The Theory of Multidimensional Persistence

Gunnar Carlsson Afra Zomorodian

(In Discrete and Computational Geometry 2009)

Abstract

Persistent homology captures the topology of a filtration aone-parameter family of increasing spaces in termsof a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topologicalentities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spacesparameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariantexists for multidimensional persistence. Instead, we propose therank invariant, a discrete invariant for the robustestimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.

1 Introduction

In this paper, we introduce the theory ofmultidimensional persistence, an extension of the concept ofpersistenthomology[9, 21]. Persistence captures the topology of afiltration, a one-parameter increasing family of spaces.Filtrations arise naturally from many processes, such as multiscale analysis of noisy datasets. Given a filtration,persistent homology provides a small description in terms of a multiset of intervals we call thebarcode. The intervalscorrespond to the lifetimes of the topological attributes.Since features have long lives, while noise is short-lived,aquick examination of the intervals enables a robust estimation of the topology of a dataset. This estimation is the keyreason for the current popularity of persistent homology for solving problems in diverse disciplines, such as shapedescription [6], denoising volumetric density data [13], detecting holes in sensor networks [8], and analyzing thestructure of natural images [2, 7]. For recent surveys, see [11, 20].

We often encounter richer structures that are described by multiple parameters. These structures may be modeledwith multifiltrations, such as the bifiltration shown in Figure 1. In previous work,we provided the theoretical founda-tions for the persistent homology of single parameter filtrations, obtaining a simple classification over fields in termsof the barcode [21]. Significantly, we showed that the barcode wascomplete, capturing all the topological informationwithin a filtration. In this paper, we show that a similar result is unattainable for multidimensional filtrations: thereexists no small complete description, like the barcode, in higher dimensions. Given this negative theoretical result,we still desire a discriminating invariant that enables detection of persistent features in a multifiltration. To this end,we propose therank invariant. In one dimension, this invariant is equivalent to the barcode and consequently com-plete. Unlike the barcode, however, the rank invariant extends to higher dimensions, where it still captures persistentfeatures, making it useful for practical applications.

1.1 Motivation

Filtrations arise naturally whenever we attempt to study the topological invariants of a space computationally. Often,our knowledge of a space is limited and imprecise. Consequently, we utilize a multiscale approach to capture theconnectivity of the space, giving us a filtration.

The first author was partially supported by NSF under grant DMS-0354543. The second author was partially supported by DARPA under grantHR 0011-06-1-0038 and by ONR under grant N 00014-08-1-0908.Both authors were partially supported by DARPA under grant HR 0011-05-1-0007.

Department of Mathematics, Stanford University, Stanford, California.Department of Computer Science, Dartmouth College, Hanover, New Hampshire.

1

Curvature

Rad

ius

Fixed 0

Fixed 0

Figure 1: A bifiltration, parameterized along curvature and radius. We can only apply persistent homology to a filtration,so we must either fix or .

Example 1 (radius) We often have a finite set of noisy samples from a subspaceX Rn, such as the point setat the bottom of the vertical box in Figure 1. If the sampling is dense enough, we should be able to compute thetopological invariants ofX directly from the points [4]. To do so, we approximate the original space as a union ofballs by placing-balls around each point. As we increase, we obtain a family of nested spaces or a filtration, asshown in the vertical box in Figure 1.

The example sketches the basic idea behind many methods for computing the topology of a point set, such asCech,Rips-Vietoris[12], or witness[7] complexes. On the other hand, we often study spaces that are filtered to begin with,and the filtrations contains important information that we wish to extract.

Example 2 (density) Suppose we have a probability density function onX Rn. We can define

X = {x X | 1/(x) }.

Clearly,X1 X2 for 1 2, so{X} is a filtration. We can obtain information about from this filtered space.For instance, the number of persistent connected components gives an estimate of the number of the modes of. Inhigher dimensions, one may uncover even more interesting structure, as was demonstrated for the nine-dimensionaldata set of33 image patches constructed by A. Lee, D. Mumford, and K. Pedersen [15] and analyzed by topologicalmethods [2, 7].

Example 3 (curvature) In prior work, we develop a methodology for obtaining compact shape descriptors formanifolds by examining the topology of derived spaces [3]. Our approach constructs thetangent complex, the closureof the tangent bundle, and filters it using curvature, as shown in the horizontal box in Figure 1. We show that thepersistence barcodes of the filtered tangent complex are useful shape descriptors.

In practice, we often have a finite set of samples from our space, giving us a filtered point set. Given a point set, wemay employ the technique in Example 1 to capture topology, constructing a filtration based on increasing the radius. But when the point set itself is filtered, our solution lies within the persistent homology along other geometricdimensions, such as density in Example 2, or curvature in Example 3. We now have multiple dimensions alongwhich our space is filtered, that is, we have amultifiltration. Of course, we could apply persistent homology alongany single dimension by fixing the value of the other parameters, as indicated by the boxes the figure [6]. However,persistent homology itself was motivated by our inability to robustly estimate values for these parameters. To eliminatethe need for fixing values, we wish to apply persistence alongall dimensions at once. Our goal is to be able to identifypersistent features by examining the entire multifiltration. We call this problemmultidimensional persistence. Variantsof this problem have appeared in other contexts, such as thefirst size homotopy groups[10].

2

1.2 Approach

To understand the structure of multidimensional persistence, we utilize a general algebraic approach consisting ofthree steps:

1. correspondence,

2. classification, and

3. andparameterization.

In the first step, we identify the algebraic structure that corresponds to our space of interest. In the second step, we ob-tain a complete classification of the structure, up to isomorphism. In the third step, we parameterize the classification.Our parameterization will be in the form of invariants. Aninvariant is a function on a set of structures (e.g. modules,graded modules, etc.) which takes values in another (typically explicitly describable) set, called theparameter set,and whose values on isomorphic structures are identical. Anexample is thetrivial invariant, which assigns the sameelement within a one element parameter set to all structuresand is therefore useless. Acompleteinvariant, on theother hand, always assigns different elements in the parameter space to structures that are not isomorphic. Completeinvariants are the most powerful type of invariant and we naturally search for them. If complete invariants do not exist,we search for incomplete invariants that have enough discriminating power to be useful.

Our goal is to obtain a useful parameterization consisting of a small set of invariants whose description is finitein size. We utilize terminology from algebraic geometry to distinguish between invariants. We seek invariants thatcorrespond to points in algebraic varieties and are not dependent on the underlying field of computation. The formercondition enables them to have finite parameterizations. The latter means that our invariant always comes from thesame set, similar to the Betti numbers, which are always integers regardless of the coefficient ring. For brevity, we callthese invariantsdiscrete, and other invariantscontinuous. Continuous invariants may be uncountable in size or dependon the underlying field of computation. Naturally, these invariants are not viable from a computational point of view.Therefore, our objective is a complete discrete invariant for multidimensional persistence. We note that our notationhas nothing to do with whether the underlying field of computation is continuous, such asR, or discrete, such asFpfor a primep.

1.3 One-Dimensional Persistence

In a previous paper, we follow the algebraic approach enumerated above and obtain a complete discrete invariant forone-dimensional persistence [21]:

1. Correspondence: We show a correspondence between the homology of a filtration in any dimension and a gradedR[t]-module, whereR[t] is the ring of polynomials with indeterminatet over ringR.

2. Classification: Over fieldsk, k[t] is a principal ideal domain, so a consequence of the standardstructure theoremfor gradedk[t]-modules gives the full classification:

n

i=1

ik[t]

m

j=1

jk[t]/(tnj ),

where denotes an-shift upward in grading.

3. Parameterization: The classification gives usn half-infinite intervals[i,) andm finite intervals[j , j+nj).The multiset ofn+m intervals is a complete discrete invariant. We call this multiset thepersistence barcode[3].

This description shows that we have achieved our stated objective completely for the casen = 1.

1.4 Contributions

In this paper, we show that multidimensional persistence has an essentially different character from its one-dimensionalversion. We devote a major portion of this paper to the following theoretical contributions:

3

We identify the algebraic structure that corresponds to multidimensional persistence to be a finitely-generatedmultigraded module over the field of multivariate polynomials.

We establish a full classification of this structure in termsof the set of the orbits of the action of an algebraicgroup on an algebraic variety.

We reveal that this classification has discrete and continuous portions. The former is canonically parameteriz-able, but the latter has no precise parameterization.

Our results imply that no complete discrete invariant exists for multidimensional persistence, unlike its one-dimensionalcounterpart. Given this negative result, we conclude the paper by describing a practical invariant:

We propose a discrete invariant, the rank invariant, that iscomputable, compact, and useful for extracting per-sistence information from multifiltrations.

We prove the rank invariant is equivalent to the persistencebarcode in one dimension, making it complete forone-dimensional persistence, the only type for which it canbe complete.

Our work has both theoretical and practical components, theformer being a full understanding of multidimensionalpersistence, and the latter being a practical invariant that is useful for computation. In Section 2, we review conceptsfrom algebra, algebraic topology, and algebraic geometry,and invent some notation. The next three sections detailthe three steps of our approach, respectively. In Section 6,we propose our discrete invariant for multidimensionalpersistence and show its completeness in one dimension.

2 Background

Let N be the set of non-negative integers, also called thenatural numbers. Foru, v Nn, we sayu . v if ui vifor 1 i n. A multisetis a set within which an element may appear multiple times, such as{a, a, b, c}. We willformalize this notion in Section 4.1. Amonomialin x1, . . . , xn is a product of the form

xv11 xv22 x

vnn ,

with vi N. We denote itxv, wherev = (v1, . . . , vn) Nn. A polynomialf in x1, . . . , xn andcoefficientsinfield k is a finite linear combination of monomials,f =

v cvxv, with cv k. We denote the set of all polynomials

k[x1, . . . , xn]. For example,5x1x22 7x31 R[x1, x2] has two non-zero coefficients:c(1,2) = 5 andc(3,0) = 7.

An algebraic varietyis the set of common zeros of a collection of polynomials. Onevariety we encounter in thispaper is theGrassmannianGrk(V ), the set ofk-dimensional subspaces of a vector spaceV . An algebraic groupis an algebraic variety endowed with group structure, so that the group operation is a morphism of the variety. Anautomorphismis an isomorphism of a mathematical object to itself. The setof all automorphisms of a set of objectsV forms theautomorphism groupAut(V ). When the objectsV are vector spaces, the automorphism group is thegeneral linear groupGL(V ), the set of invertible linear transformations onV , where the group operation is functioncomposition.

LetS be a set andG be a group. Anaction ofG onS is a binary operation : GS S such that for the identityelemente G, we havee s = s for all s S, and(g1g2) s = g1 (g2 s) for all s S andg1, g2 G. Given agroup action, we defines1 s2 iff there existsg G such thatg s1 = s2. Then, is an equivalence relation onSand partitions it. Each cell in the partition is anorbit in S underG.

An n-graded ring is a ringR equipped with a decomposition of Abelian groupsR = vRv, v Nn so thatmultiplication has the propertyRu Rv Ru+v. The set of polynomialsAn = k[x1, . . . , xn] forms thepolynomialring. An is graded byAv = kxv, v Nn and is the prototype forn-graded rings. We may visualize the 2-gradedringA2 on the integer gridN2, as shown in Figure 3(a), where each bullet is a grade that contains an element fromk.Our example polynomial5x1x22 7x

31 has non-zero elements in grades(1, 2) and(3, 0). An n-graded moduleover

ann-graded ringR is an Abelian groupM equipped with a decompositionM = v Mv, v Nn together with aR-module structure so thatRu Mv Mu+v.

4

(0,2)

(0,1)

(0,0)

(1,2)

(1,1)

(1,0)

(3,2)

(3,1)

(3,0)

(2,2)

(2,1)

(2,0)

Figure 2: A bifiltration of a triangle.

3 Correspondence

In this section, we carry out the first step of the approach outlined in Section 1.2: identifying the algebraic structureunderlying our problem. The abstraction for our input is a multifiltered space. A spaceX is multifiltered if we aregiven a family of subspaces{Xv X}vNn with inclusionsXu Xw wheneveru . w, so that the diagrams

Xu Xv1

Xv2 Xw

//

//

(1)

commute foru . v1, v2 . w. We showed an example of a bifiltration in Figure 1.In practice, our input is often a finite complexK along with a functionF : Rn K that gives a subcomplex

Kv for any valuev Rn, such as the bifiltered triangle in Figure 2. This input converts naturally to a multifilteredcomplex. Since the complex is finite, there is a finite set ofcritical coordinatesC = {vi Rn}i at which newsimplices enter the complex. ProjectingC onto each coordinate axis gives us a finite set of critical valuesCd in eachdimensiond. We now restrict ourselves to the discrete set of the Cartesian product

nd=1 Cd of the critical values,

parameterizing the resulting grid usingN in each dimension. This gives us a multifiltered complex, provided thefunctionF makes the induced diagrams (1) commute.

Given a multifiltered spaceX , the homology of each subspaceXv over a fieldk is a vector space. For instance,the bifiltered complex in Figure 2 has zeroth homology vectorspaces isomorphic to the commutative diagram

k2 k k k

k2 k3 k k

k k k k

// // //

//

OO

//

OO

//

OO OO

//

OO

//

OO

//

OO OO

where the dimension of the vector space counts the number of components of the complex, and the maps between thehomology vector spaces are induced by the inclusion maps relating the subspaces.

Definition 1 (persistence module)A persistence moduleM is a family ofk-modules{Mv}v together with homo-morphismsu,v : Mu Mv for all u . v such thatu,v v,w = u,w wheneveru . v . w.

The homology of a multifiltration in each dimension is a persistence module. To capture the structure of the maps in apersistence module, we define a multigraded module, following our treatment in the one-dimensional case [21].

5

Definition 2 (structure) Given a persistence moduleM , we define ann-graded module overAn by

(M) =

v

Mv, (2)

where thek-module structure is the direct sum structure and we requirethatxvu : Mu Mv is u,v wheneveru . v.

That is, we incorporate the relationships given by the homomorphisms into the structure of ann-graded module. Ourtreatment is consistent with, and an extension of, the one-dimensional case, where the corresponding structure is a1-graded orsingly-gradedmodule [21].

Theorem 1 (correspondence)The correspondence defines an equivalence of categories between the category offinite persistence modules overk and the category of finitely generatedn-graded modules overAn = k[x1, . . . , xn].

To recap, the homology of a finite multifiltered complex is a finite persistence module, and the structure of a persistencemodule is a finitely generatedn-graded module.

One may ask about the reverse relationship: Is every finite persistence module realizable as the homology of amultifiltration? More specifically, can we realize every such module as the homology of a finite multifiltered simplicialcomplex, since that is our usual representation of a space inpractice? The following theorem answers this question inthe affirmative.

Theorem 2 (realization) Letk = Fp for some primep, let l be a positive integer, and letM be ann-graded moduleoverk[x1, . . . , xn]. Then there is a multifiltered finite simplicial complexX so thatHk(X, k) = M as persistencemodules.

The proof is constructive and we omit it here.We end this section with an aside on our choice of input. The grid-like filtrations that we study arise naturally

in practice. Nevertheless, filtrations arising from other partial orders may also be interesting and produce algebraicinvariants. However, this would take us out of the realm of commutative algebra, perhaps into non-commutativealgebra, and definitely into another paper.

4 Classification

We have now identified the algebraic structures which correspond to our problem: finitely generatedn-graded modulesoverAn. In this section, we focus on our second task: finding a complete classification up to isomorphism of theseobjects. The general idea is to observe that the first two stages of a minimal free resolution ofM are unique up toisomorphism of free chain complexes.

4.1 Multigraded Sets and Multisets

Let n be a positive integer. By ann-graded set, we will mean a pair(X,), whereX is a set and : X Zn is amap of sets. A map of graded sets is of course a set map compatible with the reference maps in the obvious way.For anyn-gradedAn-moduleM , we associate to it ann-graded setH(M) =

vZn Mv, thehomogeneous elementsin M . We will find it useful to think of graded sets in terms ofmultisets. A multiset is a subsetL of Zn, together witha map : L N, whereN denotes the natural numbers{1, 2, . . . , n, . . .}. We note that the setZn N can be giventhe structure of ann-graded set via the projection mapZnN Zn. A multiset(L,) now specifies a graded subsetof Zn N given by{(v, n) | n (v)}. This will be a convenient way of thinking aboutn-graded sets.

Let (S, ) be any multiset whereS Nn. Then, the relation. is aquasi-partial orderon (S, ): It is reflexiveandtransitive, but notanti-symmetric, since elements appear with multiplicity.

6

-7

5

Ox1

x2

(a) 5x1x22 7x3

1

kOx1

x2

(b) Modulek

k2

k

k

Ox1

x2

(c) k-vector spaceV

Ox1

x2

(d) FreeA2-moduleF

Figure 3: 2-graded objects: (a) Polynomial5x1x22 7x3

1 in the 2-graded ringA2 = R[x1, x2] visualized onN2. (b) Field

k endowed with graded module structure. (c) Ak-vector spaceV with generators at(1, 0) and(2, 1), and two generators at(0, 1). (d) A freeA2-moduleF with same type as (b).

4.2 Free Graded Objects

Let B be anyk-algebra, wherek is a field. The usual notions of the freeB-module on a set or on ak-vector spaceadmit generalizations to the multigraded setting. By afreeAn-module on the graded set(X,), we will mean ann-gradedAn-moduleF , together with an inclusion ofn-graded sets

: (X,) H(F ) F

so that for anyn-gradedAn-moduleM and map ofn-graded sets : (X,) H(M), there is a unique homomor-phism : F M of n-gradedAn-modules so that the diagram

(X,) H(F )

H(M)

//

?

?

?

?

?

?

?

?

?

?

?

?

H()

commutes, whereH() denotes the restriction of to the homogeneous elements. Note that any homomorphism ofn-graded modules preserves homogeneous elements. It is routine to check that such modules exist, and the hypothesesguarantee that they are unique up to isomorphism. A particular example of this construction is that of a freen-gradedvector space. In this case, it is easy to show that alln-graded vector spaces are free, by analogy with the ungradedcase. For anyn-gradedAn-moduleM , we construct a newn-graded moduleM(v) for any v Zn, by settingM(v)w = Mwv. The module structure follows directly, and this module is thought of as obtained by shifting thegrading byv. Any finitely generated freen-graded module can be expressed as

iAn(vi) for some family ofvis.Similarly, every finite dimensionaln-graded vector space, such as the vector space in Figure 3(c), can be described as

i k(vi) for some choice ofvis, wherek denotes the ground field regarded as an-graded modulek wherekv = {0}for v 6= ~0, andk0 = k, as shown in Figure 3(b). Finally, one can check that if we aregiven twon-graded sets(X,) and(X , ), so that the freen-graded module (or freen-gradedk-vector space) onX is isomorphic to thecorresponding construction forX , then the two graded sets(X,) and(X , ) are also isomorphic. In other words,the graded bases for the modules are isomorphic as graded sets. We use this fact to define thetypeof ann-gradedvector spaceV as the unique multiset which is isomorphic to a graded basis for V , and denote it(V ). Similarly, wedefine(F ) for any freen-gradedAn-module.

Finally, we note that for anyn-gradedk-vector spaceV , there is freen-gradedAn-module onV . It satisfies theobvious universality property, and we will denote it byF (V ). As in the usual case, there is a canonical construction,namelyAn k V , as shown in Figure 3(d).

4.3 Automorphisms

We wish to analyze the automorphism group of a freen-graded moduleF . For any multiset, we will denote byV ()andF () a particular choice of a freen-gradedk-vector space and ann-gradedAn-module, respectively. Note that

7

any two such are canonically isomorphic, in that there is a unique basis preserving map from the one to the other. Wefirst analyze the automorphism group of ann-gradedk-vector space.

Theorem 3 LetV be ann-graded vector space overk, of type

= {(v1, n1), (v2, n2), . . . (vs, ns)}

where all thevis are distinct. Then the automorphism group is isomorphic to the product

i GLni(k).

Proof: V is isomorphic to the direct sum

i Vvi , and by the definitions, any automorphism ofV must preserve thisdirect sum decomposition. Consequently,Aut(V ) =

Aut(Vvi), andVvi = kni , giving the result.

The situation of ann-gradedAn-module is a bit more subtle, since in this case the ring has elements in degreesother than 0. We describe the structure ofAut(F ()) as an algebraic group overk. In order to do this, we will need tomake some definitions.

Definition 3 Let n be a positive integer. By ann-multifiltered k-vector space, we will mean ak-vector spaceV ,together with a familyF of subspacesFwV for everyw Zn, so that

1. If w w, thenFwV FwV .

2. There is aw0 Zn so thatFw0V = V .

3. There is aw1 Zn so thatFw1V = {0}.

For anyn-graded vector spaceV , we letFV denote theassociatedn-multifiltered vector spaceto have underlyingvector spaceV , and so that for everyw Zn, we set

FwV =

ww

Vw

It is clear what is meant by a morphism ofn-multifiltered vector spaces, namely a homomorphism of underlyingvector spaces which respects the subspaces corresponding to w for eachw Zn. It is also clear that the correspon-denceF is a functor, indeed that a morphism of then-multigraded structure can be viewed directly as a morphismofthe associated multifiltered objects.

Theorem 4 For any finite dimensionaln-multifiltered vector space(V,F), the automorphism group of(V,F) is an al-gebraic subgroup of the full automorphism groupGL(V ) of the underlying vector spaceV . For any finite dimensionaln-graded vector spaceV , Aut(V ) is an algebraic subgroup of the algebraic groupAut(FV )

Proof: Straightforward verification.

Let M denote any finitely generatedn-gradedAn-module. We define the finite dimensionalk-vector space(M) = k An M , wherek is given the module structure where all the variablesxi act trivially, i.e. by zero.We now have

Theorem 5 (automorphisms) For any finitely generated freen-gradedAn-moduleF , there is an isomorphism

Aut(F ) = Aut(F((F )))

where the second automorphism group is computed in the category ofn-multifilteredk-vector spaces. Consequently,Aut(F ) has the structure of an algebraic group in a natural way.

Proof: We writeF = iA(vi) for some finite set of vectors{vi}. We writeei for the basis element corresponding totheith factor, soei Fvi . We now have(F ) = ik(vi), and a basis for(F ) is given by the elementsei, which arethe reductions ofei in (F ). Given these basis elements, any automorphism of then-multifilteredk-vector space(F ) is determined by a choice of elementsij k, so that

(ej) =

i

ijei

8

andij = 0 whenevervj vi. Theij s also determine an automorphism of F itself via the formula

(ej) =

i

ijxvjviei

It is clear that the correspondence is actually a homomorphism fromAut(F(F )) to Aut(F ). There is ahomomorphism in the reverse directions which carries an automorphism of F to the automorphismidk An of(F ), and it is clear that these are inverse correspondences.

4.4 Free Hulls

We will wish to interpret isomorphism classes of modules in terms of isomorphism classes of two stages complexeswhich extend to a resolution of the given module. In order to relate modules with resolutions, we state the followingelementary extension of an easy consequence of Nakayamas Lemma [1]. The theorem is the extension to then-gradedcases of the similar theorem for graded modules over graded rings [17].

Theorem 6 Letf : M N be a homomorphism of finitely generated freen-gradedAn-modules, and suppose that

idk An

f : k An

M k An

N

is an isomorphism ofn-gradedk-vector spaces. Thenf is an isomorphism.

Definition 4 For a finitely generatedn-gradedAn-moduleM , afree hullfor M will be any surjective homomorphismp : F M of n-graded modules, whereF is a finitely generated freen-gradedAn-module, and such that the inducedmap

idk Ap : k

AF k

AM

is an isomorphism.

Theorem 7 Every finitely generatedn-gradedAn-module admits a free hull. Moreover, any two free hulls forM areisomorphic in the sense that ifp : F M andp : F M are both free hulls, then there is a commutative diagram

F F

M?

?

?

?

?

?

?

?

?

p

//=

p

where the horizontal arrow is an isomorphism.

Proof: This is standard in the ordinary graded (or local ring) situation and the proof here is identical, using Theorem 6in place of the standard result from the theory of local rings.

4.5 Complete Classification

We begin by defining two multiset valued invariants of finitely generatedn-gradedAn-modulesM . We define theinvariant0(M) to be((M)), i.e. the type of then-gradedk-vector space(M). This is clearly an invariant of theisomorphism class of the module. For the second invariant, select any free hullp : F M of M , and letK Fdenote the kernel ofp. Then we define1(M) to be((K)).

Theorem 8 The multiset1(M) is independent of the free hull chosen, hence is a multiset valued invariant of theisomorphism class ofM .

Proof: Given any two free hullsp : F M andp : F M , Theorem 7 asserts that there is an isomorphism : M M , so thatp = p. It is immediate that restricts to an isomorphism fromK to K = ker(p). SinceKandK are isomorphic,((K)) = ((K )), which gives the result.

9

Suppose now that we are given two multisets0 and1. We first construct a freen-gradedAn-moduleF so that((F )) = 0. Next, we letS(F, 1) denote the set of alln-gradedAn-submodulesL of F for which ((L)) = 1.Note that the automorphism group ofF acts onS(F, 1) via g K = g(K), for g Aut(F ). Next, letI(0, 1) denotethe set of isomorphism classes ofn-gradedAn-modulesM for which 0(M) = 0 and1(M) = 1. There is a mapq : S(F, 1) I(0, 1) defined by

q(K) = [F/K]

where[] denotes isomorphism class.

Theorem 9 (classification)LetF be as above, and denoteAut(F ) byGF . The mapq satisfies the formulaq(g K) =q(K), and consequently induces a mapq : GF \S(F, 1) I(0, 1). Moreover,q is a bijection.

Proof: For anyg GF , we see immediately that action byg carriesK into g(K), and that therefore we obtainan induced isomorphismg : F/K F/g(K). This shows thatq(K) = q(g K). To see thatq is surjective, itsuffices to prove the same result forq. But now, Theorem 7 shows that, there exists a surjection from : F M , if((M)) = 0. This relies on the observation that there is a unique (up to isomorphism) freen-gradedAn-moduleFwith ((F )) = 0. Nowker() is now an element inS(F ), and clearlyq(ker()) = [M ], demonstrating surjectivity.For injectivity, we suppose that we are givenK,K S(F ), and thatq(K) = q(K ), i.e. F/K = F/K . Let : F/K F/K be an isomorphism. Theorem 7 now shows that there is an automorphism of F so that thediagram

F F

F/K F/K

//

//

commutes. It is clear that carriesK isomorphically toK , which shows thatK andK represent the same orbit inI(0, 1)

5 Parameterization

Having established a complete classification of the graded modules, we now turn our attention to the third step of ourapproach: parameterizing the classification. We have shownearlier in Theorem 5 thatAut(F ) is an algebraic group.We now show that the elements ofS(0, 1) are naturally identified with the points of an algebraic variety, and furtherthat the action ofAut(F ) on them is an algebraic group action. The general picture that emerges is that this portionof the classification is a continuous invariant. To appreciate its nature, we next detail an example in two dimensions.We end this section with possible strategies for coping withthe continuous invariant.

5.1 Interpretation via Algebraic Varieties

We begin by considering anyn-gradedAn-moduleM . For everyv Zn, we consider thek-vector subspace

(IM)v =

xiMvei Mv

whereei denotes theith standard basis vector inZn. We sayv is agap for M if (IM)v 6= Mv. Let(M) denote theset of gaps ofM .

Theorem 10 If M is finitely generated, then(M) is finite. Moreover, the type of(M), ((M)), is the multiset((M), M ), where

M (v) = dimk(Mv/(IM)v) = dimk(Mv) dimk((IM)v)

Proof: It is clear that the set of gaps is contained in the set of thosev which contain an element of the generating set,which verifies the first assertion. It is immediate from the definitions that(M)v = Mv/(IM)v. It follows that thegaps are exactly thosevs for which(M)v 6= 0. Further, the type of(M) is then clearly given by the functionMdefined above.

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Theorem 11 LetF denote a freen-gradedAn-module, and letL andL denote anyn-graded submodule (note thatunlike the casen = 1, L need not be free). ThenL = L if and only if the gaps ofL andL are the same, and we haveLv = L

v for all v which are gaps of either.

Proof: Let w Zn denote a minimal element in the set{v Zn|Lv 6= Lv}. The elementw must be a gap for bothL andL, since otherwise we find that

Lw = (IL)w = (IL)w = L

w

and the equality(IL)w = (IL)w holds due to the minimality ofw.

Let = (V, ) denote any multiset, and let : V Z be any function. For any finitely generated freen-gradedAn-moduleF , letARR,(F ) denote the set of all assignmentsv Lv, wherev V , andLv is ak-linear subspaceof Fv, which satisfy the following three conditions:

1. v v = xvv

Lv Lv

2. dimk(Lv) = (v)

3. dimk(Lv/

v

1. Lv Gv for all v

2. If v v, thenLv Lv

3. dimk(Lv) = (v)

4. dimk(Lv/

v

4. dimk(Lv/

v

1. l1 becomes thex-axis,

2. l2 becomes they-axis,

3. andl3 becomes thediagonalline spanned by(1, 1).

These transformations exist asl1, l2 spank2, being non-zero and distinct, andl3 cannot be zero or either axis after thefirst two transformations. We now have a tuple(x-axis, y-axis, diagonal, 4), where4 is l4 after the transformations.While there are different matrices inGL2(k) that can transform the original tuple to this tuple, the matrices differ bymultiplication by a diagonal matrix, since the only matrices that preserve the axes and the diagonal line are diagonalmatrices. Consequently,4 is determined uniquely, and we may identify the orbitsGL2(k)\ with the lines inP1(k)with the axes and the diagonal removed. Each such line is determined by its slope which cannot be0, , or 1,according to the discussion. Therefore,GL2(k)\ can be identified withP1(k) {0, 1,} = k {0, 1}.

Now, note that this classification is dependent on the field ofcoefficientsk. If k is uncountable, so is the subspace,and in turn, the full orbit space. Ifk is a finite field, such asFp for p a prime, we get a finite solution for the subspace we have chosen, but we still have not detailed the full picture for the orbit space. However, we already see the field-dependence problem: Changing the field not only changes the classification, but also the target of the classification: Wenot only get different values, we get values from different sets altogether. This is analogous to getting Betti numbersin Z2 when computing overZ2, Betti numbers inZ3 when computing overZ3, and so on. Therefore, we cannot get adiscrete invariant for our example.

5.3 Refinement

We have illustrated that our goal obtaining a complete discrete invariant is not attainable for multigraded objects.Intuitively, the continuous invariant captures subtle second-order information about the complicated transitions in amultigraded module. This information may be worthy of studyand we end this section by suggesting possible avenuesof attack.

Our two discrete invariants may be viewed as the first two in a family of discrete invariants. We may developstandard homological algebra in the category of graded modules over ann-gradedk-algebraAn, with the resultingderived functors

An

andHomAn now being equipped with the structure of ann-gradedAn-module [19]. In particular,

the functorTorAni (M,k) makes sense and we now define a family ofn discrete invariants by

i = (

TorAni (M,k))

.

The first two invariants in the family match our two discrete invariants in the previous section. It may be interestingto study the rest of this family as each invariant will make the classification finer, as done recently by Knudson [14].However, the existence of the continuous invariant indicates that no matter how many of these invariants we include,there will still be a residual continuous component in the classification.

While the set of orbits is not a variety, we conjecture that additional structure exists in the following form. LetG = GL(F (0)) and suppose there is a family of closed subvarietiesRFn RF(0, 1) such that

1. RFn RFn+1 for all n,

2. RFn is closed under the action ofG,

3. RFn eventually becomes equal toRF(0, 1),

4. the set of orbits of theG-action onRFn RFn1 is an algebraic variety in a natural way.

This kind of structure is called anequivariant stratificationof the variety in question, with the differenceRFnRFn1being astratum. The orbit varieties are calledmoduli spacesin classification problems for which the invariant lies ina given stratum. The result is known to hold in some special cases by the work of Cohen and Orlik [5] and Terao [18].

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6 The Rank Invariant

Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence.We still desire a discriminating invariant that captures persistent information, that is, homology classes with largepersistence. This information is not contained in our two discrete invariants,0 and1, as they capture birth and deathcoordinates of the generators in the complexes. What we needlies within the relationship between the two invariantsor in the maps between the complexes. In this section, we propose and advocate a small and computable invariantthat identifies persistent features in a multifiltration. Our invariant is equivalent to persistence barcodes, and thereforecomplete, for one-dimensional filtrations.

The persistent information is contained in the relating homomorphismsu,v in Definition 1. Recall that we incor-porated these maps into a multigraded module through the action of the variables, requiring thatxvu : Mu Mv tobeu,v in Definition 2. To analyze this family of maps, we begin by defining their domains.

Definition 5 (Dn) Let N = N{} with u for all u N. LetDn Nn Nn be the subset above the diagonal,Dn = {(u, v) | u Nn, v Nn, u . v}. For (u, v), (u, v) Dn, we define(u, v) (u, v) if u . u andv . v.

It is easy to check that is a quasi-partial order onDn. With this notation, our parameterization of singly-gradedmodules in Section 1.3 is a multiset fromD1, and indicates the first pair contains the second, when the pairs areviewed as intervals.

Definition 6 (rank invariant M ) Let M be a finitely generatedn-gradedAn-module. We defineM : Dn N tobeM (u, v) = rank(xvu : Mu Mv).

The functionM is clearly a discrete invariant forM .

Lemma 1 (order-preserving) If (u, v) (u, v), thenM (u, v) M (u, v), that is,M is an order preservingfunction from(Dn,) to (N,).

Proof: Immediate using the fact that given any compositef g of linear transformations, we have

rank(f g) rank f, rank g.

We now state the rank invariants completeness in one dimension through its equivalence to barcodes. We notethat the following theorem is the converse of thek-triangle Lemma[9, 21].

Theorem 12 (completeness)The rank invariantM is complete for singly-graded modulesM .

Proof: To prove completeness, we show equivalence via a bijection between the set of barcodes and the set of rankinvariants. According to the classification theorem for a graded moduleM recalled in Section 1.3, the intervals in itsbarcode capture the lifetimes of the generators ofM . Therefore, the corresponding rank function is

()(t, s) = card{((t, s), i) | (t, s) (t, s)}.

Figure 5 illustrates this correspondence. triangle figure The barcode intervals are drawn below thet axis and therank functions domain,D1, exists above the diagonal in the(t, s)-plane. Each interval[t0, t1) has a triangular regiondefined by inequalitiest t0, s < t1, ands t, with corner vertex(t0, t1) and vertices(t0, t0) and(t1, t1) on thediagonal. Half-infinite intervals correspond to degenerate triangles, but they are handled easily, so we do not discussthem here. The rank function()(t, s) is simply the number of triangles that contain(t, s). As an aside, we note thatthe map(t, s) 7 (t, s t) gives the index-persistence figures in the previous papers [9, 21].

Clearly, we can construct each triangle from its corner by projecting the corner vertically and horizontally onto thediagonal. Moreover, there is a trivial bijection between the corner(t0, t1) and the interval[t0, t1). Given a barcode,we know how to build the rank function() by the equation above. Given a rank function, we need to identify thecorner points to build the corresponding barcode. We begin by first walking along the diagonal until the rank functionis nonzero att0 = argmint (t, t) 6= 0. By Lemma 1, the functions 7 (t0, s) is a non-increasing function, so wewalk vertically up untilt1 where(t0, t1) < (t0, t0). The point(t0, t1) is a corner, so we subtract its triangle from.The proof follows by induction.

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O

t0

t0

t1

t1

s

t

Figure 5: The intervals of a barcode are drawn below thet-axis. Each interval(t0, t1) defines a triangle as shown. Therank function()(t, s) is the number of triangles that contain(t, s).

When the module is the persistence module associated to theith homology of a multifiltration, we can define therank invariant directly in terms of the input.

Definition 7 (X,i) LetX = {Xv}vNn be a multifiltration. We defineX,i : Dn N over fieldk to

X,i(u, v) = rank(Hi(Xu, k) Hi(Xv, k)).

The functionX,i is a homeomorphism invariant of the multifiltered space, deriving its invariance from the invarianceof M . Intuitively, Theorem 12 means that the rank invariant for one-dimensional filtrations may be separated into a setof overlapping triangles whose thickness at any point is therank. These triangles, in turn, carry the same informationas a set of intervals or the barcode. Our classification theorem, on the other hand, implies that a similar result is notpossible for higher dimensions. As our example in Section 5.2 illustrates, the picture is much more complicated: It isnot possible to separate the rank invariant into overlapping regionsto extend the barcode.

The rank invariant does extend, however, as an incomplete invariant. We may utilize it to identify persistentfeatures by the following procedure. Given a rank invariant, we look for points(u, v) Dn that are far from thediagonal and have a neighborhood of constant value. The firstcondition corresponds to the persistence of the features.The second condition indicates the stability of our choice(u, v). With this procedure, the rank invariant emerges as apractical tool for reliable estimation of the Betti numbersof multifiltered spaces.

7 Conclusion

We believe the primary contribution of this paper is the theoretical description of the structure of algebraic multidimen-sional persistence: We identify the corresponding algebraic structure, classify it, and undertake its parameterization.Our theory reveals that a complete discrete invariant does not exist for multidimensional persistence, unlike its one-dimensional counterpart. A second practical contributionof our paper is the rank invariant, a tool for robust estimationof the Betti numbers. We prove that the rank invariant is equivalent to the persistent barcode in one dimension, so itis complete when it can be. Unlike the barcode, the rank invariant extends to higher dimensions as an incomplete butuseful invariant.

We have developed an algorithm for computing the rank invariant. For bifiltrations, the rank invariant is alreadyfour-dimensional, so we are examining possible interfacesfor visualizing and exploring the rank invariant. We plan toapply our work toward automatic identification of features in multifiltrations, such as the filtered tangent complex [6].

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