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含噪高维数据中稀疏主成分分析的极小极大界AI 解读

We study the problem of estimating the leading eigenvectors of a high-dimensional population covariance matrix based on independent Gaussian observations. We establish a lower bound on the minimax risk of estimators under the $l_{2}$ loss, in the joint limit as dimension and sample size increase to infinity, under various models of sparsity for the population eigenvectors. The lower bound on the risk points to the existence of different regimes of sparsity of the…

Cox模型中Lasso的预言机不等式AI 解读

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved…

满足交互项层级约束的 lassoAI 解读

We add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important. We give a precise characterization of the effect of this hierarchy constraint, prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator. A bound on this estimate reveals the amount of fitting…

高维广义线性模型的贝叶斯子集建模AI 解读

This article presents a new prior setting for high-dimensional generalized linear models, which leads to a Bayesian subset regression (BSR) with the maximum a posteriori model approximately equivalent to the minimum extended Bayesian information criterion model. The consistency of the resulting posterior is established under mild conditions. Further, a variable screening procedure is proposed based on the marginal inclusion probability, which shares the same properties of…

基于趋势平行性的高维时间序列聚类AI 解读

This article considers the problem of clustering high-dimensional time series based on trend parallelism. The underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors, a special class of nonstationary processes. For each group where the parallelism holds, I semiparametrically estimate its representative trend function and vertical shifts of group members, and establish their central limit theorems. An information criterion,…

墨西哥无尾蝙蝠鸣声音节：非平稳声学时间序列的贝叶斯函数混合模型AI 解读

We describe a new approach to analyze chirp syllables of free-tailed bats from two regions of Texas in which they are predominant: Austin and College Station. Our goal is to characterize any systematic regional differences in the mating chirps and assess whether individual bats have signature chirps. The data are analyzed by modeling spectrograms of the chirps as responses in a Bayesian functional mixed model. Given the variable chirp lengths, we compute the spectrograms on…

缺少类别标识时的辅助标记分类AI 解读

Constructing classification rules for accurate diagnosis of a disorder is an important goal in medical practice. In many clinical applications, there is no clinically significant anatomical or physiological deviation exists to identify the gold standard disease status to inform development of classification algorithms. Despite absence of perfect disease class identifiers, there are usually one or more disease-informative auxiliary markers along with feature variables…

高维数据中被选参数的经验贝叶斯置信区间AI 解读

Modern statistical problems often involve a large number of populations and hence a large number of parameters that characterize these populations. It is common for scientists to use data to select the most significant populations, such as those with the largest t statistics. The scientific interest often lies in studying and making inferences regarding these parameters, called the selected parameters, corresponding to the selected populations. The current statistical…

基于指数平方损失的稳健变量选择AI 解读

Robust variable selection procedures through penalized regression have been gaining increased attention in the literature. They can be used to perform variable selection and are expected to yield robust estimates. However, to the best of our knowledge, the robustness of those penalized regression procedures has not been well characterized. In this paper, we propose a class of penalized robust regression estimators based on exponential squared loss. The motivation for this…

无向图上的同步分组搜索与特征选择AI 解读

In high-dimensional regression, grouping pursuit and feature selection have their own merits while complementing each other in battling the curse of dimensionality. To seek a parsimonious model, we perform simultaneous grouping pursuit and feature selection over an arbitrary undirected graph with each node corresponding to one predictor. When the corresponding nodes are reachable from each other over the graph, regression coefficients can be grouped, whose absolute values…

具有预言效率的广义可加模型两步估计AI 解读

The generalized additive model (GAM) is a multivariate nonparametric regression tool for non-Gaussian responses including binary and count data. We propose a spline-backfitted kernel (SBK) estimator for the component functions and the constant, which are oracally efficient under weak dependence. The SBK technique is both computationally expedient and theoretically reliable, thus usable for analyzing high-dimensional time series. Inference can be made on component functions…

有效的模型选择后推断AI 解读

It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid “post-selection inference” by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence…

稀疏主成分分析与迭代阈值化AI 解读

Principal component analysis (PCA) is a classical dimension reduction method which projects data onto the principal subspace spanned by the leading eigenvectors of the covariance matrix. However, it behaves poorly when the number of features $p$ is comparable to, or even much larger than, the sample size $n$. In this paper, we propose a new iterative thresholding approach for estimating principal subspaces in the setting where the leading eigenvectors are sparse. Under a…

图上光滑核的低秩估计AI 解读

Let $(V,A)$ be a weighted graph with a finite vertex set $V$, with a symmetric matrix of nonnegative weights $A$ and with Laplacian $\Delta$. Let $S_{\ast}: V\times V\mapsto{\mathbb{R}}$ be a symmetric kernel defined on the vertex set $V$. Consider $n$ i.i.d. observations $(X_{j},X_{j}',Y_{j})$, $j=1,\ldots,n$, where $X_{j}$, $X_{j}'$ are independent random vertices sampled from the uniform distribution in $V$ and $Y_{j}\in{\mathbb{R}}$ is a real valued response variable…

稀疏有向无环图的 ℓ₀ 惩罚最大似然估计AI 解读

We consider the problem of regularized maximum likelihood estimation for the structure and parameters of a high-dimensional, sparse directed acyclic graphical (DAG) model with Gaussian distribution, or equivalently, of a Gaussian structural equation model. We show that the $\ell_{0}$-penalized maximum likelihood estimator of a DAG has about the same number of edges as the minimal-edge I-MAP (a DAG with minimal number of edges representing the distribution), and that it…

高维数据低维投影的条件分布AI 解读

We study the conditional distribution of low-dimensional projections from high-dimensional data, where the conditioning is on other low-dimensional projections. To fix ideas, consider a random $d$-vector $Z$ that has a Lebesgue density and that is standardized so that $\mathbb{E} Z=0$ and $\mathbb{E} ZZ'=I_{d}$. Moreover, consider two projections defined by unit-vectors $\alpha$ and $\beta$, namely a response $y=\alpha'Z$ and an explanatory variable $x=\beta'Z$. It has long…

因果推断中忠实性假设的几何结构AI 解读

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or…

大规模数据的高效高斯过程回归AI 解读

Gaussian processes are widely used in nonparametric regression, classification and spatiotemporal modelling, facilitated in part by a rich literature on their theoretical properties. However, one of their practical limitations is expensive computation, typically on the order of n3 where n is the number of data points, in performing the necessary matrix inversions. For large datasets, storage and processing also lead to computational bottlenecks, and numerical stability of…

高维二元分类变量筛选的 Kolmogorov 过滤器AI 解读

Variable screening techniques have been proposed to mitigate the impact of high dimensionality in classification problems, including t-test marginal screening (Fan & Fan, 2008) and maximum marginal likelihood screening (Fan & Song, 2010). However, these methods rely on strong modelling assumptions that are easily violated in real applications. To circumvent the parametric modelling assumptions, we propose a new variable screening technique for binary classification based on…

协变量调整的精度矩阵估计及其遗传基因组学应用AI 解读

Motivated by analysis of genetical genomics data, we introduce a sparse high-dimensional multivariate regression model for studying conditional independence relationships among a set of genes adjusting for possible genetic effects. The precision matrix in the model specifies a covariate-adjusted Gaussian graph, which presents the conditional dependence structure of gene expression after the confounding genetic effects on gene expression are taken into account. We present a…

允许高维协变量的异方差部分线性单指标模型：双重稳健与有效估计AI 解读

Summary We study the heteroscedastic partially linear single-index model with an unspecified error variance function, which allows for high dimensional covariates in both the linear and the single-index components of the mean function. We propose a class of consistent estimators of the parameters by using a proper weighting strategy. An interesting finding is that the linearity condition which is widely assumed in the dimension reduction literature is not necessary for…

超高维单指标风险率模型的独立筛选AI 解读

Summary In data sets with many more features than observations, independent screening based on all univariate regression models leads to a computationally convenient variable selection method. Recent efforts have shown that, in the case of generalized linear models, independent screening may suffice to capture all relevant features with high probability, even in ultrahigh dimension. It is unclear whether this formal sure screening property is attainable when the response is…

利用间接测量估计网络中的潜在过程AI 解读

In a communication network, point-to-point traffic volumes over time are critical for designing protocols that route information efficiently and for maintaining security, whether at the scale of an Internet service provider or within a corporation. While technically feasible, the direct measurement of point-to-point traffic imposes a heavy burden on network performance and is typically not implemented. Instead, indirect aggregate traffic volumes are routinely collected. We…

通过贝叶斯非参数仿射子空间学习进行分类AI 解读

It has become common for data sets to contain large numbers of variables in studies conducted in areas such as genetics, machine vision, image analysis and many others. When analyzing such data, parametric models are often too inflexible while nonparametric procedures tend to be non-robust because of insufficient data on these high dimensional spaces. This is particularly true when interest lies in building efficient classifiers in the presence of many predictor variables.…