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

By Frank Nielsen, Frederic Barbaresco

This e-book constitutes the refereed court cases of the second one overseas convention on Geometric technological know-how of knowledge, GSI 2015, held in Palaiseau, France, in October 2015.

The eighty complete papers provided have been rigorously reviewed and chosen from a hundred and ten submissions and are prepared into the next thematic periods:

Dimension aid on Riemannian manifolds; optimum shipping; optimum shipping and purposes in imagery/statistics; form house and diffeomorphic mappings; random geometry/homology; Hessian info geometry; topological varieties and knowledge; details geometry optimization; details geometry in photograph 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 resources for Geometric Science of Information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings

Sample text

This implies that the relative scaling of the spheres will only depend on the radius of the inner S d1 , clearly an unwanted feature. Hence, we normalize the radii with their geometric mean Dimension Reduction on Polyspheres with Application ⎛ K Ri := ri ⎝ 25 ⎞− I1 rj ⎠ j=1 (i = 1, . . , I), rescale all coordinates of the first unit sphere ∀1 ≤ k ≤ d1 + 1 : x1,k → x ˜1,k = R1 x1,k , (4) only the first di coordinates of the i-th unit sphere (i = 2, . . I) ∀1 ≤ k ≤ di : xi,k → x ˜i,k = Ri xi,k (5) and then apply the recursive operations defined in Eq.

Karcher) mean is the use of the power α of the metric instead of the square. For instance, one defines the median (α = 1) and the modes (α → 0) as the minima of the α-variance k σ α (x) = α1 i=0 distα (x, xi ). Following this idea, one could think of generalizing barycentric subspaces to the α-Fr´echet (resp. α-Karcher) barycentric subspaces. Barycentric Subspaces and Affine Spans in Manifolds 19 In fact, it turns out that the critical points of the α-variance are just elements of the affine span with weights λi = λi distα−2 (x, xi ).

D be a coordinate system on F d M . The vertical distribution is in this case spanned by the nd vector fields Djβ = ∂X j . Except β for index sums being over d instead of n terms, the situation is thus similar to the full-rank case. Note that (ξ|π∗−1 w) = (ξ|wj Dj ) = wi ξi . The cometric in coordinates is ij ˜ ij ˜ ξi ξj = ξi δ αβ Xαi Xβj + λgR ξ, ξ˜ = δ αβ Xαi ξi Xβj ξ˜j + λgR ξj = ξi W ij ξ˜j ij with W ij = δ αβ Xαi Xβj + λgR . We can then write the corresponding subRiemannian metric gF d M in terms of the adapted frame D gF d M (ξh Dh + ξhγ Dhγ ) = W ih ξh Di (7) ˜ = ξ, ξ˜ = ξi W ij ξ˜j .

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