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

现代机器学习技术的目标是设计模型和算法，使计算机能够有效地从以前未探索的海量数据中学习。这些问题被称为“无监督”，因为没有人工提供的有关数据的信息来指导机器学习过程。可以说，两种最重要的无监督机器学习工具是降维和聚类。在降维方面，该算法寻求一种简单的低维结构来捕捉数据中有趣的行为。在聚类中，该算法寻求将数据点组合在一起，形成有意义的簇。随着对越来越复杂的物理、生物和社会现象的日益高维数据的收集，同时以降维和聚类为目标的算法通常具有很高的适用性。然而，文献中的联合公式通常是特别的，并且从根本上不能操作具有缺失元素、损坏和异构性的真实数据-现代数据问题的关键机器学习挑战。这一研究项目有望在数据科学中具有广泛的适用性，并将在两个应用中得到展示：遗传学和计算机视觉。该项目中使用的联合聚类和降维公式被称为K-集聚类，它寻找被约束为具有某种低维表示的K个中心集，每个中心集代表数据中的K个簇中的一个。这个公式是K-均值、K-子空间和主成分分析的推广，它自然会导致几个新的问题实例。给定一个定义的集合几何，从两个角度来处理相应的问题实例：理解问题表述的实例的几何，以及从数据中学习那些几何模型。我们将研究三个具体的问题描述的例子：子空间聚类、变化聚类和多面体集合聚类。虽然每个例子都提出了内在的和独特的挑战，但这些只是一个更大的范例的例子，这个范例只受到一个人定义服从于对数据中的几何结构建模的集合的能力的限制。这个奖项反映了NSF的法定使命，通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.