从基因表达数据中提取信息的广泛使用方法采用基因共表达网络的构建以及随后发现网络结构的算法的应用。特别是，一个共同的目标是基因簇的计算发现，通常称为模块。当应用新的基因表达数据集上时，可以使用基因本体学富集自动评估计算模块的质量，该方法可在计算的模块中测量基因本体论项的频率并评估其统计学上的可能性。在这项工作中，我们建议基于光谱网络理论数学中相对较新的开创性工作，提出了SGC的基因聚类的新型管道。 SGC由多个新型步骤组成，这些步骤能够以无监督的方式计算高度富集的模块。但是，与所有现有框架不同，它进一步结合了一个新的步骤，该步骤在半监督聚类方法中利用基因本体学信息，进一步提高了计算模块的质量。与已经众所周知的现有框架相比，我们表明SGC导致实际数据的富集更高。特别是，在12个实际基因表达数据集中，SGC的表现优于除1个。

大多数维度降低方法采用频域表示，从基质对角线化获得，并且对于具有较高固有维度的大型数据集可能不会有效。为了应对这一挑战，相关的聚类和投影（CCP）提供了一种新的数据域策略，不需要解决任何矩阵。CCP将高维特征分配到相关的群集中，然后根据样本相关性将每个集群中的特征分为一个一维表示。引入了残留相似性（R-S）分数和索引，Riemannian歧管中的数据形状以及基于代数拓扑的持久性Laplacian进行可视化和分析。建议的方法通过与各种机器学习算法相关的基准数据集验证。

由于其数值益处增加及其坚实的数学背景，光谱聚类方法的非线性重构近来的关注。我们在$ p $ -norm中提出了一种新的直接多道谱聚类算法，以$ p \ in（1,2] $。计算图表的多个特征向量的问题$ p $ -laplacian，标准的非线性概括Graph Laplacian，被重用作为Grassmann歧管的无约束最小化问题。$ P $的价值以伪连续的方式减少，促进对应于最佳图形的稀疏解决方案载体作为$ P $接近。监测单调减少平衡图削减了我们从$ P $ -Levels获得的最佳可用解决方案的保证。我们展示了我们算法在各种人工测试案件中的算法的有效性和准确性。我们的数值和比较结果具有各种状态-Art聚类方法表明，所提出的方法在均衡的图形剪切度量和标签分配的准确性方面取得高质量的集群。此外，我们进行S面部图像和手写字符分类的束缚，以展示现实数据集中的适用性。

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。

子空间聚类是将大约位于几个低维子空间的数据样本集合集合的经典问题。此问题的当前最新方法基于自我表达模型，该模型表示样品是其他样品的线性组合。但是，这些方法需要足够广泛的样品才能准确表示，这在许多应用中可能不一定是可以访问的。在本文中，我们阐明了这个常见的问题，并认为每个子空间中的数据分布在自我表达模型的成功中起着至关重要的作用。我们提出的解决此问题的解决方案是由数据扩展在深神经网络的概括力中的核心作用引起的。我们为无监督和半监督的设置提出了两个子空间聚类框架，这些框架使用增强样品作为扩大词典来提高自我表达表示的质量。我们提出了一种使用一些标记的样品进行半监督问题的自动增强策略，该问题取决于数据样本位于多个线性子空间的联合以下事实。实验结果证实了数据增强的有效性，因为它显着提高了一般自我表达模型的性能。

Multi-view data containing complementary and consensus information can facilitate representation learning by exploiting the intact integration of multi-view features. Because most objects in real world often have underlying connections, organizing multi-view data as heterogeneous graphs is beneficial to extracting latent information among different objects. Due to the powerful capability to gather information of neighborhood nodes, in this paper, we apply Graph Convolutional Network (GCN) to cope with heterogeneous-graph data originating from multi-view data, which is still under-explored in the field of GCN. In order to improve the quality of network topology and alleviate the interference of noises yielded by graph fusion, some methods undertake sorting operations before the graph convolution procedure. These GCN-based methods generally sort and select the most confident neighborhood nodes for each vertex, such as picking the top-k nodes according to pre-defined confidence values. Nonetheless, this is problematic due to the non-differentiable sorting operators and inflexible graph embedding learning, which may result in blocked gradient computations and undesired performance. To cope with these issues, we propose a joint framework dubbed Multi-view Graph Convolutional Network with Differentiable Node Selection (MGCN-DNS), which is constituted of an adaptive graph fusion layer, a graph learning module and a differentiable node selection schema. MGCN-DNS accepts multi-channel graph-structural data as inputs and aims to learn more robust graph fusion through a differentiable neural network. The effectiveness of the proposed method is verified by rigorous comparisons with considerable state-of-the-art approaches in terms of multi-view semi-supervised classification tasks.

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.

域适应性是现代机器学习中的一种流行范式，旨在解决培训或验证数据集之间具有用于学习和测试分类器（源域）和潜在的大型未标记数据集的培训或验证数据集之间的分歧问题，其中利用了模型（目标域）（目标域）（目标域） 。任务是找到源数据集的源和目标数据集的这种常见表示，其中源数据集提供了培训的信息，因此可以最大程度地减少来源和目标之间的差异。目前，最流行的领域适应性解决方案是基于训练神经网络，这些神经网络结合了分类和对抗性学习模块，这些模块是饥饿的，通常很难训练。我们提出了一种称为域适应性主成分分析（DAPCA）的方法，该方法发现线性减少的数据表示有助于解决域适应任务。 DAPCA基于数据点对之间引入正权重，并概括了主成分分析的监督扩展。 DAPCA代表一种迭代算法，因此在每次迭代中都解决了一个简单的二次优化问题。保证算法的收敛性，并且在实践中的迭代次数很少。我们验证了先前提出的用于解决域适应任务的基准的建议算法，还显示了在生物医学应用中对单细胞法数据集进行分析中使用DAPCA的好处。总体而言，考虑到源域和目标域之间可能的差异，DAPCA可以作为许多机器学习应用程序中有用的预处理步骤。

We consider a semi-supervised $k$-clustering problem where information is available on whether pairs of objects are in the same or in different clusters. This information is either available with certainty or with a limited level of confidence. We introduce the PCCC algorithm, which iteratively assigns objects to clusters while accounting for the information provided on the pairs of objects. Our algorithm can include relationships as hard constraints that are guaranteed to be satisfied or as soft constraints that can be violated subject to a penalty. This flexibility distinguishes our algorithm from the state-of-the-art in which all pairwise constraints are either considered hard, or all are considered soft. Unlike existing algorithms, our algorithm scales to large-scale instances with up to 60,000 objects, 100 clusters, and millions of cannot-link constraints (which are the most challenging constraints to incorporate). We compare the PCCC algorithm with state-of-the-art approaches in an extensive computational study. Even though the PCCC algorithm is more general than the state-of-the-art approaches in its applicability, it outperforms the state-of-the-art approaches on instances with all hard constraints or all soft constraints both in terms of running time and various metrics of solution quality. The source code of the PCCC algorithm is publicly available on GitHub.

Parameter space reduction has been proved to be a crucial tool to speed-up the execution of many numerical tasks such as optimization, inverse problems, sensitivity analysis, and surrogate models' design, especially when in presence of high-dimensional parametrized systems. In this work we propose a new method called local active subspaces (LAS), which explores the synergies of active subspaces with supervised clustering techniques in order to carry out a more efficient dimension reduction in the parameter space. The clustering is performed without losing the input-output relations by introducing a distance metric induced by the global active subspace. We present two possible clustering algorithms: K-medoids and a hierarchical top-down approach, which is able to impose a variety of subdivision criteria specifically tailored for parameter space reduction tasks. This method is particularly useful for the community working on surrogate modelling. Frequently, the parameter space presents subdomains where the objective function of interest varies less on average along different directions. So, it could be approximated more accurately if restricted to those subdomains and studied separately. We tested the new method over several numerical experiments of increasing complexity, we show how to deal with vectorial outputs, and how to classify the different regions with respect to the local active subspace dimension. Employing this classification technique as a preprocessing step in the parameter space, or output space in case of vectorial outputs, brings remarkable results for the purpose of surrogate modelling.

最小的平方和群集（MSSC）或K-Means型聚类，传统上被认为是无监督的学习任务。近年来，使用背景知识来提高集群质量，促进聚类过程的可解释性已成为数学优化和机器学习研究的热门研究课题。利用数据群集中的背景信息的问题称为半监督或约束群集。在本文中，我们为半监控MSSC提供了一种新的分支和绑定算法，其中背景知识被包含为成对必须 - 链接和无法链接约束。对于较低的界限，我们解决了MSSC离散优化模型的Semidefinite编程宽松，并使用了用于加强界限的纤维平面程序。相反，通过使用整数编程工具，我们提出了将K-Means算法适应受约束的情况。这是第一次，所提出的全局优化算法有效地管理，以解决现实世界的情况，最高可达800个数据点，具有必要的必须 - 链接和无法链接约束以及通用数量的功能。这个问题大小大约比最先进的精确算法解决的实例大约四倍。

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.

我们讨论集群分析的拓扑方面，并表明在聚类之前推断数据集的拓扑结构可以大大增强群集检测：理论论证和经验证据表明，聚类嵌入向量，代表数据歧管的结构，而不是观察到的特征矢量他们自己是非常有益的。为了证明，我们将流形学习方法与基于密度的聚类方法DBSCAN结合了歧管学习方法UMAP。合成和真实数据结果表明，这既简化和改善了多种低维问题，包括密度变化和/或纠缠形状的群集。我们的方法简化了聚类，因为拓扑预处理始终降低DBSCAN的参数灵敏度。然后，用dbscan聚类所得的嵌入可以超过诸如spectacl和clustergan之类的复杂方法。最后，我们的调查表明，聚类中的关键问题似乎不是数据的标称维度或其中包含多少不相关的功能，而是\ textIt {可分离}群集在环境观察空间中的\ textit {可分离}，它们嵌入了它们中。 ，通常是数据特征定义的（高维）欧几里得空间。我们的方法之所以成功，是因为我们将数据投影到更合适的空间后，从某种意义上说，我们执行了群集分析。

群集集群或共识群集已成为一种强大的工具，用于提高各种聚类方法的鲁棒性和结果的稳定性。加权聚类集群自然地从集群集群中产生。加权群集集合的参数之一是聚类集群中的元素（群集或集群）具有不同的质量，或者对象或特征具有不同意义的重要性。但是，不可能直接将加权机制从分类（监督）域中应用于群集（无监督）域，因为群集本质上是一个不存在的问题。本文通过讨论不同类型的权重，确定重量值的主要方法以及将加权聚类集合与复杂数据的应用程序的主要方法概述了加权集群集群集合概述。本文提出的统一框架将有助于聚类从业者为自己的问题选择最合适的加权机制。

Graph is a highly generic and diverse representation, suitable for almost any data processing problem. Spectral graph theory has been shown to provide powerful algorithms, backed by solid linear algebra theory. It thus can be extremely instrumental to design deep network building blocks with spectral graph characteristics. For instance, such a network allows the design of optimal graphs for certain tasks or obtaining a canonical orthogonal low-dimensional embedding of the data. Recent attempts to solve this problem were based on minimizing Rayleigh-quotient type losses. We propose a different approach of directly learning the eigensapce. A severe problem of the direct approach, applied in batch-learning, is the inconsistent mapping of features to eigenspace coordinates in different batches. We analyze the degrees of freedom of learning this task using batches and propose a stable alignment mechanism that can work both with batch changes and with graph-metric changes. We show that our learnt spectral embedding is better in terms of NMI, ACC, Grassman distance, orthogonality and classification accuracy, compared to SOTA. In addition, the learning is more stable.

Selecting subsets of features that differentiate between two conditions is a key task in a broad range of scientific domains. In many applications, the features of interest form clusters with similar effects on the data at hand. To recover such clusters we develop DiSC, a data-driven approach for detecting groups of features that differentiate between conditions. For each condition, we construct a graph whose nodes correspond to the features and whose weights are functions of the similarity between them for that condition. We then apply a spectral approach to compute subsets of nodes whose connectivity differs significantly between the condition-specific feature graphs. On the theoretical front, we analyze our approach with a toy example based on the stochastic block model. We evaluate DiSC on a variety of datasets, including MNIST, hyperspectral imaging, simulated scRNA-seq and task fMRI, and demonstrate that DiSC uncovers features that better differentiate between conditions compared to competing methods.

The accuracy of k-nearest neighbor (kNN) classification depends significantly on the metric used to compute distances between different examples. In this paper, we show how to learn a Mahalanobis distance metric for kNN classification from labeled examples. The Mahalanobis metric can equivalently be viewed as a global linear transformation of the input space that precedes kNN classification using Euclidean distances. In our approach, the metric is trained with the goal that the k-nearest neighbors always belong to the same class while examples from different classes are separated by a large margin. As in support vector machines (SVMs), the margin criterion leads to a convex optimization based on the hinge loss. Unlike learning in SVMs, however, our approach requires no modification or extension for problems in multiway (as opposed to binary) classification. In our framework, the Mahalanobis distance metric is obtained as the solution to a semidefinite program. On several data sets of varying size and difficulty, we find that metrics trained in this way lead to significant improvements in kNN classification. Sometimes these results can be further improved by clustering the training examples and learning an individual metric within each cluster. We show how to learn and combine these local metrics in a globally integrated manner.

半监督学习（SSL）是使用不仅标记的示例，而且是未标记的示例学习预测模型的常见方法。尽管用于分类和回归的简单任务的SSL受到了研究社区的广泛关注，但对于具有结构依赖变量的复杂预测任务，这尚未得到适当的研究。这种情况是多标签分类和分层多标签分类任务，可能需要其他信息，可能来自未标记示例提供的描述性空间中的基础分布，以更好地面对同时预测多个类别标签的挑战性任务。在本文中，我们研究了这一方面，并​​提出了一种基于对预测性聚类树的半监督学习的（分层）多标签分类方法。我们还扩展了整体学习的方法，并提出了一种基于随机森林方法的方法。在23个数据集上进行的广泛实验评估显示了该方法的显着优势及其在其监督对应物方面的扩展。此外，该方法可保留可解释性并降低基于经典树模型的时间复杂性。

图表学习方法为解决图形所代表的复杂的现实世界问题打开了新的可能性。但是，这些应用程序中使用的许多图包括数百万节点和数十亿个边缘，并且超出了当前方法和软件实现的功能。我们提供葡萄，这是一种用于图形处理和表示学习的软件资源，能够通过使用专业和智能数据结构，算法和快速并行实现来通过大图扩展。与最先进的软件资源相比，葡萄显示出经验空间和时间复杂性的数量级的改善，以及边缘预测和节点标签预测性能的实质和统计学上的显着改善。此外，葡萄提供了来自文献和其他来源的80,000多种图，标准化界面允许直接整合第三方库，61个节点嵌入方法，25个推理模型和3个模块化管道，以允许公平且可重复的方法比较以及用于图形处理和嵌入的库。

对无监督对象发现的现有方法（UOD）不会向大大扩展到大型数据集，而不会损害其性能的近似。我们提出了一种新颖的UOD作为排名问题的制定，适用于可用于特征值问题和链接分析的分布式方法的阿森纳。通过使用自我监督功能，我们还展示了UOD的第一个有效的完全无监督的管道。对Coco和OpenImages的广泛实验表明，在每个图像中寻求单个突出对象的单对象发现设置中，所提出的LOD（大规模对象发现）方法与之相当于或更好地中型数据集的艺术（最多120K图像），比能够缩放到1.7M图像的唯一其他算法超过37％。在每个图像中寻求多个对象的多对象发现设置中，所提出的LOD平均精度（AP）比所有其他用于从20K到1.7M图像的数据的方法更好。使用自我监督功能，我们还表明该方法在OpenImages上获得最先进的UOD性能。我们的代码在HTTPS://github.com/huyvvo/lod上公开提供。