Understanding Sparse JL for Feature Hashing

TL;DR

This paper explores the use of higher sparsity levels in sparse Johnson-Lindenstrauss transforms for feature hashing, showing both theoretical advantages and empirical improvements in norm preservation over the sparsity 1 case.

Contribution

It provides a tight tradeoff analysis for sparse JL with general sparsity s, extending previous work and demonstrating benefits of s > 1 in feature vector applications.

Findings

Theoretical demonstration of improved norm preservation with s > 1.

Empirical evidence supporting the advantages of higher sparsity.

Generalization of previous tight tradeoff results for sparse JL.

Abstract

Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in $\mathbb{R}^n$ into a much lower-dimensional space $\mathbb{R}^m$, while approximately preserving Euclidean norm. These schemes can be constructed using sparse random projections, for example using a sparse Johnson-Lindenstrauss (JL) transform. A line of work introduced by Weinberger et. al (ICML '09) analyzes the accuracy of sparse JL with sparsity 1 on feature vectors with small $\ell_\infty$-to-$\ell_2$ norm ratio. Recently, Freksen, Kamma, and Larsen (NeurIPS '18) closed this line of work by proving a tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for sparse JL with sparsity $1$. In this paper, we demonstrate the benefits of using sparsity $s$ greater than $1$ in sparse JL on feature vectors. Our main result is a tight tradeoff between $\ell_\infty$-to-$\ell_2$ norm ratio and accuracy for a general sparsity $s$, that significantly generalizes the result of Freksen et. al. Our result theoretically demonstrates that sparse JL with $s > 1$ can have significantly better norm-preservation properties on feature vectors than sparse JL with $s = 1$; we also empirically demonstrate this finding.