An ideal-sparse generalized moment problem reformulation for completely positive tensor decomposition exploiting maximal cliques of multi-hypergraphs
Pengfei Huang, Minru Bai
TL;DR
This work targets scalable completely positive tensor decomposition by exploiting ideal-sparsity encoded via maximal cliques of a uniform multi-hypergraph. It introduces an algorithm to generate maximal cliques, provides a necessary CP condition, and reformulates the decomposition as an ideal-sparse generalized moment problem solved with sparse moment relaxations that come with convergence guarantees. The authors prove equivalence to the dense formulation and demonstrate substantial runtime gains in numerical experiments, including comparisons with TSSOS. The results show that hypergraph-based ideal sparsity captures a distinct sparsity structure and offers a promising route to scalable CP tensor analysis with potential broader applicability to sparse polynomial optimization problems.
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.