SciPost Physics

SciPost Phys. 12, 041 (2022) · published 26 January 2022

• doi: 10.21468/SciPostPhys.12.1.041
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Abstract

We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these models.

• Grosvenor et al., The edge of chaos: quantum field theory and deep neural networks
  SciPost Phys. 12, 081 (2022) [Crossref]
• Hartman et al., Averaged null energy and the renormalization group
  J. High Energ. Phys. 2023, 139 (2023) [Crossref]
• Lahoche et al., Functional renormalization group for multilinear disordered Langevin dynamics I Formalism and first numerical investigations at equilibrium
  J. Phys. Commun. 6, 055002 (2022) [Crossref]
• Cotler et al., Renormalization group flow as optimal transport
  Phys. Rev. D 108, 025003 (2023) [Crossref]
• Banta et al., Structures of neural network effective theories
  Phys. Rev. D 109, 105007 (2024) [Crossref]
• Berman et al., Bayesian renormalization
  Mach. Learn.: Sci. Technol. 4, 045011 (2023) [Crossref]
• Floerchinger, Exact flow equation for the divergence functional
  Physics Letters B 846, 138244 138244 (2023) [Crossref]
• Guillon et al., Mind the information gap: How sampling and clustering impact the predictability of reach‐scale channel types in California (USA)
  Earth Surf Processes Landf 49, 4610 (2024) [Crossref]
• Ran et al., Tensor Networks for Interpretable and Efficient Quantum-Inspired Machine Learning
  Intell Comput 2, 0061 (2023) [Crossref]