Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

Abstract

The present manuscript tackles the problem of learning the average of a set of symmetric positive-definite (SPD) matrices. Averages are computed via the notion of Fréchet mean, and the associated metric dispersion is interpreted as the variance of the patterns around the Fréchet mean. Also, the problem of continuous interpolation of two SPD patterns is tackled within the manuscript. The property of volume conservation of the Fréchet mean and of the considered interpolatory scheme for SPD matrices is discussed as well. The paper describes several applications where the technique could be readily exploited, including in machine learning, intelligent control, pattern classification, speech emotion classification and diffusion tensor data analysis in medicine. © 2009 Springer Science+Business Media, LLC.