On the complexity of zero-dimensional multiparameter persistence
aa r X i v : . [ m a t h . A T ] A ug ON THE COMPLEXITY OF ZERO-DIMENSIONALMULTIPARAMETER PERSISTENCE
JACEK BRODZKI, MATTHEW BURFITT, AND MARIAM PIRASHVILI
Abstract.
Multiparameter persistence is a natural extension of the well-known persistenthomology, which has attracted a lot of interest. However, there are major theoreticalobstacles preventing the full development of this promising theory.In this paper we consider the interesting special case of multiparameter persistencein zero dimensions which can be regarded as a form of multiparameter clustering. Inparticular, we consider the multiparameter persistence modules of the zero-dimensionalhomology of filtered topological spaces when they are finitely generated. Under certainassumptions, we characterize such modules and study their decompositions. In particularwe identify a natural class of representations that decompose and can be extended backto form zero-dimensional multiparameter persistence modules.Our study of this set of representations concludes that despite the restrictions, thereare still infinitely many classes of indecomposables in this set. Introduction
Persistent homology assigns numerical invariants to a topological space. In practice, thenecessary input is provided by a filtered simplicial complex created from a sample of thespace of interest [7, 16]. This leads to the formation of a persistence module, by assigninga vector space – the associated homology group – to each stage of the filtration, togetherwith the linear transformations stemming from the inclusion maps of the filtration.When the filtration is given by one parameter, i.e. when we have a nested sequenceof subcomplexes, the resulting persistence module is a representation of the so-called A n quiver. Due to a deep theorem by Gabriel in representation theory, this quiver is knownto be of finite type. This means that every representation of it can be decomposed into adirect sum of a finite set of isomorphism classes of indecomposables.This property only holds for representations of quivers whose shape is that of a Dynkinor Euclidean diagram. This is very restrictive and for most quivers the problem is knownto be extremely hard.The generalisations to multiparameter persistence modules are fraught with difficultiesdue to algebraic problems in the decomposition theory of such modules. These are quiverrepresentations over partially ordered sets with commutative restrictions. In practice, theposet considered is often a commutative n -dimensional grid. In this paper we justify con-sidering persistence modules over finite posets by showing that the algebraic problems areequivalent to N n and R n modules with a finite encoding. However, even this restriction tofinite bounded acyclic quiver representations, corresponding to multiparameter persistence,does not in any clear way simplify the problem [14, 23]. Multiparameter persistence modules Department of Mathematical Sciences, University of Southampton, Highfield, Southamp-ton, SO17 1BJ, United Kingdom
E-mail addresses : [email protected], [email protected], [email protected] .This research was supported by the EPSRC grant EP/N014189/1. also arise in theoretical works in topological data analysis. However, because the underlyingquiver is of a wild type, it becomes much more challenging to derive numerical invariants.It has been shown [11] that one cannot assign a complete discrete invariant to such modulesand in dimensions greater than zero all modules can be realised as the homology of somefiltration [18].Multiparameter persistence is strongly motivated by applications such as including den-sity variables [28] to extend the stability of persistent homology with respect to noise andto incorporate other data specific parameters, the uses of which have already been demon-strated [8, 1, 20].Studying multiparameter persistent homology in zero dimensions is equivalent to consid-ering multiparameter clustering, which is relevant to problems concerning clustering density.In fact, the origins of persistent homology lie in the study of the structure of connected com-ponents [17] and has been applied to the stability of hierarchical clustering [9, 10].The description of zero dimensional multiparameter persistence modules becomes some-what more combinatorial than that of classical quiver representation theory so that wemight begin to understand their decomposition theory. The reason for this is that theclass of persistence modules arising as zero-dimensional persistence is not necessarily stablewith respect to natural representation-theoretic transformations, for example, the reflec-tion functors. It therefore makes sense to understand the special case of zero-dimensionalmultiparameter persistence modules using more combinatorial tools.This is the question we are investigating in this paper. Restricting our attention to thezero-dimensional case allows us to consider a smaller class of representations. As a result, weare able to show that in general, the isomorphism classes of indecomposable representationsof the remaining modules are still infinite.To this end, we introduce component modules corresponding to the persistence modulesobtained from zero-dimensional multiparameter persistence and the weaker notion of semi-component modules, obtained as decompositions of component modules. We show thatprovided there is a minimal generator in the component module, it splits as an intervaland semi-component module. A situation that occurs in applications such as hierarchicalclustering. Furthermore any component module can be recovered from a semi-componentmodule up to a choice of right inverse dependent only on the placement of the minimalgenerator.A number of approaches to multiparameter persistence have been developed [13, 18, 21,22, 25, 27]. Most relevant to the content of our paper, much work has also been devotedto studying the structure of multiparameter persistence modules under restrictions. Fortwo-parameter persistent homology, Cochoy and Oudout [12] present a local conditionsthat characterises persistence module emerging as a sum of interval modules. However insimple cases on N m , Buchet and Escolar [5, 6] have given realistically plausible topologicalexamples that realise complex infinite families of indecomposable modules, in which anyone parameter persistence module can be obtained by the restriction of a two-parameterindecomposable module. Motivated by multi-filtrations arising form hierarchical clusteringBauer, Botnan and Oppermann [3] investigated two-parameter persistence modules in whichthe parameter choices induces injective or surjective maps vertically or horizontally withinthe module, giving a full classification of representation types. N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 3 Background
In this section we recall the language and relevant results on multiparameter persistenceuseful to us in subsequent sections. Unless otherwise stated assume we are working over afield K and all vector spaces over K are finite dimensional.2.1. Partially ordered sets.
Multiparameter persistence modules occur indexed over par-tially ordered sets, therefore the mathematics of theses structures plays a fundamental partin their description. We recall now notations concerning partially ordered sets used later inthis work.
Definition 2.1. A partially ordered set , a partial order on a set or a poset is a set P witha binary relation ≤ that satisfies the following. (1) For any p ∈ P , then p ≤ p (Reflexivity); (2) For any p, q ∈ P , if p ≤ q and q ≤ p then p = q (Antisymmetry); (3) For any p, q, r ∈ P , if p ≤ q and q ≤ r then p ≤ r (Transitivity). A partial order can be seen as a category with an object for each element of P and aunique morphism α : p → q whenever p ≤ q for p, q ∈ P . A morphism of partially orderedsets coincides with a morphism between the associated categories. Following this section,throughout the remainder of this paper we view a partially ordered set as a category.We use the following terminology on partial orders throughout the rest of the paper. Aposet P is finite if P is finite. Two elements p, q ∈ P are said to be comparable if either p ≤ q or q ≤ p . A poset is connected if any two elements can be written as the endpointsof a sequence of pairwise comparable elements. We say element p ∈ P is minimal if for anycomparable q ∈ P , then p ≤ q . Similarly an element p ∈ P is said to be maximal if for anycomparable q ∈ P , then q ≤ p . A poset is bounded if it has unique minimal and maximalelements. For p ≤ q such that p = q and for r ∈ P such that p ≤ r ≤ q implies r = p or r = q , then we say q covers p and we call ( p, q ) a cover relation. A graph with vertices theelement of the partial order and edges the cover relations between them is called a Hassediagram .A subset C ⊆ P is called a chain if it is totally ordered , that is if any two elements of C are comparable. A chain is called a maximal chain , if it is not contained in a largerchain. The height , h ( P ) of a partial order P is the cardinality of a maximal chain in P . An antichain is a subset of P in which no pair of elements are comparable. Given two partiallyordered sets P and Q their product is the partially ordered set on P × Q such that,( p, q ) ≤ ( p ′ , q ′ ) if and only if p ≤ p ′ in P and q ≤ q ′ in Q. Multiparameter persistence modules.
In their most general form multiparameterpersistence modules are considered indexed over an arbitrary partially ordered set.
Definition 2.2.
A (multiparameter) persistence module M over a field K is a functorfrom a poset P to the category of finite dimensional K -vector spaces and linear maps. Apersistence module M is called finite if P is finite. For simplicity from now on we call a multiparameter persistence module a persistencemodule. In the usual way, we obtain a category of persistence modules and natural trans-formations. Given a persistence module M the dimension vector of M is { dim M ( p ) } p ∈ P , ON THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE the set of dimensions of vector spaces assigned to each object of the partial order P . Sincepersistence modules are indexed over a partial order we present them as a diagram, itmakes sense to present them as Hasse diagrams, (adding direct edges). That is vertices arelabelled with the dimension of the vector space and cover relations are labelled with thecorresponding linear map with respect to a chosen basis for each vector space.The modules naturally corresponding to the barcode decomposition in one parameterpersistence (persistence over a totally order set) correspond to intervals in the order. Thenotion of an interval module is generalized easily to the multiparameter setting. Definition 2.3. A K persistence module M over a partially ordered set P is called an interval module if: (1) For every object p ∈ P , the vector space M ( p ) is isomorphic to K or . (2) For all morphisms α : q → p in P such that M ( q ) ∼ = M ( p ) ∼ = K , the linear maps M ( α ) can with respect to some basis be chosen simultaneously as the identity. (3) For morphisms α : q → p and β : p → l in P if M ( q ) ∼ = M ( l ) ∼ = K , then M ( p ) ∼ = K . Note that [2] if part (2) of the definition is weakend to require only isomorphism, thenthe definition is weakened for modules that cannot be embedded in N m for m > Definition 2.4.
A filtered topological space is a functor X from a poset P to the categoryof topological spaces and inclusions, such that the image of every object under X has finitelymany path components. We assume here that the filtered space X is a simplicial complex filtered by subcom-plexes and that the inclusions are simplicial inclusions. This can be justified by the factthat topological spaces arising form data, such as the Vietoris-Rips and Alpha complexesobtained form point cloud data are realised in this form.2.3. The theory of quiver representations.
A finite poset can be identified with a quiver . This identification is an important step in applying well-known results from thetheory of quiver representations to the setting of persistence modules. In particular thetheory of quiver representations can be applied to multiparameter persistence modules.Unless otherwise stated all the proofs of statements made in this section can be found in[26].
Quivers can be thought of as directed graphs, or rather multigraphs. In general, bothmultiple arrows between nodes and loops (i.e. arrows beginning and ending at the samenode) are allowed. However, in the case of persistence modules, we only deal with quiversthat are simple directed graphs. A representation of a quiver Q over a field K is theassociation of a K -vector space to each node of Q and a linear map between the vectorspaces to each arrow. If each vector space is finite dimensional then the representationis called finite dimensional . The direct sum of two quiver representations is the quiverrepresentations whose vector spaces and linear maps at are the direct sums of those formthe two representations. Definition 2.5.
A representation of a quiver Q is called indecomposable if it cannot beexpressed as a direct sum of at least two non-zero representations of Q . N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 5
There is an equivalence of categories between a quivers representations and left modulesover their path algebra. It is then a straightforward consequence of the classical Krull-Remak-Schmidt theorem, that a decomposition of a finite-dimensional quiver representationinto a direct sum of indecomposables is unique up to reordering and isomorphism.The following lemma appears as the most straightforward characterisation of indecom-posable quiver representations, see [4]. However here for ease of use, we state the lemma interms of persistence modules.
Lemma 2.6.
A persistence module is indecomposable if and only if all self natural trans-formations are either an isomorphism or nilpotent. In particular, given a natural transfor-mation f of persistence module M we have M = ker ( f n ) ⊕ im ( f n ) , for n large enough and where f n is the composition of f with itself n times. It is straightforward for example to show using Lemma 2.6 that all persistence modules M over P satisfy the first two conditions of Definition 2.3 are indecomposable persistencemodules, since for q ∈ Q if in M ( q ) is non-trivial then the choice of a natural endomorphism f at M ( q ) determines the choice at all other M ( p ) by naturality, hence f is either a naturalisomorphism or trivial. 3. Finite encoding
In this section we justify the abstract setting for multiparameter persistence modulesused in this work. The term finite encoding is used by Miller [22], where given a persistencemodule M over Q an encoding of M by a poset P is a poset morphism π : Q → P, together with a P -module H such that M is the pullback of H along π , which is naturallya Q -module. The encoding is finite if(1) P is finite, and(2) the vector space H p is finite dimensional for all p ∈ P .The next theorem shows that in terms of decomposition, considering finite multiparam-eter persistence modules is equivalent to considering N n and R n modules with a finiteencoding. To prove the theorem we first give the following additional terminology. Given p , . . . , p m in poset P , we call the set J the join-set of p , . . . , p n if it satisfies the followingconditions.(1) For each j ∈ J we have j ≥ p i for i = 1 , . . . , m ,(2) any k ∈ P satisfying the first condition is either incomparable to each j ∈ J , j = k ,or j ≤ k .If the join set contains a single element then this is the usual notion of a join. Theorem 3.1.
Every finite persistence module with a single maximal and minimal elementin its poset can be realised as the finite encoding of some N n module, and by extension, aright continuous R n module constant on cubical regions. We can extend the theorem to all finite multi-parameter modules if we identify them withthe module obtained by attaching a zero-dimensional vector space as the minimal and ormaximal elements of the module.
ON THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE
Proof.
Let M be a finite multiparameter persistence module over poset P and let C , C . . . , C n be a cover of P by maximal chains. We construct a poset morphism π : N n → P such thatits pullback over M is a well defined N n module for which π is a finite encoding. Let l i , for i = 1 , . . . n , be the length of the maximal chain C i . Select the l i smallest elements of axis i and assign them bijectively to the elements of C i in an order-preserving way. As the chainsare maximal, π is well defined at the origin sending it to the minimal element of P .We now extend π working inductively on elements of N n using an analogue of Cantor’sdiagonal argument on maximal antichains in N n with respect to their natural order. Let y ∈ N n on which π has not yet been defined, and let U y ⊂ N n be the set of elements m ≤ y .We let π ( y ) be an element of the join-set of the subset π ( U y ) ⊆ P . Since we assume a singlemaximal element of P , the join-set of any set of elements is always non-empty.By construction, for every y ∈ N n , π ( x ) ≤ π ( y ) for every x ∈ N n such that x ≤ y and so π is a poset map as required. Hence π is a well defined encoding of its pullback modulesand it is also finite as P is finite. (cid:3) Remark 3.2.
There are classical theoretical results in algebraic topology that justify Miller’sdefinition of finite encoding as a sensible approach to multiparameter persistence.A function that allows finite encoding is one that has constant homology on the preimagesof compact regions of the range, i.e that has regions of critical values that form submanifoldsof the range of one lesser dimension. In particular, finite encoding restricts to the usualfiniteness assumption present for one-parameter persistence. Therefore, the setting of mod-ules over an arbitrary finite partially ordered set considered here is of particular interest.We mention here two results from topology, which apply to the case of smooth functions f , f , . . . , f n : M → R on a smooth d dimensional manifold M . Sard’s Theorem [24] states that in the case when M = R m , while there might be many critical points in R m , the image, i.e. the set of thecritical values, has Lebesgue measure in R n . This is of course more general than the setof critical values being a submanifold.The Jacobi set of f , f . . . , f n is the closure of { x ∈ M | x is a critical point of f i for some i = 0 , . . . , n } . Edelsbrunner and Harer [15] worked on the structure of such critical sets for the appliedsetting. In particular they show that when n < the Jacobi set is a smoothly embeddedsub-manifold of M . Topological restrictions on zero-degree persistence modules
It is stated [11] and proved in [18], that when P = N n every finitely generated multipa-rameter persistence module over P with coefficients in F p or Q can be realised as homologyin positive degrees of a filtered simplicial complex. Using Theorem 3.1, it is clear that theseresults will also hold over an arbitrary finite partially ordered set. However, for i = 0, wesee that the situation is more restricted.Let π ( X ) be the set of (path-)connected components of the simplicial complex X . Itis an elementary fact in topology that the image of a continuous map on a (path-)connectcomponent is (path-)connected. So π is a functor from the category of simplicial complexesand continuous maps to the category of sets. It is a classical fact from algebraic topology N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 7 [19] that H ( X ) is the free vector space generated by π ( X ). Therefore, when we talk abouta basis for H ( X ), we always talk about this particular basis (unless stated otherwise). Itfollows that if f : X → Y is a continuous map, then the induced map H ( X ) → H ( Y )has the property that it takes basis elements to basis elements. Therefore, the matrixrepresenting this map consists of zeros and ones, with exactly one 1 in each column.4.1. Factoring through
Sets . The above implies that the induced maps between thehomology groups are factoring through the category of sets. This property is rather fragile,however. Standard transformations in representation theory, like reflection functors, destroyit. Take, for example, the following representation of A : K K . (cid:2) (cid:3) The reflection functor reverses the arrow, and takes this to the representation K K . (cid:20) − (cid:21) Clearly, this representation no longer satisfies our condition, as the corresponding matrixdoes not consist solely of 0 s and 1 s . In this particular case, we are dealing with a repre-sentation of the A quiver, and therefore there is a nice change of basis that makes it fitour condition (and, in fact, both of these representations decompose). However, this is nolonger necessarily the case for more complicated quivers.Geometrically, the reflection functor takes subspace embeddings to quotient maps, andvice versa. In this case, for example, the second is embedding the subspace K along thesecond diagonal in K , while the first representation corresponded to K being projectedonto the quotient space K K . So we have a short exact sequence K K K , (cid:20) − (cid:21) (cid:2) (cid:3) and the composition of the two matrices gives the zero map.4.2. Component and semi-component modules.
We first define our main object ofstudy motivated by the discussion at the beginning of the section.
Definition 4.1. A K persistence module M over a partially ordered set P is called a com-ponent module if there is a basis (called a component basis of M ) for every vector space M ( q ) , with q ∈ P , such that any morphism α : q → p in P with respect to the basis is alinear map whose matrix contains only ones and zeros with exactly one in each column. Ifthe matrices contain at most one in each column, it is called a semi-component module . Interval modules are the component modules for which M ( q ) has at most dimension onefor any q ∈ P . For any multiparameter persistence module we may define its generators.Here we give a simple characterisation. ON THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE
Definition 4.2.
An element of the component basis is called a generator if it is not in theimage of any of the linear maps M ( α ) for α : p → q in P . A (semi-)component modules issaid to have a minimal generator if, there is at least one generator in M ( p ) and maps α : p → q, for any q ∈ P such that M ( q ) contains a generator. The set of generators can be chosen to interact well with component basis.
Lemma 4.3.
The elements of a (semi-)component basis of a (semi-)component module canbe chosen so that each basis element is the image of one or more generators among allmorphisms for which it is in the domain.Proof.
The lemma follows up to changes of sign from the structure of the matrices withrespect to a component basis given in definition 4.2. (cid:3)
The next theorem demonstrates that component modules are the important modules forzero dimensional persistent homology. In particular, the component modules characterizeexactly the modules obtainable as the zero-dimensional homology of filtered topologicalspaces.
Theorem 4.4.
The zero dimensional homology of a finite filtered CW-complex over partialorder P is a finitely generated component module over P . In addition, every finitely gen-erated component module over P is the zero-dimensional homology of a some finite filteredCW-complex over P .Proof. For a finite filtered CW-complex X over a partially ordered set P , we may choose abasis of each H ( X ( q ); K ) as the set of homology classes of points in each path componentof X ( q ) for each q ∈ P . These basis are each finite since X is finite. The morphisms of thepersistence module H ( X ; K ) are induced by the the inclusions X ( α ) for α : q → p a in P .By continuity the image of a path component of X ( q ) under X ( α ) is contained in a singlepath component of X ( p ), which implies the conditions of definition 4.1. By constructionthe number, generators of H ( X ; K ) is bounded above by the number of path componentsin X , which can be no more than the number of zero simplicis and since each X ( q ) is afinite CW-complex H ( X ; K ) is finitely generated.Given a finitely generated component module M over a partially ordered set P we mayconstruct a filtered simplicial complex X such that M = H ( X ; k ) as follows. For eachgenerator g i ∈ M ( q i ) of M , there is a corresponding vertex x qi in each X ( q ) for q ∈ P witha morphism α : q i → q . In particular by Lemma 4.3, each x qi correspond to one or more v qj in the component basis. For every morphism α : q → p in P there is an edge between x qi and x qj when both have a 1 in the same row of the matrix corresponding to α with respectto the component basis. The edges can be chosen consistently because all morphism in M commute. Finally, since there were only finitely many generators of M , by constructioneach X ( q ) is a finite CW-complex. (cid:3) As a consequence of the characterisation of Theorem 4.4, we can specify the subcategoryof persistence modules of interest when restricting to zero dimensional homology.
Definition 4.5.
Over any a partial order P , define the category of zero-parameter modules to be the full subcategory of persistence modules, with objects the modules whose indecom-posables occurs as indecomposables of some component module. N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 9
Theorem 4.4 can be seen as following intuitively form properties of continuous maps be-tween path components of topological spaces, however it immediately leads to the followingsurprising consequence on the types of decomposition that can arise form the 0-dimensionalpersistent homology of filtered finite dimensional CW-complexes. This marks a significantsimplification over the general case of persistence modules.
Corollary 4.6.
Over any finite partial order set P , the number of zero-parameter modulesof a given dimension vector is finite.Proof. By Theorem 4.4 the zero-dimensional persistent homology of a filtered finite simpli-cial complex is a component module. Every component module of a fixed dimension vectorover P has a basis on each vector space, for which the linear maps of each morphisums arematrices containing only zeros and ones. Therefore there are at most finitely many suchmodules of a given dimension vector. (cid:3) Decompositions of zero dimensional persistence modules
In this section we explore the indecomposables and decompositions of persistence modulescorresponding to the zero-dimensional homology of filtered topological spaces.5.1.
An initial interval decomposition.
Not all non-interval component modules al-low interval decomposition, and those that do, do not necessarily decompose into intervalmodules as demonstrated by the following examples.
Example 5.1.
Consider the persistence module
KK K K , (cid:20) (cid:21)(cid:20) (cid:21) (cid:2) (cid:3) where the matrices represent the linear maps with respect to some component basis. Themodule is not an interval module as the central vector space is -dimensional. Moreover, itis straightforward to show that the module is indecomposable using Lemma 2.6. Notice alsothat there is no minimal generator since the two leftmost vector spaces lie in incomparableminimal positions of the partial order. Example 5.2.
Consider the component module with component basis,
K K K K K K . (cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21) (cid:2) (cid:3) We may construct a decomposition of this module using using Lemma 2.6 with idempotentendomorphism, (cid:2) (cid:3) (cid:20) − (cid:21)(cid:20) −
10 1 (cid:21) − −
11 00 1 (cid:20) −
10 1 (cid:21) (cid:2) (cid:3) . This yields a decomposition into kernel and image modules
K KK K K K (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) and KK K K . (cid:2) (cid:3)(cid:2) (cid:3) (cid:20) (cid:21)(cid:20) (cid:21) (cid:2) (cid:3) (cid:2) (cid:3) Both of these are indecomposable component modules, only the first of which is an intervalmodule. The second module can be shown to be indecomposable in the same way as Example5.2.
We will see that if a non-interval component module has a minimal generator, it alwaysdecomposes. The following theorem specialises in the one parameter case to the remark thatzero-dimensional persistent homology decomposes with a unique longest interval, providingthe filtration spaces are eventually connected.
Theorem 5.3.
Let C be a finite K component module over a partially ordered set P withat least two generators, at least one of which is minimal. Then C decomposes as a directsum of an interval and a non-trivial semi-component module over P .Proof. By Lemma 4.3 we can take generators g , . . . , g n in C ( q ) , . . . , C ( q n ) , with g i ∈ M ( q i ) and component basis v q , . . . , v q dim( C ( q )) ∈ C ( q ) , such that for each p ∈ P , any P morphism α : q i → p has C ( α )( g i ) = v pj for each i = 1 , . . . , n and j = 1 , . . . , dim( C ( p )). Let d ∈ N be the maximal dimension of C ( p ) over all p ∈ P .If d = 0 then M ( p ) = 0 for all p ∈ P , so C is the trivial module. If d = 1 then M is aninterval module. Therefore, assume now that d ≥ n ≥ N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 11 construct a natural endomorphism f on C that is neither an isomorphism nor nilpotent.Since C has a minimal generator, without loss of generality we may assume that g is aminimal generator. So, for each j = 1 , . . . , n there are morphisms α j : q → q j in P . Using the commutative condition, a natural endomorphism f of C will be determineduniquely by f q i ( g i ) for i = 1 , . . . , n . Define f q i on g i by f q i ( g i ) = ( g i − M ( α i )( g ) if i = 10 if i = 1 . (1)We now check that this extends to a natural endomorphism f on C . For any p ∈ P and v pk for k = 1 , . . . , dim( C ( p ))), let { i , . . . , i l } be the subset of { , . . . , n } such that there existsa β i j : q i j → p with β i j ( g i j ) = v pk . For each i = 1 , . . . , n since C is a component module if β i exists, β i ( g i ) cannot be zero andmust by construction be an elements of the component basis. The endomorphism f existsif and only if for all i j , each ( β i j ◦ f q ij )( g i j )agree, in which case f p ( v pk ) is the image. This is indeed the case because if for some j , i j = 1then all the ( β i j ◦ f q ij )( g i j ) are 0, otherwise ( β i j ◦ f q ij )( g i j ) are all equal to v pk − C ( α )( g ).Since n ≥
2, by construction (1) f is not an isomorphism and not nilpotent.In fact f is an idempotent, meaning f = f . To see this notice that for each i =1 , . . . , n , the linear map f q i given in equation (1) is an idempotent, since for i = 1 we have f q i ( C ( α i )( g )) = 0 by naturality of f , so f q i ( g i ) = f q i ( g i − C ( α i )( g )) = f q i ( g i )and f q = 0 otherwise. By naturality f q is determined by f q i for any other q ∈ P and so f is an idempotent.Hence by Lemma 2.6 since f is an idempotent, C can be decomposed as the direct sumof the kernel of f and the image of f . The kernel of f is the sub module generated by thesingle generator g , hence is an interval module. The image of f is generated by g i − g for i = 2 , . . . , n . Since C is a component module the restriction of the matrices with respectto the component basis on these generators is a a semi-component module. Therefore, theimage of f is a semi-component module with n − (cid:3) The theorem does not apply to semi-component modules essentially because connectedcomponents in a filtered topological space can only merge and cannot disappear. Thisproperty is not shared by semi-component modules.5.2.
Semi-component extension.
In this section we show that all semi-component mod-ules may be obtained as the decomposition of a component module. In particular, weobserve how this decomposition relates to the one derived in the previous section. We firstdemonstrate the key ideas in an example.
Example 5.4.
The component module on the right can be seen to decompose into the sumof the semi-component module shown on the left together with an interval module givenby the entire underlying poset. This is done by straightforward linear transformations todiagonalise each matrix. K KKK K K K K K (cid:2) (cid:3)(cid:2) (cid:3)(cid:2) (cid:3) (cid:2) (cid:3)(cid:2) (cid:3)(cid:2) (cid:3) (cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21) (cid:2) (cid:3)(cid:2) (cid:3)(cid:2) (cid:3) We can generalise the construction of this example on semi-component modules to thefollowing method.
Construction 5.5.
Take a semi-component module S and a choice of semi-componentbasis. Since with respect to this basis every linear map is represented by a matrix with atmost one in each column we may obtain form S a component module C as follows. • Add an extra dimension to every vector space in S . • Add to each matrix in S a new first column and first row. • The new first column contains a as its first entry and zeros otherwise. • After the first entry, the new first row has entries if the column already containsa and is otherwise.We call C the component extension of S . We will see that a component module resulting from a semi-component extension is thedirect sum of the extended semi-component module and an interval module. In particularall semi-component modules can be obtained from component modules by decomposition.First we make the following observations.By choosing a basis of the vector space of each vertex of a quiver Q , the entries ofthe matrices associated to the arrows give a point in the representation space of quiverrepresentations of a given dimension vector. The action of an elementary matrix operationsat a vertex on the representation space is easily determined. Remark 5.6.
Given a quiver representation, we may choose a basis for each K -vector spaceat a node, realising the linear map associated to each arrow between nodes as a matrix ofthe appropriate dimensions. A decomposition of a quiver representation corresponds to achoice of basis at each node such that the matrices of each arrow are simultaneously blockmatrices of the same type. A change of basis at a node corresponds to a square matrix B .The the change of basis at this node effects each matrix T of an arrow into the node, byreplacing it with T B and each matrix S of an arrow out of the node, by replacing it with B − S . In particular: N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 13 (1)
Performing an elementary row operation on the matrix of some arrow, simultane-ously performs the reverse of the elementary column operation of the same type onall matrices of arrows with the same target. (2)
Performing an elementary column operation on the matrix of some arrow, simulta-neously performs the reverse of the elementary row operation of the same type onall matrices of arrows with the same source.
To obtain our theorem on semi-component extension we first prove the following lemmaand to this end we give some additional terminology. For each n ≥ n ], theset { , , , . . . , n } with the usual total order. Definition 5.7.
Given a finite partially ordered set P , a map f : P → [ h ( P ) − is calleda pre-grading of P if f is a poset map with the additional property that if p, q ∈ P when p < q then f ( p ) < f ( q ) . Usually P is called a graded poset if there is a map f : P → N , that in addition to theproperty above also satisfies, f ( p ) = f ( q ) + 1 whenever p covers q . Not all posets have agrading. For example a bounded poset has a grading only when all maximal chains havethe same size. Lemma 5.8.
Any partially ordered set with finite height has a pre-grading. In particularthe preimage of each element of the pre-grading function will be an antichain.Proof.
Given a finite partial order P , we can construct a pre-grading function f by inductionon [ h ( P ) −
1] as follows. Define the preimage of f at 0 as all minimal elements of P . Minimalelements form an antichain as they are incomparable by definition, making f well definedon these elements of P .Assume inductively that the preimage of f has been defined on all elements with preimageless than k ∈ [ h ( P ) − S ⊆ P be all elements which cover f − ( k − S ′ ⊆ S to be the subset such that for any p, q ∈ S , if p ≤ q then q / ∈ S ′ . Hence S ′ is an antichainand we may choose the preimage of f at k to be S ′ . By construction of f all elementsof P are assigned a value by f equal to one less than the length of the smallest maximalchain in which they and a minimal element of P are contained, therefore f is a well definedpre-grading. The inductive construction will terminate at h ( P ) − f is uniquelydetermined on each maximal chain. (cid:3) Theorem 5.9.
Any semi-component module S over a finite partial order P arises as asummand of a component module C in the following way C = S ⊕ I, where I is an interval module over P .Proof. Given a semi-component module with a component basis, take its component exten-sion with respect to this basis, giving us a component module and corresponding componentbasis. We now outline a procedure to decompose this component basis through elementaryrow and column operations on the corresponding matrices of its linear maps representingcover relations in the underlying partial order.Take some pre-grading of the partial order as defined in Definition 5.7, which existsby Lemma 5.8. Beginning from cover relation matrices whose source is of pre-grade 0 inthe partial order, we may perform column operations subtracting the first column fromevery other column. Since these matrices must represent linear maps whose domain is a minimal element of the partial order, by Remark 5.6 these operations have no effect onother matrices. All zero columns of these matrices that were zero in the semi-componentmodule are zero and the other columns have a − − (cid:3) Remark 5.10.
Given a p ∈ P such that all generators lie in some p ≤ q ∈ P , we can stillderive a corresponding semi-component extension satisfying Theorem 5.9 if Construction5.5 were changed to leave all s ≤ p as zero dimensional vector space and the extension ofDefinition . is applied as before to all r ≥ p . Together Theorems 5.3 and 5.9 show that semi-component modules play an importantrole in understanding the indecomposables of component modules as highlighted by thefollowing corollary.
Corollary 5.11.
Let C be a component module over a partial order P with a minimalgenerator. Then there is a unique decomposition C = S ⊕ I, where S is a semi-component module and I an interval over P . In particular C is thecomponent extension of S for some choice of minimal generator in the sense of Remark5.10.Proof. The interval module I obtained in the proof of Theorem 5.3 is uniquely determinedup to isomorphism as the module with dimension 1 in every position greater than a minimalgenerator and identity maps between them. Hence the uniqueness of S follows from theKrull-Remak-Schmidt theorem.Using the proof of Theorem 5.9 extension, the extension of S with the minimal generatorin the position it occurred in C decomposes as S ⊕ I and is therefore C . (cid:3) In other words the corollary states under its conditions that up maintaining the positionof the minimal generator the decomposition of theorem 5.3 and component extension areinverses.5.3.
Decompositions of semi-component modules.
Although Theorems 5.9 and 5.3show semi-component modules form an interesting sub-class of modules arsing from de-compositions of component modules, they are not all indecomposable (see Example 5.1),do not contain all their indecomposable modules and still contain a large number of possi-ble indecomposables. We first give an example showing that semi-component modules dodecompose into two non-semi-component modules.
N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 15
Example 5.12.
The following semi-component module has indecomposable decomposition, K K K K K (cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21)(cid:20) (cid:21) = K K KK K (cid:2) (cid:3)(cid:2) (cid:3)h √ i(cid:2) (cid:3) ⊕ K K K . K K (cid:2) (cid:3)(cid:2) (cid:3)(cid:2) (cid:3)h √ i Both of the decomposed modules can easily be seen to be incomparable and non-semi-componentby the argument given in the example of [11, § . Using Lemma 2.6 the decomposition isobtained using an idempotent endomorphism given by linear endomorphisms A = 5 √ √ √ − − −
11 0 √ − − √ − B = 5 √ " √ − − √ − C = 5 √ " √ − √ , where the first matrix is for the central vector space, the second matrix for the top andbottom vector spaces and the last matrix for the left and right vector spaces. The choice ofbasis of the kernel and image of A are * − √ √ , − √ + and * √ − , √ − − + , respectively. The chosen bases of the four kernels and the four images of B and C in ordertop, left, bottom and right are " √ , " √ − , " − √ , " √ − and " √ − , " − √ − , " √ − , " √ − respectively. The next example demonstrates that even in a relatively simple case there can still beinfinitely many indecomposable semi-component modules over a given partial order.
Example 5.13.
Let us consider the following family of semi-component representations ofthe -star quiver: K m K m K m K m K m (cid:2) I (cid:3)(cid:2) I (cid:3)(cid:2) I I (cid:3)(cid:2)
I J m ( λ ) (cid:3) where J m ( λ ) is the Jordan matrix λ · · · λ · · · ... ... ... . . . ... · · ·
10 0 0 · · · λ , and we use the value λ = 0 so that our representations are all semi-component modules.Then it is easy to show that all these representations are indecomposable for every m ∈ N .This shows that, in general, there are infinitely many semi-component representations ofquivers that are not of the Dynkin type. In fact, all indecomposables of this quiver can beclassified, see [5] . References [1] A. Adcock, D. Rubin, and G. Carlsson. Classification of hepatic lesions using the matching metric.
Computer Vision and Image Understanding , 121, Oct 2012.[2] H. Asashiba, M. Buchet, E. G. Escolar, K. Nakashima, and M. Yoshiwaki. On Interval Decomposabilityof 2D Persistence Modules. arXiv e-prints , December 2018.[3] U. Bauer, M. B. Botnan, S. Oppermann, and J. Steen. Cotorsion torsion triples and the representationtheory of filtered hierarchical clustering.
Advances in Mathematics , 369:107171, 08 2020.[4] M. Brion. Representations of quivers, geometric methods in representation theory. i.
S´emin. CongrSoc.Math , 24:103–144, 2012.[5] M. Buchet and E. G. Escolar. Realizations of indecomposable persistence modules of arbitrarily largedimension. In , volume 99, pages151–1513, Germany, 6 2018. Schloss Dagstuhl, Leibniz-Zentrum f¨u Informatik GmbH.[6] M. Buchet and G. E. Escolar. Every 1d persistence module is a restriction of some indecomposable 2dpersistence module.
Journal of Applied and Computational Topology , 4:387–424, May 2020.[7] G. Carlsson. Topology and data.
Bull. Amer. Math. Soc. (N.S.) , 46(2):255–308, 2009.[8] G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian. On the local behavior of spaces of naturalimages.
International Journal of Computer Vision , 76(1):1–12, Jan 2008.[9] G. Carlsson and F. M´emoli. Characterization, stability and convergence of hierarchical clustering meth-ods.
Journal of Machine Learning Research , 11:1425–1470, 04 2010.
N THE COMPLEXITY OF ZERO-DIMENSIONAL MULTIPARAMETER PERSISTENCE 17 [10] G. Carlsson and F. M´emoli. Multiparameter hierarchical clustering methods. In Hermann Locarek-Junge and Claus Weihs, editors,
Classification as a Tool for Research , pages 63–70, Berlin, Heidelberg,2010. Springer Berlin Heidelberg.[11] G. Carlsson and A. Zomorodian. The theory of multidimensional persistence.
Discrete & ComputationalGeometry , 42(1):71–93, Jul 2009.[12] J. Cochoy and S. Oudot. Decomposition of exact pfd persistence bimodules. working paper or preprint,May 2016.[13] R. Corbet, U. Fugacci, M. Kerber, C. Landi, and B. Wang. A kernel for multi-parameter persistenthomology.
ArXiv , abs/1809.10231, 2018.[14] P. Donovan and M.R. Freislich. The representation theory of finite graphs and associated algebras.
Carleton Math , Lecture Notes 5, 1973.[15] H. Edelsbrunner and J. Harer. Jacobi sets of multiple morse functions.
Foundations of ComputationalMathematics - FoCM , Jan 2004.[16] H. Edelsbrunner and D. Morozov. Persistent homology: theory and practice.
European Congress ofMathematics , pages 31–50, Jan 2014.[17] P. Frosini. A distance for similarity classes of submanifolds of a euclidean space.
Bulletin of the Aus-tralian Mathematical Society , 42(3):407415, 1990.[18] H. A. Harrington, N. Otter, H. Schenck, and U. Tillmann. Stratifying multiparameter persistent ho-mology. arXiv e-prints , Aug 2017.[19] A. Hatcher.
Algebraic topology . Cambridge University Press, Cambridge, 2002.[20] B. Keller, M. Lesnick, and T. L. Willke. Persistent homology for virtual screening, Aug 2018.[21] M. Lesnick. The theory of the interleaving distance on multidimensional persistence modules.
Founda-tions of Computational Mathematics , 15(3):613–650, Jun 2015.[22] E. Miller. Data structures for real multiparameter persistence modules. arXiv e-prints , page 107,Spetember 2017.[23] L. A. Nazarova. Representations of quivers of infinite type.
Izv. Akad. Nauk SSSR , Ser. mat. 37:752–791,1973.[24] A. Sard. The measure of the critical values of differentiable maps.
Bull. Amer. Math. Soc. , 48:883–890,1942.[25] M. Scolamiero, W. Chach´olski, A. Lundman, R. Ramanujam, and S. ¨Oberg. Multidimensional persis-tence and noise.
Foundations of Computational Mathematics , 17(6):1367–1406, Dec 2017.[26] D. Simson and A. Skowronski.
Elements of the Representation Theory of Associative Algebras , volume 3of
London Mathematical Society Student Texts . Cambridge University Press, 2007.[27] J. Skryzalin and G. Carlsson. Numeric invariants from multidimensional persistence.
Journal of Appliedand Computational Topology , 1(1):89–119, Sep 2017.[28] L. Wasserman. Topological data analysis.