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Title: Well-tempered Hamiltonian Monte Carlo on active-space Authors:  Ritabrata Dutta - Warwick University (United Kingdom) [presenting]
Antonietta Mira - University of Lugano (Switzerland)
Abstract: When the gradient of the log-target distribution is available, Hamiltonian Monte Carlo (HMC) has been proved to be an efficient simulation algorithm. However, HMC performs poorly when the target is high-dimensional and it has multiple isolated modes. To alleviate these problems we propose to perform HMC on a locally and continuously tempered target distribution. This tempering is based on an efficient approach to simulate molecular dynamics in high-dimensional space, known as well-tempered meta-dynamics. The tempering we suggest is performed locally and only along the directions of the maximum changes in the target which we identify as the active space of the target. The active space is the span of the eigenfunctions corresponding to the dominant eigenvalues of the expected Hessian matrix of the log-target. To capture the state dependent non-linearity of the target, we iteratively estimate the active space from the most recent batch of samples obtained from the target. Finally, we suggest a re-weighting scheme to provide importance weights for the samples drawn from the continuously-tempered distribution. We illustrate the performance of this scheme for target distributions with complex geometry and multiple modes on high-dimensional spaces in comparison with traditional HMC with No-U-Turn-Sampler (NUTS).