Abstract Tensor networks (TNs) have seen an increase in applications in recent years. While they were originally developed to model many-body quantum systems, their usage has expanded into the field of machine learning. This work adds to the growing range of applications by focusing on planning by combining the generative modeling capabilities of matrix product states and the action selection algorithm provided by active inference. Their ability to deal with the curse of dimensionality, to represent probability distributions, and to dynamically discover hidden variables make matrix product states specifically an interesting choice to use as the generative model in active inference, which relies on ‘beliefs’ about hidden states within an environment. We evaluate our method on the T-maze and Frozen Lake environments, and show that the TN-based agent acts Bayes optimally as expected under active inference.
Abstract Tensor networks (TNs) have seen an increase in applications in recent years. While they were originally developed to model many-body quantum systems, their usage has expanded into the field of machine learning. This work adds to the growing range of applications by focusing on planning by combining the generative modeling capabilities of matrix product states and the action selection algorithm provided by active inference. Their ability to deal with the curse of dimensionality, to represent probability distributions, and to dynamically discover hidden variables make matrix product states specifically an interesting choice to use as the generative model in active inference, which relies on ‘beliefs’ about hidden states within an environment. We evaluate our method on the T-maze and Frozen Lake environments, and show that the TN-based agent acts Bayes optimally as expected under active inference.
Probabilistic inference is a fundamental task in modern machine learning. Recent advances in tensor network (TN) contraction algorithms have enabled the development of better exact inference methods. However, many common inference tasks in probabilistic graphical models (PGMs) still lack corresponding TN-based adaptations. In this paper, we advance the connection between PGMs and TNs by formulating and implementing tensor-based solutions for the following inference tasks: (A) computing the partition function, (B) computing the marginal probability of sets of variables in the model, (C) determining the most likely assignment to a set of variables, (D) the same as (C) but after having marginalized a different set of variables, and (E) generating samples from a learned probability distribution using a generalized method. Our study is motivated by recent technical advances in the fields of quantum circuit simulation, quantum many-body physics, and statistical physics. Through an experimental evaluation, we demonstrate that the integration of these quantum technologies with a series of algorithms introduced in this study significantly improves the performance efficiency of existing methods for solving probabilistic inference tasks. Published by the American Physical Society 2024
Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models such as restricted Boltzmann machines and neural networks. In this Letter, we present autoregressive matrix product states (AMPS), a tensor network model combining matrix product states from quantum many-body physics and autoregressive modeling from machine learning. Our model enjoys the exact calculation of normalized probability and unbiased sampling. We demonstrate the performance of our model using two applications, generative modeling on synthetic and real-world data, and reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor network models and the restricted Boltzmann machines, and is competitive with state-of-the-art neural network models.
The tensor network (TN) has recently triggered extensive interests in developing machine learning models defined in quantum Hilbert space. Here, we propose a generative TN classification (GTNC) model for supervised machine learning. The strategy is first to map the classical data onto the states in a many-body Hilbert space and then to capture these states with tensor network schemes. We adopt the TN in the form of matrix product states as an example to implement GTNC where the testing images are classified by comparing the fidelities between different states. Our results show that GTNC has a very impressive performance on benchmark datasets in comparison to several well-known machine learning models. The advantage of GTNC is reflected from the facts that the samples are naturally clustering in the many-body Hilbert space, and it relies much less on hyperparameters. These characters make GTNC an adaptive and universal quantum-inspired method, which would have important applications in quantum computation and quantum information.
Tensor network (TN) techniques - often used in the context of quantum many-body physics - have shown promise as a tool for tackling machine learning (ML) problems. The application of TNs to ML, however, has mostly focused on supervised and unsupervised learning. Yet, with their direct connection to hidden Markov chains, TNs are also naturally suited to Markov decision processes (MDPs) which provide the foundation for reinforcement learning (RL). Here we introduce a general TN formulation of finite, episodic and discrete MDPs. We show how this formulation allows us to exploit algorithms developed for TNs for policy optimisation, the key aim of RL. As an application we consider the issue - formulated as an RL problem - of finding a stochastic evolution that satisfies specific dynamical conditions, using the simple example of random walk excursions as an illustration.
Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.
Recently it has been shown that tensor networks (TNs) have the ability to represent the expected return of a single-agent finite Markov decision process (FMDP). The TN represents a distribution model, where all possible trajectories are considered. When extending these ideas to a multi-agent setting, distribution models suffer from the curse of dimensionality: the exponential relation between the number of possible trajectories and the number of agents. The key advantage of using TNs in this setting is that there exists a large number of established optimisation and decomposition techniques that are specific to TNs, that one can apply to ensure the most efficient representation is found. In this report, these methods are used to form a TN that represents the expected return of a multi-agent reinforcement learning (MARL) task. This model is then applied to a 2 agent random walker example, where it was shown that the policy is correctly optimised using a DMRG technique. Finally, I demonstrate the use of an exact decomposition technique, reducing the number of elements in the tensors by 97.5%, without experiencing any loss of information.