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

统计数据是从大数据中提取有意义信息的核心。它的主要任务包括评估和不确定性评估。后者在大数据分析中至关重要，有助于做出合理的决策。对于前者，深度学习机器中使用的方法，如谷歌的Brain、AlphaGo和微软的Cortana背后的方法，需要理解。这一研究项目旨在为这些领域的统计实践和理论架起桥梁。它旨在为大数据的线性建模提供可访问的不确定性度量，并基于数学分析得出对深度学习如何工作的见解。这项研究项目开发和分析了数据科学从业者容易使用的线性和非线性高维统计推理方法。在线性情况下，发展和分析了基于成熟的Bootstrap、Lasso、部分岭和随机投影法的推理方法。在非线性情况下，它首先原则性地解释深度学习在图像分类和语音识别等实际问题上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高维模型下通过分析和模拟手段进行研究。将探索两层神经网络的生成模型(或分层非线性模型)，以通过分析和模拟研究来理解深度学习并将其与其他方法进行比较。深度学习作为一种一般的有监督的学习方法，其改进是通过从大脑连接研究中强制实施具有生物学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.