On JEVD of semi-definite positive matrices and CPD of nonnegative tensors

Abstract

In this paper, we mainly address the problem of Joint EigenValue Decomposition (JEVD) subject to nonnegative constraints on the eigenvalues of the matrices to be diagonalized. An efficient method based on the Alternating Direction Method of Multipliers (ADMM) is designed. ADMM provides an elegant approach for handling nonnegativity constraints, while taking advantage of the structure of the objective function. Numerical tests on simulated matrices show the interest of the proposed method for low Signal-to-Noise Ratio (SNR) values when the similarity transformation matrix is ill-conditioned. The ADMM was recently used for the Canonical Polyadic Decomposition (CPD) of nonnegative tensors leading to the ADMoM algorithm. We show through computer results that DIAG+, a semi-algebraic CPD method using our ADMM-based JEVD+ algorithm, will give a better estimation of factors than ADMoM in the presence of swamps. DIAG+ also appears to be less time-consuming than ADMoM when low-rank tensors of high dimensions are considered.