Regression on Manifolds: Estimation of the Exterior Derivative
Anil Aswani, Peter Bickel, Claire Tomlin

Citation
Anil Aswani, Peter Bickel, Claire Tomlin. "Regression on Manifolds: Estimation of the Exterior Derivative". Annals of Statistics, 2010; To appear.

Abstract
Collinearity and near-collinearity of predictors cause difficulties when doing nonparametric regression on manifolds. In such scenar ios, variable selection becomes untenable because of mathematical difficulties concerning the existence and numerical stability of the regression coefficients. In addition, once computed, the regression coefficients are difficult to interpret, because a gradient does not exist for functions on manifolds. Fortunately, there is an extension of the gradient to functions on manifolds; this extension is known as the exterior derivative of a function. It is the natural quantity to estimate, because it is a mathematically well-defined quantity with a geometrical interpretation. We propose a set of novel estimators using a regularization scheme for the regression problem which considers the geometrical intuition of the exterior derivative. The advantage of this regularization scheme is that it allows us to add lasso-type regularization to the regression problem, which enables lasso-type regressions in the presence of collinearities. Finally, we consider the "large p, small n" problem in our context and show the consistency and variable selection abilities of our estimators.

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• HTML

  Anil Aswani, Peter Bickel, Claire Tomlin. <a
  href="http://chess.eecs.berkeley.edu/pubs/603.html"
  >Regression on Manifolds: Estimation of the Exterior
  Derivative</a>, <i>Annals of
  Statistics</i>,  2010; To appear.

• Plain text

  Anil Aswani, Peter Bickel, Claire Tomlin. "Regression
  on Manifolds: Estimation of the Exterior Derivative".
  <i>Annals of Statistics</i>,  2010; To appear.

• BibTeX

  @article{AswaniBickelTomlin10_RegressionOnManifoldsEstimationOfExteriorDerivative,
      author = {Anil Aswani and Peter Bickel and Claire Tomlin},
      title = {Regression on Manifolds: Estimation of the
                Exterior Derivative},
      journal = {Annals of Statistics},
      year = {2010},
      note = {To appear},
      abstract = {Collinearity and near-collinearity of predictors
                cause difficulties when doing nonparametric
                regression on manifolds. In such scenar ios,
                variable selection becomes untenable because of
                mathematical difficulties concerning the existence
                and numerical stability of the regression
                coefficients. In addition, once computed, the
                regression coefficients are difficult to
                interpret, because a gradient does not exist for
                functions on manifolds. Fortunately, there is an
                extension of the gradient to functions on
                manifolds; this extension is known as the exterior
                derivative of a function. It is the natural
                quantity to estimate, because it is a
                mathematically well-defined quantity with a
                geometrical interpretation. We propose a set of
                novel estimators using a regularization scheme for
                the regression problem which considers the
                geometrical intuition of the exterior derivative.
                The advantage of this regularization scheme is
                that it allows us to add lasso-type regularization
                to the regression problem, which enables
                lasso-type regressions in the presence of
                collinearities. Finally, we consider the "large p,
                small n" problem in our context and show the
                consistency and variable selection abilities of
                our estimators.},
      URL = {http://chess.eecs.berkeley.edu/pubs/603.html}
  }
  

Posted by Anil Aswani on 17 Jun 2009.
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