105 research outputs found
Ergodic SDEs on submanifolds and related numerical sampling schemes
In many applications, it is often necessary to sample the mean value of
certain quantity with respect to a probability measure {\mu} on the level set
of a smooth function , .
A specially interesting case is the so-called conditional probability measure,
which is useful in the study of free energy calculation and model reduction of
diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate
the mean value is to compute the time average along an infinitely long
trajectory of an ergodic diffusion process on the level set whose invariant
measure is {\mu}. Motivated by the previous work of Ciccotti, Leli\`evre, and
Vanden-Eijnden [11], as well as the work of Leli\`evre, Rousset, and Stoltz
[33], in this paper we construct a family of ergodic diffusion processes on the
level set of whose invariant measures coincide with the given one. For
the conditional measure, in particular, we show that the corresponding SDEs of
the constructed ergodic processes have relatively simple forms, and, moreover,
we propose a consistent numerical scheme which samples the conditional measure
asymptotically. The numerical scheme doesn't require computing the second
derivatives of and the error estimates of its long time sampling
efficiency are obtained.Comment: 45 pages. Accepted versio
Monte Carlo on manifolds in high dimensions
We introduce an efficient numerical implementation of a Markov Chain Monte
Carlo method to sample a probability distribution on a manifold (introduced
theoretically in Zappa, Holmes-Cerfon, Goodman (2018)), where the manifold is
defined by the level set of constraint functions, and the probability
distribution may involve the pseudodeterminant of the Jacobian of the
constraints, as arises in physical sampling problems. The algorithm is easy to
implement and scales well to problems with thousands of dimensions and with
complex sets of constraints provided their Jacobian retains sparsity. The
algorithm uses direct linear algebra and requires a single matrix factorization
per proposal point, which enhances its efficiency over previously proposed
methods but becomes the computational bottleneck of the algorithm in high
dimensions. We test the algorithm on several examples inspired by soft-matter
physics and materials science to study its complexity and properties
Robust Bayesian Inference on Riemannian Submanifold
Non-Euclidean spaces routinely arise in modern statistical applications such
as in medical imaging, robotics, and computer vision, to name a few. While
traditional Bayesian approaches are applicable to such settings by considering
an ambient Euclidean space as the parameter space, we demonstrate the benefits
of integrating manifold structure into the Bayesian framework, both
theoretically and computationally. Moreover, existing Bayesian approaches which
are designed specifically for manifold-valued parameters are primarily
model-based, which are typically subject to inaccurate uncertainty
quantification under model misspecification. In this article, we propose a
robust model-free Bayesian inference for parameters defined on a Riemannian
submanifold, which is shown to provide valid uncertainty quantification from a
frequentist perspective. Computationally, we propose a Markov chain Monte Carlo
to sample from the posterior on the Riemannian submanifold, where the mixing
time, in the large sample regime, is shown to depend only on the intrinsic
dimension of the parameter space instead of the potentially much larger ambient
dimension. Our numerical results demonstrate the effectiveness of our approach
on a variety of problems, such as reduced-rank multiple quantile regression,
principal component analysis, and Fr\'{e}chet mean estimation
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