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Zhang Q, Li B, Xue L. Nonlinear sufficient dimension reduction for distribution-on-distribution regression. J MULTIVARIATE ANAL 2024; 202:105302. [PMID: 38525479 PMCID: PMC10956811 DOI: 10.1016/j.jmva.2024.105302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/26/2024]
Abstract
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.
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Affiliation(s)
- Qi Zhang
- Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA
| | - Bing Li
- Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA
| | - Lingzhou Xue
- Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA
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2
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Li B, Wang Z, Liu Z, Tao Y, Sha C, He M, Li X. DrugMetric: quantitative drug-likeness scoring based on chemical space distance. Brief Bioinform 2024; 25:bbae321. [PMID: 38975893 PMCID: PMC11229036 DOI: 10.1093/bib/bbae321] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2024] [Revised: 05/20/2024] [Accepted: 06/27/2024] [Indexed: 07/09/2024] Open
Abstract
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.
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Affiliation(s)
- Bowen Li
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
| | - Zhen Wang
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
- College of Electrical and Information Engineering, Hunan University, Changsha, 410082 Hunan, China
| | - Ziqi Liu
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
- Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou, 310024 Zhejiang, China
| | - Yanxin Tao
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
| | - Chulin Sha
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
| | - Min He
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
- College of Electrical and Information Engineering, Hunan University, Changsha, 410082 Hunan, China
| | - Xiaolin Li
- Hangzhou Institute of Medicine, Chinese Academy of Sciences, Hangzhou, 310018 Zhejiang, China
- ElasticMind Inc, Hangzhou, 310018 Zhejiang, China
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3
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Quan J, Li Y, Wang L, He R, Yang S, Guo L. EEG-based cross-subject emotion recognition using multi-source domain transfer learning. Biomed Signal Process Control 2023. [DOI: 10.1016/j.bspc.2023.104741] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2023]
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4
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Wang FY, Zhu JX. Limit theorems in Wasserstein distance for empirical measures of diffusion processes on Riemannian manifolds. ANNALES DE L'INSTITUT HENRI POINCARÉ, PROBABILITÉS ET STATISTIQUES 2023. [DOI: 10.1214/22-aihp1251] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Affiliation(s)
- Feng-Yu Wang
- Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
| | - Jie-Xiang Zhu
- Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
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5
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Wang FY. Convergence in Wasserstein distance for empirical measures of semilinear SPDEs. ANN APPL PROBAB 2023. [DOI: 10.1214/22-aap1807] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/24/2023]
Affiliation(s)
- Feng-Yu Wang
- Center for Applied Mathematics, Tianjin University
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6
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Della Maestra L, Hoffmann M. The LAN property for McKean–Vlasov models in a mean-field regime. Stoch Process Their Appl 2023. [DOI: 10.1016/j.spa.2022.10.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/04/2022]
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7
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Rosen J. Tightness for thick points in two dimensions. ELECTRON J PROBAB 2023. [DOI: 10.1214/23-ejp910] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023]
Affiliation(s)
- Jay Rosen
- Department of Mathematics, College of Staten Island, CUNY, Staten Island, NY
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8
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Dai X, Zhao L, He R, Du W, Zhong W, Li Z, Qian F. Data-driven crude oil scheduling optimization with a distributionally robust joint chance constraint under multiple uncertainties. Comput Chem Eng 2023. [DOI: 10.1016/j.compchemeng.2023.108156] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023]
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9
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Persistence in a large network of sparsely interacting neurons. J Math Biol 2022; 86:16. [PMID: 36534174 DOI: 10.1007/s00285-022-01844-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2021] [Revised: 09/26/2022] [Accepted: 11/22/2022] [Indexed: 12/23/2022]
Abstract
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.
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10
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Niles-Weed J, Rigollet P. Estimation of Wasserstein distances in the Spiked Transport Model. BERNOULLI 2022. [DOI: 10.3150/21-bej1433] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Affiliation(s)
- Jonathan Niles-Weed
- Courant Institute of Mathematical Sciences & Center for Data Science, New York University, 251 Mercer Street, New York, NY 10012-1185, USA
| | - Philippe Rigollet
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
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11
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Affiliation(s)
- Johannes C.W. Wiesel
- Columbia University, Department of Statistics, 1255 Amsterdam Avenue, New York, NY 10027, USA
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12
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Riekert A. Convergence rates for empirical measures of Markov chains in dual and Wasserstein distances. Stat Probab Lett 2022. [DOI: 10.1016/j.spl.2022.109605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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13
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Huesmann M, Mattesini F, Trevisan D. Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.09.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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14
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Rate of convergence for particle approximation of PDEs in Wasserstein space. J Appl Probab 2022. [DOI: 10.1017/jpr.2021.102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Abstract
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.
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15
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Goldman M, Huesmann M. A fluctuation result for the displacement in the optimal matching problem. ANN PROBAB 2022. [DOI: 10.1214/21-aop1562] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Michael Goldman
- Laboratoire Jacques-Louis Lions (LJLL), CNRS, Université de Paris
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16
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Niles-Weed J, Berthet Q. Minimax estimation of smooth densities in Wasserstein distance. Ann Stat 2022. [DOI: 10.1214/21-aos2161] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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17
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Padoan SA, Rizzelli S. Consistency of Bayesian inference for multivariate max-stable distributions. Ann Stat 2022. [DOI: 10.1214/21-aos2160] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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18
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Chassagneux JF, Szpruch L, Tse A. Weak quantitative propagation of chaos via differential calculus on the space of measures. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1725] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | | | - Alvin Tse
- School of Mathematics, University of Edinburgh
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19
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Erny X. Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients. Stoch Process Their Appl 2022. [DOI: 10.1016/j.spa.2022.04.012] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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20
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Wang P, Ji H, Liu L. Consistent fusion method with uncertainty elimination for distributed multi-sensor systems. Inf Sci (N Y) 2022. [DOI: 10.1016/j.ins.2022.03.011] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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21
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Imaizumi M, Ota H, Hamaguchi T. Hypothesis Test and Confidence Analysis with Wasserstein Distance on General Dimension. Neural Comput 2022; 34:1448-1487. [PMID: 35534006 DOI: 10.1162/neco_a_01501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2021] [Accepted: 02/01/2022] [Indexed: 11/04/2022]
Abstract
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.
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Affiliation(s)
- Masaaki Imaizumi
- University of Tokyo Meguro, Tokyo 153-0041, Japan.,RIKEN Center for Advanced Intelligence Project, Chuo, Tokyo, 103-0027, Japan
| | - Hirofumi Ota
- Rutgers University, Piscataway, NJ 08854. U.S.A.
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22
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Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization. SUSTAINABILITY 2022. [DOI: 10.3390/su14084501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
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.
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23
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Ghosal P, Sen B. Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing. Ann Stat 2022. [DOI: 10.1214/21-aos2136] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Promit Ghosal
- Department of Mathematics, Massachusetts Institute of Technology
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24
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Motte M, Pham H. Mean-field Markov decision processes with common noise and open-loop controls. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1713] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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25
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26
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Probabilistic analysis of replicator–mutator equations. ADV APPL PROBAB 2022. [DOI: 10.1017/apr.2021.22] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
AbstractThis 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.
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27
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Measure estimation on manifolds: an optimal transport approach. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01118-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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28
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29
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Backhoff J, Bartl D, Beiglböck M, Wiesel J. Estimating processes in adapted Wasserstein distance. ANN APPL PROBAB 2022. [DOI: 10.1214/21-aap1687] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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30
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Joint access point fuzzy rough set reduction and multisource information fusion for indoor Wi-Fi positioning. Neural Comput Appl 2022. [DOI: 10.1007/s00521-021-05934-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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31
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Journel L, Monmarché P. Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: Uniform estimates in a compact soft case. ESAIM-PROBAB STAT 2022. [DOI: 10.1051/ps/2021017] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
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.
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32
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Galeati L, Harang FA, Mayorcas A. Distribution dependent SDEs driven by additive continuous noise. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp756] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Lucio Galeati
- Institute of Applied Mathematics, University of Bonn, 53115 Endenicher Allee 60, Bonn, Germany
| | - Fabian A. Harang
- Department of Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316, Oslo, Norway
| | - Avi Mayorcas
- Centre for Mathematical Sciences, Wilberforce Rd, Cambridge CB3 0WA, UK
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33
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Clarté G, Diez A, Feydy J. Collective proposal distributions for nonlinear MCMC samplers: Mean-field theory and fast implementation. Electron J Stat 2022. [DOI: 10.1214/22-ejs2091] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Affiliation(s)
- Grégoire Clarté
- Department of Computer Science, University of Helsinki, FCAI
| | - Antoine Diez
- Department of Mathematics, Imperial College London, UK
| | - Jean Feydy
- HeKA team, Inria Paris, F-75012 Paris, France
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34
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Laurière M, Tangpi L. Backward propagation of chaos. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp777] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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35
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Lacker D. Quantitative approximate independence for continuous mean field Gibbs measures. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp743] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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36
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Manole T, Balakrishnan S, Wasserman L. Minimax confidence intervals for the Sliced Wasserstein distance. Electron J Stat 2022. [DOI: 10.1214/22-ejs2001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Tudor Manole
- Department of Statistics and Data Science, Carnegie Mellon University
| | | | - Larry Wasserman
- Department of Statistics and Data Science, Carnegie Mellon University
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37
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Reygner J, Touboul A. Reweighting samples under covariate shift using a Wasserstein distance criterion. Electron J Stat 2022. [DOI: 10.1214/21-ejs1974] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Adrien Touboul
- CERMICS, Ecole des Ponts, Marne-la-Vallée, France IRT SystemX, Paris-Saclay, France
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38
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Fontbona J, Muñoz-Hernández F. Quantitative mean-field limit for interacting branching diffusions. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp874] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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39
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Bartl D, Drapeau S, Obłój J, Wiesel J. Sensitivity analysis of Wasserstein distributionally robust optimization problems. Proc Math Phys Eng Sci 2021; 477:20210176. [PMID: 35153602 PMCID: PMC8670962 DOI: 10.1098/rspa.2021.0176] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2021] [Accepted: 11/09/2021] [Indexed: 11/12/2022] Open
Abstract
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.
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Affiliation(s)
- Daniel Bartl
- Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
| | - Samuel Drapeau
- School of Mathematical Sciences & Shanghai Advanced Institute of Finance, Shanghai Jiao Tong University, 211 West Huaihai Road, Shanghai 200030, People’s Republic of China
| | - Jan Obłój
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
| | - Johannes Wiesel
- Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA
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40
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Bobkov SG, Ledoux M. A simple Fourier analytic proof of the AKT optimal matching theorem. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1656] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Michel Ledoux
- Institut de Mathématiques de Toulouse, Université de Toulouse—Paul-Sabatier
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41
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On the capacity of deep generative networks for approximating distributions. Neural Netw 2021; 145:144-154. [PMID: 34749027 DOI: 10.1016/j.neunet.2021.10.012] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2021] [Revised: 10/01/2021] [Accepted: 10/14/2021] [Indexed: 11/22/2022]
Abstract
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.
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42
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Benth FE, Di Nunno G, Schroers D. Copula measures and Sklar's theorem in arbitrary dimensions. Scand Stat Theory Appl 2021. [DOI: 10.1111/sjos.12559] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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43
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Szpruch Ł, Tse A. Antithetic multilevel sampling method for nonlinear functionals of measure. ANN APPL PROBAB 2021. [DOI: 10.1214/20-aap1614] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
| | - Alvin Tse
- School of Mathematics, University of Edinburgh
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Feng W, Feng Y, Zhang Q. Multistage distributionally robust optimization for integrated production and maintenance scheduling. AIChE J 2021. [DOI: 10.1002/aic.17329] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Affiliation(s)
- Wei Feng
- State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering Zhejiang University Hangzhou China
| | - Yiping Feng
- State Key Laboratory of Industrial Control Technology, College of Control Science and Engineering Zhejiang University Hangzhou China
| | - Qi Zhang
- Department of Chemical Engineering and Materials Science University of Minnesota Minneapolis Minnesota USA
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Nonparametric estimation for interacting particle systems: McKean–Vlasov models. Probab Theory Relat Fields 2021. [DOI: 10.1007/s00440-021-01044-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Srivastava S, Xu Y. Distributed Bayesian Inference in Linear Mixed-Effects Models. J Comput Graph Stat 2021. [DOI: 10.1080/10618600.2020.1869025] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Sanvesh Srivastava
- Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA
| | - Yixiang Xu
- Marketing Group, Haas School of Business, University of California, Berkeley, Berkeley, CA
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Bailleul I, Catellier R, Delarue F. Propagation of chaos for mean field rough differential equations. ANN PROBAB 2021. [DOI: 10.1214/20-aop1465] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Frikha N, Li L. Well-posedness and approximation of some one-dimensional Lévy-driven non-linear SDEs. Stoch Process Their Appl 2021. [DOI: 10.1016/j.spa.2020.10.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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