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Activity Number:
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490
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Type:
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Invited
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Date/Time:
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Thursday, August 2, 2007 : 8:30 AM to 10:20 AM
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Sponsor:
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IMS
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| Abstract - #307850 |
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Title:
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Sparsity Oracle Inequalities for the Lasso
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Author(s):
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Marten Wegkamp*+ and Alexandre Tsybakov and Florentina Bunea
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Companies:
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Florida State University and Paris VI and Florida State University
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Address:
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, Tallahassee, FL, 32306-4330,
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Keywords:
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Lasso ; Sparsity ; Mutual coherence ; Oracle inequalities ; Empirical risk minimization
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Abstract:
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We study oracle properties of the lasso estimator in a nonparametric regression setting with random design. Our goal is to estimate the regression function via linear combinations of elements in some dictionary and to estimate its performance in terms of the L2 risk. An oracle that knows the regression function would be able to tell us in advance the sparsest approximating submodel with the smallest risk. We show that the lasso estimator behaves like the oracle. The results are valid even when the dimension of the initial model is (much) larger than the sample size and the regression matrix is not positive definite. Instead we introduce a new concept of local mutual coherence. Our results can be applied to high-dimensional linear regression, to nonparametric adaptive regression estimation and to the problem of aggregation of arbitrary estimators.
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