arXiv subjects

Espen Bernton

Publications and source records attributed to Espen Bernton.

7 recordsLinked to original sources

arXiv2022

Stability of Entropic Optimal Transport and Schrödinger Bridges

We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the use of $c$-cyclical monotonicity in classical optimal transport. As a consequence of stability, we obtain the wellposedness of the solution in this geometric sense, even when all transports have infinite cost. More generally, our results apply to a class of static Schrödinger bridge problems including entropic optimal transport.

math.OC↗

arXiv2022

Entropic Optimal Transport: Geometry and Large Deviations

We study the convergence of entropically regularized optimal transport to optimal transport. The main result is concerned with the convergence of the associated optimizers and takes the form of a large deviations principle quantifying the local exponential convergence rate as the regularization parameter vanishes. The exact rate function is determined in a general setting and linked to the Kantorovich potential of optimal transport. Our arguments are based on the geometry of the optimizers and inspired by the use of $c$-cyclical monotonicity in classical transport theory. The results can also be phrased in terms of Schrödinger bridges.

math.OC↗

arXiv2019

Schrödinger Bridge Samplers

Consider a reference Markov process with initial distribution $π_{0}$ and transition kernels $\{M_{t}\}_{t\in[1:T]}$, for some $T\in\mathbb{N}$. Assume that you are given distribution $π_{T}$, which is not equal to the marginal distribution of the reference process at time $T$. In this scenario, Schrödinger addressed the problem of identifying the Markov process with initial distribution $π_{0}$ and terminal distribution equal to $π_{T}$ which is the closest to the reference process in terms of Kullback--Leibler divergence. This special case of the so-called Schrödinger bridge problem can be solved using iterative proportional fitting, also known as the Sinkhorn algorithm. We leverage these ideas to develop novel Monte Carlo schemes, termed Schrödinger bridge samplers, to approximate a target distribution $π$ on $\mathbb{R}^{d}$ and to estimate its normalizing constant. This is achieved by iteratively modifying the transition kernels of the reference Markov chain to obtain a process whose marginal distribution at time $T$ becomes closer to $π_T = π$, via regression-based approximations of the corresponding iterative proportional fitting recursion. We report preliminary experiments and make connections with other problems arising in the optimal transport, optimal control and physics literatures.

stat.CO↗

arXiv2019

On parameter estimation with the Wasserstein distance

Statistical inference can be performed by minimizing, over the parameter space, the Wasserstein distance between model distributions and the empirical distribution of the data. We study asymptotic properties of such minimum Wasserstein distance estimators, complementing results derived by Bassetti, Bodini and Regazzini in 2006. In particular, our results cover the misspecified setting, in which the data-generating process is not assumed to be part of the family of distributions described by the model. Our results are motivated by recent applications of minimum Wasserstein estimators to complex generative models. We discuss some difficulties arising in the approximation of these estimators and illustrate their behavior in several numerical experiments. Two of our examples are taken from the literature on approximate Bayesian computation and have likelihood functions that are not analytically tractable. Two other examples involve misspecified models.

stat.ME↗

arXiv2019

Langevin Monte Carlo and JKO splitting

Algorithms based on discretizing Langevin diffusion are popular tools for sampling from high-dimensional distributions. We develop novel connections between such Monte Carlo algorithms, the theory of Wasserstein gradient flow, and the operator splitting approach to solving PDEs. In particular, we show that a proximal version of the Unadjusted Langevin Algorithm corresponds to a scheme that alternates between solving the gradient flows of two specific functionals on the space of probability measures. Using this perspective, we derive some new non-asymptotic results on the convergence properties of this algorithm.

stat.CO↗

arXiv2019

Approximate Bayesian computation with the Wasserstein distance

A growing number of generative statistical models do not permit the numerical evaluation of their likelihood functions. Approximate Bayesian computation (ABC) has become a popular approach to overcome this issue, in which one simulates synthetic data sets given parameters and compares summaries of these data sets with the corresponding observed values. We propose to avoid the use of summaries and the ensuing loss of information by instead using the Wasserstein distance between the empirical distributions of the observed and synthetic data. This generalizes the well-known approach of using order statistics within ABC to arbitrary dimensions. We describe how recently developed approximations of the Wasserstein distance allow the method to scale to realistic data sizes, and propose a new distance based on the Hilbert space-filling curve. We provide a theoretical study of the proposed method, describing consistency as the threshold goes to zero while the observations are kept fixed, and concentration properties as the number of observations grows. Various extensions to time series data are discussed. The approach is illustrated on various examples, including univariate and multivariate g-and-k distributions, a toggle switch model from systems biology, a queueing model, and a Lévy-driven stochastic volatility model.

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arXiv2015

Locally weighted Markov chain Monte Carlo

We propose a weighting scheme for the proposals within Markov chain Monte Carlo algorithms and show how this can improve statistical efficiency at no extra computational cost. These methods are most powerful when combined with multi-proposal MCMC algorithms such as multiple-try Metropolis, which can efficiently exploit modern computer architectures with large numbers of cores. The locally weighted Markov chain Monte Carlo method also improves upon a partial parallelization of the Metropolis-Hastings algorithm via Rao-Blackwellization. We derive the effective sample size of the output of our algorithm and show how to estimate this in practice. Illustrations and examples of the method are given and the algorithm is compared in theory and applications with existing methods.

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