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广义韧性与稳健统计AI 解读

Robust statistics traditionally focuses on outliers, or perturbations in total variation distance. However, a dataset could be maliciously corrupted in many other ways, such as systematic measurement errors and missing covariates. We consider corruption in either TV or Wasserstein distance, and show that robust estimation is possible whenever the true population distribution satisfies a property called generalized resilience, which holds under moment or hypercontractive…

学习排列混合模型：成对比较组与组合矩方法AI 解读

In applications such as rank aggregation, mixture models for permutations are frequently used when the population exhibits heterogeneity. In this work, we study the widely used Mallows mixture model. In the high-dimensional setting, we propose a polynomial-time algorithm that learns a Mallows mixture of permutations on n elements with the optimal sample complexity that is proportional to logn, improving upon previous results that scale polynomially with n. In the high-noise…

线性最小二乘的精确极小极大风险与样本协方差矩阵的下尾AI 解读

We consider random-design linear prediction and related questions on the lower tail of random matrices. It is known that, under boundedness constraints, the minimax risk is of order d/n in dimension d with n samples. Here, we study the minimax expected excess risk over the full linear class, depending on the distribution of covariates. First, the least squares estimator is exactly minimax optimal in the well-specified case, for every distribution of covariates. We express…

双成分高斯混合模型中的精确最优恢复AI 解读

In this paper, we study the problem of clustering in the Two component Gaussian mixture model when the centers are separated by some Δ>0. We present a nonasymptotic lower bound for the corresponding minimax Hamming risk improving on existing results. We also propose an optimal, efficient and adaptive procedure that is minimax rate optimal. The rate optimality is moreover sharp in the asymptotics when the sample size goes to infinity. Our procedure is based on a variant of…

加性模型下的随机连续臂老虎机：极小极大遗憾与自适应算法AI 解读

We consider d-dimensional stochastic continuum-armed bandits with the expected reward function being additive β-Hölder with sparsity s for 0<β<∞ and 1≤s≤d. The rate of convergence O˜(s·T β+12β+1) for the minimax regret is established where T is the number of rounds. In particular, the minimax regret does not depend on d and is linear in s. A novel algorithm is proposed and is shown to be rate-optimal, up to a logarithmic factor of T. The problem of adaptivity is also…

大样本协方差矩阵尖峰特征值与线性谱统计量的渐近独立性AI 解读

We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the…

对抗训练分类准确率的精确统计分析AI 解读

Despite the wide empirical success of modern machine learning algorithms and models in a multitude of applications, they are known to be highly susceptible to seemingly small indiscernible perturbations to the input data known as adversarial attacks. A variety of recent adversarial training procedures have been proposed to remedy this issue. Despite the success of such procedures at increasing accuracy on adversarially perturbed inputs or robust accuracy, these techniques…

最大似然估计鞍点近似的渐近精度AI 解读

The saddlepoint approximation gives an approximation to the density of a random variable in terms of its moment generating function. When the underlying random variable is itself the sum of n unobserved i.i.d. terms, the basic classical result is that the relative error in the density is of order 1/n. If instead the approximation is interpreted as a likelihood and maximised as a function of model parameters, the result is an approximation to the maximum likelihood estimate…

未知方差下的分布式自适应高斯均值估计：交互协议有助于适应AI 解读

Distributed estimation of a Gaussian mean with unknown variance under communication constraints is studied. Necessary and sufficient communication costs under different types of distributed protocols are derived for any estimator that is adaptively rate-optimal over a range of possible values for the variance. Communication-efficient and statistically optimal procedures are developed. The analysis reveals an interesting and important distinction among different types of…

尖峰随机矩阵模型中的最优信号检测：似然比检验与线性谱统计量AI 解读

We study signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on signal components that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to-noise ratios are below certain bounds. In addition, the log-likelihood…

惩罚回归学习动态治疗方案的泛化误差界AI 解读

A dynamic treatment regime (DTR) is a sequence of decision rules, one per stage of intervention, that maps up-to-date patient information to a recommended treatment. Discovering an appropriate DTR for a given disease is a challenging issue especially when a large set of prognostic variables are observed. To address this problem, we propose penalized regression-based learning methods with l1 penalty to estimate the optimal DTR that would maximize the expected outcome if…

利用自适应筛选程序检测多个可重复信号AI 解读

Replicability is a fundamental quality of scientific discoveries: we are interested in those signals that are detectable in different laboratories, different populations, across time etc. Unlike meta-analysis which accounts for experimental variability but does not guarantee replicability, testing a partial conjunction (PC) null aims specifically to identify the signals that are discovered in multiple studies. In many contemporary applications, for example, comparing…

基于不变性的随机化检验的一致性AI 解读

Invariance-based randomization tests—such as permutation tests, rotation tests, or sign changes—are an important and widely used class of statistical methods. They allow drawing inferences under weak assumptions on the data distribution. Most work focuses on their type I error control properties, while their consistency properties are much less understood. We develop a general framework to study the consistency of invariance-based randomization tests, assuming the data is…

网络干扰下处理效应估计的随机图渐近理论AI 解读

The network interference model for treatment effect estimation places experimental units at the vertices of an undirected exposure graph, such that treatment assigned to one unit may affect the outcome of another unit if and only if these two units are connected by an edge. This model has recently gained popularity as means of incorporating interference effects into the Neyman–Rubin potential outcomes framework; and several authors have considered estimation of various…

二阶差分估计量的中心极限定理及波动率与强度应用AI 解读

In this paper, we introduce a general method for estimating the quadratic covariation of one or more spot parameter processes associated with continuous time semimartingales, and present a central limit theorem that has this class of estimators as one of its applications. The class of estimators we introduce, that we call Two-Scales Quadratic Covariation (TSQC) estimators, is based on sums of increments of second differences of the observed processes, and the intervals over…

变换群不变分布的极限定理AI 解读

A distributional symmetry is invariance of a distribution under a group of transformations. Exchangeability and stationarity are examples. We explain that a result of ergodic theory implies a law of large numbers for such invariant distributions: If the group satisfies suitable conditions, expectations can be estimated by averaging over subsets of transformations, and these estimators are strongly consistent. We show that, if a mixing condition holds, the averages also…

随机向量独立性的普遍一致、完全分布自由秩检验AI 解读

Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct…

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…

具有对称不变性的高斯模型空间中的模型选择AI 解读

We consider multivariate centered Gaussian models for the random variable Z=(Z1,…,Zp), invariant under the action of a subgroup of the group of permutations on {1,…,p}. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter Σ and also the analytic expression of the normalizing constant of the Diaconis–Ylvisaker conjugate prior for the precision parameter K=Σ−1. We…

基于计数数据的 Hawkes 过程谱估计AI 解读

This paper presents a parametric estimation method for ill-observed linear stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, methods based on the likelihood are not feasible. We show that spectral estimation based on Whittle’s method is adapted to this case and provides consistent and asymptotically normal estimators, provided a mild moment condition on the reproduction function. Simulated…

稀疏纵向观测的内蕴黎曼函数型数据分析AI 解读

A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from…

Boosting 与最小 ℓ1 范数插值分类器的精确高维渐近理论AI 解读

This paper establishes a precise high-dimensional asymptotic theory for boosting on separable data, taking statistical and computational perspectives. We consider a high-dimensional setting where the number of features (weak learners) p scales with the sample size n, in an overparametrized regime. Under a class of statistical models, we provide an exact analysis of the generalization error of boosting when the algorithm interpolates the training data and maximizes the…

形状约束下密度水平集估计的数据自适应方法AI 解读

Given a random sample of points from some unknown density, we propose a method for estimating density level sets, for a positive threshold t, under the r-convexity assumption. This shape condition generalizes the convexity property and allows to consider level sets with more than one connected component. The main problem in practice is that r is an unknown geometric characteristic of the set related to its curvature, which may depend on t. A stochastic algorithm is proposed…

大型 VAR 中的协整AI 解读

The paper analyzes cointegration in vector autoregressive processes (VARs) for the cases when both the number of coordinates, N, and the number of time periods, T, are large and of the same order. We propose a way to examine a VAR of order 1 for the presence of cointegration based on a modification of the Johansen likelihood ratio test. The advantage of our procedure over the original Johansen test and its finite sample corrections is that our test does not suffer from…