On Fast Johnson-Lindernstrauss Embeddings of Compact Submanifolds of
$\mathbb{R}^N$ with Boundary
- URL: http://arxiv.org/abs/2110.04193v1
- Date: Fri, 8 Oct 2021 15:27:52 GMT
- Title: On Fast Johnson-Lindernstrauss Embeddings of Compact Submanifolds of
$\mathbb{R}^N$ with Boundary
- Authors: Mark A. Iwen, Benjamin Schmidt, Arman Tavakoli
- Abstract summary: We consider the probability that a random matrix $A in mathbbRm times N$ will serve as a bi-Lipschitz function $A: mathcalM rightarrow mathbbRm$ with bi-Lipschitz constants close to one.
We present a new class of highly structured distributions for embedding sufficiently low-dimensional submanifolds of $mathbbRN$.
- Score: 0.4125187280299246
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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.
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