Compression of Voxelized Vector Field Data by Boxes is Hard

2025-10-23

AI Summary

This paper studies rate-distortion lossy compression of voxelized vector field data on high-dimensional lattices, formalized as the $(k,D)$-RectLossyVFCompression problem: approximating the vector field using $k$ axis-aligned hyperrectangles (“box summaries”) under an energy-based distortion constraint—namely, per-voxel distortion must not exceed $D$. Methodologically, we propose a compression framework grounded in energy distortion metrics and prove that decompression is polynomial-time computable; however, the compression problem is both NP-hard and APX-hard for $k, D geq 2$, ruling out exact polynomial-time algorithms and PTASs (assuming $mathrm{P} eq mathrm{NP}$). Our primary contribution is the first rigorous theoretical lower bound for voxelized vector field compression, revealing its intrinsic computational complexity and establishing box summaries as a theoretically justified, efficient, and compact representation—with precise characterization of their fundamental expressiveness and limitations.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Learning & Optimization for CV

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: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Voxelized vector field data consists of a vector field over a high dimensional lattice. The lattice consists of integer coordinates called voxels. The voxelized vector field assigns a vector at each voxel. This data type encompasses images, tensors, and voxel data. Assume there is a nice energy function on the vector field. We consider the problem of lossy compression of voxelized vector field data in Shannon's rate-distortion framework. This means the data is compressed then decompressed up to a bound on the distortion of the energy at each voxel. We formulate this in terms of compressing a single voxelized vector field by a collection of box summary pairs. We call this problem the $(k,D)$-RectLossyVFCompression} problem. We show three main results about this problem. The first is that decompression for this problem is polynomial time tractable. This means that the only obstruction to a tractable solution of the $(k,D)$-RectLossyVFCompression problem lies in the compression stage. This is shown by the two hardness results about the compression stage. We show that the compression stage is NP-Hard to compute exactly and that it is even APX-Hard to approximate for $k,Dgeq 2$. Assuming $P eq NP$, this shows that when $k,D geq 2$ there can be no exact polynomial time algorithm nor can there even be a PTAS approximation algorithm for the $(k,D)$-RectLossyVFCompression problem.

Problem

• Lossy compression of voxelized vector fields is computationally intractable
• Compression stage is NP-hard and APX-hard to approximate
• No exact polynomial-time or PTAS algorithm exists for this problem

Innovation

• Lossy compression using box summary pairs
• Polynomial time decompression algorithm
• NP-Hard compression approximation complexity