Developing statistical and topological learning methodologies for high-dimensional complex data

自然过程可以产生如此复杂和高维的数据模式，以至于人类思维无法将其可视化。高维数据的例子包括但不限于社会/传感器网络、语义、DNA序列、基因组研究和医学图像。 持久同调是计算拓扑学的一个特殊分支，它研究滤子的拓扑特征的演化，滤子是一族单参数的嵌套空间族。它可以与传统的统计方法以及机器学习技术相结合，并已被证明在区分信号和噪声之间的差异方面是有效的。持久同调的基本思想类似于统计学中的导数显著过零和计算机视觉中的尺度空间理论。然而，在所有这三个学科中，只考虑了一个参数：持久同调中函数的高度，统计学中的带宽，以及计算机视觉中的分辨率。在许多情况下，有必要让几个参数同时变化。例如，在核密度估计中，有两个参数：函数的高度和带宽。多参数过滤的持久同调仍然没有解决：我们的主要研究目标之一是研究双滤波的持久同调。 我们提出了以下四个研究目标：(1)高维拓扑特征的计算、可视化和解释；(2)将二维持久性应用于随机场、函数数据分析和多元回归分析；(3)将二维持久同调引入到聚类分析和机器学习技术中，如支持向量机；以及(4)开发将结合我们新开发的技术的统计和拓扑学习工具。 建议的方法将在几个方面得到应用：可视化和解释社会网络、语义、分子、DNA序列和脑图像中的高维拓扑特征；比较儿童阻塞性睡眠呼吸暂停(OSA)患者和正常受试者的头面部形状和上呼吸道；基于100多个变量根据儿童OSA患者的严重程度对其进行聚类和/或分类；将序贯分析应用于(1)-(4)中的所有组合方法，其动机源于对儿童OSA患者进行的临床试验。序贯分析与拓扑学和机器学习方法相结合，可能有助于尽早终止临床试验。 总而言之，我们的建议包括三个主要的科学学科--统计学、计算拓扑学和机器学习--并朝着将这些研究领域的强大技术结合在一起的方向迈出了一步。

Natural processes can yield data patterns so complex and high-dimensional that they cannot be visualized by the human mind. Examples of high-dimensional data include, but are not limited to, social/sensor networks, semantics, DNA sequences, genomic studies, and medical images. Persistent homology is a particular branch of computational topology which studies the evolution of topological features of a filtration, a one-parameter family of nested spaces. It can be combined with traditional statistical methods as well as machine learning techniques and has been shown to be effective in discerning the differences between signal and noise. The fundamental idea of persistent homology is analogous to significant zero crossings of derivatives in statistics and scale-space theory in computer vision. In all three disciplines, however, only one parameter has been considered: the height of the function in persistent homology, bandwidth in statistics, and the scale of resolution in computer vision. In many situations, it is necessary to let several parameters vary simultaneously. For instance, in kernel density estimations, there are two parameters: the height of the function and the bandwidth. Persistent homology of multi-parameter filtrations remains unsolved: one of our main research goals is to study the persistent homology of bifiltrations. We propose the following four research goals: (1) computation, visualization, and interpretation of high-dimensional topological features; (2) development of two-dimensional persistence to be applied to random fields, functional data analysis, and multivariate regression analysis; (3) incorporation of two-dimensional persistent homology into cluster analysis and techniques in machine learning, such as support vector machine; and (4) development of statistical and topological learning tools that will incorporate our newly-developed techniques. The proposed methods will be applied in several ways: visualizing and interpreting high-dimensional topological features in social networks, semantics, molecules, DNA sequences, and brain images; comparing the craniofacial shapes and upper airways of pediatric obstructive sleep apnea (OSA) patients and normative subjects; clustering and/or classifying pediatric patients in terms of their OSA severity based on over one hundred variables; applications of sequential analysis to all of the combined methods in (1)-(4), the motivation for which stemming from clinical trials performed on pediatric OSA patients. Sequential analysis, in combination with topological and machine learning methodologies, could be conducive to early termination of the clinical trials. Altogether, our proposal encompasses three major scientific disciplines--statistics, computational topology, and machine learning--and serves as a step towards combining powerful techniques from each of these research areas.