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Seminar Details

Date 9-10-2007
Time 14:30
Room/Location Sala conferenze-3 piano
Title Sparse Recovery in Learning Theory
Speaker Dott. Vladimir Koltchinskii
Affiliation Georgia Institute of Technology
Link http://www.gatech.edu/
Abstract A problem of learning of a target function based on its noisy observations at random points via penalized empirical risk minimization with convex loss function and convex complexity penalty will be discussed. This includes a number of regression and large margin classification problems. We will consider two specific settings. In one of them, the target function belongs to the linear span of a very large dictionary of given functions. In a more general case, it belongs to the linear span of a large number of reproducing kernel Hilbert spaces. Aggregation problems for large ensembles of kernel machines and statistical problems for additive models in regression and classification can be studied in this framework. It is assumed that the target function has a "sparse representation" in the "dictionary" and the goal is to recover the function with an error that depends mainly on the "sparsity" of the problem (even if the size of the "dictionary" is very large). An approach based on $\ell_1$-type complexity penalization will be discussed. We prove several probabilistic bounds showing the relationship between the sparsity of the empirical solution and the sparsity of the target function and provide oracle inequalities on the excess risk and the $L_2$-error that depend on the sparsity of the problem. (Some parts of this talk are based on a joint work and on discussions with Philippe Rigollet, Alexandre Tsybakov and Ming Yuan).
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