# Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra

arXiv: https://arxiv.org/abs/2204.10177 · HTML: https://mlsanity.com/paper/2204.10177/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 55 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 4 of 4)

- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Multiscale Sparse Microcanonical Models](https://mlsanity.com/paper/1801.02013/) (2018)
- [Maximum Entropy Models from Phase Harmonic Covariances](https://mlsanity.com/paper/1911.10017/) (2019)
- [Intermittent process analysis with scattering moments](https://mlsanity.com/paper/1311.4104/) (2013)

## Built on by (most cited first, 7 of 7)

- [NuTime: Numerically Multi-Scaled Embedding for Large-Scale Time-Series Pretraining](https://mlsanity.com/paper/2310.07402/) (2023)
- [Scattering Spectra Models for Physics](https://mlsanity.com/paper/2306.17210/) (2023)
- [Predicting partially observable dynamical systems via diffusion models with a multiscale inference scheme](https://mlsanity.com/paper/2511.19390/) (2025)
- [Multi-scale clustering and source separation of InSight mission seismic data](https://mlsanity.com/paper/2305.16189/) (2023)
- [Hierarchic Flows to Estimate and Sample High-dimensional Probabilities](https://mlsanity.com/paper/2405.03468/) (2024)
- [Separation Capacity of Scattering Networks](https://mlsanity.com/paper/2606.30822/) (2026)
- [Mean-Field Microcanonical Gradient Descent](https://mlsanity.com/paper/2403.08362/) (2024)

## Abstract (arXiv)

We introduce the wavelet scattering spectra which provide 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 correlation across time and scales of wavelet coefficients and their modulus. This correlation matrix is nearly diagonalized by a second wavelet transform, which defines the scattering spectra. We show that this vector of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. We prove that self-similar processes have scattering spectra which are scale invariant. This property can be tested statistically on a single realization and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering spectra coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.