Average case analysis of sparse recovery from combined fusion frame measurements

Sparse representations have emerged as a powerful tool in signal and information processing, culminated by the success of new acquisition and processing techniques such as Compressed Sensing (CS). Fusion frames are very rich new signal representation methods that use collections of subspaces instead of vectors to represent signals. These exciting fields have been recently combined to introduce a new sparsity model for fusion frames. Signals that are sparse under the new model can be compressively sampled and uniquely reconstructed in ways similar to sparse signals using standard CS. The combination provides a promising new set of mathematical tools and signal models useful in a variety of applications. With the new model, a sparse signal has energy in very few of the subspaces of the fusion frame, although it does not need to be sparse within each of the subspaces it occupies. In this paper we demonstrate that although a worst-case analysis of recovery under the new model can often be quite pessimistic, an average case analysis is not and provides significantly more insight. Using a probability model on the sparse signal we show that under very mild conditions the probability of recovery failure decays exponentially with increasing dimension of the subspaces.