409 research outputs found
The Restricted Isometry Property of Subsampled Fourier Matrices
A matrix satisfies the restricted isometry
property of order with constant if it preserves the
norm of all -sparse vectors up to a factor of . We prove
that a matrix obtained by randomly sampling rows from an Fourier matrix satisfies the restricted
isometry property of order with a fixed with high
probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math.,
2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).Comment: 16 page
An Improved Lower Bound for Sparse Reconstruction from Subsampled Hadamard Matrices
We give a short argument that yields a new lower bound on the number of
subsampled rows from a bounded, orthonormal matrix necessary to form a matrix
with the restricted isometry property. We show that a matrix formed by
uniformly subsampling rows of an Hadamard matrix contains a
-sparse vector in the kernel, unless the number of subsampled rows is
--- our lower bound applies whenever . Containing a sparse vector in the kernel precludes not only
the restricted isometry property, but more generally the application of those
matrices for uniform sparse recovery.Comment: Improved exposition and added an autho
Stable image reconstruction using total variation minimization
This article presents near-optimal guarantees for accurate and robust image
recovery from under-sampled noisy measurements using total variation
minimization. In particular, we show that from O(slog(N)) nonadaptive linear
measurements, an image can be reconstructed to within the best s-term
approximation of its gradient up to a logarithmic factor, and this factor can
be removed by taking slightly more measurements. Along the way, we prove a
strengthened Sobolev inequality for functions lying in the null space of
suitably incoherent matrices.Comment: 25 page
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
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