41 research outputs found
Continuum limit of total variation on point clouds
We consider point clouds obtained as random samples of a measure on a
Euclidean domain. A graph representing the point cloud is obtained by assigning
weights to edges based on the distance between the points they connect. Our
goal is to develop mathematical tools needed to study the consistency, as the
number of available data points increases, of graph-based machine learning
algorithms for tasks such as clustering. In particular, we study when is the
cut capacity, and more generally total variation, on these graphs a good
approximation of the perimeter (total variation) in the continuum setting. We
address this question in the setting of -convergence. We obtain almost
optimal conditions on the scaling, as number of points increases, of the size
of the neighborhood over which the points are connected by an edge for the
-convergence to hold. Taking the limit is enabled by a transportation
based metric which allows to suitably compare functionals defined on different
point clouds
Existence of Ground States of Nonlocal-Interaction Energies
We investigate which nonlocal-interaction energies have a ground state
(global minimizer). We consider this question over the space of probability
measures and establish a sharp condition for the existence of ground states. We
show that this condition is closely related to the notion of stability (i.e.
-stability) of pairwise interaction potentials. Our approach uses the direct
method of the calculus of variations.Comment: This version is to appear in the J Stat Phy
Nonlocal Wasserstein Distance: Metric and Asymptotic Properties
The seminal result of Benamou and Brenier provides a characterization of the
Wasserstein distance as the path of the minimal action in the space of
probability measures, where paths are solutions of the continuity equation and
the action is the kinetic energy. Here we consider a fundamental modification
of the framework where the paths are solutions of nonlocal (jump) continuity
equations and the action is a nonlocal kinetic energy. The resulting nonlocal
Wasserstein distances are relevant to fractional diffusions and Wasserstein
distances on graphs. We characterize the basic properties of the distance and
obtain sharp conditions on the (jump) kernel specifying the nonlocal transport
that determine whether the topology metrized is the weak or the strong
topology. A key result of the paper are the quantitative comparisons between
the nonlocal and local Wasserstein distance
Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics
Motivated by the challenge of sampling Gibbs measures with nonconvex
potentials, we study a continuum birth-death dynamics. We improve results in
previous works [51,57] and provide weaker hypotheses under which the
probability density of the birth-death governed by Kullback-Leibler divergence
or by divergence converge exponentially fast to the Gibbs equilibrium
measure, with a universal rate that is independent of the potential barrier. To
build a practical numerical sampler based on the pure birth-death dynamics, we
consider an interacting particle system, which is inspired by the gradient flow
structure and the classical Fokker-Planck equation and relies on kernel-based
approximations of the measure. Using the technique of -convergence of
gradient flows, we show that on the torus, smooth and bounded positive
solutions of the kernelized dynamics converge on finite time intervals, to the
pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we
provide quantitative estimates on the bias of minimizers of the energy
corresponding to the kernelized dynamics. Finally we prove the long-time
asymptotic results on the convergence of the asymptotic states of the
kernelized dynamics towards the Gibbs measure.Comment: significant mathematical changes with more rigor on gradient flow
Average-distance problem for parameterized curves
We consider approximating a measure by a parameterized curve subject to
length penalization. That is for a given finite positive compactly supported
measure , for and we consider the functional where , is an
interval in , , and is the distance of to .
The problem is closely related to the average-distance problem, where the
admissible class are the connected sets of finite Hausdorff measure , and to (regularized) principal curves studied in statistics. We obtain
regularity of minimizers in the form of estimates on the total curvature of the
minimizers. We prove that for measures supported in two dimensions the
minimizing curve is injective if or if has bounded density.
This establishes that the minimization over parameterized curves is equivalent
to minimizing over embedded curves and thus confirms that the problem has a
geometric interpretation