Zhang Yikun, Chen Yen-Chi
Department of Statistics, University of Washington, Seattle, WA 98195, USA.
Inf inference. 2022 Apr 9;12(1):210-311. doi: 10.1093/imaiai/iaac005. eCollection 2023 Mar.
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.
本文研究子空间约束均值漂移（SCMS）算法的线性收敛性，该算法是一种用于识别由核密度估计器定义的密度脊的著名算法。通过论证SCMS算法是具有自适应步长的子空间约束梯度上升（SCGA）算法的一种特殊变体，我们推导了这种SCGA算法的线性收敛性。虽然现有研究主要集中在欧几里得空间中的密度脊，但我们将密度脊和SCMS算法推广到方向数据。特别是，我们建立了方向数据密度脊的稳定性定理，并证明了我们提出的方向SCMS算法的线性收敛性。
