Patrizio Frosini | Università di Bologna (original) (raw)

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Papers by Patrizio Frosini

Comparison between multidimensional persistent Betti numbers is often based on the multidimension... more Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the abov...

Group equivariant non-expansive operators have been recently proposed as basic components in topo... more 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 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 F . As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant nonexpansive operators in the considered manifold.

Nature Machine Intelligence

The aim of this paper is to provide a general mathematical framework for group equivariance in th... more 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.

Topological Dynamics and Topological Data Analysis

Lecture Notes in Computer Science

Topological data analysis is a new approach to processing digital data, focusing on the fact that... more Topological data analysis is a new approach to processing digital data, focusing on the fact that topological properties are quite important for efficient data comparison. In particular, persistent topology and homology are relevant mathematical tools in TDA, and their study is attracting more and more researchers. As a matter of fact, in many applications data can be represented by continuous real-valued functions defined on a topological space X, and persistent homology can be efficiently used to compare these data by describing the homological changes of the sub-level sets of those functions. However, persistent homology is invariant under the action of the group Homeo(X) of all selfhomeomorphisms of X, while in many cases an invariance with respect to a proper subgroup G of Homeo(X) is preferable. Interestingly, it has been recently proved that this restricted invariance can be obtained by applying G-invariant non-expansive operators to the considered functions. As a consequence, in order to proceed along this line of research we need methods to build G-invariant non-expansive operators. According to this perspective, in this paper we prove some new results about the algebra of GINOs.

Journal of Applied and Computational Topology

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persis... more 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 a property of 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.

Homology, Homotopy and Applications

Discrete & Computational Geometry, 2016

In many applications concerning the comparison of data expressed by R m-valued functions defined ... more In many applications concerning the comparison of data expressed by R m-valued functions defined on a topological space X, the invariance with respect to a given group G of self-homeomorphisms of X is required. While persistent homology is quite efficient in the topological and qualitative comparison of this kind of data when the invariance group G is the group Homeo(X) of all self-homeomorphisms of X, this theory is not tailored to manage the case in which G is a proper subgroup of Homeo(X), and its invariance appears too general for several tasks. This paper proposes a way to adapt persistent homology in order to get invariance just with respect to a given group of self-homeomorphisms of X. The main idea consists in a dual approach, based on considering the set of all G-invariant non-expanding operators defined on the space of the admissible filtering functions on X. Some theoretical results concerning this approach are proven and an experiment is presented, illustrating the application of the proposed technique to compare 1D-signals, when the invariance is expressed by the group of affinities, the group of orientation-preserving affinities, the group of isometries, the group of translations and the identity group.

Storage and Retrieval for Image and Video Databases, 1991

Frosini, Patrizio (1996) Una rapida escursione fra le distanze naturali di taglia. Atti della Acc... more Frosini, Patrizio (1996) Una rapida escursione fra le distanze naturali di taglia. Atti della Accademia Peloritana dei Pericolanti, LXXIII. pp. 26-40. ... PDF - Requires Adobe Acrobat Reader or other PDF viewer. ... In questo lavoro si illustra il concetto di distanza naturale di taglia con ...

ATTI DEL SEMINARIO MATEMATICO E FISICO UNIVERSITA DI MODENA, 1999

Applied Mathematics Letters, 2002

Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with... more Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of loo(R). An approximation result is also proved.