由方向持久图重构一元函数
Reconstruction of univariate functions from directional persistence diagrams
摘要：
我们描述了一种方法，用于近似一个单变量的功能$f$使用持久性图的子水平集的$f$从高度功能在不同的方向。我们提供算法的分段线性的情况下，光滑的情况下。三个方向足以从方向持久图的集合中定位分段线性连续函数的所有局部最大值和最小值，而在具有非退化临界点的光滑函数的情况下需要五个方向。我们的近似功能的持久性图的方式是由机器学习中的重要性归因的研究，其中一个寻求减少信号功能的临界点的数量，而没有显着的信息的损失的神经网络分类器的动机。
英文摘要：
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.
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