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It has long been thought that high-dimensional data encountered in many practical machine learning tasks have low-dimensional structure, i.e., the manifold hypothesis holds.
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Analysis and Geometry of Markov Diffusion Operators
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Probability in high dimension
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Dimensionality estimation without distances
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Bickel, Peter J, Li, Bo, et al · 2007
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Manifold-adaptive dimension estimation
Farahmand, Amir Massoud, Szepesvári, Csaba, & Audibert, Jean-Yves. 2007 · 2007
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Random projection trees and low dimensional manifolds
Dasgupta, Sanjoy, & Freund, Yoav. 2008 · 2008
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Finding the homology of submanifolds with high confidence from random samples
Niyogi, Partha, Smale, Stephen, & Weinberger, Shmuel. 2008 · 2008
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Optimal transport: old and new
Villani, Cédric. 2008 · 2008
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A characterization of dimension free concentration in terms of transportation inequalities
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Camastra, Francesco, & Staiano, Antonino. 2016 · 2016
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Testing the manifold hypothesis
Fefferman, Charles, Mitter, Sanjoy, & Narayanan, Hariharan. 2016 · 2016
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Mathematical foundations of infinite-dimensional statistical models
Giné, Evarist, & Nickl, Richard. 2016 · 2016
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Adaptive metric dimensionality reduction
Gottlieb, Lee-Ad, Kontorovich, Aryeh, & Krauthgamer, Robert. 2016 · 2016
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Wasserstein generative adversarial networks
Arjovsky, Martin, Chintala, Soumith, & Bottou, Léon. 2017 · 2017
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Fitting a putative manifold to noisy data
Fefferman, Charles, Ivanov, Sergei, Kurylev, Yaroslav, Lassas, Matti, & Narayanan, Hariharan. 2018 · 2018
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Introduction to Riemannian manifolds
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Liang, Tengyuan. 2018 · 2018
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Estimating the reach of a manifold
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The reach, metric distortion, geodesic convexity and the variation of tangent spaces
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Adaptive nonparametric clustering
Efimov, Kirill, Adamyan, Larisa, & Spokoiny, Vladimir. 2019 · 2019
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Minimax Rates for Estimating the Dimension of a Manifold
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Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
Weed, Jonathan, Bach, Francis, et al · 2019
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Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation
Chen, Minshuo, Liao, Wenjing, Zha, Hongyuan, & Zhao, Tuo. 2020 · 2020
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Adaptive Approximation and Generalization of Deep Neural Network with Intrinsic Dimensionality
Nakada, Ryumei, & Imaizumi, Masaaki. 2020 · 2020
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Functions with average smoothness: structure, algorithms, and learning
Ashlagi, Yair, Gottlieb, Lee-Ad, & Kontorovich, Aryeh. 2021 · 2021
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