Lower Memory Oblivious (Tensor) Subspace Embeddings with Fewer Random Bits: Modewise Methods for Least Squares

TL;DR

This paper introduces faster, more storage-efficient modewise Johnson-Lindenstrauss subspace embeddings for large vectors and tensors, with improved space complexity and fewer random bits needed, applicable to low-rank tensor and subspace embeddings.

Contribution

Proposes new modewise JL embeddings that are faster to generate, require fewer random bits, and are applicable to tensor and subspace embeddings with improved space complexity bounds.

Findings

New modewise JL embeddings are faster and easier to store.

Embeddings achieve small distortions with fewer random bits.

Numerical experiments confirm effectiveness in compression and least squares.

Abstract

In this paper new general modewise Johnson-Lindenstrauss (JL) subspace embeddings are proposed that are both considerably faster to generate and easier to store than traditional JL embeddings when working with extremely large vectors and/or tensors. Corresponding embedding results are then proven for two different types of low-dimensional (tensor) subspaces. The first of these new subspace embedding results produces improved space complexity bounds for embeddings of rank-$r$ tensors whose CP decompositions are contained in the span of a fixed (but unknown) set of $r$ rank-one basis tensors. In the traditional vector setting this first result yields new and very general near-optimal oblivious subspace embedding constructions that require fewer random bits to generate than standard JL embeddings when embedding subspaces of $\mathbb{C}^N$ spanned by basis vectors with special Kronecker structure. The second result proven herein provides new fast JL embeddings of arbitrary $r$-dimensional subspaces $\mathcal{S} \subset \mathbb{C}^N$ which also require fewer random bits (and so are easier to store - i.e., require less space) than standard fast JL embedding methods in order to achieve small $\epsilon$-distortions. These new oblivious subspace embedding results work by $(i)$ effectively folding any given vector in $\mathcal{S}$ into a (not necessarily low-rank) tensor, and then $(ii)$ embedding the resulting tensor into $\mathbb{C}^m$ for $m \leq C r \log^c(N) / \epsilon^2$. Applications related to compression and fast compressed least squares solution methods are also considered, including those used for fitting low-rank CP decompositions, and the proposed JL embedding results are shown to work well numerically in both settings.