1,365 research outputs found
Compressed sensing reconstruction using Expectation Propagation
Many interesting problems in fields ranging from telecommunications to
computational biology can be formalized in terms of large underdetermined
systems of linear equations with additional constraints or regularizers. One of
the most studied ones, the Compressed Sensing problem (CS), consists in finding
the solution with the smallest number of non-zero components of a given system
of linear equations for known
measurement vector and sensing matrix . Here, we
will address the compressed sensing problem within a Bayesian inference
framework where the sparsity constraint is remapped into a singular prior
distribution (called Spike-and-Slab or Bernoulli-Gauss). Solution to the
problem is attempted through the computation of marginal distributions via
Expectation Propagation (EP), an iterative computational scheme originally
developed in Statistical Physics. We will show that this strategy is
comparatively more accurate than the alternatives in solving instances of CS
generated from statistically correlated measurement matrices. For computational
strategies based on the Bayesian framework such as variants of Belief
Propagation, this is to be expected, as they implicitly rely on the hypothesis
of statistical independence among the entries of the sensing matrix. Perhaps
surprisingly, the method outperforms uniformly also all the other
state-of-the-art methods in our tests.Comment: 20 pages, 6 figure
Robust Linear Regression Analysis - A Greedy Approach
The task of robust linear estimation in the presence of outliers is of
particular importance in signal processing, statistics and machine learning.
Although the problem has been stated a few decades ago and solved using
classical (considered nowadays) methods, recently it has attracted more
attention in the context of sparse modeling, where several notable
contributions have been made. In the present manuscript, a new approach is
considered in the framework of greedy algorithms. The noise is split into two
components: a) the inlier bounded noise and b) the outliers, which are
explicitly modeled by employing sparsity arguments. Based on this scheme, a
novel efficient algorithm (Greedy Algorithm for Robust Denoising - GARD), is
derived. GARD alternates between a least square optimization criterion and an
Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers.
The case where only outliers are present has been studied separately, where
bounds on the \textit{Restricted Isometry Property} guarantee that the recovery
of the signal via GARD is exact. Moreover, theoretical results concerning
convergence as well as the derivation of error bounds in the case of additional
bounded noise are discussed. Finally, we provide extensive simulations, which
demonstrate the comparative advantages of the new technique
Statistical Compressive Sensing of Gaussian Mixture Models
A new framework of compressive sensing (CS), namely statistical compressive
sensing (SCS), that aims at efficiently sampling a collection of signals that
follow a statistical distribution and achieving accurate reconstruction on
average, is introduced. For signals following a Gaussian distribution, with
Gaussian or Bernoulli sensing matrices of O(k) measurements, considerably
smaller than the O(k log(N/k)) required by conventional CS, where N is the
signal dimension, and with an optimal decoder implemented with linear
filtering, significantly faster than the pursuit decoders applied in
conventional CS, the error of SCS is shown tightly upper bounded by a constant
times the k-best term approximation error, with overwhelming probability. The
failure probability is also significantly smaller than that of conventional CS.
Stronger yet simpler results further show that for any sensing matrix, the
error of Gaussian SCS is upper bounded by a constant times the k-best term
approximation with probability one, and the bound constant can be efficiently
calculated. For signals following Gaussian mixture models, SCS with a piecewise
linear decoder is introduced and shown to produce for real images better
results than conventional CS based on sparse models
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