720 research outputs found
Deterministic Constructions of Binary Measurement Matrices from Finite Geometry
Deterministic constructions of measurement matrices in compressed sensing
(CS) are considered in this paper. The constructions are inspired by the recent
discovery of Dimakis, Smarandache and Vontobel which says that parity-check
matrices of good low-density parity-check (LDPC) codes can be used as
{provably} good measurement matrices for compressed sensing under
-minimization. The performance of the proposed binary measurement
matrices is mainly theoretically analyzed with the help of the analyzing
methods and results from (finite geometry) LDPC codes. Particularly, several
lower bounds of the spark (i.e., the smallest number of columns that are
linearly dependent, which totally characterizes the recovery performance of
-minimization) of general binary matrices and finite geometry matrices
are obtained and they improve the previously known results in most cases.
Simulation results show that the proposed matrices perform comparably to,
sometimes even better than, the corresponding Gaussian random matrices.
Moreover, the proposed matrices are sparse, binary, and most of them have
cyclic or quasi-cyclic structure, which will make the hardware realization
convenient and easy.Comment: 12 pages, 11 figure
Worst Configurations (Instantons) for Compressed Sensing over Reals: a Channel Coding Approach
We consider the Linear Programming (LP) solution of the Compressed Sensing
(CS) problem over reals, also known as the Basis Pursuit (BasP) algorithm. The
BasP allows interpretation as a channel-coding problem, and it guarantees
error-free reconstruction with a properly chosen measurement matrix and
sufficiently sparse error vectors. In this manuscript, we examine how the BasP
performs on a given measurement matrix and develop an algorithm to discover the
sparsest vectors for which the BasP fails. The resulting algorithm is a
generalization of our previous results on finding the most probable
error-patterns degrading performance of a finite size Low-Density Parity-Check
(LDPC) code in the error-floor regime. The BasP fails when its output is
different from the actual error-pattern. We design a CS-Instanton Search
Algorithm (ISA) generating a sparse vector, called a CS-instanton, such that
the BasP fails on the CS-instanton, while the BasP recovery is successful for
any modification of the CS-instanton replacing a nonzero element by zero. We
also prove that, given a sufficiently dense random input for the error-vector,
the CS-ISA converges to an instanton in a small finite number of steps. The
performance of the CS-ISA is illustrated on a randomly generated matrix. For this example, the CS-ISA outputs the shortest instanton (error
vector) pattern of length 11.Comment: Accepted to be presented at the IEEE International Symposium on
Information Theory (ISIT 2010). 5 pages, 2 Figures. Minor edits from previous
version. Added a new reference
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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