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Sampling via Measure Transport: An Introduction

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Handbook of Uncertainty Quantification

Abstract

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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Notes

  1. 1.

    See [61] for a discussion on the asymptotic equivalence of the K–L divergence and Hellinger distance in the context of transport maps.

  2. 2.

    The lexicographic order on \(\mathbb{R}^{n}\) is defined as follows. For \(x,y \in \mathbb{R}^{n}\), we define \(x\preceq y\) if and only if either x = y or the first nonzero coordinate in y − x is positive [32]. \(\preceq\) is a total order on \(\mathbb{R}^{n}\). Thus, we define T to be a monotone increasing function if and only if \(x\preceq y\) implies \(T(x)\preceq T(y)\). Notice that monotonicity can be defined with respect to any order on \(\mathbb{R}^{n}\) (e.g., \(\preceq\) need not be the lexicographic order). There is no natural order on \(\mathbb{R}^{n}\) except when n = 1. It is easy to verify that for a triangular function T, monotonicity with respect to the lexicographic order is equivalent to the following: the kth component of T is a monotone function of the kth input variable.

  3. 3.

    Roots can be found using, for instance, Newton’s method. When a component of the inverse transport is parameterized using polynomials, however, then a more robust root-finding approach is to use a bisection method based on Sturm sequences (e.g., [63]).

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Room 37-451, 77 Massachusetts Avenue, 02139, Cambridge, MA, USA

    Youssef Marzouk, Matthew Parno & Alessio Spantini

  2. D. E. Shaw Group, 1166 Avenue of the Americas, New York, NY, USA

    Tarek Moselhy

  3. U. S. Army Cold Regions Research and Engineering Laboratory, 03755, Hanover, NH, USA

    Matthew Parno

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  1. Youssef Marzouk
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  2. Tarek Moselhy
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  3. Matthew Parno
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  4. Alessio Spantini
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Correspondence to Youssef Marzouk .

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Editors and Affiliations

  1. Viterbi School of Engineering, University of Southern California, Los Angeles, California, USA

    Roger Ghanem

  2. Los Alamos National Laboratory, Los Alamos, New Mexico, USA

    David Higdon

  3. California Institute of Technology , Pasadena, California, USA

    Houman Owhadi

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Marzouk, Y., Moselhy, T., Parno, M., Spantini, A. (2016). Sampling via Measure Transport: An Introduction. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-11259-6_23-1

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