11,156 research outputs found
Measure What Should be Measured: Progress and Challenges in Compressive Sensing
Is compressive sensing overrated? Or can it live up to our expectations? What
will come after compressive sensing and sparsity? And what has Galileo Galilei
got to do with it? Compressive sensing has taken the signal processing
community by storm. A large corpus of research devoted to the theory and
numerics of compressive sensing has been published in the last few years.
Moreover, compressive sensing has inspired and initiated intriguing new
research directions, such as matrix completion. Potential new applications
emerge at a dazzling rate. Yet some important theoretical questions remain
open, and seemingly obvious applications keep escaping the grip of compressive
sensing. In this paper I discuss some of the recent progress in compressive
sensing and point out key challenges and opportunities as the area of
compressive sensing and sparse representations keeps evolving. I also attempt
to assess the long-term impact of compressive sensing
Nonlinear Compressive Particle Filtering
Many systems for which compressive sensing is used today are dynamical. The
common approach is to neglect the dynamics and see the problem as a sequence of
independent problems. This approach has two disadvantages. Firstly, the
temporal dependency in the state could be used to improve the accuracy of the
state estimates. Secondly, having an estimate for the state and its support
could be used to reduce the computational load of the subsequent step. In the
linear Gaussian setting, compressive sensing was recently combined with the
Kalman filter to mitigate above disadvantages. In the nonlinear dynamical case,
compressive sensing can not be used and, if the state dimension is high, the
particle filter would perform poorly. In this paper we combine one of the most
novel developments in compressive sensing, nonlinear compressive sensing, with
the particle filter. We show that the marriage of the two is essential and that
neither the particle filter or nonlinear compressive sensing alone gives a
satisfying solution.Comment: Accepted to CDC 201
Learning Model-Based Sparsity via Projected Gradient Descent
Several convex formulation methods have been proposed previously for
statistical estimation with structured sparsity as the prior. These methods
often require a carefully tuned regularization parameter, often a cumbersome or
heuristic exercise. Furthermore, the estimate that these methods produce might
not belong to the desired sparsity model, albeit accurately approximating the
true parameter. Therefore, greedy-type algorithms could often be more desirable
in estimating structured-sparse parameters. So far, these greedy methods have
mostly focused on linear statistical models. In this paper we study the
projected gradient descent with non-convex structured-sparse parameter model as
the constraint set. Should the cost function have a Stable Model-Restricted
Hessian the algorithm produces an approximation for the desired minimizer. As
an example we elaborate on application of the main results to estimation in
Generalized Linear Model
Nonlinear Basis Pursuit
In compressive sensing, the basis pursuit algorithm aims to find the sparsest
solution to an underdetermined linear equation system. In this paper, we
generalize basis pursuit to finding the sparsest solution to higher order
nonlinear systems of equations, called nonlinear basis pursuit. In contrast to
the existing nonlinear compressive sensing methods, the new algorithm that
solves the nonlinear basis pursuit problem is convex and not greedy. The novel
algorithm enables the compressive sensing approach to be used for a broader
range of applications where there are nonlinear relationships between the
measurements and the unknowns
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
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