Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computation of the optimal error exponent function for fixed-length lossy source coding in discrete memoryless sources

Yutaka Jitsumatsu

TL;DR

The main contribution of this paper is the development of a parametric expression that is in perfect agreement with the inverse function of the Marton exponent, which has two layers and is convex optimization and can be computed efficiently.

Abstract

Marton's optimal error exponent for the lossy source coding problem is defined as a non-convex optimization problem. This fact had prevented us to develop an efficient algorithm to compute it. This problem is caused by the fact that the rate-distortion function $R(Δ|P)$ is potentially non-concave in the probability distribution $P$ for a fixed distortion level $Δ$. The main contribution of this paper is the development of a parametric expression that is in perfect agreement with the inverse function of the Marton exponent. This representation has two layers. The inner layer is convex optimization and can be computed efficiently. The outer layer, on the other hand, is a non-convex optimization with respect to two parameters. We give a method for computing the Marton exponent based on this representation.

Computation of the optimal error exponent function for fixed-length lossy source coding in discrete memoryless sources

TL;DR

The main contribution of this paper is the development of a parametric expression that is in perfect agreement with the inverse function of the Marton exponent, which has two layers and is convex optimization and can be computed efficiently.

Abstract

Marton's optimal error exponent for the lossy source coding problem is defined as a non-convex optimization problem. This fact had prevented us to develop an efficient algorithm to compute it. This problem is caused by the fact that the rate-distortion function is potentially non-concave in the probability distribution for a fixed distortion level . The main contribution of this paper is the development of a parametric expression that is in perfect agreement with the inverse function of the Marton exponent. This representation has two layers. The inner layer is convex optimization and can be computed efficiently. The outer layer, on the other hand, is a non-convex optimization with respect to two parameters. We give a method for computing the Marton exponent based on this representation.
Paper Structure (23 sections, 15 theorems, 63 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 15 theorems, 63 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Sub-optimality of Blahut's exponent
  3. Paper organization
  4. Main Results
  5. Parametric expression of the inverse function of the Marton exponent
  6. Computation Method for $R_{\rm M}(E|\Delta, P)$
  7. Comparison with the inverse function of the Blahut exponent
  8. Computational cost
  9. Numerical Examples
  10. Proof of Theorem \ref{['theorem2']}
  11. A parametric expression for the rate-distortion function
  12. Minimax theorem
  13. The second Lagrange multiplier
  14. Proofs
  15. Proof of Theorem \ref{['theorem_inverse_Blahut']}
  16. ...and 8 more sections

Key Result

Lemma 1

For any $P\in \mathcal{P(X)}$, distortion measure $d(x,y)$, $R\geq 0$, and $\Delta\geq0$, $E_{\rm B}(R|\Delta,P)$ is a lower convex envelope of $E_{\rm M}(R|\Delta,P)$.

Figures (8)

  • Figure 1: The rate-distortion function $R(\Delta|Q_{\lambda})$ as a function of $\lambda$ of Ahlswede's counterexample with parameters #1 in Table \ref{['table1']}. The unit of rate $R$ is the bit.
  • Figure 2: Marton's and Blahut's error exponents are illustrated as functions of $R$ for Ahlswede's counterexample with parameters #1 in Table \ref{['table1']}.
  • Figure 3: $R_{\rm M}(E | \Delta , P)$ for Ahlswede's counterexample of Fig. \ref{['fig.1']}.
  • Figure 4: $E=0.5$
  • Figure 5: $E=1.0$
  • ...and 3 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Theorem 4: Sion Sion1958
  • Definition 1
  • Lemma 4
  • Lemma 5
  • ...and 7 more