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Learning in high dimensions, noise, sparsity and treelets

Wednesday 9th January 2008 - 16:30 to 17:30
INI Seminar Room 1
Session Chair: 
Peter Bickel

In recent years there is growing practical need to perform learning (classification,regression, etc) in high dimensional settings where p>>n. Consequently instead of the standard limit $n\to\infty$, learning algorithms are typically analyzed in the joint limit $p,n\to\infty$. In this talk we present a different approach, that keeps $p,n$ fixed, but considers noise as a small parameter. This resulting perturbation analysis reveals the importance of a robust low dimensional representation of the noise-free signals, the possible failure of simple variable selection methods and the key role of sparsity for the success of learning in high dimensions. We also discuss sparsity in a-priori unknown basis and a possible data-driven adaptive construction of such basis, called treelets. We present a few applications of our analysis, mainly to error-in-variables linear regression problems, principal component analysis, and rank determination.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons