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(2017) GSI2017

A Riemanian Approach to Blob Detection in Manifold-Valued Images Aleksei Shestov, Mikhail Kumskov GSI2017
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This paper is devoted to the problem of blob detection in manifold-valued images. Our solution is based on new de nitions of blob response functions. We de ne the blob response functions by means of curvatures of an image graph, considered as a submanifold. We call the proposed framework Riemannian blob detection. We prove that our approach can be viewed as a generalization of the grayscale blob detection technique. An expression of the Riemannian blob response functions through the image Hessian is derived. We provide experiments for the case of vector-valued images on 2D surfaces: the proposed framework is tested on the task of chemical compounds classi cation.
A Riemanian Approach to Blob Detection in Manifold-Valued Images
A Riemanian Approach to Blob Detection in Manifold-Valued Images (slides) GSI2017
Natural Langevin Dynamics for Neural Networks Gaétan Marceau-Caron, Yann Ollivier GSI2017
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One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGLD is a standard stochastic gradient descent to which is added a controlled amount of noise, specifically scaled so that the parameter converges in law to the posterior distribution [WT11,TTV16]. The posterior predictive distribution can be approximated by an ensemble of samples from the trajectory. Choice of the variance of the noise is known to impact the practical behavior of SGLD: for instance, noise should be smaller for sensitive parameter directions. Theoretically, it has been suggested to use the inverse Fisher information matrix of the model as the variance of the noise, since it is also the variance of the Bayesian posterior [PT13,AKW12,GC11]. But the Fisher matrix is costly to compute for large-dimensional models. 
Here we use the easily computed Fisher matrix approximations for deep neural networks from [MO16,Oll15]. The resulting natural Langevin dynamics combines the advantages of Amari’s natural gradient descent and
Fisher-preconditioned Langevin dynamics for large neural networks. 
Small-scale experiments on MNIST show that Fisher matrix preconditioning brings SGLD close to dropout as a regularizing technique.
Natural Langevin Dynamics for Neural Networks
A variational formulation for uid dynamics with irreversible processes François Gay-Balmaz, Hiroaki Yoshimura GSI2017
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In this paper, we present a variational formulation for heat conducting viscous uids, which extends the Hamilton principle of continuum mechanics to include irreversible processes. This formulation follows from the general variational description of nonequilibrium thermodynamics introduced in [3, 4] for discrete and continuum systems. It relies on the concept of thermodynamic displacement. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of the entropy production associated to the irreversible processes involved.
A variational formulation for  uid dynamics with irreversible processes
Natural Langevin Dynamics for Neural Networks (slides) GSI2017
Warped metrics for location-scale models Salem Said, Yannick Berthoumieu GSI2017
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This paper argues that a class of Riemannian metrics, called warped metrics, plays a fundamental role in statistical problems involving location-scale models. The paper reports three new results : i) the Rao-Fisher metric of any location-scale model is a warped metric, provided that this model satis es a natural invariance condition, ii) the analytic expression of the sectional curvature of this metric, iii) the exact analytic solution of the geodesic equation of this metric. The paper applies these new results to several examples of interest, where it shows that warped metrics turn location-scale models into complete Riemannian manifolds of negative sectional curvature. This is a very suitable situation for developing algorithms which solve problems of classification and on-line estimation. Thus, by revealing the connection between warped metrics and location-scale models, the present paper paves the way to the introduction of new ecient statistical algorithms.
Warped metrics for location-scale models
Women in Science : 150th Marie Curie Birthday Alice Barbara Tumpach GSI2017
Information Geometry Under Monotone Embedding. Part II: Geometry Jan Naudts, Jun Zhang GSI2017
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The rho-tau embedding of a parametric statistical model defines both a Riemannian metric, called “rho-tau metric”, and an alpha family of rho-tau connections. We give a set of equivalent conditions for such a metric to become Hessian and for the ±1-connections to be dually flat. Next we argue that for any choice of strictly increasing functions ρ(u) and τ (u) one can construct a statistical model which is Hessian and phi-exponential. The metric derived from the escort expectations is conformally equivalent with the rho-tau metric.
Information Geometry Under Monotone Embedding. Part II: Geometry
Information Geometry Under Monotone Embedding. Part II: Geometry (slides) GSI2017
The Cramér-Rao inequality on singular statistical models (slides) GSI2017
The Cramér-Rao inequality on singular statistical models Hong Van Le, Jürgen Jost, Lorenz Schwachhöfer GSI2017
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We introduce the notions of essential tangent space and reduced Fisher metric and extend the classical Cramér-Rao inequality to 2-integrable (possibly singular) statistical models for general ϕ-estimators, where ϕ is a V-valued feature function and V is a topological vector space. We show the existence of a ϕ-efficient estimator on strictly singular statistical models associated with a finite sample space and on a class of infinite dimensional exponential models that have been discovered by Fukumizu. We conclude that our general Cramér-Rao inequality is optimal.
The Cramér-Rao inequality on singular statistical models
Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms Michel Nguiffo Boyom, Mohammad Hasan Shahid, Mohammed Jamali, Mohd Aquib GSI2017
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Statistical manifolds are abstract generalizations of statistical models introduced by Amari [1] in 1985. Such manifolds have been studied in terms of information geometry which includes the notion of dual connections, called conjugate connection in affine geometry. Recently, Furuhata [5] de ned and studied the properties of holomorphic statistical space forms.
In this paper, we obtain the generalized Wintgen type inequality for Lagrangian submanifolds in holomorphic statistical space forms. We also obtain condition under which the submanifold becomes minimal or H is some scalar multiple of H*.
Generalized Wintegen type inequality for Lagrangian submanifolds in holomorphic Statistical space forms
Extremal curves in Wasserstein space Giovanni Conforti, Michele Pavon GSI2017
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We show that known Newton-type laws for Optimal Mass Transport, Schrodinger Bridges and the classic Madelung fluid can be derived from variational principles on Wasserstein space. The second order differential equations are accordingly obtained by annihilating the first variation of a suitable action.
Extremal curves in Wasserstein space
Anisotropic Edge-based Balloon Eikonal Active Contours (slides) GSI2017
Shape analysis on Lie groups and homogeneous spaces (sllides) GSI2017
Shape analysis on Lie groups and homogeneous spaces Alexander Schmeding, Elena Celledoni, Markus Eslitzbichler, Sølve Eidnes GSI2017
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In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT).
We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometry of the natural Lie
group actions on these spaces.
Shape analysis on Lie groups and homogeneous spaces
Anomaly detection in network traffic with a relationnal clustering criterion Damien Nogues GSI2017
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Unsupervised anomaly detection is a very promising technique for intrusion detection. Among many other approaches, clustering algorithms have often been used to perform this task. However, to describe network traffic, both numerical and categorical variables are commonly used. So most clustering algorithms are not very well-suited to such data. Few clustering algorithms have been proposed for such heterogeneous data. Many approaches do not possess suitable complexity.
In this article, using Relational Analysis, we propose a new, unified clustering criterion. This criterion is based on a new similarity function for values in a lattice, which can then be applied to both numerical and categorical variables. Finally we propose an optimisation heuristic of this criterion and an anomaly score which outperforms many state of the art solutions.
Anomaly detection in network traffic with a relationnal clustering criterion
Log-Determinant Divergences Between Positive De nite Hilbert-Schmidt Operators Hà Quang Minh GSI2017
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The current work generalizes the author's previous work on the infinite-dimensional Alpha Log-Determinant (Log-Det) divergences and Alpha-Beta Log-Det divergences, de ned on the set of positive definite unitized trace class operators on a Hilbert space, to the entire Hilbert manifold of positive de nite unitized Hilbert-Schmidt operators. 
This generalization is carried out via the introduction of the extended Hilbert-Carleman determinant for unitized Hilbert-Schmidt operators, in addition to the previously introduced extended Fredholm determinant for unitized trace class operators. The resulting parametrized family of Alpha-Beta Log-Det divergences is general and contains many divergences between positive de nite unitized Hilbert-Schmidt operators as special cases, including the infinite-dimensional generalizations of the affine-invariant Riemannian distance and symmetric Stein divergence.
Log-Determinant Divergences Between Positive Denite Hilbert-Schmidt Operators
Bootstrapping Descriptors for Non-Euclidean Data Benjamin Eltzner, Stephan Huckemann GSI2017
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For data carrying a non-Euclidean geometric structure it is natural to perform statistics via geometric descriptors. Typical candidates are means, geodesics, or more generally, lower dimensional subspaces, which carry specific structure. Asymptotic theory for such descriptors is slowly unfolding and its application to statistical testing usually requires one more step: Assessing the distribution of such descriptors.
To this end, one may use the bootstrap that has proven to be a very successful tool to extract inferential information from small samples. In this communication we review asymptotics for descriptors of manifold valued data and study a non-parametric bootstrap test that aims at a high power, also under the alternative.
Bootstrapping Descriptors for Non-Euclidean Data
Log-Determinant Divergences Between Positive De nite Hilbert-Schmidt Operators 5SLLIDES° GSI2017
Anisotropic Edge-based Balloon Eikonal Active Contours Da Chen, Laurent Cohen GSI2017
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In this paper, we propose a new edge-based active contour model for image segmentation and curve evolution by an asymmetric Finsler metric and the corresponding minimal paths. We consider the edge anisotropy information and the balloon force term to build a Finsler metric comprising of a symmetric quartic term and an asymmetric linear term. Unlike the traditional geodesic active contour model where the curve evolution is carried out by the level set framework, we search for a more robust optimal curve by solving an Eikonal partial differential equation (PDE) associated to the Finsler metrics. Moreover, we present an interactive way for geodesics extraction and closed contour evolution. Compared to the level set-based geodesic active contour model, our model is more robust to spurious edges, and also more efficient in numerical solution.
Anisotropic Edge-based Balloon Eikonal Active Contours
Density estimation for Compound Cox processes on hyperspheres Florent Chatelain, Jonathan H. Manton, Nicolas Le Bihan GSI2017
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Cox multiple scattering processes on hyperspheres are a class of doubly stochastic Poisson processes that can be used to describe scattering phenomenon in Physics (optics, micro-waves, acoustics, etc.). In this article, we present an EM (Expectation Maximization) technique to estimate the concentration parameter of a Compound Cox process with values on hyperspheres. The proposed algorithm is based on an approximation formula for multiconvolution of von Mises Fisher densities on spheres of any dimension.
Density estimation for Compound Cox processes on hyperspheres
An Approach to Dynamical Distance Geometry Antonio Mucherino, Douglas Gonçalves GSI2017
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We introduce the dynamical distance geometry problem (dynDGP), where vertices of a given simple weighted undirected graph are to be embedded at different times t. Solutions to the dynDGP can be seen as motions of a given set of objects. In this work, we focus our attention on a class of instances where motion inter-frame distances are not available, and reduce the problem of embedding every motion frame as a static distance geometry problem. Some preliminary computational experiments are presented.
An Approach to Dynamical Distance Geometry
Particle observers for contracting dynamical systems Jean-Jacques Slotine, Silvère Bonnabel GSI2017
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In the present paper we consider a class of partially observed dynamical systems. As in the Rao-Blackwellized particle lter (RBPF) paradigm (see e.g., [Doucet et al. (2000)]), we assume the state x can be broken into two sets of variables x = (z; r) and has the property that conditionally on z the system's dynamics possess geometrical contraction properties, or is amenable to such a system by using a nonlinear observer whose dynamics possess contraction properties. Inspired by the RBPF we propose to use particles to approximate the r variable and to use a simple copy of the dynamics (or an observer) to estimate the rest of the state. 
This has the bene ts of 1- reducing the computational burden (a particle filter would sample the variable x also), which is akin to the interest of the RBPF, 2- coming with some indication of stability stemming from contraction (actual proofs of stability seem difficult), and 3- the obtained filter is well suited to systems where the dynamics of x conditionally on z is precisely known and the dynamics governing the evolution of z is quite uncertain.
Particle observers for contracting dynamical systems
Fast method to fit a C1 piecewise-Bézier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere S2? Laurent Jacques, Pierre-Antoine Absil, Pierre-Yves Gousenbourger GSI2017
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We propose an analysis of the quality of the tting method proposed in [7]. This method ts smooth paths to manifold-valued data points using C1 piecewise-Bezier functions. This method is based on the principle of minimizing an objective function composed of a data-attachment term and a regularization term chosen as the mean squared acceleration of the path. However, the method strikes a tradeo between speed and accuracy by following a strategy that is guaranteed to yield the optimal curve only when the manifold is linear. In this paper, we focus on the sphere S2. We compare the quality of the path returned by the algorithms from [7] with the path obtained by minimizing, over the same search space of C1 piecewise-Bezier curves, a nite-di erence approximation of the objective function by means of a derivative-free manifold-based optimization method.
Fast method to fit a C1 piecewise-Bézier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere S2?
Fast method to fit a C1 piecewise-Bézier function to manifold-valued data points: how suboptimal is the curve obtained on the sphere S2? (slides) GSI2017