Estimation for ultra-high dimensional factor model: a pivotal variable detection based approach

For factor model, the involved covariance matrix often has no row sparse structure because the common factors may lead some variables to strongly associate with many others. Under the ultra-high dimensional paradigm, this feature causes existing methods for sparse covariance matrix in the literature not directly applicable. In this paper, for general covariance matrix, a novel approach to detect these variables that is called the pivotal variables is suggested. Then, two-stage estimation procedures are proposed to handle ultra-high dimensionality in factor model. In these procedures, pivotal variable detection is performed as a screening step and then existing approaches are applied to refine the working model. The estimation efficiency can be promoted under weaker assumptions on the model structure. Simulations are conducted to examine the performance of the new method and a real dataset is analysed for illustration

Deep-gKnock: nonlinear group-feature selection with deep neural network

Feature selection is central to contemporary high-dimensional data analysis. Grouping structure among features arises naturally in various scientific problems. Many methods have been proposed to incorporate the grouping structure information into feature selection. However, these methods are normally restricted to a linear regression setting. To relax the linear constraint, we combine the deep neural networks (DNNs) with the recent Knockoffs technique, which has been successful in an individual feature selection context. We propose Deep-gKnock (Deep group-feature selection using Knockoffs) as a methodology for model interpretation and dimension reduction. Deep-gKnock performs model-free group-feature selection by controlling group-wise False Discovery Rate (gFDR). Our method improves the interpretability and reproducibility of DNNs. Experimental results on both synthetic and real data demonstrate that our method achieves superior power and accurate gFDR control compared with state-of-the-art methods

Theory of Bergman Spaces in the Unit Ball of CnC^n

There has been a great deal of work done in recent years on weighted Bergman spaces \apa on the unit ball \bn of \cn, where $0<p<\infty$ and $\alpha>-1$. We extend this study in a very natural way to the case where $\alpha$ is {\em any} real number and $0<p\le\infty$. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^2$, and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.Comment: 83 pages, revised from 2005 manuscrip

Pareto optimal multi-robot motion planning

This paper studies a class of multi-robot coordination problems where a team of robots aim to reach their goal regions with minimum time and avoid collisions with obstacles and other robots. A novel numerical algorithm is proposed to identify the Pareto optimal solutions where no robot can unilaterally reduce its traveling time without extending others'. The consistent approximation of the algorithm in the epigraphical profile sense is guaranteed using set-valued numerical analysis. Experiments on an indoor multi-robot platform and computer simulations show the anytime property of the proposed algorithm; i.e., it is able to quickly return a feasible control policy that safely steers the robots to their goal regions and it keeps improving policy optimality if more time is given

Stochastic Primal-Dual Coordinate Method with Large Step Size for Composite Optimization with Composite Cone-constraints

We introduce a stochastic coordinate extension of the first-order primal-dual method studied by Cohen and Zhu (1984) and Zhao and Zhu (2018) to solve Composite Optimization with Composite Cone-constraints (COCC). In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for COCC. We obtain almost surely convergence and O(1/t) expected convergence rate in this work. The high probability complexity bound is also derived in this paper.Comment: arXiv admin note: substantial text overlap with arXiv:1804.0080

On genetic correlation estimation with summary statistics from genome-wide association studies

Genome-wide association studies (GWAS) have been widely used to examine the association between single nucleotide polymorphisms (SNPs) and complex traits, where both the sample size n and the number of SNPs p can be very large. Recently, cross-trait polygenic risk score (PRS) method has gained extremely popular for assessing genetic correlation of complex traits based on GWAS summary statistics (e.g., SNP effect size). However, empirical evidence has shown a common bias phenomenon that even highly significant cross-trait PRS can only account for a very small amount of genetic variance (R^2 often <1%). The aim of this paper is to develop a novel and powerful method to address the bias phenomenon of cross-trait PRS. We theoretically show that the estimated genetic correlation is asymptotically biased towards zero when complex traits are highly polygenic/omnigenic. When all p SNPs are used to construct PRS, we show that the asymptotic bias of PRS estimator is independent of the unknown number of causal SNPs m. We propose a consistent PRS estimator to correct such asymptotic bias. We also develop a novel estimator of genetic correlation which is solely based on two sets of GWAS summary statistics. In addition, we investigate whether or not SNP screening by GWAS p-values can lead to improved estimation and show the effect of overlapping samples among GWAS. Our results may help demystify and tackle the puzzling "missing genetic overlap" phenomenon of cross-trait PRS for dissecting the genetic similarity of closely related heritable traits. We illustrate the finite sample performance of our bias-corrected PRS estimator by using both numerical experiments and the UK Biobank data, in which we assess the genetic correlation between brain white matter tracts and neuropsychiatric disorders.Comment: 50 page

Stochastic Primal-Dual Coordinate Method for Nonlinear Convex Cone Programs

Block coordinate descent (BCD) methods and their variants have been widely used in coping with large-scale nonconstrained optimization problems in many fields such as imaging processing, machine learning, compress sensing and so on. For problem with coupling constraints, Nonlinear convex cone programs (NCCP) are important problems with many practical applications, but these problems are hard to solve by using existing block coordinate type methods. This paper introduces a stochastic primal-dual coordinate (SPDC) method for solving large-scale NCCP. In this method, we randomly choose a block of variables based on the uniform distribution. The linearization and Bregman-like function (core function) to that randomly selected block allow us to get simple parallel primal-dual decomposition for NCCP. The sequence generated by our algorithm is proved almost surely converge to an optimal solution of primal problem. Two types of convergence rate with different probability (almost surely and expected) are also obtained. The probability complexity bound is also derived in this paper

C-metric like vacuum with non-negative cosmological constant in five dimensions

We present and analyze an exact 5-dimensional vacuum solution of Einstein equation with non-negative cosmological constant written in a C-metric like coordinate. The metric does not contain any black hole horizons in it, but has two acceleration horizons and a static patch in between. The coordinate system, horizon geometry and global structures are analyzed in detail, and in the case of vanishing cosmological constant, a simple exterior geometric interpretation is given. The metric possesses a spacelike Killing coordinate $\phi$ besides the timelike coordinate $t$, along which the spacetime can be dimensionally reduced via Kaluza-Klein mechanism and interpreted as Einstein gravity coupled to a 4 dimensional Liouville field (and a Maxwell field as well if a boost operation is performed before the Kaluza-Klein reduction).Comment: PDFLaTeX with 3 PDF figure

Three dimensional central configurations in H3 and S3

We show that each central configuration in the three-dimensional hyperbolic sphere is equivalent to one central configuration on a particular two- dimensional hyperbolic sphere. However, there exist both special and ordinary central configurations in the three-dimensional sphere that are not confined to any two-dimensional sphere.Comment: 10 pages, 2 figure

Half of an antipodal spherical design

We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$