Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

TL;DR

This paper introduces ideal-sparsity for the generalized moment problem, enabling more efficient and often tighter bounds in matrix factorization rank problems by reformulating the problem using graph-based sparsity without chordality constraints.

Contribution

It proposes a novel ideal-sparsity approach that improves the efficiency and tightness of moment relaxations in matrix factorization rank bounds, surpassing traditional correlative sparsity methods.

Findings

Ideal-sparse hierarchies provide equal or tighter bounds than dense hierarchies.

The ideal-sparse approach is computationally faster than dense methods.

No chordal graph assumption is needed for ideal-sparsity.

Abstract

We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.