arXiv - STAT - Statistics Theory最新文献_第8页

Likelihood Geometry of the Squared Grassmannian 平方格拉斯曼的似然几何

arXiv - STAT - Statistics Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03730

Hannah Friedman

引用次数: 0

Error bounds of Median-of-means estimators with VC-dimension 具有 VC 维度的均值中值估计器的误差边界

arXiv - STAT - Statistics Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03410

Yuxuan Wang, Yiming Chen, Hanchao Wang, Lixin Zhang

引用次数: 0

The Geometry and Well-Posedness of Sparse Regularized Linear Regression 稀疏正则化线性回归的几何学和良好假设性

arXiv - STAT - Statistics Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03461

Jasper Marijn Everink, Yiqiu Dong, Martin Skovgaard Andersen

引用次数: 0

Convergence Rates for the Maximum A Posteriori Estimator in PDE-Regression Models with Random Design 具有随机设计的 PDE 回归模型中最大后验估计器的收敛率

arXiv - STAT - Statistics Theory Pub Date : 2024-09-05 DOI: arxiv-2409.03417

Maximilian Siebel

{"title":"Convergence Rates for the Maximum A Posteriori Estimator in PDE-Regression Models with Random Design","authors":"Maximilian Siebel","doi":"arxiv-2409.03417","DOIUrl":"https://doi.org/arxiv-2409.03417","url":null,"abstract":"We consider the statistical inverse problem of recovering a parameter\u0000$thetain H^alpha$ from data arising from the Gaussian regression problem\u0000begin{equation*} Y = mathscr{G}(theta)(Z)+varepsilon end{equation*} with nonlinear forward\u0000map $mathscr{G}:mathbb{L}^2tomathbb{L}^2$, random design points $Z$ and\u0000Gaussian noise $varepsilon$. The estimation strategy is based on a least\u0000squares approach under $VertcdotVert_{H^alpha}$-constraints. We establish\u0000the existence of a least squares estimator $hat{theta}$ as a maximizer for a\u0000given functional under Lipschitz-type assumptions on the forward map\u0000$mathscr{G}$. A general concentration result is shown, which is used to prove\u0000consistency and upper bounds for the prediction error. The corresponding rates\u0000of convergence reflect not only the smoothness of the parameter of interest but\u0000also the ill-posedness of the underlying inverse problem. We apply the general\u0000model to the Darcy problem, where the recovery of an unknown coefficient\u0000function $f$ of a PDE is of interest. For this example, we also provide\u0000corresponding rates of convergence for the prediction and estimation errors.\u0000Additionally, we briefly discuss the applicability of the general model to\u0000other problems.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"68 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Bulk Spectra of Truncated Sample Covariance Matrices 截断样本协方差矩阵的总体频谱

arXiv - STAT - Statistics Theory Pub Date : 2024-09-04 DOI: arxiv-2409.02911

Subhroshekhar Ghosh, Soumendu Sundar Mukherjee, Himasish Talukdar

{"title":"Bulk Spectra of Truncated Sample Covariance Matrices","authors":"Subhroshekhar Ghosh, Soumendu Sundar Mukherjee, Himasish Talukdar","doi":"arxiv-2409.02911","DOIUrl":"https://doi.org/arxiv-2409.02911","url":null,"abstract":"Determinantal Point Processes (DPPs), which originate from quantum and\u0000statistical physics, are known for modelling diversity. Recent research [Ghosh\u0000and Rigollet (2020)] has demonstrated that certain matrix-valued $U$-statistics\u0000(that are truncated versions of the usual sample covariance matrix) can\u0000effectively estimate parameters in the context of Gaussian DPPs and enhance\u0000dimension reduction techniques, outperforming standard methods like PCA in\u0000clustering applications. This paper explores the spectral properties of these\u0000matrix-valued $U$-statistics in the null setting of an isotropic design. These\u0000matrices may be represented as $X L X^top$, where $X$ is a data matrix and $L$\u0000is the Laplacian matrix of a random geometric graph associated to $X$. The main\u0000mathematically interesting twist here is that the matrix $L$ is dependent on\u0000$X$. We give complete descriptions of the bulk spectra of these matrix-valued\u0000$U$-statistics in terms of the Stieltjes transforms of their empirical spectral\u0000measures. The results and the techniques are in fact able to address a broader\u0000class of kernelised random matrices, connecting their limiting spectra to\u0000generalised Marv{c}enko-Pastur laws and free probability.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Smoothed Robust Phase Retrieval 平滑稳健相位检索

arXiv - STAT - Statistics Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01570

Zhong Zheng, Lingzhou Xue

{"title":"Smoothed Robust Phase Retrieval","authors":"Zhong Zheng, Lingzhou Xue","doi":"arxiv-2409.01570","DOIUrl":"https://doi.org/arxiv-2409.01570","url":null,"abstract":"The phase retrieval problem in the presence of noise aims to recover the\u0000signal vector of interest from a set of quadratic measurements with infrequent\u0000but arbitrary corruptions, and it plays an important role in many scientific\u0000applications. However, the essential geometric structure of the nonconvex\u0000robust phase retrieval based on the $ell_1$-loss is largely unknown to study\u0000spurious local solutions, even under the ideal noiseless setting, and its\u0000intrinsic nonsmooth nature also impacts the efficiency of optimization\u0000algorithms. This paper introduces the smoothed robust phase retrieval (SRPR)\u0000based on a family of convolution-type smoothed loss functions. Theoretically,\u0000we prove that the SRPR enjoys a benign geometric structure with high\u0000probability: (1) under the noiseless situation, the SRPR has no spurious local\u0000solutions, and the target signals are global solutions, and (2) under the\u0000infrequent but arbitrary corruptions, we characterize the stationary points of\u0000the SRPR and prove its benign landscape, which is the first landscape analysis\u0000of phase retrieval with corruption in the literature. Moreover, we prove the\u0000local linear convergence rate of gradient descent for solving the SRPR under\u0000the noiseless situation. Experiments on both simulated datasets and image\u0000recovery are provided to demonstrate the numerical performance of the SRPR.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Demystified: double robustness with nuisance parameters estimated at rate n-to-the-1/4 解密：以 n 比 1/4 的速率估算滋扰参数的双重稳健性

arXiv - STAT - Statistics Theory Pub Date : 2024-09-03 DOI: arxiv-2409.02320

Judith J. Lok

引用次数: 0

Deconvolution of repeated measurements corrupted by unknown noise 对受未知噪声干扰的重复测量进行解卷积

arXiv - STAT - Statistics Theory Pub Date : 2024-09-03 DOI: arxiv-2409.02014

Jérémie Capitao-Miniconi, Elisabeth Gassiat, Luc Lehéricy

引用次数: 0

Formalizing the causal interpretation in accelerated failure time models with unmeasured heterogeneity 将具有未测量异质性的加速故障时间模型中的因果解释形式化

arXiv - STAT - Statistics Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01983

Mari Brathovde, Hein Putter, Morten Valberg, Richard A. J. Post

{"title":"Formalizing the causal interpretation in accelerated failure time models with unmeasured heterogeneity","authors":"Mari Brathovde, Hein Putter, Morten Valberg, Richard A. J. Post","doi":"arxiv-2409.01983","DOIUrl":"https://doi.org/arxiv-2409.01983","url":null,"abstract":"In the presence of unmeasured heterogeneity, the hazard ratio for exposure\u0000has a complex causal interpretation. To address this, accelerated failure time\u0000(AFT) models, which assess the effect on the survival time ratio scale, are\u0000often suggested as a better alternative. AFT models also allow for\u0000straightforward confounder adjustment. In this work, we formalize the causal\u0000interpretation of the acceleration factor in AFT models using structural causal\u0000models and data under independent censoring. We prove that the acceleration\u0000factor is a valid causal effect measure, even in the presence of frailty and\u0000treatment effect heterogeneity. Through simulations, we show that the\u0000acceleration factor better captures the causal effect than the hazard ratio\u0000when both AFT and proportional hazards models apply. Additionally, we extend\u0000the interpretation to systems with time-dependent acceleration factors,\u0000revealing the challenge of distinguishing between a time-varying homogeneous\u0000effect and unmeasured heterogeneity. While the causal interpretation of\u0000acceleration factors is promising, we caution practitioners about potential\u0000challenges in estimating these factors in the presence of effect heterogeneity.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A sparse PAC-Bayesian approach for high-dimensional quantile prediction 用于高维量化预测的稀疏 PAC-Bayesian 方法

arXiv - STAT - Statistics Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01687

The Tien Mai

{"title":"A sparse PAC-Bayesian approach for high-dimensional quantile prediction","authors":"The Tien Mai","doi":"arxiv-2409.01687","DOIUrl":"https://doi.org/arxiv-2409.01687","url":null,"abstract":"Quantile regression, a robust method for estimating conditional quantiles,\u0000has advanced significantly in fields such as econometrics, statistics, and\u0000machine learning. In high-dimensional settings, where the number of covariates\u0000exceeds sample size, penalized methods like lasso have been developed to\u0000address sparsity challenges. Bayesian methods, initially connected to quantile\u0000regression via the asymmetric Laplace likelihood, have also evolved, though\u0000issues with posterior variance have led to new approaches, including\u0000pseudo/score likelihoods. This paper presents a novel probabilistic machine\u0000learning approach for high-dimensional quantile prediction. It uses a\u0000pseudo-Bayesian framework with a scaled Student-t prior and Langevin Monte\u0000Carlo for efficient computation. The method demonstrates strong theoretical\u0000guarantees, through PAC-Bayes bounds, that establish non-asymptotic oracle\u0000inequalities, showing minimax-optimal prediction error and adaptability to\u0000unknown sparsity. Its effectiveness is validated through simulations and\u0000real-world data, where it performs competitively against established\u0000frequentist and Bayesian techniques.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0