We develop a method for calculating the persistence landscapes of affine fractals using the parameters of the corresponding transformations. Given an iterated function system of affine transformations that satisfies a certain compatibility condition, we prove that there exists an affine transformation acting on the space of persistence landscapes, which intertwines the action of the iterated function system. This latter affine transformation is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. We present several examples of the theory as well as confirm the main results through simulations.
Affine fractals are the invariant set of an iterated function system (IFS) consisting of affine transformations acting on Euclidean space. Well-known examples of such fractals are Cantor sets and Mandelbrojt sets. Affine fractals, as subsets of Euclidean space, possess topological properties that can be extracted through the methods of algebraic topology. In particular, these subsets of Euclidean space can be associated with a persistence landscape , which is a sequence of functions that encode geometric properties of the set based on Euclidean distances. These distances give rise to a family of homology groups derived from a filtration of complexes. The homology groups in turn produce a persistence module from which the persistence landscapes are defined.
Interest in studying fractals using the tools of algebraic topology has occurred recently. In , it was shown that persistence homology can be used to distinguish fractals of the same Hausdorff dimension. In , the authors describe a relationship between the Hausdorff dimension of fractals and the persistence intervals of Betti numbers.
Our main result (Theorem 4) concerns the calculation of the persistence diagrams and landscapes of affine fractals. We prove that, under a certain compatibility condition, there exists an affine transformation , which is defined by the parameters of the IFS. This transformation acts on the space of persistence landscapes. Moreover, it is a strict contraction and its unique fixed point is the persistence landscape of the affine fractal. Consequently, the persistence landscape of the fractal can be computed via the limiting process of repeated applications of to any initial input. We also prove, under an additional assumption on the IFS, that intertwines the action of the iterated function system (Theorem 5).
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 around each point and increase . 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 in the persistence diagram represents a loop, whose two coordinates correspond to the radius when the hole is formed ( ) and when it gets filled in ( ). The persistence diagram provides a multiscale summary, encoding geometric and topological features of the set .
Unfortunately, however, barcodes do not possess a vector space structure, and so the quantitative analysis and precise comparison can be difficult . 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 . Persistence landscapes have been used to study protein binding , phase transitions , audio signals , and microstructures in materials science .
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.
Suppose is a set of maps acting on a metric space . We say that is invariant for if .
For the maps , we denote the compositions (i.e. iterations) of the maps by:
In , Hutchinson laid out the main relationship between fractals and iterated function systems (IFS); this relationship is the foundation of our results. Recall that is Lipschitz if for all , there exists such that
The Lipschitz constant of is the infimum of all such . We say that is a contraction if it has Lipschitz constant less than 1. Hutchinson’s theorem is as follows.
Let be a complete metric space and a finite set of contraction maps on X. Then there exists a unique closed bounded set A such that
Furthermore, A is compact and is the closure of the set of fixed points of finite compositions of members of . Moreover, for a closed bounded , in the Hausdorff metric.
Recall that the Hausdorff distance between two sets is given by
For us, the maps consist of affine transformations acting on . 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 , where . The followings are examples of affine fractals in our class:
The classical middle-third Cantor set in ;
The Sierpinski gasket in ;
The Sierpinski carpet in ;
The Menger sponge (or Sierpinski cube) in .
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 acting on and a nonempty , we define the sequence . We will typically consider , which consists of finitely many points in , and therefore by Hutchinson’s theorem, converges in the Hausdorff metric to the fractal generated by . Indeed, we will show that choosing 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 that satisfies a certain compatibility condition and appropriate initialization , there exists an affine transformation acting on the space of persistence landscapes such that for every , the following diagram commutes:
Here, is the persistence landscape of and associates with its persistence landscape. We will show that 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 . Consequently, the persistence landscape of the fractal is obtained by
for any initialization .
2 Persistent homology
In this section, we briefly review some standard facts from algebraic topology and persistent homology as well as establish our notation. Excellent resources for (simplicial) homology can be found in [26,27], and [4,28,29] provide a good introduction to persistent homology.
2.1 Simplicial complexes
For a simplicial complex , the p-skeleton of K, denoted by , is the subcomplex consisting of all simplices of dimension less than or equal to . The set of all -simplices is denoted . Thus, the set of vertices can be written as . Recall that if is a -simplex, then every point can be expressed as a convex combination , where and .
Given two simplicial complexes and , we say that is a vertex map if for any -simplex in , is a simplex in . Thus, vertex maps send the vertices of simplices in to simplices in . We do not require to be injective, so with a strict inequality if for some . Given a vertex map , we can extend it to a map by
where . In this way, we say that is the simplicial map induced by .
The first step in our goal of computing topological properties of affine fractals will be to construct their Cěch complexes. If , for any , we can define the Cěch complex to be
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 and is defined by
In comparing equations (3) and (4), we see that the 1-simplices in are the same as those in , but it is not necessary that . 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 , 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 complex of affine fractals and their approximations.
2.2 A review of homology
Given a simplicial complex , an Abelian group , and a nonnegative integer , we define the group of p-chains with coefficients in G to be formal -linear combinations of -simplices of and denote it . A typical element of is a finite formal sum of the form , where and . The differential (or boundary) of a -simplex is expressed as follows:
where is the -simplex obtained by omitting from . Extending to a -linear homomorphism to all of gives . 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: for all . Hence, . We denote the p-cycles by , and the p-boundaries by .
The p th simplicial homology group of a simplicial complex is expressed as follows:
We let denote the collection of homology groups for all dimensions .
Moving forward, we assume . This choice of coefficients is common in persistent homology and simplifies many of the computations, e.g., the factors of appearing in equation (5) vanish.
For simplicial complexes and , is a simplicial map if is continuous and maps each simplex of linearly onto a simplex of . Define a homomorphism by first defining
and extend the homomorphism to the rest of linearly. A standard fact in algebraic topology is that further induces a map on homology for every . Furthermore, if is a simplicial homeomorphism, then is an isomorphism .
Let be a finite point cloud. Let be a similitude with scaling constant . Let , , and let . Then is a vertex map and induces a simplicial homeomorphism f between L and . Thus, induces an isomorphism .
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.
Let be a finite subset and be a collection of similitudes on . Let equal the scaling constant of the similitude . Define
Then for all dimensions , and all
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 is a simplicial complex with subcomplexes and such that . 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 of a simplicial complex is a collection of subcomplexes , for an indexing set such that whenever ,
The inclusion map induces a homomorphism . The collection of groups , together with the collection of homomorphisms form the -persistent homology of .
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 -vector spaces , and a collection of linear transformations between these spaces that satisfy
We will often refer to the persistence module as simply . By setting and as the homomorphism induced by inclusion, we see that every simplicial filtration gives rise to a persistence module.
If , then the collection of simplicial complexes
form a simplicial filtration of , the full simplicial complex on . For , the inclusion mapping induces a homomorphism . We let denote the persistence module resulting from the filtration of the Cěch complex of . We adopt the convention that
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 , it was shown that the persistence module can be decomposed into interval modules as long as each 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 with the interval module decomposition
and we define the persistence diagram of to be the multiset
where is the set of diagonal points each counted with infinite multiplicity.
We will denote a persistence diagram as , where is the collection of distinct birth and death times and is defined as the multiplicity of the birth death time .
To ensure that the persistence modules we consider have a decomposition into interval modules, we require that is q-tame .
The persistence module indexed over is said to be q-tame if
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 . A prior stability result was presented by Cohen-Steiner et al. in , which required a more restrictive tameness condition.
Persistence modules resulting from a Cěch filtration built on a finite point cloud are -tame. Moreover, when is compact, is -tame  for all . As a result, we obtain the following:
For any affine iterated function system, the invariant set and any finite approximation of , possess a well-defined persistence diagram.
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 that is -tame, its persistence landscape is a function by
where 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 , which is defined as follows:
There is an alternative definition given in  that allows us to relate the persistence landscape of to its persistence diagram. If the persistence module is represented as a persistence diagram , then we can define the functions
and for all , ,
We use to denote the th largest element of a set. Note that since is a multiset, certain birth death pairs 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 is the persistence landscape obtained from and is the persistence landscape obtained from , we define the persistence landscape distance between and to be
Suppose X is a metric space, and have well-defined persistence diagrams. Then for all ,
Let be a metric space and suppose such that their Cěch complexes have well-defined persistence diagrams. Then
where and are the persistence landscapes of S and T, respectively.
3 The persistence landscapes of affine fractals
3.1 The persistence landscape of the Cantor set
We begin with calculating the persistence landscape of the middle-third Cantor set . This calculation illustrates the main ideas of our construction, while also highlighting some of the technical requirements on the IFS. Recall that is generated by the IFS given by the collection of contraction maps , where
For convenience, we suppose that acts on the metric space with the standard metric. We define the sequence of approximations to as follows:
We also define
The length of each closed interval in is . Note that since the points in are equal to the end points in the disjoint closed intervals that make up , and we have for all , we find
As a consequence of this and Lemma 2.3, we obtain the following convergence result.
Let be the sequence of persistence landscapes generated from the sequence of point clouds as in equation (14), and let be the persistence landscape of . Then, we have that
Knowing that the limit exists, we would still like to have a formula for the persistence landscape of . To help us determine this formula, we will use a two-step approach. First, we will find an affine operator such that for all sufficiently large, maps the persistence landscape of to the persistence landscape of . We will then show that the persistence landscape of is the fixed point of and use to compute the fixed point.
To describe persistence landscapes, we will use the hat functions defined in equation (11). Recall that persistence landscapes are defined by letting , where equals the 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
We define for each by , 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 is 1. Thus, when computing persistence landscape of , the first function in the sequence will be . We can see that is an affine contraction. Indeed, if we define by , where
and then we see that is linear with . Further, for all , , where .
Let be the sequence of persistence landscape generated from the sequence of point clouds as in equation (14), and let be the persistence landscape of . Let be given by equation (15). Then, for all , .
In other words, the commutative diagram in equation (1) holds.
For , we denote . It easy to check for using direct computation. Since and we see that , for , and for all . We can also see that
By the definition of , we see that . In general, for , let
as depicted in Figure 1. For all , we see that
Since , we know that . Moreover,
Let denote the persistence diagram of . Note that since has distinct points, if , then . We assume without the loss of generality that for . This means , and for , .
We claim that for , implies that with . To help us do this, we first prove that for , the diagram
commutes, where and are homomorphisms induced by the inclusions and , respectively, and and are isomporphisms induced by as in Corollary 1. Indeed, for , we may use coset notation to write for some . Thus,
On the other hand,
Therefore, , which implies diagram (17) commutes.
To prove the first claim, suppose with , and then there are distinct classes that die at . By assumption, there exists such that , but , for . Let . Since the diagram in (17) commutes, if and only if . Thus, , but for .
For , let be the homomorphism induced by the natural inclusion. By replacing with , and with in the argument mentioned earlier, we can see that there also exists such that , but for . Since is not path connected to in for , it follows that
For , the homomorphism induced by the obvious inclusion satisfies
Thus, for every class with death time , there exist two distinct classes in with death time , which implies with . This is equivalent to what we claimed.
It follows from our claim that
By (16), we also see that . Since we are working with the zero-dimensional homology of a Cěch complex, we also have , since the space is nonempty. This accounts for all elements of . For convenience, we reindex and repeating elements according to their multiplicity and ordering them by decreasing death times so that
Applying the definition in (12), we see that , which is defined by
Since and , we easily check that satisfies
Since is Lipschitz with constant , we see that has a unique fixed point and that unique fixed point is , which is the persistence landscape function of . The explicit formula for can be found using by repeatedly applying to any vector in . We find that the persistence landscape of is , where
We see from the illustration that the persistence landscape exhibits its own version of self-similarity. This is a reflection of the fact that the fractal contains several scaled copies of itself. Indeed, since scaling a subset of Euclidean space results in a proportional scaling of its persistence landscape, we should expect the persistence landscape of a fractal to contain a subsequence, which is a scaled copy of itself. The number of scaled copies, which corresponds to the number of generators of the IFS, is also reflected as a multiplicity in the persistence landscape.
3.2 Affine fractals with well-separated images and extreme points
The proof of Theorem 2 suggests that a more general result exists for an IFS satisfying certain properties. The two main ingredients that enable our calculations in the proof include (1) a judicious choice for the initial approximation and (2) a compatibility condition of the images of the maps in . We refer to this condition as well-separated images, and formalize this condition in Definition 4. We first consider the choice of . Before proceeding, we will need to introduce some notation and definitions. Unless stated otherwise, we assume is an IFS consisting only of similitudes on with the form
We let denote the invariant set of . For a set , we let Conv( ) denote the convex hull of . We let denote the set of extreme points of Conv( ). As we shall see, for the affine fractal , choosing is a good choice of initialization of our algorithm. This corresponds to the choice we made for the middle third Cantor set ; see also the example in Section 4.4 for further evidence of this assertion.
Since the maps in are contractions, each map has a unique fixed element. Indeed, it is easy to calculate that for , the fixed point is . We let denote the set of these fixed points. Theorem A guarantees that . The following result tells us that we can easily find .
Suppose is the invariant set for some IFS consisting of similitudes of the form in equation (19). Then .
Let . Clearly, . Assume with . We first observe that for , since , we know that This implies that for ,
If , then for some we have
and for any ,
Since for all , this implies that . Since was arbitrary, this implies that the union of these images, . From Theorem A, we have
in the Hausdorff metric. We claim that . Indeed, choose . For all , it follows from equation (20) that there exists such that for some ,
Since for all , we have . Since is a closed set and , this implies , which proves the claim. Thus, we have the sequence of containment:
By definition, this implies . By the Krein-Milman theorem, it follows that
For any , , it follows that , where . Since is an extreme point, for some , . Therefore, .□
One key property of that was used in the proof of Theorem 2 was that at each scale, the set could be partitioned into a left and right set. The two halves were a significant distance away from each other, and each half was a scaled down version of the previous scale. This property can be described in terms of the IFS, and because of its usefulness, we will define it formally.
Let be an IFS with invariant set . We say that has well-separated images (or satisfies the well-separated condition) if
This definition may apply to any IFS, not only those of the form given in equation (19). Note that on the left-hand side of the inequality, we have the usual Euclidean distance, not the Hausdorff distance.
3.3 Main results
By using the well-separated condition and the ideas in Section 3.1, we are now ready to elucidate the relationship between an IFS and the persistence landscape of its invariant set in more generality. Our main focus is on IFS with well-separated images having the form in equation (19), but many of the following results do not require these assumptions. We will use the same two-step approach that we used with . We first identify a contraction on that has a fixed point equal to the persistence landscape of interest and then use the operator to find a formula for the persistence landscape. Theorem A implies that if is any IFS consisting of contractions then iteratively applying to any compact set creates a sequence of compact sets, , that converges to the invariant set in the Hausdorff metric. Therefore, as a consequence of Theorem B:
Let be an IFS on consisting of contractions with invariant set . Let be a compact set and define the sequence of compact sets by
Then, for any ,
We remark that the statement of Theorem 3 only mentions the Cěch filtration, which applies to any dimension of homology. Also note that the hypothesis makes no assumptions on except that it consists of contractions.
Having established that there is a sequence of persistence landscapes that converge to the persistence landscape of the invariant set, we now seek a contraction on whose fixed point is the landscape of interest. Just like we did earlier with , we will approximate by a finite set and compare the persistence landscapes of and to determine the operator.
Let be an IFS consisting of similitudes all with scaling constants . Let be the invariant set of . For any and any , there exists a finite set such that the followings hold:
Choose . Since is compact, there exists such that . Since and , (a) is satisfied. Since , we have . To show the other containment, choose . Since , there exists , such that . Thus, for some , , which implies
Thus, . Thus, (b) is satisfied. Applying Theorem B, we know for any compact set ,
In light of equation (22), (c) follows from (a). Similarly, (d) follows from (b).□
For a disconnected set , we say that is -connected if it cannot be expressed as a union of two nonempty sets such that . We say that is an -component of if is -connected and
We let denote the number of distinct -components of .
Note that equals the number of connected components of , which is precisely the rank of . Clearly, this number is nonincreasing with respect to .
If we assume is an IFS of similitudes with well-separated images and the scaling constant for each equals , then we can define a sequence of distances
by letting , and for ,
We will make use of the fact that when has well-separated images,
This also means that
Suppose is compact, and is finite with . Then for all , we have
Suppose first that is an -component of . This means is connected and . Let . Since , we know that . We have and . Indeed, if , then and , which implies . Therefore, . With this claim, we have established that
Since is nonempty and compact, it contains at least one distinct -component of . Since was an arbitrary -component of , this implies
For the other inequality, suppose now that is an -component of . Let . Choose and . By assumption, there exists with . Since , we have for all , . Since , there exists , with , which implies . Therefore, . Hence,
This implies that . From this bound, we see that if we partition into its distinct -components , then by letting , we see that . We also have that each contains at least one distinct -component of . Thus,