Localized Statistical Learning with Kernels

统计机器学习方法，特别是正则化核方法，已经在许多数据分析应用中被证明是成功的，并且现在还享有良好的理论基础。近年来，所谓的大数据应用，通常需要对巨大的，高维的数据集进行统计分析，数据质量往往不确定，越来越成为研究的焦点。不幸的是，标准的正则化内核方法很难与样本大小进行缩放，因此不适合大数据场景。的 这个项目的目标是通过建立理论上和经验上有充分依据的数据分解方法来解决这个严重的问题，以减少基于内核的学习方法的计算需求。在经验方面，我们设想的方法，可以在一个合理的时间内在一个单一的桌面上处理数百万个样本的高维度。在理论方面，我们寻求，不像大多数以前的尝试，加快基于内核的学习方法，这些分解approaches.In一个简单的，我们的目标包括四个部分：（i）识别面向空间的数据分解策略，大大减少了计算需求，而不牺牲泛化性能。 这与文献中提出的从原始数据集进行随机二次抽样的几种最近的方法形成鲜明对比。(ii)对成功的分解策略进行严格的数学分析，其中包括普遍一致性，快速，如果可能的话，最佳学习率和统计鲁棒性。(iii)对计算要求和泛化性能之间的权衡的描述。(iv)软件原型，以证明这些方法适用于大数据情况。由于所考虑的分解策略的空间性质以及对它们的理论分析的强烈关注，我们谈到了带内核的本地化统计学习（LSLK）。

Statistical machine learning approaches, and in particular regularized kernel methods, have been proven successful in many data analysis applications and additionally enjoy a nowadays well-founded theory. In recent years, so-called big data applications, which typically require the statistical analysis of huge, high-dimensional data sets with often uncertain data quality, moved more and more into the research focus. Unfortunately, standard regularized kernel methods poorly scale with the sample size, and are therefore not appropriate for big data scenarios. The goal of this project is to address this serious issue by establishing theoretically and empirically well-founded data-decomposition approaches for decreasing the computational requirements of kernel-based learning methods. On the empirical side we envision methods that can handle millions of samples in high dimensions on a single desktop within a reasonable period of time. On the theoretical side, we seek, unlike most previous attempts for speeding up kernel-based learning methods, strong guarantees for these decomposition approaches.In a nutshell, our objectives consist of four parts:(i) Identification of spatially oriented data decomposition strategies that reduce the computational requirements significantly without sacrificing generalization performance. This is in sharp contrast to several recent approaches in the literature in which random subsampling from the original data set is proposed.(ii) A rigorous mathematical analysis of successful decomposition strategies, which includes universal consistency, fast and, if possible, optimal learning rates, and statistical robustness.(iii) A description of the trade-off between computational requirements and generalization performance.(iv) Software prototypes to demonstrate that these methods are applicable in big data situations. Because of the spatial nature of the considered decomposition strategies and the strong focus on their theoretical analysis, we speak of localized statistical learning with kernels (LSLK).