Nonlinear least squares iterative solver developed earlier by the author is applied for numerical solution of special system of multilinear equations arising in the problem of canonical tensor decomposition. The proposed algorithm is based on easily parallelizable computational kernels such as matrix-vector multiplications and elementary vector operations and therefore has a potential for a quite efficient implementation on modern high-performance computers. The results of numerical testing presented for certain examples of large scale dense and medium size sparse 3D tensors found in existing literature seem very competitive with respect to computational costs involved.

Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition

An improved version of derivative-free nonlinear least squares iterative solver developed earlier by the author is described. First, we apply a regularization technique to stabilize the evaluation of search directions similar to the one used in the Levenberg-Marquardt methods. Second, we propose several modified designs for the rectangular preconditioning matrix, in particular a sparse adaptive techniques avoiding the use of pseudorandom sequences. The resulting algorithm is based on easily parallelizable computational kernels such as dense matrix factorizations and elementary vector operations thus having a potential for an efficient implementation on modern high-performance computers. Numerical results are presented for several standard test problems as well as for some special complex-valued cases to demonstrate the effectiveness of the proposed improvements to method. 

Igor E. Kaporin

Papers

Nonlinear Least Squares Solver for Evaluating Canonical Tensor Decomposition

A Derivative-Free Nonlinear Least Squares Solver