Regression on manifolds: Estimation of the exterior derivative

TL;DR

This paper introduces a differential geometric approach to regression on manifolds that addresses collinearity issues by estimating the exterior derivative, with regularization techniques that enhance stability and extend to high-dimensional settings.

Contribution

It develops a novel regression method based on exterior derivatives and differential geometry, improving estimation under collinearity and enabling regularization and high-dimensional extensions.

Findings

Regularization improves estimation accuracy.

Method handles collinearity and near-collinearity effectively.

Extensions to high-dimensional data are feasible.

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.