{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T13:59:35Z","timestamp":1768312775633,"version":"3.49.0"},"reference-count":90,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,8,7]],"date-time":"2018-08-07T00:00:00Z","timestamp":1533600000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National R&amp;D Program of China","award":["2017YFA0302901"],"award-info":[{"award-number":["2017YFA0302901"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11190024"],"award-info":[{"award-number":["No. 11190024"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11474331"],"award-info":[{"award-number":["No. 11474331"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 11774398"],"award-info":[{"award-number":["No. 11774398"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Ministry of Science and Technology of China","award":["2016YFA0300603"],"award-info":[{"award-number":["2016YFA0300603"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We compare and contrast the statistical physics and quantum physics inspired approaches for unsupervised generative modeling of classical data. The two approaches represent probabilities of observed data using energy-based models and quantum states, respectively. Classical and quantum information patterns of the target datasets therefore provide principled guidelines for structural design and learning in these two approaches. Taking the Restricted Boltzmann Machines (RBM) as an example, we analyze the information theoretical bounds of the two approaches. We also estimate the classical mutual information of the standard MNIST datasets and the quantum R\u00e9nyi entropy of corresponding Matrix Product States (MPS) representations. Both information measures are much smaller compared to their theoretical upper bound and exhibit similar patterns, which imply a common inductive bias of low information complexity. By comparing the performance of RBM with various architectures on the standard MNIST datasets, we found that the RBM with local sparse connection exhibit high learning efficiency, which supports the application of tensor network states in machine learning problems.<\/jats:p>","DOI":"10.3390\/e20080583","type":"journal-article","created":{"date-parts":[[2018,8,7]],"date-time":"2018-08-07T11:20:23Z","timestamp":1533640823000},"page":"583","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":87,"title":["Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7652-9344","authenticated-orcid":false,"given":"Song","family":"Cheng","sequence":"first","affiliation":[{"name":"Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0538-689X","authenticated-orcid":false,"given":"Jing","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China"},{"name":"Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA"}]},{"given":"Lei","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,8,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. 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