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Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to its cusps follows readily. The main ingredient is a uniformly bilipschitz model for the quotient of ℍ 3 by a Kleinian surface group.

We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.

Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasible method to discover these communities. The stochastic blockmodel [Social Networks 5 (1983) 109–137] is a social network model with well-defined communities; each node is a member of one community. For a network generated from the Stochastic Blockmodel, we bound the number of nodes \"misclustered\" by spectral clustering. The asymptotic results in this paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the stochastic blockmodel, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a \"population\" normalized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original.

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, that is, sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator −d2dx2+x+2βbx′-\\frac {d^2}{dx^2} + x + \\frac {2}{\\sqrt {\\beta }} b_x^{\\prime } restricted to the positive half-line, where bx′b_x^{\\prime } is white noise. In doing so we extend the definition of the Tracy-Widom(β\\beta) distributions to all β>0\\beta >0 and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converges to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

We consider a random planar map M n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m n be the vertex set of M n , which is equipped with the graph distance d gr . Both when q ≥ 4 is an even integer and when q = 3, there exists a positive constant c q such that the rescaled metric spaces (m n , c q n -1/4 d gr ) converge in distribution in the Gromov—Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

We consider sequences of graphs (G n ) and define various notions of convergence related to these sequences including \"left-convergence,\" defined in terms of the densities of homomorphisms from small graphs into G n , and \"right-convergence,\" defined in terms of the densities of homomorphisms from G n into small graphs. We show that right-convergence is equivalent to left-convergence, both for simple graphs G n , and for graphs G n with nontrivial nodeweights and edgeweights. Other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics, such as maximum cuts and Szemerédi partitions, and fundamental notions from statistical physics, like energies and free energies. We thereby relate local and global properties of graph sequences. Quantitative forms of these results express the relationships among different measures of similarity of large graphs.

This paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and Bregman divergence losses. A major focus is on the derivation of a rate sharp minimax lower bound. The problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems. Standard techniques fail to yield good results, and new tools are thus needed. We first develop a lower bound technique that is particularly well suited for treating \"two-directional\" problems such as estimating sparse covariance matrices. The result can be viewed as a generalization of Le Cam's method in one direction and Assouad's Lemma in another. This lower bound technique is of independent interest and can be used for other matrix estimation problems. We then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique. A thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm. The results are then extended to the general matrix l w operator norms for 1 ≤ w ≤ ∞. In addition, we give a unified result on the minimax rate of convergence for sparse covariance matrix estimation under a class of Bregman divergence losses.

We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f(x) + Q(x, v) + g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to \"metrically regular\" problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = || JC — y||² and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with \"regular\" intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l¹ algorithm for compressive sensing and an application to rank reduction problems.