论文标题
纤维：使用矢量束的拓扑复杂数据的光纤维度降低
FibeRed: Fiberwise Dimensionality Reduction of Topologically Complex Data with Vector Bundles
论文作者
论文摘要
具有非平凡大规模拓扑的数据集可能很难嵌入具有现有维度降低算法的低维欧几里得空间中。我们建议使用向量束对拓扑复杂的数据集建模，以使基本空间解释大型拓扑，而纤维则解释了局部几何形状。这使人们可以在保留大规模拓扑的同时降低纤维的尺寸。我们将此观点形式化，作为一个应用程序，我们描述了一种算法，该算法将数据集和欧几里得空间中IT的初始表示形式一起使用，假定可以恢复其大规模拓扑的一部分，并输出一种新的表示，并通过局部线性尺寸降低沿初始的全局表示，可以整合综合的局部局部表示，并通过局部线性尺寸降低。我们在来自动态系统和化学的示例上证明了这种算法。在这些示例中，与各种基于众所周知的基于公制的降低算法相比，我们的算法能够在较低的目标维度中学习拓扑忠实的数据嵌入。
Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view, and, as an application, we describe an algorithm which takes as input a dataset together with an initial representation of it in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations, obtained through local linear dimensionality reduction, along the initial global representation. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.
