477 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
Very high quality image restoration by combining wavelets and curvelets
We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We develop a methodology to combine wavelets together these new systems to perform noise removal by exploiting all these systems simultaneously. The results of the combined reconstruction exhibits clear advantages over any individual system alone. For example, the residual error contains essentially no visually intelligible structure: no structure is lost in the reconstruction
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
Two-Photon Spiral Imaging with Correlated Orbital Angular Momentum States
The concept of correlated two-photon spiral imaging is introduced. We begin
by analyzing the joint orbital angular momentum (OAM) spectrum of correlated
photon pairs. The mutual information carried by the photon pairs is evaluated,
and it is shown that when an object is placed in one of the beam paths the
value of the mutual information is strongly dependent on object shape and is
closely related to the degree of rotational symmetry present. After analyzing
the effect of the object on the OAM correlations, the method of correlated
spiral imaging is described. We first present a version using parametric
downconversion, in which entangled pairs of photons with opposite OAM values
are produced, placing an object in the path of one beam. We then present a
classical (correlated, but non-entangled) version. The relative problems and
benefits of the classical versus entangled configurations are discussed. The
prospect is raised of carrying out compressive imaging via twophoton OAM
detection to reconstruct sparse objects with few measurements
lp-Recovery of the Most Significant Subspace among Multiple Subspaces with Outliers
We assume data sampled from a mixture of d-dimensional linear subspaces with
spherically symmetric distributions within each subspace and an additional
outlier component with spherically symmetric distribution within the ambient
space (for simplicity we may assume that all distributions are uniform on their
corresponding unit spheres). We also assume mixture weights for the different
components. We say that one of the underlying subspaces of the model is most
significant if its mixture weight is higher than the sum of the mixture weights
of all other subspaces. We study the recovery of the most significant subspace
by minimizing the lp-averaged distances of data points from d-dimensional
subspaces, where p>0. Unlike other lp minimization problems, this minimization
is non-convex for all p>0 and thus requires different methods for its analysis.
We show that if 0<p<=1, then for any fraction of outliers the most significant
subspace can be recovered by lp minimization with overwhelming probability
(which depends on the generating distribution and its parameters). We show that
when adding small noise around the underlying subspaces the most significant
subspace can be nearly recovered by lp minimization for any 0<p<=1 with an
error proportional to the noise level. On the other hand, if p>1 and there is
more than one underlying subspace, then with overwhelming probability the most
significant subspace cannot be recovered or nearly recovered. This last result
does not require spherically symmetric outliers.Comment: This is a revised version of the part of 1002.1994 that deals with
single subspace recovery. V3: Improved estimates (in particular for Lemma 3.1
and for estimates relying on it), asymptotic dependence of probabilities and
constants on D and d and further clarifications; for simplicity it assumes
uniform distributions on spheres. V4: minor revision for the published
versio
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
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