The low-rank approximation of fourth-order partial-symmetric and conjugate partial-symmetric tensor

TL;DR

This paper introduces a novel orthogonal matrix outer product decomposition for fourth-order conjugate partial-symmetric tensors, enabling exact recovery via a greedy algorithm and providing bounds on tensor rank for low-rank tensor completion.

Contribution

It presents a new matrix decomposition for CPS tensors, establishes rank bounds, and demonstrates the effectiveness of the greedy SROA algorithm for exact tensor recovery.

Findings

Exact recovery of CPS tensor decomposition using SROA

Bound on CP rank of CPS tensor via matrix rank

Application to low-rank tensor completion

Abstract

We present an orthogonal matrix outer product decomposition for the fourth-order conjugate partial-symmetric (CPS) tensor and show that the greedy successive rank-one approximation (SROA) algorithm can recover this decomposition exactly. Based on this matrix decomposition, the CP rank of CPS tensor can be bounded by the matrix rank, which can be applied to low rank tensor completion. Additionally, we give the rank-one equivalence property for the CPS tensor based on the SVD of matrix, which can be applied on the rank-one approximation for CPS tensors.