Topological Schrödinger Bridge Matching
Authors: Maosheng Yang
Venue: ICLR 2025 Spotlight (2025)
Keywords: Schrödinger Bridge
Topological Signal Distribution Matching
Topological Stochastic Dynamics
Topological Generative Models
Primary area: generative models
TL;DR: We investigate Schrödinger bridge based models for topological signal matching.

Abstract: Given two boundary distributions, the \emph{Schrödinger Bridge} (SB) problem seeks the “most likely” random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for generative modeling and distribution matching. While these methods perform well in Euclidean domains, they are not directly applicable to topological domains such as graphs and simplicial complexes, which are crucial for data defined over network entities, such as node signals and edge flows. In this work, we propose the \emph{Topological Schrödinger Bridge problem} ($\mathcal{T}$SBP) for matching signal distributions on a topological domain. We set the reference process to follow some linear tractable \emph{topology-aware} stochastic dynamics such as topological heat diffusion. For the case of Gaussian boundary distributions, we derive a \emph{closed-form} topological SB ($\mathcal{T}$SB) in terms of its time-marginal and stochastic differential. In the general case, leveraging the well-known result, we show that the optimal process follows the forward-backward topological dynamics governed by some unknowns. Building on these results, we develop $\mathcal{T}$SB-based models for matching topological signals by parameterizing the unknowns in the optimal process as \emph{(topological) neural networks} and learning them through \emph{likelihood training}. We validate the theoretical results and demonstrate the practical applications of $\mathcal{T}$SB-based models on both synthetic and real-world networks, emphasizing the role of topology. Additionally, we discuss the connections of $\mathcal{T}$SB-based models to other emerging models, and outline future directions for topological signal matching.

Published as a conference paper at ICLR 2025
TOPOLOGICAL SCHRÖDINGER BRIDGE MATCHING
Maosheng Yang
Delft University of Technology m. wan.yynnggggge 20d--delttttt.nnmm
ABSTRACT
Given two boundary distributions, the Schrödinger Bridge (SB) problem seeks the "most likely" random evolution between them with respect to a reference process. It has revealed rich connections to recent machine learning methods for genera- tive modeling and distribution matching. While these methods perform well Euclidean domains, they are not directly applicable to topological domains such as graphs and simplicial complexes, which are crucial for data defined over net- work entities, such node signals and edge flows. In this work, we propose the Topological Schrödinger Bridge problem (TSBP) for matching signal distribu- tions on a topological domain, where we set the reference process follow some linear tractable topology-aware stochastic dynamics such as topological heat dif- fusion. For the case of Gaussian boundary distributions, we derive a closed-form Gaussian topological SB in terms of its time-marginal and stochastic differential. In the general case, leveraging the well-known result, we show that the optimal topological dynamics governed by some
process follows the harward-backwand unknowns. Building on these results, we develop TSB- models for match- ing topological signals by parameterizing the unknowns in the optimal process (topological) neural networks and learning them through likelihood training. We validate the theoretical results and demonstrate the practical applications of TSB- based models on both synthetic and real-world networks, emphasizing the role topology. Additionally, discuss the connections of TT TSB-based models to other emerging models, and outline future directions for topological signal matching.
INTRODUCTION
As fundamental problem in statistics and optimization, matching distributions aims find map that transforms one distribution to another. It has found numerous applications in machine learning tasks, particularly in generative modeling, which often involves learning a transformation from data distribution to simple one (often Gaussian) for efficient sampling and inference. While various methods have been proposed, including score-based [Ho et al., 2020; Song et al., 2020b] and flow- based [Lipman al., 2022] generative models, among others [Neklyudov et al., 2023; Albergo al., 2024; Tong et al., 2024a], the Schrödinger Bridge (SB) (baee methods [De Bortoli et al., 2021; Chen al., 2022a; Liu al., 2024] provide principled framework for matching arbitrary distributions.
Inspired by Schrödinger [1931; 1932], the classical SB problem (SBP) aims to find an optimal stochastic process that evolves from an initial distribution final distribution, while minimizing the relative entropy (Kullback-Leible divergence) between the measures of the optimal process and the Wiener process [Léonard, 2014]. Alternatively, the SBP can be cast as stochastic optimal control (SOC) problem which minimizes the kinetic energy while matching the distributions through a nonlinear stochastic process [Dai Pra, 1991; Pavon Wakolbinger, 1991]. The optimal solution to this problem satisfies Schrödinger system of forwaad equations (SDEs) [Léonard, 2014]. Traditionally, SB problems have been solved by addressing the unknowns in this system using purely numerical methods [Fortet, 1940; Föllmer, 1988J. More recently, machine learning approaches have been proposed [Pavon et al., 2021; Vargas et al., 2021; Wang et al., 2021; 1; De Bortoli et al., 2021; Chen et al., 2022a] where the unknowns approximated by learnable models (e.g., Gaussian processes, neural networks) trained on data-driven objectives, such the likelihood training [Chen et al., 2022a].
(FB) stochastic differential

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However, SB-based methods have primarily focused on solving tasks in Euclidean spaces, such as time series, images [Deng et al., 2024] and point clouds. Modern learning tasks often involve data supported on irregular topological domains such graphs, simplicial and cell complexes. Arising from applications like chemical reaction networks, biological networks, power systems, social net- works [Wang al., 2022; Faskowitz et al., 2022; Bick et al., 2023], the emerging field of topological machine learning [Papamarkou et al., 2024] centers on signals supported on topological objects such as nodes and edges, which can represent sensor data or flow type data over network entities. A direct application of existing SB models to such topological data may fail due to their inability to account for the underlying topology. Thus, in this work, we investigate the SBP for topologiaal signals, with a focus node and edge signals over networks modeled as graphs and simplicial complexes. To match distributions of such topological signals, our contributions are threefold.
(i) We propose the Topological Schrödinger Bridge problem (TSBP), which seeks an optimal topological stochastic process that minimizes the relative entropy with respect reference process, while respecting the initial and final distributions. To incorporate the domain knowledge, we define the reference process to follow topology-awer aawae SDEs (TSDEs) with linear topological convolu- tion drift term, admitting tractable Gaussian transition kernels. This subsumes the commonly-used stochastic heat diffusions on graphs and simplicial complexes for networked dynamics modeling.
(ii) Focusing on the case where the end distributions are Gaussian, we find the closed-form optimal terms of stochastic interpolant time marginal, as well as its
Gaussian TSB and characterize Itô differential. This generalizes the results of Bunne et al. [2023] where the reference process is limited SDEs scalar-valued linear coefficients. For the general case, we show that, upon existing results, the optimal 7 SB adheres a pair of FB-TSDEs governed by some unknown terms (also called policies), which in turn satisfy a system driven by topological dynamics.
(iii) We propose the TSB-based model for topological signal generative modeling and matching. Specifically, we parameterize the hard-to-solve policies by some (topology-aware) learnable models (e.g.. graph/simplicial neural networks), and train them by maximizing the likelihood of the model based on Chen et al. [2022a]. We show that T based models unify the extensions of score-baser and diffusion bridge-based models [Song al., 2020b; Zhou al., 2024] for topological signals.
We validate the theoretical results and demonstrate the practical implications of SB-molsss on synthetic and real-world networks involved with brain signals, single-cell data, ocean currents, seis- mic events and traffic flows. Before concluding the paper, we extensively discuss future directions in generative modeling for topological data. Overall, our work lies in the intersection of SB theory, stochastic dynamics on topology, machine learning and generative modeling for topological signals.
Notations. We denote by X a stochustic procexs (X,)oses) as a map X 00, x] - R from the unit time interval [0, 1] (i.e., index spuce) and sample space X' (e.g., Euclidean space) to R. (state space). Here, Xx, is random variable representing the state at t. The standard n-dim Wiener process (Brownian motion) is denoted by W. Let Z = C(O, 1], IR") denote the space of all continuous R=.. valued paths on J0. 1], and let P()2) denote the space of probability measures on Ω. For a path measure P E P() describing the law of the process X, we denote by P, its time marginal that describes the distribution of Xt, i.e., if X P. then X, P,. We assume distributions of random variables are associated with measures that have a density with respect to the Lebesgue measure. We also assume locally Lipschitz smoothness the drift and diffusion coefficients of SDEs.
2 BACKGROUND
2.1 SCHRÖDINGER BRIDGE PROBLEM
Let Qw be the path measure of a Wiener process dY; = a dW, with variance o². The classical SBP [Léonard, 2014] seeks an optimal path measure IP Ω by minimizing its relative entropy DKL with respect to Qw
min DKL (PI][w) s.t. IP E P(L), P. = Po, P Pi P1,
(SBP)
where Ps and P1 the prescribed initial and final time marginals on R.. Intuitively, the SBP aims to find a stochastic process evolving from Po to P₁ that are "most likely" to a reference (a prior) process, here the Wiener process. This is in fact dynamic formulation of the entropicrregulariee
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optimal transport (OT) with quadratie transport cost [Villani, 2009]. The static formulation reads
min [ - 2 220 21100 xel((po
1
- 21112 di + o² DKL (TTlPo @ p1)
(E-OT)
where [p P1) is the set of couplings between Po and Pt, and Po @ P denotes their product measure. In this static formulation, the optimization is over the coupling of po and pl, as opposed to the full path measure in the dynamic formulation (SBP). As a - E-OT reduces to the typical 2 Z-assersttinn OT, and the associated dynamic problem is given by Benamou & Brenier [2000].
2.2 TOPOLOGICAL SIGNALS
In this work, we are interested in signals defined on a graph, simplicial complex or cell complex such a topological domain, denoted by T. If T is graph with a node set and an edge set, we may define a node signal x E R" as a collection of values associated to the nodes, where n denotes the number of nodes. Such signals often arise from sensor measurements in sensor networks, user properties in social networks, etc [Shuman et al., 2013]. Similarly, if T a simplicial 2-complex SC2 with the sets of nodes, edges, well as triangular faces (or triangles), we can define an edge flow by associating a real value to each oriented edge. Here, for an edge e = {i.j]. if we choose [i as its positive orientation, then [i. represents the opposite [Godsil & Royle, 2001]. The sign of the signal thus indicates the flow orientation relative to the chosen one. Such edge signals often represent flows of information or energy. such water flows, power flows, or transaction flows [Bick et al., 2023]. Moreover, we may consider signals on general topological objects such as higher-order simplices (or cells). If is the number of simplices, we refer to x E IR" topological signal where the i-th entry represents the signal value on the i-th simplex. In topology, these are called cochains, which are the discrete analogues to differential forms [Lim, 2020].
The emerging field of learning on graphs and topology [Barbarossa & Sardellitti, 2020; Papamarkou al., 2024] concerns such topologiaal signals, where the central idea to leverage the underlying topological structure in T. For example, the graph Laplacian or adjacency matrix (or their variants) can be used to encode the graph's ' structure, acting as a spectral operator for node signals [Chung, 1997]. Similarly, in a SC2, the Hodge Laplacian can be defined as the operator for edge flows, composed of the down and up parts, which encode the edge adjacency through a common node or triangle. respectively. Other variants of Hodge Laplacians can also be defined [Grady & Polimeni, 2010; Schaub et al., 2020]. Thus, for a topological signal x E R",, we assume a Laplacian-rype positive semidefinite, topological operator L E Rax" on T which encodes the topological structure.
In a probabilistic setting, a topological signal can be considered random, following some high-din distribution on TR" associated with the topology 'T. This allows for the application of probabilistic methods to topological signals, similar to Euclidean cases. Recent works [Borovitskiy et al., 2021; Yang et al., 2024; Alain al., 2024] have modeled node signals and edge flows using Gaussian processes (GPs) on graphs and simplicial complexes. These GPs encode the topological structure by building their covariance matrix (kernel) Σ as a matrix function of the associated Laplacian L. For example, a diffusson node GP uses the kernel Σ = exp( - 2 graph Laplacian L. Other GPs can be defined as well to model signals in specific subspaces with certain properties [Yang et al., 2024] or to jointly model the node-edge signals [Alain et al., 2024].
x² L) with a hyperparameter x and the
3 TOPOLOGICAL SCHRÖDINGER BRIDGE PROBLEM
In topological domain T, we consider a topological stochastic process X where the index space is instead the product space of [o. 1| and the set of topological objects (e.g., nodes, edges) in T, and the state space R" is the space of topological signals with ni the cardinality of the set. When we consider the node set in a graph (the edge set in a SC2), X is a stochastic process of node signals (edge flows). We assume that X follows some unknown dynamics with its law described by the path measure PP. For some prescribed initial and final time-margina then aim to obtain the optimal P by solving the Topological Schrödinger Bridge Problem (T (TSBP)::
i.c., Xo ~ Po oo and X1 ~ P1 V/,
min D (P QT), s.t. PE P(2), IPo = un P = =
(TSBP)
Here, QT is the path measure of a reference process Y which follows some prior topology-aware stochastic dynamics on T. Effectively, the solution IP to TSPP describes the "most likely" process
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Figure 1: Heat diffusion starting from a node over a graph (Left) and an edge over a SC2 (Right), followed by intermediate states, then reaching the steady states where the heat becomes uniform for the node case whereas circulating around the cycle for the edge case.
X that conforms to the prior in the sense of minimizing relative entropy with respect to QT, while respecting the initial and final distributions Vo and 1/1-
Topological stochastic dynamics. For the reference process Y. given an initial topological signal condition Yo- you assume it follows general class of topological SDEs:
dy,- [tt Y; L.) dt + 9t d...
(T (TSDE)
where ft = f(t, T.) - T2 is a time-narying drift that depends on the topological structure T through the operator I,, and gt = g(t) E is scalar diffusion coefficient. For tractability, we consider a class of linear dynamics on T with the following drift term:
[(1, Yi Y, L) = HL)Y, + az; with H2(L) - Ei-ohy(()EE
(I)
and 0₁ E R a bias term. Here, (L),,, denoted simply as Hz, is a matrix polynomial of with time-varying coefficients ha,t = (t., which is able to approximate any analytic function of L for an appropriate K by the Cayley-Hamilon theorem. The drift fi is also referred to as a topological convolution of the topological signal in the literature. With graph Laplacian L. this returns the graph convolution of a node signal [Sandryhaila Moura, 2013; 2014], and with the Hodge Lapla- cian for edges general simplices, yields the simplicial convolution (Yang et al., 2022b]. Various topological machine learning methods have been developed based on such convolutions for their expressivity and efficiency. We provide few examples of linear 7SDE, which will be used later.
Topological stochastic heat diffusion: TSDE gives the stochastic variant of heat equation on T
dY, = -cLY, dt + 9t dW,,
(TSHeat)
by setting H, = -cL, with e > When T a graph with the graph Laplacian L, this dynamics enables modeling graph- time GPs [Nikitin et al., 2022], networked dynamic system [Pereira et al., 2010: Delvenne et al., 2015; Santos et al., 2024], and social opinion dynamics [Gaitonde et al., 2021]. More importantly, in the deterministic case of gr 0, it has harmonic steady-state, reveal- ing the topological features of T. Specifically, node diffusion converges to a state that can identify the connected components (O-dim holes), while edge diffusion in SC₂ based on the Hodge Lapla- cian has a converging state of circulating around cycles (I-dim holes). We refer to Fig. I for such illustrations. In line with our goal of distribution matching for topological signals, we present three examples of T SHeat, inspired by diffusion models for generative modeling [Song et al., 2020b). Example 1 (TSHealpM). Consider a constant 9t = -g in T SHeat. This results in a mixture of topological heat diffusion and BM with variance g², which we refer to as T SHeattp.
Example (T SHeatye). For some noise scales 0 < a min < maxx consider a time-increasing 9t
/do²(1)/dl with (t) = amin Ominn(max famax/mminn' )', which drives the well-known variance exploding (VE) noising process [Song & Ermon, 2020; Song et al., 2020b). The resulting form of SHeat is
dY, = -cLY, dt + do) (t)/ dW,,
(T'SHaavE))
Example (TSHcatyp). When combined with another noising process, known as the variance pre- serving (VP) process [Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2020b), we obtain
dY, = - (2
eL) + / 33(t)dl dW,
(T SHeatvp)
where B(t) 9min + - -min)) with scales 0 3min < fmax. The drift here can be considered as an instantiation of the topological convolution H, = - ((b8(t)T + cL) in (1).
Gaussian transition kernels. The TSDE, as an Itô process, is fully characterized by its transition kernel in probabilistic sense. As a result of the linear drift (1) of TSDE, the associated transition kernel Pejo (ys ly)) (i.e., conditional distribution of Y;JY,) is Gaussian. Its mean and covariance can be computed according to Särkkä & Solin [2019, Eq. 6.7]. Let the transition matrix of the ODE dY, = H, (L)Y, dt be denoted by Vr,, = (t, s). which is given by V., = exp(J.' ! H, dr) [cf. Lemma B.5]. For brevity, we denote Van as simply V,, We then have the following lemma.
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Lemma 4 (Statistics of transition kernels). For the TSDE with the linear drift (1), its Gaussian transition kernel (yo((il80)) (o) has the mean my and the cross covariance Ki at t1 and t2:
m. V.yo + V
V V ar dr
Ve20 + Er.
(cond. mean)
0
min((1.ta) g. G.V,² dr
K4tt Vt,
V.T
(cond. cross cov)
More importantly, we may characterize them for TSHeatpm and TSHeatve in closed-forms. Both have the same mean my V,yo with ; = exp(-cId)) and their covariances are given by:
d² [exp(ccLiti tel) exp( + t-) 2c
for TSHeatgM a min In Cama) exp( -ch( + 12)) e2p min (2)) - 1]A-1, for TSHattve
K2,
(2)
++ cL. If L is singular, we perturbed I, + c/ for small e 0. We detail the derivations in Appendix B. These expressions allow for tractable solutions for TSBP and, more importantly, facilitate the construction of TSB- TSB-based learning models, which we will discuss later.
with A = In Emax Jann
3.1 TOWARDS AN OPTIMAL SOLUTION OF TSBP
To solve the classical SBP, early mathematical treatments [Fortet, 1940; Beurling, 1960; Jamison, 1975; Föllmer, 1988] lead to Schrödinger system characterizing the SB optimality. Similarly, by Disintegration of Measures, we can convert the TBPP to static problem over the joint measure Poi of the initial and final states, instead of the full path measure IP
min DEL (Poi QTOI); s.t. Poi P(R" x )R",, Po = vo, P1
(TSPP (Satttatic) static
where QToI is the joint measure of the reference process Y at 0 and 1. The TSBP static only concerns at the boundary times, unlike the (dynamic) TSPP. Using Lagrange multipliers for the arrive at Schrödinger System that is instead driven by topological
linear constraints above, dynamics (see Appendix C), differing from the classical case [Jamison, 1975; Léonard, 2014]. This can be also interpreted through the equivalent E-OT formulation of SBP
min Poi
1l 1190 - -E -,,, dP (Vo. v1) +
log(Puz) dP
(TE-OT)
RR XR=
where the transport cost is linked to the TSDE
-weihhed norm of the difference yı m-
On the other hand, this system could also be derived from the SOC view which makes more apparent connections to machine learning methods. Building on the variational formulations of the classical SBP by Dai Pra [1991]] Pavon & Wakolbinger [1991], Caluya & Halder [2021] extended the analysis to the case with general nonlinear reference process and derived the corresponding optimality condition. As it is convenient to arrive at an SOC formulation for the SBP (see Appendix C), we readily obtain the following optimality.
Proposition 5 (T SBP optimality; Caluya & Halder [2021]: Chen et al. [2022a]). The optimal solu- tion P of T SBP can be expressed as the path measure of the forward (3a), or equivalently, backward (3b), TSE:
dx, = + ge 9 dW, Xo ~ vo, Zi = 9: V Vlog
(3a)
dX, = Ift - 2. 9.. dt + gr dW,, X1 U1, 2, = g: V log (X.)
(3b)
where (3a) runs forward and (3b) runs backward with backward Wiener process. Here, yt (X,) and Φ. (X()X,)) satisfy a pair of PDEs system
backward Kolmogorov equations).
Using nonlinear Feynman- Kac formula (or applying Itō 's's formula on log it and log (r), this PDE system admits the SDEs
d s 2
1
2.11 2t dt ZJ dW, dlog = 11 -9.2. f.) 2, Ze
dt 2T dw,
(4)
Then, the optimal path measure has the time-marginal P. (X.)),
p((a)
pp(3))
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This optimality condition adapts the result from Chen al. [2022a] for TSBP. From the forward- backward TSDEs (FB-TSDEs in (3)), we see that the optimal Zi guides the forward TSDE to the final V1, and likewise Ze adjusts the reverse TSDE to return to the initial Vo. While solving the system (4) is still highly nontrivial, we highlight that the FB-TSDES (3) and (4) pave way for constructing generative models and efficient training algorithms, as demonstrated by the recent works, name a few, [Pavon et al., 2021; Vargas et al., 2021; De Bortoli et al., 2021; Chen al., 2022a]. We further discuss in detail how to build such models for topological signals in Section 5.
GAUSSIAN TOPOLOGICAL SBP
In this section, we consider the special case of TSBP where the initial and final measures are Gaus- sians, to which we refer as the Gaussian topological SBP (GTSBP). We show that there exists closed-form GTSB by following the idea in Bunne al. [2023], which focuses on limited class of reference SDEs with a scalar coefficient in the drift, instead of a convolution operator He(L). We establish the first closed-form expression on the GTSB the following theorem.
Theorem 6. Denote by P the solution to SBP with vo = NNoo Eo) and vi = =N(,, E1). Then, P is the puth measure of Markov Gaussian process whose marginal X, NNu,, Ex) admits an expression in terms of the initial and final variables, Xo, X1, as follows
X, = R + R,X, + & R441 +I,
(5)
where Z~ ~ ~N(0, 1) is standard Gaussian, independent of (Xo, X1). and
R₂ = Ku Kn',
R2 = 1 Pov Ki)] Ku Ku KakKis Kie-
(6)
Proof. We provide sketch of the proof here, with the full derivations presented in Appendix D.
1. By Disintegration of Measures, first solve the reduced static Gaussian TSBP static (i.e., TE- OT). We can then convert the problem into a classical Gaussian E-OT via a change-of-variablle closed-form formula for the latter has been recently found by Janati et al. [2020, Theorem 1]. Via an inverse transform, we can then obtain the optimal coupling Poi [i.e., the optimal GT SB
The
2. In the disintegration of SBP to its static problem, the optimum is achieved when P shares the bridge with the reference QT (i.e., P the reciprocal class of QT) [Föllmer, 1988; Léonard, 2014, Proposition 1]. The Qr-bridge, Q77 Q the conditional Gaussian formula and Lemma 4. Upon this, together with the optimal Po1 we can construct the optimal X, and the marginal P..
Yi = v). can be constructed using
At the first sight, the construction of optimal process X in (5) meets the recently proposed stochastic interpolant framework by Albergo et al. [2024, Definition 1], in that X,=0 = X1 and X,=1 X1, and Tu = Tı = O. Moreover, from (5), we can compute the marginal statistics P, in terms of its mean μ4 and covariance Σ, in closed-form as well, detailed in Corollary D.I. In the following, characterize the process X under the optimal P in terms of its Itô differential.
Theorem 7 (SDE representation). Under the optimal P, the process X admits the SDE dynamics:
ax,- fo (t, X,; L) dt 9r aW,, where fr IT
pe) + iu
with Pt; Eg the mean and covariance of X, Ief. Corollary D.1] and have
S P. - QJ + H,K,, - Ken KT
(8)
with P. (R,Zi + R,C)R, = (CR + R.,R the covariance of Xo, X1 in the optimal For.
H,Ka + g. "W, where C is T, ),
Proof. We detail the proof in Appendix D and outline sketch here. From Léonard [2014]; Caluya & Halder [2021] [ef. Theorem C.2], the optimal P is the law of an SDE in the class of (7). To determine the drift, first compute the associated infinitesimal generator by definition for some test function. Since the generator for an Itô SDE is known (dependent on the drift) [Särkkä & Solin, 2019, Eq. 5.9], can then match the two expressions and find closed-form for the drift term. n

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Theorems 6 and 7 characterize the optimal P of the G7SBP from different views. While the stochas- tie interpolant formula is intuitive and straightforward, is natural to look for the associated SDE for a Markov measure. Despite the packed variables, both results [cf. Eqs. (5) and (7)] fundamen- tally depend on the transition matrix V, and &t, t, [cf. Lemma 4], which has closed-forms (2) for T THHaatpM and SHeatyr- From broader perspective, Theorems 6 and 7 extended the existing results of Bunne et al. [2023], where the reference process has a limited drift cY₁ + ar for some scalar c. While Chen et al. [2016] aimed to solve for linear drift with a matrix coefficient, their results lead to the solution of matrix Riccati equation, which is computationally expensive.
Solution complexity. While the variables involved in Eqs. (5) and (7) involve many matrix opera- tions, we remark that (i) the underlying V, is matrix function of L and can be computed efficiently
[Higham, 2008J. Given the eigen-decomposiiion L = UAU denote by hi Sd he.r dr the integral of the scalar coefficients in H. [ef. (1)) then we have i = U exp | k=o Mi. A*)UT, where the matrix exponential can be directly computed elementwise on each diagonal element of A. (ii) The other terms depending on Ur can be computed similarly in the cigenspectrum of L.
5 FROM TSBP TO TOPOLOGICAL SIGNAL GENERATIVE MODELS
The recent SB-based generative modeling framework primarily relies on the learnable parameteri- zations of the (Z,, Z.) pair (also viewed the FB policies) in the FB-SDEs and trainable objective that approximates the SBP. Specifically. Vargas et al. [2021] and De Bortoli et al. [2021] use GPs and neural networks. respectively, to parameterize the policies, and alternatively train them using iterative proportional fitting (IPF) to solve the half-bridge problem. On the other hand, Chen et al. [2022a] derived a likelihood based on the SB optimality condition, generalizing the score matching framework [Song et al., 2020b]. Upon the proposed TSP, along with the above theoretical results, we now discuss how to build generative models for topological signals using the existing framework designed for Euclidean domains.
-SB-based model. Consider the matching task: In some topological domain T, given two sets of signal samples following initial and final distributions 1u, 1/1 on T. we aim learn Topological Schrödinger Bridge between the two distributions. From Proposition 5, the optimal TSB follows the FB-TSDEs in (3). Moreover, given a path sampled from the forward SDE (3a) with initial signal Fa. one can obtain an unbiased estimation of the log-likelihood [(x)) of the TSB model driven by the optimal policies by using (4) [Chen et al., 2022a, Theorem 4). Similarly, the log- likelihood I(x1), given a final sample Ii, can be found. This allows us to build a 7 bBsba model for topological signals, following the ideas of De Bortoli al. [2021]: Chen et al. [2022a].
We first parameterize the policies, Ze and 2, by two learnable models 2 = Ztt, x; 0) and 2 (t, 0) with parameters @ and 6, resulting in the parameterized FB-7 TB-TSDEs. Then, we can perform likelihood training by minimizing the following loss functions an alternative fashion at initial and final signal samples zo and 2
1(xo: 0)
E (xa
2 |
V.V
2
+
2 X = 30 dt,
(9a)
1(x1;0) = /
E (3))
2 11
2
20 2 X1 = x dl,
(9b)
which are, respectively, the upper bounds of the negative ligeliiols (after dropping the unre- lated terms) of the signal samples zo and given paths sampled from the FD-TSDES.
Other choice of reference TSDE. In this work, we mainly consider reference dynamics following TSeaatBM THeatyyr and TSHeatvp. For the dynamics involved with Hodge Laplacians in SC2, we may further allow heterogeneous diffusion based on the down and up parts of the Laplacian. Bunne et al. [2023] proposed to better initialize the SB model using the closed-form Gaussian SB. Likewise, we can consider the GT in (7) as a stronger prior process, which yet requires a GP approximation from signal samples. We also consider fractional Laplacian for some cases to enable a more efficient exploration of the network [Riascos& & Mateos, 2014] due to its non-local nature.
Topological neural networks (TNNs). While De Bortoli et al. [2021]; Chen et al. [2022a] applied convolutional neural networks for the Euclidean SBP, we naturally consider parameterizing the poli- cies using the emergent TNNs. For node signals, we could consider graph convolution networks
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(GCNs) [Kipf & Welling, 2017]; and likewise, for edge flows in SC2, simplicial neural networks (SNNs) [Roddenberry et al., 2021]. These topology-aware models perform convolutional learning upon the topological structure, more efficient with less parameters and better in performance.
Complexity, Like standard SB models, 7SB-based models also require simulations of the FB- TSDEs. The key difference is that these models operate over topological networks where the drift (1) involves a mattrixvvector multiplication H,Y, However, this is essentially a recursive iteration of LY,, which is efficient due to the typically sparse structure of L, reflecting the underlying topological sparsity. Moreover, our Taaapaaraeeerrized policies are also efficient for the same reason.
Connection to other models. As discussed in Chen al. [2022a], in the special case of 7, and Ze as the score function (scaled by gr), the likelihood of SB models reduces to that of the score-based models [Song et al., 2020b] when 1/1 is a simple Gaussian and the forward process is designed to reach V/. Furthermore, if the reference process is poorly designed. SB models can still guide the process to the target distribution through these learnable policies, thus generalizing score- based models. On the other hand, for FB-SSDEs, we can also obtain probubility flow ODEs [Chen et al., 2018; Song et al., 2020b] which share the same time-marginals and likelihoods, allowing for exact likelihood evaluation of the model. Training through the likelihood of these flow ODEs naturally links to flow-based models. While there are direct score-based or flow-based models for topological signals, the above discussions apply to SB-base models. We refer to Appendix E for more details where show how the variants of these models including the diffusion bridge models [Zhou et al., 2024] for topological processes can be constructed.
6 EXPERIMENTS
First, we validate the theoretical results GTSB using the syn- thetic graph in Fig. 1. Here, we aim to bridge a zero-mean graph Matérn GP vn with Σn (T + L)-1.5 and a diffusion GP i with Σ, = exp(-201) Using the TSHeateM and SHettve reference dynamics, we obtain the closed-form X, in (5), from which we further compute the covariance N [cf. Corollary D.IJ. We mea- sure the Bures- Wasssesssssrrtinn distance between E, and Σj. From the right-hand-side figure, we see that both bridges reach the target dis- tribution. The bridges exhibit distinct behaviors depending on the reference dynamics, as demonstrated by the disparate curves for TSHeatyr with = 0.01 and 10. This highlights the flexibility of T SB-models in exploring a large space of topological bridges.
14-
GTSB BM
GTSB - VE1
GTSB VE2
w3
t
We then focus on evaluating TSB-based models for topological signal generation (v1 is a Gaussian noise) and matching (general v1) in different application,, with the goal of investigating the question: whether T TBBbase models are beneficial for these tasks compared to the standard SB-based models? For this goal, we consider as the baseline SB-based models in Euclidean domains which use BM, VE and VP as reference dynamics [Chen et al., 2022a], labeled as SB-BM, SB-VE and SB SB-VP, respectively. We consider 7 SB-basd models using SHeatam. TSHatyy: and SSatty as references, labeled as TSB-BM, TSB-VE and TSB-VP, respectively. We refer Appendix F for the experimental details and additional results, as well as complexity analyses in Appendix F.2.5.
Topological signal matching. We first consider matching two sets of IMRI brain signals from the Human Connectome Project [Van Essen et al., 2013], which represent the liberal (with high energy, as the initial) and aligned (with low energy, as the final) brain activities, respectively. We use the recommended brain graph [Glasser et al., 2016] that connects 360 parcelled brain regions with edge weights denoting the connection strength. From Fig. 2, we see that a TSB-VE model learns to reach at final state with low energy indicating the aligned activity, whereas SB-VE fails.
We then consider the single-cell embryoid body data that describes cell differentiation over 5 time- points [Moon et al., 2019]. We follow the preprocessing from Tong [2023]; Tong et al. [2024a;b]. We aim to transport the initial observations to the final state. Our method relies on the affinity graph constructed from the entire set of observations (~18)). We define two normalized indicator func- tions as the boundary distributions, which specify the nodes corresponding to the data observed at the first and last timepoints. Fig. 3 shows the two-dim phate embeddings of the groundtruth and predicted data points using TSB-BM and SB-BO BM models. Here, SB-BM BM gives very noisy predictions, especially for intermediate ones, even when trained on the full dataset (see Table F.7).
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Day
Day
Energies of true (Left) final state and the predictions obtained from TS3-VE (Center) and SB-VE (Right) models.
Figure 2:
Figure 3: Phate embeddings of the single-cell data observations (Left) and predictions based on TSB- BM (Center) and SB-BM (Right) models.
Table 1: 1-Wasserstein distances for generating and matching tasks across datasets over five runs,
where * indicates using GSB-VE. and GTSB-VE for ocean currents.
Method
Seismic magnitudes
Traffic flows
Brain signals
Single-cell data
Ocean currents
SB-BM 40.5
18.69
12.08
0.33
7.21 =0:00
SB-VE
+1.49
19.04+002
17. 17.4640.14
0.33=a001
7.17=0002
SB-VP
10.06
18.22
13.41
0.33
0.83
TSB-BM
9.01
10.57
7.00
0.14
6.94
TSB-VE
7.69 2004
10.51 40.02
4.09
0.14
6.89
TSB-VE
8.40
9.92
7.67
0.14
0.53
Edge flows have been used to model vector fields upon discrete Hodge Laplacian estimate of the manifold Helmholtzian [Chen et al., 2021b]. Following the setup there, we consider the edge-based ocean current matching in a SC2 (~20k edges). With edge GP, learned by Yang et al. [2024] from drifter data, as the initial distribution modeling the currents, we synthetize a curl-free edge GP as the final one, modeling different behaviors of currents. From Fig. 4, the final curl-free state, while TSB-BM becomes more divergent, ultimately closer to the target.
that SB-BM fails to reach
For these matching tasks, evaluate the forward final predictions using I-Wasserstein distances in Table 1, showing the consistent superiority of SB--aased models over SB ones. We reasonably argue that this difference is due the improper reference in SB-based models, which overlooks the underlying topology. This highlights the role of topology using 7 SB-based models in these tasks.
Generative modeling. We model the magnitudes of yearly seismic events from IRIS node signals on a mesh graph of 576 nodes based on the geodesic distance between the vertices of an icosahedral triangulated earth surface [Moresi & Mather, 2019]. We also consider the traffic flow from PeMSD4 dataset modeled edge flows on a SC2 with 340 edges [Chen et al., 2022b]. From Table 1, we see that 7 SB-based models consistently outperform SB-based models also for signal generation tasks, highlighting the importance of topology-awan reference processes.
Effeet of policy models. From the training curves of BSBB-M//E. for ResBlock and GCN policy models the right, we see that the training converges much faster and better using GCN compared to the former. This underlines the positive effect of TNNs topo- logical signal generative modeling. We refer to Tables F.2 and F.3 for the performance metrics of other bridge models with the two policy parameterizations on both seismic and traffic datasets.
T58-BM.GO M.G
TSB. TSB--VE GCN
TSB- TSB-BM,Res Res
TSB-VE,Res
Training Stage
Effect of GTSB prior. Instead of using SHeatMM T SHeatvr the reference, here consider their corresponding closed-form SDEs (7) as the reference, imposing on the bridges a stronger prior carrying the moment information of the data samples [Bunne al., 2023]. For ocean current match- ing, we show the samples from the learned FB-7 SDEs using GTSB BM in Fig. 4, which arrives at more faithful final state compared to TSB-PM, as also evaluated in Table
t-0.0
t-0.2
t.0.
t=0.9
1=
Figure 4: Forward sampled currents using TSB-3M SB-BM (Center) and GTSB-BM (Bottom).
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7 DISCUSSION AND CONCLUSION
In this work, we demonstrated how to construct 7SB-based topological signal matching models within the likelihood training framework [Chen et al., 2022a]. We here discuss a few promising future directions based on emerging work and unexplored theoretical results.
On model training. Peluchetti [2023]; Shi et al. [2023] applied iterative Markovian fitting (IMF), as an alternative of IPF, to the classical SBP. This algorithm, trained via score matching, extends to TSB-models with 7SHeatuM or TSHeatve as the reference, thanks to their closed-form transition kernels in (2). Recent work proposed (partially) simulation-free training of SB models. Tong et al. [2024b] learns the optimal SB by flow and score matching the forward SDE upon heuristic E- OT. Korotin et al. [2024]; Gushchin et al. [2024] modeled Schrödinger potentials using Gaussian mixtures, enabling light training for the optimal drift, and Deng et al. [2024] linearized the forward policy. However, these methods require the reference dynamics to be either Wiener process or have scalar drifts. While training TSB models remain scalable w.r.t. the topology size (see Appendix F.2.5), extending these approaches to our models is worthwhile but nontrivial.
On model improving. We focused on the reference dynamics driven by topological convolution H2 up to order one. It is however worthwhile to consider more involved (potentially learnable) con- volutions impose more general priors or incorporate physics knowledge of the process. The scalar diffusion coefficient 9t could be extended as matrix-valued, enabling spatially correlated noising processes over the topology. On the other hand, SB models perform kinetic energy minimization from the SOC view. [Liu et al., 2024] considered generalized SBP by adding a cost term which can model other knowledge of the process. This broadens the applicability of SB models and TSB mod- els could benefit from this, when there are external interactions with the topological process or prior knowledge on the process, such as enforcing curl-free edge flows.
On other models. While we showed the connections of TSB to stochastic interpolants. flow- and score-based models, as well as diffusion bridges, we notice that the TSB optimality can be in- terpreted as a Wasserstein gradient flow (WGF) (sce Appendix C.2). For example, the " driven T
SB is the WGF of a functional F(v) of some measure v/ with with F(v) = c 1 (x-v(z) dz+
1 J' / log y da where D(x) := jx² T Lx is the Dirichlet energy of I and the second term is the neg- ative entropy. Thus, the --raanv-ddrvven forward TSB essentially reduces the Dirichlet energy. This may not always align with the needs of real-worll applications, which in turn motivates devel- oping topologiaal dynamies learning models via the JKO flow [Jordan al., 1998] of parametrized functional ((x) on topology, akin to approaches used in Fuclidean domains [Bunne et al., 2022].
We focused on a fixed topological domain, but it is also of interest to study the case where T itself evolves over time. The T TSP in this scenario may rely on a time-varying operator L, to guide the reference process. This is relevant for recent generative models for graphs, to name a few [Niu et al., 2020; Jo et al., 2022; Liu et al., 2023], where the graph structure, together with node features, are learned in the latent space based on diffusion models [Song et al., 2020b]. Lastly, we remark that discrete distributions on topological domains may be defined. For instance, nodes of graph can represent discrete states where node is associated with a discrete probability P.. This motivates the emerging geneartive models for discrete data [Austin et al., 2021; Ye et al., 2022; Haefeli al., 2023; Camphell et al., 2024]. For a formal treatment of matching such discrete distributions on graphs, we refer to Maas [201]] 1]; Léonard [2013]; Solomon [2018]; Chow al. [2022].
Conclusion. With the goal of matching topological signal distributions beyond Euclidean domains, introduced the TSP (topological Schrödinger bridge problem). We defined the reference pro- cess using an SDE driven by a topological convolution linear operator, which is tractable and in- cludes the commonly used heat diffusion on topological domains. When the end distributions are Gaussians, we derived a closed-fom TSB, generalizing the existing results by Bunne et al. [2023]. In general cases, we showed that the optimal process satisfies a pair of FB-TSDES governed by some optimal policies. Building upon these results, we developed TSB-based models where we parameterize the policies as (topological) neural networks and learn them from likelihood training. extending the framework of De Bortoli et al. [2021]] Chen et al. [2022a] to topological domains. We applied SBB-baded models for both topological signal generation and matching in various applica- tions, demonstrating their improved performance compared to standard SB-based models. Overall, our work lies at the intersection of the SB-based distribution matching and topological machine learning, and we hope it inspires further research in this direction.
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ACKNOWLEDGEMENTS
The author's research is funded by TU Delft AI Labs Programme and supported by professors Elvin Isufi and Geert Leus. The author is grateful for the financial support provided by the Multimedia computing group at TU Delft. Additionally, the author thanks the anonymous reviewers for their comments and valuable feedback, especially on the single-cell data experiments. Lastly, the author acknowledges the work of Bunne et al. [2023] for inspiring the proof' in Section 4.
REPRODUCIBILITY STATEMENT
For reproducibility of the theoretical results, the complete proofs of the claims can be found in the appendix, which is organized here. For reproducibility of the experiments, we refer to the GitHub repository at topological matching.
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Organizations We include several appendices with additional details, proofs, derivations and ex- periments. This work concerns with the intersection of Schrödinger bridge theory, topological signal processing and learning, stochastic dynamics (on topology) and generative modeling. For that, first introduce the necessary preliminaries on Schrödinger bridge theory in Appendix A, including needed theorems and lemmas for later. In Appendix B, first provide overview of topological signals and probabilistic methods. Then, we extensively discuss the topological stochastic dynamics based on the linear 7 SDE, as well its three instantiations: 7 SHeatgM. SHeatve and '7 SHeatvp. Here, we provide the detailed derivations on how to obtain the transition kernels of the TSDE, 7 SBP, relying on the
which are crucial for the 7SP. In Appendix C, we discuss the optimality existing results. Specifically, it includes the Schrödinger system for 7BP, an SOC formulation of the 7 SBP, the optimality, and the WGF interpretation. Appendix D proves the closed-form solution of the GTSP, along with the marginal and conditional statistics of the optimal path measure. In Appendices and F, we provide more details on the TB-base models, their connections to other models, well the experiment details and additional results.
A PRELIMINARIES ON SCHRÖDINGER BRIDGES
A.1 ON OPTIMAL TRANSPORT
Theorem A.1 (Static Gaussian OT; [Janati et al., 2020]). Let En, En be positive definite. Given Iwo Gaussian measures Pn N(po, En) and pr (M,, En), the entropic-regularzzed optimal transport
min T(DAa.)
=110 - 21ll 2 dT 21) + o DKL [ @ m))
(E-OT)
admits closed-for form solution n
Σg
Ca Σ1
π ~ N
(A.1)
[1]]
C
where
C. - ( o²1). D. =
و +
(A.2)
Remark 2. Note that while the above results are stated for positive definite covariance matrices (in order for Pu and P1 have Lebesgue density), the closed-form solution remains well-defined for positive semi-definite covariance matrices.
A.2 ON SCHRÖDINGER BRIDGE
Lemma A.3 (Léonard [2014]). For given measure TP the path space Ω, let Pay represent the conditioning of IP on paths that take values x and at t = 0 and 1, respectively. That is, pay Po = σ, X1 y]. Let Foi denote the joint probability for the values of paths at the two ends f 0, 1. Then, P can be disintegrated into
/ JE (Pr) (drr dy).
PP())
(Disintegration of Measures)
Static SBP By Disintegration of Measures, for all P. E P(S)), the relative entropy can be factorized
as
(PI( (Porl Q01)
DKL (PP dy).
(A.3)
which implies that DKL Qo1) < (Pl( Q) with equality if and only if pay Q for each (I,y) E Rn x R This allows us to reduce the (dynamic) SBP to the static one.
min DKL (Pou (P
s.t. P
P. Po: P Pt.
(SBP statie
Furthermore, it readily gives the following important theorem.
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Theorem A.4 (Föllmer [1988]; Léonard [2014]). The SBP and SBP sune admit, respectively. most one solution. If SBP has the solution I., then its joint-marginal at the end times Poi is the solution of SBP static Conversely, if solves SBP maree then the solution of SBP can be expressed as
PO [
J
tTt (PPo)oi dy),
(A.4)
which means that P. shares its bridges with Q (i.e, IP is in the reciprocal class of Q):
I Q V(z IR" RR.
(A.5)
Proposition A.S (Léonard [2014]). If the reference measure Q is Markov, then the solution IP of SBP is also Markov.
Schrödinger System
Theorem A.6 (Jamison [1975]; Léonard [2014]; Chen et al. (2021a]). Given two probability mea- sures Po. PI I. and the continuous, everywhere positive Markov kernel Pajaylaa (not necessarily associated to a scaled Brownian motion). there exists a unique pair of (up to scaling) of functions so, iPi I such that the measure Por IR" x R defined by
Poi = /
(A.6)
JREXR. xR"
has marginals Pa and pt. Moreover, the Schrödinger bridge from po to pv induces the distribution flow
P. = Pebi with (p((x)) = ( (inull))))))))
= Pro(xle)@o(dyy)
(A.7)
SOC Formulation By Girsanov's 's theorem, Dai Pra [1991]; Pavon & Wakolbinger [1991] showed an equivalent SOC formulation of the SBP which aims to minimize the kinetic energy
[T lu((,, [1M((
s.t. [dx; utt, x,) + dW,
min E wEll
(SBP
[o ~ Po: X1 ~ P1,
where U is the set of finite control functions u, =\t x)t Given the SDE constraint in SBP the associated marginal density Pt p(t, x) evolves according to the Bokker-Planckk-ee Kolmoooryy equation (FPK, Risken [1996]). This allows to arrive at an equivalent variational formulation
/ J x)(t, dr s.t. min (pe.
O.pt + V (pen)) = Z Apt. Pu Po. P1.
(aBvvr)
Given P: = pc the optimal control in SBP can be obtained by ue = o² V log St.
B STOCHASTIC DYNAMICS ON TOPOLOGICAL DOMAINS
Compared to the Euclidean domain, the dynamics on topological domains are less studied. Here provide some existing work on the dynamics on graphs and simplicial complexes. Note that our choice of the linear topological drift ft in (1) is analogous to the ideas in [Archambeau et al., 2007; Verma al., 2024] which considered linear SDEs to approximate nonlinear dynamics, enabling approximations of more complex topological dynamics.
B. PRELIMINARIES ON TOPOLOGICAL SIGNALS
Here we review the standard notions about topological signals, and we focus on the node signals and edge flows graphs and simplicial complexes. Note that we abuse some notions in this subsection.
Node signals Let G = (V, E) be an unweighted graph where V {1..... ......,.. n2 is the set of nodes and E is the set of nı edges such that if nodes i, connected, then (i.j) E E. We define real-valued functions on its node set + R, collected into a vector x = (I( (n))] E R which is referred
node signal (or graph signal). Denote the oriented node-to-edge incidence
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matrix by B1 of dimension no X n1. One obtain the graph Laplacian by L = B₁ BY which is positive semi-definit linear operator on the space IR" of node signals.
A graph GP ~~ (0, 2) assumes is random function with zero mean and graph kernel (co- variance matrix) Σ which encodes the covariance between pairs of nodes. [Borovitskly et al., 2021] constructed the diffusion and the Matérn graph GPs by extending the idea of deriving continuous GPs from SDEs, which have the kernels as follows
1s2
)), 2
Eaaéé -22 kx2 (
Eamu exp
(B.1)
where x > 0, V > 0 are hyperparameters.
Edge flows While it is possible to define edge flows graphs. we consider a more general setting for a 2C2. SC₂ generally contains V, E, T three sets, where V, E are the sets of nodes and edges, same for graphs, and 7' is the set of triangular faces (triangles) such that if (i,j), (j. k), (i. k) form a closed triangle, then f = (i, j, k) E T. Not all three pairwise connected edges are necessarily closed and included in T. For each edge and triangle, labels as their reference orientation. (Note that the orientation of a general simplex is an equivalence class of permutations of its labels. Two orientations are equivalent (resp. opposite) if they differ by an even (resp. odd) permutation [Lim, 2020].) An oriented edge, denoted as = [i is an ordering of {i.j}. This is not directed edge allowing flow only from to j. but rather an assignment of the sign of the flow: from i to j. it is positive and the reverse is negative, Likewise goes for the oriented triangle & = [i. j. k]. In SC2, we can define an edge flow by real-valued functions on its edges, collected in = [2((e) ) ..... x(ee,)] E R= which are required to be alternating, meaning that r(ē) xe)) if = [j. i] is oriented opposite to the reference e = [i, Likewise, triangle signal can be defined via an alternating function on triangles.
assume the increasing order of their node
In the same spirit as graph Laplacians operating on node functions, we can define the discrete Hodge Laplacian operating on edge flows L, La + T., == BJ B₁ B2 where B2 is the edge-to- triangle incidence matrix. The Hodge Laplacian describes the connectivity of edges where the down part La and the up part L, encode how edges are adjacent, respectively, through nodes and via triangles. Moreover, the Hodge Laplacian L is also positive semi-definite. Yang et al. [2024] generalized the graph GPs for edge flows, resulting in the diffusion and Matérn edge GPs with the kernels of the same forms as in (B.1) 1) but with the Hodge Laplacian L instead of the graph Laplacian L. Moreover, based on the combinatorial Hodge theory [Lim, 2020] that
R nt im (BJ) @ ker(L) @ im(B2)
(B.2)
where im [B,)) is the gradient space, ker (L) the harmonic space and im (2)) the curl space, one can define the two types of edge GPs living in certain Hodge subspace O E (H,G.O C} with the kernels of the forms
Eaian = of Un exp (- h - An U
22 1+ R2
(-2
E = Un
UJ
(B.3)
where (Un, An) are the eigenpairs of the Hodge Laplacian in the gradient (G), curl (C) and harmonic (H) subspaces, respectively [Yang al., 2022b]. The samples from certain Hodge GP are in the corresponding Hodge subspace, which allows us to model the edge flows with different properties.
B.2 PRELIMINARIES ON (STOCHASTIC) DIFFERENTIAL EQUATIONS
We are involved with differential equations in this work. In the following, we review some results on ordinary differential equations (ODEs) and SDEs which are required later.
B.2.1 ON ODES
Given an initial solution To R", consider linear differential system of the form
dr. = A,x; dt
(linear ODE)
with A a time- varying matrix. To solve this system in closed-form. we require an expression for the state transition matrix V (t, s) which transforms the solution at to t, x2 s)x, For the general case, the closed-form of (t, s) is not possible. In the following, we introduce an important class of matrices A, for which closed-form solution is possible [Antsaklis & Michel, 1997].
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Lemma B.1 (Closed-fori of the transition matrix of linear ODE). Given linear ODE, if for every s, t 2 O, we have
A [.. Ar dr
[]-" A, dr dr Ar
=
(B.4)
then the transition matrix is given by
(t, exp (J. Ar dr
- 1 [. Ar dr
1 _ (f," Ar (f) +....
(B.5)
2!
For the scalar case, or when Ae diagonal, or for A, = A, (B.4) always true.
Lemma B.2. For A CR (B.4) is true if and only if A,A, AgAA for all s, Lemma B.3 (Integration of matrix exponential). For nonsingular matrix A, we have
J exp(Ar) [exp(At) exp(As)]
(B.6)
B.2.2 ON SDES
Lemma B.4 (Itô isometry. Oksendal [2013]. Let W [0, x X valued Wiener process defined up to time 1, and let X [0, X is adapted to the filtration generated by W. Then
RR. denote the canonical real-
IR be a stochastic process that
E [V. x. x.aw.)
)
= =R [ X2 de
(B.7)
and
E
x xix
(B.8)
This corollary allows us to compute the covariance of two stochastic processes X, and Y, that are adapted to the same filtration.
Transition densities of SDES All Itō processes, that is, solutions to Itō SDEs, are Markov pro- cesses. This means that all Itō processes are, in probabilistic sense, completely characterized by the transition densities (from F, at time to Iy at time f, denoted by (22 = p(æ,, 8; zz. t))- The transition density is also solution to the FPK equation with degenerate (Dirac delta) initial density concentrated on 2s at time 8. We refer to Särkkä & Solin [2019, Thm 5.10J.
B.3 TRANSITION DENSITIES OF TSDE
TSDE First, we can find the transition matrix of the associated ODE TSDE. Lemma B.5. For an ODE dy: = H, (L)u, dt, the transition matrix is given by
V (t, Ves exp (", dr,
20
( 11, 1r,
(transition matrix)
with yt = Vesys. Note that Uts is symmetric since H2 is function of the symmetric L.
This is direct result from Lemmas B. and B.2 since H,H, H, for all t, r. By the definition of matrix integral, the computation of L,, is given by
K
(t r) = =
exp(( exp
I T 1*
(B.9)
where it: S hks ds are the integral of the scalar coefficients in Hy. In the following, char- acterize the transition densities the TSE, as well as the three concrete examples in Examples to 3. Then, using the formulas in Särkkä & Solin [2019, Eq. 6.7], compute the statistics of the transition kernel.
The following two lemmas compose Lemma 4.
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Lemma B.6. The transition density Ptjs (voll) of the TSDE conditioned on Y, Us is Gaussian
(els(tilbaa) (b) ~ NN(; Mess
(B.10)
with the mean and covariance, for & Z 8, as follows
m = Io + 11 f
V V. a dr, Kifs = V. (. o. dr v.
Proof. Given the transition matrix, using the transition kernel formula Särkkä Solin [2019, Eq. 6.7]. we have
Maje = iilill + [: Virer dr VesMs v, L V. "7''a7 dr,
where we use the property v V. Likewise, we have
Kd/= [- VZ V. dr V. 22 22
(f. V.
Lemma B.7. Conditioned on Yo = you the cross covariance K ((1,, of SDE at t l2 is given by
K2 cs (minn(te)))) 9 // dr V.T-
Proof. By applying the cross covariance function in Särkkä & Solin [2019, Sec 6.4), we have
Ktg,t = Cov Y (Y) = EE -EE [[O In Man dWn, ) Ira Ueirz dW )
De v Is. dr Ve, (1, (minntteees
N dr V (by Lemma B.4)
dr)
TSHeatpM
Given an initial sample yo of the random topological signal Yo, consider the SDE: dY, cLY, dt + gdW,
(7 (HHata))
Note that when L is singular, we consider perturbed version L + el with a small constant > 0.
Its steady-state distribution has zero mean and covariance matrix Σ g² L-1_ The transition matrix of the associated ODE dY, -cLY, dt is given by
2c
(B.11) Lemma B.8 The transition density (Welyy) of the THeatns conditioned on Y, ys Gaus- sian with the mean and covariance, for 1. 2 8, follows
Le exp ( (-c(t- - S)
g²
maja
VisNss
Kijs 2c
- expl-2cc((t.. [ I-1
Moreover, the conditional dynamics Y.(Yo yo has the covariance process at 11, 12
g² exp( cLL tyl) exp(-cL(t) t2)) 4-1
Kr,,
(B.12)
2c
Proof. For the conditional mean, we can obtain it directly from the transition matrix of the associated ODE. For the conditional covariance, we have
Ki/s / V(I, ((1 dr w. (['v""''rr) df w.
= guv (('xx((2cLT) dr VT
(by =xp2ex)((
g² v. eexp(2cL() exp exp((cLxs) --
(by Lemma B.3)
2c
g²
[[ - -eppl expl-2cc(( 8))] --1
(V, exp(-cL()
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To compute the covariance process of the conditional dynamics Yi for t1 I t2
yo- by definition, we have,
Vn1 (f/
K11,22 Cov(Y Y221
g dr Vis
(by Lemma 4)
W
exp(2cLT) dr 1/-
(by V,²² exp(2cLr))
1
u
(exp(2cLt)) - 1] L ---
(by Lemma B.3)
2c
g²
exp(-cl(tz 11)) exp(-cL(tt + t2))
2c
The case of 1 > t2 can be similarly derived, which completes the proof.
TSHeatye
Lemma B.9. The Gaussian transition kernel p(((s) of SHeatvr has the mean and covariance
mais toyo
Kt/s min In mmax
exp(-2cLt) (xx((2tt) exp(2As)) A
(B.13)
Omin
where Ves is the same (B.II) and A In Y. conditioned on Yo, is given by
Omax
I cL. The cross covariance between Y, and
Omin
max
K1,,2 min In
exp( cL(
(B.14)
mmin
Proof. As the associated ODE SHeatve is also a topological heat diffusion, the transition matrix is the same as V,, = exp(-cle s)) for 'T SHeatBM. By substituting o o(t) into It, we can find that
O mmax
σ mmax
= Tmin
2
(B.15)
Umin
mmin
For the covariance of the transition density, we have
Mr (f. of v,, dr y!!
Kips
for which, we need to compute the integral
[
2 dr
min
Tmax
2r
2
Tmax
exp(2cLTT) dr
Tmin
Tmin
2r
= -2 min In
Tmax
Tmax
exp(2cI.r) dr
(factor out the constant)
Omnn
Juin
2o2 2iin In
c mmax
exp
2r In
maxx
exp(22LTT dr
(by the identity exp(In x) x))
Umin
Tmin
20 2 In
0 Umax
exp(2rA) dr
(by A = In(amax/ (mmn I + cL)
Umin
Tmax
= = min
In
(exp(2At) - exp(2As))A
(by Lemma B.3)
0min
Thus, have
Kels 22 In
Tmax
exp(- exp(-2cI.t) exp(2At) exp(2As)] A-1
O
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For the cross covariance, assuming I tg. then find the covariance kernel as
%
K2,
Cov
Y2,1%a)
dr
V/2
(by Lemma 4)
27
Vi,
Omax
In
mmax
exp(2cLT) dr V (since 3. = exp(2cLT))
min
min
mmn
min
In
mmax
Ve, exp(2Att) I 1-1 1
(using the same steps as above)
mmin
min
In
Omax
exp(-cL(t1 + 12)) (exp(Atr)) - -
0min
The similar steps can be followed for 1 12, which completes the proof.
TSHeatyp For this stochastic process, a closed-form transition kernel cannot be found. Yet, we could proceed the following for numerical computations. First, we can find the closed-form transi- tion matrix of the associated ODE as
V. exp -Gb))))+
1 exp (oxpp--00 (-ebe - [- B(T)
(B.16)
where the integral can be easily obtained as
[ B(T)
[ 7 Smin) + T T3min 7 =: Bie.
[
(B.17)
This allows compute the mean of the transition kernel majs given initial solution ys- For the covariance kernel, we have
Kt/s = U, (f'(rrrr)) w.-
B(T))=2
where the integral can be expressed as
| 8()) dr 1." ['(T(Puuxx Brnin) + Penin exp
2cIr + 3-0 dr
- [[(1BBee - Buiin) Smii ((()) [ (Bmax - Smiin) ( d(. dr.
(integration by parts)
Here, we denote v(r) = exp(2cTor 3-o) dr, thus v'(r) == exp(2eI.r + Bro). which does not have simple closed-form, need compute it numerically. This gives the covariance kernel. Following the similar procedures, we can compute the cross covariance K of the conditional process Yo yo.
B.4 OTHER TOPOLOGICAL DYNAMICS
We may consider fractional Laplacian in TSHeat which allows for more efficient exploration of the network [Riascos Mateos, 201 4] due to its non-local nature.
For SHeat, we can further allow heterogeneous hear diffusion on the edge space as follows
dy = (c, C2 dt + g: dW,
(B.18)
by setting H, = -c + Lu), with C1 > O. Here, the diffusion rates are different for the different edge-adajcency types encoded in La and Lu- This in fact can be generalized using more general topological convolution operator, if L = La Lu consists of the down and up parts,
K₂
H. Σ M
Σ 1 O=
(B.19)
where h1 (t), (t) are the coefficients of the topological convolution. We refer to Yang et al. [2022b]
for more details on its expressive power compared to TSD..
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use higher-order dynamics such wave equations. Graph wave equations [Chung et al., 2007] have been used for building more expressive graph neural networks [Poli et al., 2021] and its stochastic variant for modeling graph-time GPs [Nikitin et al., 2022]. Moreover, we may allow the interactions between node and edge signals in which the dynamies is defined over the direct sum of the two spaces. While considered in this work, we refer to [Alain et al., 2024] for such cases to define topological GPs on SCs.
Beyond TSDE: Instead of first-order dynamics,
C TOWARDS THE OPTIMALITY OF TOPOLOGICAL SBP
Proposition C.1 (T (T-Schrödinger System; [Chen al., 2016; Jamison, 1975]). The optimal solution Tf SBP moti has the form Po1 - (0(2)))110(z) ( dæo dæ with &p and ip satisfying the system
(P((x))= = S (1i((1 xe) (I) dri,
(.(x))) - f
(ooo(allrr))p((((0 daa
(C.1)
12th
where (t]x()) = NNy Pels with drift in (1). Moreover, the margiaalla at can be factored as (z) = (x))
Helss K(ps) is the Gaussian transition densiry [ef. Lemma B.6] of TSDE
Proof. This is direct result of the Schrödinger system in Theorem A.6 by replacing the Markov kernel by that [ef. Lemma B.6] of the TSDE.
From this system, we see that the optimal path measure has its marginal PP, factorized into two time-marginals for and Pi, which are both governed by the TSDE.
C.I VARIATIONAL FORMULATIONS OF TSBP
By Girsanov's 's theorem, the TSPP can be formulated as the minimum energy SOC problem:
2 / [] (x)t, dt. s.t. [dX, = [S. + sbblt, X,) di 91 dW, (Xo 10, Xo v1
1
min E
(T (TSBP
where b = b(t,x,) Xx) is the control function. The SDE constraint in TSBP soe is also known as the controlled SDE, in comparison to the uncontrolled reference SDE. It further leads to the variational problem
/'' / |brl] x) dr dt, s.t. Soe + V [ve g.b:)] 1 Av. (v(0,z)) x) = vo, v x) =
min (be.
(TSBP var)
2 20 |R0
where 2(4,x)) = I., is the time-marginal of IP and follows some PDE constraint, which is the FPK equation the SDE constraint in TSBP
Theorem C.2 STSBP Optimality; Léonard [2014]; Caluya & Halder [2021]). Let sp: = =((t, and &.. = =((1,x) be the solutions to the pair of PDEs
{ ( -op fo 1 (fr + A
S.t. (p (0, uo, 20,
(C.2)
1
Then, the optimal control in TBP var is b =
V log Pt and the optimal path measure is vt = Moreover, the solution to TBP can be represented by the path measure of the following coupled (forward (forward-backward) SDEs
dX,- [fe + o V log (t., X,) dt + dW,, No vo-
(C.3a)
dx, [fc of v log Xt)) dt + gr dW,, X1 V1:
(C.3b)
where V log (t,, X2) and V log (t, X,) are the forward and backward optimal drifts, respectively.
Proof. The proof is an adaption of Caluya & Halder [2021] the topological setting. First, we make the assumptions' that g(t) is uniformly lower-bounded and f(, xr; L) satisfies Lipschitz conditions
'The nonexplosive Lipschitz condition on ft rules out the finite-tima blow up of the sample paths of the SDF, and ensures the existance and uniqueness. It, together with the uniformly guarantees the transition kernel P.ja positive and everwhere continuous.
diffusion gt.
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with at most linear growth in From the first oprder optimality conditions for the SOC formulation TSBP var. we can obtain a coupled system of nonlinear PDEs for bhy (the potential function of b,, i.c., b. Vele) and Mr which are known the Hamilton- Batttt---anoooiiielllam (HJB) and FPK equations, respectively, as well as the optimal control b; = g V log sPt- Via the Hopf-Cole transform, this system returns (C.2) [Caluya & Halder, 2021, Thm 2]. Then, by substituting the optimal control into the constraint in TSBPsoc one can obtain the forward SDE, and the backward SDE can be derived from the time-reversal of the forward SDE [Anderson, 1982; Nelson, 2020].
Using nonlinear Feynman-Kan Kac formula (or applying Itô's formula on log yr and log Φo). the PDE system (C.2) admits the SDEs [Chen et al., 2022a]
1
d log st:
dt + z dW, 2
(C.4)
2
d dlog 6t =
2 || 1 + V (9. - fo) + 2,TZ, 2, dt + 2, dW.
(C.5)
where Z₂ = Dr V log (X,) and 2, = gt log &. (Xt). This results in Proposition
C.2 WASSERSTEIN GRADIENT FLOW INTERPRETATION
The gradient flow of funcitional over the space of probability measures with Wasserstein metric, i.c. the Wasserstein gradient flow (WGF), is fundamentally linked to FPK equations [Otto, 2001; Ambrosio al., 2008]. the following, we show that solving the TSBP with T SHeatgn reference amounts to solving the WGF of some functional on a probability measure v.
Theorem C.3. Consider the TSP with the reference process THeat. The SB optimality [cf. (C.2)) respects a pair of FPK equations of the form
0pp((x)) V (cLLx(((z)) + -0 Po(x) = (o(x),
(C.6a)
depe(x) V (cLxp.(1)) g Ag po(x) = (1))) exp(2ex Lx/g²).
(C.6b)
Therefore, the Wasserstein gradiens flow of F(u) recovers the paired PDE in solving TBP
F(v) c f
1 xx LL_LL-v(ar dx +
1 1 9 [ vlog dr == (D,] + 29 S(v
(C.7)
where D(x)= = 1xx Lx is the Dirichlet energy of x and S(v) is the negative differential entropp
Proof. First, from Theorem C.2, we have the PDE system in (C.2) which can be rewritten as the pair of PDEs in (C.6) by applying Caluya & Halder [2021, Thm 3]. Both PDEs are of the following FPK form with V(x)= = eec TLx
Lx on some density Pt
1 O.pr(x) = V (pr(x) 9
(C.8)
for some initial condition. We can view Vzz) the potential energy of some function f((z). Here, we have f(() = -cLæ, = - (z). Then, from the seminal work Jordan et al. [1998], the flows generated by the PDEs in (C.6) (both of the FPK form) can be seen the gradient descent of the Lyapunov functional F(-) in the following form
1 F()=c) = RM 2n Lx (x)(()) + 3 RM
S.
1
s(-) / (.) log()) dr = cE + 2 s-))
(C.9)
with respect to the 2- Wasseesstein distance in the space P2(R") of probability measures on E= with finite second moments. Here, (-) can be & or pe.
We note that it is also possible to obtain the associated functional for the TBP with more general reference SSD based on the similar argument, but it would be more involved and lead to a time- dependent functional [Ferreira & Valencia-Guuevara 2018].
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D THE CLOSED-FORM OF GAUSSIAN TOPOLOGICAL SCHRÖDINGER BRIDGES [THEOREMS AND 7] (PROOFS AND OTHERS)
For convenience, we state the Gaussian TSBP
min DuL (P QT),
s.t. P P(L), Po Nuoo Mo), = -N(u,
(GTSBP)
and its static problem
min (Po
| @701), s.t. Poi E P(R" x R = V0, = =11
(GTSBP& static)
We also restate the main results of the Gaussian TSP.
Theorem 6. Denote by P the solution to GT SBP with Mo N Ea) and V1 = Nun, E1). Then, P is the path measure of Markov Gaussian process whose marginal X, ~ (Nu. Σ() admits an expression in terms of the initial and final variables, Xo, X1, as follows
X, RoXo R,X, + & R&11 + 11Z
(5)
where Z N(0, 1) is standard Gaussian, independent of (Xo, X1). and
R2 K v. R.V
Pg : := Cov(Y, Yi)] Kit Kei KikKi's K K1-
(6)
Corollary D.1 (Marginal Statistics). The time marginal variable X, in (5) of the optimal solution to GT SBP has the mean and covariance as follows
#t = Rapo + Rejh + Ee RaC1,
(D.la)
Σ,
R R.E, R R.CR + R.C R. +T.
(D.Ib)
where C 100 with
Ö = ー (51 D 1/2 I),
D = (45) E, 11/2 1/2//2
2
(D.2)
E Kπ 1/2 Eo Ki
Ê = KTT k--1/25,, K
Theorem (SDE representation). Under the optimal P, the process X admits the SDE dynamics:
dx, = (1, X;; L. dt + 9r dW,, where fT 2;; L.) = S xx (x 14) + jt
(7)
with Pe, Ee the mean and covariance of Xx lef. Corollary D.1] and we have
= P. Q + H₂K Kn r.
(8)
with P. (RE + R,C,,, Q. -C, + E H, Ke + VT. where C the covariance of Xo, X1 in the optimal Poi.
D.1 PRELIMINARIES FOR THE PROOF
We first introduce the following three lemmas and the definition of infinitesimal generators.
Lemma D.2 (Central identity of Quantum Field Theory [Zee, 2010]). For all matrix M > 0 and all sufficiently regular analytical function (e.g., polynomials we have
C (IR with compact support),
(2x) (det M) exz)
/
(-2x"
T Mx dr exp
1 0o M
1-
v(x)
(D.3)
2
nd
fora differential operator where exp(D) 1 D2 + .or
Lemma D.3 (Conditional Gaussians). Let (Yo. Y1) ~ N
[mo]
Ego
Eo1
Then, YolYi
[1
Σ10
Σ11
Gaussian with
EY Yi nn En EnIEiI (y m1); and Cov (VooV y) Eoo Eo1 E-1511- E10-
(D.4)
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Definition 4 (Infinitesimal generator of stochastic process). For sufficiently regular time- dependent function (tt, z) E IR x I + R, the infinitesimal generator of a stochastic process X, for o(t, 2) can be defined as
BlO(1 + h, Xx+ 2] ((1,)))
x,)
lim h-0
(D.5)
For an Itô process defined as the solution to the SDE
dX, = ft, X₂) dt + g Xi)dW,
(D.6)
with f(t, IR x R ,", the generator is given
Aco(t, x) = 0,) + f(t, x [(, + = Ap(t,a))
(D.7)
where A V2 is the Euclidean Laplacian operator.
D.2 OUTLINE OF THE PROOF
Our proofs follow the idea from Bunne et al. [2023, Theorem 3].
For Theorem We follow the following two steps:
1. We first solve the associated static SBP Specifically, we formulate the equivalent E-OT problem, which has the transport cost dependent on the transition kernel of the TSDE. By introduc- ing new variables, we can convert this involved transport cost to quadratic cost over new variables, thus, converting the GTSBP static classical Gaussian E-OT. Based the existing results [Janati et al., 2020; Mallasto et al., 2022], we can then obtain the optimal coupling over the transformed variables. The coupling
the original variables can be recovered via inverse transform.
2. From Theorem A.4, we can obtain the solution of GTSBP based on that the optimal P is in the reciprocal class Q7. specifically. by composing the static solution with the Qr-ddidge. This an optimality condition obtained from the reduction the static problem. From that, we know that the solution is a Markov Gaussian process and shares the same bridge as Q7 [ef. Proposition A.5]. This further allows us characterize the mean and covariance of the time-marginal.
For Theorem 7. Let P P(2) be finite-energy diffusion [Föllmer Wakolbinger, 1986]; that is, under P. the canonical process X has (forward) Itô differential. Furthermore. since P is in the reciprocal class of Q7, has the SDE representation the class of (7) from the SOC formulation where the drift f x I.) is to be determined. We then proceed the following two steps:
1. For the SDE (7) of the optimal process Xt, first compute its infinitesimal generator [Protter, 2005] for a test function o(t, z) E R+ x RR - R by definition using (D.5).
2. Second, we express the generator in terms of its given solution in (D.7) for the SDE (7)
Alo(t, x) dto(t, f ottt
2 V (t,,
(D.8)
By matching the generators computed in both ways, we then obtain the closed-form of the drift term.
D.3 DETAILED PROOF OF THEOREM 6
D.3.1 STEP 1: SOLVE GTSBP STATIC
First, recall that Y Yo is Gaussian process with mean m, == E(Yilyu) and covariance Ke [cf. cond. mean and cond. cross cov]. respectively. Thus, we have the transition probability density
@7110(11800) x exp
2 (y11 Vivo - 51) T (y1 - Vivo - E1)
(D.9)
2 (311 m1)² KKT (21 ma))
x exp
By introcuding the variables Y, K ( Yo + {1) and Ý K tr, we have
QT1j((1180) x exp
11 - 5o 1?
(D.10)
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Furthermore, if the joint distribution For has marginals Yo Mo and Y1 V1, ~1, then after the change of variables (Yo + Yo and Y Yi), it gives rise joint distribution Poi with marginals Yo Po NWo, Σo) and Y1 V1 = =m E1), where
po = K 2 (Vimo + Et),
=o= -==Ki v 112000 V K
(D.11)
A = K # M1,
Σ = k
That is, there an one-to-one correspondence between Puz and Poz. This allows us to expand the objective of GT SBP static in terms of mininization follows
{ R"*R"
Poi (yo v1) log (Vo v1)
DKL (Po1 (Po1l(QTo))
Qroi (No. (0o.)10)))) v1)
dyo dy:
/ log (Qoo (Yo, v1 olP (yo, un) + / log(Po1) o
2 / [10 doll ol (vo: 91) + log(For) dF + const.
/
I Toll dlP (To 11) / log(For) dF.1 + const. 2
DKL (Poz [Poz/IQ2ri (u7)))
where the second last step results from f (Fo(F dio.1 log((Por) dlo1 const.. To obtain (+), we notice that Qroi (3o. v1) = Q0 (31 (30 Qro (vo). and we have
log(QToi (vo, yn dPoi (vo.V1) /
log(QT11031 (36)) oF (yo. yi) - / log(QTo(Ba)) dP (yo. v1)
/ log Qru(vo) dPo(3o)
(og(QTio(Bulto)) dF (yo. v1)
where the last equality holds since we can remove the dependence on in the second term by inte- grating over yı, thus, appearing constant in the optimization over Poo Moreover, the expression (+) is in fact the equivalent E-OT associated to the SBP
/ l1 130 =E11 dl'oi (3o, v1) + / log((Po)) dl
min Foi 2
(TE-OT)
Note that by definition we have DRL (Pooi |PP0 & vI) f log( dlPo: J log(Po @ v1) dPoi (og(Foo) dlPo1 f log 2o dvb f log 11 du where the last two terms are constants.
Thus, solving GT GTSBPstan statie is equivalent solving the following problem
|l Jo 22 di (To, ūI) + / log(Pot) di
min Por (R()*
DKL (Poi
(D.12)
with P. = in and i2; = D1 This is a classical static Gaussian E-OT between iny and i with a = 1. The closed-form solution is given by the joint Gaussian [ef. Theorem A.IJ
jo 11
Po = -
(D.13)
δT
where
Ō= Ō=
D 1/2 1),
D= = (4E+) /451/2 " 53/22 + 1)
(D.14)
Finally, via the inverse transforms Yo = (K solution to the original problem GT statio
- E1) and Y = K K2
U1. we can obtain the
[g
Σg
C
1 ~ N
(GT SB static)
μ1
C
Σ
where C = 1 =
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D.3.2 STEP 2: FROM STATIC TO DYNAMIC VIA DISINTEGRATION FORMULA
From Theorem A.4, know that the solution P to GTSBP shares its bridges with the reference QT. We denote by Q77 the process Y conditioning on Yo = and Y1 = under QT, i.e., Yl8o. y1 Qu Q (Yo yo. Y1 yıl. It the bridge of Qr. following
Qr())
Qu (-)Q (dya dw1).
(D.15)
In the classical case of Brownian motion Y W Qw Q is often referred to as the Brownian bridge. Here, we aim to first find the Q7 -bridge, and then construct the optimal solution IP* by composing the static solution Foi with the Q [cf. (A.4) in Theorem A.4].
From the transition kernel in Lemma 4, we have the conditional distributions Y, 3o ~ N(mm K(t) and yy (M(m,, K1). Thus, the joint distribution of Y, and Yi given yo follows
me mm
Krr Kix
Kri
Yiloo N
Kn))
K11
(D.16)
Applying Lemma D.3, we know that Yı/3o: Yi 31 is Gaussian with mean
E(Yr|yo, Yi vi) m4 + Ke1 Kn (w1 -mi)
V.ya + E. Kn KinKKi (y Viyo (1) (V, KnKii 1 + Ki y1 + Ec- Kn
(D.17)
A R,y0 + Reyh + E. - R
where we recall the definitions of RG and R2 in (8), and covariance
Fe == Cov(Yi)Yo = yo. Y1 y/) Ku KinKKi' KIe-
(D.18)
Since Gaussian process is completely determined by its mean and covariance, we have
Y1
law R₂ % + R + & R4E + r Q
(D.19)
where Z ~ (0, I) is independent of Y. Now, the Disintegration of Measures and Theorem A.4
allow us to construct the solution to GT SBP by first generating (Xo, X1) P. in GT GT then connecting Xo and X1 using the Q -bridge. This is equivalent to, for Xo ~ Vo. X1 ~ 1/1 and ~ (O, I), 7 (Xo, X1), building process as
X₁ law X + R +IZ
(D.20)
which in fact is stochastic interpolant for stochastic processes over topological domains, gen- eralizing the same notion in Euclidean domains Albergo et al. [2024, Definition 1]. Note that since Q7 is stochastic process following Itô SDE, which is a Markov process, the solution P also a Markov process [cf. Proposition A.5]. Finally, we obtain the mean and covariance of the time-mann mar-mnali X;
H4 = Rejoo + Rep1 + t RI51.
(D.21)
Σ, R,E,R + R,E,R, + R,C,, + R,, R Ku KuKKi' Ku--
This concludes the proofs of Theorem 6 and Corollary D.I.
D.4 DETAILED PROOF OF THEOREM 7
D.4.1 STEP 1: COMPUTE THE INFINITESIMAL GENERATOR OF X, BY DEFINITION
For some time-varying function o(t, x:), by definition, the infinitesimal generator of X, is given by (D.5). Since X, is a Gaussian process, we could express the conditional expectation using Lemma D.3. As we are only interested in the terms that are of order O(h), we then ignore the higher-order terms. First, compute the first-order approximation of M, in (D.1)
E R. + R.E + R, + R.E +R.RR + R.C.R + +R, RT + R,, R + H0 Ko.KK))TTTK Kne -Ken ) 1r,
(D.22)
(P. + +P) - (Q: + Q,) + 0,Ku
Kie (KrIKT )
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where the last equality we recall the definitions of Pt and Q in (8). Next, denote by Σi,+hh the covariance process of X₁ evaluated at t and t + h. We can estimate E1,44h up to the first order of hh)
Eiith E - - (Xth [
= Ro + R, + R R.C R44h + K1tth - KnK K1,thh
Et + Ro (Reth- RO) RR R₂) + RRC(RRth -R
+R.CT (Reth -R,)_ + (Ki,th Ku) KKiKii (K1,th K1e)
(a) Σ, + MR RI + R, + R,CRT + R.C R. + o(h)
+ (Krrth - K(t) - Kei KIi (K1,th -Ki))
E, + M(P: Qi + +O.2 Ke. - KKiK
LaiKiiOra 0r, K / ===1)===)) + o(h)
(D.23)
where we obtain (a) by plugging in lim (R(-h - Re) = RS and lim A (Retn - Re): and
likewise, we obtain (6) by recognizing the definitions of P. and Q in (8) and using the partial derivatives
lim 1 (Ketth Ke,)) Or K to 4 52
h-0 h
(D.24)
lim (K1,th> - K1,1) 0r K K 0,Ku,. t1 > tz. A-0 h
Following the similar procedure, we can obtain a first-order approximation of Eith,t as
Ex+h.t =- E[((+++ (X+ 1
=E. + R. + RRE,R
+ R,R,, +,CRR, + o(h)
(Kithht Ku) (Keth,1 Kn)
= =E,+M(P, - -Q + Ür, Ka,,tall 0i, ko K1t) + ooh). (D.25)
where the partial derivatives should be understood as
1 (Kethit Ke,)) O2 Ke 11 1>66 l2
lim h-+0
7
(D.26)
lim (Kr+h,1 K1,1) O., Ke 0,Kr1, ti S. t2. h-0
Since Eqs. (D.22), (D.23) and (D.25) are all involved with the partial derivatives of Ke we can compute them by the closed-form of the transition matrix as
Üi, K2 = 0n Vtn
ge 2 -2
d8
T
for l1 S L/
1/
t1
v., Hes
22 v² ds
VI
gi,
2 4 11 TT
(D.27)
(b) Hi, Kestts + oh Wo -. WT
for 41 S t2
where we use the symmetry of v,, At (a) we use
a.4. exp (f, H, H,ds)) = (ods))=xx/]/ expl cxp(S H, ds) H. = V,
(D.28)
and at (b) we use the commutativity of He and V, [cf. Lemmas B. and B.5J. Similarly, we have
0e Kenes 0r2
V
2 -- ds
v
Hi for t <
0₁ Ki,12 = 0c,
Vin
g W
ds
v to
Hi, Kenta
for t > tz
(D.29)
pta
0, K2ntz = 012 V
g
ds
V T
1/
= o2, V., Vi + Knns He for 1 > 12-
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We notice that (0,K1)) 01K11 v I + H K1c. Now, by introducing in (D.23)
the variable
S - Pt Q₂ 21 (2
Kn K 11110 0t, (2 11=,
P. Q + Ku H Kn (0} v v,¹ + H)
(D.30)
= Pt- Q + KuHt - Ki (0,K (1:KKi))
we can then express the covariance process as
Er,tth = =2; + hS, + o(h),
(D.31)
and
14tht = thPE - -Q: + H Ku - (0/ V. + ) K1t) + o(h)
h(Pp Q: + H1 Ku (0rKa)KT K1.) + o(h)
(D.32)
+EithS) + o(h).
Lastly, using Lemma D.3, we see tha variable X X = x is Gaussian process with mean
N4th = = M4th + Eeth,t 2 (x - jie)
#4 + hipt + (Σ, h.S } (x - (-) (hh)
(D.33)
Mt + hip + + h.s Σx 1 (x- 14) +))+0l ooh)
= x + h(j4 - S. 24 (x - pc) + o(h)
where in (a) used Σ, Σj and covariance
Σ14, Eith E1th,E E Eg,4hh
= E, + nε, - -(I+ + h.S }r, " hSS) + oh)
(D.34)
Σ, + +5. - + hSS + h.S.) + o()
-.SS,)) ( 2 - -S..) + o(h)-
By seeing Ke as matrix function of have
ogKu = 0e { 11
2 v
ds V = HeKu + I8 + KuH,
(D.35)
This reduces (D.22) to
(P P:) (Qr + Q.) 0.Ku (0.Ki1) Kn (KndT')00
(D.36)
where the last two items appear in ST. and S., respectively. Thus, we obtain
Eeth = -($ St - -S,) ohh)
-h(1(PT + P.) (Q, QT) + O.K.. - (T. Kic (Kn (Kni
[Pt Q. -k,lt KIi KTn (T.Kiil)
(D.37)
[PTT Qi + H Kut (0,Kn) KT Ku]] } ohh)
= hgg ohh).
We can now compute E(ø(t + h, Xx+1)) = x] as follows
Eld(t + ht X Xx = x]
-.a')-exp(( (2T) (det Ee - f o(t h h,22) exp( (- - m(+n)) Diah Betth) de (D.38)
(o'
Jnn
(a) (2T) ( (det Eeth)
olt + h meth) expl
- T dē
gm
2
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where in (a) we apply a change-of- varriable I = x' -ett We further apply Lemma D.2 and arrive at
Eld(1 h, Xeth) = x]] = exp [50
E Exth )=)))))(((1)))))))))))))1 )(1 + h + petn)) 1=0
(1 0 h + o(h.o.t.)) o(t + h, + latn)lã=o (D.39)
2
olt + h. mcth) + Dolt + h, Pc+h) + o(h), (
where we expand the power series of exp(,50,, dz) and ignore the higher-ordeertermms in (a). and plug in (D.37) in (b). Recalling N444 in (D.33), we can expand the Taylor series of ((t+h, (e+.) in the second variabel
d(t + h, (cth)) = Φt (t(+t + h x + h(j + S D (o - pc)))
(D.40)
((t h, + ot + h, in + S " (x++)) ((h)
Therefore, we have
Bip(1 + h, X = x] @(t + h)) ((Vo( h, 2²) pr S STE, Ex -e)) + 1
(D.41)
2 h Ao(t h, + o(h).
Now express the infinitesimal generator of X,
ElØ(1 + h X Xx+n)Xx, = x| (((,)))
lim
h-00
h
ull + h x) u((,x)
-lim
(VV(t, No 5 14) + of At((,
(D.42)
h-0
h
=d,u(t + (Vo(t,z), S ET 'x (x)) Ad(t
D.4.2 STEP 2: MATCH THE SOLUTION OF GENERATOR FOR AN ITÔ SDE
From Léonard [2014]; Caluya & Halder [2021], we search for the optimal solution to TSBP within the class of stochastic processes following an SDE:
dx,- ft, X,) dt gr dW,
(D.43)
Recalling the solution of an infinitesimal generator in (D.7) for this SDE
1 7 Ao(t,
A,o(t, x) 0,t(t,z) fT(t, V (tt
(D.44)
then match it with the generator obtained by definition in (D.42). We observe that the two equivalent we set
(T, x) = jar + ST ET pr).
(D.45)
This concludes the proof of Theorem 7.
D.5 CONDITIONAL DISTRIBUTION X,
Corollary D.5 (Conditional distribution of X, (o)). Let X, be the stochastic process associated to the solution IP SBP Given an initial sample Do Vo. the conditional distribution v vX,,XXo = uo) ~ (Mpo. Egjo) is Gaussian with
Mejo R400 RA41 R.. (xo - 1o) + R&EI,
(D.46)
Σnjo = R. R.C -. CR, + K K11 KnkKil KIC-
Similarly, given a final sample vi, the conditional distribution v(X,|X1 = x1) N Egn) is Gaussian with
Maj1 R4æ1 Rapo R,CE (z) p1) & RAEI,
(D.47)
En1l R.E.R. R.CE CT R.T + K Kn KnITT Kic-
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Proof. First, recall the stochastic interpolant expression in (5) of the solution to GTSBP and its mean [4 and covariance Σn in (D.1). Due to the Gaussian nature of the process, we can write the joint distribution of X, and Xo
X2 Xo
N
Me
Σe
Er, Ep
(D.48)
Eo,e
where we work out the covariance between X, and Xo below
Σ., E[(X; - (Xo -oo'
R,Eo + R.Cov (X1, Cov Xo) R,Cov(Ei, Xo) Cov(Gi Xo)
R,Eo + R2 C,.Cov(E (X1, Xo)
(since & is deterministic, 1 Xo)
R.Eo + R.C
(by IP G7 statie)
Based on Lemma D.3, know that X1 zo is Gaussian with mean Mejo and covariance Eijo given by
N4 14 + -r, (zo (4) Σ, E. - En -0,t-
(D.49)
By substituting the expressions of l4 and M, from (D.1) and canceling out terms, we complete the proof for X, zo. The proof for X1 I1 follows similarly.
E TOPOLOGICAL SB GENERATIVE MODELS
Here, we provide more details the SB-baseed models. First, we give the likelihood of the model which allows for the training objective in (9), and the probability flow ODEs corresponding to the FB-TSDEs in (3). Then, discuss the variants of score-based and diffusion bridges models for topological signals, as well as their training objectives, with the goal of illustrating how BB-based models connect these models.
E. LIKELIHOOD TRAINING FOR TOPOLOGGICAL. SBP
The likelihood for the Euclidean SBP by Chen et al. [2022a] extends to the topological case. Corollary E.1 (Likelihood for TSB models; Chen et al. [2022al). Given the optimal solution of TSBP satisfying the FB-T system in (3), the likelikiihood of the TSB model at initial signal sample vn can be expressed as
1 12 V (g. fi) +111++2122
CTsa(ao) = Ellog (X1))
1
2, Ze dt (E.1)
where the expectation taken over the forward SDE in (3) with the initial condition Xo = zg. Corollary E.2 (Probability flow ODE for TSB). The following ODE characterizes the probability flow of the optimal processes of TS in (3)
1
dx, [fo + 9.
2 5.((Z 2,)] dt
(E.2)
(E22) i.e., the time marginal the path measure P is equal the and we have that for all t, P, Pt probability flow Pt of this ODE.
This is direct result from the probability flow for general SB [Chen et al., 2022a], which extends the probability flow for score-based models [Song et al., 2020b], and relates to the flo-based training.
E.2 SCORE MATCHING FOR TOPOLOGICAL SIGNALS
As discussed Section 5 and by Chen et al. [2022a]. SB-based models generalize the score-bars based models [Song et al., 2020b]. Here. provide detailed derivation how ccore-based model can be built for topological signals, specifically, on the score matching objective, since there is no direct literature this. First, show in detail that the likelihood training based (9b) returns score
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matching objective for topological signals when Z, 0 and the final Vi is simple Gaussian. The backward training objective in this case becomes
1<zu;; 0)
! (3a)
1 + ge 2 Xo 20 dt
[. X3a (3a)
1 [
+ 2 sx(00] Xo = xr] dl
where we introduce a score function s,) to approximate V log (, = zo), following 2 grs, (0). Here, Pajo is the transition kernel of the TSDE [ef. Lemma 4]. By using the trace estimator Hutchinson [1989] to compute the divergence, i.e.,
(() Eu- (0))u,,
(E.3)
and setting the weighting function A(t) == gf we then obtain the sliced score matching objective [Song et al., 2020a;b, Eq. 19] for topological signals based on TSDE, which has the form
i = arg min Eu-u(u,y)
To. A())En, En, Eu,A(( 22 sr. (0)1) + +u
sT sr(0)u)
(E.4)
and is equivalent to (an; 0). This does not require closed-form solution for the true score function v log Pejo- The associated FB-TSDEs now become the forward-backwor processes for the score- based models
ax, = St + Gr dW::
(E.5)
dx, = (J-- (0)) di + de dWW
(E.6)
Closed-form score matching For TSHeattM and TSHaatve, since we have their closed-form transition kernels in (2), we can use the direct score matching objective [Song al., 2020b, Eq. 7] to train a score-based model for topological signals
i arg min (0- A(t) Ear [|se(() - V [oggrio(x)/20011
}
(E.7)
where V log Pojo can be readily obtained based on (2).
E.3 DIFFUSION BRIDGES FOR TOPOLOGICAL SIGNALS
As discussed earlier, SB models are closely related to stochastic interpolants, flow- and score-based models. We further remark that from the TSDE, we can build the topological diffussion bridge to directly construct transport models between any topological distributions via Doob's 's h-transform [Särkkä & Solin, 2019]. This has been evidenced in Euclidean domains by converting existing diffusion processes (BM, VE, VP) to diffusion bridges so to arrive at arbitrary distributions, and training upon score matching (Heng et al., 2021; Liu et al., 2022; Delbracio & Milanfar, 2023; Li et al., 2023; Zhou et al., 2024].
Specifically, consider the TSDE. To let it arrive at final sample x1, the Doob's h-transform gives dX, [f+ log Prje (21 (21 [X.)] dt + ge aW, X1 F1, Fg 1/p
(E.8)
where (ii(T1 ((t)) is the transition kernel of the TSDE satisfying the associated backward FPK, given by Lemma B.6 (cf. Lemma 4). We can further find the time-reversal process for (E.8) [Zhou et al., 2024, Theorem 1]
dx2, = [ft - g (V log q: (21/21) v log (i)e(21))))) dt + ge dW, X1 I1
(E.9)
where 21 (are (21) is the transition kernel of the new SDE in (E.8) (instead of TSDE) conditioned on Y = F1. The goal is to learn this new score function V log (nn (x)l=1), which can be achieved by applying the score matching [Song et al., 2020b].
Given paired training samples (Zo. 21) qu1 Zhou et al. [2024, Theorem 2] considered score matching objective to learn the new score function
= arg min Ez. ((0)11a((0) - V log dep1 (2.l21)) 11
(E.
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However, we cannot directly find closed-form 4:11 in the topological case, need to sliced score matching for this case.
Note that can view that the underlying topologiaal process X now follows the new SDE pair (E.8) and (E.9) as the forward and backward processes, respectively. In this sense, the topologiaal diffusion bridge is special case of the TSB when setting the policies Z gt V log Pije (ate)
and 2, = 9r [log 20j1 (3re (x1) - V log (1 [z)) When performing learning on these policies, since Z is fixed once TSDE given, the learning boils down to training the parameterized 2/.
F ADDITIONAL EXPERIMENTS AND DETAILS
We first describe the synthetic experiment on matching Gaussian topological signal distributions based on the closed-form GT Then, we detail the generative modeling experiments conducted on real-world datasets based on -Soeml.sss.
F.1 CLOSED-FORM GTSB CORROBORATION
Graph GP matching: We build a synthetic graph with 30 nodes and 67 edges, shown in Fig. (Left). From its graph Laplacian L, we construct the initial distribution of node signals as Matérn GP with zero mean and the kernel 2. (1 + L) -1.5, and the final distribution as diffu- sion GP with zero mean and the kernel Σn (5) driven by both THeatgg and TSHeatyr. For the former, we set 0.5 and g 0.01, labeled as GTSB-SSBSSEE BM. For the latter, we consider anin 0.01 and mmax 1 with e 0.01 and 10. labeled as GTSB-VEI VE1 and GTSBGVSSVVV VE2, respectively.
(0L). We consider GTSB closed forms X, in
We then compute the covariances E, of the time marginals, which has a closed-form given by Corollary D.I, and obtain the samples based on the closed-form conditional distribution [cf. Corol- lary D.5] given an initial sample, illustrated in Fig. F.I. We also measure the (BW) distance [Bures, 1969] of Σ, and Σ₁ to evaluate the bridge quality, shown in Section 6.
Edge GP matching: We also consider matching two edge GPs which are able to model the dis- cretized edge flows in a SC2 of the vector fields defined on a 2D plane [Chen et al., 2021b]. The initial edge GP has a zero mean and divergence-free diffusion kernel with Ke 10, while the final edge GP has a zero mean and a curl-free diffusion kernel with KG = 10 [ef. Appendix B. 1.. We con- struct the closed-form GTSB X, with the TSHeatmM the reference dynamics, where c 1 and 0.01. We obtain the samples from the closed-form SDE representation (7), shown in Fig. F.2. We can see that the forward samples able reach the final state, and the backward samples are able to reach the initial state, despite some noise due to numerical simulation.
F.2 TSB-BASED GENERATIVE MODELING AND MATCHING
F.2.1 DATA
Heat flows: We use the heatflow dataset from Southeastern Australia from Mather et al. [2018], which collects the heatflow measurements with coordinates in total 294 from 1982 to 2016. Here we split the data into two parts, before and after 2010 (there is significant change in the heat flow pattern), to understand the evolution the heat flow by modeling them as initial and terminal data. That is, for this dataset, we consider the signal matching task.
Seismic magnitudes: We use the seismic event catalogue for M5.5+ (from 1990 to 2018) from IRIS which consists of 12,940 recorded earthquake events with magnitudes greater than 5.5. To process these events, we use the st py toolbox to obtain the icosahedral triangulated mesh the Earth surface [Moresi & Mather, 2019]. Using the refinement of level three, this spherical mesh has 1,922 vertices. We refer to Fig. F3 for a visualization such a mesh level one for better clarity.
Upon this mesh we first associate each earthquake event to the nearest mesh vertex based its longitude and latitude. All events are located on 576 unique vertices the mesh. Using these unique vertices, then construct 10-nearest neighbour graph based the geodesic distance between the vertices, and we use the symmetric normalized graph Laplacian. Lastly, on top this graph, we associate the yearly earthquake events its vertices and take the magnitudes node
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t=0.0
t=0.2
t=0.4
t=0.6
0.8
t=1.0
t=0.0
t=0.2
t=0.4
t=0.6
t=0.8
t=1.0
Figure F.1: Covariances of the time marginals of the GTSB driven by TSHeatgm (Top) and SHHatve (Center), well the samples conditioned on the inisital signal (Bottom).
t-0.0
t-0.20
1-0.40
t=0.60
t=1.00
Figure E.2: Forward and backward samples based on the closed-form SDE in (7) with respect to TSHeatum-
5
Figure F.3: Earth mesh (Left) and node signal sample of the earthquake magnitudes (Right).
signals, resulting in 29 such signals. Followed by this, we preprocess the magnitudes by removing mean over the years. For this dataset, we consider the signal generation task.
Traffic flows: We consider the PeMSD4 dataset which contains traffic flow in Califomia from 01- 01-2018 to 28-02-2018 over 307 sensors. convert the node data into edge flows over a SC2 with 307 nodes, 340 edges and 29 triangles, following Chen et al. [2022b], and use the normalized Hodge Laplacian. For this dataset, we consider the signal generation task.
Ocean currents: We consider the Global Lagrangian Drifter Data, which was collected by NOAA Atlantic Oceanographic and Meteorological Laboratory. The dataset itself is a 3D point cloud after converting the locations of buoys the carth-centered, earth-fied (ECEF) coordinate system. We follow the procedure Chen Meila [2021]: Chen et al. [2021b] to first sample 1,500 buoys fur- thest from each other, then construct weighted SC₂ as Vietoris-Rips (VR) complex with around
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20k edges and around 90k triangles. For this dataset, consider the signal matching task Upon the weighted Hodge Laplacian, we use edge GP learned by Yang et al. [2024] from the drifter collected data as the initial distribution and synthetize curl-free edge GP as the final distribution. These two GPs have rather different behaviors, able to model ocean currents with different behaviors and make the matching task challenging. From these GPs, we can generate the samples for training and testing in an efficient way based on eigenpairs associated to the 500 largest eigenvalues, analogous to using Karhunen-Loéve typedeecompoition for continuous GPs.
Brain /MRI signals: We consider the Human Connectome Project (HCP) [Van Essen et al.. 2013] Young Adult dataset where we model the human brain network as a graph and perform the matching task on the measured IMRI signals recorded when the subject performed different tasks. We use the HCP recommended brain atlas [Glasser et al., 2016] where each hemisphere divided into 180 cortical parcels. This results in total of 360 brain regions. We then build graph based on the physical conenction patterns between these regions where the edge weights measure the strength of the axonal connections between two regions, i.c., proportional to the inverse of the square distance [Perinelli et al., 2019]. We use the symmetric normalized graph Laplacian. In our experiments, the two sets of the fMRI signals, respectively, correspond to the liberal and aligned brain activities. The former is associated with brain regions involved in high-level cogmition, like decision making and memory, whereas the latter is associated with the sensory regions, like visual and auditory, meaning that functional signals are aligned with anatomical brain structure, as shown in Fig. F.4.
Energy brttl Gignal
Energy Terrinal Signal
Day Day d6
staled
Duy 20-27
phanel
Figure F:4: The energies of the initial (liberal) (Left) and final (aligned) brain signals (Right).
Figure F:5: Two-dim phate embedding of the single-cell data [Moon et al., 2019].
Single-cell data: We consider the single-cell embryoid body data from [Moon al., 2019], which describes the differentiation of human embryonic stem cells grown as embryoid bodies into diverse cell lineages over a period of 27 days. These cell data, X2,........ ................................ Xs. X5, are collected at 5 timepoints (day 0-3, day 6-9, day 12-15, day 18-21, day 24-27, indexed by E {1, 2, 3, 4, 5)), resulting in total 18,203 observations. We followed the preprocessing steps provided by Tor TorchCFM [TTong et al., 2024a;b]. Please refer to this link for the direct use of preprocessed data [TTnng, 2023]. Followed by this, we consider the two-dimensional phate e embedding for the data [Moon et al., 2019], resulting in the data coordinates of dimension 18, 203 x 2, as illustrated in Fig. F.5, From the preprocessing, we can build a sparse k-nearest neighbouring graph over the entire set of data observations. That is, have an adjacency matrix of dimension 18, 203 X 18, 203.
In our experiment, we aim to transport the observed data from day 0-3 to day 24-27, i.c., from f to t = 5. Thus, we build the two boundary distributions based on the normalized indicator functions, which indicate the associated nodes of the data points observed at these two timepoints. That is,
(F.1) where the sum is over the nodes associated to the first-timepoint observations in X1, as the initial distribution, and similarly, V1 := Lx,/E,e, 1x, (f) as the final one. After training the models, using the final sample X obtained from the learned TSB given the initial observations, we can obtain the predictions the five timepoints based on the sorting (from large to small) of X,-s- Specifically, given the indices after sorting, idx = arg sort( we partition them into the dis- joint sets, idx = S U S2 U.... ... U S₁ with |S,] = ne the number of observations at timepoint t for t= 1....,,. 5. We then have the prediction labels given by S; that indicate the nodes supporting the data points predicted at timepoint t. The disjointed indices in St essentially provide labeling of the whole predictions for the five timepoints. We found that using adjacency matrix as the convolution operator in the reference dynamics performs better in practice.
Vo= = 1x,/ Ejex, 1x, (7)
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F.2.2 MODEL
Models. We consider the following two sets of methods:
Euclidean SB-based models with BM, VE and VP reference processes [Chen et al., 2022a], which we refer to as SB-BM, SB-VE and SB-VP, respectively.
Topological SB-based model with 7SHeafBM. SHeatvv and TSHeatvp as the reference pro- cesses, which we refer to as TSB-BM, TSB-VE and TSB-VP, respectivly.
For some datasets, we also apply the Gaussian SB SDE solution as the reference dynamics: Eu- clidean SB-based models with the closed-form GSB SDBs (under BM and VE reference processes) respectively;
as the reference [Bunne et al., 2023], which we refer to as GSB-BBBBBBBB BM and and, topological SB-based model with the closed-form GTSB SDEs (7) (under TSHeef and TSHeatye) as the reference, which we refer to as GTSB-BM and GTSB-VE. respectively.
Improving reference dynamies. Our proposed three types of reference dynamics have fixed diffu- sion rates c. This may limit the model flexibility in capturing the dynamics of the data, which we found especially in matching ocean current data. Thus, for this task, we allow the time-varying dif- fusion rate cz. Specifically, we set it to be linearly increasing C Cuin + [Cmaa Cmin) for some Cmin, Cmax. Moreover, due to the nonlinearity of the underlying process (from a non-curl-free GP to a curl-free GP), we also consider the heterogeneous heat diffusion, as dicussed in (B.18) where the down and up diffusion rates are different.
Policy models. For the parameterization of the optimal policies (4,, 2,), we first obtain the time and signal embeddings individually. To obtain the signal embedding from the input, we consider the following two sets of models as the signal module:
multi-layer perceptron (MLP) followed by number of residual block modules where each block has three MLPs with sigmoid linear unit (SiLU) activations.
ResBlock model:
Topological neural network (TNN) model: For node signals, we consider two-layer GCNs [Kipf' & Welling, 2017] followed by one MLP; For edge flows in a SC2, consider two-layer SNNs [Roddenberry et al., 2021; Yang et al., 2022a] followed by one MLP, where each SNN layer has the linear convolution X <- XW, XW, + L,WW, with the down and up Laplacians Ld, L,, and the learnable weights Wo, Wi, W2.
To obtain the time embedding, we pass the sinusoidal positional encoding of the discretized time- point through a two-layer MLP module with SiLU activations. We then sum the two embeddings and pass it through two-layer MI.P output module with SiLU to obtain the final parameterization.
F.2.3 IMPLEMENTATION DETAILS
Our implementation is built upon the SB-framework by Chen et al. [2022a]. We Adami opti- mizer with a learning rate of 10-4 and Exponential Moving Average (EMA) with the decay rate of 0.99. For the reference processes with BM involved, we treat the noise scale g hyperparameter and optimize it by grid search. For the reference processes with VE and VP involved, we grid search the noise scales mmin; mmax and Bmin; Bmax- For BSS-based models, we grid search the optimal diffusion rate e and the noise scales involved in the T SHeat.
In computing the likelihood in (9) during training, we use the trace estimator following [Hutchinson, 1989] to compute the divergence. In generative procedures, we apply the predictor-corrector sam- pling [Song al., 2020b] to improve performance. To evaluate the models, we compute the negative log- niglikklss (NLLs) in (E.1) for generation tasks. For both generation and matching tasks, we assess the and (square rooted) Wasserstein distances between the predicted and true signals.
F.2.4 RESULTS
Heat flows: For matching the two types of heat flows, we observe from Table F.1 that: (i) TSB- based models are consistently better than SB- SB-based models; and (ii) using GCNs for policy models increases the performance by a large margin for both sets of models.
Seismic magnitudes: In generative modeling for seismic magnitudes, while we have the similar observations as for the previous datasets, from Table F.3, we also observe that the G7 models are able to achieve the best performance, and likewise GSB-based models also increase the
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t=0.0
t=0.2
1=0.9
t=1.0
1=0.0
t=0.2
h00
t=0.9
t=1,0
t-0,0
t=0.2
t-0.7
t=0.9
t=1.0
Figure F.6: Backward sampled ocean currents using TSB-BM (Top), SB-BN (Center) and CTSB-BM
(Bottom).
Table F.1: Heat flow matching results.
Table F.2: Traffic flow results.
ResBlock
GCN
Method
Method
ResBlock
SNN
Forward
Backward
Forward
Backward
S3-BM
0.82 :: 0.00
0.18 :: 0.02
SB-BM BM
-0.10 + 0.02
-0.08 = 0.03
-0.74 + 0.05
-0.72 + 0.05
83-VE
0.77 4 0.00
-0.42 : 0.01
SR-VE
-0.12 + 0.02
-0.10 + 0.01
-1.20 + 0.07
-0.95 + 0.07
83-VP
0.79 1
0.000 0.00
-0.09 1 0.01
SB-VP
-0.09 + 0.02
-0.08 = 0.02
-0.83 + 0.04
-0.66 = 0.11
TSR-BM
0.40 + 0.00 0.00
0.02 + 0.03
TSR-BM
-0.29 + 0.02
-0.27 + 0.02
-0.83 + 0.05
-0.81 + 0.05
TSB-VE
0.01 + 0.00
-0.89 = 0.02
TSB-VE
-0.31 +: 0.02
-0.29 +: 0.01
-1.26
+ 0.05
-0.97 + 0.08
TSB-VP
0.02 1 0.00
-0.32 1 0.01
TSB-VP
-0.57 1 0.02
0.02
-1,01 1 0.03
-0.92 0.03
Table F3: Seismic magnitudes results.
Table F.4: Ocean current matching results.
Method
ResBlock
GCN
Method
Foward
Backward
SB BM
2.78 + 0.01
2.71 + 0.03
SB-VE
2.97 1 0.03
2.73 1: 0.05
SB-2M
7.21 1: 0.00
7.21 1: 0.00
SB-VP
2,28 + 0.02
2.01 + 0.03
SR-VE
7.17 0.02
7.17 0.02
GSB-BM
1,86 0.02
1.83 0.05
GSB-BM
1.09
0.01
0.97 0.00
GSB-VE
1,68 0.03
1.46 1 0.07
GSB-VE
0.83 0.01
0.49 0.00
т3в-BM
2.13
1.82
ТSB-BM
0.01
3,70 0.00
TSB-VE
2.22
0.02
1.53
TSB-VE
0.00
3.60 0.00
TSB
2,00 + 0.02
1.51 0.02
GTSB-BM
1.09 0.01
0.97 1 0.00
GTSR-RM RM
1.58
+ 0.01
1.43 0.04
GTSR-VE
0.53 0.00
0.47 0.00
GTSB-VE
1.49 0.02
1.06 0.04
performance of SB-models. In Table F.5, we report the 1- and 2- 2-Wassrsstein distances between the generated samples and the true ones.
Traffic flows: In geneartive modeling of traffic flows, we observe from Table F.2 that: TSB-based models achieve smaller NLLs and using SNNs for both models improves the performance. This observation is consistent with the Wasserstein metrics reported in Table F.5.
edge GPs. note that the initial sample in the forward process in Fig. 4 and the final sample in the backward process Fig. E6 are the true samples. From these two figures, we first observe that SB-based models fail to learn the dynamics to reach the expected end states, shown in Figs. 4 and F.6. On the other hand, T TSB-based models are able to reach an end state with small curl component in the forward process, yet with some discrepancy from the true one. Moreover, the learned backward dynamics remains noisy and does not completely return to the initial state. This implies that the underlying dynamics cannot be fully captured by the Seeatt-ype reference processes. This can be largely alleviated by using G7 SB-based models, where both the forward and backward processes reach the expected states with high fidelity. This is however because we have the initial and final distributions
Ocean currents: In matching ocean currents with two types of different-behaweee
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Table F.5: Overall 1- and (square rooted) 2- 2-Wasserstein distances for generating and matching.
Seismic magnitudes
Traffic flows
Brain signals
Single-cell data
Method
Wi
W2
Wi
W2
Wi
W2
Wi
W2
SB-BM
11.73
8.29 10.04
18.691002
13.361001
12.08
8.58 1,08
1.001
0.40
SB-VE
11.49 2000
8.13
19.04
13.6
17.46
12.42.0.09
0.33
0.39
SB-VP
12.61 20.06
8.92+0.04
18.22 18.22
13.02
13,41
9.54
0.33
0,40
TSB
9.01 +000
6.37 +0.03
=0.02
7.02+
7.51 =0.08
5.51 2006
0.14
0.28
TSR-VR
7. 7.69 +004
5.44
10.51 00002
7.58+001
7.59 =0.05
5.55
0. 4..02
0.27
TSB-VP
8, 8.40
5.95
9.92++0.02
7.16+001
7.67
5.64
0. 0.14
0.22
0.0s
1.0s
-50
-50
-00
-50
-50
-50
-50
50
-50
50
-50
50
-50
0
50
-50
0
50
-50
50
X
X
X
0.0s
0.2s
1 0.45
t 0.8s
t- 0.8s
1.0s
50
50
50
50
50
-50
-50
-50
-50
60
0
50
X
X
Figure F.7: Intermediate samples of brain signals learned using SB- (Top) and TSB- VE (Bottom).
modeled by GPs, allowing for G7 SB to capture the underlying dynamics better. These observations are reflected in the square-rooted 2-Wasserstein distance results in Table F.4 between the samples given by the learned forward process and the true ones, as well as for the backward ones.
Brain [MRI signals: In matching the two brain fMRI signals, we observe from Fig. F.7 that TEB-VE- based model reaches the final state where the signals have lower energy over the brain, indicating aligned activities, whereas SB-VE fails do so. This quantatively reflected in terms of the Wasserstein metrics between the generated final samples and the groundtruth ones in Table F.5.
Ablation study on graph normalizations. Here, we compare the performance of the TSB- based models using different ways of graph Laplacian normalizations. Specifically, we consider the ran- dom walk LRw and the combinatorial Leomb graph Laplacians. For the latter, we normal- ize it by dividing the maximal cigenvalue of the Laplacian for stability. From Table F.6, no- tice that using the random walk has comparable performance with using the symmetric normal- ized Laplacian Lyy and using the combinato- rial one is worse than the other two. This is not surprising since the combinatorial one does not encode the connection strength between brain regions.
Table F.6: Ablation study results on graph nor- malizations for brain signal matching.
Method
TSB-BM
TSB-VE
TSB-VB
W., Laym
7.51 1 0.08
7.59 0.05
7.67 0.11 5.64 + 0.09
W2, Jgym
5.51 + 0.06
5.55 + 0.04
W. Law
7.51 1 0.08
7.62 : 0.09
7.65 1 0.09
W2. Law
5.52 1: 0.06
5.58 : 0.07
5.62 i 0.06
W. Lromt
8.06 + 0.05
9,21 = 0.06
9.29 + 0.05
W2, Lecmnts
5.80 1: 0.04
6.62 i 0.05
6.73 + 0.03
Single-cell data: We first measure the Wasserstein distances between the predicted single-cells and the groundtruth ones at the final timepoint, as reported in Table F.5. We here provide the predictions in the two-dim phate embedding space for the SB-BM and TSB-BM models in Fig. F.8 and the latter has a much better prediction. Moreover, from the final sample, we evaluate the predictions at the intermediate timepoints (see Appendix 12211) in Table E.7 where the performance of TSB BM is consistently better than SB-BM. Since our method relies on a graph constructed from the entire data points, we also provide the leave-one-our accuracy for S3-BM by training on the entire data points leaving out the to-be- predicced timepoint. We that while the accuracy for the final timepoint is perfect, the intermediate predictions remain poor. In contrast, topology, captures the underlying dynamics and predicts the intermediate states better.
by making use of the
41

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Published as a conference paper at ICLR 2025
Day Day 26-27
Day
Day 26-27
Day 20-27
Figure F.8: Single-cell two-dim phate embeddings of the observations (Left) and the predictions
using TSB-BI (Center) and SB-BM (Right).
Table F.:: Intermediate prediction performance on single-cell data using TSB-BM and SB BM.
Timepoint
TSB-BM BM
SB-BM
W
W2
accuracy
Wi
W2
accuracy
leave-one-out
2
0.03:0.00
0.09 10.00
0.80
0.52
0.59
0.28
0.24
3
+0.000000
0.22
0.42
0.12+200
0.21
0.23
0.20
0.00
0.16
0.45
0.19
0.34
0.26
0.26
5
0. 0..4
0.28
0.70
0.33
0.40
0.24
1
F.2.5 COMPUTATIONAL COMPLEXITY
Compared to SB-based models, the TSB-based models introduce additional topological convo- lution [ef. (1)] overhead, which however admits an efficient computation, as discussed Section 5. We here provide quantative comparison of the compelxity in terms of the training time and memory consumption. We measure them using SB-VE and TSB-W ESB-VE models on different-sized 10-nearest neighbour graphs built from Swiss roll point clouds. This comparison done in a single training stage with 2,000 iterations, running observe that in the moderate scale (< 10, 000) region, the training time and memory consumption of TSB-VE are only slightly higher than SB-VE, with negligible difference. While this overhead becomes more significant as the scale further increases, both training time and memory can be re- duced by exploiting the sparse structure (here implemented using torch tensor toospa rse) in the graph topology such that the computational overheads for for SB and TSB models remain comparable. Under the same settings, Table F.8 compares SB-BM and TSB- BM models across all datasets. The additional memory and training time introduced by TSB-Be BM remain below 4%.
single NVIDIA RTX 3080 GPU. As shown in Fig. F.9, we
Dataset
TSB-BM
SB 2M
Time SB-VE
10000
Seismic
516
512
Training time (minutes)
3.5
Time TSB-VE
50.17
51.48
8000
Memory usage (MIB)
Time TSB-VE, Sparse
$10
Traffic
$2.25
50.62
2.5
6000
5976
5892
Ocean
106.68
102.67
2.0
4000
486
468
1.5
Memory SB-VE
Brain
49.62
48.97
Memory TSB-VE
2000
1.0
Memory TSB-VE, Sparse
Single-cell
4446
4294
94.30
92.54
10000
20000
30000
40000
Table F8: Complexity (first memory (in MiB), and
Graph size (nodes)
Figure E.9; Training time and memory comparison when train- ing SB-VE and TSB-VE w.r.t. different-sized graphs.
second row: training time (in seconds)).
42


Reviews:
[{"review_id": "FxfSbplB6e", "reviewer": "Reviewer_ywBL", "summary": "The manuscript is on the Schrödinger bridge problem for special topologies like graphs. The authors define the reference process akin to the heat diffusion on said topology. For specific boundary distributions, they present closed for solutions.", "questions": "Questions and minor comments.\n\n- On line 146, and the definition of the kernel $\\Sigma = \\exp(-\\kappa^2/2L)$ is L at the numerator or denominator ? If it is like a heat kernel, it should be at the numerator.\n- On line 225 (and other sections), $\\exp(-cLt)$ is a matrix exponential ?\n- I may have missed it, but what is $\\xi_1$ on line 248 ?\n- In equations 3a and 3b, it looks like 3a goes from t=0 to t=1, and 3b goes from t=1 to t=0, but it is not clear from these equations. Do you also need to define the backward Wiener process in terms of the forward one ?\n\nPotential typo:\n- Line 133: \"concerns\"\n- Line 141: \"high-dim\"\n- Line 146 \"ndoe\"\n- line 446: \"borh\"\n\n[1] Huguet, G., Tong, A., Zapatero, M. R., Tape, C. J., Wolf, G., & Krishnaswamy, S. (2023, September). Geodesic sinkhorn for fast and accurate optimal transport on manifolds. In 2023 IEEE 33rd International Workshop on Machine Learning for Signal Processing (MLSP) (pp. 1-6). IEEE.\n\n[2] J. Solomon, F. De Goes, G. Peyre,´ et al., “Convolutional\nwasserstein distances: Efficient optimal transportation\non geometric domains,” ACM Transactions on Graphics\n(ToG), 2015\n\n[3] Charlotte Bunne, Laetitia Meng-Papaxanthos, Andreas Krause, and Marco Cuturi. Proximal Optimal Transport Modeling of Population Dynamics, February 2022. arXiv:2106.06345 [cs]. Cited\non page\n\n[4] Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid RectorBrooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models\nwith minibatch optimal transport, March 2024. arXiv:2302.00482", "limitations": "", "rating": 8, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "- The introduction and background are really a pleasure to read; the SBP is very well introduced a rigorously defined. \n- The manuscript present strong theoretical contributions and a novel method to solve the SBP.", "weaknesses": "*Experiments.*\nI believe the authors should conduct additional experiments to show the advantages and limitations of their method. There are many toy datasets or graph that could be used in addition to this one, for instance a Swiss roll (potentially rotated in higher dimensions) on which you can generate from a distribution or interpolate between distributions. I also think that reporting additional metrics for evaluation would be beneficial, for instance MMD, Wassertein etc, between predicted and ground truth distributions. \n\nIt is difficult to assess the improvement of the method on figure 2 or 3. Quantitative results along the lines of table 1 are much better. Since this data appears to have multiple timepoints, the authors could validate the quality of interpolation by training with holding out one timepoint and predicting it at inference. The same experiment, should be conducted on multiple datasets, e.g. datasets from [3,4].\n\n*On solution complexity.*\nIt is mentioned that $\\Psi_t$ can be computed efficiently, but this requires the eigendecomposition of L, which scales at O(n^3) for an $n \\times n$ L. Did the authors try other approximation methods such as Chebyshev (e.g. used in [1]) or Cholesky (e.g. used in [2]) ? These methods scale much better for on a sparse graph. Additionally,, it would be beneficial to add experiments on complexity and training time with respect to the size of the Laplacian.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hyXZ4TpenF", "reviewer": "Reviewer_oR4y", "summary": "This paper proposes to develop a method for solving the Schrödinger Bridge problem when there is an inherent topological structure to the underlying matching space.  Here, topological structure refers to discrete geometric structure, such as graphs and their higher order generalizations.  Theoretical guarantees are given and numerical demonstrations are given on both simulated and real data.", "questions": "- I am not familiar at all with the work of Deng et al. (2024) cited in in the introduction, but point clouds were cited as a Euclidean space.  Point clouds are in fact quite a challenging and general data structure (as a finite metric space) and are a key setting in which topological procedures are studied (especially in topological data analysis, which is subsequently cited in the introduction asa. basis for topological machine learning.  Wouldn't the approaches proposed in this work also apply to point clouds?\n- Would it be possible to provide a performance comparison of the TSB to SB methods on point clouds?\n- Would it be possible to have an example experiment that clearly shows what goes really wrong if the underlying topological structure is ignored and only Euclidean approaches are used?\n- Check the paper for typos (e.g., \"benefitial\" on page 8).", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, I found this paper to be a strong contribution.  I would have probably given/would have liked to give higher ratings but I am not very confident in some of the technical areas that this paper covers.\n\nThe experiments were good. They were carried out on a range of sizes where smaller ones were able to illustrate the impact of the proposed method but also showcase the computational ability of the method on larger networks.\n\nGood overview on the limitations of the work, including future directions of research that can be carried to overcome them.", "weaknesses": "I think that some of the experiments could be more illustrative in why a topological approach is needed and what is missing if the underlying topology is ignored.\n\nAlso, other details on the computational aspects appear to be missing, such as runtime, complexity, convergence analyses, etc.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "D4r8Rop4gS", "reviewer": "Reviewer_fveF", "summary": "This work aims to generalize diffusion Schrodinger Bridge generative models from Euclidean domains to general topological spaces, in order to perform topological signal matching and generation. The new method is called topological Schrodinger Bridge Matching (TSBM). The authors develop a topological version of the Schrodinger bridge using a stochastic differential equation driven by a topological convolution. In the case of graphs, this can be driven by the graph heat equation with graph Laplacian. In the case where both distributions to be matched are Gaussian distributions, they give closed form solutions that generalize previous work on Schrodinger bridges. Using this framework, the authors developed a TSB-based method for topological signal generation and matching, based on parameterizing the policy with a topological neural network. The authors also provided the results of some numerical experiments on ocean current matching and modeling seismic events, along with some synthetic experiments.\n\nA couple of typos I noticed: \" a diffusion ndoe GP\", \"Vairous topological\", \"the spacial case\", \"we showed that the optimal process satisfy a pair\"", "questions": "On a graph, there are various normalizations of the graph Laplacian. The unnormalized (combinatorial), random walk and symmetric normalized are the three most common, but there are a wide range of others depending on a continuous family of parameters. Could the authors please clarify which graph Laplacian they are using? There is a substantial difference in terms of the diffusion process generated on the graph if one uses the unnormalized versus the random walk Laplacian (with the latter being the natural choice for diffusion on graphs). I think the paper would benefit from a discussion about the type of graph Laplacian that is used, and possibly numerical experiments with different graph Laplacian normalizations.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "Deep learning on topological spaces, like graphs, is an important area of research. The paper provides an original and mathematically well-grounded method for topological signal matching and generation. The mathematics is very well-developed and clearly laid out, and the paper is very well-written.", "weaknesses": "The numerical experiments are rather limited. I would have liked to see more large scale numerical experiments to validate the method. It is not clear to me if the methods scale up to large data sets well. Some of the future work problems (i.e., \"On model training\") indicate that it might not.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pePdf59CrO", "reviewer": "Reviewer_waSt", "summary": "The paper introduces the topological Schrödinger bridge problem (TSBP), extending Schrödinger bridge (SB) methods to non-Euclidean domains, specifically graphs and simplicial complexes. Traditional SB approaches face limitations on topological domains. TSBP is proposed to address these by defining a topologically aware reference process based on topological stochastic differential equations (TSDEs). For cases with Gaussian boundary conditions, the authors derive closed-form solutions, then construct TSB-based models for generative modeling and signal matching.", "questions": "- Can the authors provide further insight into adapting TSBP for boundary conditions that are not Gaussian?\n- Could the authors expand on strategies to manage computational complexity for larger networks, especially in real-world scenarios?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The derivation of closed-form solutions for Gaussian boundaries is rigorous, and the detailed handling of topological SDEs adds significant depth to the method.\n- The paper is well-structured, guiding readers through the theoretical framework, derivations, and experimental applications, though some dense mathematical notation may hinder readability.\n- By addressing topological signal matching, this work is relevant to fields where data resides on complex networks, such as biological and social networks.", "weaknesses": "- Replicability is a major limitation. The lack of code or implementation details limits reproducibility, as replicating the results would require substantial independent effort, particularly in replicating TSDE parameterization and model training. Key training processes (e.g., likelihood optimization for neural networks on topological data) and parameter settings are not transparent, which might impede others from achieving comparable results.\n- While synthetic and specific real-world network tasks are covered, the experimental scope could be broadened. Testing across a broader range of real-world network types (e.g., biological, transportation, or communication networks) would better demonstrate the broad applicability of TSBP and strengthen the claims of generalizability. If extending the experimental coverage is impractical, the paper should provide code to enable other researchers to explore the applicability and limitations of TSBP across different types of network-structured data.\n- The primary focus on Gaussian boundary distributions restricts practical generalizability, as many real-world networked data scenarios involve non-Gaussian or mixed-distribution boundary conditions.\n-  The method’s computational demands are high, particularly in large-scale networks. Although some efficiency measures are discussed, handling matrix exponentials in large or complex network structures remains challenging.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)