关于修改的盒维数的一个结果

首先证明了修改的下盒维数的乘积公式,继而给出了R上一类随机分形集的修改的盒维数的维数性质。     

Stable Estimation in Dimension Reduction

We introduce stable estimation procedures for several aspects of a sufficient dimension-reduction matrix. We first propose a stable method for estimating structural dimension, which only selects the correct directions in the central subspace with no false positive selection. We then provide a Grassmann manifold sparse estimate for the central subspace. By using subsampling, we develop an ensemble method to obtain a stable nonsparse estimate for the central subspace. This ensemble idea is also used to stabilize the choice of the number of slices in sliced inverse methods. Theoretical results are established, and the efficacy of the proposed stable methods is demonstrated by simulation studies and the analysis of Hitters’ salary data. Supplementary materials for this article are available online.     

A Method for Dimension Reduction in Quadratic Classification Problems

This article presents a dimension-reduction method in quadratic discriminant analysis (QDA). The procedure is inspired by the geometric relation that exists between the subspaces used in sliced inverse regression (SIR) and sliced average variance estimation (SAVE). A new set of directions is constructed to improve the properties of the directions associated with the eigenvectors of the matrices usually considered for dimension reduction in QDA. Illustrative examples of application with real and simulated data are discussed.     

Dimension Reduction in Extended Quermass-Interaction Process

Many objects studied in biology, medicine or material sciences create spatial formations of random shape in which we can observe mutual interactions among those objects. In order to analyse the data composed of such patterns, we use the methods of spatial statistics. Recently, extended random-disc Quermass-interaction process was studied, simulated and consequently statistically analysed using MCMC maximum likelihood method (MCMC MLE). However, this analysis brought some problems. First, it was quite time-consuming, secondly, in some special cases, the parameter estimates may undervalue the real parameter values. In this paper, we describe how we can solve these problems by dimension reduction.     

一种基于正则参数化的降维细分算法(英文)

In our previous work, we have given an algorithm for segmenting a simplex in the n-dimensional space into rt n＋ 1 polyhedrons and provided map F which maps the n-dimensional unit cube to these polyhedrons. In this paper, we prove that the map F is a one to one correspondence at least in lower dimensional spaces （n _〈 3）. Moreover, we propose the approximating subdivision and the interpolatory subdivision schemes and the estimation of computational complexity for triangular Bézier patches on a 2-dimensional space. Finally, we compare our schemes with Goldman＇s in computational complexity and speed.     

A Posteriori Control of Dimension Reduction Errors on Long Domains

In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right‐hand side along the emphasized direction is small, the original problem can be reduced to a lower‐dimensional one that is supposed to be much easier to solve. The a‐posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right‐hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

已有降维方法的推广

SAVE,PHD和SIRⅡ已被证明是有效的降维方法,这些方法基于下面的两个假设1线性条件和常数方差条件.但是,常数方差条件非常强.当常数方差不成立时,即使线性条件成立,SAVE,PHD和SIRⅡ经常会找到中心空间之外的方向.通过去掉了常数方差条件,在较弱条件下推广了SAVE,PHD和SIRⅡ从而使得在较弱的条件下,能得到中心空间的正确估计.     

Sufficient Dimension Reduction and Graphics in Regression

In this article, we review, consolidate and extend a theory for sufficient dimension reduction in regression settings. This theory provides a powerful context for the construction, characterization and interpretation of low-dimensional displays of the data, and allows us to turn graphics into a consistent and theoretically motivated methodological body. In this spirit, we propose an iterative graphical procedure for estimating the meta-parameter which lies at the core of sufficient dimension reduction; namely, the central dimension-reduction subspace.     

Dimension Reduction in Nonparametric Kernel Discriminant Analysis

This article develops a dimension-reduction method in kernel discriminant analysis, based on a general concept of separation of populations. The ideas we present lead to a characterization of the central subspace that does not impose restrictions on the marginal distribution of the feature vector. We also give a new procedure for estimating relevant directions in the central subspace. Comparisons to other procedures are studied and examples of application are discussed.     

A Unified Framework for Linear Dimensionality Reduction in L1

For a family of interpolation norms \({\| \cdot \|_{1,2,s}}\) on \({\mathbb{R}^{n}}\), we provide a distribution over random matrices \({\Phi_s \in \mathbb{R}^{m \times n}}\) parametrized by sparsity level s such that for a fixed set X of K points in \({\mathbb{R}^{n}}\), if \({m \geq C s \log(K)}\) then with high probability, \({\frac{1}{2}\| \varvec{x} \|_{1,2,s} \leq \| \Phi_s (\varvec{x}) \|_1 \leq 2 \| \varvec{x} \|_{1,2,s}}\) for all \({\varvec{x} \in X}\). Several existing results in the literature roughly reduce to special cases of this result at different values of s: For s = n, \({\| \varvec{x} \|_{1,2,n}\equiv \| \varvec{x} \|_{1}}\) and we recover that dimension reducing linear maps can preserve the ?₁-norm up to a distortion proportional to the dimension reduction factor, which is known to be the best possible such result. For s = 1, \({\| \varvec{x} \|_{1,2,1}\equiv \| \varvec{x} \|_{2}}\), and we recover an ?₂/?₁ variant of the Johnson–Lindenstrauss Lemma for Gaussian random matrices. Finally, if \({\varvec{x}}\) is s- sparse, then \({\| \varvec{x} \|_{1,2,s} = \| \varvec{x} \|_1}\) and we recover that s-sparse vectors in \({\ell_1^n}\) embed into \({\ell_1^{\mathcal{O}(s \log(n))}}\) via sparse random matrix constructions.     

Dimension Reduction via Marginal Fourth Moments in Regression

The conditional mean of the response given the predictors is often of interest in regression problems. The central mean subspace, recently introduced by Cook and Li, allows inference about aspects of the mean function in a largely nonparametric context. We propose a marginal fourth moments method for estimating directions in the central mean subspace that might be missed by existing methods such as ordinary least squares (OLS) and principal Hessian directions (pHd). Our method, targeting higher order trends, particularly cubics, complements OLS and pHd because there is no inclusion among them. Theory, estimation and inferences as well as illustrative examples are presented.     

Dimension Reduction in Regressions With Exponential Family Predictors

We present first methodology for dimension reduction in regressions with predictors that, given the response, follow one-parameter exponential families. Our approach is based on modeling the conditional distribution of the predictors given the response, which allows us to derive and estimate a sufficient reduction of the predictors. We also propose a method of estimating the forward regression mean function without requiring an explicit forward regression model. Whereas nearly all existing estimators of the central subspace are limited to regressions with continuous predictors only, our proposed methodology extends estimation to regressions with all categorical or a mixture of categorical and continuous predictors. Supplementary materials including the proofs and the computer code are available from the JCGS website.     

大规模矩阵降维的随机逼近方法

大规模矩阵降维和分解是数据分析的核心问题之一,在工程领域应用广泛,如图像分割、文本分类、数据挖掘,然而,传统的矩阵分解方法(如SVD、谱分解)计算复杂度高,不适用于大规模矩阵处理.近些年来,随机逼近方法用来发现大规模矩阵的低维近似,有效地降低了计算复杂度,是当今的研究热点.围绕基于随机逼近的大矩阵降维方法展开论述,介绍了矩阵降维中的抽样策略、CUR分解、Nystrom方法、随机逼近方法,比较研究了这些方法的优缺点.对重要的随机逼近方法开展了一些图像试验分析.最后,进行了总结并讨论了一些方向的可行性.     

模代数的形变理论

本文构作了与模代数HA相适应的“形变复形”CH*(H,A),并利用形变复形的上同调群来刻画模代数HA的形变,证明了HA的无穷小形变的等价类是与形变复形的2阶上同调群HH2(H,A)是一一对应的,且当HH2(H,A)=0时,模代数HA为刚性的.     

Dimension Reduction for the Landau-de Gennes Model on Curved Nematic Thin Films

We use the method of \(\Gamma \)-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in Golovaty et al. (J Nonlinear Sci 25(6):1431–1451, 2015) where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustum.     

一类非线性EV模型的降维估计

考虑响应变量随机缺失情形下的非线性EV模型.给出了未知参数的降维估计,有效避免了高维核估计带来的维数灾祸问题.所构造的统计量渐近于x~2分布,所得结果可以用来构造未知参数的置信域.     

关于代数体函数Borel方向的一个结果

本文证明了下列定理设w=w(z)为|z|<∞上的υ值代数体函数,它的级满足0<ρ<∞,则存在一条方向Largz=θ     

Dimension Reduction and Optimality of the Uniform State in a Phase-Field-Crystal Model Involving a Higher-Order Functional

We study a phase-field-crystal model described by a free energy functional involving second-order derivatives of the order parameter in a periodic setting and under a fixed mass constraint. We prove a $$\Gamma $$-convergence result in an asymptotic thin-film regime leading to a reduced two-dimensional model. For the reduced model, we prove necessary and sufficient conditions for the global minimality of the uniform state. We also prove similar results for the Ohta–Kawasaki model.