On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of ℝN with Boundary

Let M be a smooth d-dimensional submanifold of ℝN with boundary that's equipped with the Euclidean (chordal) metric, and choose m ≤ N. In this paper we consider the probability that a random matrix A ∈ ℝm × N will serve as a bi-Lipschitz function A: M rightarrow ℝm with bi-Lipschitz constants close to one for three different types of distributions on the m × N matrices A, including two whose realizations are guaranteed to have fast matrix-vector multiplies. In doing so we generalize prior randomized metric space embedding results of this type for submanifolds of ℝN by allowing for the presence of boundary while also retaining, and in some cases improving, prior lower bounds on the achievable embedding dimensions m for which one can expect small distortion with high probability. In particular, motivated by recent modewise embedding constructions for tensor data, herein we present a new class of highly structured distributions on matrices which outperform prior structured matrix distributions for embedding sufficiently low-dimensional submanifolds of ℝN (with d lesssim √N) with respect to both achievable embedding dimension, and computationally efficient realizations. As a consequence we are able to present, for example, a general new class of Johnson-Lindenstrauss embedding matrices for O(logc N)-dimensional submanifolds of ℝN which enjoy O(N log (log N))-time matrix vector multiplications.