CONVERGENGE ANALYSIS ON SVD-BASED ALGORITHMS FOR TENSOR LOW RANK APPROXIMATIONS

Abstract

This thesis is to study a few problems on tensor decompositions and approximations in real space. Among other things, we revisit the classical problem of finding the best rank-R CANDECOMP/PARAFAC(CP) approximation with diff erent cases R > 1 and R = 1 respectively.

Unlike the rank-1 approximation is theoretically guaranteed to have a global optimum, general rank-R approximation (R > 1) may not exist in real space. So, there should be an added orthogonality requirement to ensure the existence of R > 1 case. In contrast to the conventional approach by the so-called alternating least squares (ALS) method that works to adjust one factor a time, proposed SVD-based algorithms improve two factors simultaneously. Convergence analysis both for the generalized Rayleigh quotient and the iterates themselves is the main contribution of this thesis. In addition, we also study the convergence property of a general framework called alternating direction methods (ADM) in this thesis.