The Theory of Multidimensional Persistence∗

Gunnar CarlssonDept. of Mathematics

Stanford UniversityStanford, California

gunnar@math.stanford.edu

Afra ZomorodianDept. of Computer Science

Dartmouth CollegeHanover, New Hampshire

afra@cs.dartmouth.edu

ABSTRACTPersistent homology captures the topology of a filtration –a one-parameter family of increasing spaces – in terms ofa complete discrete invariant. This invariant is a multisetof intervals that denote the lifetimes of the topological enti-ties within the filtration. In many applications of topology,we need to study a multifiltration: a family of spaces pa-rameterized along multiple geometric dimensions. In thispaper, we show that no similar complete discrete invariantexists for multidimensional persistence. Instead, we proposethe rank invariant, a discrete invariant for the robust esti-mation of Betti numbers in a multifiltration, and prove itscompleteness in one dimension.

Categories and Subject DescriptorsF.2.2 [Analysis of Algorithms and Problem Complex-ity]: Nonnumerical Algorithms and Problems—Computa-tions on discrete structures

General TermsAlgorithms, Theory

Keywordscomputational topology, multidimensional analysis, persis-tent homology

1. INTRODUCTIONIn this paper, we introduce the theory of multidimensional

persistence, the extension of the concept of persistent ho-mology [7, 17]. Persistence captures the topology of a fil-tration, a one-parameter increasing family of spaces. Filtra-tions arise naturally from many processes, such as multiscale

∗Research by the first author partially supported by NSFunder grant DMS-0354543, by the second author partiallysupported by DARPA under grant HR0011-06-1-0038, andby both authors partially supported by DARPA undergrants 32905.

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Curvature κ

Rad

ius ε

Fixed κ0

Fixed ε0

Figure 1: A bifiltration, parameterized along curva-ture κ and radius ǫ. We can only apply persistenthomology to a filtration, so we must either fix ǫ orκ.

analyses of noisy datasets. Given a filtration, persistent ho-mology provides a small description in terms of a multiset ofintervals we call the barcode. The intervals correspond to thelifetimes of the topological attributes. Since features havelong lives, while noise is short-lived, a quick examination ofthe intervals enables a robust estimation of the topology ofa dataset. This is the key reason for the current popularityof persistent homology for solving problems in diverse dis-ciplines, such as shape description [4], denoising volumetricdensity data [13], detecting holes in sensor networks [6], andanalyzing the structure of natural images [5].

We often encounter richer structures that are parameter-ized along multiple geometric dimensions. These structuresmay be modeled by multifiltrations, as the bifiltration shownin Figure 1. In previous work, we provided the theoreti-cal foundations for persistent homology, obtaining a simpleclassification over fields in terms of the barcode [17]. Signifi-cantly, we showed that the barcode was complete, capturingall the topological information within a filtration. In thispaper, we show that a similar result is unattainable for mul-tidimensional persistence: there exists no small completedescription, like the barcode, in higher dimensions. Giventhis negative theoretical result, we still desire a discriminat-ing invariant that enables detection of persistent features in

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a multifiltration. To this end, we propose the rank invari-ant. In one dimension, this invariant is equivalent to thebarcode and consequently complete. Unlike the barcode,however, the rank invariant extends to higher dimensions,where it still captures persistent features, making it usefulfor practical applications.

1.1 MotivationFiltrations arise naturally whenever we attempt to study

the topological invariants of a space computationally. Often,our knowledge of a space is limited and imprecise. Con-sequently, we utilize a multiscale approach to capture theconnectivity of the space, giving us a filtration.

Example 1 (radius ǫ) We often have a finite set of noisysamples from a subspace X ⊂ R

n, such as the point set atthe bottom of the vertical box in Figure 1. If the samplingis dense enough, we should be able to compute the topo-logical invariants of X directly from the points [2]. To doso, we approximate the original space as a union of ballsby placing ǫ-balls around each point. As we increase ǫ, weobtain a family of nested spaces or a filtration, as shown inthe vertical box in Figure 1.

This example states the central idea behind many methodsfor computing the topology of a point set, such as Cech,Rips-Vietoris [12], or witness [5] complexes.

Often, the space under study is filtered to begin with.And the filtration contains important information that wewish to extract.

Example 2 (density ρ) Suppose we have a probabilitydensity function δ on X ⊂ R

n. We can define

Xρ = x ∈ X | 1/δ(x) ≤ ρ.

Clearly, Xρ1 ⊆ Xρ2 for ρ1 ≤ ρ2, so Xρρ is a filtration.We can obtain information about δ from this filtered space.For instance, the number of persistent connected compo-nents gives an estimate of the number of the modes of δ.In higher dimensions, one may uncover even more interest-ing structure, as was demonstrated for the nine-dimensionalMumford dataset [5].

Example 3 (curvature κ) In prior work, we develop amethodology for obtaining compact shape descriptors formanifolds by examining the topology of derived spaces [1].Our approach constructs the tangent complex, the closure ofthe tangent bundle, and filters it using curvature, as shownin the horizontal box in Figure 1. We show that the per-sistence barcodes of the filtered tangent complex are usefulshape descriptors.

In practice, we often have a finite set of samples from ourspace, giving us a filtered point set in the last two examples.Given a point set, we may employ the technique in Exam-ple 1 to capture topology, constructing a filtration based onincreasing the radius ǫ. But when the point set itself is fil-tered, our solution lies within the persistent homology alongother geometric dimensions, such as density ρ in Example 2,or curvature κ in Example 3. We now have multiple dimen-sions along which our space is filtered, that is, we have amultifiltration. Of course, we could apply persistent homol-ogy along any single dimension by fixing the value of the

other parameters, as indicated by the boxes the figure [4].However, persistent homology itself was motivated by ourinability to robustly estimate values for these parameters.To eliminate the need for fixing values, we wish to applypersistence along all dimensions at once. Our goal is to beable to identify persistent features by examining the entiremultifiltration. We call this problem multidimensional per-sistence. Variants of this problem have appeared in othercontexts, such as the first size homotopy groups [10].

1.2 ApproachTo understand the structure of multidimensional persis-

tence, we utilize a general algebraic approach consisting ofthree steps: correspondence, classification, and parameteri-zation. In the first step, we identify the algebraic structurethat corresponds to our space of interest. In the secondstep, we obtain a complete classification of the structure,up to isomorphism. In the third step, we parameterize theclassification.

Our parameterization will be in the form of invariants. Aninvariant is a map that assigns the same object to isomor-phic structures. For example, the trivial invariant assignsthe same object to all structures and is therefore useless.A complete invariant, on the other hand, assigns differentobjects to structures that are not isomorphic. Completeinvariants are the most powerful type of invariant and wenaturally search for them. If complete invariants do not ex-ist, we search for incomplete invariants that have enoughdiscriminating power to be useful.

Our goal is to obtain a useful parameterization consist-ing of a small set of invariants whose description is finite insize. We utilize terminology from algebraic geometry to dis-tinguish between invariants. We seek invariants that corre-spond to discrete images of points in algebraic varieties andare not dependent on the underlying field of computation.The former condition enables them to have finite parameter-izations. The latter means that our invariant always comesfrom the same set, similar to the Betti numbers, which arealways integers regardless of the coefficient ring. For brevity,we call these invariants discrete, and other invariants contin-uous. Continuous invariants may be uncountable in size ordepend on the underlying field of computation. Naturally,these invariants are not viable from a computational pointof view. Therefore, our objective is a complete discrete in-variant for multidimensional persistence. We note that ournotation has nothing to do with whether the underlying fieldof computation is continuous, such as R, or discrete, such asFp for a prime p.

1.3 One-Dimensional PersistenceIn a previous paper, we follow the algebraic approach

above and obtain a complete discrete invariant forone-dimensional persistence [17]:

1. Correspondence: We show a correspondence betweenthe homology of a filtration in any dimension and agraded R[t]-module, where R[t] is the ring of polyno-mials with indeterminate t over ring R.

2. Classification: Over fields k, k[t] is a principal ideal do-main, so a consequence of the standard structure theo-rem for graded k[t]-modules gives the full

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classification:n

M

i=1

Σαik[t] ⊕

mM

j=1

Σγjk[t]/(tnj ),

where Σα denotes an α-shift upward in grading.

3. Parameterization: The classification gives us n half-infinite intervals [αi,∞) and m finite intervals [γj , γj+nj). The multiset of n + m intervals is a completediscrete invariant. We call this multiset the persistencebarcode [1].

In essence, we are able to complete all our steps for one-dimensional persistence and get everything we could possi-bly wish for.

1.4 ContributionsIn 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 fol-lowing theoretical contributions:

• We identify the algebraic structure that corresponds tomultidimensional persistence to be a finitely-generatedmultigraded module over the field of multivariate poly-nomials.

• We establish a full classification of this structure interms of the set of the orbits of the action of an alge-braic group on an algebraic variety.

• We reveal that this classification has discrete and con-tinuous portions. The former is canonically parameter-izable, but the latter has no precise parameterization.

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

• We propose a discrete invariant, the rank invariant,that is computable, compact, and useful for extractingpersistence information from multifiltrations.

• We prove the rank invariant is equivalent to the persis-tence barcode in one dimension, making it complete forone-dimensional persistence, the only type for which itcan be complete.

Our work has both theoretical and practical components,the former being a full understanding of multidimensionalpersistence, and the latter being a practical invariant thatis 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 multidimensional per-sistence and show its completeness in one dimension.

2. BACKGROUNDLet N be the set of non-negative integers, also called the

natural numbers. Intuitively, a multiset is a set within whichan element may appear multiple times, such as a, a, b, c.Formally, a multiset is a pair (S, µ), where S is the underly-ing set of elements and µ : S → N specifies the multiplicity

µ(s) of each element s ∈ S. We often characterize a multi-set via the set-theoretic definition of µ: (s, µ(s)) | s ∈ S.For the example, we get (a, 2), (b, 1), (c, 1). We define(s, i) ∈ (S, µ) iff s ∈ S and 1 ≤ i ≤ µ(s), that is, i indexesthe multiple copies of s.

For u, v ∈ Nn, we say u . v if ui ≤ vi for 1 ≤ i ≤ n. Let

(S, µ) be any multiset where S ⊆ Nn. Then, the relation

. is a quasi-partial order on (S, µ): it is reflexive and tran-sitive, but not anti-symmetric, since elements appear withmultiplicity.

A monomial in x1, . . . , xn is a product of the form

xv11 · xv22 · · ·xvnn

with vi ∈ N. We denote it xv, where v = (v1, . . . , vn) ∈ Nn.

A polynomial f in x1, . . . , xn and coefficients in field k is afinite linear combination of monomials, f =

P

v cvxv, with

cv ∈ k. We denote the set of all polynomials k[x1, . . . , xn].For example, 5x1x

22 − 7x3

1 ∈ k[x1, x2] has two non-zero coef-ficients: c(1,2) = 5 and c(3,0) = −7.

An algebraic variety is the set of common zeros of a col-lection of polynomials. One variety we encounter in this pa-per is the Grassmannian Grk(V ), the set of k-dimensionalsubspaces of a vector space V . An algebraic group is an al-gebraic variety endowed with group structure, so that thegroup operation is a morphism of the variety. The auto-morphism group GL(V ) of a system of objects V is the setof invertible linear transformations on V , where the groupoperation is function composition.

Let S be a set and G be a group. An action of G onS is a binary operation ∗ : G × S → S such that for theidentity element e ∈ G, we have e ∗ s = s for all s ∈ S, and(g1g2) ∗ s = g1 ∗ (g2 ∗ s) for all s ∈ S and g1, g2 ∈ G. Givena group action, we define s1 ∼ s2 iff there exists g ∈ G suchthat g ∗ s1 = s2. Then, ∼ is an equivalence relation on Sand partitions it. Each cell in the partition is an orbit in Sunder G.

An n-graded ring is a ring R equipped with a decompo-sition of Abelian groups R ∼= ⊕vRv, v ∈ N

n so that mul-tiplication has the property Ru · Rv ⊆ Ru+v. The set ofpolynomials An = k[x1, . . . , xn] forms the polynomial ring.An is graded by Av = kxv, v ∈ N

n and is the prototypefor n-graded rings. We may visualize the 2-graded ring A2

on the integer grid N2, as shown in Figure 3(a), where each

bullet is a grade that contains an element from k. Our exam-ple polynomial 5x1x

22 − 7x3

1 has non-zero elements in grades(1, 2) and (3, 0). An n-graded module over an n-graded ringR is an Abelian group M equipped with a decompositionM ∼= ⊕v Mv , v ∈ N

n together with a R-module structureso that Ru ·Mv ⊆Mu+v.

3. CORRESPONDENCEIn this section, we carry out the first step of the approach

enumerated in Section 1.2: identifying the algebraic struc-ture underlying our problem. The abstraction for our inputis a multifiltered space. A space X is multifiltered if we aregiven a family of subspaces Xv ⊆ Xv∈Nn with inclusions

186

(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.

Xu ⊆ Xw whenever u . w, so that the diagrams

Xu Xv1

Xv2 Xw

//

//

(1)

commute for u . v1, v2 . w. We showed an example of abifiltration in Figure 1.

In practice, our input is often a finite complex K alongwith a function F : R

n → K that gives a subcomplex Kv forany value v ∈ R

n, such as the bifiltered triangle in Figure 2.This input converts naturally to a multifiltered complex.Since the complex is finite, there is a finite set of criticalcoordinates C = vi ∈ R

ni at which new simplices enterthe complex. Projecting C onto each coordinate axis givesus a finite set of critical values Cd in each dimension d. Wenow restrict ourselves to the discrete set of the Cartesianproduct

Qn

d=1 Cd of the critical values, parameterizing theresulting grid using N in each dimension. This gives us amultifiltered complex, provided the function F makes theinduced diagrams (1) commute.

Given a multifiltered space X, the homology of each sub-space Xv over a field k is a vector space. For instance, thebifiltered complex in Figure 2 has zeroth homology vectorspaces isomorphic to

k2 k k k

k2 k3 k kk k k k

where the dimension of the vector space counts the numberof components of the complex. We also have inclusion mapsrelating the subspaces, inducing maps at the homology level.

Definition 1 (persistence module) A persistence mod-ule M is a family of k-modules Mvv together with ho-momorphisms ϕu,v : Mu → Mv for all u . v such thatϕu,v ϕv,w = ϕu,w whenever u . v . w.

The homology of a multifiltration in each dimension is apersistence module. To capture the structure of the mapsin a persistence module, we define a multigraded module,following our treatment in the one-dimensional case [17].

Definition 2 (structure) Given a persistence module M ,we define an n-graded module over An by

α(M) =M

v

Mv ,

where the k-module structure is the direct sum structure andwe require that xv−u : Mu →Mv is ϕu,v whenever u . v.

That is, we incorporate the relationships given by the homo-morphisms into the structure of an n-graded module. Ourtreatment is consistent with, and an extension of, the one-dimensional case, where the corresponding structure is a 1-graded or singly-graded module [17].

Theorem 1 (correspondence) The correspondence α de-fines an equivalence of categories between the category of fi-nite persistence modules over k and the category of finitelygenerated n-graded modules over An = k[x1, . . . , xn].

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

One may ask, however, about the reverse relationship: Isevery finite persistence module realizable as the homology ofa multifiltration? More specifically, can we realize every suchmodule as the homology of a finite multifiltered simplicialcomplex, since that is our usual representation of a spacein practice? The following theorem answers this question inthe affirmative.

Theorem 2 (realization) Every finite persistence modulemay be realized as the homology, in any dimension greaterthan zero, of a finite multifiltered space, or a finite multifil-tered simplicial complex.

The proof is constructive and we omit it here.We end this section with an aside on our choice of in-

put. The grid-like filtrations we study arise naturally inpractice. Nevertheless, filtrations arising from other par-tial orders may also be interesting and produce algebraicinvariants. However, this would take us out of the realm ofcommutative algebra, perhaps into non-commutative alge-bra, and definitely into another paper.

4. CLASSIFICATIONWe have now identified the algebraic structures that corre-

spond to our problem: finitely generated n-graded modulesover An. In this section, we focus on our second task: findinga complete classification for this structure. We begin with aclassification of finitely generated free graded objects. Next,we utilize these free objects to describe two discrete invari-ants for the modules. Finally, we examine the relationshipbetween the two invariants to complete our classification.Our approach is entirely in the spirit of the representationtheory of finite dimensional algebras [11]. We have chosento present a complete argument for clarity and explicitness.

4.1 Free Graded ObjectsIntuitively, a free object is a generalization of a vector

space: a number of generators are free to create an infinitenumber of unique elements. Consequently, a free object hasa simple structure and parameterization. In this section, wedevelop graded versions of free objects to provide discrete

187

-7

5

Ox1

x2

(a) 5x1x22 − 7x3

1

kOx1

x2

(b) Module k

k2

k

k

Ox1

x2

(c) k-vector space V

Ox1

x2

(d) Free A2-module F

Figure 3: 2-graded objects: (a) Polynomial 5x1x2

2 − 7x3

1 in the 2-graded ring A2 = k[x1, x2] visualized on N2.

(b) Field k endowed with graded module structure (Definition 3). (c) A k-vector space V with generators at(1, 0) and (2, 1), and two generators at (0, 1) (Definition 5). (d) A free A2-module F with same type as (b)(Definition 6).

invariants for our input. We begin by endowing a field kwith a graded An-module structure.

Definition 3 (k) For a field k, we define k to be the n-graded An-module with grading k0 = k and kv = 0 forv 6= 0. The An-structure is given by setting the action of allthe variables identically to zero.

We show the module k in Figure 3(b). To construct morecomplicated modules, we introduce the concept of shifting.

Definition 4 (shift) Given an n-graded object M and v ∈Nn, the shifted object M(v) is defined by M(v)u = Mu−v for

all u ∈ Nn.

In other words, the object M(v) is identical to M , but itsdirect sum decomposition is shifted upwards in grading byv. We use shifted objects to create graded vector spaces.

Definition 5 (vector space) Let ξ be a multiset of ele-ments from N

n. A finitely generated n-graded k-vector spacewith basis ξ is a finite direct sum of shifted copies of k:

V (ξ) =M

(v,i)∈ξ

k(v).

Note that we enumerate the elements with multiplicity usingour notation for multisets. Figure 3(c) displays a 2-gradedk-vector space defined by multiset

((1, 0), 1), ((0, 1), 2), ((2, 1), 1).

In a vector space, a generator’s scope is a single grade. Ina free module, we extend its scope via the action of the thevariables.

Definition 6 (free module F ) Let ξ be a multiset of el-ements from N

n. The free n-graded An-module with basis ξis the direct sum of shifted copies of An:

F (ξ) =M

(v,i)∈ξ

An(v)

=M

(v,i)∈ξ

k[x1, . . . , xn](v).

Compare this definition with the previous one. Our con-struction has the usual universal mapping property defininga free module [16]. Figure 3(d) displays the free modulewith the same defining multiset as our example vector space.Each shaded region indicates the scope of a generator in itscorner.

Lemma 1 (type ξ, isomorphism) Any finitely-generatedn-graded k-vector space may be written uniquely, up to iso-morphism, with a basis as in Definition 5. Similarly, anyfree n-graded An-module may be written uniquely, up to iso-morphism , with a basis as in Definition 6. The basis is thetype ξ(M) of the object M . Two objects of the same typeare isomorphic.

The lemma gives a full classification, establishing ξ as acomplete discrete invariant, up to isomorphism, for each ofthe two free structures. A free module has a vector space ineach grade. We use the quasi-partial order . to formalizethis next.

Lemma 2 (grade) F (ξ)v is a k-vector space with dimen-sion equal to card(u, i) ∈ ξ | u . v, where card denotescardinality.

In Figure 3(d), the dimension of Mv is simply the numberof regions that cover grade v. For example, dimM(2,1) = 4as (2, 1) is contained in all four regions. Finally, we extendthe notion of an automorphism to free graded modules byrequiring it to respect the grading.

Lemma 3 (GL.) Let µ ∈ GL(V (ξ)) be an automorphismof V (ξ). We say µ respects the grading if for any (v, i) ∈ ξ,µ(v) lies in the span of elements (v′, i′) ∈ ξ such that v′ . v.We define GL.(V (ξ)) to be the set of all such automor-phisms. Then, GL.(V (ξ)) is an (algebraic) subgroup ofGL(V (ξ)). Moreover, the automorphism group of F (ξ) isisomorphic to GL.(V (ξ)) and therefore algebraic, so we de-note it by GL(F (ξ)).

4.2 Two Discrete InvariantsIn the remainder of this section, we use M to denote our

input: a finitely generated n-graded An-module. We cannotuse the invariant ξ directly since M is not free in general.But we could look at free objects related to it. Let In =(x1, . . . , xn) be the n-graded ideal in An that is generated byx1, . . . , xn. Dividing out the elements in M with coefficientsin In, we derive a vector space that contains M ’s generators.

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Definition 7 (ξ) The space V (M) = M/InM = k ⊗An

M

is a vector space. We define ξ(M) = ξ(V (M)).

ξ(M) is our first discrete invariant for M . For the zerothhomology module for our bifiltration in Figure 2, we get:

ξ(M) = ((0, 0), 1), ((0, 1), 1), ((1, 1), 1).

The invariant ξ(M) has an intuitive meaning in the contextof one-dimensional persistence (see Section 1.3). What wecapture with ξ(M) corresponds to the left endpoints of bar-code intervals. Applying Definition 6, we may construct thefree graded module F (ξ(M)). This module has the samegenerators as M , but allows them to be free. In one dimen-sion, the construction corresponds to starting half-infiniteintervals at the left endpoints, as we have not located theright endpoints.

The invariant ξ(M) is not complete. The module F (ξ(M))is a free approximation ofM : it lacks the set of relations thatconstrain M . So, we may begin our classification by com-puting ξ(M), and then refine it by studying classificationof all modules with a fixed value of ξ(M). For this refine-ment, we use a canonical isomorphism k ⊗An F (ξ(M)) ∼=k ⊗An M = V (M).

Lemma 4 (ϕ) There exists a surjection ϕM : F (ξ(M)) →M , unique up to automorphism of F (ξ(M)), such that theinduced homomorphism

k ⊗An

ϕM : k ⊗An

F (ξ(M)) → k ⊗An

M

is the canonical isomorphism given above.

The surjection is the free hull of M , dual to the notion ofinjective hull in the literature [8, Page 628]. Its existenceis entirely analogous to the existence of the first stage of aminimal free resolution for local rings [9, Section 1B]. Thekernel kerϕM of surjection ϕM consists precisely of the re-lations defining M and is called the ideal of relations. Gen-erally, the kernel is not free, but we may use our techniquefrom Definition 7 to capture its generators. This construc-tion gives us our second discrete invariant for M : ξ(kerϕM ).Intuitively, what we capture with ξ(kerϕM ) corresponds tothe right endpoints of barcode intervals in one-dimensionalpersistence. We end this section by naming our two discreteinvariants.

Definition 8 (ξ0, ξ1) We define ξ0(M) = ξ(M) andξ1(M) = ξ(kerϕM ) as two discrete invariants for a finitelygenerated n-graded An-module M .

For the zeroth homology module for our bifiltration in Fig-ure 2, we get:

ξ1(M) = ξ(kerϕM ) = ((1, 2), 2), ((2, 1), 2).

4.3 Complete ClassificationWe now have the locations of the births and deaths of gen-

erators in M inside two multisets ξ0(M) and ξ1(M), respec-tively. The two invariants together are not complete, so wenext study the classification of all modules with fixed valuesfor the invariants. In one-dimensional persistence, we wereable to establish a significant result that we can pair birthsand deaths to get the barcode intervals [17]. To completethe classification for multidimensional persistence, we need

K = kerϕM

F0 = F (ξ0(M)) = F (ξ(M))

F1 = F (ξ1(M)) = F (ξ(K))

ϕK : F1 → K

(a) Notation

K

F1 F0 M

i

?? ??

ϕK

//ψM

// //ϕM

(b) Diagram

Figure 4: Notation and diagram for complete clas-sification.

to study the relationship between the free graded modulesassociated to our two invariants. For notational sanity, wedefine the notation in Figure 4(a). The free graded modulesF0 and F1 have our two discrete invariants as generators,and the surjection ϕK is asserted by Lemma 4. Since Kincludes in M , we have the diagram in Figure 4(b), wherewe define ψM = i ϕK , so the diagram commutes. SinceimψM = kerϕM by construction, the sequence at the bot-tom of the diagram, F1 → F0 → M , is exact. The homo-morphism ψM : F1 → F0 relates our two free modules. Tounderstand this map, we begin by modeling any relationshipbetween any two free graded modules.

Definition 9 (relation family RF, RF) Let F (ξ0) andF (ξ1) be free graded modules. A relation family RF(ξ0, ξ1)is a family Vvv∈ξ1 of vector spaces such that

1. Vv ⊆ F (ξ0)v,

2. dimVv = dimF (ξ1)v,

3. for u, v ∈ ξ1, u . v, we have θv−u(Vu) ⊆ Vv, where θwis multiplication by xw.

The collection RF(ξ0, ξ1) consists of all possible relationfamilies RF(ξ0, ξ1).

Note that in this definition, we treat ξ1 as a set, disregard-ing the multiplicities. Now, automorphisms µ ∈ GL(F (ξ0))induce automorphisms of the exact sequence at the bottomof the diagram in Figure 4(b) and therefore of M . In partic-ular, µ maps a relation family into another relation family,giving us the following.

Lemma 5 GL(F (ξ0)) is a (left) group action on the collec-tion RF(ξ0, ξ1).

Recall the homomorphism ψM : F1 → F0 from the diagram.In each grade v, ψM maps vector space (F1)v to a vectorspace within (F0)v. That is, we get a relation family.

Lemma 6 (η(ψM )) The homomorphism ψM yields a rela-tion family η(ψM ) ∈ RF(ξ0(M), ξ1(M)), where for v ∈ ξ1(M),η(ψM )v = ψM (F1)v.

189

We may now state a primary result of the paper.

Theorem 3 (classification) Let ξ0, ξ1 be multisets of ele-ments from N

n and [M ] be the isomorphism class of finitelygenerated n-graded An-modules M with ξ0(M) = ξ0 andξ1(M) = ξ1. Then, the assignment [M ] 7→ η(ψM ) is a bijec-tion from the collection of isomorphism classes to the set oforbits

RF(ξ0, ξ1)/GL(F (ξ0)). (2)

In other words, each module M is classified, up to isomor-phism, by three invariants: ξ0(M), ξ1(M), and the orbitη(ψM ) under the action of the automorphisms of the asso-ciated free graded module F (ξ0).

5. PARAMETERIZATIONHaving established a complete classification of the graded

modules, we now turn our attention to the third step ofour approach: parameterizing the classification. The twodiscrete invariants are already parameterized as multisets.The remaining invariant is the set of orbits described byTheorem 3. In this section, we examine the structure of theorbits using concepts in algebraic geometry. The generalpicture that emerges is that this portion of the classifica-tion is a continuous invariant. To appreciate its nature, wenext detail an example in two dimensions. We end this sec-tion with possible strategies for coping with the continuousinvariant.

5.1 Algebraic ActionWe begin by endowing the collection of relation families

RF(ξ0, ξ1) with the structure of an algebraic variety. Notefirst that RF(ξ0, ξ1) is a subset of the variety

Y

(v,i)∈ξ1

GrdimF (ξ1)v(F (ξ0)v) (3)

where Gr is the Grassmannian. It is now easy to verifythat the containment conditions that define the collectionRF(ξ0, ξ1) are algebraic on this variety, giving us the follow-ing.

Theorem 4 (algebraic action) RF(ξ0, ξ1) is a variety ina natural way, and the action of the algebraic groupGL(F (ξ0)) on it is an algebraic action.

Unfortunately, the set of orbits of the action of an algebraicgroup on an algebraic variety is not, in general, an algebraicvariety [14]. The number of orbits may be uncountable,giving us a continuous invariant.

5.2 The Continuous InvariantTo further appreciate the complexity of the continuous

invariant, we show its structure for a simple two-dimensionalexample. Suppose we have a set of modules for which

ξ0 = ((0, 0), 2),

ξ1 = ((3, 0), 1), ((2, 1), 1), ((1, 2), 1), ((0, 3), 1),

as visualized on N2 in Figure 5. It is easy to build a bifiltered

simplicial complex whose first homology groups correspondto this picture. At (0, 0), we have a complex composed of

k

k

k

k2

Ox1

x2k

Figure 5: Visualization of ξ0 and ξ1 on N2 for our

example, with the elements of the latter circled.

two loops, giving us k2. In each of the circled coordinates,we choose a sew a surface between the two loops such thatno two complexes are sewn the same. For example, we couldsew a cylinder at (3, 0), a punctured crosscap at (2, 1), andso on. Observe that the discrete invariants ξ0, ξ1 cannotdiscern the difference between the resulting complexes.

To obtain the classification, we apply Theorem 3. Thegenerators of F (ξ0) are co-located, so we have the full groupof automorphisms

GL(F (ξ0)) = GL(k2) = GL2(k),

where GL2(k) is the group of invertible 2 × 2 matrices withelements from k. We use Equation (3) to endow RF(ξ0, ξ1)with a variety structure. For each (v, i) ∈ ξ1, F (ξ0)v is iso-morphic to k2 and dimF (ξ1)v = 1, so GrdimF (ξ1)v

(F (ξ0)v) =

Gr1(k2) = P

1(k), where P1(k) denotes projective line, the set

of lines in k2 going through the origin. Then, the variety is

simply P1(k)

4as there are no containment conditions. The

classification is given by the orbit space

P1(k)

4/GL2(k), (4)

where elements g ∈ GL2(k) act in the evident way on thefour lines, transforming each line to another.

We claim that no discrete invariant is possible for thisbifiltration. Consider the subspace Ω of the orbit space con-taining pairwise-distinct lines. That is, we have four tupleof lines (l1, l2, l3, l4) where li 6= lj for i 6= j. The subspaceΩ is clearly invariant under the GL2(k) action and hencethe orbit space Ω/GL2(k) is a subspace of our orbit spacein Equation (4). Using matrices from GL2(k), we transformthe lines so that

1. l1 becomes the x-axis,

2. l2 becomes the y-axis,

3. and l3 becomes the diagonal line spanned by (1, 1).

These transformations exist as l1, l2 span k2, being non-zero and distinct, and l3 cannot be zero or either axis af-ter the first two transformations. We now have a tuple(x-axis, y-axis, diagonal, λ4), where λ4 is l4 after the trans-formations. While there are different matrices in GL2(k)that can transform the original tuple to this tuple, the ma-trices differ by multiplication by a diagonal matrix, sincethe only matrices that preserve the axes and the diagonalline are diagonal matrices. Consequently, λ4 is determineduniquely, and we may identify the orbits Ω/GL2(k) with thelines in P

1(k) with the axes and the diagonal removed. Each

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such line is determined by its slope which cannot be 0, ∞,or 1, according to the discussion. Therefore, Ω/GL2(k) canbe identified with P

1(k) − 0, 1,∞ = k − 0, 1.Now, note that this classification is dependent on the field

of coefficients k. If k is uncountable, so is the subspace, andin turn, the full orbit space. If k is a finite field, such as Fp forp a prime, we get a finite solution for the subspace Ω we havechosen, but we still have not detailed the full picture for theorbit space. However, we already see the field-dependenceproblem: Changing the field not only changes the classifica-tion, but also the target of the classification: We not only getdifferent values, we get values from different sets altogether.This is analogous to getting Betti numbers in Z2 when com-puting over Z2, Betti numbers in Z3 when computing overZ3, and so on. Therefore, we cannot get a discrete invariantfor our example.

5.3 RefinementWe 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 ofstudy and we end this section by suggesting possible avenuesof attack.

Our two discrete invariants may be viewed as the first twoin a family of discrete invariants. We may develop standardhomological algebra in the category of graded modules overan n-graded k-algebra An, with the resulting derived func-tors ⊗

An

and HomAn now being equipped with the structure

of an n-graded An-module [16]. In particular, the functorTorAn

i (M,k) makes sense and we now define a family of ndiscrete invariants by

ξi = ξ“

TorAni (M,k)

”

.

The first two invariants in the family match our two dis-crete invariants in Definition 8. It may be interesting tostudy the rest of this family as each invariant will make ourclassification finer. However, the existence of the continu-ous invariant indicates that no matter how many of theseinvariants we include, there will still be a residual continu-ous component in the classification.

While the set of orbits is not a variety, we conjecturethat additional structure exists in the following form. LetG = GL(F (ξ0)) and suppose there is a family of closed sub-varieties RFn ⊆ RF(ξ0, ξ1) such that

1. RFn ⊆ RFn+1 for all n,

2. RFn is closed under the action of G,

3. RFn eventually becomes equal to RF(ξ0, ξ1),

4. the set of orbits of the G-action on RFn − RFn−1 isan algebraic variety in a natural way.

This kind of structure is called an equivariant stratificationof the variety in question, with the difference RFn−RFn−1

being a stratum. The orbit varieties are called moduli spacesin classification problems for which the invariant lies in agiven stratum. The result is known to hold in some specialcases by the work of Cohen and Orlik [3] and Terao [15].

6. THE RANK INVARIANTOur study of multigraded objects shows that no com-

plete discrete invariant exists for multidimensional persis-tence. We still desire a discriminating invariant that cap-tures persistent information, that is, homology classes withlarge persistence. This information is not contained in ourtwo discrete invariants, ξ0 and ξ1, as they capture birth anddeath coordinates of the generators in the complexes. Whatwe need lies within the relationship between the two invari-ants or in the maps between the complexes. In this section,we propose and advocate a small and computable invariantthat identifies persistent features in a multifiltration. Ourinvariant is equivalent to persistence barcodes, and thereforecomplete, for one-dimensional filtrations.

The persistent information is contained in the relating ho-momorphisms ϕu,v in Definition 1. Recall that we incorpo-rated these maps into a multigraded module through theaction of the variables, requiring that xv−u : Mu → Mv tobe ϕu,v in Definition 2. To analyze this family of maps, webegin by defining their domains.

Definition 10 (Dn) Let N = N ∪ ∞ with u ≤ ∞ for all

u ∈ N. Let Dn ⊂ N

n× Nn be the subset above the diagonal,

Dn = (u, v) | u ∈ N

n, v ∈ Nn, u . v. For (u, v), (u′, v′) ∈

Dn, we define (u, v) (u′, v′) if u . u′ and v′ . v.

It is easy to check that is a quasi-partial order on Dn.

With this notation, our parameterization of singly-gradedmodules in Section 1.3 is a multiset from D

1, and indicatesthe first pair contains the second, when the pairs are viewedas intervals.

Definition 11 (rank invariant ρM ) Let M be a finitelygenerated n-graded An-module. We define ρM : D

n → N tobe ρM (u, v) = rank(xv−u : Mu →Mv).

The function ρM is clearly a discrete invariant for M .

Lemma 7 (order-preserving) If (u, v) (u′, v′), thenρM (u, v) ≤ ρM (u′, v′), that is, ρM 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 invariant’s completeness in one di-mension through its equivalence to barcodes. We note thatthe following theorem is the converse of the k-triangleLemma [7, 17].

Theorem 5 (completeness) The rank invariant ρM iscomplete for singly-graded modules M .

Proof: To prove completeness, we show equivalence via abijection ϑ between the set of barcodes and the set of rankinvariants. According to the classification theorem for agraded module M recalled in Section 1.3, the intervals inits barcode ξ capture the lifetimes of the generators of M .Therefore, the corresponding rank function is ϑ(ξ)(t, s) =card((t′, s′), i) ∈ ξ | (t, s) ⊆ (t′, s′). Figure 6 illustrates

191

O

t0

t0

t1

t1

s

t

Figure 6: The intervals of a barcode ξ are drawnbelow the t-axis. Each interval (t0, t1) defines a tri-angle as shown. The rank function ϑ(ξ)(t, s) is thenumber of triangles that contain (t, s).

this correspondence. The barcode intervals are drawn belowthe t axis and the rank function’s domain, D

1, exists abovethe diagonal in the (t, s)-plane. Each interval [t0, t1) has atriangular region defined by inequalities t ≥ t0, s < t1, ands ≥ t, with corner vertex (t0, t1) and vertices (t0, t0) and(t1, t1) on the diagonal. Half-infinite intervals correspondto degenerate triangles, but they are handled easily, so wedo not discuss them here. The rank function ϑ(ξ)(t, s) issimply the number of triangles that contain (t, s). As anaside, we note that the map (t, s) 7→ (t, s − t) gives theindex-persistence figures in the previous papers [7, 17].

Clearly, we can construct each triangle from its corner byprojecting the corner vertically and horizontally onto thediagonal. Moreover, there is a trivial bijection between thecorner (t0, t1) and the interval [t0, t1). Given a barcode ξ, weknow how to build the rank function ϑ(ξ) by the equationabove. Given a rank function ρ, we need to identify thecorner points to build the corresponding barcode. We beginby first walking along the diagonal until the rank functionis nonzero at t0 = argmint ρ(t, t) 6= 0. By Lemma 7, thefunction s 7→ ρ(t0, s) is a non-increasing function, so wewalk vertically up until t1 where ρ(t0, t1) < ρ(t0, t0). Thepoint (t0, t1) is a corner, so we subtract its triangle from ρ.The proof follows by induction.

When the module is the persistence module associated tothe ith homology of a multifiltration, we can define the rankinvariant directly in terms of the input.

Definition 12 (ρX,i) Let X = Xvv∈Nn be a multifiltra-tion. We define ρX,i : D

n → N over field k to

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

The function ρX,i is a homeomorphism invariant of the mul-tifiltered space, deriving its invariance from the invarianceof ρM . Intuitively, Theorem 5 means that the rank invari-ant for one-dimensional filtrations may be separated into aset of overlapping triangles whose thickness at any point isthe rank. These triangles, in turn, carry the same informa-tion as a set of intervals or the barcode. Our classificationtheorem, on the other hand, implies that a similar result isnot possible for higher dimensions. As our example in Sec-tion 5.2illustrates, the picture is much more complicated: It

is not possible to separate the rank invariant into overlap-ping “regions” to extend the barcode. However, the rankinvariant does extend as an incomplete invariant and wemay utilize it to identify persistent features by the follow-ing procedure. Given a rank invariant, we look for points(u, v) ∈ D

n that are far from the diagonal and have a neigh-borhood of constant value. The first condition correspondsto the persistence of the features. The second condition indi-cates the stability of our choice (u, v). With this procedure,the rank invariant emerges as a practical tool for reliableestimation of the Betti numbers of multifiltered spaces.

7. CONCLUSIONWe believe the primary contribution of this paper is the

full theoretical understanding of the structure of multidi-mensional persistence: We identify the corresponding alge-braic structure, classify it, and undertake its parameteriza-tion. Our theory reveals that a complete discrete invariantdoes not exist for multidimensional persistence, unlike itsone-dimensional counterpart. A second practical contribu-tion of our paper is the rank invariant, a tool for robustestimation of the Betti numbers. We prove that the rankinvariant is equivalent to the persistent barcode in one di-mension, so it is complete when it can be. Unlike the bar-code, the rank invariant extends to higher dimensions as anincomplete but useful invariant.

We have developed an algorithm for computing the rankinvariant. 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 featuresin multifiltrations, such as the filtered tangent complex [4].

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