PCA by Determinant Optimization has no Spurious Local Optima

Raphael Hauser (University of Oxford); Armin Eftekhari (Alan Turing Institute); Heinrich Matzinger (Georgia Institute of Technology)

Principal Component Analysis (PCA) finds the best linear representation for data and is an indispensable tool in many learning tasks. 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, and is therefore amenable to many convex solvers. We also confirm our findings numerically.

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