4,118 research outputs found
Confidence sets in sparse regression
The problem of constructing confidence sets in the high-dimensional linear
model with response variables and parameters, possibly , is
considered. Full honest adaptive inference is possible if the rate of sparse
estimation does not exceed , otherwise sparse adaptive confidence
sets exist only over strict subsets of the parameter spaces for which sparse
estimators exist. Necessary and sufficient conditions for the existence of
confidence sets that adapt to a fixed sparsity level of the parameter vector
are given in terms of minimal -separation conditions on the parameter
space. The design conditions cover common coherence assumptions used in models
for sparsity, including (possibly correlated) sub-Gaussian designs.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1170 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Detection boundary in sparse regression
We study the problem of detection of a p-dimensional sparse vector of
parameters in the linear regression model with Gaussian noise. We establish the
detection boundary, i.e., the necessary and sufficient conditions for the
possibility of successful detection as both the sample size n and the dimension
p tend to the infinity. Testing procedures that achieve this boundary are also
exhibited. Our results encompass the high-dimensional setting (p>> n). The main
message is that, under some conditions, the detection boundary phenomenon that
has been proved for the Gaussian sequence model, extends to high-dimensional
linear regression. Finally, we establish the detection boundaries when the
variance of the noise is unknown. Interestingly, the detection boundaries
sometimes depend on the knowledge of the variance in a high-dimensional
setting
Robust and Sparse Regression via -divergence
In high-dimensional data, many sparse regression methods have been proposed.
However, they may not be robust against outliers. Recently, the use of density
power weight has been studied for robust parameter estimation and the
corresponding divergences have been discussed. One of such divergences is the
-divergence and the robust estimator using the -divergence is
known for having a strong robustness. In this paper, we consider the robust and
sparse regression based on -divergence. We extend the
-divergence to the regression problem and show that it has a strong
robustness under heavy contamination even when outliers are heterogeneous. The
loss function is constructed by an empirical estimate of the
-divergence with sparse regularization and the parameter estimate is
defined as the minimizer of the loss function. To obtain the robust and sparse
estimate, we propose an efficient update algorithm which has a monotone
decreasing property of the loss function. Particularly, we discuss a linear
regression problem with regularization in detail. In numerical
experiments and real data analyses, we see that the proposed method outperforms
past robust and sparse methods.Comment: 25 page
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