Sequential unfolding SVD for low rank orthogonal tensor approximation

Abstract

This paper contributes to the field of N-way (N ≥ 3) tensor decompositions, which are increasingly popular in various signal processing applications. A novel PARATREE decomposition structure is introduced, accompanied with Sequential Unfolding SVD (SUSVD) algorithm. SUSVD applies a matrix SVD sequentially on the unfolded tensor, which is reshaped from the right hand basis vectors of the SVD of the previous mode. The consequent PARATREE model is related to the well known family of PARAFAC tensor decompositions, describing a tensor as a sum of rank-1 tensors. PARATREE is an efficient model to be used for orthogonal lower rank approximations, offering significant computational savings in algorithm implementations due to a hierarchical tree structure. The performance of the proposed algorithm is illustrated through an application of measurement noise suppression in wideband MIMO measurements.

A constructive arbitrary‐degree Kronecker product decomposition of tensors

“…We can then retrieve S from the 27 × 27 perfect shuffle matrix S 1 = S 2 = S 3 as S = Q T (S 3 ⊗ S 2 ⊗ S 1 ) Q, where Q is the permutation matrix in vec(H) = Q vec(H). The 27 × 27 permutation matrices P 1 = P 2 = P 3 that define a 3 × 3 × 3 Hankel tensor A are completely specified by the vector of indices i = [1,4,5,10,7,8,11,12,15,2,13,14,19,16,17,20,21,24,3,22,23,6,25,26,9,18,27], since vec(A)(i) = vec(A), where vec(A)(i) is Matlab notation to denote P 3 vec(A). If we now set P = Q T (P 3 ⊗ P 2 ⊗ P 1 ) Q then indeed P vec(H) = vec(H) is satisfied.…”

“…The PARATREE/TTr1SVD decomposition is another decomposition of a k ‐way tensor into orthogonal rank‐1 terms as described by Equation . The total number of terms in the TTr1SVD is upperbounded by and therefore depends on the ordering of the indices.…”

Symmetric tensor decomposition by an iterative eigendecomposition algorithm

“…This is the main motivation for the development of the STEROID algorithm. STEROID is an adaptation for symmetric tensors of our earlier developed Tensor Train rank-1 SVD (TTr1SVD) algorithm [1], which in turn was inspired by Tensor Trains [19], and was an independent derivation of PARATREE [24]. In contrast to the iterative methods mentioned above, the STEROID algorithm does not require an initial guess and the total number of terms in the decomposition follows readily from the execution of the algorithm.…”