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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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Citation formats  
  • 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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