TDA seminar, UCLA

We meet on Thursdays at 2-3pm in MS 7619C.

Winter 2020 meetings

We will discuss the paper "Inferring symbolic dynamics of chaotic flows from persistence" by G. Yalnız and N.B. Budanur, available here.

We will discuss how to extract information from persistence diagrams in some specific problems on which people in the group are working. The meeting will be fairly informal; at the beginning several people will briefly explain the problem they are working on, and then there will be plenty of time for discussions. For the moment it seems that these problems will be from dynamical systems, climate science and machine learning, but this is subject to change. Also, if any of you have an additional problem you are working on and would like to discuss, bring it along. (If you wish you can send me an email about this, but you don't have to.)

We introduce the notion of effective resistance for a simplicial network (X,R) where X is a simplicial complex and R is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a current generator \sigma, an extra top-dimensional simplex to be attached to X. We will show that the harmonic class gives rise to the current I_\sigma and the voltage V_\sigma for X\cup\sigma, satisfying Thomson's energy-minimizing principle and Ohm's law for simplicial networks.

In this talk I will give an introduction to distances between persistence diagrams. I will illustrate with some examples why the geometry of the spaces of persistence diagrams is not well suited to statistical analysis, and discuss some possible solutions to this. No prior knowledge on the subject is assumed.

This talk gives an introduction to the R package TDA, which provides some tools for Topological Data Analysis. Topological Data Analysis generally refers to utilizing topological features from data. In this talk, I will focus on persistent homology. The R package TDA provides functions to sample on various geometric objects. It also provides functions that, given some data, provide topological information about the underlying space, such as distance functions and density functions. The salient topological features of data can be quantified with persistent homology. The R package TDA provides an R interface for the efficient algorithms of the C++ libraries GUDHI, Dionysus, and PHAT for computing the persistent homology. Specifically, The R package TDA includes functions for computing the persistent homology of Rips complex, alpha complex, alpha shape complex, and a function for the persistent homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated over a grid of points or on data points. The R package TDA also provides functions for functional summaries of the persistent homology, such as the landscape function and the silhouette function. The R package TDA also provides a function for computing the confidence band that determines the significance of the features in the resulting persistence diagrams.

Past meetings

As topological data analysis is increasing in popularity, there is growing excitement in the mathematical modeling and complex systems communities about leveraging topological techniques for data-driven modeling. I will lead a discussion on the intersection of mathematical modeling and topological tools. As a jumping off point, I will present some problems from (or related to) my own work.

An observable for a collection of dataset is said to be stable if small change in the dataset results in small change in the observable. This notion of stability of observables is formalized by a stability inequality, which involves suitable metric(s) for the dataset as well as the observables. One of the most well-known example in TDA is for the persistence diagrams, where the metric for topological spaces and persistance diagram are the Gromov-Hausdorff distance and the bottleneck distance, respectively. In this talk, we review stability inequalities of some probabilistic network observables, such as the homomorphism density and conditional homomorphism density profiles, where the cut metric is involved in measuring distance between networks. The essential idea behind their proofs can be viewed as a version of the famous 'Lindeberg replacement trick'.

Persistent Homology (PH) has been used to study the topological characteristics of data across a variety of scales. In this talk, we will focus on a variety of spatial applications, as the geometric and topological features of PH are well suited to exploring data sets which are embedded in space. We will introduce two novel constructions for transforming network-based data into simplicial complexes suitable for PH computations and compare these constructions to state of the art. Additionally, we will discuss some preliminary results from applying these constructions to a variety of geographic and spatial applications, including voting data, cities and urban networks, and biological networks (i.e. spiders under the influence). We will highlight the computational performance of our constructions and discuss the implications of the PH computations for identifying and classifying certain features in our various data sets. In particular, we will talk about spatial patterns which emerge in each case, and how those patterns relate to existing scholarship in the relevant area.