Sparser Johnson-Lindenstrauss Transforms

TL;DR

This paper introduces two simple, sparse linear mappings for dimensionality reduction in Euclidean space that maintain low distortion, are computationally efficient, and improve upon previous sparse transform constructions.

Contribution

The authors present the first constructions achieving subconstant sparsity for all parameters, enhancing efficiency in Euclidean dimensionality reduction.

Findings

Achieve distortion 1+ε with high probability

Use only an O(ε)-fraction of non-zero entries per column

Match the asymptotically optimal number of rows

Abstract

We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion $1+\varepsilon$ with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas (JCSS 2003) and Dasgupta, Kumar, and Sarl\'{o}s (STOC 2010). Such distributions can be used to speed up applications where $\ell_2$ dimensionality reduction is used.