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

高维统计中的规范线性方法和分层非线性方法

• 批准号: 1613002
• 负责人: Bin Yu
• 金额: $ 60万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613002&HistoricalAwards=false
• 关键词: Canonical; Linear; Methods; Hierarchical; Non

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.

统计学是从大数据中提取有意义信息的核心。它的主要任务包括估算和不确定性评估。后者在大数据分析中对合理决策至关重要。对于前者，b谷歌的Brain、AlphaGo和微软的Cortana等深度学习机器所采用的方法令人难以理解。本研究项目旨在为这些领域的统计理论与实践搭建桥梁。它旨在为大数据的线性建模提供可访问的不确定性度量，并基于数学分析获得深度学习如何工作的见解。该研究项目开发和分析了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人员使用。在线性情况下，它开发并分析了基于成熟的bootstrap， lasso， partial ridge和随机投影方法的推理方法。在非线性情况下，它首先以原则性的方式解释深度学习在实际问题（如图像分类和语音识别）中取得的令人印象深刻的成功。特别是，这些方法的统计性质将在线性和Neyman-Rubin高维模型下，并通过分析和模拟手段进行研究。将探索一个双层神经网络（或分层非线性模型）的生成模型，通过分析和仿真研究来理解和比较深度学习与其他方法。深度学习作为一般监督学习方法的改进是通过从大脑连接研究中强制执行生物学上有意义的约束来寻求的。