Linear Convergence of the Subspace Constrained Mean Shift Algorithm: From Euclidean to Directional Data

TL;DR

This paper proves the linear convergence of the subspace constrained mean shift algorithm for both Euclidean and directional data, extending its theoretical guarantees and stability analysis to new data types.

Contribution

It generalizes the SCMS algorithm to directional data and establishes its linear convergence and stability, which was previously only known for Euclidean data.

Findings

Proves linear convergence of the directional SCMS algorithm.

Establishes stability of density ridges with directional data.

Extends theoretical analysis of SCMS to new data domains.

Abstract

This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.