Decomposing tensors via rank-one approximations

TL;DR

This paper investigates a method for tensor decomposition using successive rank-one approximations, analyzing conditions for validity and the structure of the tensor variety involved.

Contribution

It introduces a new perspective on tensor decomposition via successive rank-one approximations and characterizes when such decompositions are order-independent.

Findings

Decomposition validity depends on orthogonality in at least two factors.

The variety of tensors decomposable by this method is larger than that of odeco tensors.

Order independence occurs if and only if all singular vectors are orthogonal in at least two factors.

Abstract

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors.