CIF: CAREER: Robust, Interpretable, and Efficient Unsupervised Learning with K-set Clustering

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Modern machine learning techniques aim to design models and algorithms that allow computers to learn efficiently from vast amounts of previously unexplored data. These problems are called 'unsupervised' because no human-provided information about the data is available to guide the machine learning process. Arguably the two most important unsupervised machine learning tools are dimensionality-reduction and clustering. In dimensionality-reduction, the algorithm seeks a simple low-dimensional structure that captures the interesting behavior in the data. In clustering, the algorithm seeks to group data points together into meaningful clusters. As increasingly higher-dimensional data are collected about progressively more elaborate physical, biological, and social phenomena, algorithms that aim at both dimensionality reduction and clustering are often highly applicable. However, joint formulations in the literature are often ad-hoc and fundamentally unable to operate on real data that have missing elements, corruptions, and heterogeneity --- critical machine learning challenges for modern data problems. This research project is expected to have broad applicability in data science, and will be demonstrated in two applications: genetics and computer vision. The joint clustering and dimensionality reduction formulation used in this project, called K-set clustering, seeks K "central sets" constrained to have some low-dimensional representation, each of which represents one of K clusters in the data. The formulation is a generalization of K-means, K-subspaces, and principal component analysis, and it naturally leads to several novel problem instances. Given a defined set geometry, the corresponding problem instance is approached from two perspectives: understanding the geometry of that instance of the problem formulation, and learning those geometric models from data. Three specific examples of the problem formulation will be studied: subspace clustering, variety clustering, and polyhedral set clustering. While each example presents intrinsic and unique challenges, these are just examples of a larger paradigm that is limited only by one's ability to define sets amenable to modeling the geometric structure in data. The formulation allows for interpretable data analysis, with a framework that can readily incorporate missing data and heterogeneous data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.