Deep neural networks have spectacular applications in machine learning

but their mathematical properties are not well understood. This course

introduces a probabilist harmonic analysis framework to analyze

properties of deep network architectures. Learning then amounts to

estimate high dimensional probability distributions. Building models of

such distributions must take advantage of priors on regularities and

invariants, where harmonic analysis plays a central role. We shall

explain why it can lead to deep neural network computational

architectures, where wavelets are used to separate scales and reduce

complexity. We study applications in audio and image classification as

well as in physics modeling, particularly in quantum chemistry and

cosmology.

High-dimensional learning is closely related to statistical physics. The

factorisation of high-dimensional probability distributions across

scales can be interpreted as a renormalisation group decomposition. From

a harmonic analysis point of view, it requires to represent and

precondition operators, which are singular near phase transitions. This

preconditioning plays an important role for learning optimisation by

gradient descents. Harmonic analysis results can take care of linear

singular operators, but the presence of non-linearities raises many open

questions. We shall describe some of these problems, in relation with

physics.