通过内核矩阵或图形laplacian矩阵代表数据点的光谱方法已成为无监督数据分析的主要工具。在许多应用程序场景中，可以通过神经网络嵌入的光谱嵌入可以在数据样本上进行训练，这为实现自动样本外扩展以及计算可扩展性提供了一种有希望的方法。在Spectralnet的原始论文中采用了这种方法（Shaham等人，2018年），我们称之为Specnet1。当前的论文引入了一种名为SpecNet2的新神经网络方法，以计算光谱嵌入，该方法优化了特征问题的等效目标，并删除了SpecNet1中的正交层。 SpecNet2还允许通过通过梯度公式跟踪每个数据点的邻居来分离图形亲和力矩阵的行采样和列。从理论上讲，我们证明了新的无正交物质目标的任何局部最小化均显示出领先的特征向量。此外，证明了使用基于批处理的梯度下降法的这种新的无正交目标的全局收敛。数值实验证明了在模拟数据和图像数据集上Specnet2的性能和计算效率的提高。

内元化图亲和力矩阵的双性化归一化为基于图的数据分析中的图形laplacian方法提供了一种替代归一化方案，并且可以通过sinkhorn-knopp（SK）迭代在实践中有效地计算出来。本文证明了双性化标准化图拉普拉斯（Laplacian）与laplacian的融合，当$ n $数据点为i.i.d.从嵌入可能高维空间中的一般$ d $维歧管中取样。在$ n \ to \ infty $和内核带宽$ \ epsilon \ to 0 $的某些联合限制下，图Laplacian操作员的点融合率（2-Norm）被证明为$ O（N^{n^{ -1/（d/2+3）}）$在有限的大$ n $上，到log racture，在$ \ epsilon \ sim n^{ - 1/（d/2+3）} $时实现。当歧管数据被异常噪声损坏时，我们从理论上证明了图形laplacian点的一致性，该图与清洁歧管数据的速率匹配到与噪声矢量相互内部产物的界限成比例的附加错误项。我们的分析表明，在本文中考虑的设置下，不是精确的双性化归一化，而是大约将达到相同的一致性率。在分析的激励下，我们提出了一个近似且受约束的矩阵缩放问题，可以通过早期终止的SK迭代来解决，并适用于模拟的歧管数据既干净又具有离群的噪声。数值实验支持我们的理论结果，并显示了双形式归一化图拉普拉斯对异常噪声的鲁棒性。

当图形亲和力矩阵是由$ n $随机样品构建的，在$ d $ d $维歧管上构建图形亲和力矩阵时，这项工作研究图形拉普拉斯元素与拉普拉斯 - 贝特拉米操作员的光谱收敛。通过分析DIRICHLET形成融合并通过歧管加热核卷积构建候选本本函数，我们证明，使用高斯内核，可以设置核band band band band parame $ \ epsilon \ sim \ sim（\ log n/ n/ n）^{1/（D /2+2）} $使得特征值收敛率为$ n^{ - 1/（d/2+2）} $，并且2-norm中的特征向量收敛率$ n^{ - 1/（d+） 4）} $;当$ \ epsilon \ sim（\ log n/n）^{1/（d/2+3）} $时，eigenValue和eigenVector速率均为$ n^{ - 1/（d/2+3）} $。这些费率最高为$ \ log n $因素，并被证明是有限的许多低洼特征值。当数据在歧管上均匀采样以及密度校正的图laplacian（在两个边的度矩阵中归一化）时，结果适用于非归一化和随机漫步图拉普拉斯laplacians laplacians laplacians以及密度校正的图laplacian（其中两侧的级别矩阵）采样数据。作为中间结果，我们证明了密度校正图拉普拉斯的新点和差异形式的收敛速率。提供数值结果以验证理论。

过度参数化神经网络（NNS）的小概括误差可以通过频率偏见现象来部分解释，在频率偏置现象中，基于梯度的算法将低频失误最小化，然后再减少高频残差。使用神经切线内核（NTK），可以为训练提供理论上严格的分析，其中数据是从恒定或分段构剂概率密度绘制的数据。由于大多数训练数据集不是从此类分布中汲取的，因此我们使用NTK模型和数据依赖性的正交规则来理论上量化NN训练的频率偏差，给定完全不均匀的数据。通过用精心选择的Sobolev规范替换损失函数，我们可以进一步扩大，抑制，平衡或逆转NN训练中的内在频率偏差。

本文提出了一个无网格的计算框架和机器学习理论，用于在未知的歧管上求解椭圆形PDE，并根据扩散地图（DM）和深度学习确定点云。 PDE求解器是作为监督的学习任务制定的，以解决最小二乘回归问题，该问题施加了近似PDE的代数方程（如果适用）。该代数方程涉及通过DM渐近扩展获得的图形拉平型矩阵，该基质是二阶椭圆差差算子的一致估计器。最终的数值方法是解决受神经网络假设空间解决方案的高度非凸经验最小化问题。在体积良好的椭圆PDE设置中，当假设空间由具有无限宽度或深度的神经网络组成时，我们表明，经验损失函数的全球最小化器是大型训练数据极限的一致解决方案。当假设空间是一个两层神经网络时，我们表明，对于足够大的宽度，梯度下降可以识别经验损失函数的全局最小化器。支持数值示例证明了解决方案的收敛性，范围从具有低和高共限度的简单歧管到具有和没有边界的粗糙表面。我们还表明，所提出的NN求解器可以在具有概括性误差的新数据点上稳健地概括PDE解决方案，这些误差几乎与训练错误相同，从而取代了基于Nystrom的插值方法。

我们研究具有流形结构的物理系统的langevin动力学$ \ MATHCAL {M} \ subset \ Mathbb {r}^p $，基于收集的样品点$ \ {\ Mathsf {x} _i \} _ {_i \} _ {i = 1} ^n \ subset \ mathcal {m} $探测未知歧管$ \ mathcal {m} $。通过扩散图，我们首先了解反应坐标$ \ {\ MATHSF {y} _i \} _ {i = 1}^n \ subset \ subset \ mathcal {n} $对应于$ \ {\ {\ mathsf {x} _i _i \ \ \ \ \ _i \ \ \ \ {x} } _ {i = 1}^n $，其中$ \ mathcal {n} $是$ \ mathcal {m} $的歧义diffeomorphic，并且与$ \ mathbb {r}^\ ell $ insometryally嵌入了$ \ ell $，带有$ \ ell \ ell \ ell \ ell \ el \ ell \ el \ el \ ell \ el \ LL P $。在$ \ Mathcal {n} $上的诱导Langevin动力学在反应坐标方面捕获了缓慢的时间尺度动力学，例如生化反应的构象变化。要构建$ \ Mathcal {n} $上的Langevin Dynamics的高效稳定近似，我们利用反应坐标$ \ MATHSF {y} n effertold $ \ Mathcal {n} $上的歧管$ \ Mathcal {n} $上的相应的fokker-planck方程$。我们为此Fokker-Planck方程提出了可实施的，无条件稳定的数据驱动的有限卷方程，该方程将自动合并$ \ Mathcal {n} $的歧管结构。此外，我们在$ \ Mathcal {n} $上提供了有限卷方案的加权$ L^2 $收敛分析。所提出的有限体积方案在$ \ {\ Mathsf {y} _i \} _ {i = 1}^n $上导致Markov链，并具有近似的过渡概率和最近的邻居点之间的跳跃速率。在无条件稳定的显式时间离散化之后，数据驱动的有限体积方案为$ \ Mathcal {n} $上的Langevin Dynamics提供了近似的Markov进程，并且近似的Markov进程享有详细的平衡，Ergodicity和其他良好的属性。

在机器学习或统计中，通常希望减少高维空间$ \ mathbb {r} ^ d $的数据点样本的维度。本文介绍了一种维度还原方法，其中嵌入坐标是作为半定程序无限尺寸模拟的溶液获得的正半定核的特征向量。这种嵌入是自适应和非线性的。我们对学习内核的弱者和强烈的平滑假设讨论了这个问题。我们的方法的主要特点是在两种情况下存在嵌入坐标的样本延伸公式。该外推公式产生内核矩阵的延伸到数据相关的Mercer内核功能。我们的经验结果表明，与光谱嵌入方法相比，该嵌入方法对异常值的影响更加稳健。

Riemannian优化是解决优化问题的原则框架，其中所需的最佳被限制为光滑的歧管$ \ Mathcal {M} $。在此框架中设计的算法通常需要对歧管的几何描述，该描述通常包括切线空间，缩回和成本函数的梯度。但是，在许多情况下，由于缺乏信息或棘手的性能，只能访问这些元素的子集（或根本没有）。在本文中，我们提出了一种新颖的方法，可以在这种情况下执行近似Riemannian优化，其中约束歧管是$ \ r^{d} $的子手机。至少，我们的方法仅需要一组无噪用的成本函数$（\ x_ {i}，y_ {i}）\ in {\ mathcal {m}} \ times \ times \ times \ times \ times \ mathbb {r} $和内在的歧管$ \ MATHCAL {M} $的维度。使用样品，并利用歧管-MLS框架（Sober和Levin 2020），我们构建了缺少的组件的近似值，这些组件娱乐可证明的保证并分析其计算成本。如果某些组件通过分析给出（例如，如果成本函数及其梯度明确给出，或者可以计算切线空间），则可以轻松地适应该算法以使用准确的表达式而不是近似值。我们使用我们的方法分析了基于Riemannian梯度的方法的全球收敛性，并从经验上证明了该方法的强度，以及基于类似原理的共轭梯度类型方法。

低维歧管假设认为，在许多应用中发现的数据，例如涉及自然图像的数据（大约）位于嵌入高维欧几里得空间中的低维歧管上。在这种情况下，典型的神经网络定义了一个函数，该函数在嵌入空间中以有限数量的向量作为输入。但是，通常需要考虑在训练分布以外的点上评估优化网络。本文考虑了培训数据以$ \ mathbb r^d $的线性子空间分配的情况。我们得出对由神经网络定义的学习函数变化的估计值，沿横向子空间的方向。我们研究了数据歧管的编纂中与网络的深度和噪声相关的潜在正则化效应。由于存在噪声，我们还提出了训练中的其他副作用。

我们系统地{研究基于内核的图形laplacian（gl）的光谱}，该图在非null设置中由高维和嘈杂的随机点云构成，其中点云是从低维几何对象（如歧管）中采样的，被高维噪音破坏。我们量化了信号和噪声在信号噪声比（SNR）的不同状态下如何相互作用，并报告GL的{所产生的特殊光谱行为}。此外，我们还探索了GL频谱上的内核带宽选择，而SNR的不同状态则导致带宽的自适应选择，这与实际数据中的共同实践相吻合。该结果为数据集嘈杂时的从业人员提供了理论支持。

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.

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets.This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed-either explicitly or implicitly-to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast with O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multi-processor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs. All of the presented theoretical results are extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data.

这是针对非线性维度和特征提取方法的教程和调查论文，该方法基于数据图的拉普拉斯语。我们首先介绍邻接矩阵，拉普拉斯矩阵的定义和拉普拉斯主义的解释。然后，我们涵盖图形和光谱聚类的切割，该谱图应用于数据子空间。解释了Laplacian征收及其样本外扩展的不同优化变体。此后，我们将保留投影的局部性及其内核变体作为拉普拉斯征本征的线性特殊案例。然后解释了图嵌入的版本，这些版本是Laplacian eigenmap和局部保留投影的广义版本。最后，引入了扩散图，这是基于数据图和随机步行的方法。

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.

Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is efficiency -- given $n$ input points, most kernel-based algorithms need to materialize the full $n \times n$ kernel matrix before performing any subsequent computation, thus incurring $\Omega(n^2)$ runtime. Breaking this quadratic barrier for various problems has therefore, been a subject of extensive research efforts. We break the quadratic barrier and obtain $\textit{subquadratic}$ time algorithms for several fundamental linear-algebraic and graph processing primitives, including approximating the top eigenvalue and eigenvector, spectral sparsification, solving linear systems, local clustering, low-rank approximation, arboricity estimation and counting weighted triangles. We build on the recent Kernel Density Estimation framework, which (after preprocessing in time subquadratic in $n$) can return estimates of row/column sums of the kernel matrix. In particular, we develop efficient reductions from $\textit{weighted vertex}$ and $\textit{weighted edge sampling}$ on kernel graphs, $\textit{simulating random walks}$ on kernel graphs, and $\textit{importance sampling}$ on matrices to Kernel Density Estimation and show that we can generate samples from these distributions in $\textit{sublinear}$ (in the support of the distribution) time. Our reductions are the central ingredient in each of our applications and we believe they may be of independent interest. We empirically demonstrate the efficacy of our algorithms on low-rank approximation (LRA) and spectral sparsification, where we observe a $\textbf{9x}$ decrease in the number of kernel evaluations over baselines for LRA and a $\textbf{41x}$ reduction in the graph size for spectral sparsification.

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.

我们提出了一种基于langevin扩散的算法，以在球体的产物歧管上进行非凸优化和采样。在对数Sobolev不平等的情况下，我们根据Kullback-Leibler Divergence建立了有限的迭代迭代收敛到Gibbs分布的保证。我们表明，有了适当的温度选择，可以保证，次级最小值的次数差距很小，概率很高。作为一种应用，我们考虑了使用对角线约束解决半决赛程序（SDP）的burer- monteiro方法，并分析提出的langevin算法以优化非凸目标。特别是，我们为Burer建立了对数Sobolev的不平等现象 - 当没有虚假的局部最小值时，但在鞍点下，蒙蒂罗问题。结合结果，我们为SDP和最大切割问题提供了全局最佳保证。更确切地说，我们证明了Langevin算法在$ \ widetilde {\ omega}（\ epsilon^{ - 5}）$ tererations $ tererations $ \ widetilde {\ omega}（\ omega}中，具有很高的概率。

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。

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.