• Speaker: Hyenkyun Woo, Korea University of Technology & Education
• Title: Bregman-divergence for Legendre exponential families and data analysis
• Date: 08/24/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Bregman-divergence is a well-known generalized distance framework in various applications, such as machine learning and image processing. In this talk, by using dual structure of the Bregman-divergence associated with the subclass of convex function of Legendre function, we analyze the structure of the Legendre exponential families whose cumulant function corresponds to the conjugate convex function of Legendre type. Actually, Legendre exponential families are the extended version of the regular exponential families to include non-regular exponential families, such as the inverse Gaussian distribution. The main advantage of the proposed Bregman-divergence-based approach is that it offers systematic successive approximation tools to handle closed domain issues arising in non-regular exponential families and the statistical distribution having discrete random variables, such as Bernoulli distribution and Poisson distribution. In addition, we also introduce the generalized center-based clustering algorithm based on the Tweedie distribution. 

• Speaker: Tom Needham, Ohio State University
• Title: Gromov-Monge Quasimetrics and Distance Distributions
• Date: 09/07/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

In applications in computer graphics and computational anatomy, one seeks measure-preserving maps between shapes which preserve geometry as much as possible. Inspired by this, we define  a distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces, which has convenient lower bounds defined in terms of distance distributions. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of mm-spaces such as metric graphs or plane curves.This is joint work with Facundo Mémoli.

• Speaker: Paul Bendich, Duke University and Geometric Data Analytics
• Title: Topology and Geometry for Tracking and Sensor Fusion
• Date: 10/19/2018
• Time: 4:10 PM - 5:00 PM
• Place: C304 Wells Hall

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest. This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena. First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data. Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database. This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, and Peter Zulch.

• Speaker: Wenrui Hao, Pennsylvania State University
• Title: Computational modeling for cardiovascular risk evaluation
• Date: 11/09/2018
• Time: 4:10 PM - 5:00 PM
• Place: 1502 Engineering Building

Atherosclerosis, the leading cause of death in the United State, is a disease in which a plaque builds up inside the arteries. The LDL and HDL concentrations in the blood are commonly used to predict the risk factor for plaque growth. In this talk, I will describe a recent mathematical model that predicts the plaque formation by using the combined levels of (LDL, HDL) in the blood. The model is given by a system of partial differential equations within the plaque with a free boundary. This model is used to explore some drugs of regression of a plaque in mice, and suggest that such drugs as used for mice may also slow plaque growth in humans. Some mathematical questions, inspired by this model, will also be discussed. I will also mention briefly some related projects about abdominal aortic aneurysm (AAA) and red blood cell aggregation, which would have some potential blood biomarkers for diagnosis of AAA.