The GSVD: Where are the Ellipses?, Matrix Trigonometry, and More

Abstract

© 2020 Society for Industrial and Applied Mathematics. This paper provides an advanced mathematical theory of the generalized singular value decomposition (GSVD) and its applications. We explore the geometry of the GSVD providing a long-sought-for picture which includes a horizontal and a vertical multiaxis. We further propose that the GSVD provides natural coordinates for the Grassmann manifold. This paper proves a theorem showing how the finite generalized singular values do or do not relate to the singular values of AB\dagger . We then turn to applications, arguing that this geometrical theory is natural for understanding existing applications and recognizing opportunities for new applications. In particular the generalized singular vectors play a direct and as natural a mathematical role for certain applications as the singular vectors do for the SVD. In the same way that experts on the SVD often prefer not to cast SVD problems as eigenproblems, we propose that the GSVD, often cast as a generalized eigenproblem, is perhaps best cast in its natural setting. We illustrate this theoretical approach and the natural multiaxes (with labels from technical domains) in the context of applications where the GSVD arises: Tikhonov regularization (unregularized versus regularized), genome reconstruction (humans versus yeast), signal processing (signal versus noise), and statistical analysis such as analysis of variance and discriminant analysis (between clusters versus within clusters.) With the aid of our ellipse figure, we encourage the labeling of the natural multiaxes in any GSVD problem.