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

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.