Rank properties and computational methods for orthogonal tensor decompositions

TL;DR

This paper investigates properties of orthogonal tensor rank, proves lower semicontinuity, and introduces an augmented Lagrangian-based algorithm with orthogonalization for improved orthogonal tensor decomposition.

Contribution

It presents new theoretical insights into orthogonal tensor rank and develops a novel algorithm with orthogonalization for better decomposition accuracy.

Findings

Subtensors can have higher orthogonal rank than the original tensor

The orthogonal rank is lower semicontinuous, ensuring low-rank approximations

The proposed method outperforms existing techniques in approximation error

Abstract

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and prove the lower semicontinuity of orthogonal rank. The lower semicontinuity guarantees the existence of low orthogonal rank approximation. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method and guarantee the orthogonality by a novel orthogonalization procedure. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.