The Experts below are selected from a list of 1929 Experts worldwide ranked by ideXlab platform
A. Wilkinson - One of the best experts on this subject based on the ideXlab platform.
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Symplectomorphisms with positive Metric Entropy
arXiv: Dynamical Systems, 2019Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Ma\~n\'e. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
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Diffeomorphisms with positive Metric Entropy
Publications mathématiques de l'IHÉS, 2016Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for C 1 $C^{1}$ -generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.
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diffeomorphisms with positive Metric Entropy
Publications Mathématiques de l'IHÉS, 2016Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mane.
A. Avila - One of the best experts on this subject based on the ideXlab platform.
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Symplectomorphisms with positive Metric Entropy
arXiv: Dynamical Systems, 2019Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Ma\~n\'e. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
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Diffeomorphisms with positive Metric Entropy
Publications mathématiques de l'IHÉS, 2016Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for C 1 $C^{1}$ -generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mañé.
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diffeomorphisms with positive Metric Entropy
Publications Mathématiques de l'IHÉS, 2016Co-Authors: A. Avila, S. Crovisier, A. WilkinsonAbstract:We obtain a dichotomy for \(C^{1}\)-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and unstable spaces is dominated). This completes a program first put forth by Ricardo Mane.
Yuhong Yang - One of the best experts on this subject based on the ideXlab platform.
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Metric Entropy and sparse linear approximation of l q hulls for 0 q 1
Journal of Approximation Theory, 2013Co-Authors: Yuhong YangAbstract:Abstract Consider l q -hulls, 0 q ≤ 1 , from a dictionary of M functions in L p space for 1 ≤ p ∞ . Their precise Metric Entropy orders are derived. Sparse linear approximation bounds are obtained to characterize the number of terms needed to achieve accurate approximation of the best function in a l q -hull that is closest to a target function. Furthermore, in the special case of p = 2 , it is shown that a weak orthogonal greedy algorithm achieves the optimal approximation under an additional condition.
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Metric Entropy and sparse linear approximation of l q -hulls for 0
Journal of Approximation Theory, 2013Co-Authors: Yuhong YangAbstract:Abstract Consider l q -hulls, 0 q ≤ 1 , from a dictionary of M functions in L p space for 1 ≤ p ∞ . Their precise Metric Entropy orders are derived. Sparse linear approximation bounds are obtained to characterize the number of terms needed to achieve accurate approximation of the best function in a l q -hull that is closest to a target function. Furthermore, in the special case of p = 2 , it is shown that a weak orthogonal greedy algorithm achieves the optimal approximation under an additional condition.
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information theoretic determination of minimax rates of convergence
Annals of Statistics, 1999Co-Authors: Yuhong Yang, Andrew R. BarronAbstract:We present some general results determining minimax bounds on statistical risk for density estimation based on certain information-theoretic considerations. These bounds depend only on Metric Entropy conditions and are used to identify the minimax rates of convergence.
Wenbo V Li - One of the best experts on this subject based on the ideXlab platform.
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Small Deviations of Stable Processes via Metric Entropy
Journal of Theoretical Probability, 2020Co-Authors: Wenbo V Li, Werner LindeAbstract:Let X=(X(t))t∈T be a symMetric α-stable, 0
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Metric Entropy of high dimensional distributions
Proceedings of the American Mathematical Society, 2007Co-Authors: Ron C Blei, Wenbo V LiAbstract:Let F d be the collection of all d-dimensional probability distribution functions on [0, 1] d , d > 2. The Metric Entropy of F d under the L 2 ([0,1] d ) norm is studied. The exact rate is obtained for d = 1, 2 and bounds are given for d > 3. Connections with small deviation probability for Brownian sheets under the sup-norm are established.
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small deviations of stable processes via Metric Entropy
Journal of Theoretical Probability, 2004Co-Authors: Wenbo V Li, Werner LindeAbstract:Let X=(X(t))t∈T be a symMetric α-stable, 0<α<2, process with paths in the dual E* of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L α (S,σ) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of \(\phi (\varepsilon ) = - \log \mathbb{P}(\left\| X \right\|_{E^* } < \varepsilon )\) as e→0. In particular, we prove that a lower bound for the Metric Entropy of u implies a lower bound for φ(e) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted α-stable Levy motions, linear fractional α-stable motions and d-dimensional α-stable sheets. Our results rest upon an integral representation of L α -valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.
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approximation Metric Entropy and small ball estimates for gaussian measures
Annals of Probability, 1999Co-Authors: Wenbo V Li, Werner LindeAbstract:A precise link proved by Kuelbs and Li relates the small ball behavior of a Gaussian measure μ on a Banach space E with the Metric Entropy behavior of K μ , the unit ball of the reproducing kernel Hilbert space of μ in E. We remove the main regularity assumption imposed on the unknown function in the link. This enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete Gaussian processes. Moreover, we show that the small ball behavior of a Gaussian process is also tightly connected with the speed of approximation by finite rank processes.
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Metric Entropy and the small ball problem for gaussian measures
Journal of Functional Analysis, 1993Co-Authors: James Kuelbs, Wenbo V LiAbstract:Abstract We establish a precise link between the small ball problem for a Gaussian measure μ on a separable Banach space and the Metric Entropy of the unit ball of the Hubert space H μ generating μ. This link allows us to compute small ball probabilities from Metric Entropy results, and vice versa.
Andrew R. Barron - One of the best experts on this subject based on the ideXlab platform.
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complexity statistical risk and Metric Entropy of deep nets using total path variation
arXiv: Machine Learning, 2019Co-Authors: Andrew R. Barron, Jason M KlusowskiAbstract:For any ReLU network there is a representation in which the sum of the absolute values of the weights into each node is exactly $1$, and the input layer variables are multiplied by a value $V$ coinciding with the total variation of the path weights. Implications are given for Gaussian complexity, Rademacher complexity, statistical risk, and Metric Entropy, all of which are shown to be proportional to $V$. There is no dependence on the number of nodes per layer, except for the number of inputs $d$. For estimation with sub-Gaussian noise, the mean square generalization error bounds that can be obtained are of order $V \sqrt{L + \log d}/\sqrt{n}$, where $L$ is the number of layers and $n$ is the sample size.
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information theoretic determination of minimax rates of convergence
Annals of Statistics, 1999Co-Authors: Yuhong Yang, Andrew R. BarronAbstract:We present some general results determining minimax bounds on statistical risk for density estimation based on certain information-theoretic considerations. These bounds depend only on Metric Entropy conditions and are used to identify the minimax rates of convergence.