On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of $\mathbb{R}^N$ with Boundary
This work addresses the challenge of efficient dimensionality reduction for data on manifolds with boundaries, relevant for machine learning and data compression, though it is incremental by extending prior embedding results.
The paper tackles the problem of embedding compact submanifolds with boundary in Euclidean space into lower dimensions with small distortion, generalizing prior results to include boundaries and improving lower bounds on embedding dimensions. It introduces a new class of structured matrix distributions that outperform previous methods for low-dimensional submanifolds, achieving efficient matrix-vector multiplication in O(N log(log N)) time for O(log^c N)-dimensional cases.
Let $\mathcal{M}$ be a smooth $d$-dimensional submanifold of $\mathbb{R}^N$ with boundary that's equipped with the Euclidean (chordal) metric, and choose $m \leq N$. In this paper we consider the probability that a random matrix $A \in \mathbb{R}^{m \times N}$ will serve as a bi-Lipschitz function $A: \mathcal{M} \rightarrow \mathbb{R}^m$ with bi-Lipschitz constants close to one for three different types of distributions on the $m \times N$ 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 $\mathbb{R}^N$ 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 $\mathbb{R}^N$ (with $d \lesssim \sqrt{N}$) 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 $\mathcal{O}(\log^c N)$-dimensional submanifolds of $\mathbb{R}^N$ which enjoy $\mathcal{O}(N \log (\log N))$-time matrix vector multiplications.