Conformal inference is a versatile tool for building prediction sets in regression or classification. In this paper, we consider the false coverage proportion (FCP) in a transductive setting with a calibration sample of n points and a test sample of m points. We identify the exact, distribution-free, asymptotic distribution of the FCP when both n and m tend to infinity. This shows in particular that FCP control can be achieved by using the well-known Kolmogorov distribution, and puts forward that the asymptotic variance is decreasing in the ratio n/m. We then provide a number of extensions by considering the novelty detection problem, weighted conformal inference and distribution shift between the calibration sample and the test sample. In particular, our asymptotical results allow to accurately quantify the asymptotical behavior of the errors when weighted conformal inference is used.
共形推理是建立回归或分类预测集的通用工具。在本文中，我们考虑了在具有 n 个点的校准样本和 m 个点的测试样本的反演环境中的虚假覆盖率（FCP）。我们确定了当 n 和 m 都趋于无穷大时，FCP 的精确、无分布、渐近分布。这特别表明，使用著名的科尔莫戈罗夫分布可以实现 FCP 控制，并提出渐近方差随 n/m 之比递减。然后，我们通过考虑新颖性检测问题、加权保形推理以及校准样本和测试样本之间的分布偏移，提出了一些扩展方法。特别是，当使用加权保形推理时，我们的渐近结果可以准确量化误差的渐近行为。
{"title":"Asymptotics for conformal inference","authors":"Ulysse Gazin","doi":"arxiv-2409.12019","DOIUrl":"https://doi.org/arxiv-2409.12019","url":null,"abstract":"Conformal inference is a versatile tool for building prediction sets in\u0000regression or classification. In this paper, we consider the false coverage\u0000proportion (FCP) in a transductive setting with a calibration sample of n\u0000points and a test sample of m points. We identify the exact, distribution-free,\u0000asymptotic distribution of the FCP when both n and m tend to infinity. This\u0000shows in particular that FCP control can be achieved by using the well-known\u0000Kolmogorov distribution, and puts forward that the asymptotic variance is\u0000decreasing in the ratio n/m. We then provide a number of extensions by\u0000considering the novelty detection problem, weighted conformal inference and\u0000distribution shift between the calibration sample and the test sample. In\u0000particular, our asymptotical results allow to accurately quantify the\u0000asymptotical behavior of the errors when weighted conformal inference is used.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267764","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}
In this paper, we establish an exponential inequality for U-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound are expressed as a sum of an exponential term plus an other one involving the tail of a sum of squared norms. We start by the degenerate case. Then we provide applications to U-statistics of not necessarily degenerate fixed kernel, weighted U-statistics and incomplete U-statistics.
在本文中，我们建立了一个指数不等式，用于 i.i.d. 数据、变化内核和在可分离的希尔伯特空间中取值的 U 统计量。边界表示为一个指数项加上另一个涉及平方准则之和尾部的总和。我们首先讨论退化情况。然后，我们将其应用于不一定是退化定核的 U 统计、加权 U 统计和不完全 U 统计。
{"title":"An exponential inequality for Hilbert-valued U-statistics of i.i.d. data","authors":"Davide GiraudoIRMA","doi":"arxiv-2409.11737","DOIUrl":"https://doi.org/arxiv-2409.11737","url":null,"abstract":"In this paper, we establish an exponential inequality for U-statistics of\u0000i.i.d. data, varying kernel and taking values in a separable Hilbert space. The\u0000bound are expressed as a sum of an exponential term plus an other one involving\u0000the tail of a sum of squared norms. We start by the degenerate case. Then we\u0000provide applications to U-statistics of not necessarily degenerate fixed\u0000kernel, weighted U-statistics and incomplete U-statistics.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267767","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}
Causal inference problems often involve continuous treatments, such as dose, duration, or frequency. However, continuous exposures bring many challenges, both with identification and estimation. For example, identifying standard dose-response estimands requires that everyone has some chance of receiving any particular level of the exposure (i.e., positivity). In this work, we explore an alternative approach: rather than estimating dose-response curves, we consider stochastic interventions based on exponentially tilting the treatment distribution by some parameter $delta$, which we term an incremental effect. This increases or decreases the likelihood a unit receives a given treatment level, and crucially, does not require positivity for identification. We begin by deriving the efficient influence function and semiparametric efficiency bound for these incremental effects under continuous exposures. We then show that estimation of the incremental effect is dependent on the size of the exponential tilt, as measured by $delta$. In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of $n/delta$, instead of usual sample size $n$. Further, we establish new convergence rates and bounds on the bias of double machine learning-style estimators. Our novel analysis gives a better dependence on $delta$ compared to standard analyses, by using mixed supremum and $L_2$ norms, instead of just $L_2$ norms from Cauchy-Schwarz bounds. Finally, we show that taking $delta to infty$ gives a new estimator of the dose-response curve at the edge of the support, and we give a detailed study of convergence rates in this regime.
{"title":"Incremental effects for continuous exposures","authors":"Kyle Schindl, Shuying Shen, Edward H. Kennedy","doi":"arxiv-2409.11967","DOIUrl":"https://doi.org/arxiv-2409.11967","url":null,"abstract":"Causal inference problems often involve continuous treatments, such as dose,\u0000duration, or frequency. However, continuous exposures bring many challenges,\u0000both with identification and estimation. For example, identifying standard\u0000dose-response estimands requires that everyone has some chance of receiving any\u0000particular level of the exposure (i.e., positivity). In this work, we explore\u0000an alternative approach: rather than estimating dose-response curves, we\u0000consider stochastic interventions based on exponentially tilting the treatment\u0000distribution by some parameter $delta$, which we term an incremental effect.\u0000This increases or decreases the likelihood a unit receives a given treatment\u0000level, and crucially, does not require positivity for identification. We begin\u0000by deriving the efficient influence function and semiparametric efficiency\u0000bound for these incremental effects under continuous exposures. We then show\u0000that estimation of the incremental effect is dependent on the size of the\u0000exponential tilt, as measured by $delta$. In particular, we derive new minimax\u0000lower bounds illustrating how the best possible root mean squared error scales\u0000with an effective sample size of $n/delta$, instead of usual sample size $n$.\u0000Further, we establish new convergence rates and bounds on the bias of double\u0000machine learning-style estimators. Our novel analysis gives a better dependence\u0000on $delta$ compared to standard analyses, by using mixed supremum and $L_2$\u0000norms, instead of just $L_2$ norms from Cauchy-Schwarz bounds. Finally, we show\u0000that taking $delta to infty$ gives a new estimator of the dose-response\u0000curve at the edge of the support, and we give a detailed study of convergence\u0000rates in this regime.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267766","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}
In this thesis, we consider an $N$-dimensional Ornstein-Uhlenbeck (OU) process satisfying the linear stochastic differential equation $dmathbf x(t) = - mathbf Bmathbf x(t) dt + boldsymbol Sigma d mathbf w(t).$ Here, $mathbf B$ is a fixed $N times N$ circulant friction matrix whose eigenvalues have positive real parts, $boldsymbol Sigma$ is a fixed $N times M$ matrix. We consider a signal propagation model governed by this OU process. In this model, an underlying signal propagates throughout a network consisting of $N$ linked sensors located in space. We interpret the $n$-th component of the OU process as the measurement of the propagating effect made by the $n$-th sensor. The matrix $mathbf B$ represents the sensor network structure: if $mathbf B$ has first row $(b_1 , dots , b_N),$ where $b_1>0$ and $b_2 , dots , b_N le 0,$ then the magnitude of $b_p$ quantifies how receptive the $n$-th sensor is to activity within the $(n+p-1)$-th sensor. Finally, the $(m,n)$-th entry of the matrix $mathbf D = frac{boldsymbol Sigma boldsymbol Sigma^text T}{2}$ is the covariance of the component noises injected into the $m$-th and $n$-th sensors. For different choices of $mathbf B$ and $boldsymbol Sigma,$ we investigate whether Cyclicity Analysis enables us to recover the structure of network. Roughly speaking, Cyclicity Analysis studies the lead-lag dynamics pertaining to the components of a multivariate signal. We specifically consider an $N times N$ skew-symmetric matrix $mathbf Q,$ known as the lead matrix, in which the sign of its $(m,n)$-th entry captures the lead-lag relationship between the $m$-th and $n$-th component OU processes. We investigate whether the structure of the leading eigenvector of $mathbf Q,$ the eigenvector corresponding to the largest eigenvalue of $mathbf Q$ in modulus, reflects the network structure induced by $mathbf B.$
在本论文中，我们考虑一个 $N$ 维的奥恩斯坦-乌伦贝克（OU）过程，该过程满足线性随机微分方程 $dmathbf x(t) =- mathbf Bmathbf x(t) dt + boldsymbol Sigma d mathbf w(t)。这里，$mathbfB$是一个固定的$N times N$环形摩擦矩阵，其特征值的实部为正，$boldsymbol Sigma$是一个固定的$N times M$矩阵。我们将考虑一个受此 OU 过程控制的信号传播模型。在这个模型中，一个基本信号在由位于空间的 $N$ 链接传感器组成的网络中传播。我们将 OU 过程的第 n 个分量解释为第 n 个传感器对传播效果的测量。矩阵 $mathbf B$ 表示传感器网络结构：如果 $mathbf B$ 的第一行为 $(b_1 , dots , b_N), $ 其中 $b_1>0$ 并且 $b_2 , dots , b_N le 0, $ 那么 $b_p$ 的大小量化了 $n$-th 传感器对 $(n+p-1)$-th 传感器内活动的接受程度。最后，矩阵 $mathbf D = frac{boldsymbol Sigma Sigma^text T}{2}$ 的 $(m,n)$ 条目是注入 $m$-th 和 $n$-th 传感器的分量噪声的协方差。对于 $mathbf B$ 和 $boldsymbol Sigma$ 的不同选择，我们研究了循环分析是否能让我们恢复网络结构。粗略地说，循环分析研究的是多变量信号成分的前导-滞后动态。我们特别考虑了一个 $N times N$ 的倾斜对称矩阵 $/mathbf Q，$ 称为先导矩阵，其中 $(m,n)$-th 条目的符号捕捉了 $m$-th 和 $n$-th 分量 OU 过程之间的先导-滞后关系。我们研究了$mathbf Q的前导特征向量的结构，即对应于$mathbf Q的最大特征值的特征向量，是否反映了$mathbf B所诱导的网络结构。
{"title":"Cyclicity Analysis of the Ornstein-Uhlenbeck Process","authors":"Vivek Kaushik","doi":"arxiv-2409.12102","DOIUrl":"https://doi.org/arxiv-2409.12102","url":null,"abstract":"In this thesis, we consider an $N$-dimensional Ornstein-Uhlenbeck (OU)\u0000process satisfying the linear stochastic differential equation $dmathbf x(t) =\u0000- mathbf Bmathbf x(t) dt + boldsymbol Sigma d mathbf w(t).$ Here, $mathbf\u0000B$ is a fixed $N times N$ circulant friction matrix whose eigenvalues have\u0000positive real parts, $boldsymbol Sigma$ is a fixed $N times M$ matrix. We\u0000consider a signal propagation model governed by this OU process. In this model,\u0000an underlying signal propagates throughout a network consisting of $N$ linked\u0000sensors located in space. We interpret the $n$-th component of the OU process\u0000as the measurement of the propagating effect made by the $n$-th sensor. The\u0000matrix $mathbf B$ represents the sensor network structure: if $mathbf B$ has\u0000first row $(b_1 , dots , b_N),$ where $b_1>0$ and $b_2 , dots \u0000, b_N le 0,$ then the magnitude of $b_p$ quantifies how receptive the $n$-th\u0000sensor is to activity within the $(n+p-1)$-th sensor. Finally, the $(m,n)$-th\u0000entry of the matrix $mathbf D = frac{boldsymbol Sigma boldsymbol\u0000Sigma^text T}{2}$ is the covariance of the component noises injected into the\u0000$m$-th and $n$-th sensors. For different choices of $mathbf B$ and\u0000$boldsymbol Sigma,$ we investigate whether Cyclicity Analysis enables us to\u0000recover the structure of network. Roughly speaking, Cyclicity Analysis studies\u0000the lead-lag dynamics pertaining to the components of a multivariate signal. We\u0000specifically consider an $N times N$ skew-symmetric matrix $mathbf Q,$ known\u0000as the lead matrix, in which the sign of its $(m,n)$-th entry captures the\u0000lead-lag relationship between the $m$-th and $n$-th component OU processes. We\u0000investigate whether the structure of the leading eigenvector of $mathbf Q,$\u0000the eigenvector corresponding to the largest eigenvalue of $mathbf Q$ in\u0000modulus, reflects the network structure induced by $mathbf B.$","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267762","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}
In this paper, for the problem of heteroskedastic general linear hypothesis testing (GLHT) in high-dimensional settings, we propose a random integration method based on the reference L2-norm to deal with such problems. The asymptotic properties of the test statistic can be obtained under the null hypothesis when the relationship between data dimensions and sample size is not specified. The results show that it is more advisable to approximate the null distribution of the test using the distribution of the chi-square type mixture, and it is shown through some numerical simulations and real data analysis that our proposed test is powerful.
{"title":"Linear hypothesis testing in high-dimensional heteroscedastics via random integration","authors":"Mingxiang Cao, Hongwei Zhang, Kai Xu, Daojiang He","doi":"arxiv-2409.12066","DOIUrl":"https://doi.org/arxiv-2409.12066","url":null,"abstract":"In this paper, for the problem of heteroskedastic general linear hypothesis\u0000testing (GLHT) in high-dimensional settings, we propose a random integration\u0000method based on the reference L2-norm to deal with such problems. The\u0000asymptotic properties of the test statistic can be obtained under the null\u0000hypothesis when the relationship between data dimensions and sample size is not\u0000specified. The results show that it is more advisable to approximate the null\u0000distribution of the test using the distribution of the chi-square type mixture,\u0000and it is shown through some numerical simulations and real data analysis that\u0000our proposed test is powerful.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267763","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}
Ju-Chi YuCAMH, Julie Le BorgneRID-AGE, CHRU Lille, Anjali KrishnanCUNY, Arnaud GloaguenCNRGH, JACOB, Cheng-Ta YangNCKU, Laura A RabinCUNY, Hervé AbdiUT Dallas, Vincent Guillemot
Correspondence analysis, multiple correspondence analysis and their discriminant counterparts (i.e., discriminant simple correspondence analysis and discriminant multiple correspondence analysis) are methods of choice for analyzing multivariate categorical data. In these methods, variables are integrated into optimal components computed as linear combinations whose weights are obtained from a generalized singular value decomposition (GSVD) that integrates specific metric constraints on the rows and columns of the original data matrix. The weights of the linear combinations are, in turn, used to interpret the components, and this interpretation is facilitated when components are 1) pairwise orthogonal and 2) when the values of the weights are either large or small but not intermediate-a pattern called a simple or a sparse structure. To obtain such simple configurations, the optimization problem solved by the GSVD is extended to include new constraints that implement component orthogonality and sparse weights. Because multiple correspondence analysis represents qualitative variables by a set of binary variables, an additional group constraint is added to the optimization problem in order to sparsify the whole set representing one qualitative variable. This new algorithm-called group-sparse GSVD (gsGSVD)-integrates these constraints via an iterative projection scheme onto the intersection of subspaces where each subspace implements a specific constraint. In this paper, we expose this new algorithm and show how it can be adapted to the sparsification of simple and multiple correspondence analysis, and illustrate its applications with the analysis of four different data sets-each illustrating the sparsification of a particular CA-based analysis.
对应分析、多重对应分析及其对应的判别分析（即判别简单对应分析和判别多重对应分析）是分析多元分类数据的首选方法。在这些方法中，变量被整合为最优成分，计算为线性组合，其权重来自广义奇异值分解（GSVD），该分解整合了原始数据矩阵行和列的特定度量约束。反过来，线性组合的权重也用于解释成分，当成分 1）成对正交，2）权重值或大或小，但不是中间值时，这种解释就会变得容易--这种模式被称为简单或稀疏结构。为了获得这种简单结构，GSVD 所求解的优化问题被扩展到包括新的约束条件，以实现成分正交和权重稀疏。由于多重对应分析用一组二进制变量表示定性变量，因此在优化问题中增加了一个额外的组约束，以稀疏化代表一个定性变量的整个组。这种新算法被称为组稀疏 GSVD（gsGSVD），它通过迭代投影方案将这些约束整合到子空间的交集上，其中每个子空间都实现了特定的约束。在本文中，我们揭示了这种新算法，并展示了它如何适用于简单和多重对应分析的稀疏化，还通过分析四个不同的数据集说明了它的应用--每个数据集都说明了基于 CA 的特定分析的稀疏化。
{"title":"Sparse Factor Analysis for Categorical Data with the Group-Sparse Generalized Singular Value Decomposition","authors":"Ju-Chi YuCAMH, Julie Le BorgneRID-AGE, CHRU Lille, Anjali KrishnanCUNY, Arnaud GloaguenCNRGH, JACOB, Cheng-Ta YangNCKU, Laura A RabinCUNY, Hervé AbdiUT Dallas, Vincent Guillemot","doi":"arxiv-2409.11789","DOIUrl":"https://doi.org/arxiv-2409.11789","url":null,"abstract":"Correspondence analysis, multiple correspondence analysis and their\u0000discriminant counterparts (i.e., discriminant simple correspondence analysis\u0000and discriminant multiple correspondence analysis) are methods of choice for\u0000analyzing multivariate categorical data. In these methods, variables are\u0000integrated into optimal components computed as linear combinations whose\u0000weights are obtained from a generalized singular value decomposition (GSVD)\u0000that integrates specific metric constraints on the rows and columns of the\u0000original data matrix. The weights of the linear combinations are, in turn, used\u0000to interpret the components, and this interpretation is facilitated when\u0000components are 1) pairwise orthogonal and 2) when the values of the weights are\u0000either large or small but not intermediate-a pattern called a simple or a\u0000sparse structure. To obtain such simple configurations, the optimization\u0000problem solved by the GSVD is extended to include new constraints that\u0000implement component orthogonality and sparse weights. Because multiple\u0000correspondence analysis represents qualitative variables by a set of binary\u0000variables, an additional group constraint is added to the optimization problem\u0000in order to sparsify the whole set representing one qualitative variable. This\u0000new algorithm-called group-sparse GSVD (gsGSVD)-integrates these constraints\u0000via an iterative projection scheme onto the intersection of subspaces where\u0000each subspace implements a specific constraint. In this paper, we expose this\u0000new algorithm and show how it can be adapted to the sparsification of simple\u0000and multiple correspondence analysis, and illustrate its applications with the\u0000analysis of four different data sets-each illustrating the sparsification of a\u0000particular CA-based analysis.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267765","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}
We present a new analytical method to derive the likelihood function that has the population of parameters marginalised out in Bayesian hierarchical models. This method is also useful to find the marginal likelihoods in Bayesian models or in random-effect linear mixed models. The key to this method is to take high-order (sometimes fractional) derivatives of the prior moment-generating function if particular existence and differentiability conditions hold. In particular, this analytical method assumes that the likelihood is either Poisson or gamma. Under Poisson likelihoods, the observed Poisson count determines the order of the derivative. Under gamma likelihoods, the shape parameter, which is assumed to be known, determines the order of the fractional derivative. We also present some examples validating this new analytical method.
{"title":"Poisson and Gamma Model Marginalisation and Marginal Likelihood calculation using Moment-generating Functions","authors":"Siyang Li, David van Dyk, Maximilian Autenrieth","doi":"arxiv-2409.11167","DOIUrl":"https://doi.org/arxiv-2409.11167","url":null,"abstract":"We present a new analytical method to derive the likelihood function that has\u0000the population of parameters marginalised out in Bayesian hierarchical models.\u0000This method is also useful to find the marginal likelihoods in Bayesian models\u0000or in random-effect linear mixed models. The key to this method is to take\u0000high-order (sometimes fractional) derivatives of the prior moment-generating\u0000function if particular existence and differentiability conditions hold. In particular, this analytical method assumes that the likelihood is either\u0000Poisson or gamma. Under Poisson likelihoods, the observed Poisson count\u0000determines the order of the derivative. Under gamma likelihoods, the shape\u0000parameter, which is assumed to be known, determines the order of the fractional\u0000derivative. We also present some examples validating this new analytical method.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267805","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}
Debapratim Banerjee, Soumendu Sundar Mukherjee, Dipranjan Pal
We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition $sup_{(i, j) ne (i', j')}|mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + varepsilon)})$, where $varepsilon > 0$. It follows from Catalano et al. (2024) that the empirical spectral distribution of $n^{-1/2} X_n$ converges weakly almost surely to the standard semi-circle law. Using a F"{u}redi-Koml'{o}s-type high moment analysis, we show that the largest eigenvalue $lambda_1(n^{-1/2} X_n)$ of $n^{-1/2} X_n$ converges almost surely to $2$. This result is essentially optimal in the sense that one cannot take $varepsilon = 0$ and still obtain an almost sure limit of $2$. We also derive Gaussian fluctuation results for the largest eigenvalue in the case where the entries have a common non-zero mean. Let $Y_n = X_n + frac{lambda}{sqrt{n}}mathbf{1} mathbf{1}^top$. When $varepsilon ge 1$ and $lambda gg n^{1/4}$, we show that [ n^{1/2}bigg(lambda_1(n^{-1/2} Y_n) - lambda - frac{1}{lambda}bigg) xrightarrow{d} sqrt{2} Z, ] where $Z$ is a standard Gaussian. On the other hand, when $0 < varepsilon < 1$, we have $mathrm{Var}(frac{1}{n}sum_{i, j}X_{ij}) = O(n^{1 - varepsilon})$. Assuming that $mathrm{Var}(frac{1}{n}sum_{i, j} X_{ij}) = sigma^2 n^{1 - varepsilon} (1 + o(1))$, if $lambda gg n^{varepsilon/4}$, then we have [ n^{varepsilon/2}bigg(lambda_1(n^{-1/2} Y_n) - lambda - frac{1}{lambda}bigg) xrightarrow{d} sigma Z. ] While the ranges of $lambda$ in these fluctuation results are certainly not optimal, a striking aspect is that different scalings are required in the two regimes $0 < varepsilon < 1$ and $varepsilon ge 1$.
{"title":"Edge spectra of Gaussian random symmetric matrices with correlated entries","authors":"Debapratim Banerjee, Soumendu Sundar Mukherjee, Dipranjan Pal","doi":"arxiv-2409.11381","DOIUrl":"https://doi.org/arxiv-2409.11381","url":null,"abstract":"We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$,\u0000with zero-mean, unit variance entries satisfying the condition $sup_{(i, j)\u0000ne (i', j')}|mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + varepsilon)})$, where\u0000$varepsilon > 0$. It follows from Catalano et al. (2024) that the empirical\u0000spectral distribution of $n^{-1/2} X_n$ converges weakly almost surely to the\u0000standard semi-circle law. Using a F\"{u}redi-Koml'{o}s-type high moment\u0000analysis, we show that the largest eigenvalue $lambda_1(n^{-1/2} X_n)$ of\u0000$n^{-1/2} X_n$ converges almost surely to $2$. This result is essentially\u0000optimal in the sense that one cannot take $varepsilon = 0$ and still obtain an\u0000almost sure limit of $2$. We also derive Gaussian fluctuation results for the\u0000largest eigenvalue in the case where the entries have a common non-zero mean.\u0000Let $Y_n = X_n + frac{lambda}{sqrt{n}}mathbf{1} mathbf{1}^top$. When\u0000$varepsilon ge 1$ and $lambda gg n^{1/4}$, we show that [ n^{1/2}bigg(lambda_1(n^{-1/2} Y_n) - lambda - frac{1}{lambda}bigg)\u0000xrightarrow{d} sqrt{2} Z, ] where $Z$ is a standard Gaussian. On the other\u0000hand, when $0 < varepsilon < 1$, we have $mathrm{Var}(frac{1}{n}sum_{i,\u0000j}X_{ij}) = O(n^{1 - varepsilon})$. Assuming that\u0000$mathrm{Var}(frac{1}{n}sum_{i, j} X_{ij}) = sigma^2 n^{1 - varepsilon} (1\u0000+ o(1))$, if $lambda gg n^{varepsilon/4}$, then we have [ n^{varepsilon/2}bigg(lambda_1(n^{-1/2} Y_n) - lambda -\u0000frac{1}{lambda}bigg) xrightarrow{d} sigma Z. ] While the ranges of\u0000$lambda$ in these fluctuation results are certainly not optimal, a striking\u0000aspect is that different scalings are required in the two regimes $0 <\u0000varepsilon < 1$ and $varepsilon ge 1$.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267803","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}
We prove the large deviations principle for empirical Bures-Wasserstein barycenters of independent, identically-distributed samples of covariance matrices and covariance operators. As an application, we explore some consequences of our results for the phenomenon of dimension-free concentration of measure for Bures-Wasserstein barycenters. Our theory reveals a novel notion of exponential tilting in the Bures-Wasserstein space, which, in analogy with Cr'amer's theorem in the Euclidean case, solves the relative entropy projection problem under a constraint on the barycenter. Notably, this method of proof is easy to adapt to other geometric settings of interest; with the same method, we obtain large deviations principles for empirical barycenters in Riemannian manifolds and the univariate Wasserstein space, and we obtain large deviations upper bounds for empirical barycenters in the general multivariate Wasserstein space. In fact, our results are the first known large deviations principles for Fr'echet means in any non-linear metric space.
{"title":"Large Deviations Principle for Bures-Wasserstein Barycenters","authors":"Adam Quinn Jaffe, Leonardo V. Santoro","doi":"arxiv-2409.11384","DOIUrl":"https://doi.org/arxiv-2409.11384","url":null,"abstract":"We prove the large deviations principle for empirical Bures-Wasserstein\u0000barycenters of independent, identically-distributed samples of covariance\u0000matrices and covariance operators. As an application, we explore some\u0000consequences of our results for the phenomenon of dimension-free concentration\u0000of measure for Bures-Wasserstein barycenters. Our theory reveals a novel notion\u0000of exponential tilting in the Bures-Wasserstein space, which, in analogy with\u0000Cr'amer's theorem in the Euclidean case, solves the relative entropy\u0000projection problem under a constraint on the barycenter. Notably, this method\u0000of proof is easy to adapt to other geometric settings of interest; with the\u0000same method, we obtain large deviations principles for empirical barycenters in\u0000Riemannian manifolds and the univariate Wasserstein space, and we obtain large\u0000deviations upper bounds for empirical barycenters in the general multivariate\u0000Wasserstein space. In fact, our results are the first known large deviations\u0000principles for Fr'echet means in any non-linear metric space.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"102 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267801","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}
We consider a semi-parametric Gaussian regression model, equipped with a high-dimensional Gaussian prior. We address the frequentist validity of posterior credible sets for a vector of linear functionals. We specify conditions for a 'renormalized' Bernstein-von Mises theorem (BvM), where the posterior, centered at its mean, and the posterior mean, centered at the ground truth, have the same normal approximation. This requires neither a solution to the information equation nor a $sqrt{N}$-consistent estimator. We show that our renormalized BvM implies that a credible ellipsoid, specified by the mean and variance of the posterior, is an asymptotic confidence set. For a single linear functional, we identify such a credible ellipsoid with a symmetric credible interval around the posterior mean. We bound the diameter. We check the conditions for Darcy's problem, where the information equation has no solution in natural settings. For the Schr"odinger problem, we recover an efficient semi-parametric BvM from our renormalized BvM.
{"title":"Valid Credible Ellipsoids for Linear Functionals by a Renormalized Bernstein-von Mises Theorem","authors":"Gustav Rømer","doi":"arxiv-2409.10947","DOIUrl":"https://doi.org/arxiv-2409.10947","url":null,"abstract":"We consider a semi-parametric Gaussian regression model, equipped with a\u0000high-dimensional Gaussian prior. We address the frequentist validity of\u0000posterior credible sets for a vector of linear functionals. We specify conditions for a 'renormalized' Bernstein-von Mises theorem (BvM),\u0000where the posterior, centered at its mean, and the posterior mean, centered at\u0000the ground truth, have the same normal approximation. This requires neither a\u0000solution to the information equation nor a $sqrt{N}$-consistent estimator. We show that our renormalized BvM implies that a credible ellipsoid,\u0000specified by the mean and variance of the posterior, is an asymptotic\u0000confidence set. For a single linear functional, we identify such a credible\u0000ellipsoid with a symmetric credible interval around the posterior mean. We\u0000bound the diameter. We check the conditions for Darcy's problem, where the information equation\u0000has no solution in natural settings. For the Schr\"odinger problem, we recover\u0000an efficient semi-parametric BvM from our renormalized BvM.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"119 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142267799","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}
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