A subspaceconstrainedmeanshift (SCMS) algorithm is a non-parametric iterative technique to estimate principal curves. Principal curves, as a nonlinear generalization of principal components analysis (PCA), are smoo...
详细信息
A subspaceconstrainedmeanshift (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 meanshift (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.
We study properties of the meanshift (MS)-type algorithms for estimating modes of probability density functions (PDFs), via regarding these algorithms as gradient ascent on estimated PDFs with adaptive step sizes. We...
详细信息
We study properties of the meanshift (MS)-type algorithms for estimating modes of probability density functions (PDFs), via regarding these algorithms as gradient ascent on estimated PDFs with adaptive step sizes. We rigorously prove convergence of mode estimate sequences generated by the MS-type algorithms, under the assumption that an analytic kernel function is used. Moreover, our analysis on the MS function finds several new properties of mode estimate sequences and corresponding density estimate sequences, including the result that in the MS-type algorithm using a Gaussian kernel the density estimate monotonically increases between two consecutive mode estimates. This implies that, in the one-dimensional case, the mode estimate sequence monotonically converges to the stationary point nearest to an initial point without jumping over any stationary point.
暂无评论