Regularity properties such as the incoherence condition, the restricted isometry property, compatibility, restricted eigenvalue and iq sensitivity of covariate matrices play a pivotal role in high-dimensional regression and compressed sensing. Yet, like computing the spark of a matrix, we first show that it is NP-hard to check the conditions involving all submatrices of a given size.

The varying coefficient model is an important class of nonparametric statistical model, which allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is large, the issue of variable selection arises. In this article, we propose and investigate marginal nonparametric screening methods to screen variables in sparse ultra-high-dimensional varying coefficient models. The proposed nonparametric independence screening (NIS) selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable. The sure independent screening property is established under some mild technical conditions when the dimensionality is of nonpolynomial order, and the dimensionality reduction of NIS is quantified. To enhance the practical utility and finite sample performance, two data-driven iterative NIS (INIS) methods are

Using high-frequency data, we estimate the risk of a large portfolio with weights being the solution of an optimization problem subject to some linear inequality constraints. We propose a fully nonparametric approach as a benchmark, as well as a factor-based semiparametric approach with observable factors to attack the curse of dimensionality. We provide in-fill asymptotic distributions of the realized volatility estimators of the optimal portfolio, while taking into account the estimation error in the optimal portfolio weights as a result of the covariance matrix estimation. Our theoretical findings suggest that ignoring such an error leads to a first-order asymptotic bias which undermines the statistical inference. Such a bias is related to in-sample optimism in portfolio allocation. Our simulation results suggest satisfactory finite sample performance after bias correction, and that the factor-based approach becomes increasingly superior with a growing cross-sectional dimension. Empirically, using a large cross-section of high-frequency stock returns, we find our estimator successfully addresses the issue of in-sample optimism.

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of BoudinDesvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

In statistics and machine learning, we are interested in the eigenvectors (or singular vectors) of certain matrices (eg covariance matrices, data matrices, etc). However, those matrices are usually perturbed by noises or statistical errors, either from random sampling or structural patterns. The Davis-Kahan sin theorem is often used to bound the difference between the eigenvectors of a matrix A and those of a perturbed matrix A= A+ E, in terms of l2 norm. In this paper, we prove that when A is a low-rank and incoherent matrix, the l norm perturbation bound of singular vectors (or eigenvectors in the symmetric case) is smaller by a factor of