Stability of the Kim--Milman flow map

Aram-Alexandre Pooladian; Matthew S. Zhang; Sinho Chewi

arxiv: 2511.01154 · v2 · submitted 2025-11-03 · 🧮 math.PR · cs.LG· math.ST· stat.TH

Stability of the Kim--Milman flow map

Sinho Chewi , Aram-Alexandre Pooladian , Matthew S. Zhang This is my paper

Pith reviewed 2026-05-18 01:59 UTC · model grok-4.3

classification 🧮 math.PR cs.LGmath.STstat.TH

keywords Kim-Milman flow mapprobability flow ODErelative Fisher informationstabilitytarget measuresamplingoptimal transport

0 comments

The pith

The Kim-Milman flow map stays stable under variations in the target measure measured by relative Fisher information.

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

The paper characterizes the stability of the Kim-Milman flow map, also known as the probability flow ODE, when the target measure changes slightly. Stability here is quantified in terms of the relative Fisher information between the two target measures. If this characterization holds, then small inaccuracies in specifying the target distribution would lead to only controlled deviations in the flow trajectories. Readers interested in sampling algorithms or generative models would find this useful for understanding robustness of the generated paths.

Core claim

In this short note, we characterize stability of the Kim--Milman flow map -- also known as the probability flow ODE -- with respect to variations in the target measure in relative Fisher information. This provides a precise description of how the flow map responds to perturbations of the underlying target probability measure.

What carries the argument

The Kim-Milman flow map, which solves the probability flow ODE associated to a given target measure.

If this is right

The map from target measure to flow map is continuous in the sense induced by relative Fisher information.
Explicit bounds on the difference between two flow maps follow directly from the relative Fisher information of their targets.
Trajectories generated by the flow remain close throughout their evolution when the targets are close in relative Fisher information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This stability result could be used to analyze error when a target measure is replaced by an empirical estimate from samples.
The same approach might yield stability statements for related ODE flows in generative modeling that depend on a target distribution.
Numerical checks on simple cases such as Gaussian targets could verify the size of the stability constants.

Load-bearing premise

The target measures admit a well-defined Kim-Milman flow map and possess finite relative Fisher information with respect to each other.

What would settle it

Two target measures with small but finite relative Fisher information whose associated flow maps produce trajectories that differ by more than any bound controlled by that information.

read the original abstract

In this short note, we characterize stability of the Kim--Milman flow map -- also known as the probability flow ODE -- with respect to variations in the target measure in relative Fisher information.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This short note gives a stability bound for the Kim-Milman flow map in relative Fisher information, but leans on unstated existence assumptions for the flow itself.

read the letter

The core result here is a stability estimate showing how the Kim-Milman flow map changes when the target measure varies, with the distance controlled by relative Fisher information. It is a direct characterization rather than a restatement of earlier work on probability flows or optimal transport maps. The note keeps the argument focused and avoids unnecessary generality, which makes the bound readable for people already familiar with the flow ODE setup. The authors credit the relevant background on the Kim-Milman construction and relative Fisher information, and the derivation appears to follow standard score-function estimates without obvious circular steps. That is the useful part: a clean, targeted bound that could plug into analyses of sampling algorithms or generative models that rely on this flow. The soft spot is the handling of existence and uniqueness. The claim applies to pairs of measures with finite relative Fisher information, yet finite Fisher information alone does not automatically deliver global Lipschitz continuity or non-explosion of the ODE without extra regularity such as uniform log-concavity or moment bounds. The short-note format leaves this implicit rather than citing a precise theorem or adding a short verification for the class of measures used in the estimate. If the paper assumes the flow is already well-defined from prior literature, that should be stated explicitly. Readers working in optimal transport or continuous-time sampling will find the bound handy for perturbation arguments. It is narrow enough that most people outside those areas will skip it, but the result is honest and self-contained enough to merit referee time as a short communication. I would send it to review rather than desk-reject, with the expectation that the authors tighten the existence conditions in revision.

Referee Report

1 major / 0 minor

Summary. This short note characterizes the stability of the Kim--Milman flow map (also called the probability flow ODE) with respect to variations in the target measure, quantified in relative Fisher information.

Significance. If the central characterization holds under the stated assumptions, the result would supply a direct analytic stability bound useful for robustness analysis of probability flows in sampling and generative modeling. The paper's strength lies in its focused analytic approach rather than empirical fitting, but the short-note format limits the scope of the derivation and the class of measures treated.

major comments (1)

The stability result is stated for pairs of target measures μ, ν with finite relative Fisher information I(μ|ν) < ∞, yet the manuscript does not verify or cite conditions ensuring global existence, uniqueness, and Lipschitz continuity of the associated Kim--Milman flow map. Finite relative Fisher information bounds the L² norm of the score but is insufficient by itself to guarantee these properties without additional hypotheses (e.g., uniform log-concavity or finite second moments). This assumption is load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a key point regarding the well-posedness assumptions underlying the Kim-Milman flow map. We address the major comment below.

read point-by-point responses

Referee: The stability result is stated for pairs of target measures μ, ν with finite relative Fisher information I(μ|ν) < ∞, yet the manuscript does not verify or cite conditions ensuring global existence, uniqueness, and Lipschitz continuity of the associated Kim--Milman flow map. Finite relative Fisher information bounds the L² norm of the score but is insufficient by itself to guarantee these properties without additional hypotheses (e.g., uniform log-concavity or finite second moments). This assumption is load-bearing for the central claim.

Authors: We agree that finite relative Fisher information alone does not suffice to guarantee global existence, uniqueness, and Lipschitz continuity of the flow map. The manuscript develops the stability characterization under the standing hypothesis that the Kim-Milman flows exist and are sufficiently regular for the stated measures; this is implicit in the setup but not made explicit. In the revision we will add a short paragraph in the preliminaries clarifying the requisite conditions (for instance, uniform log-concavity or finite second moments together with standard results on the well-posedness of the probability-flow ODE) and cite the relevant literature. This will render the scope of the result transparent without changing the core statement. revision: yes

Circularity Check

0 steps flagged

No circularity: direct analytic characterization of flow-map stability

full rationale

The short note derives a stability estimate for the Kim-Milman flow map (probability flow ODE) under perturbations of the target measure measured in relative Fisher information. The derivation relies on standard ODE analysis and properties of the score function; no parameter is fitted to data and then relabeled as a prediction, no self-citation chain is invoked to justify a uniqueness theorem, and the central stability statement is not definitionally equivalent to its inputs. The existence of the flow map for measures with finite relative Fisher information is a standing assumption of the setup rather than a result derived inside the paper, which is consistent with the literature on probability-flow ODEs and does not reduce the claimed stability bound to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no visible free parameters, axioms, or invented entities; the result appears to rest on standard assumptions of the probability-flow literature.

pith-pipeline@v0.9.0 · 5549 in / 958 out tokens · 41378 ms · 2026-05-18T01:59:01.128183+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

∥T^μ_KM − T^ν_KM∥_L²(γ) ≲ √FI(ν∥μ) ... under (Θ): ∇² log Q^s [μ/γ] ≼ θ_s I
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.3 (Wibisono): d/dt FI(q^ν_t ∥ q^μ_t) ≤ −2 E[∥∇log(q^ν/q^μ)∥² (−2∇²log q^μ_t − I)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

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Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
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Stability bounds for smooth optimal transport maps and their statistical implications.arXiv preprint arXiv:2502.12326, 2025

Sivaraman Balakrishnan and Tudor Manole. Stability bounds for smooth optimal transport maps and their statistical implications.arXiv preprint arXiv:2502.12326, 2025

work page arXiv 2025
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Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

Guillaume Carlier, L´ ena¨ ıc Chizat, and Maxime Laborde. Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

work page 2024
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An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities.Comptes Rendus

Sinho Chewi and Aram-Alexandre Pooladian. An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities.Comptes Rendus. Math´ ematique, 361(G9):1471–1482, 2023

work page 2023
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Weak semiconvexity estimates for Schr¨ odinger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges.Probability Theory and Related Fields, 189(3):1045–1071, 2024

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Projected Langevin dynamics and a gradient flow for entropic optimal transport.Journal of the European Mathematical Society, 2025

Giovanni Conforti, Daniel Lacker, and Soumik Pal. Projected Langevin dynamics and a gradient flow for entropic optimal transport.Journal of the European Mathematical Society, 2025

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Tight stability bounds for entropic Brenier maps.International Mathematics Research Notices, 2025(7):rnaf078, 2025

Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. Tight stability bounds for entropic Brenier maps.International Mathematics Research Notices, 2025(7):rnaf078, 2025

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An entropic interpolation proof of the HWI inequality.Stochastic Processes and their Applications, 130(2):907–923, 2020

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On H¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54 (2):401–409, 2011

Nicola Gigli. On H¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54 (2):401–409, 2011

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Consistency trajectory models: learning probability flow ODE trajectory of diffusion

Dongjun Kim, Chieh-Hsin Lai, Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: learning probability flow ODE trajectory of diffusion. InThe Twelfth International Conference on Learning Representations, 2024

work page 2024
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A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.Mathematische Annalen, 354(3):827–862, 2012

Young-Heon Kim and Emanuel Milman. A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.Mathematische Annalen, 354(3):827–862, 2012. 13

work page 2012
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Stability of optimal transport maps on Riemannian manifolds.arXiv preprint arXiv:2504.05412, 2025

Jun Kitagawa, Cyril Letrouit, and Quentin M´ erigot. Stability of optimal transport maps on Riemannian manifolds.arXiv preprint arXiv:2504.05412, 2025

work page arXiv 2025
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A survey of the schrödinger problem and some of its connections with optimal transport

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work page arXiv 2013
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Quantitative stability of optimal transport.Notes du cours Peccot, 2025, 2025

Cyril Letrouit. Quantitative stability of optimal transport.Notes du cours Peccot, 2025, 2025

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Gluing methods for quantitative stability of optimal transport maps.arXiv preprint arXiv:2411.04908, 2024

Cyril Letrouit and Quentin M´ erigot. Gluing methods for quantitative stability of optimal transport maps.arXiv preprint arXiv:2411.04908, 2024

work page arXiv 2024
[18]

Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. InThe Eleventh International Conference on Learning Representations, 2023

work page 2023
[19]

Plugin estimation of smooth optimal transport maps.The Annals of Statistics, 52(3):966–998, 2024

Tudor Manole, Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman. Plugin estimation of smooth optimal transport maps.The Annals of Statistics, 52(3):966–998, 2024

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On the Lipschitz properties of transportation along heat flows

Dan Mikulincer and Yair Shenfeld. On the Lipschitz properties of transportation along heat flows. InGeometric aspects of functional analysis, volume 2327 of Lecture Notes in Math., pages 269–290. Springer, Cham, 2023

work page 2023
[21]

Pooladian and J

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps.arXiv preprint arXiv:2109.12004, 2021

work page arXiv 2021
[22]

Plug-in estimation of Schr¨ odinger bridges.arXiv preprint arXiv:2408.11686, 2024

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Plug-in estimation of Schr¨ odinger bridges.arXiv preprint arXiv:2408.11686, 2024

work page arXiv 2024
[23]

Minimax estimation of discontinuous optimal transport maps: the semi-discrete case

Aram-Alexandre Pooladian, Vincent Divol, and Jonathan Niles-Weed. Minimax estimation of discontinuous optimal transport maps: the semi-discrete case. In International Conference on Machine Learning, pages 28128–28150. PMLR, 2023

work page 2023
[24]

Philippe Rigollet and Austin J. Stromme. On the sample complexity of entropic optimal transport.The Annals of Statistics, 53(1):61–90, 2025

work page 2025
[25]

Diffusion Schr¨ odinger bridge matching.Advances in Neural Information Processing Systems, 36:62183–62223, 2023

Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion Schr¨ odinger bridge matching.Advances in Neural Information Processing Systems, 36:62183–62223, 2023

work page 2023
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Mixing time of the proximal sampler in relative Fisher information via strong data processing inequality.arXiv preprint arXiv:2502.05623, 2025

Andre Wibisono. Mixing time of the proximal sampler in relative Fisher information via strong data processing inequality.arXiv preprint arXiv:2502.05623, 2025. 14

work page arXiv 2025

[1] [1]

Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

Michael S. Albergo, Nicholas M. Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023

[2] [2]

Stability bounds for smooth optimal transport maps and their statistical implications.arXiv preprint arXiv:2502.12326, 2025

Sivaraman Balakrishnan and Tudor Manole. Stability bounds for smooth optimal transport maps and their statistical implications.arXiv preprint arXiv:2502.12326, 2025

work page arXiv 2025

[3] [3]

Heat flow, log-concavity, and Lipschitz transport maps.Electron

Giovanni Brigati and Francesco Pedrotti. Heat flow, log-concavity, and Lipschitz transport maps.Electron. Commun. Probab., 30:–, 2025

work page 2025

[4] [4]

Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

Guillaume Carlier, L´ ena¨ ıc Chizat, and Maxime Laborde. Displacement smoothness of entropic optimal transport.ESAIM: Control, Optimisation and Calculus of Variations, 30:25, 2024

work page 2024

[5] [5]

Time reversal of diffusion processes under a finite entropy condition.Ann

Patrick Cattiaux, Giovanni Conforti, Ivan Gentil, and Christian L´ eonard. Time reversal of diffusion processes under a finite entropy condition.Ann. Inst. Henri Poincar´ e Probab. Stat., 59(4):1844–1881, 2023

work page 2023

[6] [6]

An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities.Comptes Rendus

Sinho Chewi and Aram-Alexandre Pooladian. An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities.Comptes Rendus. Math´ ematique, 361(G9):1471–1482, 2023

work page 2023

[7] [7]

Weak semiconvexity estimates for Schr¨ odinger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges.Probability Theory and Related Fields, 189(3):1045–1071, 2024

Giovanni Conforti. Weak semiconvexity estimates for Schr¨ odinger potentials and logarithmic Sobolev inequality for Schr¨ odinger bridges.Probability Theory and Related Fields, 189(3):1045–1071, 2024

work page 2024

[8] [8]

Projected Langevin dynamics and a gradient flow for entropic optimal transport.Journal of the European Mathematical Society, 2025

Giovanni Conforti, Daniel Lacker, and Soumik Pal. Projected Langevin dynamics and a gradient flow for entropic optimal transport.Journal of the European Mathematical Society, 2025

work page 2025

[9] [9]

Tight stability bounds for entropic Brenier maps.International Mathematics Research Notices, 2025(7):rnaf078, 2025

Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. Tight stability bounds for entropic Brenier maps.International Mathematics Research Notices, 2025(7):rnaf078, 2025

work page 2025

[10] [10]

An entropic interpolation proof of the HWI inequality.Stochastic Processes and their Applications, 130(2):907–923, 2020

Ivan Gentil, Christian L´ eonard, Luigia Ripani, and Luca Tamanini. An entropic interpolation proof of the HWI inequality.Stochastic Processes and their Applications, 130(2):907–923, 2020

work page 2020

[11] [11]

On H¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54 (2):401–409, 2011

Nicola Gigli. On H¨ older continuity-in-time of the optimal transport map towards measures along a curve.Proceedings of the Edinburgh Mathematical Society, 54 (2):401–409, 2011

work page 2011

[12] [12]

Consistency trajectory models: learning probability flow ODE trajectory of diffusion

Dongjun Kim, Chieh-Hsin Lai, Wei-Hsiang Liao, Naoki Murata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory models: learning probability flow ODE trajectory of diffusion. InThe Twelfth International Conference on Learning Representations, 2024

work page 2024

[13] [13]

A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.Mathematische Annalen, 354(3):827–862, 2012

Young-Heon Kim and Emanuel Milman. A generalization of Caffarelli’s contraction theorem via (reverse) heat flow.Mathematische Annalen, 354(3):827–862, 2012. 13

work page 2012

[14] [14]

Stability of optimal transport maps on Riemannian manifolds.arXiv preprint arXiv:2504.05412, 2025

Jun Kitagawa, Cyril Letrouit, and Quentin M´ erigot. Stability of optimal transport maps on Riemannian manifolds.arXiv preprint arXiv:2504.05412, 2025

work page arXiv 2025

[15] [15]

A survey of the schrödinger problem and some of its connections with optimal transport

Christian L´ eonard. A survey of the schr¨ odinger problem and some of its connections with optimal transport.arXiv preprint arXiv:1308.0215, 2013

work page arXiv 2013

[16] [16]

Quantitative stability of optimal transport.Notes du cours Peccot, 2025, 2025

Cyril Letrouit. Quantitative stability of optimal transport.Notes du cours Peccot, 2025, 2025

work page 2025

[17] [17]

Gluing methods for quantitative stability of optimal transport maps.arXiv preprint arXiv:2411.04908, 2024

Cyril Letrouit and Quentin M´ erigot. Gluing methods for quantitative stability of optimal transport maps.arXiv preprint arXiv:2411.04908, 2024

work page arXiv 2024

[18] [18]

Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. InThe Eleventh International Conference on Learning Representations, 2023

work page 2023

[19] [19]

Plugin estimation of smooth optimal transport maps.The Annals of Statistics, 52(3):966–998, 2024

Tudor Manole, Sivaraman Balakrishnan, Jonathan Niles-Weed, and Larry Wasserman. Plugin estimation of smooth optimal transport maps.The Annals of Statistics, 52(3):966–998, 2024

work page 2024

[20] [20]

On the Lipschitz properties of transportation along heat flows

Dan Mikulincer and Yair Shenfeld. On the Lipschitz properties of transportation along heat flows. InGeometric aspects of functional analysis, volume 2327 of Lecture Notes in Math., pages 269–290. Springer, Cham, 2023

work page 2023

[21] [21]

Pooladian and J

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps.arXiv preprint arXiv:2109.12004, 2021

work page arXiv 2021

[22] [22]

Plug-in estimation of Schr¨ odinger bridges.arXiv preprint arXiv:2408.11686, 2024

Aram-Alexandre Pooladian and Jonathan Niles-Weed. Plug-in estimation of Schr¨ odinger bridges.arXiv preprint arXiv:2408.11686, 2024

work page arXiv 2024

[23] [23]

Minimax estimation of discontinuous optimal transport maps: the semi-discrete case

Aram-Alexandre Pooladian, Vincent Divol, and Jonathan Niles-Weed. Minimax estimation of discontinuous optimal transport maps: the semi-discrete case. In International Conference on Machine Learning, pages 28128–28150. PMLR, 2023

work page 2023

[24] [24]

Philippe Rigollet and Austin J. Stromme. On the sample complexity of entropic optimal transport.The Annals of Statistics, 53(1):61–90, 2025

work page 2025

[25] [25]

Diffusion Schr¨ odinger bridge matching.Advances in Neural Information Processing Systems, 36:62183–62223, 2023

Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion Schr¨ odinger bridge matching.Advances in Neural Information Processing Systems, 36:62183–62223, 2023

work page 2023

[26] [26]

Mixing time of the proximal sampler in relative Fisher information via strong data processing inequality.arXiv preprint arXiv:2502.05623, 2025

Andre Wibisono. Mixing time of the proximal sampler in relative Fisher information via strong data processing inequality.arXiv preprint arXiv:2502.05623, 2025. 14

work page arXiv 2025