A subspace constrained mean shift (SCMS) algorithm is a non-parametric iterative technique to estimate principal curves. Principal curves, as a nonlinear generalization of principal components analysis (PCA), are smooth curves (or surfaces) that pass through the middle of a data set and provide a compact low-dimensional representation of data. The SCMS algorithm combines the mean shift (MS) algorithm with a projection step to estimate principal curves and surfaces. The MS algorithm is a simple iterative method for locating modes of an unknown probability density function (pdf) obtained via a kernel density estimate. Modes of a pdf can be interpreted as zero-dimensional principal curves. These modes also can be used for clustering the input data. The SCMS algorithm generalizes the MS algorithm to estimate higher order principal curves and surfaces. Although both algorithms have been widely used in many real-world applications, their convergence for widely used kernels (e.g., Gaussian kernel) has not been sown yet. In this paper, we first introduce a modified version of the MS algorithm and then combine it with different variations of the SCMS algorithm to estimate the underlying low-dimensional principal curve, embedded in a high-dimensional space. The different variations of the SCMS algorithm are obtained via modification of the projection step in the original SCMS algorithm. We show that the modification of the MS algorithm guarantees its convergence and also implies the convergence of different variations of the SCMS algorithm. The performance and effectiveness of the proposed modified versions to successfully estimate an underlying principal curve was shown through simulations using the synthetic data.

中文翻译：

---

修正子空间约束均值漂移算法

子空间约束均值平移 (SCMS) 算法是一种用于估计主曲线的非参数迭代技术。主曲线作为主成分分析 (PCA) 的非线性推广，是穿过数据集中间并提供数据的紧凑低维表示的平滑曲线（或曲面）。SCMS 算法将均值漂移 (MS) 算法与投影步骤相结合，以估计主曲线和曲面。MS 算法是一种简单的迭代方法，用于定位通过核密度估计获得的未知概率密度函数 (pdf) 的模式。pdf 的模式可以解释为零维主曲线。这些模式也可用于对输入数据进行聚类。SCMS 算法将 MS 算法推广到估计高阶主曲线和曲面。尽管这两种算法已广泛用于许多实际应用，但它们对广泛使用的内核（例如高斯内核）的收敛性尚未播种。在本文中，我们首先介绍了 MS 算法的修改版本，然后将其与 SCMS 算法的不同变体相结合，以估计嵌入高维空间的底层低维主曲线。SCMS 算法的不同变化是通过修改原始 SCMS 算法中的投影步骤获得的。我们表明，MS 算法的修改保证了它的收敛性，也暗示了 SCMS 算法不同变体的收敛性。