Robust Differentiable SVD

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.

IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI) PREPRINT 2021

References in corpus (9)

• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• Spectral Normalization for Generative Adversarial Networks
• Perceptual Losses for Real-Time Style Transfer and Super-Resolution
• Texture Networks: Feed-forward Synthesis of Textures and Stylized Images
• Optimal whitening and decorrelation
• Universal Style Transfer via Feature Transforms
• Second-order Convolutional Neural Networks
• Backpropagation-Friendly Eigendecomposition
• Whitening and Coloring batch transform for GANs

Cited by in corpus (6)

• On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition
• Learning Curves for Decision Making in Supervised Machine Learning: A Survey
• Grouping-matrix based Graph Pooling with Adaptive Number of Clusters
• Iterative Low-rank Network for Hyperspectral Image Denoising
• JotlasNet: Joint Tensor Low-Rank and Attention-based Sparse Unrolling Network for Accelerating Dynamic MRI
• GaussED: A Probabilistic Programming Language for Sequential Experimental Design