Adaptive Sparsification Mechanisms in Signal Recovery

This thesis considers two different adaptive recovery concepts. The first concept is concerned with proximity operators intertwined between an injective linear operator with bounded range and its pseudoinverse. A prominent example in frame analysis is the proximity operator of the l1-norm, which coincides with the soft shrinkage operator and hence can be employed for sparsification. However, this will only lead to sparsity with respect to the canonical basis which in many applications is not helpful. Instead, one rather would like to have sparsity with respect to a suitable encoding, e.g. a frame. This motivates to nest the proximity operator into a frame encoding and decoding step, respectively. This immediately leads to the question whether this expression is a proximity operator itself. We can indeed show this property for arbitrary Hilbert spaces and injective operators with closed range. The second concept studied in this thesis concerns doubly sparse recovery from potentially noisy bilinear measurements. Through lifting, this can be modelled as a linear problem of the outer product of the input signals. Crucially, we will consider operators that satisfy a suitable restricted isometry property (RIP). Recovering a column- and row-sparse rank-one matrix from its RIP measurement is still a very hard problem due to the competing objectives. We follow the approach of Lee et al., which is able to guarantee successful recovery for signals with stiff bounds for the quotient of the l2- and maximum norm. This does however imply that more than half of the mass already has to be contained in the largest entry of each vector, rendering applicability impossible for many settings in practice. We solve this problem by providing a tradeoff between the peak to average power ratio and the number of required measurements.