On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of $\mathbb{R}^N$ with Boundary

Abstract

Let M be a smooth d-dimensional submanifold of R N 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 ∈ R m×N will serve as a bi-Lipschitz function A : M → R m 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 R N 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 R N (with d √ 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(log c N )-dimensional submanifolds of R N which enjoy O(N log(log N ))-time matrix vector multiplications.

Characterizing unit spheres in Euclidean spaces via reach and volume

“…The scale invariant ratio vol(M )/τ (M ) k arises in estimates for the number of metric balls in R n needed to cover M when the balls are required to be centered in M and to have equal radii (see, e.g., [1,4,8,7]). These estimates have applications in compressive sensing and mathematical data science where they are combined with probabilistic methods to estimate, e.g., the smallest dimension m < n such that M , equipped with the restriction of the metric ρ, admits a bilipshitz map to R m with bilipshitz constants close to 1.…”