Poset associahedra as sections of graph associahedra

Poset associahedra are a family of convex polytopes recently introduced by Pavel Galashin in 2021. The associahedron An is an (n-2)-dimensional convex polytope whose facial structure encodes the ways of parenthesizing an n-letter word (among several equivalent combinatorial objects). Associahedra are deeply studied polytopes that appear naturally in many areas of mathematics: algebra, combinatorics, geometry, topology... They have many presentations and generalizations. One of their incarnations is as a compactification of the configuration space of n points on a line. Similarly, the P-associahedron of a poset P is a compactification of the configuration space of order preserving maps from P to R. Galashin presents poset associahedra as combinatorial objects and shows that they can be realized as convex polytopes. However, his proof is not constructive, in the sense that no explicit coordinates are provided. The main goal of this thesis is to provide an explicit construction of poset associahedra as sections of graph associahedra, thus solving the open problem stated in Remark 1.5 of Galashin's paper.

Graph theory and geometric deep learning

La seguente tesi propone un’introduzione al geometric deep learning. Nella prima parte vengono presentati i concetti principali di teoria dei grafi ed introdotta una dinamica di diffusione su grafo, in analogia con l’equazione del calore. A seguire, iniziando dal linear classifier verranno introdotte le architetture che hanno portato all’ideazione delle graph convolutional networks. In conclusione, si analizzano esempi di alcuni algoritmi utilizzati nel geometric deep learning e si mostra una loro implementazione sul Cora dataset, un insieme di dati con struttura a grafo.

Graph-based Vision-Language Model for Multi-Modal Retrieval of Fashion Products

Artificial Intelligence is reshaping the field of fashion industry in different ways. E-commerce retailers exploit their data through AI to enhance their search engines, make outfit suggestions and forecast the success of a specific fashion product. However, it is a challenging endeavour as the data they possess is huge, complex and multi-modal. The most common way to search for fashion products online is by matching keywords with phrases in the product's description which are often cluttered, inadequate and differ across collections and sellers. A customer may also browse an online store's taxonomy, although this is time-consuming and doesn't guarantee relevant items. With the advent of Deep Learning architectures, particularly Vision-Language models, ad-hoc solutions have been proposed to model both the product image and description to solve this problems. However, the suggested solutions do not exploit effectively the semantic or syntactic information of these modalities, and the unique qualities and relations of clothing items. In this work of thesis, a novel approach is proposed to address this issues, which aims to model and process images and text descriptions as graphs in order to exploit the relations inside and between each modality and employs specific techniques to extract syntactic and semantic information. The results obtained show promising performances on different tasks when compared to the present state-of-the-art deep learning architectures.

Graph-based User Scheduling for MIMO LEO-based Satellite Communication Systems

The study of the user scheduling problem in a Low Earth Orbit (LEO) Multi-User MIMO system is the objective of this thesis. With the application of cutting-edge digital beamforming algorithms, a LEO satellite with an antenna array and a large number of antenna elements can provide service to many user terminals (UTs) in full frequency reuse (FFR) schemes. Since the number of UTs on-ground are many more than the transmit antennas on the satellite, user scheduling is necessary. Scheduling can be accomplished by grouping users into different clusters: users within the same cluster are multiplexed and served together via Space Division Multiple Access (SDMA), i.e., digital beamforming or Multi-User MIMO techniques; the different clusters of users are then served on different time slots via Time Division Multiple Access (TDMA). The design of an optimal user grouping strategy is known to be an NP-complete problem which can be solved only through exhaustive search. In this thesis, we provide a graph-based user scheduling and feed space beamforming architecture for the downlink with the aim of reducing user inter-beam interference. The main idea is based on clustering users whose pairwise great-circle distance is as large as possible. First, we create a graph where the users represent the vertices, whereas an edge in the graph between 2 users exists if their great-circle distance is above a certain threshold. In the second step, we develop a low complex greedy user clustering technique and we iteratively search for the maximum clique in the graph, i.e., the largest fully connected subgraph in the graph. Finally, by using the 3 aforementioned power normalization techniques, a Minimum Mean Square Error (MMSE) beamforming matrix is deployed on a cluster basis. The suggested scheduling system is compared with a position-based scheduler, which generates a beam lattice on the ground and randomly selects one user per beam to form a cluster.

Anomaly Detection in Random Graph

La presenti tesi ha come obiettivo lo studio di due algoritmi per il rilevamento di anomalie all' interno di grafi random. Per entrambi gli algoritmi sono stati creati dei modelli generativi di grafi dinamici in modo da eseguire dei test sintetici. La tesi si compone in una parte iniziale teorica e di una seconda parte sperimentale. Il secondo capitolo introduce la teoria dei grafi. Il terzo capitolo presenta il problema del rilevamento di comunità. Il quarto capitolo introduce possibili definizioni del concetto di anomalie dinamiche e il problema del loro rilevamento. Il quinto capitolo propone l' introduzione di un punteggio di outlierness associato ad ogni nodo sulla base del confronto tra la sua dinamica e quella della comunità a cui appartiene. L' ultimo capitolo si incentra sul problema della ricerca di una descrizione della rete in termini di gruppi o ruoli sulla base della quale incentrare la ricerca delle anomalie dinamiche.