# Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework

arXiv: https://arxiv.org/abs/2106.15666 · HTML: https://mlsanity.com/paper/2106.15666/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 29 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [BERT has a Mouth, and It Must Speak: BERT as a Markov Random Field Language Model](https://mlsanity.com/paper/1902.04094/) (2019)
- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design](https://mlsanity.com/paper/1704.01552/) (2017)
- [Enabling Multi-programming Mechanism for Quantum Computing in the NISQ Era](https://mlsanity.com/paper/2102.05321/) (2021)
- [Modeling Sequences with Quantum States: A Look Under the Hood](https://mlsanity.com/paper/1910.07425/) (2019)
- [Duality of Graphical Models and Tensor Networks](https://mlsanity.com/paper/1710.01437/) (2017)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Probabilistic Modeling with Matrix Product States](https://mlsanity.com/paper/1902.06888/) (2019)
- [Tensor network language model](https://mlsanity.com/paper/1710.10248/) (2017)
- [Quantum Tensor Networks, Stochastic Processes, and Weighted Automata](https://mlsanity.com/paper/2010.10653/) (2020)
- [Language as a matrix product state](https://mlsanity.com/paper/1711.01416/) (2017)
- [An efficient quantum algorithm for generative machine learning](https://mlsanity.com/paper/1711.02038/) (2017)

## Built on by (most cited first, 5 of 5)

- [Tensor tree learns hidden relational structures in data to construct generative models](https://mlsanity.com/paper/2408.10669/) (2024)
- [Patch-based Medical Image Segmentation using Matrix Product State Tensor Networks](https://mlsanity.com/paper/2109.07138/) (2021)
- [Tensor Network Estimation of Distribution Algorithms](https://mlsanity.com/paper/2412.19780/) (2024)
- [Efficient Probabilistic Tensor Networks](https://mlsanity.com/paper/2510.00382/) (2025)
- [On Probabilistic Inference Through Parametric Tensor Decomposition in Base Tensor Networks](https://mlsanity.com/paper/2609.23774/) (2026)

## Abstract (arXiv)

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.