Riemannian optimization on tensor products of Grassmann manifolds: Applications to generalized Rayleigh-quotients

TL;DR

This paper introduces a unified Riemannian optimization framework on tensor products of Grassmannians for diverse applications like tensor approximation, quantum entanglement, and image clustering, with new algorithms and theoretical insights.

Contribution

It develops a generalized Rayleigh-quotient on tensor Grassmannians, analyzes its geometry, and proposes two intrinsic optimization methods with theoretical guarantees.

Findings

Algorithms outperform existing methods in applications

Critical points are generically non-degenerate

Explicit conditions for Hessian non-degeneracy

Abstract

We introduce a generalized Rayleigh-quotient on the tensor product of Grassmannians enabling a unified approach to well-known optimization tasks from different areas of numerical linear algebra, such as best low-rank approximations of tensors (data compression), geometric measures of entanglement (quantum computing) and subspace clustering (image processing). We briefly discuss the geometry of the constraint set, we compute the Riemannian gradient of the generalized Rayleigh-quotient, we characterize its critical points and prove that they are generically non-degenerated. Moreover, we derive an explicit necessary condition for the non-degeneracy of the Hessian. Finally, we present two intrinsic methods for optimizing the generalized Rayleigh-quotient - a Newton-like and a conjugated gradient - and compare our algorithms tailored to the above-mentioned applications with established ones from the literature.