TDA at JSM

 ( https://ww2.amstat.org/meetings/jsm/2020/onlineprogram/ActivityDetails.cfm?SessionID=219282 )

Time: Thursday, August, 6th, 2020, 10:00 AM – 11:50 AM Eastern Daylight Time (EDT)

Place: Virtual Joint Statistical Meeting 2020 (https://ww2.amstat.org/meetings/jsm/2020/index.cfm).

Organizer(s): Chul Moon, chulm@smu.edu, Southern Methodist University

Chair(s): Hengrui Luo, luo.619@osu.edu, The Ohio State University

10:05 AM         Solution manifold and Its Statistical Applications

Speaker. Yen-Chi Chen, University of Washington

Abstract. A solution manifold is the collection of points in a d-dimensional space satisfying a system of s equations with s<d. Solution manifolds occur in several statistical problems including hypothesis testing, curved-exponential families, constrained mixture models, partial identifications, and nonparametric set estimation. We analyze solution manifolds both theoretically and algorithmically. In terms of theory, we derive five useful results: the smoothness theorem, the stability theorem (which implies the consistency of a plug-in estimator), the convergence of a gradient flow, the local center manifold theorem and the convergence of the gradient descent algorithm. To numerically approximate a solution manifold, we propose a Monte Carlo gradient descent algorithm. In the case of likelihood inference, we design a manifold constraint maximization procedure to find the maximum likelihood estimator on the manifold. We also develop a method to approximate a posterior distribution defined on a solution manifold.

10:25 AM         Persistent Topological Descriptors for Functional Brain Network

Speaker. Hyunnam Ryu, University of Georgia; Nicole Lazar, University of Georgia

Abstract. We compare the topological features of functional brain networks. In general, functional brain networks are dealt with in an elementwise manner based on the connectivity matrix as part of network data analysis. This tends to ignore the higher-order topology of the network, which can have significant implications. In recent studies, researchers have been interested in topological data analysis. Persistent homology is known to be useful for studying dynamic topological invariants hidden in complex data obtained from topological space. Analysis using persistent homology not only captures topological features that could be overlooked in the network data analysis but also addresses threshold selection problems commonly found in network data analysis.

We use persistent homology to compare the topological features of brain networks. We construct a brain network from the fMRI time series BOLD signal and calculate the persistent homology through the weighted brain network. Also, we compare the summarized topological features of different subject groups by calculating the persistence landscape.

10:45 AM         Uncovering the Holes in the Universe with Topological Data Analysis

Speaker. Jessi Cisewski-Kehe, Yale University

Abstract. The large-scale structure (LSS) of the Universe is a spatially complex web of matter that is difficult to analyze without losing potentially important information, but can help to constrain the underlying cosmological model that describes the Universe. Topological Data Analysis (TDA) is especially suitable for such weblike data and we have used this framework to visualize, define, and do inference on known (i.e., voids) and new (i.e., filament loops) cosmological structures. 

During this talk, I will discuss how TDA can be used to uncover cosmological structures.  The features on a persistence diagram represent homology group generators (connected components, loops, voids, etc.), which are not uniquely defined back in the dataset. However, having a way to visualize the generators in the dataset can be useful to better understand the data and to possibly determine the physical meaning of the structure. This led to a new procedure called “Significant Cosmic Holes in Universe” (SCHU) for defining representations of homology group generators in a cosmological survey, such as the Sloan Digital Sky Survey galaxy survey.  Cosmological voids correspond to the second homology group generators, and we also define a new class of voids based on the first homology group generators, which we call filament loops.

Persistence diagrams can also be used in hypothesis tests in order to make statistical comparisons between complicated spatial structures such as LSS.  I will present some developments using a two-sample hypothesis testing framework to distinguish LSS under different cosmological assumptions (e.g., cold dark matter vs. warm dark matter).

11:05 AM         Confidence Band for Persistent Homology

Speaker. Jisu Kim, INRIA

Abstract. Topological Data Analysis generally refers to utilizing topological features from data. For this talk, I will focus on persistent homology, which quantifies the salient topological features of data. I will present how the confidence band can be computed for determining the significance of the topological features in the persistent homology, based on the bootstrap procedure. First, I will present how the confidence band can be computed for the persistent homology of KDEs (kernel density estimators) computed on a grid. In practice, however, calculating the persistent homology of KDEs on d-dimensional Euclidean spaces requires to approximate the ambient space to a grid, which could be computationally inefficient when the dimension of the ambient space is high or topological features are in different scales. Hence, I will consider the persistent homology of KDE filtrations on Rips complexes as an alternative. I will describe how to construct an asymptotic confidence set for the persistent homology based on the bootstrap procedure. Unlike existing procedures, this method does not heavily rely on grid-approximations, scales to higher dimensions, and is adaptive to heterogeneous topological features.

11:25 AM         Discussant: Chul Moon, Southern Methodist University

11:45 AM         Floor Discussion and Follow-ups

Everyone is welcomed to register for Joint Statistical Meeting (JSM) to join our virtual session!

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