an important role is that of binary/multi-
+component droplets, particularly in situations where the different fluids have different densities. Multi-
+component droplets occur widely, from commercial alcohols such as whiskey and ouzo (Tan et al. (2019);
+Carrithers et al. (2020)) to various inks (Shargaieva et al. (2020)). While it would be certainly of interest to
+extend the analysis presente here to such droplets, a careful treatment of the internal flow would be needed,
+as the multi-component nature of the droplet significantly complicates the dynamics (Li et al. (2019)).
+Acknowledgments MRM would like to acknowledge the support of EPSRC grant EP/X035646/1.
+Declaration of Interests. The authors report no conflict of interests.
+Appendix A. Matyched asymptotic analysis in the limit of large Bo, α = O(1)
+In this appendix, we present the asymptotic solution of the solute transport problem in the limit in which
+Bo, Pe ≫ 1 and
+α = Bo−1/2Pe2/3 = O(1).
+(A 1)
+For convenience, we choose to use Pe−2/3 as our small parameter in the asymptotic expansions. Moreover, it
+transpires that it is easier to analyse the integrated mass variable formulation of the problem (2.32)–(2.35).
+A.1. Outer region
+In the droplet bulk, 1 − r is O(1), and we recall from (3.9) that the droplet free surface h is flat to all orders
+and that the velocity is given by (3.10). Upon substituting these expressions into (2.32) and (2.34), and then
+expanding M(r, t) = M0(r, t) + Pe−2/3M1(r, t) + O(Pe−4/3) as Pe → ∞, we find to leading-order
+∂M0
+∂t
++ u0
+4
+∂M0
+∂r
+= 0
+for
+0 < r, t < 1,
+M0(r, 0) = r2
+2π
+for
+0 < r < 1.
+(A 2a, b)
+This may be solved using the method of characteristics, yielding
+M0(r, t) = (1 − t)r2
+2π
++
+√1 − t(1 − √1 − t)
+π
+(1 −
+�
+1 − r2).
+(A 3)
+We see that this solution automatically satisfies the boundary condition (2.33a).
+At O(Pe−2/3), the problem for M1(r, t) is given by
+∂M1
+∂t
++ u0
+4
+∂M1
+∂r
+= −αu1
+4
+∂M0
+∂r
+for
+0 < r, t < 1,
+M1(r, 0) = 0
+for
+0 < r < 1.
+(A 4a, b)
+for 0 < r < 1, 0 < t < 1, while the initial condition is given by M1(r, 0) = αr2/π for 0 < r < 1. This may
+be solved in a similar manner using the method of characteristics, yielding
+M1(r, t) = 2ακ(r, t)
+π
+(1 − κ(r, t)) log
+�√1 − t − κ(r, t)
+1 − κ(r, t)
+�
++ α
+π (1 − (1 − κ(r, t))2),
+(A 5)
+where κ(r, t) = √1 − t(1 −
+√
+1 − r2).
+Expanding the leading-order solution (A 3) as we approach the contact line, we have
+M0(r, t) ∼
+√1 − t
+π
+− (1 − t)
+2π
+−
+�
+2(1 − t)
+π
+(1 −
+√
+1 − t)
+√
+1 − r − (1 − t)
+π
+(1 − r) + O((1 − r)3/2)
+(A 6)
+as r → 1−. Notably, this means that the leading-order outer solute mass m0 is singular at the contact line,
+which gives a strong indication of the importance of diffusive effects local to the edge of the droplet. This is
+in stark contrast to the Bo = O(1) solution, where the outer solute mass was square root bounded as r → 1−.
+A similar expansion of (A 5), yields
+M1(r, t) ∼ α√1 − t(1 − √1 − t)
+π
+log(1 − r) + α√1 − t(1 − √1 − t)
+π
+log
+�
+2(1 − t)
+(1 − √1 − t)2
+�
++
+α
+π (1 − (1 −
+√
+1 − t)2) + O(
+√
+1 − r log(1 − r))
+(A 7)
+as r → 1−. We can clearly see this will necessitate an inner expansion that contains logarithmic terms;
+similar behaviour is displayed for surface tension-dominated drops under different evaporative fluxes (Moore
+et al. 2021).
+
+24
+M. R. Moore & A. W. Wray
+Finally, if we expand the solute mass m ∼ m0 as Pe → 0 in (2.35), we find
+m0(r, t) =
+√1 − t
+π
+√
+1 − r2
+�
+1 −
+√
+1 − t(1 −
+�
+1 − r2)
+�
+.
+(A 8)
+Whilst we could proceed to O(Pe−2/3) in the solute mass expansion in the outer region, we shall not require
+this when constructing a composite profile that is valid to O(1) throughout the droplet, so we do not present
+this here.
+A.2. Inner region
+Recalling (3.11)–(3.12), (4.2) and (4.3), in order to retain a balance between the advective and diffusive
+effects in (2.32) close to the contact line, we set
+r = 1 − Pe−2/3˜r,
+u = Pe−1/3˜u,
+h = ˜h,
+M = ˜
+M,
+m = Pe2/3 ˜m
+(A 9)
+in (2.32)–(2.35). Note that we therefore have
+˜h = ˜h0 + Pe−2/3˜h1 + O(Pe−4/3),
+˜u = ˜u0 + Pe−1/3˜u1 + Pe−2/3˜u2 + O(Pe−1)
+(A 10)
+as Pe → ∞ where
+˜h0(˜r, t) = ¯h0(˜r/α, t),
+˜h1(˜r, t) = α¯h1(˜r/α, t),
+(A 11)
+and ¯h0, ¯h1 are given by (3.13)–(3.14), and
+˜u0(˜r, t) = √α¯u0(˜r/α, t),
+˜u1(˜r, t) = α¯u1(˜r/α, t),
+˜u2(˜r, t) = α3/2 ¯u2(˜r/α, t)
+(A 12)
+and ¯u0, ¯u1, ¯u2 are given by (3.15))–(3.17).
+Seeking an asymptotic expansion of the integrated mass of the form ˜
+M = ˜
+M0+Pe−1/3 ˜
+M1+Pe−2/3 log Pe−2/3 ˜
+M2+
+Pe−2/3 ˜
+M3 + o(Pe−2/3) as Pe → ∞, we find that the leading-order inner problem is given by
+∂2 ˜
+M0
+∂˜r2
++
+�
+˜u0 − 1
+˜h0
+∂˜h0
+∂˜r
+�
+∂ ˜
+M0
+∂r
+= 0,
+for
+˜r > 0, 0 < t < 1,
+(A 13)
+subject to the boundary condition
+˜
+M0(0, t) = 1/2π for 0 < t < 1 and, in order to match with the local
+expansion of leading-order-outer solution at the contact line (A 6), we must have
+˜
+M0 →
+√1 − t
+π
+− (1 − t)
+2π
+as
+˜r → ∞,
+(A 14)
+Defining the integrating factor
+I(˜r, t) =
+�
+1
+1 − e−˜r/α
+�
+exp
+�
+2
+√
+2
+(1 − t)
+� ˜r
+0
+√ξ
+1 − e−ξ/α dξ
+�
+,
+(A 15)
+we find that the solution is given by
+˜
+M0(˜r, t) = 1
+2π + B0(t)
+� ˜r
+0
+1
+I(s, t) ds,
+(A 16)
+where
+B0(t) = − 1
+π
+�
+1 −
+√
+1 − t − t
+2
+� �� ∞
+0
+1
+I(s, t) ds
+�−1
+.
+(A 17)
+We note here that the first term on the right-hand side of B0(t) is simply the leading-order accumulated
+mass at the contact line as a function of time, N(t), that is
+N(t) = 1
+4
+� t
+0
+(m0u0)(1−, τ) dτ = 1
+π
+�
+1 −
+√
+1 − t − t
+2
+�
+.
+(A 18)
+It is worth noting the similarities between (A 18) and the equivalent expression for a surface-tension domi-
+nated drop evaporating under a constant evaporative flux (Freed-Brown 2015; Moore et al. 2021).
+At O(Pe−1/3), we have
+∂2 ˜
+M1
+∂˜r2
++
+�
+˜u0 − 1
+˜h0
+∂˜h0
+∂˜r
+�
+∂ ˜
+M1
+∂r
+= 4∂ ˜
+M0
+∂t
+− ˜u1
+∂ ˜
+M0
+∂˜r
+for
+˜r > 0, 0 < t < 1,
+(A 19)
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+25
+subject to
+˜
+M1(0, t) = 0 for 0 < t < 1 and the far-field matching condition
+˜
+M1 → −
+�
+2(1 − t)
+π
+(1 −
+√
+1 − t)
+√
+˜r
+as
+˜r → ∞.
+(A 20)
+While in practice it may be easier to find
+˜
+M1(˜r, t) from (A 19)–(A 20) numerically, for posterity, we state
+that this boundary value problem has solution
+˜
+M1(˜r, t) =
+� ˜r
+0
+1
+I(s, t)
+�� s
+0
+�
+4∂ ˜
+M0
+∂t
+− ˜u1
+∂ ˜
+M0
+∂˜r
+�
+I(σ, t) dσ
+�
+ds + B1(t)
+� ˜r
+0
+1
+I(s, t) ds,
+(A 21)
+where
+B1(t) = −
+�� ∞
+0
+�
+1
+I(s, t)
+�� s
+0
+�
+4∂ ˜
+M0
+∂t
+− ˜u1
+∂ ˜
+M0
+∂˜r
+�
+I(σ, t) dσ
+�
+−
+√
+2(1 − t)∂ ˜
+M0
+∂t
+1
+√s
+�
+ds
+� �� ∞
+0
+1
+I(s, t) ds
+�−1
+,
+(A 22)
+is chosen to kill the O(1)-term in the far-field expansion of
+˜
+M1(˜r, t).
+The O(Pe−2/3 log Pe−2/3)-problem is given by
+∂2 ˜
+M2
+∂˜r2
++
+�
+˜u0 − 1
+˜h0
+∂˜h0
+∂˜r
+�
+∂ ˜
+M2
+∂r
+= 0
+for
+˜r > 0, 0 < t < 1,
+(A 23)
+subject to
+˜
+M2(0, t) = 0 for 0 < t < 1 and the far-field matching condition
+˜
+M2 → α√1 − t(1 − √1 − t)
+π
+as
+˜r → ∞.
+(A 24)
+The solution may be found in a similar manner to the leading-order problem, yielding
+˜
+M2(˜r, t) = B2(t)
+� ˜r
+0
+1
+I(s, t) ds,
+(A 25)
+where
+B2(t) = α√1 − t(1 − √1 − t)
+π
+�� ∞
+0
+1
+I(s, t) ds
+�−1
+.
+(A 26)
+Lastly, at O(Pe−2/3), we have
+∂2 ˜
+M3
+∂˜r2
++
+�
+˜u0 − 1
+˜h0
+∂˜h0
+∂˜r
+�
+∂ ˜
+M3
+∂r
+= 4∂ ˜
+M1
+∂t
+−˜u1
+∂ ˜
+M1
+∂˜r −˜u2
+∂ ˜
+M0
+∂˜r − 1
+˜h0
+�˜h1
+˜h0
+∂˜h0
+∂˜r − ∂˜h1
+∂˜r
+�
+∂ ˜
+M0
+∂˜r − ∂ ˜
+M0
+∂˜r
+=: V(˜r, t)
+(A 27)
+for ˜r > 0, 0 < t < 1, subject to
+˜
+M3(0, t) = 0 for 0 < t < 1 and the far-field condition
+˜
+M3 → −(1 − t)
+π
+˜r+
+�α√1 − t(1 − √1 − t)
+π
+� �
+log ˜r + log
+�
+2(1 − t)
+(1 − √1 − t)2
+��
++α
+π (1−(1−
+√
+1 − t)2)
+as
+˜r → ∞.
+(A 28)
+The solution is given by
+˜
+M3(˜r, t) =
+� ˜r
+0
+1
+I(s, t)
+�� s
+0
+V(σ, t)I(σ, t) dσ
+�
+ds + B3(t)
+� ˜r
+0
+1
+I(s, t) ds,
+(A 29)
+where
+B3(t) =
+�
+−
+� ∞
+1
+�
+1
+I(s, t)
+�� s
+0
+V(σ, t)I(σ, t) dσ
+�
++ (1 − t)
+π
+− α√1 − t(1 − √1 − t)
+πs
+�
+ds
+−
+� 1
+0
+1
+I(s, t)
+�� s
+0
+V(σ, t)I(σ, t) dσ
+�
+ds − (1 − t)
+π
++ α
+π (1 − (1 −
+√
+1 − t)2)
++α√1 − t(1 − √1 − t)
+π
+log
+�
+2(1 − t)
+(1 − √1 − t)2
+�� �� ∞
+0
+1
+I(s, t) ds
+�−1
+,
+(A 30)
+has been chosen to satisfy the correct far-field behaviour.
+We are now in a position to find the inner solution for the solute mass. By substituting the scalings (A 9)
+
+26
+M. R. Moore & A. W. Wray
+into (2.35), we see that
+˜m = −
+1
+1 − Pe−2/3˜r
+∂ ˜
+M
+∂˜r ,
+(A 31)
+so that expanding ˜m = ˜m0 + Pe−1/3 ˜m1 + Pe−2/3 log Pe−2/3 ˜m1 + Pe−2/3 ˜m2 as Pe → ∞, we have
+˜m0 = −∂ ˜
+M0
+∂˜r ,
+˜m1 = −∂ ˜
+M1
+∂˜r ,
+˜m2 = −∂ ˜
+M2
+∂˜r ,
+˜m3 = −∂ ˜
+M3
+∂˜r
+− ˜r∂ ˜
+M0
+∂˜r .
+(A 32)
+A.3. Composite solutions
+We now have all of the necessary components needed to construct (additive) composite solutions for com-
+parison to the numerical results.
+To construct a composite solution for the integrated mass variable, we combine the first two outer solutions
+(A 3) and (A 5), the first four inner solutions (A 16), (A 21), (A 25) and (A 29), the overlap terms given by
+(A 6)–(A 7) using Van Dyke’s matching rule Van Dyke (1964), which yields
+Mcomp(r, t) = M0(r, t) + Pe−2/3M1(r, t) + ˜
+M0
+�
+Pe2/3(1 − r), t
+�
++ Pe−1/3 ˜
+M1
+�
+Pe2/3(1 − r), t
+�
++
+Pe−2/3 log Pe−2/3 ˜
+M2
+�
+Pe2/3(1 − r), t
+�
++ Pe−2/3 ˜
+M3
+�
+Pe2/3(1 − r), t
+�
+−
+�√1 − t
+π
+− (1 − t)
+2π
+−
+�
+2(1 − t)
+π
+(1 −
+√
+1 − t)
+√
+1 − r − (1 − t)
+π
+(1 − r)+
+Pe−2/3
+�α
+π (1 − (1 −
+√
+1 − t)2) + α√1 − t(1 − √1 − t)
+π
+�
+log(1 − r) + log
+�
+2(1 − t)
+(1 − √1 − t)2
+����
+.
+(A 33)
+This composite solution is valid up to and including O(Pe−2/3) throughout the whole of the droplet.
+Similarly, for the solute mass, the equivalent composite profile is compiled by taking the first outer solution
+(A 8) as well as the first four inner solutions given by (A 32), so that, accounting for the overlap contributions,
+mcomp(r, t) = m0(r, t) + Pe2/3 ˜m0
+�
+Pe2/3(1 − r), t
+�
++ Pe1/3 ˜m1
+�
+Pe2/3(1 − r), t
+�
++
+log Pe−2/3 ˜m2
+�
+Pe2/3(1 − r), t
+�
++ ˜m3
+�
+Pe2/3(1 − r), t
+�
+−
+�
+(1 − t)(1 − √1 − t)
+√
+2π√1 − r
+− (1 − t)
+π
+.
+(A 34)
+We note that this composite solution is valid up to and including O(1) throughout the droplet.
+Appendix B. Numerical solution of the solute transport problem
+In this section, we outline the numerical scheme for solving the advection-diffusion problem (2.32)–(2.34)
+for the integrated mass variable M(r, t). As discussed previously, the integrated mass variable formulation
+is advantageous when solving numerically, since it is mass-preserving and has simple-to-implement Dirichlet
+boundary conditions.
+Our numerical method is an adaptation of that discussed in Moore et al. (2021) for the Bo = 0 regime. We
+utilize central differences with gridpoints clustered close to the contact line, where there are rapid changes
+in behaviour associated with the coffee ring. We choose a uniform grid in the variable ζ ∈ [0, 1], where
+r = 1 − ℓζ
+1 − ℓ ,
+(B 1)
+and ℓ is taken to coincide with the smallest of the two boundary layers; that is, ℓ = κ(1 − tc) where
+κ = min
+�
+Bo−1/2, Pe−2/3�
+and tc is the final computation time. Note that these boundary layers are in
+the context of large Bond number; when Bo = O(1), we have both increased the number of nodes in the
+discretization and chosen ℓ = Pe−2 to ensure we capture the diffusive boundary layer in this regime.
+Even when it is present, the secondary peak does not exhibit such extreme behaviour, with a much
+shallower profile than the primary peak, so provided that the discretization is chosen suitably small, the
+secondary peak is captured well without special considerations. The resulting system is solved using ode15s
+in MATLAB and incorporates complex step differentiation to compute the Jacobian (Shampine (2007)).
+
+Gravity can lead to multiple peaks in the early stages of coffee ring formation
+27
+The veracity of the simulations has been confirmed with stringent convergent checks alongside the excellent
+comparisons to the asymptotic results in both the order unity Bond number regime and the large Bond
+number regime (cf. figures 4, 5).
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+

YNE2T4oBgHgl3EQfvAgF/content/tmp_files/2301.04085v1.pdf.txt

@@ -0,0 +1,1609 @@
+MNRAS 000, 1–9 (2022)
+Preprint 11 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Propagating photo-z uncertainties: a functional derivative approach
+Robert Reischke⋆1
+1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB),
+German Centre for Cosmological Lensing, 44780 Bochum, Germany
+11 January 2023
+ABSTRACT
+Photometric redshifts are a key ingredient in the analysis and interpretation of large-scale
+structure (LSS) surveys. The accuracy and precision of these redshifts estimates is directly
+linked to the constraining power of photometric surveys. It is hence necessary to define preci-
+sion and accuracy requirements for the redshift calibration to not to infer biased results in the
+final analysis. For weak gravitational lensing of the LSS the photometry culminates in the es-
+timation of the source redshift distribution (SRD) in each of the tomographic bins used in the
+analysis. The focus has been on shifts of the mean of the SRDs and how well the calibration
+must be able to recover those. Since the estimated SRDs are usually given as a normalized
+histogram with corresponding errors, it would be advantageous to propagate these uncertain-
+ties accordingly to see whether the requirements of the given survey are indeed fulfilled. Here
+we propose the use of functional derivatives to calculate the sensitivity of the final observ-
+ables, e.g. the lensing angular power spectrum, with respect to the SRD at a specific redshift.
+This allows the propagation of arbitrarily shaped small perturbations to the SRD, without hav-
+ing to run the whole analysis pipeline for each realization again. We apply our method to a
+EUCLID survey and demonstrate it with SRDs of the KV450 data set, recovering previous
+results. Lastly, we note that for cosmic shear moments of order larger than two will probably
+be not relevant when propagating redshift uncertainties.
+Key words: cosmology: theory, large-scale structure of Universe, surveys, galaxies: photom-
+etry
+1
+INTRODUCTION
+Cosmic shear, the weak gravitational lensing effect imprinted on
+distant galaxies by the large-scale structure (LSS), is one of the pri-
+mal science goals for EUCLID and Rubin-LSST. The blueprint for
+these missions has been set by current stage-3 surveys, including
+the Kilo-Degree Survey (Kuijken et al. 2019; Asgari et al. 2021,
+KiDS), the Dark Energy Survey (Abbott et al. 2018; Gatti et al.
+2022, DES) or the Subaru Hyper Suprime-Cam (Hamana et al.
+2020, HST), yielding tight constraints ob the matter distribution
+in the late Universe.
+The cosmic shear signal is estimated by measuring the coher-
+ent distortion of background galaxies. Since the intrinsic elliptic-
+ity of galaxies is much larger than the lensing effect, millions of
+galaxies are required to measure a significant signal. This casts a
+complete spectroscopic survey unfeasible. Hence, one has to rely
+on redshift estimates from photometry. In order to interpret the ob-
+served ellipticity correlations, the potometric redshifts have to be
+calibrated. There are different approaches for the calibration pro-
+cedure on the market. These include the calibration with a spec-
+troscopic reference sample (possibly with re-weighting) (e.g. Lima
+et al. 2008; Newman 2008; Matthews & Newman 2010; Masters
+⋆ E-mail: reischke@astro.ruhr-uni-bochum.de
+et al. 2015; Bonnett et al. 2016; McLeod et al. 2017; Hildebrandt
+et al. 2020; Wright et al. 2020; Myles et al. 2021), using photome-
+try measurements in conjunction with clustering measurements of
+tracer populations (e.g. Sánchez & Bernstein 2019; van den Busch
+et al. 2020; Alarcon et al. 2020) and self-organising maps (Wright
+et al. 2020). It is also possible to partially self-calibrate the pho-
+tometric redshifts in weak lensing data (e.g. Schaan et al. 2020).
+In order to account for general shapes of the source-redshift dis-
+tributions (SRDs) different mixture models have been employed
+(see for example Rau et al. 2020). These Gaussian processes are
+non-parametric, but they are by definition non-linear, which makes
+their implementation in cosmology pipelines in general very diffi-
+cult. Stölzner et al. (2021) used linear fit parameters to circumvent
+this problem to self-calibrate the data, as it can be implemented in
+existing pipelines very easily.
+Currently it is best practice to propagate the redshift uncer-
+tainty in the SRDs by introducing shift parameters in the mean of
+the distribution (Hildebrandt et al. 2021; Hikage et al. 2019; Abbott
+et al. 2022). As the sensitivity of surveys rises, however, the re-
+quirements on the SRD uncertainties become larger as well. There-
+fore, the contributions from higher order cumulants of the SRD be-
+come important. As discussed above, previous works have focused
+on Gaussian mixture models to self-calibrate the cosmic shear mea-
+surement. In this paper we investigate the general sensitivity of the
+© 2022 The Authors
+arXiv:2301.04085v1  [astro-ph.CO]  10 Jan 2023
+
+2
+Reischke
+lensing power spectrum to perturbations in the SRD. In particu-
+lar we are calculating the functional derivative of the cosmic shear
+angular power spectrum with respect to the SRD at a particular
+co-moving distance. This can then be mapped to a total error in
+the cosmic shear power spectrum if a perturbation in the SRD in
+a co-moving interval is applied. We take the constraint of the nor-
+malisation of the SRD into account when calculating the functional
+derivative. Therefore we can propagate arbitrary perturbations to
+the SRDs (subject to some underlying covariance) and propagate
+them into the Cℓ of cosmic shear. This allows us to estimate the dif-
+ference in χ2 induced by the uncertainty in the SRD, without hav-
+ing to run thousands of realizations of the analysis pipeline used.
+By using a Fisher matrix for the cosmological parameters, this ∆χ2
+can then be mapped to potential biases in cosmological parameters.
+Here we studied a rather idealised scenario by working in Fourier
+space, assuming a Gaussian likelihood and ignoring intrinsic align-
+ments. The method, however, easily generalises and including these
+effects is straightforward.
+We structure the paper as follows: In Section 2 we briefly
+review cosmic shear basics and introduce the methodology used
+by calculating the functional derivative of the weak lensing angu-
+lar power spectrum. The results are presented in Section 3, where
+we apply the procedure to a survey with EUCLIDs specifications
+and to KiDS-VIKING-450 (KV450). We conclude in Section 4.
+In the appendices we also investigate the possibility of an Edge-
+worth expansion of the SRD (Appendix A), discuss photometric
+galaxy clustering (Appendix B), the distribution of the mean and
+standard deviation of the SRD in Appendix C, the general relation-
+ship to observables (Appendix D), the functional derivative of the
+non-Limber projection in Appendix E and inrinsic alignments (Ap-
+pendix F).
+2
+METHODOLOGY
+In this section we present the basic methodology of our analysis.
+In particular we describe the basics of cosmic shear and derive the
+function derivative of the lensing angular power spectrum with re-
+spect to the SRDs.
+2.1
+Cosmic shear basics
+The equation for the cosmic shear power spectrum in tomographic
+bins i and j in the Limber proejction is (Limber 1954; Loverde &
+Afshordi 2008)
+C
+κiκj
+ℓ
+=
+� χH
+0
+dχ
+χ2 W(i)
+κ (χ)W(j)
+κ (χ)Pδ
+�ℓ + 0.5
+χ
+, χ
+�
+,
+(1)
+where Pδ is the matter power spectrum, for which we use the em-
+ulated spectrum from Mead et al. (2015). W(i)
+κ (χ) is the lensing
+weight of the i-th tomographic bin as given by:
+W(i)
+κ (χ) = 3Ωm0
+2χ2
+H
+χ
+a(χ)
+� χH
+χ
+dχ′n(i)
+s (χ′)χ′ − χ
+χ′
+.
+(2)
+Here χ is the co-moving distance, a the scale factor, Ωm0 the matter
+density parameter today, χH the Hubble radius and n(i)
+s is the SRD
+in the i-th tomographic bin which builds on photo-z measurements
+and its calibration. It is normalized in each tomographic bin such
+that
+�
+dz n(i)
+s (z) = 1 =
+�
+dχ n(i)
+s (z(χ)) dz
+dχ ≡
+�
+dχ n(i)
+s (χ) .
+(3)
+Since photo-z is just an estimate of the true redshift, the estimated
+source-redshift distribution, n(i)
+s , is not exactly known. Here we in-
+vestigate two approaches:
+i) Use functional derivatives to evaluate the change of the lens-
+ing power spectrum when perturbing the n(i)
+s at different redshifts.
+Given specific survey settings and precision goals, limits on the al-
+lowed change of the n(i)
+s can be determined, which in turn can be
+mapped to changes in the cumulants or moments of the underlying
+distribution (see Section 2.2).
+ii) We expand the underlying source-redshift distribution in an
+asymptotic Edgeworth series and investigate the requirements on
+the cumulants directly in a Fisher analysis. The second approach is
+not feasible for realistic SRDs (see Appendix A).
+2.2
+Functional derivative of the lensing power spectrum
+Here we wish to investigate the sensitivity of the weak lensing
+power spectrum to the full shape of the source-redshift distribution
+using functional derivatives. In particular we start by perturbing
+n(i)
+s (χ(z)) at a certain redshift z0, such that χ0 = χ(z0). The corre-
+sponding perturbed lensing weight is thus
+∆W(i)
+κ (χ, χ0) = δW(i)
+κ (χ)
+δn(i)
+s (χ0)
+∆n(i)
+s (χ0) .
+(4)
+This expression evaluates, how the lensing weight changes if the
+source-redshift distribution is perturbed by an amount ∆n(i)
+s at the
+co-moving distance χ0 corresponding to the redshift z0.
+Ultimately, we are interested in the change of the lensing
+power spectrum, Equation (B1). First, by applying the Leibniz rule
+δC(ij)
+ℓ
+δn(a)(χ0) =
+�
+dx δC(ij)
+ℓ
+δW(a)(x)
+δW(a)(x)
+δn(a)(χ0)
+=
+�
+dx δW(a)(x)
+δn(a)(χ0)
+Pδ
+�
+ℓ+0.5
+x , x
+�
+x2
+�
+W(j)(x)δD
+ia + W(i)(x)δD
+ja
+�
+,
+(5)
+The missing ingredient is the functional derivative of the lensing
+kernel, for which we find
+δW(i)(x)
+δn(j)
+s (χ0)
+= 3Ωm0
+2χ2
+H
+x
+a(x)
+χ0 − x
+χ0
+δD
+ijΘ(χ0 − x) .
+(6)
+Θ(x) is the Heaviside function to ensure that the functional deriva-
+tive vanishes if the SRD is perturbed outside the integration bounds.
+Using Equation (4) and Equation (5) we can write the change in
+angular power spectrum ∆C(ij)
+ℓ (χ′) due to a change in the source-
+redshift distribution at co-moving distance χ0 as
+∆C(ij)
+ℓ,a (χ0) ≡
+δC(ij)
+ℓ
+δn(a)(χ0)∆n(a)(χ0)
+= 3Ωm0
+2χ2
+H
+∆n(χ0)
+�
+dx
+a(x)x
+χ0 − x
+χ0
+Pδ
+�ℓ + 0.5
+x
+, x
+�
+×
+�
+W(j)(x)δD
+ia + W(i)(x)δD
+ja
+�
+.
+(7)
+Integrating the perturbed lensing spectrum then gives the total per-
+turbation:
+∆C(ij)
+ℓ,a ≡
+�
+dχ0∆C(ij)
+ℓ,a (χ0) .
+(8)
+So far we have treated the function n(i)(z) as being completely free.
+However, the functional derivative needs to respect the constraint
+MNRAS 000, 1–9 (2022)
+
+functional photo-z
+3
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+redshift z
+1
+2
+3
+4
+n(i)
+s (z)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+Figure 1. Allowed perturbation for EUCLID to the SRD of the ten tomo-
+graphic source bins. Solid lines show the fiducial SRD, while the bands
+show the allowed perturbation to it.
+given in Equation (3), thus limiting the possible variations of n(i)(z).
+The normalization condition itself is again a function and we write
+N[n(i)
+s ] � 1 −
+�
+dz n(i)
+s (z) = 0 ,
+(9)
+this constraint can be implemented by first defining
+n(i)
+s (z) �
+f(z)
+�
+dx′ f(x′)
+(10)
+which will be normalized by construction. n(i)
+s (z) is a functional of
+f and we can now evaluate the functional derivative of C[n[ f]] as
+an unconstrained derivative but evaluated at f = n. To avoid clutter
+we ignore the sub- and superscripts in this part
+�δC[n[ f]]
+δf(x)
+� ������ f=n
+=
+�
+dx′ δC[n]
+δn(x′)
+δn(x′)
+δ f(x)
+������ f=n
+.
+(11)
+With
+δn(x′)
+δf(x) = δD(x′ − x)
+�
+dy f(y)
+−
+f(x′)
+��
+dy f(y)
+�2 ,
+(12)
+one finds
+δC[n]
+δ1n(x) ≡
+�δC[n[f]]
+δf(x)
+� ������ f=n
+= δC[n]
+δn(x) −
+�
+dy δC[n]
+δn(y) n(y) ,
+(13)
+where we denote that we want to keep the normalization fixed by
+the variation δ1. This is a very intuitive expression: the first term
+evaluates the standard functional derivative, while the second term
+corrects this variation to respect the normalization.
+2.3
+Fisher forecast
+The next step is to set some requirement on the lensing power spec-
+tra. Here we will look at the difference in the χ2, assuming a Gaus-
+sian likelihood and thus setting a lower limit on the required accu-
+racy of n(i)
+s (z). For modes aℓm with zero mean and covariance Cℓ,
+the ∆χ2 between multipoles ℓmin and ℓmax can be written as
+∆χ2(ℓmin, ℓmax) = fsky
+ℓmax
+�
+ℓ=ℓmin
+2ℓ + 1
+2
+tr
+�
+∆CℓC−1
+ℓ ∆CℓC−1
+ℓ
+�
+,
+(14)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+order of central moment n
+10−2
+100
+102
+104
+106
+108
+relative change in %
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+Figure 2. Allowed relative change in per-cent of the central moment of the
+SRD in each tomographic bin. The changes are calculated from the per-
+turbed SRD distributions as shown in Figure 1.
+note that Cℓ is the matrix with the components C(ij). The factor fsky
+takes into account the observed sky fraction. Using Equation (8) we
+rewrite the previous equation as a Riemann sum
+∆χ2(ℓmin, ℓmax) = fsky
+ℓmax
+�
+ℓ=ℓmin
+2ℓ + 1
+2
+×
+�
+r,s,i,j
+tr
+�
+δCℓ
+δ1n(i)(χr)C−1
+ℓ
+δCℓ
+δ1n(j)(χs)C−1
+ℓ
+�
+× DχrDχs∆n(i)(χr)∆n(j)(χs) ,
+(15)
+with the measure Dχr. If we define the Fisher matrix in this case
+as:
+Fαβ = fsky
+ℓmax
+�
+ℓ=ℓmin
+2ℓ + 1
+2
+tr
+� δCℓ
+δ1nα
+C−1
+ℓ
+δCℓ
+δ1nβ
+C−1
+ℓ
+�
+Dχr(α)Dχs(β) ,
+(16)
+where we labeled n(i)(χr) → nα, we recover for a difference in χ2
+using a scalar product on the finite dimensional Hilbert space of
+shifts in the redshift distribution where the Fisher matrix acts as a
+norm-inducing metric
+∆χ2 = F(∆n, ∆n) ≡ ∆nT F∆n ,
+(17)
+where ∆n is the vector containing shifts of the components nα.
+The Fisher matrix, Equation (16), describes, how well the
+shifts nα can be determined by a measurement of the angular power
+spectra Cα given certain survey settings. Clearly, if one would try to
+measure all possible perturbations, neighbouring δn(χ) are strongly
+correlated. This is, however not the question we would like to ask
+in this work. Instead, we want to look at the situation that we allow
+any perturbation ∆n, irrespective of the correlation. Therefore, by
+turning this argument around, we only use the diagonal part of the
+Fisher matrix.
+Lastly one should note that the functional derivative is strictly
+defined as a limiting process for infinitesimally small perturbation
+to the function at hand. The relation in general can be non-linear,
+but as long as relative perturbations to the function are small with
+respect to unity, these non-linear contributions are sub-dominant.
+Especially for surveys with tight requirements on the SRDs this is
+essentially always fulfilled.
+MNRAS 000, 1–9 (2022)
+
+4
+Reischke
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+redshift z
+1
+2
+3
+4
+5
+6
+n(i)
+s (z)
+KV450
+1
+2
+3
+4
+5
+tomographic bin index
+Figure 3. Allowed perturbation for KV450 to the SRD for the 5 tomo-
+graphic source bins. Solid lines show the fiducial SRD, while the bands
+show the allowed perturbation to it.
+3
+RESULTS
+3.1
+Allowed Perturbations to the Source Redshift
+Distribution
+First we will look at the allowed perturbations to the SRD by as-
+suming allowing for a total ∆χ2 of unity, corresponding to a one
+σ shift of a linear model parameter. Clearly, there are many differ-
+ent solutions ∆n that satisfy ∆χ2 = 1 subject to Equation (17). To
+show the structure of the Fisher matrix we therefore distribute the
+allowed ∆χ2 per ∆nα equally.
+We will assume EUCLID specifications for the survey as
+given in Blanchard et al. (2020) and assume ntomo = 10 tomo-
+graphic bins, a sky fraction of 0.3. Furthermore, we will collect
+multipoles between ℓmin = 10 and ℓmax = 3000. We then calculate
+the diagonal Fisher matrix from Equation (16) and distribute the er-
+rors equally as described above. This results into a possible realisa-
+tion of ∆n yielding ∆χ2 = 1 subject to the constraint Equation (3).
+Figure 1 shows the resulting perturbed SRDs. The solid lines show
+the fiducial SRD, while the shaded areas show the allowed pertur-
+bations to not cause a bias of more than 1 σ for a linear model pa-
+rameter. Lastly, the tomographic bin index is shown as a colour-bar.
+The general trend is very clear, the allowed perturbations become
+very large around a small interval ∆χ around the mean of the distri-
+butions. For most tomographic bins this coincides with the peak of
+the distribution as they are very close to Gaussian. Only for the first
+and the last bin these spikes are a bit offset since the distributions
+are a bit more asymmetric. This already confirms that the most im-
+portant part about the SRDs in cosmic shear measurements is to
+calibrate the mean redshift of each tomographic bin very well. Fur-
+thermore, we observe that the spikes tend to be narrower at higher
+redshifts, indicating that the uncertainty on the mean of the SRD
+is more important at higher redshifts. We want to stress again, that
+this is just one realization of ∆n that produces a ∆χ2 = 1, but by
+distributing the errors equally, it is possible to see, which pertur-
+bations the final measurement is most sensitive to. However, the
+uncertainties should not be used at literally value and are extreme
+values, they just give a general trend.
+Next, we use the perturbed SRDs to calculate their central mo-
+ments µn:
+µn � E[(X − E[X])n] =
+�
+p(x)(x − µ)ndx ,
+(18)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+∆χ2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+KV450
+68th
+50th
+95th percentile
+Figure 4. ∆χ2 for 106 realisations of ∆n from the CKV450
+n(χ)
+. We also show
+the 50, 68 and 95 percentiles.
+for a probability distribution function p(x) with mean µ. The per-
+turbed SRDs are used to calculate the change in the central mo-
+ments relative to the fiducial SRD. Figure 2 shows the resulting
+relative change for all tomographic bins as a function of the order
+of the central moment. Clearly, the first moment is most important
+and while the second one still needs to be known at a 10% level,
+all higher order moments are essentially unimportant. This is of
+course reminiscent of the behaviour observed in Figure 1, where
+the perturbations are such that they essentially fix the mean. It is of
+course entirely possible, that we alter the shape of the distribution
+in a different way but still achieve the desired accuracy.
+Nonetheless, the results show that for the SRD for cosmic
+shear only the mean redshift and the width are important with the
+former influencing the result way stronger (by more than an order
+of magnitude). In Appendix C we sample from the allowed changes
+in the SRD and show the relative difference of the first two mo-
+ments to illustrate their scatter.
+3.2
+Propagating Redshift Errors
+In this section we will revisit the KV450 data for the SRD (Hilde-
+brandt et al. 2020). This data set is used since it includes a covari-
+ance matrix from the direct calibration (DIR). For the clustering
+redshifts (van den Busch et al. 2020) or the self-organising maps
+(Wright et al. 2020) no bootstrap covariance was estimated so far.
+For completeness the allowed perturbations are shown in Fig-
+ure 3. Due to the lower signal-to-noise ratio of the measurement,
+the allowed perturbations are much larger than in the previous case.
+The features, however, are very similar.
+Since we are expressing everything in co-moving distance, the
+covariance matrix needs to be transformed accordingly. Let CKV450
+n(z)
+be the covariance matrix in n(z) space, the transformed covariance
+is then
+CKV450
+n(χ)
+= JTCKV450
+n(z)
+J ,
+(19)
+where J is the Jacobian with components Ji
+j = δi
+jdz/dχ. Alterna-
+tively, the Fisher matrix of the SRD perturbations can be expressed
+in redshift space by the inverse transform.
+Perturbations ∆n are now sampled from CKV450
+n(χ)
+and propa-
+gated to obtain ∆χ2 according to Equation (14). If the redshift errors
+as given in CKV450
+n(χ)
+are sufficiently small to not produce a significant
+bias in the cosmological parameters such as S 8 we expect most
+MNRAS 000, 1–9 (2022)
+
+functional photo-z
+5
+0.200
+0.225
+0.250
+0.275
+0.300
+0.325
+0.350
+0.375
+0.400
+Ωm0
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+σ8
+KV450
+1
+2
+3
+4
+5
+6
+∆χ2
+Figure 5. The black histogram shows the induced shifts by the photo-z un-
+certainty in the Ωm0-σ8-plane, derived from the ∆χ2 of Figure 4. In red
+we show the contour from the Fisher matrix for KV450 enclosing the 1σ
+confidence interval.
+realisations (i.e. 68% Hildebrandt et al. 2020) to yield ∆χ2 < 1.
+Figure 4 shows the resulting distribution in ∆χ2 for the 106 real-
+izations of ∆n for KV450. The vertical dashed lines show the 50th,
+68th and 95th percentile. It is clear from this plot that the precision
+of the SRD used in KV450 is high enough to not yield any spurious
+detection in the final parameter constraints since the 68th percentile
+is still well below unity.
+One could now further propagate these uncertainties into cos-
+mological parameters using the corresponding Fisher matrix. For a
+given shift in the SRD ∆n, the corresponding shifts in the cosmo-
+logical parameters, ∆θ can be calculated:
+∆θi = −(F−1)i
+αFα
+β∆nβ ,
+(20)
+where Greek indices run over the perturbations in the SRD, while
+Latin indices label cosmological parameters. Here we assumed the
+sum convention. Fi
+α hence is the mixed pseudo Fisher matrix:
+Fi
+α = −E
+�∂ ln L
+∂θi
+δ ln L
+δnα Dχr(α)
+�
+(21)
+and it’s inverse is a pseudo inverse. Since the inversion of this ma-
+trix is not necessarily stable we choose to go another route here.
+Since the distribution of ∆χ2 is known, we are interested in sam-
+ples of cosmological parameters with the same ∆χ2 with respect
+to the best fit value. For a Gaussian posterior in one dimension
+this would amount to a distribution such that the absolute value of
+each sample is fixed to
+√
+∆θ2. We sample from a standard Gaussian
+distribution and modify its width by
+√
+∆θ2. This Gaussian is then
+mapped into the frame of the cosmological parameters under con-
+sideration via the Cholesky decomposition of the Fisher matrix of
+the cosmological parameters. In Figure 5 we apply this procedure
+to the ∆χ2 distribution of KV450 (Figure 4). Each dot represents
+one sample of the ∆χ2 distribution with its value shown as a colour
+bar. It can be seen as the geodesic distance to the fiducial value
+for the cosmological parameters in the parameter manifold (Giesel
+et al. 2021). The red contours depict the expected 1, 2, 3σ confi-
+dence regions from the Fisher forecast for KV450. Since in the
+original analysis more than the two parameters here where used,
+we re-scale the ∆χ2 accordingly, in particular by the χ2 quantile
+function χ2
+k(p), where k = 10 is the number of parameters in the
+actual analysis analysis (Hildebrandt et al. 2017) and p = 0.68.
+This is done in order to obtain a fair comparison. It is clear from
+0.794
+0.796
+0.798
+0.800
+0.802
+0.804
+0.806
+S8
+0
+25
+50
+75
+100
+125
+150
+175
+200
+KV450
+Figure 6. Induced scatter on the S 8 = σ8
+√Ωm0/0.3 parameter. This is
+directly derived from the samples of Figure 5. The scatter is roughly 15
+per-cent of the statistical error budget reported in Hildebrandt et al. (2017,
+2020).
+the plot, that all samples for the photometric redshift distribution lie
+well within the 1σ contour. Furthermore, it should be noted that we
+are considering a very idealised forecast with two free parameters
+and no systematics here. The procedure, however, can be general-
+ized to any number of parameters. Furthermore, one can apply the
+same analysis to a full Monte-Carlo-Markov-Chain (MCMC) by
+matching those samples which are ∆χ2 away from the maximum
+likelihood of the MCMC. Lastly, the samples from Figure 5 can be
+mapped to S 8 = σ8
+√Ωm0/0.3. Figure 6 shows the resulting his-
+togram of the scatter due to the photo-z uncertainties. Comparing
+this to ∆S 8 = 0.076 at 68% confidence (Hildebrandt et al. 2020)
+shows that the scatter induced by the redshift uncertainties as sam-
+pled from the KV450 SRD covariance have a small effect on the
+overall error budget. In Hildebrandt et al. (2017) a Fisher matrix
+method for the shifts of the mean of the SRDs was investigated as
+a a source of systematics, which found similar results to the once
+presented here. The main difference between the two methods is
+that we allow for general perturbations to the redshift distribution
+(provided there correlation is given). Generalizing the procedure
+in Hildebrandt et al. (2017) to moments higher than the variance
+is bound to fail (see Appendix A). However, we would also con-
+clude that even for EUCLID, the analysis of the first two moments
+is probably sufficient.
+In appendix C the mean and standard deviation of each SRD
+in the five tomographic bins are shown for the realisations used in
+this section as sampled from the DIR covariance matrix. Figure C1
+shows a very similar behaviour to what we found in fig. 2. In par-
+ticular this is that the mean scatters less at higher redshifts, while
+the standard deviation scatters roughly equally for most of the bins.
+We close the section with a general discussion about the usage
+of ∆χ2 or directly uncertainties in the parameters. It is in general
+advantageous to make accuracy assessments for the SRD using the
+∆χ2 and not by inverting the Fisher matrix for the parameters of
+interests to obtain the shift values for those. The reason for this is
+that ∆χ2 is an invariant quantity, while shifts in parameter space are
+dependent on the specific model choice. The only caveat in the ∆χ2
+is that the number of parameter must be taken into account, this is,
+however, much easier than calculating the Fisher matrix.
+MNRAS 000, 1–9 (2022)
+
+6
+Reischke
+4
+CONCLUSIONS
+In this paper we have analysed the dependence of the cosmic shear
+angular power spectrum on the SRD. This has be done by employ-
+ing functional derivatives of the cosmic shear Cℓ with respect to
+the SRD at a fixed co-moving distance χ0. By integrating over the
+introduced error we estimated the ∆χ2 introduced by arbitrary un-
+certainties in the SRD. We applied our method to a cosmic shear
+survey with EUCLID specifications and KV450 since a covariance
+of the SRD estimate was given. Our main findings can be sum-
+marised as follows:
+(i) Allowed perturbations of the SRD are such that they preserve
+the mean of the underlying distribution. If they do, they can be
+rather larger, even for a survey like EUCLID. This is in line with
+the common practice of using only shifted means of the underlying
+redshift distribution.
+(ii) In order to achieve the accuracy required for EUCLID, the
+mean of the redshift distribution needs to be determined between
+1 and 0.01 per-cent, depending on the tomographic bin under con-
+sideration. The variance of the SRD is still important at the 10 per-
+cent level. There is still some sensitivity left in the skewness, but
+all other moments are not relevant.
+(iii) We performed a simplistic analysis of the KV450 SRDs to
+check whether they fulfill the requirements and found that the un-
+certainties, in this very idealised scenario, only yield biases up to
+1σ in the final constraints. In a full analysis, this bias would be even
+smaller. Thus confirming the redshift calibration used in KV450.
+(iv) Even for EUCLID it is most likely not necessary to inves-
+tigate moments of the redshift distribution n > 2. This conclusion
+could change for different settings and self-calibration methods.
+(v) The procedure outlined here has the advantage to be very
+cheap computationally, since the functional derivatives only need
+to be computed once. It is then only a matter of sampling from the
+underlying SRD and to propagate these perturbations with the pre-
+viously calculated functional derivative. It is hence not necessary
+to push thousands of realisations of the SRD through the analysis
+pipeline.
+The method outlined here can thus be used to analyse whether a
+perturbation in the SRD still fulfills the requirements of a given
+experiment so that no biases of model parameters are introduced. It
+allows for arbitrary perturbations to the SRD without requiring a fit
+to the actual distribution. We intend to apply the presented method
+to the updated SRDs of KiDS in the future.
+For the interested reader the appendices Appendix A - Ap-
+pendix E discuss various aspects of the analysis which could be
+refined in future work. In particular we look at the Edgeworth ex-
+pansion of the SRD in Appendix A, i.e. an expansion in the cu-
+mulants of the underlying SRDs. However, we find that, even for
+a realistic setting, the Edgeworth expansion cannot reproduce the
+original SRDs if cumulants n > 2 are considered.
+Data Availability: The data underlying this article will be
+shared on reasonable request to the corresponding author.
+ACKNOWLEDGMENTS
+RR would like to thank Hendrik Hildebrandt and Björn Malte
+Schäfer for insightful discussions and comments on the manuscript.
+RR is supported by the European Research Council (Grant No.
+770935).
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+APPENDIX A: EDGEWORTH EXPANSION
+In this section we employ an Edgeworth expansion for the photo-
+z distribution. The Edgeworth expansion is an asymptotic expan-
+sion (in contrast to the Gram-Charlier expansion). Starting from
+the characteristic function (Blinnikov & Moessner 1998).
+ϕ(j)
+Z (t) = En( j)(z)
+�
+eitZ�
+,
+(A1)
+i.e. the Fourier transform of the probability density n( j)(z). With the
+definition of the moments ˜µn, the Taylor expansion of the charac-
+teristic function is
+ϕ(j)
+Z (t) = 1 +
+∞
+�
+n1
+˜µn
+n! (it)n .
+(A2)
+The logarithm of the characterstic function is the cumulant, κn, gen-
+erating function
+κn = 1
+in
+dn
+dtn log ϕ( j)
+Z (t)
+�����t=0
+.
+(A3)
+MNRAS 000, 1–9 (2022)
+
+functional photo-z
+7
+Using this definition one can relate the cumulants to the moments
+κn = n!
+�
+{km}
+(−1)r−1(r − 1)!
+n
+�
+m=1
+1
+km!
+� ˜µm
+m!
+�km
+,
+(A4)
+where {km} denotes the set of all solutions to the Diophantine equa-
+tion
+n
+�
+a=1
+aka − n = 0 .
+(A5)
+If a distribution is then expanded as a asymptotic series around a
+normal distribution one finds
+n(z) =
+1
+√2πκ2
+exp
+�
+−(z − κ1)2
+2κ2
+�
+×
+�
+1 +
+∞
+�
+s=1
+κs/2
+2
+�
+{km}
+Hes+2r
+�������
+z
+κ1/2
+2
+�������
+s
+�
+m=1
+1
+km!
+�
+λm+2
+(m + 2)!
+�km �
+≡ nG(z)(1 + Eg(z)),
+(A6)
+where λn � κn/κn/2
+2 . We are now interested in the sensitivity of the
+distribution with respect to its cumulants. Here the cases n = 1, 2
+are a bit special:
+∂n(z)
+∂κ1
+= n(z)z − κ1
+κ2
+(A7)
+and for κ2
+∂n(z)
+∂κ2
+=
+1
+2κ1/2
+2
+�������n(z)
+�(z − κ1)2
+κ2
+− 1
+�
++ nG(z)∂Eg(z)
+∂κ1/2
+2
+������� ,
+(A8)
+where
+∂Eg(z)
+∂κ1/2
+2
+=
+∞
+�
+s=1
+�
+{km}
+κs/2
+2 P(s, {km})
+�
+He2r+s
+�������
+z
+κ1/2
+2
+�������
+×
+�������
+s
+κ1/2
+2
+−
+s
+�
+a
+ka(2a + 2)
+κka(a+1)+1/2
+2
+������� − (2r + s)He2r+s−1
+�������
+z
+κ3/2
+2
+�������
+z
+κ2
+�
+,
+(A9)
+where we also defined the product:
+P(s, {km}) �
+s
+�
+m=1
+1
+km!
+�
+κm+2
+(m + 2)!κ2m+2
+2
+�km
+.
+(A10)
+For all cumulants with n ≥ 3 one finds:
+∂n(z)
+∂κn
+= nG(z)
+∞
+�
+s=1
+�
+{km}
+κs/2
+2 P(s, {km})He2r+s
+�������
+z
+κ1/2
+2
+�������
+kn−2
+κn
+.
+(A11)
+It should be noted, however, that the Edgeworth expansion is not a
+convergent series but rather an asymptotic expansion. One therefore
+needs to check whether the expansion is a good approximation of
+the underlying distribution.
+In this case one can define the ordinary Fisher matrix using
+partial derivatives:
+Fκ(i)
+m κ( j)
+n = fsky
+ℓmax
+�
+ℓ=ℓmin
+2ℓ + 1
+2
+tr
+� ∂Cℓ
+∂κ(i)
+m
+C−1
+ℓ
+∂Cℓ
+∂κ(j)
+n
+C−1
+ℓ
+�
+,
+(A12)
+where κ(i)
+m is the m-th cumulant of the source-redshift distribution in
+the i-th tomographic bin.
+Figure A1 shows the fiducial redshift distributions for EU-
+CLID and their Edgeworth expanded approximations as solid and
+dashed lines respectively. The top plot uses the expansion up to κ3,
+while the bottom plot sums contributions up to κ6. For all but the
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+z
+0
+1
+2
+3
+4
+p(z)
+order = 3
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+z
+0
+1
+2
+3
+4
+p(z)
+order = 6
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+Figure A1. SRD for EUCLID in all 10 tomographic bins. Solid lines rep-
+resent the fiducial SRD, while dashed lines represent their respective Edge-
+worth expansion. Cumulants up to order n = 3 , 6 are used respectively.
+first and last tomographic bin, the Edgeworth series is a good ap-
+proximation. This is expected as they are essentially Gaussian and
+therefore κn ≈ 0 for n > 2. The first tomographic bin experiences
+boundary effects at z = 0 and is therefore slightly skewed. This ef-
+fect is even larger for the last tomographic bin, which has a very
+long tail to high redshifts. While the first bin can still be described
+by the Edgeworth expansion and the series converges, the 10th bin
+shows negative probability in the Edgeworth series already at third
+order. The situation becomes worst if higher order cumulants are
+included. This goes to show that even for such an idealized case as
+the EUCLID forecast, the use of the Edgeworth expansion can be
+very dangerous.
+For the case n = 2 we show the Pearson correlation coeffi-
+cient of the joint covariance matrix between the first three cumu-
+lants in each tomographic bin and four cosmological parameters in
+Figure A2. We observe some correlations between the first and sec-
+ond moment of each tomographic bin. There is a very strong cor-
+relation between first and second moment of two different redshift
+bins. Furthermore, one can see that parameters controlling the am-
+plitude of the lensing spectrum are anti-correlated with the mean.
+We want to stress again, however, that the expansion, even in this
+case, is not convergent and results obtained with n > 2 have thus to
+be taken with care.
+MNRAS 000, 1–9 (2022)
+
+8
+Reischke
+κ(0)
+1
+κ(0)
+2
+κ(1)
+1
+κ(1)
+2
+κ(2)
+1
+κ(2)
+2
+κ(3)
+1
+κ(3)
+2
+κ(4)
+1
+κ(4)
+2
+κ(5)
+1
+κ(5)
+2
+κ(6)
+1
+κ(6)
+2
+κ(7)
+1
+κ(7)
+2
+κ(8)
+1
+κ(8)
+2
+κ(9)
+1
+κ(9)
+2
+Ωm0
+σ8
+w0
+wa
+κ(0)
+1
+κ(0)
+2
+κ(1)
+1
+κ(1)
+2
+κ(2)
+1
+κ(2)
+2
+κ(3)
+1
+κ(3)
+2
+κ(4)
+1
+κ(4)
+2
+κ(5)
+1
+κ(5)
+2
+κ(6)
+1
+κ(6)
+2
+κ(7)
+1
+κ(7)
+2
+κ(8)
+1
+κ(8)
+2
+κ(9)
+1
+κ(9)
+2
+Ωm0
+σ8
+w0
+wa
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+rij
+Figure A2. Pearson correlation coefficient for the joint covariance matrix of the first two cumulants of the EUCLID like survey and four cosmological
+parameters.
+APPENDIX B: PHOTOMETRIC GALAXY CLUSTERING
+For photometric galaxy clustering, the procedure can be simply
+adopted by changing the weight function (up to galaxy bias, which
+we absorb in the power spectrum). Again by using the Limber pro-
+jection:
+C
+gig j
+ℓ
+=
+� χH
+0
+dχ
+χ2 W(i)
+g (χ)W(j)
+g (χ)Pgg
+�ℓ + 0.5
+χ
+, χ
+�
+,
+(B1)
+with the galaxy power spectrum Pgg and corresponding weights
+given by:
+W(i)
+g (χ) = n(i)
+g (χ) ,
+(B2)
+therefore the functional derivative takes the very simple form
+δC
+gig j
+ℓ
+δna(χ0) =
+Pgg
+�
+ℓ+0.5
+χ , χ
+�
+χ2
+�
+n(j)(x)δD
+ia + n(i)(x)δD
+ja
+�
+.
+(B3)
+APPENDIX C: DISTRIBUTION OF THE MEAN AND
+VARIANCE
+We show the relative difference of the mean redshift and the stan-
+dard deviation of the SRD for each tomographic. As before we dis-
+tinguish between the EUCLID’s survey settings and KV450. In par-
+ticular we sample from the diagonal covariance obtained from the
+functional Fisher matrix as described in Section 3 for the former,
+while we use the DIR covariance for the latter.
+The top plots of Figure C1 show the distribution of the mean
+and the standard deviation and show generally good agreement with
+Figure 2, that is that the mean must be known below the per-cent
+level for most bins, while the standard deviation needs to be de-
+termined by roughly 10 per-cent. It should be noted that Figure 2
+considers the extreme case where we exactly look at the envelope
+shown in Figure 1.
+Finally, the buttom two plots show the same for KV450, where
+we find much wider errors on mean and standard deviation, few per-
+cent and a few ten per-cent respectively. The general trend, how-
+ever, is the same - high redshift bins are more important than lower
+redshift bins.
+APPENDIX D: RELATIONSHIPS TO OBSERVABLES
+Real surveys usually do not use the angular power spectra as a fi-
+nal statistic. This is for example due to incomplete sky coverage,
+masking effects, variable depth or simply the dimensionality of the
+data vector. All these factors require a sufficient summary statis-
+tic. Very commonly used ones are the correlation function or band
+powers (or similarly pseudo-Cℓ). All of these are essentially linear
+transformations of the pure angular power spectrum Cℓ and assume
+the following general form:
+O[Cℓ] =
+�
+dℓCℓWO(ℓ) ,
+(D1)
+MNRAS 000, 1–9 (2022)
+
+functional photo-z
+9
+−0.004
+−0.002
+0.000
+0.002
+0.004
+¯z/⟨¯z⟩ − 1
+0
+100
+200
+300
+400
+500
+600
+700
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+−0.20 −0.15 −0.10 −0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+σz/⟨σz⟩ − 1
+0
+10
+20
+30
+40
+50
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+tomographic bin index
+−0.15
+−0.10
+−0.05
+0.00
+0.05
+0.10
+0.15
+¯z/⟨¯z⟩ − 1
+0
+10
+20
+30
+40
+50
+KV450
+1
+2
+3
+4
+5
+tomographic bin index
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+σz/⟨σz⟩ − 1
+0
+2
+4
+6
+8
+10
+12
+14
+16
+KV450
+1
+2
+3
+4
+5
+tomographic bin index
+Figure C1. Distribution of the relative deviation of the mean and the variance of the SRD, n(i)
+s (z). Top: For the EUCLID survey settings with realisations from
+the inverse of the diagonal Fisher matrix used in Figure 1. Bottom: For KV450 using the samples generated from the DIR bootstrap covariance.
+where O is some observable of interest and WO(ℓ) is the associ-
+ated kernel defining the transformation. Again by the chain rule,
+the functional derivative of this new observable with respect to the
+SRD is readily available:
+δO[Cℓ]
+δn(χ0) =
+�
+dx
+δO
+δCℓ(x)
+δCℓ(x)
+δn(χ0) ,
+(D2)
+where we dropped all the indices for less clutter. For band powers,
+Cl, this would for example assume the following form:
+δCl[Cℓ]
+δn(χ0) = 1
+Nl
+�
+dℓℓS ℓ
+δCℓ
+δn(χ0) ,
+(D3)
+where S ℓ is the band power response function and Nl is the normal-
+isation. For the two-point correlation function ξ± one finds:
+δξ±(θ)
+δn(χ0) = 1
+2π
+�
+dℓℓJ0,4(ℓθ) δCℓ
+δn(χ0) .
+(D4)
+APPENDIX E: NON-LIMBER Cℓ
+The Limber projection used for the Cℓ is not valid on large angu-
+lar scales, where it must be replaced by the full expression. In full
+generality, for any tracers i and j of the matter density
+Ci j
+ℓ = 2
+π
+�
+dk k2Iℓ,k,i[ni]Iℓ,k,j[ni] ,
+(E1)
+where the functional Ik,i[ni] is given by
+Iℓ,k,i[ni] =
+�
+dχWi[ni]
+�
+Pii(k, χ)jℓ(χk) .
+(E2)
+Here Pii is the auto power spectrum of the tracer i and Wi is its
+associated weight. Thus we find:
+δCij
+ℓ
+δna =
+�
+dk k2
+�
+Iℓ,k,i
+δIℓ,k,j
+δna δD
+ja + Iℓ,k,j
+δIℓ,k,i
+δna δD
+ia
+�
+,
+(E3)
+where
+δIℓ,k,i
+δni
+=
+�
+dχδWi
+δn
+�
+Pii(k, χ) jℓ(χk) .
+(E4)
+The derivative of the weight function is calculated as before.
+APPENDIX F: INTRINSIC ALGINMENTS
+In this work, we have ignored intrinsic alignments (IA). Its in-
+clusion is, however,straight forward by noting that the IA angular
+power spectrum is simply given by
+CII
+ℓ =
+� χH
+0
+dχ
+χ2 n(i)
+s (χ)n(j)
+s (χ)PII
+�ℓ + 0.5
+χ
+, χ
+�
+,
+(F1)
+where PII is the IA power spectrum, which summarises the reac-
+tion of galaxy shapes to the ambient LSS on the two-point level.
+The functional derivative there proceeds in the same way as in Ap-
+pendix B. For the GI term of intrinsic alignments, one proceeds as
+before for cosmic shear (compare Section 2).
+MNRAS 000, 1–9 (2022)
+

_NE1T4oBgHgl3EQfCwKv/content/tmp_files/2301.02869v1.pdf.txt

@@ -0,0 +1,453 @@
+ 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+Deep Learning-Based UAV Aerial Triangulation without Image Control Points 
+ 
+Jiageng Zhong1, Ming Li1, Jiangying Qin1, Hanqi Zhang1 
+1State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing, 
+Wuhan University, Wuhan 430079 China,  
+Email: zhongjiageng@whu.edu.cn, lisouming@whu.edu.cn, jy_qin@whu.edu.cn, hqzhang@whu.edu.cn 
+ 
+KEY WORDS: aerial triangulation; Unmanned Aerial Vehicle; Convolutional Neural Network; image matching 
+ 
+ABSTRACT: The emerging drone aerial survey has the advantages of low cost, high efficiency, and flexible use. 
+However, UAVs are often equipped with cheap POS systems and non-measurement cameras, and their flight attitudes 
+are easily affected. How to realize the large-scale mapping of UAV image-free control supported by POS faces many 
+technical problems. The most basic and important core technology is how to accurately realize the absolute orientation 
+of images through advanced aerial triangulation technology. In traditional aerial triangulation, image matching 
+algorithms are constrained to varying degrees by preset prior knowledge. In recent years, deep learning has developed 
+rapidly in the field of photogrammetric computer vision. It has surpassed the performance of traditional handcrafted 
+features in many aspects. It has shown stronger stability in image-based navigation and positioning tasks, especially 
+it has better resistance to unfavorable factors such as blur, illumination changes, and geometric distortion. Based on 
+the introduction of the key technologies of aerial triangulation without image control points, this paper proposes a 
+new drone image registration method based on deep learning image features to solve the problem of high mismatch 
+rate in traditional methods. It adopts SuperPoint as the feature detector, uses the superior generalization performance 
+of CNN to extract precise feature points from the UAV image, thereby achieving high-precision aerial triangulation. 
+Experimental results show that under the same pre-processing and post-processing conditions, compared with the 
+traditional method based on the SIFT algorithm, this method achieves suitable precision more efficiently, which can 
+meet the requirements of UAV aerial triangulation without image control points in large-scale surveys.  
+ 
+1.  INTODUCTION 
+ 
+The UAV-based aerial triangulation using POS (position and orientation system) is extremely rapid and can easily 
+cover the survey area. The POS provides the measurement of the position and orientation of the camera so that each 
+image and pixel can be georeferenced to the Earth without the need for image control points, and the most important 
+and most commonly used data is the position data collected from Global Navigation Satellite Systems (GNSS). 
+Although the drone aerial survey has the advantages of low cost and high efficiency, it is still a problem to achieve 
+high accuracy. One important factor that would affect global accuracy is the precision of feature matching which 
+directly influences the precision of the entire registration. Therefore, accurate features extraction is the basic and key 
+technique in aerial triangulation. 
+ 
+The feature extraction consists of keypoint detection and description and has been used in computer vision task for a 
+long time. The keypoint or feature can be described as a specific meaningful structure, but it is not clear what are the 
+relevant keypoints for an arbitrary input image (Mukherjee and et al., 2015). The function of a feature detector is to 
+detect keypoints and their corresponding descriptors. In the past decades, feature detectors have been an activate area 
+of research. Among the many detectors, SIFT (Scale-Invariant Feature Transform) (Lowe, 2004) is the most 
+representative and influential one. SIFT aims to solve the image rotation, affine transformations, intensity, and 
+viewpoint change in matching features (Karami and et al., 2017). It generally includes two major steps. It firstly 
+convolves the image with Gaussian filters at various scales and finds scale invariant keypoints via estimating a scale 
+space extreme. Then, for each keypoint, the local image descriptor is computed based on image gradient magnitude 
+and orientation (Lowe, 2004). And there are many other SIFT-like detectors such as SURF (Speed up Robust Feature) 
+(Bay and et al., 2008) and ORB (Oriented FAST and Rotated BRIEF) (Rublee and et al., 2011) which are more 
+efficient than SIFT. 
+ 
+With the rapid development of deep learning methods and the increasing demand for better feature detection, new 
+feature detectors emerged, most of which are based on convolutional neural network. Different from the classical 
+algorithms, deep learning approaches can learn abstract image features from high-dimensional data in an end-to-end 
+fashion instead of relying on handcrafted features such as distinctive corners (Zeiler and Fergus, 2014).  As 
+convolutional neural networks learn features based on supervision, their performance heavily relies on ground truth 
+information (Bojanić and et al.,2019). In other word, the key is usually a large dataset of 2D ground truth locations 
+labeled by human annotators. Unlike these approaches, a novel detector named SuperPoint (DeTone and et al., 2018) 
+is supervised by itself and works well for matching tasks.  
+
+ 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+ 
+In this paper, a new aerial survey method without image control points is proposed, and SuperPoint is applied to 
+replace traditional feature detector in the aerial triangulation flow. Through the multiple perspectives experiment, it 
+is shown that the new method is able to achieve suitable precision more efficiently, compared to those based on classic 
+feature detectors. 
+ 
+2.  METHODOLOGY 
+ 
+2.1 Big Picture 
+ 
+Referring COLMAP’s incremental Structure-from-Motion pipeline (Schonberger and Frahm, 2016), our method can 
+be divided into three stages as shown in Figure 1. The first stage is to prepare UAV images and corresponding position 
+data which contains latitude and longitude location information. Note that the position data should be converted in 
+Gauss-Kruger Projection. 
+ 
+ 
+Figure 1. A flow chart of our aerial survey method 
+ 
+The second stage is correspondence search which finds overlap in the input images and identifies projections of the 
+same points in overlapping images. For each image, the first step is to extract features which are invariant under 
+radiometric and geometric changes. In traditional aerial triangulation, SIFT (Lowe, 2004) is applied mostly, here it is 
+replaced by SuperPoint which can also output L2-normalized fixed length descriptors. As for feature matching, a 
+matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search is adopted to improve 
+the matching accuracy. Based on matched features, images that cover the same scene part are discovered. So the 
+output of the second stage is a set of potentially overlapping image pairs and their associated feature correspondences 
+(Schonberger and Frahm, 2016). In addition, there are typically geometric verification that uses projective geometry 
+to verify the matches. 
+ 
+The third stage is mainly carried out in four steps. Based on the output of the second stage, new images can be 
+registered. Next, as newly registered images lead to increase scene coverage, new scene points can be triangulated 
+and added to the scene structure. Then, considering the possible problem of error accumulation in reconstruction, BA 
+(Bundle Adjustment) (Triggs and et al., 1999) is applied to refine camera parameters and point parameters via 
+minimizing the reprojection error. Finally, the outliers are filtered. This iterative strategy can significantly improve 
+completeness and accuracy (Schonberger and Frahm, 2016).  
+ 
+Through the workflows above, the aerial survey without image control points is finished. Specifically, all the images 
+are aligned and a set of sparse point cloud of the survey area is formed. 
+ 
+
+Images
+GNSSMeasurement
+SuperPointFeatureExtraction
+Feature Matching
+Features
+ImageRegistration
+OutlierFiltering
+Reconstruction
+Triangulation
+Bundle Adjustment 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+2.2 SuperPoint Network 
+ 
+SuperPoint is an encoder-decoder architecture and is a fully-convolutional neural network which operates on a full-
+sized image. Its structure is shown in Figure 2. Its shared encoder processes the input image at first and branches into 
+two decoders, one for interest point detection and the other for interest point description. This strategy is quite different 
+from traditional system which first detects keypoints and then calculates the descriptors.  
+ 
+ 
+Figure 2. Structure of SuperPoint Network (DeTone and et al., 2018) 
+ 
+It should be noted that Superpoint adopts self-supervised training strategy. It firstly trains a base detector called 
+MagicPoint based on supervision from a synthetic dataset, where keypoints can be determined unambiguously. Then 
+the detector capacity is expanded to real images using homographic adaption. Finally, a keypoint descriptor is 
+computed by an additional subnetwork. 
+ 
+3.  EXPERIMENTS 
+ 
+3.1 Data Preparation 
+ 
+In this section, we present experiment results of the traditional method (based on SIFT) and our method for 
+comparison. All experiments in this paper are based on two datasets collected from Chongqing. Each dataset contains 
+a UAV image sequence and corresponding POS data of a scene. The GSD (ground sample distance) of the aerial 
+images is 0.2 m, and the GNSS standard horizontal and vertical precision is 1 cm and 3 cm respectively. In addition, 
+there are also ground known points for precision check. To be more specific, for Scene 1 and Scene 2, there are 24 
+images and 6 images respectively. The heading overlap and side overlap rate were set to 80% and 60% respectively. 
+These can meet the specification of topographic mapping at small scale.  
+ 
+3.2 System Runtime 
+ 
+The run-time of SuperPoint and SIFT is measured using a RTX 2060 GPU. The SuperPoint architecture is 
+implemented with Pytorch deep learning library (Paszke and et al., 2019). The average run-time of different 
+algorithms is measured as shown in Table 1. As the inference of the deep model is done in a single forward propagation 
+step, the run-time of a single forward pass is measured to be about 148 ms. And SIFT takes about 368 ms to process 
+one image. It can be seen that SuperPoint executes more efficiently than SIFT and may be applied in real time 
+surveying. 
+ 
+Table 1.  Mean execution times 
+Algorithm 
+SuperPoint 
+SIFT 
+Run-time 
+148 ms 
+368 ms 
+ 
+3.3 Feature Extraction and Matching 
+ 
+Several comparative experiments are carried out for qualitative and quantitative evaluation of the performance of the 
+keypoint detector and descriptor generator on the datasets.  
+ 
+The appearance and distribution of keypoints from different detectors are intuitively demonstrated in Figure 3, and 
+the image is from the dataset of Scene 1. It can be observed that SuperPoint produces fewer feature points compared 
+to SIFT, and its location distribution is more dispersive. For instance, there is a tract of farmland in the top left of 
+
+InterestPointDecoder
+W
+Conv
+x
+W/8
+Input
+Encoder
+Softmax
+Reshape
+W
+H/8
+H
+65
+1
+DescriptorDecoder
+W
+Conv
+H
+W/8
+D
+Bi-Cubic
+L2
+1
+Interpolate
+Norm
+H/8
+H
+D
+D 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+images, SIFT can hardly extract keypoints, and SuperPoint can extract more well-distributed keypoints. Instead, for 
+tree or residential areas that have rich texture information, there are more dense points produced by SIFT. 
+ 
+ 
+ 
+(1) SuperPoint 
+(2) SIFT 
+Figure 3. The appearance and distribution of keypoints from different detectors 
+ 
+ 
+(1) SuperPoint 
+ 
+(2) SIFT 
+Figure 4. Qualitative result of matching 
+ 
+Then, to evaluate the performance of the descriptors, the extracted features are matched by a matcher that combines 
+Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search. The ratio test checks if matches are ambiguous 
+and should be removed, because the probability that a match is correct can be determined by taking the ratio of 
+distance from the closest neighbor to the distance of the second closest (Lowe, 2004). A qualitative example of 
+SuperPoint versus SIFT is shown in Figure 4 and the distance ratio is set to 0.7. SuperPoint tends to produce a larger 
+number of correct matches which densely cover the image while there are several mismatches in the result of SIFT.  
+ 
+The statistics analysis of the quality of descriptors is performed. Figure 5 shows the match rates and mismatch rates 
+under different distance ratios for real image data. The match rate is defined as the ratio of matched keypoints to all 
+keypoints, and the mismatch rate is defined as the ratio of keypoints which are matched falsely to matched keypoints. 
+
+ 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+Figure 5(a) and 5(b) shows that SuperPoint has higher match rates and lower mismatch rates in most cases. For SIFT 
+detector, there are too many mismatches to estimate the pose when the distance ratio is greater than 0.8. The ratio is 
+typically set between 0.5 to 0.8 in application, and in this range, SuperPoint can achieve zero mismatch. Therefore, it 
+is reasonable to suppose that SuperPoint is able to produce better descriptors.  
+ 
+ 
+ 
+(a) Match rate 
+(b) Mismatch rate 
+Figure 5. The statistical results of feature matching 
+ 
+Table 2.  Relative orientation error 
+Algorithm 
+SuperPoint 
+SIFT 
+Reprojection Error 
+0.1454 pixel 
+0.1008 pixel 
+ 
+A relative orientation process recreates relative translation and angular relationships between two successive 
+overlapping images (Tjahjadi and Agustina, 2019). In this paper, the reprojection error of relative orientation is used 
+as the metric for evaluating the quality of keypoints. Table 2 displays the errors of the image pair in Figure 4. SIFT 
+performs better on this metric as SuperPoint has a higher reprojection error. This is likely due to the fact that SIFT 
+performs extra sub-pixel localization, while SuperPoint does not perform this step.  
+ 
+3.4 Aerial Triangulation 
+ 
+Using our new aerial survey method described in Section 2, the aerial triangulation without image control points is 
+carried out on datasets of Scene 1 and Scene 2. The reprojection errors in bundle adjustment are displayed in Table 3, 
+and the camera position errors are displayed in Table 4. Due to extra sub-pixel localization, the SIFT-based method 
+reaches higher precision. As for camera position, our method has slightly higher precision, presumably because the 
+keypoints extracted by SuperPoint distribute more evenly. 
+ 
+Table 3. Reprojection errors in bundle adjustment 
+ 
+Our Method 
+SIFT 
+Scene 1 
+0.387 pixel 
+0.332 pixel 
+Scene 2 
+0.412 pixel 
+0.353 pixel 
+ 
+Table 4. Camera position errors 
+ 
+ERROR 
+Our Method 
+SIFT 
+Scene 1 
+X error 
+0.603 m 
+0.530 m 
+Y error 
+0.716 m 
+1.004 m 
+Z error 
+0.202 m 
+0.166 m 
+XY error 
+0.936 m 
+1.136 m 
+XYZ error 
+0.958 m 
+1.148 m 
+Scene 2 
+X error 
+0.451 m 
+0.425 m 
+Y error 
+0.570 m 
+0.422 m 
+Z error 
+0.791 m 
+1.043 m 
+XY error 
+0.727 m 
+0.599 m 
+XYZ error 
+1.074 m 
+1.203 m 
+
+MatchRate
+0.7
+SuperPoint
+0.6
+SIFT
+0.5
+0.2
+0.1
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Ratio of distances (closest/next closest)MismatchRate
+0.14
+SuperPoint
+0.12
+SIFT
+0.1
+0.04
+0.02
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Ratio of distances (closest/next closest) 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+Table 5. Error of the checkpoint 
+ERROR 
+Our Method 
+SIFT 
+X error 
+-1.821 m 
+-2.444 m 
+Y error 
+-2.217 m 
+-1.635 m 
+Z error 
+-3.755 m 
+-3.925 m 
+XY error 
+2.870 m 
+2.940 m 
+XYZ error 
+4.726 m 
+4.904 m 
+ 
+A known point in Scene 2 is used as the checkpoint for precision check, and Table 5 displays the checkpoint error. 
+The comparison result is consistent with Table 4. The results of experiments illustrate that our method is likely to 
+reach higher precision compared to traditional SIFT-based method, which confirms that learned representations for 
+descriptor matching outperform hand-tuned representations. 
+ 
+4.  CONCLUSION 
+ 
+This paper presents a new aerial survey method without image control points, which adopts SuperPoint as feature 
+detector. A series of comparative experiments illustrate that our method has obvious advantage in efficiency, keypoint 
+distribution and matching quality, and it can achieve suitable precision. So, it can be concluded that our method is 
+capable to meet the application requirements of aerial triangulation. Future work will comprehensively evaluate the 
+performance of our method with more experiment. 
+ 
+This paper has shown that the deep learning method outperforms traditional methods in many aspects, therefore, we 
+can consider that deep learning-based aerial survey would have an expected future. 
+ 
+ACKNOWLEDGEMENTS 
+ 
+This research was funded by the National Key R&D Program of China, grant numbers 2018YFB0505400, the 
+National Natural Science Foundation of China (NSFC), grant number 41901407 and the LIESMARS Special 
+Research Funding. 
+ 
+REFERENCES 
+ 
+Bay, H., Ess, A., Tuytelaars, T. and Van Gool, L. 2008. Speeded-up robust features (SURF). Computer vision and 
+image understanding, 110(3), pp.346-359. 
+ 
+Bojanić, D., Bartol, K., Pribanić, T., Petković, T., Donoso, Y. D. and Mas, J. S. 2019. On the comparison of classic 
+and deep keypoint detector and descriptor methods. In: 2019 11th International Symposium on Image and Signal 
+Processing and Analysis (ISPA), pp. 64-69. 
+ 
+DeTone, D., Malisiewicz, T. and Rabinovich, A. 2018. Superpoint: Self-supervised interest point detection and 
+description. In: Proceedings of the IEEE conference on computer vision and pattern recognition workshops. pp. 224-
+236. 
+ 
+Karami, E., Prasad, S. and Shehata, M. 2017. Image matching using SIFT, SURF, BRIEF and ORB: performance 
+comparison for distorted images. arXiv preprint arXiv:1710.02726. 
+ 
+Lowe, D. G. 2004. Distinctive image features from scale-invariant keypoints. International journal of computer vision, 
+60(2), pp.91-110. 
+ 
+Mukherjee, D., Wu, Q. J. and Wang, G. 2015. A comparative experimental study of image feature detectors and 
+descriptors. Machine Vision and Applications, 26(4), pp.443-466. 
+ 
+Paszke, A., Gross, S. and et al. 2019. Pytorch: An imperative style, high-performance deep learning library. Advances 
+in neural information processing systems, 32, pp.8026-8037. 
+ 
+Rublee, E., Rabaud, V., Konolige, K. and Bradski, G. 2011. ORB: An efficient alternative to SIFT or SURF. In: 2011 
+International conference on computer vision. pp. 2564-2571. 
+ 
+
+ 
+ 
+The 42nd Asian Conference on Remote Sensing (ACRS2021) 
+22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam 
+ 
+Schonberger, J. L. and Frahm, J. M. 2016. Structure-from-motion revisited. In: Proceedings of the IEEE conference 
+on computer vision and pattern recognition. pp. 4104-4113. 
+ 
+Tjahjadi, M. E. and Agustina, F. 2019. Fast and stable direct relative orientation of UAV-based stereo pair. 
+International Journal of Advances in Intelligent Informatics, 5(1), pp.24-39. 
+ 
+Triggs, B., McLauchlan, P. F., Hartley, R. I. and Fitzgibbon, A. W. 1999. Bundle adjustment—a modern synthesis. 
+In: International workshop on vision algorithms. pp. 298-372. 
+ 
+Zeiler, M. D. and Fergus, R. 2014. Visualizing and understanding convolutional networks. In: European conference 
+on computer vision. pp. 818-833. 
+

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+page_content='The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam  Deep Learning-Based UAV Aerial Triangulation without Image Control Points  Jiageng Zhong1, Ming Li1, Jiangying Qin1, Hanqi Zhang1 1State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing, Wuhan University, Wuhan 430079 China, Email: zhongjiageng@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
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+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='cn, jy_qin@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='cn, hqzhang@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='cn  KEY WORDS: aerial triangulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Unmanned Aerial Vehicle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Convolutional Neural Network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' image matching  ABSTRACT: The emerging drone aerial survey has the advantages of low cost, high efficiency, and flexible use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' However, UAVs are often equipped with cheap POS systems and non-measurement cameras, and their flight attitudes are easily affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' How to realize the large-scale mapping of UAV image-free control supported by POS faces many technical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The most basic and important core technology is how to accurately realize the absolute orientation of images through advanced aerial triangulation technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In traditional aerial triangulation, image matching algorithms are constrained to varying degrees by preset prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In recent years, deep learning has developed rapidly in the field of photogrammetric computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It has surpassed the performance of traditional handcrafted features in many aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It has shown stronger stability in image-based navigation and positioning tasks, especially it has better resistance to unfavorable factors such as blur, illumination changes, and geometric distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Based on the introduction of the key technologies of aerial triangulation without image control points, this paper proposes a new drone image registration method based on deep learning image features to solve the problem of high mismatch rate in traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  It adopts SuperPoint as the feature detector, uses the superior generalization performance of CNN to extract precise feature points from the UAV image, thereby achieving high-precision aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Experimental results show that under the same pre-processing and post-processing conditions, compared with the traditional method based on the SIFT algorithm, this method achieves suitable precision more efficiently, which can meet the requirements of UAV aerial triangulation without image control points in large-scale surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' INTODUCTION  The UAV-based aerial triangulation using POS (position and orientation system) is extremely rapid and can easily cover the survey area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The POS provides the measurement of the position and orientation of the camera so that each image and pixel can be georeferenced to the Earth without the need for image control points, and the most important and most commonly used data is the position data collected from Global Navigation Satellite Systems (GNSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Although the drone aerial survey has the advantages of low cost and high efficiency, it is still a problem to achieve high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' One important factor that would affect global accuracy is the precision of feature matching which directly influences the precision of the entire registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Therefore, accurate features extraction is the basic and key technique in aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The feature extraction consists of keypoint detection and description and has been used in computer vision task for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The keypoint or feature can be described as a specific meaningful structure, but it is not clear what are the relevant keypoints for an arbitrary input image (Mukherjee and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The function of a feature detector is to detect keypoints and their corresponding descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In the past decades, feature detectors have been an activate area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Among the many detectors, SIFT (Scale-Invariant Feature Transform) (Lowe, 2004) is the most representative and influential one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' SIFT aims to solve the image rotation, affine transformations, intensity, and viewpoint change in matching features (Karami and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It generally includes two major steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It firstly convolves the image with Gaussian filters at various scales and finds scale invariant keypoints via estimating a scale space extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Then, for each keypoint, the local image descriptor is computed based on image gradient magnitude and orientation (Lowe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' And there are many other SIFT-like detectors such as SURF (Speed up Robust Feature) (Bay and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2008) and ORB (Oriented FAST and Rotated BRIEF) (Rublee and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2011) which are more efficient than SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  With the rapid development of deep learning methods and the increasing demand for better feature detection, new feature detectors emerged, most of which are based on convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Different from the classical algorithms, deep learning approaches can learn abstract image features from high-dimensional data in an end-to-end fashion instead of relying on handcrafted features such as distinctive corners (Zeiler and Fergus, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' As convolutional neural networks learn features based on supervision, their performance heavily relies on ground truth information (Bojanić and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=',2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In other word, the key is usually a large dataset of 2D ground truth locations labeled by human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Unlike these approaches, a novel detector named SuperPoint (DeTone and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2018) is supervised by itself and works well for matching tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam  In this paper, a new aerial survey method without image control points is proposed, and SuperPoint is applied to replace traditional feature detector in the aerial triangulation flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Through the multiple perspectives experiment, it is shown that the new method is able to achieve suitable precision more efficiently, compared to those based on classic feature detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' METHODOLOGY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='1 Big Picture  Referring COLMAP’s incremental Structure-from-Motion pipeline (Schonberger and Frahm, 2016), our method can be divided into three stages as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The first stage is to prepare UAV images and corresponding position data which contains latitude and longitude location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Note that the position data should be converted in Gauss-Kruger Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' A flow chart of our aerial survey method  The second stage is correspondence search which finds overlap in the input images and identifies projections of the same points in overlapping images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' For each image, the first step is to extract features which are invariant under radiometric and geometric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In traditional aerial triangulation, SIFT (Lowe, 2004) is applied mostly, here it is replaced by SuperPoint which can also output L2-normalized fixed length descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' As for feature matching, a matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search is adopted to improve the matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Based on matched features, images that cover the same scene part are discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' So the output of the second stage is a set of potentially overlapping image pairs and their associated feature correspondences (Schonberger and Frahm, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In addition, there are typically geometric verification that uses projective geometry to verify the matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The third stage is mainly carried out in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Based on the output of the second stage, new images can be registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Next, as newly registered images lead to increase scene coverage, new scene points can be triangulated and added to the scene structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Then, considering the possible problem of error accumulation in reconstruction, BA (Bundle Adjustment) (Triggs and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 1999) is applied to refine camera parameters and point parameters via minimizing the reprojection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Finally, the outliers are filtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' This iterative strategy can significantly improve completeness and accuracy (Schonberger and Frahm, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Through the workflows above, the aerial survey without image control points is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Specifically, all the images are aligned and a set of sparse point cloud of the survey area is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Images  GNSSMeasurement  SuperPointFeatureExtraction  Feature Matching  Features  ImageRegistration  OutlierFiltering  Reconstruction  Triangulation  Bundle Adjustment  The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='2 SuperPoint Network  SuperPoint is an encoder-decoder architecture and is a fully-convolutional neural network which operates on a full- sized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Its structure is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Its shared encoder processes the input image at first and branches into two decoders, one for interest point detection and the other for interest point description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' This strategy is quite different from traditional system which first detects keypoints and then calculates the descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Structure of SuperPoint Network (DeTone and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2018)  It should be noted that Superpoint adopts self-supervised training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It firstly trains a base detector called MagicPoint based on supervision from a synthetic dataset, where keypoints can be determined unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Then the detector capacity is expanded to real images using homographic adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Finally, a keypoint descriptor is computed by an additional subnetwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' EXPERIMENTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='1 Data Preparation  In this section, we present experiment results of the traditional method (based on SIFT) and our method for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' All experiments in this paper are based on two datasets collected from Chongqing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Each dataset contains a UAV image sequence and corresponding POS data of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The GSD (ground sample distance) of the aerial images is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='2 m, and the GNSS standard horizontal and vertical precision is 1 cm and 3 cm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In addition, there are also ground known points for precision check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' To be more specific, for Scene 1 and Scene 2, there are 24 images and 6 images respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The heading overlap and side overlap rate were set to 80% and 60% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' These can meet the specification of topographic mapping at small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='2 System Runtime  The run-time of SuperPoint and SIFT is measured using a RTX 2060 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The SuperPoint architecture is implemented with Pytorch deep learning library (Paszke and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The average run-time of different algorithms is measured as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' As the inference of the deep model is done in a single forward propagation step, the run-time of a single forward pass is measured to be about 148 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' And SIFT takes about 368 ms to process one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It can be seen that SuperPoint executes more efficiently than SIFT and may be applied in real time surveying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Mean execution times Algorithm SuperPoint SIFT Run-time 148 ms 368 ms  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='3 Feature Extraction and Matching  Several comparative experiments are carried out for qualitative and quantitative evaluation of the performance of the keypoint detector and descriptor generator on the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The appearance and distribution of keypoints from different detectors are intuitively demonstrated in Figure 3, and the image is from the dataset of Scene 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' It can be observed that SuperPoint produces fewer feature points compared to SIFT, and its location distribution is more dispersive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' For instance, there is a tract of farmland in the top left of  InterestPointDecoder  W  Conv  x  W/8  Input  Encoder  Softmax  Reshape  W  H/8  H  65  1  DescriptorDecoder  W  Conv  H  W/8  D  Bi Cubic  L2  1  Interpolate  Norm  H/8  H  D  D  The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam  images, SIFT can hardly extract keypoints, and SuperPoint can extract more well-distributed keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Instead, for tree or residential areas that have rich texture information, there are more dense points produced by SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  (1) SuperPoint (2) SIFT Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The appearance and distribution of keypoints from different detectors  (1) SuperPoint  (2) SIFT Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Qualitative result of matching  Then, to evaluate the performance of the descriptors, the extracted features are matched by a matcher that combines Lowe’s ratio test matcher (Lowe, 2004) and Nearest Neighbor search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The ratio test checks if matches are ambiguous and should be removed, because the probability that a match is correct can be determined by taking the ratio of distance from the closest neighbor to the distance of the second closest (Lowe, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' A qualitative example of SuperPoint versus SIFT is shown in Figure 4 and the distance ratio is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' SuperPoint tends to produce a larger number of correct matches which densely cover the image while there are several mismatches in the result of SIFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The statistics analysis of the quality of descriptors is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Figure 5 shows the match rates and mismatch rates under different distance ratios for real image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The match rate is defined as the ratio of matched keypoints to all keypoints, and the mismatch rate is defined as the ratio of keypoints which are matched falsely to matched keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  The 42nd Asian Conference on Remote Sensing (ACRS2021) 22-24th November, 2021 in Can Tho University, Can Tho city, Vietnam  Figure 5(a) and 5(b) shows that SuperPoint has higher match rates and lower mismatch rates in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' For SIFT detector, there are too many mismatches to estimate the pose when the distance ratio is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The ratio is typically set between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='8 in application, and in this range, SuperPoint can achieve zero mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Therefore, it is reasonable to suppose that SuperPoint is able to produce better descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  (a) Match rate (b) Mismatch rate Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The statistical results of feature matching  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Relative orientation error Algorithm SuperPoint SIFT Reprojection Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='1454 pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='1008 pixel  A relative orientation process recreates relative translation and angular relationships between two successive overlapping images (Tjahjadi and Agustina, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' In this paper, the reprojection error of relative orientation is used as the metric for evaluating the quality of keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Table 2 displays the errors of the image pair in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' SIFT performs better on this metric as SuperPoint has a higher reprojection error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' This is likely due to the fact that SIFT performs extra sub-pixel localization, while SuperPoint does not perform this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='4 Aerial Triangulation  Using our new aerial survey method described in Section 2, the aerial triangulation without image control points is carried out on datasets of Scene 1 and Scene 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The reprojection errors in bundle adjustment are displayed in Table 3, and the camera position errors are displayed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Due to extra sub-pixel localization, the SIFT-based method reaches higher precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' As for camera position, our method has slightly higher precision, presumably because the keypoints extracted by SuperPoint distribute more evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Reprojection errors in bundle adjustment  Our Method  SIFT  Scene 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
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+page_content='074 m  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
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+page_content=' Error of the checkpoint ERROR Our Method SIFT X error -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
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+page_content='940 m XYZ error 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='726 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='904 m  A known point in Scene 2 is used as the checkpoint for precision check, and Table 5 displays the checkpoint error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The comparison result is consistent with Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' The results of experiments illustrate that our method is likely to reach higher precision compared to traditional SIFT-based method, which confirms that learned representations for descriptor matching outperform hand-tuned representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' CONCLUSION  This paper presents a new aerial survey method without image control points, which adopts SuperPoint as feature detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' A series of comparative experiments illustrate that our method has obvious advantage in efficiency, keypoint distribution and matching quality, and it can achieve suitable precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' So, it can be concluded that our method is capable to meet the application requirements of aerial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content=' Future work will comprehensively evaluate the performance of our method with more experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  This paper has shown that the deep learning method outperforms traditional methods in many aspects, therefore, we can consider that deep learning-based aerial survey would have an expected future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This research was funded by the National Key R&D Program of China, grant numbers 2018YFB0505400, the National Natural Science Foundation of China (NSFC), grant number 41901407 and the LIESMARS Special Research Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE1T4oBgHgl3EQfCwKv/content/2301.02869v1.pdf'}
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+Addressing the Selection Bias in Voice Assistance: Training Voice Assistance
+Model in Python with Equal Data Selection
+KASHAV PIYA, Augustana College, USA
+SRIJAL SHRESTHA, Augustana College, USA
+CAMERAN FRANK, Augustana College, USA
+ESTEPHANOS JEBESSA, Augustana College, USA
+TAUHEED KHAN MOHD, Augustana College, USA
+In recent times, voice assistants have become a part of our day-to-day lives, allowing information retrieval by voice synthesis, voice
+recognition, and natural language processing. These voice assistants can be found in many modern-day devices such as Apple, Amazon,
+Google, and Samsung. This project is primarily focused on Virtual Assistance in Natural Language Processing. Natural Language
+Processing is a form of AI that helps machines understand people and create feedback loops. This project will use deep learning
+to create a Voice Recognizer and use Commonvoice and data collected from the local community for model training using Google
+Colaboratory. After recognizing a command, the AI assistant will be able to perform the most suitable actions and then give a response.
+The motivation for this project comes from the race and gender bias that exists in many virtual assistants. The computer industry
+is primarily dominated by the male gender, and because of this, many of the products produced do not regard women. This bias has an
+impact on natural language processing. This project will be utilizing various open-source projects to implement machine learning
+algorithms and train the assistant algorithm to recognize different types of voices, accents, and dialects. Through this project, the goal
+to use voice data from underrepresented groups to build a voice assistant that can recognize voices regardless of gender, race, or accent.
+Increasing the representation of women in the computer industry is important for the future of the industry. By representing
+women in the initial study of voice assistants, it can be shown that females play a vital role in the development of this technology. In
+line with related work, this project will use first-hand data from the college population and middle-aged adults to train voice assistant
+to combat gender bias.
+Additional Key Words and Phrases: Voice Assistance, Machine Learning, Virtual Assistance, Artificial Intelligence, Selection Bias,
+Sample Population, Python 3.10, Pyttsx3, PyTorch, JSON
+1
+INTRODUCTION
+The first-ever voice-activated consumer product was released to the public in 1922. It was known as “Radio Rex.” This
+product was a toy that had a doghouse with a dog inside it. When someone said “Rex” next to the dog house, the dog
+would jump out of the doghouse. This voice-activated toy was created even before modern computers existed.[1]
+Authors’ addresses: Kashav Piya, kashavpiya19@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Srijal Shrestha,
+srijalshrestha18@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Cameran Frank, cameranfrank18@augustana.edu,
+Augustana College, 639 38th St, Rock Island, Illinois, USA, 61201; Estephanos Jebessa, estephanosjebessa19@augustana.edu, Augustana College, 639 38th
+St, Rock Island, Illinois, USA, 61201; Tauheed Khan Mohd, tauheedkhanmohd@augustana.edu, Augustana College, 639 38th St, Rock Island, Illinois, USA,
+61201.
+Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not
+made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components
+of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to
+redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.
+© 2023 Association for Computing Machinery.
+Manuscript submitted to ACM
+Manuscript submitted to ACM
+1
+arXiv:2301.00646v1  [eess.AS]  20 Dec 2022
+
+2
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+Since the development of that toy, there has been a considerable amount of development in voice recognition, natural
+language processing, and machine learning. A voice assistant, also known as an intelligent personal assistant or a
+connected speaker, is based on natural language speech recognition. Recently it has had a rise in popularity and has
+been marketed and used by Apple, Amazon, Google, and Samsung. Now voice assistants are widely found in most
+modern-day devices that a person would use.
+Voice assistants are multi-purposed one of their main purposes was for a search to be carried out using a voice
+command entered by the user as an input. They are also known to be used for information retrieval by voice synthesis.
+They use a variety of voice recognition techniques, language processing algorithms, and voice synthesis to listen to
+specific voice commands that may include wake words, tasks, and queries, and return relevant information or perform
+a specific function as requested by the user. These assistants can be software-based which allows them to be integrated
+into a wide range of devices such as laptops, mobile devices, and speakers, or can be specifically designed into a
+standalone device like Amazon Echo or Amazon Alexa Wall Clock. [2]
+These voice assistants work like a charm and are quite fascinating, making one might ask themselves what goes on
+behind the hood of these amazing innovations or how do they work the way they do?
+To answer the above query, in short voice assistants use artificial intelligence and voice recognition to deliver the
+result that the user is looking for efficiently, and precisely. The user provides a command to the voice assistant that
+is called intent. Through voice recognition, these intentions can be understood by our virtual assistants. Here, voice
+recognition allows the speaker to speak into a device that takes the analog signal from the speaker and changes it into
+a digital signal which is then processed by the computer to match it with words or phrases and then recognize the
+command. Machine learning also has a huge part to play in this as the computer needs to be taught to recognize the
+speaker’s words by feeding it a database of words and syllables in each language to match it with digital signals. This
+process is known as pattern recognition. Additionally, these devices gather a lot of information from the commands
+that they received previously to improve upon themselves using machine learning.[3]
+There are multiple approaches to voice assistants, specifically two types: task-oriented and knowledge-oriented.
+Most voice assistants these days can combine both the task-oriented as well as a knowledge-oriented workflow to
+complete all the tasks that a user may ask the voice assistant to carry out. A task-oriented approach will most likely ask
+something as simple as filling out a form, whereas a knowledge-based approach may include answering questions such
+as who the President of the United States of America is or finding out what engine is in a Ford F50 which is a technical
+specification of a product. [4]
+The task-oriented approach/workflow is pretty much self-explanatory as it uses goals and tasks to achieve whatever
+the user wants or needs. This approach usually requires the voice assistant to use a different application such as time,
+weather, web browser, and music apps, to help complete its tasks. Some examples would be, asking a voice assistant to
+set a reminder to take medicine at 6 PM, playing music using Spotify, etc. This approach does not require the virtual
+voice assistant to search massive databases for knowledge. These tasks are often known as skills. And various assistants
+allow for different skills to be installed according to the user’s preferences. [5]
+Whereas a knowledge-oriented approach/workflow requires the use of analytical data to complete the tasks and help
+the users to complete their tasks. [6] Unlike a task-oriented approach, this approach focuses on using online databases
+to get related information in addition to already recorded knowledge to help users to complete tasks. An example of a
+knowledge-based approach would be if a user asked for a question that would require searching the internet such as
+what is the capital of the state of Illinois or who invented the telephone?
+Manuscript submitted to ACM
+
+Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+3
+Furthermore, there are two types of artificial intelligence (AI); In general, there is a weak AI and there is a strong
+AI.[7] There are many types of machines such as Siri, Alexa, Cortana, and Bixby that can only perform certain tasks
+that have been defined by the user while making the AI. These types of AIs are called Weak AI. And some machines or
+systems have a mind of their own and can make decisions or take actions on their own without human interference.
+These types of machines are called Strong AI. After learning the differences between strong and weak AI, the voice
+assistant this project is opting for is an example of weak AI.
+This projects Virtual Voice Assistant will include a variety of features such as greeting the users, fetching information
+about a person, an object, or anything else in general from the internet, providing the time, opening web browsers,
+playing music, and so on. It might also include additional features such as opening the web camera to take pictures,
+forecasting the weather, logging off from your personal computer, telling you a joke, and many other features.
+The field of Virtual Assistance has many avenues to consider from providing help in technology to connecting
+people through the usage of technology. When studying what exactly a Virtual Assistant is, the field that was decided
+on was Virtual Assistance in natural language processing, which means the technology can understand people more
+accurately. Natural Language Processing is a form of AI that gives machines the ability to not just read but to understand
+and interpret human language. With NLP, machines can make sense of the written or spoken text and perform tasks
+including speech recognition, sentiment analysis, and automatic text summarization [8]. Therefore, not only does
+natural language processing help humans it also helps with machine learning, in the sense that NLP will continue to
+provide more data to better that analysis of speech and create a feedback loop.
+The English language is an extremely hard language to understand and speak, especially if English is not your first
+language. Not only is the usage of English vernacular hard to comprehend and execute, but there is also a form of a
+language barrier in the different dialects that people possess. A person’s dialect can be a communication inhibitor in
+many languages, not just in English, but in this project, the focus is on the English language [9].
+For this project, synthetic voices were originally used instead of human voices, in which data was collected. A
+synthetic voice is a pre-recorded voice produced through text to speech whereas the human voice is pre-recorded.
+‘Its use involves recording, in advance, a text read aloud by a human being.’ The usage of a synthetic voice will give
+flexibility as they have a high capacity in reading textual context and can generate voice constantly. But “there is a
+disadvantage in expressing social signs as it cannot express emotions, intentions, and attitudes through modulation of
+the voice” [10].
+The computer industry is primarily dominated by the male gender and because of this extreme one-sided representa-
+tion in the field a multitude of the products that are produced do not regard women. One of the fields that this bias
+impacts is natural language processing. Because of the lack of females in the industry, the identification percentage
+for female voices is lower than that of male voices, therefore resulting in the analysis and research of this topic of
+selection bias of voice assistance [11]. Data augmentation by controlling the gender attribute is an effective technique
+in mitigating gender bias in NLP processes.
+2
+RELATED WORK
+2.1
+How Does Voice Assistant Work?
+Voice assistance has now been defined is, but how does it work? A voice assistant uses speech recognition along with
+other identification of speech components to help the machine process the voice. Then, the speech is rendered into its
+textual representation based on extracted patterns. Following that, this program isolates the most important words
+Manuscript submitted to ACM
+
+4
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+Fig. 1. How Does Voice Assistant Work?
+or the action also known as anticipated intent. If the intent is not clear, the voice assistant is programmed to ask
+more questions. It then retrieves information by API calls to access the relevant knowledge base. Finally, it relays the
+information back to the human user through text to speech or fulfills the necessary action. Voice assistants rely on
+Natural Language Processing and other machine learning algorithms to perform the best and overcome the challenges
+that it faces. [12]
+2.2
+Speech Recognition
+How does a voice assistant understand what a person is talking about? These voice assistants use fast decoding
+algorithms that allow real-time continuous speech recognition systems to provide instant responses.[13] Speech
+Recognition has been a part of our day-to-day life for more than 10 years now as it seems to become more and more
+advanced. The beauty of speech-to-text models goes unnoticed in this process as the machines make it look so seamless.
+Most speech recognition algorithms use a mix of Natural Language Processing and Deep Learning techniques to parse
+through the user query, get an appropriate response and present it back to the user in whichever form the user desires.
+All the big companies such as Google’s Assistant, Amazon’s Alexa, Apple’s Siri, and others use the same techniques but
+just with some different variations.[14]
+Manuscript submitted to ACM
+
+Voice Assistant
+5.
+Human User
+NLP
+What is the
+APIs
+weather for
+3.
+tomorrow?
+4.Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+5
+2.2.1
+Signal Processing for Speech Recognition. Audio signals are any object that vibrates to produce sound waves.
+When an object vibrates, the air molecules oscillate to and from their rest position and transmits its energy to the
+neighboring molecules. This results in the transmission of energy from one molecule to another which in turn produces
+a sound wave. [15]
+There are a few terms that should be familiar when talking about sound and signal processing.
+• Amplitude: Amplitude refers to the maximum displacement of the air molecules from the rest position
+• Crest and Trough: The crest is the highest point in the wave whereas trough is the lowest point
+• Wavelength: The distance between two successive crests or troughs is known as a wavelength
+• Cycle: Every audio signal traverses in the form of cycles. One complete upward movement and downward
+movement of the signal form a cycle
+• Frequency: Frequency refers to how fast a signal is changing over a period
+Additionally, there are two different types of signals: Digital and Analog Signal. For digital signal is a discrete
+representation of a signal over a period. Here, the finite number of samples exists between any two-time intervals
+whereas the analog signal is a continuous representation of a signal over a period which implies that there will be an
+infinite number of samples between any two given time intervals. [16]
+For this project, audio signals are needed for the Voice Assistant, so there is a question about how it stores a
+signal that has an infinite number of samples since they are analog signals. This project will incorporate changing the
+memory-hogging analog signal using complex techniques to convert it into digital signals to make it more convenient
+to work with them.
+To convert a signal from analog to digital, a technique called sampling the signal should be used which selects a
+certain number of samples per second from the analog signal which makes storing and processing the signal memory
+efficient. In analog to digital sampling "an input signal is converted from some continuously varying physical value (e.g.
+pressure in air, or frequency or wavelength of light), by some electro-mechanical device into a continuously varying
+electrical signal." [17]
+2.2.2
+Feature Extraction Techniques for Audio signals. For this projects model to use the audio signals, the features
+must be extracted from the audio such as time domain and frequency domain. The audio signal is representing by the
+amplitude as a function of time. The features here are the amplitudes which are recorded at different time intervals. On
+the other hand, in the frequency domain, the audio signal is represents amplitude as a function of frequency where the
+features are amplitudes that are recorded at different frequencies.
+2.3
+Papers Regarding this Topic
+In machine learning when one trains their model, there is always a chance for problems when applying the model to the
+population. This problem arises in big data when the population is too large for machines with limited computability.
+To deal with this problem of big data, a sample population can be used. A sample population refers to a subset of the
+population that represents the whole population. [18] This project will apply machine learning to samples population.
+But there is a chance for selection bias, “selection bias is a systematic error that results in differences between a study
+population and a target population; selection bias primarily affects the external validity of the results of a study”.
+This means that the results thought to be true for the sample population may not be true for the actual population.
+Depending on the population sample, if there are too many true results it will just show true no matter what. Sample
+population can technically be wrong from the population just because of the sample collected.
+Manuscript submitted to ACM
+
+6
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+Likewise, voice assistants use sample data to analyze a user’s speech. “Currently speech recognition has significant
+race and gender biases. It is just another form of AI that performs worse for women and non-white people. Currently, it
+is designed to understand white male voices well” [19]. This is a significant problem because these days voice assistants
+are a crucial part of people’s lives from setting alarms to transportation. Low accuracy in voice recognition would mean
+severe consequences in people’s life [19]. In the paper Empirical Analysis of Bias in Voice-based Personal Assistants,
+they try to check the accuracy of relevant voice assistants such as Google Assistant and Siri. They look at the different
+accents for the Brazilian Portuguese, and how accuracy was off for certain accents than other ones. They also found
+variation in the quality of recognition based on gender. [20]
+So, taking the two paper’s ideas, to tackle this problem and move forward in this topic, data was gathered and training
+our model not only on prevalent male voice data sets but also gathering voices manually and from Common-Voice
+data sets by Mozilla for female and minority voices. This allows us to create samples proportionally to demographic
+indicators of the country. The process for machine learning will be as follows:
+• Filter the words that the user says
+• Digitize the user’s speech into a format that the machine can read
+• Analyze the user’s speech for meaning
+• Decide what the user needs based on previous input and algorithms”
+As talked above, data sets for voice recognition must be sampled so that each voice has an equal probability to be
+included in the sample. This allows for less racial and gender bias for the models to work with. So, this will result in
+everyone having their voice heard.
+Additionally the paper “Dangerous Skills: Understanding and Mitigating Security Risks of Voice-Controlled Third-
+Party Functions on Virtual Personal Assistant Systems” [21] and “Your Voice Assistant is Mine: How to Abuse Speakers
+to Steal Information and Control Your Phone” talks about the development of voice assistant in speech recognition and
+IoTs and with its development comes more vulnerabilities. They talk about voice-based remote attacks and permission
+bypassing. These problems are extremely dangerous and can be misused to expose people’s information. So, these
+papers advise incorporating the ideas of not allowing zero permission and context-aware information collection and
+analysis features to the voice assistant. Not allowing those ideas will allow focusing on forcing restrictions on specific
+operations that a particular process can perform [22].
+3
+EXPERIMENTAL SETUP
+The data used is partially from Common-Voice, which is an online open source of voice recorders in multiple languages.
+However, since this project is specifically for the English language the data set collected was English. However, this
+data set, as expected has more male voices than female voices. Therefore, to even out the distribution between the
+voices represented the rest of the data was collected via outreach into the Augustana College community, contacting
+over 200 female students (ages ranging from 18 - 24 years old) and receiving about 65 voices samples with 4-7 minutes
+of voice recording from each sample. The data set collected was then doubled, converted, and combined into .wav files.
+After the data collection and manipulated, it was then applied to a gender recognition model in order to numerically
+identify via the frequency of each voice; according to ASHA (American Speech Language and Hearing Association) the
+average range for an adult woman is 165 to 255 Hz and the average range for adult males is 85 to 155 Hz [23]. There is
+inherent technical problem down to the fact that females generally have higher pitched voices. Female voices tend to be
+Manuscript submitted to ACM
+
+Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+7
+quieter and sound more “breathy”. Female more easily masked by noise, like a fan or traffic in the background, which
+makes it harder for speech recognition systems.
+In order to run the data through training model for speech recognition. The full data set from Common-Voice was
+not needed in fact there only segments of the a statement is used, therefore it was necessary to parse the data into
+smaller form.
+4
+FRAMEWORK
+For Machine Learning, Python is known to be the most common language used due to its versatility, readability, and
+abundant packages. So, for this project, it will be using Python 3.10 as the main programming language. This project
+will be using the following python packages or libraries:
+• Speech Recognition (Will only be used initially to create and test the basic functionalities of the project and will
+be later replaced by this projects own model): It helps understand what the human is saying and converts the
+speech into text.
+• Pyttsx3: It is a simple text to speech conversion library in python which will be used to give this projects Voice
+Assistant a voice.
+• Wikipedia: This package is a Python that extracts information and data from Wikipedia which is a multilingual
+online encyclopedia used by many people.
+• Capture: It helps to capture images from your camera.
+• Datetime: It is an inbuilt module in python that works with date and time.
+• Os: It provides functions to interact with the operating system.
+• Web Browser: It is an in-build package from Python that allows you to extract information from the web.
+• JSON: It is a module that helps to store and exchange data.
+• PyJokes: It gives the user a random joke.
+• Pyaudio: It allows python to play and record audio between different platforms.
+• Pywhatkit: It can access YouTube in order to play a video.
+• Librosa: It is a package for analysing sound, audio and music.
+• Soundfile: This package can help read and write sound files.
+• Numpy: Numpy provides a powerful N-dimensional array object, sophisticated functions, tolls for integrating
+C/C++ and Fortran code, useful linear algebra, Fourier transform, and random number capabilities, and much
+more.
+• BeautifulSoup: Beautiful Soup is a Python library for pulling data out of HTML and XML files.
+• Pandas: pandas is a fast, powerful, flexible and easy to use open source data analysis and manipulation tool, built
+on top of the Python programming language.
+This project will be using all these modules and packages to create basic functionalities of the Voice Assistant. For
+example, this project will use the Wikipedia package to get information regarding a person or a company. This function
+is used to grab the first few sentences from its corresponding Wikipedia page which is generally the broad introduction
+to the subject.
+Then this project will have a Voice Recognizer which will replace the existing Speech Recognition module as well
+as the Google API. The program uses deep learning to create this projects voice recognition using PyTorch. The data
+is from open sources Mozilla Common-Voice as well as a few more from Kaggle to train this projects model and add
+Manuscript submitted to ACM
+
+8
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+voices from volunteers to help balance the data to reduce the biases while training for the wake word as well as the
+general voice recognition itself. This project utilizes Google Colaboratory to train this projects model. Finally, if time
+permits, the goal is to build a simple User Interface either with Flask or with other python libraries itself for a visual
+element to the program.
+5
+METHODS
+As stated in the experimental setup, the Common-Voice data set planned to be used for the training model has recognized
+46% male voices and 16% female voices. Therefore, the goal of this project is to come up with an effective solution to
+this disparity.
+When training a speech recognition model, different types of voice data can be used. Depending on the type of
+interaction one’s looking to build, and how robust that interaction should be, different types of voice data might
+be required. Although there are several easily available sources of speech data, such as public speech corpora or
+pre-packaged data sets, it is almost always necessary to cooperate with a data services provider to collect your own
+speech data, either remotely or in person. When you gather one’s own data, it is easy to tailor the speech data set to
+include variables such as language, speaker demographics, audio requirements, and collection size. [24]
+The Bureau of Labor Statistics (BLS) projects computer science research jobs will grow 19 percent by 2026. However,
+in the United States, the percentage of women that receive a bachelor’s degree in Computer Science is still only 18
+percent. There is currently a high demand for computer scientists in the professional industry but despite this fact, this
+industry remains male-dominated, in the United States. For example, in this year’s Senior Inquire class for Computer
+Science, there is a 1:6 female to male ratio [25].
+Computer technology first emerged during World War II and continuing into the 1960s, women made up most of the
+computing workforce. However, by 1970 women only accounted for about 14 percent of bachelor’s in computer science.
+In 1984 that number rose to 37 percent. The percentage of women in computer science has since declined to 18 percent.
+It was around the same time personal computers started showing up in homes. According to NPR, personal computers
+were marketed almost exclusively to men and families were more likely to buy computers for boys than girls.
+Computers are now commonplace in both classrooms and on individuals as personal assistance. It is hard, however, to
+explain the exact reason why females are not as present in this major. There are organizations now that are researching
+and improving ways to increase the potential of more females in the computer science major. It is said that one of the
+reasons why women tend to trend away from the computer science field is because of marketing of the industry in the
+past tailored to those with the geek persona and the social innuendos of what being a geek used to mean. [26]
+One of the reasons why females should be represented in the computer industry is increasing the inclusion of women
+is a sound business strategy. “A study by Deloitte found that women’s choices account for up to 85 percent of buying
+decisions nationwide, and that diversity drives innovation. Though it is still commonplace to find boards and project
+teams without a female member, the integration of female perspectives will naturally lead to higher revenues and a
+better understanding of consumer marketplaces.” [27]
+Therefore, the purpose of this project is to strive to increase the attractiveness of the potential that computer science
+avenues provide for women, by increasing the representation of females in the initial study of voice assistants. If females
+can realize that they possess a more essential in the initial makeup that products that are generated there is a possibility
+that females will be more inclined into joining the computer science industry, by showing them that they do in fact
+play an essential role in the make of technology.
+Speech recognition data can be classified into three categories:
+Manuscript submitted to ACM
+
+Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+9
+• Controlled: Scripted speech data
+• Semi-controlled: Scenario-based speech data
+• Natural: Unscripted or conversational speech data
+For this project, the primary type of voice data used is Semi-controlled data and natural unscripted conversational
+speech data. For semi-controlled data, When developers need a natural sampling of different ways to ask for the same
+thing or a greater diversity of command intentions, scenario-based voice data is collected (i.e. asking for different
+things). As a result, scenario-based speech data adds variety to what is said as well as how it is said.
+On the other hand, The most "natural" kind of speech is unscripted or conversational speech data, which is a recording
+of a conversation between two or more speakers. Unscripted speech data, like spontaneous speech, occurs in a variety
+of formats in the real world. For example, this information could be captured in the form of phone conversations or
+recordings of individuals conversing in a busy room. If a developer is looking for conversational data on a given topic
+(for example, music), two speakers might be asked to conduct a conversation about it.
+Fig. 2. Train-Loss Graph
+To compensate for this gender disparity, data augmentation will be done on the voice data-set in order to artificially
+increase the diversity of the data-set and to increase the data-set size. This technique is performed by changing the
+pitch, speed, injecting noise, and/or adding reverb to the audio data.
+The model was trained with about 30,000 separate lines of data and was trained over 7 epochs with a learning rate of
+5e-1 and batch sizes of 15.
+The train loss graph shows that the speech recognition model has more than enough data to train the model and is
+using more than enough data making the model almost over-fit which is also supported by the test loss graph.
+As, shown by the train loss graph, the test loss graph also gives the same conclusion that the speech recognition
+model might be slightly over fitting since the train loss is very low, compared the test loss which is much higher than
+the train loss itself. However, the test loss is no too huge which makes the model usable.
+Finally, after half of the training, the learning rate started slowing down as the model kept getting more and more
+training with each epochs.
+Manuscript submitted to ACM
+
+train_loss
+：
+8
+6
+4
+2
+0
+0
+ 5k
+10k
+15k
+20k
+25k
+30k10
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+Fig. 3. Test-Loss Graph
+Fig. 4. Training Learning Graph
+6
+RESULTS
+The goal of this project is to create a voice-generated virtual assistant using python that can work similarly to the modern
+popular voice assistants such as Siri, Alexa, or Bixby. This projects personal AI voice assistant can understand voice
+commands using speech recognition in Python and will be able to perform multiple tasks, using the pre-programmed
+functions built into it as well as training data.
+At the end of the project, this project will have AI voice assistant to be able to recognize voices and their commands
+and perform the most suitable actions. This process of voice recognition is done by breaking down audio into individual
+sounds, then converting them into a digital format that will be using Machine Learning algorithms and models to find
+the word for that sound. The words will then be used by the voice assistant and see if anything related to it has been
+pre-programmed and if not, it will be able to perform a most suitable action. The assistant will be speech-enabled, so
+Manuscript submitted to ACM
+
+test_loss
+1.4
+1.2
+1
+0.8
+0.6
+5k
+10k
+15k
+20k
+25ktrain_learning_rate
+500μ
+400μ
+300μ
+200μ
+100μ
+0
+0
+10k
+20k
+30kAddressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+11
+after recognizing a statement or a question, it will call the necessary function to execute the task then give a response
+based on the algorithm.
+However, the project will incorporate some features such as sending text messages, sending emails, opening songs,
+predicting the time, telling jokes, surfing the internet for information, forecasting weather, and many more. The final
+project will not be limited to the features listed above and will possess more as well as having a clearer vision of what
+this projects personal voice AI can do.
+Fig. 5. Cer graph
+Fig. 6. Wer graph
+For the model that was trained, the Wer(Word Error Rate) was reduced to about 50% which is not so great when
+compared to other models made by large scale companies, such as Google’s 4.9% WER and Microsoft’s 5.1%. Also from
+Manuscript submitted to ACM
+
+cer
+ :
+0.4
+0.3
+0.2
+5k
+10k
+15k
+20k
+25kwer
+：
+0.9
+0.8
+0.7
+0.6
+0.5
+5k
+10k
+15k
+20k
+25k12
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+the trained model the Cer(Character Error Rate) to about 20% which means one out of every 5 characters were predicted
+incorrectly which is not the best.
+Fig. 7. Pitch Estimator graph
+Part of this project was to see if the computer could correctly analyze whether a voice is female. Using the data
+set collected from Augustana College where every of the 62 voice samples are females. Using the frequency ranges
+mentioned earlier in the paper reported by ASHA as the boundaries for determination the computer categorized that 34
+out the 62 voice samples are female voices meaning that the computer correctly estimated about 55% of the samples.
+Figure 7 is a sample of one of the correctly categorized voices samples showing the frequency boundary ranges for both
+male and female voices with the red line be the voice samples calculated frequency. This results proves the hypothesis
+that computers simple have a hard time distinguishing female voice samples regardless of the frequency ranges.
+One of the biggest challenges is training the assistant to recognize different types of voices, accents, and dialects. To
+counteract this problem by researching effective machine learning algorithms to implement to train the necessary voice
+data. This project will utilize various open-source projects to complete the voice recognizer which then can incorporate
+into this project own voice assistant.
+One of the primary goals for this project is to build and train a model that utilizes more voice data from underrepre-
+sented groups, as there is a huge gender and race bias in most virtual assistants. This statement above can be implied
+because the training data used in some of the original voice assistants consists mainly of Caucasian males and Asian
+males. The female group as well as people from other backgrounds that might have different accents or dialects are
+extremely underrepresented. These models store the public data to try and improve the accuracy of the voice recognizer
+in their voice assistants, but will try to incorporate these data from the beginning. In the end, the voice assistant will be
+able to recognize voices regardless of their gender, race, or accent.
+Another goal is the addition of a two factor authentication for added security to the user and their data. The two
+factor authentication will be set to access the history of all the commands provided by the user. The collection of data is
+done for the functionality of the virtual assistant. As well as allowing the user to keep track of the commands they
+have used. The goal is to have this authentication app ready for the final project, however this is now part of a future
+endeavor. A downside is that the voice recognition may not be viable, because it requires each user to train their voice
+specifically to the virtual assistant. This process will also require a lot more storage of data and processing on the
+programmers part.
+This project demonstrates that gender and race biases exist in a lot of virtual assistants and tries to fix this problem by
+training more voice data from underrepresented groups. Males, especially white and Asian men, are disproportionately
+Manuscript submitted to ACM
+
+pitch estimation
+Measured Frequency
+ Female min Freqency
+20 
+Female max Freqency
+--- Male min Freqency
+Male max Freqency
+5 
+0
+100
+200
+OOE
+400
+500
+frequency (Hz)Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+13
+over represented in Computer Science education and careers. Because of the male-dominated employment milieu, many
+women abandon Computer Science careers. Researchers have found many hurdles to women in Computer Science
+courses at the academic level, although efforts to address these issues have differed depending on geographic region or
+educational level.
+To combat this over-representation, gender depiction standards, as expressed through look, name, and voice, as well
+as a review procedure, are critical changes that must be introduced. Users are predisposed to draw gender from a voice,
+and often ascribe one to voices that are supposed to be neutral, as the makers of the gender-neutral voice assistant Q
+discovered [28].
+7
+DISCUSSION
+Virtual Assistants take in voice commands and process that data into commands or useful information. In this age, many
+developed countries can already be considered as aged societies [29]. This causes lots of changes such as a discrepancy
+in the proportion of seniors with the rest of the population. So, assistance technology such as a virtual assistant can
+help with the problems associated with it. In the article “Home-Assistant Robot for an Aging Society” they provide
+information on how assistance technology helps in labor support, healthy lifestyle support, and household and care
+support. In consideration to virtual assistants, they give a list of activities done by an IRT home assistant robot that
+models 3D space in a vector space and uses voice recognition with reinforcement learning to do several tasks:
+• Performing Chores in a Home Environment
+• Deformable Object Detection through image-based learning
+• Geometrical Object Modeling and Its Application to Position Estimation
+• Manipulation of Daily Tools and Appliances
+• Integration of Basic Tasks Into Sequential Behavior
+• Failure Detection and Recovery
+The IRT home assistant robot works with voice commands to learn tasks. One can guide it to perform physical
+activities such as carrying items, pushing objects, collecting items, and sweeping. For example, it can take images of
+clothing and with vector data, learn about wrinkles in clothing to pick up clothes. The robot with more data can do the
+task without any commands. Likewise, it uses environment recognition to do similar tasks at several locations of the
+house using commands. The implementation of this assistance technology can increase the ease and productivity of
+performing home activities. These assistants not only help old people but also physically disabled and blind people as it
+does not require learning and typing where they can just speak and ask questions to the assistant.
+In the expected results, this project discussed some of the outcomes of this projects virtual assistant. Again, it is
+still in the beginning phase, however, the main goal is to make it so that the assistant can do tasks like Siri or Alexa.
+“Virtual assistants are regularly used to make online interfaces more user friendly. It also generates positive responses
+from Internet users leading to a more interpersonal shopping experience, greater pleasure, and customer flow” [30].
+In line with the related work, this project will be aiming to tackle the problem of the under-representation of women’s
+voices and dialects for training. This project will be collecting first-hand data through participatory surveys and training
+our model based on that. Then this project will use data from online databases to further train our voice assistant.
+By doing this, this project can avoid the gender bias present in voice assistants. To confront the bias for dialects, this
+project will also be collecting voice command data from a diverse audience. This helps create a proportional sample
+that will help represent the college population and middle-aged adults that is the target.
+Manuscript submitted to ACM
+
+14
+Kashav Piya, Srijal Shrestha, Cameran Frank, Estephanos Jebessa, and Tauheed Khan Mohd
+Currently the project is not a proper application. It was presented with a GUI (Graphic User Interface) to access
+this project. In the future the goal is to implement an startup application with a better GUI that will run as soon as
+you turn your computer on. Since the application at the moment is only a GUI that will only receive commands there
+are no security risks. The program was also implemented so that the voice assistant will only listen when you press
+the listen button. In the future the plan is to make it so you can use your voice in order to activate it from the start by
+using a wake word. Doing so will make it easier to use but the goal is for the voice assistant to not collect sensitive
+information when the wake word is used by accident. To counter that the plan is to add a mute feature for the assistant
+to not allow it to listen. The future plan is also to add an indicator for when the voice assistant is actively listening
+as well as keeping logs of when listening occur ed and a text alert through a mobile device. But when developing the
+full application there will be a lot of potential security risks to consider. The plan to add several measures to manage
+the risks. One of the ways would be to personalize the voice assistant to one account unless one links another one.
+The future plan also includes adding a feature to ask for two-factor authentication in order to access the device use
+and history. This will help mitigate potential risks of identity theft or user impersonation. It will also add an extra
+layer of security to protect sensitive data from theft. The two-factor authorization that is used will be a pattern and/or
+pass-code and a token, or biometric data and a token. For the token, it will have a token generator or an authentication
+app. For the pattern and/or pass-code, this project will make sure that people can input it through their phones. As for
+biometric data, this project will add a part to the voice assistant so that it will learn the user’s tone and pitch if allowed.
+As for the future of the project, the voice assistant will keep on building up with more features aimed for making
+the day to day activities easier for the user. Our research on the speech recognition will continue to grow as well as
+trying on new alternatives such as adding more layers to our neural network, using audio books to train our model,
+and using various other architectures available such as ctcdecode which can only be used in Linux environment which
+will improve our word error rate dramatically. The end goal of this project is ambitious considering that there are many
+speech recognition systems out already that perform very well, and have resources that cannot be compared to ours,
+but there will be improvements in the speech recognition system with more training and additional libraries.
+Manuscript submitted to ACM
+
+Addressing the Selection Bias in Voice Assistance: Training Voice Assistance Model in Python with Equal Data Selection
+15
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+World Wide Web Conference, pp. 533–538, 2019.
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+pp. 11–5, 2021.
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+Manuscript submitted to ACM
+

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@@ -0,0 +1,1405 @@
+arXiv:2301.01359v1  [math.NT]  3 Jan 2023
+PROOFS OF MODULO 11 AND 13 CYLINDRIC KANADE-RUSSELL CONJECTURES
+FOR A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+ALI KEMAL UNCU
+Abstract. We present proofs of two new families of sum-product identities arising from the cylindric parti-
+tions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for
+the proofs were first conjectured by Kanade–Russell in the spirit of Andrews–Schilling–Warnaar identities of
+the A2 Rogers–Ramanujan type. We follow the footsteps of Kanade–Russell while we alter the computations
+heavily to accomplish our goals.
+1. Introduction
+There is an ever-growing synergy between number theory, combinatorics, q-series, and affine Lie algebras
+that led to groundbreaking techniques and beautiful mathematical discoveries. Among these are the Rogers–
+Ramanujan type identities where an infinite q-series is equal to a infinite product with a modular structure.
+First appeared at the intersection of number theory and combinatorics, the Rogers–Ramanujan identities
+have been of great interest. These sum-product identities have been studied, proved and generalized in many
+different ways over the years [3, 6, 13, 14, 17, 22, 24, 25, 37]. These identities also naturally arose in many
+other fields including mathematical physics [10], representation theory of affine Lie algebras and vector operator
+algebras [31, 32], knot theory in relation to the colored Jones polynomials [8], and algebraic geometry [15] over
+the years.
+For some non-negative integer L and formal variables a and q, let q-Pochhammer symbol be (a; q)L :=
+(1 − a)(1 − aq) . . . (1 − aqL−1), and (a; q)∞ := limL→∞(a; q)L, θ(a; q) := (a, q/a; q)∞, and for a1, . . . , ak some
+formal variables, define the shorthand notation θ(a1, a2, . . . , ak; q) := θ(a1; q)θ(a2; q) . . . θ(ak; q).
+The Rogers–Ramanujan identities are as follows [38].
+Theorem 1.1 (Rogers–Ramanujan identities).
+(1.1)
+�
+n≥0
+qn2
+(q; q)n
+=
+1
+θ(q; q5)
+and
+�
+n≥0
+qn2+n
+(q; q)n
+=
+1
+θ(q2; q5).
+The reciprocal q-Pochhammer products on the right-hand side of (1.1) has the ±1 and ±2 residue classes
+modulo 5, respectively. We call these modulo 5 identities.
+A composition c of n is a finite list of non-negative integers that sum up to n. A partition is a composition
+where no element of the list (called parts) are zero and the list elements are ordered in a non-increasing order.
+We define the size of a composition π as the sum of all its parts and denote this by |π|. We denote the number
+of parts in a composition π by #(π). A composition (resp. partition) with size n is called “a composition
+(resp. partition) of n.” The empty list is considered as the unique composition/partition of 0 with 0 parts.
+For example, (2, 0, 2) is a composition with 3 parts and (1, 1, 1, 1), (4, 3, 1), and (2, 2) are partitions of 4, 8,
+and 4, respectively.
+MacMahon [33] and Schur [39] gave combinatorial interpretations to Rogers–Ramanujan identities inde-
+pendently.
+Date: January 5, 2023.
+2010 Mathematics Subject Classification. Primary 05A15; Secondary 05A17, 05A19, 11B65, 11P84, 17B65, 68R05.
+Key words and phrases. Cylindric partitions, Partition identities, Rogers–Ramanujan identities, Andrews–Schilling–Warnaar
+identities.
+Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF grant P-34501N.
+1
+
+2
+ALI KEMAL UNCU
+Theorem 1.2 (Combinatorial interpretaton of Rogers–Ramanujan identities). Let i = 0 or 1. For every
+natural number n, the number of partitions of n such that the difference between two consecutive parts is at
+least 2 and the the smallest part is strictly greater than i is equal to the number of partitions of n into parts
+congruent to ±(1 + i) mod 5.
+Gordon [24] presented a wide generalization of Theorem 1.2 to all odd modulus ≥ 5.
+Theorem 1.3 (Gordon’s identities, 1961). Let r and i be integers such that r ≥ 2 and 1 ≤ i ≤ r. The number
+of partitions π = (π1, π2, . . . , πs) of n such that πj − πj+r−1 ≥ 2 for all j with at most i− 1 1s appears as parts
+in π are equal to the number of partitions of n whose parts are not congruent to 0, ±i mod 2r + 1.
+The Rogers–Ramanujan identities correspond to the cases r = i = 2 and r = 2, i = 1.
+Andrews found the q-series counterpart to Gordon’s identities [3].
+Theorem 1.4 (Andrews–Gordon identities, 1974). Let r ≥ 2 and 1 ≤ i ≤ r be two integers. We have
+(1.2)
+�
+n1≥···≥nr−1≥0
+qn2
+1+···+n2
+r−1+ni+···+nr−1
+(q; q)n1
+�
+n1
+n1 − n2
+�
+q
+· · ·
+�
+nr−2
+nr−2 − nr−1
+�
+q
+= θ(qi; q2r+1)(q2r+1; q2r+1)∞
+(q; q)∞
+,
+where for two integers n and m,
+�m + n
+m
+�
+q
+:=
+
+
+
+
+
+(q; q)m+n
+(q; q)m(q; q)n
+for m, n ≥ 0,
+0
+otherwise,
+is the classical q-binomial coefficient.
+Note that the Rogers–Ramanujan identities are the particular case of (1.2) where r = i = 2, and r = 2 and
+i = 1. Interested readers can get a great overview of the history of the Rogers–Ramanujan identities, their
+significance, and some generalizations in the recent book of Sills [40].
+The identities (1.2) can be proven by the Bailey machinery coming from the world of q-series. This powerful
+mechanism starts with a pair of q-expressions, called a Bailey pair, that satisfies a pre-defined relation and
+modifies this pair iteratively (using Bailey lemma or one of its generalizations) to make a new Bailey pair (see
+[2, 4, 9, 40]). That way, by starting with the pair related to Rogers–Ramanujan identities, a whole infinite
+chain of identities (1.2) can be acquired. The identities (1.2) are certain characters related to affine Lie algebra
+A(1)
+1 , and we thus refer to them as A1 Rogers–Ramanujan identities. The original Bailey mechanism was later
+extended to An−1 for general n [34, 35]. However,these works did not yield An−1 Rogers–Ramanujan identities.
+In their influential paper, Andrews, Schilling and Warnaar [7] were able to describe an A2 Bailey lemma
+and the associated Bailey machinery. They found several infinite families of identities, One of their modulo 7
+identities is as follows.
+Theorem 1.5 (Andrews–Schilling–Warnaar, 1999).
+(1.3)
+�
+r1,s1≥0
+qr2
+1−r1s1+s2
+1+r1+s1
+(q; q)r1
+�2r1
+s1
+�
+q
+=
+1
+θ(q2, q3, q3; q7).
+Andrews–Schilling–Warnaar found several very general families of sum-product identities. Of particular
+interest to representation theory, the product-sides of these identities are character formulas of the W3 algebra
+multiplied by an extra factor (q; q)−1
+∞ [21]. These formulas do not yield manifestly positive sum-sides for the
+character formulas because of this extra factor.
+For example, one of Andrews–Schilling–Warnaar’s modulo 10 identities after clearing the extra factor
+(q; q)−1
+∞ is as follows.
+Theorem 1.6 (Andrews–Schilling–Warnaar, 1999).
+(1.4)
+(q, q)∞
+�
+r1≥r2≥0
+s1≥s2≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2+r1+r2+s1+s2
+(q; q)r1−r2(q; q)r2(q; q)s1−r2(q; q)s2(q; q)r2+s2+1
+=
+1
+θ(q2, q3, q3, q4, q4, q5; q10)
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+3
+Recall the Euler’s Pentagonal Number Theorem [5]
+(1.5)
+(q, q)∞ =
+∞
+�
+i=−∞
+(−1)iqi(3i+1)/2.
+Although it is easy to see that the right-hand side of (1.4) has positive coefficients, in light of (1.5) this is not
+directly visible on the left-hand side. In contrast, both sides of (1.3) are manifestly positive. The manifestly
+positive sum representations give insight to the structure of certain modules for the affine Lie algebra A(1)
+2 .
+These mentioned character of standard modules for the affine Lie algebra A(1)
+2 . Interested readers can find
+more on this connection in [7, 29, 28, 31, 32].
+Recently, the discovery of manifestly positive identities of these character formulas through a scheme with
+combinatorial roots attracted the attention and led to many new Rogers–Ramanujan type identities.
+In 1997, Gessel and Krattenthaler [23] defined cylindric partitions in context of non-intersecting lattice
+paths. Borodin [11] gave univariate product formulas for the generating functions of the number of cylindric
+partitions. Foda and Welsh [21] proved the A1 Rogers–Ramanujan identities using the combinatorics of cylin-
+dric partitions. This led to Corteel’s combinatorial proof of the Rogers–Ramanujan identities using cylindric
+partitions [17]. In 2019, Corteel and Welsh [18] derived functional equations for the bivariate generating func-
+tions for the number the number of cylindric partitions using the largest part statistic. While doing so, they
+also gave a new proof of Andrews–Schilling–Warnaar’s modulo 7 A2 Rogers–Ramanujan identities (including
+(1.3)) and a fifth missing identity which was originally conjectured by Feigin–Foda–Welsh [20].
+All these
+modulo 7 identities have manifestly positive sum sides. [18] has been the catalyst for the recent developments.
+Ablinger and the author [1] implemented the Corteel–Welsh’s cylindric partitions related functional equations
+in their symbolic computation implementation qFunctions to be able to exploit this combinatorial idea using
+formal manipulation and computer algebra techniques. Corteel, Dousse and the author [19] later proved the
+modulo 8 identities that arise from the cylindric partitions paradigm with the help of this implementation.
+One of such identities is as follows (see Theorem 1.6 in [19]).
+Theorem 1.7 (Corteel–Dousse–U., 2021).
+(1.6)
+�
+r1≥s1≥r2≥0
+r1≥s2≥0
+qr2
+1−r1s1+s2
+1+r2
+2+s2
+2+s1s2+r1+r2+s1+s2
+(q; q)r1
+�r1
+s1
+�
+q
+�r1
+s2
+�
+q
+�s1
+r2
+�
+q
+=
+1
+θ(q2, q3, q3, q4, q4, q5; q10)
+Unlike (1.4), (1.6) has a manifestly positive sum-side. Shortly after [19], in late 2021, Warnaar [42] come
+up with many beautiful conjectures for manifestly positive sum-sides related to higher moduli (not divisible
+by 3). In 2022, Tsuchioka [41] proved manifestly positive sum-sides for modulus 6 using finite-automata and
+automated proofs. He was also able to analyze the structure of relevant level 3 standard modules for the affine
+Lie algebra A(1)
+2 .
+In a different vein, Bridges and the author studied weighted versions of cylindric partitions as well as
+cylindric partitions into distinct parts in [12].
+Earlier in 2022, Kanade and Russell [29] aimed (and succeeded) at conjecturing A2 Rogers–Ramanujan
+type identities in the form of Andrews–Schilling–Warnaar instead of aiming for manifestly positive sum-sides.
+They were able to make explicit claims for each modulus ≥ 5. They proved the cases for moduli 5, 6, 7, 8 and
+10. Their exploration came to an end due to the increasing computational difficulties.
+In this paper, we approach the conjectures of Kanade–Russell by changing the computational techniques
+used. We prove all modulo 11 and 13 A2 Rogers–Ramanujan identities coming from the cylindric partitions
+paradigm. Two such identities are as follows:
+
+4
+ALI KEMAL UNCU
+Theorem 1.8.
+�
+r1≥r2≥r3≥0
+s1≥s2≥s3≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2+r2
+3+r3s3+s2
+3+r1+r2+r3+s1+s2+s3
+(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1
+=
+1
+(q; q)∞
+1
+θ(q2, q3, q3, q4, q4, q5, q5; q11).
+Theorem 1.9.
+�
+r1≥r2≥r3≥0
+s1≥s2≥s3≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2+r2
+3−r3s3+s2
+3+r1+r2+r3+s1+s2+s3
+(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1
+=
+1
+(q; q)∞
+1
+θ(q2, q3, q3, q4, q4, q5, q5, q6, q6; q13).
+The organization of this paper is as follows. In Section 2, we introduce cylindric partitions, the relevant
+results and the conjectures of Kanade–Russell of which we prove some cases of. Section 3 is dedicated to
+rewording the conjectures and the description ot the proof methodology. In Sections 4 and 5 we present
+the proofs of the modulo 11 and 13 A2 Rogers–Ramanujan identities in Andrews–Schilling–Warnaar form,
+respectively. We outline some natural questions and mathematical challenges that arise from this work in
+Section 6. Section 7 is reserved for a discussion on how the computerized proofs have been carried in earlier
+work [19, 29] and this paper and what future improvements can be done to take us further mathematically.
+Acknowledgement
+The author would like to thank the workshop on cylindric partitions group that came together in November
+2022 in Linz for all the stimulating discussions. In particular, the author would like to thank Shashank Kanade
+for suggesting that the researchers working on cylindric partitions should come together and join forces in the
+first place, and for all his comments on this manuscript.
+The author would also like to thank Christian
+Koutschan his encouragement of the author in the necessary implementations.
+Research of the author is partly supported by EPSRC grant number EP/T015713/1 and partly by FWF
+grant P-34501N.
+2. Necessary definitions
+We shall start with the definition of a cylindric partition.
+Definition 2.1. A cylindric partition is made up of a composition c = (c1, c2, . . . , cr) called profile with r
+parts, and a vector π = (π(1), π(1), . . . , π(r)) consisting of r partitions π(i) = (π(i)
+1 , π(i)
+2 , . . . ), that satisfy the
+inequalities
+π(i)
+j
+≥ π(i+1)
+j+ci+1
+and
+π(r)
+j
+≥ π(1)
+j+c1.
+For example, the vector partition π = {(1, 1, 1, 1), (4, 3, 1), (2, 2)} together with the profile (2, 0, 2) is a
+cylindric partition. Note that the same vector partition can also satisfy the cylindric partition inequalities
+with different profiles. For example, π is also a cylindric partition for profiles (2, 0, 0), (2, 0, 1), etc. We can
+define the total size of a cylindric partition π as the sum of all the sizes of the partitions included. We denote
+the total size, once again, by |π|. Let c be a composition and let Pc be the set of all vector partitions that are
+cylindric partitions with profile c.
+For a given profile c, let Pc be the set of all cylindric partitions with profile c. Let
+Fc(z, q) :=
+�
+π∈Pc
+zmax(π)q|π|,
+the bivariate generating function for the number of cylindric partitions where the exponents of z and q are
+keeping record of the largest parts size and the total of the parts in π, respectively. Borodin [11] showed that
+when z = 1, Fc(z, q) generating functions have product formula.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+5
+Theorem 2.2 (Borodin, 2007). Let r and l be positive integers, and let c = (c1, c2, . . . , cr) be a composition
+of l. Define m := r + l and s(i, j) := ci + ci+1 + · · · + cj. Then,
+(2.1)
+Fc(1, q) =
+1
+(qm; qm)∞
+r
+�
+i=1
+r
+�
+j=i
+ci
+�
+k=1
+1
+(qk+j−i+s(i+1,j); qm)∞
+r
+�
+i=2
+i�
+j=2
+ci
+�
+k=1
+1
+(qm−k+j−i−s(j,i−1); qm)∞
+.
+Focusing on replacing the largest part in a given cylindric partition, Corteel–Welsh [18] defined a q-difference
+equation for Fc(z, q). This functional equation relates Fc(z, q) with other generating functions Fc∗(z, q) where
+#(c) = #(c∗) and |c| = |c∗|. Let c = (c1, . . . , cr) (with the convention that c0 = cr) be a given composition
+and define Ic to be the set of indices for the non-zero entries in c. Given a non-empty subset J ⊆ Ic, the
+composition c(J) = (c1(J), . . . , cr(J)) is defined by:
+(2.2)
+ci(J) :=
+
+
+
+
+
+ci − 1
+if i ∈ J and i − 1 /∈ J,
+ci + 1
+if i /∈ J and i − 1 ∈ J,
+ci
+otherwise.
+Then the explicit q-difference equation Fc(z, q) satisfies is as follows.
+Theorem 2.3 (Corteel–Welsh, 2019). For any profile c,
+(2.3)
+Fc(z, q) =
+�
+∅⊂J⊆Ic
+(−1)|J|−1 Fc(J)(zq|J|, q)
+(1 − zq|J|)
+,
+with the initial conditions Fc(0, q) = Fc(z, 0) = 1.
+Let c be a profile and c′ be a cyclic shift of c. There is a clear one-to-one correspondence between cylindric
+partitions in Pc and Pc′ by cyclically shifting the vector of partitions counted in Pc. This is enough to see that
+the generating functions for these sets of cylindric partitions are equal, i.e. Fc(z, q) = Fc′(z, q). Therefore, we
+can cyclically shift the profiles and lower the number of (seemingly different) generating functions that appear
+in the coupled system of q-difference equations.
+We can also normalize (2.3) and get an equivalent q-difference equation. For example, let
+Gc(z, q) := (zq; q)∞Fc(z, q).
+The equation (2.3) is equivalent to
+(2.4)
+Gc(z, q) =
+�
+∅⊂J⊆Ic
+(−1)|J|−1(zq; q)|J|−1Gc(J)(zq|J|, q),
+with the initial conditions Gc(0, q) = Gc(z, 0) = 1. This q-difference equation (2.4) with polynomial coeffi-
+cients, in practice, played a central role in the proofs of modulo 7 and modulo 8 cylindric partition with 3-part
+profile identities in [18] and [19], respectively. Weighted versions of (2.3) and (2.4) are later presented in [12].
+In [29], Kanade–Russell decided to change the initial conditions of (2.4) slightly. While this does not change
+the q-difference equations, this lead to the conjectural discovery of explicit formulas for most of these 3-part
+profile cylindric partition generating functions. Let
+(2.5)
+Hc(z, q) := (zq; q)∞
+(q; q)∞
+Fc(z, q).
+Then Hc(z, q) satisfies the same q-difference equation as Gc(z, q), namely
+(2.6)
+Hc(z, q) =
+�
+∅⊂J⊆Ic
+(−1)|J|−1(zq; q)|J|−1Hc(J)(zq|J|, q),
+with the initial conditions Hc(0, q) = 1/(q; q)∞ and Hc(z, 0) = 1.
+From this point forward we only focus on cylindric partition profiles with 3-parts.
+Let k ≥ 2, let
+(2.7)
+ρ = (ρ1, ρ2, . . . , ρk−1) ∈ Zk−1, and σ = (σ1, σ2, . . . , σk−1) ∈ Zk−1
+
+6
+ALI KEMAL UNCU
+and define
+S3k−1(z; ρ|σ) =
+�
+r,s∈Zk−1
+≥0
+zr1
+q
+�k−1
+i=1 r2
+i −risi+s2
+i +ρiri+σisi
+�k−2
+i=1 (q; q)ri−ri+1(q; q)si−si+1
+q2rk−1sk−1
+(q; q)rk−1(q; q)sk−1(q; q)rk−1+sk−1+1
+,
+(2.8)
+S3k(z; ρ|σ) =
+�
+r,s∈Zk−1
+≥0
+zr1
+q
+�k−1
+i=1 r2
+i −risi+s2
+i +ρiri+σisi
+�k−2
+i=1 (q; q)ri−ri+1(q; q)si−si+1
+1
+(q; q)rk−1+sk−1(q; q)rk−1+sk−1+1
+�rk−1 + sk−1
+rk−1
+�
+q3,
+(2.9)
+S3k+1(z; ρ|σ) =
+�
+r,s∈Zk−1
+≥0
+zr1
+q
+�k−1
+i=1 r2
+i −risi+s2
+i +ρiri+σisi
+�k−2
+i=1 (q; q)ri−ri+1(q; q)si−si+1
+1
+(q; q)rk−1(q; q)sk−1(q; q)rk−1+sk−1+1
+.
+(2.10)
+Let
+(2.11)
+ei = (0, 0, . . . , 0
+�
+��
+�
+i
+, 1, 1, . . ., 1) ∈ Zk−1
+and
+δi := (δij)1≤j≤k−1 ∈ Zk−1.
+It is easy to see that
+(2.12)
+Sm(zqn; ρ|σ) = Sm(z; ρ + nδ1|σ)
+for any n ∈ Z.
+Kanade–Russell conjectured that for any fixed k ≥ 3, the H(c1,c2,c3)(z, q) can be expressed as linear combi-
+nations of the Sm(z; ρ|σ) functions. Precisely they claimed the following.
+Conjecture 2.4 (Kanade–Russell, 2022). Let k ≥ 3 and |c|+3 = m = 3k+{−1, 0, 1}. Using cyclis symmetries
+asusme that c1 ≥ c2, c3. If c2, c3 ≤ k − 1, then
+(2.13)
+H(c1,c2,c3)(z, q) =
+
+
+
+Sm(z; ec2|ec3) − qSm(z; ec2−1|ec3−1),
+c2, c3 > 0,
+Sm(z; ec2|e0),
+c3 = 0,
+Sm(z; e0|ec3) − q(1 − z)Sm(z; e0 + δ0|ec3−1),
+c2 = 0, c3 ̸= 0,
+where ei and δi be defined as in (2.11).
+The explicit claims that Conjecture 2.4 provide do not cover all the functions Hc(z, q) with |c| + 3 = m.
+It does provide enough claims to recover explicit expression claims. How to find the conjectural Sm(z; ρ|σ)
+equivalents of the other Hc(z, q) that appear in the coupled q-difference equation system is explained in [29].
+The profiles related to the functions to be recovered are called ”under-the-line” profiles by Kanade–Russell.
+We will also call these profiles as such while we ignore to explain anything about the line. These under-the-line
+profile related functions can have shifts of z in the Sm(z; ρ|σ) language. These shifts are inherited from the
+q-difference equations (2.6). One can use (2.12) to clear all the shifts in z. Therefore, from now on in all our
+expressions we will translate any z shift of Sm(z; ρ|σ) using (2.12) and this way ignore any and all shifts in z.
+To further emphasize this moving forward on we suppress the variable z from our notation and write
+Sm(ρ|σ) := Sm(z; ��|σ).
+Proof of Conjecture 2.4 (and its extension to all 3-part profiles with total m − 3) requires one to show that
+the expressions in Sm(ρ|σ) are the correct expressions for the respective Hc(z; q) functions. This can be done
+by showing that the expressions in Sm(ρ|σ) satisfies the same recurrence relation specified by (2.6) and the
+initial conditions of the expression holds. In [29], it is already proven that for c2, c3 ≤ k − 1, the conjectural
+formulas of (2.13) all satisfy the necessary initial conditions
+(2.14)
+Hc(z, 0) = 1
+and
+Hc(0, z) = 1/(q; q)∞.
+It was noted in the [29, Lemma 9.1, Lemma 9.2] that Sm(ρ|σ) functions satisfy the following list of recur-
+rences.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+7
+Lemma 2.5 (Kanade–Russell, 2022). Let k ≥ 3, let m = 3k + {−1, 0, 1} and let δi := (δij)1≤j≤k−1 ∈ Zk−1,
+where δij is the Kronecker delta function. The following recurrence relations follow for all 1 ≤ i ≤ k − 2,
+Sm(ρ|σ) − Sm(ρ + δi − δi+1|σ) − zqi+�i
+j=1 ρjSm(ρ + 2
+i
+�
+j=1
+δj|σ −
+i
+�
+j=1
+δj) = 0,
+(R(i)
+1 (ρ|σ))
+Sm(ρ|σ) − Sm(ρ|σ + δi − δi+1) − zqi+�i
+j=1 σjSm(ρ −
+i
+�
+j=1
+δj|σ + 2
+i
+�
+j=1
+δj) = 0.
+(R(i)
+2 (ρ|σ))
+i. If m ≡ −1 (mod 3),
+a) and if σk−1 = 0, then
+(R3(ρ|σ))
+Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−1|σ + δk−2 + δk−1) = 0.
+b) and if ρk−1 = 0, then
+(R4(ρ|σ))
+Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1) + qSm(ρ + δk−2 + δk−1|σ + δk−1) = 0.
+ii. If m ≡ 0 (mod 3), then
+Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1)
+(R3(ρ|σ))
+− zqk−1+�k−1
+j=1 ρjSm(ρ + 2
+k−1
+�
+j=1
+δj|σ −
+k−1
+�
+j=1
+δj)
+− qk−1+�k−1
+j=1 σjSm(ρ −
+k−2
+�
+j=1
+δj + 2δk−1|σ + 2
+k−1
+�
+j=1
+δj) = 0.
+Sm(ρ|σ) − (1 + q)Sm(ρ + δk−1|σ + δk−1) + qSm(ρ + 2δk−1|σ + 2δk−1)
+(R4(ρ|σ))
+− zqk−1+�k−1
+j=1 ρjSm(ρ + 2
+k−1
+�
+j=1
+δj|σ −
+k−2
+�
+j=1
+δj + 2δk−1)
+− qk−1+�k−1
+j=1 σjSm(ρ −
+k−1
+�
+j=1
+δj|σ + 2
+k−1
+�
+j=1
+δj) = 0.
+iii. If m ≡ 1 (mod 3), then
+Sm(ρ|σ) − Sm(ρ|σ + δk−1) − qSm(ρ + δk−1|σ + 2δk−1)
+(R3(ρ|σ))
++ qSm(ρ + δk−1|σ + 2δk−1) − qk−1+�k−1
+j=1 σjSm(ρ −
+k−1
+�
+j=1
+δj|σ + 2
+k−1
+�
+j=1
+δj) = 0
+Sm(ρ|σ) − Sm(ρ + δk−1|σ) − qSm(ρ + δk−1|σ + δk−1)
+(R4(ρ|σ))
++ qSm(ρ + 2δk−1|σ + δk−1) − zqk−1+�k−1
+j=1 ρjSm(ρ + 2
+k−1
+�
+j=1
+δj|σ −
+k−1
+�
+j=1
+δj) = 0.
+Then they made the following claim (see [29, Conjecture 9.3]).
+Conjecture 2.6 (Kanade–Russell, 2022). In each modulus m ≥ 5, the relations (R(i)
+1 (ρ|σ))-(R4(ρ|σ)) are
+enough to prove recurrences necessary for the proof of Conjecture 2.4.
+We find this conjecture highly sensible. For all m ≥ 5, the explicit Sm’s are 2⌊m/3⌋-fold sums. Same is
+true for the number of distinct functional equations (R(i)
+1 (ρ|σ))-(R4(ρ|σ)). One can easily check that these
+relations are distinct by comparing the first two terms in each left-hand side. Each second term corresponds to
+a canonical shift in one of the summation variables. One would expect to see every relation that the Sm(ρ|σ)
+functions satisfy to be translated and recovered as a combination the relations (R(i)
+1 (ρ|σ))-(R4(ρ|σ)). Hence,
+
+8
+ALI KEMAL UNCU
+if the claims of Conjecture 2.4 are correct, for any fixed profile c the coupled q-difference equations (2.6)
+written using the explicit claims of (2.13) (together with the “under-the-line” expressions) can be recovered
+as a combination of the relations (R(i)
+1 (ρ|σ))-(R4(ρ|σ)).
+3. Proof Methodology
+Conjecture 2.6 can be rephrased as a set inclusion question. Let m = 3k + {−1, 0, 1} with k ≥ 3, ρ and σ
+as in (2.7) and 1 ≤ i ≤ k − 2. Define
+(3.1)
+Im := ⟨R(i)
+1 (ρ|σ), R(i)
+2 (ρ|σ), R3(ρ|σ), R4(ρ|σ)⟩,
+the ideal generated by the left-hand sides of the recurrences (R(i)
+1 (ρ|σ))-(R4(ρ|σ)) as polynomials in the ring
+Z((q, z))[Sm(ρ|σ)]. Here which R3(ρ|σ) and R4(ρ|σ) to be included in Im is to be understood by the residue class
+of m modulo 3. Recall that ρ and σ are integer vectors with k −1 entries. Therefore the ring Z((q, z))[Sm(ρ|σ)]
+is a formal polynomial ring defined on a countable set of variables.
+For any given fixed m = 3k + {−1, 0, 1} with k ≥ 3, let the set of all the coupled system of q-difference
+equations (2.6) for the profiles c with |c| + 3 = m be Hm. Any relation in Hm can be written in Sm(ρ|σ)
+functions using the Conjecture 2.4 (and the paragraph below it). Let Sm be the set of all relations in Hm
+written in the conjectural Sm(ρ|σ) form.
+Now we can write Conjecture 2.6 in its equivalent form:
+Conjecture 3.1. Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed.
+∀h ∈ Sm, we have h ∈ Im.
+The infinite set {R(i)
+1 (ρ|σ), R(i)
+2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : 1 ≤ i ≤ k − 2, ρ, σ ∈ Zk−1} that spans Im has
+non-trivial relations within itself and not all the elements of this set are generators of Im. However, we do not
+know an exact pattern of which elements are related at the moment. Nevertheless, it is easy to understand
+that Im is generated by infinitely many elements since ρ and σ ∈ Zk−2.
+On the other hand, for any fixed m, the Sm(ρ|σ) functions that appear within the formulas from Sm
+make up a finite list. One can easily find explicit bounds for the entries of vectors ρ and σ such that every
+Sm(ρ|σ) that appear in Sm is within the bounds. This observation suggests that instead of attempting to
+prove Conjecture 3.1, we can instead go after a stronger conjecture that is more suitable for computations.
+To that end, let [N] := {−N, . . . , −1, 0, 1, . . ., N} and we define
+Im,N := ⟨{R(i)
+1 (ρ|σ), R(i)
+2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1}⟩ ⊂ Im.
+With this definition we form the stronger conjecture:
+Conjecture 3.2. Let m = 3k + {−1, 0, 1} with k ≥ 3 be fixed. There is some N ∈ N such that
+∀h ∈ Sm, we have h ∈ Im,N.
+Since Im,N ⊂ Im, it is clear that Conjecture 3.2 implies Conjecture 3.1.
+Finally we transferred the open problems into a linear algebra setting, and we can approach it as such.
+Let m and N be fixed. we can order all the Sm(ρ|σ) that appears in the spanning set of Im,N and write in
+a column vector ⃗s. Then the matrix A is uniquely defined by
+A⃗s = ⃗0A,
+where ⃗0A is the colum vector with the same number of rows as A. Every row of A, corresponds to a functional
+relation Rj(σ|ρ) ∈ {R(i)
+1 (ρ|σ), R(i)
+2 (ρ|σ), R3(ρ|σ), R4(ρ|σ) : ρ, σ ∈ [N]k−1} and every column of A corresponds
+to the coefficients of Sm(ρ|σ). Also observe that A is a finite dimensional matrix with entries in Z[q, z].
+One can use Gaussian elimination on A. Any non-trivial relation within the functional relations (R(i)
+1 (ρ|σ))-
+(R4(ρ|σ)) within the defining bounds of A would yield 0 rows. Let B be the matrix consisting of non-zero
+rows of A after the Gaussian elimination is performed. It should still be clear that
+B⃗s = ⃗0B.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+9
+Moreover, the ideal Im,N is generated by the equations that appear in B⃗s = ⃗0.
+Therefore, for any element of h ∈ Sm one can check whether that element is in Im,N by simply writing that
+relation as a row vector ⃗h (with respect to the vector ⃗s, i.e.
+vech is defined by h := [⃗h⃗s = 0]), add the row vector ⃗h to B and perform Gaussian elimination to this new
+matrix. If the Gaussian elimination yields a zero row, this means that ⃗h is a linear combination of rows in B,
+or equivalently this means h ∈ Im,N. If no zero row appears, then h ̸∈ Im,N.
+This approach is clearly algorithmic. Furthermore, termination of the algorithm and a definitive answer
+among the termination are both guaranteed. Top it all up, the explicit combination of (R(i)
+1 (ρ|σ))-(R4(ρ|σ))
+functional equations that is equivalent to a given h ∈ Sm is also easy to find. One only needs to use an
+augmented version of A where one more column is added to keep track of the name of the relations (R(i)
+1 (ρ|σ))-
+(R4(ρ|σ)) while doing the row reductions.
+After these considerations, proof of Conjectures 3.2 (and consequently Conjectures 2.4, 2.6, and 3.1) comes
+down to experimentally identifying an N and being able to perform the Gaussian elimination calculations.
+4. Modulo 11 Identities
+Let m = 11 (= 3k − 1 with k = 4), for this family of identities ρ and σ ∈ Z3. There are a total of 15
+essentially unique 3 part compositions of 8 that appear in the coupled q-difference system (2.6). Conjecture 2.4
+suggests that the following sum representations for Hc(z, q) hold for all but one of these:
+(4.1)
+H(8,0,0)(z, q)
+= S11((1, 1, 1) | (1, 1, 1)),
+H(7,1,0)(z, q)
+= S11((0, 1, 1) | (1, 1, 1)),
+H(7,0,1)(z, q)
+= S11((1, 1, 1) | (0, 1, 1)) − q(1 − z)S11((2, 1, 1) | (1, 1, 1)),
+H(6,2,0)(z, q)
+= S11((0, 0, 1) | (1, 1, 1)),
+H(6,1,1)(z, q)
+= S11((0, 1, 1) | (0, 1, 1)) − qS11((1, 1, 1) | (1, 1, 1)),
+H(6,0,2)(z, q)
+= S11((1, 1, 1) | (0, 0, 1)) − q(1 − z)S11((2, 1, 1) | (0, 1, 1)),
+H(5,3,0)(z, q)
+= S11((0, 0, 0) | (1, 1, 1)),
+H(5,2,1)(z, q)
+= S11((0, 0, 1) | (0, 1, 1)) − qS11((0, 1, 1) | (1, 1, 1)),
+H(5,1,2)(z, q)
+= S11((0, 1, 1) | (0, 0, 1)) − qS11((1, 1, 1) | (0, 1, 1)),
+H(5,0,3)(z, q)
+= S11((1, 1, 1) | (0, 0, 0)) − q(1 − z)S11((2, 1, 1) | (0, 0, 1)),
+H(4,3,1)(z, q)
+= S11((0, 0, 0) | (0, 1, 1)) − qS11((0, 0, 1) | (1, 1, 1)),
+H(4,2,2)(z, q)
+= S11((0, 0, 1) | (0, 0, 1)) − qS11((0, 1, 1) | (0, 1, 1)),
+H(4,1,3)(z, q)
+= S11((0, 1, 1) | (0, 0, 0)) − qS11((1, 1, 1) | (0, 0, 1)),
+H(3,3,2)(z, q)
+= S11((0, 0, 0) | (0, 0, 1)) − qS11((0, 0, 1) | (0, 1, 1)).
+Only H(4,4,0)(z, q) misses a claimed formula and that can be recovered by the q-difference equations (2.6).
+We know that H(4,4,0)(z, q) satisfies
+(4.2)
+H(4,4,0)(z, q) + (1 − qz)H(4,1,3)(q2z, q) − H(4,3,1)(qz, q) − H(5,0,3)(qz, q) = 0.
+Using the conjectured series equivalents (4.1) of H(4,1,3)(z, q), H(4,3,1)(z, q) and H(5,0,3)(z, q), we see that
+H(4,4,0)(z, q) = −(S11(qz; (0, 0, 0)|(0, 1, 1)) − qS11(qz; (0, 0, 1)|(1, 1, 1)))
++ (1 − qz)(S11(q2z; (0, 1, 1)|(0, 0, 0)) − qS11(q2z; (1, 1, 1)|(0, 0, 1)))
+(4.3)
+− (S11(qz; (1, 1, 1)|(0, 0, 0)) − q(1 − z)S11(qz(2, 1, 1)|(0, 0, 1))).
+Notice that we used the shifts in the variable z in (4.3). We clear these shifts by employing (2.12). This yields
+an explicit claim for H(4,4,0)(z, q):
+(4.4)
+H(4,4,0)(z, q) = S11((1, 0, 0) | (0, 1, 1)) − qS11((1, 0, 1) | (1, 1, 1)) + qzS11(2, 1, 1) | (0, 0, 0)),
+with no shifts in z, where the S11(ρ|σ) functions fit the forms in Lemma 2.5.
+We can also see that H(4,4,0)(z, q) satisfies the necessary initial conditions (2.14). The initial condition
+H(4,4,0)(z, 0) = 1 is immediate by (4.4) and (2.8). We can also see that H(4,4,0)(0, q) = 1/(q; q)∞ by plugging
+
+10
+ALI KEMAL UNCU
+in z = 0 in (4.2) and using the initial conditions of the other proven initial conditions (2.14) for the functions
+in (4.2).
+Our proof routine explained in Section 3 can start once all the normalized generating functions Hc(z, q)’s
+are (conjecturally) translated in the S11((a1, a2, a3)|(b1, b2, b3)) language. It is easy to see that the following
+four recurrences,
+H(8,0,0)(z, q) − H(7,1,0)(qz, q) = 0,
+H(7,0,1)(z, q) − H(6,1,1)(qz, q) + (1 − qz)H(7,1,0)(q2z, q) − H(8,0,0)(qz, q) = 0,
+H(6,0,2)(z, q) − H(5,1,2)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) − H(7,0,1)(qz, q) = 0,
+H(5,0,3)(z, q) − H(4,1,3)(qz, q) + (1 − qz)H(5,1,2)(q2z, q) − H(6,0,2)(qz, q) = 0,
+trivializes to 0 = 0 once the terms on the left-hand sides are written in S11((a1, a2, a3)|(b1, b2, b3)) using (4.1)
+and (2.12). Therefore, these relations are trivially in I11, the ideal generated by the functional relations of the
+S11((a1, a2, a3)|(b1, b2, b3)) series.
+Recall that we used the coupled q-difference equation (4.2) to make an explicit claim for H(4,4,0)(z, q).
+Hence, the functional relation of H(4,4,0)(z, q) also trivializes to 0 = 0 once written in the claimed S11(ρ|σ)
+forms. The very claim (4.4) is instrumental in proving that the q-difference equations satisfied by H(5,3,0)(z, q),
+H(4,3,1)(z, q), and H(4,1,3)(z, q) in S11((a1, a2, a3)|(b1, b2, b3)) language are elements of I11.
+Next, we look at the q-difference equation satisfied by H(7,1,0)(z, q) from (2.6):
+H(7,1,0)(z, q) − H(7,0,1)(qz, q) − H(6,2,0)(qz, q) + (1 − qz)H(6,1,1)(q2z, q) = 0.
+After the use of (4.1) and (2.12), we see that this q-difference equation is equivalent to the following conjectural
+form
+S11((0, 1, 1)|(1, 1, 1)) − S11((1, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 1, 1)) = 0.
+This is nothing but the relation R(1)
+1 (0, 1, 1)|(1, 1, 1) of (R(i)
+1 (ρ|σ)) given in Lemma 2.5. Hence, this relation is
+also within I11 and covered by the relations of S11((a1, a2, a3)|(b1, b2, b3)).
+As a second explicit example, consider the q-difference equation satisfied by H(6,1,1)(z, q),
+H(6,1,1)(z, q) − H(7,1,0)(qz, q) − H(6,0,2)(qz, q) − H(5,2,1)(qz, q) + (1 − qz)H(7,0,1)(q2z, q)
++ (1 − qz)H(6,2,0)(q2z, q) + (1 − qz)H(5,1,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,1,1)(q3z, q) = 0.
+Using employing (4.1) and (2.12), we see get the (conjecturally) equivalent form
+S11((0, 1, 1)|(0, 1, 1)) − S11((1, 0, 1)|(0, 1, 1)) − S11((1, 1, 1)|(1, 1, 1))
++ (1 − qz)S11((2, 0, 1)|(1, 1, 1)) − qzS11((2, 1, 1)|(0, 0, 1)) + q2z(1 − qz)S11((3, 1, 1)|(0, 1, 1)) = 0.
+This relation can be checked to be the side-by-side additions of
+R(1)
+1 ((0, 1, 1)|(0, 1, 1)) − (1 − qz)R(1)
+1 ((1, 1, 1)|(1, 1, 1)) + qzR(1)
+2 ((2, 1, 1)|(−1, 1, 1)) ∈ I11.
+We can one-by-one write down the remaining 8 recurrences, their S11(ρ|σ) equivalents, and what combina-
+tion of (R(i)
+1 (ρ|σ))-(R4(ρ|σ)) is equivalent to the functional equations in the S11(ρ|σ). This way we prove that
+these relations are all included in the ideal I11. We need to say that these relations gets messier, pages long and
+not hand-verifiable. Printing these would be a waste of page/paper and instead we keep these in the digital
+realm for interested readers to check it easily, or print on paper as they wish. To that end, similar to how it was
+handled in [29], we include text files M11RecHXYZ Explicit.txt in the ancillary files portion of ArXiv and on
+the author’s website [36]. Here XYZ is to be replaced by the relevant profile’s digits such as 620 for the profile
+(6, 2, 0). One can check that the elements of I11 given in these text files are equivalent to the q-difference
+equations (2.6) satisfied by H(X,Y,Z)(z, q) after they are translated to S11(ρ|σ) form using (4.1), (4.4) and
+(2.12). The functional equation names are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X
+and Y to be replaced by 1 or 2 to denote R(Y )
+X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for
+Z to be replaced by 3 or 4 to denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+11
+The definitions of these functional equations can be found in Lemma 2.5 for m = 11. A guide document that
+explicitly lists each R functional relation for modulo 10 is given in M11R text file. One also can see that the
+largest entry within ρ = (a1, a2, a3) and σ = (b1, b2, b3) of the relations (R(i)
+1 (ρ|σ))-(R4(ρ|σ)) for the modulo
+11 case given in the additional documents is 6. This proves the following theorem and its corollary.
+Theorem 4.1. Conjecture 3.2 is correct for m = 11 and N = 6.
+Corollary 4.2. Conjecture 3.1 is correct for m = 11.
+Corollary 4.2 is equivalent to the following theorem:
+Theorem 4.3. The claimed expressions (4.1) and (4.4) hold.
+Observe that Theorem 4.3 adds a new supporting case to Corollary 2.6.
+Now that the main conjectures are proven for the modulus 11 cases, we can specialize z = 1 and see the 15
+sum-product identities coming from the cylindric partitions paradigm.
+Theorem 4.4. The following identities hold
+�
+r1≥r2≥r3≥0
+s1≥s2≥s3≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2+r2
+3+r3s3+s2
+3 pc(r1, r2, r3, s1, s2, s3, q)
+(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1
+(4.5)
+=
+1
+(q; q)∞
+1
+θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7; q11),
+where the polynomials pc(r1, r2, r3, s1, s2, s3, q) and the 7-tuples (i1, i2, i3, i4, i5, i6, i7) for each profile is given
+in the following table:
+Profile c
+pc(r1, r2, r3, s1, s2, s3, q)
+(i1, i2, i3, i4, i5, i6, i7)
+(8, 0, 0)
+qr1+r2+r3+s1+s2+s3
+(2, 3, 3, 4, 4, 5, 5)
+(7, 1, 0)
+qr2+r3+s1+s2+s3
+(1, 2, 3, 4, 4, 5, 5)
+(7, 0, 1)
+qr1+r2+r3+s2+s3
+(1, 2, 3, 4, 4, 5, 5)
+(6, 2, 0)
+qr3+s1+s2+s3
+(1, 2, 2, 3, 4, 5, 5)
+(6, 1, 1)
+qr2+r3+s2+s3(1 − qr1+s1+1)
+(1, 1, 3, 3, 4, 5, 5)
+(6, 0, 2)
+qr1+r2+r3+s3
+(1, 2, 2, 3, 4, 5, 5)
+(5, 3, 0)
+qs1+s2+s3
+(1, 2, 2, 3, 3, 4, 5)
+(5, 2, 1)
+qr3+s2+s3(1 − qr2+s1+1)
+(1, 1, 2, 3, 4, 4, 5)
+(5, 1, 2)
+qr2+r3+s3(1 − qr1+s2+1)
+(1, 1, 2, 3, 4, 4, 5)
+(5, 0, 3)
+qr1+r2+r3
+(1, 2, 2, 3, 3, 4, 5)
+(4, 3, 1)
+qs2+s3(1 − qr3+s1+1)
+(1, 1, 2, 3, 3, 4, 5)
+(4, 2, 2)
+qr3+s3(1 − qr2+s2+1)
+(1, 1, 2, 2, 4, 4, 5)
+(4, 1, 3)
+qr2+r3(1 − qr1+s3+1)
+(1, 1, 2, 3, 3, 4, 5)
+(3, 3, 2)
+qs3(1 − qr3+s2+1)
+(1, 1, 2, 2, 3, 5, 5)
+(4, 4, 0)
+qr1(qs2+s3 − qr3+s1+s2+s3+1 + qr1+r2+r3+1)
+(1, 2, 2, 3, 3, 4, 4)
+In Theorem 4.4, we chose to put the profile (4, 4, 0) related sum-product identity under a line in the table.
+This is to indicate that this identity is not a direct claim made by combining (2.4) with z = 1 and (2.1). We
+first recovered a formula for H(4,4,0)(z, q) as a combination of S11((a1, a2, a3)|(b1, b2, b3)) series and then made
+this claim. This line also has the added benefit that it aligns us with Kanade–Russell’s language as this is the
+sum-product identity related to the under-the-line Hc(z, q) function, which we chose not to directly define.
+The sum sides are the expressions (4.1) and (4.4) with z = 1 written explicitly using (2.8) with k = 4. The
+product sides follow from (2.5) with z = 1 followed by (2.1). The product related to the first profile, (8, 0, 0),
+on the table is presented in the introduction as Theorem 1.8.
+Observe that the products that appear on the right-hand side of (4.5) related to the profiles (c1, c2, c3) and
+(c1, c3, c2) are the same. The symmetry for the generating functions have been observed and noted before, for
+example in [19, Corollary 2.2]. This symmetry is visible on the sum side of Theorem 4.4 too. One can get
+
+12
+ALI KEMAL UNCU
+the “other” sum by merely replacing the variable ‘r’s and ‘s’s. Note that this is a byproduct of setting z = 1
+and this similarity does not exist on the sum side for generic z. In that light, this theorem consisting of 15
+sum-product identities actually provide a total of 10 essentially unique sum-product identities.
+We also note that among these identities the ones related to profiles (8, 0, 0), (6, 1, 1), (4, 2, 2), (3, 3, 2) are
+the i = 1, . . . , 4 cases of (5.28), respectively, and (5, 3, 0) and (6, 2, 0) are the σ = 0 and 1 cases of (5.29),
+respectively, of [7, Theorem 5.3].
+5. Modulo 13 Identities
+Similar to Section 4, we start by listing the explicit claims of Conjecture 2.4 for the modulus m = 13 family.
+(5.1)
+H(10,0,0)(z, q)
+= S13((1, 1, 1)|(1, 1, 1)),
+H(9,1,0)(z, q)
+= S13((0, 1, 1)|(1, 1, 1)),
+H(9,0,1)(z, q)
+= S13((1, 1, 1)|(0, 1, 1)) − q(1 − z)S13((2, 1, 1)|(1, 1, 1)),
+H(8,2,0)(z, q)
+= S13((0, 0, 1)|(1, 1, 1)),
+H(8,1,1)(z, q)
+= S13((0, 1, 1)|(0, 1, 1)) − qS13((1, 1, 1)|(1, 1, 1)),
+H(8,0,2)(z, q)
+= S13((1, 1, 1)|(0, 0, 1)) − q(1 − z)S13((2, 1, 1)|(0, 1, 1)),
+H(7,3,0)(z, q)
+= S13((0, 0, 0)|(1, 1, 1)),
+H(7,2,1)(z, q)
+= S13((0, 0, 1)|(0, 1, 1)) − qS13((0, 1, 1)|(1, 1, 1)),
+H(7,1,2)(z, q)
+= S13((0, 1, 1)|(0, 0, 1)) − qS13((1, 1, 1)|(0, 1, 1)),
+H(7,0,3)(z, q)
+= S13((1, 1, 1)|(0, 0, 0)) − q(1 − z)S13((2, 1, 1)|(0, 0, 1)),
+H(6,3,1)(z, q)
+= S13((0, 0, 0)|(0, 1, 1)) − qS13((0, 0, 1)|(1, 1, 1)),
+H(6,2,2)(z, q)
+= S13((0, 0, 1)|(0, 0, 1)) − qS13((0, 1, 1)|(0, 1, 1)),
+H(6,1,3)(z, q)
+= S13((0, 1, 1)|(0, 0, 0)) − qS13((1, 1, 1)|(0, 0, 1)),
+H(5,3,2)(z, q)
+= S13((0, 0, 0)|(0, 0, 1)) − qS13((0, 0, 1)|(0, 1, 1)),
+H(5,2,3)(z, q)
+= S13((0, 0, 1)|(0, 0, 0)) − qS13((0, 1, 1)|(0, 0, 1)),
+H(4,3,3)(z, q)
+= S13((0, 0, 0)|(0, 0, 0)) − qS13((0, 0, 1)|(0, 0, 1)).
+There are six profiles that are not covered by Conjecture 2.4. Once again, using (2.6) explicit claims for
+the normalized generating functions related to the number of cylindric partitions with these profiles can be
+recovered. We make the claims in the following succession.
+First we look at the q-difference equation (2.6) that H(7,3,0):
+(5.2)
+H(7,3,0)(z, q) − H(7,2,1)(qz, q) − H(6,4,0)(qz, q) + (1 − qz)H(6,3,1)(q2z, q) = 0.
+By writing the S13((a1, a2, a3)|(b1, b2, b3)) equivalents for the functions in (5.1) and using (2.12), we get
+H(6,4,0)(z, q) = S13((−1, 0, 0)|(1, 1, 1)) − S13((0, 0, 1)|(0, 1, 1)) + qS13((0, 1, 1)|(1, 1, 1))
+(5.3)
++ (1 − z)S13((1, 0, 0)|(0, 1, 1)) − q(1 − z)S13((1, 0, 1)|(1, 1, 1)).
+Note that we did not use the q-difference equation of H(6,4,0)(z, q) to make a claim for its formula. In
+Section 4, there was only a single missing formula. That allowed us to use the q-difference equation for that
+very function and get a formula in S11((a1, a2, a3)|(b1, b2, b3))’s with no backwards shifts (i.e. z �→ z/q, which
+also reflects as negative indices in the first variable a1). This may not be possible in general. The q-difference
+equation H(6,4,0)(z, q) satisfies is
+(5.4)
+H(6,4,0)(z, q) − H(6,3,1)(qz, q) − H(5,5,0)(qz, q) + (1 − qz)H(5,4,1)(q2z, q) = 0.
+The conjectural formulas (5.1) does not cover H(5,5,0)(z, q). Hence, we cannot fully translate H(6,4,0)(z, q) to
+a formula made up of S13((a1, a2, a3)|(b1, b2, b3)) series. Nevertheless, as also noted in [29], we can recover
+formulas for all the missing functions using other recurrences and backwards shifts in a1.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+13
+In fact, the recurrence (5.4) and (5.3) can be put together to claim a formula for H(5,5,0)(z, q). After the
+similar considerations we claim
+H(5,5,0)(z, q) = S13((−2, 0, 0)|(1, 1, 1)) − S13((−1, 0, 1)|(0, 1, 1)) + qS13((−1, 1, 1)|(1, 1, 1))
+(5.5)
++ (1 − z/q − z)S13((0, 0, 0)|(0, 1, 1)) − (q − z − qz)S13((0, 0, 1)|(1, 1, 1))
+− q(1 − z)zS13((1, 0, 0)|(1, 1, 1)) − (1 − z)S13((1, 0, 1)|(0, 0, 1))
++ q(1 − z)S13((1, 1, 1)|(0, 1, 1)) + (1 − z)(1 − qz)S13((2, 0, 0)|(0, 0, 1))
+− q(1 − z)(1 − qz)S13((2, 0, 1)|(0, 1, 1)).
+We point out that the coefficients of the claimed H(5,5,0)(z, q) formula now can be seen to have a Laurent
+polynomial. This is a byproduct of the backwards shifts in z.
+Using the q-difference equation for H(6,3,1)(z, q),
+H(6,3,1)(z, q) − H(7,3,0)(qz, q) − H(6,2,2)(qz, q) − H(5,4,1)(qz, q) + (1 − qz)H(7,2,1)(q2z, q)
+(5.6)
++ (1 − qz)H(6,4,0)(q2z, q) + (1 − qz)H(5,3,2)(q2z, q) − (1 − qz)(1 − q2z)H(6,3,1)(q3z, q) = 0,
+(5.1) and (5.3) we claim that
+H(5,4,1)(z, q) = S13((−1, 0, 0)|(0, 1, 1)) − qS13((−1, 0, 1)|(1, 1, 1)) − zS13((0, 0, 0)|(1, 1, 1))
+(5.7)
+− S13((0, 0, 1)|(0, 0, 1)) + qS13((0, 1, 1)|(0, 1, 1)) + (1 − z)S13((1, 0, 0)|(0, 0, 1))
+− q(1 − z)S13((1, 0, 1)|(0, 1, 1)).
+Using the q-difference equation for H(5,3,2)(z, q),
+H(5,3,2)(z, q) − H(6,3,1)(qz, q) − H(5,2,3)(qz, q) − H(4,4,2)(qz, q) + (1 − qz)H(6,2,2)(q2z, q)
+(5.8)
++ (1 − qz)H(5,4,1)(q2z, q) + (1 − qz)H(4,3,3)(q2z, q) − (1 − qz)(1 − q2z)H(5,3,2)(q3z, q) = 0,
+(5.1) and (5.7) we claim that
+H(4,4,2)(z, q) = S13((−1, 0, 0)|(0, 0, 1)) − qS13((−1, 0, 1)|(0, 1, 1)) − zS13((0, 0, 0)|(0, 1, 1))
+(5.9)
+− S13((0, 0, 1)|(0, 0, 0)) + qzS13((0, 0, 1)|(1, 1, 1)) + qS13((0, 1, 1)|(0, 0, 1))
++ (1 − z)S13((1, 0, 0)|(0, 0, 0)) − qz(1 − z)S13((1, 0, 0)|(1, 1, 1))
+− q(1 − z)S13((1, 0, 1)|(0, 0, 1)).
+Then, by the q-difference equation for H(5,2,3)(z, q),
+H(5,2,3)(z, q) − H(6,2,2)(qz, q) − H(5,1,4)(qz, q) − H(4,3,3)(qz, q) + (1 − qz)H(6,1,3)(q2z, q)
+(5.10)
++ (1 − qz)H(5,3,2)(q2z, q) + (1 − qz)H(4,4,2)(q2z, q) − (1 − qz)(1 − q2z)H(5,2,3)(q3z, q) = 0,
+together with (5.1) and (5.9) we claim that
+H(5,1,4)(z, q) = S13((−1, 0, 1)|(0, 0, 0)) − qS13((−1, 1, 1)|(0, 0, 1)) − S13((0, 0, 0)|(0, 0, 0))
+(5.11)
++ (1 − z)S13((0, 0, 0)|(0, 0, 1)) − (1 − q)S13((0, 0, 1)|(0, 0, 1))
+− q(1 − z)S13((0, 0, 1)|(0, 1, 1)) + qS13((0, 1, 1)|(0, 1, 1))
++ (1 − z)S13((1, 0, 0)|(0, 0, 1)) − q(1 − z)zS13((1, 0, 0)|(0, 1, 1))
+− (1 − z)S13((1, 0, 1)|(0, 0, 0)) − q(1 − z)S13((1, 0, 1)|(0, 1, 1))
++ q2(1 − z)zS13((1, 0, 1)|(1, 1, 1)) + (1 − z)S13((1, 1, 1)|(0, 0, 0))
++ q(1 − z)S13((1, 1, 1)|(0, 0, 1)) + (1 − z)(1 − qz)S13((2, 0, 0)|(0, 0, 0))
+− q2z(1 − z)(1 − qz)S13((2, 0, 0)|(1, 1, 1)) − (1 − z)(1 − qz)S13((2, 0, 1)|(0, 0, 0))
+− q(1 − z)(1 − qz)S13((2, 0, 1)|(0, 0, 1)) − q2z(1 − z)S13((2, 1, 1)|(0, 0, 1)).
+
+14
+ALI KEMAL UNCU
+Finally, by replacing the formulas in (5.1) and (5.11) in
+(5.12)
+H(6,0,4)(z, q) − H(7,0,3)(qz, q) − H(5,1,4)(qz, q) + (1 − qz)H(6,1,3)(q2z, q) = 0
+we conjecture that
+H(6,0,4)(z, q) = S13((0, 0, 1)|(0, 0, 0)) − qS13((0, 1, 1)|(0, 0, 1)) − S13((1, 0, 0)|(0, 0, 0))
+(5.13)
++ (1 − qz)S13((1, 0, 0)|(0, 0, 1)) − (1 − q)S13((1, 0, 1)|(0, 0, 1))
+− q(1 − qz)S13((1, 0, 1)|(0, 1, 1)) + qS13((1, 1, 1)|(0, 1, 1))
++ (1 − qz)S13((2, 0, 0)|(0, 0, 1)) − q2z(1 − qz)S13((2, 0, 0)|(0, 1, 1))
+− (1 − qz)S13((2, 0, 1)|(0, 0, 0)) − q(1 − qz)S13((2, 0, 1)|(0, 1, 1))
++ q3z(1 − qz)S13((2, 0, 1)|(1, 1, 1)) + S13((2, 1, 1)|(0, 0, 0))
++ q(1 − qz)S13((2, 1, 1)|(0, 0, 1)) + (1 − qz)(1 − q2z)S13((3, 0, 0)|(0, 0, 0))
+− q3z(1 − qz)(1 − q2z)S13((3, 0, 0)|(1, 1, 1)) − (1 − qz)(1 − q2z)S13((3, 0, 1)|(0, 0, 0))
+− q(1 − qz)(1 − q2z)S13((3, 0, 1)|(0, 0, 1)) − q3z(1 − qz)S13((3, 1, 1)|(0, 0, 1)).
+We can prove that the later claimed H(6,4,0)(z, q), H(5,5,0)(z, q), H(5,4,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q),
+and H(6,0,4)(z, q) the initial conditions Hc(0, q) = 1/(q; q)∞ and Hc(z, 0) = 1 in the succession from (5.2),
+(5.4), (5.6), (5.8), (5.10), and (5.12), respectively. To prove the Hc(0, q) = 1/(q; q)∞ initial condition we need
+to first shift z �→ z/q in all but the last of the functional equations.
+The q-difference equations for H(10,0,0)(z, q), H(9,0,1)(z, q), H(8,0,2)(z, q), and H(7,0,3)(z, q) becomes tau-
+tologies once translated into S13 form using (5.1) and (2.12). The q-difference equations for H(7,3,0)(z, q),
+H(6,4,0)(z, q), H(6,3,1)(z, q), H(5,3,2)(z, q), H(5,2,3)(z, q), and H(6,0,4)(z, q) are the recurrences used to define the
+missing Hc(z, q) functions in the modulo 13 family (see (5.2), (5.4), (5.6), (5.8), (5.10), and (5.12), resp.).
+Hence, these equations also trivializes once the relevant functions are written in their claimed S13 forms using
+(5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) together with (2.12).
+After the considerations above, we end up with 10 non-trivial coupled q-difference equations to prove.
+Showing that the q-difference equations’ in the claimed S13((a1, a2, a3)|(b1, b2, b3)) belong to the ideal I13,
+which is generated by the relations of S13((a1, a2, a3)|(b1, b2, b3))s (see Lemma 2.5), is done by the method
+outlined in Section 3. Explicit linear combination of (R(i)
+1 (ρ|σ))-(R4(ρ|σ)) equivalents of these 12 functional
+equations in S13 form can, once again, be found in the ancillary files portion of ArXiv and on the author’s
+website [36] under the file names M13RecHXYZ Explicit.txt. Here XYZ is to be replaced by the relevant
+profile’s digits such as 910 for the profile (9, 1, 0). One can check that the elements of I13 given in these
+text files are equivalent to the q-difference equations (2.6) satisfied by H(X,Y,Z)(z, q) after they are translated
+to S11(ρ|σ) form using (5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) and (2.12). The recurrence names
+are reflected in the text as RX[{Y},{{a1,a2,a3},{b1,b2,b3}}] for X and Y to be replaced by 1 or 2 to
+denote R(Y )
+X ((a1, a2, a3)|(b1, b2, b3)), or RZ[{{a1,a2,a3},{b1,b2,b3}}] for Z to be replaced by 3 or 4 to
+denote R3((a1, a2, a3)|(b1, b2, b3)) and R4((a1, a2, a3)|(b1, b2, b3)), respectively. Finally, a guide document that
+explicitly lists each R functional relation for modulo 10 is given in M13R text file.
+This tedious, error prone and impossible by hand calculation proves the following theorem and its corollary.
+Theorem 5.1. Conjecture 3.2 is correct for m = 13 and N = 6.
+Corollary 5.2. Conjecture 3.1 is correct for m = 13.
+Corollary 5.2 is equivalent to the following theorem:
+Theorem 5.3. The claimed expressions of (5.1), (5.3), (5.5), (5.7), (5.9), (5.11), and (5.13) hold.
+As before, Theorem 5.3 adds another new witness to Corollary 2.6, and increases our confidence in it.
+Now that the main conjectures are proven for the modulus 13 cases, we can set z = 1 and see the 22
+sum-product identities coming from the cylindric partitions paradigm.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+15
+Theorem 5.4. The following identities hold
+�
+r1≥r2≥r3≥0
+s1≥s2≥s3≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2+r2
+3−r3s3+s2
+3 pc(r1, r2, r3, s1, s2, s3, q)
+(q; q)r1−r2(q; q)r2−r3(q; q)r3(q; q)s1−s2(q; q)s2−s3(q; q)s3(q; q)r3+s3+1
+(5.14)
+=
+1
+(q; q)∞
+1
+θ(qi1, qi2, qi3, qi4, qi5, qi6, qi7, qi8, qi9; q13),
+where the polynomials pc(r1, r2, r3, s1, s2, s3, q) and the 9-tuples (i1, i2, i3, i4, i5, i6, i7, i8, i9) for each profile is
+given in the following table:
+Profile c
+pc(r1, r2, r3, s1, s2, s3, q)
+(i1, i2, i3, i4, i5, i6, i7, i8, i9)
+(10, 0, 0)
+qr1+r2+r3+s1+s2+s3
+(2, 3, 3, 4, 4, 5, 5, 6, 6)
+(9, 1, 0)
+qr2+r3+s1+s2+s3
+(1, 2, 3, 4, 4, 5, 5, 6, 6)
+(9, 0, 1)
+qr1+r2+r3+s2+s3
+(1, 2, 3, 4, 4, 5, 5, 6, 6)
+(8, 2, 0)
+qr3+s1+s2+s3
+(1, 2, 2, 3, 4, 5, 5, 6, 6)
+(8, 1, 1)
+qr2+r3+s2+s3(1 − qr1+s1+1)
+(1, 1, 3, 3, 4, 5, 5, 6, 6)
+(8, 0, 2)
+qr1+r2+r3+s3
+(1, 2, 2, 3, 4, 5, 5, 6, 6)
+(7, 3, 0)
+qs1+s2+s3
+(1, 2, 2, 3, 3, 4, 5, 6, 6)
+(7, 2, 1)
+qr3+s2+s3(1 − qr2+s1+1)
+(1, 1, 2, 3, 4, 4, 5, 6, 6)
+(7, 1, 2)
+qr2+r3+s3(1 − qr1+s2+1)
+(1, 1, 2, 3, 4, 4, 5, 6, 6)
+(7, 0, 3)
+qr1+r2+r3
+(1, 2, 2, 3, 3, 4, 5, 6, 6)
+(6, 3, 1)
+qs2+s3(1 − qr3+s1+1)
+(1, 1, 2, 3, 3, 4, 5, 5, 6)
+(6, 2, 2)
+qr3+s3(1 − qr2+s2+1)
+(1, 1, 2, 2, 4, 4, 5, 5, 6)
+(6, 1, 3)
+qr2+r3(1 − qr1+s3+1)
+(1, 1, 2, 3, 3, 4, 5, 5, 6)
+(5, 3, 2)
+qs3(1 − qr3+s2+1)
+(1, 1, 2, 2, 3, 4, 5, 5, 6)
+(5, 2, 3)
+qs3(1 − qr3+s2+1)
+(1, 1, 2, 2, 3, 4, 5, 5, 6)
+(4, 3, 3)
+(1 − qr3+s3+1)
+(1, 1, 2, 2, 3, 3, 5, 6, 6)
+(6, 4, 0)
+qs2+s3(q−r1+s1 − qr3 + qr2+r3+s1+1)
+(1, 2, 2, 3, 3, 4, 4, 5, 6)
+(6, 0, 4)
+qr3 − qr2+r3+s3+1 − qr1 + q2r1+r2+r3 + qr1+r2+r3+s2+s3+1
++(1 − q)qr1+s3(1 − qr3 − qr3+s3+1)
+−(1 − q)q2r1(qr3 − qs3 − qr2+r3+s3+1 + qr1+r2+r3+s3+3
++qs2+s3+2 + qr3+s2+s3+1 − qr3+s1+s2+s3+3)
++(1 − q)(1 − q2)qr3(1 − qr3 − qr3+s3+1 + qs1+s2+s3+3)
+(1, 2, 2, 3, 3, 4, 4, 5, 6)
+(5, 5, 0)
+q−2r1+s1+s2+s3 − q−r1+r3+s2+s3 − qs2+s3−1 + qr3+s1+s2+s3
++q−r1+r2+r3+s1+s2+s3+1
+(1, 2, 2, 3, 3, 4, 4, 5, 5)
+(5, 4, 1)
+q−r1+s2+s3 − q−r1+r3+s1+s2+s3+1 − qs1+s2+s3 − qr3+s3
++qr2+r3+s2+s3+1
+(1, 1, 2, 3, 3, 4, 4, 5, 6)
+(5, 1, 4)
+q−r1+r3 − q−r1+r2+r3+s3+1 + qr2+r3+s2+s3+1 − (1 − q)qr3+s3 − 1
+(1, 1, 2, 3, 3, 4, 4, 5, 6)
+(4, 4, 2)
+q−r1+s3 − q−r1+r3+s2+s3+1 − qs2+s3 − qr3 + qr3+s1+s2+s3+1
++qr2+r3+s3+1
+(1, 1, 2, 2, 3, 4, 4, 6, 6)
+Once we ignore the symmetries between variables r and s, Theorem 5.4 proves 16 essentially unique sum-
+product identities. It can easily be seen that within the under-the-line identities, we do not see these sym-
+metries. The product related to the first profile, (10, 0, 0), on the table is presented in the introduction as
+Theorem 1.9.
+We also note that among these identities the ones related to profiles (10, 0, 0), (8, 1, 1), (6, 2, 2), (4, 3, 3) are
+the i = 1, . . . , 4 cases of (5.22), respectively, and (7, 3, 0) and (8, 2, 0) are the σ = 0 and 1 cases of (5.23),
+respectively, of [7, Theorem 5.1].
+
+16
+ALI KEMAL UNCU
+6. Future Directions
+There are many mathematical questions that arose from the recent studies on cylindric partitions. It is
+relevant to mention some of the future directions we plan to pursue.
+The approach outlined in [29] and in this paper attempts to prove sum-representations for all the normalized
+generating function Hc(z, q) in one stroke for any fixed |c| where #(c) = 3. The proof requires hefty calculations
+after the under-the-line sums are recovered. Then by setting z = 1 and using (2.1), we prove sum-product
+identities for all profiles within a cylindric partition system for a fixed modulus, again in one stroke. Therefore,
+to prove A2 Rogers–Ramanujan identities we first prove a more general and more complicated combinatorial
+connection with a free variable z. The success of this method depends on the completion of these calculations,
+which is virtually impossible by hand.
+Warnaar [43] mentioned that he build the necessary theory of the Bailey machinery for profiles with 3
+parts. This machinery will allow us to prove one sum-product identity at a time. This is wonderful to hear
+and a great advancement in mathematics. Sadly, it comes with its own short-comings. Warnaar acknowledged
+that this Bailey machinery can not prove any under-the-line identity at the moment. It can only find the
+sum-product relation related to the z = 1 specializations of Conjecture 2.13. This is similar to the situation of
+the original Andrews–Schilling–Warnaar paper, where for example at the modulo 7 case the Bailey machinery
+there couldn’t reach the under-the-line identity related to the profile (2, 2, 0), which was later proven in [18].
+Be that as it may, we plan to investigate ways to simplify calculations necessary to prove the identities as
+a whole in one stroke for the free z case by adding the extra information we gather from Warnaar’s results.
+At the very least, for the z = 1 specialization, we should pursue ways to prove under-the-line identities using
+the Bailey-machinery-proven over-the-line identities.
+There are other sum-product identities that are not visible through the cylindric partitions paradigm.These
+identities do not have a related cylindric partition profiles attached to them either. Similar to the under-the-
+line identities, we discover and prove these sum representations using the proven relations in the cylindric
+partitions system. For example, there are the following two modulo 10 examples similar to (1.4):
+�
+r1≥r2≥0
+s1≥s2≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2 qs1+s2(1 + qr1+r2+1)
+(q; q)r1−r2(q; q)s1−s2(q; q)r2(q; q)s2(q; q)r2+s2+1
+=
+1
+(q; q)∞
+1
+θ(q, q, q3, q4, q4, q4; q10),
+(6.1)
+�
+r1≥r2≥0
+s1≥s2≥0
+qr2
+1−r1s1+s2
+1+r2
+2−r2s2+s2
+2 qs1+s2(1 − qr1+r2+1)
+(q; q)r1−r2(q; q)s1−s2(q; q)r2(q; q)s2(q; q)r2+s2+1
+=
+1
+(q; q)∞
+1
+θ(q2, q2, q2, q3, q3, q3; q10).
+(6.2)
+All the products associated to principal characters of modulo 10 A2 Rogers–Ramanujan identities are covered
+by the products that appear in (2.1). The identities (6.1) and (6.2) are outside of this system and appear, so
+to say, on the dark-side of the cylinder. We hope to find a cylindric partition interpretation of these identities
+in the future. Nevertheless, we plan to present the proofs of these theorems using q-theoretic means in an
+upcoming paper.
+It is still highly relevant to find manifestly positive sum representations for any one of the identities men-
+tioned here. We are looking for ways to see the positivity of the series coefficients. In [7], Andrews–Schilling–
+Warnaar suggests applying hypergeometric transformations to eliminate the (q; q)∞ factor that appear in the
+identities (such as (1.4)) to get a manifestly positive representation. That suggestion is limited and might not
+be widely applicable, especially for the under-the-line identities.
+In the study of symmetric cylindric partitions [12] another two fundamental modulo 8 partition theoretic
+identity families, namely G¨ollnitz–Gordon and little G¨ollnitz identities, showed up.
+The G¨ollnitz–Gordon
+identities are known to be related to the level 2 modules of affine Lie algebra A(2)
+5
+[27]. This raises new
+questions of whether, similar to the symmetric partitions paradigm, we can also relate symmetric cylindric
+partitions to character formulas of some affine Lie algebras. The product formula analogous to (2.1) for the
+count of symmetric cylindric partitions’ is present in [12]. At the moment, the relation of these products’ to
+affine Lie algebra character formulas are fuzzy, and there are no general conjectural series representations for
+symmetric cylindric partitions either. We plan to study these objects further.
+
+MOD 11 AND 13 A2 ROGERS–RAMANUJAN TYPE IDENTITIES
+17
+Finally, we plan to pursue sum representations of any generating functions for cylindric partitions with
+profiles of more than 3 parts.
+The product representation (2.1) and the functional equations (2.3) apply
+regardless of the size and length of the profiles.
+So far, we are only able to prove and conjecture sum
+representations for the profiles with up to 3 parts.
+7. Comments on Computations
+In the computerized proofs of [19], we make extensive use of [1] and [30]. Those proofs had three main
+steps. Finding a recurrence relation (over the exponent of z) for claimed sum formulas of the (normalized)
+generating functions of cylindric partitions, uncoupling the q-difference equation system laid out by the (2.3)
+to get a recurrence satisfied by the coefficient of the z’s in the true generating functions of cylindric partitions,
+comparing recurrences (taking greatest common divisors of recurrences as operators if needed) and showing
+that both sequences satisfy the same recurrences with the same initial conditions. Once the critical mass
+of proved identities were reached the rest of the identities were shown by series manipulations guided by
+(2.3). That way we showed that all the claimed sum and the true combinatorial generating function were the
+same. This proof required two hefty algorithms, namely Creative Telescoping algorithm and Gr¨obner bases
+calculations, to find the recurrence of a given hypergeometric sum dependent of a discrete variable and to
+uncouple a coupled system of recurrences, respectively.
+We tried using the same method to prove some claims Warnaar [42] made for cylindric partitions with 3
+part profiles where the modulus is not divisible by 3. Then we quickly saw that the Creative Telescoping
+calculations were not terminating (in any definition of reasonable time). This is due to the increasing number
+of nested summations in these conjectures. However, uncoupling of recurrences could still be performed.
+Kanade–Russell’s approach [29] to prove that the claimed series representations for the bivariate generating
+functions of cylindric partitions are the true generating functions is a fresh take on things. It is somehow
+backwards compared to the proofs of [19], in the sense that we first extend our conjectural identities using the
+explicit conjectures of Conjecture 2.13 and series manipulations, then prove all these conjectural identities by
+showing that the coupled relations are satisfied and that we still satisfy the initial conditions. The key idea
+of reducing coupled q-difference equation with the functional relations of the claimed hypergeometric sums
+was also used in [16] in a different context. Moreover, this approach replaces (the old bottle-neck) Creative
+Telescoping with the contiguous relations of Lemma 2.5. However, rewriting the coupled relations of (2.3) in
+the new language as a linear combination of terms in the ideal Im (see Section 3) with coefficients in Z((q, z)) is
+highly non-trivial. Kanade [26] mentioned that they found these linear combinations by first making an ansatz
+for a single case at a time and then solving for undetermined coefficients. The identification of the minimal
+necessary ansatz is impossible. They also mentioned that each hard-case proof of modulo 10 calculations took
+about 8 hours to terminate on a home computer. With the matrix reduction approach of this paper, we are
+order of 2 faster in the modulo 10 cases. This is basically because once we reduce a matrix, we can use it
+repeadetly for all the functional relations, whereas the previous approach needs to make a single ansatz and
+solve if for all cases individually. It is with this speed upgrade that we could prove the new modulo 11 and
+modulo 13 cases. On the other hand, modulo 9 and modulo 12 cases are still open. This is likely due to the
+extra degree of complication the q-binomial coefficients in (2.9)’s introduce. As the order of the recurrences
+the Sm(ρ|σ) satisfy increases, the systems we need to reduce also become larger.
+Mathematica’s Gaussian elimination function RowReduce is adamant in calculating the reduced row echelon
+form of matrices.
+This is not only not necessary, it also overcomplicates the calculations by introducing
+large rational function expressions for upper triangular coefficients. This forced us to implementing our own
+Gaussian elimination algorithm within the Mathematica computer algebra system. This basic implementation
+sorts, performs row elimination of a matrix with entries in a polynomial ring with integer coefficients, such as
+Z[q, z], while not introducing rational functions, and it terminates when a row echelon matrix (a triangular
+system of equations) is reached. This function will be made a part of the impending next version release of
+qFunctions package. As a side note, we implemented a naive parallelization of this elimination but we have
+not seen any benefits of splitting calculations yet.
+We should also acknowledge that there are at least two crucial optimizations waiting to be implemented to
+aid proos of families in cylindric partitions scheme and other similar schemes. First task that should be done
+
+18
+ALI KEMAL UNCU
+is to keep track of nullified relations and to remove the contributions of the nullspace in later calculations. To
+put it in concrete terms, at the moment we do not know if N = 6 is the minimal number to prove Theorems 4.1
+and/or 5.1. We know that it is a sufficient number. By removing any and all nullified relations we would only
+see a minimal representation (dependent on the choice of N) of these recurrences as elements in the ideals Im,
+and that can give us an idea of what the optimal bound for N is supposed to be in general. The second pending
+addition is dynamic extension of the matrix to be reduced. At the moment, we fix an N experimentally hoping
+that it is enough to show that the relations of interest are in the nullspace of this matrix. This is in the same
+spirit of making a fixed ansatz. Row reduction as a preprocessing step helps for the repeated calculations.
+Having an echelon system boosts the speed of later calculations immensely. If the chosen N is not enough,
+then we need to pick a larger N and start all over. This requires performing row reduction of the matrix for
+N once more as a subproblem. This should be changed by extending the already triangularized matrix for N
+to N + 1 and doing the row reduction again for only the added relations. The incrementality of the matrix
+would also carry us to the minimal necessary N for any given m (assuming that Conjecture 3.2 is correct)
+naturally.
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+Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science, Altenberg-
+erstraße 69, A-4040 Linz, Austria
+Email address: akuncu@ricam.oeaw.ac.at
+University of Bath, Faculty of Science, Department of Computer Science, Bath, BA2 7AY, UK
+Email address: aku21@bath.ac.uk
+