211 research outputs found
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
Estimation with Norm Regularization
Analysis of non-asymptotic estimation error and structured statistical
recovery based on norm regularized regression, such as Lasso, needs to consider
four aspects: the norm, the loss function, the design matrix, and the noise
model. This paper presents generalizations of such estimation error analysis on
all four aspects compared to the existing literature. We characterize the
restricted error set where the estimation error vector lies, establish
relations between error sets for the constrained and regularized problems, and
present an estimation error bound applicable to any norm. Precise
characterizations of the bound is presented for isotropic as well as
anisotropic subGaussian design matrices, subGaussian noise models, and convex
loss functions, including least squares and generalized linear models. Generic
chaining and associated results play an important role in the analysis. A key
result from the analysis is that the sample complexity of all such estimators
depends on the Gaussian width of a spherical cap corresponding to the
restricted error set. Further, once the number of samples crosses the
required sample complexity, the estimation error decreases as
, where depends on the Gaussian width of the unit norm
ball.Comment: Fixed technical issues. Generalized some result
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