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arxiv: 2601.11919 · v2 · pith:SDLTSK34new · submitted 2026-01-17 · 💻 cs.IT · math.IT

Rate-Distortion-Classification Representation Theory for Bernoulli Sources

Nam Nguyen , Thinh Nguyen , Bella Bose This is my paper

Pith reviewed 2026-05-21 15:56 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords rate-distortion-classificationBernoulli sourceHamming distortionlinear programuniversal encodertask-oriented compressionone-shot tradeoff
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The pith

Closed-form one-shot RDC and DRC tradeoffs are derived for Bernoulli sources with Hamming distortion under binary symmetric classification coupling.

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

The paper examines task-oriented lossy compression for Bernoulli sources by introducing rate-distortion-classification representations. It first obtains closed-form characterizations of the one-shot RDC tradeoff and its dual DRC version using a common-randomness formulation. A representation-based approach then reduces the achievable distortion-classification region for any fixed representation to the solution of a linear program that traces its lower boundary. Finally the work supplies computable lower and upper bounds on the minimum asymptotic rate needed by universal encoders that must serve an entire family of distortion-classification operating points.

Core claim

Building on the one-shot common-randomness formulation, the paper derives closed-form characterizations of the one-shot RDC and the dual DRC tradeoffs for a Bernoulli source under Hamming distortion when the binary classification variable is coupled to the source by a binary symmetric model. It further characterizes the achievable distortion-classification region induced by any fixed representation by deriving the lower boundary of that region via a linear program, and it obtains computable lower and upper bounds on the minimum asymptotic rate required by universal encoders that support a family of DC operating points, thereby quantifying the associated rate penalty.

What carries the argument

The linear program that computes the lower boundary of the achievable distortion-classification region induced by any fixed representation.

If this is right

  • The closed-form RDC and DRC expressions permit exact evaluation of the one-shot tradeoffs without iterative optimization.
  • Any chosen representation yields an explicitly computable lower boundary for its achievable distortion-classification region through the linear program.
  • The derived bounds quantify the exact rate penalty incurred when a single encoder must support an entire family of distortion-classification operating points.

Where Pith is reading between the lines

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

  • The closed-form results could guide the construction of practical encoders when classification accuracy and reconstruction fidelity must be traded against each other.
  • The linear-program formulation may extend to other discrete memoryless sources once the binary-symmetric coupling assumption is relaxed.
  • Bounds on universal rates suggest a concrete way to measure the overhead of building flexible representations that remain useful across changing task requirements.

Load-bearing premise

The binary classification variable is coupled to the source via a binary symmetric model.

What would settle it

A direct numerical computation of the one-shot RDC tradeoff for concrete parameter values that deviates from the claimed closed-form expression would refute the characterization.

Figures

Figures reproduced from arXiv: 2601.11919 by Bella Bose, Nam Nguyen, Thinh Nguyen.

Figure 2
Figure 2. Figure 2: One-shot setting with common randomness. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Task-oriented lossy compression framework. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of Theorem 2. R(B) (D, C) versus D with given C = 0.8, qX = 0.3, qS1 = 0.2. (4) is feasible if C ≥ Hb(qS1 ). Moreover, under common randomness, R (B) (D, C) =    Hb(qX)(qX−D) qX , 0 ≤ D < qX(C−Hb(qS1 ) Hb(m)−Hb(qS1 ) Hb(qX)[Hb(m)−C] Hb(m)−Hb(qS1 ) , qX[C−Hb(qS1 )] Hb(m)−Hb(qS1 ) ≤ D ≤ 1 0, C ≥ Hb (m) and qX < D ≤ 1. where m = (1−qX)(1−qS1 ) +qXqS1 and Hb(.) denotes the binary entropy fu… view at source ↗
Figure 5
Figure 5. Figure 5: RDC curves for a fixed C: R(B) (D, C) and R(∞) (D, C) versus D, for C = 0.9, qX = 0.3, and qS1 = 0.2. variable S ∼ pS, the corresponding one-shot distortion mini￾mization problem can be expressed as D∗ (R, C) = min pU , pXˆ|X,U E[∆(X, Xˆ)] (5a) s.t. H(Xˆ|U, X) = 0, I(X, U) = 0, (5b) H(Xˆ|U) ≤ R, H(S|Xˆ) ≤ C. (5c) where pU,X,Xˆ = pU pX pXˆ|U,X [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of Theorem 4. R(B) (D, C) versus D with given C = 0.8, qX = 0.3, qS1 = 0.2. Theorem 4. Let X ∼ Bern(qX) be a Bernoulli source and let S be a binary task variable jointly distributed with X via S = X ⊕ S1, where S ∼ Bern(qS) and S1 ∼ Bern(qS1 ) (0 ≤ qX, qS, qS1 ≤ 1 2 ). Assume the Hamming distortion measure. Then the optimization problem (5) is feasible if C ≥ H(qS1 ). Moreover, under common ra… view at source ↗
Figure 9
Figure 9. Figure 9: The universal encoder representation framework. [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lower boundary of achievable region, Π(pZ|X) [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DRC curves for a fixed R: D(∞) [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
read the original abstract

We study task-oriented lossy compression through the lens of rate-distortion-classification (RDC) representations. The source is Bernoulli, the distortion measure is Hamming, and the binary classification variable is coupled to the source via a binary symmetric model. Building on the one-shot common-randomness formulation, we first derive closed-form characterizations of the one-shot RDC and the dual distortion-rate-classification (DRC) tradeoffs. We then use a representation-based viewpoint and characterize the achievable distortion-classification (DC) region induced by a fixed representation by deriving its lower boundary via a linear program. Finally, we study universal encoders that must support a family of DC operating points and derive computable lower and upper bounds on the minimum asymptotic rate required for universality, thereby yielding bounds on the corresponding rate penalty. Numerical examples are provided to illustrate the achievable regions and the resulting universal RDC/DRC curves.

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

0 major / 4 minor

Summary. The manuscript develops a rate-distortion-classification (RDC) representation theory for Bernoulli sources under Hamming distortion, where the binary classification variable is coupled to the source through a binary-symmetric channel. Using a one-shot common-randomness formulation, it derives closed-form expressions for the one-shot RDC and dual distortion-rate-classification (DRC) tradeoffs. It then characterizes the achievable distortion-classification (DC) region for a fixed representation via the lower boundary of a linear program and derives computable lower and upper bounds on the minimum asymptotic rate required for universal encoders that must support a family of DC operating points, along with the associated rate penalty. Numerical examples illustrate the regions and curves.

Significance. If the derivations hold, the paper supplies explicit, computable characterizations of RDC and DRC tradeoffs together with an LP formulation of the DC region and bounds on the universal-rate penalty. These results are grounded in standard information-theoretic optimization and provide concrete tools for analyzing task-oriented compression that jointly controls distortion and classification performance. The explicit forms and numerical illustrations strengthen the contribution for the Bernoulli/Hamming case, which serves as a canonical setting for further extensions.

minor comments (4)
  1. The abstract states that closed-form characterizations are obtained, but the manuscript should explicitly state the range of parameters (e.g., crossover probability of the binary-symmetric coupling and distortion levels) for which the closed forms remain valid without case distinctions.
  2. In the LP formulation of the DC region (presumably §4), the decision variables and the objective should be written with explicit dependence on the representation mapping so that the boundary computation is reproducible from the given expressions.
  3. The universal-rate bounds are presented as computable; the manuscript would benefit from a short algorithmic description or pseudocode showing how the lower and upper bounds are evaluated for a given family of DC points.
  4. Notation for the one-shot common-randomness auxiliary variables should be introduced once and used consistently across the RDC, DRC, and universal sections to avoid reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its explicit characterizations for the Bernoulli/Hamming setting, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from an explicitly stated source model (Bernoulli with Hamming distortion and binary-symmetric coupling to the classification variable) and proceeds through standard one-shot common-randomness optimization to obtain closed-form RDC/DRC expressions, followed by an LP characterization of the DC region and computable bounds on the universal rate. These steps are internally consistent once the coupling model is granted and do not reduce any claimed prediction or first-principles result to a fitted parameter or self-referential definition by construction. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are exhibited in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are detailed in the provided text.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Cross-Domain Lossy Compression via Constrained Minimum Entropy Coupling

    cs.IT 2026-05 unverdicted novelty 7.0

    Rate-constrained minimum entropy coupling enables cross-domain lossy compression with classification preservation, providing closed-form solutions for Bernoulli sources and neural implementations for MNIST and SVHN tasks.

Reference graph

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32 extracted references · 32 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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    implies H(S| ˆX)≥H(S|X) =H(X⊕S 1|X) =H(S 1). Hence, feasibility of the classification constraint requiresC≥ H(S1). We now evaluateH(S| ˆX, U=u)for each mapping. ForU= 1: ˆX=X H(S| ˆX, U= 1) =H(S|X) =H(X⊕S 1|X) =H(S 1) =H b(qS1). ForU= 2: ˆX= 1−X H(S| ˆX, U= 2) =H(S|X) =H(S 1) =H b(qS1). ForU= 3: ˆX= 0 S=X⊕S 1 ⇒P(S= 0) = (1−q X)(1−q S1) +q X qS1 , H(S| ˆX,...

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