Two talks will be presented :

**Alexandre AKSENOV**

**Title.** Introduction to strongly b-multiplicative sequences and rarefaction.

**Abstract.** The class of “strongly b-multiplicative sequences” is the class of numerical sequences which are closest to the classical Thue-Morse sequence. This sequence was defined at first as a source of examples of “counter-intuitive objects”, i.e. in theoretical computer science. Today, these objects can be called “transcendental”, “chaotic” or “fractal”.

In this talk we will see how they objects can be linked to a (seemingly) different domain: Number Theory.

The talk is planned as popular, and the quantity of technical details is intended to be reasonnable.

**Marco CONGEDO**

**Title.** Power Means and Mean Fields in the Riemannian Manifold of Symmetric Positive Matrices”

**Abstract.** Averaging is possibly the most common and most important operation in statistics and signal processing. This talk is concerned about recent advances in averaging of symmetric positive-definite (SPD) matrices.

In brain-computer interfaces we have introduced the use of the geometric mean on the Riemannian manifold of SDP matrices and we have found that a simple minimum distance to mean classifier outperforms state-of-the-art classifiers. The power means of SPD matrices with exponent p in the interval [-1, 1] interpolate continuously in between the Harmonic (p = -1) and the Arithmetic mean (p = 1), while the geometric mean corresponds to their limit evaluated at p → 0. In this talk we present a new fixed point algorithm for estimating means along the interval [-1, 1]\{0}. The convergence rate of the proposed algorithm for p = ±0.5 does not deteriorate with the number or dimension of matrices given as input, which is very useful in practical applications. Along the whole interval it is also robust with respect to the dispersion of the points on the manifold (i.e., noise). Thus, the proposed algorithm allows the efficient estimation of the whole family of power means, including the geometric mean. Finally, we will introduce the concept of Mean Fields, a sampling of mean in the interval [-1, 1], with application in Riemannian signal classification and detection.