A Constrained BA Algorithm for Rate-Distortion and Distortion-Rate Functions

TL;DR

This paper introduces a modified Blahut-Arimoto algorithm that efficiently computes rate-distortion and distortion-rate functions for specific targets, accelerating convergence and broadening applicability.

Contribution

A novel modification of the BA algorithm using root-finding for direct RD function computation, improving speed and versatility.

Findings

Converges to RD and DR solutions at rate O(1/n).

Achieves $ ext{O}(rac{MN ext{log}N}{ ext{varepsilon}})$ complexity for $ ext{varepsilon}$-approximate solutions.

Demonstrates significant acceleration over original BA in numerical experiments.

Abstract

The Blahut-Arimoto (BA) algorithm has played a fundamental role in the numerical computation of rate-distortion (RD) functions. This algorithm possesses a desirable monotonic convergence property by alternatively minimizing its Lagrangian with a fixed multiplier. In this paper, we propose a novel modification of the BA algorithm, wherein the multiplier is updated through a one-dimensional root-finding step using a monotonic univariate function, efficiently implemented by Newton's method in each iteration. Consequently, the modified algorithm directly computes the RD function for a given target distortion, without exploring the entire RD curve as in the original BA algorithm. Moreover, this modification presents a versatile framework, applicable to a wide range of problems, including the computation of distortion-rate (DR) functions. Theoretical analysis shows that the outputs of the modified algorithms still converge to the solutions of the RD and DR functions with rate $O(1/n)$, where $n$ is the number of iterations. Additionally, these algorithms provide $\varepsilon$-approximation solutions with $O\left(\frac{MN\log N}{\varepsilon}(1+\log |\log \varepsilon|)\right)$ arithmetic operations, where $M,N$ are the sizes of source and reproduced alphabets respectively. Numerical experiments demonstrate that the modified algorithms exhibit significant acceleration compared with the original BA algorithms and showcase commendable performance across classical source distributions such as discretized Gaussian, Laplacian and uniform sources.