Persistent homology analyzes topological and geometric features of data by reducing it into so-called persistence diagrams. Then, it is significantly important to explicitly detect subsets of the data which generate those topological and geometric features (e.g., material designs, and geometric analysis of atomic structures and functions of molecules). This problem can be formulated as an inverse problem using sparse linear optimization on homological algebra.
literature
A minimum generator of 1st homology embedded in hemoglobin
Example of a volume-optimal cycle