Convergence Properties of an Isospectral Saddle-Point Dynamics

Abstract

Abstract We introduce a saddle-point dynamics, evolving isospectrally on the product of two homogeneous space of symmetric matrices, each orthogonally equivalent to a positive diagonal matrix. This dynamics is a saddle-point version of the so-called double-bracket flow and corresponds to a minimax version of the Frobenius norm minimization problem considered in (Brockett, 1989, 1991). We study the set of equilibria of this dynamics and prove its asymptotic convergence properties under suitable conditions. We also demonstrate how, under appropriate conditions, this dynamics provides a learning strategy for a two-player zero-sum game, where the players strategy set is the set of permutations.