  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}.
