Guarantees for the Kronecker Fast Johnson-Lindenstrauss Transform Using a Coherence and Sampling Argument

TL;DR

This paper provides an alternative proof that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is a valid Johnson-Lindenstrauss transform for Kronecker vectors, using coherence and sampling arguments, and compares it with other sketching methods.

Contribution

It offers a new proof of KFJLT's properties, introduces a different embedding dimension bound, and demonstrates its effectiveness through numerical experiments.

Findings

KFJLT is a Johnson-Lindenstrauss transform for Kronecker vectors

New bound on embedding dimension improves previous results

KFJLT performs competitively compared to other sketch techniques

Abstract

In the recent paper [Jin, Kolda & Ward, arXiv:1909.04801], it is proved that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is, in fact, a Johnson-Lindenstrauss transform, which had previously only been conjectured. In this paper, we provide an alternative proof of this, for when the KFJLT is applied to Kronecker vectors, using a coherence and sampling argument. Our proof yields a different bound on the embedding dimension, which can be combined with the bound in the paper by Jin et al. to get a better bound overall. As a stepping stone to proving our result, we also show that the KFJLT is a subspace embedding for matrices with columns that have Kronecker product structure. Lastly, we compare the KFJLT to four other sketch techniques in numerical experiments on both synthetic and real-world data.