A Sublinear Algorithm for Sparse Reconstruction with l2/l2 Recovery Guarantees

Compressed Sensing aims to capture attributes of a sparse signal using very few measurements. Cand`{e}s and Tao showed that sparse reconstruction is possible if the sensing matrix acts as a near isometry on all $boldsymbol{k}$-sparse signals. This property holds with overwhelming probability if the entries of the matrix are generated by an iid Gaussian or Bernoulli process. There has been significant recent interest in an alternative signal processing framework; exploiting deterministic sensing matrices that with overwhelming probability act as a near isometry on $boldsymbol{k}$-sparse vectors with uniformly random support, a geometric condition that is called the Statistical Restricted Isometry Property or StRIP. This paper considers a family of deterministic sensing matrices satisfying the StRIP that are based on srm codes (binary chirps) and a $boldsymbol{k}$-sparse reconstruction algorithm with sublinear complexity. In the presence of stochastic noise in the data domain, this paper derives bounds on the $boldsymbol{ell_2}$ accuracy of approximation in terms of the $boldsymbol{ell_2}$ norm of the measurement noise and the accuracy of the best $boldsymbol{k}$-sparse approximation, also measured in the $boldsymbol{ell_2}$ norm. This type of $boldsymbol{ell_2 /ell_2}$ bound is tighter than the standard $boldsymbol{ell_2 /ell_1}$ or $boldsymbol{ell_1/ ell_1}$ bounds.

Efficient and Robust Compressed Sensing using High-Quality Expander Graphs

Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any $n$-dimensional vector that is $k$-sparse (with $kll n$) can be fully recovered using $O(klogfrac{n}{k})$ measurements and only $O(klog n)$ simple recovery iterations. In this paper we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only $O(k)$ recovery iterations are required, which is a significant improvement when $n$ is large. In fact, full recovery can be accomplished by at most $2k$ very simple iterations. The number of iterations can be made arbitrarily close to $k$, and the recovery algorithm can be implemented very efficiently using a simple binary search tree. We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the recovery time complexity. Finally we will show how our analysis extends to give a robust algorithm that finds the position and sign of the $k$ significant elements of an almost $k$-sparse signal and then, using very simple optimization techniques, finds in sublinear time a $k$-sparse signal which approximates the original signal with very high precision.