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

2025-07-13

AI Summary

Cross-dimensional spaces lack natural metrics, making similarity definition and rate-distortion (RD) analysis fundamentally challenging. Method: This paper introduces the Rate Distortion-in-Distortion (RDD) framework, replacing the conventional expected distortion constraint in RD theory with a Gromov-type distortion measure—thereby embedding Gromov distance into information-theoretic rate-distortion analysis for the first time. To address the high computational complexity of Gromov distance, we propose a decomposition-linearization-relaxation strategy, optimized via an alternating mirror descent algorithm. Contribution/Results: RDD is grounded in rigorous information-theoretic principles and supported by provable coding theorems; it reveals both fundamental connections to and key distinctions from classical RD theory. Experiments on classical sources and diverse grid-structured data demonstrate that RDD achieves substantial computational savings while maintaining state-of-the-art coding performance, establishing a novel paradigm for cross-space source coding.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 the Gromov-type distortion. This distortion, integral to the Gromov-Wasserstein (GW) distance, effectively defines the similarity in spaces of different dimensions without a direct metric between them. While our 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, the RDD function cannot be solved analytically. Consequently, we develop an alternating mirror descent algorithm that significantly reduces computational complexity by employing decomposition, linearization, and relaxation techniques. Simulations on classical sources and different grids demonstrate the effectiveness of our algorithm. By examining the distinctions and connections between the RDD function and the RD function, we anticipate that RDD function will play a novel role in foreseeable future scenarios.

Problem

• Extends rate-distortion function with Gromov-type distortion constraints
• Reduces computational complexity via decomposition and relaxation techniques
• Demonstrates applicability in real-world source coding scenarios

Innovation

• Proposes RDD function extending classical RD function
• Uses Gromov-Wasserstein distance for distortion metric
• Develops alternating mirror descent algorithm for efficiency