Rademacher Chaos, Random Eulerian Graphs and The Sparse Johnson-Lindenstrauss Transform

TL;DR

This paper improves the sparsity and randomness requirements of Johnson-Lindenstrauss transforms using Rademacher chaos and Eulerian graph properties, enabling faster dimension reduction for sparse data.

Contribution

It introduces new bounds on the sparsity of the transform matrix and connects chaos moments to Eulerian graph properties, advancing the theoretical understanding of sparse JL transforms.

Findings

Reduced the non-zero entries per column in the transform matrix.

Established a new tail bound on Rademacher chaos using Eulerian graph analysis.

Connected chaos moments to combinatorial properties of random multigraphs.

Abstract

The celebrated dimension reduction lemma of Johnson and Lindenstrauss has numerous computational and other applications. Due to its application in practice, speeding up the computation of a Johnson-Lindenstrauss style dimension reduction is an important question. Recently, Dasgupta, Kumar, and Sarlos (STOC 2010) constructed such a transform that uses a sparse matrix. This is motivated by the desire to speed up the computation when applied to sparse input vectors, a scenario that comes up in applications. The sparsity of their construction was further improved by Kane and Nelson (ArXiv 2010). We improve the previous bound on the number of non-zero entries per column of Kane and Nelson from $O(1/\epsilon \log(1/\delta)\log(k/\delta))$ (where the target dimension is $k$, the distortion is $1\pm \epsilon$, and the failure probability is $\delta$) to $$ O\left({1\over\epsilon} \left({\log(1/\delta)\log\log\log(1/\delta) \over \log\log(1/\delta)}\right)^2\right). $$ We also improve the amount of randomness needed to generate the matrix. Our results are obtained by connecting the moments of an order 2 Rademacher chaos to the combinatorial properties of random Eulerian multigraphs. Estimating the chance that a random multigraph is composed of a given number of node-disjoint Eulerian components leads to a new tail bound on the chaos. Our estimates may be of independent interest, and as this part of the argument is decoupled from the analysis of the coefficients of the chaos, we believe that our methods can be useful in the analysis of other chaoses.