Stochastics and Statistics Seminar

Henry L. Rietz, the first president of IMS, published his book “Mathematical Statistics” in 1927. One review wrote in 1928: “Professor Rietz has developed this theory so skillfully that the ’workers in other fields’, provided only that they have a passing familiarity with the grammar of mathematics, can secure a satisfactory understanding of the points involved.” In this lecture, I would like to promote the good tradition of mathematical statistics as expressed in Rietzs book in order to gain insight…

Consider an n by n linear system Ax=b. If the right-hand side of the system is known up to a certain error, then in process of the solution, this error gets amplified by the condition number of the matrix A, i.e. by the ratio of its largest and smallest singular values. This observation led von Neumann and his collaborators to consider the condition number of a random matrix and conjecture that it should be of order n. This conjecture was…

Eigendecomposition of quadratic forms guaranteed by the spectral theorem is the foundation for many important algorithms in computer science, data analysis, and machine learning. In this talk I will discuss our recent work on generalizations from quadratic forms to a broad class of functions based on an analogue of the spectral decomposition in an orthogonal basis. We call such functions ``orthogonally decomposable". It turns out that many inferential problems of recent interest including orthogonal tensor decompositions, Independent Component Analysis (ICA),…

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Consider the problem of estimating the entries of a large matrix, when the observed entries are noisy versions of a small random fraction of the original entries. This problem has received widespread attention in recent times. I will describe a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure". Surprisingly, this simple estimator achieves the minimax error rate up to a constant factor. The method is applied to…

Abstract: We consider the problem of influence maximization in fixed networks, for both stochastic and adversarial contagion models. In the stochastic setting, nodes are infected in waves according to linear threshold or independent cascade models. We establish upper and lower bounds for the influence of a subset of nodes in the network, where the influence is defined as the expected number of infected nodes at the conclusion of the epidemic. We quantify the gap between our upper and lower bounds…

Prediction problems typically assume the training data are independent samples, but in many modern applications samples come from individuals connected by a network. For example, in adolescent health studies of risk-taking behaviors, information on the subjects’ social networks is often available and plays an important role through network cohesion, the empirically observed phenomenon of friends behaving similarly. Taking cohesion into account should allow us to improve prediction. Here we propose a regression-based framework with a network penalty on individual node…

We will present some results and some open problems related to shotgun assembly of graphs for random generating models.Shotgun assembly of graphs is the problem of recovering a random graph or a randomly labelled graphs from small pieces. This problem generalizes the theoretically elegant and practically important problem of shotgun assembly of DNA sequences. The general problem of shotgun assembly presents novel problems in random graphs, percolation, and random constraint satisfaction problems. Based on joint works with Nathan Ross, with…

Abstract: In sparse principal components analysis (PCA), the task is to infer a sparse, low-rank matrix from noisy observations. Johnstone and Lu proposed the popular “spiked covariance” model, wherein the population distribution is equivariant with the exception of a single direction, called the spike. Assuming that the spike direction is sparse in some basis, they also proposed a simple scheme to estimate its support based on the diagonal entries of the sample covariance. Indeed, later information-theoretic analysis demonstrated that the…

Abstract: In this talk, we will study higher-order accuracy of the weighted bootstrap procedure for estimation of a distribution of a sum of independent random vectors with bounded fourth moments, on the set of all Euclidean balls. Our approach is based on Berry-Esseen type inequality which extends the classical normal approximation bound. These results justify in non-asymptotic setting that the weighted bootstrap can outperform Gaussian (or chi-squared) approximation in accuracy w.r.t. dimension and sample size. In addition, the presented results lead…