Toward a Unified Theory of Sparse Dimensionality Reduction in Euclidean Space Jean Bourgain and Jelani Nelson

Let Φ ∈ Rm×n be a sparse Johnson-Lindenstrauss transform [KN] with s non-zeroes per column. For T a subset of the unit sphere, ε ∈ (0, 1/2) given, we study settings for m, s required to ensure E Φ sup x∈T ∣∣‖Φx‖22 − 1∣∣ < ε, i.e. so that Φ preserves the norm of every x ∈ T simultaneously and multiplicatively up to 1 + ε. In particular, our most general theorem shows that it suffices to set m = Ω̃(γ 2(T ) + 1) and s = Ω̃(1) as long as s,m additionally satisfy a certain tradeoff condition that is governed by the geometry of T (and as we show for several examples of interest, is easy to verify). Here γ2 is Talagrand’s functional [Tal05], and we write f = Ω̃(g) to mean f ≥ Cg(ε−1 logn) for some constants C, c > 0. Our result can be seen as an extension to sparse Φ of the works [KM05, Gor88, MPTJ07] which were concerned with dense Φ having i.i.d. (sub)gaussian entries. Our work introduces a theory that qualitatively unifies several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based methods for obtaining matrices satisfying the restricted isometry property.