8,378 research outputs found
Sparse Recovery with Very Sparse Compressed Counting
Compressed sensing (sparse signal recovery) often encounters nonnegative data
(e.g., images). Recently we developed the methodology of using (dense)
Compressed Counting for recovering nonnegative K-sparse signals. In this paper,
we adopt very sparse Compressed Counting for nonnegative signal recovery. Our
design matrix is sampled from a maximally-skewed p-stable distribution (0<p<1),
and we sparsify the design matrix so that on average (1-g)-fraction of the
entries become zero. The idea is related to very sparse stable random
projections (Li et al 2006 and Li 2007), the prior work for estimating summary
statistics of the data.
In our theoretical analysis, we show that, when p->0, it suffices to use M=
K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered in
one scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N.
If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16K
log N. This means the design matrix can be indeed very sparse at only a minor
inflation of the sample complexity.
Interestingly, as p->1, the required number of measurements is essentially M
= 2.7K log N, provided g= 1/K. It turns out that this result is a general
worst-case bound
Efficient and Robust Compressed Sensing Using Optimized 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 can be fully recovered using O(klog n) 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 reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(nlog(n/k))). We also show that by tolerating a small penal- ty 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 time complexity and the simplicity of recovery. 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 a k-sparse signal which is close to the best k-term approximation of the original signal
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
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