arXiv subjects

Barbara Giunti

Publications and source records attributed to Barbara Giunti.

17 recordsLinked to original sources

arXiv2026

Comparing Chatbot Performance Enhanced with Persistent Homology

Chatbots have become increasingly prevalent across various domains, offering automated assistance in many areas, especially mental health support. The training is done using extremely large datasets, which are sometimes not available in very specific domains. Moreover, it would sometimes be ideal to train the chatbot with personal information about the patients, which, of course, cannot be done on shared servers since it would violate patient confidentiality. Hence, being able to improve the performance of a chatbot, possibly trained locally and on a restricted dataset, without having to increase the dataset itself, would be extremely beneficial. In this work, we will enhance the input datasets using persistent homology (PH) vectorizations computed from the raw datasets themselves. Then we will compare, across several metrics, the performance of multiple chatbot models with or without the PH enhancement. Our experiments suggest that, while at times the PH enhancement is not particularly beneficial, it sometimes brings remarkable advantages for virtually no cost.

cs.LG↗

arXiv2026

Canopies: A Generalization of Vines and Vineyards for Parameterized Persistence

In this paper, we provide a new construction for studying parameterized persistence, called a canopy. We give two versions of this construction: the A-canopy, retaining all information about points on the diagonal of the persistence diagram; and the D-canopy, encoding the information of the "standard" persistence diagram. We do this by making a simple but major modification in the persistence bundle representation information: namely, rather than tracking a point in the persistence diagram, we instead track some choice of pairs of simplices that created said point. This viewpoint is a combinatorial version of tracking the chain complex information rather than just the output of persistence. We show how to construct the canopies from any filtered filtration function, proving, using the algebraic structure of filtered chain complexes, that different choices of pairs result in homeomorphic structures. Finally, we showcase the power of our approach by using canopies to define vines even in the presence of points with multiplicity; to discuss monodromy; and to obtain some immediate results linking non-trivial monodromy in the persistent homology transform with the existence of non-Hausdorff points in the canopy.

cs.CG↗

arXiv2026

Minimal intersection radius for $n$ growing, non-homogeneous ellipsoids in $\mathbb{R}^d$

In this paper, we compute the Minimal Intersection Radius (MIR) of growing, non-homogeneous ellipsoids in arbitrary ambient dimension. We provide a geometric method to find the MIR using techniques from convex optimization, a secondary method using second-order cone programs, and show that the MIR can be phrased as an LP-type problem, where the computation from convex optimization acts as a certificate. We implement these methods and benchmark them using different convex solvers, and with or without the LP setting. We also provide a comparison with similar but different problems that appeared in the literature, and show that finding the MIR is, in general, not equivalent to finding the minimal enclosing ellipsoid.

math.OC↗

arXiv2026

SuPerPoV: Score and evolution of the stratospheric polar vortex via persistent homology

Classifying the stratospheric polar vortex provides predictability for surface weather on extended-range timescales. However, providing a scientifically sound classification is challenging: all the definitions proposed in over 60 years of study depend on empirically chosen parameters and yield different results when one of them changes. Moreover, as they are based on static thresholds, it is not straightforward to use them to study the spatiotemporal evolution of the vortex. Here, we introduce SuPerPoV, a score system that computes displacement and split ratios of the polar vortex using tools from topological data analysis, thus providing a sound classification of the polar vortex. The scores are computed by adapting superlevel set persistence and comparing prominent features. Our definition is entirely threshold-free and implemented open source. The scores generally recovers previous definitions and are output for a user-defined number of days, thus showing the evolution of the event. SuPerPoV offers a paradigm shift in the study of the polar vortex, hopefully bringing a deeper understanding of the polar vortex and related extreme events, such as sudden stratospheric warmings.

math.AT↗

arXiv2025

The persistence of painting styles

Art is a deeply personal and expressive medium, where each artist brings their own style, technique, and cultural background into their work. Traditionally, identifying artistic styles has been the job of art historians or critics, relying on visual intuition and experience. However, with the advancement of mathematical tools, we can explore art through more structured lens. In this work, we show how persistent homology (PH), a method from topological data analysis, provides objective and interpretable insights on artistic styles. We show how PH can, with statistical certainty, differentiate between artists, both from different artistic currents and from the same one, and distinguish images of an artist from an AI-generated image in the artist's style.

cs.CV↗

arXiv2024

Decomposing the Persistent Homology Transform of Star-Shaped Objects

In this paper, we study the geometric decomposition of the degree-$0$ Persistent Homology Transform (PHT) as viewed as a persistence diagram bundle. We focus on star-shaped objects as they can be segmented into smaller, simpler regions known as ``sectors". Algebraically, we demonstrate that the degree-$0$ persistence diagram of a star-shaped object in $\R^2$ can be derived from the degree-$0$ persistence diagrams of its sectors. Using this, we then establish sufficient conditions for star-shaped objects in $\R^2$ so that they have ``trivial geometric monodromy''. Consequently, the PHT of such a shape can be decomposed as a union of curves parameterized by $\bS^1$, where the curves are given by the continuous movement of each point in the persistence diagrams that are parameterized by $\bS^{1}$. Finally, we discuss the current challenges of generalizing these results to higher dimensions.

math.AT↗

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.

math.AT↗

arXiv2023

Pruning vineyards: updating barcodes and representative cycles by removing simplices

The barcode of a filtration and its representative cycles encode rich information often useful in data analysis. However, obtaining them can be computationally expensive. Therefore, it is useful to have methods that update them if the associated filtration undergoes small changes. There are already efficient algorithms updating a barcode if simplices exchange entrance order or are added, but not if simplices are removed. We provide an implementation to update a reduced boundary matrix when simplices in the filtration are removed. Our algorithm, the Simplicial Removal Update Procedure (SiRUP), intrinsically updates also the representative cycles, and is compatible with the clearing optimizations. We show that the complexity of our algorithm is lower than recomputing the barcode from scratch and that the number of executed matrix column additions is minimal, with both theoretical and experimental methods.

math.AT↗

arXiv2023

Topological Data Analysis in smart manufacturing: State of the art and futuredirections

Topological Data Analysis (TDA) is a discipline that applies algebraic topology techniques to analyze complex, multi-dimensional data. Although it is a relatively new field, TDA has been widely and successfully applied across various domains, such as medicine, materials science, and biology. This survey provides an overview of the state of the art of TDA within a dynamic and promising application area: industrial manufacturing and production, particularly within the Industry 4.0 context. We have conducted a rigorous and reproducible literature search focusing on TDA applications in industrial production and manufacturing settings. The identified works are categorized based on their application areas within the manufacturing process and the types of input data. We highlight the principal advantages of TDA tools in this context, address the challenges encountered and the future potential of the field. Furthermore, we identify TDA methods that are currently underexploited in specific industrial areas and discuss how their application could be beneficial, with the aim of stimulating further research in this field. This work seeks to bridge the theoretical advancements in TDA with the practical needs of industrial production. Our goal is to serve as a guide for practitioners and researchers applying TDA in industrial production and manufacturing systems. We advocate for the untapped potential of TDA in this domain and encourage continued exploration and research.

cs.LG↗

arXiv2023

DONUT -- Creation, Development, and Opportunities of a Database

DONUT is a database of papers about practical, real-world uses of Topological Data Analysis (TDA). Its original seed was planted in a group chat formed during the HIM Spring School on Applied and Computational Algebraic Topology in April 2017. This document describes the creation, curation, and maintenance process of the database.

cs.DL↗

arXiv2022

Keeping it sparse: Computing Persistent Homology revisited

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

cs.CG↗

arXiv2022

Decomposing filtered chain complexes: geometry behind barcoding algorithms

In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called interval spheres. In this paper, we provide an algorithm to decompose filtered chain complexes into such interval spheres. This algorithm provides geometric insights into various aspects of the standard persistence algorithm and two of its run-time optimizations. Moreover, since it works for any filtered chain complexes, our algorithm can be applied in more general cases. As an application, we show how to decompose filtered kernels with it.

math.AT↗

arXiv2021

Expected Complexity of Persistence Barcode Computation via Matrix Reduction

We study the algorithmic complexity of computing the persistence barcode of a randomly generated filtration. We provide a general technique to bound the expected complexity of reducing the boundary matrix in terms of the density of its reduced form. We apply this technique finding upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on \v{C}ech, Vietoris--Rips and Erd\H{o}s--R\'enyi filtrations after matrix reduction, thus obtaining bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. Our fill-in bounds for \v{C}ech and Vietoris--Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erd\H{o}s--R\'enyi filtration realising the worst-case fill-in and computation.

math.AT↗

arXiv2021

Amplitudes in persistence theory

The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a general framework to compare distances and invariants in multiparameter persistence, where there is no natural choice of invariants and distances between them. We define amplitudes, monotone, and subadditive invariants that arise from assigning a non-negative real number to objects of an abelian category. We then present different ways to associate distances to such invariants, and we provide a classification of classes of amplitudes relevant to topological data analysis. In addition, we study the the relationships as well as the discriminitative power of such amplitude distances arising in topological data analysis scenarios.

math.AT↗

arXiv2021

Notes on pivot pairings

We present a row reduction algorithm to compute the barcode decomposition of persistence modules. This algorithm dualises the standard persistence one and clarifies the symmetry between clear and compress optimisations.

cs.CG↗

arXiv2020

Invariants for tame parametrised chain complexes

We set the foundations for a new approach to Topological Data Analysis (TDA) based on homotopical methods at chain complexes level. We present the category of tame parametrised chain complexes as a comprehensive environment that includes several cases that usually TDA handles separately, such as persistence modules, zigzag modules, and commutative ladders. We extract new invariants in this category using a model structure and various minimal cofibrant approximations. Such approximations and their invariants retain some of the topological, and not just homological, aspects of the objects they approximate.

math.AT↗

arXiv2017

Dynamical systems on graphs through the signless Laplacian matrix

There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree characteristics, using the signless Laplacian matrix. We expose the theoretical results about the eigenvalue of the matrix and how they are related to the dynamical system. Then, we perform numerical computations on real-like graphs and observe the resulting system. Comparing the theoretical and numerical results we found a perfect consistency. Furthermore, we define a metric which takes in account the "rigidity" of the graph and enables us to relate all together the topological properties of the graph, the signless Laplacian matrix and the dynamical system.

math.CO↗