Toward a Unified Theory of Sparse Dimensionality Reduction in Euclidean Space

Let Φ∈Rm x n be a sparse Johnson-Lindenstrauss transform [52] with column sparsity s. For a subset T of the unit sphere and ε∈(0,1/2), we study settings for m,s to ensure EΦ supx∈ T |Φ x|22 - 1| < ε, i.e. so that Φ preserves the 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 randomized linear algebra, compressed sensing, manifold learning, and constrained least squares problems such as the Lasso.