An important problem in statistics, machine learning, and modern signal processing is to recover information of limited complexity, or, more specifically, of low-dimensional structure, from seemingly few data. Often, this amounts to recover a sparse signal, i.e., a signal which is non-zero at few locations only, by solving an under-determined system of linear equations. A by now well known example is compressive sampling [CW08], a signal processing technique for efficiently acquiring and

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... ructing certain signals from far fewer measurements than necessary for reconstruction with traditional methods. Compressive sampling relies on the insight that many real-world signal have a sparse representation in some basis. In this thesis, we will use ideas from sparse signal processing to cluster high dimensional data points, to identify sparse linear operators, and to recover sparse signals with certain block-structure. In the first part of this thesis, we consider the problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers. The number of subspaces, their dimensions, and their orientations are assumed unknown. This problem is known as subspace clustering and has applications in, e.g., unsupervised learning, image processing, disease detection and computer vision [Vid11]. We propose a simple lowcomplexity subspace clustering algorithm, termed thresholding-based subspace clustering (TSC), which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points.