Author: Anass Bellachehab, Jérémie Jakubowicz
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_75
Video: http://www.youtube.com/watch?v=wHWz3og1E5Q
Slides: Bellachehab_Gossip in CAT(K).pdf
Presentation: https://www.see.asso.fr/node/14297
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
In the context of sensor networks, gossip algorithms are a popular, well established technique, for achieving consensus when sensor data are encoded in linear spaces. Gossip algorithms also have several extensions to non linear data spaces. Most of these extensions deal with Riemannian manifolds and use Riemannian gradient descent. This paper, instead, studies gossip in a broader CAT(k) metric setting, encompassing, but not restricted to, several interesting cases of Riemannian manifolds. As it turns out, convergence can be guaranteed as soon as the data lie in a small enough ball of a mere CAT(k) metric space. We also study convergence speed in this setting and establish linear rates of convergence.

Author: Tudor Ratiu
Institution: Section de Mathematiques, Faculté des Sciences de Base EPFL, Lausanne, Switzerland
Website: http://cag.epfl.ch/page-39504-en.html
Video: http://www.youtube.com/watch?v=Dwk4U9jrGPM
Slides: Ratiu_symmetry methods.pdf
Presentation: https://www.see.asso.fr/node/14276
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
The goal of these lectures is to show the influence of symmetry in various aspects of theoretical mechanics. Canonical actions of Lie groups on Poisson manifolds often give rise to conservation laws, encoded in modern language by the concept of momentum maps. Reduction methods lead to a deeper understanding of the dynamics of mechanical systems. Basic results in singular Hamiltonian reduction will be presented. The Lagrangian version of reduction and its associated variational principles will also be discussed. The understanding of symmetric bifurcation phenomena in for Hamiltonian systems are based on these reduction techniques. Time permitting, discrete versions of these geometric methods will also be discussed in the context of examples from elasticity.

References

• Demoures, F., Gay-Balmaz, F., Ratiu, T.S.: Multisymplectic variational integrators and space/time symplecticity, Communications in Analysis and Applications (2015), to appear
• Gay-Balmaz, F., Ratiu, T.S.: The geometric structure of complex fluids, Advances in Applied Mathematics, 42 (2009), 176--275
• Gay-Balmaz, F., Marsden, J.E., Ratiu, T.S.: Reduced variational formulations in free boundary continuum mechanics, Journ. Nonlinear Sci., 22 (2012), 463-497
• Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)
• Marsden, J.E, Ratiu, T.S.: Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, second edition, Springer Verlag, (1998)
• Marsden, J.E, Misiolek, G., Ortega, J.-P., Perlmutter, M., Ratiu, T.S.: Hamiltonian Reduction by Stages, Springer Lecture Notes in Mathematics, 1913, Springer-Verlag, New York (2007)
• Marsden J.E, West M.: Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357–514.
• Ortega, J.-P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction, Progress in Mathematics 222, Birkh"auser, Boston (2004)

Bio:

• BA in Mathematics, University of Timisoara, Romania, 1973
• MA in Applied Mathematics, University of Timisoara, Romania, 1974
• Ph.D. in Mathematics, University of California, Berkeley, 1980
• T.H. Hildebrandt Research Assistant Professor, University of Michigan, Ann Arbor, USA 1980-1983
• Associate Professor of Mathematics, University of Arizona, Tuscon, USA 1983-1988
• Professor of Mathematics, University of California, Santa Cruz, USA, 1988-2001
• Chaired Professor of Mathematics, Ecole Polytechnique Federale de Lausanne, Switzerland, 1998 - present
• Professor of Mathematics, Skolkovo Institute of Science and Technonology, Moscow, Russia, 2014 - present

Author: Dena Asta
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_83
Video: Not available
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiX0lnY3YzZDV1Y0E
Presentation: https://www.see.asso.fr/node/14314
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
We introduce a novel kernel density estimator for a large class of symmetric spaces and prove a minimax rate of convergence as fast as the minimax rate on Euclidean space. We prove a minimax rate of convergence proven without any compactness assumptions on the space or Hölder-class assumptions on the densities. A main tool used in proving the convergence rate is the Helgason-Fourier transform, a generalization of the Fourier transform for semisimple Lie groups modulo maximal compact subgroups. This paper obtains a simplified formula in the special case when the symmetric space is the 2-dimensional hyperboloid.

Author: Florence Nicol, Stephane Puechmorel
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_82
Video: http://www.youtube.com/watch?v=EerzzjhGEGs
Slides: Puechmorel_entropy minimizing curves.pdf
Presentation: https://www.see.asso.fr/node/14313
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
Air traffic management (ATM) aims at providing companies with a safe and ideally optimal aircraft trajectory planning. Air traffic controllers act on flight paths in such a way that no pair of aircraft come closer than the regulatory separation norm. With the increase of traffic, it is expected that the system will reach its limits in a near future: a paradigm change in ATM is planned with the introduction of trajectory based operations. This paper investigate a mean of producing realistic air routes from the output of an automated trajectory design tool. For that purpose, an entropy associated with a system of curves is defined and a mean of iteratively minimizing it is presented. The network produced is suitable for use in a semi-automated ATM system with human in the loop.

Author: Emmanuel Chevallier, Ivar Farup, Jesús Angulo
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_81
Video: http://www.youtube.com/watch?v=gYLcRA4q2W0
Slides: Chevallier_Color Histograms.pdf
Presentation: https://www.see.asso.fr/node/14312
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
We address here the problem of perceptual colour histograms. The Riemannian structure of perceptual distances is measured through standards sets of ellipses, such as Macadam ellipses. We propose an approach based on local Euclidean approximations that enables to take into account the Riemannian structure of perceptual distances, without introducing computational complexity during the construction of the histogram.

Author: Emmanuel Chevallier, Frédéric Barbaresco, Jesús Angulo
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_80
Video: http://www.youtube.com/watch?v=lDIEU2vVubY
Slides: Chevallier_Probability Density Estimation Radar.pdf
Presentation: https://www.see.asso.fr/node/14311
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
The two main techniques of probability density estimation on symmetric spaces are reviewed in the hyperbolic case. For computational reasons we chose to focus on the kernel density estimation and we provide the expression of Pelletier estimator on hyperbolic space. The method is applied to density estimation of reflection coefficients derived from radar observations.

Author: Alexis Decurninge
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_13
Video: http://www.youtube.com/watch?v=lF0PbFujrGs
Slides: Decurninge_MultivariateLmoments.pdf
Presentation: https://www.see.asso.fr/node/14310
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
Univariate L-moments are expressed as projections of the quantile function onto an orthogonal basis of univariate polynomials. We present multivariate versions of L-moments expressed as collections of orthogonal projections of a multivariate quantile function on a basis of multivariate polynomials. We propose to consider quantile functions defined as transports from the uniform distribution on [0; 1] d onto the distribution of interest and present some properties of the subsequent L-moments. The properties of estimated L-moments are illustrated for heavy-tailed distributions.

Author: Jean-Michel Loubes, Thibaut Le Gouic
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_12
Video: http://www.youtube.com/watch?v=lzGxDemDCkc
Slides: LeGouic_Barycenter Wasserstein spaces.pdf
Presentation: https://www.see.asso.fr/node/14309
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
We study barycenters in the Wasserstein space Pp(E) of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure ℙ on Pp(E) as its Fréchet mean. The paper establishes its existence and states consistency with respect to ℙ. We thus extends previous results on ℝ d , with conditions on ℙ or on the sequence converging to ℙ for consistency.

Author: Reiner Lenz
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_11
Video: http://www.youtube.com/watch?v=ghrJtRmzTXE
Slides: Lenz_ generalized Pareto distributions.pdf
Presentation: https://www.see.asso.fr/node/14308
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
We introduce the generalized Pareto distributions as a statistical model to describe thresholded edge-magnitude image filter results. Compared to the more commonWeibull or generalized extreme value distributions these distributions have at least two important advantages, the usage of the high threshold value assures that only the most important edge points enter the statistical analysis and the estimation is computationally more efficient since a much smaller number of data points have to be processed. The generalized Pareto distributions with a common threshold zero form a two-dimensional Riemann manifold with the metric given by the Fisher information matrix. We compute the Fisher matrix for shape parameters greater than -0.5 and show that the determinant of its inverse is a product of a polynomial in the shape parameter and the squared scale parameter. We apply this result by using the determinant as a sharpness function in an autofocus algorithm. We test the method on a large database of microscopy images with given ground truth focus results. We found that for a vast majority of the focus sequences the results are in the correct focal range. Cases where the algorithm fails are specimen with too few objects and sequences where contributions from different layers result in a multi-modal sharpness curve. Using the geometry of the manifold of generalized Pareto distributions more efficient autofocus algorithms can be constructed but these optimizations are not included here.

Author: Julien Rabin, Nicolas Papadakis
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_10
Video: http://www.youtube.com/watch?v=k8obU3o5LEo
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiZFV3TFZWYXV5NEE
Presentation: https://www.see.asso.fr/node/14307
Creative Commons Attribution-ShareAlike 4.0 International

Abstract:
Optimal transport (OT) is a major statistical tool to measure similarity between features or to match and average features. However, OT requires some relaxation and regularization to be robust to outliers. With relaxed methods, as one feature can be matched to several ones, important interpolations between different features arise. This is not an issue for comparison purposes, but it involves strong and unwanted smoothing for transfer applications. We thus introduce a new regularized method based on a non-convex formulation that minimizes transport dispersion by enforcing the one-to-one matching of features. The interest of the approach is demonstrated for color transfer purposes.