50 research outputs found
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
On the Construction of Group Equivariant Non-Expansive Operators via Permutants and Symmetric Functions
Group Equivariant Operators (GEOs) are a fundamental tool in the research on neural networks, since they make available a new kind of geometric knowledge engineering for deep learning, which can exploit symmetries in artificial intelligence and reduce the number of parameters required in the learning process. In this paper we introduce a new method to build non-linear GEOs and non-linear Group Equivariant Non-Expansive Operators (GENEOs), based on the concepts of symmetric function and permutant. This method is particularly interesting because of the good theoretical properties of GENEOs and the ease of use of permutants to build equivariant operators, compared to the direct use of the equivariance groups we are interested in. In our paper, we prove that the technique we propose works for any symmetric function, and benefits from the approximability of continuous symmetric functions by symmetric polynomials. A possible use in Topological Data Analysis of the GENEOs obtained by this new method is illustrated
On the topological theory of Group Equivariant Non-Expansive Operators
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
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.Comment: Added references. Extended Section 7. Added 3 figures. Corrected
typos. 42 pages, 7 figure
A compactness theorem in group invariant persistent homology
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
Algebra esterna e grassmanniane
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 proprietĂ e l'introduzione del concetto di Grassmanniana