Luis Scoccola

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All published works (29)

Multi-parameter Persistence Modules are Generically Indecomposable

2025-02-26

Abstract Algebraic persistence studies persistence modules (typically, linear representations of the poset $\textbf{R}^{n}$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of … Abstract Algebraic persistence studies persistence modules (typically, linear representations of the poset $\textbf{R}^{n}$ with $n \geq 1$) and the algebraic relationships between persistence modules that are interleaved. The notion of $\varepsilon $-interleaving between persistence modules is a generalization of the notion of isomorphism (recovering isomorphism when $\varepsilon = 0$), which can be used to quantify how far any two persistence modules are from being isomorphic. An emblematic example of this kind of study is the algebraic stability theorem, which strengthens the Krull–Schmidt property of one-parameter persistence modules (representations of $\textbf{R}$) by generalizing isomorphism to interleaving: If a pair of one-parameter persistence modules is $\varepsilon $-interleaved, then there exists a partial matching between the indecomposable summands of the two modules such that matched indecomposables are $\varepsilon $-interleaved and unmatched indecomposables are $\varepsilon $-interleaved with the zero module. Our first main result implies that the obvious extension of the algebraic stability theorem to the case of multi-parameter persistence modules (representations of $\textbf{R}^{n}$ with $n \geq 2$) fails spectacularly: Any finitely presentable multi-parameter persistence module can be approximated arbitrarily well by an indecomposable module. Our second main result states that modules that are sufficiently close to an indecomposable decompose as a direct sum of an indecomposable and a nearly trivial module. We derive from these two results several consequences about the interplay between the algebraic and the topological properties of multi-parameter persistence modules. These results provide strong motivation for approaching multi-parameter persistence in a way that does not rely on directly decomposing modules by indecomposables.

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Decomposing zero-dimensional persistent homology over rooted tree quivers

2024-11-28

Riju Bindua , Thomas Brüstle , Luis Scoccola

Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with … Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is fundamental in persistence theory, where it is known as persistent homology, and where the category is typically a poset. Persistence theory is particularly successful when the poset is a finite linearly ordered set, owing to the fact that in this case its category of representations is of finite type. We show that when the poset is a rooted tree poset (a poset with a maximum and whose Hasse diagram is a tree) the additive closure of the category of representations obtainable as zero-dimensional persistent homology is of finite type, and give a quadratic-time algorithm for decomposition into indecomposables. In doing this, we give an algebraic characterization of the additive closure in terms of Ringel's tree modules, and show that its indecomposable objects are the reduced representations of Kinser.

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Computing Betti tables and minimal presentations of zero-dimensional persistent homology

2024-10-29

Dmitriy Morozov , Luis Scoccola

The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and … The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes change. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of $0$-dimensional homology (which is relevant for clustering and graph classification) the Betti tables of a bigraded module can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that a minimal presentation of $0$-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

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On the bottleneck stability of rank decompositions of multi-parameter persistence modules

2024-06-21

Magnus Bakke Botnan , Steffen Oppermann , Steve Oudot +1 more

A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations-often referred to as persistence modules.One such invariant is the … A significant part of modern topological data analysis is concerned with the design and study of algebraic invariants of poset representations-often referred to as persistence modules.One such invariant is the minimal rank decomposition, which encodes the ranks of all the structure morphisms of the persistence module by a single ordered pair of rectangle-decomposable modules, interpreted as a signed barcode.This signed barcode generalizes the concept of persistence barcode from one-parameter persistence to any number of parameters, raising the question of its bottleneck stability.We show in this paper that the minimal rank decomposition is not stable under the natural notion of signed bottleneck matching between signed barcodes.We remedy this by turning our focus to the rank exact decomposition, a related signed barcode induced by the minimal projective resolution of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings.As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules.We also give a bound for the size of the rank exact decomposition that is polynomial in the size of the usual minimal projective resolution, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure related to the upsets of the indexing poset.This set of results combines concepts from topological data analysis and from the representation theory of posets, and we believe is relevant to both areas.Contents 1. Introduction 1 2. Background and notation 6 3. Bottleneck instability of minimal rank decompositions 11 4. Bottleneck stability of hook-decomposable modules 12 5. Global dimension of the rank exact structure and size of the rank exact decomposition 16 6.Bottleneck stability of the rank exact decomposition and universality 25 7.Other exact structures 27 Appendix A. Proofs of results from previous work 29 References 32

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Differentiability and Optimization of Multiparameter Persistent Homology

2024-06-12

Luis Scoccola , Siddharth Setlur , David Loiseaux +2 more

Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in … Real-valued functions on geometric data -- such as node attributes on a graph -- can be optimized using descriptors from persistent homology, allowing the user to incorporate topological terms in the loss function. When optimizing a single real-valued function (the one-parameter setting), there is a canonical choice of descriptor for persistent homology: the barcode. The operation mapping a real-valued function to its barcode is differentiable almost everywhere, and the convergence of gradient descent for losses using barcodes is relatively well understood. When optimizing a vector-valued function (the multiparameter setting), there is no unique choice of descriptor for multiparameter persistent homology, and many distinct descriptors have been proposed. This calls for the development of a general framework for differentiability and optimization that applies to a wide range of multiparameter homological descriptors. In this article, we develop such a framework and show that it encompasses well-known descriptors of different flavors, such as signed barcodes and the multiparameter persistence landscape. We complement the theory with numerical experiments supporting the idea that optimizing multiparameter homological descriptors can lead to improved performances compared to optimizing one-parameter descriptors, even when using the simplest and most efficiently computable multiparameter descriptors.

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On the Stability of Multigraded Betti Numbers and Hilbert Functions

2024-01-23

Steve Oudot , Luis Scoccola

.Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory—it completely determines the Hilbert function of the module and the isomorphism … .Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory—it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution—as well as in practice—it is easy to visualize, and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the two-parameter case, a 1-Wasserstein stability result for Hilbert functions with respect to the 1-presentation distance of Bjerkevik and Lesnick.Keywordsmultigraded Betti numberstopological data analysisstability theoremsfree resolutionsMSC codes55N3162R40

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

2023-12-12

David Loiseaux , Luis Scoccola , Mathieu Carrière +2 more

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The Hurewicz theorem in homotopy type theory

2023-07-25

J. Daniel Christensen , Luis Scoccola

We prove the Hurewicz theorem in homotopy type theory, i.e., that for X a pointed, (n -1)-connected type (n ≥ 1) and A an abelian group, there is a natural … We prove the Hurewicz theorem in homotopy type theory, i.e., that for X a pointed, (n -1)-connected type (n ≥ 1) and A an abelian group, there is a natural isomorphism πn(X) ab ⊗ A ∼ = Hn(X; A) relating the abelianization of the homotopy groups with the homology.We also compute the connectivity of a smash product of types and express the lowest non-trivial homotopy group as a tensor product.Along the way, we study magmas, loop spaces, connected covers and prespectra, and we use 1-coherent categories to express naturality and for the Yoneda lemma.As homotopy type theory has models in all ∞-toposes, our results can be viewed as extending known results about spaces to all other ∞-toposes.

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Rectification of interleavings and a persistent Whitehead theorem

2023-05-09

Edoardo Lanari , Luis Scoccola

The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance … The homotopy interleaving distance, a distance between persistent spaces, was introduced by Blumberg and Lesnick and shown to be universal, in the sense that it is the largest homotopy-invariant distance for which sublevel-set filtrations of close-by real-valued functions are close-by. There are other ways of constructing homotopy-invariant distances, but not much is known about the relationships between these choices. We show that other natural distances differ from the homotopy interleaving distance in at most a multiplicative constant, and prove versions of the persistent Whitehead theorem, a conjecture of Blumberg and Lesnick that relates morphisms that induce interleavings in persistent homotopy groups to stronger homotopy-invariant notions of interleaving.

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Approximate and discrete Euclidean vector bundles

2023-01-01

Luis Scoccola , José A. Perea

Abstract We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle … Abstract We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon> 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

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Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

2023-01-01

David Loiseaux , Luis Scoccola , Mathieu Carrière +2 more

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on … Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

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FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles

2022-01-01

Luis Scoccola , José A. Perea

Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, … Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an application, we describe an algorithm which takes as input a dataset together with an initial representation of it in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations, obtained through local linear dimensionality reduction, along the initial global representation. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.

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On the bottleneck stability of rank decompositions of multi-parameter persistence modules

2022-01-01

Magnus Bakke Botnan , Steffen Oppermann , Steve Oudot +1 more

The notion of rank decomposition of a multi-parameter persistence module was introduced as a way of constructing complete and discrete representations of the rank invariant of the module. In particular, … The notion of rank decomposition of a multi-parameter persistence module was introduced as a way of constructing complete and discrete representations of the rank invariant of the module. In particular, the minimal rank decomposition by rectangles of a persistence module, also known as the generalized persistence diagram, gives a uniquely defined representation of the rank invariant of the module by a pair of rectangle-decomposable modules. This pair is interpreted as a signed barcode, with the rectangle summands of the first (resp. second) module playing the role of the positive (resp. negative) bars. The minimal rank decomposition by rectangles generalizes the concept of persistence barcode from one-parameter persistence, and, being a discrete invariant, it is amenable to manipulations on a computer. However, we show that it is not bottleneck stable under the natural notion of signed bottleneck matching between signed barcodes. To remedy this, we turn our focus to the signed barcode induced by the Betti numbers of the module relative to the so-called rank exact structure, which we prove to be bottleneck stable under signed matchings. As part of our proof, we obtain two intermediate results of independent interest: we compute the global dimension of the rank exact structure on the category of finitely presentable multi-parameter persistence modules, and we prove a bottleneck stability result for hook-decomposable modules, which are in fact the relative projective modules of the rank exact structure. We also bound the size of the multigraded Betti numbers relative to the rank exact structure in terms of the usual multigraded Betti numbers, we prove a universality result for the dissimilarity function induced by the notion of signed matching, and we compute, in the two-parameter case, the global dimension of a different exact structure that is related to the upsets of the indexing poset.

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Generic Multi-Parameter Persistence Modules are Nearly Indecomposable

2022-01-01

Ulrich Bauer , Luis Scoccola

We study the interplay between algebraic and metric properties of multi-parameter persistence modules (linear representations of the poset $\mathbf{R}^n$ with $n \geq 2$). In the context of topological data analysis … We study the interplay between algebraic and metric properties of multi-parameter persistence modules (linear representations of the poset $\mathbf{R}^n$ with $n \geq 2$). In the context of topological data analysis and inference, a fundamental property of one-parameter persistence modules (linear representations of $\mathbf{R}$) is that the operation of decomposing them into their indecomposable summands is Lipschitz continuous, under suitable choices of metrics on modules and on collections of indecomposables. This is far from true of multi-parameter persistence modules: We show that any finitely presentable multi-parameter persistence module can be approximated arbitrarily well by an indecomposable module, and that modules that are sufficiently close to an indecomposable decompose as a direct sum of an indecomposable and a nearly trivial module. These results have several consequences of interest, which provide strong motivation for approaching multi-parameter persistence in a way that does not rely on directly decomposing modules by indecomposables.

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Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

2022-01-01

Luis Scoccola , Hitesh Gakhar , Johnathan Bush +4 more

The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is … The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra--Lenstra--Lov\'asz algorithm from computational number theory.

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On the stability of multigraded Betti numbers and Hilbert functions

2021-01-01

Steve Oudot , Luis Scoccola

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and … Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution -- as well as in practice -- it is easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the $2$-parameter case, a $1$-Wasserstein stability result for Hilbert functions with respect to the $1$-presentation distance of Bjerkevik and Lesnick.

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Approximate and discrete Euclidean vector bundles

2021-01-01

Luis Scoccola , José A. Perea

We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, … We introduce $\varepsilon$-approximate versions of the notion of Euclidean vector bundle for $\varepsilon \geq 0$, which recover the classical notion of Euclidean vector bundle when $\varepsilon = 0$. In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon$-approximate vector bundles can be used to represent classical vector bundles when $\varepsilon > 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data, and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

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The Integers as a Higher Inductive Type

2020-07-01

Thorsten Altenkirch , Luis Scoccola

We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its … We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

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The Integers as a Higher Inductive Type

2020-05-26

Thorsten Altenkirch , Luis Scoccola

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Stable and consistent density-based clustering.

2020-05-18

Alexander Rolle , Luis Scoccola

We present a multiscale, consistent approach to density-based clustering that satisfies stability theorems -- in both the input data and in the parameters -- which hold without distributional assumptions. The … We present a multiscale, consistent approach to density-based clustering that satisfies stability theorems -- in both the input data and in the parameters -- which hold without distributional assumptions. The stability in the input data is with respect to the Gromov--Hausdorff--Prokhorov distance on metric probability spaces and interleaving distances between (multi-parameter) hierarchical clusterings we introduce. We prove stability results for standard simplification procedures for hierarchical clusterings, which can be combined with our approach to yield a stable flat clustering algorithm. We illustrate the stability of the approach with computational examples. Our framework is based on the concepts of persistence and interleaving distance from Topological Data Analysis.

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Nilpotent types and fracture squares in homotopy type theory

2020-05-01

Luis Scoccola

Abstract We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of … Abstract We develop the basic theory of nilpotent types and their localizations away from sets of numbers in Homotopy Type Theory. For this, general results about the classifying spaces of fibrations with fiber an Eilenberg–Mac Lane space are proven. We also construct fracture squares for localizations away from sets of numbers. All of our proofs are constructive.

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Localization in Homotopy Type Theory

2020-02-11

Egbert Rijke , J. Daniel Christensen , Luis Scoccola +1 more

We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural … We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L'$. Furthermore, we prove results establishing that $L'$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.

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Rectification of interleavings and a persistent Whitehead theorem

2020-01-01

Edoardo Lanari , Luis Scoccola

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The Integers as a Higher Inductive Type

2020-01-01

Thorsten Altenkirch , Luis Scoccola

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The Hurewicz theorem in Homotopy Type Theory

2020-01-01

J. Daniel Christensen , Luis Scoccola

We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected type $(n \geq 1)$ and $A$ an abelian group, there is a natural isomorphism … We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected type $(n \geq 1)$ and $A$ an abelian group, there is a natural isomorphism $π_n(X)^{ab} \otimes A \cong \tilde{H}_n(X; A)$ relating the abelianization of the homotopy groups with the homology. We also compute the connectivity of a smash product of types and express the lowest non-trivial homotopy group as a tensor product. Along the way, we study magmas, loop spaces, connected covers and prespectra, and we use $1$-coherent categories to express naturality and for the Yoneda lemma. As homotopy type theory has models in all $\infty$-toposes, our results can be viewed as extending known results about spaces to all other $\infty$-toposes.

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Stable and consistent density-based clustering via multiparameter persistence

2020-01-01

Alexander Rolle , Luis Scoccola

We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the correspondence-interleaving … We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the correspondence-interleaving distance, a metric for hierarchical clusterings that we introduce. Taking certain one-parameter slices of degree-Rips recovers well-known methods for density-based clustering, but we show that these methods are unstable. However, we prove that degree-Rips, as a multiparameter object, is stable, and we propose an alternative approach for taking slices of degree-Rips, which yields a one-parameter hierarchical clustering algorithm with better stability properties. We prove that this algorithm is consistent, using the correspondence-interleaving distance. We provide an algorithm for extracting a single clustering from one-parameter hierarchical clusterings, which is stable with respect to the correspondence-interleaving distance. And, we integrate these methods into a pipeline for density-based clustering, which we call Persistable. Adapting tools from multiparameter persistent homology, we propose visualization tools that guide the selection of all parameters of the pipeline. We demonstrate Persistable on benchmark datasets, showing that it identifies multi-scale cluster structure in data.

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Spaces of Clusterings.

2019-02-04

Alexander Rolle , Luis Scoccola

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Visualization tools for parameter selection in cluster analysis

2019-01-01

Alexander Rolle , Luis Scoccola

We propose an algorithm, HPREF (Hierarchical Partitioning by Repeated Features), that produces a hierarchical partition of a set of clusterings of a fixed dataset, such as sets of clusterings produced … We propose an algorithm, HPREF (Hierarchical Partitioning by Repeated Features), that produces a hierarchical partition of a set of clusterings of a fixed dataset, such as sets of clusterings produced by running a clustering algorithm with a range of parameters. This gives geometric structure to such sets of clustering, and can be used to visualize the set of results one obtains by running a clustering algorithm with a range of parameters.

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Localization in Homotopy Type Theory

2018-01-01

J. Daniel Christensen , Morgan Opie , Egbert Rijke +1 more

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Common Coauthors

Coauthor	Papers Together
Steve Oudot	7
Magnus Bakke Botnan	4
José A. Perea	4
Alexander Rolle	4
J. Daniel Christensen	4
Thorsten Altenkirch	3
David Loiseaux	3
Mathieu Carrière	3
Ulrich Bauer	2
Steffen Oppermann	2
Morgan Opie	2
Edoardo Lanari	2
Egbert Rijke	2
Thomas Brüstle	1
Tatum Rask	1
Nikolas Schonsheck	1
Riju Bindua	1
Siddharth Setlur	1
Ling Zhou	1
Dmitriy Morozov	1
Hitesh Gakhar	1
Johnathan Bush	1

Commonly Cited References

Interactive Visualization of 2-D Persistence Modules

2015-01-01

Michael Lesnick , Matthew Wright

The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this … The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes of 1-D affine slices of a 2-D persistence module $M$. It also computes and visualizes the dimension of each vector space in $M$ and the bigraded Betti numbers of $M$. At the heart of our computational approach is a novel data structure based on planar line arrangements, on which we can perform fast queries to find the barcode of any slice of $M$. We present an efficient algorithm for constructing this data structure and establish bounds on its complexity.

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On approximation of 2D persistence modules by interval-decomposables

2023-08-24

Hideto Asashiba , Emerson G. Escolar , Ken Nakashima +1 more

In this work, we propose a new invariant for 2D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank … In this work, we propose a new invariant for 2D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a 2D persistence module M, we propose an "interval-decomposable replacement" δ⁎(M) (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that M is interval-decomposable if and only if δ⁎(M) is equal to M in the split Grothendieck group. Furthermore, even for modules M not necessarily interval-decomposable, δ⁎(M) preserves the dimension vector and the rank invariant of M. In addition, we provide an algorithm to compute δ⁎(M) (a high-level algorithm in the general case, and a detailed algorithm for the size 2×n case).

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The Theory of Multidimensional Persistence

2009-04-23

Gunnar Carlsson , Afra Zomorodian

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An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

2021-09-29

Frédéric Chazal , Bertrand Michel

With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for … With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis ( tda ) is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data. It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques. This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts.

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On the Stability of Interval Decomposable Persistence Modules

2021-05-11

Håvard Bakke Bjerkevik

Abstract The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to … Abstract The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n -dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

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Accelerated Hierarchical Density Based Clustering

2017-11-01

Leland McInnes , John Healy

We present an accelerated algorithm for hierarchical density based clustering. Our new algorithm improves upon HDBSCAN*, which itself provided a significant qualitative improvement over the popular DBSCAN algorithm. The accelerated … We present an accelerated algorithm for hierarchical density based clustering. Our new algorithm improves upon HDBSCAN*, which itself provided a significant qualitative improvement over the popular DBSCAN algorithm. The accelerated HDBSCAN* algorithm provides comparable performance to DBSCAN, while supporting variable density clusters, and eliminating the need for the difficult to tune distance scale parameter. This makes accelerated HDBSCAN* the default choice for density based clustering. Library available at: https://github.com/scikit-learn-contrib/hdbscan

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Stratifying Multiparameter Persistent Homology

2019-01-01

Heather A. Harrington , Nina Otter , Hal Schenck +1 more

A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal … A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications, data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian, we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes, and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

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Multidimensional Interleavings and Applications to Topological Inference

2012-01-01

Michael Lesnick

This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations … This work concerns the theoretical foundations of persistence-based topological data analysis. We develop theory of topological inference in the multidimensional persistence setting, and directly at the (topological) level of filtrations rather than only at the (algebraic) level of persistent homology modules. Our main mathematical objects of study are interleavings. These are tools for quantifying the similarity between two multidimensional filtrations or persistence modules. They were introduced for 1-D filtrations and persistence modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce generalizations of the definitions of interleavings given by Chazal et al. and use these to define pseudometrics, called interleaving distances, on multidimensional filtrations and multidimensional persistence modules. We present an in-depth study of interleavings and interleaving distances. We then use them to formulate and prove several multidimensional analogues of a topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These results hold directly at the level of filtrations; they yield as corollaries corresponding results at the module level.

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Homological approximations in persistence theory

2022-12-12

Benjamin Blanchette , Thomas Brüstle , Eric J. Hanson

Abstract We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data … Abstract We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by “spread modules,” which are sometimes called “interval modules” in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the “single-source” spread modules gives rise to a new invariant which is finer than the rank invariant.

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Decomposition of persistence modules

2019-10-30

Magnus Bakke Botnan , William Crawley-Boevey

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result … We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of fundamental structure theorems for persistence modules.

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Eilenberg-MacLane spaces in homotopy type theory

2014-07-14

Daniel R. Licata , Eric Finster

Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs … Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-MacLane spaces. For an abelian group G, an Eilenberg-MacLane space K(G,n) is a space (type) whose nth homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.

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The Structure and Stability of Persistence Modules

2016-01-01

Frédéric Chazal , Vin de Silva , Marc Glisse +1 more

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The fiber of the persistence map for functions on the interval

2018-12-01

Justin Curry

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Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

2022-05-26

Michael Lesnick , Matthew Wright

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes … Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\operatorname{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gröbner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of 2-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

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The Theory of the Interleaving Distance on Multidimensional Persistence Modules

2015-03-23

Michael Lesnick

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Algebraic Homotopy Interleaving Distance

2021-01-01

Nicolas Berkouk

Homological algebra of modules over posets

2020-01-01

Ezra Miller

Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and … Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue is how to define finiteness to replace the noetherian hypothesis which fails. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both theoretical and computational purposes: it guarantees finite presentations and resolutions of various sorts, all related by a syzygy theorem, amenable to algorithmic manipulation. Tameness and its homological theory are new even in the finitely generated discrete setting of $\mathbb{N}^n$-gradings, where tame is materially weaker than noetherian. In the context of persistent homology of filtered topological spaces, especially with multiple real parameters, the algebraic theory of tameness yields topologically interpretable data structures in terms of birth and death of homology classes.

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The Rank Invariant Stability via Interleavings

2018-01-01

Claudia Landi

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Approximation by interval-decomposables and interval resolutions of persistence modules

2023-04-06

Hideto Asashiba , Emerson G. Escolar , Ken Nakashima +1 more

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The homotopy theory of type theories

2018-08-30

Krzysztof Kapulkin , Peter LeFanu Lumsdaine

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Computing Persistent Homology

2004-11-19

Afra Zomorodian , Gunnar Carlsson

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On the Stability of Multigraded Betti Numbers and Hilbert Functions

2024-01-23

Steve Oudot , Luis Scoccola

Corrections and Supplementaries to My Paper concerning Krull-Remak-Schmidt’s Theorem

1950-06-01

Gorô Azumaya

It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in … It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.

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Edit Distance and Persistence Diagrams over Lattices

2022-04-11

Alexander McCleary , Amit Patel

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 … 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 M\"obius 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.

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Semantics of higher inductive types

2019-06-17

Peter LeFanu Lumsdaine , Michael Shulman

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development … Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy theory within type theory, as well as in formalizing ordinary set-level mathematics in type theory. In this article, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has *weakly stable typal initial algebras* for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specializes to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction, and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed $(\infty,1)$-category is presented by some model category to which our results apply.

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Basic Bundle Theory and K-Cohomology Invariants

2007-12-08

Dale Husemöller , M. Joachim , Branislav Jurčo +1 more

Representations of Compact Lie Groups

1985-01-01

Theodor Bröcker , Tammo tom Dieck

Curvatures of left invariant metrics on lie groups

1976-09-01

John Milnor

Detecting strange attractors in turbulence

1981-01-01

Floris Takens

Exposition and Interpretation of the Topology of Neural Networks

2019-12-01

Rickard Brüel Gabrielsson , Gunnar Carlsson

Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize … Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize to unseen data. We use topological data analysis to show that the information encoded in the weights of a CNN can be organized in terms of a topological data model and demonstrate how such information can be interpreted and utilized. We show that the weights of convolutional layers at depths from 1 through 13 learn simple global structures. We also demonstrate the change of the simple structures over the course of training. In particular, we define and analyze the spaces of spatial filters of convolutional layers and show the recurrence, among all networks, depths, and during training, of a simple circle consisting of rotating edges, as well as a less recurring unanticipated complex circle that combines lines, edges, and non-linear patterns. We also demonstrate that topological structure correlates with a network's ability to generalize to unseen data and that topological information can be used to improve a network's performance. We train over a thousand CNN's on MNIST, CIFAR-10, SVHN, and ImageNet.

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Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices

2010-12-02

Sylvia Serfaty , Luigi Ambrosio , Edoardo Mainini

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Riemannian Geometry and Geometric Analysis

1998-01-01

Jürgen Jost

Cohomology of Cryo-Electron Microscopy

2017-01-01

Ke Ye , Lek‐Heng Lim

The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images. In this paper, we show that the … The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images. In this paper, we show that the basic premise of cryo-EM---patching together 2-dimensional projections to reconstruct a 3-dimensional object---is naturally one of Čech cohomology with SO(2)-coefficients. We deduce that every cryo-EM reconstruction problem corresponds to an oriented circle bundle on a simplicial complex, allowing us to classify cryo-EM problems via principal bundles. In practice, the 2-dimensional images are noisy and a main task in cryo-EM is to denoise them. We will see how the aforementioned insights can be used towards this end.

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Noncommutative Noetherian Rings

2001-02-27

John McConnell , J. C. Robson

Articles on the history of mathematics can be written from many dierent perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from … Articles on the history of mathematics can be written from many dierent perspectives. Some aim to survey a more or less wide landscape, and require the observer to watch from afar as theories develop and movements are born or become obsolete. At the other extreme, there are those that try to shed light on the history of particular theorems and on the people who created them. This article belongs to this second category. It is an attempt to explain Goldie’s theorems on quotient rings in the context of the life and times of the man who discovered them. 1. Fractions Fractions are at least as old as civilisation. The Egyptian scribes of 3,000 years ago were very skilful in their manipulation as attested by many ancient papyri. To the Egyptians and Mesopotamians, fractions were just tools to find the correct answer to practical problems in land surveying and accounting. However, the situation changed dramatically in Ancient Greece. To the Greek philosophers, number meant positive integer, and 1 was ‘the unity’, and as such, had to be indivisible. So how could ‘half’ be a number, since ‘half the unity’ did not make sense? Possibly as a consequence of that, the Greek mathematicians thought of fractions in terms of ratios of integers, rather than numbers. After the demise of Greek civilisation, mathematicians reverted to the more prosaic view that fractions were numbers. Indeed, for the next thousand years everyone seemed happy to compute with all sorts of ‘numbers’ without worrying much about what a number was really supposed to be. It was the need for a sound foundation for the infinitesimal calculus that put mathematicians face to face with the nature of numbers. The movement began in the 18th century, but its first fruits were only reaped in the 19th century in the movement that became known as the arithmetization of analysis. In short, mathematicians felt quite sure that they knew their integers very well; so they thought that by constructing the real numbers in terms of positive integers they would place the latter in a sure foundation. The most extreme version of this credo

PARTIALLY ORDERED SETS OF INFINITE TYPE

1975-10-31

Л. А. Назарова

In this paper a necessary and sufficient condition is given that a partially ordered set have tame type, i.e. that it admit classification of its indecomposable representations. Bibliography: 16 titles. In this paper a necessary and sufficient condition is given that a partially ordered set have tame type, i.e. that it admit classification of its indecomposable representations. Bibliography: 16 titles.

Twisty Takens: a geometric characterization of good observations on dense trajectories

2019-08-16

Boyan Xu , Christopher J. Tralie , Alice Antia +2 more

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Persistent homotopy theory

2020-01-01

John F. Jardine

Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are … Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are viewed as systems (or functors), which are defined on a parameter space. The category of systems of spaces admits a partial homotopy theory that is based on controlled equivalences, suitably defined, that are the output of homotopy stability results.

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The Geometry of Synchronization Problems and Learning Group Actions

2019-05-28

Tingran Gao , Jacek Brodzki , Sayan Mukherjee

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Nilpotent types and fracture squares in homotopy type theory

2020-05-01

Luis Scoccola

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Sheaves are the canonical data structure for sensor integration

2016-12-05

Michael Robinson

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Assignments to sheaves of pseudometric spaces

2020-06-02

Michael Robinson

An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related … An assignment to a sheaf is the choice of a local section from each open set in the sheaf's base space, without regard to how these local sections are related to one another. This article explains that the consistency radius -- which quantifies the agreement between overlapping local sections in the assignment -- is a continuous map. When thresholded, the consistency radius produces the consistency filtration, which is a filtration of open covers. This article shows that the consistency filtration is a functor that transforms the structure of the sheaf and assignment into a nested set of covers in a structure-preserving way. Furthermore, this article shows that consistency filtration is robust to perturbations, establishing its validity for arbitrarily thresholded, noisy data.

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Estimation of intrinsic dimensionality of samples from noisy low-dimensional manifolds in high dimensions with multiscale SVD

2009-08-01

Anna Little , Jason D. Lee , Yoon-Mo Jung +1 more

The problem of estimating the intrinsic dimensionality of certain point clouds is of interest in many applications in statistics and analysis of high-dimensional data sets. Our setting is the following: … The problem of estimating the intrinsic dimensionality of certain point clouds is of interest in many applications in statistics and analysis of high-dimensional data sets. Our setting is the following: the points are sampled from a manifold M of dimension k, embedded in Ropf <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sup> , with k Lt D, and corrupted by D-dimensional noise. When M is a linear manifold (hyperplane), one may analyse this situation by SVD, hoping the noise would perturb the rank k covariance matrix. When M is a nonlinear manifold, SVD performed globally may dramatically overestimate the intrinsic dimensionality. We discuss a multiscale version SVD that is useful in estimating the intrinsic dimensionality of nonlinear manifolds.

Exact computation of the matching distance on 2-parameter persistence modules

2018-01-01

Michael Kerber , Michael Lesnick , Steve Oudot

The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known … The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.

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Homotopy Type Theory: Univalent Foundations of Mathematics

2013-01-01

Peter Aczel , Benedikt Ahrens , Thorsten Altenkirch +7 more

Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of … Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.

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Differential Forms in Algebraic Topology

1982-01-01

Raoul Bott , Loring W. Tu

Gauge Symmetries and Fibre Bundles

1983-01-01

A. P. Balachandran , G. Marmo , B.-S. Skagerstam +1 more

Barcodes: The persistent topology of data

2007-10-26

Robert Ghrist

This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool … This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—

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The reach, metric distortion, geodesic convexity and the variation of tangent spaces

2019-06-01

Jean‐Daniel Boissonnat , André Lieutier , Mathijs Wintraecken

In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound … In this paper we discuss three results. The first two concern general sets of positive reach: we first characterize the reach of a closed set by means of a bound on the metric distortion between the distance measured in the ambient Euclidean space and the shortest path distance measured in the set. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the reach and the distance between the two points.

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Local formulae for Stiefel-Whitney classes

1996-12-01

D. A. McLaughlin

Vector Bundles and Gromov–Hausdorff Distance

2009-08-25

Marc A. Rieffel

Abstract We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will … Abstract We show how to make precise the vague idea that for compact metric spaces that are close together for Gromov–Hausdorff distance, suitable vector bundles on one metric space will have counterpart vector bundles on the other. Our approach employs the Lipschitz constants of projection-valued functions that determine vector bundles. We develop some computational techniques, and we illustrate our ideas with simple specific examples involving vector bundles on the circle, the two-torus, the two-sphere, and finite metric spaces. Our topic is motivated by statements concerning “monopole bundles” over matrix algebras in the literature of theoretical high-energy physics.

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