63,620 research outputs found
A General Framework of Dual Certificate Analysis for Structured Sparse Recovery Problems
This paper develops a general theoretical framework to analyze structured
sparse recovery problems using the notation of dual certificate. Although
certain aspects of the dual certificate idea have already been used in some
previous work, due to the lack of a general and coherent theory, the analysis
has so far only been carried out in limited scopes for specific problems. In
this context the current paper makes two contributions. First, we introduce a
general definition of dual certificate, which we then use to develop a unified
theory of sparse recovery analysis for convex programming. Second, we present a
class of structured sparsity regularization called structured Lasso for which
calculations can be readily performed under our theoretical framework. This new
theory includes many seemingly loosely related previous work as special cases;
it also implies new results that improve existing ones even for standard
formulations such as L1 regularization
General empirical Bayes wavelet methods and exactly adaptive minimax estimation
In many statistical problems, stochastic signals can be represented as a
sequence of noisy wavelet coefficients. In this paper, we develop general
empirical Bayes methods for the estimation of true signal. Our estimators
approximate certain oracle separable rules and achieve adaptation to ideal
risks and exact minimax risks in broad collections of classes of signals. In
particular, our estimators are uniformly adaptive to the minimum risk of
separable estimators and the exact minimax risks simultaneously in Besov balls
of all smoothness and shape indices, and they are uniformly superefficient in
convergence rates in all compact sets in Besov spaces with a finite secondary
shape parameter. Furthermore, in classes nested between Besov balls of the same
smoothness index, our estimators dominate threshold and James-Stein estimators
within an infinitesimal fraction of the minimax risks. More general block
empirical Bayes estimators are developed. Both white noise with drift and
nonparametric regression are considered.Comment: Published at http://dx.doi.org/10.1214/009053604000000995 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Estimation of sums of random variables: Examples and information bounds
This paper concerns the estimation of sums of functions of observable and
unobservable variables. Lower bounds for the asymptotic variance and a
convolution theorem are derived in general finite- and infinite-dimensional
models. An explicit relationship is established between efficient influence
functions for the estimation of sums of variables and the estimation of their
means. Certain ``plug-in'' estimators are proved to be asymptotically efficient
in finite-dimensional models, while ``'' estimators of Robbins are proved
to be efficient in infinite-dimensional mixture models. Examples include
certain species, network and data confidentiality problems.Comment: Published at http://dx.doi.org/10.1214/009053605000000390 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Discussion: One-step sparse estimates in nonconcave penalized likelihood models
Discussion of ``One-step sparse estimates in nonconcave penalized likelihood
models'' [arXiv:0808.1012]Comment: Published in at http://dx.doi.org/10.1214/07-AOS0316C the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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