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Rate-Distortion-Perception Function of Bernoulli Vector Sources

Praneeth Kumar Vippathalla, Mihai-Alin Badiu, Justin P. Coon

TL;DR

This work obtains an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source using the total variation distance.

Abstract

In this paper, we consider the rate-distortion-perception (RDP) trade-off for the lossy compression of a Bernoulli vector source, which is a finite collection of independent binary random variables. The RDP function quantifies in a way the efficient compression of a source when we impose a distortion constraint that limits the dissimilarity between the source and the reconstruction and a perception constraint that restricts the distributional discrepancy of the source and the reconstruction. In this work, we obtain an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source. The solution can be described by partitioning the set of distortion and perception levels $(D,P)$ into three regions, where in each region the optimal distortion and perception levels we allot to the components have a similar nature. Finally, we introduce the RDP function for graph sources and apply our result to the Erdős-Rényi graph model.

Rate-Distortion-Perception Function of Bernoulli Vector Sources

TL;DR

This work obtains an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source using the total variation distance.

Abstract

In this paper, we consider the rate-distortion-perception (RDP) trade-off for the lossy compression of a Bernoulli vector source, which is a finite collection of independent binary random variables. The RDP function quantifies in a way the efficient compression of a source when we impose a distortion constraint that limits the dissimilarity between the source and the reconstruction and a perception constraint that restricts the distributional discrepancy of the source and the reconstruction. In this work, we obtain an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source. The solution can be described by partitioning the set of distortion and perception levels into three regions, where in each region the optimal distortion and perception levels we allot to the components have a similar nature. Finally, we introduce the RDP function for graph sources and apply our result to the Erdős-Rényi graph model.
Paper Structure (7 sections, 6 theorems, 32 equations, 2 figures, 1 table)

This paper contains 7 sections, 6 theorems, 32 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. RDP Function of a Bernoulli Vector Source
  3. Main Results
  4. Proof of Theorem \ref{['thm:closed_form']}
  5. Rate-Distortion-Perception Function for Graphs
  6. Inhomogeneous Erdős-Rényi (ER) model
  7. Conclusion

Key Result

Theorem 1

For a Bernoulli vector source with parameters $q_i \leq 1/2$, $i \in [n]$, the rate-distortion-perception function is given by where $R(d,p,q)$ is the RDP function of a Bernoulli source with probability $q$, which is as defined in rdpber1 and rdpber2.

Figures (2)

  • Figure 1: A partition of $\mathbb{R}^2_{+}:=\{(D,P): D\geq 0, P\geq 0\}$.
  • Figure 2: A partition of $d_{i}\geq 0$ and $p_{i}\geq 0$

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Corollary 2
  • proof
  • Lemma 2
  • ...and 1 more