arXiv subjects

Luis Scoccola

Publications and source records attributed to Luis Scoccola.

At least 19 recordsLinked to original sources

arXiv2026

An Algebraic Introduction to Persistence

We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and finite dimensional algebras, Morse theory and other areas of geometry, as well as topological inference and topological data analysis -- often via persistent homology. In some of these contexts, the category of poset representations of interest admits a metric structure given by the so-called interleaving distance. Persistence studies the algebraic properties of these poset representations and their behavior under perturbations in the interleaving distance. We survey fundamental results in the area, applications to pure and applied mathematics, advanced topics such as multiparameter persistence, as well as theoretical challenges and open questions.

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arXiv2026

A metrically complete and Krull--Schmidt space of multiparameter persistence modules

We show that the observable category of q-tame multiparameter persistence modules satisfies good metric and algebraic properties: it forms a complete metric space with respect to the interleaving distance, and it is Krull--Schmidt in the sense that every object admits an essentially unique decomposition into indecomposables. Moreover, we show that these metric and algebraic structures are compatible: two objects are at distance zero if and only if they are isomorphic. We argue that the observable category of q-tame multiparameter persistence modules is the right setup for multiparameter persistence by showing that many of the categories already considered in the literature form full subcategory of this category. We also characterize precompact sets in terms of finite representation type of certain discretizations, and show that the image of several of the main constructions in multiparameter persistence is precompact.

math.RT↗

arXiv2025

Stabilization of the Spread-Global Dimension

Motivated by constructions from applied topology, there has been recent interest in the homological algebra of linear representations of posets, particularly in the context of homological algebra relative to non-standard exact structures. A prominent example is the spread exact structure on the category of representations of a fixed poset, in which the indecomposable projectives are the spread representations (that is, the indicator representations of convex and connected subsets). The spread-global dimension is known to be finite for finite posets and not uniformly bounded on the collection of all Cartesian products between two arbitrary finite total orders. It was conjectured in [AENY23] that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite total order and an arbitrary finite total order. We provide a positive answer to this conjecture and, more generally, prove that the spread-global dimension is uniformly bounded on the collection of all Cartesian products between a fixed finite poset and an arbitrary finite total order. In doing so, we also establish the existence of finite spread-resolutions for finitely presented representations of arbitrary grid posets.

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arXiv2025

Counts and end-curves in two-parameter persistence

Given a finite dimensional, bigraded module over the polynomial ring in two variables, we define its two-parameter count, a natural number, and its end-curves, a set of plane curves. These are two-dimensional analogues of the notions of bar-count and endpoints of singly-graded modules over the polynomial ring in one variable, from persistence theory. We show that our count is the unique one satisfying certain natural conditions; as a consequence, several inclusion-exclusion formulas in two-parameter persistence yield the same positive number, which equals our count, and which in turn equals the number of end-curves, giving geometric meaning to this count. We show that the end-curves determine the classical Betti tables by showing that they interpolate between generators, relations, and syzygies. Using the band representations of a certain string algebra, we show that the set of end-curves admits a canonical partition, where each part forms a closed curve on the plane; we call this the boundary of the module. As an invariant, the boundary is neither weaker nor stronger than the rank invariant, but, in contrast to the rank invariant, it is a complete invariant on the set of spread-decomposable representations. Our results connect several lines of work in multiparameter persistence, and their extension to modules over the real-exponent polynomial ring in two variables relates to two-dimensional Morse theory.

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arXiv2025

Cover Learning for Large-Scale Topology Representation

Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation or infer local geometry by constructing a graph on the input data. More recently, topological data analysis popularized the use of simplicial complexes to represent data topology with two main methodologies: topological inference with geometric complexes and large-scale topology visualization with Mapper graphs -- central to these is the nerve construction from topology, which builds a simplicial complex given a cover of a space by subsets. While successful, these have limitations: geometric complexes scale poorly with data size, and Mapper graphs can be hard to tune and only contain low dimensional information. In this paper, we propose to study the problem of learning covers in its own right, and from the perspective of optimization. We describe a method for learning topologically-faithful covers of geometric datasets, and show that the simplicial complexes thus obtained can outperform standard topological inference approaches in terms of size, and Mapper-type algorithms in terms of representation of large-scale topology.

cs.LG↗

arXiv2024

Decomposing zero-dimensional persistent homology over rooted tree quivers

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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arXiv2024

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

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 changes. 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 zero-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of zero-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

cs.CG↗

arXiv2024

Differentiability and Optimization of Multiparameter Persistent Homology

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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arXiv2023

Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures

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.

cs.LG↗

arXiv2022

Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction

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.

cs.CG↗

arXiv2022

Multi-Parameter Persistence Modules are Generically Indecomposable

Algebraic persistence studies persistence modules (typically, linear representations of the poset $\mathbf{R}^n$ 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 $\mathbf{R}$) 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 $\mathbf{R}^n$ 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.

math.RT↗

arXiv2022

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

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 multi-parameter 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.

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arXiv2022

FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with 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.

cs.CG↗

arXiv2021

On the stability of multigraded Betti numbers and Hilbert functions

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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arXiv2021

Approximate and discrete Euclidean vector bundles

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 \v{C}ech 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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arXiv2020

Rectification of interleavings and a persistent Whitehead theorem

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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arXiv2020

The Hurewicz theorem in Homotopy Type Theory

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 $\pi_n(X)^{ab} \otimes A \cong \tilde{H}_n(X; A)$ 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.

math.AT↗

arXiv2020

The Integers as a Higher Inductive Type

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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