Regression on manifolds: Estimation of the exterior derivative

Details Regression on manifolds: Estimation of the exterior derivative Regression on manifolds: Estimation of the exterior derivative

2011-03-08

arxiv.org/abs/1103.1457v1

Annals of Statistics 2011, Vol. 39, No. 1, 48-81

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/10-AOS823 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

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.

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