An official website of the United States government
Distributional data have become increasingly prominent in modern signal processing, highlighting the necessity of computing optimal transport (OT) maps across multiple probability distributions. Nevertheless, recent studies on neural OT methods predominantly focused on the efficient computation of a single map between two distributions. To address this challenge, we introduce a novel approach to learning transport maps for new empirical distributions. Specifically, we employ the transformer architecture to produce embeddings from distributional data of varying length; these embeddings are then fed into a hypernetwork to generate neural OT maps. Various numerical experiments were conducted to validate the embeddings and the generated OT maps.
more »
« less
Autoregressive optimal transport models
Abstract Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.
more »
« less
- PAR ID:
- 10413077
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 85
- Issue:
- 3
- ISSN:
- 1369-7412
- Page Range / eLocation ID:
- p. 1012-1033
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Motivated by the statistical analysis of the discrete optimal transport problem, we prove distributional limits for the solutions of linear programs with random constraints. Such limits were first obtained by Klatt, Munk, & Zemel (2022), but their expressions for the limits involve a computationally intractable decomposition of R^m into a possibly exponential number of convex cones. We give a new expression for the limit in terms of auxiliary linear programs, which can be solved in polynomial time. We also leverage tools from random convex geometry to give distributional limits for the entire set of random optimal solutions, when the optimum is not unique. Finally, we describe a simple, data-driven method to construct asymptotically valid confidence sets in polynomial time.more » « less
-
ABSTRACT The prediction of extreme events in time series is a fundamental problem arising in many financial, scientific, engineering, and other applications. We begin by establishing a general Neyman–Pearson‐type characterization of optimal extreme event predictors in terms of density ratios. This yields new insights and several closed‐form optimal extreme event predictors for additive models. These results naturally extend to time series, where we study optimal extreme event prediction for both light‐ and heavy‐tailed autoregressive and moving average models. Using a uniform law of large numbers for ergodic time series, we establish the asymptotic optimality of an empirical version of the optimal predictor for autoregressive models. Using multivariate regular variation, we obtain an expression for the optimal extremal precision in heavy‐tailed infinite moving averages, which provides theoretical bounds on the ability to predict extremes in this general class of models. We address the important problem of predicting solar flares by applying our theory and methodology to a state‐of‐the‐art time series consisting of solar soft x‐ray flux measurements. Our results demonstrate the success and limitations in solar flare forecasting of long‐memory autoregressive models and long‐range‐dependent, heavy‐tailed FARIMA models.more » « less
-
Numerical solutions of stochastic problems require the representation of random functions in their definitions by finite dimensional (FD) models, i.e., deterministic functions of time and finite sets of random variables. It is common to represent the coefficients of these FD surrogates by polynomial chaos (PC) models. We propose a novel model, referred to as the polynomial chaos translation (PCT) model, which matches exactly the marginal distributions of the FD coefficients and approximately their dependence. PC- and PCT- based FD models are constructed for a set of test cases and a wind pressure time series recorded at the boundary layer wind tunnel facility at the University of Florida. The PCT-based models capture the joint distributions of the FD coefficients and the extremes of target times series accurately while PC-based FD models do not have this capability.more » « less
-
Abstract Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separable. This is true in arbitrary dimension, and for families of distributions generated through perturbations of shifts and scalings of a fixed distribution. We also prove conditions under which the $L^2$ distance of the LOT embedding between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only compute $$N$$ optimal transport maps to define the $N^2$ pairwise distances between $$N$$ distributions. We demonstrate the benefits of LOT on a number of distribution classification problems.more » « less
