Compressive sampling based signal separation.

Today's society is characterized by an abundance of data that is generated at an unprecedented velocity. However, much of this data is immediately thrown away by compression or information extraction. In a compressed sensing (CS) setting the inherent sparsity in many datasets is exploited by avoiding the acquisition of superfluous data in the first place. We combine this technique with tensors, or multiway arrays of numerical values, which are higher-order generalizations of vectors and matrices. As the number of entries scales exponentially in the order, tensor problems are often large-scale. We show that the combination of simple, low-rank tensor decompositions with CS effectively alleviates or even breaks the so-called curse of dimensionality.

After discussing the larger data fusion optimization framework for coupled and constrained tensor decompositions, we investigate three categories of CS type algorithms to deal with large-scale problems. First, we look into sample-based algorithms that require only a subset of the tensor entries at a time and discuss (constrained) incomplete tensor techniques and randomized block sampling. Second, we exploit the inherent structure due to, e.g., sparsity or compression, by defining the structured tensor framework, which allows constraints and coupling to be incorporated trivially. Thanks to the new concept of implicit tensorization, deterministic blind source separation techniques become feasible in a large-scale setting. By formulating tensor updating in the framework, we derive new algorithms to track a time-varying decomposition or to update the decomposition when new data arrives at a possibly high rate. Finally, we present a technique to decompose tensors that are given implicitly as the solution of a linear system of equations. We present a single-step approach to compute the solution using algebraic and optimization-based algorithms and derive generic uniqueness conditions.

Numerous applications such as blind separation of convolutive mixtures, classification of hazardous gasses and modeling the melting temperature of alloys involving gigabytes or terabytes of data, are handled throughout this thesis. A final part of this thesis is dedicated to two specific applications. First, we show how tensor models enable the simulation of microstructure evolution allowing the investigation of promising multicomponent alloys using a smooth model of a sparsely sampled tensor. Second, we formulate face recognition as an implicitly given tensor decomposition to improve the accuracy.