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We prove a generalisation of Fernique’s theorem which applies to a class of (measurable) functionals on abstract Wiener spaces by using the isoperimetric inequality. Our motivation comes from rough path theory where one deals with iterated integrals of Gaussian processes (which are generically not Gaussian). Gaussian integrability with explicitly given constants for variation and Hölder norms of the (fractional) Brownian rough path, Gaussian rough paths and the Banach space valued Wiener process enhanced with its Lévy area [Ledoux, Lyons, Qian. “Lévy area of Wiener processes in Banach spaces”, Ann. Probab., 30(2):546–578, 2002] then all follow from applying our main theorem.

The volume contains the proceedings of the international workshop on Concentration, Functional Inequalities and Isoperimetry, held at Florida Atlantic University in Boca Raton, Florida, from October 29-November 1, 2009. The interactions between concentration, isoperimetry and functional inequalities have led to many significant advances in functional analysis and probability theory. Important progress has also taken place in combinatorics, geometry, harmonic analysis and mathematical physics, to name but a few fields, with recent new applications in random matrices and information theory. This book should appeal to graduate students and researchers interested in the fascinating interplay between analysis, probability, and geometry.

In this paper, we give three different characterizations for the boundedness and compactness of generalized weighted composition operators from the space of bounded analytic function to the logarithmic Bloch space.

This volume contains the proceedings of the International Workshop on Tropical and Idempotent Mathematics, held at the Independent University of Moscow, Russia, from August 26-31, 2012. The main purpose of the conference was to bring together and unite researchers and specialists in various areas of tropical and idempotent mathematics and applications. This volume contains articles on algebraic foundations of tropical mathematics as well as articles on applications of tropical mathematics in various fields as diverse as economics, electroenergetic networks, chemical reactions, representation theory, and foundations of classical thermodynamics. This volume is intended for graduate students and researchers interested in tropical and idempotent mathematics or in their applications in other areas of mathematics and in technical sciences.

This article discusses minimal model explanations, which we argue are distinct from various causal, mechanical, difference-making, and so on, strategies prominent in the philosophical literature. We contend that what accounts for the explanatory power of these models is not that they have certain features in common with real systems. Rather, the models are explanatory because of a story about why a class of systems will all display the same large-scale behavior because the details that distinguish them are irrelevant. This story explains patterns across extremely diverse systems and shows how minimal models can be used to understand real systems.

Estimation of structure, such as in variable selection, graphical modelling or cluster analysis, is notoriously difficult, especially for high dimensional data. We introduce stability selection. It is based on subsampling in combination with (high dimensional) selection algorithms. As such, the method is extremely general and has a very wide range of applicability. Stability selection provides finite sample control for some error rates of false discoveries and hence a transparent principle to choose a proper amount of regularization for structure estimation. Variable selection and structure estimation improve markedly for a range of selection methods if stability selection is applied. We prove for the randomized lasso that stability selection will be variable selection consistent even if the necessary conditions for consistency of the original lasso method are violated. We demonstrate stability selection for variable selection and Gaussian graphical modelling, using real and simulated data.

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive.

This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [J. Assoc. Comput. Mach., 12 (1965), pp. 547–560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anderson mixing; however, it seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities, this method has received relatively little attention from these communities over the years. A recent paper by H. Fang and Y. Saad [Numer. Linear Algebra Appl., 16 (2009), pp. 197–221] has clarified a remarkable relationship of Anderson acceleration to quasi-Newton (secant updating) methods and extended it to define a broader Anderson family of acceleration methods. In this paper, our goals are to shed additional light on Anderson acceleration and to draw further attention to its usefulness as a general tool. We first show that, on linear problems, Anderson acceleration without truncation is \"essentially equivalent\" in a certain sense to the generalized minimal residual (GMRES) method. We also show that the Type 1 variant in the Fang—Saad Anderson family is similarly essentially equivalent to the Arnoldi (full orthogonalization) method. We then discuss practical considerations for implementing Anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications.

We study an instance of high-dimensional inference in which the goal is to estimate a matrix Θ* ∈ ℝ m₁ × m₂ on the basis of N noisy observations. The unknown matrix Θ* is assumed to be either exactly low rank, or \"near\" low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider a standard M-estimator based on regularization by the nuclear or trace norm over matrices, and analyze its performance under high-dimensional scaling. We define the notion of restricted strong convexity (RSC) for the loss function, and use it to derive nonasymptotic bounds on the Frobenius norm error that hold for a general class of noisy observation models, and apply to both exactly low-rank and approximately low rank matrices. We then illustrate consequences of this general theory for a number of specific matrix models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes and recovery of low-rank matrices from random projections. These results involve nonasymptotic random matrix theory to establish that the RSC condition holds, and to determine an appropriate choice of regularization parameter. Simulation results show excellent agreement with the high-dimensional scaling of the error predicted by our theory.