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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.

Author: Gianni Franchi, Jesús Angulo
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_44
Video: http://www.youtube.com/watch?v=Xwav5x4AWEY
Slides: Franchi_Quantization hyperspectral image.pdf
Presentation: https://www.see.asso.fr/node/14306
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Abstract:
A technique of spatial-spectral quantization of hyperspectral images is introduced. Thus a quantized hyperspectral image is just summarized by K spectra which represent the spatial and spectral structures of the image. The proposed technique is based on α-connected components on a region adjacency graph. The main ingredient is a dissimilarity metric. In order to choose the metric that best fit the hyperspectral data manifold, a comparison of different probabilistic dissimilarity measures is achieved.

Author: Jesús Angulo
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_43
Video: http://www.youtube.com/watch?v=L96rEpsPm4s
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiNklpWThBRHBBRVk
Presentation: https://www.see.asso.fr/node/14305
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Abstract:
Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

Author: Frank Nielsen, Olivier Schwander
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_42
Video: http://www.youtube.com/watch?v=-SQVbgQ5wuo
Slides: Schwander_bag-of-components.pdf
Presentation: https://www.see.asso.fr/node/14304
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Abstract:
Practical estimation of mixture models may be problematic when a large number of observations are involved: for such cases, online versions of Expectation-Maximization may be preferred, avoiding the need to store all the observations before running the algorithms. We introduce a new online method well-suited when both the number of observations is large and lots of mixture models need to be learned from different sets of points. Inspired by dictionary methods, our algorithm begins with a training step which is used to build a dictionary of components. The next step, which can be done online, amounts to populating the weights of the components given each arriving observation. The usage of the dictionary of components shows all its interest when lots of mixtures need to be learned using the same dictionary in order to maximize the return on investment of the training step. We evaluate the proposed method on an artificial dataset built from random Gaussian mixture models.

Author: Aqsa Shabbir, Geert Verdoolaege
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_41
Video: http://www.youtube.com/watch?v=rKBmPBPg0sU
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTiUHc1X0d6SUM1eEk
Presentation: https://www.see.asso.fr/node/14303
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Abstract:
We present a new texture discrimination method for textured color images in the wavelet domain. In each wavelet subband, the correlation between the color bands is modeled by a multivariate generalized Gaussian distribution with fixed shape parameter (Gaussian, Laplacian). On the corresponding Riemannian manifold, the shape of texture clusters is characterized by means of principal geodesic analysis, specifically by the principal geodesic along which the cluster exhibits its largest variance. Then, the similarity of a texture to a class is defined in terms of the Rao geodesic distance on the manifold from the texture’s distribution to its projection on the principal geodesic of that class. This similarity measure is used in a classification scheme, referred to as principal geodesic classification (PGC). It is shown to perform significantly better than several other classifiers.

Author: Lionel Bombrun, Salem Said, Yannick Berthoumieu
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_40
Video: http://www.youtube.com/watch?v=sZuednf5tZc
Slides: https://drive.google.com/open?id=0B0QKxsVtOaTieUFtU3JPb28yb1E
Presentation: https://www.see.asso.fr/node/14302
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Abstract:
The current paper introduces new prior distributions on the zero-mean multivariate Gaussian model, with the aim of applying them to the classification of covariance matrices populations. These new prior distributions are entirely based on the Riemannian geometry of the multivariate Gaussian model. More precisely, the proposed Riemannian Gaussian distribution has two parameters, the centre of mass ˉY and the dispersion parameter σ. Its density with respect to Riemannian volume is proportional to exp(−d2(Y;ˉY)), where d2(Y;ˉY) is the square of Rao’s Riemannian distance. We derive its maximum likelihood estimators and propose an experiment on the VisTex database for the classification of texture images.

Author: Michele Pavon, Tryphon Georgiou, Yonxin Chen
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_9
Video: http://www.youtube.com/watch?v=gxFFKApXJZU
Slides: Pavon_optimal mass transport.pdf
Presentation: https://www.see.asso.fr/node/14301
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Abstract:
We present an overview of our recent work on implementable solutions to the Schrödinger bridge problem and their potential application to optimal transport and various generalizations.

Author: Benoit Huyot, Jean-François Marcotorchino, Yves Mabiala
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_8
Video: http://www.youtube.com/watch?v=sYBXemYu3XY
Slides: Huyot_ Optimal transport independence vs indetermination.pdf
Presentation: https://www.see.asso.fr/node/14300
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Abstract:
This article leans on some previous results already presented in [10], based on the Fréchet’s works,Wilson’s entropy and Minimal Trade models in connectionwith theMKPtransportation problem (MKP, stands for Monge-Kantorovich Problem). Using the duality between “independance” and “indetermination” structures, shown in this former paper, we are in a position to derive a novel approach to design a copula, suitable and efficient for anomaly detection in IT systems analysis.

Author: Christian Leonard
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_7
Video: http://www.youtube.com/watch?v=EF3JsimmzH8
Slides: Leonard_geometric aspect schrodinger problem.pdf
Presentation: https://www.see.asso.fr/node/14299
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Absract:
This note presents a short review of the Schrödinger problem and of the first steps that might lead to interesting consequences in terms of geometry. We stress the analogies between this entropy minimization problem and the renowned optimal transport problem, in search for a theory of lower bounded curvature for metric spaces, including discrete graphs.

Author: Alfred Galichon, Scott Kominers, Simon Weber
DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_6
Video: http://www.youtube.com/watch?v=YSu3ECW7pz8
Slides: Galichon_Topics Equilibrium transportation.pdf
Presentation: https://www.see.asso.fr/node/14298
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Abstract:
In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the “nonlinear Bernstein-Schrödinger system”, which is well-known in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.