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Statistics and Computing

, Volume 25, Issue 5, pp 893–911 | Cite as

High-dimensional regression with gaussian mixtures and partially-latent response variables

  • Antoine Deleforge
  • Florence Forbes
  • Radu Horaud
Article

Abstract

The problem of approximating high-dimensional data with a low-dimensional representation is addressed. The article makes the following contributions. An inverse regression framework is proposed, which exchanges the roles of input and response, such that the low-dimensional variable becomes the regressor, and which is tractable. A mixture of locally-linear probabilistic mapping model is introduced, that starts with estimating the parameters of the inverse regression, and follows with inferring closed-form solutions for the forward parameters of the high-dimensional regression problem of interest. Moreover, a partially-latent paradigm is introduced, such that the vector-valued response variable is composed of both observed and latent entries, thus being able to deal with data contaminated by experimental artifacts that cannot be explained with noise models. The proposed probabilistic formulation could be viewed as a latent-variable augmentation of regression. Expectation-maximization (EM) procedures are introduced, based on a data augmentation strategy which facilitates the maximum-likelihood search over the model parameters. Two augmentation schemes are proposed and the associated EM inference procedures are described in detail; they may well be viewed as generalizations of a number of EM regression, dimension reduction, and factor analysis algorithms. The proposed framework is validated with both synthetic and real data. Experimental evidence is provided that the method outperforms several existing regression techniques.

Keywords

Regression Latent variable Mixture models Expectation-maximization Dimensionality reduction 

Notes

Acknowledgments

The authors wish to thank the anonymous reviewers for their constructive remarks and suggestions which helped organizing and improving this article.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Antoine Deleforge
    • 1
  • Florence Forbes
    • 1
  • Radu Horaud
    • 1
  1. 1.INRIA Grenoble Rhône-AlpesMontbonnot Saint-MartinFrance

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