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Series: Stochastics Seminar

When considering smooth functionals of dependent data, block bootstrap methods have enjoyed considerable success in theory and application. For nonsmooth functionals of dependent data, such as sample quantiles, the theory is less well-developed. In this talk, I will present a general theory of consistency and optimality, in terms of achieving the fastest convergence rate, for block bootstrap distribution estimation for sample quantiles under mild strong mixing assumptions. The case of density estimation will also be discussed. In contrast to existing results, we study the block bootstrap for varying numbers of blocks. This corresponds to a hybrid between the subsampling bootstrap and the moving block bootstrap (MBB). Examples of `time series’ models illustrate the benefits of optimally choosing the number of blocks. This is joint work with Stephen M.S. Lee (University of Hong Kong) and Alastair Young (Imperial College London).

Series: Stochastics Seminar

We present a unified framework for sequential low-rank matrix completion and estimation, address the joint goals of uncertainty quantification (UQ) and statistical design. The first goal of UQ aims to provide a measure of uncertainty of estimated entries in the unknown low-rank matrix X, while the second goal of statistical design provides an informed sampling or measurement scheme for observing the entries in X. For UQ, we adopt a Bayesian approach and assume a singular matrix-variate Gaussian prior the low-rank matrix X which enjoys conjugacy. For design, we explore deterministic design from information-theoretic coding theory. The effectiveness of our proposed methodology is then illustrated on applications to collaborative filtering.

Series: Stochastics Seminar

The study of graph-partition problems such as Maxcut, max-bisection and
min-bisection have a long and rich history in combinatorics and theoretical
computer science. A recent line of work studies these problems on sparse random
graphs, via a connection with mean field spin glasses. In this talk, we will look
at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erdös-Rényi and random regular graphs.
Based on joint work with Aukosh Jagannath.

Series: Stochastics Seminar

We consider the $\textit{linearly transformed spiked model}$, where observations $Y_i$ are noisy linear transforms of unobserved signals of interest $X_i$: $$Y_i = A_i X_i + \varepsilon_i,$$ for $i=1,\ldots,n$. The transform matrices $A_i$ are also observed. We model $X_i$ as random vectors lying on an unknown low-dimensional space. How should we predict the unobserved signals (regression coefficients) $X_i$? The naive approach of performing regression for each observation separately is inaccurate due to the large noise. Instead, we develop optimal linear empirical Bayes methods for predicting $X_i$ by "borrowing strength'' across the different samples. Our methods are applicable to large datasets and rely on weak moment assumptions. The analysis is based on random matrix theory. We discuss applications to signal processing, deconvolution, cryo-electron microscopy, and missing data in the high-noise regime. For missing data, we show in simulations that our methods are faster, more robust to noise and to unequal sampling than well-known matrix completion methods. This is joint work with William Leeb and Amit Singer from Princeton, available as a preprint at arxiv.org/abs/1709.03393.

Series: Stochastics Seminar

This talk concerns to spectral gap of random regular graphs. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by
providing a tight upper bound for the non trivial eigenvalues of its adjacency operator, proving Alon's Conjecture for this family of graphs. Also, we use a spectral algorithm to recover hidden communities in a random network model we call regular stochastic block model. Our proofs rely on a technique introduced recently by Massoullie, which we developed for random
regular graphs.

Series: Stochastics Seminar

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the
edges with common distribution F. For which F is there an infinite self-avoiding path with
finite total weight? This question arises in first-passage percolation, the study of the
random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2
(there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there
is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably
more difficult due to the presence of finite paths of zero-weight edges on all scales. I will
discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient
conditions on F for the existence of an infinite finite-weight path. The methods involve
comparing the model to another one, invasion percolation, and showing that geodesics
in first-passage percolation have the same first order travel time as optimal paths in an
embedded invasion cluster.

Series: Stochastics Seminar

We discuss two recent results concerning disease modeling on networks. The infection is assumed to spread via contagion (e.g., transmission over the edges of an underlying network). In the first scenario, we observe the infection status of individuals at a particular time instance and the goal is to identify a confidence set of nodes that contain the source of the infection with high probability. We show that when the underlying graph is a tree with certain regularity properties and the structure of the graph is known, confidence sets may be constructed with cardinality independent of the size of the infection set. In the scenario, the goal is to infer the network structure of the underlying graph based on knowledge of the infected individuals. We develop a hypothesis test based on permutation testing, and describe a sufficient condition for the validity of the hypothesis test based on automorphism groups of the graphs involved in the hypothesis test. This is joint work with Justin Khim (UPenn).

Series: Stochastics Seminar

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

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.