The persistent homology of a sampled map: from a viewpoint of quiver representations

Abstract

This paper is intended to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations. The direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the underlying map if the sample is sufficiently dense. Compared to existing methods using eigenspace functors, our filtration analysis represents an important advantage that no prior information related to the eigenvalues of the underlying map is required. Some numerical examples are given to demonstrate the effectiveness of our method.