A Lattice-Theoretic Perspective on the Persistence Map

We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our description is local, which offers the potential to speed up inverse computations, and brings classical tools in combinatorics to bear on an active area of research in topological data analysis (TDA).

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1 Background on the Inverse Problem

Merge trees play a central role in topological data analysis (TDA). One can apply persistent homology to a merge tree to obtain an “adjacency free” description of a merge tree in terms of its barcode, we call this association of a barcode to a merge tree the persistence map. Characterizing precisely how many merge trees map to the same barcode was studied in [curry2017fiber, TRN, trees_barcodesII] and has yielded significant connections to geometric group theory, combinatorics, and statistics. Understanding the fiber of the persistence map is crucial for understanding how noise in data propagates to noise in persistent homology.

In [TRN, trees_barcodesII] a combinatorial version of this inverse problem was considered; see Figure 1. A combinatorial merge tree is a binary, rooted, combinatorial tree with birth-ordered labels on the leaves and death-ordered labels on the internal nodes. Every barcode with finite-length bars whose left (birth) endpoints are distinct and whose right (death) endpoints are distinct can be encoded by a combinatorial barcode if the birth endpoint is matched with the death endpoint. Equivalently, a combinatorial barcode is the graph of a permutation of .

In this abstract, we characterize the persistence map from combinatorial merge trees to combinatorial barcodes in terms of monotone maps between two lattices: the subset lattice and the partition lattice. We show that a maximal chain in the subset and partition lattices corresponds to a combinatorial barcode and combinatorial merge tree respectively, and that one may incrementally construct solutions to the inverse problem using this correspondence.

Figure 1: Figure from [trees_barcodesII], expressing the combinatorial inverse problem.

2 A Lattice Version of the Inverse Problem

Let be a poset. Recall that a lattice is a poset equipped with meets and joins. A totally ordered subset is called a chain. An interval is a subset where if and , then . A path is a chain that is also an interval. A path is based at if the lowest element in is . If has a unique lowest element (e.g. a lattice), we write as the poset of paths based at , which is a poset via containment of paths. There is a unique surjective map sending a path to its endpoint. Furthermore, if is a monotone map of posets, there is a unique map such that . We call the lift of .

[Subset Lattice] Let and consider , the set of all subsets of , including the empty set , equipped with the partial order of “being a subset of”. This forms the subset lattice of , with and being the meet and join of , respectively. The poset of paths in based at is .

[Partition Lattice] A partition of the set is a collection of disjoint subsets of whose union is . A partition refines a partition , written , if every subset of is equal to a union of elements of . We denote the lattice of partitions of by . The poset of paths based at is

We can filter a combinatorial barcode with bars into sets where is the set of pairs . We refer to as a partial (combinatorial) barcode. The set of all partial barcodes with at most bars forms a poset by containment, which we denote by . Similarly, a partial (combinatorial) merge tree is a filtration of a combinatorial merge tree with leaves by subgraphs where is the full subgraph supported on the set of leaf nodes and all internal nodes with label less than or equal to . Partial merge trees also forms a poset by subgraph containment, denoted ; see Figure 2. The persistence map between combinatorial merge trees and barcodes extends to a map from to , which we also call the persistence map.

The poset of partial merge trees and barcodes are isomorphic to and , respectively. Furthermore, there is a monotone map whose lift is naturally isomorphic to the persistence map from .

Proof.

Every partial merge tree defines a path , where is the partition of the leaf node labels induced by connected components in the graph ; see Figure 2. Every partial barcode defines a path , where is the set of birth labels whose deaths occur by time . These specify the isomorphisms.

Define as follows: Let be a partition of . For each , let . Let . This map is monotone, since if , then the latter partition is obtained by collapsing parts of the first, which can only add elements to . It is easy to see that this map is also surjective. This lifts to a natural map , defined on paths.

The maximal element (endpoint) of a path corresponds to a partition that indexes the leaf labels of the connected components of , the stage in a partial merge tree. The Elder Rule [curry2017fiber] of persistent homology maps each of the to as encodes the oldest leaf node, which goes unpaired by the persistence algorithm. The image is the union of leaf node labels that have been killed by stage . The combinatorial barcode is encoded by the successive differences between and . ∎

3 Future Work

Theorem 2 is still in need of a full geometric description that accounts for actual positions and lengths of bars in a barcode and edges in a merge tree. In [barcode_coxeter] a novel coordinatization of barcode space was given based on the relation with the symmetric group. However, a similar picture for merge tree space that uses the connection with the partition lattice is unknown. Additionally, the lattice structure on these “skeletonizations” of barcode and merge tree space has not been fully explored. As noted in [gulen2022diagrams, mccleary2020edit, patel2018generalized], Möbius inversion provides another way of summarizing topological changes in a filtration, which suggests that inverse problems, lattice theory, and Möbius inversion may occupy a rich intersection of ideas.

References