Alternating mahalanobis distance minimization for accurate and well-conditioned cp decomposition

Canonical polyadic decomposition (CPD) is prevalent in chemometrics, signal processing, data mining, and many more fields. While many algorithms have been proposed to compute the CPD, alternating least squares (ALS) remains one of the most widely used algorithms for computing the decomposition. Recent works have introduced the notion of eigenvalues and singular values of a tensor and explored applications of eigenvectors and singular vectors in signal processing, data analytics, and various other fields. We introduce a new formulation for deriving singular values and vectors of a tensor by considering the critical points of a function differently from previous works. Computing these critical points in an alternating manner motivates an alternating optimization algorithm which corresponds to the ALS algorithm in the matrix case. However, for tensors with order greater than or equal to 3, it minimizes an objective function which is different from the commonly used least squares loss. Alternating optimization of this new objective leads to simple updates to the factor matrices with the same asymptotic computational cost as ALS. We show that a subsweep of this algorithm can achieve a superlinear convergence rate for exact CPD when the known rank is not larger than the mode lengths of the input tensor. We verify our theoretical arguments experimentally. We then view the algorithm as optimizing a Mahalanobis distance with respect to each factor with the ground metric dependent on the other factors. This perspective allows us to generalize our approach to interpolate between updates corresponding to the ALS and the new algorithm to manage the tradeoff between stability and fitness of the decomposition. Our experimental results show that for approximating synthetic and real-world tensors, this algorithm and its variants converge to a better conditioned decomposition with comparable and sometimes better fitness as compared to the ALS algorithm.