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An Elementary Introduction to Modern Convex Geometry
 in Flavors of Geometry
, 1997
"... Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality and It ..."
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Cited by 171 (3 self)
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Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality and Its Extensions 25 Lecture 6. Convolutions and Volume Ratios: The Reverse Isoperimetric Problem 32 Lecture 7. The Central Limit Theorem and Large Deviation Inequalities 37 Lecture 8. Concentration of Measure in Geometry 41 Lecture 9. Dvoretzky's Theorem 47 Acknowledgements 53 References 53 Index 55 Preface These notes are based, somewhat loosely, on three series of lectures given by myself, J. Lindenstrauss and G. Schechtman, during the Introductory Workshop in Convex Geometry held at the Mathematical Sciences Research Institute in Berkeley, early in 1996. A fourth series was given by B. Bollobas, on rapid mixing and random volume algorithms; they are found els
Gaussian processes for ordinal regression
 Journal of Machine Learning Research
, 2004
"... We present a probabilistic kernel approach to ordinal regression based on Gaussian processes. A threshold model that generalizes the probit function is used as the likelihood function for ordinal variables. Two inference techniques, based on the Laplace approximation and the expectation propagation ..."
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Cited by 117 (4 self)
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We present a probabilistic kernel approach to ordinal regression based on Gaussian processes. A threshold model that generalizes the probit function is used as the likelihood function for ordinal variables. Two inference techniques, based on the Laplace approximation and the expectation propagation algorithm respectively, are derived for hyperparameter learning and model selection. We compare these two Gaussian process approaches with a previous ordinal regression method based on support vector machines on some benchmark and realworld data sets, including applications of ordinal regression to collaborative filtering and gene expression analysis. Experimental results on these data sets verify the usefulness of our approach.
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
, 2001
"... A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian ..."
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Cited by 87 (9 self)
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A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.
H.T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
"... We consider N × N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. We study the connection between eigenvalue statistics on microscopic energy scales η ≪ 1 and (de)localization properties of the eigen ..."
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Cited by 80 (13 self)
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We consider N × N Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. We study the connection between eigenvalue statistics on microscopic energy scales η ≪ 1 and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order η ∼ logN/N. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales η ≫ N −2/3. We show that most eigenvectors are fully delocalized in the sense that their ℓ pnorms are comparable with N 1/p−1/2 for p ≥ 2, and we obtain the weaker bound N 2/3(1/p−1/2) for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.
Aggregation and Social Choice: A Mean Voter Theorem
 Econometrica
, 1991
"... COWLES FOUNDATION DISCUSSION PAPER NO. 938 ..."
Spectral Statistics of ErdősRényi Graphs I: Local Semicircle Law
, 2011
"... We consider the ensemble of adjacency matrices of ErdősRényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least ..."
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Cited by 52 (18 self)
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We consider the ensemble of adjacency matrices of ErdősRényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N), the density of eigenvalues of the ErdősRényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N −1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ℓ ∞norms of the ℓ 2normalized eigenvectors are at most of order N −1/2 with a very high probability. The estimates in this paper will be used in the companion paper [13] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN ≫ N 2/3.
On Homogenization and Scaling Limit of Some Gradient Perturbations of a Massless Free Field
, 1995
"... : We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) co ..."
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Cited by 47 (6 self)
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: We study the continuum scaling limit of some statistical mechanical models defined by convex Hamiltonians which are gradient perturbations of a massless free field. By proving a central limit theorem for these models, we show that their long distance behavior is identical to a new (homogenized) continuum massless free field. We shall also obtain some new bounds on the 2point correlation functions of these models. 1. Introduction and Statement of the Main Results In this article, we study the long distance behavior of (lattice) statistical mechanical models defined by convex Hamiltonians H(') which are gradient perturbations of a massless free Gaussian. Under certain assumptions (see (H1) and (H2) below), we shall prove a central limit theorem for these models and show that their behavior at long distances is governed by suitable continuum massless Gaussians. The main idea is that these statistical mechanical models can be expressed (following Helffer and Sjostrand [8, 12]) in te...
The Cauchy Process and the Steklov Problem
"... Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimen ..."
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Cited by 40 (16 self)
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Let X t be a Cauchy process in R , d 1. We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the "Mixed Steklov Problem." Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (1, 1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric #stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not rarely available for more general symmetric #stable processes. #
Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes
 Comm. Math. Phys
"... Abstract. We consider the following eigenvalue optimization problem: Given a bounded domain Ω ⊂ R n and numbers α ≥ 0, A ∈ [0, Ω], find a subset D ⊂ Ω of area A for which the first Dirichlet eigenvalue of the operator − ∆ + αχD is as small as possible. We prove existence of solutions and investiga ..."
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Cited by 37 (9 self)
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Abstract. We consider the following eigenvalue optimization problem: Given a bounded domain Ω ⊂ R n and numbers α ≥ 0, A ∈ [0, Ω], find a subset D ⊂ Ω of area A for which the first Dirichlet eigenvalue of the operator − ∆ + αχD is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than Ω; on the other hand, for convex Ω reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.