论文标题
推断凸传感数据的固有维度的拓扑方法
A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data
论文作者
论文摘要
我们考虑了一个常见的测量范式，其中仿射空间的未知子集由未知的连续准串联函数测量。考虑到测量数据，可以确定该空间的维度吗？在本文中，我们开发了一种在自然通用假设下通过准凸功能从测量中推断数据的固有维度的方法。 维度推断问题仅取决于传感器函数引起的测量空间点排序的离散数据。我们介绍了Dowker复合物过滤的结构，该络合物与准分子函数的测量相关。然后，这些复合物的拓扑特征用于推断固有维度。我们证明，在自然通用假设下，可以保证在大数据限制中获得正确的固有维度的收敛定理。我们还说明了该方法在模拟中的可用性。
We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one determine the dimension of this space? In this paper, we develop a method for inferring the intrinsic dimension of the data from measurements by quasi-convex functions, under natural generic assumptions. The dimension inference problem depends only on discrete data of the ordering of the measured points of space, induced by the sensor functions. We introduce a construction of a filtration of Dowker complexes, associated to measurements by quasi-convex functions. Topological features of these complexes are then used to infer the intrinsic dimension. We prove convergence theorems that guarantee obtaining the correct intrinsic dimension in the limit of large data, under natural generic assumptions. We also illustrate the usability of this method in simulations.
