2016
DOI: 10.1016/j.acha.2016.01.002
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Sparse recovery via differential inclusions

Abstract: In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, i.e. finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics Bregman ISS and Linearized Bregman ISS. A well-known shortcoming of LASSO and any convex regularization ap… Show more

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Cited by 48 publications
(65 citation statements)
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“…Then there exists an index j ∈ I τ and σ j ∈ {±1}, as well as a sequence β k →β such that sgn(u β k ,α1,j ) = sgn(u β k ,α2,j ) = σ j for all k and some α l ≤ α 1 < α 2 ≤ α u . As a consequence, the KKT conditions (5) imply that…”
Section: Support Tilingmentioning
confidence: 99%
See 1 more Smart Citation
“…Then there exists an index j ∈ I τ and σ j ∈ {±1}, as well as a sequence β k →β such that sgn(u β k ,α1,j ) = sgn(u β k ,α2,j ) = σ j for all k and some α l ≤ α 1 < α 2 ≤ α u . As a consequence, the KKT conditions (5) imply that…”
Section: Support Tilingmentioning
confidence: 99%
“…Indeed, provided with the signal's support, the signal entries can be easily recovered with optimal statistical rate [3]. Therefore, support recovery has been a topic of active and fruitful research in the last years [4,5,6]. One typically considers linear observation model problems of the form (1) Au…”
mentioning
confidence: 99%
“…The dynamics of discrete, iterative numerical algorithms can often be better understood from their continuous time limit, typically described by ordinary differential equations. This perspective has been proven fruitful in the analysis of many deterministic optimization algorithms [30,31,32,33,34]. An analogy of this type of continuous-timelimit analysis for SGD algorithms is provided by the diffusion approximation [18,20]: in any finite time interval, the distribution of X n defined by the SGD dynamics (4) is close to the distribution of the solution of the following SDE at time t = nη:…”
Section: Main Contribution: Long-time Weak Approximation For Sgd Via Sdementioning
confidence: 99%
“…Such a minimizer is unbiased when sign consistency is reached, hence is statistically more accurate than any convex regularized estimator such as LASSO. For more details, we refer the readers to see [33] and references therein.…”
Section: Parsimonious Paths Of Multi-level Models With Linearized Brementioning
confidence: 99%