Open Access
February 2011 Regression on manifolds: Estimation of the exterior derivative
Anil Aswani, Peter Bickel, Claire Tomlin
Ann. Statist. 39(1): 48-81 (February 2011). DOI: 10.1214/10-AOS823


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.


Download Citation

Anil Aswani. Peter Bickel. Claire Tomlin. "Regression on manifolds: Estimation of the exterior derivative." Ann. Statist. 39 (1) 48 - 81, February 2011.


Published: February 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1209.62063
MathSciNet: MR2797840
Digital Object Identifier: 10.1214/10-AOS823

Primary: 58A10 , 62G08
Secondary: 62G20 , 62J07

Keywords: collinearity , Manifold , Model selection , Nonparametric regression , regularization

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 1 • February 2011
Back to Top