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Modified Subspace Constrained Mean Shift Algorithm

Author

Listed:
  • Youness Aliyari Ghassabeh

    (University of Toronto)

  • Frank Rudzicz

    (University of Toronto
    Li Ka Shing Knowledge Institute, St Michael’s Hospital
    Surgical Safety Technologies Inc
    Vector Institute for Artificial Intelligence)

Abstract

A subspace constrained mean shift (SCMS) algorithm is a non-parametric iterative technique to estimate principal curves. Principal curves, as a nonlinear generalization of principal components analysis (PCA), are smooth curves (or surfaces) that pass through the middle of a data set and provide a compact low-dimensional representation of data. The SCMS algorithm combines the mean shift (MS) algorithm with a projection step to estimate principal curves and surfaces. The MS algorithm is a simple iterative method for locating modes of an unknown probability density function (pdf) obtained via a kernel density estimate. Modes of a pdf can be interpreted as zero-dimensional principal curves. These modes also can be used for clustering the input data. The SCMS algorithm generalizes the MS algorithm to estimate higher order principal curves and surfaces. Although both algorithms have been widely used in many real-world applications, their convergence for widely used kernels (e.g., Gaussian kernel) has not been sown yet. In this paper, we first introduce a modified version of the MS algorithm and then combine it with different variations of the SCMS algorithm to estimate the underlying low-dimensional principal curve, embedded in a high-dimensional space. The different variations of the SCMS algorithm are obtained via modification of the projection step in the original SCMS algorithm. We show that the modification of the MS algorithm guarantees its convergence and also implies the convergence of different variations of the SCMS algorithm. The performance and effectiveness of the proposed modified versions to successfully estimate an underlying principal curve was shown through simulations using the synthetic data.

Suggested Citation

  • Youness Aliyari Ghassabeh & Frank Rudzicz, 2021. "Modified Subspace Constrained Mean Shift Algorithm," Journal of Classification, Springer;The Classification Society, vol. 38(1), pages 27-43, April.
    • Handle: RePEc:spr:jclass:v:38:y:2021:i:1:d:10.1007_s00357-019-09353-1
    DOI: 10.1007/s00357-019-09353-1
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    References listed on IDEAS

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    1. Delicado, Pedro, 2001. "Another Look at Principal Curves and Surfaces," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 84-116, April.
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