Thursday, April 20, 2017 - 15:05 , Location: Skiles 006 , Lutz Warnke , School of Mathematics, GaTech , Organizer: Christian Houdre
We consider rooted subgraph extension counts, such as (a) the number of triangles containinga given vertex, or (b) the number of paths of length three connecting two given vertices. In 1989 Spencer gave sufficient conditions for the event that whp all roots of the binomial random graph G(n,p) have the same asymptotic number of extensions, i.e., (1 \pm \epsilon) times their expected number. Perhaps surprisingly, the question whether these conditions are necessary has remained open. In this talk we briefly discuss our qualitative solution of this problem for the `strictly balanced' case, and mention several intriguing questions that remain open (which lie at the intersection of probability theory + discrete mathematics, and are of concentration inequality type). Based on joint work in progress with Matas Sileikis
Thursday, April 13, 2017 - 15:05 , Location: Skiles 006 , Christian Houdré , School of Mathematics, Georgia Institute of Technology , Organizer: Christian Houdre
I will revisit the classical Stein's method, for normal random variables, as well as its version for Poisson random variables and show how both (as well as many other examples) can be incorporated in a single framework.
Scalings and saturation in infinite-dimensional control problems with applications to stochastic partial differential equationsFriday, April 7, 2017 - 13:05 , Location: Skiles 270 , David Herzog , Iowa State University , firstname.lastname@example.org , Organizer: Michael Damron
We discuss scaling methods which can be used to solve low mode control problems for nonlinear partial differential equations. These methods lead naturally to a infinite-dimensional generalization of the notion of saturation, originally due to Jurdjevic and Kupka in the finite-dimensional setting of ODEs. The methods will be highlighted by applying them to specific equations, including reaction-diffusion equations, the 2d/3d Euler/Navier-Stokes equations and the 2d Boussinesq equations. Applications to support properties of the laws solving randomly-forced versions of each of these equations will be noted.
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 , email@example.com , 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.