Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module
Micka\"el Buchet, Emerson G. Escolar
TL;DR
This paper proves that every 1D persistence module with finite support can be realized as a restriction of an indecomposable 2D persistence module, expanding understanding of the structure and visualization of multidimensional persistence modules.
Contribution
It provides a constructive proof that any finite-support 1D persistence module is a restriction of some indecomposable 2D persistence module, with new examples and generalizations to higher dimensions.
Findings
Any 1D finite support persistence module is a restriction of an indecomposable 2D module.
Existence of indecomposable 2D modules with support containing holes.
Finite-rectangle-decomposable nD modules are restrictions of (n+1)D indecomposable modules.
Abstract
A recent work by Lesnick and Wright proposed a visualisation of D persistence modules by using their restrictions onto lines, giving a family of D persistence modules. We give a constructive proof that any D persistence module with finite support can be found as a restriction of some indecomposable D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable D persistence modules whose support has holes as well as an indecomposable D persistence module containing all D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable D persistence module can be found as a restriction of some indecomposable D persistence module.
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Every 1D Persistence Module is a Restriction of Some Indecomposable 2D Persistence Module
Mickaël Buchet
Emerson G. Escolar
Abstract
A recent work by Lesnick and Wright proposed a visualisation of D persistence modules by using their restrictions onto lines, giving a family of D persistence modules. We give a constructive proof that any D persistence module with finite support can be found as a restriction of some indecomposable D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable D persistence modules whose support has holes as well as an indecomposable D persistence module containing all D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable D persistence module can be found as a restriction of some indecomposable D persistence module.
Keywords Representation theory, Multidimensional persistence
Mathematics Subject Classification (2010) 16G20, 55N99
1 Introduction
In the theory of persistent homology [11], D persistence modules can be summarized and easily visualized using the so-called persistence diagrams, which led to successful applications of topological data analysis. From a mathematical standpoint, the existence of the persistence diagram is a direct consequence of the fact that, viewed as representations of the underlying quiver , D persistence modules can be uniquely decomposed into indecomposable representations that are intervals [12]. The endpoints of the intervals give the birth and death indices of the topological features.
In the multidimensional persistence [7], multidimensional persistence modules can be studied as representations of an underlying quiver which is a commutative D grid. The indecomposable representations are no longer intervals and, more devastatingly, cannot be easily listed. More precisely, there is no complete discrete invariant that can describe all indecomposable D persistence modules when [7]. In the representation-theoretic language, the (large enough) commutative grid is of wild representation type.
Lesnick and Wright [14] proposed an interactive visualisation of D persistence modules using restrictions onto lines on the grid. For each line, the restriction is a D persistence module and thus can be summarized by a persistence diagram. They took the approach of exploring restrictions onto all possible lines of non-negative slope in order to visualize and obtain insight into the structure of a D persistence module. Note that considering the barcodes over all such lines is equivalent to considering the rank invariant [7] of the D persistence module.
For most of this work, we only consider persistence modules with finite support. For the sake of simplicity, unless stated otherwise, all persistence modules will have finite support. The only exception is the infinite D persistence module we build in Section 6.
In this work, we look into the possible outcomes of the restriction onto a single line of a D persistence module that is indecomposable. It was hoped that, being the building blocks of D persistence modules, the D indecomposables should have restrictions that are “simple” or coming from a restricted set of possibilities. However, we show that this is not the case.
Our main result is a constructive proof that any D persistence module can be obtained via the restriction onto a line of an indecomposable D persistence module (of large enough, but finite, support). We show the following Theorem (see Sec. 2 for precise definitions).
Theorem 1.1**.**
Let . Then there exists an indecomposable with finite support such that is a line restriction of .
Theorem 1.1 can be seen as another expression of how complicated the indecomposable D persistence modules are. Stated another way, the indecomposable D persistence modules collectively contain all possible D persistence modules via line restrictions. Another, less rigorous, interpretation is that by just looking at one line restriction of some D persistence module (not necessarily indecomposable), and without any other information about the original D persistence module , we cannot infer much about the decomposition structure of . More precisely, for each decomposition (with not necessarily indecomposable), it is possible that such that are indecomposable, restricts to , and restricts to [math] on the line. Note that our construction of an indecomposable is not the only choice of such an .
Using our construction for Theorem 1.1, we are able to build several objects of particular interest. First, we are able to exhibit indecomposable D persistence modules whose support can have an arbitrary finite number of holes. We believe that their existence was not obvious and that providing a concrete example will help deepen the understanding of D persistence modules. We also address the question of what is the smallest number of vertices needed in the support of an indecomposable D persistence module with at least one hole. Second, we build a single indecomposable D persistence module with infinite support that contains all possible D persistence modules with finite support as line restrictions.
Finally, we extend Theorem 1.1 to finite-rectangle-decomposable D persistence modules. In particular, we show that any finite-rectangle-decomposable D persistence module can be obtained via a “hyperplane restriction” of an indecomposable D persistence module. The construction directly allows us to build indecomposable D persistence modules with an arbitrary finite number of D holes.
We introduce basic definitions in Section 2 and tools in Section 3. We then describe our construction of an indecomposable D persistence module given and its properties in Section 4. In Section 5, we exhibit examples of D indecomposables with holey support obtained as a result of our construction. There, we also find the smallest number of vertices needed in the support of an indecomposable D persistence module with a hole, via an ad hoc construction. In Section 6, we build the indecomposable D persistence module with infinite support containing all possible D persistence modules with finite support as line restrictions. In Section 7 we discuss the extension to finite-rectangle-decomposable D persistence modules.
2 Background
We use the formalism of both category theory and representation theory, while providing clear intuition as much as possible. For convenience, we mainly use the representation theory of posets, but the representation theory of bound quivers is also used. If needed, we refer the reader to [3] for more details on the representation theory of bound quivers, and to [15] for the category theory.
2.1 The poset and D persistence modules
A poset (a partially ordered set) can be viewed as a category with objects , and morphisms determined as follows. There is a unique morphism if and only if . We abuse notation and write to also mean this unique morphism. Composition follows from transitivity: if there are morphisms and , then and , implying . Thus, there is a unique morphism , which is the composition of the two morphisms. The identity of object is clearly the morphism .
We define to be the set . By , we mean the poset category of with the usual order . We shall see below that the category is intimately related to the quiver
[TABLE]
which can be thought of as the Hasse diagram of .
Let be a field, which we fix throughout this work. Recall that a -category is a category where the set of morphisms between any two objects has -vector space structure, and composition is -bilinear.
Let be a poset category and a -category. A -valued representation of is a functor . That is, it assigns an object of to each , and to each morphism in (i.e. denoted by in the sequel) a morphism in , satisfying the following properties: is the identity on , and .
A morphism between two -valued representations of is nothing but a natural transformation. That is, given and , a morphism is a family of morphisms in such that
[TABLE]
commutes for all in . The set of morphisms from to is denoted by . Morphisms from to itself are called endomorphisms, and we use the notation . Composition
{F}$${G}$${H}$$\scriptstyle{\eta}$$\scriptstyle{\nu}
is the obvious one: is defined by . The category of -valued representations of shall be denoted by .
For the case , the category of finite-dimensional -vector spaces, we use the notation for the category of -valued representations, also called -linear representations. Unless specified otherwise, representation shall mean a -linear representation. The support of is the set .
It is known that is an additive -category. The direct sum of two representations and is given by for all . The zero representation is the functor [math] that takes everything in to zero vector spaces and zero maps. A representation is said to be indecomposable if implies or is [math].
For example, a representation of is an assignment of the following data: to each object , a finite-dimensional -vector space ; to each pair , a -linear map so that the composition
[TABLE]
is equal to for each , and is the identity for each .
Thus, is completely determined by vector spaces for and linear maps for . This corresponds to the usual notion of a representation of the quiver given in Diagram (2.1). In fact, , where the latter is the category of finite-dimensional -linear representations of the quiver . We shall freely use this identification. A D persistence module is simply a representation of .
A fundamental result (Krull-Schmidt Theorem) is that every representation in can be uniquely decomposed into a finite direct sum of indecomposable representations, up to isomorphism and permutation of terms. Furthermore, it is known [12] that any indecomposable representation of is isomorphic to an interval representation for some . The interval representation is defined by:
[TABLE]
Combining the above two facts, we get that for each ,
[TABLE]
uniquely for some positive integers and for with all pairs distinct. In other words, the D persistence module is, up to isomorphism, entirely determined by the multiset consisting of pairs with multiplicities for all . This can be drawn as a multiset of points in and is called the persistence diagram of the D persistence module .
In practice, the persistence module is often obtained as the sequence of homology vector spaces and induced homology maps of a sublevel set filtration parametrized over . In this case, the underlying poset is instead. However, some mild tameness assumptions on the filtration or persistence module guarantee the existence of a persistence diagram [8, 9]. Moreover, if there are only a finite number of “critical points”, then we can consider the persistence module as a representation of .
2.2 Commutative grids and line restrictions
Given , we will construct an indecomposable persistence module over an equioriented commutative D grid of size . The size of the grid will depend on itself, and so may be larger than . To avoid having to specify sizes in our definition of line restrictions, we use an infinite discrete commutative grid.
First, we extend to an infinite line by the inclusion of in the poset . Thus, is a full subcategory of . We use the following definition for the equioriented commutative D grids.
Definition 2.1** (Equioriented commutative D grids).**
We define to be the poset category with if and only if and . 2. 2.
is the poset category , which is the full subcategory of with set of objects .
To see why there is a “commutative” in the name, we provide as an example . A representation is completely determined by the data:
[TABLE]
where by functoriality,
[TABLE]
That is, a commutativity relation is automatically satisfied.
We call representations of as D persistence modules. If has finite total dimension, then it has finite support, and can be viewed (after translation) as a representation of some finite D grid .
Alternatively, is isomorphic to the representation category of the equioriented D grid quiver bound by commutativity relations. See for example, [1], for precise definitions.
Definition 2.2** (Line).**
A line is a functor
[TABLE]
such that if , then .
Note that the definition above does not correspond exactly with a geometric “line”. For example, the functor defined on objects by is a line. However, it geometrically intersects points on the D grid that are not part of its image, for example the point . Our result does not rely on this peculiarity.
Given a line , we get an induced functor between representation categories via composition:
[TABLE]
and with effect on morphisms also by composition. That is, if is a morphism between representations of (a natural transformation), then defines the morphism .
Definition 2.3** (Line Restriction).**
Let . We say that is a line restriction of if there is a line such that .
3 Tools
3.1 Stacking
To prove Theorem 1.1, for each we construct an indecomposable D persistence module such that is a line restriction of . In order to easily construct D persistence modules, we stack together representations of (for some longer ). This stacking procedure can be theoretically justified by the following lemma.
Lemma 3.1** (Stacking).**
There is an isomorphism of categories
[TABLE]
In words, representations of the commutative grid can be viewed as -valued representations of a D grid (the vertical line ) by collapsing each row to an object of . In this interpretation, the representations of are drawn as rows, and of them are stacked in a line, with a morphism from the th row to the th row for .
This can also be seen as “currying” a functor in . More generally, this is analogous to tensor-hom adjunction in -enriched functor categories, as in Section 2.3 of [13]. Note that the poset categories , , and are not -enriched, but we can consider instead their -linearizations , , and . The analogy follows by considering -linear functors from the -linearizations instead of functors from the poset categories and noting that . We provide an elementary proof of Lemma 3.1 in Appendix A.
The stacking lemma is an obvious extension of an idea in the paper [2], where representations of commutative ladders, which are commutative D grids of size , are viewed as morphisms in . This is essentially stacking two representations of .
Objects in the category are -valued representations of . Thus, each arrow is given the data of a morphism in since is -valued, instead of a morphism between vector spaces as when -valued. In the next subsection, we study morphisms in and provide a convenient way to describe them.
3.2 Matrix Formalism
Let us quickly review the matrix formalism introduced in the paper [2], which allows us to write a morphism in in a matrix form. In this work, we only need to consider and equal to a direct sum of the interval representations, and this simplifies the presentation below.
Let be a morphism in with and given as below:
[TABLE]
Note that the intervals for (respectively for ) may contain duplicates. This is not a problem.
Then can be written in a matrix form relative to these decompositions, with entries given by :
[TABLE]
where and are the evident inclusions and projections, respectively. Instead of a scalar, each entry in the matrix form is a morphism from to . The following lemma tells us the possible morphisms between intervals.
Lemma 3.2** (Lemma 1 of [2]).**
Let be interval representations of .
The dimension of as a -vector space is either [math] or . 2. 2.
There exists a canonical basis for each nonzero such that
[TABLE] 3. 3.
In fact,
[TABLE]
Going back to our morphism in matrix form, the entry at is given by
[TABLE]
for some scalar if the dimension of is nonzero; otherwise is always [math]. As a shorthand, we hide the morphisms whenever we display the matrix form of morphisms in .
3.3 Auslander-Reiten quivers
We give a quick review of Auslander-Reiten quivers, which provide a very convenient framework to visualize our construction. By definition, the vertices of the Auslander-Reiten quiver are the indecomposable representations, and arrows are determined by the irreducible morphisms between indecomposables. See the book [3], for example, for a more detailed discussion.
For our purposes, we only need to consider the Auslander-Reiten quiver of , which takes on a relatively simple form. It is given by
[TABLE]
where and where we adopt the convention of placing the simple representations (lifespan [math] intervals) on a horizontal line, with everything else below.
To visualize our construction, we shall abstractly represent the Auslander-Reiten quiver of by a triangle:
[TABLE]
Furthermore, we will need to freely extend the length of the underlying poset to some . We will thus consider the Auslander-Reiten quiver of as being embedded inside that of , like so:
[TABLE]
where the top line keeps track of the simple representations, with at the leftmost and at the rightmost.
4 Main Construction
Before proving our main result, we introduce the following concept, which is essential in our proof for indecomposability. This is an offshoot of the idea behind the algebraic construction in [5], where certain configurations of indecomposables were found.
Definition 4.1** (Vertical).**
A collection of interval representations is said to be vertical if and only if all the pairs are distinct and there exists a constant such that for all .
If a collection of intervals is vertical, then they are all located along a vertical line on the Auslander-Reiten quiver, as follows:
[TABLE]
because of the way we illustrate them. The following (almost trivial) observation about vertical intervals forms an essential part of our construction. Note that the matrix form below is written using the matrix formalism as reviewed in Subsection 3.2.
Lemma 4.2**.**
Let be a representation of such that is vertical. Then the matrix form of any morphism is diagonal:
[TABLE]
for some scalars (recall our shorthand of hiding the morphisms ).
Proof.
Since is vertical, there exists as in the definition and distinct numbers such that and for all .
For , it follows from part 3 of Lemma 3.2 that
[TABLE]
since the inequality cannot be satisfied. Otherwise, we have
[TABLE]
implying and , and so . This is a contradiction since and are supposed to be distinct. Thus, off-diagonal terms are always zero. Terms on the diagonal are some constant times , which are identity maps. ∎
We reproduce here our Theorem 1.1 and provide a proof.
See 1.1
Proof.
Without loss of generality, we identify . The proof takes the form of a construction process composed of three parts: separate-and-shift, verticalize, and coning.
Separate-and-shift
We separate our intervals into distinct death-indices with the extra condition that these indices are large enough so that the next step is possible.
More precisely, we choose distinct integers such that
for all and 2. 2.
for all pairs and .
The first condition is clearly possible to satisfy. For the second condition, choose distinct satisfying the first, and let and define . Then
[TABLE]
for all pairs , showing that both conditions can be satisfied by distinct integers .
We then choose an integer such that for all . For example, will do. We have the inclusion , so that we can consider as a representation of , essentially by padding [math]’s from vertices to . Further constructions will take place in .
By construction, there exists a nonzero morphism for each , which are in fact surjections. We form the representation and the morphism
[TABLE]
(written in the matrix formalism). The diagonal entries can indeed be nonzero because of the existence of the nonzero morphisms.
The matrix form given in Diagram (4.1) is not the identity matrix and does not represent the identity morphism. The entries are morphisms between intervals, and we are hiding the factors for simplicity.
Pictorially, we think of as a multiset of points on the Auslander-Reiten quiver of , which is considered as a subquiver of . Separate-and-shift moves each point (treating multiple copies of the same point as different) to a different death-index . We illustrate this procedure as follows:
[TABLE]
where the rightmost point appears with multiplicity , and each copy is brought to a different death index.
Verticalize
We move our separated-and-shifted intervals into a vertical by increasing birth indices. The “shift” part (the second condition) of the previous step ensures that the increased birth indices do not go beyond the death indices.
Let , the maximum of the midpoints of the intervals . Note that by second condition of the previous step, for all . We define for all , which can be visualized as “pivoting” around to get . Note that while may be a half-integer, so defined is always an integer.
It follows that
[TABLE]
for all pairs . This inequality is very important for our construction, and we will refer again to this fact later.
In particular, we see that for all . Thus, the intervals are non-empty and distinct (since are), and is vertical with common midpoint as given. Furthermore, Ineq. (4.2) says that the common midpoint shared by all serves as a wall separating all birth indices and all death indices .
Since , there exists a nonzero for each , which are in fact inclusions. We form the representation and the morphism
[TABLE]
We illustrate this step by the following figure:
[TABLE]
Coning
There exists
[TABLE]
with nonzero morphisms to all by using Lemma 3.2 and the fact that for all from Ineq. (4.2). We can thus construct the morphism
[TABLE]
with nonzero entries.
Constructed object and proof of indecomposability
Our construction is summarized in the following picture:
[TABLE]
and we obtain , a -valued representation of
[TABLE]
determined by
[TABLE]
where , is obtained by separate-and-shift, by verticalization (of ), and by coning.
Let us show that is indecomposable. Any is given by a collection of maps making the diagram
[TABLE]
commutative.
Since the summands of are vertical, has matrix form
[TABLE]
for some scalars by Lemma 4.2.
By commutativity of the second and third boxes in Diagram (4.5), the scalars in the matrix forms of and are nonzero only along the diagonal and with scalars along the diagonal given by . This involves some technical details, which we delay to Lemma 4.3, where Ineq. (4.2) is essential.
By the commutativity of the first box in Diagram (4.5), we get . Thus, is local and so is indecomposable.
We have shown that is indecomposable. By vertical stacking Lemma 3.1, this can be viewed as an indecomposable representation of the commutative grid . This can be viewed (by padding with zeroes), as an indecomposable representation of with finite support.
The original representation is obtained by line restriction of to the horizontal line that runs along the th row. ∎
Finally, let us prove the technical details used to prove the form of the endomorphisms of .
Lemma 4.3** (Propagation of nonzeroes).**
Let and such that for all , and let be the morphism given in matrix form with only nonzeros in the diagonal given by the inclusions , as above. Suppose that
[TABLE]
is commutative. Then the matrix form of is . 2. 2.
Let and such that for all , and let be the morphism given in matrix form with only nonzeros in the diagonal given by the projections , as above. Suppose that
[TABLE]
is commutative. Then the matrix form of is .
Before giving the proof, we give an example where the conclusion of part 1 fails to hold if the condition for all is not satisfied. If is vertical, then this condition is automatically satisfied, so we construct an example where this is not the case. We let and The death-indices are fixed at and , while and . Note that . The commutative diagram (4.7) becomes
[TABLE]
Note that is the identity morphism.
The composition is
[TABLE]
However, the composition is
[TABLE]
where the lower-left entry is equal to [math] since the composition . This can be illustrated by the following.
[TABLE]
The birth index has exceeded the smallest death index , and so the composition “falls through” and fails to be nonzero.
The commutativity of the diagram implies the equality of Eq. (4.8) and Eq. (4.9), from which we obtain and , but is not determined. Thus, we cannot conclude that the matrix form of is the same diagonal.
Proof of Lemma 4.3.
The matrix formalism hides the morphisms , and the composition of two nonzero composable can possibly be zero. What is important for this lemma is that the compositions should be nonzero where needed, so that the commutativity relation forces the scalars to be zero off-diagonal and the same as along the diagonal.
The composition is equal to
[TABLE]
The entries with nonzero are given by by Lemma 3.2 (these are the entries where are present). For each such entry, the corresponding morphism in the composition is given by
[TABLE]
Using Lemma 3.2 and the fact that , we see that this composition is nonzero.
Thus, has scalars in its matrix form the same as (nonzeros cannot go to zeros). Equating with the other composition in the commutative diagram, for all and for , as required. 2. 2.
The composition is equal to
[TABLE]
The entries with nonzero are given by by Lemma 3.2. In these entries, the composition
[TABLE]
is also nonzero by Lemma 3.2 and the fact that . Then the conclusion is obtained by a similar argument as above.
∎
Dual Construction
We note that a dual construction is possible. Instead of constructing to the left, we can proceed to the right and obtain
[TABLE]
where is obtained by separate-and-shift choosing smaller (possibly negative) birth-indices that are distinct, and by verticalization, and by coning. Note that in this case, the correct condition to use for separate-and-shift is to have all distinct with for all and for all pairs . Then we use and for verticalization. Coning is given by . Here, we can check that for all pairs , which allows the ’s in the matrix form of the morphism to , and enables the dual version of propagation of nonzeros.
Remark 4.4**.**
Note that the primal and dual constructions can be combined into one, yielding the following object
[TABLE]
of , where and are obtained from separate-and-shift to the right and left, and from verticalization, and and from coning. This corresponds to a D persistence module by Stacking Lemma 3.1.
We note that the support of the obtained module is contained in a rectangle, with the lower right and upper left corners both supporting one-dimensional vector spaces. Moreover its endomorphism ring is isomorphic to . In general, we call D persistence modules with these properties candy modules as they are wrapped like candies with two handles we can manipulate. These modules will play a central role in the construction of section 6.
5 Consequences on the homology of supports
One consequence of our construction is the easy construction of examples of indecomposable representations of (indecomposable D persistence modules) whose support has a “hole”. We contrast this with interval representations, a restricted class of representations that has been studied before [4, 10]. By definition, an interval representation associates to each vertex a vector space with dimension at most , its internal linear maps are always identity between non-zero vector spaces, and its support satisfies a convexity and connectedness condition. In particular, the support of any interval representation of always has a distinctive staircase shape [1].
5.1 Indecomposables with holes in support
Note that while the technical details of our construction use nonnegative indices, it still works after shifting all indices. Let , whose form is chosen for symmetry. We follow Remark 4.4 and obtain by the primal and dual construction:
[TABLE]
corresponding to the D persistence module
[TABLE]
where all but one zero vector space outside the support are omitted for the sake of clarity. Maps between identical vector spaces are to be understood as the identity maps. Then by the arguments in the proof for Theorem 1.1 and the dual construction, we see that , and thus is indecomposable. Moreover its support has a hole as the central vector space is [math].
One way to formally define the notion of the hole in the support is to consider the cubical complex induced by the support of a D persistence module as follows. Recall that the support of is the set of points . We then build a cubical complex by adding an edge between and whenever and and are adjacent on the D grid. Whenever we have four points for some , we add a unit square with those corner vertices. This is the clique-cubical complex of . We then consider holes in the support as being non-trivial classes in the -dimensional homology of this cubical complex. We say that the support of a D persistence module has holes if the -dimensional homology of the cubical complex built on the support has linearly independent classes. It is clear that the representation given in Diagram (5.1) has a hole in this sense.
Our construction can then be used to generate indecomposable D persistence modules having an arbitrary finite number of holes in its support.
Corollary 5.1**.**
Let . There exists an indecomposable D persistence module whose support has exactly holes.
Proof.
Let be as given, and fix .
Note that has exactly connected components. Again we define the indecomposable representation
[TABLE]
by using the construction in Theorem 1.1 and its dual. By the Stacking Lemma 3.1, is viewed as a D persistence module.
By construction, each representation has connected support because all of the intervals composing have a common index (depending on ) where they are non-zero. Moreover, the right extremities of , and are aligned, and thus the cubical complex of the support of is contractible. Similarly the left extremities of , and are also aligned, and the cubical complex of the support of is also contractible.
Finally, the support of is included in both the support of and , ensuring that every neighboring pair of connected components in the support of can be used to generate a non-trivial -homology class in the cubical complex of . There are such pairs, and it is clear that has exactly holes in its support.
Below, we illustrate the clique cubical complex of the support of .
[TABLE]
Note that this realizes a topological suspension of the support of . ∎
5.2 A minimal construction
The construction used to prove Theorem 1.1 works for general D persistence modules , but does not consider any size minimality for particular cases of . For example, the persistence module in Diagram (5.1) is not the smallest example of an indecomposable D persistence module whose support has a hole. Indeed, using only the primal construction to obtain the bottom half of Diagram (5.1), and “capping off” the top of with an interval would have yielded an example with smaller support.
Before discussing a minimal example in Theorem 5.4, let us study a local construction that we use repeatedly for its proof. Consider the family of D persistence modules whose support is contained in a small “cross” shape. The small cross shape is the vertices , , , , (filled-in circles) and filled-in arrows , , , :
[TABLE]
where the unfilled circles and grayed-out arrows are not part of the support.
Consider . Since the upper-left and lower-right vertex are not part of the support, associates the zero vector space to those vertices. By commutativity relations, the compositions and must be zero. These zero relations are denoted by the dashed lines. Thus, we can also treat as a representation of the corresponding bound quiver.
Lemma 5.2**.**
There is no indecomposable whose support contains both the bottom and right vertices ( and ) or both the left and top vertices ( and ).
We provide a proof in the Appendix.
Remark 5.3**.**
Note that the quiver formed by the filled-in vertices and arrows in Diagram (5.2), together with the relations and , generates what is known as a string algebra. Lemma 5.2 in fact follows immediately from a known characterization of indecomposable representations of string algebras as the string modules and band modules [6].
The important consequence of Lemma 5.2 is that given any D persistence module on such a cross shape with zero relations as in Diagram (5.2), bases for its vector spaces can be chosen that allows for a decomposition into the direct sum of two representations such that and , or symmetrically with and .
Theorem 5.4**.**
The minimal number of vertices in the support of an indecomposable D persistence module whose support has a hole is equal to . For example, the D persistence module
[TABLE]
realizes this minimum.
Proof.
The support of contains exactly vertices and has a hole. An elementary computation shows that the endomorphism ring of is isomorphic to , and thus is indecomposable. It is then sufficient to prove that there is no indecomposable D persistence module whose support has a hole and contains or less vertices. The existence of a hole requires at least vertices in the support.
vertices
Suppose that is an indecomposable D persistence module whose support has exactly vertices and has a hole. Then is of the form:
[TABLE]
By commutativity, both and are zero maps. We can thus construct the following:
[TABLE]
where and are the induced maps on the respective coimages. It is a simple matter to check that . Since the support of has exactly vertices, and are nonzero, and so and are nonzero. This contradicts the fact that is indecomposable.
In order to prepare for the remaining cases, let us rephrase this argument using Lemma 5.2 and its proof. First, we view as a representation of
[TABLE]
We then restrict our attention to the “partial crosses” centered at and , and use a similar argument as in the proof of Lemma 5.2. By choices of bases, we obtain summands where the vector spaces at vertices and are not both nonzero, and where the vector spaces at vertices and are not both nonzero.
These choices can be done simultaneously for the two partial crosses, as they share no vertices. In general, such a “local” decomposition does not induce a decomposition of a representation as a whole.
However, in this case, we can build the following two subrepresentations and of . For , we take the decomposition summands over the two crosses which are non-zero at vertices and , and then add the vector space at vertex with the linear maps from to and to . Using Lemma 5.2 ensures that we can simply fix the vector spaces of at , and to be [math], and that the result is indeed a subrepresentation. Note that we obtain the same as defined previously. Symmetrically, we obtain , and it is a simple matter to show that .
vertices
A support with vertices and containing a hole can only be obtained by adding one vertex somewhere on the outside of Diagram (5.3). Depending on where these vertices are added, an appropriate zero relation needs to be imposed in order to ensure the embeddability in the equioriented commutative D grid. For example, with the additional vertex in the support, the underlying bound quiver becomes
[TABLE]
where we still have partial crosses centered at and . The same argument as the one in the case of vertices, using these two partial crosses, shows that a D persistence module with such a support cannot be indecomposable.
Note that the extra vector space at vertex is handled in two different ways depending on the position of . If is part of one of the partial crosses at or then it is automatically split in a direct sum of vector spaces, one being part of and the other part of by the use of Lemma 5.2.
If is not part of one of the crosses (for example Diagram (5.4)), then the vector space on its adjacent vertex ( in Diagram (5.4)) is split into a part in and a part in . By construction, at least one of those must be [math]. The vector space at vertex together with the map connecting it to the rest of the diagram can be added to the subrepresentation with the not necessarily zero part. For example, in Diagram (5.4), the subrepresentation is [math] at vertex . The vector space and linear map of associated to can thus be added to the obtained in the vertices case while guaranteeing that is still a subrepresentation.
vertices
The case with vertices can be obtained in two ways.
First, the hole in Diagram (5.3) can be made wider by adding two vertices (and corresponding arrows) to the support, for example between and and between and . This means that the D persistence module will need to encompass two [math] vector spaces in its center. The proof for this case is similar to the one for vertices.
The second case is obtained when we add two vertices (and corresponding arrows) to Diagram (5.3) on the outside. Depending on where these vertices are added, zero relations or commutativity relations need to be imposed in order to ensure the embeddability in the equioriented commutative D grid, as we shall see below.
Thus we further subdivide this case. First, in the case that the two new vertices are not adjacent, we only need to deal with new zero relations. For example, both vertices can be attached at corner , as below:
[TABLE]
The argument using the partial crosses can be used as before.
A second possibility is that the two new vertices are adjacent, for example:
[TABLE]
where we need to impose the commutative relation in the right square.
Even in this case, a variant of the argument in Lemma 5.2 is still feasible. Suppose a D persistence module has Diagram (5.5) as its support. We construct subrepresentations and . For , its vector spaces at vertices , , , and are obtained as before, using the same partial cross argument around vertex .
We set the vector space of at vertex as the image of under the linear map of from to . Similarly, the vector space is the image of under the composition of maps from to to .
By commutativity and the zero relation, is in the kernel of the map from to . Thus, and can be set to [math], along with and as before, while guaranteeing that is indeed a subrepresentation of .
Similarly, a subrepresentation that is [math] on , , can be constructed such that . As before, we note that is nonzero at and is nonzero at , showing that is in fact decomposable.
Finally, it is possible to have adjacent new vertices without inducing a commutative relation either by having an arrow elongating a path such as:
[TABLE]
where the arguments from the 9 vertex case apply. Otherwise the new arrow points in the opposite direction of the existing one such as:
[TABLE]
This case is a bit more involved. However a careful computation on the subpart induced by vertices leads to a decomposition similar to the one of Lemma 5.2, enabling the same arguments to follow as in the other cases.
In all cases considered, a persistence module with support of the prescribed shape and number of vertices cannot be indecomposable. This shows the claimed result. ∎
6 A representation to contain them all
In Section 4, we showed that for every D persistence module in , we can build an indecomposable D persistence module containing it as a line restriction. We can go further and build an indecomposable D persistence module with infinite support that contains all D persistence modules with finite support as line restrictions.
Recall that the primal and dual construction for Theorem 1.1 starting from a D persistence module results in a D candy module (see Remark 4.4) of the form
[TABLE]
Given two candy modules and , we define the following concatenation operation. Let be the rightmost vertex in the lowest interval of . We translate such that the leftmost vertex of the top interval of (denoted ) is located below and to the right of :
[TABLE]
Note that the vector spaces at both and are . On the vertex below (and to the left of ), we associate the vector space , and put identity maps on the arrows to and . The concatenation is the D persistence module defined as being equal to and restricted to their respective supports, over , and with identity maps over the arrows from to and to , and zero elsewhere.
Lemma 6.1**.**
Given and two candy modules, is a candy module.
Proof.
It is easy to see that is a well-defined D persistence module, as no commutativity relations are violated by our construction. Indeed the supports of and are disjoint, and there are no paths from one to the other. Moreover, the addition of the vector space over and the two identity maps only create paths from to the upper row of and the right column of .
Next, we check that is indeed a candy module. Its top left vertex corresponds to the top left vertex of and is associated to the vector space , while its bottom right vertex corresponds to the bottom right vertex of and also supports .
By the property of candy modules, . Let be an endomorphism of . The restriction of to the support of , respectively and , is defined by a unique scalar , respectively , . Therefore there exist two non-zero constants and such that restricted at vertex , respectively and is the multiplication by the scalar , respectively and . The commutativity conditions along the arrows from to and from to implies that . Note that no other conditions are required for . Therefore is completely determined by one scalar , and the endomorphism ring of is isomorphic to . ∎
Lemma 6.2**.**
The set of isomorphism classes of D persistence modules of finite total dimension is a countable set.
Proof.
Note that a representative for a class can be decomposed as
[TABLE]
in a unique canonical way. Indeed the intervals in an indecomposable decomposition can be ordered using the lexicographic order on . We thus have a unique representative for each class and denote by the set of all those representatives.
We define to be the set of all elements of whose total dimension is less than and whose support is contained in . Every set is finite and . Therefore the set is countable and is in bijection with the set of all isomorphism classes of D persistence modules of finite total dimension.
Note that the sequence is in fact a filtration of as . ∎
Theorem 6.3**.**
There exists an indecomposable D persistence module such that any D persistence module of finite total dimension is a line restriction of .
Proof.
As a consequence of Lemma 6.2 the set of all non-zero D persistence modules up to isomorphism and translation is a countable set. We can enumerate its elements as a sequence . For each of these , we apply our dual and primal construction to obtain a candy module , as in Remark 4.4. Note that every has exactly seven rows and that its endomorphism ring is , see the proof of Theorem 1.1 for details.
We concatenate to obtain a D persistence module as below:
[TABLE]
The proof of Lemma 6.1 naturally extends to the setting of a countable concatenation, and therefore is a well-defined persistence module with . Thus is an indecomposable D persistence module.
Note that the line just above the module is [math] so also contains the zero D persistence module as a line restriction. By construction all non-zero D persistence modules with finite total dimension appear as a line restriction to the middle row of the corresponding candy module . Every vertex supports a finite dimensional vector space so is locally finite. Obviously has infinite support and vector spaces with arbitrarily high dimension. ∎
7 Rectangle-decomposable D persistence modules
Denote elements of as -tuples . We consider the poset , where if and only if for all .
Recall that by D persistence module we mean a functor in . Throughout this section, persistence module will mean persistence module of finite support: is a finite set.
Definition 7.1**.**
Let . The finite rectangle from to , denoted , is the D persistence module defined by
[TABLE]
The support of is exactly the points on the D grid contained in the rectangle
[TABLE]
We denote this set of points on the D grid by .
A more general treatment would involve possibly unbounded rectangles (See [4] for example). However, we only treat D persistence modules of finite support, and this finiteness is essential to our proof of Theorem 7.7.
Definition 7.2**.**
An D persistence module is said to be finite-rectangle-decomposable if and only if
[TABLE]
for some elements for .
Note that we assume finite supports, and only discuss finite sums. The finite rectangles do not need to be distinct.
The following lemma generalizes Lemma 3.2 to finite rectangles.
Lemma 7.3**.**
Let be finite rectangular D persistence modules.
The dimension of as a -vector space is either [math] or . 2. 2.
There exists a canonical basis for each nonzero such that
[TABLE] 3. 3.
In fact,
[TABLE]
Proof.
Lemma 3.3 and Corollary 3.4 of [4]. ∎
Thus, for morphisms between finite-rectangle-decomposable D persistence modules, we have a similar matrix formalism as in the D case. As before, we adopt the convention of hiding the morphisms for simplicity.
Definition 7.4** (Vertical).**
A collection of finite rectangles is said to be vertical if and only if all the pairs are distinct and there exists a constant such that for all .
In the definition above, the arithmetic operations are performed component-wise. The following is the generalized version of Lemma 4.2.
Lemma 7.5**.**
Let be finite-rectangle-decomposable D persistence module whose set of rectangles is vertical. Then the matrix form of any morphism is diagonal:
[TABLE]
for some scalars .
Proof.
Similar to the proof of Lemma 4.2, and using Lemma 7.3. ∎
Definition 7.6** (Hyperplane Restriction).**
Let be a positive integer.
A hyperplane is a functor that is injective on objects. 2. 2.
Let be an D persistence module. We say that is a hyperplane restriction of the D persistence module if there is a hyperplane such that .
Note that in the proof of Theorem 1.1, we used the Auslander-Reiten quiver of in order to visualize the movements of the intervals. However, it was not essential to the proof. Above, we have provided generalizations of the essential tools to finite rectangles. Thus, the following theorem can be proved similarly as Theorem 1.1, except we no longer draw the Auslander-Reiten quiver.
Theorem 7.7**.**
Let be a finite-rectangle-decomposable D persistence module. Then there exists an indecomposable D persistence module with finite support such that is a hyperplane restriction of .
Proof.
The proof is similar to the proof of Theorem 1.1. Below is a quick outline.
Without loss of generality, let . We then separate-and-shift to distinct such that for all and for all pairs and . Take
[TABLE]
where the is taken component-wise. This will serve as the shared midpoint for verticalization, by defining . Again, we have
[TABLE]
for all pairs , . Finally, coning is done with the finite rectangle
[TABLE]
We get the object , a -valued representation of :
[TABLE]
where , is obtained by separate-and-shift, by verticalization (of ), and by coning. By stacking, can be viewed as a representation of , id est, an D persistence module. Note that the D persistence module obtained from clearly has finite support. The original D persistence module is clearly a hyperplane restriction by a natural inclusion .
It can be shown that , proving the indecomposability of . Note that a generalization of Lemma 4.3 (propagation of nonzeros) to finite-rectangle-decomposables is needed, but the proof is similar to that of Lemma 4.3. ∎
Note that, as before, the construction in the proof of the theorem has a dual, which we denote by . A direct consequence of this extension is the construction of indecomposable D persistence modules whose support has an arbitrary number of D holes. Here, D holes in the support can be formalized as linearly independent classes of the -dimensional homology of the clique cubical complex of the support.
Corollary 7.8**.**
For any , there exists an indecomposable D persistence module whose support has exactly D holes and no lower dimensional hole.
Proof.
The proof is a direct adaptation of the proof of Corollary 5.1.
Clearly, it is possible to build an D persistence module that is rectangle decomposable and contains exactly D holes in its support and no lower dimensional hole, for example by building with rectangles the boundaries of D hypercubes.
Note that the construction of Theorem 7.7 can be dualized in the same way as the one of Theorem 1.1. As in the proof of Corollary 5.1, we obtain the D persistence module obtained by combining the primal and dual construction with in the middle. The D persistence module has support with exactly D holes and is indecomposable.
To see this, by considering the construction as working in the up and down direction, we can interpret it as stacking D slices: , , and below for the primal construction, and , , and above for the dual construction. Each of those new slices has contractible support, and they are connected along a contractible intersection, so that has contractible support (and similarly for the dual part).
The construction with primal and dual parts modifies the support to become a double cone with apexes and , in other words its topological suspension. Such operation transforms each D hole into an D hole for every , without creating other topological features. In our case this results in the requested topology. ∎
Gluing the constructions in all dimensions using an operation similar to the concatenation of Section 6, we obtain the following more interesting result.
Theorem 7.9**.**
Given a positive dimension and a sequence of non-negative integers, there exists an indecomposable D persistence module whose support has for its Betti number for every .
Proof.
Using Corollary 7.8, we first build a sequence of indecomposable persistence modules for that satisfy the condition for exactly the th Betti number. More precisely, each is an indecomposable D persistence module whose support has exactly -dimensional holes and no other holes. In other words, its Betti numbers are in dimension [math], in dimension and [math] otherwise.
By construction, has two other important properties. First, . Second, the support of has two extremal vertices, along which we string together these persistence modules. These extremal vertices are minimal and maximal under the following partial order: the usual order for each of the first coordinates and the opposite order for the last coordinate (along which we stack). In symbols, we define if and only if for all and . For example, . Under this order and by construction, the support of has a minimal element and a maximal element and both of them host a one-dimensional vector space. Note that this order is only introduced to simplify the specification of these extremal points. We are not changing the underlying poset of which is a representation.
We now build an indecomposable persistence module with the required property inductively. Let . Clearly, is an indecomposable D persistence module with and whose support has Betti numbers . Furthermore, (a one-dimensional -vector space), by construction.
Assume that we have built , an indecomposable D persistence module with , Betti numbers , and with maximal element in its support where .
We embed into the -dimensional grid by adding an extra [math] coordinate at the end of all indices, and denote it by . We translate the D persistence module in the grid such that the maximal element (which is after embedding) and (after this translation) are adjacent on the D grid, with the only difference coming from the last coordinate.
The D persistence module is defined to be equal to over its support and equal to the translated over its support, with the identity linear map for the arrow from to , and zero elsewhere.
First, is obtained by assembling the two modules and . To see that this is indeed a D persistence module, we check that no nontrivial commutativity condition exists between the two parts as no pair of vertices in the support of and are comparable except for those on the same slice as . In other words, no path in the grid, from the support of to the support of , exists that does not go through the newly added arrow.
The support of is obtained by linking the support of and by a single arrow. The Betti numbers of are therefore the sum of the Betti numbers of and for all positive dimensions and the sum minus one for . Hence, has Betti numbers .
The shifted index is maximal for .
The endomorphism ring of is isomorphic to . Indeed, let be an endomorphism of . By hypotheses, is uniquely defined by two scalar values and corresponding to the parts and . However, we have an extra condition due to the arrow linking the two parts of the module. An elementary computation shows that this condition implies that for some constant , and therefore every endomorphism is uniquely defined by one scalar.
By induction satisfies all the desired conditions and therefore is an indecomposable D persistence module whose support has Betti numbers . ∎
8 Discussion
The construction presented in this paper shows, for any given D persistence module with finite support, how to build a D indecomposable module containing it as a line restriction. In fact we also built a single D indecomposable with infinite support that contains all D persistence modules with finite support as line restrictions.
We then showed that our main construction naturally extends to any finite-rectangle-decomposable D persistence module, showing that each can be found in some indecomposable D persistence module as a hyperplane restriction. As a side result, we were also able, given arbitrary numbers for , to build an indecomposable D persistence module whose cubical complex has Betti number for all dimensions .
Our construction can be extended to persistence modules indexed over that can be decomposed into a finite direct sum of finite intervals. This extension requires a careful handling of the various types of intervals (open, closed and half-open), and separate-and-shift requires an extra enlarging of death indices (the second condition should be a strict inequality).
It is natural to ask whether or not our construction can be made functorial. In particular, suppose that for each , we fix exactly one construction as in Eq. (4.4) (recall that there are many choices in the construction; fix one choice for each). If we let be the horizontal line at index (th row), clearly , where is viewed as an object in . Indeed, this is what we proved in Theorem 1.1.
Then the question is as follows. For each morphism , can we give a morphism so that becomes a functor? More strictly, is it possible to give the values so that is (or at least isomorphic to) the identity functor? Unfortunately, it seems like our construction cannot be made into a functor in such a nice way.
Our methods relying on the structure of D interval (D finite-rectangle) representations do not extend trivially to general D persistence modules. We do not yet know whether or not statements similar to our main Theorem 1.1 and Theorem 7.7 can be made for general D persistence modules, or indeed, for interval-decomposable D persistence modules with .
Acknowledgements
We would like to thank Peter Bubenik for suggesting the extension of our main result to finite-rectangle-decomposable D persistence modules, and Tomoo Yokoyama for raising the question which we studied in Section 6.
Appendix A Appendix
Here, we reproduce Lemma 3.1 and Lemma 5.2 and provide their proofs.
See 3.1
Proof.
Let us construct an isomorphism
[TABLE]
with inverse
[TABLE]
Effect of on objects.
For , define the representation by the following. On , we set
[TABLE]
which can be checked to be a functor from to , and is thus an object in . This is the restriction of to row . On in , we set
[TABLE]
which is the collection of maps indexed by with value at :
[TABLE]
That each is a morphism (a natural transformation) in follows from the functoriality of . In detail, we have for each ,
[TABLE]
commutes since .
Finally, it can be checked that itself is a functor , and thus is an object of .
Effect of on morphisms.
For a morphism in , let us construct . This is a collection of maps for , where is given by restriction to the th row:
[TABLE]
This th row is the collection with
[TABLE]
For each , that is a natural transformation follows from naturality of .
Then we need to check that the collection in fact defines a natural transformation. This also follows from naturality of .
Functoriality of .
Next, is clearly a functor: so that is the identity, and
[TABLE]
Effect of on objects.
In the other direction, let now . Let us define . We put for . For the pairs ,
[TABLE]
should be a morphism from to , which we define to be either composition in
[TABLE]
It does not matter which path is taken, as they are equal by naturality of .
Then it can be checked that is a functor, and thus an object in .
Effect of on morphisms.
For a morphism in , let for . That is a morphism can be checked by using naturality of , as follows. For in , consider the diagram
[TABLE]
where the composition of the top row is and the bottom row is . The commutativity of the left square follows from naturality of for fixed , and that of the right square follows from naturality of and then evaluating at . This shows the commutativity of Diagram (A.1), and thus is indeed a natural transformation.
Functoriality of .
is clearly the identity, and
[TABLE]
Inverse.
By the definitions given
[TABLE]
and is given by the composition of the top row in
[TABLE]
By functoriality of , we see from the above commutative diagram that
[TABLE]
Thus, as functors.
On morphisms in , we see that
[TABLE]
and so . Thus, is equal to the identity functor.
In the other direction, it can be checked that for functors , and similarly for morphisms . Thus, is also an identity functor. ∎
See 5.2
Proof.
Write as:
[TABLE]
with the relations and which entails and .
Using the isomorphism theorems, we can decompose as:
[TABLE]
A similar decomposition can be obtained for due to the symmetry of the structure. Similarly, we can decompose as:
[TABLE]
and symmetrically . The middle vector space decomposes into summands:
[TABLE]
Note that the maps induced by , , and on these decompositions restricted to one of the summands in its domain has image fully contained in at most one summand of its codomain. Furthermore, every summand in the codomain contains the image of at most one summand of the domain. In other words, this decomposition of the vector spaces induces a direct sum decomposition of . Note that we did not necessarily compute an indecomposable decomposition of , as this decomposition already suffices for our purposes.
Finally, inspecting the form of the decomposition gives the claimed result. ∎
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