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A generative model and a generalized trust region Newton method for noise reduction

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  • Seppo Pulkkinen
  • Marko Mäkelä
  • Napsu Karmitsa

Abstract

In practical applications related to, for instance, machine learning, data mining and pattern recognition, one is commonly dealing with noisy data lying near some low-dimensional manifold. A well-established tool for extracting the intrinsically low-dimensional structure from such data is principal component analysis (PCA). Due to the inherent limitations of this linear method, its extensions to extraction of nonlinear structures have attracted increasing research interest in recent years. Assuming a generative model for noisy data, we develop a probabilistic approach for separating the data-generating nonlinear functions from noise. We demonstrate that ridges of the marginal density induced by the model are viable estimators for the generating functions. For projecting a given point onto a ridge of its estimated marginal density, we develop a generalized trust region Newton method and prove its convergence to a ridge point. Accuracy of the model and computational efficiency of the projection method are assessed via numerical experiments where we utilize Gaussian kernels for nonparametric estimation of the underlying densities of the test datasets. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Seppo Pulkkinen & Marko Mäkelä & Napsu Karmitsa, 2014. "A generative model and a generalized trust region Newton method for noise reduction," Computational Optimization and Applications, Springer, vol. 57(1), pages 129-165, January.
  • Handle: RePEc:spr:coopap:v:57:y:2014:i:1:p:129-165
    DOI: 10.1007/s10589-013-9581-4
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    References listed on IDEAS

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    1. Michael E. Tipping & Christopher M. Bishop, 1999. "Probabilistic Principal Component Analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 611-622.
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    Cited by:

    1. Pulkkinen, Seppo, 2015. "Ridge-based method for finding curvilinear structures from noisy data," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 89-109.

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