Toward a unified theory of sparse dimensionality reduction in Euclidean space

Abstract

Let \(\Phi \in \mathbb{R}^{m×n}\) be a sparse Johnson-Lindenstrauss transform [KN14] with s non-zeroes per column. For a subset T of the unit sphere, \(\epsilon \in (0,1/2)\) given, we study settings for m,s required to ensure \(\underset {\Phi_{x \in T}} {\mathbb{E} sup} \mid || \Phi x ||^2_2 - 1 \mid < \epsilon\), i.e.\ so that \(\Phi\) preserves the norm of every \(x \in T\) simultaneously and multiplicatively up to \(1+\epsilon\). We introduce a new complexity parameter, which depends on the geometry of T, and show that it suffices to choose s and m such that this parameter is small. Our result is a sparse analog of Gordon's theorem, which was concerned with a dense \(\Phi\) having i.i.d. gaussian entries. We qualitatively unify several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based restricted isometries. Our work also implies new results in using the sparse Johnson-Lindenstrauss transform in numerical linear algebra, classical and model-based compressed sensing, manifold learning, and constrained least squares problems such as the Lasso.