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Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space

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Pith's one-line read The paper proves an exact single-letter formula for the distortion-rate-perception tradeoff in lossy source coding with limited common randomness, under squared error and squared Wasserstein-2 perception.

desk verdict A likely correct and valuable generalization of RDP theory to finite common randomness; the achievability proof has a repairable derandomization gap that should be fixed before acceptance. read the letter →

arxiv 2504.17236 v1 pith:QB7TSH77 submitted 2025-04-24 cs.IT cs.LGmath.IT

Xiqiang Qu , Jun Chen , Lei Yu , Xiangyu Xu This is my paper · ORCID
classification cs.ITcs.LGmath.IT MSC 94A2994A34
keywords rate-distortion-perceptiontradeoffcommonrandomnessWasserstein-2distancelossysourcecodingMMSEestimateGaussianuniversalrepresentationssoft-coveringlemma
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 targets the fundamental tradeoff in lossy compression when the reconstruction must match the source distribution at the whole-sequence level, not just symbol by symbol. In that setting the amount of common randomness shared by encoder and decoder changes what is achievable, and before this paper only extreme cases were understood: no common randomness, unlimited common randomness, or perfect perception. The paper proposes a single-letter optimization over one representation distribution and two couplings, and proves that it equals the true optimum for every square-integrable source under squared-error distortion and squared Wasserstein-2 perception. It then evaluates the formula explicitly for Gaussian sources, yielding closed-form distortion curves and thresholds that show when extra common randomness stops helping. A sympathetic reader would care because the result turns an infinite-blocklength coding problem into a finite-dimensional optimization and clarifies when one encoder can serve every perception target.

What carries the argument

The load-bearing mechanism is the quadratic structure that makes both the distortion and the perception costs expectations of the same squared Euclidean distance. Given any representation $W$, the optimal reconstruction from $W$ under a perception budget $P$ is not a deterministic function of $W$; it is a convex combination of the MMSE estimate $\tilde S=\mathbb{E}[S\mid W]$ and a fresh source-distributed sample $S'$ produced from $\tilde S$ by optimal transport, with coefficient $1-\sqrt{P}/W_2(p_S,p_{\tilde S})$ on the fresh sample and $\sqrt{P}/W_2(p_S,p_{\tilde S})$ on the estimate when the budget binds. This identity splits total distortion into the MMSE error plus a perception shortfall term $[(\sqrt{\mathbb{E}[\|S-\tilde S\|^2]}-\sqrt{P})_+]^2$. In the coding theorem, the two couplings $\mu$ and $\nu$ do different work: $\mu$ measures the squared-error cost of the representation, while $\nu$ measures how close the representation's distribution is to the source in Wasserstein distance. The achievability proof couples both costs through a soft-covering lemma adapted to the Wasserstein-2 metric, which guarantees that a random codebook of size $2^{nR}$ placed on the representation distribution approximately realizes $p_{\tilde X}^n$ after conditioning.

What would settle it

Take a concrete source and finite blocklength, and compute the joint law of the two random costs $(f(C),g(C))$ that Appendix C averages over the random codebook; if the mean of that pair is not a convex combination of two codebook realizations, the two-codebook derandomization is impossible. Alternatively, for a non-Gaussian square-integrable source, evaluate $D(R,C,P)$ by numerical optimization and compare it with large-blocklength simulations of the proposed coding scheme; a systematic gap would refute Theorem 1.

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Extended reading notes

Core claim

The central discovery is that the asymptotic problem collapses to a finite-dimensional two-coupling optimization. For a source $X$ with $\mathbb{E}[\|X\|^2]<\infty$, the paper defines $D(R,C,P)$ as the infimum over a representation variable $\tilde X$ and two couplings $\mu,\nu$ between $p_X$ and $p_{\tilde X}$ of $$\mathbb{E}_\mu[\|X-\tilde X\|^2] + \bigl[(\sqrt{\mathbb{E}_\nu[\|X-\tilde X\|^2]}-\sqrt{P})_+\bigr]^2,$$ subject to the MMSE condition $\mathbb{E}_\mu[X\mid \tilde X]=\tilde X$, the rate constraint $I_\mu(X;\tilde X)\le R$, and the combined-rate constraint $I_\nu(X;\tilde X)\le R+C$. Theorem 1 asserts that this quantity equals $D^*(R,C,P)$, the minimum distortion achievable by any length-$n$ system with code rate $R$, common-randomness rate $C$, and sequence-level perception constraint $\frac1n W_2^2(p_{X^n},p_{\hat X^n})\le P$. The proof builds a random codebook with distribution $p_{\tilde X}^n$, uses $\mu$ to control the MMSE error and $\nu$ to control the Wasserstein distance between the source and the representation, and finishes with a linear-interpolation decoder that optimally trades perception against distortion.

Load-bearing premise

The achievability proof assumes in Appendix C that the randomness used to generate the codebook can be removed by blending two codebook realizations with one time-sharing weight so that both the expected squared-error cost and the expected Wasserstein cost are matched exactly; matching two expectations simultaneously generally requires three realizations, so this step is not justified as written.

Editorial extensions

If this is right

  • The formula recovers all previously known extreme cases: $C=0$ gives $D(R)+[(\sqrt{D(R)}-\sqrt{P})_+]^2$, $C=\infty$ gives the standard distortion-rate-perception function $D(R,P)$, and $P=0$ matches the output-constrained lossy source coding characterization.
  • For a scalar Gaussian source, $D^*(R,C,P)=\gamma 2^{-2R} + [(\sqrt{\gamma(2-2^{-2R}-2\psi(R,R+C))}-\sqrt{P})_+]^2$ with $\psi(a,b)=\sqrt{(1-2^{-2a})(1-2^{-2b})}$, and closed-form thresholds on $C$ or $R$ identify when the perception constraint is inactive.
  • The scalar Gaussian minimizer does not depend on $P$, so one asymptotically optimal coded representation works for every perception level; this extends known universality results beyond the unlimited-common-randomness case.
  • For vector Gaussian sources, the evaluation reduces to a waterfilling-style rate allocation, and in general the optimal allocation depends on $P$ when $C>0$, so asymptotically universal representations need not exist with limited common randomness.
  • The infimum in the single-letter formula is attained, and $D(R,C,P)$ is decreasing, convex, and continuous in $(R,C,P)$, so the tradeoff surface is well-behaved for numerical computation.

Reading between the lines

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

  • Editorial extension: if the two-codebook derandomization in Appendix C cannot be repaired by a three-codebook argument, the formula may still be true but the achievability proof as written would need revision; the fix is a proof-technical question that a counterexample to the support step would settle.
  • The perception-inactive threshold for scalar Gaussians suggests a practical design rule: a system can stop spending common randomness once $C$ reaches the saturation point in (81), and further seed rate is wasted; this rule could be tested by implementing the described codebook scheme on Gaussian-like data.
  • The vector-source non-universality result points to a qualitative conclusion not stated in the paper: with intermediate common randomness, the encoder must know the perception target, so practical perception-aware codecs with shared randomness should be trained target-dependently rather than as universal representations.
  • A natural next step, not taken in the paper, is to solve the coupling optimization numerically for small-support non-Gaussian sources and compare the resulting $D(R,C,P)$ with finite-blocklength simulations of the coding scheme; agreement would indicate that the quadratic-Wasserstein structure, rather than Gaussianity, is what drives tractability.
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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 lossy source coding of a square-integrable source under squared error distortion and squared Wasserstein-2 perception, with limited common randomness at rate C. The main contribution is a claimed single-letter formula D(R,C,P) for the minimal distortion D*(R,C,P), defined as an infimum over two couplings µ and ν sharing the same reconstruction marginal p_{X̃}, with mutual information constraints R and R+C and an MMSE condition on µ (Theorem 1). The authors also provide explicit evaluations for scalar Gaussian (Theorem 2) and vector Gaussian (Theorem 3) sources, and discuss universal representations. The proof of Theorem 1 is based on a new soft-covering lemma for Wasserstein-2 distance and an optimal interpolation decoder.

Significance. If the main theorem is correct, the result is a significant advance: it unifies previously known extreme cases (C=0, C=∞, P=0) and yields closed-form expressions for Gaussian sources, which are likely to be useful in applications. The Wasserstein soft-covering lemma (Lemma 3) is an independently useful technical tool. The discussion of universal representations, especially the negative result for vector Gaussian sources, is conceptually interesting. However, the proof of Theorem 1 contains a gap in the derandomization step, so the stated results are not fully established in the current manuscript.

major comments (1)
  1. [Appendix C, Step 3 (Derandomization)] The derandomization step invokes the support lemma to claim that two codebook realizations c0 and c1 and a scalar λ ∈ [0,1] can match both E_{p,C}[f(C)] and E_{q,C}[g(C)] simultaneously. Since the pair (E[f], E[g]) is a point in the convex hull of {(f(c), g(c))} ⊂ R², Carathéodory's theorem guarantees a representation by at most three points, not two. No argument is provided that the set of attainable pairs lies on a line segment or that two points suffice. Consequently, the existence of a deterministic codebook satisfying both (147) and (148) is not established. This is a load-bearing gap in the achievability proof of Theorem 1. A repair appears feasible—for example, by using three codebooks with time-sharing or by a probabilistic argument showing that a single typical codebook satisfies both constraints—but the current proof is incomplete.
minor comments (4)
  1. [Appendix C, Eq. (141)] In equation (141), the second expectation in the distortion expression should be E_ν[·], not E_µ[·], and the term [(·)⁺] is missing the square; the correct form follows the definition in (30).
  2. [Appendix C, Step 3] The phrase 'a scalar a number λ' contains a typo; it should read 'a scalar λ'.
  3. [Appendix D] The proof of Theorem 3 is quite terse; expanding the derivation of (201)–(204), including the entropy lower bounds and the verification of the sum constraints, would improve readability and verifiability.
  4. [Appendix A] There are minor typos, e.g., 'probablility' and 'distriution'; a careful proofread is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the coding theorem and the Gaussian formula are derived from an independent optimization definition plus external soft-covering and Wasserstein-interpolation lemmas; the authors' self-citations appear only in background and consistency checks and are not load-bearing.

full rationale

The central result is not circular. D(R,C,P) is defined independently of the operational D*(R,C,P), and Theorem 1 is proved by a random-coding achievability argument whose soft-covering lemma (Appendix C-C) is based on external results [23, 24], together with a converse that constructs the single-letter couplings (mu, nu) from any feasible operational code. The Gaussian formula (Theorem 2) is obtained by explicitly evaluating E_mu and E_nu for chosen bivariate Gaussian couplings and proving matching lower bounds via differential-entropy inequalities; no fitted parameter is renamed as a prediction. The prior work by the present authors ([4], [17], [18]) is used for background, comparisons, and consistency checks, not as the load-bearing step: for example, the C=infinity consistency is re-derived from the external result [12], and the P=0 and C=0 cases are verified directly rather than imported. The only notable defect is a correctness gap, not a circularity: in Appendix C, Step 3, the paper states "By the support lemma, there exist two realizations c0 and c1 of the codebook C, along with a scalar a number lambda..." to match two expected values simultaneously, which in R^2 would generally require up to three points by Caratheodory's theorem; this is a repairable proof gap and does not make the derivation equivalent to its inputs. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the optimization variables r_l are optimized, not fitted. No new theoretical entities are introduced; the paper relies on known couplings, interpolation schemes, and soft-covering lemmas.

assumptions (4)
  • standard math Soft-covering lemma for total variation (Hayashi 2015; Cuff 2013) yields exponential convergence of pY^n|C to pY^n when R > I(X;Y).
    Used in Lemma 3; it is a cited known result.
  • standard math Distortion-perception interpolation theorem and MMSE lower bound from Freirich et al. [12].
    Provides the lower bound (15) and interpolation construction (16).
  • domain assumption The source has finite second moment E[||X||^2] < ∞.
    Stated in the problem definition; needed for the Wasserstein-2 distance and tightness arguments.
  • domain assumption For Gaussian sources, the source is zero-mean Gaussian with covariance γ or Γ; for the vector case, components are independent after diagonalization.
    Theorems 2 and 3 are specific to Gaussian sources.

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Pith. "Pith review of Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space." pith.science (2026). https://pith.science/paper/QB7TSH77

@misc{pith2026250417236,
  author       = {Pith},
  title        = {Pith review of: Rate-Distortion-Perception Theory for the Quadratic Wasserstein Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QB7TSH77}},
  note         = {Machine review of arXiv:2504.17236}
}
read the original abstract

We establish a single-letter characterization of the fundamental distortion-rate-perception tradeoff with limited common randomness under the squared error distortion measure and the squared Wasserstein-2 perception measure. Moreover, it is shown that this single-letter characterization can be explicitly evaluated for the Gaussian source. Various notions of universal representation are also clarified.

Figures

Figures reproduced from arXiv: 2504.17236 by the authors.

Figure 1
Figure 1. Illustration of D∗(R, C, P) as a function of R, for P = 0.1 and different values of C, with pX = N (0, 1). 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Illustration of D∗ (R, C, P) as a function of R, for C = 1 and different values of P, with pX = N (0, 1). which highlights the regime where increasing C alone does not further reduce D∗ (R, C, P). It is also possible to express (79) as a threshold on R: R ≥    1 2 log 2P γ +2−2(C−1)+ q ( 2P γ +2−2(C−1)) 2+4(1−2−2(C−1))( 4P γ − P 2 γ2 ) 2( 4P γ − P 2 γ2 ) ! , P ∈ (0, γ), 0, P ≥ γ. (82) To gain a more concrete un… view at source ↗

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