Reeb graphs have become an important tool in topological data analysis for the purpose of visualizing continuous functions on complex spaces, as they yield a simplified discrete structure. Originally developed in relation to Morse theory [Reeb1946], these objects are used extensively for shape comparison, constructing skeletons of data sets, surface simplification, and visualization; for more details on these and more applications, we refer to recent surveys on the topic [Biasotti2008, yan2021scalar]. While some information is lost in the construction of the Reeb graph, such simplified structures allow for more efficient methods to analyze and compare data sets.
More precisely, given a topological space and a real valued function , the pair is known as an -space; e.g. see Figure 1. The Reeb graph of is then obtained by collapsing each connected component in a level set into single point, and collecting the points together using the quotient topology. The result is a 1-dimensional stratified space (i.e. a graph) along with an induced real-valued function; we will refer to this pair as .
Given the many algorithms available to compute these objects efficiently [Doraiswamy2009, Parsa2012, Harvey2010, Gueunet2019], the Reeb graph is a practical, simplified structure which can be used for tasks ranging from simplification to visualization. Thus, there is a practical need for ways to compare and analyze Reeb graphs. Many possible options have been studied recently [DiFabio2016, Bauer2020, deSilva2016, chambers2020family, Bauer2014, Carriere2017, Cohen-Steiner2009a, Bauer2015b, Bauer2014, deSilva2016, Bauer2020]. In this work, we focus on recent work [deSilva2016, chambers2020family] which introduces the concept of smoothing a Reeb graph as a byproduct of the interleaving distance. This smoothing operation generates a new Reeb graph for every which simplifies the topological structure of the graph. In particular, it continuously removes small loops, which are often viewed as noise in the input data.
Despite being defined via category theory, the resulting construction can be viewed from a completely combinatorial viewpoint. An algorithm for constructing the smoothed Reeb graph was provided in [deSilva2016], and has been extended to the more recently introduced truncated smoothing functor as well [chambers2020family]. Further, it has been shown that given a Reeb graph with critical points , the smoothed Reeb graph has critical set contained in [deSilva2016]. However, no exact combinatorial characterization of the graph changes has been proven in the literature prior to this paper.
We note that -space data can also be studied via its persistent homology, an algebraic method for measuring topological features of shapes and functions [Oudot2017a, Dey2021]. In fact, that there is a close relationship between critical values of the Reeb graph and points in the extended persistence diagram [Agarwal2006, CohenSteiner2006], which will be a key component of our work. Because of this relationship, one might expect there to be an available inverse map; i.e. given an extended persistence diagram, can we reconstruct the Reeb graph uniquely? The answer, in general, is no since many different Reeb graphs can have the same persistence diagram. Despite difficulties, consideration of these inverse problems are increasingly of interest in topological data analysis as a whole [curry2019left, Oudot2020, Curry2021], as they can provide insight into how much information is lost in the process of computing such a topological signature, be it a persistence diagram or a Reeb graph. However, they are notoriously difficult, as key information is inevitably lost when going from a space to such a signature.
1.1 Outline of our results
The main contribution of this paper is to initiate the study of inverse problems on vineyards realized by Reeb graphs, using the smoothing and truncated smoothing functors to determine when such an inverse can be determined. After introducing necessary background and notation in Section 2, in Section 3 we explicitly enumerate all changes to a Reeb graph under Reeb graph smoothing, and then generalize this framework for the more recently developed truncated smoothing operation. In Section 4, we use this characterization to solve the inverse problem in a restricted setting for Reeb graphs using smoothing and truncated smoothing, by showing that certain paths in persistence diagram space (commonly known as vineyards [Cohen-Steiner2006]) can be realized by a path in the space of Reeb graphs. Namely, using our characterization of how truncated smoothing affects the combinatorial structure of the Reeb graph, we are able to determine sufficient restrictions on a time varying set of persistence diagrams, so that as long as an initial Reeb graph is specified, we can determine a set of time varying Reeb graphs which realize the vineyard. We conclude in Section 5 by discussing several possible future directions motivated by our work.
2 Background and definitions
2.1 Reeb graphs
Let X be a topological space, and let be continuous real-valued function. The pair is referred to as an -space (or a scalar field, depending upon the setting). The level set of at is the set . We define an equivalence relation on by if and only if and and are in the path same connected component of the levelset . The Reeb graph is the quotient space , with an induced function inherited from the -space given by . (We will generally abuse notation slightly and call both functions .)
For the purposes of our work, we will often divorce the idea of a Reeb graph from the -space it came from. In particular, given reasonable assumptions on the input -space, the Reeb graph is indeed a finite graph, so we will assume that our graphs have this property. We will assume further that all Reeb graphs are constructible, defined as follows.
An -space is constructible if it is homeomorphic to one constructed in the following manner. We are given a finite set of critical points , and a collection of spaces and . Further, we specify left attaching maps for and right attaching maps . We then define to be the quotient space
with respect to the identifications and . We define the function to be the projection on the second factor.
A constructible -space is a Reeb graph if all the and are discrete, finite sets of points.
In particular, a Reeb graph can be encoded by the combinatorial data of a finite graph , and a function defined on the vertices . This can be extended to the edges linearly, and in this case, we require that no adjacent vertices have the same function value. We call the number of edges incident to that have higher values of its up-degree, and define the term down-degree symmetrically. We always assume that a vertex of both up- and down-degree 1 is replaced with the relevant edge; this way, every vertex in the Reeb graph is a critical point. A vertex is local maximum (local minimum) if it has up-degree (down-degree ). Likewise, a vertex is an up-split (down-split) if it has up-degree (down-degree) strictly larger than . Note that a vertex can be both an up-split and a down-split; or an up-split and a local minimum, etc., although this does not happen for a generic Morse function on a constructible -space
In this paper, we will primarily restrict our attention to Reeb graphs with fairly strong genericity assumptions in order to simplify proofs. Our notion of genericity will involve two distinct criteria. The first is when we have more than one critical point at a given function value; if no such pair exists, we say the graph is function-generic. The second possible violation is if the vertex is of a type not seen in the case of Morse functions on manifolds. The four kinds of vertices which can appear in this case, with their (down-, up-) degrees specified, are local minima (0,1); up-forks (1,2); down-forks (2,1); and local maxima (1,0). If a Reeb graph has only vertices of these four types, it is called Morse-generic. If a Reeb graph is both function- and Morse-generic, we simply call it generic.
In the case of Morse-generic Reeb graphs, we have a strong characterizations of the critical points. We say a down-fork is an ordinary down fork if the two lower branches of are contained in different connected components of of the open sublevel set . Otherwise, we say is an essential down fork. The ordinary and essential up-forks are defined in the same way, using the open super-level set .
2.2 Smoothing and truncated smoothing
We now turn our attention to the (geometric) definition of smoothing given in [deSilva2016], and the truncated smoothing given in [chambers2020family]. Both smoothing and truncated smoothing were studied in the context of comparing two Reeb graphs, where the focus was on defining distances between the graphs. Both allow for the definition of an interleaving distance with desirable theoretical properties [deSilva2016, chambers2020family]. While these distances partially motivate our study, we do not directly use these interleaving distances in our work, but rather focus on the two operations themselves.
Let be a Reeb graph and let . Define by . We define the -smoothing to be the Reeb graph of ; we denote the corresponding quotient map by . Further, we denote the induced map ; i.e. is defined so that the diagram
commutes. See Fig. 2 for an example.
It is worth noting that smoothing is a functor, with further structure that we will not utilize in this paper; see [deSilva2016, deSilva2018] for details. We will focus on the combinatorial properties of smoothing and the map in our work, and will characterize how a Reeb graph changes under as varies. We begin with the following lemma that shows does not change the connected components; this result is implicitly referenced in prior work [deSilva2016] but never explicitly proven, so we include a proof here for completeness.
The induced map gives an isomorphism
Note that because , functoriality says that we need only show that and are isomorphisms as .
From Theorem 23.6 of Munkres [munkres2000topology], a finite Cartesian product of connected spaces is connected. So for each connected component , is connected, and thus is an isomorphism.
The map is surjective because by definition of a quotient map, is surjective. We next show is injective. By Munkres Exercise 23.11 [munkres2000topology], since our quotient map has connected, then for each connected component of , is connected. This implies that two connected components of cannot map to the same connected component of under without a contradiction. Thus is injective, finishing the proof. ∎
Truncated smoothing is a more recently developed variation of the smoothing functor, which intuitively smooths and then “chops off” tails in the smoothed graph [chambers2020family]. More formally, define a path in the Reeb graph to be a map . This path is monotone increasing (resp. decreasing) if (resp. ) for all . We also call these up- and down-paths, respectively. The height of a monotone path is . Let be the set of points of that do not have a height up-path and let be the set of points of that do not have a height down-path. The truncation of Reeb graph is defined by
so that we keep only the subgraph of that consists of the points that have both up-path and down-path of height . The truncated smoothing of Reeb graph is defined to be
See Fig. 3 for examples.
The truncated smoothing operation inherits many of the useful properties of regular smoothing. First, is a functor and we have a map for any . Note that we abuse notation and write since this map is a restriction of the map .111This map is denoted in [chambers2020family].. We note also that , since it is shown in [chambers2020family] that if and is connected then is connected.
2.3 Persistent Homology
In this section, we give a brief introduction to extended persistent homology. We assume that the reader is familiar with standard homology and direct the reader to [munkres2000topology, Hatcher] for further details, as well as to [Oudot2015] for more details on (extended) persistent homology.
First, given any filtration of topological spaces
For the expert, we note that we assume that our homology is computed over a field so that the resulting spaces are indeed vector spaces. We define a persistence module to be a sequence of vector spaces and linear transformations of the form
An interval module is a persistence module where is the trivial vector space for , is the 1-dimensional vector space otherwise, and the linear maps are isomorphisms when possible and 0 otherwise. By [Zomorodian2004] (see also [Oudot2015, Thm. 1.9]), any pointwise finite dimensional persistence module can be written as a direct sum of interval modules
This representation is unique up to isomorphism, so we can draw the information as a persistence diagram: a multiset of points in the upper half plane. For technical reasons related to computing distances, we also consider every diagram to have trivial points on the diagonal . We often refer to a point in the persistence diagram as representing a class born at and dying entering .
Given a constructible -space with critical set , the (standard) persistence diagram is defined to be the above decomposition for the filtration
for the (closed) sublevel sets . Here, the maps on homology are induced by inclusion for . When necessary, we specify the dimension of homology used by calling this a -dimensional diagram.
In the case of a Reeb graph , the only interesting portions of the standard persistence diagrams are that local minima of the form give rise to points in the 0-dimensional diagram with an infinite class for the global minimum of each connected component; and essential down forks of the form give rise to points in the 1-dimensional persistence diagrams (i.e., the loop is born at the top). Clearly, there is more obvious homological structure in these Reeb graphs, so we instead turn to the extended persistence diagram [Agarwal2006, CohenSteiner2008], defined as follows.
We extend our ordinary persistence module by using the relative homology of the super level sets. Write for the super-level sets. Note that ; is a discrete set consisting of the points with function value at the global maxima; and . Further, , so we have a map . Thus, we build the extended persistence module
We can decompose the resulting persistence module as before, but represent the intervals slightly differently by associating them to the indices and for where the interval starts and ends, while encoding whether these endpoints happen in the first or second half of the persistence module by putting them in a different sub-diagram. In a Reeb graph, the resulting extended persistence diagram can be decomposed into four sub-diagrams [Agarwal2006].
First, there are classes which are born and die in the first half of the module; we represent these by so-called ordinary persistence points since they correspond to finite-lifetime points which would show up in the traditional persistence diagram. In the case of a Reeb graph, these only appear in dimension 0, and the ordinary sub-diagram, , contains a point with for each non-infinite bar in the traditional persistence diagram.
The second type of points come from bars in the persistence module which are entirely contained in the second half of the diagram. We represent an interval lasting from at in the extended persistence diagram, noting that so these points are always below the diagonal. In the case of Reeb graphs, these kinds of points can only appear in dimension 1, so we call this the relative sub-diagram, .
The last kind of intervals that appear in the persistence module are those that begin at and end at ; we then include a point in the extended diagram at . In this case, we can have either or so points can be both above and below the diagonal in the diagram. In the case of Reeb graphs, these kind of intervals can appear in both dimensions 0 and 1, so we have the extended sub diagrams and .
In the case of a Morse-generic Reeb graph, we have a complete pairing of the vertices with respect to the extended persistence diagram. Each point in the diagram corresponds to a connected component of the Reeb graph, born at the global minimum value in the first half of the module, and dying at the global max value in the second half of the module. Each point in the diagram corresponds to a loop in the Reeb graph, born in the first half of the persistence module at the highest function value vertex in the loop, and dying in the second half of the persistence module at the lowest function value vertex. Note that these are always essential up- and down-forks. Points in and correspond to pairs of vertices: local minima with ordinary down forks in the first case and local maxima with ordinary up forks in the second case. See Figure 4 for an illustration where symbols corresponds to the pairing induced by the extended persistence diagram.
When we lose the Morse-generic assumption, this pairing still is present but is a bit more subtle. In particular, it is possible to have a vertex in the Reeb graph which contributes to more than one type of persistence point. More precisely, an up-split with up edges will contribute to more than one different persistence points; down-splits behave analogously. For this reason we will state our main theorem in the context of generic Reeb graphs, although with some bookkeeping it can be modified to the case of non-generic Reeb graphs.
2.4 Bottleneck distance
The bottleneck distance first arose in the context of persistent homology, where it was used to assess stability in persistence diagrams [Steiner2005]. Intuitively, this distance between diagrams considers all pairings of points of the same dimension and type in the two diagrams and calculates the maximum distance among all matched pairs under the norm. Note that points are also allowed to match to the diagonal to compensate for the potential of having a different number of off-diagonal points in the diagrams under consideration. In a sense, we can think of this as overlaying the extended persistence diagrams of both graphs and then matching points either to a point of the same type, or to the diagonal.
Let be two Reeb graphs. We define the bottleneck distance between their extended persistence diagrams and as
where is a bijection between the multiset of points of and (including points on the diagonal), and the bijection must match points of the same type ( to , to , etc.)
See Fig. 5 for a visual representation of the bottleneck matching between two persistence diagrams.
3 Geometric analysis of smoothing
In this section, we analyze the smoothing operation more carefully and precisely characterize the behavior of the critical set during the smoothing based on which part of the extended persistence diagram it contributes to. Geometrically, this implies that smoothing eliminates cycles whose height is less than , and that smoothing eventually results in the graph becoming a forest. The proof of this theorem depends upon a more detailed geometric classification of how smoothing affects down- and up-forks.
We first need the following lemma, proven in [deSilva2016], which specifies properties of the projection map works in smoothing:
[deSilva2016, Lemma 4.22] Let be the projection map onto the first factor. Then the map restricts to a homotopy equivalence .
This lemma allows us to completely characterize vertices in the smoothed Reeb graph by looking at inverse images from our original Reeb graph rather than needing to work with the thickened version, . In particular, we can investigate at function value to characterize when there is a vertex at that value, and what its up- and down-degrees are.
For the sake of notation, denote by and the closed and open intervals of width , respectively. Taking the limit of Lem. 3.1 over intervals as , we have that ; this gives the points in at the function value . Because is constructible, for a small enough the points in immediately below are elements of , while the points immediately above are elements of . Intuitively, the points at are regular if there is exactly one up and one down path emanating from them; deviating from this results in that point being a vertex.
To determine which elements of these sets correspond to a vertex, we can keep track of the attaching maps as follows. First, note that by assumption, and intersect the same collection of critical values of , thus by constructibility, the map induced by inclusion is an isomorphism. The point of passing to this larger interval is that now and are both contained in it, so we have the diagram
In , the number of points at is in the bottom set, the lower edges at left, and the upper edges at right. So and give attaching information for the lower and upper edges, respectively. Further, there is a vertex at function value if or (or both) are not isomorphisms. See Fig. 6 for an example of these interval representations. We will use this setup repeatedly in the next section to characterize vertices and attaching information to understand how the vertices move in the smoothed version of the Reeb graph.
3.1 Smoothing Reeb Graphs
Next, we consider how to completely determine the combinatorial structure of . In fact, prior work considered this: [deSilva2016, Corollary 4.25] claims that the set of critical points of , where is the set of critical points of the Reeb graph . In fact, there is some nuance to this issue, as is a subset of , but (assuming we reduce vertices of degree 2 and keep only vertices of degree 3 or more as critical values) it is never equal to the set .
In this paper, we work exclusively with generic Reeb graphs and values which keep the graph generic as well. However, even with a generic input Reeb graph, particular choices of can result in non-generic , as smoothing intuitively will move vertices to new function values where they have the potential to ”bump into” other vertices. Our proof can be adapted to work for non-generic Reeb graphs and the attaching information described in the last section still completely determines the combinatorial changes; however, we limit our proof to only generic graphs in order to simplify the case analysis.
For a generic Reeb graph , we completely characterize the critical set of as follows:
Let be a generic Reeb graph with vertex set . We denote the critical set and assume that the vertices are sorted so that . Let be a value such that for any . Let be the subset of vertices where each contributes to a point in which has lifetime at most . Then there is a bijection
and is generic.
We will explicitly construct the map , and show it is a bijection. Given a vertex with , this is done by finding a vertex in with either function value or depending on the type of , and showing that this matching between the vertex sets is unique.
First, assume is a local minimum. Building the diagram of Eq. 2 with with corresponding maps and , we see that the connected component of in is not in the image of . Thus, there is a vertex in with , and we define . Further, the above construction gives us that the down-degree of is zero. Noting that was chosen sufficiently small in Eq. 2 so that there is no additional vertex between and , we also have that is an isomorphism. Therefore, the up-degree of is 1, implying that is itself a local minimum vertex.
We can use the symmetric argument with in the case where is a local maximum to find that its connected component is not in the image of , and thus there is a local maximum vertex at height in . We again define
Next, assume that is a down fork which is not part of a pair with lifetime less than ; i.e. . In this case, we start by building Eqn. 2 with . Consider the two lower edges of , and the points and on these edges at height . We first show that and are in different connected components of . If they are in the same connected component, there is a path in connecting the two points, let be the vertex with lowest function value on this path. By choice of , we know that there is no vertex in the interval , so the height difference between and is at most . However, in this case, extended persistence would pair the vertex (the lowest possible vertex) with as a pair which has lifetime at most . This means , contradicting our assumption.
As we now know that and are in two different connected components of , there are two elements of whose image under is the component containing . Because there are no vertices of in by our choice of , there is a single component in mapping to the component of under . Thus, we have a vertex in at height with down degree 2 and up degree 1. We set . See Fig. 6 for an example.
By symmetry, we can use this same argument with for an up fork vertex, so that for any up fork in , there is an up fork in at height , and set .
Now that has been defined for all required vertices, show that is a bijection. For surjectivity, let be a vertex in at height . By assumption on our Reeb graphs, this must have at least one of the up or down degree 0 or 2 (i.e., not both degree 1). Assume first that the down degree is 0 and again setup the diagram of Eqn. 2. Because of vertex , we must have that there is a connected component which is not in the image of ; but this is exactly the requirement for having a local minimum vertex at height in , and so . A symmetric argument can be made for local maxima.
Similarly, assume is a vertex in at height which has down degree 2. Then there are two connected components in whose image under is the component represented by , and thus there is a vertex in at height with . Again, a symmetric argument can be used in the case of up-degree 2; thus is surjective.
For injectivity, we note that by assumption, our Reeb graph is generic, meaning that we cannot have two vertices at the same height. If there was a vertex at height and two vertices with , this would imply that one must be at height and the other at height ; without loss of generality assume these are and respectively. But then our value , where and are critical values, contradicting our assumptions on . Thus is a bijection, and is generic. ∎
Our next proposition proves that does not affect the persistence pairing of critical points in the Reeb graph.
Given a generic Reeb graph, for vertices which are paired under extended persistence, and are paired and of the same type in the extended persistence diagram of .
We break this into cases based on the four types of paired points in the extended persistence diagram. In each case, we assume we have paired vertices of , and . For notation, we assume , , and .
From prior work describing extended persistence of Reeb graphs [Agarwal2006, Cohen-Steiner2009a, Cole-McLaughlin2003] we have a simple pairing for all the vertices. Namely, the global maximum in any component is paired with its corresponding global minimum ( points). A down fork is paired with the highest up fork that spans a loop with it in the graph, if one exists ( points). Any remaining downfork is paired with the higher of the two minima for the two components in its sublevel set ( points); similarly any remaining upfork is paired with the lower of the two maxima of the two components of its superlevel set ( points).
If the pair is in , we have a point in the diagram. Further, and are the global minimum and maximum respectively of a connected component of . From Theorem 3.2, we know that has vertices and at height and . Moreover, any other critical point in this component of the original graph has value , so any critical point in the related component of is between and . This implies and are the global maximum and minimum of a component of the smoothed Reeb graph, and hence they are paired by extended persistence in .
If the pair is in , we have a point in the diagram, where is a local minimum, and is an ordinary down fork. Let and be the connected components of below . Let and be the minimum function value vertices of these two connected components respectively. Since is paired with , this implies that . After smoothing by , by Theorem 3.2 we know that has local minimum at the critical value , local minimum at , and downfork at . Consider the connected component(s) below in . Note that by construction, the Reeb quotient map of the thickened space maintains connected components, so
gives an isomorphism.
The sets and must be disconnected in because and are disconnected in . Thus, by the isomorphism , they must map to different connected components below . First, this implies that must be an essential downfork. Second, we must have and in these connected components, and they must be the minimum function value vertices of each of the connected components. We know that , so this implies that is paired with .
The argument for is the same as that of , with super- and sublevel sets switched. Thus our final case is when and are paired in . We have already shown that all points in , and from stay paired under , so all that remains is to be sure and cannot pair with any other points from . Since is the highest up fork that spans a loop with , there are exactly two connected components in which attach to and in via the inclusion maps into . Therefore, by Lemma 3.1, we have the commutative diagram
and thus there are exactly two connected components in which attach to and in via the inclusion maps. Therefore, and will remain paired in . ∎
Again, the prior result is only proven here for generic graphs, but the proof can be adapted to work in non-generic graphs as well. In that case, a vertex of the graph simply corresponds to multiple critical values, so that any vertex of degree in the graph will appear in persistence pairs in the diagram; the case analysis correspondingly becomes more complex.
Finally, we arrive at the true main result of this section, where we can use the characterization of the movement of the critical points to keep track of movement in the persistence diagram.
Consider a generic Reeb graph , a value , and the bijection as in Theorem 3.2. For every point in the persistence diagram of , the corresponding point in the persistence diagram of located (depending on its type) as follows:
3.2 Truncating Smoothed Reeb Graphs
We next consider truncated smoothing, and prove an analogous characterization of its impact on the persistence diagram. We know that , since their groups are isomorphic [chambers2020family]. We begin by proving an analogous result on .
Given and then induces an isomorphism .
Consider a loop in with critical values and . By Theorem 3.2, the loop either disappears under the smoothing functor (if its height is less than ) or is still present in the smoothed graph (under the bijection ). By Prop. 4.3 and Lemma 4.4 of [chambers2020family], for , truncation of the smoothed graph will not reach any essential fork. Hence each loop maps to a unique loop in truncated graph, under the natural inclusion from , and the resulting isomorphism follows since no new loops can be created under truncation. ∎
The goal of the following proposition is to see how truncation affects the four types of persistence points (, , and ) after smoothing, with assumptions to ensure that truncation does not change the topology of a given graph. See Fig. 8 for a visualization of the theorem, which constrains the effect of truncated smoothing on the different points of the persistence diagram.
Fix . Suppose is a generic Reeb graph. For every point in the persistence diagram of the Reeb graph, the corresponding point in the persistence diagram of the truncated smoothed Reeb graph is
if the new point is on the same side of the diagonal as , and the point is removed completely if not.
By Lemma 3.5 and our assumptions on , we know that truncation on the smoothed graph moves any critical point that a local minimum up by , any local maximum down by , and leaves up- and down-forks unchanged. The proof then follows from the same bookkeeping as in Corollary 3.4, after tracking these new critical values. ∎
4 Inverse problems on Reeb graphs
We are now in a position to make use of this characterization to provide a solution to the time varying inverse problem in a particular restricted setting. In the most general setting, our problem can be stated as follows. We assume we are given a path in persistence diagram space, colloquially known as a vineyard, which is data . We assume this path is a continuous map under the bottleneck distance, and that we have also been provided with an initial Reeb graph . The goal is to find a path in Reeb graph space, continuous with respect to the interleaving distance [deSilva2016], , for which and .
We restrict our view to essentially creating a notion of piecewise linear paths, defined by linear interpolation between the path defined at discrete times. For the sake of notation, we will denote , and for the Reeb graph path we will construct, . Geodesics in persistence diagram space are defined by matchings arising from the bottleneck distance computation. While a minimum cost matching is not unique in this setting, say we have a matching , then the geodesic between diagrams and is given by sliding each point at constant speed along the line between and .
Following Prop. 3.6, we define the map on diagrams by defining a map on each point as follows:
Then the map is defined by
We can abuse notation and also view this construction as a matching on the points of related diagrams themselves. Given an and , all points of the same type (, , etc) move in the same direction. Thus, for a given direction vector , which depends on , , and point type, we can define an updated single-type diagram by:
Given this updated diagram, we have a matching defined by:
With this setup, we begin with the following theorem, which proves directly that our truncated smoothing map in fact is at least locally bottleneck optimal, motivating our use of this operation for inverse problems.
For with magnitude
and a diagram with only points of a single type, the matching induced by achieves the bottleneck distance between and .
Note that by assumption, no points of have been removed in , since the magnitude of cannot reach the diagonal. As a result, the matching is a bijection between only the off-diagonal points in the diagrams sending to .
We will prove this theorem by contradiction. Assume that is not a bottleneck matching; in particular, this means that . So there exists another matching that achieves bottleneck distance. Specifically, for all , . If any point matches to the diagonal under , then by assumption on , we have that , contradicting our assumption. So we may assume every point will match to an off-diagonal point under .
We can think of the matchings and as being defined as edge sets and in a bipartite graph with vertex sets given by the off-diagonal points of and , respectively. Then we can consider the symmetric difference of the matchings as the set of pairs for which or but not both. By [West2001, Lem. 3.1.9], every component of the symmetric difference of two (not necessarily perfect) matchings is a path or an even cycle. Because and are perfect matchings, every vertex in has degree 0 or degree 2; hence, each component must be an even cycle, since a path requires degree 1 vertices.
Choose one of these cycles . We will find a pair in this cycle for which , thus showing the required contradiction. The choice of will be dependent on the direction of , however, so we split our work into four cases depending on the angle of : (R) ; (U) ; (L) ; and (D) .
Assume case (R), so is essentially going to the right. In this case, the change in the -axis determines the bottleneck matching distance for . Let be the point in cycle with smallest -coordinate. Define , and enumerate the cycle starting from , so with for and . Because we chose for its -coordinate and by assumption on the direction of , we know that where denotes -coordinate. Then in this case, we have
giving us the desired contradiction. See Fig. 10 for an example.
The remaining cases follow the same structure as (R) with mild tweaks. For case (L), we set to be the point with largest -coordinate. For the cases (U) and (D), we choose to be the point with smallest and largest -value, respectively. ∎
Next, we turn our attention to when a sequence of diagrams (i.e. a vineyard) is realizable by Reeb graphs using smoothing and truncation operations.
A sequence of input diagrams is admissible if for every , there exists a pair with , such that .
Assume is admissible and Reeb graph is given so that . Then there is a sequence of Reeb graphs for which .
The proof proceeds by induction. For , and since the ’s are admissible, by Proposition 3.6 we have a pair such that is the diagram of i.e . We can then define the first Reeb graph to be , which has the property that by constructing the truncated smoothing.
Next we for any , we assume we have a Reeb graph