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多维凸回归：最小二乘估计量的次优性AI 解读

Under the usual nonparametric regression model with Gaussian errors, Least Squares Estimators (LSEs) over natural subclasses of convex functions are shown to be suboptimal for estimating a d-dimensional convex function in squared error loss when the dimension d is 5 or larger. The specific function classes considered include: (i) bounded convex functions supported on a polytope (in random design), (ii) Lipschitz convex functions supported on any convex domain (in random…

样本扩增的统计复杂性AI 解读

The “sample amplification” problem formalizes the following question: Given n i.i.d. samples drawn from an unknown distribution P, when is it possible to produce a larger set of n+m samples which cannot be distinguished from n+m i.i.d. samples drawn from P? In this work, we provide a firm statistical foundation for this problem by deriving generally applicable amplification procedures, lower bound techniques and connections to existing statistical notions. Our techniques…

维度增长的非参数交互模型中的深度神经网络AI 解读

Deep neural networks have achieved tremendous success due to their representation power and adaptation to low-dimensional structures. Their potential for estimating structured regression functions has been recently established in the literature. However, most of the studies require the input dimension to be fixed and consequently ignore the effect of dimension on the rate of convergence and hamper their applications to modern big data with high dimensionality. In this paper,…

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偏斜 Bernstein–von Mises 定理与偏斜模态近似AI 解读

Gaussian deterministic approximations are routinely employed in Bayesian statistics to ease inference when the target posterior is intractable. Although these approximations are justified, in asymptotic regimes, by Bernstein-von Mises type results, in practice the expected Gaussian behavior might poorly represent the actual shape of the target posterior, thus affecting approximation accuracy. Motivated by these considerations, we derive an improved class of closed-form and…

基于球极投影的马尔可夫链蒙特卡洛AI 解读

High-dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves results in empirically observed “stickiness” and poor theoretical mixing properties—lack of geometric ergodicity. In this paper, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and…

四区制分段回归模型的统计推断AI 解读

Segmented regression models offer model flexibility and interpretability as compared to the global parametric and the nonparametric models, and yet are challenging in both estimation and inference. We consider a four-regime segmented model for temporally dependent data with segmenting boundaries depending on multivariate covariates with nondiminishing boundary effects. A mixed integer quadratic programming algorithm is formulated to facilitate the least square estimation of…

通过迭代投影估计张量因子模型AI 解读

Tensor time series, which is a time series consisting of tensorial observations, has become ubiquitous. It typically exhibits high dimensionality. One approach for dimension reduction is to use a factor model structure, in a form similar to Tucker tensor decomposition, except that the time dimension is treated as a dynamic process with a time dependent structure. In this paper, we introduce two approaches to estimate such a tensor factor model by using iterative orthogonal…

时间一致中心极限定理与渐近置信序列AI 解读

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS)—sequences of…

张量对张量回归：黎曼优化、过参数化与统计计算鸿沟的相互作用AI 解读

We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression…

利用正则变差随机向量的凸组合估计谱测度AI 解读

The extremal dependence structure of a regularly varying random vector X is fully described by its limiting spectral measure. In this paper, we investigate how to recover characteristics of the measure, such as extremal coefficients, from the extremal behaviour of convex combinations of components of X. Our considerations result in a class of new estimators of moments of the corresponding combinations for the spectral vector. We show asymptotic normality by means of a…

非独立成分分析AI 解读

A seminal result in the ICA literature states that for AY=ε, if the components of ε are independent and at most one is Gaussian, then A is identified up to sign and permutation of its rows (Signal Process. 36 (1994)). In this paper we study to which extent the independence assumption can be relaxed by replacing it with restrictions on higher order moment or cumulant tensors of ε. We document new conditions that establish identification for several nonindependent component…

通过树计数高效检验网络相关性AI 解读

We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the cooccurrences of signed trees for a family of nonisomorphic trees. When the two networks are Erdős–Rényi random graphs G(n,q) that are either independent or correlated with correlation coefficient ρ, our test runs in n2+o(1) time and succeeds with high probability as n→∞, provided that nmin{q,1−q}≥n−o(1) and…

用于模型检验的高斯过程方法AI 解读

This paper proposes a Gaussian process (GP) approach for testing conditional moment restrictions. Tests are based on squared Neyman orthogonal function-parametric processes integrated with respect to a GP distribution. This methodology leads to a general unified framework of kernel-based tests having the following properties: (i) bootstrap tests are easy to implement in the presence of nuisance parameters (they are simple quadratic forms, and there is no need to reestimate…

多层随机区块模型的计算与统计阈值AI 解读

We study the problem of community recovery and detection in multi-layer stochastic block models, focusing on the critical network density threshold for consistent community structure inference. Using a prototypical two-block model, we reveal a computational barrier for such multilayer stochastic block models that does not exist for its single-layer counterpart: When there are no computational constraints, the density threshold depends linearly on the number of layers.…

格点数据的多元趋势滤波AI 解读

We study a multivariate version of trend filtering, called Kronecker trend filtering or KTF, for the case in which the design points form a lattice in d dimensions. KTF is a natural extension of univariate trend filtering (Int. J. Comput. Vis. 70 (2006) 214–255; SIAM Rev. 51 (2009) 339–360; Ann. Statist. 42 (2014) 285–323), and is defined by minimizing a penalized least squares problem whose penalty term sums the absolute (higher-order) differences of the parameter to be…

弱条件下时间序列二阶参数的同时统计推断AI 解读

Strict stationarity is an assumption commonly used in time-series analysis in order to derive asymptotic distributional results for second-order statistics, like sample autocovariances and sample autocorrelations. Focusing on weak stationarity, this paper derives the asymptotic distribution of the maximum of sample autocovariances and sample autocorrelations under weak conditions by using Gaussian approximation techniques. The asymptotic theory for parameter estimators…

基于概率核嵌入的椭圆分布非参数检验AI 解读

Elliptical distribution is a basic assumption underlying many multivariate statistical methods. For example, in sufficient dimension reduction and statistical graphical models, this assumption is routinely imposed to simplify the data dependence structure. Before applying such methods, we need to decide whether the data are elliptically distributed. Currently existing tests either focus exclusively on spherical distributions, or rely on bootstrap to determine the null…

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…

非平稳时间序列的最优速率高斯逼近与显式构造AI 解读

Statistical inference for time series such as curve estimation for time-varying models or testing for existence of change-point have garnered significant attention. However, these works are generally restricted to the assumption of independence and/or stationarity at its best. The main obstacle is that the existing Gaussian approximation results for non-stationary processes only provide an existential proof and thus they are difficult to apply. In this paper, we provide two…

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环境不变线性最小二乘AI 解读

This paper considers a multi-environment linear regression model in which data from multiple experimental settings are collected. The joint distribution of the response variable and covariates may vary across different environments, yet the conditional expectations of the response variable, given the unknown set of important variables, are invariant. Such a statistical model is related to the problem of endogeneity, causal inference, and transfer learning. The motivation…

复合假设下有检验能力的p值与e值的存在性AI 解读

Given a composite null P and composite alternative Q, when and how can we construct a p-value whose distribution is exactly uniform under the null, and stochastically smaller than uniform under the alternative? Similarly, when and how can we construct an e-value whose expectation exactly equals one under the null, but its expected logarithm under the alternative is positive? We answer these basic questions, and other related ones, when P and Q are convex polytopes (in the…

考虑相关结构的高维两样本均值新检验AI 解读

This paper is concerned with high-dimensional two-sample mean problems, which receive considerable attention in recent literature. To utilize the correlation information among variables for enhancing the power of two-sample mean tests, we consider the setting in which the precision matrix of high-dimensional data possesses a linear structure. Thus, we first propose a new precision matrix estimation procedure with considering its linear structure, and further develop…

有限总体中的分位数过程及其应用AI 解读

The weak convergence of the quantile processes, which are constructed based on different estimators of the finite population quantiles, is shown under various well-known sampling designs based on a superpopulation model. The results related to the weak convergence of these quantile processes are applied to find asymptotic distributions of the smooth L-estimators and the estimators of smooth functions of finite population quantiles. Based on these asymptotic distributions,…

Wasserstein生成对抗网络作为极小极大最优分布估计量AI 解读

We provide nonasymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure μ with support in Rp. The probability μ is considered to be the push forward of the Lebesgue measure on the d-dimensional torus Td by a map g⋆:Td→Rp of smoothness β+1. Measuring the…