1,013 research outputs found
Numerical methods for a Kohn-Sham density functional model based on optimal transport
In this paper, we study numerical discretizations to solve density functional
models in the "strictly correlated electrons" (SCE) framework. Unlike previous
studies our work is not restricted to radially symmetric densities. In the SCE
framework, the exchange-correlation functional encodes the effects of the
strong correlation regime by minimizing the pairwise Coulomb repulsion,
resulting in an optimal transport problem. We give a mathematical derivation of
the self-consistent Kohn-Sham-SCE equations, construct an efficient numerical
discretization for this type of problem for N = 2 electrons, and apply it to
the H2 molecule in its dissociating limit. Moreover, we prove that the SCE
density functional model is correct for the H2 molecule in its dissociating
limit.Comment: 22 pages, 6 figure
A Numerical Method to solve Optimal Transport Problems with Coulomb Cost
In this paper, we present a numerical method, based on iterative Bregman
projections, to solve the optimal transport problem with Coulomb cost. This is
related to the strong interaction limit of Density Functional Theory. The first
idea is to introduce an entropic regularization of the Kantorovich formulation
of the Optimal Transport problem. The regularized problem then corresponds to
the projection of a vector on the intersection of the constraints with respect
to the Kullback-Leibler distance. Iterative Bregman projections on each
marginal constraint are explicit which enables us to approximate the optimal
transport plan. We validate the numerical method against analytical test cases
Learning Arbitrary Statistical Mixtures of Discrete Distributions
We study the problem of learning from unlabeled samples very general
statistical mixture models on large finite sets. Specifically, the model to be
learned, , is a probability distribution over probability
distributions , where each such is a probability distribution over . When we sample from , we do not observe
directly, but only indirectly and in very noisy fashion, by sampling from
repeatedly, independently times from the distribution . The problem is
to infer to high accuracy in transportation (earthmover) distance.
We give the first efficient algorithms for learning this mixture model
without making any restricting assumptions on the structure of the distribution
. We bound the quality of the solution as a function of the size of
the samples and the number of samples used. Our model and results have
applications to a variety of unsupervised learning scenarios, including
learning topic models and collaborative filtering.Comment: 23 pages. Preliminary version in the Proceeding of the 47th ACM
Symposium on the Theory of Computing (STOC15
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