TY - JOUR
T1 - Open problem
T2 - 28th Conference on Learning Theory, COLT 2015
AU - Banerjee, Arindam
AU - Chen, Sheng
AU - Sivakumar, Vidyashankar
N1 - Publisher Copyright:
© 2015 A. Agarwal & S. Agarwal.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2015
Y1 - 2015
N2 - 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 (sub- Gaussian) 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 sub- Gaussian distributions? An answer to the question will guide the design of high-dimensional estimators for heavy tailed problems.
AB - 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 (sub- Gaussian) 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 sub- Gaussian distributions? An answer to the question will guide the design of high-dimensional estimators for heavy tailed problems.
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M3 - Conference article
AN - SCOPUS:84984682118
SN - 1532-4435
VL - 40
JO - Journal of Machine Learning Research
JF - Journal of Machine Learning Research
IS - 2015
Y2 - 2 July 2015 through 6 July 2015
ER -