arXiv subjects

Francesca Tombari

Publications and source records attributed to Francesca Tombari.

12 recordsLinked to original sources

arXiv2026

Persistent Homology and Equivariance in Data Analysis: A Topological Introduction

This new book is intended as a first elementary introduction to Topological Data Analysis for mathematics students seeking a rigorous account of the foundations of persistent homology, as well as for computer scientists interested in its theoretical underpinnings. The exposition is as self-contained as possible: all the required background is recalled when needed, and only a few standard results are cited without proof. One section of the book, devoted to monodromy in biparameter persistence (Section 4.4), requires more advanced knowledge of algebraic topology. Persistent homology can be introduced from different perspectives, reflecting the variety of mathematical languages that have shaped its development over the years. Some approaches emphasize the algebraic foundations of the theory, while others highlight its topological essence. In this book, we adopt the latter viewpoint - the one that historically marked the birth of the subject - because we believe it offers both conceptual clarity and pedagogical effectiveness, making it particularly suitable for undergraduate and early graduate students. This book differs from existing introductory texts in several respects. First, it adopts a functional viewpoint: rather than representing data as finite (pseudo-)metric spaces, it treats them as functions encoding the information to be analyzed. This interpretative framework allows data to be viewed as measurable objects and highlights the role of observers and their equivariances in the analysis process. Second, this perspective provides a natural bridge between Topological Data Analysis and machine learning through the theory of Group Equivariant Non-Expansive Operators (GENEOs), which offers a mathematically grounded framework for incorporating symmetries and invariances into learning systems.

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arXiv2026

Stratified Morse Theory for Cell Complexes

We develop a version of discrete Morse theory for finite regular CW complexes equipped with an auxiliary stratification. The key construction is the halo of a cell, which contains all those faces in the boundary that enter closed sublevelsets precisely when the threshold reaches that cell's value. The complement of this halo in the boundary, called the shadow, is always a subcomplex. A stratified discrete Morse function requires Forman's conditions on each stratum together with the requirement that closures of paired cells admit filtered collapses onto their shadows. We establish fundamental Morse lemmas: filtered collapses across regular intervals, and controlled attachments at critical values. For functions satisfying only the stratum-wise Forman condition, we construct an upper envelope on the barycentric subdivision whose local Morse data decomposes into horizontal and vertical components. This yields a simplicial analogue of the standard tangential-normal splitting of local Morse data in the sense of Goresky and MacPherson.

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arXiv2025

The Convex Matching Distance in Multiparameter Persistence

We introduce the convex matching distance, a novel metric for comparing functions with values in the real plane. This metric measures the maximal bottleneck distance between the persistence diagrams associated with the convex combinations of the two function components. Similarly to the traditional matching distance, the convex matching distance aggregates the information provided by two real-valued components. However, whereas the matching distance depends on two parameters, the convex matching distance depends on only one, offering improved computational efficiency. We further show that the convex matching distance can be more discriminative than the traditional matching distance in certain cases, although the two metrics are generally not comparable. Moreover, we prove that the convex matching distance is stable and characterize the coefficients of the convex combination at which it is attained. Finally, we demonstrate that this new aggregation framework benefits from the computational advantages provided by the Pareto grid, a collection of curves in the plane whose points lie in the image of the Pareto critical set associated with functions assuming values on the real plane. Experimental validation on MNIST digits, synthetic shapes, and chaotic attractors suggests that the convex matching distance provides a reliable and efficient alternative to the matching distance, at a significantly lower computational cost.

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arXiv2024

Gromov-Hausdorff Distance for Directed Spaces

The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The directed Gromov-Hausdorff distance measures the distance between two extended metric spaces, where the new metric, defined on the same underlying space, is induced by the length of zigzag paths. This distance is then computed by isometrically embedding the directed metric spaces into an ambient directed space equipped with the zigzag distance. Analogously to the classical Gromov-Hausdorff distance, we also propose alternative formulations based on the distortion of d-maps and d-correspondences. However, unlike the classical case, these directed distances are not equivalent.

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arXiv2024

Koszul complexes and relative homological algebra of functors over posets

Under certain conditions, Koszul complexes can be used to calculate relative Betti diagrams of vector space-valued functors indexed by a poset, without the explicit computation of global minimal relative resolutions. In relative homological algebra of such functors, free functors are replaced by an arbitrary family of functors. Relative Betti diagrams encode the multiplicities of these functors in minimal relative resolutions. In this article we provide conditions under which grading the chosen family of functors leads to explicit Koszul complexes whose homology dimensions are the relative Betti diagrams, thus giving a scheme for the computation of these numerical descriptors.

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arXiv2024

Abelian and model structures on tame functors

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a structure theorem for cofibrant objects in the category of tame functors indexed by realizations of posets of dimension $1$ with values in the category of chain complexes in an abelian category whose all objects are projectives. Moreover, we introduce a general technique to generate indecomposable objects in the abelian category of functors indexed by finite posets.

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arXiv2023

Matching distance via the extended Pareto grid

One of the most animated themes of multidimensional persistence is the comparison between invariants. The matching distance between persistent Betti numbers functions (or rank invariants), is among the most studied metrics in this context, particularly in 2-parameter persistence. The main reason for this interest is that, in the 2-parameter case, the foliation method allows for an effective computation of the matching distance, based on filtering the space along lines of positive slope. Our work provides a qualitative analysis, based on a construction called extended Pareto grid, of the filtering lines that actually contribute to the computation of the matching distance. Under certain genericity assumptions, we show that these lines must either be horizontal, vertical, of slope 1 or belong to a finite collection of special lines associated with discontinuity phenomena.

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arXiv2023

Geometry of the matching distance for 2D filtering functions

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for $\mathbb{R}^2$-valued regular functions defined on a Riemannian closed manifold. In particular, we prove that in this case the matching distance is realised either at special values or at values corresponding to vertical, horizontal or slope 1 lines.

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arXiv2023

A topological model for partial equivariance in deep learning and data analysis

In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. If the set of transformations acting is a group, then we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces. We define pseudo-metrics on them and show some properties of the resulting spaces. In particular, we show how such spaces have convenient approximation and convexity properties.

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arXiv2021

Realisations of posets and tameness

We introduce a construction called realisation which transforms posets into posets. We show that realisations share several key features with upper semilattices. For example, we define local dimensions of points in a poset and show that these numbers for realisations behave in a similar way as they do for upper semilattices. Furthermore, similarly to upper semilattices, realisations have well-behaved discrete approximations which are suitable for capturing homological properties of functors indexed by them. These discretisations are convenient and effective for describing tameness of functors. Homotopical and homological properties of tame functors, particularly those indexed by realisations, are discussed, with emphasis on the use of Koszul complexes to compute Betti diagrams of minimal free resolutions of tame functors indexed by upper semilattices and realisations.

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arXiv2020

Landscapes of data sets and functoriality of persistent homology

The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a finite domain. Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms. Different choices of the set of endomorphisms encode different symmetries of the data set. We describe various category structures on such enriched data sets and prove some of their properties such as decompositions and morphism formations. We also describe a data structure, based on coloured directed graphs, which is convenient to encode the mentioned enrichment. We show that persistent homology preserves only some aspects of these landscapes of enriched data sets however not all. In other words persistent homology is not a functor on the entire category of enriched data sets. Nevertheless we show that persistent homology is functorial locally. We use the concept of equivariant operators to capture some of the information missed by persistent homology.

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arXiv2020

Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.

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