498 research outputs found
Successive Concave Sparsity Approximation for Compressed Sensing
In this paper, based on a successively accuracy-increasing approximation of
the norm, we propose a new algorithm for recovery of sparse vectors
from underdetermined measurements. The approximations are realized with a
certain class of concave functions that aggressively induce sparsity and their
closeness to the norm can be controlled. We prove that the series of
the approximations asymptotically coincides with the and
norms when the approximation accuracy changes from the worst fitting to the
best fitting. When measurements are noise-free, an optimization scheme is
proposed which leads to a number of weighted minimization programs,
whereas, in the presence of noise, we propose two iterative thresholding
methods that are computationally appealing. A convergence guarantee for the
iterative thresholding method is provided, and, for a particular function in
the class of the approximating functions, we derive the closed-form
thresholding operator. We further present some theoretical analyses via the
restricted isometry, null space, and spherical section properties. Our
extensive numerical simulations indicate that the proposed algorithm closely
follows the performance of the oracle estimator for a range of sparsity levels
wider than those of the state-of-the-art algorithms.Comment: Submitted to IEEE Trans. on Signal Processin
Comparison of several reweighted l1-algorithms for solving cardinality minimization problems
Reweighted l1-algorithms have attracted a lot of attention in the field of
applied mathematics. A unified framework of such algorithms has been recently
proposed by Zhao and Li. In this paper we construct a few new examples of
reweighted l1-methods. These functions are certain concave approximations of
the l0-norm function. We focus on the numerical comparison between some new and
existing reweighted l1-algorithms. We show how the change of parameters in
reweighted algorithms may affect the performance of the algorithms for finding
the solution of the cardinality minimization problem. In our experiments, the
problem data were generated according to different statistical distributions,
and we test the algorithms on different sparsity level of the solution of the
problem. Our numerical results demonstrate that the reweighted l1-method is one
of the efficient methods for locating the solution of the cardinality
minimization problem
Minimization of multi-penalty functionals by alternating iterative thresholding and optimal parameter choices
Inspired by several recent developments in regularization theory,
optimization, and signal processing, we present and analyze a numerical
approach to multi-penalty regularization in spaces of sparsely represented
functions. The sparsity prior is motivated by the largely expected
geometrical/structured features of high-dimensional data, which may not be
well-represented in the framework of typically more isotropic Hilbert spaces.
In this paper, we are particularly interested in regularizers which are able to
correctly model and separate the multiple components of additively mixed
signals. This situation is rather common as pure signals may be corrupted by
additive noise. To this end, we consider a regularization functional composed
by a data-fidelity term, where signal and noise are additively mixed, a
non-smooth and non-convex sparsity promoting term, and a penalty term to model
the noise. We propose and analyze the convergence of an iterative alternating
algorithm based on simple iterative thresholding steps to perform the
minimization of the functional. By means of this algorithm, we explore the
effect of choosing different regularization parameters and penalization norms
in terms of the quality of recovering the pure signal and separating it from
additive noise. For a given fixed noise level numerical experiments confirm a
significant improvement in performance compared to standard one-parameter
regularization methods. By using high-dimensional data analysis methods such as
Principal Component Analysis, we are able to show the correct geometrical
clustering of regularized solutions around the expected solution. Eventually,
for the compressive sensing problems considered in our experiments we provide a
guideline for a choice of regularization norms and parameters.Comment: 32 page
Inference for Generalized Linear Models via Alternating Directions and Bethe Free Energy Minimization
Generalized Linear Models (GLMs), where a random vector is
observed through a noisy, possibly nonlinear, function of a linear transform
arise in a range of applications in nonlinear
filtering and regression. Approximate Message Passing (AMP) methods, based on
loopy belief propagation, are a promising class of approaches for approximate
inference in these models. AMP methods are computationally simple, general, and
admit precise analyses with testable conditions for optimality for large i.i.d.
transforms . However, the algorithms can easily diverge for general
. This paper presents a convergent approach to the generalized AMP
(GAMP) algorithm based on direct minimization of a large-system limit
approximation of the Bethe Free Energy (LSL-BFE). The proposed method uses a
double-loop procedure, where the outer loop successively linearizes the LSL-BFE
and the inner loop minimizes the linearized LSL-BFE using the Alternating
Direction Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP,
is similar in structure to the original GAMP method, but with an additional
least-squares minimization. It is shown that for strictly convex, smooth
penalties, ADMM-GAMP is guaranteed to converge to a local minima of the
LSL-BFE, thus providing a convergent alternative to GAMP that is stable under
arbitrary transforms. Simulations are also presented that demonstrate the
robustness of the method for non-convex penalties as well
On Known-Plaintext Attacks to a Compressed Sensing-based Encryption: A Quantitative Analysis
Despite the linearity of its encoding, compressed sensing may be used to
provide a limited form of data protection when random encoding matrices are
used to produce sets of low-dimensional measurements (ciphertexts). In this
paper we quantify by theoretical means the resistance of the least complex form
of this kind of encoding against known-plaintext attacks. For both standard
compressed sensing with antipodal random matrices and recent multiclass
encryption schemes based on it, we show how the number of candidate encoding
matrices that match a typical plaintext-ciphertext pair is so large that the
search for the true encoding matrix inconclusive. Such results on the practical
ineffectiveness of known-plaintext attacks underlie the fact that even
closely-related signal recovery under encoding matrix uncertainty is doomed to
fail.
Practical attacks are then exemplified by applying compressed sensing with
antipodal random matrices as a multiclass encryption scheme to signals such as
images and electrocardiographic tracks, showing that the extracted information
on the true encoding matrix from a plaintext-ciphertext pair leads to no
significant signal recovery quality increase. This theoretical and empirical
evidence clarifies that, although not perfectly secure, both standard
compressed sensing and multiclass encryption schemes feature a noteworthy level
of security against known-plaintext attacks, therefore increasing its appeal as
a negligible-cost encryption method for resource-limited sensing applications.Comment: IEEE Transactions on Information Forensics and Security, accepted for
publication. Article in pres
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