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In this thesis we aim to provide a general topological and geometrical framework for group equivariance in the machine learning context. A crucial part of this framework is a synergy between persistent homology and the theory of group actions. In our approach, instead of focusing on data, we focus on suitable operators defined on the functions that represent the data. In particular, we define group equivariant non-expansive operators (GENEOs), which are maps between function spaces endowed with the actions of groups of transformations. We investigate the topological, geometric and metric properties of the space of GENEOs. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of GENEOs and proving some results about our model. 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 some new methods to build different classes of GENEOs in order to populate and approximate the space of GENEOs. Moreover, we define a suitable Riemannian structure on manifolds of GENEOs making available the use of gradient descent methods.

Nella tesi viene fornita una costruzione dell'algebra esterna di un K-spazio vettoriale, alcune conseguenze principali come la derivazione in maniera traspente del determinante di e alcune sue proprietà e l'introduzione del concetto di Grassmanniana.

In this thesis we present a new result concerning the theory of group invariant persistent homology. This theory adapts persistent homology in the presence of the action on a space of functions Phi of a subgroup G of the group H of all self-homeomorphisms of a topological space X. Its model is based on a space of suitable operators defined on Phi. After describing the mathematical setting and recalling some basic results, we prove that the space of these operators is compact with respect to a suitable topology. In order to prove this result, we require that Phi, G, X are compact.

The aim of this article is to describe a new perspective on functoriality ofpersistent homology and explain its intrinsic symmetry that is often overlooked. A dataset for us is a finite collection of functions, called measurements, with a finite domain. Sucha data set might contain internal symmetries which are effectively captured by the actionof a set of the domain endomorphisms. Different choices of the set of endomorphismsencode different symmetries of the data set. We describe various category structureson such enriched data sets and prove some of their properties such as decompositionsand morphism formations. We also describe a data structure, based on coloured directedgraphs, which is convenient to encode the mentioned enrichment. We show that persistenthomology preserves only some aspects of these collection of enriched data sets however notall. In other words persistent homology is not a functor on the entire category of enricheddata sets. Nevertheless we show that persistent homology is functorial locally. We usethe concept of set equivariant operator (SEO) to capture some of the information missedby persistent homology. Moreover, we provide examples and give ways to construct suchSEOs., QC 20230524

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for ℝ2-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., QC 20230525

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for ℝ2-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., QC 20230525

The aim of this article is to describe a new perspective on functoriality ofpersistent homology and explain its intrinsic symmetry that is often overlooked. A dataset for us is a finite collection of functions, called measurements, with a finite domain. Sucha data set might contain internal symmetries which are effectively captured by the actionof a set of the domain endomorphisms. Different choices of the set of endomorphismsencode different symmetries of the data set. We describe various category structureson such enriched data sets and prove some of their properties such as decompositionsand morphism formations. We also describe a data structure, based on coloured directedgraphs, which is convenient to encode the mentioned enrichment. We show that persistenthomology preserves only some aspects of these collection of enriched data sets however notall. In other words persistent homology is not a functor on the entire category of enricheddata sets. Nevertheless we show that persistent homology is functorial locally. We usethe concept of set equivariant operator (SEO) to capture some of the information missedby persistent homology. Moreover, we provide examples and give ways to construct suchSEOs., QC 20230524

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for ℝ2-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., QC 20230525

The aim of this article is to describe a new perspective on functoriality ofpersistent homology and explain its intrinsic symmetry that is often overlooked. A dataset for us is a finite collection of functions, called measurements, with a finite domain. Sucha data set might contain internal symmetries which are effectively captured by the actionof a set of the domain endomorphisms. Different choices of the set of endomorphismsencode different symmetries of the data set. We describe various category structureson such enriched data sets and prove some of their properties such as decompositionsand morphism formations. We also describe a data structure, based on coloured directedgraphs, which is convenient to encode the mentioned enrichment. We show that persistenthomology preserves only some aspects of these collection of enriched data sets however notall. In other words persistent homology is not a functor on the entire category of enricheddata sets. Nevertheless we show that persistent homology is functorial locally. We usethe concept of set equivariant operator (SEO) to capture some of the information missedby persistent homology. Moreover, we provide examples and give ways to construct suchSEOs., QC 20230524

The aim of this article is to describe a new perspective on functoriality ofpersistent homology and explain its intrinsic symmetry that is often overlooked. A dataset for us is a finite collection of functions, called measurements, with a finite domain. Sucha data set might contain internal symmetries which are effectively captured by the actionof a set of the domain endomorphisms. Different choices of the set of endomorphismsencode different symmetries of the data set. We describe various category structureson such enriched data sets and prove some of their properties such as decompositionsand morphism formations. We also describe a data structure, based on coloured directedgraphs, which is convenient to encode the mentioned enrichment. We show that persistenthomology preserves only some aspects of these collection of enriched data sets however notall. In other words persistent homology is not a functor on the entire category of enricheddata sets. Nevertheless we show that persistent homology is functorial locally. We usethe concept of set equivariant operator (SEO) to capture some of the information missedby persistent homology. Moreover, we provide examples and give ways to construct suchSEOs., QC 20230524

In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for ℝ2-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., QC 20230525

In this thesis we present a new result concerning the theory of group invariant persistent homology. This theory adapts persistent homology in the presence of the action on a space of functions Phi of a subgroup G of the group H of all self-homeomorphisms of a topological space X. Its model is based on a space of suitable operators defined on Phi. After describing the mathematical setting and recalling some basic results, we prove that the space of these operators is compact with respect to a suitable topology. In order to prove this result, we require that Phi, G, X are compact.

Nella tesi viene fornita una costruzione dell'algebra esterna di un K-spazio vettoriale, alcune conseguenze principali come la derivazione in maniera traspente del determinante di e alcune sue proprietà e l'introduzione del concetto di Grassmanniana.