A Fixed-Point Algorithm for Estimating Power Means of Positive Definite Matrices

Marco Congedo (1) , Ronald Phlypo (1) , Alexandre Barachant (2)

1 GIPSA-VIBS - GIPSA - Vision and Brain Signal Processing
2 Weill Medical College of Cornell University [New York]

The estimation of means of data points lying on the Riemannian manifold of symmetric positive-definite (SPD) matrices is of great utility in classification problems and is currently heavily studied. The power means of SPD matrices with exponent p in the interval [-1, 1] interpolate in between the Harmonic (p =-1) and the Arithmetic mean (p = 1), while the Geometric (Karcher) mean corresponds to their limit evaluated at 0. In this article we present a simple fixed point algorithm for estimating means along this whole continuum. The convergence rate of the proposed algorithm for p = ±0.5 deteriorates very little with the number and dimension of points given as input. Along the whole continuum it is also robust with respect to the dispersion of the points on the manifold. Thus, the proposed algorithm allows the efficient estimation of the whole family of power means, including the geometric mean .

Mots clés: Power Mean; Geometric Mean; High Dimension; Riemannian Manifold; Symmetric Positive-Definite Matrix

Domaines: Traitement du signal et de l'image [eess.SP]; Machine Learning [stat.ML]

Marco Congedo, Ronald Phlypo, Alexandre Barachant. A Fixed-Point Algorithm for Estimating Power Means of Positive Definite Matrices. EUSIPCO 2016 - 24th European Signal Processing Conference, IEEE, Aug 2016, Budapest, Hungary. pp.2016-2010. ⟨hal-01363817⟩