TY - JOUR

T1 - Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

AU - Korda, Milan

AU - Laurent, Monique

AU - Magron, Victor

AU - Steenkamp, Andries

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2024/5

Y1 - 2024/5

N2 - We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. In this setting, one optimizes over a measure restricted to be supported on the variety of an ideal generated by quadratic bilinear monomials. We show that this restriction 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 corresponding to the quadratic bilinear constraints. 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.

AB - We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. In this setting, one optimizes over a measure restricted to be supported on the variety of an ideal generated by quadratic bilinear monomials. We show that this restriction 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 corresponding to the quadratic bilinear constraints. 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.

KW - Completely positive rank

KW - Generalized moment problem

KW - Matrix factorization rank

KW - Nonnegative rank

KW - Polynomial optimization

KW - Semidefinite programming

KW - Sparsity

UR - http://www.scopus.com/inward/record.url?scp=85164161074&partnerID=8YFLogxK

U2 - 10.1007/s10107-023-01993-x

DO - 10.1007/s10107-023-01993-x

M3 - Article

SN - 0025-5610

VL - 205

SP - 703

EP - 744

JO - Mathematical Programming

JF - Mathematical Programming

IS - 1-2

ER -