Shining light on data

Experimental sciences have come to depend heavily on our ability to organize, interpret and analyze high-dimensional datasets produced from observations of a large number of variables governed by natural processes. Natural laws, conservation principles, and dynamical structure introduce intricate inter-dependencies among these observed variables, which in turn yield geometric structure, with fewer degrees of freedom, on the dataset. We show how fine-scale features of this structure in data can be extracted from discrete approximations to quantum mechanical processes given by data-driven graph Laplacians and localized wavepackets. This leads to a novel, yet natural uncertainty principle for data analysis induced by limited data. We illustrate some applications to learning with algorithms on several model examples and real-world datasets.