Optimal sampling and recovery for multilinear signals and systems

Tufts University, Medford MA

Investigators

Abstract

This research involves the development and validation of a novel framework for signals and systems defined in dimensions greater than two. Such multilinear signals can be viewed as high dimensional data tensors, and the associated systems as multidimensional linear operators acting on this class of signals. These mathematical objects are ubiquitous in science and engineering and of growing practical importance. They arise in time varying tomographic problems such as medical imaging, hyperspectral remote sensing, and seismic data acquisition and processing for rapid exploration for oil and gas, to name but a few applications. More recently multilinear signals have become prominent in the context of data analytics for social networks, online big-data management, data visualization, predictive modeling for disease propagation and forecasting of complex events. Effective multilinear computation and processing requires generalization of the traditional 2-D factorization methods to higher dimensions. To date, tensor factorization techniques are either optimal in terms of compactness of data or operator representation but NP-hard to evaluate, or on the other extreme computationally feasible but can be severely non-compact thereby loosing structural information. This adversely affects the efficiency in sampling (tensor data compression) and computational costs in recovery of multilinear data using regularized inversion algorithms. In this context, the investigators develop fundamentally new approaches for tensor decompositions, which, while being computationally feasible, are also provably compact. Specifically, the investigators will develop: Algorithms for handling large scale processing for multilinear signals and systems; Optimal sampling and recovery of multilinear data as non-trivial extension of the theory of compressive sensing; and Tensor representation based, regularized inversion models and algorithms for multilinear inverse problems.