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Sample complexity of Schr\"odinger potential estimation

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Pith's one-line read This paper establishes a high-probability excess-KL bound for empirical Schrödinger-potential estimation that decreases as O(log^2 n / n) when the target is realizable, without requiring compact support.

desk verdict Real theorem, oversold rate: the true Schrödinger potential for a canonical Gaussian target violates Assumption 3, so the advertised O(log^2/n) realizable rate is empty for the targets the abstract highlights. read the letter →

arxiv 2506.03043 v1 pith:FLLN2B2N submitted 2025-06-03 cs.LG math.STstat.MLstat.TH

Nikita Puchkin , Iurii Pustovalov , Yuri Sapronov , Denis Suchkov , Alexey Naumov , Denis Belomestny This is my paper
classification cs.LGmath.STstat.MLstat.TH MSC 62G0562G2060J60
keywords SchrödingerbridgepotentialestimationempiricalriskminimizationKullback-LeiblerdivergenceOrnstein-Uhlenbeckprocesssamplecomplexitynon-asymptoticboundsgenerativemodeling
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 studies how many samples are needed to estimate the Schrödinger potential that steers a reference diffusion to match a target endpoint distribution. It proves that the empirical Kullback-Leibler minimizer over a finite-dimensional class of log-potentials has excess KL risk bounded by an expression that decays like D $log^{2}$ n / n when the target lies in the class, and more generally by an oracle inequality with the same rate. This holds while allowing both the initial and target distributions to have unbounded support, requiring only that the target density be bounded and sub-Gaussian and that admissible log-potentials grow at most quadratically. The result matters because Schrödinger-bridge generative models need finite-sample guarantees, and previous analyses either required compact supports, log-concavity, or only gave slower $n^{{-1/2}}$ rates.

What carries the argument

The central object is the Ornstein-Uhlenbeck operator T_t, defined by T_t g(x) = E[g(X_t) | X_0 = x] for the reference OU process, and the Doob h-transform representation rho^psi_T(y) = integral q(y|x) $e^{{psi(y)}}$ (T_T e^psi(x))^{-1} rho_0(x) dx. The proof builds an epsilon-net over the parameter class, applies a Bernstein inequality for unbounded sub-exponential losses, and shows that the variance of the log-loss satisfies a Bernstein-type condition controlled by the KL divergence plus 1/n. The key technical lemmas bound the smoothness of the terminal log-density with respect to the log-potential through an explicit estimate involving K(T) = 1 + O($\sqrt$(d) $e^{{-bT}}$), which is what turns the epsilon-net approximation into a uniform high-probability bound.

What would settle it

Choose a known target density inside the Gaussian-mixture class from Proposition E.1, simulate n i.i.d. samples, compute the empirical KL minimizer, and check numerically whether the excess KL decays like $log^{2}$ n / n. If instead the observed rate is $n^{{-1/2}}$, the theorem's realizable rate would contradict the data; alternatively, constructing a target whose true log-potential grows faster than quadratically would place it outside the theorem's scope and should break the indicated rate.

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Extended reading notes

Core claim

Theorem 1 demonstrates that the excess KL divergence between the true terminal density rho*_T and the terminal density produced by the empirical risk minimizer, minus the best achievable risk inside the class, is with high probability O( $\sqrt$( Upsilon(n,delta) * inf_psi KL(rho*_T, rho^psi_T) ) + Upsilon(n,delta) ), where Upsilon(n,delta) is of order (Lambda d + M + d) (d + log(RLn/delta) + (M or log Lambda) $\sqrt$(d) $e^{{-bT}}$) D log n / n. In the realizable case, when the class Psi contains the true log-potential, the excess KL risk is O($log^{2}$ n / n). The paper claims this is the first non-asymptotic guarantee of this kind that permits unbounded support for both rho_0 and rho*_T and does not require the target density to be bounded away from zero.

Load-bearing premise

All stated rates rest on the assumption that the admissible log-potential class satisfies a two-sided quadratic growth condition (bounded above by M and below by -Lambda times a squared norm, minus M) together with a Lipschitz finite-dimensional parametrization; if the true Schrödinger potential cannot be approximated by such a class, the realizable O($log^{2}$ n / n) rate does not follow.

Editorial extensions

If this is right

  • If the true Schrödinger potential belongs to the admissible class, then n samples suffice for endpoint KL error O(log^2 n / n), a rate comparable to parametric density estimation.
  • The theorem applies to targets with unbounded support, so it covers realistic sub-Gaussian data distributions without truncated or compactly supported assumptions.
  • The bound holds with high probability uniformly over the whole class, giving a finite-sample confidence-style guarantee suitable for model selection.
  • Unlike plug-in Sinkhorn estimators whose errors blow up as the terminal time approaches T, the bound directly controls the KL divergence between the endpoint marginals.
  • The dimension D of the parameterization enters only logarithmically through Upsilon(n,delta), so high-dimensional potential classes are not automatically penalized by an exponential factor.

Reading between the lines

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

  • The argument relies mainly on the Gaussian transition kernel of the reference process, so the same style of proof may extend to other diffusion priors with explicitly known Gaussian or log-concave transition densities.
  • The paper leaves the approximation error inf_psi KL(rho*_T, rho^psi_T) implicit; for concrete classes such as Gaussian mixtures or neural networks, an additional approximation-theoretic analysis is needed to convert the oracle bound into an absolute rate.
  • A testable consequence is that the Bernstein-type variance bound is the engine of the fast rate: if a potential class violates the quadratic growth control, one should expect the n^{-1/2} behavior to return even in the realizable case.
  • The O(log^2 n / n) realizable rate suggests that Schrödinger-potential estimation behaves like well-specified parametric density estimation; a minimax analysis over natural smoothness classes would clarify how sharp this rate is.
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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

1 major / 4 minor

Summary. The paper studies the sample complexity of estimating a Schrödinger potential from n i.i.d. observations of the terminal marginal density ρ*_T of a Schrödinger bridge problem with an Ornstein-Uhlenbeck prior. The proposed estimator is the empirical Kullback-Leibler risk minimizer over a finite-dimensional class Ψ of admissible log-potentials satisfying a two-sided quadratic growth bound and a Lipschitz-type parameterization. The main result, Theorem 1, is a non-asymptotic oracle inequality: with probability at least 1−2δ, the excess KL risk of the estimated terminal density is bounded by C(√(Υ(n,δ)·inf_{ψ∈Ψ} KL(ρ*_T,ρψ_T)) + Υ(n,δ)), where Υ(n,δ) is of order (Λd+M+d)(d+log(RLn/δ)+...)D log n/n. In the realizable case, the authors conclude that the excess risk is O(log²n/n). The proof relies on sub-exponential tail bounds for the log-likelihood ratio, a Bernstein-type variance bound, an ε-net argument, and detailed semigroup estimates for the Ornstein-Uhlenbeck operator. The appendix contains complete proofs of all technical lemmas, including an explicit treatment of the Gaussian case.

Significance. If the main theorem is correct, the paper provides a nontrivial statistical guarantee for Schrödinger bridge estimation in a setting with unbounded marginals, improving on earlier analyses that assume compact support or log-concavity. The oracle-inequality form is the right target for this problem, and the proof is unusually detailed: the Orlicz-norm argument, the variance decomposition, and the Ornstein-Uhlenbeck semigroup lemmas are all stated with explicit parameter dependence. The paper also honestly acknowledges that approximation error is outside its scope. The main advertised consequence, however, is the fast O(log²n/n) rate in the realizable case; the compatibility of that claim with the assumptions is the principal point that needs clarification before the paper can be accepted.

major comments (1)
  1. [§4, Theorem 1 and §E, Proposition E.1] The advertised realizable O(log²n/n) rate has a narrower scope than the abstract and the contribution paragraph suggest. For the canonical example ρ0=ρ*_T=N(0,I_d), an OU prior with Σ=I, m=0, and bT large, Proposition E.1 gives an explicit log-potential whose quadratic coefficient is approximately −1/2 + 1/(1−e^{-2bT}) ≈ +1/2. Hence ψ*(x)→+∞ along every direction, violating Assumption 3's uniform upper bound ψ≤M. Since the Schrödinger bridge solution is unique up to a constant, no admissible ψ∈Ψ satisfying Assumptions 3–4 can produce exactly the terminal density N(0,I_d); the realizable case is empty for this target, and Theorem 1 does not yield O(log²n/n) for the total KL risk. The sentence in §4 stating that for Gaussian marginals the log-potential 'satisfies Assumption 3' is therefore too broad; it holds only under the additional covariance condition S_T ⪯ (1−e^{-2bT})/(2b)I mentioned in Appendix E. I ask the authors either to add an explicit discussion of when the realizable case is compatible with Assumptions 3–4, or to qualify the fast-rate claims in the abstract and contribution section.
minor comments (4)
  1. [Theorem 1 statement] The theorem states that the hidden constant behind ≲ depends on Σ, m, b, and v only, but Step 1 of Appendix A says that the hidden constant in the Orlicz-norm bound also depends on ρmax from Assumption 2; either include ρmax in the theorem statement or show that it can be absorbed.
  2. [Theorem 1 and Assumption 3] The notation (M∨logΛ) is undefined when Λ=0, which is allowed by Assumption 3; replace it with an expression such as M+log(M∨Λ∨1) or add a convention for logΛ in that case.
  3. [Appendix A, Step 6] The displayed maximum in Step 6 contains the factor exp{12K(T)e^{-bT}||Σ_T^{-1}||(...)}, while Step 5 derived the bound with exp{12e^{-bT}||Σ_T^{-1}||(...)}; please align the two displays or clarify why the additional K(T) factor appears.
  4. [Equation (3) and Section 3] It would improve readability to state explicitly that hψ(y,T)=e^{ψ(y)}, so that the integrand in (3) is exactly the terminal density ρψ_T(y); this identity is used throughout the proof but is not written next to the definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the oracle inequality is derived from stated assumptions via concentration and OU-semigroup estimates, and the O(log^2 n/n) rate is explicitly conditional on realizability or approximation quality rather than being assumed.

full rationale

The paper's main result, Theorem 1, is an oracle inequality for the empirical KL minimizer: it bounds KL(rho*_T, rho_hat_T) - inf_{psi in Psi} KL(rho*_T, rho^psi_T) by a quantity involving Upsilon(n,delta) and the oracle approximation term inf KL. The derivation chain in Appendix A uses Assumptions 3 and 4 only to control Orlicz norms, variances, and Holder-type continuity of log rho^psi_T with respect to psi; these are genuine analytic inputs, not restatements of the conclusion. The advertised O(log^2 n/n) rate is explicitly described as conditional: the paper states 'In the realizable case (that is, rho*_T in {rho^psi_T : psi in Psi}) the right-hand side in Theorem 1 becomes O(log^2 n/n),' and earlier 'the excess risk may decrease as fast as O(log^2 n/n) provided that the class of log-potentials Psi is rich enough to approximate the target density rho*_T.' The paper also explicitly defers approximation error: 'we focus on the statistical error leaving study of the approximation out of the scope of the present paper.' That limitation narrows the claim's scope but is not circularity. The skeptic's concern that the true Schrodinger potential for N(0,I_d) under an OU prior can violate the upper-bound part of Assumption 3 for large T is a correctness or scope issue about whether the assumptions cover the advertised example, not a step in which a prediction is equivalent to its inputs by construction. No fitted parameter is renamed as a prediction, no load-bearing argument reduces to a self-citation, and no uniqueness theorem is imported from the authors' prior work. The derivation is self-contained given Assumptions 1-4 and standard concentration and Gaussian-semigroup results.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The theory is self-contained conditional on explicit regularity assumptions about the target and the potential class. No numbers are fitted to data, and no new physical or mathematical entities are introduced. The main costs are the growth and smoothness assumptions on Psi and the large-horizon condition, not hidden free parameters.

assumptions (7)
  • domain assumption The Schrödinger bridge problem is well-posed and the optimal process is a Doob h-transform with boundary potentials nu0 and nuT (Dai Pra 1991, Theorem 3.2).
    Used in Section 1 to represent the optimal drift through h_psi and to define the class of endpoint densities rho^psi_T.
  • domain assumption The reference process is a multivariate Ornstein-Uhlenbeck process with Gaussian transition kernels.
    Assumption 1 and Section 3; all proof steps exploit the explicit Gaussian form of T_t and the OU semigroup.
  • domain assumption The target density is bounded and sub-Gaussian (Assumption 2).
    Needed to control sub-exponential tails and exponential moments in Steps 1, 3, and 4 of Appendix A.
  • domain assumption Log-potentials satisfy -Lambda ||Sigma^{-1/2}(x-m)||^2 - M <= psi(x) <= M and T_infinity psi = 0 (Assumption 3).
    The entire sub-exponential and variance analysis of log(rho*_T / rho^psi_T) depends on this two-sided growth control.
  • domain assumption The class Psi is a finite-dimensional smooth parametric family with Lipschitz constant growing as (1 + ||x||^2) (Assumption 4).
    This makes the epsilon-net argument in Steps 2 and 4 of Appendix A work and contributes the D log n / n term.
  • domain assumption The large horizon condition bT >= (5 + log d) or log(160b(v^2 or 1)||Sigma^{-1}||) holds.
    Used in Lemma B.2 and Step 4 of Appendix A to ensure exponential moments of the OU-kernel factors are finite.
  • standard math Standard concentration and Gaussian-semigroup results from Vershynin, Lecue-Mitchell, Rigollet-Hutter, and the included Lemmas D.1-D.3 are valid.
    Bernstein inequalities for sub-exponential variables, sub-Gaussian norm tails, chi-square Lp bounds, and Gronwall-type inequalities are invoked throughout the proof.

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Pith. "Pith review of Sample complexity of Schr\"odinger potential estimation." pith.science (2026). https://pith.science/paper/FLLN2B2N

@misc{pith2026250603043,
  author       = {Pith},
  title        = {Pith review of: Sample complexity of Schr\"odinger potential estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FLLN2B2N}},
  note         = {Machine review of arXiv:2506.03043}
}
abstract

We address the problem of Schr\"odinger potential estimation, which plays a crucial role in modern generative modelling approaches based on Schr\"odinger bridges and stochastic optimal control for SDEs. Given a simple prior diffusion process, these methods search for a path between two given distributions $\rho_0$ and $\rho_T^*$ requiring minimal efforts. The optimal drift in this case can be expressed through a Schr\"odinger potential. In the present paper, we study generalization ability of an empirical Kullback-Leibler (KL) risk minimizer over a class of admissible log-potentials aimed at fitting the marginal distribution at time $T$. Under reasonable assumptions on the target distribution $\rho_T^*$ and the prior process, we derive a non-asymptotic high-probability upper bound on the KL-divergence between $\rho_T^*$ and the terminal density corresponding to the estimated log-potential. In particular, we show that the excess KL-risk may decrease as fast as $O(\log^2 n / n)$ when the sample size $n$ tends to infinity even if both $\rho_0$ and $\rho_T^*$ have unbounded supports.

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Works this paper leans on

38 extracted references · 24 canonical work pages

  1. [1]

    Baptista, A.-A

    R. Baptista, A.-A. Pooladian, M. Brennan, Y. Marzouk, and J. Niles-Weed. Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional B renier maps. arXiv preprint arXiv:2411.07154, 2024

    arXiv 2024

  2. [2]

    Beurling

    A. Beurling. An automorphism of product measures. Ann. Math. (2), 72: 0 189--200, 1960. ISSN 0003-486X. doi:10.2307/1970151

    work page doi:10.2307/1970151 1960

  3. [3]

    Bunne, Y.-P

    C. Bunne, Y.-P. Hsieh, M. Cuturi, and A. Krause. The S chr \"o dinger bridge between gaussian measures has a closed form. In International Conference on Artificial Intelligence and Statistics, pages 5802--5833. PMLR, 2023

    work page 2023

  4. [4]

    Y. Chen, T. Georgiou, and M. Pavon. Entropic and displacement interpolation: a computational approach using the H ilbert metric. SIAM Journal on Applied Mathematics, 76 0 (6): 0 2375--2396, 2016

    work page 2016

  5. [5]

    Y. Chen, T. T. Georgiou, and M. Pavon. Stochastic control liaisons: R ichard S inkhorn meets G aspard M onge on a S chr \"o dinger bridge. Siam Review, 63 0 (2): 0 249--313, 2021

    work page 2021

  6. [6]

    Chiarini, G

    A. Chiarini, G. Conforti, G. Greco, and L. Tamanini. A semiconcavity approach to stability of entropic plans and exponential convergence of S inkhorn's algorithm, 2024. URL https://arxiv.org/abs/2412.09235

    arXiv 2024

  7. [7]

    o dinger potentials and logarithmic S obolev inequality for S chr \

    G. Conforti. Weak semiconvexity estimates for S chr \"o dinger potentials and logarithmic S obolev inequality for S chr \"o dinger bridges, 2024. URL https://arxiv.org/abs/2301.00083

    arXiv 2024

  8. [8]

    Conforti, A

    G. Conforti, A. Durmus, and G. Greco. Quantitative contraction rates for S inkhorn algorithm: beyond bounded costs and compact marginals, 2024. URL https://arxiv.org/abs/2304.04451

    arXiv 2024

Show all 38 references
  1. [9]

    P. Dai Pra. A stochastic control approach to reciprocal diffusion processes. Applied Mathematics and Optimization, 23 0 (1): 0 313--329, 1991. doi:10.1007/BF01445134

    1991 doi

  2. [10]

    De Bortoli, J

    V. De Bortoli, J. Thornton, J. Heng, and A. Doucet. Diffusion S chr \"o dinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34: 0 17695--17709, 2021

    2021

  3. [11]

    Eckstein

    S. Eckstein. Hilbert's projective metric for functions of bounded growth and exponential convergence of S inkhorn's algorithm. Probability Theory and Related Fields, pages 1--37, 2025

    2025

  4. [12]

    R. Fortet. R \'e solution d'un syst \`e me d' \'e quations de M . Schr \"o dinger . J. Math. Pures Appl. (9), 19: 0 83--105, 1940. ISSN 0021-7824

    1940

  5. [13]

    Genevay, G

    A. Genevay, G. Peyr \'e , and M. Cuturi. Learning generative models with S inkhorn divergences. In International Conference on Artificial Intelligence and Statistics, pages 1608--1617. PMLR, 2018

    2018

  6. [14]

    Gushchin, S

    N. Gushchin, S. Kholkin, E. Burnaev, and A. Korotin. Light and optimal S chr\"odinger bridge matching. In Forty-first International Conference on Machine Learning, 2024 a . URL https://doi.org/10.48550/arXiv.2402.03207

    2024 doi arXiv

  7. [15]

    Gushchin, D

    N. Gushchin, D. Selikhanovych, S. Kholkin, E. Burnaev, and A. Korotin. Adversarial S chr \"o dinger bridge matching. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024 b . URL https://arxiv.org/abs/2405.14449

    2024 arXiv

  8. [16]

    B. Jamison. Reciprocal processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 30 0 (1): 0 65--86, 1974

    1974

  9. [17]

    Janati, B

    H. Janati, B. Muzellec, G. Peyr\' e , and M. Cuturi. Entropic optimal transport between unbalanced gaussian measures has a closed form. In Advances in Neural Information Processing Systems, volume 33, pages 10468--10479. Curran Associates, Inc., 2020. URL https://proceedings.n...

    2020

  10. [18]

    Korotin, N

    A. Korotin, N. Gushchin, and E. Burnaev. Light S chr\"odinger bridge. In The Twelfth International Conference on Learning Representations, 2024. URL https://doi.org/10.48550/arXiv.2310.01174

    2024 doi arXiv

  11. [19]

    Lecu \'e and C

    G. Lecu \'e and C. Mitchell. Oracle inequalities for cross-validation type procedures. Electronic Journal of Statistics, 6 0 (none): 0 1803 -- 1837, 2012. doi:10.1214/12-EJS730. URL https://doi.org/10.1214/12-EJS730

    2012 doi

  12. [20]

    C. Leonard. A survey of the S chr\" o dinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems, 34 0 (4): 0 1533--1574, 2014

    2014

  13. [21]

    G.-H. Liu, A. Vahdat, D.-A. Huang, E. A. Theodorou, W. Nie, and A. Anandkumar. I ^2 SB: image-to-image S chr\"odinger bridge. In Fortieth International Conference on Machine Learning, 2023. URL https://doi.org/10.48550/arXiv.2302.05872

    2023 doi arXiv

  14. [22]

    H. D. March and P. Henry-Labordere. Building arbitrage-free implied volatility: S inkhorn's algorithm and variants, 2023. URL https://arxiv.org/abs/1902.04456

    2023 arXiv

  15. [23]

    Pavon, G

    M. Pavon, G. Trigila, and E. G. Tabak. The data-driven S chr\"odinger bridge. Communications on Pure and Applied Mathematics, 74 0 (7): 0 1545--1573, 2021

    2021

  16. [24]

    Peluchetti

    S. Peluchetti. Diffusion bridge mixture transports, S chr \"o dinger bridge problems and generative modeling. Journal of Machine Learning Research, 24 0 (374): 0 1--51, 2023

    2023

  17. [25]

    Pooladian and J

    A.-A. Pooladian and J. Niles-Weed. Plug-in estimation of S chr\"odinger bridges. arXiv preprint arXiv:2408.11686, 2024

    2024 arXiv

  18. [26]

    Rapakoulias, A

    G. Rapakoulias, A. R. Pedram, and P. Tsiotras. G o W ith the F low: F ast D iffusion for G aussian M ixture M odels. Preprint. ArXiv:2412.09059v3, 2024

    2024

  19. [27]

    Rigollet and J.-C

    P. Rigollet and J.-C. H\"utter. High-dimensional statistics. Preprint. ArXiv:2310.19244, 2023

    2023 arXiv

  20. [28]

    Schmidt-Hieber

    J. Schmidt-Hieber. Nonparametric regression using deep neural networks with R e LU activation function. The Annals of Statistics, 48 0 (4): 0 1875--1897, 2020

    2020

  21. [29]

    odinger. \

    E. Schr\"odinger. \"U ber die U mkehrung der N aturgesetze. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, pages 144--153, 1932

    1932

  22. [30]

    Y. Shi, V. De Bortoli, A. Campbell, and A. Doucet. Diffusion S chr\"odinger bridge matching. arXiv preprint arXiv:2303.16852, 2023

    2023 arXiv

  23. [31]

    M. G. Silveri, A. O. Durmus, and G. Conforti. Theoretical guarantees in KL for diffusion flow matching. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL https://openreview.net/forum?id=ia4WUCwHA9

    2024

  24. [32]

    Sinkhorn

    R. Sinkhorn. Diagonal equivalence to matrices with prescribed row and column sums. The American Mathematical Monthly, 74 0 (4): 0 402--405, 1967. ISSN 00029890, 19300972. URL http://www.jstor.org/stable/2314570

    1967

  25. [33]

    A. Stromme. Sampling from a S chr \"o dinger bridge. In F. Ruiz, J. Dy, and J.-W. van de Meent, editors, Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, volume 206 of Proceedings of Machine Learning Research, pages 4058--4067. PMLR, ...

    2023

  26. [34]

    Tzen and M

    B. Tzen and M. Raginsky. Theoretical guarantees for sampling and inference in generative models with latent diffusions. In Conference on Learning Theory, pages 3084--3114. PMLR, 2019

    2019

  27. [35]

    Vargas, P

    F. Vargas, P. Thodoroff, A. Lamacraft, and N. Lawrence. Solving S chr \"o dinger bridges via maximum likelihood. Entropy, 23 0 (9): 0 1134, 2021

    2021

  28. [36]

    Vershynin

    R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2018

    2018

  29. [37]

    G. Wang, Y. Jiao, Q. Xu, Y. Wang, and C. Yang. Deep generative learning via S chr\"odinger bridge. In International conference on machine learning, pages 10794--10804. PMLR, 2021

    2021

  30. [38]

    Zhang and Y

    Q. Zhang and Y. Chen. Path integral sampler: A stochastic control approach for sampling. In International Conference on Learning Representations, 2022. URL https://doi.org/10.48550/arXiv.2111.15141

    2022 doi arXiv

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