SOS Tensor Decomposition: Theory and Applications
Haibin Chen, Guoyin Li, Liqun Qi

TL;DR
This paper investigates sum-of-squares tensor decompositions for structured tensors, establishing theoretical bounds, properties, and applications such as eigenvalue computation and positive definiteness testing.
Contribution
It provides new theoretical insights into SOS tensor decomposition, including bounds on SOS-rank and SOS-width, and demonstrates practical algorithms for large-scale tensor applications.
Findings
Several classes of structured tensors have SOS tensor decompositions.
Upper bounds for SOS-rank and SOS-width are established.
Applications include eigenvalue computation and positive definiteness testing.
Abstract
In this paper, we examine structured tensors which have sum-of-squares (SOS) tensor decomposition, and study the SOS-rank of SOS tensor decomposition. We first show that several classes of even order symmetric structured tensors available in the literature have SOS tensor decomposition. These include positive Cauchy tensors, weakly diagonally dominated tensors, -tensors, double -tensors, quasi-double -tensors, -tensors, -tensors, absolute tensors of positive semi-definite -tensors and extended -tensors. We also examine the SOS-rank of SOS tensor decomposition and the SOS-width for SOS tensor cones. The SOS-rank provides the minimal number of squares in the SOS tensor decomposition, and, for a given SOS tensor cone, its SOS-width is the maximum possible SOS-rank for all the tensors in this cone. We first deduce an upper bound for general tensors that have SOS…
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Taxonomy
TopicsTensor decomposition and applications · Parallel Computing and Optimization Techniques · Electromagnetic Scattering and Analysis
