  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.
