Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices II: Computation and Applications

Abstract

The extended Davis-Kahan theorem makes use of polynomial matrix transformations to produce bounds at least as tight as the standard Davis-Kahan theorem. The optimization problem of finding transformation parameters resulting in optimal bounds from the extended Davis-Kahan theorem is presented for affine transformations. It is demonstrated how globally optimal bound values can be computed automatically using fractional programming theory. Two different solution approaches, the Charnes-Cooper transformation and Dinkelbach's algorithm are reviewed. Our implementation of the extended Davis--Kahan theorem is used to calculate bound values in three significant examples. First, a pairwise comparison is made of the spaces spanned by the eigenvectors of the graph shift operator matrices corresponding to different stochastic block model graphs. Second our bound is calculated on the distance of the spaces spanned by eigenvectors of the graph shift operators and their corresponding generating matrices in the stochastic blockmodel, and, third, on the sample and population covariance matrices in a spiked covariance model. Our extended bound values, using affine transformations, not only outperform the standard Davis-Kahan bounds in all examples where both theorems apply, but also demonstrate good performance in several cases where the standard Davis-Kahan theorem cannot be used.