A semi-algebraic framework for approximate CP decompositions via joint matrix diagonalization and generalized unfoldings

Abstract

The Canonical Polyadic (CP) decomposition of R-way arrays is a powerful tool in multilinear algebra. Algorithms to compute an approximate CP decomposition from noisy observations are often based on Alternating Least Squares (ALS) which may require a large number of iterations to converge. To avoid this drawback we investigate semi-algebraic approaches that algebraically reformulate the CP decomposition into a set of simultaneous matrix diagonalization (SMD) problems. In particular, we propose a SEmi-algebraic framework for approximate CP decompositions via SImultaneous matrix diagonalization (SMD) and generalized unfoldings (SECSI-GU). SECSI-GU combines the benefits of two existing semi-algebraic approaches based on SMDs: the SECSI framework which selects the model estimate from multiple candidates obtained by solving multiple SMDs and the “Semi-Algebraic Tensor Decomposition” (SALT) algorithm which considers a “generalized” unfolding of the tensor in order to enhance the identifiability for tensors with R > 3 dimensions. The resulting SECSI-GU framework offers a large number of degrees of freedom to flexibly adapt the performance-complexity trade-off. As we show in numerical simulations, it outperforms SECSI and SALT for tensors with R > 3 dimensions.