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

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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.