General Persistence
4.5 Levelset zigzag persistence
Now, we consider a special type of zigzag persistence stemming from a function over a topological space. In standard persistence, growing sublevel sets of the function constitute the filtration over which the persistence is defined. In levelset zigzag persistence, we replace the sublevel sets with level sets andinterval setsand the maps going from the level sets to the adjacent interval sets give rise to a zigzag filtration. To produce a zigzag filtration corresponding to a level set persistence, we consider a PL-function on the underlying space of a simplicial complex and then convert a zigzag sequence of subspaces (level and interval sets) into subcomplexes. This is similar to what we did while considering the standard persistence for a PL-function in Sections 3.1 and 3.5.
Before we focus on a PL-function, let us consider a more general real-valued continuous function f :X →Ron a topological spaceX. We need a restriction on f that keeps all homology groups being considered to be finite. For a real value s∈Rand an intervalI ⊆R, we denote the level set f−1(s) byX=sand theinterval set f−1(I) byXI.
Definition 4.14 (Critical, regular value). An open interval I ⊆ R is called aregular interval if there exist a topological space Y and a homeomorphismΦ : Y × I → XI so that f ◦Φis the projection ontoIandΦextends to a continuous function ¯Φ:Y×I¯→XI¯where ¯Iis the closure of I. We assume that f is ofMorse type[63] meaning that each levelset X=shas finitely-generated homology groups and there are finitely many values called critical a0 = −∞ < a1 < · · · <
an < an+1 = +∞, so that each interval (ai,ai+1) is a maximal interval that is regular. A value s∈(ai,ai+1) is then called aregular value.
The original construction [63] of level set (henceforth written levelset) zigzag persistence picks regular valuess0,s1, . . . ,snso that eachsi∈(ai,ai+1). Then, thelevelset zigzag filtrationof
f is defined as follows:
X[s0,s1]←-· · ·,→X[si−1,si]←-X=si ,→X[si,si+1]←-· · ·,→X[sn−1,sn].
This construction relies on a choice of regular values and there is no canonical choice. As we work on simplicial complexes, different regular values can result in different complexes in the filtration. Therefore, we adopt the following alternative definition of a levelset zigzag filtrationX, which does not rely on a choice of regular values:
X:X(a0,a2)←-· · ·,→X(ai−1,ai+1)←-X(ai,ai+1) ,→X(ai,ai+2)←-· · ·,→X(an−1,an+1). (4.15) The space of the type X(ai−1,ai+1)contains a critical valueai and hence is called acritical space.
For a similar reason a space of the type X(ai,ai+1) is calledregular spacewhich does not contain any critical value. Considering the homology groups of the spaces, we get the zigzag persistence module:
HpX:Hp(X(a0,a2))← · · · →Hp(X(ai−1,ai+1))←Hp(X(ai,ai+1))→Hp(X(ai,ai+2))← · · · →Hp(X(an−1,an+1)).
Note thatX(ai,ai+1)deformation retracts toX=si andX(ai−1,ai+1)deformation retracts toX[si−1,si], so the zigzag modules induced by the two diagrams are isomorphic, i.e., equivalent at the persis-tent homology level. See Figure 4.10 for an example of a levelset zigzag filtration.
a2 a3 a4
a1 X(a
0,a2) X(a
1,a2) X(a
1,a3) X(a
2,a3)
: · · ·
Figure 4.10: A torus with four critical values. The real-valued function is the height function over the horizontal line. The first several subspaces in the levelset zigzag diagram are given and the remaining ones are symmetric. Empty dot indicates that the point is not included.
Generation of barcode for levelset zigzag. The interval decomposition of the moduleHpX gives the barcode for the zigzag persistence. However, the endpoints of the bars may belong to either the index of a critical or regular space. If it belongs to a critical spaceX(ai−1,ai+1), we map it to the critical value ai. Otherwise, if it belongs to a regular space X(ai,ai+1), we map it to the regular value si. After this conversion, still the bars do not end solely in critical values. We modify the endpoints further. In keeping with the understanding that even the levelset homology classes do not change in the regular spaces, we convert an endpointsito an adjacent critical value and make the bar (interval module) open at that critical value. Precisely we modify the bars as (i) [ai,aj] ⇔ [ai,aj], (ii) [ai,sj] ⇔ [ai,aj+1) (iii) [si,aj] ⇔ (ai,aj] (ii) [si,sj] ⇔ (ai,aj+1). As in the case of standard zigzag filtration, the intervals in (i)-(iv) are referred asclosed-closed, closed-open,open-closed, andopen-openbars respectively. Our goal is to compute these four types of bars for a PL-function where the spaceXis the underlying space of a simplicial complexK.
4.5.1 Simplicial levelset zigzag filtration
We now turn to asimplicial versionof the construction we just described. For a given complex K, letX =|K|and f :X →Rbe a PL-function defined by interpolating values on the vertices of K(Definition 3.2). We also assume f to begeneric, that is, no two vertices ofKhave the same function value.
We know that f can have critical values only atK’s vertices (Section 3.5.1). We call these vertices critical and call other vertices regular. Let v1, . . . ,vn be all the critical vertices of f with values a1 < · · · < an, and let a0 = −∞, an+1 = +∞ be two additional critical values.
For two critical values ai < aj, let X(i,j) := X(ai,aj) and K(i,j) be the complex σ ∈ K| ∀v ∈ σ, f(v) ∈ (ai,aj) . Then, the space and simplicial levelset zigzag filtrationX and K of f are defined respectively as:
X:X(0,2)←-· · ·,→X(i−1,i+1) ←-X(i,i+1) ,→X(i,i+2)←-· · ·,→X(n−1,n+1) (4.16)
K:K(0,2)←-· · ·,→K(i−1,i+1)←-K(i,i+1),→K(i,i+2)←-· · ·,→K(n−1,n+1) (4.17) A complex of the formK(i,i+1)in the filtration is called aregular complexand a complex of the formK(i,i+2)is called acritical complex. Note that while we can expect the space and simplicial
ai ai+1 ai ai+1
Figure 4.11: Simplicial zigzag filtration is made equivalent to space filtration by subdivision.
levelset zigzag filtrations for a finely tessellated complex to be equivalent, this is not always the case. For example, in Fig-ure 4.11, let K0 be the complex on the left;
K(i,i0 +1)
(thick edges) is not homotopy equivalent to|K0|(i,i+1), and hence the simplicial levelset zigzag filtration is not equivalent to the space one. We ob-serve that the non-equivalence is caused by the two central trian-gles which contain more than one critical value. A subdivision of the two central triangles in the complexK00on the right, where no triangles contain more than one critical value, renders |K00|(i,i+1) deformation retracting to
K(i,i00+1)
. Based on the above observa-tion, we formulate the following property, which guarantees that
the module of the simplicial levelset zigzag filtration remain isomorphic to that of the space one.
Definition 4.15. A complexKis calledcompatible with the levelsetsof a PL-function f :|K| →R if for every simplexσofKand its convex hull|σ|, function values of points in|σ|contain at most one critical value of f.
Given a PL-function f on a complex K, one can makeK compatible with the levelsets of f by subdividingKwith barycentric subdivisions; see e.g. [103].
Proposition 4.14. Let K be compatible with the levelsets of f , and let X = |K|; one has that X(ai,aj) deformation retracts to
K(i,j)
for any two critical values ai < aj. Therefore, the zigzag modules induced by the space and the simplicial levelset zigzag filtrations are isomorphic.
Our goal is to compute the four types of bars for the zigzag filtrationXfrom its simplicial versionK. For this, we makeKsimplex-wise and call itF. First,Fstarts and ends with the same original complexes inK. Second, whenever an inclusion inKis expanded so that one simplex is added at a time, the addition follows the order of the simplices’ function values. Formally, for the inclusionK(i,i+1) ,→K(i,i+2)inK, letu1 =vi+1,u2, . . . ,uk be all the vertices with function values in [ai+1,ai+2) such that f(u1) < f(u2)< · · ·< f(uk); then, the lower stars ofu1, . . . ,ukare added in sequence by F. Note that for eachuj ∈
u1, . . . ,uk , we do not restrict how simplices in the lower star ofujare added. For the inclusionK(i−1,i+1)←- K(i,i+1)inK, everything is reversed, i.e., vertices are ordered in decreasing function values and upper stars are added. With this expansion, the zigzag filtrationKin Eqn. (4.17) is converted to a filtrationFshown below where a dashed arrow indicates insertions of one or more simplices and a solid arrow indicates a single simplex insertion. In particular, we indicate that the backward inclusionK(i−1,i+1) c- K(i,i+1)is expanded into a simplex-wise filtration.
F:· · ·,dK(i−1,i+1)←-· · · ←-K`−1←-K`←-· · · ←-K(i,i+1),dK(i,i+2)c-· · · (4.18) After expanding all forward and backward inclusions to make them simplex-wise, we obtain a zigzag filtration whose complexes can be indexed by 0,1, . . . ,nas we assume next.
4.5.2 Barcode for levelset zigzag filtration
One can compute the barcode for the zigzag filtrationFin Eqn. (4.18) that is derived from the original zigzag filtration K in Eqn. (4.17). There is one technicality that we need to take care
of. To apply the algorithm in Section 4.3.2, we need the input zigzag filtration to begin with an empty complex. The filtrationFas constructed from expandingKhas the first complexK(0,2)that is non-empty. So, as before, we expandK(0,2)simplex-wise and beginFwith an empty complex.
We assume below this is the case forF.
The bars in the barcode forFdo not necessarily coincide with the four types of bars forK with endpoints only in critical values. However, we can read the bars forKfrom the bars ofF.
First, assume thatFis indexed as
F:∅= K0↔K1↔ · · · ↔Kn−1↔Kn.
This means that a complexKj, j>0, is of the four categories, (i) it is a complex in the expansion of the backward inclusionK(i−1,i+1)c- K(i,i+1), (ii) it is a complex in the expansion of the forward inclusionK(i,i+1) ,dK(i,i+2), (iii) it is a regular complexK(i,i+1)for somei> 0, (iv) it is a critical complexK(i−1,i+1)for somei>0. The types of complexes where the endpoints of a bar [b,d] for F are located determine the bars forK and henceXwhich can be of four types: closed-closed [ai,aj],closed-open[ai,aj),open-closed(ai,aj], andopen-open(ai,aj).
Let [b,d] be a bar forF. If both KbandKd appear in the expansion of a forward inclusion K(i,i+1) ,d K(i,i+2), we ignore the bar because it is an artificial bar created due to expanding the filtration K into the filtration F. Similarly, we ignore the bar if both Kb and Kd appear in the expansion of a backward inclusionK(i−1,i+1) c-K(i,i+1). We explain other cases below.
(Case 1.)Kbis either a regular complexK(i,i+1)or in the expansion ofK(i−1,i+1)c-K(i,i+1): the complexKb is a subcomplex of the critical complexK(i−1,i+1)which stands for the critical value ai. So, the endbis mapped toaiand made open because the class for the bar [b,d] does not exist inK(i−1,i+1).
(Case 2.)Kbis either the critical complexK(i,i+2)or in the expansion ofK(i,i+1),dK(i,i+2): the complex is a subcomplex of the critical complexK(i,i+2) which stands for the critical valueai+1. So, the endbis mapped toai+1and is closed because the class for [b,d] is alive inK(i,i+2)
(Case 3.)Kdis the critical complexK(i−1,i+1)or is in the expansion of the backward inclusion K(i−1,i+1) c-K(i,i+1): the complex is a subcomplex of the critical complexK(i−1,i+1) which stands for the critical valueai. So, the enddis mapped toai and made closed because the class for the bar [b,d] exists inK(i−1,i+1).
(Case 4.)Kdis either the regular complexK(i,i+1)or in the expansion ofK(i,i+1),dK(i,i+2): the complex is a subcomplex of the critical complexK(i,i+2) which stands for the critical valueai+1. So, the enddis mapped toai+1and is open because the class for [b,d] is not alive inK(i,i+2). 4.5.3 Correspondence to sublevel set persistence
Standard persistence as we have seen already is defined by considering the sublevel sets of f, that is,X[0,i] = f−1[s0,si]= f−1(−∞,si] wheresi ∈(ai,ai+1) is a regular value. We get the following sublevel set diagram:
X:X[0,0]→X[0,1]→ · · · →X[0,n].
Then, considering f to be a PL-function onX = |K|, we have already seen in Section 3.5 thatX can be converted to a simplicial filtrationKshown below whereK[0,i]={σ∈K| f(σ)≤ai}. This filtration can further be converted into a simplex-wise filtration which can be used for computing Dgmp(K) forp≥0.
K:K[0,0] →K[0,1] →K[0,2]· · · →K[0,n]
The bars for this case have the form [ai,aj) whereajcan bean+1 =∞. Each such bar is closed at the left endpoint because the homology class being born exists atK[0,i]. However, it is open at the right endpoint because it does not exist atK[0,j].
One can see that there are two types of bars in the sublevel set persistence, one of the type [ai,aj), j ≤ n, which is bounded on the right, and the other of the type [ai,∞) = [ai,an+1) which is unbounded on the right. The unbounded bars are the infinite bars we introduced in Section 3.2.1. They correspond to the essential homology classes sinceHp(K)L
i[ai,∞). The work of [59, 63] imply that both types of barcodes of the standard persistence can be recovered from those of the levelset zigzag persistence as the theorem below states:
Theorem 4.15. LetKandK0denote the filtrations for the sublevel sets and level sets respectively induced by a continuous function f on a topological space with critical values a0,a1,· · · ,an+1
where a0=−∞and an+1=∞. For every p≥0,
1. [ai,aj), j,n+1is a bar forDgmp(K)iffit is so forDgmp(K0),
2. [ai,an+1) is a bar forDgmp(K)iffeither[ai,aj]is a closed-closed bar forDgmp(K0) for some aj >ai, or(aj,ai)is an open-open bar forDgmp−1(K0)for some aj <ai.
4.5.4 Correspondence to extended persistence
There is another persistence considered in the literature under the nameextended persistence[103], and it turns out that there is a correspondence between extended persistence and level set zigzag persistence. For a real-valued function f :X →R, letX[0,i]denote the sublevel set f−1[s0,si] as before andX[i,n]denote the superlevel set f−1[si,sn]. Then, a persistence module that considers the sublevel set filtration first and then juxtaposes it with a filtration of quotient spaces of X as shown below gives the notion of extended persistence:
X:X[0,0],→ · · ·,→X[0,n] ,→(X[0,n],X[n,n]),→ · · ·,→(X[0,n],X[0,n]).
Observe that each inclusion map between two quotient spaces induces a linear map in their relative homology groups. One can read that the above sequence arises by first growing the space to the full space X[0,n]with sublevel sets and then shrinking it by quotienting with the superlevel sets.
Again, taking f : X →Ras a PL-function onX = |K|, we get the simplicial extended filtration whereK[0,i]={σ∈K| f(σ)≤ai}andK[i,n] ={σ∈K|f(σ)≥ai}.
E:K[0,0],→ · · ·,→K[0,n],→(K[0,n],K[n,n]),→ · · ·,→(K[0,n],K[0,n]).
The decomposition of the persistence moduleHpEarising out ofEprovides the bars in Dgmp(E).
For the first part of the sequence, the endpoints of the bars are designated with respective function
valuesaias before. For the second part, the birth or death point of a bar is designated asan+iif its class either is born in (K[0,n],K[i,n]) or dies entering into (K[0,n],K[i,n]) respectively for 0 ≤i≤ n.
We leave the proof of the following theorem as an exercise; see also [63].
Theorem 4.16. Let K and E denote the simplicial levelset zigzag filtration and the extended filtration of a PL-function f :|K| →R. Then, for every p≥0,
1. [ai,aj)is a bar forDgmp(K)iffit is a bar forDgmp(E),
2. (ai,aj]is a bar forDgmp(K)iff[an+j,an+i)is a bar forDgmp+1(E), 3. [ai,aj]is a bar forDgmp(K)iff[ai,an+j)is a bar forDgmp(E), 4. (ai,aj)is a bar forDgmp(K)iff[aj,an+i)is a bar forDgmp+1(E).
Clearly, for two persistence modulesHpEandHpE0arising out of two extended filtrationsE andE0, the stability of persistence diagrams holds, that is,db(DgmpE,DgmpE0)=dI(HpE,HpE0) (Theorem 3.11).
4.6 Notes and Exercises
Computation of persistent homology induced by simplicial towers generalizing filtrations were considered in the context of TDA by Dey, Fan, Wang [122]. They gave two approaches to compute persistence diagrams for such towers, one by converting a tower to a zigzag filtration which we described in Section 4.4 and the other by considering annotations in combination with the link conditions allowing edge collapses without altering homotopy types which is described in Section 4.2.1. The first approach apparently increases the size of the filtration which motivated the second approach. Kerber and Schreiber showed that indeed the first approach can be leveraged to produce filtrations instead of zigzag filtrations and without blowing up sizes [210].
The concept of zigzag modules obtained from a zigzag filtration by taking the homology groups and linear maps induced by inclusions is closely related to quiver theory due to Gabriel [163]
which was brought to the attention of TDA community by Carlsson and de Silva [62]. They were the first to propose the concept of zigzag persistence and its computation [62]. They observed that any zigzag module can be decomposed into a set of other zigzag modules where the forward non-zero maps are only injective and the backward non-zero maps are only surjective. Although they did not compute this decomposition, they used its existence to design an algorithm for com-puting the interval decomposition of a given zigzag module. Later, with Morozov, they used these concepts to present an O(n3) algorithm for computing the persistence of a simplex-wise zigzag filtration withn arrows [63]. Milosavljevi´c et al. [234] improved the algorithm for any zigzag filtration withnarrows to have a time complexity ofO(nω+n2log2n), whereω∈[2,2.373) is the exponent for matrix multiplication. Maria and Oudot [228] presented a different algorithm where they showed how a filtration of the last complex in the prefix of a zigzag filtration can help com-puting the persistence incrementally. The algorithm in this chapter draws upon these approaches though is presented quite differently. Indeed, adaptation of the presented approach on graphs led to recent near-linear time algorithms for zigzag persistence on graphs [127].
Given a real valued function f :X →Ron a topological spaceX, the level sets at the critical and intermediate values give rise to a levelset zigzag filtration as shown in Section 4.5. Carlsson, de Silva, and Morozov [63] introduced this set up and observed the decomposition of the zigzag module into interval modules with open or closed ends. The four types of bars arising out of this zigzag module give more information than the standard sublevel set persistence which only outputs open and infinite bars. It was observed in [59] that the open-open and closed-closed bars indeed capture the infinite bars of the sublevel set persistence with an appropriate dimension shift. Theorem 4.15 summarizes this connection. The extended persistence originally proposed for surfaces [5] and later extended for filtrations [103] also computes all four types of bars, but they are described differently using the persistence diagrams rather than open and closed ends.
Exercises
1. Show that the inequality in Proposition 4.1 cannot be improved to equality by giving a counterexample.
2. Prove Proposition 4.5.
3. Prove Proposition 4.6.
4. Prove Proposition 4.7.
5. Prove Proposition 4.8.
6. For computing the persistence of a simplicial tower, we checked the link condition in all dimensions. Argue that it is sufficient to check the condition only for three relevant dimen-sions.
7. LetK be a triangulated 2-manifold of genus gwithout boundary. Consider the following tasks:
• Compute the genusgby the formula 2−2g=#vertices−#edges+#triangles.
• Compute a spanning treeT of the 1-skeleton ofK, and a spanning treeT∗of the dual graph none of whose edges are dual to any edge inT.
• Annotate the edges inT with zero vector of length g, index the edges not inT and whose duals are not in T∗ ase1, . . . ,e2g. Annotateei with a vector that has the ith entry 1 and all other entries 0.
• Propagate systematically the annotation to the rest of the edges.
Complete the above approach with a proof of correctness into an algorithm that computes the annotation for edges inO(gn) time ifKhasnsimplices.
8. Do we get the same barcode if we run the zigzag persistence algorithm given in Sec-tion 4.3.1 and the standard persistence algorithm on a non-zigzag filtraSec-tion? If so, prove it.
If not, show the difference and suggest a modification to the zigzag persistence algorithm so that the both output become the same.
9. Suppose that a persistence module {Vi fi
→ Vi+1} is presented with the linear maps fi as matrices whose columns and rows are fixed bases of Vi andVi+1 respectively. Design an algorithm to compute the barcode for the input module. Do the same when the input module is a zigzag tower.
10. ([127]) We have seen that for graphs a near-linear time algorithm exists for computing non-zigzag persistence. Design a near-linear time algorithm for computing non-zigzag persistence for graphs.
11. Consider a PL-function f :|K| →R.
(a) Design an algorithm to compute the barcode of−f from a level set zigzag filtration of f.
(b) Show that f and−f produce the same closed-closed and open-open bars for the lev-elset zigzag filtration.
(c) In general, given a zigzag filtrationF, consider the filtration F0 = −F in opposite direction from right to left. What is the relation between the barcodes of these two filtrations?
12. We computed persistence of zigzag towers by first converting it into a zigzag filtration and then using the algorithm in section 4.3 to compute the bars. Design an algorithm that skips the intermediate conversion to a filtration.
13. Design an algorithm for computing the extended persistence from a given PL-function on an input simplicial complex.
14. ([63]) Prove Theorem 4.16.