Maximum Entropy Models from Phase Harmonic Covariances

Maximum Entropy Models from Phase Harmonic Covariances Sixin Zhang 1, 4, St ephane Mallat 1, 2,3 1 ENS, PSL University, Paris, France 2 Coll ege de France, Paris, France 3 Flatiron Institute, New York, USA 4 Center for Data Science, Peking University, Beijing, China November 25, 2019 Abstract We define maximum entropy models of non-Gaussian stationary random vectors from covariances of nonlinear representations. These representations are calculated by multiplying the phase of Fourier or wavelet coefficients with harmonic integers, which amounts to compute a windowed Fourier transform along their phase. Rectifiers in neural networks compute such phase windowing. The covariance of these harmonic coefficients capture dependencies of Fourier and wavelet coefficients across frequencies, by canceling their random phase. We introduce maximum entropy models conditioned by such covariances over a graph of local interactions. These models are approximated by transporting an initial maximum ...