[Paper] Rate-Distortion Bounds for Heterogeneous Random Fields on Finite Lattices

Source: arXiv - 2603.09833v1

Overview

The paper extends classic rate‑distortion theory—traditionally limited to infinite, memoryless sources—to the messy reality of scientific data: high‑dimensional, spatially correlated fields that vary in statistical properties across a domain and are processed in fixed‑size tiles. By deriving finite‑blocklength achievability and converse bounds for such heterogeneous random fields on finite lattices, the authors provide the first principled way to predict how tile size, correlation, and heterogeneity affect compression performance in modern lossy scientific compressors (e.g., SZ, ZFP, MGARD).

Key Contributions

• Finite‑blocklength framework for heterogeneous fields – Introduces a source model that captures piecewise‑stationary second‑order statistics on a finite lattice with explicit tiling constraints.
• Non‑asymptotic achievability and converse bounds – Extends Kostina‑Verdú style bounds to the heterogeneous, tiled setting under an excess‑distortion probability criterion.
• Second‑order (dispersion) expansion – Quantifies how spatial correlation, region geometry, heterogeneity, and tile size jointly influence the required bitrate beyond the classic first‑order rate‑distortion function.
• Closed‑form expressions for practical tile shapes – Provides analytic formulas for common tile geometries (e.g., cubic, rectangular) that can be plugged into compressor design tools.
• Guidelines for tile‑size selection – Shows how to balance compression efficiency against memory and parallelism constraints in high‑performance computing (HPC) pipelines.

Methodology

1. Source Modeling – The data domain is represented as a finite lattice (grid) partitioned into regions. Within each region the field is assumed second‑order stationary (constant mean, covariance that depends only on relative position). Different regions may have distinct covariance matrices, capturing heterogeneity.
2. Tiling Incorporation – The lattice is further divided into fixed‑size tiles (the units actually processed by compressors). Tiles may cut across region boundaries; the model accounts for the resulting mixed statistics inside a tile.
3. Excess‑Distortion Criterion – Instead of average distortion, the authors bound the probability that distortion exceeds a target (D). This aligns with scientific error‑budget requirements (e.g., “no more than 1 % of points exceed 10⁻⁴ error”).
4. Achievability Construction – They design a random coding scheme that first decorrelates each tile using a region‑aware linear transform (e.g., Karhunen‑Loève), then quantizes the transformed coefficients with a Gaussian test channel. The scheme respects tile independence, matching real compressors.
5. Converse Argument – Using information‑theoretic inequalities (e.g., change‑of‑measure, Berry‑Esseen CLT), they prove that any encoder respecting the tile constraints must incur at least the derived bitrate.
6. Second‑Order Expansion – By applying a refined central‑limit analysis to the sum of per‑tile information densities, they extract a dispersion term that scales with (\sqrt{n}) (where (n) is tile size) and explicitly depends on the covariance eigenvalues and region geometry.

Results & Findings

Practical Implications

• Informed Tile‑Size Selection – Developers can use the closed‑form second‑order formulas to pick tile dimensions that meet a given bitrate or error budget without costly trial‑and‑error runs.
• Adaptive Compression Pipelines – The framework suggests a low‑overhead way to dynamically adjust tile sizes per region (e.g., larger tiles in smooth areas, smaller in highly variable zones) to maximize overall compression efficiency.
• Benchmarking & Standardization – The non‑asymptotic bounds provide a rigorous baseline against which new scientific compressors can be compared, moving beyond ad‑hoc PSNR or compression‑ratio metrics.
• Hardware‑Aware Optimization – Since tile size directly impacts cache usage and parallel workload distribution on GPUs/accelerators, the theory helps balance memory bandwidth constraints with compression quality.
• Error‑Budget Guarantees – By framing distortion as an excess‑probability, the results align with scientific reproducibility requirements, enabling compressors to offer provable guarantees on the fraction of out‑lier errors.

Limitations & Future Work

• Gaussian Assumption – The analysis assumes each region follows a Gaussian field; extending to heavy‑tailed or non‑linear phenomena (e.g., turbulence) remains open.
• Static Tiling – While the model captures tile‑size effects, it does not yet address overlapping or adaptive tiling strategies that some compressors employ.
• Higher‑Dimensional Correlations – The second‑order expansion treats each tile independently; capturing long‑range dependencies across tiles would require a more complex multi‑tile joint analysis.
• Implementation Overhead – The optimal linear transforms suggested by the theory may be computationally expensive; future work could explore low‑cost approximations that retain most of the theoretical gain.

Overall, this work bridges a long‑standing gap between information theory and the practical realities of high‑performance scientific data compression, giving developers a mathematically grounded toolkit for designing faster, more efficient, and error‑aware compressors.

Authors

• Sujata Sinha
• Vishwas Rao
• Robert Underwood
• David Lenz
• Sheng Di
• Franck Cappello
• Lingjia Liu

Paper Information

• arXiv ID: 2603.09833v1
• Categories: cs.IT, cs.DC
• Published: March 10, 2026
• PDF: Download PDF