In many applied problems we are interested in comparing two -valued functions defined on a topological space, up to a certain group of tranformations. As an example, we can think of the case of taking pictures of two objects and from every possible oriented direction (at a constant distance) and comparing the sets of images we get. In such a case the image
taken from the oriented direction of a unit vectorcan be approximated by a point in . This point describes a matrix , which represents the grey levels on a grid discretizing the image . Our global measurement is a function , taking each oriented direction to the vector describing the matrix , associated with the picture that we get from that oriented direction. In this case the position of the examined objects cannot be predetermined but we can control the direction of the camera that takes the pictures. As a consequence, two different sets of pictures (described by two different functions ) can be considered similar if an orientation-preserving rigid motion of exists, such that the picture of taken from the oriented direction of the unit vector is similar to the picture of taken from the oriented direction of the unit vector , for every . Formally speaking, the two different sets of pictures can be considered similar if is small, where denotes the group of orientation-preserving isometries of and is the max-norm.
The previous example illustrates the use of the following definition, where represents the set of all continuous functions from to . These functions are called -dimensional filtering functions on the topological space .
In this paper we will assume that the space is triangulable. This assumption allows to guarantee that the persistent Betti number functions (PBNFs) are finite without using any tameness assumption (cf. Theorem 2.3 in CeDFFe13 ). The assumption that the PBNFs are finite is necessary to our treatment. We could weaken the assumption that is triangulable and consider a compact and locally contractible subspace of (cf. CaLa11 ), but we preferred to refer to an assumption that is usual for the community interested in persistent homology.
Let be a triangulable space. Let be a subgroup of the group of all homeomorphisms . The pseudo-distance defined by setting
is called the natural pseudo-distance associated with the group .
The previous definition generalizes the concept of natural pseudo-distance studied in FrMu99 ; DoFr04 ; DoFr07 ; DoFr09 ; Fa11 to the case , and is a particular case of the general setting described in FrLa11 . The case that is a proper subgroup of is also examined in Ca10 ; CaDiLa12 , and in Fr90 for the case of the group of diffeomorphisms (in an infinite dimensional setting).
The pseudo-distance is difficult to compute. Fortunately, if , persistent homology can be used to obtain lower bounds for . For example, if we denote by the matching distance between the -th persistent Betti number functions and of the functions and , we have that (cf. BiCeFrGiLa08 ; CeDFFe13 ).
In literature concerning persistent homology, the expression matching distance (a.k.a. bottleneck distance) usually denotes a metric between persistence diagrams. However, each persistence diagram represents just one persistent Betti number function, provided that two persistent Betti number functions are considered equivalent if they differ in a subset of their domain that has a vanishing measure. As a consequence, the matching distance can be seen as a metric between persistent Betti number functions. In this paper we shall use the expression matching distance in this sense.
A natural question arises: How could we obtain a lower bound for in the general case ? Does an analogue of the concept of persistent Betti number function exist, suitable for getting a lower bound for ? Since , one could think of using the classical lower bounds for the natural pseudo-distance in order to get lower bounds for the pseudo-distance . Before proceeding we illustrate two examples, showing that in some cases this choice is not useful.
Let us consider an experimental setting where a robot is in the middle of a room, measuring its distance from the surrounding walls by a sensor, for each oriented direction. This measurement can be formalized by a function , where equals minus the distance from the wall in the oriented direction represented by the unit vector , for each . Figure 1 represents two instances and of the function for two different shapes of the room. Let denote the group of orientation-preserving rigid motions of . We observe that a homeomorphism exists, such that and . It follows that , so that the direct application of classical persistent homology does not give a positive lower bound for , while we will see that .
Let us consider the functions from the unit disk to the real numbers, representing images of the letters . For each letter , the function describes the grey level at each point of the topological space , with reference to the considered instance of the letter (see Figure 2). Black and white correspond to the values and , respectively (so that light grey corresponds to a value close to ). It is easy to recognize that for each pair with a homeomorphism exists such that the max-distance between the functions vanishes. This is due to the fact that the letters are homeomorphic to each other. It follows that vanishes. As a consequence, the distance between the classical persistence diagrams of and vanishes, too. This proves that the direct application of classical persistent homology is not of much use in this example.
One could think of solving the problem described in the two previous examples by using other filtering functions. Unfortunately, this is not always easy to do. To make this point clear, think of acquiring data by magnetic resonance imaging (MRI). Asking for further filtering functions means asking for new measurements, of similar or different kind. This approach could be expensive or impractical. Furthermore, choosing the data we have to manage is not allowed, in many applications.
Moreover, in the fortunate case that we can choose the filtering function, another difficulty arises. It consists in the fact that shape comparison is usually based on judgements of experts, expressed by invariance properties. As an example, the expert can say that rotation and scaling are not important in the considered field of research. On one hand, we observe that it is not easy to translate the invariance properties expressed by the expert into the choice of a filtering function. On the other hand, it is quite natural to try to directly insert the information given by the expert into our theoretical setting. In this paper we will show that we can do that. Indeed, we can adapt persistent homology in order to obtain the invariance with respect to the action of a given group of homeomorphisms. This allows us to obtain a theory that can give a positive lower bound for , in the previous examples (and in many similar cases, where a direct application of classical persistent homology is not of much use).
We are going to describe this idea in the next section.
2 Adapting persistent homology to the group
This section is devoted to the introduction of some abstract definitions and the statement of a general result. In the next sections we will show how these concepts can be put into effect.
Shape comparison is commonly based on comparing properties (usually described by -valued functions) with respect to the action of a transformation group. Let us interpret these concepts in a homological setting. Before proceeding, let us fix a chain complex over a field (so that each group of -chains is a vector space). We consider the partial order on defined by setting if and only if for every .
Let be a chain complex over a field . Assume a function is given, such that
takes the null chain to the -tuple , for every ;
for every ;
for every , , ;
for every with , and every .
We shall say that is a filtering function on the chain complex .
Let be a chain complex over a field . Let us assume that a group is given, such that acts linearly on each vector space and its action commutes with , i.e., for every (in particular, every is a chain isomorphism from to ). The chain complex will be said a -chain complex. We shall call the group the -th homology group associated with the -chain complex .
We observe that the group acts on the kernel and image whose quotient is the group . As a consequence, also acts on the homology group.
In the previous definition we do not specify how the action of on each vector space is chosen, confining ourselves to assume that this action is linear and commutes with . In the next section, will be the singular chain complex of a triangulable space over a field , and will be assumed to be a subgroup of . In that section, the action of each on each singular simplex in will be given by the usual composition of functions. For more details about -complexes and equivariant homology we refer the interested reader to Br72 ; Il73 ; tD87 ; Wi75 .
Now, let us assume that is a -chain complex, endowed with a filtering function . For every we can consider the chain subcomplex of defined by setting and restricting to . is a subcomplex of because of the properties in Definition 2.1 (in particular, ). We observe that will not be a -chain complex, since , in general. For the sake of simplicity, we will use the symbol in place of .
The chain complex will be called the chain subcomplex of associated with the value , with respect to the filtering function .
We refer to KaMiMr04 for the definition of chain subcomplex.
Now we can define the concept of the -th persistent homology group of , with respect to .
If and (i.e., for every index ), we can consider the inclusion of the chain complex into the chain complex . Such an inclusion induces a homomorphism . We shall call the group the -th persistent homology group of the -chain complex , computed at the point with respect to the filtering function . The rank of this group will be called the -th persistent Betti number function (PBNF) of the -chain complex , computed at the point with respect to the filtering function .
The key property of is the invariance expressed by the following result.
If is a chain isomorphism from to and with , the groups and are isomorphic.
We define a map in the following way. Let us consider an element . By definition, a cycle exists, such that is the equivalence class of in . We observe that and the equivalence class of in belongs to . We set .
If is another cycle such that , then a chain exists, such that . We observe that . The inequality (see Definition 2.1) implies that . As a consequence, . These equalities follow from the fact that is a chain isomorphism. This proves that is well defined.
Let , with . We observe that . From the linearity of , it follows that , for every . Hence, we have that . Therefore, is linear.
Furthermore, if then , so that a chain exists, such that . Moreover, . It follows that , because of Definitions 2.1 and the fact that is a chain isomorphism. As a consequence, . This proves that is injective.
Finally, is surjective. In order to prove this, we observe that if with the homomorphism induced by the inclusion , then a chain exists such that . We have that and .
Therefore is an isomorphism. ∎
The previous theorem justifies the name -invariant persistent homology, showing that the PBNFs of a -chain complex do not change if we replace the filtering function with the function , for .
3 Stability of the PBNFs with respect to
In the previous section we have introduced some abstract definitions and a theorem. In this section we will show how we can obtain structures conforming to the previously described properties.
Let and be a triangulable space and its singular chain complex over a field , respectively.
Assume that a subgroup of the group of all homeomorphisms and a continuous function are chosen. For every , let us set . Let us consider the action of on defined by setting for every and every singular simplex in , and extending this action linearly on . We recall that, by definition, every singular -simplex in is a continuous function from the standard -simplex into .
Now, assume that a -chain subcomplex of the singular chain complex is given (we will show in the next section how this subcomplex can be constructed). We observe that, for every topological subspace of , is a chain complex over the field . The symbol denotes the chain complex where is the vector space of the singular -chains in that belong to .
In order to avoid “wild” chain complexes, we also make this assumption (see Remark 3.2 below):
If and are two closed subsets of with , then a topological subspace of exists such that and the homology group is finitely generated for every non-negative integer .
Let us consider the set of all (distinct) singular -simplexes in . Obviously, if is not a finite topological space, will be an infinite (usually uncountable) set. Then we can endow the chain complex with a filtering function in the following way. If equals the null chain in , we set . If is a non-null singular -chain, we can write with , for every index , and for . This representation is said to be reduced. In this case we set , with each equal to the maximum of on the union of the images of the singular simplexes . In other words, is the smallest vector such that the corresponding sublevel set contains the image of each singular simplex involved in the reduced representation of that we have considered. We observe that this representation is unique up to permutations of its summands, so that is well defined. Furthermore, the properties in Definition 2.1 are fulfilled. We shall say that the function is induced by .
An elementary introduction to singular homology can be found in Ha02 .
For every the -th persistent Betti number function of the -chain complex , endowed with the filtering function , is finite at each point in its domain.
Since and is continuous, we have that the set is closed and contained in the interior of the closed set . Property implies that a topological subspace of exists such that and is finitely generated. The inclusions induce the homomorphisms . Since , we obtain that also is finitely generated. ∎
We stress the importance of the assumption . It allows us to avoid chain complexes like the one where the -chains are all the usual singular -chains of and the only -chain is the singular zero -chain of . Obviously, this is a -chain complex for any subgroup of . In this case, for any pair of distinct points of the topological space , there is no singular -chain whose boundary is the singular -chain (here, for the sake of simplicity, we are not distinguishing the singular -simplexes from their images in ). Since the boundary homomorphism from -chains to -chains is zero, no non-zero -chain is a boundary. Hence the homology group is not finitely generated, in general, and the property does not hold. For example, it does not hold for , independently of the regularity of the space (unless is a finite set). As a consequence, the proof that we gave for Proposition 3.1 does not work, and it is easy to check that its statement is false for the chain complex we have just described. This is the reason for which the finiteness results proven in CeDFFe13 and CaLa11 for classical persistent homology cannot be directly applied to -invariant persistent homology, without assuming property . Finally, we observe that is not as much an assumption about the regularity of the topological space , but rather an assumption about the regularity of the -chain complex.
From now on, in order to avoid technicalities that are not relevant in this paper, we shall consider two PBNFs equivalent if they differ in a subset of their domain that has a vanishing measure.
A standard way of comparing two classical persistent Betti number functions is the matching distance , a.k.a. bottleneck distance (cf. EdHa08 ; CeDFFe13 ). It is important to observe that, in order to define it, we need the finiteness of the persistent Betti number functions (cf. CoEdHa07 ). This distance can be applied without any modification to the case of the persistent Betti number functions of the -chain complex , because of the finiteness stated in Proposition 3.1.
The following theorem shows that the matching distance between persistent Betti number functions of the -chain complex is a lower bound for the natural pseudo-distance . In other words, a small change of the filtering function with respect to produces just a small change of the corresponding persistent Betti number function with respect to . This property allows the use of PBNFs in real applications, where the presence of noise is unavoidable.
For every , let us consider the -th persistent Betti number functions , of the -chain complex , endowed with the filtering functions and induced by and , respectively. Then
We can proceed by mimicking step by step the proof of stability for ordinary persistent Betti number functions (cf. CeDFFe13 ). This is possible because that proof depends only on properties of PBNFs that are shared by both classical persistent Betti number functions and persistent Betti number functions of a -chain complex endowed with a filtering function, once we have proven that the PBNFs are finite (Proposition 3.1). It is sufficient to replace the group with the group , and the homology groups of each sublevel set with the homology groups of the -chain complex . Since the only difference in the proof consists in the need to show that -invariant persistent Betti number functions are finite in order to be allowed to use the matching distance , we refer the reader interested in the technical details to CeDFFe13 . ∎
4.1 A first application of our method
In this subsection we illustrate how -invariant persistent homology can be used to discriminate between the rooms described in Example 1.3, showing that no rotation of changes the function into .
In order to manage this problem we can consider the chain complex whose -chains are all the singular -chains for which the following property holds:
If a singular simplex appears in a reduced representation of with respect to the basis of , then the antipodal simplex appears in that representation with the same multiplicity of , where is the antipodal map .
In other words, in we accept by definition only the singular chains in that can be written in the form . It easy to check that is a -chain subcomplex of the complex .
Every rotation commutes with the antipodal map and is a chain isomorphism from to . Moreover, it is easy to verify that the properties in Definition 2.2 are fulfilled, for and . The chains in will be called symmetric chains.
We can prove that the property holds for the -chain complex that we have defined. Let and be two closed subsets of with . Let us set equal to the -dilation111The -dilation of a subset of a metric space is the set of points of that have a distance strictly less than from . On we consider the metric induced by the Euclidean metric in . of in , choosing so small that the . We observe that the set is open and . Moreover, is the union of a finite family of pairwise disjoint open arcs, having the property that if then also (possibly, ). Now, let us consider the topological quotient space obtained by taking all unordered pairs of antipodal points in . We have that is homeomorphic to the union of a finite family of pairwise disjoint open arcs of (possibly, ), and hence the -th homology group is finitely generated. A chain isomorphism from to exists, taking each -chain to the chain given by the singular simplex , defined by setting for every . induces an isomorphism from to . Therefore also is finitely generated. Property follows by observing that .
Referring to Example 1.3, let us consider the birth of the first homology class in the homology groups and , respectively, when the parameter increases. While the group becomes non-trivial when reaches the value , the group becomes non-trivial when reaches a value . This is due to the fact that the sublevel set contains two pairs of antipodal points, while the sublevel set contains no pair of antipodal points (see Figure 3). In other words, the only points at infinity in the persistence diagrams associated with the -th persistent homology groups of the -chain subcomplex of with respect to and are and , respectively.
It follows that the matching distance between the -th persistent Betti number functions of the -chain complex with respect to the filtering functions and is at least . By applying Theorem 3.3, we obtain the inequality . In other words, -invariant persistent homology gives a non-trivial lower bound for , while the matching distance between the classical persistent Betti number functions with respect to the filtering functions and vanishes.
The interested reader can find the -th persistent Betti number functions and of the -chain complex in Figure 4. We notice that the persistent Betti number functions and of the -chain complex coincide. Indeed, and take the same absolute maximum . Hence both the groups and becomes non-trivial (and equal to ) when reaches the same value . After that change, no further change happens. As a consequence, the persistent Betti number functions in degree of the -chain complex with respect to the filtering functions and coincide.
As an alternative approach to the problem of comparing two filtering functions , the reader could think of using the well known concept of equivariant homology (cf. Wi75 ). In other words, in the case that acts freely on , one could think of considering the topological quotient space , endowed with the filtering functions that take each orbit of the group to the maximum of and on , respectively. We observe that this approach would not be of help in the case illustrated in Example 1.3, since the quotient of is just a singleton. As a consequence, if we considered two filtering functions with , the persistent homology of the induced functions would be the same. For more details about -complexes and equivariant homology we refer the interested reader to Br72 ; Il73 ; tD87 .
4.2 A generalization of our technique
The approach that we have illustrated in the previous subsection can be generalized to triangulable spaces different from and invariance groups that are different from the group of rotations. The main idea consists in looking for another subgroup of such that
is finite (i.e. );
for every and every .
Due to the finiteness of , the property 2 implies that the restriction to of the conjugacy action of each is a permutation of .
The legitimate -chains in our chain complex are defined to be the linear combinations of “elementary” singular chains that can be written as , where is a singular -simplex in . Because of the property 2 and the linearity of the action of each , is another legitimate chain in our chain complex , so that results to be a -chain complex. In Example 1.3, we have chosen , where is the antipodal simmetry. We recall that the filtering function induces a filtering function on the set of legitimate chains, where is the smallest vector such that the corresponding sublevel set contains the image of each singular simplex involved in a reduced representation of , for every non-null chain .
However, we stress the fact that our approach is far more general. Indeed, in both Examples 1.3 and 1.4, if we set equal to the (Abelian and finite) group generated by the reflections with respect to the coordinate axes, we could choose equal to the group generated by the counterclockwise rotation of radians (where denotes a fixed natural number greater than ). It is interesting to observe that in this case, if the homeomorphism reverses the orientation, then the conjugacy action is not the identity, since it takes each homeomorphism to its inverse . Furthermore, .
On the basis of the remarks that we have made in this subsection, we can give another example concerning our adaptation of persistent homology to invariance groups. Let us consider , and the two sets , . Let us consider also the topological space , with the topology (and the metric) induced by its embedding in . From the topological point of view, is the disjoint union of two copies of .
Let be the group of all isometries that can be represented (with a little abuse of notation) as for every , where is an isometry of . In plain words, these are the isometries that act similarly on and . Assume that we are interested in the comparison of continuous functions from to with respect to the group . In order to proceed, we have to choose a group verifying the properties 1 and 2 in this subsection. For instance, we can set equal to the group , generated by the map