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Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets

T0 review · 2 major / 2 minor · reviewed 2026-05-16 · grok-4.3

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Pith's one-line read Rate-distortion infima are attained for lower semi-continuous distortions on locally compact Polish alphabets.

desk verdict This paper supplies existence theorems for rate-distortion on noncompact alphabets via one-point compactification and concentration-compactness. read the letter →

arxiv 2601.07246 v3 submitted 2026-01-12 cs.IT math.IT

Jiayang Zou , Luyao Fan , Jiayang Gao , Jia Wang This is my paper · ORCID
classification cs.ITmath.IT
keywords rate-distortiontheorylowersemi-continuousdistortionnoncompactalphabetsexistenceofoptimaldistributionsPolishspacesone-pointcompactificationconcentration-compactness
checked against Cost.FunctionalEquation
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 establishes existence of optimal reconstruction distributions for rate-distortion problems when the reproduction alphabet is noncompact. The authors prove two complementary mechanisms under lower semi-continuity: one-point compactification works for bounded distortions, while concentration-compactness handles unbounded coercive distortions. These results apply directly to alphabets such as the real line that classical compactness-based theorems cannot cover. The paper supplies counterexamples showing the conditions are close to sharp.

What carries the argument

Lower semi-continuity of the distortion function on a locally compact Polish reproduction alphabet, which permits one-point compactification for bounded cases or concentration-compactness for coercive unbounded cases to produce minimizing reconstruction distributions.

What would settle it

An explicit locally compact Polish alphabet together with a lower semi-continuous distortion for which no minimizing reconstruction distribution exists would falsify the attainability claim.

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Extended reading notes

Core claim

When the reproduction alphabet is a locally compact Polish space and the distortion function is lower semi-continuous, the rate-distortion infimum is attained. Bounded distortions yield attainment through the one-point compactification argument. Unbounded coercive distortions yield attainment through a concentration-compactness argument. Several counterexamples confirm that these attainability conditions are nearly necessary.

Load-bearing premise

The reproduction alphabet is a locally compact Polish space.

Editorial extensions

If this is right

  • The rate-distortion function is achieved by some reconstruction distribution whenever the stated conditions hold.
  • Optimal rate-distortion performance is attainable without requiring the reproduction alphabet to be compact.
  • Existence extends to common noncompact alphabets such as the real line under bounded or coercive distortions.
  • The two mechanisms together give a unified existence theorem for lower semi-continuous distortions.

Reading between the lines

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

  • The results justify direct rate-distortion analysis of continuous-valued sources without preliminary discretization.
  • Similar existence arguments may apply to related quantities such as channel capacity under matching distortion constraints.
  • Numerical optimization procedures for finding optimal distributions gain theoretical support in these noncompact settings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper establishes existence of optimal reproduction distributions attaining the rate-distortion infimum for general sources when the reproduction alphabet is a locally compact Polish space and the distortion measure is lower semi-continuous. It provides two complementary mechanisms: one-point compactification for bounded distortions, and concentration-compactness for unbounded coercive distortions. Several counterexamples are given to show that the attainability results are close to sharp, yielding a unified existence theorem beyond the classical compact-alphabet setting.

Significance. If the central claims hold, the results meaningfully extend classical rate-distortion attainability theorems (which rely on compactness and continuity) to noncompact alphabets under only lower semi-continuity. The explicit proof strategies, use of standard topological tools, and sharpness counterexamples constitute a solid contribution to information theory on general alphabets, with potential applicability to source coding problems involving unbounded or noncompact reproduction spaces.

major comments (2)
  1. [unbounded coercive distortions section] Unbounded coercive case (concentration-compactness argument): pointwise coercivity (lim_{y→∞} d(x,y)=∞ for each fixed x) does not automatically imply tightness of minimizing sequences {Y_n} when the source μ itself has escaping mass. Without uniformity in x, Markov-type bounds on E_μ[d(X,Y_n)] may fail to control mass escape in Y, so weak convergence in P(Y) is not guaranteed and the attainment step remains open. This directly affects the central existence claim for the unbounded case.
  2. [bounded distortions section] The one-point compactification argument for bounded distortions is less exposed but should still verify that the extended distortion remains lower semi-continuous at the point at infinity and that the infimum is preserved under the compactification.
minor comments (2)
  1. [introduction] Notation for the rate-distortion functional and the sets of admissible reproduction measures should be introduced with explicit definitions early in the paper to improve readability.
  2. [counterexamples section] A few counterexamples could benefit from explicit numerical values or simple concrete alphabets (e.g., R with quadratic distortion) to make the sharpness claims more transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the insightful comments and the recommendation for major revision. We address each major comment below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [unbounded coercive distortions section] Unbounded coercive case (concentration-compactness argument): pointwise coercivity (lim_{y→∞} d(x,y)=∞ for each fixed x) does not automatically imply tightness of minimizing sequences {Y_n} when the source μ itself has escaping mass. Without uniformity in x, Markov-type bounds on E_μ[d(X,Y_n)] may fail to control mass escape in Y, so weak convergence in P(Y) is not guaranteed and the attainment step remains open. This directly affects the central existence claim for the unbounded case.

    Authors: We thank the referee for highlighting this potential gap. We agree that pointwise coercivity requires additional care to ensure tightness of the minimizing sequences. In the revised manuscript, we will strengthen the argument by first using the tightness of the source measure μ (guaranteed since the alphabet is Polish) to truncate the source to a compact set with high probability. On this compact set, the lower semi-continuity of the distortion allows us to establish effective uniformity of the coercivity for the purpose of applying Markov's inequality to control the escape of mass in Y_n. We will include this detailed step in the concentration-compactness proof. This addresses the concern and confirms the attainment result under the stated conditions. revision: yes

  2. Referee: [bounded distortions section] The one-point compactification argument for bounded distortions is less exposed but should still verify that the extended distortion remains lower semi-continuous at the point at infinity and that the infimum is preserved under the compactification.

    Authors: We thank the referee for this constructive comment. We will revise the bounded distortions section to explicitly verify that the one-point compactification preserves the lower semi-continuity of the distortion measure at the added point at infinity. Specifically, we will show that the extended distortion function is lower semi-continuous on the compactified space. Furthermore, we will demonstrate that the rate-distortion infimum remains unchanged under this compactification, as measures assigning positive mass to the infinity point can be approximated by sequences of measures supported on the original alphabet without increasing the mutual information or the expected distortion. These verifications will be added to the manuscript. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: standard topological arguments for existence

full rationale

The paper derives existence of rate-distortion minimizers via one-point compactification (bounded case) and concentration-compactness (unbounded coercive case) on locally compact Polish spaces under lower semi-continuity. These steps invoke classical results from topology and measure theory without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations. All claims rest on external, independently verifiable properties of the spaces and distortion functions rather than reducing to the paper's own inputs by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on two domain assumptions about the alphabet and distortion; no free parameters or invented entities are introduced.

assumptions (2)
  • domain assumption The reproduction alphabet is a locally compact Polish space.
    Invoked to enable one-point compactification and concentration-compactness arguments on the space.
  • domain assumption The distortion function is lower semi-continuous.
    Key regularity condition used to guarantee attainment of the infimum in both cases.

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Pith. "Pith review of Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets." pith.science (2026). https://pith.science/paper/2601.07246

@misc{pith2026260107246,
  author       = {Pith},
  title        = {Pith review of: Rate-distortion Theory with Lower Semi-continuous Distortion on Noncompact Alphabets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2601.07246}},
  note         = {Machine review of arXiv:2601.07246}
}
read the original abstract

In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions on noncompact alphabets. Classical attainability results typically rely on compactness of the reproduction alphabet together with continuity of the distortion function, which may fail in many noncompact settings. We identify two complementary existence mechanisms under lower semi-continuity on locally compact Polish alphabets. For bounded distortions, we prove that the rate-distortion infimum is attained via the one-point compactification argument. For unbounded coercive distortions, we establish existence via concentration-compactness. We also give several counterexamples showing that our attainability results are close to sharp. Our results provide a unified and transparent existence theorem for rate-distortion problems with lower semi-continuous distortions.

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    For eachm, lety m ∈ Xsatisfy Qm(R)≤ Z BR(ym) dµm + 1 m ,lim m→∞ Qm(R) =Q(R), where the choice ofy m may depend onϵ. ThenZ BR0 (xm) dµm + Z BR(ym) dµm ≥Q m(R0) +Q m(R)− 2 m →Q(R 0) +Q(R) > 1 2 + 1 2 = 1 = Z X dµm,(15) for all sufficiently largem. Hence the ballsB R0(xm)andB R(y...

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