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多项式时间内达到极小极大最优的排序恢复AI 解读

We consider the seriation problem, whose goal is to recover a hidden ordering from a noisy observation of a permuted Robinson matrix. We establish sharp minimax rates under average-Lipschitz conditions that strictly extend the bi-Lipschitz framework of Giraud, Issartel and Verzelen (Electron. J. Stat. (2023) 17 1587–1662). We further design a polynomial-time algorithm that attains these optimal rates, thereby resolving two open questions raised in Giraud, Issartel and…

矩形矩阵 Schatten 范数的最优估计AI 解读

We consider the twin problems of estimating the effective rank and the Schatten norms ‖A‖s of a rectangular p×q matrix A from noisy observations. When s is an even integer, we introduce a polynomial-time estimator of ‖A‖s that achieves the minimax rate (pq)1/4. Interestingly, this optimal rate does not depend on the underlying rank of the matrix A. When s is not an even integer, the optimal rate is much slower. A simple thresholding estimator of the singular values achieves…

最优变点检测与定位AI 解读

Given a times series Y in Rn, with a piecewise constant mean and independent components, the twin problems of change-point detection and change-point localization, respectively amount to detecting the existence of times where the mean varies and estimating the positions of those change-points. In this work, we tightly characterize optimal rates for both problems and uncover the phase transition phenomenon from a global testing problem to a local estimation problem.…

众包问题中的最优排列估计AI 解读

Motivated by crowdsourcing applications, we consider a model where we have partial observations from a bivariate isotonic n×d matrix with an unknown permutation π∗ acting on its rows. Focusing on the twin problems of recovering the permutation π∗ and estimating the unknown matrix, we introduce a polynomial-time procedure achieving the minimax risk for these two problems, this for all possible values of n, d, and all possible sampling efforts. Along the way we establish that,…

未知零假设分布下的错误发现率控制：能否模拟理想程序？AI 解读

Classical multiple testing theory prescribes the null distribution, which is often too stringent an assumption for nowadays large scale experiments. This paper presents theoretical foundations to understand the limitations caused by ignoring the null distribution, and how it can be properly learned from the same data set, when possible. We explore this issue in the setting where the null distributions are Gaussian with unknown rescaling parameters (mean and variance) whereas…

伴随异常值筛选的最小效应估计AI 解读

We introduce one-sided versions of Huber’s contamination model, in which corrupted samples tend to take larger values than uncorrupted ones. Two intertwined problems are addressed: estimation of the mean of the uncorrupted samples (minimum effect) and selection of the corrupted samples (outliers). Regarding estimation of the minimum effect, we derive the minimax risks and introduce estimators that are adaptive with respect to the unknown number of contaminations. The optimal…

模型辅助变量聚类：极小极大最优恢复与算法AI 解读

The problem of variable clustering is that of estimating groups of similar components of a $p$-dimensional vector $X=(X_{1},\ldots ,X_{p})$ from $n$ independent copies of $X$. There exists a large number of algorithms that return data-dependent groups of variables, but their interpretation is limited to the algorithm that produced them. An alternative is model-based clustering, in which one begins by defining population level clusters relative to a model that embeds notions…

高斯向量模型中稀疏度的自适应估计AI 解读

Consider the Gaussian vector model with mean value $\theta$. We study the twin problems of estimating the number $\Vert \theta \Vert_{0}$ of nonzero components of $\theta$ and testing whether $\Vert \theta \Vert_{0}$ is smaller than some value. For testing, we establish the minimax separation distances for this model and introduce a minimax adaptive test. Extensions to the case of unknown variance are also discussed. Rewriting the estimation of $\Vert \theta \Vert_{0}$ as a…

稀疏性约束下线性泛函的最优自适应估计AI 解读

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector $\theta \in\mathbb{R}^{d}$ belongs to a class of $s$-sparse vectors with unknown $s$. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when $s$ is unknown. Furthermore, we address the issue of…

稀疏混合模型中的检测与特征选择AI 解读

We consider Gaussian mixture models in high dimensions, focusing on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive minimax rates for the problems of testing and of variable selection. We find these rates to depend crucially on the knowledge of the covariance matrices and on whether the mixture is symmetric or not. We establish the performance of various procedures, including the top sparse eigenvalue of the…

网络模型的预言不等式与稀疏 graphon 估计AI 解读

Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. We consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network. Taking the stochastic block model as an approximation, we construct estimators of network connection probabilities—the ordinary block constant least squares estimator, and its restricted version.…

摘要暂缺

稠密随机网络中的社群检测AI 解读

We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $N$ nodes. Under the null hypothesis, the graph is a realization of an Erdős–Rényi graph with probability $p_{0}$. Under the (composite) alternative, there is an unknown subgraph of $n$ nodes where the probability of connection is $p_{1}>p_{0}$. We derive a…

函数型线性模型的极小极大自适应检验AI 解读

We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional principal component analysis. Interestingly, the procedures are completely data-driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local…

平稳高斯场的自适应估计AI 解读

We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection…

高维高斯线性模型的拟合优度检验AI 解读

Let (Y, (Xi)1≤i≤p) be a real zero mean Gaussian vector and V be a subset of {1, …, p}. Suppose we are given n i.i.d. replications of this vector. We propose a new test for testing that Y is independent of (Xi)i∈{1, …, p}∖V conditionally to (Xi)i∈V against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of X or the variance of Y and applies in a high-dimensional setting. It straightforwardly extends to test the…