稳定密度脊：子空间约束均值漂移的一致性与收敛性
Stable Density Ridges: Consistency and Convergence of Subspace Constrained Mean Shift
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中文总结 AI 辅助
本文针对子空间约束均值漂移（SCMS）算法的理论缺陷，提出基于动力系统的稳定脊概念，构建广义SCMS框架并证明其收敛性与效率，解决了原始算法的理论不足与计算复杂度问题。
中文摘要 AI 辅助
子空间约束均值漂移（SCMS）算法是一种用于提取密度脊的流行非参数方法，密度脊可作为高维数据的低维表示。文献中普遍认为SCMS轨迹收敛于经典密度脊，我们称之为“静态脊”，其通过密度梯度及密度Hessian矩阵的特征值与特征向量定义。本文证明该假设并不普遍成立，因为静态定义未考虑算法底层向量场连续流过程中尾随特征空间的旋转。为解决此问题，我们提出范式转变，引入“稳定脊”这一新型几何结构，其基于动力系统视角及投影密度梯度的雅可比矩阵定义。我们证明稳定脊是SCMS算法的真实理论目标。在此基础上，我们开发采用固定步长的广义SCMS框架，确立其对稳定脊的均匀R-线性收敛性及拓扑满射性，进一步推导以豪斯多夫距离衡量的稳定脊估计收敛速率。最后，我们揭示原始SCMS算法存在多项式时间计算复杂度问题，该问题由步长通过均值漂移算子隐式耦合到平滑带宽导致，并证明我们的广义框架如何提供统计一致且更高效的解决方案。
英文摘要
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.
