PCA by Determinant Optimization has no Spurious Local Optima

Abstract

Principal component analysis (PCA) is an indispensable tool in many learning tasks that finds the best linear representation for data. Classically, principal components of a dataset are interpreted as the directions that preserve most of its "energy", an interpretation that is theoretically underpinned by the celebrated Eckart-Young-Mirsky Theorem. There are yet other ways of interpreting PCA that are rarely exploited in practice, largely because it is not known how to reliably solve the corresponding non-convex optimisation programs. In this paper, we consider one such interpretation of principal components as the directions that preserve most of the "volume" of the dataset. Our main contribution is a theorem that shows that the corresponding non-convex program has no spurious local optima. We apply a number of solvers for empirical confirmation.