Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework

Abstract

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.

arXiv comment18 pages, 11 figures

References in corpus (13)

• Adam: A Method for Stochastic Optimization
• Quantum Computing in the NISQ era and beyond
• Area laws for the entanglement entropy - a review
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Are biological systems poised at criticality?
• A generative modeling approach for benchmarking and training shallow quantum circuits
• Unsupervised Generative Modeling Using Matrix Product States
• Perfect Sampling with Unitary Tensor Networks
• Enhancing Generative Models via Quantum Correlations
• Probabilistic Modeling with Matrix Product States
• An efficient quantum algorithm for generative machine learning
• Tensor network language model
• Language as a matrix product state

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