Compression for Quadratic Similarity Queries: Finite Blocklength and Practical Schemes

TL;DR

This paper investigates practical compression schemes for quadratic similarity queries, providing finite blocklength bounds, numerical evaluations, and implementation insights using spherical codes and lattices.

Contribution

It introduces a finite blocklength achievability bound based on shape-gain quantization and analyzes practical implementation with wrapped spherical codes and lattices.

Findings

Numerical results converge to asymptotic limits

Provides nonasymptotic bounds on compression performance

Demonstrates practical schemes using wrapped spherical codes

Abstract

We study the problem of compression for the purpose of similarity identification, where similarity is measured by the mean square Euclidean distance between vectors. While the asymptotical fundamental limits of the problem - the minimal compression rate and the error exponent - were found in a previous work, in this paper we focus on the nonasymptotic domain and on practical, implementable schemes. We first present a finite blocklength achievability bound based on shape-gain quantization: The gain (amplitude) of the vector is compressed via scalar quantization and the shape (the projection on the unit sphere) is quantized using a spherical code. The results are numerically evaluated and they converge to the asymptotic values as predicted by the error exponent. We then give a nonasymptotic lower bound on the performance of any compression scheme, and compare to the upper (achievability) bound. For a practical implementation of such a scheme, we use wrapped spherical codes, studied by Hamkins and Zeger, and use the Leech lattice as an example for an underlying lattice. As a side result, we obtain a bound on the covering angle of any wrapped spherical code, as a function of the covering radius of the underlying lattice.