BIGDATA: F: DKA: Learning a Union of Subspaces from Big and Corrupted Data

该项目开发了在高维数据中自动发现多个低维结构的理论和算法，并评估了这些算法在图像聚类中的应用。所开发的技术增强了我们处理来自多个来源和模式的大数据问题的能力，并提高了如何解释大量复杂高维数据的知识。本项目开发的技术可以显着拓宽稀疏表示理论中现有结果对子空间聚类问题的适用性，这些问题在图像处理（例如图像去噪、压缩、表示和分割）、计算机视觉（例如运动分割和人脸聚类）和动力系统（例如混合系统识别）中得到了广泛的应用。本研究开发了可证明正确且可扩展的算法，用于从大数据和损坏数据中学习低维子空间的并集。该算法基于所谓的数据的自表达性，即一个未损坏的数据点可以通过其他未损坏数据点的仿射组合很好地近似。研究表明，通过对系数施加结构化稀疏和低秩先验，可以发现数据中的多个结构。在未损坏数据的情况下，研究小组研究了可能实现完美聚类的数据条件。在数据被异常值破坏的情况下，研究小组研究了完美聚类和排除异常值的条件。对于缺失条目的数据，研究小组研究了完美聚类和数据补全的条件。该项目还开发了高效和可扩展的算法，这些算法受益于分布式和高性能计算，用于解决各种子空间聚类问题。这些算法能够解决计算机视觉中的大规模问题，包括图像聚类。

This project develops theory and algorithms for automatically discovering multiple low-dimensional structures in high-dimensional data, and evaluates these algorithms in image clustering applications. The developed techniques enhance our ability to handle big data problems from multiple sources and modalities, and advance the knowledge on how to interpret massive amounts of complex high-dimensional data. The techniques developed in this project can significantly broaden the applicability of existing results in sparse representation theory to subspace clustering problems, which have found widespread applications in image processing (e.g., image denoising, compression, representation, and segmentation), computer vision (e.g., motion segmentation and face clustering) and dynamical systems (e.g., hybrid system identification). This research develops provably correct and scalable algorithms for learning a union of low-dimensional subspaces from big and corrupted data. The algorithms are based on the so-called self-expressiveness property of the data, which states that an uncorrupted data point can be well approximated by an affine combination of other uncorrupted data points. This research shows that by imposing a structured sparse and low-rank prior on the coefficients, one can discover multiple structures in the data. In the case of uncorrupted data, the research team studies conditions on the data under which a perfect clustering is possible. In the case of data corrupted by outliers, the research team studies conditions under which perfect clustering and outlier rejection are possible. In the case of data with missing entries, the research team studies conditions under which perfect clustering and data completion are possible. The project also develops efficient and scalable algorithms that benefit from distributed and high-performance computing for solving the various subspace clustering problems. These algorithms enable solving large-scale problems in computer vision, including image clustering.