我们提供了一个理论框架来研究我们称之为单发概括的现象。这种现象是指算法在单个任务中执行转移学习的能力，这意味着它正确地对训练集中具有单个示例的测试点进行了分类。我们提出了一个简单的数据模型，并使用它以两种方式研究这种现象。首先，我们证明了一种非反应基碱线 - 基于最近的邻分类的内核方法无法执行单发概括，而与核的选择无关，并且训练集的大小。其次，我们从经验上表明，我们数据模型最直接的神经网络体系结构几乎完美地执行了单发概括。这种鲜明的差异使我们相信，单发概括机制是对神经网络的经验成功的部分原因。

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。

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.

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.

分析大型随机矩阵的浓度是多种领域的常见任务。给定独立的随机变量，许多工具可用于分析随机矩阵，其条目在变量中是线性的，例如基质 - 伯恩斯坦不平等。但是，在许多应用中，我们需要分析其条目是变量中多项式的随机矩阵。这些自然出现在光谱算法的分析中，例如霍普金斯等人。 [Stoc 2016]，Moitra-Wein [Stoc 2019]；并根据正方形层次结构的总和（例如Barak等。 [FOCS 2016]，Jones等。 [焦点2021]。在这项工作中，我们基于Paulin-Mackey-Tropp（概率Annals of Poylibity of Poyliby of 2016]，我们提出了一个通用框架来获得此类界限。 Efron-Stein不等式通过另一个简单（但仍然是随机）矩阵的范围来界定随机矩阵的规范，我们将其视为通过“区分”起始矩阵而引起的。通过递归区分，我们的框架减少了分析更简单的矩阵的主要任务。对于Rademacher变量，这些简单的矩阵实际上是确定性的，因此，分析它们要容易得多。对于一般的非拉多巴纳变量，任务减少到标量浓度，这要容易得多。此外，在多项式矩阵的设置中，我们的结果推广了Paulin-Mackey-Tropp的工作。使用我们的基本框架，我们在文献中恢复了简单的“张量网络”和“密集图矩阵”的已知界限。使用我们的一般框架，我们得出了“稀疏图矩阵”的边界，琼斯等人最近才获得。 [焦点2021]使用痕量功率方法的非平地应用，并且是其工作中的核心组成部分。我们希望我们的框架对涉及非线性随机矩阵浓度现象的其他应用有帮助。

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。

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.

我们根据计算一个扎根于每个顶点的某个加权树的家族而构成的相似性得分提出了一种有效的图形匹配算法。对于两个erd \ h {o} s-r \'enyi图$ \ mathcal {g}（n，q）$，其边缘通过潜在顶点通信相关联，我们表明该算法正确地匹配了所有范围的范围，除了所有的vertices分数外，有了很高的概率，前提是$ nq \ to \ infty $，而边缘相关系数$ \ rho $满足$ \ rho^2> \ alpha \ ailpha \大约0.338 $，其中$ \ alpha $是Otter的树木计数常数。此外，在理论上是必需的额外条件下，可以精确地匹配。这是第一个以显式常数相关性成功的多项式图匹配算法，并适用于稀疏和密集图。相比之下，以前的方法要么需要$ \ rho = 1-o（1）$，要么仅限于稀疏图。该算法的症结是一个经过精心策划的植根树的家族，称为吊灯，它可以有效地从同一树的计数中提取图形相关性，同时抑制不同树木之间的不良相关性。

A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonably-sized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Explicit bounds are also derived demonstrating that learning multiple tasks within an environment of related tasks can potentially give much better generalization than learning a single task.

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.

我们研究了小组测试问题，其目标是根据合并测试的结果，确定一组k感染的人，这些k含有稀有疾病，这些人在经过测试中至少有一个受感染的个体时返回阳性的结果。团体。我们考虑将个人分配给测试的两个不同的简单随机过程：恒定柱设计和伯努利设计。我们的第一组结果涉及基本统计限制。对于恒定柱设计，我们给出了一个新的信息理论下限，这意味着正确识别的感染者的比例在测试数量越过特定阈值时会经历急剧的“全或全或无所不包”的相变。对于Bernoulli设计，我们确定解决相关检测问题所需的确切测试数量（目的是区分小组测试实例和纯噪声），改善Truong，Aldridge和Scarlett的上限和下限（2020）。对于两个小组测试模型，我们还研究了计算有效（多项式时间）推理程序的能力。我们确定了解决检测问题的低度多项式算法所需的精确测试数量。这为在少量稀疏度的检测和恢复问题中都存在固有的计算统计差距提供了证据。值得注意的是，我们的证据与Iliopoulos和Zadik（2021）相反，后者预测了Bernoulli设计中没有计算统计差距。

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M . Can we complete the matrix and recover the entries that we have not seen?We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys m ≥ C n 1.2 r log n for some positive numerical constant C, then with very high probability, most n × n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.

Suppose we are given an $n$-dimensional order-3 symmetric tensor $T \in (\mathbb{R}^n)^{\otimes 3}$ that is the sum of $r$ random rank-1 terms. The problem of recovering the rank-1 components is possible in principle when $r \lesssim n^2$ but polynomial-time algorithms are only known in the regime $r \ll n^{3/2}$. Similar "statistical-computational gaps" occur in many high-dimensional inference tasks, and in recent years there has been a flurry of work on explaining the apparent computational hardness in these problems by proving lower bounds against restricted (yet powerful) models of computation such as statistical queries (SQ), sum-of-squares (SoS), and low-degree polynomials (LDP). However, no such prior work exists for tensor decomposition, largely because its hardness does not appear to be explained by a "planted versus null" testing problem. We consider a model for random order-3 tensor decomposition where one component is slightly larger in norm than the rest (to break symmetry), and the components are drawn uniformly from the hypercube. We resolve the computational complexity in the LDP model: $O(\log n)$-degree polynomial functions of the tensor entries can accurately estimate the largest component when $r \ll n^{3/2}$ but fail to do so when $r \gg n^{3/2}$. This provides rigorous evidence suggesting that the best known algorithms for tensor decomposition cannot be improved, at least by known approaches. A natural extension of the result holds for tensors of any fixed order $k \ge 3$, in which case the LDP threshold is $r \sim n^{k/2}$.

We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

我们提出了改进的算法，并为身份测试$ n $维分布的问题提供了统计和计算下限。在身份测试问题中，我们将作为输入作为显式分发$ \ mu $，$ \ varepsilon> 0 $，并访问对隐藏分布$ \ pi $的采样甲骨文。目标是区分两个分布$ \ mu $和$ \ pi $是相同的还是至少$ \ varepsilon $ -far分开。当仅从隐藏分布$ \ pi $中访问完整样本时，众所周知，可能需要许多样本，因此以前的作品已经研究了身份测试，并额外访问了各种有条件采样牙齿。我们在这里考虑一个明显弱的条件采样甲骨文，称为坐标Oracle，并在此新模型中提供了身份测试问题的相当完整的计算和统计表征。我们证明，如果一个称为熵的分析属性为可见分布$ \ mu $保留，那么对于任何使用$ \ tilde {o}（n/\ tilde {o}），有一个有效的身份测试算法Varepsilon）$查询坐标Oracle。熵的近似张力是一种经典的工具，用于证明马尔可夫链的最佳混合时间边界用于高维分布，并且最近通过光谱独立性为许多分布族建立了最佳的混合时间。我们将算法结果与匹配的$ \ omega（n/\ varepsilon）$统计下键进行匹配的算法结果补充，以供坐标Oracle下的查询数量。我们还证明了一个计算相变：对于$ \ {+1，-1，-1 \}^n $以上的稀疏抗抗铁磁性模型，在熵失败的近似张力失败的状态下，除非RP = np，否则没有有效的身份测试算法。

现代神经网络通常以强烈的过度构造状态运行：它们包含许多参数，即使实际标签被纯粹随机的标签代替，它们也可以插入训练集。尽管如此，他们在看不见的数据上达到了良好的预测错误：插值训练集并不会导致巨大的概括错误。此外，过度散色化似乎是有益的，因为它简化了优化景观。在这里，我们在神经切线（NT）制度中的两层神经网络的背景下研究这些现象。我们考虑了一个简单的数据模型，以及各向同性协变量的矢量，$ d $尺寸和$ n $隐藏的神经元。我们假设样本量$ n $和尺寸$ d $都很大，并且它们在多项式上相关。我们的第一个主要结果是对过份术的经验NT内核的特征结构的特征。这种表征意味着必然的表明，经验NT内核的最低特征值在$ ND \ gg n $后立即从零界限，因此网络可以在同一制度中精确插值任意标签。我们的第二个主要结果是对NT Ridge回归的概括误差的表征，包括特殊情况，最小值-ULL_2 $ NORD插值。我们证明，一旦$ nd \ gg n $，测试误差就会被内核岭回归之一相对于无限宽度内核而近似。多项式脊回归的误差依次近似后者，从而通过与激活函数的高度组件相关的“自我诱导的”项增加了正则化参数。多项式程度取决于样本量和尺寸（尤其是$ \ log n/\ log d $）。

我们派生并分析了一种用于估计有限簇树中的所有分裂的通用，递归算法以及相应的群集。我们进一步研究了从内核密度估计器接收级别设置估计时该通用聚类算法的统计特性。特别是，我们推出了有限的样本保证，一致性，收敛率以及用于选择内核带宽的自适应数据驱动策略。对于这些结果，我们不需要与H \“{o}连续性等密度的连续性假设，而是仅需要非参数性质的直观几何假设。

Motivated by alignment of correlated sparse random graphs, we introduce a hypothesis testing problem of deciding whether or not two random trees are correlated. We obtain sufficient conditions under which this testing is impossible or feasible. We propose MPAlign, a message-passing algorithm for graph alignment inspired by the tree correlation detection problem. We prove MPAlign to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We then conjecture that graph alignment is not feasible in polynomial time when the associated tree detection problem is impossible. If true, this conjecture together with our sufficient conditions on tree detection impossibility would imply the existence of a hard phase for graph alignment, i.e. a parameter range where alignment cannot be done in polynomial time even though it is known to be feasible in non-polynomial time.

在过去十年中，图形内核引起了很多关注，并在结构化数据上发展成为一种快速发展的学习分支。在过去的20年中，该领域发生的相当大的研究活动导致开发数十个图形内核，每个图形内核都对焦于图形的特定结构性质。图形内核已成功地成功地在广泛的域中，从社交网络到生物信息学。本调查的目标是提供图形内核的文献的统一视图。特别是，我们概述了各种图形内核。此外，我们对公共数据集的几个内核进行了实验评估，并提供了比较研究。最后，我们讨论图形内核的关键应用，并概述了一些仍有待解决的挑战。

我们调查与高斯的混合的数据分享共同但未知，潜在虐待协方差矩阵的数据。我们首先考虑具有两个等级大小的组件的高斯混合，并根据最大似然估计导出最大切割整数程序。当样品的数量在维度下线性增长时，我们证明其解决方案实现了最佳的错误分类率，直到对数因子。但是，解决最大切割问题似乎是在计算上棘手的。为了克服这一点，我们开发了一种高效的频谱算法，该算法达到最佳速率，但需要一种二次样本量。虽然这种样本复杂性比最大切割问题更差，但我们猜测没有多项式方法可以更好地执行。此外，我们收集了支持统计计算差距存在的数值和理论证据。最后，我们将MAX-CUT程序概括为$ k $ -means程序，该程序处理多组分混合物的可能性不平等。它享有相似的最优性保证，用于满足运输成本不平等的分布式的混合物，包括高斯和强烈的对数的分布。