Learning-Based Low-Rank Approximations

TL;DR

This paper presents a learning-based approach to low-rank matrix approximation that replaces random sketching matrices with learned ones, significantly improving approximation accuracy across various datasets.

Contribution

It introduces a novel learning-based algorithm for low-rank approximation that optimizes sketching matrices using training data, outperforming traditional random methods.

Findings

Learned sketch matrices reduce approximation error by up to tenfold.

Mixed matrices with trained and random rows maintain performance and guarantees.

Theoretical analysis provides approximation and generalization bounds for the learning approach.

Abstract

We introduce a "learning-based" algorithm for the low-rank decomposition problem: given an $n \times d$ matrix $A$, and a parameter $k$, compute a rank-$k$ matrix $A'$ that minimizes the approximation loss $\|A-A'\|_F$. The algorithm uses a training set of input matrices in order to optimize its performance. Specifically, some of the most efficient approximate algorithms for computing low-rank approximations proceed by computing a projection $SA$, where $S$ is a sparse random $m \times n$ "sketching matrix", and then performing the singular value decomposition of $SA$. We show how to replace the random matrix $S$ with a "learned" matrix of the same sparsity to reduce the error. Our experiments show that, for multiple types of data sets, a learned sketch matrix can substantially reduce the approximation loss compared to a random matrix $S$, sometimes by one order of magnitude. We also study mixed matrices where only some of the rows are trained and the remaining ones are random, and show that matrices still offer improved performance while retaining worst-case guarantees. Finally, to understand the theoretical aspects of our approach, we study the special case of $m=1$. In particular, we give an approximation algorithm for minimizing the empirical loss, with approximation factor depending on the stable rank of matrices in the training set. We also show generalization bounds for the sketch matrix learning problem.