Barycenters of Toeplitz matrices and application in clustering

Abstract

This paper presents two innovative centering notions, the p-barycenter and the Lp-center of mass, for Toeplitz matrices. The p-barycenter employs a distance function that relies on symbol functions, while the Lp-center of mass is based on the Riemannian distance on the manifold of positive definite matrices. Our proposed methods extend the k-means machine learning algorithm to Toeplitz matrices, thereby enabling potential applications in various fields, including signal processing. Furthermore, when p = 2, one of the resulting objects is the geometric mean or Karcher mean, which is also a Toeplitz matrix. These centering notions have great potential for enhancing the performance of clustering algorithms on Toeplitz matrices and can be applied in areas such as image processing, audio signal processing, and time series analysis.