72 research outputs found
Gaussian Mixtures Based IRLS for Sparse Recovery With Quadratic Convergence
In this paper, we propose a new class of iteratively
re-weighted least squares (IRLS) for sparse recovery problems.
The proposed methods are inspired by constrained maximum-likelihood
estimation under a Gaussian scale mixture (GSM) distribution
assumption. In the noise-free setting, we provide sufficient
conditions ensuring the convergence of the sequences generated by
these algorithms to the set of fixed points of the maps that rule their
dynamics and derive conditions verifiable a posteriori for the convergence
to a sparse solution. We further prove that these algorithms
are quadratically fast in a neighborhood of a sparse solution.
We show through numerical experiments that the proposed
methods outperform classical IRLS for l_p-minimization with p\in(0,1]
in terms of speed and of sparsity-undersampling tradeoff
and are robust even in presence of noise. The simplicity and the
theoretical guarantees provided in this paper make this class of algorithms
an attractive solution for sparse recovery problems
Quadratically fast IRLS for sparse signal recovery
We present a new class of iterative algorithms for sparse recovery problems that combine iterative support detection and estimation. More precisely, these methods use a two state Gaussian scale mixture as a proxy for the signal model and can be interpreted both as iteratively reweighted least squares (IRLS) and Expectation/Maximization (EM) algorithms for the constrained maximization of the log-likelihood function. Under certain conditions, these methods are proved to converge to a sparse solution and to be quadratically fast in a neighborhood of that sparse solution, outperforming classical IRLS for lp-minimization. Numerical experiments validate the theoretical derivations and show that these new reconstruction schemes outperform classical IRLS for lp-minimization with p\in(0,1] in terms of rate of convergence and sparsity-undersampling tradeoff
Distributed estimation from relative measurements of heterogeneous and uncertain quality
This paper studies the problem of estimation from relative measurements in a
graph, in which a vector indexed over the nodes has to be reconstructed from
pairwise measurements of differences between its components associated to nodes
connected by an edge. In order to model heterogeneity and uncertainty of the
measurements, we assume them to be affected by additive noise distributed
according to a Gaussian mixture. In this original setup, we formulate the
problem of computing the Maximum-Likelihood (ML) estimates and we design two
novel algorithms, based on Least Squares regression and
Expectation-Maximization (EM). The first algorithm (LS- EM) is centralized and
performs the estimation from relative measurements, the soft classification of
the measurements, and the estimation of the noise parameters. The second
algorithm (Distributed LS-EM) is distributed and performs estimation and soft
classification of the measurements, but requires the knowledge of the noise
parameters. We provide rigorous proofs of convergence of both algorithms and we
present numerical experiments to evaluate and compare their performance with
classical solutions. The experiments show the robustness of the proposed
methods against different kinds of noise and, for the Distributed LS-EM,
against errors in the knowledge of noise parameters.Comment: Submitted to IEEE transaction
Robust computation of linear models by convex relaxation
Consider a dataset of vector-valued observations that consists of noisy
inliers, which are explained well by a low-dimensional subspace, along with
some number of outliers. This work describes a convex optimization problem,
called REAPER, that can reliably fit a low-dimensional model to this type of
data. This approach parameterizes linear subspaces using orthogonal projectors,
and it uses a relaxation of the set of orthogonal projectors to reach the
convex formulation. The paper provides an efficient algorithm for solving the
REAPER problem, and it documents numerical experiments which confirm that
REAPER can dependably find linear structure in synthetic and natural data. In
addition, when the inliers lie near a low-dimensional subspace, there is a
rigorous theory that describes when REAPER can approximate this subspace.Comment: Formerly titled "Robust computation of linear models, or How to find
a needle in a haystack
Large Scale Variational Inference and Experimental Design for Sparse Generalized Linear Models
Many problems of low-level computer vision and image processing, such as
denoising, deconvolution, tomographic reconstruction or super-resolution, can
be addressed by maximizing the posterior distribution of a sparse linear model
(SLM). We show how higher-order Bayesian decision-making problems, such as
optimizing image acquisition in magnetic resonance scanners, can be addressed
by querying the SLM posterior covariance, unrelated to the density's mode. We
propose a scalable algorithmic framework, with which SLM posteriors over full,
high-resolution images can be approximated for the first time, solving a
variational optimization problem which is convex iff posterior mode finding is
convex. These methods successfully drive the optimization of sampling
trajectories for real-world magnetic resonance imaging through Bayesian
experimental design, which has not been attempted before. Our methodology
provides new insight into similarities and differences between sparse
reconstruction and approximate Bayesian inference, and has important
implications for compressive sensing of real-world images.Comment: 34 pages, 6 figures, technical report (submitted
Applying Compactness Constraints to Differential Traveltime Tomography
Tomographic imaging problems are typically ill-posed and often require the use of regularization techniques
to guarantee a stable solution. Minimization of a weighted norm of model length is one commonly
used secondary constraint. Tikhonov methods exploit low-order differential operators to select for solutions
that are small, flat, or smooth in one or more dimensions. This class of regularizing functionals
may not always be appropriate, particularly in cases where the anomaly being imaged is generated by
a non-smooth spatial process. Timelapse imaging of flow-induced velocity anomalies is one such case;
flow features are often characterized by spatial compactness or connectivity. By performing inversions
on differenced arrival time data, the properties of the timelapse feature can be directly constrained. We
develop a differential traveltime tomography algorithm which selects for compact solutions i.e. models
with a minimum area of support, through application of model-space iteratively reweighted least squares.
Our technique is an adaptation of minimum support regularization methods previously explored within
the potential theory community. We compare our inversion algorithm to the results obtained by traditional
Tikhonov regularization for two simple synthetic models; one including several sharp localized
anomalies and a second with smoother features. We use a more complicated synthetic test case based on
multiphase flow results to illustrate the efficacy of compactness constraints for contaminant infiltration
imaging. We conclude by applying the algorithm to a CO[subscript 2] sequestration monitoring dataset acquired
at the Frio pilot site. We observe that in cases where the assumption of a localized anomaly is correct,
the addition of compactness constraints improves image quality by reducing tomographic artifacts and
spatial smearing of target features.Massachusetts Institute of Technology. Earth Resources Laborator
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