An NPDo Approach for Tensor Block-Diagonalization

TL;DR

This paper introduces an NPDo method for Partial Tensor Block-Diagonalization, optimizing tensor block structures via orthonormal transformations with proven convergence and demonstrated efficiency.

Contribution

It proposes a novel NPDo approach for tensor block-diagonalization that generalizes Tucker decomposition and tensor SVD, with guaranteed convergence and practical effectiveness.

Findings

NPDo approach converges globally to a stationary point.

Numerical experiments show the method's efficiency.

The approach generalizes Tucker decomposition and tensor SVD.

Abstract

This paper is concerned with Partial Tensor Block-Diagonalization of a multiway tensor by orthonormal matrices so that the extracted block-diagonal part optimally represents the tensor. The basic idea is to maximize the block-diagonal part via the tensor's mode-multiplications by orthonormal matrices. For that reason, it will be referred to Principal Tensor Block-Diagonalization (PTBD), which contains the Tucker decomposition (TD) of a tensor as a special case with just one block. Also as a special case is the approximate dominant tensor SVD in which each block-size is 1-by-1. An NPDo approach is proposed to optimize the block-diagonal part for computing \ptbd. It is shown the NPDo approach combined with Gauss-Seidel-type updating is globally convergent to a stationary point while the objective increases monotonically. Numerical experiments are presented to illustrate the efficiency of the NPDo approach.