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Kernel \(\ell ^1\)-norm principal component analysis for denoising

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Abstract

In this paper we describe a method for denoising data using kernel principal component analysis (KPCA) that is able to recover preimages of the intrinsic variables in the feature space using a single line search along the gradient descent direction of its squared projection error. This method combines a projection-free preimage estimation algorithm with an \(\ell ^1\)-norm KPCA. These two stages provide distinct advantages over other KPCA preimage methods in the sense that they are insensitive to outliers and computationally efficient. The method can improve the results of a range of unsupervised learning tasks, such as denoising, and clustering. Numerical experiments in the Amsterdam Library of Object Images demonstrate that the proposed method performs better in terms of mean squared error than the \(\ell ^2\)-norm analogue, as well as in synthetic data. The proposed method is applied to different datasets and the results are reported.

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Data availability

The image datasets generated during and/or analysed during the current study are available in the Amsterdam Library of Object Images repository at https://aloi.science.uva.nl.

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Acknowledgements

The authors would like to express their sincere gratitude to anonymous reviewers for their valuable insights. Algorithm 1 is implemented in C using key functionality in the Intel oneAPI Math Kernel Library to achieve performance on Intel CPU architectures. The code is openly accessible and can be forked from the repository at https://github.com/lingxpca/kl1pca.git.

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Author notes

  1. Anh Bui and Paul Brooks have contributed equally to this work.

Authors and Affiliations

  1. Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, 1015 Floyd Avenue, Richmond, VA, 23284, USA

    Xiao Ling, Anh Bui & Paul Brooks

  2. Department of Information Systems, Virginia Commonwealth University, 301 West Main Street, Richmond, VA, 23284, USA

    Anh Bui

Authors
  1. Xiao Ling
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  2. Anh Bui
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  3. Paul Brooks
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Correspondence to Xiao Ling.

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Ling, X., Bui, A. & Brooks, P. Kernel \(\ell ^1\)-norm principal component analysis for denoising. Optim Lett 18, 2133–2148 (2024). https://doi.org/10.1007/s11590-023-02051-3

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