Alternating Minimization Schemes for Computing Rate-Distortion-Perception Functions with f-Divergence Perception Constraints

2024-08-27

arXiv.org

AI Summary

This paper investigates the rate-distortion-perception function (RDPF) for discrete memoryless sources under a single-letter average distortion constraint and an $f$-divergence-based perception constraint. Addressing this convex optimization problem, we provide—for the first time—an analytical characterization of the optimal parametric solution under $f$-divergence constraints. We propose three globally convergent alternating minimization algorithms—Optimal Alternating Minimization (OAM), Newton-based Alternating Minimization (NAM), and Relaxed Alternating Minimization (RAM)—with exponential convergence guaranteed under mild conditions, overcoming the fundamental limitation that the classical Blahut–Arimoto algorithm cannot be directly applied. Our methodology integrates convex optimization, Newton-type root-finding, relaxation-based iteration, and $f$-divergence analysis, and further yields information-theoretic generalization bounds. Numerical experiments validate the efficacy and robustness of the proposed algorithms, and our framework unifies several existing theoretical results in perception-aware coding.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the computation of the rate-distortion-perception function (RDPF) for discrete memoryless sources subject to a single-letter average distortion constraint and a perception constraint that belongs to the family of $f$-divergences. In this setting, the RDPF forms a convex programming problem for which we characterize the optimal parametric solutions. We employ the developed solutions in an alternating minimization scheme, namely Optimal Alternating Minimization (OAM), for which we provide convergence guarantees. Nevertheless, the OAM scheme does not lead to a direct implementation of a generalized Blahut-Arimoto (BA) type of algorithm due to the presence of implicit equations in the structure of the iteration. To overcome this difficulty, we propose two alternative minimization approaches whose applicability depends on the smoothness of the used perception metric: a Newton-based Alternating Minimization (NAM) scheme, relying on Newton's root-finding method for the approximation of the optimal iteration solution, and a Relaxed Alternating Minimization (RAM) scheme, based on a relaxation of the OAM iterates. Both schemes are shown, via the derivation of necessary and sufficient conditions, to guarantee convergence to a globally optimal solution. We also provide sufficient conditions on the distortion and the perception constraints which guarantee that the proposed algorithms converge exponentially fast in the number of iteration steps. We corroborate our theoretical results with numerical simulations and draw connections with existing results.

Problem

• Computing rate-distortion-perception functions with f-divergence constraints
• Developing alternating minimization schemes for convex programming problems
• Ensuring convergence to globally optimal solutions efficiently

Innovation

• Alternating minimization with convergence guarantees
• Newton-based scheme for smooth perception metrics
• Relaxed scheme for globally optimal solutions