光谱方法通过图矩阵上的特征向量计算在图中提供了一个可拖动的全局框架。 HyperGraph数据（其中实体在任意大小的边缘上相互作用）对矩阵表示构成了挑战，因此对光谱聚类构成了挑战。我们研究了基于超透明型非背带操作员的非均匀超图的光谱聚类。在审查了该操作员及其基本属性的定义之后，我们证明了Ihara-Bass类型的定理，该定理允许在较小的矩阵上进行特征Pair计算，通常可以更快地计算。然后，我们通过线性化信念传播提出了一种交替的算法，用于在超图随机块模型中推断，该算法涉及光谱聚类的步骤，再次使用非背部跟踪操作员。我们提供与该算法相关的证明，这些算法既正式又扩展了几个先前的结果。我们对光谱方法的极限和超图随机块模型中的可检测性提出了几种猜想，并通过对我们研究的操作员的特征因的不接受分析来支持它们。我们在真实和合成数据中执行实验，这些实验证明了当不同尺寸的相互作用带有有关群集结构的不同信息时，超图方法比基于图的方法的好处。

随机块模型（SBM）是一个随机图模型，其连接不同的顶点组不同。它被广泛用作研究聚类和社区检测的规范模型，并提供了肥沃的基础来研究组合统计和更普遍的数据科学中出现的信息理论和计算权衡。该专着调查了最近在SBM中建立社区检测的基本限制的最新发展，无论是在信息理论和计算方案方面，以及各种恢复要求，例如精确，部分和弱恢复。讨论的主要结果是在Chernoff-Hellinger阈值中进行精确恢复的相转换，Kesten-Stigum阈值弱恢复的相变，最佳的SNR - 单位信息折衷的部分恢复以及信息理论和信息理论之间的差距计算阈值。该专着给出了在寻求限制时开发的主要算法的原则推导，特别是通过绘制绘制，半定义编程，（线性化）信念传播，经典/非背带频谱和图形供电。还讨论了其他块模型的扩展，例如几何模型和一些开放问题。

The stochastic block model (SBM) is a random graph model with planted clusters. It is widely employed as a canonical model to study clustering and community detection, and provides generally a fertile ground to study the statistical and computational tradeoffs that arise in network and data sciences.This note surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational thresholds, and for various recovery requirements such as exact, partial and weak recovery (a.k.a., detection). The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial recovery, the learning of the SBM parameters and the gap between information-theoretic and computational thresholds.The note also covers some of the algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, linearized belief propagation, classical and nonbacktracking spectral methods. A few open problems are also discussed.

假设$ g $是根据所谓的HyperGraph随机块模型（HSBM）产生的，我们考虑了稀疏$ Q $均匀的HyperGraph $ G $中的社区检测问题。我们证明，基于非折线操作员的光谱方法具有很高的概率，可以降低到Angelini等人猜想的广义kesten-Stigum检测阈值。我们表征了稀疏HSBM的非背带操作员的频谱，并使用Ihara-Bass公式为超图提供有效的尺寸降低程序。结果，可以将稀疏HSBM的社区检测减少为$ 2N \ times 2n $非正态矩阵的特征向量问题，该矩阵从邻接矩阵和超级格雷普的学位矩阵中构建。据我们所知，这是第一种可证明，有效的光谱算法，它可以根据一般对称概率张量生成$ K $块的HSBMS阈值。

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.

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.

最近有一项激烈的活动在嵌入非常高维和非线性数据结构的嵌入中，其中大部分在数据科学和机器学习文献中。我们分四部分调查这项活动。在第一部分中，我们涵盖了非线性方法，例如主曲线，多维缩放，局部线性方法，ISOMAP，基于图形的方法和扩散映射，基于内核的方法和随机投影。第二部分与拓扑嵌入方法有关，特别是将拓扑特性映射到持久图和映射器算法中。具有巨大增长的另一种类型的数据集是非常高维网络数据。第三部分中考虑的任务是如何将此类数据嵌入中等维度的向量空间中，以使数据适合传统技术，例如群集和分类技术。可以说，这是算法机器学习方法与统计建模（所谓的随机块建模）之间的对比度。在论文中，我们讨论了两种方法的利弊。调查的最后一部分涉及嵌入$ \ mathbb {r}^ 2 $，即可视化中。提出了三种方法：基于第一部分，第二和第三部分中的方法，$ t $ -sne，UMAP和大节。在两个模拟数据集上进行了说明和比较。一个由嘈杂的ranunculoid曲线组成的三胞胎，另一个由随机块模型和两种类型的节点产生的复杂性的网络组成。

我们提出了对学度校正随机块模型（DCSBM）的合适性测试。该测试基于调整后的卡方统计量，用于测量$ n $多项式分布的组之间的平等性，该分布具有$ d_1，\ dots，d_n $观测值。在网络模型的背景下，多项式的数量（$ n $）的数量比观测值数量（$ d_i $）快得多，与节点$ i $的度相对应，因此设置偏离了经典的渐近学。我们表明，只要$ \ {d_i \} $的谐波平均值生长到无穷大，就可以使统计量在NULL下分配。顺序应用时，该测试也可以用于确定社区数量。该测试在邻接矩阵的压缩版本上进行操作，因此在学位上有条件，因此对大型稀疏网络具有高度可扩展性。我们结合了一个新颖的想法，即在测试$ K $社区时根据$（k+1）$ - 社区分配来压缩行。这种方法在不牺牲计算效率的情况下增加了顺序应用中的力量，我们证明了它在恢复社区数量方面的一致性。由于测试统计量不依赖于特定的替代方案，因此其效用超出了顺序测试，可用于同时测试DCSBM家族以外的各种替代方案。特别是，我们证明该测试与具有社区结构的潜在可变性网络模型的一般家庭一致。

Pre-publication draft of a book to be published byMorgan & Claypool publishers. Unedited version released with permission. All relevant copyrights held by the author and publisher extend to this pre-publication draft.

A common approach to modeling networks assigns each node to a position on a low-dimensional manifold where distance is inversely proportional to connection likelihood. More positive manifold curvature encourages more and tighter communities; negative curvature induces repulsion. We consistently estimate manifold type, dimension, and curvature from simply connected, complete Riemannian manifolds of constant curvature. We represent the graph as a noisy distance matrix based on the ties between cliques, then develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We apply our approach to data-sets from economics and neuroscience.

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.

社区检测是网络科学中最重要的方法领域之一，在过去的几十年里引起了大量关注的方法之一。该区域处理网络的自动部门到基础构建块中，目的是提供其大规模结构的概要。尽管它的重要性和广泛的采用普及，所谓的最先进和实际在各种领域实际使用的方法之间存在明显的差距。在这里，我们试图通过根据是否具有“描述性”或“推论”目标来划分现有方法来解决这种差异。虽然描述性方法在基于社区结构的直观概念的网络中找到模式的模式，但是推理方法阐述了精确的生成模型，并尝试将其符合数据。通过这种方式，他们能够为网络形成机制提供见解，并以统计证据支持的方式与随机性的单独结构。我们审查如何使用推论目标采用描述性方法被陷入困境和误导性答案，因此应该一般而言。我们认为推理方法更通常与更清晰的科学问题一致，产生更强大的结果，并且应该是一般的首选。我们试图消除一些神话和半真半假在实践中使用社区检测时，努力改善这些方法的使用以及对结果的解释。

我们开发了一种高效的随机块模型中的弱恢复算法。该算法与随机块模型的Vanilla版本的最佳已知算法的统计保证匹配。从这个意义上讲，我们的结果表明，随机块模型没有稳健性。我们的工作受到最近的银行，Mohanty和Raghavendra（SODA 2021）的工作，为相应的区别问题提供了高效的算法。我们的算法及其分析显着脱离了以前的恢复。关键挑战是我们算法的特殊优化景观：种植的分区可能远非最佳意义，即完全不相关的解决方案可以实现相同的客观值。这种现象与PCA的BBP相转变的推出效应有关。据我们所知，我们的算法是第一个在非渐近设置中存在这种推出效果的鲁棒恢复。我们的算法是基于凸优化的框架的实例化（与平方和不同的不同），这对于其他鲁棒矩阵估计问题可能是有用的。我们的分析的副产物是一种通用技术，其提高了任意强大的弱恢复算法的成功（输入的随机性）从恒定（或缓慢消失）概率以指数高概率。

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.

对于从分布$ \ mu $采样的图值数据，根据选择度量计算样品矩。在这项工作中，我们为图表集合了由$ \ ell_2 $规范定义的伪金属，相应的邻接矩阵的特征值之间。我们使用此伪度量标准和图值数据集的各个样本矩来推断分布的参数$ \ hat {\ mu} $，并将其解释为$ \ mu $的近似值。我们通过实验验证复杂的分布$ \ mu $可以很好地近似地使用这种方法。

网络可能具有弱信号和严重程度的异质性，并且可能在一次出现时非常稀疏，但在另一个发生中非常致密。得分（Jin，2015）是最近网络社区检测的方法。它适应严重的程度异质性，并适应不同水平的稀疏性，但它对具有弱信号的网络的性能尚不清楚。在本文中，我们认为，在广泛的网络设置中，我们允许弱信号，严重程度异质性和广泛的网络稀疏性，得分实现了完善的聚类，并且在汉明集群中具有所谓的“指数率”错误。证据对网络邻接矩阵的领先特征向量进行了最新的进出方程。理论分析向我们保证，在弱信号设置中，得分继续运行，但它不排除分数可以进一步提高的可能性，以在实际应用中具有更好的性能，特别是对于具有弱信号的网络。作为纸张的第二份贡献，我们提出得分+作为改进的分数版本。我们调查了8个网络数据集的得分+，发现它优于几种代表性的方法。特别是，对于具有相对强烈的信号的6个数据集，得分+具有与得分相似的性能，但对于2个数据集（Simmons，Caltech）具有可能弱信号，得分+的误差率较低。得分+提出了几个变化以得分。我们使用理论和数值研究的混合物仔细解释每个变化的基本原理。

光谱嵌入是可用于获得图形节点的矢量表示的过程。本文提出了称为随机点产品图的潜在网络模型的概括，以允许将这些载体表示的解释为潜在位置估计。需要泛化异化连接（例如，“对立面”）并更普遍地应对负特征值。我们表明，是否使用邻接或归一化的拉普拉斯矩阵，光谱嵌入产生均匀一致的潜在估计，渐近高斯误差（最高可识别性）。标准和混合会员随机块模型是特殊情况，其中潜在的位置只需要k $不同的向量值，代表社区，或以$（k-1）$ - simplex与那些顶点一起生活。在随机块模型下，我们的理论建议使用高斯混合模型（而不是$ k $ -means），并且根据混合成员资格，拟合封闭单纯x的最小卷，此前仅在非负面明确假设下支持的现有建议。在网络安全示例中，在网络安全示例中证明了链路预测（在随机点产品图中）的经验改进（在随机点产品图中），以及露出更丰富的潜在结构（比标准或混合隶属块模型的位置）。

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.

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.

在这项工作中，我们研究了具有对抗性节点损坏的随机块模型中社区发现的问题。我们的主要结果是一种有效的算法，该算法可以忍受$ \ epsilon $ - 损坏和达到错误$ o（\ epsilon） + e^{ - \ frac {c} {2} {2}（1 \ pm o（1））} $其中$ c =（\ sqrt {a} - \ sqrt {b}）^2 $是信噪比，$ a/n $和$ b/n $是互发和intra-intra-intra-社区连接概率分别。这些界限基本上与无损坏的SBM的最小值相匹配。我们还为$ \ mathbb {z} _2 $ -Synchronization提供了可靠的算法。我们算法的核心是一个新的半决赛程序，它使用全局信息来鲁棒提高粗糙聚类的准确性。此外，我们表明我们的算法是双重的，因为它们在更具挑战性的噪声模型中起作用，该模型将对抗性腐败与无限制的单调变化混合在一起，从半随机模型中。