Canonical Linear Methods and Hierarchical Non-Linear Methods in High-Dimensional Statistics

统计是从大数据中提取有意义信息的核心。其主要任务包括估计和不确定性评估。后者在大数据分析中对于合理决策至关重要。对于前者，深度学习机器中使用的方法，例如谷歌的大脑和AlphaGo以及微软的Cortana背后的方法，都需要理解。本研究项目旨在弥合这些领域的统计实践和理论。它旨在为大数据的线性建模提供可访问的不确定性度量，并基于数学分析深入了解深度学习的工作原理。该研究项目开发和分析线性和非线性高维统计推断方法，这些方法易于数据科学从业者使用。在线性的情况下，它开发和分析推理方法的基础上建立良好的引导，套索，部分岭，随机投影方法。在非线性的情况下，它采取了第一步，以原则性的方式解释了深度学习在图像分类和语音识别等实际问题中取得的令人印象深刻的成功。特别是，这些方法的统计特性将在线性和Neyman-Rubin高维模型下进行研究，并通过分析和模拟手段。将探索双层神经网络（或分层非线性模型）的生成模型，以通过分析和模拟研究来理解和比较深度学习与其他方法。作为一种通用的监督学习方法，深度学习的改进是通过从大脑连接研究中实施生物学上有意义的约束来寻求的。

Statistics is at the heart of extracting meaningful information from big data. Its primary tasks include estimation and uncertainty assessment. The latter is crucial in big data analysis for sound decision making. For the former, the methods employed in deep learning machines, such as those behind Google's Brain and AlphaGo and Microsoft's Cortana, beg understanding. This research project is intended to bridge practice and theory of statistics in these areas. It aims to provide accessible uncertainty measures for linear modeling of big data and to derive insights into how deep learning works, based on mathematical analysis. This research project develops and analyzes linear and non-linear high-dimensional statistical inferential methods that are easily accessible by practitioners in data science. In the linear case, it develops and analyzes inferential methods based on well-established bootstrap, lasso, partial ridge, and random projection methods. In the non-linear case, it takes the first steps to explain in a principled manner the impressive success of deep learning in practical problems such as image classification and speech recognition. In particular, statistical properties of these methods will be studied under linear and Neyman-Rubin high dimensional models, and via analytical and simulation means. A generative model of a two-layer neural network (or hierarchical non-linear model) will be explored to understand and compare deep learning with other methods, analytically and through simulation studies. Improvements over deep learning as a general supervised learning method are sought by enforcing biologically meaningful constraints from brain connectivity research.