Georgia Institute of Technology College of Sciences

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.

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.

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.

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