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Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data.

Yikun Zhang1, Yen-Chi Chen1

  • 1Department of Statistics, University of Washington, Seattle, WA 98195, USA.

Information and Inference : a Journal of the IMA
|February 10, 2023
PubMed
Summary

This study proves the linear convergence of the subspace constrained mean shift (SCMS) algorithm for density ridge identification. The research extends the SCMS algorithm to directional data, establishing its stability and convergence.

Keywords:
49Q1262G0562H11directional dataoptimization on a manifoldridgessubspace constrained mean shift

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Analysis

Background:

  • The subspace constrained mean shift (SCMS) algorithm is widely used for identifying density ridges.
  • Existing research primarily focuses on Euclidean spaces, limiting applications for directional data.

Purpose of the Study:

  • To analyze the linear convergence of the SCMS algorithm.
  • To generalize the SCMS algorithm and the concept of density ridges to directional data.
  • To establish the stability of density ridges in directional spaces.

Main Methods:

  • The SCMS algorithm is presented as a variant of the subspace constrained gradient ascent (SCGA) algorithm.
  • Linear convergence is derived for the SCGA algorithm with an adaptive step size.
  • The study develops a directional SCMS algorithm for directional data analysis.

Main Results:

  • The linear convergence of the SCGA algorithm is proven.
  • A stability theorem for density ridges with directional data is established.
  • The proposed directional SCMS algorithm demonstrates linear convergence.

Conclusions:

  • The study successfully extends the SCMS algorithm to directional data.
  • The findings provide theoretical guarantees for the convergence and stability of the directional SCMS algorithm.
  • This generalization opens new avenues for density ridge analysis in various fields involving directional data.