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(2017) GSI2017

Self-similar Geometry for Ad-Hoc Wireless Networks: Hyperfractals (slides) GSI2017
Affine-Invariant Orders on the Set of Positive-Defi nite Matrices Cyrus Mostajeran, Rodolphe Sepulchre GSI2017
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We introduce a family of orders on the set S+of positive definite matrices of dimension n derived from the homogeneous geometry of S+induced by the natural transitive action of the general linear group GL(n). The orders are induced by affne-invariant cone fields, which arise naturally from a local analysis of the orders that are compatible with the homogeneous structure of S+. We then revisit the well-known Löwner-Heinz theorem and provide an extension of this classical result derived using differential positivity with respect to affne-invariant cone elds.
Affine-Invariant Orders on the Set of Positive-Definite Matrices
Affine-Invariant Orders on the Set of Positive-Defi nite Matrices (slides) GSI2017
Von Mises-like probability density functions on surfaces Florence Nicol, Stephane Puechmorel GSI2017
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Directional densities were introduced in the pioneering work of von Mises, with the de nition of a rotationally invariant probability distribution on the circle. It was further generalized to more complex objects like the torus or the hyperbolic space. The purpose of the present work is to give a construction of equivalent objects on surfaces with genus larger than or equal to 2, for which an hyperbolic structure exists.
Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more
angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a
quotient of the hyperbolic plane.
Von Mises-like probability density functions on surfaces
Von Mises-like probability density functions on surfaces (slides) GSI2017
A topological view on forced oscillations and control of an inverted pendulum Ivan Polekhin GSI2017
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We consider a system of a planar inverted pendulum in a gravitational eld. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is constrained to be strictly above the pivot point (the rod cannot fall `below the horizon'). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions.
A topological view on forced oscillations and control of an inverted pendulum
A topological view on forced oscillations and control of an inverted pendulum (slides) GSI2017
Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation Fabrice Silva, Thomas Hélie GSI2017
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Port Hamiltonian systems (PHS) are open passive systems that full a power balance: they correspond to dynamical systems composed of energy-storing elements, energy-dissipating elements and external ports, endowed with a geometric structure (called Dirac structure) that encodes conservative interconnections. This paper presents a minimal PHS model of the full vocal apparatus. Elementary components are: (a) an ideal subglottal pressure supply, (b) a glottal ow in a mobile channel, (c) vocal-folds, (d) an acoustic resonator reduced to a single mode. Particular attention is paid to the energetic consistency of each component, to passivity and to the conservative interconnection. Simulations are presented. They show the ability of the model to produce a variety of regimes, including self-sustained oscillations. Typical healthy or pathological conguration laryngeal congurations are explored.
Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation
Self-oscillations of a vocal apparatus: a port-Hamiltonian formulation (slides) GSI2017
Geometry of Policy Improvement Guido Montùfar, Johannes Rauh GSI2017
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We investigate the geometry of optimal memoryless time independent decision making in relation to the amount of information that the acting agent has about the state of the system. We show that the expected long term reward, discounted or per time step, is maximized by policies that randomize among at most k actions whenever at most k world states are consistent with the agent's observation. Moreover, we show that the expected reward per time step can be studied in terms of the expected discounted reward. Our main tool is a geometric version of the policy improvement lemma, which identi es a polyhedral cone of policy changes in which the state value function increases for all states.
Geometry of Policy Improvement
Bounding the convergence time of local probabilistic evolution Alain Sarlette, Francesco Ticozzi, Simon Apers GSI2017
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Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry [1, 2], graph theory [4, 3] and Markov chain analysis [8, 5, 6]. In this paper we use isoperimetric inequalities to construct a bound on the convergence time of any local probabilistic evolution that leaves its limit distribution invariant. We illustrate how this general result leads to new bounds on convergence times beyond the explicit Markovian setting, among others on quantum dynamics.
Bounding the convergence time of local probabilistic evolution
Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups Vitaly Mikheev GSI2017
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A method of the solution of the main problem of homogeneous spaces thermodynamics for non-compact Lie groups is presented in the work. The method originates from formalism of non-commutative Fourier analysis based on method of coadjoint orbits. A formula that allows eciently evaluate heat kernel and statistic sum on non-compact Lie group is obtained. The algorithm of construction of high temperature heat kernel expansion is also discussed.
Method of orbits of co-associated representation in thermodynamics of the Lie non-compact groups
Constructing Universal, Non-Asymptotic Con dence Sets for Intrinsic Means on the Circle Matthias Glock, Thomas Hotz GSI2017
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We construct con dence sets for the set of intrinsic means on the circle based on i.i.d. data which guarantee coverage of the entire latter set for nite sample sizes without any further distributional assumptions.
Simulations demonstrate its applicability even when there are multiple intrinsic means.
Constructing Universal, Non-Asymptotic Condence Sets for Intrinsic Means on the Circle
On the Error Exponent of a Random Tensor with Orthonormal Factor Matrices Frank Nielsen, Rémy Boyer GSI2017
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In signal processing, the detection error probability of a random quantity is a fundamental and often dicult problem. In this work, we assume that we observe under the alternative hypothesis a noisy
tensor admitting a Tucker decomposition parametrized by a set of orthonormal factor matrices and a random core tensor of interest with fixed multilinear ranks. The detection of the random entries of the core tensor is hard to study since an analytic expression of the error probability is not tractable. To cope with this diculty, the Cherno Upper Bound (CUB) on the error probability is studied for this tensor-based detection problem. The tightest CUB is obtained for the minimal error exponent value, denoted by s*, that requires a costly numerical optimization algorithm. An alternative strategy to upper bound the error probability is to consider the Bhattacharyya Upper Bound (BUB) by prescribing s* = 1/2. In this case, the costly numerical optimization step is avoided but no guarantee exists on the tightness optimality of the BUB. In this work, a simple analytical expression of s? is provided with respect to the Signal to Noise Ratio (SNR). Associated to a compact expression of the CUB, an easily tractable expression of the tightest CUB is provided and studied. A main conclusion of this work is that the BUB is the tightest bound at low SNRs but this property is no longer true at higher SNRs.
On the Error Exponent of a Random Tensor with Orthonormal Factor Matrices
Women, diversity, science and Unconscious Bias Nina Miolane GSI2017
On Mixture and Exponential Connection by Open Arcs Barbara Trivellato, Marina Santacroce, Paola Siri GSI2017
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Results on mixture and exponential connections by open arcs are revised and used to prove additional duality properties of statistical models.
On Mixture and Exponential Connection by Open Arcs
Deformed exponential bundle: the linear growth case Giovanni Pistone, Luigi Montrucchio GSI2017
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Vigelis and Cavalcante extended the Naudts' deformed exponential families to a generic reference density. Here, the special case of Newton's deformed logarithm is used to construct an Hilbert statistical bundle for an infinite dimensional class of probability densities.
Deformed exponential bundle: the linear growth case
(Para-)Holomorphic Connections for Information Geometry Jun Zhang, Sergey Grigorian GSI2017
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On a statistical manifold (M, g, ∇), the Riemannian metric g is coupled to an (torsion-free) affine connection ∇, such that ∇g is totally symmetric; {∇g} is said to form “Codazzi coupling”. This leads ∇*, the g-conjugate of ∇, to have same torsion as that of ∇. In this paper, we investigate how statistical structure interacts with L in an almost Hermitian and almost para-Hermitian manifold (M,g,L) where L denotes, respectively, an almost complex structure J with J2 = - id or an almost para-complex structure K with K2 = - id. Starting with ∇L, the L-conjugate of ∇, we investigate the interaction of (generally
torsion-admitting) ∇ with L, and derive a necessary and sufficient condition (called “Torsion Balancing” condition) for L to be integrable, hence making (M,g,L)  (para-)Hermitian, and for ∇to be (para-)holomorphic.
We further derive that ∇L is (para-)holomorphic if and only if ∇is, and that ∇* is (para-)holomorphic if and only if ∇ is (para-)holomorphic and Codazzi coupled to g. Our investigations provide concise conditions to extend statistical manifolds to (para-)Hermitian manifolds.
(Para-)Holomorphic Connections for Information Geometry
Maximum likelihood estimation of Riemannian metrics from Euclidean data (slides) GSI2017
Classif ication Of Totally Umbilical CR-Statistical Submanifolds In Holomorphic Statistical Manifolds With Constant Holomorphic Curvature Aliya Naaz Siddiqui, Michel Nguiffo Boyom, Mohammad Hasan Shahid, Wan Ainun Mior Othman GSI2017
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In 1985, Amari [1] introduced an interesting manifold, i.e., statistical manifold in the context of information geometry. The geometry of such manifolds includes the notion of dual connections, called conjugate connections in affine geometry, it is closely related to affine geometry. A statistical structure is a generalization of a Hessian one, it connects Hessian geometry. 
In the present paper, we study CR-statistical submanifolds in holomorphic statistical manifolds. Some results on totally umbilical CR-statistical submanifolds with respect to - and - in holomorphic statistical manifolds with constant holomorphic curvature are obtained.
Classification Of Totally Umbilical CR-Statistical Submanifolds In Holomorphic Statistical Manifolds With Constant Holomorphic Curvature
Surface Matching Using Normal Cycles Joan Alexis Glaunès, Pierre Roussillon GSI2017
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In this article we develop in the case of triangulated meshes the notion of normal cycle as a dissimilarity measure introduced in [13].
Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. We derive the computational setting for discrete surfaces, using the Large
Deformation Diffeomorphic Metric Mapping framework as model for deformations.
We present experiments on real data and compare with the varifolds approach.
Surface Matching Using Normal Cycles
Process comparison combining signal power ratio and Jeffrey's divergence between unit-power signals Eric Grivel, Leo Legrand GSI2017
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Jeffrey's divergence (JD), the symmetric Kullback-Leibler (KL) divergence, has been used in a wide range of applications. In recent works, it was shown that the JD between probability density functions of k successive samples of autoregressive (AR) and/or moving average (MA) processes can tend to a stationary regime when the number k of variates increases. The asymptotic JD increment, which is the difference between two JDs computed for k and k-1 successive variates tending to a nite constant value when k increases, can hence be useful to compare the random processes. However, interpreting the value of the asymptotic JD increment is not an easy task as it depends on too many parameters, i.e. the AR/MA parameters and the driving-process variances. In this paper, we propose to compute the asymptotic JD increment between the processes that have been normalized so that their powers are equal to 1. Analyzing the resulting JD on the one hand and the ratio between the original signal powers on the other hand makes the interpretation easier. Examples are provided to illustrate the relevance of this way to operate with the JD.
Process comparison combining signal power ratio and Jeffrey's divergence between unit-power signals
Maximum likelihood estimation of Riemannian metrics from Euclidean data Georgios Arvanitidis, Lars Kai Hansen, Søren Hauberg GSI2017
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Euclidean data often exhibit a nonlinear behavior, which may be modeled by assuming the data is distributed near a nonlinear submanifold in the data space. One approach to find such a manifold is to estimate a Riemannian metric that locally models the given data.
Data distributions with respect to this metric will then tend to follow the nonlinear structure of the data. In practice, the learned metric rely on parameters that are hand-tuned for a given task. We propose to estimate
such parameters by maximizing the data likelihood under the assumed distribution. This is complicated by two issues: (1) a change of parameters imply a change of measure such that different likelihoods are incomparable; (2) some choice of parameters renders the numerical calculation of distances and geodesics unstable such that likelihoods cannot be evaluated. As a practical solution, we propose to (1) re-normalize likelihoods with respect to the usual Lebesgue measure of the data space, and (2) to bound the likelihood when its exact value is unattainable. We provide practical algorithms for these ideas and illustrate their use on synthetic data, images of digits and faces, as well as signals extracted from EEG scalp measurements.
Maximum likelihood estimation of Riemannian metrics from Euclidean data
Uniform observability of linear time-varying systems and application to robotics problems (slides) GSI2017
Information Distances in Stochastic Resolution Analysis Radmila Pribić GSI2017
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A stochastic approach to resolution is explored that uses information distances computed from the geometry of data models characterized by the Fisher information in cases with spatial-temporal measurements for multiple parameters. Stochastic resolution includes probability of resolution at signal-to-noise ratio (SNR) and separation of targets. The probability of resolution is assessed by exploiting different information distances in likelihood ratios. Taking SNR into account is especially relevant in compressive sensing (CS) due to its fewer measurements. Our stochastic resolution is also compared with actual resolution from sparse-signal processing that is nowadays a major part of any CS sensor. Results demonstrate the suitability of the proposed analysis due to its ability to include crucial impacts on the performance guarantees: array configuration or sensor design, SNR, separation and probability of resolution.
Information Distances in Stochastic Resolution Analysis
Information Distances in Stochastic Resolution Analysis (slides) GSI2017
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics Geert Verdoolaege GSI2017
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We present a new regression method called geodesic least squares (GLS), which is particularly robust against data and model uncertainty. It is based on minimization of the Rao geodesic distance on a probabilistic manifold. We apply GLS to Tully-Fisher scaling of the total baryonic mass vs. the rotation velocity in disk galaxies and we show the excellent robustness properties of GLS for estimating the coefficients and the tightness of the scaling.
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics
Geodesic Least Squares Regression on the Gaussian Manifold with an Application in Astrophysics (slides) GSI2017