Core Concepts
The solutions of discrete schemes for computing the rate-distortion function of continuous sources converge to the solution of the original continuous problem.
Abstract
The paper presents a rigorous mathematical analysis to establish the convergence of solutions from discrete schemes to the original continuous rate-distortion optimization problem. The key ideas are:
- Constructing a series of finite-dimensional spaces to approximate the infinite-dimensional space of the probability measure for the reproduction variable.
- Leveraging functional space analysis and estimation techniques to prove the desired convergence property.
- Showing that the solutions of the discrete problems converge to those of the original continuous problems, independent of the specific numerical algorithms used for the discrete problems.
- Providing complexity analyses for the Blahut-Arimoto (BA) algorithm and the Constrained BA (CBA) algorithm in terms of the required arithmetic operations to achieve a given accuracy.
- Discussing the importance of the convergence result, as it addresses the issue that the solution of a discrete problem does not necessarily imply convergence to the solution of the original continuous problem, especially for non-linear problems like the rate-distortion optimization.
Stats
The paper does not provide any specific numerical data or statistics. The analysis focuses on establishing the theoretical convergence properties of the discrete schemes.
Quotes
"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."
"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."