Updating Zigzag Persistence and Maintaining Representatives over Changing Filtrations
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Computing persistence over changing filtrations give rise to a stack of 2D persistence diagrams where the birth-death points are connected by the so-called 'vines'. We consider computing these vines over changing filtrations for zigzag persistence. We observe that eight atomic operations are sufficient for changing one zigzag filtration to another and provide an update algorithm for each of them. As with the zigzag persistence algorithms for a static filtration, these updates are implemented with the maintenance of representatives. Since finding consistent representatives for zigzag persistence is more involved, the updates for the zigzag case are more costly than their counterparts in the non-zigzag case. As motivations, we identify some potential use of our update algorithms including the case of dynamic point cloud data, where a vineyard of zigzag persistence diagrams captures changing homological features across distance and time.
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