  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.
