Overview
I am a mathematician whose main research focus lies in adapting theory from ostensibly pure areas of mathematics, such as topology, geometry, and abstract algebra, into tools that can be broadly used in many data-centered applications.
My initial training was in a recently-emerging field called topological data analysis (TDA). I have been responsible for several essential and widely-used elements of its theoretical toolkit, with a particular focus on building TDA methodology for use on stratified spaces. Some of this work involves the creation of efficient algorithms, but much of it centers around theorem-proof mathematics, using proof techniques not only from algebraic topology, but also from computational geometry, from probability, and from abstract algebra.
Recently, I have done foundational work on TDA applications in several areas, including to neuroscience, to multi-target tracking, to multi-modal data fusion, and to a probabilistic theory of database merging. I am also becoming involved in efforts to integrate TDA within deep learning theory and practice.
I typically teach courses that connect mathematical principles to machine learning, including upper-level undergraduate courses in topological data analysis and more general high-dimensional data analysis, as well as a sophomore level course (joint between pratt and math) that serves as a broad introduction to machine learning and data analysis concepts.
Current Appointments & Affiliations
Recent Publications
Convolutional persistence transforms
Journal Article Journal of Applied and Computational Topology · November 1, 2024 In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local mot ... Full text CiteTopological Decompositions Enhance Efficiency of Reinforcement Learning
Conference IEEE Aerospace Conference Proceedings · January 1, 2024 Coordinating multiple sensors can be expressed as a reinforcement learning [RL] problem. Deep RL has excelled at observation processing (for example using convolution networks to process gridded data), but it suffers from sample inefficiency. To address th ... Full text CiteImplications of data topology for deep generative models
Journal Article Frontiers in Computer Science · January 1, 2024 Many deep generative models, such as variational autoencoders (VAEs) and generative adversarial networks (GANs), learn an immersion mapping from a standard normal distribution in a low-dimensional latent space into a higher-dimensional data space. As such, ... Full text CiteRecent Grants
Geometric and Topological Methods for Multi-Modal Data Analysis and Fusion
ResearchCo-Principal Investigator · Awarded by Air Force Office of Scientific Research · 2018 - 2023BIGDATA: F: DKA: CSD: Topological Data Analysis and Machine-Learning with Community-Accepted Features
ResearchCo-Principal Investigator · Awarded by National Science Foundation · 2014 - 2019Topological Signal Analysis for Multi-Modal Data Analysis
ResearchPrincipal Investigator · Awarded by Geometric Data Analytics, Inc. · 2016 - 2017View All Grants