Topological Persistence is devoted to the study of stable properties of sublevel sets of topological spaces and, in the course of its development, has revealed itself to be a suitable framework when dealing with applications in the field of Shape Analysis and Comparison. Since the beginning of the 1990s research on this subject has been carried out under the name of Size Theory, studying the concept of size function, a mathematical tool able to describe the qualitative properties of a shape in a quantitative way. More precisely, the main idea is to model a shape by a topological space endowed with a continuous function , called measuring function. Such a function is chosen according to applications and can be seen as a descriptor of the features considered relevant for shape characterization. Under these assumptions, the size function associated with the pair is a descriptor of the topological attributes that persist in the sublevel sets of induced by the variation of
. According to this approach, the problem of comparing two shapes can be reduced to the simpler comparison of the related size functions. Since their introduction, these shape descriptors have been widely studied and applied in quite a lot of concrete applications concerning Shape Comparison and Pattern Recognition (cf., e.g.,[4, 8, 15, 34, 35, 36]). From a more theoretical point of view, the notion of size function plays an essential role since it is strongly related to the one of natural pseudodistance. This is another key tool of Size Theory, defining a (dis)similarity measure between compact and locally connected topological spaces endowed with measuring functions (see  for historical references and [16, 18, 19] for a detailed review about the concept of natural pseudodistance). Indeed, size functions provide easily computable lower bounds for the natural pseudodistance (cf. [12, 13, 17]).
Approximately ten years after the introduction of Size Theory, Persistent Homology re-proposed similar ideas from the homological point of view (cf. ; for a survey on this topic see ). In this context, the notion of size function coincides with the dimension of the -th persistent homology group, i.e. the -th rank invariant .
We refer the interested reader to Appendix A for more information about the relationship existing between Size Theory and Persistent Homology.
The study of Topological Persistence is capturing more and more attention in the mathematical community, with particular reference to the multidimensional setting (see [21, 29]). When dealing with size functions, the term multidimensional
means that the measuring functions are vector-valued, and has no reference to the dimension of the topological space under study. However, while the basic properties of a size functionare now clear when it is associated with a measuring function taking values in , very little is known when takes values in . More precisely, some questions about the structure of size functions associated with -valued measuring functions need further investigation, with particular reference to the localization of their discontinuities. Indeed, this last research line is essential in the development of efficient algorithms allowing us to apply Topological Persistence to concrete problems in the multidimensional context.
In this paper we start to fill this gap by proving a new result on the discontinuities of the so-called multidimensional size functions, showing that they can be located only at points with at least one pseudocritical or special coordinate (Theorem 2.11 and Theorem 2.13). This is proved by using an approximation technique and the theoretical machinery developed in , improving the comprehension of multidimensional Topological Persistence and laying the basis for its computation.
This paper is organized in two sections. In Section 1 the basic results about multidimensional size functions are recalled, while in Section 2 our main theorems are proved.
1. Preliminary Results on Size Theory
The main idea in Size Theory is to study a given shape by performing a geometrical/topological exploration of a suitable topological space , with respect to some properties expressed by an -valued continuous function defined on . Following this approach, Size Theory introduces the concept of size function as a stable and compact descriptor of the topological changes occurring in the lower level sets as varies in .
In this section we recall some basic definitions and results about size functions, confining ourselves to those that will be useful in the rest of this paper. For a deeper investigation on these topics, the reader is referred to [2, 3, 28]. For further details about Topological Persistence in the multidimensional setting, see [7, 28].
In proving our new results we need to assume that is a closed Riemannian manifold. However, we prefer to report here the basic concepts of Size Theory in their classical formulation, i.e. by supposing that is a non-empty compact and locally connected Hausdorff space. We shall come back to the case of a Riemannian manifold later.
In the context of Size Theory, any pair , where is a continuous function, is called a size pair. The function is said to be a -dimensional measuring function. The relations and are defined in as follows: for and , we write (resp. ) if and only if (resp. ) for every index . Furthermore, is equipped with the usual -norm: . Now we are ready to introduce the concept of size function for a size pair . We shall denote the open set by , while will be the closure of . For every -tuple , the set will be defined as .
For every -tuple , we shall say that two points are -connected if and only if a connected subset of exists, containing and .
We shall call the (-dimensional) size function associated with the size pair the function , defined by setting equal to the number of equivalence classes in which the set is divided by the -connectedness relation.
In other words, is equal to the number of connected components in containing at least one point of . The finiteness of this number is a consequence of the compactness and local connectedness of (cf. ).
In the following, we shall refer to the case of measuring functions taking value in by using the term “-dimensional”. Before going on, we introduce the following notations: when is fixed, the symbol will be used to denote the function that takes each -tuple to the value . An analogous convention will hold for the symbol .
From Remark 1.3 it can be immediately deduced that for every fixed the function is non–decreasing with respect to , while for every fixed the function is non–increasing.
1.1. The particular case
In this section we will discuss the specific framework of measuring functions taking values in , namely the -dimensional case. Indeed, Size Theory has been extensively developed in this setting (cf. ), showing that each -dimensional size function admits a compact representation as a formal series of points and lines of (cf. ). Due to this representation, a suitable matching distance between -dimensional size functions can be easily introduced, proving that these descriptors are stable with respect to such a distance [11, 13]. Moreover, the role of -dimensional size functions is crucial in the approach to the -dimensional case proposed in .
Following the notations used in the literature about the case , the symbols , , , , will be replaced respectively by , , , , .
When dealing with a (-dimensional) measuring function , the size function associated with gives information about the pairs , where is defined by setting for .
Figure 1 shows an example of a size pair and the associated -dimensional size function.
On the left (Figure 1) one can find the considered size pair , where is the curve depicted by a solid line, and is the ordinate function. On the right (Figure 1) the associated -dimensional size function is given. As can be seen, the domain is divided into bounded and unbounded regions, in each of which the -dimensional size function takes a constant value. The displayed numbers coincide with the values of in each region. For example, let us now compute the value of at the point . By applying Remark 1.3 in the case , it is sufficient to count how many of the three connected components in the sublevel contain at least one point of . It can be easily verified that .
Following the -dimensional framework, the problem of comparing two size pairs can be easily translated into the simpler one of comparing the related -dimensional size functions. In , the matching distance has been formally proven to be the most suitable distance between these descriptors. The definition of is based on the observation that -dimensional size functions can be compactly described by a formal series of points and lines lying on the real plane, called respectively proper cornerpoints and cornerpoints at infinity (or cornerlines) and defined as follows:
For every point with , consider the number defined as the minimum, over all the positive real numbers with , of
When this finite number, called multiplicity of , is strictly positive, the point will be called a proper cornerpoint for .
For every line with equation , consider the number defined as the minimum, over all the positive real numbers with , of
When this finite number, called multiplicity of , is strictly positive, the line will be called a cornerpoint at infinity (or cornerline) for .
The fundamental role of proper cornerpoints and cornerpoints at infinity is explicitly shown in the following Representation Theorem, claiming that their multiplicities completely and univocally determine the values of -dimensional size functions.
For the sake of simplicity, each line of equation will be identified to a point at infinity with coordinates .
Theorem 1.7 (Representation Theorem).
For every , it holds that
In plain words, the Representation Theorem 1.7 claims that the value equals the number of cornerpoints lying above and on the left of . By means of this theorem we are able to compactly represent -dimensional size functions as formal series of cornerpoints and cornerlines (An example is given by Figure 2 and Figure 2).
Each discontinuity point for is such that either is a discontinuity point for , or is a discontinuity point for , or both these conditions hold.
We are now able to introduce the matching distance . Before going on, we observe that the Representation Theorem 1.7 allows us to reduce the problem of comparing -dimensional size functions into the comparison of the related multisets of cornerpoints. Indeed, the matching distance can be seen as a measure of the cost of transporting the cornerpoints of a -dimensional size function into the cornerpoints of another one, with respect to a functional depending on the -distance between two matched cornerpoints and on their -distance from the diagonal . An example of matching between two formal series is given by Figure 2.
Let us now define more formally the matching distance . Assume that two -dimensional size functions , are given. Consider the multiset (respectively ) of cornerpoints for (resp. ), counted with their multiplicities and augmented by adding the points of the diagonal counted with infinite multiplicity. If we denote by the set extended by the points at infinity of the kind , i.e. , the matching distance is then defined as
where varies among all the bijections between and and
for every , and with the convention about that when , , , , and .
In plain words, the pseudometric measures the pseudodistance between two points and as the minimum between the cost of moving one point onto the other and the cost of moving both points onto the diagonal, with respect to the max-norm and under the assumption that any two points of the diagonal have vanishing pseudodistance (we recall that a pseudodistance is just a distance missing the condition , i.e. two distinct elements may have vanishing distance with respect to ).
An application of the matching distance is given by Figure 2. As can be seen by this example, different -dimensional size functions may in general have a different number of cornerpoints. Therefore allows a proper cornerpoint to be matched to a point of the diagonal: this matching can be interpreted as the destruction of a proper cornerpoint. Moreover, we stress that the matching distance is stable with respect to perturbations of the measuring functions, as the following Matching Stability Theorem states:
Theorem 1.10 (Matching Stability Theorem).
If , are two size pairs with , then it holds that .
For a proof of the previous theorem and more details about the matching distance the reader is referred to [12, 13] (see also  for the analogue of the matching distance in Persistent Homology and its stability).
1.1.1. Coordinates of cornerpoints and discontinuity points
Following the related literature (see also  for the case of measuring functions with a finite number of critical homological values), it can be easily deduced that, if finite, both the coordinates of a cornerpoint for a -dimensional size function are critical values of the measuring function , under the assumption that is . However, to the best of our knowledge, this result has never been explicitly proved until now. Therefore, for the sake of completeness we formalize here this statement, that will be used in Section 2:
Let be a closed Riemannian manifold, and let be a measuring function. Then if is a proper cornerpoint for , it follows that both and are critical values of . If is a cornerpoint at infinity for , it follows that is a critical value of .
We confine ourselves to prove the former statement, since the proof of the latter is analogous.
First of all, let us remark that there exists a closed Riemannian manifold that is -diffeomorphic to through a -diffeomorphism (cf. [30, Thm. 2.9]). Set . Obviously, the size functions associated with the size pairs and coincide. Therefore, is also a cornerpoint for .
We observe that the claim of our theorem holds for a closed Riemannian manifold endowed with a Morse measuring function (see [25, Thm. 2.2]). Now, for every real value it is possible to find a Morse measuring function such that and : We can obtain by considering first the smooth measuring function given by the convolution of and an opportune “regularizing” function, and then a Morse measuring function approximating in the previous measuring function (cf. [32, Corollary 6.8]). Therefore, from the Matching Stability Theorem 1.10 it follows that for every we can find a cornerpoint for the size function with and as critical values for . Passing to the limit for we obtain that both and are critical values for . The claim follows by observing that, since and have the same critical values, both and are also critical values for . ∎
Let be a closed Riemannian manifold, and let be a measuring function. Let also be a discontinuity point for . Then at least one of the following statements holds:
is a discontinuity point for and is a critical value for ;
is a discontinuity point for and is a critical value for .
1.2. Reduction to the 1-dimensional case
We are now ready to review the approach to multidimensional Size Theory proposed in . In that work, the authors prove that the case can be reduced to the -dimensional framework by a change of variable and the use of a suitable foliation. In particular, they show that there exists a parameterized family of half-planes in such that the restriction of a -dimensional size function to each of these half-planes can be seen as a particular -dimensional size function. The motivations at the basis of this approach move from the fact that the concepts of proper cornerpoint and cornerpoint at infinity, defined for -dimensional size functions, appear not easily generalizable to an arbitrary dimension (namely the case ). As a consequence, at a first glance it does not seem possible to obtain the multidimensional analogue of the matching distance and therefore it is not clear how to generalize the Matching Stability Theorem 1.10. On the other hand, all these problems can be bypassed by means of the results we recall in the rest of this subsection.
For every unit vector of such that for , and for every vector of such that , we shall say that the pair is admissible. We shall denote the set of all admissible pairs in by . Given an admissible pair , we define the half-plane of by the following parametric equations:
for , with .
The following proposition implies that the collection of half-planes given in Definition 1.13 is actually a foliation of .
For every there exists one and only one admissible pair such that .
Now we can show the reduction to the -dimensional case.
Theorem 1.15 (Reduction Theorem).
Let be an admissible pair, and be defined by setting
Then, for every the following equality holds:
In the following, we shall use the symbol in the sense of the Reduction Theorem 1.15.
In plain words, the Reduction Theorem 1.15 states that each multidimensional size function corresponds to a -dimensional size function on each half-plane of the given foliation. It follows that each multidimensional size function can be represented as a parameterized family of formal series of points and lines, following the description introduced in Subsection 1.1 for the case . Indeed, it is possible to associate a formal series with each admissible pair , with describing the -dimensional size function . Therefore, on each half-plane , the matching distance and the Matching Stability Theorem 1.10 can be applied. Moreover, the family turns out to be a complete descriptor for , since two multidimensional size functions coincide if and only if the corresponding parameterized families of formal series coincide.
Before proceeding, we now introduce an example showing how the Reduction Theorem 1.15 works.
In consider the set and the unit sphere of equation . Let also be the continuous function, defined as . In this setting, consider the size pairs and where , , and and are respectively the restrictions of to and . In order to compare the size functions and , we are interested in studying the foliation in half-planes , where with , and with . Any such half-plane is represented by
with , . Figure 3 shows the size functions and , for and , i.e. and .
With this choice, we obtain that and . Therefore, Theorem 1.15 implies that, for every , we have
The matching distance is equal to , i.e. the cost of moving the point of coordinates onto the point of coordinates , computed with respect to the -norm. The points and are representative of the characteristic triangles of the size functions and , respectively. Note that the matching distance computed for and induces a pseudodistance. This means that, even by considering just one half-plane of the foliation, it is possible to effectively compare multidimensional size functions. We conclude by observing that and . In other words, the multidimensional size functions, with respect to , are able to discriminate the cube and the sphere, while both the -dimensional size functions, with respect to and , cannot do that. This higher discriminatory power of multidimensional size functions gives a further motivation for their definition and use.
The next result proves the stability of with respect to the choice of the half-planes of the foliation. Indeed, the next proposition states that small enough changes in with respect to the -norm induce small changes of with respect to the matching distance.
If is a size pair, and is a real number with , then for every admissible pair with , it holds that
Analogously, it is possible to prove (cf. [2, Prop. 2]) that is stable with respect to the chosen measuring function, i.e. that small enough changes in with respect to the -norm induce small changes of with respect to the matching distance.
2. Main Results
In this section we are going to prove some new results about the discontinuities of multidimensional size functions. In order to do that, we will confine ourselves to the case of a size pair , where is a closed Riemannian -manifold.
From now to Theorem 2.11 we shall assume that an admissible pair is fixed, considering the -dimensional size function , where . We shall say that and are the (-dimensional) measuring function and the size function corresponding to the half-plane , respectively.
The main results of this section are stated in Theorem 2.11 and Theorem 2.13, showing a necessary condition for a point to be a discontinuity point for the size function , under the assumption that is and , respectively. For the sake of clarity, we will now provide a sketch of the arguments that will lead us to the proof of our main results.
Theorem 2.11 is a generalization in the -dimensional setting of Corollary 1.12, stating that each discontinuity point for a -dimensional size function , related to a measuring function , is such that at least one of its coordinates is a critical value for . We recall that Corollary 1.12 directly descends from the Representation Theorem 1.7 and from Theorem 1.11, according to which each finite coordinate of a cornerpoint for has to be a critical value for . Our first goal is to prove that a modified version of this last statement holds for the -dimensional size function corresponding to the half-plane . The reason for such an adaptation is that the -dimensional measuring function is not (even in case is ), and therefore we need to generalize the concepts of critical point and critical value by introducing the definitions of -pseudocritical point and -pseudocritical value for a function (Definition 2.1). These notions, together with an approximation in of the function by functions, are used to prove that, if , each finite coordinate of a cornerpoint for has to be an -pseudocritical value for (Theorem 2.3). Next, we show (Proposition 2.4) that a correspondence exists between the discontinuity points of and the ones of . Theorem 2.3 and Proposition 2.4 lead us to the relation (Theorem 2.7) between the discontinuity points for , lying on the half-plane , and the -pseudocritical values for . This last result is refined in Theorem 2.11 under the assumption that is , providing a necessary condition for discontinuities of that does not depend on the half-planes of the foliation. This can be done by introducing the concepts of pseudocritical point and pseudocritical value for an -valued function (Definition 2.8), and considering a suitable projection . The necessary condition given in Theorem 2.11 is finally generalized to the case of continuous measuring functions (Theorem 2.13), once more by means of an approximation technique, and the notions of special point and special value.
Before going on, we need the following definition:
Assume that . For every , set . We shall say that is an -pseudocritical point for if the convex hull of the gradients , , contains the null vector, i.e. for every there exists a real value such that , with and . If is an -pseudocritical point for , the value will be called an -pseudocritical value for .
The concept of -pseudocritical point is strongly connected, via the function introduced in Definition 2.1, with the notion of generalized gradient introduced by F. H. Clarke . For a point , the condition of being -pseudocritical for corresponds to the one of being “critical” for the generalized gradient of [9, Prop. 2.3.12]. However, in this context we prefer to adopt a terminology highlighting the dependence on the considered half-plane.
We can now state our first result.
Assume that . If is a proper cornerpoint of , then both and are -pseudocritical values for . If is a cornerpoint at infinity of , then is an -pseudocritical value for .
We confine ourselves to proving the former statement, since the proof of the latter is analogous. The idea is to show that our thesis holds for a function approximating the measuring function in , and verify that this property passes to the limit. Let us now set and choose such that , for every . Consider the function sequence , , where and : Such a sequence converges uniformly to the function . Indeed, for every and for every index we have that
Let us now consider a proper cornerpoint of the size function . By the Matching Stability Theorem 1.10 it follows that it is possible to find a large enough and a proper cornerpoint of the -dimensional size function (associated with the size pair ) such that is arbitrarily close to . Since is a proper cornerpoint of , it follows from Theorem 1.11 that its coordinates are critical values of the function . By focusing our attention on the abscissa of (analogous considerations hold for the ordinate of ) it follows that there exists with and (in respect to local coordinates of the -manifold )