Multiple Testing and Variable Selection along Least Angle Regression's path

In this article, we investigate multiple testing and variable selection using Least Angle Regression (LARS) algorithm in high dimensions under the Gaussian noise assumption. LARS is known to produce a piecewise affine solutions path with change points referred to as knots of the LARS path. The cornerstone of the present work is the expression in closed form of the exact joint law of K-uplets of knots conditional on the variables selected by LARS, namely the so-called post-selection joint law of the LARS knots. Numerical experiments demonstrate the perfect fit of our finding. Our main contributions are three fold. First, we build testing procedures on variables entering the model along the LARS path in the general design case when the noise level can be unknown. This testing procedures are referred to as the Generalized t-Spacing tests (GtSt) and we prove that they have exact non-asymptotic level (i.e., Type I error is exactly controlled). In that way, we extend a work from (Taylor et al., 2014) where the Spacing test works for consecutive knots and known variance. Second, we introduce a new exact multiple false negatives test after model selection in the general design case when the noise level can be unknown. We prove that this testing procedure has exact non-asymptotic level for general design and unknown noise level. Last, we give an exact control of the false discovery rate (FDR) under orthogonal design assumption. Monte-Carlo simulations and a real data experiment are provided to illustrate our results in this case. Of independent interest, we introduce an equivalent formulation of LARS algorithm based on a recursive function.Comment: 62 pages; new: FDR control and power comparison between Knockoff, FCD, Slope and our proposed method; new: the introduction has been revised and now present a synthetic presentation of the main results. We believe that this introduction brings new insists compared to previous version

Shadowing by non uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context

A Rice method proof of the Null-Space Property over the Grassmannian

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell\_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure

Cyclic and ruled Lagrangian surfaces in complex Euclidean space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-space. We also describe ruled Lagrangian surfaces. Finally we characterize those cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces

A second order cone formulation of continuous CTA model

The final publication is available at link.springer.comIn this paper we consider a minimum distance Controlled Tabular Adjustment (CTA) model for statistical disclosure limitation (control) of tabular data. The goal of the CTA model is to find the closest safe table to some original tabular data set that contains sensitive information. The measure of closeness is usually measured using l1 or l2 norm; with each measure having its advantages and disadvantages. Recently, in [4] a regularization of the l1 -CTA using Pseudo-Huber func- tion was introduced in an attempt to combine positive characteristics of both l1 -CTA and l2 -CTA. All three models can be solved using appro- priate versions of Interior-Point Methods (IPM). It is known that IPM in general works better on well structured problems such as conic op- timization problems, thus, reformulation of these CTA models as conic optimization problem may be advantageous. We present reformulation of Pseudo-Huber-CTA, and l1 -CTA as Second-Order Cone (SOC) op- timization problems and test the validity of the approach on the small example of two-dimensional tabular data set.Peer ReviewedPostprint (author's final draft

Parametric Representations of Facial Expressions on PDE-Based Surfaces

NoParameterisation of facial expressions on PDE surface representations of human faces are presented in this work. Taking advantage of the boundary-value approach inherent to Bloor-Wilson PDE method, facial expressions are achieved by manipulating the original boundary curves. Such curves are responsible for generating a surface representation of a human face in its neutral configuration, so that regions on these curves represent a given facial expression in a fast and realistic manner. Additionally, the parameterisation proposed here is carried out by applying different mathematical transformations to the affected curves according to the corresponding facial expression. Full analytic expressions parameterising some of the most common facial expressions such as smiling and eyebrow raising are in this work. Some graphical examples of these facial expressions are used to illustrate the results obtained using Bloor-Wilson PDE method as the foundations of the parameterisation scheme proposed here. Thus, it is shown that an efficient, intuitive and realistic parameterisation of facial expressions is attainable using Bloor-Wilson PDE method in along with a suitable mathematical expression

PDE Face: A Novel 3D Face Model

YesWe introduce a novel approach to face models, which exploits the use of Partial Differential Equations (PDE) to generate the 3D face. This addresses some common problems of existing face models. The PDE face benefits from seamless merging of surface patches by using only a relatively small number of parameters based on boundary curves. The PDE face also provides users with a great degree of freedom to individualise the 3D face by adjusting a set of facial boundary curves. Furthermore, we introduce a uv-mesh texture mapping method. By associating the texels of the texture map with the vertices of the uv mesh in the PDE face, the new texture mapping method eliminates the 3D-to-2D association routine in texture mapping. Any specific PDE face can be textured without the need for the facial expression in the texture map to match exactly that of the 3D face model

PDE-based Facial Animation: Making the Complex Simple

YesDirect parameterisation is among the most widely used facial animation techniques but requires complicated ways to animate face models which have complex topology. This paper develops a simple solution by introducing a PDE-based facial animation scheme. Using a PDE face model means we only need to animate a group of boundary curves without using any other conventional surface interpolation algorithms. We describe the basis of the method and show results from a practical implementation.EPSR

Intrinsic Parameters of GRB990123 from Its Prompt Optical Flash and Afterglow

We have constrained the intrinsic parameters, such as the magnetic energy density fraction ($\epsilon_{B}$), the electron energy density fraction ($\epsilon_e$), the initial Lorentz factor ($\Gamma_0$) and the Lorentz factor of the reverse external shock ($\Gamma_{rs}$), of GRB990123, in terms of the afterglow information (forward shock model) and the optical flash information (reverse shock model). Our result shows: 1) the inferred values of $\epsilon_e$ and $\epsilon_B$ are consistent with the suggestion that they may be universal parameters, comparing to those inferred for GRB970508; 2) the reverse external shock may have become relativistic before it passed through the ejecta shell. Other instrinsic parameters of GRB990123, such as energy contained in the forward shock $E$ and the ambient density $n$ are also determined and discussed in this paper.Comment: 5 pages, MN LaTeX style, a few changes made according to referee's suggestions, references up dated, MNRAS accepte