Georgia Institute of Technology College of Sciences

For a random (complex) entire function, what can we say about the behavior of the zero set of its N-th derivative, as N goes to infinity? In this talk, we shall discuss the result of repeatedly differentiating a certain class of random entire functions whose zeros are the points of a Poisson process of intensity 1 on the real line. We shall also discuss the asymptotic behavior of the coefficients of these entire functions. Based on joint work with Robin Pemantle.

Detecting change-points from high-dimensional streaming data is a fundamental problem that arises in many big-data applications such as video processing, sensor networks, and social networks. Challenges herein include developing algorithms that have low computational complexity and good statistical power, that can exploit structures to detecting weak signals, and that can provide reliable results over larger classes of data distributions. I will present two aspects of our recent work that tackle these challenges: (1) developing kernel-based methods based on nonparametric statistics; and (2) using sketching of high-dimensional data vectors to reduce data dimensionality. We also provide theoretical performance bounds and demonstrate the performance of the algorithms using simulated and real data.

We consider the problem of recovering a set of locations given observations of the direction between pairs of these locations. This recovery task arises from the Structure from Motion problem, in which a three-dimensional structure is sought from a collection of two-dimensional images. In this context, the locations of cameras and structure points are to be found from epipolar geometry and point correspondences among images. These correspondences are often incorrect because of lighting, shadows, and the effects of perspective. Hence, the resulting observations of relative directions contain significant corruptions. To solve the location recovery problem in the presence of corrupted relative directions, we introduce a tractable convex program called ShapeFit. Empirically, ShapeFit can succeed on synthetic data with over 40% corruption. Rigorously, we prove that ShapeFit can recover a set of locations exactly when a fraction of the measurements are adversarially corrupted and when the data model is random. This and subsequent work was done in collaboration with Choongbum Lee, Vladislav Voroninski, and Tom Goldstein.

Low-rank structure commonly arises in many applications including genomics, signal processing, and portfolio allocation. It is also used in many statistical inference methodologies such as principal component analysis. In this talk, I will present some recent results on recovery of a high-dimensional low-rank matrix with rank-one measurements and related problems including phase retrieval and optimal estimation of a spiked covariance matrix based on one-dimensional projections. I will also discuss structured matrix completion which aims to recover a low rank matrix based on incomplete, but structured observations.