Collaborative Research: Probabilistic models and geometry for high dimensional data

与高维基因组数据相关的推理问题为现代统计学、机器学习和数据挖掘研究提供了基本挑战。在这一领域取得成功的方法对模型施加了约束，这些模型包含简单性、平滑性或鲁棒性的概念。约束通常形式化为贝叶斯方法的先验或机器学习方法的几何标准。这个建议的核心是开发和关联的重要性的几何基础的数据概率建模。该计划的具体研究重点是：1）利用几何假设解决高维模型中的模型不确定性和变量选择问题; 2）在预测建模中使用辅助或未标记数据的贝叶斯框架; 3）非参数贝叶斯核模型的理论，方法和计算; 4）从正则化和几何角度对高维数据进行非线性降维的新方法。该提案为功能基因组学中的统计建模提供了理论、方法和计算工具。现代分子生物学产生了规模和复杂性迅速升级的数据-高通量基因组学数据，遗传和序列信息，蛋白质组学和代谢组学数据，以及其他形式的更传统的生物医学或临床信息。对这些数据进行建模，以预测预后、诊断和通路失调的表型，以及了解相关变量及其关联，是现代统计学、机器学习和数据挖掘研究的基本挑战。这些方法的发展将对其他几个科学领域产生影响，包括生物学、工程学、环境和健康科学以及社会科学。

The inference problems associated with high-dimensional genomic data offer fundamental challenges for modern statistics, machine learning, and data-mining research. Methods that have had success in this domain impose constraints on models incorporating notions of simplicity, smoothness, or robustness. The constraints are often formalized either as priors for Bayesian methods or as geometric criteria for machine learning methods. The heart of this proposal is to develop and relate the importance of the geometry underlying the data to probabilistic modeling. The specific research foci of the proposal are: 1) The exploitation of geometric assumptions for problems of model uncertainty and variable selection in high-dimensional models; 2) A Bayesian framework for the use of ancillary or unlabeled data in predictive modeling; 3) Theory, methods and computation for nonparametric Bayesian kernel models; 4) Novel methods for nonlinear dimension reduction for high-dimensional data from regularization and geometric perspectives.The proposal develops theory, methods and computational tools for statistical modeling motivated by applications in functional genomics. Modern molecular biology has generated data of a rapidly escalating scale and complexity -- high-throughput genomics data, genetic and sequence information, proteomic and metabolomic data, and other forms of more traditional biomedical or clinical information. Modeling this data for predictive phenotypes of prognosis, diagnosis, and pathway deregulation as well as understanding relevant variables and their associations are fundamental challenges for modern statistics, machine learning, and data-mining research. These methodological developments will have impact on several other scientific areas including biology, engineering, environmental and health science, and social sciences.