149 research outputs found
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
Nonuniform Sparse Recovery with Subgaussian Matrices
Compressive sensing predicts that sufficiently sparse vectors can be
recovered from highly incomplete information. Efficient recovery methods such
as -minimization find the sparsest solution to certain systems of
equations. Random matrices have become a popular choice for the measurement
matrix. Indeed, near-optimal uniform recovery results have been shown for such
matrices. In this note we focus on nonuniform recovery using Gaussian random
matrices and -minimization. We provide a condition on the number of
samples in terms of the sparsity and the signal length which guarantees that a
fixed sparse signal can be recovered with a random draw of the matrix using
-minimization. The constant 2 in the condition is optimal, and the
proof is rather short compared to a similar result due to Donoho and Tanner
Sparsity and Parallel Acquisition: Optimal Uniform and Nonuniform Recovery Guarantees
The problem of multiple sensors simultaneously acquiring measurements of a
single object can be found in many applications. In this paper, we present the
optimal recovery guarantees for the recovery of compressible signals from
multi-sensor measurements using compressed sensing. In the first half of the
paper, we present both uniform and nonuniform recovery guarantees for the
conventional sparse signal model in a so-called distinct sensing scenario. In
the second half, using the so-called sparse and distributed signal model, we
present nonuniform recovery guarantees which effectively broaden the class of
sensing scenarios for which optimal recovery is possible, including to the
so-called identical sampling scenario. To verify our recovery guarantees we
provide several numerical results including phase transition curves and
numerically-computed bounds.Comment: 13 pages and 3 figure
Sampling by blocks of measurements in compressed sensing
Various acquisition devices impose sampling blocks of measurements. A typical example is parallel magnetic resonance imaging (MRI) where several radio-frequency coils simultaneously acquire a set of Fourier modulated coefficients. We study a new random sampling approach that consists in selecting a set of blocks that are predefined by the application of interest. We provide theoretical results on the number of blocks that are required for exact sparse signal reconstruction. We finish by illustrating these results on various examples, and discuss their connection to the literature on CS
Necessary and sufficient conditions of solution uniqueness in minimization
This paper shows that the solutions to various convex minimization
problems are \emph{unique} if and only if a common set of conditions are
satisfied. This result applies broadly to the basis pursuit model, basis
pursuit denoising model, Lasso model, as well as other models that
either minimize or impose the constraint , where
is a strictly convex function. For these models, this paper proves that,
given a solution and defining I=\supp(x^*) and s=\sign(x^*_I),
is the unique solution if and only if has full column rank and there
exists such that and for . This
condition is previously known to be sufficient for the basis pursuit model to
have a unique solution supported on . Indeed, it is also necessary, and
applies to a variety of other models. The paper also discusses ways to
recognize unique solutions and verify the uniqueness conditions numerically.Comment: 6 pages; revised version; submitte
RIPless compressed sensing from anisotropic measurements
Compressed sensing is the art of reconstructing a sparse vector from its
inner products with respect to a small set of randomly chosen measurement
vectors. It is usually assumed that the ensemble of measurement vectors is in
isotropic position in the sense that the associated covariance matrix is
proportional to the identity matrix. In this paper, we establish bounds on the
number of required measurements in the anisotropic case, where the ensemble of
measurement vectors possesses a non-trivial covariance matrix. Essentially, we
find that the required sampling rate grows proportionally to the condition
number of the covariance matrix. In contrast to other recent contributions to
this problem, our arguments do not rely on any restricted isometry properties
(RIP's), but rather on ideas from convex geometry which have been
systematically studied in the theory of low-rank matrix recovery. This allows
for a simple argument and slightly improved bounds, but may lead to a worse
dependency on noise (which we do not consider in the present paper).Comment: 19 pages. To appear in Linear Algebra and its Applications, Special
Issue on Sparse Approximate Solution of Linear System
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