Download Geometric Science of Information: Second International by Frank Nielsen, Frederic Barbaresco PDF

By Frank Nielsen, Frederic Barbaresco

This booklet constitutes the refereed court cases of the second one foreign convention on Geometric technological know-how of data, GSI 2015, held in Palaiseau, France, in October 2015.

The eighty complete papers offered have been conscientiously reviewed and chosen from one hundred ten submissions and are equipped into the next thematic classes:

Dimension relief on Riemannian manifolds; optimum shipping; optimum delivery and functions in imagery/statistics; form house and diffeomorphic mappings; random geometry/homology; Hessian details geometry; topological types and knowledge; info geometry optimization; info geometry in picture research; divergence geometry; optimization on manifold; Lie teams and geometric mechanics/thermodynamics; computational info geometry; Lie teams: novel statistical and computational frontiers; geometry of time sequence and linear dynamical platforms; and Bayesian and knowledge geometry for inverse problems.

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Extra info for Geometric Science of Information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings

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K ) that do not sum up to zero. To define the moments of that distribution, we have to take care that the Riemannian log and distance functions are not smooth at the cut-locus of the points {xi }. Definition 1 ((k + 1)-Pointed Riemannian Manifold). Let {x0 , . . xk } ∈ Mk+1 be a set of k + 1 distinct points in the Riemannian manifold M and C(x0 , . . xk ) = ∪ki=0 C(xi ) be the union of the cut loci of these points. We call (k + 1)-pointed manifold M∗ (x0 , . . xk ) = M/C(x0 , . . xk ) the submanifold of the non-cut points of the points.

However, these methods cannot be adapted to a backward analysis and they are not symmetric in the parametrization of the subspaces. We propose in this paper a new and more general type of family of subspaces in manifolds: barycentric subspaces are implicitly defined as the locus of points which are weighted means of k + 1 reference points. Depending on the generalization of the mean that we use, we obtain the Fr´echet/Karcher barycentric subspaces (FBS/KBS) or the affine span (with exponential barycenter).

Recall that a logarithmic scale is well suited for lengths, linearizing ratios, such that the geometric mean corresponds to the arithmetic mean on a logarithmic scale. Hence, it is natural to define the mean radius of each individual sphere by the geometric mean of the radii of the data points, cf. [11]. There is yet another subtlety to be dealt with. In Eq. (3), simply multiplying the coordinates of S d2 and S d1 by the corresponding radii, implies that all coordinates of the sphere S d2 +d1 are in particular scaled with the radius of the outer sphere S d2 .

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