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(2017) GSI2017

Sigma Point Kalman Filtering on Matrix Lie Groups Applied to the SLAM Problem David Evan Zlotnik, James Richard Forbes GSI2017
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This paper considers sigma point Kalman ltering on matrix Lie groups. Sigma points that are elements of a matrix Lie group are generated using the matrix exponential. Computing the mean and covariance using the sigma points via weighted averaging and effective use of the matrix natural logarithm, respectively, is discussed. The specific details of estimating landmark locations, and the position and attitude of a vehicle relative to the estimated landmark locations, is considered.
Sigma Point Kalman Filtering on Matrix Lie Groups Applied to the SLAM Problem
Multi-Scale Activity Estimation with Spatial Abstractions GSI2017
Torsional Newton-Cartan Geometry Athanasios Chatzistavrakidis, Eric Bergshoe, Jan Rosseel, Luca Romano GSI2017
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Having in mind applications to Condensed Matter Physics, we perform a null-reduction of General Relativity in d + 1 spacetime dimensions thereby obtaining an extension to arbitrary torsion of the twistless-torsional Newton-Cartan geometry. We shortly discuss the implementation of the equations of motion.
Torsional Newton-Cartan Geometry
Multi-Scale Activity Estimation with Spatial Abstractions Florian T. Pokorny, Majd Hawasly, Subramanian Ramamoorthy GSI2017
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Estimation and forecasting of dynamic state are fundamental to the design of autonomous systems such as intelligent robots. State-ofthe art algorithms, such as the particle lter, face computational limitations when needing to maintain beliefs over a hypothesis space that is made large by the dynamic nature of the environment. We propose an algorithm that utilises a hierarchy of such lters, exploiting a ltration arising from the geometry of the underlying hypothesis space. In addition to computational savings, such a method can accommodate the availability of evidence at varying degrees of coarseness.We show, using synthetic trajectory datasets, that our method achieves a better normalised error in prediction and better time to convergence to a true class when compared against baselines that do not similarly exploit geometric structure.
Multi-Scale Activity Estimation with Spatial Abstractions
About the de nition of port variables for contact Hamiltonian systems Arjan van der Schaft, Bernhard Maschke GSI2017
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Extending the formulation of reversible thermodynamical transformations to the formulation of irreversible transformations of open thermodynamical systems di erent classes of nonlinear control systems has been de ned in terms of control Hamiltonian systems de ned on a contact manifold. In this paper we discuss the relation between the de nition of variational control contact systems and the input-output contact systems . We have rst given an expression of the variational control contact systems in terms of a nonlinear control systems. Secondly we have shown that the conservative input-output contact systems are a subclass of the contact variational systems with integrable output dynamics.
About the denition of port variables for contact Hamiltonian systems
A stochastic look at geodesics on the sphere Jean-Claude Zambrini, Marc Arnaudon GSI2017
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We describe a method allowing to deform stochastically the completely integrable (deterministic) system of geodesics on the sphere S2 in a way preserving all its symmetries.
A stochastic look at geodesics on the sphere
About the de nition of port variables for contact Hamiltonian systems (slides) GSI2017
Weighted Closed Form Expressions Based on Escort Distributions for Rényi Entropy Rates of Markov Chains Loïck Lhote, Philippe Regnault, Valérie Girardin GSI2017
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For Markov chains with transition probabilities pij , the Shannon entropy rate is well-known to be equal to the sum of the −Σj pij log pij weighted by the stationary distribution. This expression derives from the chain rule specific to Shannon entropy. For an ergodic Markov chain, the stationary distribution is the limit of the marginal distributions of the chain. 
Here a weighted expression for the R´enyi entropy rate is derived from a chain rule, that involves escort distributions of the chain, specific to Rényi entropy. Precisely, the rate is equal to the logarithm of a weighted sum of the −Σj ps ij , where s is the parameter of Rényi entropy. The weighting distribution is the normalized left eigenvector of the maximum eigenvalue of the matrix (psij). This distribution, that plays the role of the stationary distribution for Shannon entropy, is shown to be the limit of marginals of escort distributions of the chain.
Weighted Closed Form Expressions Based on Escort Distributions for Rényi Entropy Rates of Markov Chains
Positive Signal Spaces and the Mehler-Fock Transform Reiner Lenz GSI2017
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Eigenvector expansions and perspective projections are used to decompose a space of positive functions into a product of a half-axis and a solid unit ball. This is then used to construct a conical coordinate system where one component measures the distance to the origin, a radial measure of the distance to the axis and a unit vector describing the position on the surface of the ball. A Lorentz group is selected as symmetry group of the unit ball which leads to the Mehler-Fock transform as the Fourier transform of functions depending an the radial coordinate only. The theoretical results are used to study statistical properties of edge magnitudes computed from databases of image patches. The constructed radial values are independent of the orientation of the incoming light distribution (since edge-magnitudes are used), they are independent of global intensity changes (because of the perspective projection) and they characterize the second order statistical moment properties of the image patches. Using a large database of images of natural scenes it is shown that the generalized extreme value distribution provides a good statistical model of the radial components. Finally, the visual properties of textures are characterized using the Mehler-Fock transform of the probability density function of the generalized extreme value distribution.
Positive Signal Spaces and the Mehler-Fock Transform
Quantification of Model Risk: Data Uncertainty (slides) GSI2017
Sasakian statistical manifolds II Hitoshi Furuhata GSI2017
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Sasakian statistical manifolds II
Translations in the exponential Orlicz space with Gaussian weight (slides) GSI2017
Analysis of Optimal Transport Related Misfi t Functions in Seismic Imaging (slides) GSI2017
Analysis of Optimal Transport Related Mis fit Functions in Seismic Imaging Björn Engquist, Yunan Yang GSI2017
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We analyze different misfit functions for comparing synthetic and observed data in seismic imaging, for example, the Wasserstein metric and the conventional least-squares norm.We revisit the convexity and insensitivity to noise of the Wasserstein metric which demonstrate the robustness of the metric in seismic inversion. Numerical results illustrate that full waveform inversion with quadratic Wasserstein metric can often effectively overcome the risk of local minimum trapping in the optimization part of the algorithm. A mathematical study on Fréchet derivative with respect to the model parameters of the objective functions further illustrates the role of optimal transport maps in this iterative approach. In this context we refer to the objective function as misfit. A realistic numerical example is presented.
Analysis of Optimal Transport Related Misfit Functions in Seismic Imaging
Translations in the exponential Orlicz space with Gaussian weight Giovanni Pistone GSI2017
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We study the continuity of space translations on non-parametric exponential families based on the exponential Orlicz space with Gaussian reference density.
Translations in the exponential Orlicz space with Gaussian weight
Normalization and ϕ-function: definition and consequences (slides) GSI2017
Normalization and ϕ-function: definition and consequences Charles Casimiro Cavalcante, Francisca L. J. Vieira, Luiza Helena Félix de Andrade, Rui F. Vigelis GSI2017
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It is known from the literature that a ϕ-function may be used to construct the ϕ -families of probability distributions. In this paper, we assume that one of the properties in the definition of ϕ -function is not satisfied and we analyze the behavior of the normalizing function near the boundary of its domain. As a consequence, we find a measurable function that does not belong to the Musielak–Orlicz class, but the normalizing function applied to this found function converges to a finite value near the boundary of its domain. We conclude showing that this change in the definition of ϕ -function affects the behavior of the normalizing function.
Normalization and ϕ-function: definition and consequences
Quantification of Model Risk: Data Uncertainty Carlos Vázquez, Pedro Pablo Perez-Velasco, Zuzana Krajcovicova GSI2017
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Worldwide regulations oblige financial institutions to manage and address model risk (MR) as any other type of risk. MR quantification is essential not only in meeting these requirements but also for institution’s basic internal operative. 
In (5) the authors introduce a framework for the quantification of MR based on information geometry. The framework is applicable in great generality and accounts for different sources of MR during the entire lifecycle of a model. The aim of this paper is to extend the framework in (5) by studying its relation with the uncertainty associated to the data used for building the model.We define a metric on the space of samples in order to measure the data intrinsic distance, providing a novel way to probe the data for insight, allowing us to work on the sample space, gain business intuition and access tools such as perturbation methods.
Quantification of Model Risk: Data Uncertainty
Optimization in the Space of Smooth Shapes Kathrin Welker GSI2017
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The theory of shape optimization problems constrained by partial differential equations is connected with the differential-geometric structure of the space of smooth shapes.
Optimization in the Space of Smooth Shapes
On the existence of paths connecting probability distributions (slides) GSI2017
On the existence of paths connecting probability distributions Charles Casimiro Cavalcante, Luiza Helena Félix de Andrade, Rui F. Vigelis GSI2017
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We introduce a class of paths defined in terms of two deformed exponential functions. Exponential paths correspond to a special case of this class of paths. Then we give necessary and sufficient conditions for any two probability distributions being path connected.
On the existence of paths connecting probability distributions
Optimal Transport to Rényi Entropies Olivier Rioul GSI2017
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Recently, an optimal transportation argument was proposed by the author to provide a simple proof of Shannon's entropy-power inequality. Interestingly, such a proof could have been given by Shannon himself in his 1948 seminal paper. In fact, by 1948 Shannon established all the ingredients necessary for the proof and the transport argument takes the form of a simple change of variables.
In this paper, the optimal transportation argument is extended to Rényi entropies in relation to Shannon's entropy-power inequality and to a reverse version involving a certain conditional entropy. The transportation argument turns out to coincide with Barthe's proof of sharp direct and reverse Young's convolutional inequalities and can be applied to derive

recent Rényi entropy-power inequalities.

Optimal Transport to Rényi Entropies
Parallel transport in shape analysis: a scalable numerical scheme Alexandre Bône, Benjamin Charlier, Maxime Louis, Stanley Durrleman GSI2017
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The analysis of manifold-valued data requires efficient tools from Riemannian geometry to cope with the computational complexity at stake. This complexity arises from the always-increasing dimension of the data, and the absence of closed-form expressions to basic operations such as the Riemannian logarithm. In this paper, we adapt a generic numerical scheme recently introduced for computing parallel transport along geodesics in a Riemannian manifold to finite-dimensional manifolds of diffeomorphisms. We provide a qualitative and quantitative analysis of its behavior on high-dimensional manifolds, and investigate an application with the prediction of brain structures progression.
Parallel transport in shape analysis: a scalable numerical scheme
Self-similar Geometry for Ad-Hoc Wireless Networks: Hyperfractals Dalia Popescu, Philippe Jacquet GSI2017
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In this work we study a Poisson patterns of xed and mobile nodes on lines designed for 2D urban wireless networks. The particularity of the model is that, in addition to capturing the irregularity and variability of the network topology, it exploits self-similarity, a characteristic of urban wireless networks. The pattern obeys to "Hyperfractal" measures which show scaling properties corresponding to an apparent dimension larger than 2. The hyperfractal pattern is best suitable for capturing the traffic over the streets and highways in a city. The scaling e ect depends on the hyperfractal dimensions. Assuming propagation on axes, we prove results on the scaling of routing metrics and connectivity graph.
Self-similar Geometry for Ad-Hoc Wireless Networks: Hyperfractals