590 research outputs found
On Verifiable Sufficient Conditions for Sparse Signal Recovery via Minimization
We propose novel necessary and sufficient conditions for a sensing matrix to
be "-good" - to allow for exact -recovery of sparse signals with
nonzero entries when no measurement noise is present. Then we express the error
bounds for imperfect -recovery (nonzero measurement noise, nearly
-sparse signal, near-optimal solution of the optimization problem yielding
the -recovery) in terms of the characteristics underlying these
conditions. Further, we demonstrate (and this is the principal result of the
paper) that these characteristics, although difficult to evaluate, lead to
verifiable sufficient conditions for exact sparse -recovery and to
efficiently computable upper bounds on those for which a given sensing
matrix is -good. We establish also instructive links between our approach
and the basic concepts of the Compressed Sensing theory, like Restricted
Isometry or Restricted Eigenvalue properties
A fast and accurate first-order algorithm for compressed sensing
This paper introduces a new, fast and accurate algorithm
for solving problems in the area of compressed sensing,
and more generally, in the area of signal and image reconstruction
from indirect measurements. This algorithm
is inspired by recent progress in the development of novel
first-order methods in convex optimization, most notably
Nesterov’s smoothing technique. In particular, there is a
crucial property thatmakes thesemethods extremely efficient
for solving compressed sensing problems. Numerical
experiments show the promising performance of our
method to solve problems which involve the recovery of
signals spanning a large dynamic range
On the linear independence of spikes and sines
The purpose of this work is to survey what is known about the linear
independence of spikes and sines. The paper provides new results for the case
where the locations of the spikes and the frequencies of the sines are chosen
at random. This problem is equivalent to studying the spectral norm of a random
submatrix drawn from the discrete Fourier transform matrix. The proof involves
depends on an extrapolation argument of Bourgain and Tzafriri.Comment: 16 pages, 4 figures. Revision with new proof of major theorem
Analysis of Basis Pursuit Via Capacity Sets
Finding the sparsest solution for an under-determined linear system
of equations is of interest in many applications. This problem is
known to be NP-hard. Recent work studied conditions on the support size of
that allow its recovery using L1-minimization, via the Basis Pursuit
algorithm. These conditions are often relying on a scalar property of
called the mutual-coherence. In this work we introduce an alternative set of
features of an arbitrarily given , called the "capacity sets". We show how
those could be used to analyze the performance of the basis pursuit, leading to
improved bounds and predictions of performance. Both theoretical and numerical
methods are presented, all using the capacity values, and shown to lead to
improved assessments of the basis pursuit success in finding the sparest
solution of
A Counterexample for the Validity of Using Nuclear Norm as a Convex Surrogate of Rank
Rank minimization has attracted a lot of attention due to its robustness in
data recovery. To overcome the computational difficulty, rank is often replaced
with nuclear norm. For several rank minimization problems, such a replacement
has been theoretically proven to be valid, i.e., the solution to nuclear norm
minimization problem is also the solution to rank minimization problem.
Although it is easy to believe that such a replacement may not always be valid,
no concrete example has ever been found. We argue that such a validity checking
cannot be done by numerical computation and show, by analyzing the noiseless
latent low rank representation (LatLRR) model, that even for very simple rank
minimization problems the validity may still break down. As a by-product, we
find that the solution to the nuclear norm minimization formulation of LatLRR
is non-unique. Hence the results of LatLRR reported in the literature may be
questionable.Comment: accepted by ECML PKDD201
(k,q)-Compressed Sensing for dMRI with Joint Spatial-Angular Sparsity Prior
Advanced diffusion magnetic resonance imaging (dMRI) techniques, like
diffusion spectrum imaging (DSI) and high angular resolution diffusion imaging
(HARDI), remain underutilized compared to diffusion tensor imaging because the
scan times needed to produce accurate estimations of fiber orientation are
significantly longer. To accelerate DSI and HARDI, recent methods from
compressed sensing (CS) exploit a sparse underlying representation of the data
in the spatial and angular domains to undersample in the respective k- and
q-spaces. State-of-the-art frameworks, however, impose sparsity in the spatial
and angular domains separately and involve the sum of the corresponding sparse
regularizers. In contrast, we propose a unified (k,q)-CS formulation which
imposes sparsity jointly in the spatial-angular domain to further increase
sparsity of dMRI signals and reduce the required subsampling rate. To
efficiently solve this large-scale global reconstruction problem, we introduce
a novel adaptation of the FISTA algorithm that exploits dictionary
separability. We show on phantom and real HARDI data that our approach achieves
significantly more accurate signal reconstructions than the state of the art
while sampling only 2-4% of the (k,q)-space, allowing for the potential of new
levels of dMRI acceleration.Comment: To be published in the 2017 Computational Diffusion MRI Workshop of
MICCA
Guaranteed clustering and biclustering via semidefinite programming
Identifying clusters of similar objects in data plays a significant role in a
wide range of applications. As a model problem for clustering, we consider the
densest k-disjoint-clique problem, whose goal is to identify the collection of
k disjoint cliques of a given weighted complete graph maximizing the sum of the
densities of the complete subgraphs induced by these cliques. In this paper, we
establish conditions ensuring exact recovery of the densest k cliques of a
given graph from the optimal solution of a particular semidefinite program. In
particular, the semidefinite relaxation is exact for input graphs corresponding
to data consisting of k large, distinct clusters and a smaller number of
outliers. This approach also yields a semidefinite relaxation for the
biclustering problem with similar recovery guarantees. Given a set of objects
and a set of features exhibited by these objects, biclustering seeks to
simultaneously group the objects and features according to their expression
levels. This problem may be posed as partitioning the nodes of a weighted
bipartite complete graph such that the sum of the densities of the resulting
bipartite complete subgraphs is maximized. As in our analysis of the densest
k-disjoint-clique problem, we show that the correct partition of the objects
and features can be recovered from the optimal solution of a semidefinite
program in the case that the given data consists of several disjoint sets of
objects exhibiting similar features. Empirical evidence from numerical
experiments supporting these theoretical guarantees is also provided
Adaptive Measurement Network for CS Image Reconstruction
Conventional compressive sensing (CS) reconstruction is very slow for its
characteristic of solving an optimization problem. Convolu- tional neural
network can realize fast processing while achieving compa- rable results. While
CS image recovery with high quality not only de- pends on good reconstruction
algorithms, but also good measurements. In this paper, we propose an adaptive
measurement network in which measurement is obtained by learning. The new
network consists of a fully-connected layer and ReconNet. The fully-connected
layer which has low-dimension output acts as measurement. We train the
fully-connected layer and ReconNet simultaneously and obtain adaptive
measurement. Because the adaptive measurement fits dataset better, in contrast
with random Gaussian measurement matrix, under the same measuremen- t rate, it
can extract the information of scene more efficiently and get better
reconstruction results. Experiments show that the new network outperforms the
original one.Comment: 11pages,8figure
Toward a unified theory of sparse dimensionality reduction in Euclidean space
Let be a sparse Johnson-Lindenstrauss
transform [KN14] with non-zeroes per column. For a subset of the unit
sphere, given, we study settings for required to
ensure i.e. so that preserves the norm of every
simultaneously and multiplicatively up to . We
introduce a new complexity parameter, which depends on the geometry of , and
show that it suffices to choose and such that this parameter is small.
Our result is a sparse analog of Gordon's theorem, which was concerned with a
dense having i.i.d. Gaussian entries. We qualitatively unify several
results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and
Fourier-based restricted isometries. Our work also implies new results in using
the sparse Johnson-Lindenstrauss transform in numerical linear algebra,
classical and model-based compressed sensing, manifold learning, and
constrained least squares problems such as the Lasso
Compressed sensing and robust recovery of low rank matrices
In this paper, we focus on compressed sensing and recovery schemes for low-rank matrices, asking under what conditions a low-rank matrix can be sensed and recovered from incomplete, inaccurate, and noisy observations. We consider three schemes, one based on a certain Restricted Isometry Property and two based on directly sensing the row and column space of the matrix. We study their properties in terms of exact recovery in the ideal case, and robustness issues for approximately low-rank matrices and for noisy measurements
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