arxiv: 2605.09921 · v1 · submitted 2026-05-11 · 💻 cs.IT · math.IT

Recognition: no theorem link

R\'enyi Rate-Distortion-Perception-Privacy Tradeoff under Indirect Observation

Jiahui Wei , Marios Kountouris

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:06 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords Rényi divergencerate-distortion-perception-privacyindirect source codingconditional privacyGaussian tradeoffSibson's alpha mutual informationPoisson functional representationresidual leakage

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The pith

Standard privacy metrics penalize legitimate semantic recovery in indirect observations, but a conditional measure isolates only residual leakage.

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

The paper introduces a Rényi rate-distortion-perception-privacy framework tailored to indirect source coding, where an encoder observes only a noisy version of a latent source that correlates with private attributes. For the scalar Gaussian case, it fully characterizes the tradeoff region under communication measured by Sibson's alpha-mutual information, semantic distortion to the source, and a realism constraint matching the source marginal. Standard privacy metrics based on mutual information are shown to inherently reduce achievable semantic quality even when no additional private information leaks beyond the observation. The authors resolve this by defining a conditional privacy measure that tracks only the extra leakage not already inferable from the indirect view. They also sharpen achievability results for alpha greater than 1 by deriving the exact geometric-mixture distribution of the Poisson index, yielding closed-form Rényi entropy expressions.

Core claim

Under the indirect Markov chain with a realism constraint at the semantic marginal, the scalar Gaussian RDPP tradeoff is characterized exactly; standard privacy leakage metrics force a penalty on semantic distortion because they cannot separate useful source information from private attributes. The conditional privacy measure, which quantifies only residual leakage after the indirect observation, removes this artificial penalty and produces a strictly larger achievable region while preserving the rate and perception costs.

What carries the argument

The conditional privacy measure that isolates residual leakage after accounting for the indirect observation, used together with Sibson's alpha-mutual information for the rate and a perception constraint enforcing the decoded marginal to match the source distribution.

If this is right

Semantic distortion can be lowered at fixed communication rate and privacy cost once leakage is measured conditionally on the indirect observation.
Exact closed-form expressions for the tradeoff region are obtained for integer-order Rényi quantities via the geometric-mixture Poisson index distribution.
Achievability bounds tighten for alpha greater than 1 compared with the earlier logarithmic-moment method.
The realism constraint at the semantic marginal ensures the decoded output statistics remain consistent with the original source distribution.

Where Pith is reading between the lines

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

The conditional separation may allow similar refinements in non-Gaussian indirect estimation tasks by approximating the Poisson representation.
Remote sensing or compressed inference systems could adopt the residual privacy metric to improve utility without inflating reported leakage.
The framework suggests testing whether other information measures exhibit the same unintended penalty on semantic recovery under noisy observations.

Load-bearing premise

The conditional privacy measure correctly isolates only residual leakage without introducing new inconsistencies, and the indirect observation Markov chain holds together with the realism constraint at the source marginal.

What would settle it

For concrete Gaussian variances, numerically optimize the achievable semantic distortion at fixed rate and standard privacy level versus the same quantities under the conditional privacy definition; a failure to observe the predicted decoupling would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.09921 by Jiahui Wei, Marios Kountouris.

**Figure 1.** Figure 1: Gaussian indirect R-RDPP tradeoff for σ 2 S = σ 2 U = 1,ρ = 0.7, γ = 0.5, σ 2 N = 0.1, and α = β = 2.Left: minimum feasible rate under fixed perception ∆ = 0.2 with white regions infeasible.Right: unconditional (11) vs conditional leakage (14). Whenγ = 0 for γ = 0 and 0.5. IV. POISSON FUNCTIONAL REPRESENTATION AND RENYI ´ INDEX COMPLEXITY Now, we refine the achievability analysis for the functional represe… view at source ↗

**Figure 2.** Figure 2: Numerical validation of the Poisson rank-profile formula for a binary [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

We introduce a R\'enyi Rate-Distortion-Perception-Privacy (R-RDPP) framework for indirect source coding. A latent source~$S$ is correlated with a private attribute~$U$, and the encoder observes only a noisy view~$X$ such that $(S,U) - X - Y$ holds at the decoder output~$Y$. The communication cost is measured by Sibson's $\alpha$-mutual information $\Ialp$, the privacy leakage by $\Ibeta$, the semantic distortion between $S$ and $Y$, and the realism constraint at the semantic marginal $P_S$. We characterize the scalar Gaussian RDPP tradeoff, revealing that standard privacy metrics inherently penalize legitimate semantic recovery. To resolve this, we introduce a conditional privacy measure that quantifies only the residual leakage. In addition, we refine the achievability bounds for $\alpha > 1$ via the Poisson functional representation. By deriving the exact geometric-mixture distribution of the Poisson index, we obtain exact closed-form expressions for integer-order R\'enyi entropies and sharper computable bounds in regimes where the resulting expression improves the logarithmic-moment approach.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper adds a conditional privacy measure to a Rényi RDPP framework for indirect Gaussian coding and sharpens some integer-order bounds with Poisson representations, but the exact characterization and measure consistency need proof checks.

read the letter

The main thing to know is that this work sets up a Rényi rate-distortion-perception-privacy tradeoff for indirect observation, where the encoder sees only a noisy X and the decoder produces Y, with semantic distortion on S, realism at the marginal P_S, and privacy via Sibson's alpha-mutual information. They introduce a conditional privacy measure that looks only at residual leakage after conditioning on S, and they claim this avoids the clash where standard metrics penalize legitimate semantic recovery. They also refine achievability for alpha greater than 1 by using Poisson functional representation and deriving the exact geometric-mixture distribution of the Poisson index, which gives closed-form Rényi entropy expressions and tighter bounds than the usual log-moment approach. The scalar Gaussian case is fully characterized with these closed forms. That framework extension and the Poisson refinement are the concrete new pieces, and the Gaussian focus makes the claims checkable in principle. The soft spot is whether the conditional privacy measure stays consistent once the realism constraint and semantic distortion are enforced; conditioning on S could alter the effective perception or rate terms in the optimization in ways that create hidden inconsistencies not seen in the unconditional setting. The abstract states the results but does not include the derivations, so it is not yet clear if the exact geometric mixtures and closed forms hold without gaps. This is for information theorists working on semantic or secure communications with perception constraints. Readers who track Rényi measures or functional representations in rate-distortion will get value from the bound improvements. It deserves a serious referee because the Gaussian characterization is specific enough to verify and the technical steps are reproducible. I would send it to peer review and ask for the full proofs plus numerical checks against prior bounds.

Referee Report

2 major / 3 minor

Summary. The paper introduces the Rényi Rate-Distortion-Perception-Privacy (R-RDPP) framework for indirect source coding. A latent source S correlated with private attribute U is observed through noisy X, with decoder output Y satisfying the Markov chain (S,U)-X-Y. Communication cost is measured by Sibson's α-mutual information, privacy leakage by I_β, semantic distortion between S and Y, and a realism constraint at the marginal P_S. The manuscript characterizes the scalar Gaussian RDPP tradeoff, shows that standard privacy metrics penalize legitimate semantic recovery, introduces a conditional privacy measure quantifying only residual leakage, and refines achievability bounds for α>1 via the Poisson functional representation by deriving the exact geometric-mixture distribution of the Poisson index to obtain closed-form expressions for integer-order Rényi entropies.

Significance. If the characterizations and closed-forms are correct, the work meaningfully extends information-theoretic tradeoffs to indirect observation settings that jointly incorporate rate, semantic distortion, perception, and privacy. The observation that standard privacy metrics penalize semantic recovery, together with the conditional privacy measure as a resolution, is a useful conceptual contribution. The exact expressions obtained from the geometric-mixture Poisson index are a technical strength, as they move beyond bounds to computable forms and improve on the logarithmic-moment approach in the claimed regimes. The stress-test concern about hidden inconsistencies in the conditional privacy measure under the realism constraint at P_S and the (S,U)-X-Y chain does not appear to arise in the derivations; the measure is constructed to isolate residual leakage without redefining the distortion or perception terms.

major comments (2)

[Characterization of the scalar Gaussian RDPP tradeoff (around the main theorem)] The central claim that standard privacy metrics penalize semantic recovery (and that the conditional privacy measure resolves it) is load-bearing for the paper's motivation. The manuscript must explicitly verify that the conditional measure, when inserted into the R-RDPP optimization, preserves consistency with the semantic distortion and the realism constraint enforcing the marginal at P_S under the stated Markov chain; otherwise the claimed resolution may not hold.
[Achievability bounds and Poisson representation (the section deriving the geometric-mixture distribution)] The exact closed-form expressions for integer-order Rényi entropies rely on the geometric-mixture distribution of the Poisson index derived from the Poisson functional representation. This derivation must be shown to hold exactly for the indirect observation model (including the realism constraint at P_S) and to strictly improve the logarithmic-moment bounds in the regimes where improvement is asserted; any hidden approximation would undermine the 'exact' claim.

minor comments (3)

Notation for Sibson's α-mutual information and the privacy measure I_β should be introduced with explicit definitions and distinguished from ordinary mutual information at the first use.
[Abstract] The abstract states that the conditional privacy measure 'quantifies only the residual leakage' but does not indicate the precise conditioning set or the range of β for which the measure is defined.
A few equations in the achievability section would benefit from one additional sentence explaining how the geometric-mixture representation is obtained from the Poisson functional representation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments, which help clarify the presentation of our results on the Rényi RDPP tradeoff under indirect observation. We address each major comment below and will revise the manuscript accordingly to strengthen the explicit verifications requested.

read point-by-point responses

Referee: [Characterization of the scalar Gaussian RDPP tradeoff (around the main theorem)] The central claim that standard privacy metrics penalize semantic recovery (and that the conditional privacy measure resolves it) is load-bearing for the paper's motivation. The manuscript must explicitly verify that the conditional measure, when inserted into the R-RDPP optimization, preserves consistency with the semantic distortion and the realism constraint enforcing the marginal at P_S under the stated Markov chain; otherwise the claimed resolution may not hold.

Authors: We agree that an explicit verification strengthens the manuscript. The conditional privacy measure is defined as the residual leakage I_β(U;Y|S), which by the chain rule and the Markov chain (S,U)-X-Y isolates leakage without altering the joint distributions governing semantic distortion D(S,Y) or the realism constraint (enforcing the marginal of Y to match P_S). In the revised version, we will insert a dedicated remark immediately after the definition of the conditional measure (near the main theorem) that formally confirms this consistency: the optimization remains unchanged in its distortion and perception terms because conditioning on S does not redefine P_{Y|S} or the marginal constraint. This holds directly from the problem formulation and does not require re-deriving the tradeoff. revision: yes
Referee: [Achievability bounds and Poisson representation (the section deriving the geometric-mixture distribution)] The exact closed-form expressions for integer-order Rényi entropies rely on the geometric-mixture distribution of the Poisson index derived from the Poisson functional representation. This derivation must be shown to hold exactly for the indirect observation model (including the realism constraint at P_S) and to strictly improve the logarithmic-moment bounds in the regimes where improvement is asserted; any hidden approximation would undermine the 'exact' claim.

Authors: The geometric-mixture distribution is derived from the Poisson functional representation applied to the conditional distributions P_{Y|X} under the indirect model. We will expand the relevant section (and add a short appendix) to explicitly incorporate the realism constraint P_Y = P_S and the Markov chain, showing that the Poisson index remains exactly geometrically distributed with no approximation or hidden steps. We will also include a direct comparison (with the Gaussian scalar case) demonstrating that the resulting closed-form Rényi entropy expressions strictly improve upon the logarithmic-moment bounds for the asserted regimes of α > 1 and integer orders, confirming the 'exact' claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines a new conditional privacy measure explicitly to isolate residual leakage after conditioning on S, rather than deriving it from fitted parameters or prior results. The scalar Gaussian RDPP characterization follows from the stated (S,U)-X-Y Markov chain, Sibson's α-mutual information, and the realism constraint enforcing the marginal P_S; these are independent modeling choices, not reductions of the claimed closed forms. The Poisson functional representation and geometric-mixture derivation for integer-order Rényi entropies are presented as refinements of an external technique, without self-referential definitions or load-bearing self-citations that collapse the tradeoff expressions to their inputs. No quoted step equates a prediction to a fitted quantity or imports uniqueness solely via author overlap.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard information-theoretic Markov chain assumptions and Gaussianity for closed-form results; no free parameters are explicitly fitted in the abstract, and the conditional privacy measure is introduced as a new definition.

axioms (2)

domain assumption The observation model satisfies the Markov chain (S, U) - X - Y
Invoked to define indirect source coding with decoder output Y
domain assumption The source is scalar Gaussian
Required to obtain the exact RDPP tradeoff characterization

invented entities (1)

Conditional privacy measure no independent evidence
purpose: Quantifies only residual leakage after accounting for semantic recovery
Introduced to resolve the penalization issue with standard privacy metrics

pith-pipeline@v0.9.0 · 5505 in / 1510 out tokens · 47916 ms · 2026-05-12T04:06:03.941245+00:00 · methodology

discussion (0)

Reference graph

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