Breaking symmetry to save symmetry: asymmetric PDMP momentum sampling in statistical physics

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• Michael Faulkner (University of Warwick)
• Tuesday 26 November 2024, 11:25-12:15
• Seminar Room 1, Newton Institute.

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SSDW04 - Monte Carlo sampling: beyond the diffusive regime

Sampling algorithms are commonplace in statistics and machine learning – in particular, in Bayesian computation – and have been used for decades to enable inference, prediction and model comparison in many different settings.  They are also widely used in statistical physics, where many popular sampling algorithms first originated, including the famous Metropolis algorithm [1].  The algorithm has led to huge success in both fields, but typically exhibits slow mixing when faced with broken symmetry in statistical physics — as well as strong autocorrelations in the broad regions of probability mass found at transitions into symmetry-broken thermodynamic phases.  More recent developments in both fields [2, 3] have led, however, to state-of-the-art sampling algorithms that augment the state space with auxiliary momenta, which leads to ballistic-style dynamics that drive the system through the original state space.  This makes these piecewise deterministic Markov processes (PDMPs) the optimal candidate for tackling both issues.  In particular, breaking symmetry on momentum space has been shown to recover symmetry on the original state space [4].  The concept also accelerated mixing in the seminal work that solved the two-dimensional melting transition [5].  This talk gives a brief introduction to broken symmetry in statistical physics, before moving on to Metropolis and PDMP sampling of smooth probability distributions.  We then explain the (non-proven!) freedom to break symmetry on PDMP momentum space, before showcasing its recovery of symmetry on the original state space of an important model from statistical physics [4].  We finish by discussing the implications for dealing with strong autocorrelations at symmetry-broken phase transitions.  This talk uses concepts developed in a recent paper on the shared structure of the two research fields [6].   [1] Metropolis et al., J. Chem. Phys. 21 1087 (1953) [2] Bierkens & Roberts, Ann. Appl. Probab. 27, 846 (2017) [3] Bernard, Krauth & Wilson, Phys. Rev. E 80 056704 (2009) [4] Faulkner, Phys. Rev. B 109 , 085405 (2024) [5] Bernard & Krauth, Phys. Rev. Lett. 107, 155704 (2011) [6] Faulkner & Livingstone, Statist. Sci. 39, 137 (2024)

This talk is part of the Isaac Newton Institute Seminar Series series.

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