Apprentissage topologique

ECTS : 3

Volume horaire : 24

Description du contenu de l'enseignement :

Course Content

1. Topological Foundations

• From distances to topology: Metric spaces, open balls, neighborhoods, and the transition to abstract topological spaces.
• Invariants and deformations: Connectivity, holes, homotopy, and why topology is more fundamental than geometry for learning.
• Discrete and continuous structures: Graphs, hypergraphs, simplicial complexes, CW-complexes, and their geometric realizations.

2. Mathematical Tools for Topological Analysis

• Combinatorial topology: Simplices, complexes, cochains, and associated matrices (incidence, adjacency, Laplacian).
• Algebraic topology: Homology, cohomology, and applications in Topological Data Analysis (TDA).

3. Deep Learning on Topological Structures

• Message passing: Convolutions, attention, and diffusion on graphs and higher-order structures.
• Spectral representations: Normalized Laplacian, Fourier analysis on graphs, and implications for learning.
• Limitations and extensions: Oversmoothing, heterophily, sheaf theory, and Ricci curvature for more robust models.

4. Applications and Implementation

• Graphs and hypergraphs: Social networks, knowledge graph embeddings (TransE), community detection.
• Simplicial complexes: 3D mesh processing, molecular classification, and topological signal processing.
• Practical tools: Hands-on experience with libraries like TopoNetX to prototype topological models.

5. Critical Review and Perspectives

• State of the art: GCN, GAT, Simplicial Neural Networks, and foundational graph models.
• Open challenges: Topological identification capacity, spatio-temporal structures, and integration with generative AI.

Compétence à acquérir :

• Understand and apply core topological concepts (metric spaces, topological spaces, topological invariants) to data science and machine learning problems.
• Model and analyze discrete (graphs, hypergraphs, simplicial complexes) and continuous (manifolds, abstract topological spaces) structures within a unified framework.
• Design and interpret topological data representations (embeddings, Laplacian matrices, homology) for deep learning tasks.
• Implement message-passing algorithms (GNNs, attention mechanisms, diffusion) and spectral analysis on topological structures.
• Evaluate the limitations of existing models (oversmoothing, heterophily) and propose solutions inspired by algebraic or differential topology (sheaf theory, Ricci curvature).
• Apply these tools to real-world problems: social networks, 3D mesh processing, molecular classification, dimensionality reduction, and structured data generation.
• Engage with recent literature in topological learning, including advances in Topological Data Analysis (TDA), manifold learning, and topological neural networks.