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arxiv: 2601.11862 · v2 · submitted 2026-01-17 · 💻 cs.IT · math.IT

On the R\'enyi Rate-Distortion-Perception Function and Functional Representations

Jiahui Wei , Marios Kountouris This is my paper

Pith reviewed 2026-05-16 13:57 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords Rényi rate-distortion-perceptionSibson mutual informationfunctional representation lemmaGaussian sourcephase transitionheavy-tailed codebookperception constraint
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The pith

Rényi rate-distortion-perception functions for scalar Gaussian sources admit closed-form expressions, with perception constraints defining a feasible interval for reproduction variance and functional representations showing a phase shift in

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

This paper extends the rate-distortion-perception framework into the Rényi regime by replacing ordinary mutual information with Sibson's α-mutual information to characterize the minimal rate under simultaneous distortion and perception constraints. For scalar Gaussian sources it obtains explicit formulas for that minimal rate and shows that the perception requirement forces the reproduction variance to lie inside a closed interval. It also proves a Rényi version of the strong functional representation lemma that bounds the cost of encoding shared randomness. The analysis uncovers a sharp change in representation complexity: when 0.5 < α < 1 the optimal codebook must have heavy polynomial tails whose decay is controlled by an α-divergence of order α+1, whereas for α > 1 the codebook can be reduced to finite support.

Core claim

The central claim is that the Rényi rate-distortion-perception function of a scalar Gaussian source possesses a closed-form expression in which the perception constraint restricts the allowable reproduction variance to an interval; moreover, a Rényi generalization of the strong functional representation lemma holds, under which the coding cost of shared randomness is bounded by the α-divergence of order α+1 for 0.5 < α < 1 (forcing heavy-tailed codebooks) and collapses to a finite-support representation for α > 1.

What carries the argument

Sibson's α-mutual information together with the Rényi-generalized strong functional representation lemma that governs the minimal cost and tail behavior of optimal functional representations under combined distortion and perception constraints.

If this is right

  • The perception constraint restricts reproduction variance to a specific feasible interval for scalar Gaussian sources.
  • For 0.5 < α < 1 the optimal functional representation requires a codebook with heavy-tailed polynomial decay governed by α-divergence of order α+1.
  • For α > 1 the optimal representation can be realized with a finite-support codebook.
  • The results supply explicit bounds on the rate needed to compress shared randomness under the Rényi notion of mutual information.

Where Pith is reading between the lines

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

  • Practical perceptual compression systems that adopt Rényi measures may therefore require qualitatively different random-number generators on either side of α = 1.
  • Analogous phase transitions in representation complexity may appear in other rate-distortion settings that replace Shannon mutual information with a parameterized divergence.
  • Direct simulation of small-dimensional Gaussian vectors and enumeration of minimal codebook cardinalities for varying α would provide an immediate numerical check of the predicted transition.

Load-bearing premise

The source is scalar Gaussian and Sibson's α-mutual information correctly captures the fundamental limits under the combined distortion and perception constraints.

What would settle it

An explicit calculation or numerical optimization that produces a lower rate than the claimed closed-form Rényi RDP expression for a chosen Gaussian variance, distortion, perception level and α, or that shows the minimal codebook support size fails to switch from infinite to finite at the stated α threshold.

Figures

Figures reproduced from arXiv: 2601.11862 by Jiahui Wei, Marios Kountouris.

Figure 1
Figure 1. Figure 1: Contour plots of the Gaussian R-RDP function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical Validation of Renyi-SFRL ( ´ α = 0.6 and α = 2). Left: Histogram of the selected codebook indices K (log scale), showing heavy￾tailed behavior for α < 1 and finite support for α > 1. Right: Convergence of the empirical α-moment E[K0.6 ] (E[log(K)] for α = 2, blue), remaining strictly below the theoretical bound (red). V. CONCLUSION In this work, we generalize the RDP framework to the Renyi ´ regi… view at source ↗
read the original abstract

We extend the Rate-Distortion-Perception (RDP) framework to the R\'enyi information-theoretic regime, utilizing Sibson's $\alpha$-mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the R\'enyi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a R\'enyi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for $0.5<\alpha < 1$, the coding cost is bounded by the $\alpha$-divergence of order $\alpha+1$, necessitating a codebook with heavy-tailed polynomial decay; conversely, for $\alpha > 1$, the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual 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.

Referee Report

3 major / 2 minor

Summary. The manuscript extends the Rate-Distortion-Perception (RDP) framework to the Rényi regime using Sibson's α-mutual information to characterize fundamental limits under distortion and perception constraints. For scalar Gaussian sources X ~ N(0, σ²), it derives closed-form expressions for the Rényi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. It also establishes a Rényi-generalized Strong Functional Representation Lemma and identifies a phase transition in optimal functional representations: for 0.5 < α < 1 the coding cost is bounded by the α-divergence of order α+1 requiring heavy-tailed polynomial decay codebooks, while for α > 1 the representation collapses to finite support.

Significance. If the closed-form expressions and phase-transition analysis hold, the work supplies concrete benchmarks for Rényi RDP under joint distortion-perception constraints and clarifies how α affects the complexity of shared-randomness representations. The generalization of the Strong Functional Representation Lemma is a notable technical contribution that could inform subsequent studies of functional representations under generalized mutual information measures.

major comments (3)
  1. [Section 3 (closed-form Rényi RDP for Gaussian sources)] The closed-form derivation for the Rényi RDP function (Section 3) restricts the reproduction to Y ~ N(0, v) with v in the perception-induced interval and claims optimality. However, Sibson's α-mutual information I_α(X;Y) = min_Q D_α(P_{XY} || P_X Q) is minimized over arbitrary conditionals P_{Y|X}; for 0.5 < α < 1 the α-divergence favors heavier tails, so it is not immediate that a non-Gaussian marginal cannot achieve strictly lower I_α while meeting the same distortion and perception bounds. A proof that the Gaussian marginal is optimal (or an explicit argument that any better non-Gaussian Y would violate the constraints) is required.
  2. [Section 5 (Rényi-generalized Strong Functional Representation Lemma and phase transition)] The phase-transition claim for functional representations (Section 5) inherits the above gap: the statement that the coding cost is bounded by the α-divergence of order α+1 (necessitating heavy-tailed codebooks) for 0.5 < α < 1, versus finite support for α > 1, rests on the optimality of the Gaussian reproduction used to obtain the closed-form RDP. Without a supporting argument that the minimizing conditional yields a Gaussian marginal, the complexity classification cannot be asserted.
  3. [Section 3, paragraph following Eq. (14)] The feasible interval for reproduction variance induced by the perception constraint is stated to follow from D_α(P_Y || P_X) or an analogous quantity, yet the precise definition of the perception functional and the derivation that it produces a closed interval for v are not accompanied by an error analysis or verification that the interval remains non-empty for all admissible distortion levels.
minor comments (2)
  1. [Section 2] Notation for Sibson's α-mutual information and the Rényi divergence should be introduced with an explicit reference to the original definitions (Sibson 1969) in the preliminaries section to avoid ambiguity for readers unfamiliar with the α-regime.
  2. [Abstract and Section 3] The abstract claims 'closed-form expressions' but the manuscript would benefit from an explicit statement of the final formula for the Rényi RDP function (including the dependence on α, D, and the perception parameter) in a single displayed equation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment below and will revise the manuscript accordingly to strengthen the technical arguments.

read point-by-point responses

  1. Referee: [Section 3 (closed-form Rényi RDP for Gaussian sources)] The closed-form derivation for the Rényi RDP function (Section 3) restricts the reproduction to Y ~ N(0, v) with v in the perception-induced interval and claims optimality. However, Sibson's α-mutual information I_α(X;Y) = min_Q D_α(P_{XY} || P_X Q) is minimized over arbitrary conditionals P_{Y|X}; for 0.5 < α < 1 the α-divergence favors heavier tails, so it is not immediate that a non-Gaussian marginal cannot achieve strictly lower I_α while meeting the same distortion and perception bounds. A proof that the Gaussian marginal is optimal (or an explicit argument that any better non-Gaussian Y would violate the constraints) is required.

    Authors: We agree that a rigorous proof of optimality for the Gaussian marginal is required, particularly for 0.5 < α < 1 where the α-divergence may favor heavier tails. The current derivation obtains closed forms by restricting to Gaussian Y, but does not explicitly prove this achieves the global minimum over all conditionals. We will add an appendix providing this proof, leveraging the Gaussian source and properties of the α-divergence to show that any non-Gaussian Y satisfying the constraints cannot yield strictly lower I_α. revision: yes

  2. Referee: [Section 5 (Rényi-generalized Strong Functional Representation Lemma and phase transition)] The phase-transition claim for functional representations (Section 5) inherits the above gap: the statement that the coding cost is bounded by the α-divergence of order α+1 (necessitating heavy-tailed codebooks) for 0.5 < α < 1, versus finite support for α > 1, rests on the optimality of the Gaussian reproduction used to obtain the closed-form RDP. Without a supporting argument that the minimizing conditional yields a Gaussian marginal, the complexity classification cannot be asserted.

    Authors: The phase-transition analysis relies on the closed-form Rényi RDP derived under the Gaussian assumption in Section 3. We will revise Section 5 to explicitly reference the new optimality proof added in response to the first comment, thereby justifying the complexity classification (heavy-tailed codebooks for 0.5 < α < 1 and finite support for α > 1). revision: yes

  3. Referee: [Section 3, paragraph following Eq. (14)] The feasible interval for reproduction variance induced by the perception constraint is stated to follow from D_α(P_Y || P_X) or an analogous quantity, yet the precise definition of the perception functional and the derivation that it produces a closed interval for v are not accompanied by an error analysis or verification that the interval remains non-empty for all admissible distortion levels.

    Authors: We will expand the paragraph following Eq. (14) to provide the precise definition of the perception functional (based on D_α(P_Y || P_X)) and a complete derivation of the closed interval for v. This will include an explicit verification that the interval is non-empty for all admissible distortion levels, along with any necessary error bounds or analysis to confirm the interval properties. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from definitions with no reduction to inputs

full rationale

The paper derives closed-form Rényi RDP expressions and the generalized Strong Functional Representation Lemma directly from the definitions of Sibson's α-mutual information, the distortion constraint, and the perception constraint (D_α(P_Y || P_X)) for scalar Gaussian sources. The feasible interval for reproduction variance follows from these definitions without any fitted parameters renamed as predictions or self-definitional equations. The phase-transition claims for functional representations (heavy-tailed vs finite support) are obtained by bounding the α-divergence of order α+1 and analyzing support properties, with no load-bearing self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors themselves. The central results remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard definition of Sibson's α-mutual information and the Gaussian source model; no free parameters are fitted to data and no new entities are postulated.

axioms (1)
  • domain assumption Sibson's α-mutual information characterizes the fundamental limits under distortion and perception constraints in the Rényi regime
    Invoked to extend the RDP framework and derive the closed-form expressions.

pith-pipeline@v0.9.0 · 5460 in / 1414 out tokens · 56284 ms · 2026-05-16T13:57:40.406871+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Channel coding rate in the finite blocklength regime,

    Y . Polyanskiy, H. V . Poor, and S. Verdu, “Channel coding rate in the finite blocklength regime,”IEEE Transactions on Information Theory, vol. 56, no. 5, pp. 2307–2359, 2010

    work page 2010

  2. [2]

    Cover and J

    T. Cover and J. Thomas,Elements of information theory. Wiley- Interscience, 2006

    work page 2006

  3. [3]

    On measures of entropy and information,

    A. R ´enyi, “On measures of entropy and information,” inProceedings of the fourth Berkeley symposium on mathematical statistics and probabil- ity, volume 1: contributions to the theory of statistics, vol. 4. University of California Press, 1961, pp. 547–562

    work page 1961

  4. [4]

    A coding theorem and R ´enyi’s entropy,

    L. Campbell, “A coding theorem and R ´enyi’s entropy,”Information and Control, vol. 8, no. 4, pp. 423–429, 1965

    work page 1965

  5. [5]

    Generalized cutoff rates and Renyi’s information measures,

    I. Csiszar, “Generalized cutoff rates and Renyi’s information measures,” IEEE Transactions on Information Theory, vol. 41, no. 1, pp. 26–34, 1995

    work page 1995

  6. [6]

    An inequality on guessing and its application to sequential decoding,

    E. Arikan, “An inequality on guessing and its application to sequential decoding,”IEEE Transactions on Information Theory, vol. 42, no. 1, pp. 99–105, 1996

    work page 1996

  7. [7]

    Guessing subject to distortion,

    E. Arikan and N. Merhav, “Guessing subject to distortion,”IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1041–1056, 1998

    work page 1998

  8. [8]

    Encoding tasks and R ´enyi entropy,

    C. Bunte and A. Lapidoth, “Encoding tasks and R ´enyi entropy,”IEEE Transactions on Information Theory, vol. 60, no. 9, pp. 5065–5076, 2014

    work page 2014

  9. [9]

    Sibsonα-mutual information and its variational representations,

    A. R. Esposito, M. Gastpar, and I. Issa, “Sibsonα-mutual information and its variational representations,” 2025

    work page 2025

  10. [10]

    Communication complexity of exact sampling under R ´enyi information,

    S. Hill, F. Alajaji, and T. Linder, “Communication complexity of exact sampling under R ´enyi information,” 2025

    work page 2025

  11. [11]

    Information radius,

    R. Sibson, “Information radius,”Zeitschrift f ¨ur Wahrscheinlichkeitsthe- orie und Verwandte Gebiete, vol. 14, pp. 149–160, 1969

    work page 1969

  12. [12]

    Information measures and capacity of orderαfor discrete memoryless channels,

    S. Arimoto, “Information measures and capacity of orderαfor discrete memoryless channels,”Topics in Information Theory, 1977

    work page 1977

  13. [13]

    Testing against independence and a R ´enyi information measure,

    A. Lapidoth and C. Pfister, “Testing against independence and a R ´enyi information measure,” in2018 IEEE Information Theory Workshop (ITW), 2018, pp. 1–5

    work page 2018

  14. [14]

    Correlation detection and an op- erational interpretation of the R ´enyi mutual information,

    M. Hayashi and M. Tomamichel, “Correlation detection and an op- erational interpretation of the R ´enyi mutual information,”Journal of Mathematical Physics, vol. 57, no. 10, p. 102201, 10 2016

    work page 2016

  15. [15]

    The perception–distortion tradeoff,

    Y . Blau and T. Michaeli, “The perception–distortion tradeoff,” inIEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018

    work page 2018

  16. [16]

    On the computation of the gaussian rate–distortion–perception function,

    G. Serra, P. A. Stavrou, and M. Kountouris, “On the computation of the gaussian rate–distortion–perception function,”IEEE Journal on Selected Areas in Information Theory, vol. 5, pp. 314–330, 2024

    work page 2024

  17. [17]

    Strong functional representation lemma and applications to coding theorems,

    C. T. Li and A. E. Gamal, “Strong functional representation lemma and applications to coding theorems,”IEEE Transactions on Information Theory, vol. 64, no. 11, pp. 6967–6978, 2018

    work page 2018

  18. [18]

    A unified framework for one-shot achievability via the poisson matching lemma,

    C. T. Li and V . Anantharam, “A unified framework for one-shot achievability via the poisson matching lemma,”IEEE Transactions on Information Theory, vol. 67, no. 5, pp. 2624–2651, 2021

    work page 2021

  19. [19]

    Non-asymptotic achievable rate- distortion region for indirect wyner-ziv source coding,

    J. Wei, P. Mary, and E. Dupraz, “Non-asymptotic achievable rate- distortion region for indirect wyner-ziv source coding,” in2025 IEEE Information Theory Workshop (ITW), 2025, pp. 1–6

    work page 2025

  20. [20]

    A coding theorem for the rate-distortion- perception function,

    L. Theis and A. B. Wagner, “A coding theorem for the rate-distortion- perception function,” inNeural Compression Workshop at ICLR, 2021

    work page 2021

  21. [21]

    α-mutual information,

    S. Verd ´u, “α-mutual information,” in2015 Information Theory and Applications Workshop (ITA), 2015, pp. 1–6

    work page 2015

  22. [22]

    Optimal young’s inequality and its converse: a simple proof,

    F. Barthe, “Optimal young’s inequality and its converse: a simple proof,” Geometric & Functional Analysis GAFA, vol. 8, pp. 234–242, 1997

    work page 1997

  23. [23]

    R ´enyi divergence and Kullback-Leibler divergence,

    T. van Erven and P. Harremos, “R ´enyi divergence and Kullback-Leibler divergence,”IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 3797–3820, 2014

    work page 2014

  24. [24]

    Rethinking lossy compression: The rate–distortion–perception tradeoff,

    Y . Blau and T. Michaeli, “Rethinking lossy compression: The rate–distortion–perception tradeoff,” inInternational Conference on Ma- chine Learning (ICML), 2019

    work page 2019

  25. [25]

    On an extremum problem of information theory,

    I. Csisz ´ar, “On an extremum problem of information theory,”Studia Scientiarum Mathematicarum Hungarica, vol. 9, no. 1, pp. 57–71, 1974

    work page 1974

  26. [26]

    Convergence of random processes and limit theorems in probability theory,

    Y . V . Prokhorov, “Convergence of random processes and limit theorems in probability theory,”Theory of Probability & Its Applications, vol. 1, no. 2, pp. 157–214, 1956

    work page 1956

  27. [27]

    Villaniet al.,Optimal transport: old and new

    C. Villaniet al.,Optimal transport: old and new. Springer, 2008, vol. 338

    work page 2008

  28. [28]

    J. F. C. Kingman,Poisson processes. Clarendon Press, 1992, vol. 3

    work page 1992

  29. [29]

    Pointwise redundancy in one-shot lossy compression via poisson functional representation,

    C. T. Li, “Pointwise redundancy in one-shot lossy compression via poisson functional representation,” 2024. APPENDIXA PROOF OFTHEOREM1 Sibson’s mutual information [11] admits the following norm representation Iα(X;Y) = 1 α−1 log∥h∥ 1/α,(21) whereh(y) := R pX(x)pY|X (y|x)αdx. Achievability (direct calculation with an affine Gaussian test channel) Fixc∈Rand...

    work page 2024

  30. [30]

    The optimal distribution maximizing R ´enyi entropy under theα-moment constraint has infinite support with a poly- nomially decaying tail

    Under this condition, the Lagrange multipliersλandµcan be chosen to satisfy both the moment constraint and the normalization constraint. The optimal distribution maximizing R ´enyi entropy under theα-moment constraint has infinite support with a poly- nomially decaying tail. This implies that representations in theα <1regime are inherently heavy-tailed, r...

    work page