An ideal-sparse generalized moment problem reformulation for completely positive tensor decomposition exploiting maximal cliques of multi-hypergraphs

TL;DR

This paper introduces a novel reformulation of the completely positive tensor decomposition problem using ideal-sparsity and maximal cliques, leading to faster computations and new sparsity structures.

Contribution

It proposes an algorithm for generating maximal cliques, reformulates the problem into an ideal-sparse moment problem, and studies the convergence of the proposed hierarchies.

Findings

The ideal-sparse reformulation is computationally faster.

The approach exploits a new sparsity structure different from existing methods.

Numerical results confirm improved efficiency and effectiveness.

Abstract

In this paper, we consider the completely positive tensor decomposition problem with ideal-sparsity. First, we propose an algorithm to generate the maximal cliques of multi-hypergraphs associated with completely positive tensors. This also leads to a necessary condition for tensors to be completely positive. Then, the completely positive tensor decomposition problem is reformulated into an ideal-sparse generalized moment problem. It optimizes over several lower dimensional measure variables supported on the maximal cliques of a multi-hypergraph. The moment-based relaxations are applied to solve the reformulation. The convergence of this ideal-sparse moment hierarchies is studied. Numerical results show that the ideal-sparse problem is faster to compute than the original dense formulation of completely positive tensor decomposition problems. It also illustrates that the new reformulation utilizes sparsity structures that differs from the correlative and term sparsity for completely positive tensor decomposition problems.