A Revisit to Rate-distortion Theory via Optimal Weak Transport

Jia Wang; Jiayang Gao; Jiayang Zou; Luyao Fan

REVIEW 3 major objections 4 minor 48 references

A Revisit to Rate-distortion Theory via Optimal Weak Transport

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims to find a new parametric form of the rate-distortion function using optimal weak transport, tying it to Schrödinger bridge equations and rederiving when the Shannon lower bound is achieved.

desk verdict Worth a careful referee: the OWT reformulation is genuinely useful and the main theorems look right, but Theorem 7's proof has an unjustified minimax swap and Theorem 8's proof is too terse; the stress-test's falsity claim does not hold up. read the letter →

arxiv 2501.09362 v3 pith:SQQ52R47 submitted 2025-01-16 cs.IT math.IT

Jiayang Zou , Luyao Fan , Jiayang Gao , Jia Wang This is my paper · ORCID

classification cs.ITmath.IT MSC 94A3449Q22

keywords rate-distortiontheoryoptimalweaktransportSchrödingerbridgeproblemShannonlowerboundparametricrepresentationabstractalphabetsentropic

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

This paper tries to show that the rate-distortion function $R(D)$ — the minimum coding rate needed to keep expected distortion at or below $D$ — can be derived from optimal weak transport, a recently introduced generalization of optimal transport in which moving a point $x$ costs a function of the whole conditional distribution of the reconstruction, not just of a single destination. Working on abstract alphabets modeled as Polish spaces (complete separable metric spaces), it derives a parametric representation of $R(D)$ that ties the function to the Schrödinger bridge problem: for any $\beta$ in the subdifferential of $R$ at $D$ (the set of its supporting slopes), the function equals an infimum over reconstruction measures $\nu$ of an explicit functional built from $e^{-\beta\rho(x,y)}$ plus a correction term $L(\nu,\beta)$. When an optimal reconstruction measure exists, the correction vanishes and the optimal joint distribution takes the product-like form $d\pi^* \propto e^{-\beta\rho(x,y)} d\mu d\nu^*$, yielding a Shannon-lower-bound-style formula. As a byproduct, the paper rederives the known achievability conditions for the Shannon lower bound under squared-error distortion without invoking variational calculus. A sympathetic reader would care because the representation offers a new, more geometric handle on $R(D)$ and points numerical methods developed for Schrödinger bridges at rate-distortion computation.

What carries the argument

The carrying object is the rate-distortion function rewritten as a weak-transport problem: the paper decomposes $R(D) = \inf_{\nu} \inf_{\pi \in \Pi(\mu,\nu), E_\pi \rho \le D} I(X;Y)$ and then relaxes the distortion constraint to $J(\nu,\beta) = \inf_{\pi \in \Pi(\mu,\nu)} [ I(X;Y) + \beta (E_\pi \rho - D) ]$. The workhorse identity expresses $J(\nu,\beta)$ as relative entropy against the reference measure $\gamma = K e^{-\beta\rho(x,y)} d\mu d\nu$, so the inner minimization becomes an entropic optimal transport problem whose optimizer has the multiplicative density $d\pi^*/d\gamma = f(x)g(y)$ by the Schrödinger bridge characterization (Lemma 1). That multiplicative structure is what produces the correction term $L(\nu,\beta)$, assembled from the $g(y)$ solving the Schrödinger equations. Existence of the inner minimizer is supplied by the refined weak-transport existence theorem under the paper's moment and lower-semicontinuity assumptions, and the final representation is assembled by restricting $\beta$ to the subdifferential $\partial R(D)$ of the convex rate-distortion function.

What would settle it

Take a binary source with Hamming distortion at a distortion level where a subgradient $\beta$ is known, compute the claimed infimum-over-$\nu$ expression in Theorem 7, and compare the result with the exactly known $R(D)$ from the classical alternating-minimization algorithm for rate-distortion; any gap would expose the $\inf_\nu \sup_\beta$ swap as the failing step. Alternatively, find any source obeying the paper's assumptions for which an optimal reconstruction $\nu^*$ exists but the optimal joint distribution is not proportional to $e^{-\beta\rho(x,y)} d\mu d\nu^*$, which would refute Theorem 8.

Watch

Extended reading notes

Core claim

The central claim is Theorem 7: under the paper's assumptions on the source, the loss function, and finite moments, the rate-distortion function admits the parametric representation $R(D) = \inf_{\nu} \{ -\int \log(\int e^{-\beta\rho(x,y)} d\nu) d\mu - \beta D + L(\nu,\beta) \}$ for every $\beta$ in the subdifferential $\partial R(D)$, where $L(\nu,\beta)$ is a correction term defined through the function $g(y)$ that solves the Schrödinger bridge equations for the reference measure $\gamma = K e^{-\beta\rho} d\mu d\nu$. The route is to rewrite $R(D)$ as an infimum over reconstruction measures $\nu$ of a weak-transport problem, relax the distortion constraint with a Lagrange multiplier $\beta$, and then use the known structure of Schrödinger-bridge optimizers — $d\pi^*/d\gamma = f(x)g(y)$ — to evaluate the inner minimization. Theorem 8 adds that if an optimal reconstruction $\nu^*$ exists, the optimal joint distribution is $d\pi^* = e^{-\beta\rho(x,y)} (\int e^{-\beta\rho(x,y)} d\nu^*)^{-1} d\mu d\nu^*$, so that $L(\nu^*,\beta) = 0$ and $R(D) = -\int_X \log(\int_Y e^{-\beta\rho(x,y)} d\nu^*) d\mu - \beta D$, the form anticipated by the Shannon lower bound. As a stated payoff, Corollary 2 reproduces earlier achievability conclusions for the Shannon lower bound: for quadratic distortion on $\mathbb{R}^n$, the RD function coincides with the bound when the support of the optimal reproduction has an accumulation point; when the bound is not achieved, that support consists of isolated singularities, and a bounded such support forces the reconstruction alphabet to be finite and discrete.

Load-bearing premise

The load-bearing step is the interchange of the infimum over reconstruction measures and the supremum over the Lagrange multiplier — the $\inf_\nu \sup_\beta = \sup_\beta \inf_\nu$ step inside the chain (31) — for which the paper cites no minimax theorem and proves no convex-concave structure; if that swap fails, the parametric representation of $R(D)$ does not follow from the surrounding estimates.

Editorial extensions

If this is right

For any abstract source satisfying the paper's assumptions, $R(D)$ can be written as an explicit infimum over reconstruction measures of a functional of $e^{-\beta\rho(x,y)}$, for each $\beta$ in the subdifferential of $R$ at $D$.
Where an optimal reconstruction exists, the optimal test channel has the form $d\pi^* \propto e^{-\beta\rho(x,y)} d\mu d\nu^*$, tying rate-distortion-optimal channels directly to Schrödinger bridges.
The Shannon lower bound is achieved for squared-error distortion when the optimal reproduction's support has an accumulation point; failing that, the support is made of isolated singularities, and a bounded such support forces a finite discrete reproduction alphabet.
The connection suggests that algorithms developed for Schrödinger bridge problems can be brought to bear on computing rate-distortion functions.

Reading between the lines

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

If the representation in Theorem 7 survives numerical checks, the correction term $L(\nu,\beta)$ can be read as the price of using a non-optimal reconstruction measure; that suggests an alternating scheme that solves Schrödinger bridge equations at fixed $\nu$ and then updates $\nu$, with $L(\nu,\beta)$ as a convergence certificate.
A natural test case is finite alphabets, where the classical alternating-minimization algorithm gives $R(D)$ exactly: formula (24) should collapse to the familiar fixed-point equations of that algorithm, and checking that would anchor the whole framework.
The same weak-transport perspective could in principle be applied to other information-theoretic quantities that are infima over couplings with one free marginal, such as the capacity-cost function or the information bottleneck, giving each a Schrödinger-bridge-style representation.
The scope of Theorem 8 is left open because it assumes an optimal reconstruction exists; identifying source classes where that existence is provable, such as Gaussian sources or compact alphabets, would settle how widely the Shannon-lower-bound-style formula holds.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Zou et al. recast rate-distortion in the optimal weak transport framework and derive a parametric representation with a correction term L(ν,β) that vanishes when the Shannon lower bound is tight. That correction is the paper's real contribution; it connects the RD function to the Schrödinger bridge problem in a way that is not just cosmetic. The paper is also honest about reproducing Rose's SLB achievability, and the OWT existence theorems are used cleanly. I think the work deserves a serious referee.

The soft spot is the proof writing, not the conclusions. Theorem 7's proof contains an inf/sup interchange in the chain (31) that is not justified by any minimax theorem. That said, the epsilon argument in the same appendix actually proves L(β_D)=R(D)+β_D D without the interchange, so the representation can be salvaged. The authors should restructure the proof to avoid the unjustified step. Theorem 8's proof is terse: the step from a global minimizer to the tilted channel via Csiszár's Lemma 1.4 is correct in substance, but it should be spelled out that the global-minimizer property, not merely the fixed-marginal minimization, is what forces g to be constant.

I disagree with the stress-test's claim that Theorem 8 is false. In the finite-alphabet Hamming examples it cites, the balance equation has interior solutions for the relevant β (close to the source distribution when β is large), so the tilted form does hold. The step (35) is not circular; it follows from the subgradient inequality and the existence of an optimal π⋆ with Eρ=D. The stress-test is right that the proof's step (39) is compressed, but the X-marginal condition forces the normalization and the Y-marginal condition is exactly the balance that the global minimizer satisfies.

This paper is for researchers working on rate-distortion theory and its connections to transport; it may inspire numerical algorithms via Schrödinger bridges. The citation pattern is solid and includes the classical work. Assumption 6 is stronger than Csiszár's, as the authors concede, but it is reasonable. I'd send it to peer review with a request to fix the proof structure. I would not cite it in my own work until the proofs are cleaned up.

Referee Report

3 major / 4 minor

Summary. This paper attempts to reformulate classical rate-distortion theory in the language of optimal weak transport. The authors introduce an OWT formulation of the rate-distortion Lagrangian, invoke existence theorems for weak transport and the static Schrödinger problem, and propose a parametric representation of R(D) (Theorem 7) involving an infimum over reconstruction measures and a correction term L(ν,β) defined through Schrödinger equations. They then claim (Theorem 8) that when an optimal reconstruction exists, the optimal joint distribution takes the Schrödinger form dπ* = e^{-βρ}/(∫e^{-βρ}dν*) dµ×dν*, yielding the Shannon lower bound, and they use this to reproduce K. Rose's theorem on SLB achievability (Corollary 2).

Significance. The topic is of potential interest: a rigorous connection between rate-distortion theory and Schrödinger bridges could yield new structural and computational insights. The paper correctly applies several external results (the weak-transport existence theorems of Backhoff-Veraguas et al. and Csiszár's parametric representation), and the manipulation of the Lagrangian via weak transport is instructive. However, the central new claims are not reliable as stated: Theorem 8 is false, Theorem 7 includes an unjustified minimax interchange, and Corollary 2 is asserted without proof. The advertised connection to Schrödinger bridges is therefore not established.

major comments (3)

[Appendix F, Eq. (39), Theorem 8] The proof of Theorem 8 uses Csiszár's Lemma 1.4 to write dπ* = α(x)e^{-βρ(x,y)} dµ dν* and then identifies α(x) with (∫e^{-βρ(x,y)}dν*)^{-1}. This identification is equivalent to the balance equation ∫ e^{-βρ(x,y)} / ∫ e^{-βρ(x,y')} dν*(y') dµ(x) = 1 for ν*-a.e. y, which is precisely the tightness condition for the Shannon lower bound. Nothing in the existence of an optimal reconstruction for the rate-distortion problem forces this balance. A concrete counterexample is any non-uniform finite-alphabet source with at least three symbols under Hamming distortion: it satisfies Assumptions 1–6, admits an optimal reconstruction for every D, yet R(D) strictly exceeds the expression in (25) for small D. Hence Theorem 8 and the Schrödinger-bridge representation (24)–(25) are false as stated.
[Appendix E, Eq. (31), Theorem 7] The chain of equalities in Eq. (31) interchanges the infimum over ν and the supremum/maximum over β — including 'inf_ν sup_β' to 'sup_β inf_ν', and 'inf_ν max_{β∈∂R(D)}' to 'max_{β∈∂R(D)} inf_ν' — without any minimax theorem or verification of a convex-concave saddle-point structure. The displayed equalities are therefore unsupported. Although the final statement 'R(D)=inf_ν J(ν,β) for β∈∂R(D)' can be obtained directly from Csiszár's subgradient inequality and the definition of R(D), the theorem as stated includes the stronger inf-max equality, which is not established by the given argument.
[Section IV and Corollary 2] Corollary 2, which is advertised as a main byproduct reproducing K. Rose's results without variational calculus, is not proved in the manuscript. The text merely says 'By (24) along with K. Rose's methods of using the completeness of Hermite polynomials, we can reproduce the following conclusion,' with no derivation supplied. Rose's argument is highly nontrivial, and the premise (24) is furnished by the false Theorem 8, so the claimed reproduction is invalid as it stands.

minor comments (4)

[Section II-B and throughout] The spelling 'Schödinger' should be 'Schrödinger' in several places, and there are numerous typographical artifacts (e.g., '/greaterorequalslant' and 'heorem 2') that should be corrected in a revised manuscript.
[Theorems 3 and 4] Both theorems are cited as 'Theorem 2.1 in [7]', which appears inconsistent; the numbering in Csiszár's paper should be checked and the two results distinguished.
[Theorem 7, definition of L(ν,β)] The definition of L(ν,β) is elliptical: it refers to 'g(y) satisfying the Schrödinger equations (7)', but g is determined only up to a multiplicative constant and the domain of the infimum over ν is not made explicit. A self-contained definition would improve readability.
[Appendix D, Proposition 1] The proof of Proposition 1 applies the Wasserstein triangle inequality and bounds W_t(µ,ν) by c^{-1/t} D^{1/t}; the argument assumes D is finite and should state this explicitly.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 8's Schrödinger-bridge/SLB conclusion is inserted, not derived: Appendix F silently identifies Csiszár's α(x) with the Gibbs normalization.

other [Theorem 8, Appendix F, displayed equation (39); cf. main-text assertion before Theorem 8]
"Furthermore, by Lemma 1.4 in [7], we have dπ⋆ = α(x)e^{−βρ(x,y)}dµ × dν⋆ = e^{−βρ(x,y)} / ∫_Y e^{−βρ(x,y)}dν⋆ dµ × dν⋆. (39)"

The quoted step reduces the theorem's conclusion to a rewriting. Lemma 1.4 in [7], as invoked, provides the factorization dπ⋆ = α(x)e^{−βρ}dµdν⋆ together with an integral constraint on α, but it does not identify α(x) pointwise with the Gibbs normalization (∫ e^{−βρ(x,y)}dν⋆(y))^{-1}. That identification is exactly the balance equation ∫ e^{−βρ(x,y)}(∫ e^{−βρ(x,y′)}dν⋆(y′))^{-1}dµ(x)=1 for ν⋆-a.e. y, i.e., the condition that the Shannon lower bound is tight. This balance is the content of (24)–(25), not a consequence of the existence of an optimal reconstruction ν⋆ nor of Lemma 1.4. Thus the proof assumes the target identity in the final equality of (39).

full rationale

The paper is mostly a genuine re-derivation using external mathematics: Theorem 7 builds on Gozlan et al.'s weak-transport formulation and on the Backhoff-Veraguas–Pammer existence/semicontinuity theorems, and Corollary 2 reproduces Rose's own Hermite-polynomial method; reference [42] is a self-citation but is not load-bearing. The unchecked inf/sup interchange in eq. (31) is a correctness/rigor gap rather than a circular reduction, so it does not raise the circularity score by itself. The genuinely circular step is localized to Theorem 8 and Appendix F. Eq. (39) claims that Csiszár's Lemma 1.4 gives dπ⋆ = α(x)e^{−βρ}dµdν⋆ and then immediately rewrites α(x) as (∫ e^{−βρ}dν⋆)^{-1}. The lemma, as used, supplies only the factorization plus the ν⋆-a.e. integral condition; the pointwise Gibbs form is equivalent to the balance equation and therefore to SLB tightness, which is exactly (24)–(25). The proof therefore inserts the central conclusion rather than deriving it. Because this clean Schrödinger-bridge/SLB formula is the paper's advertised payoff, the circularity is substantive, although the surrounding OWT existence material and the classical RD lemmas remain independent and non-circular.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical or mathematical entities. It relies on the external optimal weak transport existence theory and on classical rate-distortion properties from Csiszár. The specific assumptions A4-A6 are ad hoc to this paper and are stronger than the classical Csiszár conditions, which limits the claimed generality.

assumptions (8)

domain assumption Assumption 1: inf_y ρ(x,y) = 0 for all x.
Standard normalization of the loss function, taken from Csiszár's setup.
domain assumption Assumption 2: existence of random variables ξ, η with I(ξ;η) < ∞ and Eρ(ξ,η) < ∞.
Ensures the rate-distortion function is not identically infinite.
domain assumption Assumption 3: there exists a finite set B ⊂ Y with ∫ ρ(x,B) dµ < ∞.
Used to guarantee finiteness of R(D) above D_min, following Csiszár.
ad hoc to paper Assumption 4: (x,p) → ∫ ρ(x,y) dp is jointly lower semicontinuous on X × P(Y).
Needed for the first weak transport existence theorem; fails for squared Euclidean distance, as the paper notes.
ad hoc to paper Assumption 5: (x,p) → ∫ ρ(x,y) dp is jointly lower semicontinuous on X × P_t(Y).
Weaker than Assumption 4 and allows ρ(x,y) = ||x-y||^t; supports the refined existence theorem.
ad hoc to paper Assumption 6: X = Y are Polish, µ ∈ P_t(X), and ρ(x,y) ≥ c · d(x,y)^t for some c > 0.
Stronger than Csiszár's assumptions; used to prove the reconstruction distribution lies in P_t(Y) (Proposition 1).
standard math Existence and semicontinuity theorems for weak transport (Theorem 1 from [20], Theorem 2 from [21]).
External results invoked to guarantee minimizers of J(ν,β).
standard math Schrödinger bridge structure of entropic optimal transport minimizers (Lemma 1 from [21]).
External result used to derive the product form of the optimal coupling density.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Revisit to Rate-distortion Theory via Optimal Weak Transport." pith.science (2026). https://pith.science/paper/SQQ52R47

@misc{pith2026250109362,
  author       = {Pith},
  title        = {Pith review of: A Revisit to Rate-distortion Theory via Optimal Weak Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQQ52R47}},
  note         = {Machine review of arXiv:2501.09362}
}

read the original abstract

This paper revisits the rate-distortion theory from the perspective of optimal weak transport, as recently introduced by Gozlan et al. While the conditions for optimality and the existence of solutions are well-understood in the case of discrete alphabets, the extension to abstract alphabets requires more intricate analysis. Within the framework of weak transport problems, we derive a parametric representation of the rate-distortion function, thereby connecting the rate-distortion function with the Schr\"odinger bridge problem, and establish necessary conditions for its optimality. As a byproduct of our analysis, we reproduce K. Rose's conclusions regarding the achievability of Shannon lower bound concisely, without reliance on variational calculus.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 43 canonical work pages

[1]

A mathematical theory of communication,

C. E. Shannon, “A mathematical theory of communication, ” The Bell System Technical Journal , vol. 27, no. 3, pp. 379–423, 1948

1948
[2]

Coding theorems for a discrete source with a ﬁdelity criterion,

——, “Coding theorems for a discrete source with a ﬁdelity criterion,” International Convention Record , vol. 7, pp. 325–350, 1959

work page 1959
[3]

Berger, Rate-distortion theory: A mathematical basis for data com- pression

T. Berger, Rate-distortion theory: A mathematical basis for data com- pression. Englewood Cliffs: Prentice-Hall, 1971

work page 1971
[4]

Computation of channel capacity and rate-di stortion func- tions,

R. Blahut, “Computation of channel capacity and rate-di stortion func- tions,” IEEE Trans. Inf. Theory , vol. 18, no. 4, pp. 460–473, 1972

work page 1972
[5]

R. E. Blahut, Principles and practice of information theory . Addison- Wesley Longman Publishing Co., Inc., 1987

work page 1987
[6]

T. M. Cover and J. A. Thomas, Elements of Information Theory , 2nd ed. New Y ork, NY , USA: Wiley, 2006

work page 2006
[7]

On an extremum problem of information theor y,

I. Csiszár, “On an extremum problem of information theor y,” Studia Scientiarum Mathematicarum Hungarica , vol. 9, no. 1, pp. 57–71, 1974

work page 1974
[8]

An algorithm for computing the capacity of a rbitrary discrete memoryless channels,

S. Arimoto, “An algorithm for computing the capacity of a rbitrary discrete memoryless channels,” IEEE Trans. Inf. Theory , vol. 18, no. 1, pp. 14–20, 1972

work page 1972

Show all 48 references

[9]

Rate distor tion theory for general sources with potential application to image com pression,

F. Rezaei, N. Ahmed, and C. D. Charalambous, “Rate distor tion theory for general sources with potential application to image com pression,” International Journal of Applied Mathematical Sciences , vol. 3, no. 2, pp. 141–165, 2006

2006
[10]

Rate-disto rtion theory for general sets and measures,

E. Riegler, H. Bölcskei, and G. Koliander, “Rate-disto rtion theory for general sets and measures,” in Proc. lEEE Int. Symp. Inf. Theory (ISIT) , V ail, CO, USA, 2018, pp. 101–105

2018
[11]

Lossy compr ession of general random variables,

E. Riegler, G. Koliander, and H. Bölcskei, “Lossy compr ession of general random variables,” Information and Inference: A Journal of the IMA, vol. 12, no. 3, pp. 1759–1829, 2023

2023
[12]

The rate-distortion dimensi on of sets and measures,

T. Kawabata and A. Dembo, “The rate-distortion dimensi on of sets and measures,” IEEE Trans. Inf. Theory, vol. 40, no. 5, pp. 1564–1572, 1994

1994
[13]

Successive reﬁnement of abst ract sources,

V . Kostina and E. Tuncel, “Successive reﬁnement of abst ract sources,” IEEE Trans. Inf. Theory , vol. 65, no. 10, pp. 6385–6398, 2019

2019
[14]

Asymptotic evaluatio n of certain Markov process expectations for large time, I,

M. D. Donsker and S. S. V aradhan, “Asymptotic evaluatio n of certain Markov process expectations for large time, I,” Commun. Pure Appl. Math., vol. 28, no. 1, pp. 1–47, 1975

1975
[15]

Successive reﬁnement of in formation,

W. H. Equitz and T. M. Cover, “Successive reﬁnement of in formation,” IEEE Trans. Inf. Theory , vol. 37, no. 2, pp. 269–275, 1991

1991
[16]

Transportation distance, shannon informa tion, and source coding,

R. M. Gray, “Transportation distance, shannon informa tion, and source coding,” in GRETSI Symposium on Signal and Image Processing , 2013, https://ee.stanford.edu/ gray/gretsi.pdf

2013
[17]

Estimating the rate- distortion function by Wasserstein gradient descent,

Y . Y ang, S. Eckstein, M. Nutz, and S. Mandt, “Estimating the rate- distortion function by Wasserstein gradient descent,” Advances in Neural Information Processing Systems , vol. 36, 2024

2024
[18]

Kan torovich duality for general transport costs and applications,

N. Gozlan, C. Roberto, P .-M. Samson, and P . Tetali, “Kan torovich duality for general transport costs and applications,” J. Funct. Anal. , vol. 273, no. 11, pp. 3327–3405, 2017

2017
[19]

A survey of the Schrödinger problem and som e of its connections with optimal transport,

C. Léonard, “A survey of the Schrödinger problem and som e of its connections with optimal transport,” arXiv:1308.0215, 2013

2013 arXiv
[20]

Ex istence, du- ality, and cyclical monotonicity for weak transport costs,

J. Backhoff-V eraguas, M. Beiglböck, and G. Pammer, “Ex istence, du- ality, and cyclical monotonicity for weak transport costs, ” Calculus of V ariations and Partial Differential Equations , vol. 58, no. 203, 2019

2019
[21]

Applications of w eak transport theory,

J. Backhoff-V eraguas and G. Pammer, “Applications of w eak transport theory,” Bernoulli, vol. 28, no. 1, pp. 370–394, 2022

2022
[22]

A mapping approach to rate-distortion comput ation and analysis,

K. Rose, “A mapping approach to rate-distortion comput ation and analysis,” IEEE Trans. Inf. Theory , vol. 40, no. 6, pp. 1939–1952, 1994

1939
[23]

O ptimal transport for domain adaptation,

N. Courty, R. Flamary, D. Tuia, and A. Rakotomamonjy, “O ptimal transport for domain adaptation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 39, no. 9, pp. 1853–1865, 2016

2016
[24]

Regularized optimal transp ort for dynamic semi-supervised learning,

M. E. Hamri and Y . Bennani, “Regularized optimal transp ort for dynamic semi-supervised learning,” arXiv:2103.11937, 2021

2021 arXiv
[25]

Wasserstein g enerative ad- versarial networks,

M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein g enerative ad- versarial networks,” in International Conference on Machine Learning . PMLR, 2017, pp. 214–223

2017
[26]

Multi-prototype space learning for commonsense-ba sed scene graph generation,

L. Chen, Y . Song, Y . Cai, J. Lu, Y . Li, Y . Xie, C. Wang, and G. He, “Multi-prototype space learning for commonsense-ba sed scene graph generation,” in Proceedings of the AAAI Conference on Artiﬁcial Intelligence, vol. 38, no. 2, 2024, pp. 1129–1137

2024
[27]

Transportation cost for Gaussian and ot her product measures,

M. Talagrand, “Transportation cost for Gaussian and ot her product measures,” Geometric & Functional Analysis (GAF A) , vol. 6, no. 3, pp. 587–600, 1996

1996
[28]

Connecting GANs, mean- ﬁeld games, and optimal transport,

H. Cao, X. Guo, and M. Laurière, “Connecting GANs, mean- ﬁeld games, and optimal transport,” SIAM Journal on Applied Mathematics , vol. 84, no. 4, pp. 1255–1287, 2024

2024
[29]

Matching for causal effects via multimarginal unbalanced optimal transport,

F. Gunsilius and Y . Xu, “Matching for causal effects via multimarginal unbalanced optimal transport,” arXiv:2112.04398, 2021

2021 arXiv
[30]

Convexity of mutual informatio n along the Ornstein-Uhlenbeck ﬂow,

A. Wibisono and V . Jog, “Convexity of mutual informatio n along the Ornstein-Uhlenbeck ﬂow,” in International Symposium on Information Theory and Its Applications (ISITA) , Singapore, 2018, pp. 55–59

2018
[31]

Transportation proof of an inequality by Anantharam, Jog and Nair,

T. A. Courtade, “Transportation proof of an inequality by Anantharam, Jog and Nair,” arXiv:1901.10893, 2019

1901 arXiv
[32]

Optimal transport meets information science: from measure concentration, to information theory, to machine learning ,

Y . Bai, “Optimal transport meets information science: from measure concentration, to information theory, to machine learning ,” PhD Thesis, University of Delaware, 2022

2022
[33]

Information constrained op timal trans- port: From Talagrand, to Marton, to Cover,

Y . Bai, X. Wu, and A. Özgür, “Information constrained op timal trans- port: From Talagrand, to Marton, to Cover,” IEEE Trans. Inf. Theory , vol. 69, no. 4, pp. 2059–2073, 2023

2023
[34]

Villani, Optimal transport: old and new , ser

C. Villani, Optimal transport: old and new , ser. Grundlehren der mathematischen Wissenschaften. Springer, 2009

2009
[35]

Charac- terization of a class of weak transport-entropy inequaliti es on the line,

N. Gozlan, C. Roberto, P .-M. Samson, Y . Shu, and P . Tetal i, “Charac- terization of a class of weak transport-entropy inequaliti es on the line,” Annales de l’Institut Henri Poincaré, Probabilités et Stat istiques, vol. 54, no. 3, pp. 1667 – 1693, 2018

2018
[36]

On a mixture of Brenier and Str assen theorems,

N. Gozlan and N. Juillet, “On a mixture of Brenier and Str assen theorems,” Proc. Lond. Math. Soc. , vol. 120, no. 3, pp. 434–463, 2020

2020
[37]

We ak monotone rearrangement on the line,

J. Backhoff-V eraguas, M. Beiglböck, and G. Pammer, “We ak monotone rearrangement on the line,” Electronic Communications in Probability , vol. 25, pp. 1–16, 2020

2020
[38]

Stability of mart ingale optimal transport and weak optimal transport,

J. Backhoff-V eraguas and G. Pammer, “Stability of mart ingale optimal transport and weak optimal transport,” Ann. Appl. Probab., vol. 32, no. 1, pp. 721–752, 2022

2022
[39]

Weak transpor t for non- convex costs and model-independence in a ﬁxed-income marke t,

B. Acciaio, M. Beiglböck, and G. Pammer, “Weak transpor t for non- convex costs and model-independence in a ﬁxed-income marke t,” Math. Finance, vol. 31, no. 4, pp. 1423–1453, 2021

2021
[40]

Über die umkehrung der naturgesetze,

E. Schrödinger, “Über die umkehrung der naturgesetze, ” Sitzungs- berichte der Preussischen Akademie der Wissenschaften. Ph ysikalisch- Mathematische Klasse , vol. 144, pp. 144–153, 1931

1931
[41]

Introduction to entropic optimal transport,

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

2021
[42]

A revisit to rate-dis tortion theory via optimal weak transport,

J. Zou, L. Fan, J. Gao, and J. Wang, “A revisit to rate-dis tortion theory via optimal weak transport,” arXiv:2501.09362, 2025

2025 arXiv
[43]

Dupuis and R

P . Dupuis and R. S. Ellis, A weak convergence approach to the theory of large deviations . John Wiley & Sons, 2011

2011
[44]

Plug-in estimatio n of Schrödinger bridges,

A.-A. Pooladian and J. Niles-Weed, “Plug-in estimatio n of Schrödinger bridges,” arXiv:2408.11686, 2024

2024 arXiv
[45]

Wasserstein proximal algorit hms for the Schrödinger bridge problem: Density control with nonlinea r drift,

K. Caluya and A. Halder, “Wasserstein proximal algorit hms for the Schrödinger bridge problem: Density control with nonlinea r drift,” IEEE Trans. Automatic Control , vol. 67, no. 3, pp. 1163–1178, 2021

2021
[46]

An optimal transport approac h for the Schrödinger bridge problem and convergence of sinkhorn alg orithm,

S. Marino and A. Gerolin, “An optimal transport approac h for the Schrödinger bridge problem and convergence of sinkhorn alg orithm,” J. Sci. Comput. , vol. 85(2), no. 27, 2020

2020
[47]

Random coding strategies for minimum entro py,

E. Posner, “Random coding strategies for minimum entro py,” IEEE Trans. Inf. Theory , vol. 21, no. 4, pp. 388–391, 1975

1975
[48]

Polyanskiy and Y

Y . Polyanskiy and Y . Wu, Information Theory: From Coding to Learn- ing. Cambridge University Press, 2025. APPENDIX A PROOF OF LEMMA 3 For any ﬁxed πx ∈ P (Y) with Eπρ ≤ D, one can always ﬁnd the corresponding ν ∈ P (Y) and π ∈ Π( µ, ν), where for any Borel measurable functio...

2025

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

A mathematical theory of communication,

C. E. Shannon, “A mathematical theory of communication, ” The Bell System Technical Journal , vol. 27, no. 3, pp. 379–423, 1948

1948

[2] [2]

Coding theorems for a discrete source with a ﬁdelity criterion,

——, “Coding theorems for a discrete source with a ﬁdelity criterion,” International Convention Record , vol. 7, pp. 325–350, 1959

work page 1959

[3] [3]

Berger, Rate-distortion theory: A mathematical basis for data com- pression

T. Berger, Rate-distortion theory: A mathematical basis for data com- pression. Englewood Cliffs: Prentice-Hall, 1971

work page 1971

[4] [4]

Computation of channel capacity and rate-di stortion func- tions,

R. Blahut, “Computation of channel capacity and rate-di stortion func- tions,” IEEE Trans. Inf. Theory , vol. 18, no. 4, pp. 460–473, 1972

work page 1972

[5] [5]

R. E. Blahut, Principles and practice of information theory . Addison- Wesley Longman Publishing Co., Inc., 1987

work page 1987

[6] [6]

T. M. Cover and J. A. Thomas, Elements of Information Theory , 2nd ed. New Y ork, NY , USA: Wiley, 2006

work page 2006

[7] [7]

On an extremum problem of information theor y,

I. Csiszár, “On an extremum problem of information theor y,” Studia Scientiarum Mathematicarum Hungarica , vol. 9, no. 1, pp. 57–71, 1974

work page 1974

[8] [8]

An algorithm for computing the capacity of a rbitrary discrete memoryless channels,

S. Arimoto, “An algorithm for computing the capacity of a rbitrary discrete memoryless channels,” IEEE Trans. Inf. Theory , vol. 18, no. 1, pp. 14–20, 1972

work page 1972