# Maximum Entropy Models from Phase Harmonic Covariances

arXiv: https://arxiv.org/abs/1911.10017 · HTML: https://mlsanity.com/paper/1911.10017/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Nov 2019.

## Built on by (most cited first, 13 of 13)

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- [Deep Learning based Optical Image Super-Resolution via Generative Diffusion Models for Layerwise in-situ LPBF Monitoring](https://mlsanity.com/paper/2409.13171/) (2024)
- [On the Nash equilibrium of moment-matching GANs for stationary Gaussian processes](https://mlsanity.com/paper/2203.07136/) (2022)
- [Mallat Scattering Transformation based surrogate for MagnetoHydroDynamics](https://mlsanity.com/paper/2302.10243/) (2023)
- [Statistical Component Separation for Targeted Signal Recovery in Noisy Mixtures](https://mlsanity.com/paper/2306.15012/) (2023)
- [Mean-Field Microcanonical Gradient Descent](https://mlsanity.com/paper/2403.08362/) (2024)

## Abstract (arXiv)

The covariance of a stationary process X is diagonalized by a Fourier transform. It does not take into account the complex Fourier phase and defines Gaussian maximum entropy models. We introduce a general family of phase harmonic covariance moments, which rely on complex phases to capture non-Gaussian properties. They are defined as the covariance of hatH (L X), where L is a complex linear operator and hatH  is a non-linear phase harmonic operator which multiplies the phase of each complex coefficient by integers. The operator hatH (L X) can also be calculated from rectifiers, which relates hatH (L X) to neural network coefficients. If L is a Fourier transform then the covariance is a sparse matrix whose non-zero off-diagonal coefficients capture dependencies between frequencies. These coefficients have similarities with high order moment, but smaller statistical variabilities because hatH (L X) is Lipschitz. If L is a complex wavelet transform then off-diagonal coefficients reveal dependencies across scales, which specify the geometry of local coherent structures. We introduce maximum entropy models conditioned by these wavelet phase harmonic covariances. The precision of these models is numerically evaluated to synthesize images of turbulent flows and other stationary processes.