Stable Volumes for Persistent Homology

This paper proposes a stable volume and its variant, a stable sub-volume for more reliable data analysis using persistent homology. In the previous research, an optimal cycle and similar ideas are proposed to identify the homological structure corresponding to each birth-death pair in a persistence diagram. It is helpful for data analysis using persistent homology. However, the result is unstable against noises. Stable volumes and stable sub-volumes are proposed to solve the problem. In a special case, we prove that a stable volume is the robust part of an optimal volume against noises. We implement stable volumes and sub-volumes on HomCloud, persistent homology based data analysis software. This paper also shows some examples of stable volumes and sub-volumes.