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Large deviations for scaled families of Schr\"odinger bridges with reflection

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Pith's one-line read Schrödinger bridges with reflected Brownian reference satisfy a large deviation principle, quantifying their exponential convergence to the quadratic optimal transport plan.

desk verdict A real but modest extension of the Bernton–Ghosal–Nutz LDP to reflected Schrödinger bridges, held up by one fixable gap in the uniform heat-kernel lower bound. read the letter →

arxiv 2506.03999 v4 pith:5CCXUK2H submitted 2025-06-04 math.PR

Viktor Nilsson , Pierre Nyquist This is my paper
classification math.PR MSC 60F1049Q2293C10
keywords largedeviationsSchrödingerbridgesentropicoptimaltransportreflecteddiffusionscyclicalmonotonicityNeumannheatkernelSkorokhodmapgenerativemodeling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a large deviation principle (LDP) for sequences of Schrödinger bridges / entropic optimal transport plans in the small-noise limit, when the costs vary with the noise level and converge uniformly to a fixed cost function. This extends the fixed-cost LDP of [2] and answers an open problem posed there. The main application is to reflected Brownian motion on bounded convex domains: the paper proves the associated cost functions converge uniformly to the quadratic cost, so reflected Schrödinger bridges satisfy the LDP. The result matters because it quantifies, at exponential scale, how fast entropic transport plans converge to the limiting optimal transport plan, a question directly relevant to diffusion Schrödinger bridge generative models on constrained domains.

What carries the argument

The argument is carried by four pieces. (1) Cyclical invariance (2.10): each EOT minimizer $\pi_\eta$ is characterized by a density identity involving the cost differences $c_\eta(x_i,y_i) - c_\eta(x_i,y_{i+1})$, which is the exact bridge between the plan's mass and the rate function. (2) The rate function $I$ in (4.4), read as the maximal cost improvement obtainable by permuting partners in a cycle. (3) A slack-variable modification of the fixed-cost lemmas of [2] that absorbs the uniform error $\|c_\eta - c\|_\infty$, converting the fixed-cost LDP into the varying-cost version. (4) For the reflected application: the Neumann heat-kernel upper bound of [8] and a matching lower bound derived from the continuity of the Skorokhod map on the bounded convex domain, giving uniform control on $-\eta \log p^r_\eta(1,x,y)$.

What would settle it

Using the explicit density (5.6) for $D=[0,1]$, compute $\sup_{x,y\in[0,1]} | -\eta\log p^r_\eta(1,x,y) - |x-y|^2/2 |$ for decreasing $\eta$; the theorem asserts this error goes to zero uniformly, so a persistent positive error near the boundary at any fixed $\eta$ would falsify the uniform convergence that feeds the large deviation principle.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 4.2: under Assumptions 2.1 and 4.1, if the cost functions $c_\eta$ converge uniformly to $c$ as $\eta \downarrow 0$, then the entropic optimal transport plans $\pi_\eta$ satisfy a weak-type LDP with rate function $I(x,y) = \sup_{k\ge 2} \sup_{(x_i,y_i)\subseteq \Gamma} \sup_{\sigma\in\Sigma(k)} \sum_{i=1}^k [c(x_i,y_i) - c(x_i,y_{\sigma(i)})]$, where $\Gamma$ is the support of the limiting plan. The rate function can also be written as $c - (-\psi \oplus \psi^c)$ on the support for any Kantorovich potential $\psi$. Theorem 5.2 shows that for reflected Brownian motion on an open bounded convex domain, the costs $c_\eta(x,y) = -\eta \log p^r_\eta(1,x,y)$ converge uniformly to $c(x,y) = |x-y|^2/2$, so the LDP applies to reflected Schrödinger bridges. Under compact supports the weak-type bound upgrades to a full LDP (Corollary 4.3).

Load-bearing premise

The load-bearing premise is that a single shrink of the domain by one common margin lets every pair of start and end points stay within that margin of the shrunk set, which is what makes the reflected heat-kernel lower bound uniform across all pairs.

Editorial extensions

If this is right

  • Reflected Schrödinger bridge plans on any open bounded convex domain converge to the quadratic optimal transport plan at an exponential rate governed by the rate function $I$.
  • The LDP is full (not merely weak-type) whenever the supports of the two marginals are compact, giving the usual large-deviation upper and lower bounds on all open and closed sets.
  • The rate function is computable from the limit cost $c$ and the Kantorovich potential $\psi$ as $c - (-\psi \oplus \psi^c)$, so the exponential asymptotics are explicit for the quadratic-cost case.
  • The uniform-convergence hypothesis transfers the entire large-deviation machinery of [2] from one fixed cost to any family of costs that stabilize uniformly, which is the partial answer to the open problem posed there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same theorem should apply to other reference dynamics whose transition densities admit matching Gaussian-type upper and lower bounds, e.g., reflected diffusions with bounded drift on bounded convex domains; verifying uniform convergence of $-\eta \log q_\eta(1,x,y)$ would be the only missing step.
  • The rate function $I$ admits a sample-based interpretation in generative modeling: for a learned transport plan, evaluating the cycle-cost sums against the OT plan gives a quantitative measure of how far the model is from being optimal, which could serve as a diagnostic for diffusion Schrödinger bridge training.
  • A path-space LDP for the dynamic reflected Schrödinger bridge remains open; because the static plans satisfy the LDP and the bridge interpolation is the reference process's own bridges, a path-space statement would need a control on the full bridge laws, and the Gaussian-bridge argument used for Brownian motion in [19] will not carry over to reflected Brownian motion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proves a large deviation principle (LDP) for sequences of static Schrödinger bridges / entropic optimal transport plans when the cost functions c_η converge uniformly to a limit c as the noise scale η↓0 (Theorem 4.2), generalizing the fixed-cost LDP of Bernton–Ghosal–Nutz [2]. It then shows that for reflected Brownian motion on a bounded convex domain, the associated cost c_η(x,y) = −η log p^r_η(1,x,y) converges uniformly to |x−y|^2/2 (Theorem 5.2), so the LDP applies to reflected Schrödinger bridges. The upper-bound half of Theorem 4.2 is proved in detail via cyclical-invariance arguments; the lower-bound half is deferred to [2]. Theorem 5.2 relies on heat-kernel upper and lower bounds, the latter proved via Skorokhod-map and Brownian-bridge estimates.

Significance. If the proof gaps are fixed, this would be a meaningful contribution: it addresses an open problem from [2], extends LDPs for entropic optimal transport to non-fixed cost functions, and would provide the first LDP for reflected Schrödinger bridges. The general theorem (Theorem 4.2) is clean, the rate function is explicitly derived from the limit cost with no fitted parameters, and the analysis of reflected heat kernels in Section 5 is nontrivial. The manuscript is transparent about which results are deferred to [2] and which analytic bounds are imported from [8] and [23]. The main claims are clearly stated and the overall strategy is sound, but the two gaps described below are load-bearing.

major comments (2)
  1. [Section 4, Corollary 4.11] The lower-bound half of Theorem 4.2 is not proved. The text states that Proposition 4.10 and Corollary 4.11 are "identical to [2]" and omits their proofs. This cannot be literally correct: Corollary 4.11 concerns the sequence π_η with varying costs c_η, whereas the corresponding result in [2] is proved for a fixed cost c. Adapting the proof requires controlling the difference c_η − c in the cyclical-invariance identity (2.10)/(3.5) and in the density estimates, exactly as done for the upper bound in Lemmas 4.6 and 4.8. Since the lower bound is essential to the LDP of Theorem 4.2, the authors should provide the proof or a detailed adaptation rather than a citation.
  2. [Section 5, Proposition 5.7] The claimed uniform lower bound (5.11) is not established. In the final Chapman–Kolmogorov step (display 5.24), the parameter ε′ is "chosen so that dist(x, D_{−ε′}) ≤ ε" for each pair (x,y), and the constants ᾱ, β̃ absorb the volume factor vol(B(0, ε′/2)) and terms containing ε′. If ε′ varies with (x,y), the constants inherit this dependence, so the asserted uniformity in x,y does not follow. The proof also uses |y − y_{ε′}| ≤ ε (display 5.25) without stating that ε′ is chosen for both coordinates. The fix is to select a single ε′ > 0, depending only on ε and D, such that sup_{x∈D} dist(x, D_{−ε′}) ≤ ε (e.g., ε′ = ε/2 for bounded convex D), and to prove the required exhaustion property of the inner parallel sets D_{−ε′}. Without this, Theorem 5.2 is unsupported.
minor comments (4)
  1. [Section 4, Theorem 4.2] In the display (4.4), the notation "(xi,yi)^k_{2=1}⊆Γ" is a typo; it should be "(x_i,y_i)^k_{i=2}⊆Γ".
  2. [Section 4, Lemma 4.6] There is a typo "TThis establishes" in the proof, and the final sentence should read "lim inf_{η↓0} η log π^k_η(A) ≥ −δ′" (the minus sign is missing in the sentence after (4.13)).
  3. [Section 2.1] The text contains minor typos: "Kullback-Liebler" should be "Kullback-Leibler", and "the the strict convexity" has a duplicated article.
  4. [Section 5.1] The statement "it is straightforward to show that c_η converges uniformly" would benefit from explicitly naming the mode of convergence (uniform on D×D) before referencing Figure 1, since Figure 1 only demonstrates pointwise slices.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LDP derivation is self-contained modulo external standard results, and the reflected-heat-kernel uniformity concern is a correctness gap, not a circular step.

full rationale

The claimed derivation chain is not circular. The sequence cη is explicitly constructed from transition densities via cη(x,y) = −η log p^r_η(1,x,y) (Section 5, display 5.4), and the limiting cost c(x,y) = |x−y|^2/2 is obtained from independent upper and lower heat-kernel bounds (Proposition 5.3 from Davies [8] and Proposition 5.7 proved in the paper). Theorem 4.2 is a modification of Bernton–Ghosal–Nutz [2], an external published result, not a self-citation; the paper restates the needed lemmas, adjusts them for non-constant costs, and the omitted Proposition 4.10 and Corollary 4.11 are explicitly credited to [2], which is independent support. No parameter is fitted to the object being predicted: the rate function I is defined directly from the limiting cost c and the weak limit π, and the uniform convergence condition cη→c is established rather than assumed in the main application. The known Brownian quadratic cost is not renamed but recovered from the Neumann transition density. The only substantive concern in Section 5 is a possible uniformity gap in the final step of Proposition 5.7, where ε′ is chosen per pair (x,y) around display (5.24); this is a missing geometric estimate, not a circular step, because it does not make any theorem equal to its input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on three explicit domain assumptions (2.1, 4.1, 5.1) and on standard results from optimal transport, heat-kernel theory, and the Skorokhod problem. No free parameters are fitted and no new entities are postulated. The least explicit premise is the uniform geometric approximation of D by D_{-ε'} in Proposition 5.7, which is true for bounded convex domains but not stated.

assumptions (6)
  • domain assumption Assumption 2.1 (existence of a finite-entropy coupling for all (R, μ, ν) and finite EOT objective).
    Stated in Section 2.1 and used to guarantee existence and uniqueness of optimizers for the SB and EOT problems throughout the paper.
  • domain assumption Assumption 4.1: uniqueness of Kantorovich potentials up to additive constants on X0.
    Needed for the lower-bound half of the LDP (Proposition 4.10 and Corollary 4.11 in [2]); asserted to hold for the quadratic limit cost on connected convex domains with ν ≪ Lebesgue.
  • domain assumption Assumption 5.1: the differential entropy H(ν) = -∫ log(dν/dλ)dν is finite.
    Used in Section 5 to reduce the reflected SB problem to the EOT formulation with cη(x,y) = -η log p^r_η(1,x,y).
  • domain assumption Uniform geometric approximation of D by inner sets D_{-ε'}: for each ε > 0 there exists ε' > 0 such that every x ∈ D is within distance ε of D_{-ε'}.
    This unstated premise is used in the final Chapman-Kolmogorov step of Proposition 5.7 to make the lower-bound constants independent of x,y; it is true for bounded convex domains but not proved in the text.
  • standard math Neumann heat-kernel upper bound for bounded regions with the extension property (Davies [8, Thm 3.2.9]).
    Quoted as Proposition 5.3 and used to obtain the matching upper bound in Theorem 5.2.
  • standard math Lions-Sznitman: existence, uniqueness, and uniform continuity of the Skorokhod map on compact subsets of path space for bounded convex domains.
    Used in Proposition 5.7 to control reflected paths near Brownian bridges; cited as [23].

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Cite this review

Pith. "Pith review of Large deviations for scaled families of Schr\"odinger bridges with reflection." pith.science (2026). https://pith.science/paper/5CCXUK2H

@misc{pith2026250603999,
  author       = {Pith},
  title        = {Pith review of: Large deviations for scaled families of Schr\"odinger bridges with reflection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CCXUK2H}},
  note         = {Machine review of arXiv:2506.03999}
}
read the original abstract

In this paper, we show a large deviation principle for certain sequences of static Schr\"{o}dinger bridges, typically motivated by a scale-parameter decreasing towards zero, extending existing large deviation results to cover a wider range of reference processes. Our results provide a theoretical foundation for studying convergence of such Schr\"{o}dinger bridges to their limiting optimal transport plans. Within generative modeling, Schr\"{o}dinger bridges, or entropic optimal transport problems, constitute a prominent class of methods, in part because of their computational feasibility in high-dimensional settings. Recently, Bernton et al. established a large deviation principle, in the small-noise limit, for fixed-cost entropic optimal transport problems. In this paper, we address an open problem posed by Bernton et al. and extend their results to hold for Schr\"{o}dinger bridges associated with certain sequences of more general reference measures with enough regularity in a similar small-noise limit. These can be viewed as sequences of entropic optimal transport plans with non-fixed cost functions. Using a detailed analysis of the associated Skorokhod maps and transition densities, we show that the new large deviation results cover Schr\"{o}dinger bridges where the reference process is a reflected diffusion on bounded convex domains, corresponding to recently introduced model choices in the generative modeling literature.

Figures

Figures reproduced from arXiv: 2506.03999 by the authors.

Figure 1
Figure 1. Uniform convergence of cη on D = [0, 1], computed using (5.6) on the slice x = 0.3. 5.1. Examples with explicit transition densities. For D = [0, ∞) with reflec￾tion at 0, the Skorokhod map is given by Γf(t) = f(t) − (infs≤t f(s) ∧ 0). One can show that the transition density of ΓW is (5.5) p+(t, x, y) := p(t, x, y) + p(t, −x, y), For D = [0, 1], an explicit expression for the Skorokhod map is also available, see [2… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Reflected Schr\"odinger Bridge Matching

    cs.LG 2026-07 accept novelty 6.0 of 10

    Reflected Schrödinger bridges on the unit cube can be learned by α-IMF using an exact reflected Brownian bridge sampler and a factorized sum-of-Gaussians score, with domain guarantees and essentially free overhead.

Reference graph

Works this paper leans on

45 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [2]

    Bernton, P

    E. Bernton, P. Ghosal, and M. Nutz. Entropic optimal transport: Geometry and large devi- ations. Duke Math. J., 171(16):3363–3400, 2022

    work page 2022

  2. [8]

    E. B. Davies.Heat kernels and spectral theory. Cambridge University Press, 1989. 26 V. NILSSON AND P. NYQUIST

    work page 1989

  3. [23]

    Lions and A.-S

    P.-L. Lions and A.-S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math., 37(4):511–537, 1984

    work page 1984

  4. [1]

    M. S. Albergo and E. Vanden-Eijnden. Building normalizing flows with stochastic inter- polants. InInternational Conference on Learning Representations, 2023

    work page 2023

  5. [3]

    Budhiraja and P

    A. Budhiraja and P. Dupuis.Analysis and approximation of rare events, Springer, New York, 2019

    work page 2019

  6. [4]

    K. F. Caluya and A. Halder. Reflected Schrödinger bridge: Density control with path con- straints. In2021 American Control Conference (ACC), 1137–1142, IEEE, 2021

    work page 2021

  7. [5]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud. Neural ordinary differ- ential equations.Advances in Neural Information Processing Systems, 31, 2018

    work page 2018

  8. [6]

    Chen, G.-H

    T. Chen, G.-H. Liu, and E. A. Theodorou. Likelihood training of Schrödinger bridge using forward-backward SDEs theory. InInternational Conference on Learning Representations, 2022

    work page 2022

Show all 45 references
  1. [7]

    M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport.Advances in Neural Information Processing Systems, 26, 2013

    2013

  2. [9]

    De Bortoli, J

    V. De Bortoli, J. Thornton, J. Heng, and A. Doucet. Diffusion Schrödinger bridge with applications to score-based generative modeling.Advances in Neural Information Processing Systems, 34:17695–17709, 2021

    2021

  3. [10]

    Dembo and O

    A.. Dembo and O. Zeitouni.Large deviations techniques and applications. 2nd ed. Springer, New York, NY, 1998

    1998

  4. [11]

    W. Deng, Y. Chen, N. Tianjiao Yang, H. Du, Q. Feng, and R. T. Q. Chen. Reflected Schrödinger bridge for constrained generative modeling. InProceedings of the Fortieth Con- ference on Uncertainty in Artificial Intelligence, 1055–1082. PMLR, 2024

    2024

  5. [12]

    Dupuis and H

    P. Dupuis and H. Ishii. SDEs with oblique reflection on nonsmooth domains.Ann. Probab., 21(1):554–580, 1993

    1993

  6. [13]

    Dupuis and K

    P. Dupuis and K. Ramanan. Convex duality and the Skorokhod problem. I.Probab. Theory Related Fields, 115:153–195, 1999

    1999

  7. [14]

    Ghosal, M

    P. Ghosal, M. Nutz, and E. Bernton. Stability of entropic optimal transport and Schrödinger bridges. J. Funct. Anal., 283(9):109622, 2022

    2022

  8. [15]

    Grathwohl, R

    W. Grathwohl, R. T. Q. Chen, J. Bettencourt, I. Sutskever, and D. Duvenaud. Ffjord: Free- form continuous dynamics for scalable reversible generative models. InInternational Confer- ence on Learning Representations, 2019

    2019

  9. [16]

    M Harrison and M I Reiman

    J. M Harrison and M I Reiman. Reflected brownian motion on an orthant.Ann. Probab., 9(2):302–308, 1981

    1981

  10. [17]

    J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    2020

  11. [18]

    Karatzas and S

    I. Karatzas and S. E. Shreve.Brownian motion and stochastic calculus. 2nd ed., Springer, New York, NY, 1998

    1998

  12. [19]

    K. Kato. Large deviations for dynamical Schrödinger problems.Preprint, arXiv:2402.05100, 2024

    2024 arXiv

  13. [20]

    L. Kruk, J. Lehoczky, K. Ramanan, and S. E. Shreve. An explicit formula for the Skorokhod map on [0, a].Ann. Probab., 35(5):1740–1768, 2007

    2007

  14. [21]

    C. Léonard. From the Schrödinger problem to the Monge–Kantorovich problem.J. Funct. Anal., 262(4):1879–1920, 2012

    1920

  15. [22]

    C. Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst., 34(4):1533–1574, 2014

    2014

  16. [24]

    Lipman, R

    Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, and M. Le. Flow matching for generative modeling. NeurIPS tutorial; arXiv preprint arXiv:2210.02747, 2022

    2022 arXiv

  17. [25]

    G.-H. Liu, A. Vahdat, D.-A. Huang, E. A. Theodorou, W. Nie, and A. Anandkumar. I2SB: Image-to-Image Schrödinger Bridge. InInternational Conference on Machine Learning, 2023

    2023

  18. [26]

    X. Liu, C. Gong, and Q. Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. InInternational Conference on Learning Representations, 2023

    2023

  19. [27]

    Lou and S

    A. Lou and S. Ermon. Reflected diffusion models. InInternational Conference on Machine Learning, 22675–22701, PMLR, 2023

    2023

  20. [28]

    T. Mikami. Monge’s problem with a quadratic cost by the zero-noise limit ofh-path processes. Probab. Theory Related Fields, 129(2):245–260, 2004

    2004

  21. [29]

    M. Nutz. Introduction to entropic optimal transport.Lecture notes, Columbia University, 2021

    2021

  22. [30]

    Peyré, M

    G. Peyré, M. Cuturi. Computational optimal transport: With applications to data science. Foundations and Trends® in Machine Learning, 11(5-6):355–607, 2019

    2019

  23. [31]

    Pilipenko.An introduction to stochastic differential equations with reflection, vol 1

    A. Pilipenko.An introduction to stochastic differential equations with reflection, vol 1. Uni- versitätsverlag Potsdam, 2014

    2014

  24. [32]

    Multisample flow matching: Straightening flows with minibatch couplings

    A.-A.Pooladian, H.Ben-Hamu, C.Domingo-Enrich, B.Amos, Y.Lipman, andR.T.Q.Chen. Multisample flow matching: Straightening flows with minibatch couplings. InInternational Conference on Machine Learning, 2023

    2023

  25. [33]

    S. T. Rachev and L. Rüschendorf. Mass transportation problems: Applications. Springer Science & Business Media, 2006

    2006

  26. [34]

    S. T. Rachev and L. Rüschendorf. Mass Transportation Problems: Volume 1: Theory . Springer Science & Business Media, 2006

    2006

  27. [35]

    Birkhäuser/Springer, Cham, 2015

    Filippo Santambrogio.Optimal transport for applied mathematicians. Birkhäuser/Springer, Cham, 2015. LARGE DEVIATIONS FOR SCALED SCHRÖDINGER BRIDGES WITH REFLECTION 27

    2015

  28. [36]

    Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique

    Erwin Schrödinger. Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. Henri Poincaré, 2: 269–310, 1932

    1932

  29. [37]

    Y. Shi, V. De Bortoli, A. Campbell, and A. Doucet. Diffusion Schrödinger bridge matching. Advances in Neural Information Processing Systems, 36:62183–62223, 2023

    2023

  30. [38]

    Sohl-Dickstein, E

    J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli. Deep unsupervised learn- ing using nonequilibrium thermodynamics. InInternational Conference on Machine Learn- ing, 2256–2265, PMLR, 2015

    2015

  31. [39]

    Song and S

    Y. Song and S. Ermon. Generative modeling by estimating gradients of the data distribution. Advances in Neural Information Processing Systems, 32, 2019

    2019

  32. [40]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations.International Conference on Learning Representations, 2021

    2021

  33. [41]

    A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks, G. Wolf, and Y. Ben- gio. Improving and generalizing flow-based generative models with minibatch optimal trans- port. Transactions on Machine Learning Research, 3, 2024

    2024

  34. [42]

    Vargas, P

    F. Vargas, P. Thodoroff, A. Lamacraft, and N. Lawrence. Solving Schrödinger bridges via maximum likelihood.Entropy, 23(9):1134, 2021

    2021

  35. [43]

    Villani.Optimal transport: Old and new, Springer-Verlag, Berlin, 2009

    C. Villani.Optimal transport: Old and new, Springer-Verlag, Berlin, 2009

    2009

  36. [44]

    P. Vincent. A connection between score matching and denoising autoencoders.Neural com- putation, 23(7):1661–1674, 2011

    2011

  37. [45]

    G. Wang, Y. Jiao, Q. Xu, Y. Wang, and C. Yang. Deep generative learning via Schrödinger bridge. InInternational Conference on Machine Learning, 10794–10804, PMLR, 2021. Current address: Department of Mathematics, KTH, 100 44 Stockholm, Sweden

    2021

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