Revisiting the Principal Component Analysis: New Insights and Algorithms

Dr. Armin Eftekhari (Alan Turing Institute, London)

Frank Adams 2,

This talk consists of two parts. In the first part, we discuss an
alternative formulation of the principal component analysis (PCA)
using determinants and present the corresponding version of the
Eckart-Young-Mirsky Theorem. This alternative formulation of PCA is
known but not used in practice because it is not clear how to solve
the underlying non-convex optimisation program. Our main contribution
here is to show that this non-convex program has no spurious local
optima; it therefore behaves like a convex program and is amenable to
a variety of convex solvers. We apply a number of these solvers and
find that it often provides a competitive alternative to the state of
the art in computing principal components of data. These findings also
pave the way for entirely new approaches to sparse PCA and nonnegative
matrix factorization.

In the second part of the talk, we present MOSES, a streaming and
memory-limited algorithm for PCA in scenarios where (high-dimensional)
data is presented sequentially and limited storage is available. We
study the performance of MOSES in a deterministic and then a
stochastic setup similar to the spiked covariance model. We also find
that MOSES empirically improves over the state of the art.

This is a joint work with Raphael Hauser at Oxford University.

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