Topological Data Analysis



Special Session on Topological Data Analysis in Korean Society for Industrial and Applied Mathematics (KSIAM) Spring Conference 2019 (May 17-18 2019, Yonsei University, Seoul)



Motivation: Topological data analysis including persistent homology has seen tremendous surge in recent years. We would like to showcase how topological methods can be used in solving complex biomedical problems.

Organizer: Moo K. Chung (http://www.stat.wisc.edu/~mchung), Associate Professor, Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison
Contact information: mkchung@wisc.edu, 1-608-217-2452


Invited Speakers:

Seonhee Lim, Professor
Department of Mathematical Sciences, Seoul National University, Korea
http://www.math.snu.ac.kr/~lim/research.html

Title: Volume entropy and informaton flow in brain networks: case of tinnitus in subjects with hearing loss

Abstract: We use volume entropy and the weights on vertices and edges, as well as other type of entropy-like invariants in modeling the brain networks of various populations including  tinnitus and hearing loss and that of mild epilepsy using EEG. Volume entropy of a metric graph, a global measure of information, measures the exponential growth rate of the number of network paths. Capacity of nodes and edges, a local measure of information, represents the stationary distribution of information propagation in brain networks. Full abstract


Hyekyoung Lee, Research Professor
Department of Nuclear Medicine, Seoul National University Hospital, Korea
https://sites.google.com/site/hkleebrain/home

Title: Network dissimilarity based on harmonic holes

Abstract: Persistent homology has been applied to brain network analysis for finding the shape of brain networks across multiple thresholds. In the persistent homology, the shape of networks is often quantified by the sequence of k-dimensional holes and Betti numbers. The Betti numbers are more widely used than holes themselves in topological brain network analysis. However, the holes show the local connectivity of networks, and they can be very informative features in analysis. In this talk, I show a new method of measuring network differences based on the dissimilarity measure of harmonic holes (HHs). The HHs, which represents the substructure of brain networks, are extracted by the Hodge Laplacian of brain networks. I also show the most contributed HHs to the network difference based on the HH dissimilarity. In clinical application, the proposed method is applied to clustering the networks of 4 groups, normal control (NC), stable and progressive mild cognitive impairment (MCI), and Alzheimer's disease (AD). The results show that the clustering performance of the proposed method was better than that of network distances based on only the global change of topology. The talk is in part based on  arXiv:1811:04355. Full abstract


Christopher Bresten, Research Assistant Professor
Department of Data Science, Ajou University, Korea

Title: Topological data analysis for classification of vascular flows: overcoming irregular geometry

Abstract: Classification of vascular flows on regular domains has been developed in previous work using topological data analysis.  This work is to develop a methodology for projecting the velocity field from irregular, curved vascular geometries to a co-ordinate system that makes the velocity fields more consistent with a regular geometry. This is done with the goal of applying theoretical results developed with simulations on regular geometries to the geometries present in actual human anatomy. Joint work with Jae-Hun Jung, Department of Data Science, Ajou University. Full abstract


Moo K. Chung, Associate Professor
Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, USA

Title: Random walks in the permutation group

Abstract: The permutation test is an often used test procedure in brain imaging studies. Unfortunately, generating every possible permutation for large-scale brain image datasets such as HCP and ADNI with hundreds images is not practical. Many previous attempts at speeding up the permutation test rely on various approximation strategies such as estimating the tail distribution with known parametric distributions. In this study, we show how to accelerate the permutation test without any type of approximate strategies by exploiting the underlying topological structure of the permutation group. The method is applied to large number of diffusion tensor images in determining the  heritability of structural brain network. The talk is in part based on arXiv 1812.06696. Full abstract