论文检索

张量分解与稀疏对数线性模型AI 解读

Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. We derive several results…

Monge–Kantorovich 深度、分位数、秩与符号AI 解读

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks and signs, based on canonical transportation maps between a distribution of interest on $\mathbb{R}^{d}$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge–Kantorovich depth, specializes to halfspace depth for $d=1$ and in the case of spherical distributions, but for more general distributions, differs from the latter in the…

稀疏高维模型中假设检验与置信域的一般理论AI 解读

We consider the problem of uncertainty assessment for low dimensional components in high dimensional models. Specifically, we propose a novel decorrelated score function to handle the impact of high dimensional nuisance parameters. We consider both hypothesis tests and confidence regions for generic penalized M-estimators. Unlike most existing inferential methods which are tailored for individual models, our method provides a general framework for high dimensional inference…

EM 算法的统计保证：从总体到有限样本分析AI 解读

The EM algorithm is a widely used tool in maximum-likelihood estimation in incomplete data problems. Existing theoretical work has focused on conditions under which the iterates or likelihood values converge, and the associated rates of convergence. Such guarantees do not distinguish whether the ultimate fixed point is a near global optimum or a bad local optimum of the sample likelihood, nor do they relate the obtained fixed point to the global optima of the idealized…

对数凹密度估计的全局收敛速率AI 解读

The estimation of a log-concave density on $\mathbb{R}^{d}$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $n$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk…

摘要暂缺

对《高维聚类中的重要特征主成分分析》的讨论AI 解读

We would like to congratulate the authors for an interesting paper and a novel proposal for clustering high-dimensional Gaussian mixtures with a diagonal covariance matrix.The proposed two-stage procedure first selects features based on the Kolmogorov-Smirnov statistics and then applies a spectral clustering method to the post-selected data.A rigorous theoretical analysis for the clustering error is given and the results are supported by a competitive performance in…

《高维聚类的影响特征主成分分析》讨论AI 解读

I commend Jin and Wang on a very interesting paper introducing a novel approach to feature selection within clustering and a detailed analysis of its clustering performance under a Gaussian mixture model.I shall divide my discussion into several parts: (i) prior work on feature selection and clustering; (ii) theoretical aspects; (iii) practical aspects; and finally (iv) some questions and directions for future research.On feature selection and clustering.Jin and Wang write…

摘要暂缺

We study the problem of variable selection in convex nonparametric regression. Under the assumption that the true regression function is convex and sparse, we develop a screening procedure to select a subset of variables that contains the relevant variables. Our approach is a two-stage quadratic programming method that estimates a sum of one-dimensional convex functions, followed by one-dimensional concave regression fits on the residuals. In contrast to previous methods for…

用于超高维无模型特征选择的边际切片逆回归AI 解读

Model-free variable selection has been implemented under the sufficient dimension reduction framework since the seminal paper of Cook [Ann. Statist. 32 (2004) 1062–1092]. In this paper, we extend the marginal coordinate test for sliced inverse regression (SIR) in Cook (2004) and propose a novel marginal SIR utility for the purpose of ultrahigh dimensional feature selection. Two distinct procedures, Dantzig selector and sparse precision matrix estimation, are incorporated to…

高维聚类中的重要特征主成分分析AI 解读

We consider a clustering problem where we observe feature vectors $X_{i}\in R^{p}$, $i=1,2,\ldots,n$, from $K$ possible classes. The class labels are unknown and the main interest is to estimate them. We are primarily interested in the modern regime of $p\gg n$, where classical clustering methods face challenges. We propose Influential Features PCA (IF-PCA) as a new clustering procedure. In IF-PCA, we select a small fraction of features with the largest Kolmogorov–Smirnov…

高维加性模型估计的极小极大最优收敛率AI 解读

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal a behavior universal to this class of high dimensional problems. In the sparse regime when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression and, therefore, there is no additional cost to entertain a…

多重检验的族级分离率AI 解读

Starting from a parallel between some minimax adaptive tests of a single null hypothesis, based on aggregation approaches, and some tests of multiple hypotheses, we propose a new second kind error-related evaluation criterion, as the core of an emergent minimax theory for multiple tests. Aggregation-based tests, proposed for instance by Baraud [Bernoulli 8 (2002) 577–606], Baraud, Huet and Laurent [Ann. Statist. 31 (2003) 225–251] or Fromont and Laurent [Ann. Statist. 34…

高维贝叶斯变量选择的计算复杂度AI 解读

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a…

单调轨迹动力学的非参数估计AI 解读

We study a class of nonlinear nonparametric inverse problems. Specifically, we propose a nonparametric estimator of the dynamics of a monotonically increasing trajectory defined on a finite time interval. Under suitable regularity conditions, we show that in terms of $L^{2}$-loss, the optimal rate of convergence for the proposed estimator is the same as that for the estimation of the derivative of a function. We conduct simulation studies to examine the finite sample…

若干弱可识别有限混合模型的参数估计收敛速率AI 解读

We establish minimax lower bounds and maximum likelihood convergence rates of parameter estimation for mean-covariance multivariate Gaussian mixtures, shape-rate Gamma mixtures and some variants of finite mixture models, including the setting where the number of mixing components is bounded but unknown. These models belong to what we call “weakly identifiable” classes, which exhibit specific interactions among mixing parameters driven by the algebraic structures of the class…

次高斯均值估计量AI 解读

We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a nonasymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.

非对称最小二乘回归的高维推广及其应用AI 解读

Asymmetric least squares regression is an important method that has wide applications in statistics, econometrics and finance. The existing work on asymmetric least squares only considers the traditional low dimension and large sample setting. In this paper, we systematically study the Sparse Asymmetric LEast Squares (SALES) regression under high dimensions where the penalty functions include the Lasso and nonconvex penalties. We develop a unified efficient algorithm for…

二次支撑风险下的一致模型选择准则AI 解读

In this paper, we study asymptotic properties of model selection criteria for high-dimensional regression models where the number of covariates is much larger than the sample size. In particular, we consider a class of loss functions called the class of quadratically supported risks which is large enough to include the quadratic loss, Huber loss, quantile loss and logistic loss. We provide sufficient conditions for the model selection criteria, which are applicable to the…

通过图示干预层级统一因果推断AI 解读

Identifying causal parameters from observational data is fraught with subtleties due to the issues of selection bias and confounding. In addition, more complex questions of interest, such as effects of treatment on the treated and mediated effects may not always be identified even in data where treatment assignment is known and under investigator control, or may be identified under one causal model but not another. Increasingly complex effects of interest, coupled with a…

Peter Hall 在高维数据与分类研究中的工作AI 解读

In this article, I summarise Peter Hall’s contributions to high-dimensional data, including their geometric representations and variable selection methods based on ranking. I also discuss his work on classification problems, concluding with some personal reflections on my own interactions with him. This article complements [Ann. Statist. 44 (2016) 1821–1836; Ann. Statist. 44 (2016) 1837–1853; Ann. Statist. 44 (2016) 1854–1866 and Ann. Statist. 44 (2016) 1867–1887], which…

Peter Hall对反卷积研究的主要贡献AI 解读

Peter Hall died in Melbourne on January 9, 2016. He was an extremely prolific researcher and contributed to many different areas of statistics. In this paper, I talk about my experience with Peter and I summarise his main contributions to deconvolution, which include measurement error problems and problems in image analysis.

Peter Hall 对非参数函数估计与建模的贡献AI 解读

Peter Hall made wide-ranging and far-reaching contributions to nonparametric modeling. He was one of the leading figures in the developments of nonparametric techniques with over 300 published papers in the field alone. This article gives a selective overview on the contributions of Peter Hall to nonparametric function estimation and modeling. The focuses are on density estimation, nonparametric regression, bandwidth selection, boundary corrections, inference under shape…