  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.
