  This paper aims 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, and 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 dense enough. Compared to existing methods
using eigenspace functors, our filtration analysis has an advantage that no
prior information on the eigenvalues of the underlying map is required. Some
numerical examples are given to illustrate the effectiveness of our method.
