在许多应用程序(例如运动锦标赛或推荐系统)中,我们可以使用该数据,包括一组$ n $项目(或玩家)之间的成对比较。目的是使用这些数据来推断每个项目和/或其排名的潜在强度。此问题的现有结果主要集中在由单个比较图$ g $组成的设置上。但是,存在成对比较数据随时间发展的场景(例如体育比赛)。这种动态设置的理论结果相对有限,是本文的重点。我们研究\ emph {翻译同步}问题的扩展,到动态设置。在此设置中,我们给出了一系列比较图$(g_t)_ {t \ in \ mathcal {t}} $,其中$ \ nathcal {t} \ subset [0,1] $是代表时间的网格域,对于每个项目$ i $和time $ t \ in \ mathcal {t} $,有一个关联的未知强度参数$ z^*_ {t,i} \ in \ mathbb {r} $。我们的目标是恢复,以$ t \在\ Mathcal {t} $中,强度向量$ z^*_ t =(z^*_ {t,1},\ cdots,z^*_ {t,n}) $从$ z^*_ {t,i} -z^*_ {t,j} $的噪声测量值中,其中$ \ {i,j \} $是$ g_t $中的边缘。假设$ z^*_ t $在$ t $中顺利地演变,我们提出了两个估计器 - 一个基于平滑度的最小二乘方法,另一个基于对合适平滑度操作员低频本质空间的投影。对于两个估计器,我们为$ \ ell_2 $估计错误提供有限的样本范围,假设$ g_t $已连接到\ mathcal {t} $中的所有$ t \网格尺寸$ | \ MATHCAL {T} | $。我们通过有关合成和真实数据的实验来补充理论发现。
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在许多应用中,我们获得了流畅的函数的嘈杂模态样本的访问,其目标是鲁棒地解开样本,即估计该功能的原始样本。在最近的工作中,Cucuringu和Tyagi通过首先将它们代表在单元复杂圆上,然后解决平滑度规则化最小二乘问题 - Laplacian的平滑度适用的Proximity Graph的平滑度$ G $ - ON单位圆的产品歧管。这个问题是二次受约束的二次程序(QCQP),其是非凸显的,因此提出解决其球形放松导致信任区域子问题(TRS)。就理论担保而言,派生$ \ ell_2 $错误界限(trs)。然而,这些界限通常弱,并且没有真正证明由(TRS)进行的去噪。在这项工作中,我们分析(TRS)以及(QCQP)的不受约束的放松。对于这些估算器,我们在高斯噪声的设置中提供了一种精致的分析,并导出了噪音制度,其中他们可否证明模数观察W.R.T $ \ ell_2 $常规。分析在$ G $是任何连接的图形中的常规设置中进行。
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本文研究了基于Laplacian Eigenmaps(Le)的基于Laplacian EIGENMAPS(PCR-LE)的主要成分回归的统计性质,这是基于Laplacian Eigenmaps(Le)的非参数回归的方法。 PCR-LE通过投影观察到的响应的向量$ {\ bf y} =(y_1,\ ldots,y_n)$ to to changbood图表拉普拉斯的某些特征向量跨越的子空间。我们表明PCR-Le通过SoboLev空格实现了随机设计回归的最小收敛速率。在设计密度$ P $的足够平滑条件下,PCR-le达到估计的最佳速率(其中已知平方$ l ^ 2 $ norm的最佳速率为$ n ^ { - 2s /(2s + d) )} $)和健美的测试($ n ^ { - 4s /(4s + d)$)。我们还表明PCR-LE是\ EMPH {歧管Adaptive}:即,我们考虑在小型内在维度$ M $的歧管上支持设计的情况,并为PCR-LE提供更快的界限Minimax估计($ n ^ { - 2s /(2s + m)$)和测试($ n ^ { - 4s /(4s + m)$)收敛率。有趣的是,这些利率几乎总是比图形拉普拉斯特征向量的已知收敛率更快;换句话说,对于这个问题的回归估计的特征似乎更容易,统计上讲,而不是估计特征本身。我们通过经验证据支持这些理论结果。
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Bradley-terry-luce(BTL)模型是一种流行的统计方法,用于使用成对比较估算项目集合的全局排名。为了确保准确的排名,必须在$ \ ell _ {\ infty} $损失中获得模型参数的精确估计。该任务的难度取决于给定项目对成对比较图的拓扑。但是,除了很少有良好的情况外,例如完整和ERD \“ OS-r \'enyi比较图,对$ \ ell_中BTL模型参数的最大似然估计量mLE的性能鲜为人知。 {\ infty} $ - 在更通用的图形拓扑下的损失。在本文中,我们在$ \ ell _ {\ infty} $估计错误的btl mLE估计误差上得出了小说的一般上限,该错误明确取决于比较的代数连接性图,跨项目和样本复杂性的最大性能差距。我们证明,与使用不同的损失函数以及更受限制的假设和图形拓扑获得的已知结果相比,派生的界限性能很好,并且在某些情况下相比更为敏锐。我们将结果仔细比较我们的结果与我们的结果进行比较。 Yan等人(2012年),它在精神上最接近我们的工作。我们进一步提供了$ \ ell _ {\ infty} $下的最小值下限 - 错误几乎与一类足够常规的图形拓扑相匹配。最后。 ,我们St udy,我们的$ \ ell _ {\ infty} $的含义是高效(离线)锦标赛设计的界限。我们通过各种示例和模拟来说明和讨论我们的发现。
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Bradley-terry-luce(BTL)模型是一个基准模型,用于个人之间的成对比较。尽管最近在几种流行程序的一阶渐近学上进行了最新进展,但对BTL模型中不确定性定量的理解基本上仍然不完整,尤其是当基础比较图很少时。在本文中,我们通过重点关注两个估计量的估计器来填补这一空白:最大似然估计器(MLE)和频谱估计器。使用统一的证明策略,我们在基础比较图的最稀少的可能的制度(最多达到某些多同源因​​素)中,为两个估计量提供了尖锐而均匀的非反应膨胀。这些扩展使我们能够获得:(i)两个估计器的有限维中心限制定理; (ii)构建个人等级的置信区间; (iii)$ \ ell_2 $估计的最佳常数,这是由MLE实现的,但不是由光谱估计器实现的。我们的证明是基于二阶剩余矢量的自洽方程和新的两次分析分析。
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我们研究了趋势过滤的多元版本,称为Kronecker趋势过滤或KTF,因为设计点以$ D $维度形成格子。 KTF是单变量趋势过滤的自然延伸(Steidl等,2006; Kim等人,2009; Tibshirani,2014),并通过最大限度地减少惩罚最小二乘问题,其罚款术语总和绝对(高阶)沿每个坐标方向估计参数的差异。相应的惩罚运算符可以编写单次趋势过滤惩罚运营商的Kronecker产品,因此名称Kronecker趋势过滤。等效,可以在$ \ ell_1 $ -penalized基础回归问题上查看KTF,其中基本功能是下降阶段函数的张量产品,是一个分段多项式(离散样条)基础,基于单变量趋势过滤。本文是Sadhanala等人的统一和延伸结果。 (2016,2017)。我们开发了一套完整的理论结果,描述了$ k \ grone 0 $和$ d \ geq 1 $的$ k ^ {\ mathrm {th}} $ over kronecker趋势过滤的行为。这揭示了许多有趣的现象,包括KTF在估计异构平滑的功能时KTF的优势,并且在$ d = 2(k + 1)$的相位过渡,一个边界过去(在高维对 - 光滑侧)线性泡沫不能完全保持一致。我们还利用Tibshirani(2020)的离散花键来利用最近的结果,特别是离散的花键插值结果,使我们能够将KTF估计扩展到恒定时间内的任何偏离晶格位置(与晶格数量的大小无关)。
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我们提出了一种凸锥程序,可推断随机点产品图(RDPG)的潜在概率矩阵。优化问题最大化Bernoulli最大似然函数,增加核规范正则化术语。双重问题具有特别良好的形式,与众所周知的SemideFinite程序放松MaxCut问题有关。使用原始双功率条件,我们绑定了原始和双解决方案的条目和等级。此外,我们在轻微的技术假设下绑定了最佳目标值并证明了略微修改模型的概率估计的渐近一致性。我们对合成RDPG的实验不仅恢复了自然集群,而且还揭示了原始数据的下面的低维几何形状。我们还证明该方法在空手道俱乐部图表和合成美国参议图中恢复潜在结构,并且可以扩展到最多几百个节点的图表。
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这项调查旨在提供线性模型及其背后的理论的介绍。我们的目标是对读者进行严格的介绍,并事先接触普通最小二乘。在机器学习中,输出通常是输入的非线性函数。深度学习甚至旨在找到需要大量计算的许多层的非线性依赖性。但是,这些算法中的大多数都基于简单的线性模型。然后,我们从不同视图中描述线性模型,并找到模型背后的属性和理论。线性模型是回归问题中的主要技术,其主要工具是最小平方近似,可最大程度地减少平方误差之和。当我们有兴趣找到回归函数时,这是一个自然的选择,该回归函数可以最大程度地减少相应的预期平方误差。这项调查主要是目的的摘要,即线性模型背后的重要理论的重要性,例如分布理论,最小方差估计器。我们首先从三种不同的角度描述了普通的最小二乘,我们会以随机噪声和高斯噪声干扰模型。通过高斯噪声,该模型产生了可能性,因此我们引入了最大似然估计器。它还通过这种高斯干扰发展了一些分布理论。最小二乘的分布理论将帮助我们回答各种问题并引入相关应用。然后,我们证明最小二乘是均值误差的最佳无偏线性模型,最重要的是,它实际上接近了理论上的极限。我们最终以贝叶斯方法及以后的线性模型结束。
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This paper concerns with statistical estimation and inference for the ranking problems based on pairwise comparisons with additional covariate information such as the attributes of the compared items. Despite extensive studies, few prior literatures investigate this problem under the more realistic setting where covariate information exists. To tackle this issue, we propose a novel model, Covariate-Assisted Ranking Estimation (CARE) model, that extends the well-known Bradley-Terry-Luce (BTL) model, by incorporating the covariate information. Specifically, instead of assuming every compared item has a fixed latent score $\{\theta_i^*\}_{i=1}^n$, we assume the underlying scores are given by $\{\alpha_i^*+{x}_i^\top\beta^*\}_{i=1}^n$, where $\alpha_i^*$ and ${x}_i^\top\beta^*$ represent latent baseline and covariate score of the $i$-th item, respectively. We impose natural identifiability conditions and derive the $\ell_{\infty}$- and $\ell_2$-optimal rates for the maximum likelihood estimator of $\{\alpha_i^*\}_{i=1}^{n}$ and $\beta^*$ under a sparse comparison graph, using a novel `leave-one-out' technique (Chen et al., 2019) . To conduct statistical inferences, we further derive asymptotic distributions for the MLE of $\{\alpha_i^*\}_{i=1}^n$ and $\beta^*$ with minimal sample complexity. This allows us to answer the question whether some covariates have any explanation power for latent scores and to threshold some sparse parameters to improve the ranking performance. We improve the approximation method used in (Gao et al., 2021) for the BLT model and generalize it to the CARE model. Moreover, we validate our theoretical results through large-scale numerical studies and an application to the mutual fund stock holding dataset.
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众所周知,许多网络系统,例如电网,大脑和舆论动态社交网络,都可以遵守保护法。这种现象的例子包括电网中的基尔乔夫法律和社交网络中的意见共识。网络系统中的保护定律可以建模为$ x = b^{*} y $的平衡方程,其中$ b^{*} $的稀疏模式捕获了网络的连接,$ y,x \在\ mathbb {r}^p $中分别是节点上“电势”和“注入流”的向量。节点电位$ y $会导致跨边缘的流量,并且在节点上注入的流量$ x $是网络动力学的无关紧要的。在几个实用的系统中,网络结构通常是未知的,需要从数据估算。为此,可以访问节点电位$ y $的样本,但只有节点注射$ x $的统计信息。在这个重要问题的激励下,我们研究了$ n $ y $ y $ y $ y $ y $ y $ y $ y $ b^{*} $稀疏结构的估计,假设节点注射$ x $遵循高斯分布,并带有已知的发行协方差$ \ sigma_x $。我们建议在高维度中为此问题的新$ \ ell_ {1} $ - 正则最大似然估计器,网络的大小$ p $大于样本量$ n $。我们表明,此优化问题是目标中的凸,并接受了独特的解决方案。在新的相互不一致的条件下,我们在三重$(n,p,d)$上建立了足够的条件,对于$ b^{*} $的精确稀疏恢复是可能的; $ d $是图的程度。我们还建立了在元素最大,Frobenius和运营商规范中回收$ b^{*} $的保证。最后,我们通过对拟议估计量对合成和现实世界数据的性能进行实验验证来补充这些理论结果。
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We consider the problem of estimating a multivariate function $f_0$ of bounded variation (BV), from noisy observations $y_i = f_0(x_i) + z_i$ made at random design points $x_i \in \mathbb{R}^d$, $i=1,\ldots,n$. We study an estimator that forms the Voronoi diagram of the design points, and then solves an optimization problem that regularizes according to a certain discrete notion of total variation (TV): the sum of weighted absolute differences of parameters $\theta_i,\theta_j$ (which estimate the function values $f_0(x_i),f_0(x_j)$) at all neighboring cells $i,j$ in the Voronoi diagram. This is seen to be equivalent to a variational optimization problem that regularizes according to the usual continuum (measure-theoretic) notion of TV, once we restrict the domain to functions that are piecewise constant over the Voronoi diagram. The regression estimator under consideration hence performs (shrunken) local averaging over adaptively formed unions of Voronoi cells, and we refer to it as the Voronoigram, following the ideas in Koenker (2005), and drawing inspiration from Tukey's regressogram (Tukey, 1961). Our contributions in this paper span both the conceptual and theoretical frontiers: we discuss some of the unique properties of the Voronoigram in comparison to TV-regularized estimators that use other graph-based discretizations; we derive the asymptotic limit of the Voronoi TV functional; and we prove that the Voronoigram is minimax rate optimal (up to log factors) for estimating BV functions that are essentially bounded.
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本文为信号去噪提供了一般交叉验证框架。然后将一般框架应用于非参数回归方法,例如趋势过滤和二元推车。然后显示所得到的交叉验证版本以获得最佳调谐的类似物所熟知的几乎相同的收敛速度。没有任何先前的趋势过滤或二元推车的理论分析。为了说明框架的一般性,我们还提出并研究了两个基本估算器的交叉验证版本;套索用于高维线性回归和矩阵估计的奇异值阈值阈值。我们的一般框架是由Chatterjee和Jafarov(2015)的想法的启发,并且可能适用于使用调整参数的广泛估算方法。
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We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.
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对于高维和非参数统计模型,速率最优估计器平衡平方偏差和方差是一种常见的现象。虽然这种平衡被广泛观察到,但很少知道是否存在可以避免偏差和方差之间的权衡的方法。我们提出了一般的策略,以获得对任何估计方差的下限,偏差小于预先限定的界限。这表明偏差差异折衷的程度是不可避免的,并且允许量化不服从其的方法的性能损失。该方法基于许多抽象的下限,用于涉及关于不同概率措施的预期变化以及诸如Kullback-Leibler或Chi-Sque-diversence的信息措施的变化。其中一些不平等依赖于信息矩阵的新概念。在该物品的第二部分中,将抽象的下限应用于几种统计模型,包括高斯白噪声模型,边界估计问题,高斯序列模型和高维线性回归模型。对于这些特定的统计应用,发生不同类型的偏差差异发生,其实力变化很大。对于高斯白噪声模型中集成平方偏置和集成方差之间的权衡,我们将较低界限的一般策略与减少技术相结合。这允许我们将原始问题与估计的估计器中的偏差折衷联动,以更简单的统计模型中具有额外的对称性属性。在高斯序列模型中,发生偏差差异的不同相位转换。虽然偏差和方差之间存在非平凡的相互作用,但是平方偏差的速率和方差不必平衡以实现最小估计速率。
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In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be obtained by sampling a distribution on a union of manifolds $\mathcal{M} = \mathcal{M}_1 \cup\dots \cup \mathcal{M}_N$ that may intersect with each other and that may have different dimensions. We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their corresponding graph Laplacians to capture the right geometric information to solve the MMC problem. Precisely, we provide high probability error bounds for the spectral approximation of a tensorized Laplacian on $\mathcal{M}$ with a suitable graph Laplacian built from the observations; the recovered tensorized Laplacian contains all geometric information of all the individual underlying manifolds. We provide an example of a family of similarity graphs, which we call annular proximity graphs with angle constraints, satisfying these sufficient conditions. We contrast our family of graphs with other constructions in the literature based on the alignment of tangent planes. Extensive numerical experiments expand the insights that our theory provides on the MMC problem.
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Network data are ubiquitous in modern machine learning, with tasks of interest including node classification, node clustering and link prediction. A frequent approach begins by learning an Euclidean embedding of the network, to which algorithms developed for vector-valued data are applied. For large networks, embeddings are learned using stochastic gradient methods where the sub-sampling scheme can be freely chosen. Despite the strong empirical performance of such methods, they are not well understood theoretically. Our work encapsulates representation methods using a subsampling approach, such as node2vec, into a single unifying framework. We prove, under the assumption that the graph is exchangeable, that the distribution of the learned embedding vectors asymptotically decouples. Moreover, we characterize the asymptotic distribution and provided rates of convergence, in terms of the latent parameters, which includes the choice of loss function and the embedding dimension. This provides a theoretical foundation to understand what the embedding vectors represent and how well these methods perform on downstream tasks. Notably, we observe that typically used loss functions may lead to shortcomings, such as a lack of Fisher consistency.
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在因果推理和强盗文献中,基于观察数据的线性功能估算线性功能的问题是规范的。我们分析了首先估计治疗效果函数的广泛的两阶段程序,然后使用该数量来估计线性功能。我们证明了此类过程的均方误差上的非反应性上限:这些边界表明,为了获得非反应性最佳程序,应在特定加权$ l^2 $中最大程度地估算治疗效果的误差。 -规范。我们根据该加权规范的约束回归分析了两阶段的程序,并通过匹配非轴突局部局部最小值下限,在有限样品中建立了实例依赖性最优性。这些结果表明,除了取决于渐近效率方差之外,最佳的非质子风险除了取决于样本量支持的最富有函数类别的真实结果函数与其近似类别之间的加权规范距离。
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In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying overparameterized models. One of the central phenomena in this regime is the ability of the model to interpolate noisy data, but still have test error lower than the amount of noise in that data. arXiv:1906.11300 characterized for which covariance structure of the data such a phenomenon can happen in linear regression if one considers the interpolating solution with minimum $\ell_2$-norm and the data has independent components: they gave a sharp bound on the variance term and showed that it can be small if and only if the data covariance has high effective rank in a subspace of small co-dimension. We strengthen and complete their results by eliminating the independence assumption and providing sharp bounds for the bias term. Thus, our results apply in a much more general setting than those of arXiv:1906.11300, e.g., kernel regression, and not only characterize how the noise is damped but also which part of the true signal is learned. Moreover, we extend the result to the setting of ridge regression, which allows us to explain another interesting phenomenon: we give general sufficient conditions under which the optimal regularization is negative.
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近似消息传递(AMP)是解决高维统计问题的有效迭代范式。但是,当迭代次数超过$ o \ big(\ frac {\ log n} {\ log log \ log \ log n} \时big)$(带有$ n $问题维度)。为了解决这一不足,本文开发了一个非吸附框架,用于理解峰值矩阵估计中的AMP。基于AMP更新的新分解和可控的残差项,我们布置了一个分析配方,以表征在存在独立初始化的情况下AMP的有限样本行为,该过程被进一步概括以进行光谱初始化。作为提出的分析配方的两个具体后果:(i)求解$ \ mathbb {z} _2 $同步时,我们预测了频谱初始化AMP的行为,最高为$ o \ big(\ frac {n} {\ mathrm {\ mathrm { poly} \ log n} \ big)$迭代,表明该算法成功而无需随后的细化阶段(如最近由\ citet {celentano2021local}推测); (ii)我们表征了稀疏PCA中AMP的非反应性行为(在尖刺的Wigner模型中),以广泛的信噪比。
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随机奇异值分解(RSVD)是用于计算大型数据矩阵截断的SVD的一类计算算法。给定A $ n \ times n $对称矩阵$ \ mathbf {m} $,原型RSVD算法输出通过计算$ \ mathbf {m mathbf {m} $的$ k $引导singular vectors的近似m}^{g} \ mathbf {g} $;这里$ g \ geq 1 $是一个整数,$ \ mathbf {g} \ in \ mathbb {r}^{n \ times k} $是一个随机的高斯素描矩阵。在本文中,我们研究了一般的“信号加上噪声”框架下的RSVD的统计特性,即,观察到的矩阵$ \ hat {\ mathbf {m}} $被认为是某种真实但未知的加法扰动信号矩阵$ \ mathbf {m} $。我们首先得出$ \ ell_2 $(频谱规范)和$ \ ell_ {2 \ to \ infty} $(最大行行列$ \ ell_2 $ norm)$ \ hat {\ hat {\ Mathbf {M}} $和信号矩阵$ \ Mathbf {M} $的真实单数向量。这些上限取决于信噪比(SNR)和功率迭代$ g $的数量。观察到一个相变现象,其中较小的SNR需要较大的$ g $值以保证$ \ ell_2 $和$ \ ell_ {2 \ to \ fo \ infty} $ distances的收敛。我们还表明,每当噪声矩阵满足一定的痕量生长条件时,这些相变发生的$ g $的阈值都会很清晰。最后,我们得出了近似奇异向量的行波和近似矩阵的进入波动的正常近似。我们通过将RSVD的几乎最佳性能保证在应用于三个统计推断问题的情况下,即社区检测,矩阵完成和主要的组件分析,并使用缺失的数据来说明我们的理论结果。
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