Simple Analysis of Sparse, Sign-Consistent JL

TL;DR

This paper provides a simplified, combinatorics-free analysis of sparse, sign-consistent Johnson-Lindenstrauss distributions, matching previous bounds and revealing new dimension-sparsity tradeoffs, with broader implications for moment bounds in theoretical CS.

Contribution

It introduces a simple, combinatorics-free proof technique for sparse, sign-consistent JL, improving understanding of dimension and sparsity bounds and their tradeoffs.

Findings

Achieves the same dimension and sparsity bounds as previous complex analyses.

Introduces new dimension-sparsity tradeoff results.

Develops a novel moment bound for quadratic forms of Rademacher variables.

Abstract

Allen-Zhu, Gelashvili, Micali, and Shavit construct a sparse, sign-consistent Johnson-Lindenstrauss distribution, and prove that this distribution yields an essentially optimal dimension for the correct choice of sparsity. However, their analysis of the upper bound on the dimension and sparsity requires a complicated combinatorial graph-based argument similar to Kane and Nelson's analysis of sparse JL. We present a simple, combinatorics-free analysis of sparse, sign-consistent JL that yields the same dimension and sparsity upper bounds as the original analysis. Our analysis also yields dimension/sparsity tradeoffs, which were not previously known. As with previous proofs in this area, our analysis is based on applying Markov's inequality to the pth moment of an error term that can be expressed as a quadratic form of Rademacher variables. Interestingly, we show that, unlike in previous work in the area, the traditionally used Hanson-Wright bound is not strong enough to yield our desired result. Indeed, although the Hanson-Wright bound is known to be optimal for gaussian degree-2 chaos, it was already shown to be suboptimal for Rademachers. Surprisingly, we are able to show a simple moment bound for quadratic forms of Rademachers that is sufficiently tight to achieve our desired result, which given the ubiquity of moment and tail bounds in theoretical computer science, is likely to be of broader interest.