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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
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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].
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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:
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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.
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(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].
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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.
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-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
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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
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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.
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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.
15
-
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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