The Fiber of the Persistence Map
In this paper we study functions on the interval that have the same persistent homology. By introducing an equivalence relation modeled after topological conjugacy, which we call graph-equivalence, a precise enumeration of functions with the same persistent homology is given, inviting comparisons with Arnold’s Calculus of Snakes. The equivalence classes used here are indexed by chiral merge trees, which are binary merge trees where a left-right ordering of the children of each vertex is given. Enumeration of merge trees and chiral merge trees with the same persistence makes essential use of the Elder Rule (a criterion for pairing critical points), which is given a new proof here as well.
1. Introduction and Acknowledgements
Let be a piece-wise linear function on a finite simplicial complex. Persistence is a new type of geometry that generalizes Morse theory by quantifying the lifetimes of homological features of , when filtered by sub-level sets of . The lifetime of a homology class is captured using an interval in and the collection of the lifetimes of all homology classes is captured using a collection of intervals called the barcode or the persistence diagram associated to —the latter being described in terms of a configuration of points in the extended plane . The persistence map is the map that takes functions on to their associated persistence diagrams.
Although persistence has proved remarkably useful in data science [2, 7, 10], the analytical study of the persistence of functions has received little attention. We mention two candidate problems in this vein.
The Realization Problem: What is the image of the persistence map for each topological space and particular type of function ?
The Closed Formula Problem: Is it possible to determine the persistence diagram associated to a semi-algebraic set that is filtered by a semi-algebraic function purely in terms of the equations involved?
In this paper we take up the first question in the simplest possible case—functions on the interval—and go further by giving a complete characterization of the fiber of the persistence map. Our main result, Theorem 6.11, proves that, after imposing a graph-equivalence relation on functions with local minima at and , the fiber of the persistence map is finite and given by the formula
We note that our graph-equivalence relation is a restricted version of topological conjugacy, so in some respects this work is similar to Arnold’s Calculus of Snakes . However, unlike Arnold’s work, the quantities depend on the particular arrangement of points in the persistence diagram , so the number of realizations is not purely a function of the number of critical points, but also requires specifying the number of nested, paired critical points.
This nesting of paired critical points is captured via the Elder Rule, which provides a way of extracting persistent using persistent . Originally described using the merge tree associated to a function , the Elder Rule also promises to deliver the decomposition of a persistent vector space (Definition 2.4) freely generated by a persistent set (Definition 2.2) into indecomposables—a decomposition that exists in the finite-dimensional case by Crawley-Boevey’s Theorem 2.8. In the restricted setting of Morse sets (Definition 3.5), we prove that this promise holds and give a new proof of the Elder Rule in Theorem 3.8.
In pursuit of our main theorem, Theorem 6.11, we do other things as well: We count the number of merge trees that have the same barcode in Theorem 4.8, and introduce the notion of a chiral merge tree, which is a merge tree with a handedness decorating each of its edges.
The first real step forward on this problem happened in conversations with Hans Riess, who was an undergraduate at Duke at the time. It was there that the author developed a method for constructing at least one function realizing any suitable barcode. A full characterization of the fiber of the persistence map was obtained after conversations with Yuliy Baryshnikov at ICERM and with John Harer at Duke. Yuliy Baryshnikov introduced the concept of the chiral merge tree to the author, which the author used to create multiple realizations of a single barcode. Finally, John Harer helped cinch the counting result by pointing out the restriction that the Elder Rule imposed on chiral merge trees. The paper might not exist in its present form if these conversations had not occurred.
2. The Persistence Map
Let denote the partially ordered set of real numbers, where if and only if is a non-negative real number. We also view as a category with one object for every real number and with a unique morphism whenever .
Let be any category. A persistence module valued in is a functor
In more detail, a persistence module valued in specifies an object of for every time and a morphism for every relation . This collection of morphisms must satisfy and for all . We denote the category of persistence modules valued in by .
We now note some important special cases of this definition and the corresponding special language.
2.1. Persistent Sets
Definition 2.2 (Persistent Sets).
Let denote the category of sets and set maps. A persistent set is a functor
Example 2.3 (Persistent Path Components).
Suppose is a topological space and is a function. We can study the persistent set that assigns to every value the set of path components and to every pair of numbers the map on path components.
2.2. Persistent Vector Spaces
When every set has the structure of a vector space and when every map is a linear transformation of vector spaces, then we have the notion of a persistent vector space, which is called a persistence module in the literature.
Definition 2.4 (Persistent Vector Spaces).
Let denote the category of -vector spaces and -linear transformations. A persistent vector space is a functor
We often omit mention of the field since it is arbitrary for this paper.
Remark 2.5 (Terminology).
To the author’s knowledge, the terms “persistent set” and “persistent vector space” are being used here for the first time. The reasons for bringing this new terminology into circulation is twofold: Firstly, the pronounciation of “persistent set” sounds better than “persistence set,” which would be the logical shortening of “persistence module valued in .” Secondly, an attempt to maintain grammatical consistency suggests that we use “persistent vector spaces” following “persistent homology,” rather than “persistence modules.”
Let be a persistent set. We get the persistent vector space freely generated by , written , by letting be the vector space freely generated by and letting the shift maps of be the linear maps freely generated by shift maps of .
Let be an interval, i.e. if and , then . Associated to any interval in is a persistent vector space that assigns to every and assigns the zero vector space to . For pairs of numbers , both of which are in , then the associated shift map is the identity map, otherwise it is the zero map. In the literature this is called the interval module .
Note that if the interval has , then the interval module cannot be freely generated by any persistent set.
2.3. Persistent Homology
Although this narrative is anachronistic, persistent homology can be viewed as resting on the following technical result of Crawley-Boevey, which provides a finite characterization of any finite-dimensional persistent vector space.
Theorem 2.8 ([5, Thm. 1.1.]).
Any pointwise finite-dimensional persistent vector space (i.e. persistence module) has a uniquely associated direct sum decomposition into interval modules.
Interval modules are indecomposable representations of the poset . In the setting where our persistent vector spaces are pullbacks of representations of -type quivers, the above result follows from earlier work of Gabriel .
The intervals that appear in the direct sum decomposition of a persistent vector space guaranteed by Theorem 2.8 define the barcode associated to , written . In general, a barcode is any multi-set of intervals ; here is the number of repetitions of the interval . We will often work in the generic setting where or , and drop this extra information. We denote the set of all barcodes by .
Remark 2.11 (Persistence Diagrams).
Given a barcode , we can associate a collection of points in the extended plane . This is done by taking each interval to the coordinate-pair where and . This is called the persistence diagram  and it encodes the rank function of a persistent vector space as follows: If is the barcode associated to a persistent vector space, then the rank of the map is given by the number of points in the persistence diagram up and to the left of the point . It should be noted that persistence was initially defined in terms of images of the shift maps, see , and the language of barcodes appeared somewhat later . We pass back and forth between these two perspectives freely, using whichever representation works best in the moment, but we remark that work of Amit Patel  indicates that persistence diagrams can be defined more generally than barcodes.
Associated to a function is the persistent homology in degree , which is the persistent vector space
Here denotes the homology functor in degree taken with coefficients in a field .
Since is the vector space freely generated by , we can think of as the persistent vector space freely generated by the persistent set described in Example 2.3.
Suppose is a compact simplicial complex and let denote the space of continuous functions on , equipped with the sup-norm. The persistence map in degree is the map that takes each function to its persistent homology in degree and then its uniquely associated barcode:
This map is actually -Lipschitz, after one equips the category of persistent vector spaces with the interleaving distance, see  for an overview. Moreover, the last map, which takes a persistent vector space to its barcode, is an isometry once the space of barcodes is equipped with the bottleneck distance . However, we will not make explicit use of these metrics, and only refer to them in passing, e.g. Remark 4.6 and Remark 4.10.
3. The Elder Rule for Morse Sets
In this section we give a treatment of the Elder Rule that differs in flavor from the description given in [7, p. 150]. In our language, the Elder Rule allows us to associate a barcode directly to a persistent set , without first considering the persistent vector space that freely generates and by applying Theorem 2.8. The fact that these two ways of associating a barcode to a persistent set agree is the content of Theorem 3.8.
Our treatment makes use of standard constructions in the theory of partially-ordered sets. To that end, we recall some standard results and terminology associated to posets. An up-set is any set where and jointly imply that . A principal up-set is any set of the form . A chain in is a subset of that is totally-ordered upon restriction of the partial order to . A chain is maximal if whenever we have two chains , then .
Let be a persistent set. Consider the set . We define a partial order on by declaring whenever is non-negative and .
Any functor can be thought of as a sheaf of sets in the Alexandrov topology. This is the topology whose basis is given by principal up-sets . The construction above is a special case of the observation that the étalé space associated to a sheaf over a poset is also a poset.
Recall that a map of posets is one that is order-preserving. Equivalently, a map of posets is a continuous map in the Alexandrov topology.
Let be a map of posets. Whenever we write we mean . If , then we say is supported at . The support of is defined to be
Note that need not be closed in with the Euclidean or the Alexandrov topology.
We now define what it means for a chain to be older than another chain.
Let be a map of posets and let and be two chains in . We say that is older than if there exists a such that for all .
We view the Elder Rule as a method for partitioning the poset associated to a persistent set into chains. In order to describe this method inductively, and without special case analysis, we introduce the notion of a Morse set, which is a special case of a constructible persistence module valued in , see [13, Def. 2.2]. Intuitively speaking, a Morse set is an abstraction of the persistent set of path components associated to a Morse function on a compact, connected manifold; cf. Example 2.3.
Recall that a constant functor is a functor that assigns to every object in its domain of definition a single object and sends every morphism to the identity map on that object.
Let denote the category of finite sets and set maps. A Morse set is a functor along with a finite sequence of times so that
is naturally isomorphic to the constant functor with value for ,
for all the fiber has cardinality 0,1, or 2, and
each element is contained in a unique, oldest maximal chain in the associated poset defined in Definition 3.1.
Moreover, we assume that the set of times is the minimal set if times making the above statements true. We let denote the set of isomorphism classes of Morse sets.
Definition 3.6 (The Elder Rule).
Let be a Morse set, constructible with respect to the times . The Elder Rule gives the following inductive chain decomposition of the poset described in Definition 3.1. For any poset map , we define .
Let be the poset and let be the unique, oldest maximal chain containing the element .
Let , i.e. the poset with the chain removed.
By the fourth and fifth hypotheses of Definition 3.5, has a unique element that is contained in a unique, oldest chain .
Let to be the set of intervals associated to the chains via projection along . This defines the barcode associated to by the Elder Rule.
In Figure 2 we have the same persistent set , but with two different partitions into chains. By considering the projection of the chain decomposition, we get two different barcodes and , depicted to the left and right, respectively, in Figure 2.
If one considers the persistent vector spaces and generated using the intervals in their respective barcodes, one can see an important distinction between the two: Only has the property that it is isomorphic to —the persistent vector space freely generated by the persistent set . To see that note that the linear map from to is zero for and non-zero for .
We now prove that the Elder Rule is true. To the author’s knowledge there is no precise proof of the Elder Rule in the literature. Although the correctness of the Elder Rule is in many cases guaranteed by the correctness of the persistence algorithm, we provide a proof of a different flavor.
Theorem 3.8 (The Elder Rule).
Let be a Morse set. Let be the persistent vector space freely generated by . If is the persistent vector space associated to the barcode generated by the Elder Rule, then
Note that for each , a basis for is given by the chains supported at . This basis is also well-adapted to the shift maps internal to , because always takes to , with the understanding that if in , then in . With this basis the shift map is diagonal for every .
We now turn our attention to the persistent vector space . The shift maps do not take chains to chains; rather, they take certain up-sets associated to chains to up-sets with a re-indexing rule. To introduce some more notation, let ; this is the up-set associated to the chain . Note that spans for all , again with the understanding that if in , then in . The only way in which the set fails to be a basis is when there exist chains and with associated upsets and that intersect at time and then the same vector appears twice in the set . We correct this by adopting the convention that whenever we choose the lower index . With this choice of basis for all , one can see that the linearized shift maps take each to where .
Using the convention that and hence , the change of basis map taking to is defined on chains supported at by
where is the number of chains in the decomposition of given by the Elder Rule in Definition 3.6. We now need to show that for all and that
If , then there is nothing to check. If , then we distinguish two cases based on . If and intersect at , then . To see this, note that if , then there is an older chain passing through , and hence is not supported at . In this case, the left hand side of Equation 1 reads
This in turn agrees with the right hand side of Equation 1:
The other case, where , implies that , which is exactly what is defined to be. This completes the proof. ∎
4. Merge Trees and the Elder Map
The poset in Definition 3.1 serves as a total space for any persistent set . When is constructible, e.g. Morse, we can associate a different total space to called the merge tree associated to . Below we define what a merge tree is in the abstract, without reference to a function on a space or a persistent set.
A merge tree consists of the following data:
A connected, locally-finite, contractible, one-dimensional cell complex with a distinguished edge without compact closure. In other words is a compact rooted tree with , the one vertex incident to , as the distinguished vertex.
A continuous map , where has the Euclidean topology, that restricts to an orientation-preserving homeomorphism between each open edge and , where has the restriction of the standard orientation on .
Note that the second condition implies that every edge has a length, given by its diameter . Also, we can define the child of a vertex as any vertex connected by an edge to with . We say two merge trees and are isomorphic if there is a map of trees making .
Every Morse set has an associated merge tree .
We first form the disjoint union
and then we impose the equivalence relation that is identified with if and only if . This defines the total space . Projection onto the second factor defines the map . ∎
It is clear that the hypotheses of a Morse set make into a binary tree. We set this aside as a special definition.
If is the merge tree associated to a Morse set, then we call it a Morse tree. Let denote the set of isomorphism classes of Morse trees.
The set of Morse sets and the set of Morse trees are isomorphic.
Note that Morse sets constitute a subcategory of and Morse trees constitute a subcategory of the over category . The lemma follows from the more general statement that these subcategories are equivalent, but we content ourselves with an argument at the level of objects.
Consider the merge tree associated to a Morse set . Naturally associated to is a persistent set . Since every sub-level set deformation retracts onto , which is , we have that . ∎
Lemma 4.4 implies that to every Morse tree we can associate a barcode, via the Elder Rule.
Recall that denotes the set of all possible barcodes. The Elder Rule defines the Elder map from the set of (isomorphism classes of) Morse Trees to the set of barcodes :
Remark 4.6 (The Elder Map is Lipschitz).
The Elder map is also -Lipschitz, and hence continuous, by using the interleaving distance on merge trees .
Remark 4.7 (A Sheaf-Theoretic Aside).
Let be a Morse tree. Let be the constant sheaf on . Consider the pushforward along the map , which is constructible with respect to the stratification of given by the set of times used in the definition of a Morse set. Using results of , we can describe this sheaf in terms of a zig-zag module, which in this case is a sequence of vector spaces and maps of the form
We note that maps pointing to the left are always invertible, so we can regard this is as a persistent vector space. Consequently, appealing to Theorem 3.8, the Elder Rule provides a decomposition of the pushforward of the constant sheaf along into indecomposable sheaves over .
We now describe the fiber of the Elder map .
Let be a barcode where and where for we have a strict containment with . We denote the set of intervals in containing by and let denote the cardinality of this set. The number of merge trees realizing the barcode is
We construct a Morse tree inductively from and enumerate all the possible choices along the way. We note that the intervals in are already ordered by their right-hand endpoints via its index. This is why we start by setting to be the interval , equipped with the Euclidean topology.
To construct from we must select a place to attach the interval to . Note that every interval has a right-hand endpoint that is to the right of , and hence has already been used in the construction of . Once we select an —and we note that there were choices—we can define
where . The equivalence relation identifies the point with the right-hand endpoint . Once , we will have constructed a tree using the intervals in . The map is simply the map that takes each to .
Note that every possible merge tree constructed in this way picks out a unique isomorphism class. To see this, observe that if we have chosen , then by hypothesis . Attaching at produces an edge of length , which is an isomorphism invariant. This completes the proof of the theorem. ∎
Remark 4.10 (Stratifying the Elder Map).
By using the persistence diagram perspective, we can think of barcodes in terms of configurations of points in . Note that the formula given in Theorem 4.8 defines a constructible function, which is constant on strata in a stratification of the space of barcodes . This stratification is defined using the number of points in the persistence diagram and the containment relations used to define , which can be phrased in terms of linear inequalities. For example, in Figure 4 we see two barcodes, viewed as persistence diagrams, lying in different strata. If we moved the upper most point in the right persistence diagram out of the shadow of the lower point, then we would move from one piece of the stratification of to another piece. Strata appear to be indexed by isomorphism classes of the containment relation poset associated to , which is the poset on given by containment of intervals. Whether every finite poset on elements with a unique maximal element can be realized as the containment relation poset of a barcode satisfying the hypotheses of Theorem 4.8 is unclear at the moment, but may follow by embedding the Hasse diagram in the plane so that comparable elements are always up and to the left.
5. Ordered Persistent Sets and Chiral Merge Trees
Suppose is a continuous map to , in the Euclidean topology. For every real number we note that can be written as a disjoint union of closed intervals. Because is totally ordered, we can order the intervals appearing in from left to right. This implies that the persistent path components of are actually organized by a richer structure.
Let denote the category of totally-ordered sets and order-preserving maps. An ordered persistent set is any functor .
If is a continuous map, then set is totally ordered using the left-to-right ordering of the intervals making up the pre-image .
This, of course, begets the notion of an ordered merge tree and it’s generic version, the chiral merge tree.
An ordered merge tree is a merge tree with the additional data of specifying a total ordering of the edges connecting a vertex to its children.
If a vertex has two children and , then an order amounts to an assignment of left and right to the two children, where we use the convention that . If every vertex in has at most two children, then we call an ordered merge tree a chiral merge tree. Let denote the set of isomorphism classes of chiral merge trees.
Suppose is a piece-wise linear (PL) function where every critical point has a distinct critical value. Here a critical point means that is either a local minimum or a local maximum, which is characterized by the existence of an open neighborhood for which either or for all . The merge tree associated to then has the structure of a chiral merge tree since at most two intervals merge when crossing a critical value associated to a local maximum.
Defining the Chiral Elder Map via the composition
we have the following corollary of Theorem 4.8.
Let be a barcode satisfying the hypotheses of Theorem 4.8. The number of chiral merge trees realizing is
The proof follows the proof of Theorem 4.8 with the exception that instead of there being possibilities for attaching to , there are now possibilities, since we may attach to the left or to the right of every interval appearing in . ∎
Consider the barcode
Corollary 5.5 predicts that there are chiral merge trees realizing .
6. The Persistence Map for Functions on the Interval
Using the theory already developed, we can now characterize the fiber of the persistence map for suitably nice functions on the interval. Obviously, the fiber of the persistence map
is uncountable. However, if we introduce an equivalence relation called graph-equivalence and impose boundary conditions on the functions, then the fiber of the persistence map becomes finite and is indexed by chiral merge trees; see Figure 5.
We say two continuous functions are graph-equivalent if there is an orientation preserving homeomorphism such that .
As the name suggests, if two functions are graph-equivalent then their graphs and are homeomorphic in a level-set and orientation-preserving way. To see this, note that
This implies that the sub-level sets of and are homeomorphic for every , so in particular the persistent vector spaces
are isomorphic, so this equivalence relation is constant on fibers of the persistence map. However, the fact that is orientation-preserving also makes the following true.
If two functions are graph-equivalent, then they have isomorphic chiral merge trees.
Corollary 6.2 is one direction in a bijection that connects graph-equivalence classes of functions with chiral merge trees. In order to go the other way, from a chiral merge tree (CMT) to a function, we must delve deeper into the structure of CMTs by proving several technical lemmas.
A chiral merge tree has a naturally associated total order on its vertices.
As noted in Definition 4.1 a CMT is rooted at the vertex . Consequently, for any vertex in there is a unique shortest path from to . We can represent the path as either a string of vertices or a string of edges . Since every edge is labeled with an element of the set , we can associate to a unique sequence of letters where each indicates whether is a left incoming edge () or a right incoming edge () to Note that the sequence also uniquely determines the path and hence the vertex , with the number indicating the depth of in the tree.
Suppose we have a vertex , represented by a sequence , and another vertex , represented by a sequence . Since in general , we append a sequence of empty characters to whatever string is shortest. By using the rule that we can use the lexicographical ordered to order and . ∎
As an example, suppose we have three vertices, represented by , and . The ordering described would put .
For a general ordered merge tree it’s not clear how to specify a total ordering on the vertices. In particular, at a vertex with an odd number of children, it is not clear how to order this vertex with respect to its children. More concretely, this question amounts to a choice between orderings of the form and , neither of which is canonical.
One of the properties of this ordering is that leaf nodes are exactly the odd numbers between and .
Let be a chiral merge tree. Suppose is the number of vertices in . In the total ordering of the vertices provided by Lemma 6.3, the leaf nodes correspond to when is odd.
We use a recursive description of the enumeration given in Lemma 6.3. Our input is a full binary tree with every vertex labeled with a name. The algorithm takes in a binary tree with a distinguished vertex . If has children, then we call the algorithm again on the left sub-tree. Print the name of . Call the algorithm on the right sub-tree.
Note that the algorithm does not stop the recursive call until has no children, i.e. it is a leaf node. Thus the first node name that is printed is a leaf node. Popping out of this first, deepest level of recursion, we must print the name of the parent, which is the second name printed. The assumption that the binary tree is full guarantees that the next call on the right subtree is not empty, so another leaf node is printed next. This implies that leaf nodes are always printed at odd numbers.
We now explain why the order of names called in the algorithm above gives the enumeration described in Lemma 6.3. Suppose is a string of ’s and is the string describing the leaf node just printed by the algorithm above. In the lexicographical ordering the immediate successor of is . The immediate successor of is , where the number of ’s appearing to the right of is maximal for the given tree, thereby implying in the algorithm above that this vertex’s name is the next to be printed. ∎
We can now prove a reconstruction result.
To every chiral merge tree with at least three vertices there is a PL function whose chiral merge tree is .
Note that the number of nodes in a chiral merge tree must be odd. We apply Lemma 6.3 to obtain an ordering of the vertices, which we label as . By definition of a CMT, we also have real values associated to each of the vertices. To each vertex we can associate a pair of coordinates with and . Now connect each to with a straight line. This defines a PL function .
We check that the chiral merge tree of is . As the proof of Lemma 6.6 shows, when is odd represents a leaf node and represents an ancestral (parent, grandparent…) node, so . This implies that the line connecting to has positive slope when is odd and negative slope when is even. As a consequence, when is odd, the point is a local minimum and the point is a local maximum of the function . Note that the -value of each of these maxima and minima are ordered by the usual order on . Let denote the th vertex of the merge tree associated to . Note that the vertices have the same -order (in-order) and -values as the abstract chiral merge tree . This implies that the merge tree determined by is the same as . ∎
We say a continuous function is Morse-like if it is graph-equivalent to a PL function where every critical point is isolated and has a distinct critical value. In other words the graph of consists of a finite number of line segments, each of which have non-zero slope.
In general the converse of Corollary 6.2 is not true, but it is in a special case.
Suppose and are two Morse-like functions having both and as local minima. If and have isomorphic chiral merge trees, then and are graph equivalent.
Since the functions and are Morse-like and since graph-equivalence is transitive and preserves CMTs, it suffices to consider the case where and are PL functions. Let denote the common chiral merge tree of and and let denote the PL function associated to constructed in Proposition 6.7. We now show that is graph-equivalent to , which, by arguing symmetrically, must be graph-equivalent to . First we note that both and have critical points, which we label and . The only way for this not to be the case is if had a local max at either or , which we ruled out by hypothesis. Consequently, each and are the same type of critical point with the same critical value. It is thus easy to define an affine map making . Concatenating these affine maps together and using the fact that and we can define a PL homeomorphism take to . This proves that and are graph-equivalent. Arguing with in place of proves that and are graph-equivalent. Transitivity of graph-equivalence proves that and are graph-equivalent. ∎
Let denote the set of Morse-like functions on the interval with local minima at and , modulo graph-equivalence. Let denote the set of isomorphism classes of chiral merge trees. The map
is a bijection onto its image.
We can now have reached our main result.
Let and be the sets described in Corollary 6.10. If is a barcode where , with for all , and where every left-hand endpoint is distinct, then the number of graph-equivalence classes of functions in realizing is
where we have identified with the composition of followed by the Chiral Elder map .
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