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The physics of energy-based models

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Abstract

Energy-based models (EBMs) are experiencing a resurgence of interest in both the physics community and the machine learning community. This article provides an intuitive introduction to EBMs, without requiring any background in machine learning, connecting elementary concepts from physics with basic concepts and tools in generative models, and finally giving a perspective where current research in the field is heading. This article, in its original form, was written as an online lecture note in HTML and Javascript and contains interactive graphics. We recommend the reader to also visit the interactive version.

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Notes

  1. The entropy of a system is given by \(S = \sum _x -P(x)\log P(x)\), where P(x) is the probability of a state x. If all states are equally likely, the entropy is maximal. If only one state \(P(x) = 1\) then the entropy is minimal.

  2. By definition, being at thermal equilibrium with a bath of temperature T means that the system is also at tempearature T. The temperature determines the average energy of a system. With large T, the probability of high energy states increases, and so does the average energy.

  3. For any distribution in the exponential family, a statistic T(x) is sufficient if we can write the probability p(x) as

    $$\begin{aligned}p(x) = \exp \left( \alpha (\theta ) T(x) + A(\theta ) \right) ,\end{aligned}$$

    where \(\alpha (\theta )\) is a vector-valued function and \(A(\theta )\) is a scalar, which for a Boltzmann distribution is related to the partition function as \(A(\theta ) = \log (1/Z)\) (Li et al. 2013).

  4. There are other conventions for the Ising energy function in the literature, where the signs of the energy terms change. For example,

    $$\begin{aligned} E(\sigma ) = \sum \limits _i b_i \sigma _i + \sum \limits _{ij} w_{ij} \sigma _i\sigma _j. \end{aligned}$$

    We follow the convention introduced above.

  5. Non-commuting operators do not have a common eigenfunction or eigenstate. Since these eigenfunctions in quantum mechanics (for hermitian operators) are orthogonal, non-commutating operators lead to non-orthogonal eigenfunctions or eigenstates and therefore there is no measurement operator that can reliably distinguish these non-orthogonal states. This gives rise to the uncertainty principle. The no-cloning theorem follows a similar argument. Orthogonal states can be cloned, but non-orthogonal ones cannot.

  6. Droplet is a local low-energy cluster of spins where the distribution is disconnected from the rest of the system.

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Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Patrick Huembeli

  2. Xanadu, Toronto, Canada

    Juan Miguel Arrazola & Nathan Killoran

  3. Google AI Quantum, Venice, USA

    Masoud Mohseni

  4. University of Toronto, Toronto, Canada

    Peter Wittek

  5. Creative Destruction Lab, Toronto, Canada

    Peter Wittek

  6. Vector Institute for Artificial Intelligence, Toronto, Canada

    Peter Wittek

  7. Perimeter Institute for Theoretical Physics, Waterloo, Canada

    Peter Wittek

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  1. Patrick Huembeli
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  2. Juan Miguel Arrazola
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  4. Masoud Mohseni
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  5. Peter Wittek
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Correspondence to Patrick Huembeli.

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Huembeli, P., Arrazola, J.M., Killoran, N. et al. The physics of energy-based models. Quantum Mach. Intell. 4, 1 (2022). https://doi.org/10.1007/s42484-021-00057-7

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