Maximum Entropy Models from Phase Harmonic Covariances

We define maximum entropy models of non-Gaussian stationary random vectors from covariances of non-linear 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 entropy measure with a gradient descent. The precision of wavelet phase harmonic models is numerically evaluated over turbulent flows and other non-Gaussian stationary processes.