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1. Embedding Empirical Distributions for Computing Optimal Transport Maps

   Jiang, Mingchen; Xu, Peng; Ye, Xichen; Chen, Xiaohui; Yang, Yun; Chen, Yifan (June 2025, IEEE International Symposium on Information Theory (ISIT))

   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. 

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2. On the Optimal Prediction of Extreme Events in Heavy‐Tailed Time Series With Applications to Solar Flare Forecasting

   https://doi.org/10.1111/jtsa.12820  

   Verma, Victor; Stoev, Stilian; Chen, Yang (February 2025, Journal of Time Series Analysis)

   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. 

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3. Harnessing Vision Models for Time Series Analysis: A Survey

   https://doi.org/10.24963/ijcai.2025/1178  

   Ni, Jingchao; Zhao, Ziming; Shen, ChengAo; Tong, Hanghang; Song, Dongjin; Cheng, Wei; Luo, Dongsheng; Chen, Haifeng (September 2025, International Joint Conferences on Artificial Intelligence Organization)

   Time series analysis has evolved from traditional autoregressive models to deep learning, Transformers, and Large Language Models (LLMs). While vision models have also been explored along the way, their contributions are less recognized due to the predominance of sequence modeling. However, challenges such as the mismatch between continuous time series and LLMs’ discrete token space, and the difficulty in capturing multivariate correlations, have led to growing interest in Large Vision Models (LVMs) and Vision-Language Models (VLMs). This survey highlights the advantages of vision models over LLMs in time series analysis, offering a comprehensive dual-view taxonomy that answers key research questions like how to encode time series as images and how to model imaged time series. Additionally, we address pre- and post-processing challenges in this framework and outline future directions for advancing the field. 

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4. The Brownian transport map

   https://doi.org/10.1007/s00440-024-01286-0  

   Mikulincer, Dan; Shenfeld, Yair (May 2024, Probability Theory and Related Fields)

   Abstract Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan–Lovász–Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors. 

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5. An approximation theory framework for measure-transport sampling algorithms

   https://doi.org/10.1090/mcom/4013  

   Baptista, Ricardo; Hosseini, Bamdad; Kovachki, Nikola; Marzouk, Youssef; Sagiv, Amir (July 2025, Mathematics of Computation)

   This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling—a central task in statistical inference and generative modeling. We provide a priori error estimates in the continuum limit, i.e., when the measures (or their densities) are given, but when the transport map is discretized or approximated using a finite-dimensional function space. Our analysis relies on the regularity theory of transport maps and on classical approximation theory for high-dimensional functions. A third element of our analysis, which is of independent interest, is the development of new stability estimates that relate the distance between two maps to the distance (or divergence) between the pushforward measures they define. We present a series of applications of our framework, where quantitative convergence rates are obtained for practical problems using Wasserstein metrics, maximum mean discrepancy, and Kullback–Leibler divergence. Specialized rates for approximations of the popular triangular Knöthe–Rosenblatt maps are obtained, followed by numerical experiments that demonstrate and extend our theory. 

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