论文标题
从定向持久图的单变量函数重建
Reconstruction of univariate functions from directional persistence diagrams
论文作者
论文摘要
我们描述了一种使用来自不同方向的高度函数的$ f $ $ f $的持续图近似单变量函数$ f $的方法。我们为分段线性盒和光滑外壳提供算法。三个方向足以从其定向持续图集合中找到分段线性连续函数的所有局部最大值和最小值，而在具有非分类临界点的平滑函数的情况下，需要五个方向。 我们通过持久图的函数近似是由对机器学习中重要性归因的研究进行的，在该研究中，人们试图减少信号功能的关键点的数量，而没有明显的神经网络分类器信息损失。
We describe a method for approximating a single-variable function $f$ using persistence diagrams of sublevel sets of $f$ from height functions in different directions. We provide algorithms for the piecewise linear case and for the smooth case. Three directions suffice to locate all local maxima and minima of a piecewise linear continuous function from its collection of directional persistence diagrams, while five directions are needed in the case of smooth functions with non-degenerate critical points. Our approximation of functions by means of persistence diagrams is motivated by a study of importance attribution in machine learning, where one seeks to reduce the number of critical points of signal functions without a significant loss of information for a neural network classifier.
