Thursday, April 6, 2017 - 15:05 , Location: Skiles 006 , Zhou Fan , Stanford University , Organizer: Christian Houdre
Spectral algorithms are a powerful method for detecting low-rank structure in dense random matrices and random graphs. However, in certain problems involving sparse random graphs with bounded average vertex degree, a naive spectral analysis of the graph adjacency matrix fails to detect this structure. In this talk, I will discuss a semidefinite programming (SDP) approach to address this problem, which may be viewed both as imposing a delocalization constraint on the maximum eigenvalue problem and as a natural convex relaxation of minimum graph bisection. I will discuss probabilistic results that bound the value of this SDP for sparse Erdos-Renyi random graphs with fixed average vertex degree, as well as an extension of the lower bound to the two-group stochastic block model. Our upper bound uses a dual witness construction that is related to the non-backtracking matrix of the graph. Our lower bounds analyze the behavior of local algorithms, and in particular imply that such algorithms can approximately solve the SDP in the Erdos-Renyi setting. This is joint work with Andrea Montanari.
Thursday, March 30, 2017 - 15:05 , Location: Skiles 006 , Sumit Mukherjee , Columbia University , Organizer: Mayya Zhilova
We consider the problem of studying the limiting distribution of the number of monochromatic two stars and triangles for a growing sequence of graphs, where the vertices are colored uniformly at random. We show that the limit distribution of the number of monochromatic two stars is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree 1 or 2. Further, we show that any limit distribution for the number of monochromatic two stars has an expansion of this form. In the triangle case the problem is more challenging, as in this case the class of limit distributions can involve terms with products of Poisson random variables. In this case, we deduce a necessary and sufficient condition on the sequence of graphs such that the number of monochromatic triangles is asymptotically Poisson in distribution and in the first two moments. This work is joint with Bhaswar B. Bhattacharya at University of Pennsylvania.
Thursday, March 16, 2017 - 15:05 , Location: Skiles 006 , Stas Minsker , University of Southern California , Organizer: Christian Houdre
Estimation of the covariance matrix has attracted significant attention of the statistical research community over the years, partially due to important applications such as Principal Component Analysis. However, frequently used empirical covariance estimator (and its modifications) is very sensitive to outliers, or ``atypical’’ points in the sample. As P. Huber wrote in 1964, “...This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance…” Motivated by Tukey's question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We will present extensions of our approach to matrix-valued U-statistics, as well as applications such as the matrix completion problem. Part of the talk will be based on a joint work with Xiaohan Wei.
Thursday, March 9, 2017 - 15:05 , Location: Skiles 006 , Victor-Emmanuel Brunel , MIT , Organizer: Mayya Zhilova
Determinantal point processes (DPPs) have attracted a lot of attention in probability theory, because they arise naturally in many integrable systems. In statistical physics, machine learning, statistics and other fields, they have become increasingly popular as an elegant mathematical tool used to describe or to model repulsive interactions. In this talk, we study the geometry of the likelihood associated with such processes on finite spaces. Interestingly, the local behavior of the likelihood function around its global maxima can be very different according to the structure of a specific graph that we define for each DPP. Finally, we discuss some statistical consequences of this fact, namely, the asymptotic accuracy of a maximum likelihood estimator.
Thursday, March 2, 2017 - 15:05 , Location: Skiles 006 , Vu-Lan Nguyen , Harvard University , Organizer: Christian Houdre
As a general fact, directed polymers in random environment are localized in the so called strong disorder phase. In this talk, based on a joint with Francis Comets, we will consider the exactly solvable model with log gamma environment,introduced recently by Seppalainen. For the stationary model and the point to line version, the localization can be expressed as the trapping of the endpoint in a potential given by an independent random walk.
Thursday, February 23, 2017 - 15:05 , Location: Skiles 006 , David Sivakoff , Ohio State University , firstname.lastname@example.org , Organizer: Michael Damron
Excitable media are characterized by a local tendency towards synchronization, which can lead to waves of excitement through the system. Two classical discrete, deterministic models of excitable media are the cyclic cellular automaton and Greenberg-Hastings models, which have been extensively studied on lattices, Z^d. One is typically interested in whether or not sites are excited (change states) infinitely often (fluctuation vs fixation), and if so, whether the density of domain walls between disagreeing sites tends to 0 (clustering). We introduce a new comparison process for the 3-color variants of these models, which allows us to study the asymptotic rate at which a site gets excited. In particular, for a class of infinite trees we can determine whether the rate is 0 or positive. Using this comparison process, we also analyze a new model for pulse-coupled oscillators in one dimension, introduced recently by Lyu, called the firefly cellular automaton (FCA). Based on joint works with Lyu and Gravner.
Wednesday, February 22, 2017 - 14:05 , Location: Skiles 005 , Grigoris Paouris , Texas A&M , email@example.com , Organizer: Galyna Livshyts
Please note the special time! This is Stochastic & Analysis seminars joint.
Motivated by the investigation on the dependence on ``epsilon" in the Dvoretzky's theorem, I will show some refinements of the classical concentration of measure for convex functions. Applications to convexity will be presented if time permits. The talk will be based on joint works with Peter Pivovarov and Petros Valettas.
Thursday, February 9, 2017 - 15:05 , Location: Skiles 006 , Christopher Hoffman , University of Washington , firstname.lastname@example.org , Organizer: Michael Damron
First-passage percolation is a classical random growth model which comes from statistical physics. We will discuss recent results about the relationship between the limiting shape in first passage percolation and the structure of the infinite geodesics. This incudes a solution to the midpoint problem of Benjamini, Kalai and Schramm. This is joint work with Gerandy Brito and Daniel Ahlberg.
Thursday, February 2, 2017 - 15:05 , Location: Skiles 006 , Sohail Bahmani , ECE, GaTech , email@example.com , Organizer: Christian Houdre
We propose a new convex relaxation for the problem of solving (random) quadratic equations known as phase retrieval. The main advantage of the proposed method is that it operates in the natural domain of the signal. Therefore, it has significantly lower computational cost than the existing convex methods that rely on semidefinite programming and competes with the recent non-convex methods. In the proposed formulation the quadratic equations are relaxed to inequalities describing a "complex polytope". Then, using an *anchor vector* that itself can be constructed from the observations, a simple convex program estimates the ground truth as an (approximate) extreme point of the polytope. We show, using classic results in statistical learning theory, that with random measurements this convex program produces accurate estimates. I will also discuss some preliminary results on a more general class of regression problems where we construct accurate and computationally efficient estimators using anchor vectors.
Thursday, January 26, 2017 - 15:05 , Location: Skiles006 , Vladimir Koltchinskii , Georgia Tech , Organizer: Christian Houdre
We study the problem of estimation of a linear functional of the eigenvector of covariance operator that corresponds to its largest eigenvalue (the first principal component) based on i.i.d. sample of centered Gaussian observations with this covariance. The problem is studied in a dimension-free framework with its complexity being characterized by so called "effective rank" of the true covariance. In this framework, we establish a minimax lower bound on the mean squared error of estimation of linear functional and construct an asymptotically normal estimator for which the bound is attained. The standard "naive" estimator (the linear functional of the empirical principal component) is suboptimal in this problem. The talk is based on a joint work with Richard Nickl.