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.
Exploiting Chordal Sparsity for Globally Optimal Estimation with Factor Graphs
College of Computing, Georgia Institute of Technology（佐治亚理工学院计算机学院）；Robotics Institute, University of Toronto（多伦多大学机器人研究所）；Department of Mechanical Engineering, Carnegie Mellon University（卡内基梅隆大学机械工程系）
Dynamic Topic Analysis in Academic Journals using Convex Non-negative Matrix Factorization Method
Northeastern University（东北大学）；China Criminal Police University（中国刑事警察学院）；Carnegie Mellon University（卡内基梅隆大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
