arXiv subjects

Patrizio Frosini

Publications and source records attributed to Patrizio Frosini.

At least 19 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.

math.AT↗

arXiv2026

An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning

Geometric and Topological Deep Learning are rapidly growing research areas that enhance machine learning through the use of geometric and topological structures. Within this framework, Group Equivariant Non-Expansive Operators (GENEOs) have emerged as a powerful class of operators for encoding symmetries and designing efficient, interpretable neural architectures. Originally introduced in Topological Data Analysis, GENEOs have since found applications in Deep Learning as tools for constructing equivariant models with reduced parameter complexity. GENEOs provide a unifying framework bridging Geometric and Topological Deep Learning and include the operator computing persistence diagrams as a special case. Their theoretical foundations rely on group actions, equivariance, and compactness properties of operator spaces, grounding them in algebra and geometry while enabling both mathematical rigor and practical relevance. While a previous representation theorem characterized linear GENEOs acting on data of the same type, many real-world applications require operators between heterogeneous data spaces. In this work, we address this limitation by introducing a new representation theorem for linear GENEOs acting between different perception pairs, based on generalized T-permutant measures. Under mild assumptions on the data domains and group actions, our result provides a complete characterization of such operators. We also prove the compactness and convexity of the space of linear GENEOs. We further demonstrate the practical impact of this theory by applying the proposed framework to improve the performance of autoencoders, highlighting the relevance of GENEOs in modern machine learning applications.

math.RT↗

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.

math.AT↗

arXiv2025

Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators

As sixth generation (6G) wireless networks evolve, accurate signal-to-interference-noise ratio (SINR) maps are becoming increasingly critical for effective resource management and optimization. However, acquiring such maps at high resolution is often cost-prohibitive, creating a severe data scarcity challenge. This necessitates machine learning (ML) approaches capable of robustly reconstructing the full map from extremely sparse measurements. To address this, we introduce a novel reconstruction framework based on Group Equivariant Non-Expansive Operators (GENEOs). Unlike data-hungry ML models, GENEOs are low-complexity operators that embed domain-specific geometric priors, such as translation invariance and rotational equivariance, directly into their structure. This provides a strong inductive bias, enabling effective reconstruction from very few samples. Our key insight is that for network management, preserving the topological structure of the SINR map, such as the geometry of coverage holes and interference patterns, is often more critical than minimizing pixel-wise error. We validate our approach on realistic ray-tracing-based urban scenarios, evaluating performance with both statistical metrics (mean squared error (MSE)) and, crucially, a topological metric (1-Wasserstein distance). Results show that our method achieves superior statistical and topological accuracy across diverse urban scenarios. Compared to the best-performing baselines, GENEO reduces MSE up to 45% and decreases the 1-Wasserstein distance up to 54%. Crucially, these performance gains are maintained even under the most extreme tested conditions, such as a 1% sampling rate with a 30% measurement error, and when measurements are spatially biased. This demonstrates the practical advantage of GENEOs for creating structurally accurate SINR maps that are more reliable for downstream network optimization tasks.

cs.LG↗

arXiv2025

GENEOnet: Statistical analysis supporting explainability and trustworthiness

Group Equivariant Non-Expansive Operators (GENEOs) have emerged as mathematical tools for constructing networks for Machine Learning and Artificial Intelligence. Recent findings suggest that such models can be inserted within the domain of eXplainable Artificial Intelligence (XAI) due to their inherent interpretability. In this study, we aim to verify this claim with respect to GENEOnet, a GENEO network developed for an application in computational biochemistry by employing various statistical analyses and experiments. Such experiments first allow us to perform a sensitivity analysis on GENEOnet's parameters to test their significance. Subsequently, we show that GENEOnet exhibits a significantly higher proportion of equivariance compared to other methods. Lastly, we demonstrate that GENEOnet is on average robust to perturbations arising from molecular dynamics. These results collectively serve as proof of the explainability, trustworthiness, and robustness of GENEOnet and confirm the beneficial use of GENEOs in the context of Trustworthy Artificial Intelligence.

cs.LG↗

arXiv2025

Mathematical Foundation of Interpretable Equivariant Surrogate Models

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive Operators (GENEOs) transformations. The central concept involves quantifying the distance between GEOs by measuring the non-commutativity of specific diagrams. Additionally, the paper proposes a definition of interpretability of GEOs according to a complexity measure that can be defined according to each user preferences. Moreover, we explore the formal properties of this framework and show how it can be applied in classical machine learning scenarios, like image classification with convolutional neural networks.

stat.ML↗

arXiv2024

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.

math.AT↗

arXiv2024

A novel approach to graph distinction through GENEOs and permutants

The theory of Group Equivariant Non-Expansive Operators (GENEOs) was initially developed in Topological Data Analysis for the geometric approximation of data observers, including their invariances and symmetries. This paper departs from that line of research and explores the use of GENEOs for distinguishing $r$-regular graphs up to isomorphisms. In doing so, we aim to test the capabilities and flexibility of these operators. Our experiments show that GENEOs offer a good compromise between efficiency and computational cost in comparing $r$-regular graphs, while their actions on data are easily interpretable. This supports the idea that GENEOs could be a general-purpose approach to discriminative problems in Machine Learning when some structural information about data and observers is explicitly given.

cs.LG↗

arXiv2023

On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators

Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold.

math.DG↗

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.

math.AT↗

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.

stat.ML↗

arXiv2023

Low-Resource White-Box Semantic Segmentation of Supporting Towers on 3D Point Clouds via Signature Shape Identification

Research in 3D semantic segmentation has been increasing performance metrics, like the IoU, by scaling model complexity and computational resources, leaving behind researchers and practitioners that (1) cannot access the necessary resources and (2) do need transparency on the model decision mechanisms. In this paper, we propose SCENE-Net, a low-resource white-box model for 3D point cloud semantic segmentation. SCENE-Net identifies signature shapes on the point cloud via group equivariant non-expansive operators (GENEOs), providing intrinsic geometric interpretability. Our training time on a laptop is 85~min, and our inference time is 20~ms. SCENE-Net has 11 trainable geometrical parameters and requires fewer data than black-box models. SCENE--Net offers robustness to noisy labeling and data imbalance and has comparable IoU to state-of-the-art methods. With this paper, we release a 40~000 Km labeled dataset of rural terrain point clouds and our code implementation.

cs.CV↗

arXiv2022

Generalized Permutants and Graph GENEOs

In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices or edges. This is done by showing how the general concept of GENEO can be used to transform graphs and to give information about their structure. This requires the introduction of the new concepts of generalized permutant and generalized permutant measure and the mathematical proof that these concepts allow us to build GENEOs between graphs. An experimental section concludes the paper, illustrating the possible use of our operators to extract information from graphs. This paper is part of a line of research devoted to developing a compositional and geometric theory of GENEOs for Geometric Deep Learning.

math.CO↗

arXiv2022

A probabilistic result on impulsive noise reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators

In recent years, group equivariant non-expansive operators (GENEOs) have attracted attention in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by $L$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$.

cs.CG↗

arXiv2022

On the finite representation of group equivariant operators via permutant measures

The study of $G$-equivariant operators is of great interest to explain and understand the architecture of neural networks. In this paper we show that each linear $G$-equivariant operator can be produced by a suitable permutant measure, provided that the group $G$ transitively acts on a finite signal domain $X$. This result makes available a new method to build linear $G$-equivariant operators in the finite setting.

math.GR↗

arXiv2022

GENEOnet: A new machine learning paradigm based on Group Equivariant Non-Expansive Operators. An application to protein pocket detection

Nowadays there is a big spotlight cast on the development of techniques of explainable machine learning. Here we introduce a new computational paradigm based on Group Equivariant Non-Expansive Operators, that can be regarded as the product of a rising mathematical theory of information-processing observers. This approach, that can be adjusted to different situations, may have many advantages over other common tools, like Neural Networks, such as: knowledge injection and information engineering, selection of relevant features, small number of parameters and higher transparency. We chose to test our method, called GENEOnet, on a key problem in drug design: detecting pockets on the surface of proteins that can host ligands. Experimental results confirmed that our method works well even with a quite small training set, providing thus a great computational advantage, while the final comparison with other state-of-the-art methods shows that GENEOnet provides better or comparable results in terms of accuracy.

q-bio.BM↗

arXiv2019

Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

The aim of this paper is to provide a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. We define group-equivariant non-expansive operators (GENEOs), which are maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialise and compose operators. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of non-expansive operators. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show examples on the MNIST and fashion-MNIST datasets. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators, and show how the selected and sampled operators can be used to perform both classical metric learning and an effective initialisation of the kernels of a convolutional neural network.

cs.LG↗

arXiv2018

On the geometrical properties of the coherent matching distance in 2D persistent homology

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.

math.AT↗