Neural Networks for Principal Component Analysis: A New Loss Function Provably Yields Ordered Exact Eigenvectors

Reza Oftadeh, Jiayi Shen, Zhangyang Wang, Dylan Shell

Keywords: Principal Component Analysis, Autoencoder, Neural Network

TL;DR: A new loss function for PCA with linear autoencoders that provably yields ordered exact eigenvectors

Abstract: In this paper, we propose a new loss function for performing principal component analysis (PCA) using linear autoencoders (LAEs). Optimizing the standard L2 loss results in a decoder matrix that spans the principal subspace of the sample covariance of the data, but fails to identify the exact eigenvectors. This downside originates from an invariance that cancels out in the global map. Here, we prove that our loss function eliminates this issue, i.e. the decoder converges to the exact ordered unnormalized eigenvectors of the sample covariance matrix. For this new loss, we establish that all local minima are global optima and also show that computing the new loss (and also its gradients) has the same order of complexity as the classical loss. We report numerical results on both synthetic simulations, and a real-data PCA experiment on MNIST (i.e., a 60,000 x784 matrix), demonstrating our approach to be practically applicable and rectify previous LAEs' downsides.