63 research outputs found
Discretizing Distributions with Exact Moments: Error Estimate and Convergence Analysis
The maximum entropy principle is a powerful tool for solving underdetermined
inverse problems. This paper considers the problem of discretizing a continuous
distribution, which arises in various applied fields. We obtain the
approximating distribution by minimizing the Kullback-Leibler information
(relative entropy) of the unknown discrete distribution relative to an initial
discretization based on a quadrature formula subject to some moment
constraints. We study the theoretical error bound and the convergence of this
approximation method as the number of discrete points increases. We prove that
(i) the theoretical error bound of the approximate expectation of any bounded
continuous function has at most the same order as the quadrature formula we
start with, and (ii) the approximate discrete distribution weakly converges to
the given continuous distribution. Moreover, we present some numerical examples
that show the advantage of the method and apply to numerically solving an
optimal portfolio problem.Comment: 20 pages, 14 figure
Accelerated gradient descent method for functionals of probability measures by new convexity and smoothness based on transport maps
We consider problems of minimizing functionals of probability
measures on the Euclidean space. To propose an accelerated gradient descent
algorithm for such problems, we consider gradient flow of transport maps that
give push-forward measures of an initial measure. Then we propose a
deterministic accelerated algorithm by extending Nesterov's acceleration
technique with momentum. This algorithm do not based on the Wasserstein
geometry. Furthermore, to estimate the convergence rate of the accelerated
algorithm, we introduce new convexity and smoothness for based on
transport maps. As a result, we can show that the accelerated algorithm
converges faster than a normal gradient descent algorithm. Numerical
experiments support this theoretical result.Comment: 31 page
Monte Carlo construction of cubature on Wiener space
In this paper, we investigate application of mathematical optimization to
construction of a cubature formula on Wiener space, which is a weak
approximation method of stochastic differential equations introduced by Lyons
and Victoir (Cubature on Wiener Space, Proc. R. Soc. Lond. A 460, 169--198).
After giving a brief review of the cubature theory on Wiener space, we show
that a cubature formula of general dimension and degree can be obtained through
a Monte Carlo sampling and linear programming. This paper also includes an
extension of stochastic Tchakaloff's theorem, which technically yields the
proof of our main result.Comment: 24 pages; the organization is modified and Proposition 13 is newly
adde
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