We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus
of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturing structure in discrete probability distributions, this behavior also renders the direct application of standard PGM tools impossible. We aim to bridge this gap by introducing a hybrid PGM-TN formalism that integrates quantum-like correlations into PGM models in a principled manner, using the physically-motivated concept of decoherence. We first prove that applying decoherence to the entirety of a BM model converts it into a discrete UGM, and conversely, that any subgraph of a discrete UGM can be represented as a decohered BM. This method allows a broad family of probabilistic TN models to be encoded as partially decohered BMs, a fact we leverage to combine the representational strengths of both model families. We experimentally verify the performance of such hybrid models in a sequential modeling task, and identify promising uses of our method within the context of existing applications of graphical models.
In [7], a notion of constant scalar curvature metrics on piecewise flat manifolds is defined. Such metrics are candidates for canonical metrics on discrete manifolds. In this paper, we define a class of vertex transitive metrics on certain triangulat
ions of $mathbb{S}^3$; namely, the boundary complexes of cyclic polytopes. We use combinatorial properties of cyclic polytopes to show that, for any number of vertices, these metrics have constant scalar curvature.
