Abstract: In this talk, we discuss optimal hypothesis testing for distinguishing a stochastic block model from an Erdos--Renyi random graph when the average degree grows to infinity with the graph size. We show that linear spectral statistics based on Chebyshev polynomials of the adjacency matrix can approximate signed cycles of growing lengths when the graph is sufficiently dense. The signed cycles have been shown by Banerjee (2018) to determine the likelihood ratio statistic asymptotically. In this way one achieves sharp…

Abstract: Many contemporary large-scale applications, from genomics to advertising, involve linking a response of interest to a large set of potential explanatory variables in a nonlinear fashion, such as when the response is binary. Although this modeling problem has been extensively studied, it remains unclear how to effectively select important variables while controlling the fraction of false discoveries, even in high-dimensional logistic regression, not to mention general high-dimensional nonlinear models. To address such a practical problem, we propose a new…

Abstract: We discuss a recent approach to bias reduction in a problem of estimation of smooth functionals of high-dimensional parameters of statistical models. In particular, this approach has been developed in the case of estimation of functionals of covariance operator Σ : Rd d → Rd of the form f(Σ), B based on n i.i.d. observations X1, . . . , Xn sampled from the normal distribution with mean zero and covariance Σ, f : R → R being a…

Abstract: The prototypical high-dimensional statistics problem entails finding a structured signal in noise. Many of these problems exhibit an intriguing phenomenon: the amount of data needed by all known computationally efficient algorithms far exceeds what is needed for inefficient algorithms that search over all possible structures. A line of work initiated by Berthet and Rigollet in 2013 has aimed to explain these gaps by reducing from conjecturally hard problems in computer science. However, the delicate nature of average-case reductions has…

Abstract: In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple…

Andrea Montanari Professor, Department of Electrical Engineering, Department of Statistics Stanford University This lecture is in conjunction with the LIDS Student Conference. Abstract: Let A be n × n symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing xT Ax over binary vectors with ±1 entries. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this…