Constrained Density Estimation

Laurence, Peter; Pignol, Ricardo J.; Tabak, Esteban G.

doi:10.1007/978-1-4614-7248-3_10

Cited by 7 publications

(3 citation statements)

References 2 publications

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“…i D1 : Similar ideas have appeared in different contexts to compare measures and sample sets, for instance, in the MaxEnt framework of [7] and in the constrained density estimation technique of [24].…”

Section: The Sample-based Formulationmentioning

confidence: 95%

Sample‐Based Optimal Transport and Barycenter Problems

Kuang

Tabak

2019

Comm Pure Appl Math

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A methodology is developed for the numerical solution to the sample‐based optimal transport and Wasserstein barycenter problems. The procedure is based on a characterization of the barycenter and of the McCann interpolants that permits the decomposition of the global problem under consideration into various local problems where the distance among successive distributions is small. These local problems can be formulated in terms of feature functions and shown to have a unique minimizer that solves a nonlinear system of equations. Both the theoretical underpinnings of the methodology and its practical implementation are developed, and illustrated with synthetic and real data sets. © 2019 Wiley Periodicals, Inc.

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Section: The Sample-based Formulationmentioning

confidence: 95%

Sample‐Based Optimal Transport and Barycenter Problems

Kuang

Tabak

2019

Comm Pure Appl Math

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“…Moving from polynomial approximations to nonparametric approaches, [84,45,85] introduce schemes for multivariate density estimation based on progressively transforming a given set of samples to a (joint) standard Gaussian by composing a sequence of monotone maps. The maps are typically chosen to be rather simple in form (e.g., sigmoid-type functions of one variable).…”

Section: Related Workmentioning

confidence: 99%

Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

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We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling-i.e., a transport map-between a complex "target" probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

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“…The strategy proposed here, similar to the one developed for constrained density estimation in [19], alternates between two kinds of steps: one that follows the direction of descent of y D as before, and one that finds a direction along which the original cost decreases while the value of y D does not deteriorate. Thus, for this second kind of steps, we consider the two objective functions…”

Section: Enforcing the Optimality Of The Final Mapmentioning

confidence: 99%

Data‐Driven Optimal Transport

Trigila

Tabak

2015

Comm Pure Appl Math

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The problem of optimal transport between two distributions ρ(x) and μ(y) is extended to situations where the distributions are only known through a finite number of samples {xi} and {yj}. A weak formulation is proposed, based on the dual of the Kantorovich formulation, with two main modifications: replacing the expected values in the objective function by their empirical means over the {xi} and {yj}, and restricting the dual variables u(x) and v(y) to a suitable set of test functions adapted to the local availability of sample points. A procedure is proposed and tested for the numerical solution of this problem, based on a fluidlike flow in phase space, where the sample points play the role of active Lagrangian markers. © 2016 Wiley Periodicals, Inc.

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Constrained Density Estimation

Cited by 7 publications

References 2 publications

Sample‐Based Optimal Transport and Barycenter Problems

Sample‐Based Optimal Transport and Barycenter Problems

Sampling via Measure Transport: An Introduction

Data‐Driven Optimal Transport

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