Stein’s method in high dimensions with applications

Adrian Röllin

Displaying similar documents to “Stein’s method in high dimensions with applications”

Estimating composite functions by model selection

Yannick Baraud, Lucien Birgé (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the problem of estimating a function $s$ on ${[- 1, 1]}^{k}$ for large values of $k$ by looking for some best approximation of $s$ by composite functions of the form $g \circ u$ . Our solution is based on model selection and leads to a very general approach to solve this problem with respect to many different types of functions $g, u$ and statistical frameworks. In particular, we handle the problems of approximating $s$ by additive functions, single and multiple index models, artificial neural networks, mixtures...

Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality

Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2013)

Journal of the European Mathematical Society

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We considerably improve upon the recent result of [37] on the mixing time of Glauber dynamics for the 2D Ising model in a box of side $L$ at low temperature and with random boundary conditions whose distribution $P$ stochastically dominates the extremal plus phase. An important special case is when $P$ is concentrated on the homogeneous all-plus configuration, where the mixing time $T_{M I X}$ is conjectured to be polynomial in $L$ . In [37] it was shown that for a large enough inverse-temperature $β$ and...

Gaussian approximation of Gaussian scale mixtures

Gérard Letac, Hélène Massam (2020)

Kybernetika

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For a given positive random variable $V > 0$ and a given $Z \sim N (0, 1)$ independent of $V$ , we compute the scalar $t_{0}$ such that the distance in the $L^{2} (ℝ)$ sense between $Z V^{1 / 2}$ and $Z \sqrt{t_{0}}$ is minimal. We also consider the same problem in several dimensions when $V$ is a random positive definite matrix.

Logarithmically improved blow-up criterion for smooth solutions to the Leray- $α$ -magnetohydrodynamic equations

Ines Ben Omrane, Sadek Gala, Jae-Myoung Kim, Maria Alessandra Ragusa (2019)

Archivum Mathematicum

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In this paper, the Cauchy problem for the $3 D$ Leray- $α$ -MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray- $α$ -MHD model in terms of the magnetic field $B$ only in the framework of homogeneous Besov space with negative index.

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

Balcar's theorem on supports

Lev Bukovský (2018)

Commentationes Mathematicae Universitatis Carolinae

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In A theorem on supports in the theory of semisets [Comment. Math. Univ. Carolinae 14 (1973), no. 1, 1–6] B. Balcar showed that if $σ \subseteq D \in M$ is a support, $M$ being an inner model of ZFC, and $𝒫 (D ∖ σ) \cap M = r ` ` σ$ with $r \in M$ , then $r$ determines a preorder " $⪯$ " of $D$ such that $σ$ becomes a filter on $(D, ⪯)$ generic over $M$ . We show that if the relation $r$ is replaced by a function $𝒫 (D ∖ σ) \cap M = f_{- 1} (σ)$ , then there exists an equivalence relation " $\sim$ " on $D$ and a partial order on $D / \sim$ such that $D / \sim$ is a complete Boolean algebra, $σ / \sim$ is a generic filter and ${[f (u)]}_{\sim} = - \sum (u / \sim)$ for...

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

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Consider the classical $(2 + 1)$ -dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $ℤ^{2}$ . The model describes a crystal surface by assigning a non-negative integer height $η_{x}$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp (- β ℋ (η))$ , where $β$ is the inverse-temperature and $ℋ (η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough...

Scale-free percolation

Maria Deijfen, Remco van der Hofstad, Gerard Hooghiemstra (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We formulate and study a model for inhomogeneous long-range percolation on $ℤ^{d}$ . Each vertex $x \in ℤ^{d}$ is assigned a non-negative weight $W_{x}$ , where ${(W_{x})}_{x \in ℤ}^{d}$ are i.i.d. random variables. Conditionally on the weights, and given two parameters $α, λ g t; 0$ , the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{x y} = 1 - exp {- λ W_{x} W_{y} {/ | x - y |}^{α}}$ . The parameter $λ$ is the percolation parameter, while $α$ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether...

Uniform mixing time for random walk on lamplighter graphs

Júlia Komjáthy, Jason Miller, Yuval Peres (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$ . The lamplighter chain $X^{⋄}$ associated with $X$ is the random walk on the wreath product $𝒢^{⋄} = 𝐙_{2} ≀ 𝒢$ , the graph whose vertices consist of pairs $(\underset{̲}{f}, x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of $𝐙_{2} = {0, 1}$ and $x$ is a vertex in $𝒢$ . There is an edge between $(\underset{̲}{f}, x)$ and $(\underset{̲}{g}, y)$ in $𝒢^{⋄}$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_{z} = g_{z}$ for all $z \neq x, y$ . In each step, $X^{⋄}$ moves from a configuration $(\underset{̲}{f}, x)$ by updating $x$ to $y$ using the transition rule of $X$ and then...

Nonconventional limit theorems in averaging

Yuri Kifer (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider “nonconventional” averaging setup in the form $\frac{d X^{ε} (t)}{d t} = ε B (X^{ε} (t)$ , $𝛯 (q_{1} (t)), 𝛯 (q_{2} (t)), ..., 𝛯 (q_{ℓ} (t)))$ where $𝛯 (t)$ , $t \geq 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_{j} (t) = α_{j} t$ , $α_{1} l t; α_{2} l t; \dots l t; α_{k}$ and $q_{j}$ , $j = k + 1, ..., ℓ$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

$𝒞^{k}$ -regularity for the $\bar{\partial}$ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let $D$ be a $𝒞^{d}$ $q$ -convex intersection, $d \geq 2$ , $0 \leq q \leq n - 1$ , in a complex manifold $X$ of complex dimension $n$ , $n \geq 2$ , and let $E$ be a holomorphic vector bundle of rank $N$ over $X$ . In this paper, $𝒞^{k}$ -estimates, $k = 2, 3, \dots, \infty$ , for solutions to the $\bar{\partial}$ -equation with small loss of smoothness are obtained for $E$ -valued $(0, s)$ -forms on $D$ when $n - q \leq s \leq n$ . In addition, we solve the $\bar{\partial}$ -equation with a support condition in $𝒞^{k}$ -spaces. More precisely, we prove that for a $\bar{\partial}$ -closed form $f$ in $𝒞_{0, q}^{k} (X ∖ D, E)$ , $1 \leq q \leq n - 2$ , $n \geq 3$ , with compact support and for $ε$ with $0 < ε < 1$ there...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let $Ω$ be a bounded domain of class $C^{2}$ in $ℝ$ N and let $K$ be a compact subset of $\partial Ω$ . Assume that $q \geq (N + 1) / (N - 1)$ and denote by $U_{K}$ the maximal solution of $- Δ u + u^{q} = 0$ in $Ω$ which vanishes on $\partial Ω ∖ K$ . We obtain sharp upper and lower estimates for $U_{K}$ in terms of the Bessel capacity $C_{2 / q, q^{'}}$ and prove that $U_{K}$ is $σ$ -moderate. In addition we describe the precise asymptotic behavior of $U_{K}$ at points $σ \in K$ , which depends on the “density” of $K$ at $σ$ , measured in terms of the capacity $C_{2 / q, q^{'}}$ .

Distortion and spreading models in modified mixed Tsirelson spaces

S. A. Argyros, I. Deliyanni, A. Manoussakis (2003)

Studia Mathematica

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The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], $θ_{n + m} \geq θ ₙ θ ₘ$ and $l i m_{n} θ ₙ^{1 / n} = 1$ , admits an $ℓ ₁^{ω}$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $l i m_{n} θ ₙ^{1 / n} = 1$ , such that, for every n ∈ ℕ, $| | \sum_{k = 1}^{m} x_{k} | | \geq θ ₙ \sum_{k = 1}^{m} | | x_{k} | |$ for every ₙ-admissible block sequence ${(x_{k})}_{k = 1}^{m}$ of vectors in X, then there exists c...

$C^{*}$ -points vs $P$ -points and $P^{♭}$ -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space $X$ , the point $p \in X$ is called a $C^{*}$ -point if every real-valued continuous function on $C ∖ {p}$ can be extended continuously to $p$ . Every point in an extremally disconnected space is a $C^{*}$ -point. A classic example is the space $𝐖^{*} = ω_{1} + 1$ consisting of the countable ordinals together with $ω_{1}$ . The point $ω_{1}$ is known to be a $C^{*}$ -point as well as a $P$ -point. We supply a characterization of $C^{*}$ -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...