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

• E. Escolar and Y. Hiraoka. Optimal Cycles for Persistent Homology via Linear Programming. Optimization in the Real World --Towards Solving Real-World Optimization Problems--. Mathematics for Industry, Springer (2015).
• I. Obayashi. Volume Optimal Cycle: Tightest representative cycle of a generator in persistent homology. SIAM Journal on Applied Algebra and Geometry 2(4), 508--534 (2018). [ link ]