Fast binary embeddings with Gaussian circulant matrices: improved bounds

TL;DR

This paper improves theoretical bounds for fast binary embeddings using Gaussian circulant matrices, removing previous assumptions and introducing a more efficient method for sparse data.

Contribution

It provides improved variance bounds for binary Gaussian circulant embeddings, eliminating the need for data spread assumptions, and proposes a faster embedding method for sparse data.

Findings

Improved variance bounds for binary Gaussian circulant embeddings.

Removal of well-spreadness assumptions in variance analysis.

Introduction of a faster embedding method for sparse data.

Abstract

We consider the problem of encoding a finite set of vectors into a small number of bits while approximately retaining information on the angular distances between the vectors. By deriving improved variance bounds related to binary Gaussian circulant embeddings, we largely fix a gap in the proof of the best known fast binary embedding method. Our bounds also show that well-spreadness assumptions on the data vectors, which were needed in earlier work on variance bounds, are unnecessary. In addition, we propose a new binary embedding with a faster running time on sparse data.