Persistence landscapes of affine fractals

1.1 Persistence landscapes

Persistent homology is a relatively new approach to studying topological spaces. In the context of data science, persistent homology can be applied to a data set to complement traditional statistical approaches by studying the geometry of the data. We employ the tools of persistent homology, including persistence landscapes, to analyze affine fractals.

Persistent homology typically begins with a set of points, equipped with a pairwise notion of distance. We place a metric ball of radius r around each point and increase r . We are interested in the topological properties of the union of these balls as a function of r. Typical properties of interest include connectedness, loops or holes, and voids of the union, and importantly the radii at which these appear or disappear. While this type of information may seem crude, a surprising amount of insight about the underlying set of points can be extracted in this way.

The data of changing topological properties are conveniently summarized in what is known as a persistence diagram, a multiset of points in the plane. If we focus on loops of the union, then each point ( b , d ) in the persistence diagram represents a loop, whose two coordinates correspond to the radius when the hole is formed ( r = b ) and when it gets filled in ( r = d ). The persistence diagram provides a multiscale summary, encoding geometric and topological features of the set [4].

Unfortunately, however, barcodes do not possess a vector space structure, and so the quantitative analysis and precise comparison can be difficult [5]. To remedy this, we map the barcodes to some feature space (a Banach space in our case) using a well-studied feature map known as a persistence landscape. The mapping from barcodes to landscapes is reversible, and so this vectorization scheme loses no information [1]. Persistence landscapes have been used to study protein binding [6], phase transitions [7], audio signals [8], and microstructures in materials science [9].

See Section 2 for a full discussion of persistent homology and persistence landscapes of metric spaces.

1.2 Affine fractals

A fractal, for our purposes, is a set that has a self-similarity property. The middle-third Cantor set is the canonical example of a self-similar set. Fractals are commonly studied objects in many contexts. Cantor sets, in particular, appear in the context of analysis [10,11], number theory [12,13], probability [14,15,16], geometry [17,18], physics [19,20,21], and harmonic analysis [22,23,24].

In this paper, we consider specifically the class of affine fractals, which are generated by iterated function systems consisting of affine transformations. By this, we mean that the fractal is the invariant set for the iterated function system.

For the maps Ψ , we denote the compositions (i.e. iterations) of the maps by:

In [25], Hutchinson laid out the main relationship between fractals and iterated function systems (IFS); this relationship is the foundation of our results. Recall that ψ : X → X is Lipschitz if for all x , y ∈ X , there exists C > 0 such that

The Lipschitz constant of ψ is the infimum of all such C . We say that ψ is a contraction if it has Lipschitz constant less than 1. Hutchinson’s theorem is as follows.

Recall that the Hausdorff distance between two sets A , B ⊂ X is given by

For us, the maps Ψ consist of affine transformations acting on R d . Moreover, we assume that the linear part of the maps are scalars that are common to all of the maps. Therefore, our maps have the form ψ j ( x → ) = c ( x → + b → j ) , where c ∈ ( 0 , 1 ) . The followings are examples of affine fractals in our class:

1. The classical middle-third Cantor set in R ;

2. The Sierpinski gasket in R 2 ;

3. The Sierpinski carpet in R 2 ;

4. The Menger sponge (or Sierpinski cube) in R 3 .

Specifically, the Cantor set is the invariant set for the IFS with generators

Likewise, the Sierpinski carpet is the invariant set for the IFS with generators

For an IFS Ψ acting on R d and a nonempty S 0 ⊂ R d , we define the sequence S n + 1 = Ψ ( S n ) . We will typically consider S 0 , which consists of finitely many points in R d , and therefore by Hutchinson’s theorem, { S n } converges in the Hausdorff metric to the fractal generated by Ψ . Indeed, we will show that choosing S 0 to be the extreme points of the convex hull of the fractal is ideal in establishing our algorithm for calculating the persistence landscapes of the fractal.

One of our main results is to prove that for a fixed affine IFS Ψ that satisfies a certain compatibility condition and appropriate initialization S 0 , there exists an affine transformation ℒ acting on the space of persistence landscapes such that for every n ∈ N , the following diagram commutes:

Here, f n is the persistence landscape of S n and γ associates with S n its persistence landscape. We will show that ℒ is a strict contraction on the set of persistence landscapes, and so it possesses a unique fixed point. That fixed point will be the persistence landscape for the fractal generated by Ψ . Consequently, the persistence landscape f of the fractal is obtained by

for any initialization f 0 .

2.1 Simplicial complexes

For a simplicial complex K , the p-skeleton of K, denoted by K ( p ) , is the subcomplex consisting of all simplices of dimension less than or equal to p . The set of all p -simplices is denoted K p . Thus, the set of vertices can be written as K 0 . Recall that if σ = [ u 0 , u 1 , … , u p ] is a p -simplex, then every point x ∈ σ can be expressed as a convex combination x = ∑ i α i u i , where 0 ≤ α i ≤ 1 and ∑ i α i = 1 .

Given two simplicial complexes K and L , we say that φ : K 0 → L 0 is a vertex map if for any p -simplex [ x 0 , … , x p ] in K , [ φ ( x 0 ) , … , φ ( x p ) ] is a simplex in L . Thus, vertex maps send the vertices of simplices in K to simplices in L . We do not require φ to be injective, so dim [ φ ( x 0 ) , … , φ ( x p ) ] ≤ p with a strict inequality if φ ( x j ) = φ ( x k ) for some j , k ∈ { 0 , 1 , … , p } . Given a vertex map φ : K 0 → L 0 , we can extend it to a map f : K → L by

where x = ∑ j = 0 p α j x j . In this way, we say that f is the simplicial map induced by φ .

The first step in our goal of computing topological properties of affine fractals will be to construct their Cěch complexes. If X ⊂ R d , for any ε > 0 , we can define the Cěch complex to be

where B ( x , r ) is the ball of radius r centered at x . A consequence of the Nerve theorem [28,30] is that for a finite set X ⊂ R d , { x : d ( x , X ) ≤ ε / 2 } is homotopy equivalent to Cěch ( X , ε ) .

There is another popular variant in persistent homology for associating a topological space with a set, known as the Vietoris-Rips complex. The Vietoris-Rips complex for X ⊂ R d and ε > 0 is defined by

In comparing equations (3) and (4), we see that the 1-simplices in Cěch ( X , ε ) are the same as those in VR ( X , ε ) , but it is not necessary that Cěch ( X , ε ) = VR ( X , ε ) . Furthermore, verifying the existence of a point in the intersection of equation (3) often requires much more work than verifying the pairwise condition of equation (4). For this reason, together with recent advances in computational efficiency in software [31], applications of persistent homology tend to rely on Vietoris-Rips complexes. The two complexes are related by a well-known result [32, Thm. 2.5].

Our focus will be on the Cěch complex of affine fractals and their approximations.

2.2 A review of homology

Given a simplicial complex K , an Abelian group G , and a nonnegative integer p ≥ 0 , we define the group of p-chains with coefficients in G to be formal G -linear combinations of p -simplices of K and denote it C p ( K ; G ) . A typical element of C p ( K ; G ) is a finite formal sum of the form ∑ i g i σ i , where σ i ∈ K p and g i ∈ G . The differential (or boundary) of a p -simplex σ = [ u 0 , u 1 , … , u p ] is expressed as follows:

where [ u 0 , … , u ˆ j , … , u p ] is the ( p − 1 ) -simplex obtained by omitting u ˆ j from σ . Extending ∂ p to a G -linear homomorphism to all of C p ( K ; G ) gives ∂ p : C p ( K ; G ) → C p − 1 ( K ; G ) . This gives rise to the simplicial chain complex of K with coefficients in G

with the essential property that any two successive compositions equal the trivial map: ∂ j ∂ j + 1 = 0 for all j ≥ 0 . Hence, im ( ∂ j + 1 ) ⊂ ker ( ∂ j ) . We denote the p-cycles by Z p ( K ) = ker ( ∂ p ) ⊂ C p ( K ) , and the p-boundaries by B p ( K ) = im ( ∂ p + 1 ) ⊂ C p ( K ) .

Moving forward, we assume G = Z 2 . This choice of coefficients is common in persistent homology and simplifies many of the computations, e.g., the factors of ( − 1 ) j appearing in equation (5) vanish.

For simplicial complexes K and L , f : K → L is a simplicial map if f is continuous and f maps each simplex of K linearly onto a simplex of L . Define a homomorphism f # : C p ( K ) → C p ( L ) by first defining

and extend the homomorphism to the rest of C p ( K ) linearly. A standard fact in algebraic topology is that f # further induces a map on homology f ∗ : H p ( K ) → H p ( L ) for every p . Furthermore, if f is a simplicial homeomorphism, then f ∗ : H ∗ ( K ) → H ∗ ( L ) is an isomorphism [27].

A particularly nice feature of homology groups is that the homology group of a space is isomorphic to the direct sum of the homology groups of the path components [26, Prop. 2.6]. This directly leads to the following lemma.

Lemma 2 implies that given a finite point cloud and a collection of similitudes, so long as the images of the point cloud under those similitudes are sufficiently far apart, the homology group resulting from the union of those images is easily related to the homology groups resulting from the original point cloud.

Another tool we will use for the computation of homology groups is known as the Mayer-Vietoris sequence. Suppose K is a simplicial complex with subcomplexes K 1 and K 2 such that K = K 1 ∪ K 2 . The Mayer-Vietoris sequence is the long exact sequence:

For a detailed explanation of the maps in the sequence and a proof of its exactness, see [27, Chapter 3].

2.3 Persistent homology

A simplicial filtration K • of a simplicial complex K is a collection of subcomplexes { K i } i ∈ I , for an indexing set I ⊂ R such that whenever s ≤ t ,

The inclusion map K s → K t induces a homomorphism H ∗ ( K s ) → H ∗ ( K t ) . The collection of groups { H p ( K s ) } s ∈ I , together with the collection of homomorphisms { H p ( K s ) → H p ( K t ) } s ≤ t form the p th -persistent homology of K • .

Understanding the structure of persistent homology is difficult as presented. Instead, we turn to a generalization of this structure known as a persistence module. A persistence module is a collection of Z 2 -vector spaces V = { V t } t ∈ I , and a collection of linear transformations between these spaces M = { M s , t : V s → V t } s ≤ t that satisfy

We will often refer to the persistence module ( V , M ) as simply M . By setting V s = H p ( K s ) and M s , t : H p ( K s ) → H p ( K t ) as the homomorphism induced by inclusion, we see that every simplicial filtration K • gives rise to a persistence module.

If X ⊂ R d , then the collection of simplicial complexes

form a simplicial filtration of Cěch ( X , ∞ ) , the full simplicial complex on X . For 0 ≤ s ≤ t , the inclusion mapping Cěch ( X , s ) → Cěch ( X , t ) induces a homomorphism H p Cěch ( X , s ) → H p Cěch ( X , t ) . We let H p Cěch ( X ) denote the persistence module resulting from the filtration of the Cěch complex of X . We adopt the convention that

for s < 0 .

A persistence diagram is a multiset of birth and death times derived from a decomposition of a persistence module. Interval modules form the basic building blocks of persistence modules. Interval modules are indecomposable. In [33], it was shown that the persistence module ( V , M ) can be decomposed into interval modules as long as each V t is finite dimensional. Moreover, an application of [34, Thm. 1], implies that this decomposition is unique, up to reordering. Thus, if we have a persistence module M with the interval module decomposition

and we define the persistence diagram of M to be the multiset

where Δ is the set of diagonal points { ( x , x ) ∣ x ∈ R } each counted with infinite multiplicity.

We will denote a persistence diagram as D = ( D , μ ) , where D = { ( b λ , d λ ) } λ is the collection of distinct birth and death times and μ : D → N 0 is defined μ ( b λ , d λ ) as the multiplicity of the birth death time ( b λ , d λ ) .

To ensure that the persistence modules we consider have a decomposition into interval modules, we require that M is q-tame [35].

Assuming a persistence module is q-tame guarantees us not only a well-defined persistence diagram, but also stability with respect to the bottleneck distance [36]. A prior stability result was presented by Cohen-Steiner et al. in [37], which required a more restrictive tameness condition.

Persistence modules resulting from a Cěch filtration built on a finite point cloud X are q -tame. Moreover, when X ⊂ R d is compact, H p Cěch ( X ) is q -tame [38] for all p . As a result, we obtain the following:

While persistence diagrams are an effective representation of a persistence module, they are not conducive to statistical analysis. Persistence landscapes address this issue by embedding persistence diagrams into a Banach space. Given a persistence module M that is q -tame, its persistence landscape is a function f : N × R → R ¯ by

where R ¯ denotes the extended real numbers, [ − ∞ , ∞ ] . We have that all persistence landscapes are elements of

We compute the distance between two persistence landscapes using the standard norm on L ∞ ( N × R ) , which is defined as follows:

There is an alternative definition given in [39] that allows us to relate the persistence landscape of M to its persistence diagram. If the persistence module M is represented as a persistence diagram D = { ( a i , b i ) } i ∈ I , then we can define the functions

and for all k ∈ N , t ∈ R ,

We use kmax to denote the k th largest element of a set. Note that since D is a multiset, certain birth death pairs ( a i , b i ) can appear more than once.

It is common for many stability results to write the distance between two persistence landscapes in terms of their corresponding persistence modules. If f = { f ( k ) } k = 1 ∞ is the persistence landscape obtained from M and g = { g ( k ) } k = 1 ∞ is the persistence landscape obtained from M ′ , we define the persistence landscape distance between M and M ′ to be

By combining the stability results from [1,38,36], we obtain the following:

3.1 The persistence landscape of the Cantor set

We begin with calculating the persistence landscape of the middle-third Cantor set C . This calculation illustrates the main ideas of our construction, while also highlighting some of the technical requirements on the IFS. Recall that C is generated by the IFS given by the collection of contraction maps Ψ = { ψ 0 , ψ 2 } , where

For convenience, we suppose that Ψ acts on the metric space X = [ 0 , 1 ] with the standard metric. We define the sequence of approximations to C as follows:

We also define

The length of each closed interval in C n is 1 / 3 n . Note that since the points in S n are equal to the end points in the disjoint closed intervals that make up C n , and we have S n ⊂ C ⊂ C n for all n ∈ N , we find

As a consequence of this and Lemma 2.3, we obtain the following convergence result.

Knowing that the limit exists, we would still like to have a formula for the persistence landscape of C . To help us determine this formula, we will use a two-step approach. First, we will find an affine operator ℒ : L ∞ ( N × R ) → L ∞ ( N × R ) such that for all n sufficiently large, ℒ maps the persistence landscape of S n to the persistence landscape of S n + 1 . We will then show that the persistence landscape of C is the fixed point of ℒ and use ℒ to compute the fixed point.

To describe persistence landscapes, we will use the hat functions defined in equation (11). Recall that persistence landscapes h ∈ L ∞ ( N × R ) are defined by letting h = { h ( j ) } j = 1 ∞ , where h ( j ) equals the j th largest hat function supported on an interval whose end points equal one of the birth-death pairs from the persistence diagram. We also note that the hat functions scale nicely so that for t ∈ R

We define ℒ for each g ∈ L ∞ ( N × R ) by ℒ g = h = { h ( j ) } j = 1 ∞ , where

We adopt the convention that the maximum death time will be equal to the diameter of the invariant set, which in the case of C is 1. Thus, when computing persistence landscape of C , the first function in the sequence will be τ ( 0 , 1 ) . We can see that ℒ is an affine contraction. Indeed, if we define T : L ∞ ( N × R ) → L ∞ ( N × R ) by T g = h , where

and then we see that T is linear with ‖ T ‖ = 1 3 . Further, for all f ∈ L ∞ ( N × R ) , ℒ f = v + T f , where v = { τ ( 0 , 1 ) , 0 , 0 , 0 , … } .

In other words, the commutative diagram in equation (1) holds.

Proof

For n ∈ N , we denote f n = { f n ( j ) } j = 1 ∞ . It easy to check for n = 1 using direct computation. Since S 1 = { 0 , 1 } and S 2 = { 0 , 1 / 3 , 2 / 3 , 1 } we see that f 2 ( 1 ) = τ ( 0 , 1 ) , f 2 ( j ) = τ ( 0 , 3 − 1 ) for 2 ≤ j ≤ 4 , and f 2 ( j ) = 0 for all j > 4 . We can also see that

f 1 = { τ ( 0 , 1 ) , 0 , 0 , 0 , 0 , … } .

By the definition of ℒ , we see that ℒ f 1 = f 2 . In general, for n ∈ N , let

L n = ϕ 0 ( S n ) , R n = ϕ 2 ( S n ) ,

as depicted in Figure 1. For all n ∈ N , we see that

S n + 1 = L n ∪ R n .

Since S n ⊂ [ 0 , 1 ] , we know that L n ∩ R n = ∅ . Moreover,

(16) d ( L n , R n ) = min x ∈ L n , y ∈ R n ∣ x − y ∣ = 1 3 .

Let D n = ( D n , μ n ) denote the persistence diagram of H 0 Cěch ( S n ) . Note that since S n has 2 n distinct points, if D n = { ( 0 , d j ) } j = 1 k , then ∑ j = 1 k μ n ( 0 , d j ) = 2 n . We assume without the loss of generality that d j < d j + 1 for j ∈ { 1 , … , k } . This means d k = 1 , and for j < k , d j < 1 .

We claim that for j < k , ( 0 , d j ) ∈ D n implies that 0 , d j 3 ∈ D n + 1 with μ n + 1 ( 0 , d j / 3 ) = 2 μ n ( 0 , d j ) . To help us do this, we first prove that for 0 < ε < 1 , the diagram

(17)

commutes, where M 0 , ε and P 0 , ε are homomorphisms induced by the inclusions Cěch ( S n , 0 ) → Cěch ( S n , ε ) and Cěch ( L n , 0 ) → Cěch L n , ε 3 , respectively, and ϕ 0 ∗ 0 and ϕ 0 ∗ ε are isomporphisms induced by ϕ 0 as in Corollary 1. Indeed, for γ ∈ H 0 Cěch ( S n , 0 ) , we may use coset notation to write γ = [ x ] B 0 ( Cěch ( S n , 0 ) ) for some x ∈ S n . Thus,

P 0 , ε ϕ 0 ∗ 0 γ = P 0 , ε x 3 B 0 Cěch ( L n , 0 ) = x 3 B 0 Cěch L n , ε 3 .

On the other hand,

ϕ 0 ∗ ε M 0 , ε γ = ϕ 0 ∗ [ x ] B 0 Cěch ( S n , ε ) = x 3 B 0 Cěch L n , ε 3 .

Therefore, ϕ 0 ∗ ε M 0 , ε = P 0 , ε ϕ 0 ∗ 0 , which implies diagram (17) commutes.

To prove the first claim, suppose ( 0 , d j ) ∈ D n with j < k , and then there are μ n ( 0 , d j ) distinct classes γ ∈ H 0 Cěch ( S n , 0 ) that die at d j . By assumption, there exists γ ∈ H 0 Cěch ( S n , 0 ) such that γ ∈ ker M 0 , d j , but γ ∉ ker M 0 , t , for t < d j . Let γ − = ϕ 0 ∗ 0 γ . Since the diagram in (17) commutes, γ − ∈ ker P 0 , ε if and only if γ ∈ ker M 0 , ε . Thus, γ − ∈ ker P 0 , d j , but γ − ∉ ker P 0 , t for t < d j .

For s ≤ t , let Q s , t : H 0 Cěch R n , s 3 → H 0 Cěch R n , t 3 be the homomorphism induced by the natural inclusion. By replacing L n with R n , and x 3 with x + 2 3 in the argument mentioned earlier, we can see that there also exists γ + ∈ H 0 Cěch ( R n , 0 ) such that γ + ∈ ker Q 0 , d j , but γ + ∉ ker Q 0 , t for t < d j . Since Cěch ( L n , ε ) is not path connected to Cěch ( R n , ε ) in Cěch ( S n + 1 , ε ) for ε < 1 3 , it follows that

H 0 Cěch ( S n + 1 , ε ) ≅ H 0 Cěch ( L n , ε ) ⊕ H 0 Cěch ( R n , ε ) .

For 0 ≤ s ≤ t < 1 3 , the homomorphism F s , t : H 0 Cěch S n + 1 , s 3 → H 0 Cěch S n + 1 , t 3 induced by the obvious inclusion satisfies

ker F s , t ≅ ker P s , t ⊕ ker Q s , t .

Thus, for every class γ ∈ H 0 Cěch ( S n , 0 ) with death time d j < 1 , there exist two distinct classes in H 0 Cěch ( S n + 1 ) with death time d j 3 , which implies 0 , d j 3 ∈ D n + 1 with μ n + 1 0 , d j 3 = 2 μ n ( 0 , d j ) . This is equivalent to what we claimed.

It follows from our claim that

0 , d j 3 j = 1 k − 1 ⊂ D n + 1

and

∑ j = 1 k − 1 μ n + 1 0 , d j 3 = 2 ∑ j = 1 k − 1 μ n ( 0 , d j ) = 2 ( 2 n − 1 ) = 2 n + 1 − 2 .

By (16), we also see that ( 0 , 1 / 3 ) ∈ D n + 1 . Since we are working with the zero-dimensional homology of a Cěch complex, we also have ( 0 , 1 ) ∈ D n + 1 , since the space is nonempty. This accounts for all 2 n + 1 elements of D n + 1 . For convenience, we reindex D n and D n + 1 repeating elements according to their multiplicity and ordering them by decreasing death times so that

D n = { ( 0 , 1 ) , ( 0 , d ( 2 ) ) , ( 0 , d ( 3 ) ) , … , ( 0 , d ( 2 n ) ) }

and

D n + 1 = { ( 0 , 1 ) , ( 0 , 1 / 3 ) , ( 0 , d ( 2 ) / 3 ) , ( 0 , d ( 2 ) / 3 ) , … , ( 0 , d ( 2 n ) / 3 ) , ( 0 , d ( 2 n ) / 3 ) } .

Applying the definition in (12), we see that f n = { f n ( j ) } j = 1 ∞ , which is defined by

f n ( j ) = τ ( 0 , d ( j ) ) , for j ∈ { 1 , 2 , … , 2 n } , and f n ( j ) = 0 for j > 2 n .

Since d ( 1 ) = 1 and d ( 2 ) = 1 / 3 , we easily check that f n + 1 = { f n + 1 ( j ) } j = 1 ∞ satisfies

f n + 1 ( 1 ) = τ ( 0 , 1 ) , f n + 1 ( 2 ) ( t ) = τ ( 0 , 1 / 3 ) ( t ) = 1 3 f n ( 1 ) ( 3 t ) f n + 1 ( 2 j + 1 ) ( t ) = f n + 1 ( 2 j + 2 ) ( t ) = τ ( 0 , d ( j ) / 3 ) ( t ) = 1 3 f j ( n ) ( 3 t ) for j ∈ N .

Therefore, ℒ f n = f n + 1 .□