Robust Differentiable SVD

TL;DR

This paper introduces a new differentiable SVD method that provides more stable and accurate gradients for eigendecomposition in deep networks, improving tasks like image classification and style transfer.

Contribution

It presents a theoretically grounded Taylor expansion approach to compute SVD gradients, avoiding iterative errors of previous methods.

Findings

More accurate gradients improve deep network training stability.

Enhanced eigendecomposition benefits image classification and style transfer.

The method outperforms previous approaches in stability and accuracy.

Abstract

Eigendecomposition of symmetric matrices is at the heart of many computer vision algorithms. However, the derivatives of the eigenvectors tend to be numerically unstable, whether using the SVD to compute them analytically or using the Power Iteration (PI) method to approximate them. This instability arises in the presence of eigenvalues that are close to each other. This makes integrating eigendecomposition into deep networks difficult and often results in poor convergence, particularly when dealing with large matrices. While this can be mitigated by partitioning the data into small arbitrary groups, doing so has no theoretical basis and makes it impossible to exploit the full power of eigendecomposition. In previous work, we mitigated this using SVD during the forward pass and PI to compute the gradients during the backward pass. However, the iterative deflation procedure required to compute multiple eigenvectors using PI tends to accumulate errors and yield inaccurate gradients. Here, we show that the Taylor expansion of the SVD gradient is theoretically equivalent to the gradient obtained using PI without relying in practice on an iterative process and thus yields more accurate gradients. We demonstrate the benefits of this increased accuracy for image classification and style transfer.