his advanced course explores the convergence of geometric learning, graph neural networks (GNNs), and time-variant data analysis, with a strong emphasis on anomaly detection in complex and dynamic systems. The course begins with the foundations of geometric machine learning, focusing on the representation of data in non-Euclidean domains, such as graphs and manifolds. It then progresses to cutting-edge developments in graph neural networks, including architectures adapted to temporal and heterogeneous graph data. The course also introduces techniques for time-series modeling, both classical and modern (e.g., machine learning and deep learning approaches), and culminates in a comprehensive module on anomaly detection, covering both statistical and learning-based methods. Throughout the course, students will engage with real-world datasets and hands-on programming exercises, gaining both theoretical insights and practical expertise.

By the end of the course, students will be able to: Knowledge and Understanding • Grasp the mathematical underpinnings of geometric learning and its applications to manifolds, symmetry groups, and invariant representations. • Understand the architecture and functioning of graph neural networks, including: • Graph Convolutional Networks (GCN) • Graph Attention Networks (GAT) • Graph Isomorphism Networks (GIN) • Temporal GNNs (T-GNN, EvolveGCN, TGAT) • Comprehend the principles of time-series modeling, including: • Classical models: ARIMA, VAR, Kalman Filters • Machine learning approaches: Random Forests, Gradient Boosting • Deep learning methods: RNNs, LSTMs, TCNs • Master the theory and methods for anomaly detection, including: • Statistical tests and control charts • Distance- and density-based approaches • Autoencoders and hybrid deep learning models • Methods specific to temporal and graph data Practical and Computational Skills • Implement GNNs and time-series models using PyTorch Geometric, DGL, scikit-learn, stats models, and TensorFlow/Keras. • Conduct feature engineering and data preprocessing for time-variant and graph-structured data. • Evaluate model performance using metrics such as ROC-AUC, F1-score, precision-recall curves, and time-aware diagnostics. • Develop anomaly detection pipelines for real-time monitoring and post-hoc analysis. Application and Critical Thinking • Apply learned techniques to domains such as: • Cybersecurity (e.g., intrusion detection) • Finance (e.g., fraud detection, volatility shifts) • Healthcare (e.g., patient monitoring) • Industrial systems (e.g., predictive maintenance) • Critically evaluate and compare different modeling choices for real-world problems involving non-Euclidean, temporal, and dynamic data.

Course Prerequisites • Linear Algebra, Probability and Statistics • Programming in Python • Fundamentals of Machine Learning • Basic knowledge of graph theory and differential geometry (beneficial but not strictly required) These prerequisites are typically covered in the first-year coursework of the Master’s in Data Science and Engineering.

Module 1: Introduction to Geometric Learning • Symmetries and group actions in data • Invariant and equivariant representations • Manifold learning: Isomap, LLE, Diffusion Maps • Gauge theories and local geometries Module 2: Graph Neural Networks (GNNs) • Message-passing neural networks (MPNNs) • GCN, GAT, GIN • Heterogeneous and dynamic graphs • Temporal GNNs: T-GCN, TGAT, EvolveGCN • Applications: node classification, link prediction, graph classification Module 3: Time-Variant Data and Time Series Modeling • Nature and challenges of temporal data • Classical time series analysis: stationarity, autocorrelation, seasonality • Forecasting with ARIMA, VAR, Kalman Filters • Machine learning and deep learning for time series: RNN, LSTM, GRU, Transformer Module 4: Anomaly Detection in Time and Graph Data • Definitions and types of anomalies: point, contextual, collective • Statistical methods: z-scores, PCA, control charts • Machine learning-based methods: Isolation Forest, One-Class SVM • Deep learning-based methods: Autoencoders, LSTM-AE, GANs • Graph-based anomaly detection: subgraph detection, spectral methods, GAD models

• Class materials, datasets, and code notebooks will be made available on the course GitHub repository. • Guest lectures and seminars by experts in AI and complex systems may be included. • The course can be extended to include a mini-research project for PhD students.

• 60 hours of lectures • 20 hours of hands-on sessions, including coding labs, case studies, and project work • Continuous assessment through assignments and a final project or oral exam

Geometric Learning Bronstein et al., Geometric Deep Learning (2024) Graph Neural Networks Hamilton, Graph Representation Learning (2020) Time Series Analysis Hyndman & Athanasopoulos, Forecasting: Principles and Practice (open-access) Anomaly Detection Aggarwal, Outlier Analysis (2017) Visual/No-code Tools Berthold et al., KNIME: The Konstanz Information Miner

Slides; Dispense; Libro di testo; Esercizi; Esercizi risolti; Esercitazioni di laboratorio; Esercitazioni di laboratorio risolte; Materiale multimediale ;

Modalitΰ di esame: Elaborato scritto individuale;

Exam: Individual essay;

... Mini project report describing their workflow, rationale, and findings. (0-16 points) Oral discussion of their conclusions and questions on the theoretical aspects of the methodologies they implied in their project. (0-16 points) Weight: 50% mini project/50% oral exam. The laude is assigned to students with a total score higher than 30 points.

Gli studenti e le studentesse con disabilitΰ o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unitΰ Special Needs, al fine di permettere al/la docente la declinazione piω idonea in riferimento alla specifica tipologia di esame.