The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation

We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE and LASSO, as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the sign permutahedron. Further considerations based this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE model to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE models.Comment: new title, minor change

Wheel-regolith interactions on small-body surfaces

We conduct experiments using a single-wheel testbed and simulations using the Soft-Sphere Discrete Element Method to study wheel-regolith interactions on small-body surfaces. We analyze wheel sinkage and traction on different surface materials and we discuss the influence that lowgravity has on rover maneuverability

The influence of gravity on granular impacts II. A gravity-scaled collision model for slow interactions

Slow interactions on small body surfaces occur both naturally and through human intervention. The resettling of grains and boulders following a cratering event, as well as observations made during small body missions, can provide clues regarding the material properties and the physical evolution of a surface. In order to analyze such events, it is necessary to understand how gravity influences granular behavior. In this work, we study slow impacts into granular materials for different collision velocities and gravity levels. Our objectives are to develop a model that describes penetration depth in terms of the dimensionless Froude number and to use this model to understand the relationship between collision behavior, collision velocity, and gravity. We use the soft-sphere discrete element method to simulate impacts into glass beads under gravitational accelerations ranging from 9.81 m/s^2 to 0.001 m/s^2. We quantify collision behavior using the peak acceleration, the penetration depth, and the collision duration of the projectile, and we compare the collision behavior for impacts within a Froude number range of 0 to 10. The measured penetration depth and collision duration for low-velocity collisions are comparable when the impact parameters are scaled by the Froude number, and the presented model predicts the collision behavior well within the tested Froude number range. If the impact Froude number is low (0 < Fr < 1.5), the collision occurs in a regime that is dominated by a depth-dependent quasi-static friction force. If the impact Froude number is high enough (1.5 < Fr < 10), the collision enters a second regime that is dominated by inertial drag. The presented collision model can be used to constrain the properties of a granular surface material using the penetration depth measurement from a single impact event. If the projectile size, the collision velocity, the gravity level, and the final penetration depth are known and the material density is estimated, then the internal friction angle of the material can be deduced

IDEFIX, the MMX Rover – in-situ science on Phobos

IDEFIX is a small rover, contributed by the Centre National d’Etudes Spatiales (CNES) and the German Aerospace Center (DLR) to JAXA´s Martian Moons eXploration (MMX) mission which will investigate the Martian moons Phobos and Deimos [1,2]. IDEFIX will be delivered to the surface of Phobos to perform in-situ science but also to serve as a scout, gathering data to prepare the landing of the main spacecraft. The MMX Rover will deliver information on the regolith properties by high resolution imaging (NavCams and WheelCams), measurement of the spectral properties in the thermal infrared as well as the thermophysical properties using a radiometer (miniRAD) and Raman spectroscopy (RAX) [3,4]

Relevance of Phobos in-situ science for understanding asteroids

The origin of the martian moons, Phobos and Deimos is under debate since a very long time. There exist arguments and counter arguments that they may be captured asteroids. Other models favor, e.g., a massive impact at Mars as their origin [1]. The Martian Moons eXploration (MMX) mission by the Japan Aerospace Exploration Agency, JAXA, is going to explore both Martian moons remotely, but also return samples from Phobos, and deliver a small Rover to its surface [2,3]. This rover, provided by CNES and DLR, with contributions from INTA and the University of Tokyo has a payload of four scientific instruments, analyzing the physical, dynamical and mineralogical properties of Phobos´ surface. Parallels to asteroids of a similar size are eminent and the results will help deciphering the origin of Phobos [4]

Représentation parcimonieuse et procédures de tests multiples : application à la métabolomique

Let Y be a Gaussian vector distributed according to N (m,sigma²Idn) and X a matrix of dimension n x p with Y observed, m unknown, sigma and X known. In the linear model, m is assumed to be a linear combination of the columns of X In small dimension, when n ≥ p and ker (X) = 0, there exists a unique parameter Beta* such that m = X Beta*; then we can rewrite Y = Beta* + Epsilon. In the small-dimensional linear Gaussian model framework, we construct a new multiple testing procedure controlling the FWER to test the null hypotheses Beta*i = 0 for i belongs to [[1,p]]. This procedure is applied in metabolomics through the freeware ASICS available online. ASICS allows to identify and to qualify metabolites via the analyse of RMN spectra. In high dimension, when n < p we have ker (X) ≠ 0 consequently the parameter Beta* described above is no longer unique. In the noiseless case when Sigma = 0, implying thus Y = m, we show that the solutions of the linear system of equation Y = X Beta having a minimal number of non-zero components are obtained via the lalpha with alpha small enough.Considérons un vecteur gaussien Y de loi N (m,sigma²Idn) et X une matrice de dimension n x p avec Y observé, m inconnu, Sigma et X connus. Dans le cadre du modèle linéaire, m est supposé être une combinaison linéaire des colonnes de X. En petite dimension, lorsque n ≥ p et que ker (X) = 0, il existe alors un unique paramètre Beta* tel que m = X Beta* ; on peut alors réécrire Y sous la forme Y = X Beta* + Epsilon. Dans le cadre du modèle linéaire gaussien en petite dimension, nous construisons une nouvelle procédure de tests multiples contrôlant le FWER pour tester les hypothèses nulles Beta*i = 0 pour i appartient à [[1,p]]. Cette procédure est appliquée en métabolomique au travers du programme ASICS qui est disponible en ligne. ASICS permet d'identifier et de quantifier les métabolites via l'analyse des spectres RMN. En grande dimension, lorsque n < p on a ker (X) ≠ 0, ainsi le paramètre Beta* décrit précédemment n'est pas unique. Dans le cas non bruité lorsque Sigma = 0, impliquant que Y = m, nous montrons que les solutions du système linéaire d'équations Y = X Beta avant un nombre de composantes non nulles minimales s'obtiennent via la minimisation de la "norme" lAlpha avec Alpha suffisamment petit

Sparse representation and multiple testing procedures : application to metabolimics

Considérons un vecteur gaussien Y de loi N (m,sigma²Idn) et X une matrice de dimension n x p avec Y observé, m inconnu, Sigma et X connus. Dans le cadre du modèle linéaire, m est supposé être une combinaison linéaire des colonnes de X. En petite dimension, lorsque n ≥ p et que ker (X) = 0, il existe alors un unique paramètre Beta* tel que m = X Beta* ; on peut alors réécrire Y sous la forme Y = X Beta* + Epsilon. Dans le cadre du modèle linéaire gaussien en petite dimension, nous construisons une nouvelle procédure de tests multiples contrôlant le FWER pour tester les hypothèses nulles Beta*i = 0 pour i appartient à [[1,p]]. Cette procédure est appliquée en métabolomique au travers du programme ASICS qui est disponible en ligne. ASICS permet d'identifier et de quantifier les métabolites via l'analyse des spectres RMN. En grande dimension, lorsque n < p on a ker (X) ≠ 0, ainsi le paramètre Beta* décrit précédemment n'est pas unique. Dans le cas non bruité lorsque Sigma = 0, impliquant que Y = m, nous montrons que les solutions du système linéaire d'équations Y = X Beta avant un nombre de composantes non nulles minimales s'obtiennent via la minimisation de la "norme" lAlpha avec Alpha suffisamment petit.Let Y be a Gaussian vector distributed according to N (m,sigma²Idn) and X a matrix of dimension n x p with Y observed, m unknown, sigma and X known. In the linear model, m is assumed to be a linear combination of the columns of X In small dimension, when n ≥ p and ker (X) = 0, there exists a unique parameter Beta* such that m = X Beta*; then we can rewrite Y = Beta* + Epsilon. In the small-dimensional linear Gaussian model framework, we construct a new multiple testing procedure controlling the FWER to test the null hypotheses Beta*i = 0 for i belongs to [[1,p]]. This procedure is applied in metabolomics through the freeware ASICS available online. ASICS allows to identify and to qualify metabolites via the analyse of RMN spectra. In high dimension, when n < p we have ker (X) ≠ 0 consequently the parameter Beta* described above is no longer unique. In the noiseless case when Sigma = 0, implying thus Y = m, we show that the solutions of the linear system of equation Y = X Beta having a minimal number of non-zero components are obtained via the lalpha with alpha small enough