Riemannian Optimization on Tensor Products of Grassmann Manifolds: Applications to Generalized Rayleigh-Quotients

摘要

We introduce a generalized Rayleigh-quotient on the tensor product ofGrassmannians enabling a unified approach to well-known optimization tasks fromdifferent areas of numerical linear algebra, such as best low-rankapproximations of tensors (data compression), geometric measures ofentanglement (quantum computing) and subspace clustering (image processing). Webriefly discuss the geometry of the constraint set, we compute the Riemanniangradient of the generalized Rayleigh-quotient, we characterize its criticalpoints and prove that they are generically non-degenerated. Moreover, we derivean explicit necessary condition for the non-degeneracy of the Hessian. Finally,we present two intrinsic methods for optimizing the generalizedRayleigh-quotient - a Newton-like and a conjugated gradient - and compare ouralgorithms tailored to the above-mentioned applications with established onesfrom the literature.