🤖 AI Summary
This work proposes a model-class-agnostic structural learning method to identify latent probabilistic structures directly from empirical probability tensors. Leveraging the vanishing binomials of toric models as algebraic signatures, the approach introduces minimal invariant constraints (MICs) as fundamental units that generalize classical notions of independence. By exploiting the correspondence between ideals and varieties, the method performs signature matching within an enumerable class of Kronecker-stack configuration matrices. This study represents the first systematic application of algebraic statistics to structure discovery in computational linguistics, demonstrating effectiveness on both synthetic and large-scale real-world linguistic data. The identified rank-one structures correspond to interpretable word sets, thereby establishing a novel pathway for applying algebraic statistics in this domain.
📝 Abstract
Algebraic statistics characterizes statistical models through polynomial constraints, but it has mainly been used for analytically specified model classes. This paper studies the inverse problem: identifying probabilistic structure from vanishing binomials observed in empirical probability tensors. We treat the vanishing binomials of a toric model as its algebraic signature, and turn the ideal-variety correspondence of algebraic statistics into an operational procedure for structural learning that identifies a model by signature matching without parameter estimation. By restricting attention to a computationally tractable class of configuration matrices, which we call {\it the Kronecker-stack class}, we make these signatures explicitly enumerable. Within this class we define minimum invariant constraint (MIC) as the atomic unit characterizing each signature and generalizing the notion of independence. We tested this approach employing MICs on synthetic data as well as on corpus-scale real language data. The results suggested the utility of the method, revealing that the identified rank-one structures correspond to interpretable sets of words. These results open up a new avenue for applying algebraic statistics to computational linguistics.