Projects

Spectral relaxations of persistent rank invariants

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TopologyPersistenceLinear Algebra

We introduce a framework for constructing families of continuous relaxations of the persistent rank invariant for persistence modules indexed over the real line. Applications to multi-parameter persistence, parameter optimization, and shape classification are also presented.

Using the fact that the persistent rank invariant determines the persistence diagram and vice versa, we introduce a framework for constructing families of continuous relaxations of the persistent rank invariant for persistence modules indexed over the real line. Like the rank invariant, these families obey inclusion-exclusion, are derived from simplicial boundary operators, and encode all the information needed to construct a persistence diagram. Unlike the rank invariant, these spectrally-derived families enjoy a number of stability and continuity properties typically reserved for persistence diagrams, such as smoothness and differentiability over the positive semi-definite cone. By leveraging its relationship with combinatorial Laplacian operators, we find the non-harmonic spectra of our proposed relaxation encode valuable geometric information about the underlying space, prompting several avenues for geometric data analysis. As these Laplacian operators are trace-class operators, we also find the corresponding relaxation can be efficiently approximated with a randomized algorithm based on the stochastic Lanczos quadrature method. We investigate the utility of our relaxation with applications in topological data analysis and machine learning, such as parameter optimization and shape classification.

Publications

• Piekenbrock, Matthew, and Jose A. Perea. “Spectral families of persistent rank invariants.” Computational Persistence Workshop (2023). (link)

Software

• pbsig unorganized code containing the experiments
• primate package code for implicit matrix trace estimation
• simplextree (python package)

Related materials

• YouTube link for CompPers23 talk

Move Schedules: Fast persistence computations in coarse dynamic settings

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TopologyAlgorithmsPersistence

Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are quadratically many such transpositions, this maintenance procedure exhibits limited scalability and often is too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations that requires only subquadratic time and linear space to construct.

Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles are derived. Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are quadratically such transpositions, this maintenance procedure exhibits limited scalability and often is too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations that requires only superlinear time and linear space to construct. By reduction to a particular longest common subsequence problem, we show the storage needed to employ this strategy is actually sublinear in expectation. Exploiting this connection, we show experimentally that the decrease in operations to compute diagrams across a family of filtrations, is proportional to the difference between the expected quadratic number of states and the proposed sublinear coarsening. Applications to video data, dynamic metric space data, and multi-parameter persistence are also presented.

Publications

• Piekenbrock, Matthew, and Jose A. Perea. “Move Schedules: Fast persistence computations in coarse dynamic settings.” arXiv preprint arXiv:2104.12285 (2021).

Software

• Move scheduling code
• simplextree (python package)

Efficient Multiscale Simplicial Complex Generation for Mapper

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ClusteringTopologyR Package

The primary result of the Mapper framework is the geometric realization of a simplicial complex, depicting topological relationships and structures suitable for visualizing, analyzing, and comparing high dimensional data...

The primary result of the Mapper framework is the geometric realization of a simplicial complex, depicting topological relationships and structures suitable for visualizing, analyzing, and comparing high dimensional data. As an unsupervised tool that may be used for exploring or modeling heterogeneous types of data, Mapper naturally relies on a number of parameters which explicitly control the quality of the resulting construction; one such critical parameter controls the entire relational component of the output complex. In practice, there is little guidance on what values may provide 'better' or more 'stable' sets of simplices. In this effort, we provide a new algorithm that enables efficient computation of successive mapper realizations with respect to this crucial parameter. Our results not only enhances the exploratory/confirmatory aspect of Mapper, but also give tractability to recent theoretical extensions to Mapper related to persistence and stability.

Publications

• Multiscale Mapper paper (never finished!)

Software

• Mapper (R Package)
• simplextree (R Package)
• Vignette on using Mapper for shape recognition

Automating Point of Interest Discovery in Geospatial Contexts

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ClusteringGeospatial analysisNetwork modeling

With the rapid development and widespread deployment of sensors dedicated to location-acquisition, new types of models have emerged to predict macroscopic patterns that manifest in large data sets representing "significant" group behavior. Partially due to the immense scale of geospatial data, current approaches to discover these macroscopic patterns are primarily driven by inherently heuristic detection methods. Although useful in practice, the inductive bias adopted by such mainstream detection schemes is often unstated or simply unknown. Inspired by recent theoretical advances in efficient non-parametric density level set estimation techniques, in this research effort we describe a semi-supervised framework for automating point of interest discovery in geospatial contexts. We outline the flexibility and utility of our approach through numerous examples, and give a systematic framework for incorporating semisupervised information while retaining finite-sample estimation guarantees.

With the rapid development and widespread deployment of sensors dedicated to location-acquisition, new types of models have emerged to predict macroscopic patterns that manifest in large data sets representing "significant" group behavior. Partially due to the immense scale of geospatial data, current approaches to discover these macroscopic patterns are primarily driven by inherently heuristic detection methods. Although useful in practice, the inductive bias adopted by such mainstream detection schemes is often unstated or simply unknown. Inspired by recent theoretical advances in efficient non-parametric density level set estimation techniques, in this research effort we describe a semi-supervised framework for automating point of interest discovery in geospatial contexts. We outline the flexibility and utility of our approach through numerous examples, and give a systematic framework for incorporating semisupervised information while retaining finite-sample estimation guarantees.

Publications

• Piekenbrock, Matthew J. Discovering Intrinsic Points of Interest from Spatial Trajectory Data Sources. Masters thesis. Wright State University, 2018. link
• Piekenbrock, Matthew, and Derek Doran. “Intrinsic point of interest discovery from trajectory data.” arXiv:1712.05247 (2017) (doi: https://doi.org/10.48550/arXiv.1712.05247)
• Matthew Maurice, Matt Piekenbrock, and Derek Doran. Waminet: An open source library for dynamic geospace analysis using WAMI. In IEEE International Symposium on Multimedia, pages 445–448. IEEE, 2015.

Bringing High Performance Density-based Clustering to R

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ClusteringR PackageHigh performance computing

Density-based clustering techniques have become extremely popular in the past decade. It's often conjectured that the reason for the success of these methods is due to their ability of identify 'natural groups' in data. These groups are often non-convex (in terms of shape), deviating the typical premise of 'minimal variance' that underlies parametric, model-based approaches, and often appear in very large data sets. As the era of 'Big Data' continues to rise in popularity, it seems that typical notions having access to scalable, easy-to-use, and scalable implementations of these density-based methods is paramount. In this research effort, we provide fast, state-of-the-art density-based algorithms in the form of an open-source package in R. We also provide several related density-based clustering tools to help bring make state of the art density-based clustering accessible to people with large, computationally difficult problems.

Density-based clustering techniques have become extremely popular in the past decade. It's often conjectured that the reason for the success of these methods is due to their ability of identify 'natural groups' in data. These groups are often non-convex (in terms of shape), deviating the typical premise of 'minimal variance' that underlies parametric, model-based approaches, and often appear in very large data sets. As the era of 'Big Data' continues to rise in popularity, it seems that typical notions having access to scalable, easy-to-use, and scalable implementations of these density-based methods is paramount. In this research effort, we provide fast, state-of-the-art density-based algorithms in the form of an open-source package in R. We also provide several related density-based clustering tools to help bring make state of the art density-based clustering accessible to people with large, computationally difficult problems.

Publications

• Hahsler, Michael, Matthew Piekenbrock, and Derek Doran. “dbscan: Fast Density Based Clustering in R”, Journal of Statistical Software, 2018. (https://doi.org/10.18637/jss.v091.i01)

Software

• dbscan (R Package)
• Vignette on using HDBSCAN

Towards Autonomous Aerial Refueling: Massive Parallel Iterative Closest Point

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GeometryPoint registrationHigh performance computing

The Iterative Closest Point (ICP) problem is now a well-studied problem that seeks to align a given query point cloud to a fixed reference point cloud. The ICP problem computationally is dominated by the first phase, a pairwise distance minimization. The ''brute-force'' approach, an embarrassingly parallel problem amenable to GPU-acceleration..

The Iterative Closest Point (ICP) problem is now a well-studied problem that seeks to align a given query point cloud to a fixed reference point cloud. The ICP problem computationally is dominated by the first phase, a pairwise distance minimization. The 'brute-force' approach, an embarrassingly parallel problem amenable to GPU-acceleration, involves calculating the pairwise distance from every point in the query set to every point in the reference set. This however still requires linear runtime complexity per thread, rendering the trivial solution unsuitable for e.g. real-time applications. Alternative spatial indexing data structures utilizing branch-and-bound (B&B) properties have been proposed as a means of reducing the algorithmic complexity of the ICP problem, however they were originally developed for serial applications: it is well known that direct conversion to their parallel equivalents often results in slower runtime performance than GPU-employed brute-force approaches due to frequent suboptimal memory access patterns and conditional computations. In this application-motivated effort, we propose a novel two-step method which exposes the intrinsic parallelism of the ICP problem, yet retains a number of the B&B properties. Our solution involves an O(log n) approximate search, followed by fast vectorized search we call the Delaunay Traversal, which we show empirically finishes in O(k) time on average, where k << n, and is demonstrated to generally exhibit extremely small growth factors on average. We demonstrate the superiority of our method compared to the traditional B&B and brute-force implementations using a variety of benchmark data sets, and demonstrate its usefulness in the context of Autonomous Aerial Refueling

Publications

• J. Robinson, M. Piekenbrock, L. Burchett, et. al. Parallelized Iterative Closest Point for Autonomous Aerial Refueling. In International Symposium on Visual Computing (pp. 593-602). Springer International Publishing. (2016, December) (doi: 10.1007/978-3-319-50835-1_53)
• Piekenbrock, M., Robinson, J., Burchett, L., Nykl, S., Woolley, B., & Terzuoli, A. (2016, July). Automated aerial refueling: Parallelized 3D iterative closest point: Subject area: Guidance and control. In Aerospace and Electronics Conference (NAECON) and Ohio Innovation Summit (OIS), 2016 IEEE National (pp. 188-192). IEEE. (doi: 10.1109/NAECON.2016.7856797)

Employment

1. Graduate Research Assistant

   Perea Lab

   Fall 2019 - Present

   MSU/NEU

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   Topological Data AnalysisLinear AlgebraMachine Learning

   Motivated by my previous work on the foundations of density-based clustering, I focused on implementing and extending the Mapper algorithm, a popular and very general method which has been used successfully for data analysis.

   After beginning my doctoral research at Michigan State University (Fall 19’), I transferred to Northeastern University (Boston, MA) in Fall 2021 after my advisor (Jose Perea) accepted a joint appointment offer at the Khoury College of Computer Sciences.

   
   

   My doctoral research focused on applications of topological theory to various common machine learning applications. In particular, much of my time was spent on accelerating the persistence algorithm in time-varying settings, codeveloping a topological dimensionality reduction using fiber bundle theory, and on studying a spectral-relaxations of the persistent rank invariant.

2. SCaN Intern

   National Aeronautics and Space Administration

   Summer 2022

   John H. Glenn Research Center at Lewis Field, OH

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   Space networkingGraph TheoryFlow algorithms

   Towards enabling delay-tolerant satellite communications in uncertain space environments, I was re-hired back at NASA as part of the Space Communications and Navigation (SCaN) program to expand the algorithmic theory on time-dependent routing.

   Job Description

   Towards enabling delay-tolerant satellite communications in uncertain space environments, I was re-hired back at NASA as part of the Space Communications and Navigation (SCaN) program to expand the algorithmic theory on time-dependent routing. My research focused on incorporating additional geometric assumptions into routing models built for of delay- and disruption-tolerant networks, particularly in the low Earth orbit regime.
3. Research Associate

   Oak Ridge Institute for Science and Education

   Fall 2017, Fall 2018 - Fall 2019

   Air Force Research Laboratory, WPAFB

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   TopologyMapperR package

   Motivated by my previous work on the foundations of density-based clustering, I focused on implementing and extending the Mapper algorithm, a popular and very general method which has been used successfully for data analysis.

   Job Description

   In 2017, I joined with a local research group under Dr. Ryan Kramer as part of AFRL’s Human Performance Wing to explore and expand the intersection between algorithms in TDA and machine learning. My time there was focused on implementing and extending the Mapper algorithm, a topological method that reframes common data analysis tasks as problems of analyzing level sets on topological spaces (introduction available here).

   
   

   My research was centered around enabling the efficient construction of mappers in multiscale settings and on understanding the connections the Mapper algorithm had to other existing constructions, such as Reeb graphs, nerve complexes, and hierarchical clustering. Beyond learning the theory and developing the software, I also applied Mapper to various geospatial and image analysis tasks, such as video segmentation, object recognition, and clustering.

   Software

   • Mapper (R Package)
   • simplextree (R Package)
   • Vignette on using Mapper for shape recognition
4. Graduate Research Assistant

   Web and Complex Systems Group

   Spring 2016 - Fall 2018

   Wright State University

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   ClusteringNetwork analysisMachine Learning

   Motivated by my previous work on the foundations of density-based clustering, I focused on implementing and extending the Mapper algorithm, a popular and very general method which has been used successfully for data analysis.

   My graduate research aimed at modeling real-world traffic network networks at a macroscopic scale. The high-level goal of the project was to model dynamic network representations extracted from raw positioning/track information via random (distributional) network models. On the software side, the project involved:

   • Density-based clustering (R/Rcpp)
   • Geospatial Point of Interest (POI) detection / Nonparameteric distribution modeling (R/C++)
   • Spatio-temporal network models (R)

   Research topics involved during this time include density-based clustering algorithms, cluster validation measures, non-parametric density estimation techniques, Markov Chain Monte Carlo (MCMC) optimization techniques, and random graph modeling (stochastic block models).

5. LERCIP Intern

   National Aeronautics and Space Administration

   Summer 2018

   John H. Glenn Research Center at Lewis Field, OH

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   Experimental designMachine learningMaterial science

   Towards accelerating the design and discovery materials for use in extreme environments, I was hired by Dr. Steven Arnold under NASAs 10-week LERCIP program to apply Machine Learning to a specific Material Science problem.

   Job Description:

   I was hired by Dr. Steven Arnold under NASAs 10-week LERCIP program to use machine learning to accelerate the simulation-based design of materials and structures through multiscale modeling, in line with NASA’s 2040 vision.

   Publications:

   • Stuckner, J., Piekenbrock, M., Arnold, S. M., & Ricks, T. M. (2021). Optimal experimental design with fast neural network surrogate models. Computational Materials Science, 200, 110747.
   • Arnold, S. M., Piekenbrock, M., Ricks, T. M., & Stuckner, J. (2020). Multiscale analysis of composites using surrogate modeling and information optimal designs. In AIAA Scitech 2020 Forum (p. 1863).

   Software

   • Experimental optimal design code
6. Student Participant

   Google Summer of Code

   Summer 2017

   R Project for Statistical Computing

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   ClusteringLearning theoryR package

   Towards unifying recent developments related the theory and utility of density-based clustering, this project involved a mixture of research and code development which culminated in the form of an R package for estimating the empirical cluster tree.

   I submitted a successful funding proposal under the Google Summer of Code (GSOC) Initiative to the R Project for Statistical Computing to explore, develop, and unify recent developments related the theory of density-based clustering (see the project page). This involved a mixture of research and code development which culminated in the form of an R package for estimating the cluster tree, a hierarchical summary of the level-sets of a density function. There was also a WSU newsroom piece that describes the proposal in a non-technical way.

7. Civilian Research Assistant

   Oak Ridge Institute for Science and Education

   Spring 2014 - Spring 2015

   Air Force Institute of Technology, WPAFB

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   OptimizationGraph theoryFlow algorithms

   Towards the end of my undergraduate degree, my contract at the [Air Force Institute of Technology](https://www.afit.edu/) (AFIT) was extended under ORISE, where I continued working with the LOREnet group under Dr. Andrew Terzuoli

   Job Description

   Towards the end of my undergraduate degree, my contract at the Air Force Institute of Technology (AFIT) was extended under ORISE, where I continued working with the LOREnet group under Dr. Andrew Terzuoli. During this time I primarily worked with Dr. Scott Nykl on the development of a novel Iterative Closest Point algorithm amenable to massive parallelization, implemented in C++/CUDA, for the purposes of enabling real-time tracking of aircraft in the context of Autonomous Aerial Refueling.

   Publications

   • J. Robinson, M. Piekenbrock, L. Burchett, et. al. Parallelized Iterative Closest Point for Autonomous Aerial Refueling. In International Symposium on Visual Computing (pp. 593-602). Springer International Publishing. (2016, December) (doi: 10.1007/978-3-319-50835-1_53)
   • Piekenbrock, M., Robinson, J., Burchett, L., Nykl, S., Woolley, B., & Terzuoli, A. (2016, July). Automated aerial refueling: Parallelized 3D iterative closest point: Subject area: Guidance and control. In Aerospace and Electronics Conference (NAECON) and Ohio Innovation Summit (OIS), 2016 IEEE National (pp. 188-192). IEEE. (doi: 10.1109/NAECON.2016.7856797)

   Projects

   Example projects included, but were not limited too:

   • Researching hierarchical markov model for predicting web navigation patterns
   • Parallelizing existing atmospheric absorption routines with OpenCL
   • Coding a nonlinear optimization algorithm in ANSI-C, and making it callable from MATLAB via MEX
8. Civilian Research Assistant

   Southwestern Ohio Council For Higher Education

   December 2013 - June 2014

   Air Force Institute of Technology, WPAFB

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   OptimizationGraph theoryFlow algorithms

   As my first experience doing undergraduate research, I worked in a heavily multi-disciplinary team called the Low Orbitals Radar and Electromagnetism group, where I worked on a diverse set of projects involving computational, statistical, or physics-based requirements

   Job Description

   Under the guidance of Dr. Andrew Terzuoli, I was hired at the Air Force Institute of Technology (AFIT) to do research in a multi-disciplinary team called the Low Orbitals Radar and Electromagnetism (LOREnet) group, where I worked on a diverse set of projects involving computational, statistical, or physics-based requirements. As my first research-oriented experience, I primarily assisted graduate students with programmatic or computationally-intensive tasks.

   Projects

   Example projects included, but were not limited too:

   • Implementing an unsplittable flow approximation algorithm (C++ and Python)
   • Creating a conversion tool between Oracle’s Abstract Data Type and XMLType (Java)
   • Developing a prototype UI for searching and viewing 3d models (JavaScript+three.js)

Education

1. Doctorate in CS (Pursuing)

   Khoury College of Computer Sciences

   Northeastern University, 2021-Present

   Advisor: Jose Perea

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   Click for teaching experience, coursework taken, and other details...

   Teaching:

   • Graduate teaching assistant - Data Mining Techniques (CS 6220 / DS 5230), Summer 2023
   • Graduate teaching assistant - Machine Learning (CS 6140/4420), Spring 2023
   • Graduate teaching assistant - Unsupervised Learning (CS 6220 / DS 5230), Fall 2022

   Coursework (GPA: 3.83):

   • Formal Verification, Modeling, & Synthesis
   • Network Visualization
2. Doctorate in CMSE (Transferred)

   Computational Mathematics, Science and Engineering

   Michigan State University, 2019-2021

   Advisor: Jose Perea

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   Click for teaching experience, coursework taken, and other details...

   Coursework (GPA: 3.83):

   • Numerical Linear Algebra (CMSE 823)
   • Numerical Differential Equations (CMSE 821)
   • Math Foundations of Data Science (CMSE 890)
   • Topological Methods for the Analysis of Data (CMSE 890)
   • Parallel Computing (CMSE 822)
   • Geometry and Topology II (MTH 869)
   • Mathematical foundations of analysis (CMSE 890)
   • Algebra I (MTH 818)
3. Masters of Science in CS

   College of Engineering and Computer Science

   Wright State University, 2015-2018

   Advisor: Derek Doran

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   Click for teaching experience, coursework taken, and other details...

   Coursework (GPA: 3.88):

   • Network Science
   • Machine Learning
   • Information Theory
   • Applied Stochastic Processes
   • Algorithm Design and Analysis
   • Empirical Analysis
   • Advanced Programming Languages
   • Distributed Computing
4. Bachelor of Science in CS (+STT)

   College of Engineering and Computer Science

   Wright State University, 2010-2015

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   Click for teaching experience, coursework taken, and other details...

   Coursework (GPA: 3.42, in-major):

   • Applied Statistics I & II
   • Optimization Techniques
   • Foundations of AI
   • Computational Tools for Data Analysis
   • Theoretical Statistics
   • Linear Algebra