Seminars and Colloquia by Series

Chaining, interpolation, and convexity

Series
Stochastics Seminar
Time
Thursday, January 14, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ramon van Handel – Princeton University
A significant achievement of modern probability theory is the development of sharp connections between the boundedness of random processes and the geometry of the underlying index set. In particular, the generic chaining method of Talagrand provides in principle a sharp understanding of the suprema of Gaussian processes. The multiscale geometric structure that arises in this method is however notoriously difficult to control in any given situation. In this talk, I will exhibit a surprisingly simple but very general geometric construction, inspired by real interpolation of Banach spaces, that is readily amenable to explicit computations and that explains the behavior of Gaussian processes in various interesting situations where classical entropy methods are known to fail. (No prior knowledge of this topic will be assumed in the talk.)

ShapeFit: Exact location recovery from corrupted pairwise directions

Series
Stochastics Seminar
Time
Thursday, December 3, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Paul Hand – Rice University –
We consider the problem of recovering a set of locations given observations of the direction between pairs of these locations. This recovery task arises from the Structure from Motion problem, in which a three-dimensional structure is sought from a collection of two-dimensional images. In this context, the locations of cameras and structure points are to be found from epipolar geometry and point correspondences among images. These correspondences are often incorrect because of lighting, shadows, and the effects of perspective. Hence, the resulting observations of relative directions contain significant corruptions. To solve the location recovery problem in the presence of corrupted relative directions, we introduce a tractable convex program called ShapeFit. Empirically, ShapeFit can succeed on synthetic data with over 40% corruption. Rigorously, we prove that ShapeFit can recover a set of locations exactly when a fraction of the measurements are adversarially corrupted and when the data model is random. This and subsequent work was done in collaboration with Choongbum Lee, Vladislav Voroninski, and Tom Goldstein.

Convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Series
Stochastics Seminar
Time
Thursday, November 19, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Konstantin Tikhomirov – University of Alberta
Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko-Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries. Based on joint work with Djalil Chafai.

Recovery of High-Dimensional Low-Rank Matrices

Series
Stochastics Seminar
Time
Thursday, November 12, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tony Cai – Wharton School, University of Pennsylvania
Low-rank structure commonly arises in many applications including genomics, signal processing, and portfolio allocation. It is also used in many statistical inference methodologies such as principal component analysis. In this talk, I will present some recent results on recovery of a high-dimensional low-rank matrix with rank-one measurements and related problems including phase retrieval and optimal estimation of a spiked covariance matrix based on one-dimensional projections. I will also discuss structured matrix completion which aims to recover a low rank matrix based on incomplete, but structured observations.

More on Logarithmic sums of convex bodies

Series
Stochastics Seminar
Time
Thursday, November 5, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christos Saraoglou – Kent State University
We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesgue measure in dimension n would imply the log-BMI and, therefore, the B-conjecture for any even log-concave measure in dimension n. As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension n, there is a density f_n, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density f_n. As byproduct of our methods, we study possible log-concavity of the function t -> |(K+_p\cdot e^tL)^{\circ}|, where p\geq 1 and K, L are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

Recent Berry-Esseen bounds obtained with Stein's method and Poincare inequalities, with Geometric applications

Series
Stochastics Seminar
Time
Thursday, October 29, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Raphael Lachieze-Rey – University of Southern California
Recently, new general bounds for the distance to the normal of a non-linear functional have been obtained, both with Poisson input and with IID points input. In the Poisson case, the results have been obtained by combining Stein's method with Malliavin calculus and a 'second-order Poincare inequality', itself obtained through a coupling involving Glauber's dynamics. In the case where the input consists in IID points, Stein's method is again involved, and combined with a particular inequality obtained by Chatterjee in 2008, similar to the second-order Poincar? inequality. Many new results and optimal speeds have been obtained for some Euclidean geometric functionals, such as the minimal spanning tree, the Boolean model, or the Voronoi approximation of sets.

Longest Subsequences Problems and Maximal Eigenvalues of Gaussian Random Matrices

Series
Stochastics Seminar
Time
Thursday, October 22, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Houdre – School of Mathematics, Georgia Tech
This is survey talk where, both for random words and random permutations, I will present a panoramic view of the subject ranging from classical results to recent breakthroughs. Throughout, equivalencies with some directed last passage percolation models with dependent weights will be pointed out.

Convex regularization for low rank tensor estimation

Series
Stochastics Seminar
Time
Thursday, October 8, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ming Yuan – University of Wisconsin
Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. We introduce a general framework of convex regularization for low rank tensor estimation.

Gaussian fluctuations for linear statistics of Wigner matrices

Series
Stochastics Seminar
Time
Thursday, October 1, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe Sosoe – Harvard University
In the 1970s, Girko made the striking observation that, after centering, traces of functions of large random matrices have approximately Gaussian distribution. This convergence is true without any further normalization provided f is smooth enough, even though the trace involves a number of terms equal to the dimension of the matrix. This is particularly interesting, because for some rougher, but still natural observables, like the number of eigenvalues in an interval, the fluctuations diverge. I will explain how such results can be obtained, focusing in particular on controlling the fluctuations when the function is not very regular.

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