Scale Dependencies and Self-Similarity Through Wavelet Scattering Covariance

Abstract

We introduce a scattering covariance matrix which provides non-Gaussian models of time-series having stationary increments. A complex wavelet transform computes signal variations at each scale. Dependencies across scales are captured by the joint covariance across time and scales of complex wavelet coefficients and their modulus. This covariance is nearly diagonalized by a second wavelet transform, which defines the scattering covariance. We show that this set of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes, including fractional Brownian motions, Poisson, multifractal random walks and Hawkes processes. We prove that self-similar processes have a scattering covariance matrix which is scale invariant. This property can be estimated numerically and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering covariance coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.

Non-Gaussian modelling and statistical denoising of Planck dust polarisation full-sky maps using scattering transforms

“…One the other hand, other successful sets of scattering statistics, which give better syntheses on regular 2D grids, could be used. For instance, the wavelet phase harmonics (Allys et al 2020;Jeffrey et al 2022) and their recent multi-channel extensions (Régaldo-Saint Blancard et al 2022), or the more recent representations built from wavelet scattering covariances (Morel et al 2022;Cheng et al 2022). However, the main challenge is to make these improvements feasible given the computational and memory costs of the FoCUS algorithm.…”

Non-Gaussian modelling and statistical denoising of Planck dust polarization full-sky maps using scattering transforms

“…One the other hand, other successful sets of scattering statistics, which give better syntheses on regular 2D grids, could be used. For instance, the Wavelet Phase Harmonics (Allys et al 2020;Jeffrey et al 2022) or the more recent representations built from Wavelet Scattering Covariances (Morel et al 2022;Cheng et al 2022). However, the main challenge is to make such improvements feasible given the computational and memory costs of the FoCUS algorithm.…”

Unearthing InSights into Mars: unsupervised source separation with limited data

“…Since we are operating in a limited-data regime, we cannot afford self-supervised learning with Transformers in order to obtain high-performing features. Instead, we propose to use wavelet scattering covariances (Morel et al, 2022) as means to transfer data to a suitable representation space for source separation. This transform is based on scattering networks (Bruna & Mallat, 2013) that provide interpretable representation of data and are able to characterize a wide range of non-Gaussian properties of multiscale stochastic processes (Morel et al, 2022)-a type of signals that are considered in this paper.…”

“…Instead, we propose to use wavelet scattering covariances (Morel et al, 2022) as means to transfer data to a suitable representation space for source separation. This transform is based on scattering networks (Bruna & Mallat, 2013) that provide interpretable representation of data and are able to characterize a wide range of non-Gaussian properties of multiscale stochastic processes (Morel et al, 2022)-a type of signals that are considered in this paper. The wavelet scattering covariance generally does not require any pretraining and its weights, i.e., wavelets in the scattering network, are often chosen beforehand (see Seydoux et al (2020) for a data-driven wavelet choice) according to the time-frequency properties of data.…”

“…In order to achieve this, we embed inductive biases into our approach through the use of domain knowledge from time series analysis and signal processing via the scattering transform (Bruna & Mallat, 2013). As a means of capturing non-Gaussian and multiscale characteristics of the sources, we extract second-order information of scattering coefficients, known as the wavelet scattering covariance representation (Morel et al, 2022). We perform source separation by solving an optimization problem over the unknown sources that entails minimizing multiple carefully selected and normalized loss functions in the wavelet scattering covariance representations space.…”