Abstract—We present an efficient global optimization algorithm for exponential family principal component analysis (PCA) and associated low-rank matrix factorization problems. Exponential family PCA has been shown to improve the results of standard PCA on non-Gaussian data. Unfortunately, the widespread use of exponential family PCA has been hampered by the existence of only local optimization procedures. The prevailing assumption has been that the non-convexity of the problem prevents an efficient global optimization approach from being developed. Fortunately, this pessimism is unfounded. We present a reformulation of the underlying optimization problem that preserves the identity of the global solution while admitting an efficient optimization procedure. The algorithm we develop involves only a sub-gradient optimization of a convex objective plus associated eigenvector computations. (No general purpose semidefinite programming solver is required.) The lowrank constraint is exactly preserved, while the method can be kernelized through a consistent approximation to admit a fixed non-linearity. We demonstrate improved solution quality with the global solver, and also add to the evidence that exponential family PCA produces superior results to standard PCA on non-Gaussian data. I.

This survey paper is intended for the graduate students and researchers who are interested in Operations Research, have solid understanding of linear optimization but are not familiar with Semidefinite Programming (SDP). Here, I provide a very gentle introduction to SDP, some entry points for further look into the SDP literature, and brief introductions to some selected well-known applications which may be attractive to such audience and in turn motivate them to learn more about semidefinite optimization.