Decorated merge trees for persistent topology

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent spaces. DMTs combine both $$\pi _0$$ and $$H_n$$ information into single data structure that distinguishes filtrations homology cannot distinguish alone. Three variants on DMTs, which emphasize category theory, representation theory persistence barcodes, respectively, offer different advantages in terms of computation. Two notions distance—an interleaving distance bottleneck distance—for are defined hierarchy stability results refine generalize existing is proved here. To overcome some the computational complexity inherent these distances, we provide use Gromov-Wasserstein couplings to compute optimal tree alignments combinatorial version our can be tractably estimated. We introduce frameworks generating, visualizing comparing derived from synthetic real data. Example applications include comparison point clouds, interpretation sliding window embeddings time series, visualization topological features segmented brain tumor images topology-driven graph alignment.