By Shun-ichi Amari (auth.), Frank Nielsen, Frédéric Barbaresco (eds.)

This booklet constitutes the refereed complaints of the 1st foreign convention on Geometric technology of data, GSI 2013, held in Paris, France, in August 2013. The approximately a hundred papers provided have been conscientiously reviewed and chosen from a variety of submissions and are geared up into the subsequent thematic periods: Geometric facts on Manifolds and Lie teams, Deformations suit areas, Differential Geometry in sign Processing, Relational Metric, Discrete Metric areas, Computational details Geometry, Hessian info Geometry I and II, Computational features of data Geometry in records, Optimization on Matrix Manifolds, optimum delivery idea, chance on Manifolds, Divergence Geometry and Ancillarity, Entropic Geometry, Tensor-Valued Mathematical Morphology, Machine/Manifold/Topology studying, Geometry of Audio Processing, Geometry of Inverse difficulties, Algebraic/Infinite dimensional/Banach details Manifolds, details Geometry Manifolds, and Algorithms on Manifolds.

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**Sample text**

New York (2002) 37. : Sobolev spaces, 2nd edn. Pure and Applied Mathematics, vol. 140. Elsevier/Academic Press, Amsterdam (2003) 38. : Statistical mechanics: A short treatise. Texts and Monographs in Physics. Springer, Berlin (1999) 39. : Pseudo-Boolean optimization. Discrete Appl. Math. 123(1-3), 155–225 (2002); Workshop on Discrete Optimization, DO 1999 (Piscataway, NJ) 40. : Dual connections in nonparametric classical information geometry. Technical Report math-ph/0104031 v1, arXiv (2001) 41.

The real function u → Kp (u) is inﬁnitely Fr´echet diﬀerentiable according to Prop. 3. The derivative is computed as 1 1 u − Kp (u) ) = 2 2 u=0 1 1 1 1 u − Kp (u) w − dw Kp (u) = w, exp 2 2 2 2 u=0 √ and ﬁnally applying the isometry 12 w → 12 w p. √ For each p ∈ P> deﬁne Ip : Hp P> u → pu ∈ T√p S. We can use the isometry Ip and the isometry Uyx of Prop. 10 to build an isometry dw (u → exp √ q Uqp = Iq−1 ◦ U√p ◦ Ip : Hp P> → Hq P> . as in the diagram (11). √ q U√ p T√p S O Uqp u = q −1/2 U√p (p1/2 u) (11) v→q−1/2 v u→p1/2 u Hp P > Substituting u = √ q / T√q S Uqp / Hq P > √ √ √ pu, x = p, y = q in (6), u − (1 + (x · y)) −1 (x + y)(u · y) = √ pu − 1 + √ pq dμ −1 √ √ ( p + q)( √ pu − 1 + Eq p q √ pqu dμ) = −1 √ √ ( p + q)Eq p u q so that Uqp u = p u − 1 + Eq q −1 p q 1+ p q Eq p u q (12) Proposition 13 1.

Pistone The exponential family pθ ∝ e−θx is deﬁned for θ > 0. The random variable u(x) = ex/2 has second moment ∞ ∞ (u(x))2 pθ (x) μ(dx) ∝ 0 e−(θ−1)x μ(dx), 0 which is ﬁnite for θ ≥ 1 and inﬁnite for 0 < θ < 1. We are going to show that the Hp ’s are actually isomorphic as Hilbert spaces and that our Hilbert bundle can be viewed as a push-back of the tangent bundle of the unit sphere of L2 (μ). In turn, this construction provides a derivation of the metric connection, see [6, VIII §4]. Connections on statistical manifolds are a key ingredient of Amari’s theory [4], while the non parametric version has been done in [9] and [23] in the Lp case, commutative and non commutative, respectively.