Hamiltonian Estimations by Conditional Renormalisation Group and Convolution Nets

Estimating high-dimensional probability distributions and physical Hamiltonians from data is an old outstanding problem. It is typically unstable, specially near phase transitions. We revisit this topic with models resulting from multiscale harmonic analysis and neural networks. We show that renormalisation group calculations in wavelet orthonormal bases amount to precondition the Hamiltonian estimation. Stable estimations are shown on the phi^4 model and Cosmological weak lensing. Hamiltonian models are obtained from conditional probabilities which specify interactions across scales. Multiscale models of turbulences are computed, with a deep network whose filters are wavelets. Hamiltonian estimation is also closely related to classification. ResNet accuracy is obtained on ImageNet with wavelet filters, by learning the potential functions that approximate Hamiltonians.

Stéphane Mallat is an applied mathematician, Professor at the Collège de France on the chair of Data Sciences. He is a member of the French Academy of sciences, of the Academy of Technologies and a foreign member of the US National Academy of Engineering. He was a Professor at the Courant Institute of NYU in New York for 10 years, then at Ecole Polytechnique and Ecole Normale Supérieure in Paris. He also was the co-founder and CEO of a semiconductor start-up company.