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Bounded Manifold Completion
Pattern Recognition ( IF 8 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.patcog.2020.107661
Kelum Gajamannage , Randy Paffenroth

Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to detect the existence of a low-dimensional manifold of a given dimension that lies within a set of bounds derived from a given point cloud. A matrix representing the appropriately defined distances on a low-dimensional manifold is low-rank, and our method is based on current techniques for recovering a partially observed matrix from a small set of fully observed entries that can be implemented as a low-rank Matrix Completion (MC) problem. MC methods are currently used to solve challenging real-world problems, such as image inpainting and recommender systems, and we leverage extent efficient optimization techniques that use a nuclear norm convex relaxation as a surrogate for non-convex and discontinuous rank minimization. Our proposed method provides several advantages over current nonlinear dimensionality reduction techniques, with the two most important being theoretical guarantees on the detection of low-dimensional embeddings and robustness to non-uniformity in the sampling of the manifold. We validate the performance of this approach using both a theoretical analysis as well as synthetic and real-world benchmark datasets.

中文翻译:

有界流形完成

非线性降维或等效地,使用低维非线性流形逼近高维数据是一个活跃的研究领域。在本文中,我们将提出一种主题不同的方法来检测给定维度的低维流形的存在,该流形位于从给定点云导出的一组边界内。表示在低维流形上适当定义的距离的矩阵是低秩的,我们的方法基于当前的技术,用于从一小组完全观察到的条目中恢复部分观察到的矩阵,这些矩阵可以实现为低秩矩阵完成(MC)问题。MC 方法目前用于解决具有挑战性的现实世界问题,例如图像修复和推荐系统,我们利用范围有效的优化技术,使用核范数凸松弛作为非凸和不连续秩最小化的替代。我们提出的方法与当前的非线性降维技术相比具有几个优点,其中最重要的两个是低维嵌入检测的理论保证和对流形采样中非均匀性的鲁棒性。我们使用理论分析以及合成和现实基准数据集来验证这种方法的性能。最重要的两个是检测低维嵌入的理论保证和对流形采样中非均匀性的鲁棒性。我们使用理论分析以及合成和现实基准数据集来验证这种方法的性能。最重要的两个是检测低维嵌入的理论保证和对流形采样中非均匀性的鲁棒性。我们使用理论分析以及合成和现实基准数据集来验证这种方法的性能。
更新日期:2021-03-01
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