Scholars@Duke
Paul L Bendich

Paul L Bendich

Research Professor of Mathematics
Mathematics
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Selected Publications

Convolutional persistence transforms

Journal Article Journal of Applied and Computational Topology · January 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

Topological 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 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

Topological Simplification of Signals for Inference and Approximate Reconstruction

Conference IEEE Aerospace Conference Proceedings · January 1, 2023 As Internet of Things (loT) devices become both cheaper and more powerful, researchers are increasingly finding solutions to their scientific curiosities both financially and com-putationally feasible. When operating with restricted power or communications ... 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 Parallax: A Geometric Specification for Deep Perception Models

Conference Advances in Neural Information Processing Systems · January 1, 2023 For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and ex ... 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

From Geometry to Topology: Inverse Theorems for Distributed Persistence

Conference Leibniz International Proceedings in Informatics, LIPIcs · June 1, 2022 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

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 Cite

A Fast and Robust Method for Global Topological Functional Optimization

Conference Proceedings of Machine Learning Research · January 1, 2021 Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for top ... 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

Graph Spectral Embedding for Parsimonious Transmission of Multivariate Time Series

Conference IEEE Aerospace Conference Proceedings · March 1, 2020 We propose a graph spectral representation of time series data that 1) is parsimoniously encoded to user-demanded resolution; 2) is unsupervised and performant in data-constrained scenarios; 3) captures event and event-transition structure within the time ... Full text Cite

Machine learning in/with information fusion for infrastructure understanding, panel summary

Conference Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2020 During the 2019 SPIE DSS conference, panelists were invited to highlight the trends and use of artificial intelligence and machine learning (AI/ML) for information fusion. The common themes between the panelists include leveraging AI/ML coordinated with In ... 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

Multi-Scale Geometric Summaries for Similarity-Based Sensor Fusion

Conference IEEE Aerospace Conference Proceedings · March 1, 2019 In this work, we address fusion of heterogeneous sensor data using wavelet-based summaries of fused self-similarity information from each sensor. The technique we develop is quite general, does not require domain specific knowledge or physical models, and ... Full text Cite

Topology, geometry, and machine-learning for tracking and sensor fusion

Conference Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2019 Cite

Upstream fusion of multiple sensing modalities using machine learning and topological analysis: An initial exploration

Conference IEEE Aerospace Conference Proceedings · June 25, 2018 This paper presents a processing pipeline for fusing 'raw' and / or feature-level multi-sensor data - upstream fusion - and initial results from this pipeline using imagery, radar, and radio frequency (RF) signals data to determine which tracked object, am ... Full text Cite

Geometric Cross-Modal Comparison of Heterogeneous Sensor Data

Conference Proceedings of the 39th IEEE Aerospace Conference · March 2018 In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by detectors at va ... Open Access 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

Convolutional persistence transforms

Journal Article Journal of Applied and Computational Topology · January 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

Topological 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 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

Topological Simplification of Signals for Inference and Approximate Reconstruction

Conference IEEE Aerospace Conference Proceedings · January 1, 2023 As Internet of Things (loT) devices become both cheaper and more powerful, researchers are increasingly finding solutions to their scientific curiosities both financially and com-putationally feasible. When operating with restricted power or communications ... 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 Parallax: A Geometric Specification for Deep Perception Models

Conference Advances in Neural Information Processing Systems · January 1, 2023 For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multiscale geometric structure. Our proofs and ex ... 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

From Geometry to Topology: Inverse Theorems for Distributed Persistence

Conference Leibniz International Proceedings in Informatics, LIPIcs · June 1, 2022 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

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 Cite

A Fast and Robust Method for Global Topological Functional Optimization

Conference Proceedings of Machine Learning Research · January 1, 2021 Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for top ... 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

Graph Spectral Embedding for Parsimonious Transmission of Multivariate Time Series

Conference IEEE Aerospace Conference Proceedings · March 1, 2020 We propose a graph spectral representation of time series data that 1) is parsimoniously encoded to user-demanded resolution; 2) is unsupervised and performant in data-constrained scenarios; 3) captures event and event-transition structure within the time ... Full text Cite

Machine learning in/with information fusion for infrastructure understanding, panel summary

Conference Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2020 During the 2019 SPIE DSS conference, panelists were invited to highlight the trends and use of artificial intelligence and machine learning (AI/ML) for information fusion. The common themes between the panelists include leveraging AI/ML coordinated with In ... 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

Multi-Scale Geometric Summaries for Similarity-Based Sensor Fusion

Conference IEEE Aerospace Conference Proceedings · March 1, 2019 In this work, we address fusion of heterogeneous sensor data using wavelet-based summaries of fused self-similarity information from each sensor. The technique we develop is quite general, does not require domain specific knowledge or physical models, and ... Full text Cite

Topology, geometry, and machine-learning for tracking and sensor fusion

Conference Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2019 Cite

Upstream fusion of multiple sensing modalities using machine learning and topological analysis: An initial exploration

Conference IEEE Aerospace Conference Proceedings · June 25, 2018 This paper presents a processing pipeline for fusing 'raw' and / or feature-level multi-sensor data - upstream fusion - and initial results from this pipeline using imagery, radar, and radio frequency (RF) signals data to determine which tracked object, am ... Full text Cite

Geometric Cross-Modal Comparison of Heterogeneous Sensor Data

Conference Proceedings of the 39th IEEE Aerospace Conference · March 2018 In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by detectors at va ... Open Access 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

Geometric models for musical audio data

Conference Leibniz International Proceedings in Informatics, LIPIcs · June 1, 2016 We study the geometry of sliding window embeddings of audio features that summarize perceptual information about audio, including its pitch and timbre. These embeddings can be viewed as point clouds in high dimensions, and we add structure to the point clo ... Full text Open Access Cite

Geometric Models for Musical Audio Data

Conference Proceedings of the 32st International Symposium on Computational Geometry (SOCG) · June 2016 Link to item 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

Cover Song Identification with Timbral Shape Sequences

Conference 16th International Society for Music Information Retrieval (ISMIR) · October 1, 2015 We introduce a novel low level feature for identifying cover songs which quantifies the relative changes in the smoothed frequency spectrum of a song. Our key insight is that a sliding window representation of a chunk of audio can be viewed as a time-order ... Open Access Link to item Cite

Multi-scale local shape analysis and feature selection in machine learning applications

Conference Proceedings of the International Joint Conference on Neural Networks · September 28, 2015 We introduce a method called multi-scale local shape analysis for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse ... Full text Open Access Cite

Feature-aided multiple hypothesis tracking using topological and statistical behavior classifiers

Conference Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2015 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 that uses elementary topological data analysis coupled with b ... Full text 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

Local homology transfer and stratification learning

Conference Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms · January 1, 2012 The objective of this paper is to show that point cloud data can under certain circumstances be clustered by strata in a plausible way. For our purposes, we consider a stratified space to be a collection of manifolds of different dimensions which are glued ... 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

Persistent homology under non-uniform error

Conference Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) · November 22, 2010 Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be r ... 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

Stratification learning through homology inference

Report · January 1, 2010 We develop a topological approach to stratification learning. Given point cloud data drawn from a stratified space, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a strat ... Cite