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Manifold Learning in Regression Tasks

  • Alexander Bernstein
  • Alexander Kuleshov
  • Yury YanovichEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9047)

Abstract

The paper presents a new geometrically motivated method for non-linear regression based on Manifold learning technique. The regression problem is to construct a predictive function which estimates an unknown smooth mapping f from q-dimensional inputs to m-dimensional outputs based on a training data set consisting of given ‘input-output’ pairs. The unknown mapping f determines q-dimensional manifold M(f) consisting of all the ‘input-output’ vectors which is embedded in (q+m)-dimensional space and covered by a single chart; the training data set determines a sample from this manifold. Modern Manifold Learning methods allow constructing the certain estimator M* from the manifold-valued sample which accurately approximates the manifold. The proposed method called Manifold Learning Regression (MLR) finds the predictive function fMLR to ensure an equality M(fMLR) = M*. The MLR simultaneously estimates the m×q Jacobian matrix of the mapping f.

Keywords

Nonlinear regression Dimensionality reduction Manifold learning Tangent bundle manifold learning Manifold learning regression 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander Bernstein
    • 1
    • 2
  • Alexander Kuleshov
    • 1
    • 2
  • Yury Yanovich
    • 1
    • 2
    Email author
  1. 1.Kharkevich Institute for Information Transmission Problems RASMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia

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