78,080 research outputs found
Transform-based particle filtering for elliptic Bayesian inverse problems
We introduce optimal transport based resampling in adaptive SMC. We consider
elliptic inverse problems of inferring hydraulic conductivity from pressure
measurements. We consider two parametrizations of hydraulic conductivity: by
Gaussian random field, and by a set of scalar (non-)Gaussian distributed
parameters and Gaussian random fields. We show that for scalar parameters
optimal transport based SMC performs comparably to monomial based SMC but for
Gaussian high-dimensional random fields optimal transport based SMC outperforms
monomial based SMC. When comparing to ensemble Kalman inversion with mutation
(EKI), we observe that for Gaussian random fields, optimal transport based SMC
gives comparable or worse performance than EKI depending on the complexity of
the parametrization. For non-Gaussian distributed parameters optimal transport
based SMC outperforms EKI
A Modica-Mortola approximation for branched transport
The M^\alpha energy which is usually minimized in branched transport problems
among singular 1-dimensional rectifiable vector measures with prescribed
divergence is approximated (and convergence is proved) by means of a sequence
of elliptic energies, defined on more regular vector fields. The procedure
recalls the Modica-Mortola one for approximating the perimeter, and the
double-well potential is replaced by a concave power
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Approximation by finitely supported measures
Given a compactly supported probability measure on a Riemannian manifold, we
study the asymptotic speed at which it can be approximated (in Wasserstein
distance of any exponent p) by finitely supported measure. This question has
been studied under the names of ``quantization of distributions'' and, when
p=1, ``location problem''. When p=2, it is linked with Centroidal Voronoi
Tessellations.Comment: v2: the main result is extended to measures defined on a manifold.
v3: references added. 25 pp. To appear in ESAIM:COC
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