BIGDATA: Small: DA: Dynamical diffusion map methods for high dimensional data

George Mason University, Fairfax VA

Investigators

Abstract

The planned research aims to radically transform the current state of the art of dimension reduction, through the development of new approaches to interpretation, resolution, and feature extraction for high-dimensional dynamical data. These extensions fit into a larger program of data analysis which seeks to find the intrinsic geometry tailored to each type of data set. The development of diffusion maps first established a new way to recover geometry from generic data sets. Recently, we showed how to recover the intrinsic geometry for observations of a dynamical system, meaning data sets that have the additional structure of a time ordering. In each case the focus on discovery of the intrinsic geometry for the particular structure of the data resulted in significant practical benefits in the form of new algorithms for dimensionality reduction and noise reduction. With this progress as a foundation, the proposal greatly expands the impact of this effort in two important directions: (1) Overcoming critical weaknesses in the current dynamical diffusion map approach, by (a) extensions to allow drift and anisotropy in the Laplace-Beltrami operator represented by the diffusion map, and (b) merging diffusion maps with discrete exterior calculus through the Hodge star operator to handle higher-dimensional dynamics; and (2) development of an automatically-constructed, data-adapted harmonic (or wavelet) basis in order to capture essential features of spatiotemporal data. Our data-adapted construction starts with an a priori spatial structure and then combines this with the data itself to form the data-adapted spatial geometry. We then propose a novel method of using the diffusion geometry, improved with the results of (1), to find symmetries in the adapted geometry that represent intrinsic features of the data. The project involves a series of investigations in the development of computational dynamical systems theory and methods, with the goal of significantly changing the way massive spatiotemporal data sets are analyzed. The algorithms resulting from the study represent a distinctly new form of time-scale and space-scale separation that can break high-dimensional dynamical data into parts adapted to the dynamics. This new approach will handle a wide range of spatiotemporal inputs, such as high-frame-rate videos of physical experiments, spatially and temporally irregular geophysical databases such as oceanographic, weather or climate time series, econometric and logistical databases, and multivariate measurements on complex dynamic networks such as biological connectomes. The data comes from problems spanning diverse areas of sciences and engineering, with special focus on physical and biological systems. These modern high-resolution data sets are particularly vulnerable to the curse of dimensionality, making current parametric statistical techniques impractical due to exponential increases in model complexity and data requirements. Our approach implicitly eliminates redundancies and selects features of interest in an automatic data-adapted way, reducing the data requirements for statistically significant analyses to feasible levels. Educational impacts include integration of the research topics into undergraduate and graduate teaching, and the enhancement of research infrastructure through joint research with collaborators in physics, bioengineering, biology, and medicine.