On Convergence of Discrete Schemes for Computing the Rate-Distortion Function of Continuous Source

TL;DR

This paper rigorously analyzes the convergence of discrete numerical schemes to compute the rate-distortion function of continuous sources, bridging the gap between discrete approximations and the original continuous optimization problem.

Contribution

It provides a rigorous mathematical framework proving that solutions from discretized schemes converge to the true continuous solutions, addressing a gap in existing literature.

Findings

Discrete schemes converge to continuous solutions

Finite-dimensional approximations effectively model the continuous problem

Addresses a gap in convergence analysis for rate-distortion computation

Abstract

Computing the rate-distortion function for continuous sources is commonly regarded as a standard continuous optimization problem. When numerically addressing this problem, a typical approach involves discretizing the source space and subsequently solving the associated discrete problem. However, existing literature has predominantly concentrated on the convergence analysis of solving discrete problems, usually neglecting the convergence relationship between the original continuous optimization and its associated discrete counterpart. This neglect is not rigorous, since the solution of a discrete problem does not necessarily imply convergence to the solution of the original continuous problem, especially for non-linear problems. To address this gap, our study employs rigorous mathematical analysis, which constructs a series of finite-dimensional spaces approximating the infinite-dimensional space of the probability measure, establishing that solutions from discrete schemes converge to those from the continuous problems.