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# An Efficient Low-Rank Tensors Representation for Algorithms in Complex Probabilistic Graphical Models

*Proceedings of the 10th International Conference on Probabilistic Graphical Models*, PMLR 138:173-184, 2020.

#### Abstract

Probabilistic Graphical Models form a class of compact
representations of high-dimensional probability distributions by
decomposing these distributions in a set of multivariate factors
(potentials). Every exact algorithm (for probabilistic inference, MAP,
etc.) operates on a specific representation of these potentials. However
complex probabilistic models often lead to very large potentials which
dramatically impact both the space and time complexities of these
algorithms and which can make inference in complex models intractable.
In this paper we propose a new approach based on low-rank tensor
representation to approximate and operate with these potentials. The
low-rank tensor representation is used for the approximation of
potentials with controlled precision and an important reduction in the
number of parameters. Every operator used in such algorithms
(multiplication, addition, projection, etc.) can be defined within this
representation, leading to an approximation framework where every
algorithm for PGMs can be easily implemented. As an instance of this
framework, we present a classical message passing algorithm in Bayesian
networks using the tensor train format. By reducing significantly the
computational complexity and the memory usage, the proposed approach
makes probabilistic inference much more scalable. These results are
illustrated by experiments on dynamic Bayesian networks and classical
Bayesian networks performed using a Python implementation with
TensorFlow, T3F and pyAgrum.