RDD Function: A Tradeoff Between Rate and Distortion-in-Distortion

TL;DR

This paper introduces the RDD function, extending classical rate-distortion theory using Gromov-type distortion, and develops an efficient algorithm to compute it, with promising applications in source coding.

Contribution

The paper proposes the RDD function based on Gromov-Wasserstein distance and develops an alternating mirror descent algorithm for its computation.

Findings

Algorithm reduces computational complexity

Numerical results validate effectiveness

Potential applications in source coding

Abstract

In this paper, we propose a novel function named Rate Distortion-in-Distortion (RDD) function as an extension of the classical rate-distortion (RD) function, where the expected distortion constraint is replaced by a Gromov-type distortion. This distortion, integral to the Gromov-Wasserstein (GW) distance, effectively defines the similarity in spaces of possibly different dimensions even without a direct metric between them. While the RDD function qualifies as an informational RD function, encoding theorems substantiate its status as an operational RD function, thereby underscoring its potential applicability in real-world source coding. Due to the high computational complexity associated with Gromov-type distortion, in general, the RDD function cannot be evaluated analytically. Consequently, we develop an alternating mirror descent algorithm that significantly reduces computational complexity by employing decomposition, linearization, and relaxation techniques. Numerical results on classical sources and different grids demonstrate the effectiveness of the developed algorithm. By exploring the relationship between the RDD function and the RD function, we suggest that the RDD function may have potential applications in future scenarios.