Restricted Eigen Condition for Heavy Tailed Designs
published: Aug. 20, 2015, recorded: July 2015, views: 1633
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The restricted eigenvalue (RE) condition characterizes the sample complexity of accurate recovery in the context of high-dimensional estimators such as Lasso and Dantzig selector (Bickel et al., 2009). Recent work has shown that random design matrices drawn from any thin-tailed (subGaussian) distributions satisfy the RE condition with high probability, when the number of samples scale as the square of the Gaussian width of the restricted set (Banerjee et al., 2014; Tropp, 2015). We pose the equivalent question for heavy-tailed distributions: Given a random design matrix drawn from a heavy-tailed distribution satisfying the small-ball property (Mendelson, 2015), does the design matrix satisfy the RE condition with the same order of sample complexity as subGaussian distributions? An answer to the question will guide the design of high-dimensional estimators for heavy tailed problems.
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