An Orthogonal Equivalence Theorem for Third Order Tensors

TL;DR

This paper establishes an orthogonal invariance property for third order tensors under T-product operations, defining tensor ranks and singular values based on a specific Fourier-based mapping, with implications for tensor approximation.

Contribution

It introduces an orthogonal equivalence theorem for third order tensors, defining invariant tensor ranks and singular values via the Kilmer-Martin mapping.

Findings

Tensor tubal rank and T-rank are invariant under T-product with orthogonal tensors.

Properties of singular values and T-rank are analyzed.

Best T-rank one approximation properties are discussed.

Abstract

In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.