Figure 11 - Persistence Diagrams with Linear Machine Learning Models

(a) shows the reconstructed persistence diagram from the learned vector w and (b) shows the positive and negative areas of (a) with a certain threshold. Recall that, from the 0/1 assignment, the generators in the blue (resp. red) area contributes to classifying into PPP (resp. GPP). From the learned persistence diagram, we observe that the red area is located on the region with large birth values. This is consistent to the fact that GPP has a repulsive interaction, and hence it prevents the point cloud from constructing rings with small birth values. (c) and (d) show the death positions of the generators in the blue and red areas of (b) with the same colors, where (c) (resp. (d)) corresponds to PPP (resp. GPP). Similarly to the discussion in Figure 5, these death positions express characteristic geometric features used for learnings more explicitly. We remark that PPP and GPP can also be distinguished by using other descriptors such as average nearest neighbor distances. An advantage of our method is that we do not need any prior knowledge, providing us with more universal method compared to problem-specific descriptors. In fact, the analysis using average nearest neighbor distance can be realized by the 0th persistence diagram.