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Rate of convergence of over-parametrized deep neural network regression estimates learned by stochastic gradient descent 随机梯度下降法学习的超参数化深度神经网络回归估计的收敛速度
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2026-01-03 DOI: 10.1007/s10463-025-00967-4
Michael Kohler, Adam Krzyżak
{"title":"Rate of convergence of over-parametrized deep neural network regression estimates learned by stochastic gradient descent","authors":"Michael Kohler,&nbsp;Adam Krzyżak","doi":"10.1007/s10463-025-00967-4","DOIUrl":"10.1007/s10463-025-00967-4","url":null,"abstract":"<div><p>Nonparametric regression with random design is considered. The <span>(L_2)</span> error with integration with respect to the design measure is used as error criterion. Over-parametrized deep neural network estimates are defined with logistic activation function where all parameters are learned by stochastic gradient descent. It is shown that the estimates achieve a nearly optimal rate of convergence in case that the regression function is (<i>p</i>, <i>C</i>)–smooth. In case that the regression function satisfies a projection pursuit model or more generally a hierarchical composition model the estimate achieves a rate of convergence which does not depend on the input dimension.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 4","pages":"545 - 601"},"PeriodicalIF":1.4,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148357740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction to: Hidden AR process and adaptive Kalman filter 修正：隐藏AR过程和自适应卡尔曼滤波
IF 0.6 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-12-20 DOI: 10.1007/s10463-025-00974-5
Yury A. Kutoyants
{"title":"Correction to: Hidden AR process and adaptive Kalman filter","authors":"Yury A. Kutoyants","doi":"10.1007/s10463-025-00974-5","DOIUrl":"10.1007/s10463-025-00974-5","url":null,"abstract":"","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 1","pages":"175 - 175"},"PeriodicalIF":0.6,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the differentiability of (phi)-projections in the discrete finite case 离散有限情况下(phi) -投影的可微性
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-11-20 DOI: 10.1007/s10463-025-00959-4
G. Geenens, I. Kojadinovic, T. Martini
{"title":"On the differentiability of (phi)-projections in the discrete finite case","authors":"G. Geenens,&nbsp;I. Kojadinovic,&nbsp;T. Martini","doi":"10.1007/s10463-025-00959-4","DOIUrl":"10.1007/s10463-025-00959-4","url":null,"abstract":"<div><p>In the case of finite measures on finite spaces, we state conditions under which <span>(phi)</span>-projections are continuously differentiable. When the set on which one wishes to <span>(phi)</span>-project is convex, we show that the required assumptions are implied by easily verifiable conditions. In particular, for input probability vectors and a rather large class of <span>(phi text {-})</span>divergences, we obtain that <span>(phi)</span>-projections are continuously differentiable when projecting on a set defined by linear equalities. The obtained results are applied to <span>(phi)</span>-projection estimators (that is, minimum <span>(phi text {-})</span>divergence estimators). A first application, rooted in robust statistics, concerns the computation of the influence functions of such estimators. In a second set of applications, we derive their asymptotics when projecting on parametric sets of probability vectors, on sets of probability vectors generated from distributions with certain moments fixed and on Fréchet classes of bivariate probability arrays.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 5","pages":"805 - 838"},"PeriodicalIF":1.4,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148782651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The space of positive transition measures on a Markov chain 马尔可夫链上正转移测度的空间
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-11-19 DOI: 10.1007/s10463-025-00965-6
Naomichi Nakajima
{"title":"The space of positive transition measures on a Markov chain","authors":"Naomichi Nakajima","doi":"10.1007/s10463-025-00965-6","DOIUrl":"10.1007/s10463-025-00965-6","url":null,"abstract":"<div><p>Information geometry of Markov chains has been studied using the dually flat structure of the space of transition probabilities. Although applications of this structure have been investigated, few attempts have examined its statistical meaning. In this paper, we construct a foundation for investigating the statistical meaning based on Amari’s theory of <i>positive measures</i>. For the space of discrete distributions, Amari has introduced <i>the space of positive measures</i> by removing the constraint condition and investigated the extended space by finding the Bregman and <i>F</i>-divergence suitably. According to this, we introduce an extension of the space of transition probabilities equipped with suitable <i>F</i>-divergence for a given Markov chain. We regard it as <i>the space of positive transition measures on a Markov chain</i>, and study its dually flat structure. This provides new insight into the geometry of Markov chains and may lead to the development of the theory of Markov embeddings.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 5","pages":"873 - 888"},"PeriodicalIF":1.4,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unveiling low-dimensional patterns induced by convex non-differentiable regularizers 揭示由凸不可微正则子诱导的低维模式
IF 0.6 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-11-04 DOI: 10.1007/s10463-025-00957-6
Ivan Hejný, Jonas Wallin, Małgorzata Bogdan, Michał Kos
{"title":"Unveiling low-dimensional patterns induced by convex non-differentiable regularizers","authors":"Ivan Hejný,&nbsp;Jonas Wallin,&nbsp;Małgorzata Bogdan,&nbsp;Michał Kos","doi":"10.1007/s10463-025-00957-6","DOIUrl":"10.1007/s10463-025-00957-6","url":null,"abstract":"<div><p>This paper explores the asymptotic distributions of low-dimensional patterns in linear regression with regularizers such as Lasso, Elastic Net, Generalized Lasso, and SLOPE, as the number of observations <i>n</i> grows and the penalty increases at rate <span>(sqrt{n})</span>. While the asymptotic distribution of rescaled estimation errors is well-understood, convergence of patterns lacks proof in the literature, even for Lasso. We provide a proof using the Hausdorff distance for subdifferentials. We also derive the limiting probability of recovering the true model pattern, which approaches 1 when the penalty scaling diverges and the regularizer-specific asymptotic irrepresentability condition is satisfied. We propose two-step procedures that asymptotically recover model patterns, regardless of the irrepresentability condition. Our theory shows that Fused Lasso cannot reliably recover its clustering pattern for independent regressors, but this can be resolved by concavifying its penalty coefficients. Simulation studies compare the asymptotic properties of Lasso, Fused Lasso, and SLOPE. </p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 3","pages":"479 - 514"},"PeriodicalIF":0.6,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147924153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Building a theoretical foundation for combining negative controls and replicates 建立阴性对照与重复相结合的理论基础
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-11-04 DOI: 10.1007/s10463-025-00962-9
Jiming Jiang, Johann A. Gagnon-Bartsch, Terence P. Speed
{"title":"Building a theoretical foundation for combining negative controls and replicates","authors":"Jiming Jiang,&nbsp;Johann A. Gagnon-Bartsch,&nbsp;Terence P. Speed","doi":"10.1007/s10463-025-00962-9","DOIUrl":"10.1007/s10463-025-00962-9","url":null,"abstract":"<div><p>Studies using assays to quantify the expression of thousands of genes on tens to thousands of cell samples have been carried out for over 20 years. Such studies involve unwanted variation, often called batch effects, associated with the cell samples and the assay process. Removing this unwanted variation is essential before the measurements can be used to address the questions that motivated the studies. Combining the results of replicate assays with measurements on negative control genes to estimate the unwanted variation and remove it has proved to be effective at this task. The main goal of this paper is to present asymptotic theory that explains this effectiveness. The approach can be widened by using pseudo-replicate sets of pseudo-samples, for use with studies having no replicate assays. Theory covering this case is also presented. The established theory is supported by results of simulation studies and a real-data example.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 4","pages":"653 - 688"},"PeriodicalIF":1.4,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148357739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An information criterion for robust estimation with unnormalized statistical models 非归一化统计模型鲁棒估计的信息准则
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-10-29 DOI: 10.1007/s10463-025-00963-8
Takashi Takenouchi, Hiroaki Sasaki
{"title":"An information criterion for robust estimation with unnormalized statistical models","authors":"Takashi Takenouchi,&nbsp;Hiroaki Sasaki","doi":"10.1007/s10463-025-00963-8","DOIUrl":"10.1007/s10463-025-00963-8","url":null,"abstract":"<div><p>Noise contrastive estimation (NCE) is a popular approach for parameter estimation of unnormalized statistical models. NCE is based on a maximum likelihood estimation framework for a classification task, which makes the parameter estimation of unnormalized models sensitive to outlier noise included in the dataset. To cope with this problem, a robust version of NCE called <span>(gamma)</span>-NCE (GNCE) has been proposed. In this paper, we investigate asymptotic statistical properties of GNCE and propose an information criterion for GNCE, which enables us to select an appropriate model even when the dataset is contaminated by outlier noise.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 5","pages":"783 - 804"},"PeriodicalIF":1.4,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Combining variable screening methods for model averaging in high-dimensional data analysis 结合变量筛选方法进行高维数据分析模型平均
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-10-29 DOI: 10.1007/s10463-025-00961-w
Zhihao Zhao, Yuhong Yang, Li Wen
{"title":"Combining variable screening methods for model averaging in high-dimensional data analysis","authors":"Zhihao Zhao,&nbsp;Yuhong Yang,&nbsp;Li Wen","doi":"10.1007/s10463-025-00961-w","DOIUrl":"10.1007/s10463-025-00961-w","url":null,"abstract":"<div><p>Both model averaging (MA) and variable screening have been subjects of extensive research. The literature has introduced various methods for variable screening, typically relying on a single approach, such as marginal correlation. However, the performances of such methods may be very poor outside of their specifically applicable scenarios, and it is often difficult to know which scenarios are proper in real data analysis. In this study, we propose a method called <i>Recommendation and Trimmed Mean Scoring</i> (RTMS) for variable screening, which enhances reliability and stability of variable screening by integrating rankings from multiple screening methods. We introduce a practical concept of <i>approximate consistency in ranking</i> of variables, establishing a theoretical property of the RTMS method in both hard sparse and gradually decaying coefficient scenarios. Simulation and empirical results demonstrate that the RTMS method outperforms individual ranking methods in terms of variable screening and predictive performance of MA as well.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 4","pages":"689 - 720"},"PeriodicalIF":1.4,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148357741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Threshold detection under a semiparametric regression model 半参数回归模型下的阈值检测
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-10-24 DOI: 10.1007/s10463-025-00958-5
Graciela Boente, Florencia Leonardi, Daniela Rodriguez, Mariela Sued
{"title":"Threshold detection under a semiparametric regression model","authors":"Graciela Boente,&nbsp;Florencia Leonardi,&nbsp;Daniela Rodriguez,&nbsp;Mariela Sued","doi":"10.1007/s10463-025-00958-5","DOIUrl":"10.1007/s10463-025-00958-5","url":null,"abstract":"<div><p>Linear regression models have been extensively considered in the literature. However, in some practical applications they may not be appropriate all over the range of the covariate. In this paper, a more flexible model is proposed, introducing a response variable that is a function of the covariate plus a random error term. This function is assumed linear for large values of the covariate, specifically for values greater than a certain threshold. The penalized procedure estimates this threshold, focusing on a semiparametric approach with no parametric model assumed for the regression function for values smaller than the threshold. Consistency properties for the threshold estimator and the estimators of the linear model’s parameters are derived under mild assumptions. A numerical study explores small sample properties and underscores the importance of penalization. Real dataset analyses demonstrate the utility of penalized estimators.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 5","pages":"743 - 782"},"PeriodicalIF":1.4,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized high-dimensional tensor learning with nuclear norm regularization 核范数正则化的广义高维张量学习
IF 1.4 4区 数学
Annals of the Institute of Statistical Mathematics Pub Date : 2025-10-21 DOI: 10.1007/s10463-025-00960-x
Weilin Shen, Yao Wang, Xuehu Zhu
{"title":"Generalized high-dimensional tensor learning with nuclear norm regularization","authors":"Weilin Shen,&nbsp;Yao Wang,&nbsp;Xuehu Zhu","doi":"10.1007/s10463-025-00960-x","DOIUrl":"10.1007/s10463-025-00960-x","url":null,"abstract":"<div><p>Generalized linear models (GLMs) are widely used in diverse domains but are limited to vector or matrix data, overlooking modern high-dimensional structures such as videos and color images. To address this gap, the storage of information in multi-way arrays, known as tensors, has become a common practice. Inspired by the proposed tensor singular value decomposition (t-SVD), this paper introduces a generalized tensor regression learning framework that incorporates a tensor nuclear norm regularization. This framework elegantly extends the capabilities of traditional GLMs, which are tailored for vector and matrix data, to handle low-rank, high-dimensional tensors for regression and classification tasks. We also rigorously establish theoretical guarantees, including the convergence rate of the proposed estimator, to ensure statistical validity. Numerical experiments on synthetic datasets confirm its efficiency and superiority over matrix-based methods, while applications to image classification further demonstrate its practical effectiveness.</p></div>","PeriodicalId":55511,"journal":{"name":"Annals of the Institute of Statistical Mathematics","volume":"78 5","pages":"721 - 741"},"PeriodicalIF":1.4,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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