Fast Differentiable Matrix Square Root

TL;DR

This paper introduces two efficient methods for computing the differentiable matrix square root, significantly speeding up the process compared to traditional SVD and Newton-Schulz approaches, with applications in vision tasks.

Contribution

Proposes two novel, faster methods using Matrix Taylor Polynomial and Matrix Pade Approximants for differentiable matrix square root computation.

Findings

Methods outperform SVD and Newton-Schulz in speed

Achieve competitive or better accuracy in vision tasks

Applicable to batch normalization and vision transformers

Abstract

Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor Polynomial (MTP), and the other method is to use Matrix Pad\'e Approximants (MPA). The backward gradient is computed by iteratively solving the continuous-time Lyapunov equation using the matrix sign function. Both methods yield considerable speed-up compared with the SVD or the Newton-Schulz iteration. Experimental results on the de-correlated batch normalization and second-order vision transformer demonstrate that our methods can also achieve competitive and even slightly better performances. The code is available at \href{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}{https://github.com/KingJamesSong/FastDifferentiableMatSqrt}.