TOPOLOGICAL DATA ANALYSIS
3.5 Reeb graphs and Reeb spaces .1 Reeb graphs.1Reeb graphs
for 0≤i≤ j≤n.
The ranks of these groups are called thepth persistent Betti numbers βi,pj=rankHi,pj.
Note that the persistent homology groupsHi,pj consist of homology classes ofKithat still exist inKj, andHi,ip =Hp(Ki).
A homology classα∈Hp(Ki)isbornatKiif it is not in the image of the map induced by the inclusion Ki−1 ⊂ Ki. We sayα dies entering Kj if the image of the map induced byKi−1 ⊂ Kj−1 does not contain the image ofαbut the image of the map induced byKi−1 ⊂ Kj does. Thepersistence ofαisaj−ai or j−i, depending on the application. A class isessentialif it does not die within the filtration.
Persistence diagrams
A persistence diagram is a multiset of points in the extended planeR2∞. Letµi,pjbe the number ofp-dimensional classes born atKiand dying enter- ing Kj. Then the pth persistence diagram of the filtration of (K, f), denoted dgmp(f), consists of points(ai,aj), each with multiplicityµi,pj.
3.5 Reeb graphs and Reeb spaces
Geometric Reeb graphs
The geometric Reeb graph provides an intuitive summary of the structure of the level sets of a pair(X, f).
Definition 3.10(geometric Reeb graph [20]). Let X be a topological space, and let f : X→Rbe a continuous real-valued function. For x,x0 ∈ X, we writex ∼f x0 if xand x0 belong to the same path-component of a level set f−1(a), for some a ∈ R. The quotient space X/ ∼f is the (geometric) Reeb graphof(X, f).
Example 3.11 (height function). We illustrate the Reeb graph of a surface given by the height function in Figure 3.1.
Figure 3.1: Reeb graph of height function.
In general, the quotient space X → X/ ∼f may be poorly behaved. Thus, we restrict ourselves to a class ofconstructiblepairs(X, f).
Definition 3.12(R-Top[20]). LetR-Topbe the category whose objects con- sist of pairs (X, f), where X is a topological space and f : X → R is a continuous map, and whose morphismsφ : (X, f) → (Y,g)are continuous
mapsφ: X→Ysuch that the following diagram commutes:
X φ //
f
Y
g
R
Definition 3.13(R-Topc[20]). We say that an object ofR-Topisconstructible if it is isomorphic to some(X, f)constructed in the following manner: given a finite setA ={a0,· · · ,an}, listed in increasing order,
• specify a locally path-connected spaceVifor each 0≤i≤n,
• specify a locally path-connected spaceEi, for each 0≤i<n, and
• specify continuous mapsli : Ei→Viandri :Ei →Vi+1. LetXbe the quotient space
G
i
Vi× {ai}
G
G
i
Ei×[ai,ai+1]
/∼,
with(li(x),ai) ∼(x,ai)and(ri(x),ai+1) ∼(x,ai+1)for alliand allx∈Ei, and let f : X→Rbe the projection onto the second factor.
We denote the full subcategory of constructible(X, f)∈R-TopbyR-Topc. Examples of constructible(X, f)include Morse functions on compact man- ifolds and piecewise linear functions on compact polyhedra.
Abstract Reeb graphs
In many applications, it is important to ensure that two Reeb graphs are similar when the functions they arise from are similar. To do so, de Silva et al. [20] show that the data of a Reeb graph can be stored abstractly as a functor. In this setting, they define the interleaving distance between pairs of Reeb graphs and show that the interleaving distance is stable under perturbations. We summarize some of their definitions and results here and call on them later.
Definition 3.14(abstract Reeb graph [20]). LetXbe a topological space, and let f :X→ Rbe a continuous real-valued function. LetOpen(R)denote the category with objects consisting of open sets inRand arrowsI → Jbetween two objects if and only ifI⊆ J.
Theabstract Reeb graphof(X, f)is a functorF : Open(R) → Setthat maps each open set I ⊆ Rto the set of path-components of f−1(I), denoted F(I), and each arrowI ⊆ J to the set mapF(I) → F(J) induced by the inclusion
f−1(I)⊆ f−1(J), denotedF[I⊆ J].
Note that an abstract Reeb graph is an object in the category of functors SetOpen(R).
Let C : R-Top → SetOpen(R) denote the functor that maps f = (X, f)to its Reeb graphC(f) =F, with
F(I) = π0f−1(I), F[I ⊆J] =π0[f−1(I) ⊆ f−1(J)], where we letπ0denote the set of path-components of a space.
Interleaving distance between Reeb graphs
Interleavings are approximate isomorphisms. Anisomorphismbetween func- torsF,G: Open(R) →Setis a pair of families of maps
φI :F(I)→G(I), ψI :G(I)→F(I)
that are natural with respect to inclusions I ⊆ J such that φI and ψI are inverses for allI.
Anε-interleaving gives some leeway by allowing the codomains ofφI and ψI to be given by anε-expansion ofI.
Definition 3.15(ε-interleaving between Reeb graphs). For an open interval I = (a,b)⊆R, letIε = (a−ε,b+ε).
Anε-interleavingbetween two Reeb graphsF,G : Open(R) →Setis a pair of families of maps
φI : F(I) →G(Iε), ψI :G(I)→F(Iε) that are natural with respect to inclusionsI⊆ Jand satisfy
ψIε ◦φI =F[I ⊆I2ε], φIε◦ψI =G[I ⊆I2ε]
for allI.
F(I)
F[I⊆I2ε]
φI
''
G(I)
G[I⊆I2ε]
ψI
''G(Iε)
ψIε
ww
F(Iε)
φIε
ww
F(I2ε) G(I2ε)
When there exists anε-interleaving between Reeb graphs Fand G, we say thatFandGareε-interleaved.
Definition 3.16(interleaving distance). Theinterleaving distancebetween two Reeb graphsF,G: Open(R) →Setis given by
dI(F,G) =inf{ε|F,Gareε-interleaved}. (We take the infimum of an empty set to be∞.)
De Silva et al. [20] show that the interleaving distance on the space of Reeb graphs is an extended pseudometric. (It takes values in [0,∞], with dI(C(f),C(f)) =0, is symmetric, and satisfies the triangle inequality.) 3.5.2 Reeb spaces
We can also study higher-dimensional analogues of Reeb graphs, called Reeb spaces. Reeb spaces are generalizations of Reeb graphs, obtained when a real-valued map f : X→Ris replaced by a multivariate map f :X→ Rd. Munch and Wang define Reeb spaces using categorical tools and show that categorical representations of Reeb spaces and categorical representations of mapper constructions converge in the interleaving distance [19]. We summarize their definitions and some of their results here and reference them in later sections.
Geometric Reeb spaces
Definition 3.17 (geometric Reeb space [19]). Let X be a topological space, and let f : X→Rdbe a continuous function. Forx,x0 ∈X, we writex∼f x0 ifxandx0belong to the same path-component of a level set f−1(a), for some a∈Rd. The quotient spaceX/∼f is thegeometric Reeb spaceof(X, f).
Abstract Reeb spaces
Definition 3.18(abstract Reeb space [19]). LetXbe a topological space, and let f : X→Rdbe a continuous function. LetOpen(Rd)denote the category with objects consisting of open sets in Rd and arrows I → J between two objects if and only ifI ⊆ J.
Theabstract Reeb spaceof(X, f)is a functor F: Open(Rd) → Setthat maps each open setI ⊆Rdto the set of path-components of f−1(I), denotedF(I), and each arrowI ⊆ J to the set mapF(I) → F(J) induced by the inclusion
f−1(I)⊆ f−1(J), denotedF[I⊆ J].
Note that an abstract Reeb space is an object in the category of functors SetOpen(Rd).
LetRd-Topbe the category whose objects consist of pairs(X, f), whereXis a topological space and f : X→Rdis a continuous map, and whose arrows are function-preserving maps(X, f) →(Y,g).
As before, we let C:Rd-Top→SetOpen(Rd) denote the functor that maps f = (X, f)to its Reeb spaceC(f) =F, with
F(I) = π0f−1(I), F[I ⊆J] =π0[f−1(I) ⊆ f−1(J)]. Interleaving distance between Reeb spaces
Definition 3.19 (ε-interleaving between Reeb spaces [19]). For an open set I⊆Rd, let Iε ={x∈ Rd :d(x,I) < ε} denote the ε-expansion of I. An ε- interleaving between two abstract Reeb spaces F,G : Open(Rd) → Set is a pair of families of maps
φI : F(I) →G(Iε), ψI :G(I)→F(Iε) that are natural with respect to inclusionsI⊆ Jand satisfy
ψIε ◦φI =F[I ⊆I2ε], φIε◦ψI =G[I ⊆I2ε] for allI.
F(I)
F[I⊆I2ε]
φI
''
G(I)
G[I⊆I2ε]
ψI
''G(Iε)
ψIε
ww
F(Iε)
φIε
ww
F(I2ε) G(I2ε)
Where there exists anε-interleaving between Reeb spacesFand G, we say thatFandGareε-interleaved.
Definition 3.20(interleaving distance [19]). Theinterleaving distancebetween two Reeb spacesF,G: Open(Rd)→Setis given by
dI(F,G) =inf{ε|F,Gareε-interleaved}. (We take the infimum of an empty set to be∞.)
Munch and Wang [19] show that the interleaving distance on the space of abstract Reeb spaces is an extended pseudometric.
Stability of interleaving distance for Reeb spaces
De Silva et al. [20] prove a stability result for interleaving distance on the space of Reeb graphs. At the time of writing, an analogous result for Reeb spaces has not been recorded. The ideas used to prove stability for Reeb graphs carry over to the setting of Reeb spaces. We state the analogous result for Reeb spaces and give a proof here.
Proposition 3.21. Let f = (X, f),g = (Y,g)∈Rd-Top. Then dI(C(f),C(g))≤ kf −gk∞.
Proof.
Supposekf −gk∞ ≤ε.
LetI ⊆Rdbe an open set. Sincekf −gk∞ ≤ε, we have thatkf(x)−g(x)k ≤ε for allx∈X. In particular, ifx∈ f−1(I), then f(x) ∈Iand g(x) ∈Iε. Thus,
f−1(I)⊆ g−1(Iε).
Similarly, we have
g−1(I) ⊆ f−1(Iε). Thus, we can define
φI =π0[f−1(I)⊆ g−1(Iε)]
and
ψI =π0[g−1(I)⊆ f−1(Iε)].
Naturality with respect to inclusionsI⊆ Jfollows immediately.
Furthermore, since inclusions commute, we have that ψIε ◦φI =F[I ⊆I2ε], φIε◦ψI =G[I ⊆I2ε] for allI.
Thus, there is anε-interleaving betweenC(f) andC(g)for allε≥ kf −gk∞. It follows that
dI(C(f),C(g))≤ kf −gk∞.