Journal ArticleJournal 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 ...
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ConferenceIEEE 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 ...
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Journal ArticleFrontiers 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, ...
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ConferenceIEEE 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 ...
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Journal ArticleJournal 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 ...
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ConferenceAdvances 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 ...
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Journal ArticleGeophysical 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 ...
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ConferenceLeibniz 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 textCite

Journal ArticleTransactions 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 ...
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ConferenceProceedings 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 ...
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Journal ArticleProceedings 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 ...
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ConferenceIEEE 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 ...
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ConferenceProceedings 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 ...
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Journal ArticleJournal 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 ...
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ConferenceIEEE 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 textCite

ConferenceIEEE 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 textCite

ConferenceProceedings 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 ...
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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 textCite

Journal ArticleJournal 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 textCite

ConferenceIEEE 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 textCite

Journal ArticleFrontiers 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 textCite

ConferenceIEEE 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 textCite

Journal ArticleJournal 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 textCite

ConferenceAdvances 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

Journal ArticleGeophysical 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 textCite

ConferenceLeibniz 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 textCite

Journal ArticleTransactions 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 textCite

ConferenceProceedings 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

Journal ArticleProceedings 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 textCite

ConferenceIEEE 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 textCite

ConferenceProceedings 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 textCite

Journal ArticleJournal 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 itemCite

ConferenceIEEE 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 textCite

ConferenceIEEE 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 textCite

ConferenceProceedings 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 AccessLink to itemCite

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 textCite

Journal ArticleIEEE 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 ...
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ConferenceLeibniz 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 ...
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Journal ArticleAnnals 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 AccessLink to itemCite

Conference16th 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 AccessLink to itemCite

ConferenceProceedings 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 textOpen AccessCite

ConferenceProceedings 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 ...
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Journal ArticleElectronic 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 ...
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Journal ArticleHomology, 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 ...
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Journal ArticlePattern 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 ...
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ConferenceProceedings 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 ...
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Journal ArticleInverse 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 ...
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Journal ArticleFoundations 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 ...
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ConferenceLecture 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 ...
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Journal ArticleLecture 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 ...
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Journal ArticleIEEE 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 ...
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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 ...
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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 ...
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