Regression on manifolds: Estimation of the exterior derivative

Citation
Aswani, Anil et al., Regression on manifolds: Estimation of the exterior derivative, Annals of statistics , 39(1), 2011, pp. 48-81
Journal title
ISSN journal
00905364
Volume
39
Issue
1
Year of publication
2011
Pages
48 - 81
Database
ACNP
SICI code
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.