14,343 research outputs found
Activity Identification and Local Linear Convergence of Forward--Backward-type methods
In this paper, we consider a class of Forward--Backward (FB) splitting
methods that includes several variants (e.g. inertial schemes, FISTA) for
minimizing the sum of two proper convex and lower semi-continuous functions,
one of which has a Lipschitz continuous gradient, and the other is partly
smooth relatively to a smooth active manifold . We propose a
unified framework, under which we show that, this class of FB-type algorithms
(i) correctly identifies the active manifolds in a finite number of iterations
(finite activity identification), and (ii) then enters a local linear
convergence regime, which we characterize precisely in terms of the structure
of the underlying active manifolds. For simpler problems involving polyhedral
functions, we show finite termination. We also establish and explain why FISTA
(with convergent sequences) locally oscillates and can be slower than FB. These
results may have numerous applications including in signal/image processing,
sparse recovery and machine learning. Indeed, the obtained results explain the
typical behaviour that has been observed numerically for many problems in these
fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm
regularization to name only a few.Comment: Full length version of the previous short on
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
On GROUSE and Incremental SVD
GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an incremental
algorithm for identifying a subspace of Rn from a sequence of vectors in this
subspace, where only a subset of components of each vector is revealed at each
iteration. Recent analysis has shown that GROUSE converges locally at an
expected linear rate, under certain assumptions. GROUSE has a similar flavor to
the incremental singular value decomposition algorithm, which updates the SVD
of a matrix following addition of a single column. In this paper, we modify the
incremental SVD approach to handle missing data, and demonstrate that this
modified approach is equivalent to GROUSE, for a certain choice of an
algorithmic parameter
Video Compressive Sensing for Dynamic MRI
We present a video compressive sensing framework, termed kt-CSLDS, to
accelerate the image acquisition process of dynamic magnetic resonance imaging
(MRI). We are inspired by a state-of-the-art model for video compressive
sensing that utilizes a linear dynamical system (LDS) to model the motion
manifold. Given compressive measurements, the state sequence of an LDS can be
first estimated using system identification techniques. We then reconstruct the
observation matrix using a joint structured sparsity assumption. In particular,
we minimize an objective function with a mixture of wavelet sparsity and joint
sparsity within the observation matrix. We derive an efficient convex
optimization algorithm through alternating direction method of multipliers
(ADMM), and provide a theoretical guarantee for global convergence. We
demonstrate the performance of our approach for video compressive sensing, in
terms of reconstruction accuracy. We also investigate the impact of various
sampling strategies. We apply this framework to accelerate the acquisition
process of dynamic MRI and show it achieves the best reconstruction accuracy
with the least computational time compared with existing algorithms in the
literature.Comment: 30 pages, 9 figure
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