1,286 research outputs found
Discussion: The Dantzig selector: Statistical estimation when is much larger than
Discussion of ``The Dantzig selector: Statistical estimation when is much
larger than '' [math/0506081]Comment: Published in at http://dx.doi.org/10.1214/009053607000000442 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Generalized Dantzig Selector: Application to the k-support norm
We propose a Generalized Dantzig Selector (GDS) for linear models, in which
any norm encoding the parameter structure can be leveraged for estimation. We
investigate both computational and statistical aspects of the GDS. Based on
conjugate proximal operator, a flexible inexact ADMM framework is designed for
solving GDS, and non-asymptotic high-probability bounds are established on the
estimation error, which rely on Gaussian width of unit norm ball and suitable
set encompassing estimation error. Further, we consider a non-trivial example
of the GDS using -support norm. We derive an efficient method to compute the
proximal operator for -support norm since existing methods are inapplicable
in this setting. For statistical analysis, we provide upper bounds for the
Gaussian widths needed in the GDS analysis, yielding the first statistical
recovery guarantee for estimation with the -support norm. The experimental
results confirm our theoretical analysis.Comment: Updates to boun
Accuracy guarantees for L1-recovery
We discuss two new methods of recovery of sparse signals from noisy
observation based on - minimization. They are closely related to the
well-known techniques such as Lasso and Dantzig Selector. However, these
estimators come with efficiently verifiable guaranties of performance. By
optimizing these bounds with respect to the method parameters we are able to
construct the estimators which possess better statistical properties than the
commonly used ones. We also show how these techniques allow to provide
efficiently computable accuracy bounds for Lasso and Dantzig Selector. We link
our performance estimations to the well known results of Compressive Sensing
and justify our proposed approach with an oracle inequality which links the
properties of the recovery algorithms and the best estimation performance when
the signal support is known. We demonstrate how the estimates can be computed
using the Non-Euclidean Basis Pursuit algorithm
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
This paper develops a general framework for solving a variety of convex cone
problems that frequently arise in signal processing, machine learning,
statistics, and other fields. The approach works as follows: first, determine a
conic formulation of the problem; second, determine its dual; third, apply
smoothing; and fourth, solve using an optimal first-order method. A merit of
this approach is its flexibility: for example, all compressed sensing problems
can be solved via this approach. These include models with objective
functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or
a combination thereof. In addition, the paper also introduces a number of
technical contributions such as a novel continuation scheme, a novel approach
for controlling the step size, and some new results showing that the smooth and
unsmoothed problems are sometimes formally equivalent. Combined with our
framework, these lead to novel, stable and computationally efficient
algorithms. For instance, our general implementation is competitive with
state-of-the-art methods for solving intensively studied problems such as the
LASSO. Further, numerical experiments show that one can solve the Dantzig
selector problem, for which no efficient large-scale solvers exist, in a few
hundred iterations. Finally, the paper is accompanied with a software release.
This software is not a single, monolithic solver; rather, it is a suite of
programs and routines designed to serve as building blocks for constructing
complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This
version has updated reference
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