Basis-independent partial matchings induced by morphisms between persistence modules
aa r X i v : . [ m a t h . A T ] J un Basis-independent partial matchings induced by morphismsbetween persistence modules
R. Gonzalez Diaz, M. Soriano TriguerosJune 22, 2020
Department of Applied Mathematics I, University of Seville{rogodi, msoriano4}@us.es
Abstract
In this paper, we study how basis-independent partial matchings induced by morphismsbetween persistence modules (also called ladder modules) can be defined. Besides, we extendthe notion of basis-independent partial matchings to the situation of a pair of morphismswith same target persistence module. The relation with the state-of-the-art methods is alsogiven. Apart form the basis-independent property, another important property that makesour partial matchings different to the state-of-the-art ones is their linearity with respect toladder modules.
Persistent homology [9] is one of the main tools of topological data analysis. The algebraic struc-ture associated to persitent homology is persistence module [7, 12]. There are still many openquestions about persistence modules, specially multidimensional ones. Ladder modules [11] (thatis, morphisms between 1-dimensional persistence modules) are concrete cases of multidimensionalmodules and therefore interesting objects of study. In this section we will introduce such con-cept together with previous works that motivated our study. Unless stated otherwise, “persistencemodules” will refer to “1-dimensional persistence modules”.Partial matchings induced by morphisms between persistence modules are needed to answerquestions that arise in topological data analysis. For example, consider a finite filtration K ∗ ofsimplicial complexes obtained from a given dataset: K K . . . K n . The original dataset may vary, for example if it depends on time or if it is modified as part ofan experiment. Then, a new filtration L ∗ may arise. In some cases, there exists a simplicial map κ i : K i → L i between each pair of simplicial complexes ( K i , L i ) making the following diagramcommutative: L L . . . L n K K . . . K nκ κ κ n (1)Nevertheless, these simplicial maps cannot always be defined. Partial simplicial maps [10] (that is,partially defined maps which are simplicial isomorphisms in their domains) are used to model suchsituations. Let us denote a partial simplicial map between a pair of simplicial complexes ( K i , L i ) by µ i : K i L i . Actually, µ i can be expressed using simplicial maps. Consider the filtration: K ∗ ∪ µ ∗ L ∗ = K ∗ ⊔ L ∗ / ∼ µ ∗ where x ∼ µ ∗ y if y = µ ∗ ( x ) . Notice that there exists trivial injections between K ∗ , L ∗ and K ∗ ∪ µ ∗ L ∗ .1e obtain the following commutative diagram: L L . . . L n K ∪ µ L K ∪ µ L . . . K n ∪ µ n L n K K . . . K n (2)When applying the homology functor with coefficients in a field to Diagrams (1) and (2), we obtain,respectively, the commutative diagram: U U . . . U n V V . . . V nα α α n (3)that can be seen as a morphism α : V −→ U between the persistence modules V and U of length n , and U U . . . U n W W . . . W n V V . . . V nβ β β n α α α n (4)that can be seen as a pair of morphisms V α −→ W β ←− U between the persistence modules V , W and U .Since V and U are characterized up to isomorphism by their respective barcodes B ( V ) and B ( U ) (this is a consequence of Gabriel’s Theorem [13], see [12]), the following question arises: How canDiagrams (3) and (4) induce partial matchings between the multisets B ( V ) and B ( U ) ?It is proven in [11] that ladder modules with n ≤ (which can be seen as morphisms betweenpersistence modules of length n ) are finite decomposable. Such decomposition characterizes upto isomorphism the ladder module (Diagram (3)) and, in particular, the morphism involved when n ≤ . Nevertheless, ladder modules are infinite decomposable for n > .A technique for computing partial matchings induced by morphisms between persistence mod-ules of any length is introduced in [3] and extended recently in [4]. Despite being very usefulfor theoretical purposes, such techniques factorizes through the image of the given morphism andsometimes produces a different partial matching than the expected from the ladder module de-composition.Moreover, computing a partial matching induced by Diagram (4) is not straightforward. Oncewe have the matching induced by Diagram (3), we could try to create the new one composing thepartial matching between B ( V ) and B ( W ) , and the partial matching between B ( W ) and B ( U ) .Nevertheless, it is proven in [3] that there does not exist any functor from the category ofpersistence modules to the category of multisets with partial matchings as morphisms. In otherwords, composing partial matchings force us to make arbitrary basis choices. In [16], continuingprevious work [10], a persistence module K ⊂ W is defined and interpreted as the “commonpersistence” between V and U . Nevertheless no explicit relation of the persistence module K tothe persistence modules V and U is given.In this paper we will define basis-independent partial matchings such that: • They are induced by morphisms between persistence modules (Diagram (3)). • They are linear with respect to the direct sum of ladder modules (then, they do not factorizethrough the image of the given morphism.)Besides, we also extend the definition to basis-independent partial matchings induced by Diagram( ) and describe their relation with the persistence module K defined in [16] thanks to the newconcept of enriched partial matchings presented here.2 Background
In this section, we briefly recall the concept of persistence modules and all the other conceptsmentioned in the introduction section. Notice that some of the classical definitions presented herehave been slightly modified to suit our situation. Please, pay attention to the remarks in thissection, they will be used later in the proofs of the main results of the paper. In addition, someminor lemmas that may be needed later have also been added to this section. We have omittedcategorical or quiver concepts when possible to keep the paper readable by the widest audience.
From now on, we will consider vector spaces to be finite dimensional, and scalars in a fixed field F . The following elementary remarks about vector spaces will be used later. Remark 2.1.
Given three vector spaces
A, B, C , with B ⊆ A , we have: dim A / B = dim A − dim B ≥ dim A ∩ C − dim B ∩ C = dim A ∩ C / B ∩ C where dim V denotes the dimension of the vector space V . Remark 2.2.
Given a linear map α : A → B between two vector spaces A and B , we have that: dim A = dim (cid:2) α ( A ) (cid:3) + dim (cid:2) α − (0) (cid:3) , dim B = dim (cid:2) α − ( B ) (cid:3) − dim (cid:2) α − (0) (cid:3) ,B = αα − ( B ) and A ⊆ α − α ( A ) , where α − (0) denotes the kernel of α . Let us briefly introduce now the concept of zigzag, persistence and interval modules. See [5].
Definition 2.3. A zigzag module V is a sequence of vector spaces V i together with linear maps ϕ V i (called structure morphisms): V V · · · V n . ϕ V ϕ V ϕ V n − If the linear map ϕ V i goes from V i to V i +1 then ϕ V i is denoted by f V i and, if the linear map ϕ V i goesfrom V i +1 to V i then ϕ V i is denoted by g V i , that is, f V i = ϕ V i : V i −→ V i +1 and g V i = ϕ V i : V i +1 −→ V i . This way, the sequence of symbols f and g will determine the type of V . For example, thestructure of a zigzag module V of type τ = f f g will be: V V V V . f V f V g V Definition 2.4. A persistence module V of length n is a zigzag module of type τ n = f ( n − -times · · · · · · · · · f , that is, V = V V · · · V n . f V f V f V n − Remark 2.5.
For the sake of clarity, the composition f V b − ◦ f V b − ◦ . . . ◦ f V a +1 ◦ f V a will be denotedby f V a,b . In particular, observe that f V a will also be denoted by f V a,a +1 and f V a,a is the identity mapon V a . Remark 2.6.
We sometimes add the trivial spaces V = 0 and V n +1 = 0 to a given persistencemodule V of length n , together with zero maps f V : 0 → V and f V n : V n → . Then, given a, b ∈ Z ,with ≤ a ≤ b ≤ n , we have that f V a,b ( V a ) = 0 if b = n + 1 or a = 0 .Zigzag modules can be decomposed in simple parts.3 efinition 2.7. Given two zigzag modules U , V of type τ n , it is said U to be a submodule of V ,denoted U ⊆ V , if U i ⊆ V i and, ϕ V i ( U i ) ⊆ U i +1 if ϕ V i = f V i or ϕ V i ( U i ) ⊆ U i − if ϕ V i = g V i , for all i .In particular, φ U i is defined as φ V i | U i .In addition, U is said to be a summand of V if there exists another submodule W of V suchthat V i = U i ⊕ W i for all i . In that case, we say V is the direct sum of U , W , denoted V = U ⊕ W . Indecomposable modules are zigzag modules that cannot be expressed as a direct sum (exceptthemselves with the zero module). Indecomposable modules have the structure of interval modules[5] which are defined as follows.
Definition 2.8.
Given a, b, n ∈ Z , ≤ a ≤ b ≤ n , an interval module I [ a, b ] of type τ n is thefollowing zigzag module of type τ n : I I · · · I nϕ I [ a,b ]1 ϕ I [ a,b ]2 ϕ I [ a,b ] n − where, for ≤ i ≤ n , I i = (cid:26) F , if i ∈ [ a, b ] , , otherwise,and ϕ I [ a,b ] i = (cid:26) the identity map if i ∈ [ a, b − , the zero map, otherwise.Notice that, in topological data analysis, intervals are sometimes consider to be semi-open. Ifthe reader is used to that notation, please recall that in this paper [ a, b + 1) will be written as [ a, b ] .Zigzag modules can be expressed as direct sums of interval modules. Theorem 2.9 (Interval Decomposition Theorem [5]) . Let V be a zigzag module of type τ n then: V ≃ M ≤ a ≤ b ≤ n I [ a, b ] m V a,b where m V a,b is the multiplicity of the interval module I [ a, b ] . Zigzag modules are unambiguously described (up to isomorphism) by interval modules I [ a, b ] andtheir multiplicities. This information is usually represented in two equivalent ways: persistencebarcodes and persistence diagrams. Please, notice that the word “diagram” in “persistence diagram”does not have any categorical meaning. Definition 2.10.
Let V ≃ L ≤ a ≤ b ≤ n I [ a, b ] m V a,b be a zigzag modules decomposed in intervalmodules. The persistence diagram of V is the function D V :∆ + −→ Z ≥ ( a, b ) m V a,b where ∆ + = { ( a, b ) ∈ Z ≥ such that a ≤ b } . Definition 2.11.
Let V ≃ L ≤ a ≤ b ≤ n I [ a, b ] m V a,b be a zigzag modules decomposed in intervalmodules. The persistence barcode of V is a multiset formed by the intervals whose associatedinterval modules have no null multiplicity, mathematically: B V = { ([ a, b ] , m V a,b ) : m V a,b > } . The collection of all persistence barcodes will be denoted by Б . Although we will use theconcept of persistence diagrams along the paper, the concept of persistence barcodes will be usefulwhen defining enriched partial matchings in Section 4.Before finishing this subsection, let us recall the formula for computing multilplicities of theinterval modules in the decomposition of a given persistence module. Let ( a, b ) ∈ ∆ + , then: D V ( a, b ) = dim (cid:2) f V a,b ( V a ) ∩ ker f V b (cid:3) − dim (cid:2) f V a − ,b ( V a − ) ∩ ker f V b (cid:3) . (5)This expression is well-defined taking into account Remark 2.6. This formula for persistencemodules and analogous formulas for zigzag modules can be found in [5]. A multiset is a set where each element has associated a multiplicity. .4 Ladder modules and morphisms between zigzag modules In this subsection, morphisms between zigzag modules (also called 2-dimensional modules andladder modules [11]) are recalled.
Definition 2.12. A morphism α : V → U between zigzag modules of type τ n is a set of linearmaps α i : V i → U i such that the following diagram is commutative: U U · · · U n V V · · · V nϕ U ϕ U ϕ U n − ϕ V α ϕ V α ϕ V n − α n When all α i are injective (resp., surjective), we say α is injective (resp., surjective). Such diagramsare called ladder modules of type τ n .Observe that ladder modules and morphisms between zigzag modules are different point ofviews of the same concept as shown in [1]. Remark 2.13.
Consider a morphism between two interval modules I [ a, b ] α −→ I [ a ′ , b ′ ] . Then, all linear maps α i must be zero unless a ′ ≤ a ≤ b ′ ≤ b . Definition 2.14.
The image of a morphism α : V → U is the submodule of U whose vectorspaces are α i ( V i ) for i ∈ Z ≥ .There is a decomposition theorem for ladder modules of type τ n in indecomposable laddermodule when n ≤ . Nevertheless, when n > , ladder modules are, in general, representation-infinite, that is, they can not be expressed as a finite direct sum of indecomposable ladder modules. Theorem 2.15. [11] Let A be a ladder module of type τ n with n ≤ . Then, A ≃ M I ∈ Γ I m AI where m AI is the multiplicity of the indecomposable ladder module I . The set of indecomposableladder modules of type τ n with n ≤ corresponds to the finite set of vertices Γ of the Auslander-Reiten quiver [2] (Γ , Γ ) of ladder modules of type τ n with n ≤ . In this paper, we will not define the Auslander-Reiten quiver (Γ , Γ ) since it is not needed forthe understanding of our main results. Besides, the only decomposable ladder modules appearingin this paper are the ones of type τ = f , τ = f f and τ = f f f since we only deal with morphismsbetween persistence modules. To illustrate Theorem 2.15, let us describe the indecomposable laddermodules of type τ = f f (that will be represented by integer × matrices) and give an example.If an indecomposable ladder module I of type τ = f f is represented by an integer × matrixwith all entries being or , then the linear maps α i of I are the identity map when possible andthe zero map otherwise. For example, represents the following indecomposable ladder moduleof type τ = f f : F F F . Id Id
There are another indecomposable ladder modules of type τ = f f (see [11]), but only two ofthem, represented by the matrices and , are not made up of only and . Concretely,such two matrices respectively represent the following indecomposable ladder modules: F ⊕ F F F F [ ] [ 0 1 ] Id [ ] Id and F F F ⊕ F F Id [ ] Id [ 0 1 ][ 1 1 ] xample 2.16. The ladder module A = ⊕ F ⊕ F F ⊕ F ⊕ F (cid:20) (cid:21) [ 0 1 0 ] [ ] (cid:20) (cid:21) [ 1 0 ] of type τ = f f , has the following decomposition in indecomposable ladder modules: A ≃ ⊕ ⊕ Recall that the ladder module A can also be interpreted as a morphism α : V → U between twopersistence modules U and V where α : 0 → ⊕ F , α = h i : ⊕ F → ⊕ F and α = [ ] : ⊕ F → F , and V , U have the following decomposition in interval modules: V ≃ I [2 , ⊕ I [2 , and U ≃ I [1 , ⊕ I [2 , . In this subsection, we recall how persistence barcodes are related to each other via “partial match-ings”.
Definition 2.17. A partial matching between sets A and C is a partial bijection σ : A → C or,in other words, a bijection σ between the domain of σ , dom σ ⊆ A , and the image of σ , im σ ⊆ C .Given a map σ : A → C between two sets of points, the codomain and the coimage of σ aredefined, respectively, as follows:cod σ = A \ dom σ and coim σ = C \ im σ. Remark 2.18.
A representation set of the multiset A , denoted by s ( A ) , is a set obtained byenumerating the elements of A .Let us see an example. Example 2.19.
Consider the following two persistence barcodes: B V = { ([2 , , , ([3 , , } and B U = { ([1 , , , ([2 , , , ([3 , , } . Then, s ( B V ) = { [2 , , [2 , , [3 , , [3 , } and s ( B U ) = { [1 , , [2 , , [2 , , [3 , , [3 , } . A partial matching between s ( B V ) and s ( B U ) can be, for example, σ ([2 , ) = [3 , .By abuse of language, we will say “partial matchings between persistence modules V , U ” whenwe mean “partial matchings between sets s ( B V ) and s ( B U ) ”. A method for computing a partial matching induced by a morphism between persistence modulesis given in [3]. Such method is introduced in this subsection to show similarities and differenceswith our approach. First, we have to give more notations.6 otation 2.20.
Given a, b ∈ N , consider the sets { [ · , b ] ∗ } = { [ x, b ] n : x, n ∈ N } and { [ a, · ] ∗ } = { [ a, y ] n : y, n ∈ N } . For x, x ′ , y, y ′ , n, m ∈ N , we write: ( x, b ) n < ( x ′ , b ) m if (cid:26) x < x ′ , or x = x ′ and n < m. and, respectively, ( a, y ) n < ( a, y ′ ) m if (cid:26) y > y ′ , or y = y ′ and n < m. To compute a partial matching induced by an injective morphism between persistence modules α : V → U , fix b ∈ N and consider the ordered sets A = { [ · , b ] ∗ } ⊂ s ( B V ) and B = { [ · , b ] ∗ } ⊂ s ( B U ) . It is proven in [3] that the number of elements of A is less or equal than the number ofelements of B , that is, A ≤ B , so a partial matching between the two sets can be defined bymatching the i -th element of A with the i -th element of B , for ≤ i ≤ A . Putting together thepartial matchings obtained for all b ∈ N , we obtain a new partial matching between s ( B V ) and s ( B U ) denote by ι α .A similar procedure is followed in [3] when α is surjective. In this case, fix a ∈ N and considerthe sets A = { [ a, · ] ∗ } ⊆ s ( B V ) and B = { [ a, · ] ∗ } ⊆ s ( B U ) . We have that B ≤ A . Again, apartial matching between the two sets can be defined by matching the i -th element of B with the i -th element of A , for ≤ i ≤ B . Putting together the partial matchings obtained for all a ∈ N ,we obtain a new partial matching between s ( B V ) and s ( B U ) denoted by λ α .Finally, since any morphism α between persistence modules can be decomposed in a surjectiveand an injective morphism between persistence modules, a partial matching induced by α canalways be computed. Definition 2.21.
Given a morphism between persistence modules α : V → U , and its deocompo-sition α = γ ◦ β where β is surjective and γ is injective: V β −→ im α γ −→ U , the Bauer-Lesnick partial matching (or BL-matching) induced by α is the partial matching σ = λ γ ◦ ι β obtained by the composition of partial matchings ι β and λ γ : s ( B V ) λ β −→ s ( B im α ) ι γ −→ s ( B U ) . Let us see a very simple example of a BL-matching.
Example 2.22.
Let us consider the ladder module α : V → U where V = I [ a, b ] and U = I [ a ′ , b ′ ] ,being a, b, a ′ , b ′ ∈ N , a ≤ a ′ ≤ b ≤ b ′ and α i = Id when i ∈ [ a ′ , b ] . Then, I [ a, b ] β −→ im α = I [ a ′ , b ] γ −→ I [ a ′ , b ′ ] and s ( B V ) = { [ a, b ] } λ β −→ s ( B im α ) = { [ a ′ , b ] } ι γ −→ s ( B V ) = { [ a ′ , b ′ ] } where λ β (cid:0) [ a, b ] (cid:1) = [ a ′ , b ] and ι γ (cid:0) [ a ′ , b ] (cid:1) = [ a ′ , b ′ ] . Therefore, the BL-matching σ = ι γ ◦ λ β satisfies that: σ (cid:0) [ a, b ] (cid:1) = [ a ′ , b ′ ] . and it is zero otherwise. 7otice that fixed two persistence modules, the BL-matchings induced by morphisms betweenthem coincide if the images of such morphisms coincide. In other words, consider two differentmorphisms α and α between the same persistence modules. If the persistence modules im α and im α coincide then the BL-matching induced by α and the BL-matching induced by α coincide. As we will see later, this is not always the case when we compute basis-independentpartial matchings following our approach. Example 2.23.
Let us consider the following ladder module α : V → U of type τ = f f : A = ⊕ F ⊕ F F ⊕ F ⊕ F (cid:20) (cid:21) [ 0 1 0 ] f V = [ ] (cid:20) (cid:21) [ 1 0 ] Then, α = γ ◦ β where V β −→ im α γ −→ U ,β is surjective and γ is injective. The persistence module im α is: ⊕ F F [ 1 0 ] and its decomposition in interval modules is:im α ≃ I [2 , ⊕ I [2 , . To construct the BL-matching between V and U , recall that V ≃ I [2 , ⊕ I [2 , and U ≃ I [1 , ⊕ I [2 , (see Example 2.16 in page 6). Then: s ( B V ) = { [2 , , [2 , , [2 , } ,s ( B im α ) = { [2 , , [2 , } and s ( B U ) = { [1 , , [1 , , [2 , } . Therefore, λ β ([2 , ) = [2 , , λ β ([2 , ) = [2 , , λ β ([2 , x ) = ∅ ,ι γ ([2 , ) = [1 , , and ι γ ([2 , ) = [2 , . Finally, the composition σ of λ β and ι γ produces the BL-matching σ induced by α : σ ([2 , ) = [2 , , σ ([2 , ) = [1 , , σ ([2 , ) = ∅ . Observe that replacing α = h i by, for example, α = h i , will not change σ since bothim α and the decomposition of V do not change. Observe that when we transform a multiset into a set using the map s , we are making arbitrarybasis choices. If we want to define a basis-independent partial matching we should not performsuch operation. As an alternative, we can define the partial matching using multiplicities insteadof pairings. Let us define what basis-independent partial matchings mean in this paper. Definition 3.1. A basis-independent partial matching between two persistence modules V , U of length n is a function M UV : ∆ + × ∆ + → Z ≥ such that: X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ M UV ( a, b, a ′ , b ′ ) ≤ D V ( a, b ) and X ≤ b ≤ n X ≤ a ≤ b M UV ( a, b, a ′ , b ′ ) ≤ D U ( a ′ , b ′ ) . σ between s ( B V ) and s ( B U ) always induces a basis-independentpartial matching between the persistence modules V and U as follows. Assume that a partialmatching σ between s ( B V ) and s ( B U ) exists. Define M UV ( a, b, a ′ , b ′ ) = (cid:8) [ a, b ] x ∈ s ( B V ) (cid:12)(cid:12) ∃ y ∈ N such that σ ([ a, b ] x ) = [ a ′ , b ′ ] y (cid:9) . Then, X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ M UV ( a, b, a ′ , b ′ ) = (cid:8) [ a, b ] x (cid:12)(cid:12) [ a, b ] x ∈ dom σ (cid:9) ≤ D V ( a, b ) and X ≤ b ≤ n X ≤ a ≤ b M UV ( a, b, a ′ , b ′ ) = (cid:8) [ a ′ , b ′ ] x (cid:12)(cid:12) [ a ′ , b ′ ] y ∈ im σ (cid:9) ≤ D U ( a ′ , b ′ ) . In particular, given a a BL-matching, we can always compute its corresponding basis-independentpartial matching that will be denoted by M αBL . Example 3.2.
Consider the ladder module α : V → U of type τ = f f described in Example 2.16and the BL-matching σ between s ( B V ) = { [2 , , [2 , , [2 , } and s ( B U ) = { [1 , , [1 , , [2 , } given in Example 2.23 and detailed below: σ ([2 , ) = [2 , , σ ([2 , ) = [1 , , σ ([2 , ) = ∅ . Then M αBL (2 , , ,
3) = 1 , M αBL (2 , , ,
2) = 1 and M αBL ( a, b, a ′ , b ′ ) = 0 otherwise.Besides, due to the properties of the function M UV given in Definition 3.1, we always canobtain a (non-unique) partial matching between persistence modules starting from a concretebasis-independent partial matching. Nevertheless, contrary to what happens with compositionof partial matchings, the composition of basis-independent partial matchings is not well-defined.Specifically, it has been proven in [3] that no functor from the category of peristence modulesto the category of multisets with partial matchings as morphisms can exist. Nevertheless, usingbasis-independent partial matchings, we do not need any arbitrary choice of the elements of themultisets B V and B U , that is, any choice of the basis of V and U , gaining in versatility. In this subsection, we will define a basis-independent partial matching with the following charac-teristics (already mentioned in the introduction): • It is induced by a given morphism between persistence modules. • It is a linear map with respect to the direct sum of ladder modules and then it does notfactorize through the image of the given morphism.The following subspaces will play a very important role in this subsection.
Notation 3.3.
Let V be a persistence module of length n . Let a, b ∈ Z ≥ . Then, S V a,b denotes thefollowing subspace: S V a,b = (cid:26) f V a,b ( V a ) ∩ ker f V b if ≤ a ≤ b ≤ n , otherwise.Observe that S V a,b can be seen as the subspace of V a which “persists” in V b through f V a,b . Now,observe that Expression (5) in page 4 can be written as D V ( a, b ) = dim S V a,b − dim S V a − ,b . (6)Therefore, the multiplicity D V ( a, b ) can be seen as the dimension of the subspace of V a which isnot a subspace of V a − and “persists” in V b through f V a,b . This interpretation is usually called as“the elder rule” in the literature [9]. 9he following diagram can help us to figure out how we can define a basis-independent partialmatching between two persistent modules U and V induced by a morphism α : V → U : U a ′ U b ′ U b ′ +1 V a V b ′ V b V b +1 . f U a ′ ,b ′ f U b ′ f V a,b ′ α b ′ f V b ′ ,b f V b (7)But before giving such definition, we need to introduce a new notation. Definition 3.4.
Let α : V → U be a morphism between persistence modules of length n . Then, X α : Z ≥ → Z ≥ is defined as follows: X α ( a, b, a ′ , b ′ ) = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) if b ′ ≤ b ≤ n, otherwise.For the sake of simplicity, we sometimes denote X α ( a, b, a ′ , b ′ ) by ba X b ′ a ′ if there is no need toexplicitly mention the morphism α .Looking at Diagram (7), we can interpret X α as the dimension of the subspace of S U a ′ ,b ′ that“persists” in S V a,b “through” α − . In the context of zigzag homology, observe that X α is the value D [2 , of the following zigzag module: U b ′ +1 f U a ′ ,b ′ ( U a ′ ) U b ′ V b ′ f V a,b ′ ( V a ) V b V b +1 f U b ′ α b ′ f V b ′ ,b f V b Remark 3.5.
Since U = V = V n +1 = U n +1 = 0 , then ba X b ′ a ′ = 0 if any of the variables a, b, a ′ , b ′ is or n + 1 . Besides, by definition of S we have that ba X b ′ a ′ = 0 if b < a , b ′ < a ′ .To define the basis-independent partial matching induced by α , the function X α will play asimilar role to dim S in the definition of D in Expression (6) of page 9. This way, we will also makeuse of an “elder rule operator” defined as follows. Definition 3.6.
Given a function F : Z n → Z , let us define E i ( F ) as follows: E i ( F )( x , . . . , x n ) = F ( x , . . . , x i − , x i , x i +1 , . . . , x n ) − F ( x , . . . , x i − , x i − , x i +1 , . . . , x n ) where E denotes the elder rule operator . We will write E j ( E i ( F )) as E i,j ( F ) .For example, let V be a persistence module. Consider the function dim S V : ( a, b ) dim S V a,b .Then, Expression (6) in page 9 can be written as: D V ( a, b ) = E (dim S V )( a, b ) . Definition 3.7.
Let α : V → U be a morphism between persistence modules. Define M α :∆ + × ∆ + → Z ≥ as: M α = E , ( X α ) or, equivalently, M α ( a, b, a ′ , b ′ ) = ba X b ′ a ′ − ba − X b ′ a ′ − ba X b ′ a ′ − + ba − X b ′ a ′ − . The non-negative integer M α ( a, b, a ′ , b ′ ) can be interpreted as the amount of interval modules I [ a, b ] in the decomposition of V that are “sent” by α to interval modules I [ a ′ , b ′ ] in the decompositionof U . Before proving that M α is, in fact, a basis-independent partial matching, let us introducesome technical results. 10 emma 3.8. If a ≤ a ′ then ba X b ′ a ′ = ba X b ′ c for any c ∈ Z ≥ such that a ≤ c ≤ a ′ .Proof. Note that, by definition, ba X b ′ a ′ − ba X b ′ c = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U c,b ′ ) (cid:3) . Since c ≤ a ′ then f U c,b ′ ( U c ) ⊆ f U a ′ ,b ′ ( U a ) . Therefore, S U c,b ′ ⊆ S U a ′ ,b ′ then C = S V a,b ∩ f V b ′ ,b α − b ′ ( S U c,b ′ ) ⊆ A ′ = S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) . Let us see that A ′ ⊆ C . Consider x ∈ A ′ . Then, in particular, there exists y ∈ V a such that f V a,b ( y ) = x and there exists z ∈ α − b ′ ( S U a ′ ,b ′ ) such that f V b ′ ,b ( z ) = x . Then, necessarily, f V a,b ′ ( y ) = z and α b ′ f V a,b ′ ( y ) ∈ S U a ′ ,b ′ = f U a ′ ,b ′ ( U a ′ ) ∩ ker f U b ′ ,b ′ +1 . Since a ≤ c ≤ a ′ ≤ b ′ then α b ′ f V a,b ′ ( y ) = α b ′ f V c,b ′ f V a,c ( y ) = f U c,b ′ α c f V a,c ( y ) ∈ f U c,b ′ ( U a ′ ) ∩ ker f U b ′ ,b ′ +1 = S U c,b ′ . Therefore, x ∈ S V a,b ∩ f V b ′ ,b α − b ′ ( S U c,b ′ ) = C , concluding the proof. Lemma 3.9. If b ′ ≤ a then ba X b ′ a ′ = bc X b ′ a ′ for any c ∈ Z ≥ such that b ′ ≤ c ≤ a .Proof. Similar to the proof of Lemma 3.8, since b ′ ≤ c ≤ a , then S V c,b ⊆ S V a,b and therefore C = S V c,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ⊆ A = S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) . Let us prove that A ⊆ C . Note that f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) = f V c,b f V b ′ ,c α − b ′ ( S U a ′ ,b ′ ) , then f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ⊆ f V c,b ( V c ) . Then, S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ⊆ ker f V b,b +1 ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ⊆ f V b,c ( V c ) concluding that A ⊆ C .The following lemma is deduced directly from the definition of the elder rule operator. Lemma 3.10. If i = j then X a ≤ x j ≤ b E i ( F ) = E i X a ≤ x j ≤ b F . Another useful lemmas are the following.
Lemma 3.11. If ≤ i ≤ n then: X a +1 ≤ x i ≤ b E i ( F )( x , . . . , x i − , x i , x i +1 . . . , x n ) = F ( x , . . . , x i − , b, x i +1 , . . . , x n ) − F ( x , . . . , x i − , a, x i +1 , . . . , x n ) . Proof.
Writing down the sum we have X a +1 ≤ x i ≤ b E i ( F )( . . . , x i , . . . ) = F ( . . . , b, . . . ) − F ( . . . , b − , . . . )+ F ( . . . , b − , . . . ) − F ( . . . , b − , . . . ) ... + F ( . . . , a + 1 , . . . ) − F ( . . . , a, . . . ) . Cancelling addends with opposite signs, we obtain the desired result.11 emma 3.12.
Let V be a persistence module of length n . Then, for any b ∈ Z with ≤ b ≤ n andany subspace A b ⊆ V b , we have that: dim A b = X b ≤ i ≤ n dim (cid:2) f V b,i ( A b ) ∩ ker f V i,i +1 (cid:3) . Proof.
First, by Remark 2.2, we have that: dim A b = dim (cid:2) f V b,b +1 ( A b ) (cid:3) + dim (cid:2) ker f V b,b +1 (cid:3) . (8)Second, for any i ∈ Z with b ≤ i ≤ n and for any subspace A i ⊆ V i we have: dim A i = dim (cid:2) f V i,i +1 ( A i ) (cid:3) + dim (cid:2) A i ∩ ker f V i,i +1 (cid:3) (9)Then, applying recursively Property (9) to Expression (8), we have: dim A b = dim (cid:2) f V b,b +1 ( A b ) (cid:3) + dim (cid:2) ker f V b,b +1 (cid:3) = dim (cid:2) f V b,b +2 ( A b ) (cid:3) + dim (cid:2) f b,b +1 ( A b ) ∩ ker f V b +1 ,b +2 (cid:3) + dim (cid:2) ker f V b,b +1 (cid:3) = · · · = dim (cid:2) f V n,n +1 f V b,n ( A b ) (cid:3) + dim (cid:2) f V b,n ( A b ) ∩ ker f V n − ,n (cid:3) + · · · + dim (cid:2) f V b,b +1 ( A b ) ∩ ker f V b +1 ,b +2 (cid:3) + dim (cid:2) ker f V b,b +1 (cid:3) . Now, since A n +1 = 0 , f V n,n +1 f V b,n ( A b ) = 0 and f b,b is the identity, we have that ker f V b,b +1 = f V b,b ( A b ) ∩ ker f V b,b +1 . Then, dim A b = dim (cid:2) f V b,n ( A b ) ∩ ker f V n − ,n (cid:3) + · · · + dim (cid:2) f V b,b ( A b ) ∩ ker f V b,b +1 (cid:3) concluding the proof. Lemma 3.13.
Let α : V → U be a morphism between persistence modules. Let a, b ∈ Z ≥ with a ≤ b and consider a subspace A a ⊆ U a . Then: f V a,b α − a ( A a ) ⊆ α − b f U a,b ( A a ) . Proof.
By commutativity and Remark 2.2: f V a,b α − a ( A a ) ⊆ α − b α b f V a,b α − a ( A a ) = α b f U a,b α a α − a ( A a ) = α b f U a,b ( A A ) . Lemma 3.14.
Let α be a morphism between persistence modules. Then: X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) = ba X b ′ b ′ − ba − X b ′ b ′ and X ≤ a ≤ b M α ( a, b, a ′ , b ′ ) = bb X b ′ a ′ − bb X b ′ a ′ − . Proof.
Both relations can be proven in an analogous way. Let us prove the first one. UsingLemmas 3.10 and 3.11 we have: X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) = X ≤ a ′ ≤ b ′ E , ( X α )( a, b, a ′ , b ′ )= E X ≤ a ′ ≤ b ′ E ( X α )( a, b, a ′ , b ′ ) = E ( X α ( a, b, b ′ , b ′ ) − X α ( a, b, , b ′ )) . Using X α ( a, b, , b ′ ) = 0 by Remark 3.5, we have that: X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) = E ( X α ) ( a, b, b ′ , b ′ ) = ba X b ′ b ′ − ba − X b ′ b ′ . M α is, in fact, a basis-independent partial matching induced by amorphism α between persistence modules. Theorem 3.15.
Let α be a morphism between two persistence modules U and V of length n . Then, X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) ≤ dim S V a,b − dim S V a − ,b = D V ( a, b ) and X ≤ b ′ ≤ n X ≤ a ≤ b M α ( a, b, a ′ , b ′ ) ≤ dim S U a ′ ,b ′ − dim S U a ′ − ,b ′ = D U ( a ′ , b ′ ) . Proof.
Let us start with the first inequality. By Lemma 3.14, we have: X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) = X ≤ b ′ ≤ n ba X b ′ b ′ − ba − X b ′ b ′ . By definition, ba X b ′ b ′ − ba − X b ′ b ′ = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U b ′ ,b ′ ) (cid:3) − dim (cid:2) S V a − ,b ∩ f V b ′ ,b α − b ′ ( S U b ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) + dim (cid:2) S V a − ,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) . (10)Now, denote A = S V a,b ∩ f V b ′ ,b α − b ′ (0) , B = S V a − ,b ∩ f V b ′ ,b α − b ′ (0) and C = f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) . Since S V a − ,b ⊆ S V a,b then B ⊆ A . Besides, using Lemma 3.13 we have: C = f V b ′ ,b f V b ′ − ,b ′ α − b ′ − ( S U b ′ − ,b ′ − ) ⊆ f b ′ ,b α − b ′ f U b ′ − ,b ′ ( S U b ′ − ,b ′ − ) = f V b ′ ,b α − b ′ (0) . Therefore, A ∩ C = S V a,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) and B ∩ C = S V a − ,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) and from Remark 2.1 we have: − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) + dim (cid:2) S V a − ,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) ≤ − dim (cid:2) S V a,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) (cid:3) + dim (cid:2) S V a − ,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) (cid:3) . Then, applying this last result to Expression (10): X ≤ b ′ ≤ b ba X b ′ b ′ − ba − X b ′ b ′ ≤ X ≤ b ′ ≤ b dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U b ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) (cid:3) − dim (cid:2) S V a − ,b ∩ f V b ′ ,b α − b ′ ( S U b ′ ,b ′ ) (cid:3) + dim (cid:2) S V a − ,b ∩ f V b ′ − ,b α − b ′ − ( S U b ′ − ,b ′ − ) (cid:3) . Cancelling addends with opposite signs, we get: X ≤ b ′ ≤ b ba X b ′ b ′ − ba − X b ′ b ′ ≤ dim (cid:2) S V a,b ∩ f V b,b α − b ( S U b,b ) (cid:3) − dim (cid:2) S V a,b ∩ f V ,b α − ( S U , ) (cid:3) − dim (cid:2) S V a − ,b ∩ f V b,b α − b ( S U b,b ) (cid:3) + dim (cid:2) S V a − ,b ∩ f V ,b α − ( S U , ) (cid:3) . Since f V ,b and f V b,b are, respectively, the zero and the identity map, we have: X ≤ b ′ ≤ b X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) ≤ dim (cid:2) S V a,b ∩ α − b ( S U b,b ) (cid:3) − dim (cid:2) S V a − ,b ∩ α − b ( S U b,b ) (cid:3) . A = S V a,b , B = S V a − ,b and C = S U b,b , we obtain: X ≤ b ′ ≤ b X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) ≤ dim S V a,b − dim S V a − ,b = D V ( a, b ) . Besides, since ba X b ′ a ′ = 0 if b < b ′ by Remark 3.5, then: X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) = X ≤ b ′ ≤ b X ≤ a ′ ≤ b ′ M α ( a, b, a ′ , b ′ ) obtaining the desired result. For the second inequality, we will proceed in a similar way. First, bydefinition, ba X b ′ a ′ − ba X b ′ a ′ − = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b (ker α b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) (cid:3) + dim (cid:2) S V a,b ∩ f V b ′ ,b (ker α b ′ ) (cid:3) = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) (cid:3) . Now, applying Remark 2.1 to A = f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ∩ ker f V b , B = f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) ∩ ker f V b and C = S V a,b , we obtain that: dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) (cid:3) (11) ≤ dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ∩ ker f V b (cid:3) − dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) ∩ ker f V b (cid:3) . Besides, since ba X b ′ a ′ = 0 if b < b ′ , again by Remark 3.5, then: X ≤ b ≤ n X ≤ a ≤ b M α ( a, b, a ′ , b ′ ) = X b ′ ≤ b ≤ n X ≤ a ≤ b M α ( a, b, a ′ , b ′ ) . By Lemma 3.14 and Expression (11), we have: X b ′ ≤ b ≤ n X ≤ a ≤ b M α ( a, b, a ′ , b ′ ) = X b ′ ≤ b ≤ n ba X b ′ a ′ − ba X b ′ a ′ − ≤ X b ′ ≤ b ≤ n dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ∩ ker f V b (cid:3) − dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) ∩ ker f V b (cid:3) and by Lemma 3.12, X b ′ ≤ b ≤ n dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) ∩ ker f V b (cid:3) − dim (cid:2) f V b ′ ,b α − b ′ ( S U a ′ − ,b ′ ) ∩ ker f V b (cid:3) = dim (cid:2) α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) α − b ′ ( S U a ′ − ,b ′ ) (cid:3) . Finally, by Remark 2.2: dim (cid:2) α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) α − b ′ ( S U a ′ − ,b ′ ) (cid:3) = dim (cid:2) S U a ′ ,b ′ (cid:3) + dim (cid:2) ker α b ′ (cid:3) − dim (cid:2) S U a ′ − ,b ′ (cid:3) − dim (cid:2) ker α b ′ (cid:3) = dim (cid:2) S U a ′ ,b ′ (cid:3) − dim (cid:2) S U a ′ − ,b ′ (cid:3) = D U ( a ′ , b ′ ) , concluding the proof. Proposition 3.16. M α ( a, b, a ′ , b ′ ) = 0 unless a ′ ≤ a ≤ b ′ ≤ b .Proof. By Remark 3.5, if b < b ′ then ba X b ′ a ′ = 0 and so M α ( a, b, a ′ , b ′ ) = 0 . If a < a ′ , then, byLemma 3.8, ba X b ′ a ′ = ba X b ′ a ′ − and ba − X b ′ a ′ = ba − X b ′ a ′ − . Similarly, when b ′ < a , by Lemma 3.9, ba X b ′ a ′ = ba − X b ′ a ′ and ba X b ′ a ′ − = ba − X b ′ a ′ − . Then M α ( a, b, a ′ , b ′ ) = 0 in both cases. 14ecall that ladder modules of type τ = f f . . . f and morphism between persistence modulesare different points of view of the same concept. We will write M L instead of M α when we wantto focus on the point of view of ladder modules. Remark 3.17.
Denote the collection of ladder modules of type τ = f f . . . f by P LM and define ∆ = { ( a, b, a ′ , b ′ ) ∈ Z ≥ such that a ′ ≤ a ≤ b ′ ≤ b } . By the previous proposition, we can think in M as a function M : P LM × ∆ → Z ≥ such that M ( L , ∗ ) = M L ( ∗ ) . Let P M denote the collection of persistence modules, then let D denote the function: D : P M × ∆ + → Z ≥ such that D ( U , ∗ ) = D U ( ∗ ) . This way, M could be seen as a generalization of persistence barcodesfor ladder modules of type τ = f f . . . f .Before giving an example, let us prove that M is linear with respect to the direct sum of laddermodules. Proposition 3.18 (Linearity of the function M ) . Let L , L , L be ladders modules of type τ = f f . . . f such that L ≃ L ⊕ L . Then, M L = M L + M L . Proof.
It is enough to prove that ab X b ′ a ′ ( L ) = ab X b ′ a ′ ( L ) + ab X b ′ a ′ ( L ) . Notice that since L ≃ L ⊕ L we can decompose V a = ( V a ) ⊕ ( V a ) , U a ′ = ( U a ′ ) ⊕ ( U a ′ ) and α a = ( α a ) ⊕ ( α a ) . Then, S V a,b = f V a,b (( V a ) ⊕ ( V a ) ) ∩ ker f V b = (cid:0) f V a,b ( V a ) ∩ ker f V b (cid:1) ⊕ (cid:0) f V a,b ( V a ) ∩ ker f V b (cid:1) = ( S V a,b ) ⊕ ( S V a,b ) Analogously, S U a ′ ,b ′ = ( S U a ′ ,b ′ ) ⊕ ( S U a ′ ,b ′ ) . We also have f V b ′ ,b α − b ′ (cid:0) ( S U a ′ ,b ′ ) ⊕ ( S U a ′ ,b ′ ) (cid:1) = f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) ⊕ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) Since ( S U a ′ ,b ′ ) ∩ ( S U a ′ ,b ′ ) = 0 , we get: ab X b ′ a ′ ( L ) = dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) S V a,b ∩ f V b ′ ,b α − b ′ (0) (cid:3) = dim (cid:2)(cid:0) ( S V a,b ) ⊕ ( S V a,b ) (cid:1) ∩ (cid:0) f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) ⊕ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) (cid:1)(cid:3) − dim (cid:2)(cid:0) ( S V a,b ) ⊕ ( S V a,b ) (cid:1) ∩ α − b ′ (0) (cid:3) = dim (cid:2)(cid:0) ( S V a,b ) ∩ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) (cid:1) ⊕ (cid:0) ( S V a,b ) ∩ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) (cid:1)(cid:3) − dim (cid:2)(cid:0) ( S V a,b ) ∩ ( α b ′ ) − (0) (cid:1) ⊕ (cid:0) ( S V a,b ) ∩ ( α b ′ ) − (0) (cid:1)(cid:3) = dim (cid:2) ( S V a,b ) ∩ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) (cid:3) + dim (cid:2) ( S V a,b ) ∩ f V b ′ ,b ( α b ′ ) − ( S U a ′ ,b ′ ) (cid:3) − dim (cid:2) ( S V a,b ) ∩ ( α b ′ ) − (0) (cid:3) − dim (cid:2) ( S V a,b ) ∩ ( α b ′ ) − (0) (cid:3) = ab X b ′ a ′ ( L ) + ab X b ′ a ′ ( L ) . Example 3.19.
Recall that the ladder module A from Example 2.16 can be decomposed as follows: A ≃ ⊕ ⊕ = L ⊕ L ⊕ L . By Proposition 3.18, we have: M A = M L + M L + M L . In the case of L , observe that V ≃ I [2 , and U ≃ I [1 , ⊕ I [2 , . Then, S V , = F , S U , = h [ ] i , S U , = F ,f V , α − ( S U , ) = 0 and f V , α − ( S U , ) = F .
15n addition, ker α = ker α = 0 . The only non-trivial calculations for X is: X = dim (cid:2) S V , ∩ f V , α − ( S U , ) (cid:3) − dim (cid:2) S V , ∩ f V , (ker α ) (cid:3) = 1 − . Then, the resulting matching is: M L (2 , , ,
3) = X − X − X + X = 1 − − and M L ( a, b, a ′ , b ′ ) = 0 otherwise. Similar calculation gives M L (2 , , ,
2) = 1 and M L ( a, b, a ′ , b ′ ) =0 otherwise. M L ( ∗ ) is always zero. Finally, M A (2 , , ,
3) = 1 , M A (2 , , ,
2) = 1 and M A ( a, b, a ′ , b ′ ) = 0 otherwise.Let us notice that, in this case, M A = M αBL , where M αBL is the basis-independent partialmatching obtained in Example 3.2 in page 9. Nevertheless, M A and M αBL do not always coincide,as the following example shows. Example 3.20.
Consider again the ladder module A of Example 2.23 but replacing α by h i .As mentioned in that example, M A BL will not change since V , U and im α remain the same up toisomorphism. Nevertheless, now A ≃ ⊕ ⊕ and applying Proposition 3.18, we have: M A = M L + M L + M L with M L i ( ∗ ) being zero for i = 1 , , , except for M L (2 , , ,
3) = 1 and M L (2 , , ,
2) = 1 . Then M A (2 , , ,
3) = 1 , M A (2 , , ,
2) = 1 and M A ( a, b, a ′ , b ′ ) = 0 otherwise.Observe that, in this case, M A = M αBL where M αBL is the basis-independent partial matchingobtained in Example 3.2 in page 9. In particular, M A is the basis-independent partial matchingthat we might expect by looking at the decomposition of the ladder module A . Remark 3.21.
Taking into account Remark 3.17 we might ask: does M characterize up to isomor-phism ladder modules as D characterizes up to isomorphism persistence modules? Unfortunately,the answer is no. For example, ladder modules A = and B = ⊕ are not isomorphicbut M A = M B . The aim of this section is to compute a basis-independent partial matching between persistencemodules V and U from a pair of morphism V α −→ W β ←− U between persistence modules. As saidin the introduction, this problem is related to the persistence module K ⊂ W described in [16]which can be seen as the persistence submodules of U and V that “matches” and “persists” in W through the morphisms α and β . Before explaining our approach, let us introduce a motivatingexample. Example 4.1.
Consider the following diagrams:
F F . . .
F FF F . . . F F F . . .
F F
11 1 1 1 11 01 1 1 01 1 11 1 11 0 (12)16 F . . . F FF . . . F F . . .
F F
11 0 1 1 10 00 0 0 01 1 10 1 10 0 (13)In both of them, the upper and lower persistence modules consist of the interval module I [1 , n ] . Thetop and bottom interval modules “should be matched” if we compute partial matchings inducedby any of the two diagrams. Nevertheless, we should make a difference between both situations:in Diagram (13), a “minor change” of the persistence module in the middle “may remove” thematching while Diagram (12) is much more robust. A solution could be giving “persistence” to ourmatching: in the first case, the persistence of the matching should be [1 , n − and, in the secondcase, [1 , .Let us now give a new definition of partial matching that takes into account the idea of “per-sistent matchings”. Recall that we denoted the collection of persistence barcodes as Б . Definition 4.2. An enriched partial matching between two persistence modules U , V of length n is a map G UV : ∆ + × ∆ + → Б such that: G UV ( a, b, a ′ , b ′ ) = ∅ if n < a, a ′ , b, b ′ , X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ G UV ( a, b, a ′ , b ′ ) ≤ D V ( a, b ) and X ≤ b ≤ n X ≤ a ≤ b G UV ( a, b, a ′ , b ′ ) ≤ D U ( a ′ , b ′ ) . Its associated basis-independent partial matching is: M UV ( a, b, a ′ , b ′ ) = G UV ( a, b, a ′ , b ′ ) . Example 4.3.
An enriched partial matching of Diagram (12) in Example 4.1 is: G UV (1 , n, , n ) = { ([1 , n − , } and G UV ( a, b, a ′ , b ′ ) = ∅ otherwise. Our motivation in this subsection is to obtain an enriched partial matching induced by the followingdiagram: U U . . . U n W W . . . W n V V . . . V nβ f U f U β f U n − β n f W f W f W n − f V α α f V f V n − α n (14)Following a similar procedure than before, we will study the relation between the subspaces S V a,b and S U a ′ ,b ′ through the morphisms α and β . Consider the following subspaces: R ( a, b, c, d ) = ( α d (cid:16) f V a,d ( V a ) ∩ ker f V d,b +1 (cid:17) ∩ S W c,d if a, c ≤ d ≤ b , otherwise;and L ( a ′ , b ′ , c, d ) = ( β d (cid:16) f U a ′ ,d ( U a ′ ) ∩ ker f U d,b ′ +1 (cid:17) ∩ S W c,d if a ′ , c ≤ d ≤ b ′ , otherwise.17aking into account that: f V d,b (cid:0) f V a,d ( V a ) ∩ ker f V d,b +1 (cid:1) = S V a,b and f V a,d ( V a ) ∩ ker f V d,b +1 = f V a,d ( V a ) ∩ ( f V d,b ) − (cid:0) S V a,b (cid:1) , the intersection R ( a, b, c, d ) ∩ L ( a ′ , b ′ , c, d ) may be interpreted as the common subspace of S V a,b and S U a ′ ,b ′ that “persists” in the subspace S W c,d . Lemma 4.4.
The following relations hold: R ( a, b, c, d ) = f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ∩ S W c,d and L ( a ′ , b ′ , c, d ) = f W a ′ ,d β a ′ (cid:0) ker f U a ′ ,b ′ +1 (cid:1) ∩ S W c,d . Proof.
First, observe that: V a ∩ ker f V a,b +1 = ker f V a,b +1 and f V a,d ( V a ) ∩ ker f V d,b +1 = f V a,d (ker f V a,b +1 ) . Then, R ( a, b, c, d ) = α d f V a,d (cid:0) ker f V a,b +1 (cid:1) ∩ S W c,d and L ( a ′ , b ′ , c, d ) = β d f U a ′ ,d (cid:0) ker f U a ′ ,b ′ +1 (cid:1) ∩ S W c,d . By commutativity of Diagram (14), we obtain the desired result.As in the previous section, we need to “count” the dimensions before applying the elder ruleoperator E . Define the function Y : Z ≥ → Z ≥ as follows: Y ( a, b, a ′ , b ′ , c, d ) = dim( R ( a, b, c, d ) ∩ L ( a ′ , b ′ , c, d )) . Notice that if a > b ′ or a ′ > b then Y ( a, b, a ′ , b ′ , c, d ) = 0 . Now, apply the elder rule operator E to Y five times to define the enriched partial matching. Definition 4.5.
The enriched partial matching induced by morphisms V α −→ W β ←− U between persistence modules of length n , is the map G α,β : ∆ + × ∆ + → Б defined as: G α,β ( a, b, a ′ , b ′ ) = { ([ c, d ] , e c,d ) : e c,d > } where ( c, d ) ∈ ∆ + and e c,d = E , , , , ( Y )( a, b, a ′ , b ′ , c, d ) . Theorem 4.6.
Fix a pair of morphisms V α −→ W β ←− U between persistence modules of length n . The operator G α,β is, in fact, an enriched partial matching. In other words, G α,β ( a, b, a ′ , b ′ ) = ∅ if n < a, a ′ , b, b ′ , X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ G α,β ( a, b, a ′ , b ′ ) ≤ D V ( a, b ) and X ≤ b ≤ n X ≤ a ≤ b G α,β ( a, b, a ′ , b ′ ) ≤ D U ( a ′ , b ′ ) . Proof.
The property G α,β ( a, b, a ′ , b ′ ) = ∅ if n < a, a ′ , b, b ′ can be followed directly from thedefinition of G α,β . In order to prove the above inequalities, note that X ≤ b ≤ n X ≤ a ≤ b G α,β ( a, b, a ′ , b ′ ) = X ≤ b ≤ n X ≤ a ≤ b G β,α ( a ′ , b ′ , a, b ) . m = max { a, a ′ } and M = min { b, b ′ } . Observe that: G α,β ( a, b, a ′ , b ′ ) = { ([ c, d ] , e ) such that m ≤ d ≤ M and c ≤ d } = X m ≤ d ≤ M X ≤ c ≤ d e c,d = X m ≤ d ≤ M X ≤ c ≤ d E , , , , ( Y )( a, b, a ′ , b ′ , c, d ) . Notice that Y ( a, b, a ′ , b ′ , , d ) = 0 . By Lemma 3.11, we have: G α,β ( a, b, a ′ , b ′ ) = X m ≤ d ≤ M E , , , ( Y )( a, b, a ′ , b ′ , d, d ) . In particular, since a ≤ m , M ≤ b and all addends are positive, then: G α,β ( a, b, a ′ , b ′ ) ≤ X a ≤ d ≤ b E , , , ( Y )( a, b, a ′ , b ′ , d, d ) . By Remark 3.10, we have X d ≤ b ′ ≤ n X ≤ a ′ ≤ d X a ≤ d ≤ b E , , , ( Y )( a, b, a ′ , b ′ , d, d )= X d ≤ b ′ ≤ n X ≤ a ′ ≤ d E , X a ≤ d ≤ b E , ( Y ) ( a, b, a ′ , b ′ , d, d ) . Again, by Lemma 3.11 and since Y ( a, b, d, d − , d, d ) = Y ( a, b, b ′ , , d, d ) = 0 , we obtain: G α,β ( a, b, a ′ , b ′ ) ≤ X a ≤ d ≤ b E , Y ( a, b, d, n, d, d ) . By Lemma 4.4, we have: Y ( a, b, d, n, d, d ) = dim( R ( a, b, d, d ) ∩ L ( d, n, d, d ))= dim (cid:0) f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ∩ f W d,d β d (cid:0) ker f U d,n +1 (cid:1) ∩ S W d,d (cid:1) = dim (cid:0) f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ∩ β d ( U d ) ∩ S W d,d (cid:1) = dim (cid:0) f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ∩ β d ( U d ) ∩ ker f W d (cid:1) . Notice that f W a − ,d α a − (cid:0) ker f V a − ,b +1 (cid:1) ⊂ f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ,f W a,d α a (cid:0) ker f V a,b (cid:1) ⊂ f W a,d α a (cid:0) ker f V a,b +1 (cid:1) . Let A = f W a,d α a (cid:0) ker f V a,b +1 (cid:1) ∩ ker f W d ,B = f W a − ,d α a − (cid:0) ker f V a − ,b +1 (cid:1) ∩ ker f W d ,C = β d ( U d ) . Then dim( A ∩ C ) − dim( B ∩ C ) ≤ dim( A ) − dim( B ) by Remark 2.1. Again, let B ′ = f W a,d α a (cid:0) ker f V a,b (cid:1) ∩ ker f W d . Then, dim( A ∩ C ) − dim( B ′ ∩ C ) ≤ dim( A ) − dim( B ′ ) by Remark 2.1. Putting together bothinequalities, we obtain: A ∩ C ) − dim( B ∩ C ) − dim( B ′ ∩ C ) ≤ A ) − dim( B ) − dim( B ′ ) , dim( A ∩ C ) − dim( B ∩ C ) − dim( B ′ ∩ C ) ≤ dim( A ) − dim( B ) − dim( B ′ ) . (15)Therefore, E , Y ( a, b, d, n, d, d ) = Y ( a, b, d, n, d, d ) − Y ( a − , b, d, n, d, d ) − Y ( a, b − , d, n, d, d ) + Y ( a − , b − , d, n, d, d )= dim( A ∩ C ) − dim( B ∩ C ) − dim( B ′ ∩ C )+ Y ( a − , b − , d, n, d, d ) . Using Expression (15) and that Y ( a − , b − , d, n, d, d ) ≤ dim (cid:0) f W a − ,d α a − (cid:0) ker f V a − ,b (cid:1) ∩ ker f W d (cid:1) , we obtain: E , Y ( a, b, d, n, d, d ) ≤ dim (cid:0) f W a,d α a − (cid:0) ker f V a,b +1 (cid:1) ∩ ker f W d (cid:1) − dim (cid:0) f W a − ,d α a − (cid:0) ker f V a − ,b +1 (cid:1) ∩ ker f W d (cid:1) − dim (cid:0) f W a,d α a (cid:0) ker f V a,b (cid:1) ∩ ker f W d (cid:1) + dim (cid:0) f W a − ,d α a − (cid:0) ker f V a − ,b (cid:1) ∩ ker f W d (cid:1) . (16)Besides, by Lemma 3.12, we have: dim α a (cid:0) ker f V a,b +1 (cid:1) = X a ≤ i ≤ n dim (cid:2) f W a,i α a (cid:0) ker f V a,b +1 (cid:1) ∩ ker f W i (cid:3) that is equal to X a ≤ i ≤ b dim (cid:2) f W a,i α a (cid:0) ker f V a,b +1 (cid:1) ∩ ker f W i (cid:3) since f W a,i α a (cid:16) ker f V a,b +1 (cid:17) = 0 for i ≥ b + 1 . Extending this reasoning to the other cases, we have: dim f W a − ,a α a − (cid:0) ker f V a,b +1 (cid:1) = X a ≤ i ≤ b dim (cid:2) f W a − ,i α a − (cid:0) ker f V a − ,b +1 (cid:1) ∩ ker f W i (cid:3) , dim α a (cid:0) ker f V a,b (cid:1) = X a ≤ i ≤ b − dim (cid:2) f W a,i α a (cid:0) ker f V a,b (cid:1) ∩ ker f W i (cid:3) = X a ≤ i ≤ b dim (cid:2) f W a,i α a (cid:0) ker f V a,b (cid:1) ∩ ker f W i (cid:3) , dim f W a − ,a α a − (cid:0) ker f V a − ,b (cid:1) = X a ≤ i ≤ b − dim (cid:2) f W a − ,i α a − (cid:0) ker f V a − ,b (cid:1) ∩ ker f W i (cid:3) = X a ≤ i ≤ b dim (cid:2) f W a − ,i α a − (cid:0) ker f V a − ,b +1 (cid:1) ∩ ker f W i (cid:3) . Using Expression (16) and the relations above, we have: X a ≤ d ≤ b E , Y ( a, b, d, n, d, d ) ≤ dim α a (cid:0) ker f V a,b +1 (cid:1) − dim f W a − ,a α a − (cid:0) ker f V a − ,b +1 (cid:1) − dim α a (cid:0) ker f V a,b (cid:1) + dim f W a − ,a α a − (cid:0) ker f V a − ,b (cid:1) . Using the second expression of Remark 2.2 we have: dim α a (cid:0) ker f V a,b +1 (cid:1) − dim α a (cid:0) ker f V a,b (cid:1) = dim ker f V a,b +1 − dim ker f V a,b which is equal to dim S V a,b by definition. Analogously, − dim f W a − ,a α a − (cid:0) ker f V a − ,b +1 (cid:1) + dim f W a − ,a α a − (cid:0) ker f V a − ,b (cid:1) = − dim ker f V a − ,b +1 + dim ker f V a − ,b = − dim S V a − ,b X ≤ b ′ ≤ n X ≤ a ′ ≤ b ′ G α,β ( a, b, a ′ , b ′ ) ≤ X a ≤ d ≤ b E , Y ( a, b, d, n, d, d ) ≤ dim S V a,b − dim S V a − ,b = D V ( a, b ) . concluding the proof. G α,β and the persistence module K As mentioned in the introduction, our work is related to the work developed in [16]. Althoughthe persistence module K is constructed in that paper for a diagram different than Diagram (14),we can adapt such construction to our case. Notice that all the columns V i −→ W i ←− U i inDiagram (14) are zigzag modules. For those familiar with zigzag modules, observe that D [1 ,
3] =dim ( α i ( V i ) ∩ β i ( U i )) . Therefore, the vector spaces K i = α i ( V i ) ∩ β i ( U i ) encode the relation between V and U through W , but this is true only if we consider each columnseparately in the diagram. How can we extend this construction to the whole diagram? The answeris the following result. Proposition 4.7.
The persistence module K , formed by vector spaces K i and structure linearmaps f K a,b = f W a,b | K a , is well-defined and is a submodule of W .Proof. We have to prove that im f W a,b | K a ⊂ K b or equivalently, f W a,b ( α a ( V a ) ∩ β a ( U a )) ⊂ α b ( V b ) ∩ β b ( U b ) . Notice that if x ∈ α a ( V a ) ∩ β a ( U a ) then, in particular, x ∈ α a ( V a ) and there exists y ∈ V a suchthat f W a,b α a ( y ) = f W a,b ( x ) . Then, α b f V a,b ( y ) = f W a,b ( x ) by commutativity of Diagram (14). Therefore, f W a,b ( x ) ∈ α b ( V b ) . Following a similar procedure for U , we have that f W a,b ( x ) ∈ β b ( U b ) , concludingthe proof.This way, K is a persistence module obtained without fixing any basis on the given persistencemodules. Nevertheless, K gives no explicit relation between the decomposition of V and thedecomposition of U as the operator G α,β does. The relation between G α,β and K is given by thefollowing theorem that uses induced enriched partial matchings. Theorem 4.8.
Let us consider a pair of morphisms V α −→ W β ←− U between persistence modulesof length n . Let ( a, b, a ′ , b ′ , c, d ) ∈ Z ≥ such that ≤ a, a ′ , c ≤ d ≤ b, b ′ ≤ n . Then, X ≤ c ≤ d E D K ( c, d )= X d ≤ b ≤ n X ≤ a ≤ d X d ≤ b ′ ≤ n X ≤ a ′ ≤ d X ≤ c ≤ d (cid:8) ([ c, d ] , e ) ∈ G α,β ( a, b, a ′ , b ′ ) (cid:9) . In other words, the number of intervals in B K with endpoint d equals the number of intervalsin G α,β with endpoint d . Proof.
By the definition of G α,β and Y . and by Lemma 3.11 we have: X d ≤ b ≤ n X ≤ a ≤ d X d ≤ b ′ ≤ n X ≤ a ′ ≤ d (cid:8) ([ c, d ] , e ) : ([ c, d ] , e ) ∈ G α,β ( a, b, a ′ , b ′ ) (cid:9) = X d ≤ b ≤ n X ≤ a ≤ d X d ≤ b ′ ≤ n X ≤ a ′ ≤ d X ≤ c ≤ d E , , , , ( Y )( a, b, a ′ , b ′ , c, d )= Y ( d, n, d, n, d, d ) . By Lemma 3.11, we have: X ≤ c ≤ d E D K ( c, d ) = S K d,d . Y ( d, n, d, n, d, d ) = f W d,d α d (cid:0) ker f V d,n +1 (cid:1) ∩ f W d,d β d (cid:0) ker f U d,n +1 (cid:1) ∩ S W d,d = α d ( V d ) ∩ β d ( U d ) ∩ S W c,d = K d ∩ W d ∩ ker f W d,d +1 = S K d,d , concluding the proof.We conclude the section with the following final result. Corollary 4.9.
Let us consider a pair of morphisms V α −→ W β ←− U between persistence modulesof length n . Let ( a, b, a ′ , b ′ ) ∈ ∆ + × ∆ + . Then, X ≤ b ≤ n X ≤ b ′ ≤ n X ≤ a ≤ b X ≤ a ′ ≤ b ′ G α,β ( a, b, a ′ , b ′ ) = B K In other words, the sum of the cardinals of all multisets appearing in G α,β is equal to the cardinalof B K . In this paper, we have studied how a morphism between persistence modules (also called laddermodules) can induce a basis-independent partial matching between their corresponding persistencebarcodes or diagrams. We have also proved the linearity of our method with respect to direct sumof ladder modules. In addition, the concept of enriched partial matching have been introduced. Ithas been used to study the relation between persistence modules V , U through a pair of morphisms V α −→ W β ←− U . Explicit relations with other state-of-the-art tools (Bauer-Lesnick matching [3],ladder modules [11] and K [16]) have been given. Future work could follow these directions: • Implementation: although our definitions and proofs are constructive, the implicit algo-rithm can only be applied to vector spaces. It would be interesting to implement algorithmsacting directly in more common spaces like simplicial complexes. The algorithms of [8] and[6] may be good starting points. • Stability: is the induced basis-independent partial matching stable with respect to mod-ifications of the morphism α ? and is the enriched partial matching stable with respect tomorphisms α and β ? Stability theorems from [10] and [16] offer a great background toproceed. • Generalization: persistence modules can be defined in any poset and not only for finitesequences. Can we extend our partial matching to persistence modules over a real parameteror to zigzag modules? What is the exact relation of induced (enriched) basis-independentpartial matchings with multidimensional persistence modules? • Applications: we think induced (enriched) basis-independent partial matching can be usedto model real world application. One example could be given by dynamical metric spaces(see [14, 15]) where the object of study is point clouds evolving in time.
Acknowledgments:
This paper has been partially supported by Ministerio de Ciencia eInnovación (grant number: PID2019-107339GB-I00) and Universidad de Sevilla (grant number:PIF VI-PPITUS 2018).
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