Toward a unified theory of sparse dimensionality reduction in Euclidean space

Abstract

Let $${\Phi\in\mathbb{R}^{m\times n}}$$ be a sparse Johnson–Lindenstrauss transform (Kane and Nelson in J ACM 61(1):4, 2014) with s non-zeroes per column. For a subset T of the unit sphere, $${\varepsilon\in(0,1/2)}$$ given, we study settings for m, s required to ensure $$\mathop{\mathbb{E}}_\Phi \sup_{x\in T}\left|\|\Phi x\|_2^2 - 1\right| < \varepsilon,$$ i.e. so that $${\Phi}$$ preserves the norm of every $${x\in T}$$ simultaneously and multiplicatively up to $${1+\varepsilon}$$ . 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.