Single and Multiple Index Models
published: Jan. 16, 2013, recorded: December 2012, views: 4604
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Statistical estimation in the high-dimensional setting, with more variables than samples, has been the focus of considerable research over the last decade. It is now well understood that consistent estimation is still possible under such high-dimensional settings provided we impose suitable constraints on the model space. These structural constraints are however typically easier to impose when the statistical model has a finite-dimensional parametric form. A natural approach to extend these to the non-parametric setting is to work with semi-parametric models, and impose these structural constraints on the parametric component of the semi-parametric model. In this talk, we consider the semi-parametric model class of single and multiple index models. Here, the regression function is assumed to be a sum of univariate functions of linear projections of the data. We show that we can combine "classical nonparametrics" (projection pursuit regression and backfitting) with a "variational principle" based on convex optimization to address the difficult optimization problems arising in estimating such models. In particular, we show that the standard maximum-likelihood based approach is non-convex and brittle, but we can use Bregman divergences to derive a computationally tractable backfitting procedure. We demonstrate the utility of this modeling approach in a retinal modeling application.
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