The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Jean Bourgain, Sjoerd Dirksen, Jelani Nelson. 2015. "Toward a unified theory of sparse dimensionality reduction in Euclidean space." Abstract. Let Φ ∈ R m×n be a sparse Johnson-Lindenstrauss transform [KN14] with s non-zeroes per column. For a subset T of the unit sphere, ε ∈ (0, 1/2) given, we study settings for m, s required to ensure i.e. so that Φ preserves

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... norm of every x ∈ T simultaneously and multiplicatively up to 1 + ε. 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 Φ 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.