Discovering multiscale and self-similar structure with data-driven wavelets

TL;DR

This paper introduces a data-driven wavelet decomposition method that uncovers multiscale, self-similar structures in complex datasets, providing insights into hierarchical features across various scientific fields.

Contribution

The paper presents a novel analysis technique that automatically discovers localized hierarchical structures in data, applicable to diverse multiscale systems like turbulence and biological tissues.

Findings

Reveals self-similar structures in turbulence across multiple scales

Provides a model-free method for hierarchical feature extraction

Reflects inherent dataset structures regardless of complexity

Abstract

Many materials, processes, and structures in science and engineering have important features at multiple scales of time and/or space; examples include biological tissues, active matter, oceans, networks, and images. Explicitly extracting, describing, and defining such features are difficult tasks, at least in part because each system has a unique set of features. Here, we introduce an analysis method that, given a set of observations, discovers an energetic hierarchy of structures localized in scale and space. We call the resulting basis vectors a "data-driven wavelet decomposition". We show that this decomposition reflects the inherent structure of the dataset it acts on, whether it has no structure, structure dominated by a single scale, or structure on a hierarchy of scales. In particular, when applied to turbulence---a high-dimensional, nonlinear, multiscale process---the method reveals self-similar structure over a wide range of spatial scales, providing direct, model-free evidence for a century-old phenomenological picture of turbulence. This approach is a starting point for the characterization of localized hierarchical structures in multiscale systems, which we may think of as the building blocks of these systems.