Rate-Distortion Bounds for Heterogeneous Random Fields on Finite Lattices

2026-03-10

AI Summary

This work addresses the limitations of existing finite-blocklength rate-distortion theory in characterizing the performance limits of block-based compression for high-dimensional, spatially correlated, and statistically heterogeneous random fields commonly encountered in scientific computing. Focusing on the blockwise architectures widely adopted by scientific compressors, the study establishes a non-asymptotic rate-distortion framework that unifies statistical heterogeneity, domain geometry, block constraints, and finite-lattice effects. Leveraging a piecewise stationary second-order statistical model, it derives achievability and converse bounds under a constraint on the probability of exceeding a target distortion. The analysis quantifies how spatial correlation, domain shape, heterogeneity, and block size jointly influence compression rates and dispersion, thereby providing a theoretical foundation for the design of scientific data compressors.

Technology Category

Data Mining & Knowledge Management: Data CompressionMachine Learning: Other Foundations of Machine LearningConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Machine learning and data science for the WebSystems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Since Shannon's foundational work, rate-distortion theory has defined the fundamental limits of lossy compression. Classical results, derived for memoryless and stationary ergodic sources in the asymptotic regime, have shaped both transform and predictive coding architectures, as well as practical standards such as JPEG. Finite-blocklength refinements, initiated by the non-asymptotic achievability and converse bounds of Kostina and Verdu, provide precise characterizations under excess-distortion probability constraints, but primarily for memoryless or statistically homogeneous models. In contrast, error-bounded practical lossy compressors for scientific computing, such as SZ, ZFP, MGARD, and SPERR, are designed for finite, high-dimensional, spatially correlated, and statistically heterogeneous random fields. These compressors partition data into fixed-size tiles that are processed independently, making tile size a central architectural constraint. Structural heterogeneity, finite lattice effects, and tiling constraints are not addressed by existing finite-blocklength analyses. This paper introduces a finite-blocklength rate-distortion framework for heterogeneous random fields on finite lattices, explicitly accounting for the tile-based architectures used in high-performance scientific compressors. The field is modeled as piecewise homogeneous with regionwise stationary second-order statistics, and tiling constraints are incorporated directly into the source model. Under an excess-distortion probability criterion, we establish non-asymptotic achievability, converse bounds and derive a second-order expansion that quantifies the impact of spatial correlation, region geometry, heterogeneity, and tile size on the rate and dispersion.

Problem

• rate-distortion
• heterogeneous random fields
• finite lattices
• tiling constraints
• finite-blocklength

Innovation

• finite-blocklength rate-distortion
• heterogeneous random fields
• tiling constraints
• non-asymptotic bounds
• piecewise stationary statistics