Paul L Bendich

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 Cite

Implications 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 Cite

FROM GEOMETRY TO TOPOLOGY: INVERSE THEOREMS FOR DISTRIBUTED PERSISTENCE

Journal article Journal of Computational Geometry · January 1, 2023 What is the “right” topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from injective. We therefore propose ... Full text Cite

Topological Feature Tracking for Submesoscale Eddies

Journal article Geophysical Research Letters · October 28, 2022 Current state-of-the art procedures for studying modeled submesoscale oceanographic features have made a strong assumption of independence between features identified at different times. Therefore, all submesoscale eddies identified in a time series were s ... Full text Cite

PERSISTENT OBSTRUCTION THEORY FOR A MODEL CATEGORY OF MEASURES WITH APPLICATIONS TO DATA MERGING

Journal article Transactions of the American Mathematical Society Series B · February 2, 2021 Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on a product space ... Full text Cite

A Fast and Robust Method for Global Topological Functional Optimization

Journal article 24TH INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND STATISTICS (AISTATS) · 2021 Link to item Cite

Geometric fusion via joint delay embeddings

Journal article Proceedings of 2020 23rd International Conference on Information Fusion Fusion 2020 · July 1, 2020 We introduce geometric and topological methods to develop a new framework for fusing multi-sensor time series. This framework consists of two steps: (1) a joint delay embedding, which reconstructs a high-dimensional state space in which our sensors corresp ... Full text Cite

Stabilizing the unstable output of persistent homology computations

Journal article Journal of Applied and Computational Topology · November 9, 2019 We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends ... Link to item Cite

Scaffoldings and Spines: Organizing High-Dimensional Data Using Cover Trees, Local Principal Component Analysis, and Persistent Homology

Journal article · January 1, 2018 We propose a flexible and multi-scale method for organizing, visualizing, and understanding point cloud datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree for a dataset using an adaptive threshold that is ... Full text Cite

Topological and statistical behavior classifiers for tracking applications

Journal article IEEE Transactions on Aerospace and Electronic Systems · December 1, 2016 This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced. The track appraisal uses elementary topological data analysi ... Full text Cite

Persistent homology analysis of brain artery trees

Journal article Annals of Applied Statistics · 2016 New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that ... Open Access Link to item Cite

Probabilistic Fréchet means for time varying persistence diagrams

Journal article Electronic Journal of Statistics · January 1, 2015 In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In [23], Mileyko and his collaborators made the first study of the properties of the Fréchet mean in ... Full text Open Access Cite

Homology and robustness of level and interlevel sets

Journal article Homology Homotopy and Applications · April 23, 2013 Given a continuous function f: X → ℝ on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness ... Full text Cite

A point calculus for interlevel set homology

Journal article Pattern Recognition Letters · August 1, 2012 The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calcu ... Full text Cite

Improving homology estimates with random walks

Journal article Inverse Problems · December 1, 2011 This experimental paper makes the case for a new approach to the use of persistent homology in the study of shape and feature in datasets. By introducing ideas from diffusion geometry and random walks, we discover that homological features can be enhanced ... Full text Cite

Persistent Intersection Homology

Journal article Foundations of Computational Mathematics · June 1, 2011 The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection h ... Full text Cite

The robustness of level sets

Journal article Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics · November 19, 2010 We define the robustness of a level set homology class of a function f : double-struck X → ℝ as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, usi ... Full text Cite

Computing robustness and persistence for images.

Journal article IEEE transactions on visualization and computer graphics · November 2010 We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation ne ... Full text Cite

Towards Stratification Learning through Homology Inference

Journal article · August 20, 2010 A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space ... Link to item Cite