Linear Convergence of the Subspace Constrained Mean Shift Algorithm: From Euclidean to Directional Data
This work addresses the theoretical analysis of convergence for density ridge estimation algorithms, extending applicability to directional data, but it is incremental as it builds on existing SCMS methods.
The paper tackles the problem of proving linear convergence for the subspace constrained mean shift (SCMS) algorithm, showing it as a variant of subspace constrained gradient ascent with adaptive step size, and extends density ridges and SCMS to directional data, establishing stability and linear convergence for the directional version.
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.