English
资源说明：Compressed sensing has a wide range of applications that include error correction,
imaging, radar and many more. Given a sparse signal in a high dimensional
space, one wishes to reconstruct that signal accurately and efficiently from a number
of linear measurements much less than its actual dimension. Although in theory it
is clear that this is possible, the difficulty lies in the construction of algorithms that
perform the recovery efficiently, as well as determining which kind of linear measurements
allow for the reconstruction. There have been two distinct major approaches
to sparse recovery that each present different benefits and shortcomings. The first,
ℓ1-minimization methods such as Basis Pursuit, use a linear optimization problem to
recover the signal. This method provides strong guarantees and stability, but relies on
Linear Programming, whose methods do not yet have strong polynomially bounded
runtimes. The second approach uses greedy methods that compute the support of
the signal iteratively. These methods are usually much faster than Basis Pursuit, but
until recently had not been able to provide the same guarantees. This gap between
the two approaches was bridged when we developed and analyzed the greedy algorithm
Regularized Orthogonal Matching Pursuit (ROMP). ROMP provides similar
guarantees to Basis Pursuit as well as the speed of a greedy algorithm. Our more
recent algorithm Compressive Sampling Matching Pursuit (CoSaMP) improves upon
these guarantees, and is optimal in every important aspect. Recent work has also
been done on a reweighted version of the ℓ1-minimization method that improves upon
the original version in the recovery error and measurement requirements. These algorithms
are discussed in detail, as well as previous work that serves as a foundation
for sparse signal recovery.
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