Ricerc@Sapienza

Topological Data Analysis (\texttt{TDA}) is a recent and growing branch of statistics devoted to the study of the shape of the data. Motivated by the complexity of the object summarizing the topology of data, we introduce a new topological kernel that allows to extend the \texttt{TDA} toolbox to supervised learning. Exploiting the geodesic structure of the space of Persistence Diagrams, we define a geodesic kernel for Persistence Diagrams, we characterize it, and we show with an application that, despite not being positive semi--definite, it can be successfully used in regression tasks.

Multidimensional sensors represent an increasingly popular, yet challenging data source in modern statistics. Using tools from the emerging branch of Topological Data Analysis (TDA), we address two issues frequently encountered when analysing sensor data, namely their (often) high dimension and their sensibility to the reference system. We show how topological invariants provide a tool for detecting change--points which is robust with respect to both the time resolution we consider and the sensor placement.

Topological characterization of brain imaging data is gaining momentum in the statistical literature, however, the resulting representation of brain imaging data are often defined in complex spaces, not amenable for standard statistical methods. Ad hoc procedures must thus be adopted in order

Embedding spaces are one of the mainstream approaches when dealing with structured data. Granular Computing, in the last decade, emerged as a powerful paradigm for the automatic synthesis of embedding spaces that, at the same time, yield an interpretable model on the top of meaningful entities known as information granules. Usually, in these contexts, one aims at finding the smallest set of information granules in order to boost the model interpretability while keeping satisfactory performances.

Graph kernels are one of the mainstream approaches when dealing with measuring similarity between graphs, especially for pattern recognition and machine learning tasks. In turn, graphs gained a lot of attention due to their modeling capabilities for several real-world phenomena ranging from bioinformatics to social network analysis. However, the attention has been recently moved towards hypergraphs, generalization of plain graphs where multi-way relations (other than pairwise relations) can be considered.

Topological Data Analysis is a novel approach, useful whenever data can be described by topological structures such as graphs. The aim of this paper is to investigate whether such tool can be used in order to define a set of descriptors useful for pattern recognition and machine learning tasks. Specifically, we consider a supervised learning problem with the final goal of predicting proteins' physiological function starting from their respective residue contact network.

This paper investigates a novel graph embedding procedure based on simplicial complexes. Inherited from algebraic topology, simplicial complexes are collections of increasing-order simplices (e.g., points, lines, triangles, tetrahedrons) which can be interpreted as possibly meaningful substructures (i.e., information granules) on the top of which an embedding space can be built by means of symbolic histograms. In the embedding space, any Euclidean pattern recognition system can be used, possibly equipped with feature selection capabilities in order to select the most informative symbols.

In the last years, several new tools have been devised to analyze signals defined over the vertices of a graph, i.e., over a discrete domain whose structure is described by pairwise relations. In this paper, we expand these tools to the analysis of signals defined on simplicial complexes, whose domain has a structure specified by various multi-way relations. Within this framework, we show how to filter signals and how to reconstruct edge and vertex signals from a subset of observations.