Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift

This paper challenges the standard assumption that Subspace Constrained Mean Shift (SCMS) trajectories converge to the classical, static density ridge. It argues that the static definition ignores rotation of the trailing Hessian eigenspace along the algorithm’s continuous flow. The authors introduce a dynamical-systems-based “stable ridge,” defined using the Jacobian of the projected density gradient, and establish it as the theoretical target of SCMS. A generalized fixed-step SCMS framework is shown to have uniform R-linear convergence and topological surjectivity onto the stable ridge. The paper also derives Hausdorff estimation rates and identifies polynomial-time complexity in original SCMS.