Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified l/ between 0 and 1.We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. We provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled data.

The support-vector network is a new leaming machine for two-group classification problems. The machine conceptually implements the following idea: input vectors are non-linearly mapped to a very highdimension feature space. In this feature space a linear decision surface is constructed. Special properties of the decision surface ensures high generalization ability of the learning machine. The idea behind the support-vector network was previously implemented for the restricted case where the training data can be separated without errors. We here extend this result to non-separable training data.High generalization ability of support-vector networks utilizing polynomial input transformations is demonstrated. We also compare the performance of the support-vector network to various classical learning algorithms that all took part in a benchmark study of Optical Character Recognition.

Data domain description concerns the characterization of a data set. A good description covers all target data but includes no superfluous space. The boundary of a dataset can be used to detect novel data or outliers. We will present the Support Vector Data Description (SVDD) which is inspired by the Support Vector Classifier. It obtains a spherically shaped boundary around a dataset and analogous to the Support Vector Classifier it can be made flexible by using other kernel functions. The method is made robust against outliers in the training set and is capable of tightening the description by using negative examples. We show characteristics of the Support Vector Data Descriptions using artificial and real data.

这篇综述的目的是将读者介绍到图表内，以将其应用于化学信息学中的分类问题。图内核是使我们能够推断分子的化学特性的功能，可以帮助您完成诸如寻找适合药物设计的化合物等任务。内核方法的使用只是一种特殊的两种方式量化了图之间的相似性。我们将讨论限制在这种方法上，尽管近年来已经出现了流行的替代方法，但最著名的是图形神经网络。

We consider the problem of data classification where the training set consists of just a few data points. We explore this phenomenon mathematically and reveal key relationships between the geometry of an AI model's feature space, the structure of the underlying data distributions, and the model's generalisation capabilities. The main thrust of our analysis is to reveal the influence on the model's generalisation capabilities of nonlinear feature transformations mapping the original data into high, and possibly infinite, dimensional spaces.

We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semi-supervised framework that incorporates labeled and unlabeled data in a general-purpose learner. Some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases. We use properties of reproducing kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graph-based approaches) we obtain a natural out-of-sample extension to novel examples and so are able to handle both transductive and truly semi-supervised settings. We present experimental evidence suggesting that our semi-supervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework.

Neural networks with random weights appear in a variety of machine learning applications, most prominently as the initialization of many deep learning algorithms and as a computationally cheap alternative to fully learned neural networks. In the present article, we enhance the theoretical understanding of random neural networks by addressing the following data separation problem: under what conditions can a random neural network make two classes $\mathcal{X}^-, \mathcal{X}^+ \subset \mathbb{R}^d$ (with positive distance) linearly separable? We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. Crucially, the number of required neurons is explicitly linked to geometric properties of the underlying sets $\mathcal{X}^-, \mathcal{X}^+$ and their mutual arrangement. This instance-specific viewpoint allows us to overcome the usual curse of dimensionality (exponential width of the layers) in non-pathological situations where the data carries low-complexity structure. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity (based on a localized version of Gaussian mean width), which leads to sound and informative separation guarantees. We connect our result with related lines of work on approximation, memorization, and generalization.

In the era of big data, it is desired to develop efficient machine learning algorithms to tackle massive data challenges such as storage bottleneck, algorithmic scalability, and interpretability. In this paper, we develop a novel efficient classification algorithm, called fast polynomial kernel classification (FPC), to conquer the scalability and storage challenges. Our main tools are a suitable selected feature mapping based on polynomial kernels and an alternating direction method of multipliers (ADMM) algorithm for a related non-smooth convex optimization problem. Fast learning rates as well as feasibility verifications including the efficiency of an ADMM solver with convergence guarantees and the selection of center points are established to justify theoretical behaviors of FPC. Our theoretical assertions are verified by a series of simulations and real data applications. Numerical results demonstrate that FPC significantly reduces the computational burden and storage memory of existing learning schemes such as support vector machines, Nystr\"{o}m and random feature methods, without sacrificing their generalization abilities much.

所有著名的机器学习算法构成了受监督和半监督的学习工作，只有在一个共同的假设下：培训和测试数据遵循相同的分布。当分布变化时，大多数统计模型必须从新收集的数据中重建，对于某些应用程序，这些数据可能是昂贵或无法获得的。因此，有必要开发方法，以减少在相关领域中可用的数据并在相似领域中进一步使用这些数据，从而减少需求和努力获得新的标签样品。这引起了一个新的机器学习框架，称为转移学习：一种受人类在跨任务中推断知识以更有效学习的知识能力的学习环境。尽管有大量不同的转移学习方案，但本调查的主要目的是在特定的，可以说是最受欢迎的转移学习中最受欢迎的次级领域，概述最先进的理论结果，称为域适应。在此子场中，假定数据分布在整个培训和测试数据中发生变化，而学习任务保持不变。我们提供了与域适应性问题有关的现有结果的首次最新描述，该结果涵盖了基于不同统计学习框架的学习界限。

这是一门专门针对STEM学生开发的介绍性机器学习课程。我们的目标是为有兴趣的读者提供基础知识，以在自己的项目中使用机器学习，并将自己熟悉术语作为进一步阅读相关文献的基础。在这些讲义中，我们讨论受监督，无监督和强化学习。注释从没有神经网络的机器学习方法的说明开始，例如原理分析，T-SNE，聚类以及线性回归和线性分类器。我们继续介绍基本和先进的神经网络结构，例如密集的进料和常规神经网络，经常性的神经网络，受限的玻尔兹曼机器，（变性）自动编码器，生成的对抗性网络。讨论了潜在空间表示的解释性问题，并使用梦和对抗性攻击的例子。最后一部分致力于加强学习，我们在其中介绍了价值功能和政策学习的基本概念。

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.

我们考虑使用对抗鲁棒性学习的样本复杂性。对于此问题的大多数现有理论结果已经考虑了数据中不同类别在一起或重叠的设置。通过一些实际应用程序，我们认为，相比之下，存在具有完美精度和稳健性的分类器的分类器的良好分离的情况，并表明样品复杂性叙述了一个完全不同的故事。具体地，对于线性分类器，我们显示了大类分离的分布式，其中任何算法的预期鲁棒丢失至少是$ \ω（\ FRAC {D} {n}）$，而最大边距算法已预期标准亏损$ o（\ frac {1} {n}）$。这表明了通过现有技术不能获得的标准和鲁棒损耗中的间隙。另外，我们介绍了一种算法，给定鲁棒率半径远小于类之间的间隙的实例，给出了预期鲁棒损失的解决方案是$ O（\ FRAC {1} {n}）$。这表明，对于非常好的数据，可实现$ O（\ FRAC {1} {n}）$的收敛速度，否则就是这样。我们的结果适用于任何$ \ ell_p $ norm以$ p> 1 $（包括$ p = \ idty $）为稳健。

Recent advance on linear support vector machine with the 0-1 soft margin loss ($L_{0/1}$-SVM) shows that the 0-1 loss problem can be solved directly. However, its theoretical and algorithmic requirements restrict us extending the linear solving framework to its nonlinear kernel form directly, the absence of explicit expression of Lagrangian dual function of $L_{0/1}$-SVM is one big deficiency among of them. In this paper, by applying the nonparametric representation theorem, we propose a nonlinear model for support vector machine with 0-1 soft margin loss, called $L_{0/1}$-KSVM, which cunningly involves the kernel technique into it and more importantly, follows the success on systematically solving its linear task. Its optimal condition is explored theoretically and a working set selection alternating direction method of multipliers (ADMM) algorithm is introduced to acquire its numerical solution. Moreover, we firstly present a closed-form definition to the support vector (SV) of $L_{0/1}$-KSVM. Theoretically, we prove that all SVs of $L_{0/1}$-KSVM are only located on the parallel decision surfaces. The experiment part also shows that $L_{0/1}$-KSVM has much fewer SVs, simultaneously with a decent predicting accuracy, when comparing to its linear peer $L_{0/1}$-SVM and the other six nonlinear benchmark SVM classifiers.

We study the expressibility and learnability of convex optimization solution functions and their multi-layer architectural extension. The main results are: \emph{(1)} the class of solution functions of linear programming (LP) and quadratic programming (QP) is a universal approximant for the $C^k$ smooth model class or some restricted Sobolev space, and we characterize the rate-distortion, \emph{(2)} the approximation power is investigated through a viewpoint of regression error, where information about the target function is provided in terms of data observations, \emph{(3)} compositionality in the form of a deep architecture with optimization as a layer is shown to reconstruct some basic functions used in numerical analysis without error, which implies that \emph{(4)} a substantial reduction in rate-distortion can be achieved with a universal network architecture, and \emph{(5)} we discuss the statistical bounds of empirical covering numbers for LP/QP, as well as a generic optimization problem (possibly nonconvex) by exploiting tame geometry. Our results provide the \emph{first rigorous analysis of the approximation and learning-theoretic properties of solution functions} with implications for algorithmic design and performance guarantees.

考虑了建立UNKONWN地面真相函数值的样本外界限的问题。内核及其相关的希尔伯特空间是本文所采用的主要形式主义，以及一个观察模型，在该模型中，输出被有限的测量噪声损坏。噪声可以源于任何紧凑的分布，并且没有对可用数据进行独立假设。在这种情况下，我们显示计算紧密的，有限样本的不确定性范围等于求解参数四次约束线性程序。接下来，建立了我们方法的属性，并研究了其与另一种方法的关系。提出了数值实验，以说明如何在许多情况下应用理论，并将其与其他封闭形式的替代方案进行对比。

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.

支持向量机（SVM）是一种算法，该算法找到了超平面，最佳地将标记的数据点以$ \ mathbb {r} ^ n $分为正面和负类。该分离超平面裕度上的数据点称为支持向量。我们将支持向量的可能配置连接到Radon的定理，这提供了一组点可以分为两个类（正负）的保证，其凸壳相交。如果将正和负支持向量的凸壳投射到分离超平面上，则仅在超平面是最佳的，则投影在至少一个点中相交。此外，通过特定类型的一般位置，我们表明（a）支撑载体的投影凸船体在恰好一个点中相交，（b）支撑载体在扰动下稳定，（c）最多有$ n + 1 $支持向量，（d）每一个高达$ n + 1 $的支持向量是可能的。最后，我们执行研究预期的支持向量数及其配置的计算机模拟，用于随机生成的数据。我们观察到，随着该类型的随机生成的数据增加的距离增加，具有两个支持向量的配置成为最可能的配置。

我们为特殊神经网络架构，称为运营商复发性神经网络的理论分析，用于近似非线性函数，其输入是线性运算符。这些功能通常在解决方案算法中出现用于逆边值问题的问题。传统的神经网络将输入数据视为向量，因此它们没有有效地捕获与对应于这种逆问题中的数据的线性运算符相关联的乘法结构。因此，我们介绍一个类似标准的神经网络架构的新系列，但是输入数据在向量上乘法作用。由较小的算子出现在边界控制中的紧凑型操作员和波动方程的反边值问题分析，我们在网络中的选择权重矩阵中促进结构和稀疏性。在描述此架构后，我们研究其表示属性以及其近似属性。我们还表明，可以引入明确的正则化，其可以从所述逆问题的数学分析导出，并导致概括属性上的某些保证。我们观察到重量矩阵的稀疏性改善了概括估计。最后，我们讨论如何将运营商复发网络视为深度学习模拟，以确定诸如用于从边界测量的声波方程中重建所未知的WAVESTED的边界控制的算法算法。

Despite the great advances made by deep learning in many machine learning problems, there is a relative dearth of deep learning approaches for anomaly detection. Those approaches which do exist involve networks trained to perform a task other than anomaly detection, namely generative models or compression, which are in turn adapted for use in anomaly detection; they are not trained on an anomaly detection based objective. In this paper we introduce a new anomaly detection method-Deep Support Vector Data Description-, which is trained on an anomaly detection based objective. The adaptation to the deep regime necessitates that our neural network and training procedure satisfy certain properties, which we demonstrate theoretically. We show the effectiveness of our method on MNIST and CIFAR-10 image benchmark datasets as well as on the detection of adversarial examples of GT-SRB stop signs.

比较概率分布是许多机器学习算法的关键。最大平均差异（MMD）和最佳运输距离（OT）是在过去几年吸引丰富的关注的概率措施之间的两类距离。本文建立了一些条件，可以通过MMD规范控制Wassersein距离。我们的作品受到压缩统计学习（CSL）理论的推动，资源有效的大规模学习的一般框架，其中训练数据总结在单个向量（称为草图）中，该训练数据捕获与所考虑的学习任务相关的信息。在CSL中的现有结果启发，我们介绍了H \“较旧的较低限制的等距属性（H \”较旧的LRIP）并表明这家属性具有有趣的保证对压缩统计学习。基于MMD与Wassersein距离之间的关系，我们通过引入和研究学习任务的Wassersein可读性的概念来提供压缩统计学习的保证，即概率分布之间的某些特定于特定的特定度量，可以由Wassersein界定距离。