Abstract: Consider the problem of recovering a rank 1 tensor of order k that has been subject to Gaussian noise. The log-likelihood for this problem is highly non-convex. It is information theoretically possible to recover the tensor with a finite number of samples via maximum likelihood estimation, however, it is expected that one needs a polynomially diverging number of samples to efficiently recover it. What is the cause of this large statistical–to–algorithmic gap? To study this question, we investigate the…

Abstract: Dense Graphs: We consider abritrary graphs G with n vertices and minimum degree at least n. where δ > 0 is constant. If the conductance of G is sufficiently large then we obtain an asymptotic expression for the cover time CG of G as the solution to some explicit transcendental equation. Failing this, if the mixing time of a random walk on G is of a lesser magnitude than the cover time, then we can obtain an asymptotic deterministic…

Abstract: Inference in the framework of Ising models has received significant attention in Statistics and Machine Learning in recent years. In this talk we study joint estimation of the inverse temperature parameter β, and the magnetization parameter B, given one realization from the Ising model, under the assumption that the underlying graph of the Ising model is completely specified. We show that if the graph is either irregular or sparse, then both the parameters can be estimated at rate n−1/2…

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…