00696 - Advanced Mechanics

Course contents

• Refresh of Probability, Information Theory and Statistical Mechanics;
  
• Ising Models: thermodynamic states and phase transitions;
• Systems with frustration and Gauge Theory;
• Random Graphs: degree distribution, components and metrics; Configuration Model; Erdos-Renyi; Maximum Entropy Random graphs; Macroscopic Structures and Stochastic Block Model;
• Factor graphs: locally treelike graphs, Bethe Free energy;
• Belief Propagation, Message-Passing Algorithm, TAP equations;

Approfondimenti/Applicazioni

• Ising Spins: Belief Propagation vs Glauber Dynamics;
  
• Belief Propagation and community detection: detectability transitions;
  
• Coding, Transmission, Noisy Channels and Decoding;
• Image Restoration;
• Perceptron Learning and Neural Networks: critical capacity and phase transition;

Readings/Bibliography

Main references:

• M.Mézard, A.Montanari - Information, Physics, and Computation - Oxford University Press, USA
  (2009);
• Nishimori, H.: Statistical Physics of Spin Glasses and Information processing. An Introduction. Oxford
  Science Publications 2001.
• Coolen, Kuhn, Sollich, Theory of Neural InformationProcessing Systems, Oxford University Press

Suggested reading:

• Mark Newman - Networks_ An Introduction - Oxford University Press (2010);
• Decelle, A., Krzakala, F., Moore, C., & Zdeborová, L. (2011). Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Physical Review E, 84(6), 066106.

• Zdeborová, L., & Krzakala, F. (2016). Statistical physics of inference: Thresholds and
  algorithms. Advances in Physics, 65(5), 453-552.
• Gardner, E., and Derrida, B.: Optimal storage properties of neural network models. J. Phys. A: Math.
  Gen. 21, 271-284 (1988)

Teaching methods

Frontal classes

Assessment methods

The exam consists of an oral interview in order to verify the knowledge of the arguments listed in the Course Contents and the skills achieved as:

• Advanced Concepts of Applied Statistical Mechanics and Random Graph Theory;
• Ability of reading an optimization, inference, machine learning problem from the statistical mechanics perspective, designing both mathematical structure and possible solutions;
• Ability of performing a numerical experiment, running the studied algorithms on both synthetic and real data;
• Ability of deepening the analyzed topics through the most recent results in the literature;

Teaching tools

Lecture notes and updated record of lessons.

Office hours

See the website of Daniele Tantari