Sparse factorization of square matrices with application to neural attention modeling

Abstract

Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize the square matrix into a number matrices of much lower ranks. However, the low-rank constraint is a performance bottleneck if the approximated matrix is intrinsically high-rank or close to full rank. In this paper, we propose to approximate a large square matrix with a product of sparse full-rank matrices. In the approximation, our method needs only N(logN)² non-zero numbers for an N×N full matrix. Our new method is especially useful for scalable neural attention modeling. Different from the conventional scaled dot-product attention methods, we train neural networks to map input data to the non-zero entries of the factorizing matrices. The sparse factorization method is tested for various square matrices, and the experimental results demonstrate that our method gives a better approximation when the approximated matrix is sparse and high-rank. As an attention module, our new method defeats Transformer and its several variants for long sequences in synthetic data sets and in the Long Range Arena benchmarks. Our code is publicly available².

Keywords: Attention modeling; Matrix factorization; Neural networks; Sparse.