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.

翻译：自香农的开创性工作以来，率失真理论定义了有损压缩的基本极限。经典结果针对渐近条件下的无记忆和平稳遍历信源推导得出，塑造了变换编码和预测编码架构，以及JPEG等实用标准。由Kostina和Verdu的非渐近可达性与逆界所启发的有限码长细化理论，在超失真概率约束下提供了精确刻画，但主要针对无记忆或统计同质模型。相比之下，科学计算中误差有界的实用有损压缩器，如SZ、ZFP、MGARD和SPERR，是为有限、高维、空间相关且统计异构的随机场设计的。这些压缩器将数据划分为固定大小的图块并独立处理，使得图块尺寸成为核心架构约束。现有的有限码长分析并未解决结构异构性、有限格点效应和图块约束问题。本文针对有限格点上的异构随机场，引入了一个有限码长率失真框架，明确考虑了高性能科学计算压缩器中使用的基于图块的架构。该场被建模为分段同质且具有区域平稳二阶统计特性，并将图块约束直接纳入信源模型。在超失真概率准则下，我们建立了非渐近可达性、逆界，并推导了二阶展开式，以量化空间相关性、区域几何结构、异构性以及图块尺寸对速率和色散的影响。