The Annals of Statistics

Regression on manifolds: Estimation of the exterior derivative

Anil Aswani, Peter Bickel, and Claire Tomlin

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Abstract

Collinearity and near-collinearity of predictors cause difficulties when doing regression. In these cases, variable selection becomes untenable because of mathematical issues concerning the existence and numerical stability of the regression coefficients, and interpretation of the coefficients is ambiguous because gradients are not defined. Using a differential geometric interpretation, in which the regression coefficients are interpreted as estimates of the exterior derivative of a function, we develop a new method to do regression in the presence of collinearities. Our regularization scheme can improve estimation error, and it can be easily modified to include lasso-type regularization. These estimators also have simple extensions to the “large p, small n” context.

Article information

Source
Ann. Statist. Volume 39, Number 1 (2011), 48-81.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
http://projecteuclid.org/euclid.aos/1291388369

Digital Object Identifier
doi:10.1214/10-AOS823

Mathematical Reviews number (MathSciNet)
MR2797840

Zentralblatt MATH identifier
1209.62063

Subjects
Primary: 62G08: Nonparametric regression 58A10: Differential forms
Secondary: 62G20: Asymptotic properties 62J07: Ridge regression; shrinkage estimators

Keywords
Nonparametric regression manifold collinearity model selection regularization

Citation

Aswani, Anil; Bickel, Peter; Tomlin, Claire. Regression on manifolds: Estimation of the exterior derivative. Ann. Statist. 39 (2011), no. 1, 48--81. doi:10.1214/10-AOS823. http://projecteuclid.org/euclid.aos/1291388369.


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