Statistical physics methods for machine learning and traffic forecasting

This document traces back my research work done over the last 15 years or so. It is dealing with random walks, exclusion processes, queueing processes, irreversibility, all sort of cycles, belief propagation, traffic congestion, inverse Ising problem, sparse Gaussian copula, clustering and restricted Boltzmann machines. I attempt to unify this into a single document with the expectation of finding some guidelines for future work which will be discussed in the conclusion. For the moment just remark that the document has 4 parts, one to introduce material and subjects relevant to the next three parts dealing with quite distinctand not obviously directly related subjects.Part II corresponds to research done mainly at Inria Rocquencourt during aperiod which extend from 2002 to 2012. This concerns the study of stochastic processes like exclusion or queuing processes introduced in Chapter 1 and their application to microscopic road traffic (i.e. at the level of one segment) discussedin Chapter 5. The main questions of interest discussed firstly in Chapter 6 are relative to the emergence of macroscopic phenomena resulting from simple local stochastic dynamical rules, the way to relate these two and possible waysto deal with non-reversibility in absence of integrability. In Chapter 7 these models are used in an applied perspective in order to study the fluctuations of the fundamental diagram of traffic flow.Part III is overall concerned with the belief propagation algorithm and generalizations introduced in Chapter 3 and how to make it operational for traffic prediction at the level of a conurbation. This line of search started in 2007,and was developed mainly during 2009-2012 thanks to an ANR project I coordinated. An important question in this context is the inverse model problem introduced more specifically in its inverse Ising formulation in Chapter 4 which can be looked at in various ways as discussed in Chapter 9 and 10, in order to find a good trade-off between expressiveness of the model and computational tractability. My interest in this application was revived recently after getting intouch with the Sistema company who found our approach interesting and proposed us to test our method on their data as described at the end of Chapter 10. Finally part IV is dedicated to the analysis of simple non-supervised learning algorithms, which is a sub-field of Machine learning briefly introduced in Chapter 2. In Chapter 11 by considering a version of Clustering related tob elief-propagation, I discuss the question of the “true" number of clusters, which while being ill posed in principle, can actually be addressed in some cases by renormalization group considerations. In Chapter 12 we discuss the restricted Boltzmann machine, which played some time ago a central role in deep learning, in order to study the dynamics of learning under the angle of pattern formation