Publications

2024

1. 

   A Categorical Approach to Möbius Inversion via Derived Functors

   Alex Elchesen, and Amit Patel

   2024

   Abs Bib

   We develop a cohomological approach to Möbius inversion using derived functors in the enriched categorical setting. For a poset 𝑃 and a closed symmetric monoidal abelian category 𝒞, we define Möbius cohomology as the derived functors of an enriched hom functor on the category of 𝑃-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical Möbius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota’s Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra.

   @misc{elchesen2024categorical,
     title = {{A Categorical Approach to M\"obius Inversion via Derived Functors}},
     author = {Elchesen, Alex and Patel, Amit},
     journal = {arXiv preprint arXiv:2411.04362},
     year = {2024},
   }

2. JACT

   Poincaré Duality for Generalized Persistence Diagrams of (co)Filtrations

   Amit Patel, and Tatum Rask

   Applied and Computational Topology, Feb 2024

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   We dualize previous work on generalized persistence diagrams for filtrations to cofiltrations. When the underlying space is a manifold, we express this duality as a Poincaré duality between their generalized persistence diagrams. A heavy emphasis is placed on the recent discovery of functoriality of the generalized persistence diagram and its connection to Rota’s Galois Connection Theorem.

   @article{patel2022poincare,
     title = {Poincar\'e Duality for Generalized Persistence Diagrams of (co)Filtrations},
     author = {Patel, Amit and Rask, Tatum},
     year = {2024},
     month = feb,
     journal = {Applied and Computational Topology},
   }

3. FODS

   Combinatorial Persistent Homology Transform

   Brittany Terese Fasy, and Amit Patel

   Foundations of Data Science, Apr 2024

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   The combinatorial interpretation of the persistence diagram as a Möbius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a cellulation on the n-sphere to the category of combinatorial persistence diagrams. Detailed examples are provided. We hope this recasting of the PH transform will allow for the adoption of existing methods from algebraic and topological combinatorics to the study of shapes.

   @article{fasy2022persistent,
     title = {Combinatorial Persistent Homology Transform},
     author = {Fasy, Brittany Terese and Patel, Amit},
     journal = {Foundations of Data Science},
     month = apr,
     year = {2024},
     primaryclass = {math.AT},
   }

2023

1. 

   Möbius Homology

   Amit Patel, and Primoz Skraba

   Apr 2023

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   This paper introduces Möbius homology, a homology theory for representations of finite posets into abelian categories. While the connection between poset topology and Möbius functions is classical, we establish a direct connection between poset topology and Möbius inversions. More precisely, the Möbius homology categorifies the Möbius inversion because its Euler characteristic is equal to the Möbius inversion of the dimension function of the representation. We also introduce a homological version of Rota’s Galois Connection Theorem which relates the Möbius homology over two posets connected by a Galois connection. Our main application is to persistent homology over general posets. We show that under one definition, the persistence diagram is an Euler characteristic over a poset of intervals and hence Möbius homology is a categorification of the persistence diagram. This provides a new invariant for persistent homology over general posets. Finally, we use our homological Rota’s Galois Connection Theorem to prove several results about the persistence diagram.

   @misc{patel2023mobius,
     title = {M\"obius Homology},
     author = {Patel, Amit and Skraba, Primoz},
     year = {2023},
     eprint = {2307.01040},
     archiveprefix = {arXiv},
     primaryclass = {math.AT},
   }

2022

1. SIAGA

   Edit Distance and Persistence Diagrams over Lattices

   Alexander McCleary, and Amit Patel

   SIAM Journal on Applied Algebra and Geometry, Apr 2022

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   We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice, and the output is a persistence diagram defined as the Möbius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are 1-Lipschitz, making our pipeline stable. Our constructions generalize the classical persistence diagram, and in this setting, the bottleneck distance is strongly equivalent to the edit distance.

   @article{edit-distance,
     author = {McCleary, Alexander and Patel, Amit},
     journal = {SIAM Journal on Applied Algebra and Geometry},
     number = {2},
     pages = {134-155},
     title = {Edit Distance and Persistence Diagrams over Lattices},
     volume = {6},
     year = {2022},
   }

2021

1. Advances

   Persistent local systems

   Robert MacPherson, and Amit Patel

   Advances in Mathematics, Apr 2021

   Abs Bib HTML

   In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions.

   @article{MACPHERSON2021107795,
     author = {MacPherson, Robert and Patel, Amit},
     journal = {Advances in Mathematics},
     keywords = {Persistent homology, Stability, Stratified spaces, Sheaves, Cosheaves},
     pages = {107795},
     title = {Persistent local systems},
     volume = {386},
     year = {2021},
   }

2. 

   Output-sensitive Computation of Generalized Persistence Diagrams for 2-filtrations

   Dmitriy Morozov, and Amit Patel

   Apr 2021

   Abs Bib HTML

   When persistence diagrams are formalized as the Möbius inversion of the birth–death function, they naturally generalize to the multi-parameter setting and enjoy many of the key properties, such as stability, that we expect in applications. The direct definition in the 2-parameter setting, and the corresponding brute-force algorithm to compute them, requireΩ(n^4) operations, where n is the complexity of the input. But the size of the generalized persistence diagram, C, can be as low as linear (and as high as cubic). We elucidate a connection between the 2-parameter and the ordinary 1-parameter settings, which allows us to design an output-sensitive algorithm, whose running time is in O(n^2 m + Cm), where m is the number of simplices in the input.

   @misc{morozov2023outputsensitive,
     title = {Output-sensitive Computation of Generalized Persistence Diagrams for 2-filtrations},
     author = {Morozov, Dmitriy and Patel, Amit},
     year = {2021},
     eprint = {2112.03980},
     archiveprefix = {arXiv},
     primaryclass = {cs.CG},
   }

2020

1. TAC

   Classification of Constructible Cosheaves

   Justin Curry, and Amit Patel

   Theory and Applications of Categories, Apr 2020

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   In this paper we prove an equivalence theorem originally observed by Robert MacPherson. On one side of the equivalence is the category of cosheaves that are constructible with respect to a locally cone-like stratification. Our constructibility condition is new and only requires that certain inclusions of open sets are sent to isomorphisms. On the other side of the equivalence is the category of functors from the entrance path category, which has points for objects and certain homotopy classes of paths for morphisms. When our constructible cosheaves are valued in Set we prove an additional equivalence with the category of stratified coverings.

   @article{curry_patel,
     title = {Classification of Constructible Cosheaves},
     author = {Curry, Justin and Patel, Amit},
     year = {2020},
     journal = {Theory and Applications of Categories},
     number = {27},
     volume = {35},
     pages = {1012--1047},
   }

2. Abel

   Canonical Stratifications Along Bisheaves

   Amit Patel, and Vidit Nanda

   In Proceedings of the Abel Symposium 2018: Topological Data Analysis, Apr 2020

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   A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.

   @incollection{Nanda_Patel,
     title = {Canonical Stratifications Along Bisheaves},
     author = {Patel, Amit and Nanda, Vidit},
     booktitle = {Proceedings of the Abel Symposium 2018: Topological Data Analysis},
     series = {Abel Symposium},
     volume = {15},
     publisher = {Springer},
     year = {2020},
   }

3. AMS

   Bottleneck stability for generalized persistence diagrams

   Alex McCleary, and Amit Patel

   Proceedings of the American Mathematical Society. American Mathematical Society, Mar 2020

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   In this paper, we extend bottleneck stability to the setting of one dimensional constructible persistence modules valued in any skeletally small abelian category.

   @article{McCPa20,
     title = {Bottleneck stability for generalized persistence diagrams},
     author = {McCleary, Alex and Patel, Amit},
     journal = {Proceedings of the American Mathematical Society. American Mathematical Society},
     volume = {148},
     number = {7},
     pages = {3149--3161},
     month = mar,
     year = {2020},
     issn = {0002-9939, 1088-6826},
   }

2018

1. JACT

   Generalized persistence diagrams

   Amit Patel

   Journal of Applied and Computational Topology, Jun 2018

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   We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the \emphtype \mathcalA persistence diagram of a persistence module. If the category is also abelian, then we define a second \emphtype \mathcalB persistence diagram. In addition, we show that both diagrams are stable to all sufficiently small perturbations of the module. The type \mathcalB persistence diagram carries less information than the type \mathcalA persistence diagram, but it enjoys a stronger stability theorem.

   @article{Patel2018,
     title = {Generalized persistence diagrams},
     author = {Patel, Amit},
     journal = {Journal of Applied and Computational Topology},
     publisher = {Springer International Publishing},
     volume = {1},
     number = {3},
     pages = {397--419},
     month = jun,
     year = {2018},
     language = {en},
     dimensions = {false},
   }

2016

1. DCG

   Categorified Reeb Graphs

   Vin Silva, Elizabeth Munch, and Amit Patel

   Discrete & Computational Geometry, Jun 2016

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   The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.

   @article{CategReebGraph,
     author = {de Silva, Vin and Munch, Elizabeth and Patel, Amit},
     da = {2016/06/01},
     journal = {Discrete \& Computational Geometry},
     number = {4},
     pages = {854--906},
     title = {Categorified Reeb Graphs},
     volume = {55},
     year = {2016},
   }

2013

1. HHA

   Homology and robustness of level and interlevel sets

   Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, and 1 more author

   Homology, Homotopy and Applications, Jun 2013

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   Given a continuous function f : X \to \mathcalR 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 of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X = \mathcalR^3 has ramifications in the fields of medical imaging and scientific visualization.

   @article{well-groups-1d,
     title = {Homology and robustness of level and interlevel sets},
     author = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
     journal = {Homology, Homotopy and Applications},
     publisher = {International Press of Boston},
     volume = {15},
     number = {1},
     pages = {51--72},
     year = {2013},
     language = {en},
     issn = {1532-0073, 1532-0081},
   }

2012

1. APL

   Computing well diagrams for vector fields on R^n

   Frédéric Chazal, Primoz Skraba, and Amit Patel

   Applied Mathematics Letters, Jun 2012

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   Using topological degree theory, we present a fast algorithm for computing the well diagram, a quantitative property, of a vector field on Euclidean space.

   @article{CHAZAL20121725,
     author = {Chazal, Fr{\'e}d{\'e}ric and Skraba, Primoz and Patel, Amit},
     issn = {0893-9659},
     journal = {Applied Mathematics Letters},
     keywords = {Well groups, Persistence, Fixed points, Stability, Algorithms},
     number = {11},
     pages = {1725-1728},
     title = {Computing well diagrams for vector fields on $R^n$},
     volume = {25},
     year = {2012},
   }

2011

1. Top Viz

   The Stability of the Apparent Contour of an Orientable 2-Manifold

   Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel

   Jun 2011

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   The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f :M \to \mathcalR^2, is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.

   @inbook{Edelsbrunner2011,
     address = {Berlin, Heidelberg},
     author = {Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
     booktitle = {Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications},
     editor = {Pascucci, Valerio and Tricoche, Xavier and Hagen, Hans and Tierny, Julien},
     pages = {27--41},
     publisher = {Springer Berlin Heidelberg},
     title = {The Stability of the Apparent Contour of an Orientable 2-Manifold},
     year = {2011},
   }

2. FoCM

   Quantifying Transversality by Measuring the Robustness of Intersections

   Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel

   Foundations of Computational Mathematics, Jun 2011

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   By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

   @article{well-groups,
     title = {Quantifying Transversality by Measuring the Robustness of Intersections},
     author = {Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
     journal = {Foundations of Computational Mathematics},
     volume = {11},
     number = {3},
     pages = {345--361},
     month = jun,
     year = {2011},
     issn = {1615-3375, 1615-3383},
   }

2010

1. ESA

   The Robustness of Level Sets

   Paul Bendich, Herbert Edelsbrunner, Dmitriy Morozov, and 1 more author

   In Algorithms – ESA 2010, Jun 2010

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   We define the robustness of a level set homology class of a function }f: {\backslashmathbb X} \backslashto {\backslashmathbb 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, using a connection to extended persistent homology. The special case }{\backslashmathbb X} = {\backslashmathbb R}^3}has ramifications in medical imaging and scientific visualization.

   @inproceedings{robustness_level_sets,
     address = {Berlin, Heidelberg},
     author = {Bendich, Paul and Edelsbrunner, Herbert and Morozov, Dmitriy and Patel, Amit},
     booktitle = {Algorithms -- ESA 2010},
     editor = {de Berg, Mark and Meyer, Ulrich},
     isbn = {978-3-642-15775-2},
     pages = {1--10},
     publisher = {Springer Berlin Heidelberg},
     title = {The Robustness of Level Sets},
     year = {2010},
   }

2. MFCS

   Persistent Homology under Non-uniform Error

   Paul Bendich, Herbert Edelsbrunner, Michael Kerber, and 1 more author

   In Mathematical Foundations of Computer Science 2010, Jun 2010

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   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 read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.

   @inproceedings{Non-uniform,
     address = {Berlin, Heidelberg},
     author = {Bendich, Paul and Edelsbrunner, Herbert and Kerber, Michael and Patel, Amit},
     booktitle = {Mathematical Foundations of Computer Science 2010},
     editor = {Hlin{\v{e}}n{\'y}, Petr and Ku{\v{c}}era, Anton{\'\i}n},
     isbn = {978-3-642-15155-2},
     pages = {12--23},
     publisher = {Springer Berlin Heidelberg},
     title = {Persistent Homology under Non-uniform Error},
     year = {2010},
   }

3. Duke

   Reeb Spaces and the Robustness of Preimages

   Amit K. Patel

   Duke University, Jun 2010

   Bib

   @phdthesis{citekey,
     author = {Patel, Amit K.},
     title = {Reeb Spaces and the Robustness of Preimages},
     school = {Duke University},
     year = {2010},
   }

2008

1. SoCG

   Reeb Spaces of Piecewise Linear Mappings

   Herbert Edelsbrunner, John Harer, and Amit K. Patel

   In Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, Jun 2008

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   Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold.

   @inproceedings{Reeb_spaces,
     author = {Edelsbrunner, Herbert and Harer, John and Patel, Amit K.},
     title = {Reeb Spaces of Piecewise Linear Mappings},
     year = {2008},
     isbn = {9781605580715},
     publisher = {Association for Computing Machinery},
     address = {New York, NY, USA},
     booktitle = {Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry},
     pages = {242–250},
     numpages = {9},
     keywords = {combinatorial manifolds, cone neighborhoods, reeb spaces, smooth and pl topology, stratifications, algorithms, triangulations},
     location = {College Park, MD, USA},
     series = {SCG '08},
   }