Seminars and Colloquia by Series

Low-Rank Recovery: From Convex to Nonconvex Methods

Series
Job Candidate Talk
Time
Monday, February 9, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaodong LiUniversity of Pennsylvania
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.

Mathematical modeling of malaria transmission

Series
Job Candidate Talk
Time
Thursday, February 5, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Olivia ProsperDartmouth College
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.

Inversion, design of experiments, and optimal control in systems gov- erned by PDEs with random parameter functions

Series
Job Candidate Talk
Time
Tuesday, February 3, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alen AlexanderianUniversity of Texas at Austin
Mathematical models of physical phenomena often include parameters that are hard or impossible to measure directly or are subject to variability, and are thus considered uncertain. Different aspects of modeling under uncertainty include forward uncertainty propagation, statistical inver- sion of uncertain parameters, optimal design of experiments, and optimization under uncertainty. I will focus on recent advances in numerical methods for infinite-dimensional Bayesian inverse problems and optimal experimental de- sign. I will also discuss the problem of risk-averse optimization under uncertainty with applications to control of PDEs with uncertain parameters. The driving applications are systems governed by PDEs with uncertain parameter fields, such as ow in the subsurface with an uncertain permeability field, or the diffusive transport of a contaminant with an uncertain initial condition. Such problems are computationally challenging due to expensive forward PDE solves and infinite-dimensional (high-dimensional when discretized) parameter spaces.

Do Pancreatic Alpha Cells Control their Own Secretion or Follow the Orders of Other Cells?

Series
Job Candidate Talk
Time
Thursday, January 29, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Magaret WattsNIH
Diabetes is a disease of poor glucose control. Glucose is controlled by two hormones that work in opposite directions: insulin and glucagon. Pancreatic beta-cells release insulin when blood glucose is high, while pancreatic alpha-cells secrete glucagon when blood glucose is low. Both insulin and glucagon secretion are disregulated in people with diabetes. In these people, not enough insulin is secreted in response to elevated glucose levels, while the problem with glucagon secretion is two-fold: too much glucagon is secreted at high glucose levels, while not enough is secreted at low glucose levels. So far, the treatment of diabetes has focused solely on increasing insulin secretion from beta-cells. Therefore, understanding glucose regulated glucagon secretion may lead to new therapies for those with diabetes.There is an ongoing debate as to whether glucose suppresses glucagon secretion directly through an intrinsic mechanism, within the alpha-cell, or indirectly through an extrinsic mechanism. I developed a mathematical model of glucagon secretion in alpha-cells and use it to show that they can control their own secretion. However, experimental evidence shows that factors secreted by pancreatic beta- and delta- cells can also affect glucagon secretion. Therefore, I created the BAD model for pancreatic islets which contains one representative cell of each type and the cellular interactions between them. I use this model to show that these paracrine effects suppress alpha-cell heterogeneity and suggest that delta-cells play a more important role in this than beta-cells.

Some phase transitions in the stochastic block model

Series
Job Candidate Talk
Time
Thursday, January 22, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Joseph NeemanUniversity of Texas, Austin, TX
The stochastic block model is a random graph model that was originally introduced 30 years ago to model community structure in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.

Random polytopes and estimation of convex bodies

Series
Job Candidate Talk
Time
Tuesday, January 20, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Victor-Emmanuel BrunelYale University
In this talk we will discuss properties of some random polytopes. In particular, we first propose a deviation inequality for the convex hull of i.i.d. random points, uniformly distributed in a convex body. We then discuss statistical properties of this random polytope, in particular, its optimality, when one aims to estimate the support of the corresponding uniform distribution, if it is unknown.We also define a notion of multidimensional quantiles, related to the convex floating bodies, or Tukey depth level sets, for probability measures in a Euclidean space. When i.i.d. random points are available, these multidimensional quantiles can be estimated using their empirical version, similarly to the one-dimensional case, where order statistics estimate the usual quantiles.

Interlacing families: a new technique for controlling eigenvalues

Series
Job Candidate Talk
Time
Thursday, January 15, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles006
Speaker
Adam MarcusYale University
Matrices are one of the most fundamental structures in mathematics, and it is well known that the behavior of a matrix is dictated by its eigenvalues. Eigenvalues, however, are notoriously hard to control, due in part to the lack of techniques available. In this talk, I will present a new technique that we call the "method of interlacing polynomials" which has been used recently to give unprecedented bounds on eigenvalues, and as a result, new insight into a number of old problems. I will discuss some of these recent breakthroughs, which include the existence of Ramanujan graphs of all degrees, a resolution to the famous Kadison-Singer problem, and most recently an incredible result of Anari and Gharan that has led to an interesting new anomaly in computer science. This talk will be directed at a general mathematics audience and represents joint work with Dan Spielman and Nikhil Srivastava.

Tuning parameters in high-dimensional statistics

Series
Job Candidate Talk
Time
Tuesday, January 13, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Johannes LedererCornell University
High-dimensional statistics is the basis for analyzing large and complex data sets that are generated by cutting-edge technologies in genetics, neuroscience, astronomy, and many other fields. However, Lasso, Ridge Regression, Graphical Lasso, and other standard methods in high-dimensional statistics depend on tuning parameters that are difficult to calibrate in practice. In this talk, I present two novel approaches to overcome this difficulty. My first approach is based on a novel testing scheme that is inspired by Lepski’s idea for bandwidth selection in non-parametric statistics. This approach provides tuning parameter calibration for estimation and prediction with the Lasso and other standard methods and is to date the only way to ensure high performance, fast computations, and optimal finite sample guarantees. My second approach is based on the minimization of an objective function that avoids tuning parameters altogether. This approach provides accurate variable selection in regression settings and, additionally, opens up new possibilities for the estimation of gene regulation networks, microbial ecosystems, and many other network structures.

Singularity and mixing in incompressible fluid equations

Series
Job Candidate Talk
Time
Monday, January 12, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yao YaoUniversity of Wisconsin
The question of global regularity vs. finite time blow-up remains open for many fluid equations. Even in the cases where global regularity is known, solutions may develop small scales as time progresses. In this talk, I will first discuss an active scalar equation which is an interpolation between the 2D Euler equation and the surface quasi-geostrophic equation. We study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. I will also discuss a passive transport equation whose solutions are known to have global regularity, and our goal is to study how well a given initial density can be mixed if the incompressible flow satisfies some physically relevant quantitative constraints. This talk is based on joint works with A. Kiselev, L. Ryzhik and A. Zlatos.

Random lozenge tilings and Hurwitz numbers

Series
Job Candidate Talk
Time
Thursday, January 8, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jon NovakMIT
This talk will be about random lozenge tilings of a class of planar domains which I like to call "sawtooth domains." The basic question is: what does a uniformly random lozenge tiling of a large sawtooth domain look like? At the first order of randomness, a remarkable form of the law of large numbers emerges: the height function of the tiling converges to a deterministic "limit shape." My talk is about the next order of randomness, where one wants to analyze the fluctuations of tiles around their eventual positions in the limit shape. Quite remarkably, this analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double Hurwitz numbers from enumerative algebraic geometry.

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