69 research outputs found
Subspace Methods for Joint Sparse Recovery
We propose robust and efficient algorithms for the joint sparse recovery
problem in compressed sensing, which simultaneously recover the supports of
jointly sparse signals from their multiple measurement vectors obtained through
a common sensing matrix. In a favorable situation, the unknown matrix, which
consists of the jointly sparse signals, has linearly independent nonzero rows.
In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally
proposed by Schmidt for the direction of arrival problem in sensor array
processing and later proposed and analyzed for joint sparse recovery by Feng
and Bresler, provides a guarantee with the minimum number of measurements. We
focus instead on the unfavorable but practically significant case of
rank-defect or ill-conditioning. This situation arises with limited number of
measurement vectors, or with highly correlated signal components. In this case
MUSIC fails, and in practice none of the existing methods can consistently
approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC),
which improves on MUSIC so that the support is reliably recovered under such
unfavorable conditions. Combined with subspace-based greedy algorithms also
proposed and analyzed in this paper, SA-MUSIC provides a computationally
efficient algorithm with a performance guarantee. The performance guarantees
are given in terms of a version of restricted isometry property. In particular,
we also present a non-asymptotic perturbation analysis of the signal subspace
estimation that has been missing in the previous study of MUSIC.Comment: submitted to IEEE transactions on Information Theory, revised versio
A probabilistic and RIPless theory of compressed sensing
This paper introduces a simple and very general theory of compressive
sensing. In this theory, the sensing mechanism simply selects sensing vectors
independently at random from a probability distribution F; it includes all
models - e.g. Gaussian, frequency measurements - discussed in the literature,
but also provides a framework for new measurement strategies as well. We prove
that if the probability distribution F obeys a simple incoherence property and
an isotropy property, one can faithfully recover approximately sparse signals
from a minimal number of noisy measurements. The novelty is that our recovery
results do not require the restricted isometry property (RIP) - they make use
of a much weaker notion - or a random model for the signal. As an example, the
paper shows that a signal with s nonzero entries can be faithfully recovered
from about s log n Fourier coefficients that are contaminated with noise.Comment: 36 page
Relaxed Recovery Conditions for OMP/OLS by Exploiting both Coherence and Decay
We propose extended coherence-based conditions for exact sparse support
recovery using orthogonal matching pursuit (OMP) and orthogonal least squares
(OLS). Unlike standard uniform guarantees, we embed some information about the
decay of the sparse vector coefficients in our conditions. As a result, the
standard condition (where denotes the mutual coherence and
the sparsity level) can be weakened as soon as the non-zero coefficients
obey some decay, both in the noiseless and the bounded-noise scenarios.
Furthermore, the resulting condition is approaching for strongly
decaying sparse signals. Finally, in the noiseless setting, we prove that the
proposed conditions, in particular the bound , are the tightest
achievable guarantees based on mutual coherence
Sparse Power Factorization: Balancing peakiness and sample complexity
In many applications, one is faced with an inverse problem, where the known
signal depends in a bilinear way on two unknown input vectors. Often at least
one of the input vectors is assumed to be sparse, i.e., to have only few
non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and
Bresler, aims to tackle this problem. They have established recovery guarantees
for a somewhat restrictive class of signals under the assumption that the
measurements are random. We generalize these recovery guarantees to a
significantly enlarged and more realistic signal class at the expense of a
moderately increased number of measurements.Comment: 18 page
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