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Entropy 2017, 19(7), 310;

Comparing Information-Theoretic Measures of Complexity in Boltzmann Machines

Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720, USA
Departments of Computer Science and Mathematics, University of Colorado, Boulder, CO 80309, USA
Santa Fe Institute, Santa Fe, NM 87501, USA
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
Faculty of Mathematics and Computer Science, University of Leipzig, 04009 Leipzig, Germany
Author to whom correspondence should be addressed.
Received: 30 April 2017 / Revised: 19 June 2017 / Accepted: 23 June 2017 / Published: 3 July 2017
(This article belongs to the Special Issue Information Geometry II)
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In the past three decades, many theoretical measures of complexity have been proposed to help understand complex systems. In this work, for the first time, we place these measures on a level playing field, to explore the qualitative similarities and differences between them, and their shortcomings. Specifically, using the Boltzmann machine architecture (a fully connected recurrent neural network) with uniformly distributed weights as our model of study, we numerically measure how complexity changes as a function of network dynamics and network parameters. We apply an extension of one such information-theoretic measure of complexity to understand incremental Hebbian learning in Hopfield networks, a fully recurrent architecture model of autoassociative memory. In the course of Hebbian learning, the total information flow reflects a natural upward trend in complexity as the network attempts to learn more and more patterns. View Full-Text
Keywords: complexity; information integration; information geometry; Boltzmann machine; Hopfield network; Hebbian learning complexity; information integration; information geometry; Boltzmann machine; Hopfield network; Hebbian learning

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Kanwal, M.S.; Grochow, J.A.; Ay, N. Comparing Information-Theoretic Measures of Complexity in Boltzmann Machines. Entropy 2017, 19, 310.

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