Improved Sparsity Thresholds Through Dictionary Splitting

Known sparsity thresholds for basis pursuit to deliver the maximally sparse solution of the compressed sensing recovery problem typically depend on the dictionary's coherence. While the coherence is easy to compute, it can lead to rather pessimistic thresholds as it captures only limited information about the dictionary. In this paper, we show that viewing the dictionary as the concatenation of two general sub-dictionaries leads to provably better sparsity thresholds--that are explicit in the coherence parameters of the dictionary and of the individual sub-dictionaries. Equivalently, our results can be interpreted as sparsity thresholds for dictionaries that are unions of two general (i.e., not necessarily orthonormal) sub-dictionaries.Comment: IEEE Information Theory Workshop (ITW), Taormina, Italy, Oct. 2009, to appea

Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets

We present an uncertainty relation for the representation of signals in two different general (possibly redundant or incomplete) signal sets. This uncertainty relation is relevant for the analysis of signals containing two distinct features each of which can be described sparsely in a suitable general signal set. Furthermore, the new uncertainty relation is shown to lead to improved sparsity thresholds for recovery of signals that are sparse in general dictionaries. Specifically, our results improve on the well-known $(1+1/d)/2$-threshold for dictionaries with coherence $d$ by up to a factor of two. Furthermore, we provide probabilistic recovery guarantees for pairs of general dictionaries that also allow us to understand which parts of a general dictionary one needs to randomize over to "weed out" the sparsity patterns that prohibit breaking the square-root bottleneck.Comment: submitted to IEEE Trans. Inf. Theor

Compressed Sensing of Block-Sparse Signals: Uncertainty Relations and Efficient Recovery

We consider compressed sensing of block-sparse signals, i.e., sparse signals that have nonzero coefficients occurring in clusters. An uncertainty relation for block-sparse signals is derived, based on a block-coherence measure, which we introduce. We then show that a block-version of the orthogonal matching pursuit algorithm recovers block $k$-sparse signals in no more than $k$ steps if the block-coherence is sufficiently small. The same condition on block-coherence is shown to guarantee successful recovery through a mixed $\ell_2/\ell_1$-optimization approach. This complements previous recovery results for the block-sparse case which relied on small block-restricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of block-sparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.Comment: Submitted to the IEEE Trans. on Signal Processing, version 2 has updated figure