论文检索

任意噪声污染下线性模型的置信区间AI 解读

Summary We study confidence interval construction for linear regression under Huber’s contamination model, where an unknown fraction of noise variables is arbitrarily corrupted. While robust point estimation in this setting is well understood, statistical inference remains challenging, especially because the contamination proportion is not identifiable from the data. We develop a new algorithm that constructs confidence intervals for individual regression coefficients…

自适应稳健置信区间AI 解读

This paper studies the construction of adaptive confidence intervals under Huber’s contamination model when the contamination proportion is unknown. For the robust confidence interval of a Gaussian mean, we show that the optimal length of an adaptive interval must be exponentially wider than that of a nonadaptive one. An optimal construction is achieved through simultaneous uncertainty quantification of quantiles at all levels. The results are further extended beyond the…

高斯序列模型中的稀疏性与相关性AI 解读

We study estimation of an s-sparse signal in the p-dimensional Gaussian sequence model with equicorrelated observations and derive the minimax rate. A new phenomenon emerges from correlation, namely, the rate scales with respect to p−2s and exhibits a phase transition at p−2s≍p. Correlation is shown to be a blessing, provided it is sufficiently strong and the critical correlation level exhibits a delicate dependence on the sparsity level. Due to correlation, the minimax rate…

利用低阶多项式研究图函数估计的计算下界AI 解读

Graphon estimation has been one of the most fundamental problems in network analysis and has received considerable attention in the past decade. From the statistical perspective, the minimax error rate of graphon estimation has been established by (Ann. Statist. 43 (2015) 2624–2652) for both stochastic block model (SBM) and nonparametric graphon estimation. The statistical optimal estimators are based on constrained least squares and have computational complexity exponential…

基于成对比较的最优完整排序AI 解读

We consider the problem of ranking n players from partial pairwise comparison data under the Bradley–Terry–Luce model. For the first time in the literature, the minimax rate of this ranking problem is derived with respect to the Kendall’s tau distance that measures the difference between two rank vectors by counting the number of inversions. The minimax rate of ranking exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the…

前 k 名排序的部分恢复：极大似然最优性与谱方法的次优性AI 解读

Given partially observed pairwise comparison data generated by the Bradley–Terry–Luce (BTL) model, we study the problem of top-k ranking. That is, to optimally identify the set of top-k players. We derive the minimax rate with respect to a normalized Hamming loss. This provides the first result in the literature that characterizes the partial recovery error in terms of the proportion of mistakes for top-k ranking. We also derive the optimal signal to noise ratio condition…

离散结构恢复的迭代算法AI 解读

We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts and even group elements from a unified perspective. A simple iterative algorithm is proposed for discrete structure recovery, which generalizes methods including Lloyd’s algorithm and the power…

稀疏高维变点检测的极小极大速率AI 解读

We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for n independent, p-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order ploglog(8n) and has a very delicate dependence on the sample size: in a certain sparsity regime, it involves a triple iterated logarithmic factor…

随机设计非参数回归中的最优方差估计AI 解读

Consider the heteroscedastic nonparametric regression model with random design \begin{equation*}Y_{i}=f(X_{i})+V^{1/2}(X_{i})\varepsilon _{i},\quad i=1,2,\ldots ,n,\end{equation*} with $f(\cdot )$ and $V(\cdot )$ $\alpha $- and $\beta $-Hölder smooth, respectively. We show that the minimax rate of estimating $V(\cdot )$ under both local and global squared risks is of the order \begin{equation*}n^{-\frac{8\alpha \beta }{4\alpha \beta +2\alpha +\beta }}\vee n^{-\frac{2\beta }{2\beta +1}},\end{equation*}…

贝叶斯结构化线性模型的一般框架AI 解读

High dimensional statistics deals with the challenge of extracting structured information from complex model settings. Compared with a large number of frequentist methodologies, there are rather few theoretically optimal Bayes methods for high dimensional models. This paper provides a unified approach to both Bayes high dimensional statistics and Bayes nonparametrics in a general framework of structured linear models. With a proposed two-step prior, we prove a general oracle…

变分后验分布的收敛率AI 解读

We study convergence rates of variational posterior distributions for nonparametric and high-dimensional inference. We formulate general conditions on prior, likelihood and variational class that characterize the convergence rates. Under similar “prior mass and testing” conditions considered in the literature, the rate is found to be the sum of two terms. The first term stands for the convergence rate of the true posterior distribution, and the second term is contributed by…

对《通过边抽样进行网络交叉验证》的讨论AI 解读

We congratulate the authors for their exciting paper (Li et al., 2020) on the difficult problem of network model selection. The authors have proposed an interesting cross-validation procedure that selects the best network model according to the criterion ... One of the main novelties of the method is the use of matrix completion to reconstruct a full network |$\{\skew6\hat{A}_{ij}\}_{1\leq i\neq j\leq n}$| from the partial set of observations |$\{A_{ij}\}_{(i,\,j)\in\Omega}$|…

单调分段常数信号的估计AI 解读

Consider a sequence of real data points $X_{1},\ldots ,X_{n}$ with underlying means $\theta ^{*}_{1},\dots ,\theta ^{*}_{n}$. This paper starts from studying the setting that $\theta ^{*}_{i}$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the…

度修正块模型中的社区检测AI 解读

Community detection is a central problem of network data analysis. Given a network, the goal of community detection is to partition the network nodes into a small number of clusters, which could often help reveal interesting structures. The present paper studies community detection in Degree-Corrected Block Models (DCBMs). We first derive asymptotic minimax risks of the problem for a misclassification proportion loss under appropriate conditions. The minimax risks are shown…

Huber 污染模型下的稳健协方差与散布矩阵估计AI 解读

Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The…

稀疏典型相关分析：自适应估计与计算障碍AI 解读

Canonical correlation analysis is a classical technique for exploring the relationship between two sets of variables. It has important applications in analyzing high dimensional datasets originated from genomics, imaging and other fields. This paper considers adaptive minimax and computationally tractable estimation of leading sparse canonical coefficient vectors in high dimensions. Under a Gaussian canonical pair model, we first establish separate minimax estimation rates…

达到最优收敛率的 Graphon 估计AI 解读

Network analysis is becoming one of the most active research areas in statistics. Significant advances have been made recently on developing theories, methodologies and algorithms for analyzing networks. However, there has been little fundamental study on optimal estimation. In this paper, we establish optimal rate of convergence for graphon estimation. For the stochastic block model with $k$ clusters, we show that the optimal rate under the mean squared error is $n^{-1}\log k+k^{2}/n^{2}$…

稀疏典型相关分析中的极小极大估计AI 解读

Canonical correlation analysis is a widely used multivariate statistical technique for exploring the relation between two sets of variables. This paper considers the problem of estimating the leading canonical correlation directions in high-dimensional settings. Recently, under the assumption that the leading canonical correlation directions are sparse, various procedures have been proposed for many high-dimensional applications involving massive data sets. However, there…

稀疏主成分分析的速率最优后验收缩AI 解读

Principal component analysis (PCA) is possibly one of the most widely used statistical tools to recover a low-rank structure of the data. In the high-dimensional settings, the leading eigenvector of the sample covariance can be nearly orthogonal to the true eigenvector. A sparse structure is then commonly assumed along with a low rank structure. Recently, minimax estimation rates of sparse PCA were established under various interesting settings. On the other side, Bayesian…