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.
Read Online or Download Geometric Science of Information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings PDF
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Extra resources for Geometric Science of Information: Second International Conference, GSI 2015, Palaiseau, France, October 28–30, 2015, Proceedings
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 .