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Hybrid Monte Carlo methods for sampling probability measures on submanifolds

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Abstract

Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for any timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.

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Change history

  • 22 November 2019

    Shiva Darshan and Miranda Holmes–Cerfon (Courant Institute, NYU) pointed out a mistake in the projection functions to enforce the momentum constraint when rewriting the algorithm in Numerical Algorithm A of Section 3.1. Two different projection functions are actually needed, see indeed the formula for the Lagrange multiplier

  • 22 November 2019

    Shiva Darshan and Miranda Holmes���Cerfon (Courant Institute, NYU) pointed out a mistake in the projection functions to enforce the momentum constraint when rewriting the algorithm in Numerical Algorithm��A of Section��3.1. Two different projection functions are actually needed, see indeed the formula for the Lagrange multiplier

Notes

  1. This scheme is now named “Hamiltonian Monte Carlo” in the computational statistics literature.

  2. Notice that the composition of two Markov kernels which are \(\pi \)-reversible is not \(\pi \)-reversible in general; it however admits \(\pi \) as an invariant measure.

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Acknowledgements

We thank Christian Robert for pointing out the reference [36] as soon as it was preprinted on arXiv, as well as Jonathan Goodman and Miranda Cerfon–Holmes for very useful discussions. This work was funded in part by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492, as well as the Agence Nationale de la Recherche, under grant ANR-14-CE23-0012 (COSMOS). We also benefited from the scientific environment of the Laboratoire International Associé between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign.

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  1. CERMICS (ENPC), Inria, Université Paris-Est, 77455, Marne-la-Vallée, France

    Tony Lelièvre & Gabriel Stoltz

  2. SIMSART Team-Project, Inria Rennes, France

    Mathias Rousset

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  1. Tony Lelièvre
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  2. Mathias Rousset
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  3. Gabriel Stoltz
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Correspondence to Gabriel Stoltz.

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Lelièvre, T., Rousset, M. & Stoltz, G. Hybrid Monte Carlo methods for sampling probability measures on submanifolds. Numer. Math. 143, 379–421 (2019). https://doi.org/10.1007/s00211-019-01056-4

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