SPEAKERProfessor Moody T. Chu]North Carolina State University ^

TITLEPower-Like Iteration via Polar Decomposition on Riemannian Manifold for Tensor Approximation

ABSTRACT Low rank approximation is an important subject with a wide range of applications. For a high-dimensional array, most prevailing techniques for computing the low rank approximation in the Tucker format often first assemble relevant factors into matrices and then update by turns one factor matrix at a time. In order to improve two factor matrices simultaneously, a recurring theme is an interesting system of nonlinear matrix equations over a product Stiefel manifold that need be solved at every update. The solution to the system consists of orbit varieties that are invariant under the orthogonal group action, which thus imposes challenges on its analysis. Proposed in this work is a scheme similar to the power method for subspace iterations except that the polar decomposition is used as the normalization process. The iteration can be applied to both the orbits and the cross-sections. The notion of quotient manifold is employed to factor out the effect of orbital solutions. The dynamics of the iteration is completely characterized. An isometric isomorphism between the tangent spaces of two properly identified Riemannian manifolds is established, which lends a hand to the proof of convergence. (An eclectic mix of skills from tensor analysis, Riemannian geometry, matrix theory, and optimization is used in this work. This talk is meant to be expository for general audience.)