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core.ac.uk) brought to you by CORE  \n[provided by](provided by arXiv.org)[ arXiv.org](provided by arXiv.org) e-Print Archive  \narXiv :2106 . 12974v1 [ cond-mat .stat-mech] 24 Jun 2021  \nTensor networks for unsupervised machine learning  \nJing Liu, 1 Sujie Li,2, 3 Jiang Zhang, 1 and Pan Zhang2, 4, 5, ∗  \n1 School of Systems Science, Beijing Normal University  \n2 CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics,  \nChinese Academy of Sciences, Beijing 100190, China  \n3 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China  \n4 School of Fundamental Physics and Mathematical Sciences,  \nHangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China  \n5 International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China  \n(Dated: June 25, 2021)  \nModeling 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 advantages of theoretical understandings of the expressive power using entanglement properties, and as abridge connecting the classical computation and the quantum computation. Despite the great potential, however, existing tensor-network-based unsupervised models only work as a proof of principle, as their performances are much worse than the standard models such as the restricted Boltzmann machines and neural networks. In this work, we present the Autoregressive Matrix Product States (AMPS), a tensor-network-based model combining the matrix product states from quantum many-body physics and the autoregressive models from machine learning. The model enjoys exact calculation of normalized probability and unbiased sampling, as well as a clear theoretical understanding of expressive power. We demonstrate the performance of our model using two applications, the generative modeling on synthetic and real-world data, and the reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor-network-based models and the restricted Boltzmann machines, and is competitive with the state-of-the-art neural network models.  \nUnsupervised learning aims to understand the high dimensional data x with rich structures, by learning a probability distribution 􀀥(x) from the data. The central challenge is how to represent the joint distribution with an exponential number of entries using a polynomial number of parameters and how to efficiently sample from it. Indeed, even computing the normalization constant for a general joint distribution belongs to the \\# P hard problem, and has attracted enormous efforts [1, 2] in the past decades. From the point of view of computations, unsupervised learning shares the same challenges with statistical physics and quantum many-body physics, where the target distribution or the target quantum state one investigates lives in a huge space and is difficult to control. In the history of machine learning, many classical models were inspired by physics.  \nOne of the most important unsupervised model, the celebrated Boltzmann machine [3, 4], was inspired by statistical physics, where 􀀥(x) is expressed using the Boltzmann distribution with parameters encoded in the energy function 􀀚(x) ,  \n􀀥 (x) = 1􀀯 􀀴 −􀁖􀀚 (x) . (1)  \nTo efficiently sample from the Boltzmann distribution, the model is usually defined on a bipartite graph where efficient block-update Monte Carlo methods exist. This is known as the restricted Boltzmann machine (RBM) [5, 6] and has found numerous applications in machine learning and physics. Another physics-inspired model is the Born machine [7, 8], where the joint distribution of variable is modeled by squared norm of a quantum state Ψ, with  \n􀀥 (x) = 1􀀯 |Ψ(􀁇1 , 􀁇2 , · · · , 􀁇 􀀽) | 2 , (2) ","cbCaiaK1x3FfzTwl","https://ap.wps.com/l/cbCaiaK1x3FfzTwl","pdf",873530,12,"English","# Introduction\n## Unsupervised learning and challenges\n## Physics-inspired unsupervised models\n# Autoregressive Matrix Product States (AMPS)\n## Factorized autoregressive formulation\n## Exact normalization and unbiased sampling\n# Applications and results\n## Generative modeling on synthetic and real data\n## Reinforcement learning in statistical physics","[{\"question\":\"What is the main goal of the document in unsupervised machine learning?\",\"answer\":\"To model and learn high-dimensional joint probability distributions efficiently, addressing both representation and sampling challenges.\"},{\"question\":\"Why do existing tensor-network-based unsupervised models underperform?\",\"answer\":\"They typically only work as proof of principle and show significantly worse performance than standard models like RBMs and neural networks.\"},{\"question\":\"What advantages does AMPS provide over previous approaches?\",\"answer\":\"AMPS offers exact calculation of normalized probability, unbiased sampling, a clear theoretical understanding of expressiveness, and strong empirical performance across applications.\"}]","Tensor networks for unsupervised machine learning | PDF"]