Georgia Institute of Technology College of Sciences

Low-rank structures are common in modern data analysis and signal processing, and they usually play essential roles in various estimation and detection problems. It is challenging to recover the underlying low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce convex and nonconvex optimization methods for low-rank recovery by two examples. The first example is community detection in network data analysis. In the literature, it has been formulated as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However, the statistical advantages of convex optimization approaches over other competitive methods, such as spectral clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world network dataset with a heterogeneous distribution of degrees. Although SDP methods are provably effective and robust, the computational complexity is usually high and there is an issue of storage. For the problem of phase retrieval, which has various applications and can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far less storage and has much higher rate of convergence compared to convex methods.

Sir Ronald Ross’ discovery of the transmission mechanism of malaria in 1897 inspired a suite of mathematical models for the transmission of vector-borne disease, known as Ross-Macdonald models. I introduce a common formulation of the Ross-Macdonald model and discuss its extension to address a current topic in malaria control: the introduction of malaria vaccines. Following over two decades of research, vaccine trials for the malaria vaccine RTS,S have been completed, demonstrating an efficacy of roughly 50% in young children. Regions with high malaria prevalence tend to have high levels of naturally acquired immunity (NAI) to severe malaria, leading to large asymptomatic populations. I introduce a malaria model developed to address concerns about how these vaccines will perform in regions with existing NAI, discuss some analytic results and their public health implications, and reframe our question as an optimal control problem.

We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.