On the rate of convergence in Wasserstein distance of the empirical measure

Nonlinear sufficient dimension reduction for distribution-on-distribution regression

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space, while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

DrugMetric: quantitative drug-likeness scoring based on chemical space distance

The process of drug discovery is widely known to be lengthy and resource-intensive. Artificial Intelligence approaches bring hope for accelerating the identification of molecules with the necessary properties for drug development. Drug-likeness assessment is crucial for the virtual screening of candidate drugs. However, traditional methods like Quantitative Estimation of Drug-likeness (QED) struggle to distinguish between drug and non-drug molecules accurately. Additionally, some deep learning-based binary classification models heavily rely on selecting training negative sets. To address these challenges, we introduce a novel unsupervised learning framework called DrugMetric, an innovative framework for quantitatively assessing drug-likeness based on the chemical space distance. DrugMetric blends the powerful learning ability of variational autoencoders with the discriminative ability of the Gaussian Mixture Model. This synergy enables DrugMetric to identify significant differences in drug-likeness across different datasets effectively. Moreover, DrugMetric incorporates principles of ensemble learning to enhance its predictive capabilities. Upon testing over a variety of tasks and datasets, DrugMetric consistently showcases superior scoring and classification performance. It excels in quantifying drug-likeness and accurately distinguishing candidate drugs from non-drugs, surpassing traditional methods including QED. This work highlights DrugMetric as a practical tool for drug-likeness scoring, facilitating the acceleration of virtual drug screening, and has potential applications in other biochemical fields.

Persistence in a large network of sparsely interacting neurons

This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of sparse interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear mean-field process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.

Rate of convergence for particle approximation of PDEs in Wasserstein space

We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

Hypothesis Test and Confidence Analysis with Wasserstein Distance on General Dimension

We develop a general framework for statistical inference with the 1-Wasserstein distance. Recently, the Wasserstein distance has attracted considerable attention and has been widely applied to various machine learning tasks because of its excellent properties. However, hypothesis tests and a confidence analysis for it have not been established in a general multivariate setting. This is because the limit distribution of the empirical distribution with the Wasserstein distance is unavailable without strong restriction. To address this problem, in this study, we develop a novel nonasymptotic gaussian approximation for the empirical 1-Wasserstein distance. Using the approximation method, we develop a hypothesis test and confidence analysis for the empirical 1-Wasserstein distance. We also provide a theoretical guarantee and an efficient algorithm for the proposed approximation. Our experiments validate its performance numerically.

Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization

Fortet-Mourier (FM) probability metrics are important probability metrics, which have been widely adopted in the quantitative stability analysis of stochastic programming problems. In this study, we contribute to different types of convergence assertions between a probability distribution and its empirical distribution when the deviation is measured by FM metrics and consider their applications in stochastic optimization. We first establish the quantitative relation between FM metrics and Wasserstein metrics. After that, we derive the non-asymptotic moment estimate, asymptotic convergence, and non-asymptotic concentration estimate for FM metrics, which supplement the existing results. Finally, we apply the derived results to four kinds of stochastic optimization problems, which either extend the present results to more general cases or provide alternative avenues. All these discussions demonstrate the motivation as well as the significance of our study.

Probabilistic analysis of replicator–mutator equations

This paper discusses a general class of replicator–mutator equations on a multidimensional fitness space. We establish a novel probabilistic representation of weak solutions of the equation by using the theory of Fokker–Planck–Kolmogorov (FPK) equations and a martingale extraction approach. We provide examples with closed-form probabilistic solutions for different fitness functions considered in the existing literature. We also construct a particle system and prove a general convergence result to the unique solution of the FPK equation associated with the extended replicator–mutator equation with respect to a Wasserstein-like metric adapted to our probabilistic framework.

Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case

We establish the convergences (with respect to the simulation time t; the number of particles N; the timestep γ) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the d-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter (t →∞, N →∞ or γ → 0) are independent from the two others.

Sensitivity analysis of Wasserstein distributionally robust optimization problems

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first-order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide an explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least-squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.

On the capacity of deep generative networks for approximating distributions

We study the efficacy and efficiency of deep generative networks for approximating probability distributions. We prove that neural networks can transform a low-dimensional source distribution to a distribution that is arbitrarily close to a high-dimensional target distribution, when the closeness is measured by Wasserstein distances and maximum mean discrepancy. Upper bounds of the approximation error are obtained in terms of the width and depth of neural network. Furthermore, it is shown that the approximation error in Wasserstein distance grows at most linearly on the ambient dimension and that the approximation order only depends on the intrinsic dimension of the target distribution. On the contrary, when f-divergences are used as metrics of distributions, the approximation property is different. We show that in order to approximate the target distribution in f-divergences, the dimension of the source distribution cannot be smaller than the intrinsic dimension of the target distribution.