这篇论文推翻了SCMS收敛到静态脊的旧认知，重新定义了稳定脊并给出收敛保证，做密度估计理论的人值得看看。
子空间约束均值漂移(SCMS)算法常用于提取密度脊，但论文证明其轨迹并不收敛到经典静态脊。作者提出“稳定脊”概念，通过动力系统和投影密度梯度雅可比定义，证明这才是SCMS的真实目标。文章建立广义SCMS框架，使用常数步长，证明了其一致R线性收敛和到稳定脊的拓扑满射性。还推导了基于Hausdorff距离的稳定脊估计收敛速率，并指出原SCMS算法因步长与带宽耦合存在多项式时间计算复杂度。
Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift
The Subspace Constrained Mean Shift (SCMS) algorithm is a popular nonparametric method for extracting density ridges, which serve as a low-dimensional representation of high-dimensional data. It is a widely held belief in the literature that SCMS trajectories converge to the classical density ridge, which we call the "static ridge", defined via the density gradient and the eigenvalues and eigenvectors of the density's Hessian. In this paper, we demonstrate that this assumption does not hold in general, as the static definition fails to account for the rotation of the trailing eigenspace along the continuous flow of the algorithm's underlying vector field. To resolve this, we propose a paradigm shift by introducing the "stable ridge", a novel geometric structure defined through the lens of dynamical systems and the Jacobian of the projected density gradient. We prove that this stable ridge is the true theoretical target of the SCMS algorithm. Building upon this foundation, we develop a generalized SCMS framework utilizing a constant step size, establishing its uniform R-linear convergence and topological surjectivity onto the stable ridge. We further derive the rates of convergence for estimating the stable ridge in terms of the Hausdorff distance. Finally, we expose that the original SCMS algorithm suffers from polynomial-time computational complexity, which is caused by implicitly coupling the step size to the smoothing bandwidth via the Mean Shift operator, and demonstrate how our generalized framework provides a statistically consistent and more efficient solution.