Reconstruction of univariate functions from directional persistence diagrams
This work addresses the problem of function reconstruction for applications like importance attribution in machine learning, where reducing critical points without losing information is crucial, but it appears incremental as it builds on existing persistence diagram methods.
The authors tackled the problem of approximating univariate functions using persistence diagrams from directional sublevel sets, showing that three directions suffice to locate all local maxima and minima for piecewise linear functions, and five for smooth functions with non-degenerate critical points.
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.