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Ares: Approximate Representations via Efficient Sparsification -- A Stateless Approach through Polynomial Homomorphism

Dongfang Zhao

TL;DR

Ares proposes a stateless compression framework that encodes high-dimensional vectors as discrete functions and approximates them with low-degree polynomials, enabling algebraic operations directly in the compressed domain. It defines a formal polynomial metric space with an $L^2$ distance and rigorous error-growth bounds under homomorphic operations, ensuring reliability for streaming data. Empirical results show that Ares achieves faster compression and strong scalability while maintaining reconstruction accuracy comparable to PCA and Autoencoders, with notable gains on sparse, structured data. This approach offers a practical, interpretable, and scalable solution for real-time, large-scale data reduction without relying on auxiliary metadata or extensive training.

Abstract

The increasing prevalence of high-dimensional data demands efficient and scalable compression methods to support modern applications. However, existing techniques like PCA and Autoencoders often rely on auxiliary metadata or intricate architectures, limiting their practicality for streaming or infinite datasets. In this paper, we introduce a stateless compression framework that leverages polynomial representations to achieve compact, interpretable, and scalable data reduction. By eliminating the need for auxiliary data, our method supports direct algebraic operations in the compressed domain while minimizing error growth during computations. Through extensive experiments on synthetic and real-world datasets, we show that our approach achieves high compression ratios without compromising reconstruction accuracy, all while maintaining simplicity and scalability.

Ares: Approximate Representations via Efficient Sparsification -- A Stateless Approach through Polynomial Homomorphism

TL;DR

Ares proposes a stateless compression framework that encodes high-dimensional vectors as discrete functions and approximates them with low-degree polynomials, enabling algebraic operations directly in the compressed domain. It defines a formal polynomial metric space with an distance and rigorous error-growth bounds under homomorphic operations, ensuring reliability for streaming data. Empirical results show that Ares achieves faster compression and strong scalability while maintaining reconstruction accuracy comparable to PCA and Autoencoders, with notable gains on sparse, structured data. This approach offers a practical, interpretable, and scalable solution for real-time, large-scale data reduction without relying on auxiliary metadata or extensive training.

Abstract

The increasing prevalence of high-dimensional data demands efficient and scalable compression methods to support modern applications. However, existing techniques like PCA and Autoencoders often rely on auxiliary metadata or intricate architectures, limiting their practicality for streaming or infinite datasets. In this paper, we introduce a stateless compression framework that leverages polynomial representations to achieve compact, interpretable, and scalable data reduction. By eliminating the need for auxiliary data, our method supports direct algebraic operations in the compressed domain while minimizing error growth during computations. Through extensive experiments on synthetic and real-world datasets, we show that our approach achieves high compression ratios without compromising reconstruction accuracy, all while maintaining simplicity and scalability.

Paper Structure

This paper contains 49 sections, 50 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Proposed Work
  4. Contributions
  5. Related Work
  6. Linear Transformation-Based Methods
  7. Matrix Factorization Approaches
  8. Autoencoders
  9. Homomorphic Compression
  10. One-way Dimensionality Reduction
  11. Methodology
  12. Function Representation of Vectors
  13. Lower-Degree Polynomial Approximation
  14. Function-Polynomial Distance
  15. Coefficient Selection
  16. ...and 34 more sections

Figures (3)

  • Figure 1: Comparison of Compression Times for Four Algorithms on Random, URL, and Newsgroup Datasets (Target Dimension: 10). Note: The Y-axis is in logarithmic scale for better visualization of differences.
  • Figure 2: Comparison of Compression Ratios for Four Algorithms on Random, URL, and Newsgroup Datasets (Target Dimension: 10). Ares demonstrates superior compression efficiency on the sparse URL dataset but comparable performance on other datasets.
  • Figure 3: Comparison of Decompression Times for Four Algorithms on Random, URL, and Newsgroup Datasets (Target Dimension: 10). The time is measured in milliseconds (ms).