Persistent Homology for Kernels, Images, and Cokernels ∗

David Cohen-Steiner†, Herbert Edelsbrunner‡, John Harer§ and Dmitriy Morozov¶

Abstract tent homology arises from considering the corresponding sequence of homology groups, H(X0) → H(X1) → . . . → Motivated by the measurement of local homology and of H(Xm), connected from left to right by homomorphic maps functions on noisy domains, we extend the notion of persis- induced by inclusion. Persistence tracks when a homology tent homology to sequences of kernels, images, and coker- class is born and when it dies. This can also be done for nels of maps induced by inclusions in a filtration of pairs of an arbitrary sequence of vector spaces connected by homo- spaces. Specifically, we note that persistence in this context morphic maps. Motivation for studying such more general is well defined, we prove that the persistence diagrams are sequences is derived from recent investigations in computa- stable, and we explain how to compute them. tional topology. First, Bendich et al. describe a multi-scale Keywords. Persistent homology, kernels, images, cokernels, per- assessment of local homology for the purpose of reconstruct- sistence diagrams, vineyards, algorithms, stratified spaces. ing a stratified space from a point sample [2]. We will see how sequences of kernels can be used to refine their construction. Second, we use sequences of images to introduce 1 Introduction a notion of persistence that filters out noise induced by imprecise specifications of domains. This contrasts standard Natural phenomena are often modeled in terms of spaces and persistence which can handle imprecise function values but functions on these spaces. We argue that it is almost always not imprecise domains. As an application, we will approxi- more appropriate to use a multi-scale hierarchy instead of a mate the persistence diagram of a function knowing only its single space. One reason is the prevalent multi-scale orga- values at a finite set of points. The main contributions of this nization we find in nature, another is that the data we gather paper are two-fold: about nature is necessarily incomplete and requires interpo- lation. The multi-scale aspect helps bridging the gap between the data about natural phenomena and the idealized • an algorithm that computes the persistence diagrams of mathematical concepts we use for exploration. In particu- sequences of kernels, images, and cokernels in time at lar, we consider homologygroups, which are algebraic struc- most cubic in the size of the simplicial complexes rep- tures that define and count holes in a topological space [11]. resenting the data; Their multi-scale extensions are persistent homology groups introduced in [10, 13]. Similar to homology which not only • applications of the algebraic and algorithmic results to counts but also defines, persistent homology not only mea- measuring local homology and to approximating persis- sures but also creates the hierarchy. tence diagrams of noisy functions on noisy domains. In all previoussettings, the hierarchyis defined by a nested sequence of spaces, X0 ⊆ X1 ⊆ . . . ⊆ Xm, and persis-

∗This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011- Outline. The remainder of this paper is structured as fol- 05-1-0057 and by CNRS under grant PICS-3416. lows. Section 2 introduces the algebra of persistent homol- †INRIA, 2004 Route des Lucioles, BP93, Sophia-Antipolis, France. ‡Departments of Computer Science and of Mathematics, Duke Uni- ogy including its extension to sequences of kernels, images versity, Durham, Berlin Mathematical School, Berlin, Germany, and Ge- and cokernels. Section 3 explains the algorithms for com- omagic, Research Triangle Park, North Carolina, USA. puting the corresponding persistence diagrams for a nested § Departments of Mathematics and of Computer Science, Duke Univer- sequence of pairs of spaces and Section 4 proves their cor- sity, Durham, North Carolina, USA. ¶Department of Computer Science, Duke University, Durham, North rectness. Section 5 presents the two applications of our alge- Carolina, USA. braic and algorithmic results. Section 6 concludes the paper. 2 Algebra two-dimensional plane. By collecting the points for all p- dimensional classes we get the dimension p persistence dia- Beginning with a review of persistent homology, this section gram which we denote as Dgmp(p). Since birth necessarily extends this concept to sequences of kernels, images, and happens before death all points lie above the diagonal. It cokernels. It also proves that the persistence diagrams of is also possible that a class γ is born at ai but does not die these extensions are stable. since it represents a class of Xm = X. In this case, we draw γ as the point (ai, ∞) in the diagram. For technical reasons that will become clear later, we consider all points on the Persistent homology. This concept is a recent addition to diagonal to be part of the persistence diagram. Similar to classical homology theory and was originally introduced for homology groups we get a diagram for each dimension and ordered simplicial complexes [10]. We follow the exposition we write Dgm(f) for the infinite series of diagrams. Same in [6] in which we have a topological space X and a continu- as for homology groups and for the maps between them we ous function f : X → R. The sublevel set defined by a ∈ R will simplify language by ignoring the difference between a consists of all points with function value at most the thresh- −1 single diagram and an entire series. old, Xa = f (−∞,a]. We use the algebraic language of homology theory to characterize how Xa is connected, see e.g. [11]. Adding chains with modulo-2 arithmetic, we write Hp(Xa) for the dimension p homology group over Z/2Z of Kernels, images, and cokernels. For the extension of per- Xa and H(Xa) = (..., Hp(Xa), Hp+1(Xa),...) for the infi- sistence to kernels, images, and cokernels we consider two nite series obtained by collecting the groups for all dimen- functions, f : X → R and a majorizing function g : Y → R sions. Of course, only the groups for p between 0 and the defined on a subspace Y ⊆ X, that is, f(y) ≤ g(y) for all dimension of Xa are possibly non-trivial. To simplify lan- y ∈ Y ⊆ X. Assuming both functions are tame, we or- guage, we will often ignore the difference between a single der the collection of critical values of f and g and interleave homology group and the entire series. Given a ≤ b, the them with a sequence of real values si. The corresponding inclusion between the sublevel sets, Xa ⊆ Xb, induces a ho- sequences of sublevel sets give rise to two parallel sequences momorphism, f a,b : H(Xa) → H(Xb). For a = b this is an of homology groups, isomorphism and for a 0 for which fa−ε,a+ε is an isomorphism. We assume that f is tame, that is, it has only ↑ j0 ↑ j1 . . . ↑ jm finitely many critical values and every sublevel set has only

finite rank homology groups. H(Y0) → H(Y1) → . . . → H(Ym), Let a1 < a2 <...homomorphisms ji : H(Yi) → X X X , where we simplify notation by writing i = si , and a H(Xi) induced by the inclusions Yi ⊆ Xi. We call this the corresponding sequence of homology groups connected by two function setting, in contrast to the more special one func- homomorphisms, tion setting in which g is the restriction of f to Y. Moreabout the relationship between the two settings later. We are inter- H X H X H X ( 0) → ( 1) → . . . → ( m). ested in the kernels, images, and cokernels of the connecting homomorphisms, Persistence concerns itself with the history of individual homology classes within this sequence. Specifically, a class ker ji = {γ ∈ H(Yi) | ji(γ)=0 ∈ H(Xi)}; γ in H(Xi) is born at ai if it is not in the image of fi−1,i = f im ji = {ji(γ) ∈ H(Xi) | γ ∈ H(Yi)}; si−1,si . More precisely, an entire cosetis bornat ai. Further- more, if γ is born at ai we say it dies entering aj if fi,j−1(γ) cok ji = H(Xi)/im ji. is not contained in the image of fi−1,j−1 but fi,j (γ) is con- f f tained in the image of i−1,j . The images of the maps i,j are We note that the “coimage” of the map ji, in symbols referred to as persistent homology groups since they consist H(Yi)/ker ji, is isomorphic to the image of this map, and of all homology groups born at or before ai that live beyond therefore does not deserve any special attention. Figure 1 il- aj . Nothing about the definition of birth and death is spe- lustrates this construction for two contiguous spaces in both cific to homology groups. In other words, persistence makes sequences. The square defined by the four homology groups perfect sense for any sequence of vector spaces connected by commutes. It follows that the inclusion Yi ⊆ Yi+1 induces homomorphisms. a homomorphism ker ji → ker ji+1. Similarly, the inclu- It is convenient to represent the fact that γ is born at ai sion Xi ⊆ Xi+1 induces a homomorphism im ji → im ji+1 and dies entering aj by drawing the point (ai,aj ) in the and another homomorphism cok ji → cok ji+1. We thus get

2 Case ker ji im ji H(Yi) H X ( i) H(Xi+1) A birth death — B — birth birth imji imji+1 C, D — — — E — death death F death — death P birth — birth

ji ji+1 Table 1: The ﬁve cases in the two function setting relating kernels and images. Except for Case P they also occur in the one function setting.

Case C occurs when Y = X = Y is a point and X is kerji kerji+1 i−1 i−1 i i again obtained by adding an arc that forms a circle. We also H Y X ( i) H(Yi+1) retain ranks in Case D which occurs when i−1 is a circle, Yi−1 = Yi is a point on this circle, and Xi is obtained by adding a spanning disk. Case E occurs when Yi−1 = Xi−1 Figure 1: A square of four homology groups and the maps between is a circle and Y = X is obtained by adding a spanning them. The square commutes because all four maps are induced by i i disk. Case F occurs when X is a disk, Y is its bound- inclusions. i−1 i−1 ary circle, and we get Yi and Xi by adding another spanning disk to both spaces. Finally, Case P occurs when Xi−1 = Xi sequences of kernels, images, and cokernels, is a disk, Yi−1 is a point of its boundary circle, and Yi is obtained by adding the rest of the circle. This last case happens Ker (g →f) : ker j0 → ker j1 → . . . → ker jm; in the two function setting but not in the one function setting Im(g →f) : im j0 → im j1 → . . . → im jm; because it requires points that are added to the sublevel set of g strictly after they are added to the sublevel set of f. Cok(g →f) : cok j0 → cok j1 → . . . → cok jm, Similarly, the second equation eliminates two of the nine all connected from left to right by homomorphisms. Homol- combinations for images and cokernels. Another combina- ogy classes are bornand die in these sequencessame as in the tion is eliminated by im ji being a subgroup of H(Xi), hence sequences of homology groups. We can therefore deﬁne per- a death in the image implies a death in the homology group. sistent kernels, persistent images, and persistent cokernels Table 2 lists the remaining six cases. Cases A to F have been as well as construct the corresponding persistence diagrams, H X which we denote as Dgm(ker g →f), Dgm(im g →f), and Case im ji cok ji ( i) Dgm(cok g →f). A, E death — death B birth — birth C, F — birth birth Birth-death combinations. We consider the generic case D — death death in which changes happen one at a time. An event thus corre- Q — — — sponds to a birth, a death, or no change in the kernel, in the R birth death — image, and in the cokernel, giving rise to 27 different combinations. But the ranks of these groups are not independent, Table 2: The ﬁve cases in the two function setting relating cokernels that is, and images. Except for Cases Q and R they also occur in the one function setting.

rank ker ji + rank im ji = rank H(Yi); described above and the example for Case P also works for rank im ji + rankcok ji = rank H(Xi), Case Q. Case R occurs when Xi−1 = Yi = Xi is a circle Y for all i. We can therefore relate the births and deaths in the and i−1 is a point on that circle. Cases Q and R do nothap- three sequences using the births and deaths in the sequences pen in the one function setting in which every change has a non-zero effect on the rank of the homology group of X . of homology groups of the Yi and of the Xi. The ﬁrst equa- i tion eliminates two of the nine combinations for kernels and images. Another combination is eliminated by ker ji being Mapping cylinder. We reduce the two function setting to a subgroup of H(Yi), hence a death in the kernel implies a the one function setting using a construction that will be ex- death in the homology group. Table 1 lists the remaining six ploited by the algorithm described in Section 3. Speciﬁcally, ′ cases. Case A occurs for example when Yi−1 = Xi−1 = Yi the mapping cylinder of the pair Y ⊆ X is the space X = is a circle and Xi is obtained by adding a spanning disk. Case X ∪ (Y × [0, 1]) obtained by gluing Y ⊆ X to Y ×{0} ⊆ B occurs when Yi−1 = Xi−1 is a point and Yi = Xi is ob- Y × [0, 1]. This is illustrated in Figure 2. The function tained by adding an arc that completes the point to a circle. f ′ : X′ → R agrees with f on X and with g on Y′ = Y×{1},

3 ′ and the kernels, images, and cokernels of ji+1 and ji+1 Y × 1 are isomorphic. To prove that the two sequences of kernels define the same persistence diagram we still need to consider the bottom square in Figure 3. The left and right maps are isomorphisms and the square commutes by construction. Hence Dgm(ker g →f) = Dgm(ker g′ →f ′). Y × 0 X Similarly, we get Dgm(im g →f) = Dgm(im g′ →f ′) and Dgm(cok g →f) = Dgm(cok g′ →f ′) by considering the top square in Figure 3. Figure 2: The mapping cylinder of Y ⊆ X. linearly interpolating in between, that is, f ′(x) = f(x) for Stability. An important property of the persistence dia- every x ∈ X and f ′(y,t) = (1 − t)f(y)+ tg(y) for every grams is their stability originally proved in [6]. More pre- y ∈ Y and every t ∈ [0, 1]. The pair of functions f ′ and ′ ′ ′ ′ ′ cisely, the bottleneck distance between the diagrams of two g = f |Y′ induces homomorphisms j : H(Y ) → H(X ); i i i functions f and f ′′ is bounded from above by the dif- see Figure 3. The corresponding sequences of kernels, im- ′′ ference between the functions, dB(Dgm(f), Dgm(f )) ≤ ages, and cokernels are ′′ kf − f k∞. Here dB is the maximum of ku − γ(u)k∞, Ker ′ ′ ′ ′ ′ where u is a point in the diagram of f and γ is a dimension- (g →f ) : ker j0 → ker j1 → . . . → ker jm; preserving bijection between the diagrams of f and of f ′′. Im(g′ →f ′) : im j′ → im j′ → . . . → im j′ ; 0 1 m Recall that the points on the diagonal belong to the diagrams Cok ′ ′ ′ ′ ′ (g →f ) : cok j0 → cok j1 → . . . → cok jm. and can therefore be used in the effort to find a matching γ that minimizes the length of the longest edge. The proof We claim that they contain the same information as the se- of stability given in [6] can be adapted to the setting in this quences Ker(g →f), Im(g →f), and Cok(g →f). paper. Specifically, we consider the maps ja : H(Ya) → H X ′′ H Y′′ H X′′ ( a) and jaε : ( a+ε) → ( a+ε), where ε is the larger ′′

′ of the two differences between functions, kf − f k and » H X′ » H X ∞

( i) ( i+1) ′′ ′′ ′′ 7

7 Y X Y X Ç

7 Ç

Ó kg − g k , and , , , are the sublevel sets

Ó a a

Ó a ε a ε

Ó ∞ + + Ó Ó ′′ ′′

′ of g, f, g , f for thresholds a and a + ε. To adapt the proof » H(Xi) » H(Xi+1) ji+1

′ ′′

Ç Ç Y Y Ç Ç we need that the maps induced by the inclusions ⊆ ji a a+ε ′′ ′′

′ and Xa ⊆ X send the kernel of ja into the kernel of j . » j H Y′ » H Y a+ε a+ε

i ( i) ( i+1)

7 7

ji+1 7 Ó But this follows from the commutativity of the diagram

Ó » H Y » H Y ( i) ( i+1) H X H X′′ ( a) → ( a+ε)

′′ Figure 3: The diagram of homology groups of two contiguous sub- ↑ ja ↑ ja+ε level sets of f,g,f ′,g′. The diagram commutes because ten of the H Y H Y′′ twelve maps are induced by inclusions, and the left and right maps ( a) → ( a+ε). of the bottom square are induced by the inverses of the mapping ′ ′ cylinder retractions restricted to Yi and Yi+1 respectively. Y′′ Y X′′ Similarly, we need that the inclusions a ⊆ a+ε and a ⊆ X ′′ a+ε send ker ja into ker ja+ε which follows by symmetry. With this property the original proof of stability goes through MAPPING CYLINDER LEMMA. The pairs of functions and we refer to [6] for details. The arguments for the images f,g and f ′,g′ deﬁne the same persistence diagrams for and the cokernels are the same and we state the results. kernels, images, and cokernels: Dgm(grp g →f) = Dgm(grp g′ →f ′) for grp ∈{ker , im , cok }. STABILITY THEOREM. Let f,f ′′ : X → R and g,g′′ : X X′ Y R ′′ ′′ PROOF. We note that i is a deformation retract of i. This → with f(y) ≤ g(y) and f (y) ≤ g (y) for every H X H X′ Y X ′′ ′′ implies that the map from ( i) to ( i) induced by the y ∈ ⊆ and ε = max{kf − f k∞, kg − g k∞}. Then X X′ inclusion i ⊆ i is an isomorphism. Similarly, the map the bottleneck distance between the persistence diagrams is H Y H Y′ from ( i) to ( i) implied by the inverse of the retrac- boundedfrom aboveby the difference between the functions: tion is an isomorphism. The maps ji : H(Yi) → H(Xi) ′ H Y′ H X′ ′′ ′′ and ji : ( i) → ( i) are also induced by inclusions dB (Dgm(grp g →f), Dgm(grp g →f )) ≤ ε, which implies that the left square in the diagram of Figure 3 commutes. It follows that the pairs of kernels, images, and for grp ∈ {ker , im , cok }, provided f, g, f ′′, and g′′ are ′ ′ continuous and tame and there is a triangulation of X in cokernels are isomorphic, ker ji ≃ ker ji, im ji ≃ im ji, ′ which Y arises as a subcomplex. and cok ji ≃ cok ji. Similarly, the right square commutes

4 3 Algorithms class. Equivalently, column i of V stores a cycle and column i of R is zero. Symmetrically, we call σi negative if its addi- In this section, we describe the algorithms for computing the tion to Ki−1 gives death to a homology class. Equivalently, persistence diagrams of the sequences of kernels, images, column i of V stores a chain that is not a cycle and column and cokernels. At their core is the reduction of a matrix as i of R is non-zero. The significance of the lowest one in this introduced in [8] which we describe first. column of R is that the negative σi is paired with the positive σℓ, with ℓ = low(i), which gives birth to the class that Matrix reduction. To get a finite representationof the data σi kills. we now substitute a function on a simplicial complex for the The persistence diagrams of f can be obtained from the continuous function on a topological space. In particular, we reduced matrix. Specifically, each lowest one, ℓ = low(i), assume a (finite) simplicial complex K and an injective func- corresponds to a pair of simplices, σℓ, σi, and we draw the R tion f : K → that maps each simplex to a real number. point (f(σℓ),f(σi)) in the diagram whose dimension is the The only additional requirement is f(σ)

Equivalently, the reduced matrix is obtained by multiplying K K K the incidence matrix from the right with an upper-triangular L L matrix, R = DV , such that the map from the non-zero columns of R to the row indices of their lowest ones is in- = K jective. As proved in [8], R is not unique but the map is. By K-L K-L

Rim Dim Vim

= Figure 5: Matrices computed in the reduction of the incidence matrix of K with reordered rows. The matrix Vim is upper-triangular with all ones in the diagonal.

R D V Step 3 Delete some of the columns from Vim and reorder Figure 4: The reduced matrix equals the incidence matrix times the the rows to get a new matrix Dker . Specifically, keep chain matrix. All three are upper-triangular. the columns that represent cycles and remove all others. Furthermore, reorder the rows so they correspond to the construction, the rows of R and V correspond to individual simplices in L, ordered by g, followed by the simplices simplices, same as the rows of D, but the columns of R and in K − L, ordered by f. Finally, reduce the new matrix V correspond to chains. Specifically, column i of R stores to get Rker = Dker Vker . the boundary of the chain stored in column i of V . We call Step 4 Starting again with Df , replace some of the columns σi positive if its addition to Ki−1 gives birth to a homology to get a new matrix Dcok . Specifically, substitute

5 columns in Vg that represent cycles for the correspond- Death. A simplex τ gives death in Cok(g →f) iff τ is neg- ing columns in Df , adding zeros to compensate for the ative in Rf and the lowest one in its column in Rim simplices in K − L, which are missing in Vg. Reduce corresponds to a simplex in K − L. In this case, the the new matrix to get Rcok = Dcok Vcok . lowest one in the column of τ in Rcok corresponds to a simplex σ that gives birth in Cok(g →f). Then (σ, τ) is We note that reducing Df is redundant because the type a pair. information it furnishes is also available from Rim . We still use Rf because this clarifies which information is used The dimension p persistence diagram consists of all points where. (f(σ),f(τ)) encoding pairs of p- and (p+1)-simplices identified in the Death case as well as points (f(σ), ∞) encod- Births, deaths, and pairs. We use the reduced matrices to ing unpaired p-simplices identified in the Birth case. The 3 compute the persistence diagrams of the sequences of ker- running time of the three algorithms is O(m ), same as the nels, images, and cokernels. Specifically, Rf and Rg (and in original persistence algorithm given in [10]. Furthermore, it is possible to extend the algorithm in [8] so that it main- one case Rim ) decide which simplices give birth and which tains the reduced matrices in time O( ) per transposition of give death and Rker , Rim , Rcok determine how the births m match up with the deaths. We begin with the sequence of contiguous simplices in the ordered sequences; details can kernels and recall the relevant Cases A and F in Table 1. be found in Appendix A. This is the method of choice for computing the vineyard of a pair of 1-parameter families of Algorithm for kernels: functions f and g, as they arise in applications considered in Birth. A simplex σ gives birth in Ker(g →f) iff σ ∈ K − L, Section 5. σ is negative in Rf , and the lowest one in its column in R corresponds to a simplex in L. im 4 Correctness Death. A simplex τ gives death in Ker(g →f) iff τ ∈ L, τ is negative in Rg, and τ is positive in Rf . In this case, We prove the correctness of the algorithms inductively, by the lowest one in the column of τ in Rker corresponds considering one simplex at a time. For each index i, we con- to a simplex σ ∈ K − L that gives birth in Ker(g →f). sider the actual births, deaths, and pairs that occur in the se- Then (σ, τ) is a pair. quences up to ji, and the computed births, deaths, and pairs reported by the algorithm working on the simplices up to σi. A dimension p homology class is given birth to in the kernel To prove that the corresponding sets are the same at the end, bya -simplexand it dies at the hand of another - (p+1) (p+1) for i = m, we show they are the same throughout, for all i. simplex. The dimension p persistence diagram thus consists We do this in two steps, first proving that the algorithms are of all points (f(σ),f(τ)) encoding pairs of (p+1)-simplices necessary and second that they are sufficient. In other words, identified in the Death case as well as all points (f(σ), ∞) we first prove that the computed information is correct and encoding unpaired (p + 1)-simplices identified in the Birth second that it is complete. case. We continue with the sequence of images and recall the relevant Cases A, B, E in Tables 1 and 2. Preparation. We begin with a few preliminary observa- Algorithm for images: tions. Recall that Tables 1 and 2 list the possible combina- Birth. A simplex σ gives birth in Im(g →f) iff σ ∈ L and σ tions of births and deaths under the simplifying assumption is positive in Rg. that each group has at most one change happeningat any one time. This is indeed the situation if we add individual sim- Death. A simplex τ gives death in Im(g →f) iff τ is neg- plices to a growing complex. We can therefore use the two ative in R and the lowest one in its column in R f im tables in the correctness proof, but since we only consider corresponds to a simplex σ ∈ L. Then (σ, τ) is a pair. the one function setting, we can further simplify and com- Note that the Death case splits into Case A with τ ∈ K − L bine them into Table 3. We get Ki by adding σi to Ki−1. If and Case E with τ ∈ L. The dimension p persistence dia- H H gram consists of all points (f(σ),f(τ)) encoding pairs of p- Case ker ji im ji cok ji (Li) (Ki) σi and (p + 1)-simplices identified in the Death case as well as A birth death — — death K − L points (f(σ), ∞) encoding unpaired p-simplices identified B — birth — birth birth L in the Birth case. We continue with the sequence of coker- C — — birth — birth K − L D — — death — death K − L nels and recall the relevant Cases C, F and D in Table 2. E — death — death death L Algorithm for cokernels: F death — birth death birth L Birth. A simplex σ gives birth in Cok(g →f) iff σ is positive Table 3: In the one function setting there are six cases in which the in Rf and it is either in K − L or negative in Rg. addition of σi changes the kernel, the image, or the cokernel.

6 σi ∈ L then Li = Li−1 ∪{σi} else Li = Li−1. In Cases reordering the rows, moving simplices in L up and simplices B, E, F, the addition of σi changes the homology of Li−1, in K − L down. The reordering maintains the relative order which can only happen if σi ∈ L. In the remaining three of the simplices in K − L. This implies (v). cases, the addition of σi changes the homology of Ki−1 but Finally, we prove (vi) by contradiction, assuming the low- not that of Li−1, which can only happenif σi ∈ K −L. Note est one in Rker [i] corresponds to a simplex σl ∈ L. Since also that the change in the homology of Ki−1 is unambigu- Rf [i]=0 we have Rim = 0. It follows that column i of ous in all cases, that is, σi is positive in Cases B, C, F, and Vim is part of Dker , after reordering the rows. Because of negative in Cases A, D, E. We note that each death is paired the upper triangular structure of Vim , the diagonal ones may with a unique birth but some births remain unpaired until the be moved by the reordering but they are not cancelled in the very end. It is convenient to rephrase the pairing condition reduction. By assumption, the lowest one in Rker [i] corre- in a form that is most directly useful in the argument below. sponds to a simplex in L which can therefore only be σi, that We say a cycle appears in grp jl if the class it represents is is, l = i. But then Rker [i] stores a cycle in L, a contradiction born at that group, where grp ∈ {ker , im , cok } as usual. to Rg[i] 6=0. We note that the cycle might exist in the complex before it appears in the group. Inductive step. We assume inductively that the actual and DEATH LEMMA. Let l

(ii) Rf [i]=0 iff Rim [i]=0. This cycle ﬁrst appears in im jl and it is zero in Ki and therefore also in im j . Furthermore, there is no i′ < i for (iii) If σ ∈ L and R [i] 6= 0 then the lowest one in R [i] i i f im which there is a chain that ﬁrst appears in im j and is zero corresponds to a simplex in L. l in im ji′ . Otherwise, (σl, σi′ ) would be a pair and the lowest ′ (iv) The columns of Rker are all non-zero. one in Rim [i ] would correspond to σl, by inductive hypoth- (v) If σi ∈ K − L then the lowest one in Rker [i] corre- esis. But then Rim could be reduced further, a contradiction. sponds to σi. By the Death Lemma, σi gives rise to an actual death and (σl, σi) is an actual pair in the sequence of images. (vi) If σi ∈ L, Rg[i] 6= 0, and Rf [i]=0 then the lowest one in Rker [i] corresponds to a simplex in K − L. Cokernels. The algorithm for the cokernels reports a birth iff Rf [i]=0 and either σi ∈ K − L or else σi ∈ L and PROOF. Except for added zeros the columns of a simplex R [i] 6= 0. In this case, we have indeed a new class in the in L are the same in D and in D . This implies (i). The g g f cokernel, namely the one represented by V [i]. reordering of rows does not change the rank of the matrix. f The algorithm reports a death iff R [i] 6= 0 and the low- This implies (ii). By Observation (i), R [i] 6= 0 follows f g est one in R [i] corresponds to a simplex in K − L. By from σ ∈ L and R [i] 6= 0. Since the columns of σ in im i f i Observation (iii), this implies σ ∈ K − L. Letting σ cor- D and D are the same, except for added zeros as be- i l g im respond to the lowest one in R [i], the algorithm reports fore, the lowest one in R [i] cannot be lower than that in cok im (σ , σ ) as a pair. In this case, the reduced column is a sum R [i]. This implies (iii). The matrix V is upper triangular l i g im of boundaries in K and cycles in L , with a diagonal of ones. It thus has full rank and so does i i which consists of a subset of the columns in . This Dker Vim Rcok [i] = X Dcok [k]. implies (iv). We get Dker from this subset of columns by σk ∈Vcok [i]

7 This cycle appears ﬁrst in cok jl and it is zero in Ki and Case 1.2 Rf [i] 6= 0. Let σl correspond to the lowest ′ therefore also in cok ji. Furthermore, there is no index i

8 ′ ′ Hp(Xi,k) → Hp(Xi′,k′ ), for ﬁxed i

5.1 Local Homology We begin with the application of kernels to measuring the local homology of a space in Rn at a point not necessarily

in the space. Following earlier work, we assume that the Radius Radius z space is not known other then indirectly through a finite set of points sampled in or near the space. Death Death Birth Birth Measuring local homology. Bendich et al. study the re- construction of a stratified space S from a finite point sample U in Rn [2]. Specifically, they use persistence to define Figure 6: The vineyard of the chain of loops defined using the se- a multi-scale version of the local homology of S at a point Rn S quence (1) on the left and the sequence (2) of kernels on the right. z ∈ . Let α be the sets of points at Euclidean distance To compare the two diagrams we see that the kernel sequence re- S Sα at most α from , the set of points at distance at least α flects the left half of the left vineyard across the diagonal plane and S from , and Br the closed ball of radius r centered at z. They corrects for the removed absolute homology groups. express the homology within a fixed distance r of z in terms of the persistence diagram of the sequence between the vineyard of S and that of a finite point sample 0 → H(Sα ∩ Br) → . . . → H(Br) U ⊆ Rn similar to the Local Homology Inference and the → H(B , Sα ∩ B ) → . . . → 0, (1) r r Inverse LHI Theorems in [2] can be proved using the same where α first goes up, from 0 to ∞, and then down, from ∞ methods. Details are omitted. to 0. The first half of the sequence captures the development of the absolute homology of Sα within Br, and it can be shown that the second half captures the development of the relative homology of Sα within the pair (Br,∂Br). We note that cycles that lie entirely inside the ball are captured twice, once in each half. Finally, Bendich et al. vary r from 0 to Computing local homology. Following the prior work, we ∞ and this way obtain a vineyard that expresses the local use the Delaunay triangulations of U restricted to the ball homology of S at z under the 2-parameter variation of α and Br and to the ball without the interior of the power cell of r. They also prove relationships between this vineyard and the point z, Z0(r) = Br − int Z(r); see [2, Section 6] the similarly defined vineyard of a finite set of points U ⊆ for details. We use distance functions and Delaunay trian- Rn sampled near S. gulations for clear and solid theoretical footing. For prac- In this paper, we propose to substitute a sequence of ker- tical high-dimensional implementations, like [2], these re- nels for (1). Specifically, let X = Br, Y = ∂Br, and let sults need to be extended to Vietoris-Rips [12] and Witness f : X → R, g : Y → R map each point to its Euclidean dis- complexes [9]. Let Kα = Del(U|Uα ∩ Br) and Lα = −1 tance from S. For each value α we write Xα = f (−∞, α], Del(U|Uα ∩ Z0(r)) be the Delaunay triangulations that are −1 Yα = g (−∞, α] and let jα : H(Yα) → H(Xα) be the further restricted to the set of points at distance at most α map induced by the inclusion Yα ⊆ Xα. Assuming f and g from U. Let furthermore are both tame we have a finite set of critical values and thus a finite sequence of kernels, iα : H(Uα ∩ Z0(r)) → H(Uα ∩ Br); Ker(g →f) : ker j0 → ker j1 → . . . → ker jm, (2) lα : H(Lα) → H(Kα) which traces the evolution of the relative homology classes in (1) that have a non-zero boundary in Y = ∂Br. The Sta- bility Theorem in Section 2 implies that varying r from 0 to be the maps between the homology groups induced by in- ∞ gives a vineyard. It tracks a homologyclass as long as the clusion. To justify the use of the restricted Delaunay trian- boundary of the ball with radius r intersects all its represen- gulations, we need to show that the persistence diagrams of tatives. The interval of radii expresses relevant size informa- the kernels of these maps are the same. This requires that tion, namely how far away from z the class starts and ends. the following diagram is well-defined, commutative, and its

9 vertical maps are isomorphisms whenever α ≤ α′: ideal function f˜ : X → R can be approximately recovered knowing only a noisy approximation f of f˜. We claim that ker j → ker j ′ α α the persistence for images can be used to furthermore ﬁlter out the topological noise induced by the domain itself. ↓ iα ↓ iα′

ker iα → ker iα′ Stability. We assume an unknown ideal domain given as the zero sublevel set of the unknownideal function h˜ : Rn → −1 ↑ hα ↑ hα′ R, that is, X˜ = h˜ (−∞, 0]. On this domain we consider another unknown ideal function but because we will ker lα → ker lα′ . vary the domain we assume it is defined on the entire am- bient space, f˜ : Rn → R. Can we estimate the persis- Consider the following diagram whose maps are all induced tence diagram of the restriction f˜| : X˜ → R knowing by inclusions except for the lower vertical maps which are X˜ ˜ ˜ motivated by the Nerve Subdivision Lemma [2, Section 8]. only noisy approximations h,f of h, f? We give an af- firmative answer under mild requirements on the functions. jα H(Uα ∩ ∂Br) → H(Uα ∩ Br) To describe these requirements we use superscripts for sublevel sets of h and h˜ and subscripts for sublevel sets of f ˜ Xu −1 X −1 ↓ ↓ and f, that is, = h (−∞,u], a = f (−∞,a], Xu X Xu X˜ u X˜ X˜ u a = a ∩ and similarly for , a, a . Writing iα ε = kh − h˜k we require that h˜ is smooth and the norm H(Uα ∩ Z (r)) → H(Uα ∩ Br) ∞ 0 of its gradient is bounded away from 0 where this is relevant, that is, k∇h˜k ≥ µ > 0 on X˜ 2ε − X˜ −2ε. Further- ↑ ↑ ˜ more, we write δ = kf − fk∞ and require that f is Lips- chitz with constant κ, that is, |f(x) − f(y)|≤ κkx − yk for H lα H (Lα) → (Kα). all x, y ∈ Rn. The top square commutes. The bottom square also com- We note that the requirement on h˜ implies a homotopy mutes since the horizontalmaps are induced by inclusion and between the identity on X˜ 2ε and a retraction ̺ from X˜ 2ε to 0 the vertical map for Lα is the restriction of the one for Kα. X˜ . To construct the homotopy we consider the integral lines Therefore the kernel diagram is well-defined. Since the ker- of the vector field −∇h˜ starting at points x ∈ X˜ 2ε. Let nels are subgroups of the domains of their defining maps, the ̺(x) be the first point on the curve starting at x that satisfies kernel diagram is a restriction of a diagram considered in [2, h˜(̺(x)) = h˜(x) − 2ε or h˜(̺(x)) ≤−2ε. In words, ̺ moves Section 8]. The analysis there implies that it commutes and the fringe outside the boundary of X˜ to the fringe inside that its vertical maps are isomorphisms, as required. boundary and it moves the fringe inside that boundary to the The Stability Theorem of Section 2 implies that the kernel boundary of X˜ −2ε. Since the gradient has norm no smaller persistence diagrams change continuously with the radius of than µ > 0 along the integral line, the retraction ̺ : X˜ 2ε → X˜ 2ε the restricted ball Br. We construct the implied vineyard by is well defined. By construction, there is a homotopy maintaining the ordering of the simplices and the reduced between the identity on X˜ 2ε and ̺ that moves points by at matrices. However, unlike with relative homology in [2] we most 2ε/µ. We are now ready to state our result. cannot use excision to maintain different orderings of a static complex. We need to be able to handle insertion of sim- STABILITY THEOREM FOR NOISY DOMAINS. Let ˜ Rn R ˜ ˜ plices into the ordering when the power cell of the point z h, h : → with ε = kh − hk∞, h smooth, and expands with r to include new simplices. Fortunately, the the norm of the gradient satisfying k∇h˜k ≥ µ > 0 on new simplices are paired amongst each other so that these X˜ 2ε − X˜ −2ε. Furthermore, let f, f˜ : Rn → R with ˜ updates can be done in linear time per insertion. Once a sim- δ = kf − fk∞ and f Lipschitz with constant κ. Then plex is inserted, its position in the ordering can be described by a continuous function; see Appendices A and B of [2]. dB(Dgm(f˜|X˜ ), Dgm(im f|X−ε →f|Xε )) ≤ 2κε/µ + δ It therefore suffices to maintain the decompositions in the four steps of the algorithm under transpositions of contigu- provided the restrictions of f˜ to X˜ and of f to X˜, Xε, X−ε ous simplices. Details on how to perform these operations are continuous and tame and there exists a triangulation of are given in Appendix A. Xε in which X−ε and X˜ arise as subcomplexes.

5.2 Noisy Domains PROOF. By the Stability Theorem for ordinary persistence we have dB(Dgm(f˜|X˜ ), Dgm(f|X˜ )) ≤ δ. It remains to Persistent homology has proven to be well-suited for dealing show that the bottleneck distance between Dgm(f|X˜ ) and with noisy functions. Indeed, the stability of persistence di- Dgm(im f|X−ε →f|Xε ) is bounded from above by c = agrams implies that the topological features of an unknown 2κε/µ.

10 H X˜ 0 Writing Fa for ( a), the diagrams Dgm(f|X˜ ) is ob- that the µ-reach as recently introduced in [4] is a notion of tained from the sequence formed by the maps Fa → Fb feature size that permits the treatment of non-smoothobjects. X˜ 0 X˜ 0 induced by the inclusion a ⊆ b for all a ≤ b. Simi- Under this assumption, the Stability Theorem for Noisy Do- H X−ε H Xε larly, writing Ja forthe imageof themap ( a ) → ( a), mains implies that it is possible to estimate the persistence the diagram Dgm(im f|X−ε →f|Xε ) is obtained from the se- diagram of a function f˜restricted to the 2ε-offset of S know- quence of maps Ja → Jb again induced by inclusion and for ing only a Lipschitz function f that approximates f˜ and the all a ≤ b. To adaptthe proofofstability givenin [6], we need point set U that samples S. to connect these two sequences by maps φa : Fa−c → Ja To approximate the persistence diagram of f˜ restricted to R 2ε and ψa : Ja−c → Fa, for all a ∈ , in such a way S itself, we exploit the existence of an isotopy ι from S to that the diagram formed by the two sequences together with an arbitrarily small offset Sη of the shape such that the points the new maps commutes. We construct the new maps us- move by less than 2ε/µ during the deformation. The construction of ι is similar to the construction of the homotopy between the identity and ̺ described above. The isotopy im- −1 H X˜ 0 2ε η

( a−c) plies that Dgm(f|S ) equals Dgm(f ◦ ι |S ) which in turn

j j

j η

j is c-close to Dgm(f|S ) by stability. Hence, the latter dia-

Øj Ø gram can also be estimated from f and U with an accuracy

» » » » H X˜ −2ε » H X−ε H X˜ 0 H X H X˜ 2ε

( a ) ( a ) ( a) ( a) ( a ) of 2c. Perhaps surprisingly, Dgm(f|Sη ) may not converge to

j j

j S

j S as goes to ; see [5] for examples of shapes

j Dgm(f| ) η 0

Ø j that lack this convergence property. However, convergence H X˜ 0 ( a+c) holds for sufficiently regular spaces S, such as smooth sub- manifolds or geometrically realized simplicial complexes. In these cases we can estimate Dgm(f|S) with precision 2c. Figure 7: The diagram used to define the maps φa and ψa. The In the more realistic case in which f is only known at a sequence Fa−c → Fa → Fa+c is drawn vertically from top to bottom and Ja can be seen as the image of the composition of two finite set of points, U, a valid approach replaces f by the ¯ horizontal maps. All maps except for the diagonal ones are induced function f that is constant on the Voronoi cells of the points by inclusion. and coincides with f on U. While f¯ is not continuous, it is almost everywhere continuous in a way that does not disrupt ing the homotopy between the identity and the retraction the proof of stability. Furthermore, f and f¯differ by at most ̺ : X˜ 2ε → X˜ 2ε. As mentionedearlier, the homotopymovesa 4κε on h˜−1(−∞, 4ε]. By the Stability Theorem for Noisy point by at most 2ε/µ and since f is Lipschitz with constant Domains, the persistence diagrams of the images defined by X˜ 0 X˜ −2ε X−ε ¯ ¯ κ, ̺ maps a−c to a which is included in a . The in- f and f are close. The diagram for f can be computed using H X˜ 0 H X−ε the alpha shape filtration of U. We thus get a practical al- duced homomorphism from ( a−c) to ( a ) composed H X−ε H Xε gorithm for estimating the persistence diagram of functions with the induced homomorphism from ( a ) to ( a) given only at a finite set of points. gives the map φa. As shown in Figure 7, φa connects the two sequences with a shift of c = 2κε/µ. By a similar pro- cess, we construct the map ψa connecting the two sequences of vector spaces in the other direction and again with a shift 6 Discussion of c. Because ̺ is homotopic to the identity, the diagram In this paper, we consider persistent homology for sequences formed by the two sequences and the maps φ and ψ com- a a of kernels, images, and cokernels defined by a pair of topo- mutes. The remainder of the proof can be adapted directly logical spaces, Y X, and two functions, X R and from [6]. ⊆ f : → g : Y → R, with f(y) ≤ g(y) for every y ∈ Y. Since g ma- jorizes the restriction of f to Y, its sublevel sets are contained −1 −1 in those of f, Ya = g (−∞,a] ⊆ Xa = f (−∞,a], and Diagram approximation. An interesting case of the above we have homomorphisms ja : H(Ya) → H(Xa) induced by theorem arises when we consider a finite set of points, U, the inclusions. To see that persistent homology is well de- S Rn ˜ Rn R sampling an unknown shape, ⊆ . Let h : → be fined we just need to note that the diagrams the distance function of S, that is, h˜(x) = infy∈S kx − yk. Rn R Similarly, let h : → be the distance function of U H(Xa) → H(Xb) and set ε to the Hausdorff distance between S and U. For ˜ technical reasons we may have to replace h by a smooth ap- ↑ ja ↑ jb proximation, for example obtained by convolution with an infinitesimally narrow Gaussian so that the assumptions in H(Ya) → H(Yb) the theorem are satisfied. In this setting, the requirement that the norm of the gradient of h˜ is bounded from below by µ is commute for any a ≤ b and thus induce homomorphic maps equivalent to the µ-reach of S exceeding 4ε. Here we recall ker ja → ker jb, im ja → im jb, and cok ja → cok jb.

11 It is worth noting that the mapping cylinder construction [2] P. BENDICH,D.COHEN-STEINER,H.EDELSBRUNNER,J.HARER described in Section 2 can be used to extend the framework AND D. MOROZOV. Inferring local homology from sampled strat- from inclusion Y X to an arbitrary continuous map ified spaces. In “Proc. 48th Ann. Sympos. Found. Comput. Sci., ⊆ j : 2007”. Y → X. If we have two functions f : X → R, and g : Y R Y X → , such that j( a) ⊆ a with the sublevel set of each [3] G. CARLSSON AND A.ZOMORODIAN. The theory of multidimen- space taken with respect to its own map, then the maps ja : sional persistence. In “Proc. 22nd Ann. Sympos. Comput. Geom., 2007”, 184–193. Ya → Xa between the sublevel sets induce homomorphisms on homology groups just the same. Constructing a mapping cylinder X′ = X ∪ Y × [0, 1] by identifying (y, 0) ∈ Y ×{0} [4] F. CHAZAL,D. COHEN-STEINER AND A.LIEUTIER. A sampling X theory for compact sets in Euclidean space. In “Proc. 22nd Ann. with j(y) ∈ , it is easy to verify that the sequences of Sympos. Comput. Geom., 2006”, 319–326. kernels, images, and cokernels induced by inclusion Ya = Y X′ a × {1} ⊆ a give the same persistence pairing as the [5] F. CHAZAL AND A.LIEUTIER. Stability and computation of topo- Rn three sequences induced by the continuous maps ja defined logical invariants of solids in . Discrete Comput. Geom. 37 (2007), above. 601–617. One can also extend the framework described in this pa- Y X Y Y [6] D. COHEN-STEINER,H.EDELSBRUNNER AND J. HARER. Stabil- per from spaces a ⊆ a to pairs of spaces ( a, a0 ) ⊆ ity of persistence diagrams. Discrete Comput. Geom. 37 (2007), 103– X X ( a, a0 ) using the cone construction exploited for the com- 120. putation of extended persistence [7]. Indeed, observing that H Y Y [7] D. COHEN-STEINER,H.EDELSBRUNNER AND J.HARER. Extend- the relative homology groups ( a, a0 ) are isomorphic to Y Y ing persistence using Poincaréand Lefschetz duality. Found. Comput. the homology groups of a with a cone on a0 rel the cone H H Y Math., to appear. point, i.e. (Ya, Ya0 ) ≃ ( a ∪ CYa0 ,ω), we can compute the persistence of kernels, images, and cokernels induced by [8] D. COHEN-STEINER,H.EDELSBRUNNER AND D. MOROZOV. the inclusions of pairs of spaces by using the algorithms of Vines and vineyards by updating persistence in linear time. In “Proc. this paper on corresponding cones. 22nd Ann. Sympos. Comput. Geom., 2006”, 119–126. The mapping cylinder construction and the cone construction can be combined to cope with arbitrary continuous maps [9] V. DE SILVA AND G. CARLSSON. Topological estimation using witness complexes. In “Proc. Sympos. Point-Based Graphics, 2004”, between pairs of spaces rather than only inclusions. 157–166. The algorithms for computing the persistence diagrams of the sequences of kernels, images, and cokernels are variants [10] H. EDELSBRUNNER,D.LETSCHER AND A.ZOMORODIAN. Topo- of the classic Smith normal form algorithm; see [11, Chapter logical persistence and simplification. Discrete Comput. Geom. 28 1.11]. More directly, we build on the algorithm in [8] which (2002), 511–533. reduces the entire incidence matrix at once, paying careful [11] J. R. MUNKRES. Elements of Algebraic Topology. Addison-Wesley, attention to the orderings of the rows and the columns which Redwood City, California, 1984. are the same and consistent with the ordering of the sublevel sets. The main difficulty in the new setting is that we [12] L. VIETORIS. Uber¨ den höheren Zusammenhang kompakter Räume have two orderings, one for K triangulating X and the other and eine Klasse von zusammenhangstreuen Abbildungen. Math. Ann. for L ⊆ K triangulating Y. We cope using the mapping 97 (1927), 454–472. cylinder construction and matrices in which the rows are re- [13] A. ZOMORODIAN AND G. CARLSSON. Computing persistent ho- ordered so that L precedes K − L. The resulting algorithms mology. Discrete Comput. Geom. 33 (2005), 249–274. run in time at most cubic in the size of K, same as the reduction algorithm in [8] as well as the classic Smith normal form algorithm for modulo-2 arithmetic. The fact that a sim- ple reordering of the rows does the trick suggests that there Appendix A may be other interesting pieces of information that can be extracted from reduced reordered matrices. Does the pairing We consider the actions necessary to maintain the matrix de- defined by the lowest ones in a reduced incidence matrix in compositions under the transposition of two contiguous sim- which columns and rows are ordered independentlyand arbi- plices σi and σi+1. The maintenance of the R = DU de- trarily have an intuitive interpretation that carries topological composition under such transpositions has been considered meaning? in [8] and their algorithm applies directly to the maintenance of Rf = Df Vf and Rg = DgVg. However, maintaining the other three decompositions is more difficult because Dker , References Dim , and Dcok are not ordinary incidence matrices. The algorithm in [8] expresses an update in terms of pre- and [1] D. ATTALI,H.EDELSBRUNNER, J. HARER AND Y. MILEYKO. post-multiplications by idempotent matrices. We observe Alpha-beta witness complexes. In “Proc. 11th Workshop Alg. Data Struct., 2007”, Springer-Verlag, Lecture Notes in Computer Science that multiplying V by the same matrices as R maintains the 4619, 386–397. equality R = DV as well as VU = I.

12 Step 2, the image. Consider ﬁrst the maintenance of If this update is necessary and k precedes l then we have a Rim = Dim Vim which is made difﬁcult by ordering the Type 1 switch in the pairing [8], that is, both transposing sim- rows and columns differently. We distinguish the case in plices are responsible for births of homology classes. If σi is which σi and σi+1 both belong to L or to K − L from the negative and σi+1 is positive in Rf , that is, we are in Case case in which one belongs to L and the other to K − L. 3 of [8], then regardless of whether the pairing switches or ′ In the former case, we let P be the transposition matrix of not, the columns of Dker remain the same. This observation ′ the rows σi and σi+1, and in the latter case we set P = I. follows from the update rule in case the pairing switches. Now we can follow the case analysis in [8] replacing the pre- This leaves the column of the negative simplex to be the sum ′ multiplication of Dim and Rim by P with P . Theonlyother of the transposing columns while leaving the column of the adjustment that we must make is in Case 1 in [8], in which positive simplex intact. The rows of Dker transpose only if both simplices σi and σi+1 are positive. Namely, we do not σi and σi+1 are in L or in K −L. However, this transposition need to check whether a collision is introduced in Rim by has no effect since Dker [i,i +1]= Vker [i,i +1]=1 if and the transposition of rows (Case 1.1 in [8]) if σi and σi+1 do only if there is a switch in the pairing in Rker . As a result, not belong to the same group, since in this case P ′ = I and we have no rows transpose. ′ ′ P Rker = (P Dker )Vker .

Step 3, the kernel. Consider second the maintenance of If σi ∈ K − L, σi+1 ∈ L and the pairing switches then the Rker = Dker Vker . Recall that the matrix Dker is obtained pair disappears from the kernel entirely: the column of Dker from Vim by keeping only the columns of positive simplices that used to represent σi+1 ∈ L now represent the simplex and reordering the rows. If both transposing simplices are σi ∈ K − L. If σi is positive and σi+1 is negative then no positive in Rker then their columns in Vker do not change. switches in the pairing may occur in Rker , so the only con- Therefore, their columns in Dker and in Rker transpose as cern is the row transposition in Dker and Rker if σi and σi+1 well as both their rows and columns in Vker . If as a result are both in L or in K−L. In thelatter case, σi remains paired Vker ceases to be upper-triangular then we can perform an with itself and is unaffected by the row transposition. In the update similar to Case 2.1 of [8] by adding the column of former case, σi+1 can never be the lowest one in a column of σi to the column of σi+1 before the transposition. Using the Rker while σi can only be the lowest one in its own column, i+1 matrix Si for this operation, we get which cannot contain σi+1. Therefore, no changes in pairing are possible, and ′ i+1 ′ i+1 P Rker S P = (P Dker P )(P Vker S P ). i i ′ ′ P Rker = (P Dker )Vker Observe that this operation may render Rker non-reduced is a proper decomposition. We observe that a Type 3 switch which can also happen as a result of a row transposition if σi in the pairing never arises in the analysis of changes to the and σi+1 both belong to L or to K−L. If Rker becomes non- i+1 decomposition in Step 3 of the algorithm. This is to be ex- reduced we can ﬁx it by multiplying again by Si which gives pected since the pairs in the kernel (image persistence is unaffected by Step 3) are always between simplices of the same ′ i+1 i+1 ′ i+1 i+1 dimension. P Rker Si PSi = (P Dker P )(P Vker Si PSi ).

This update may result in a Type 2 switch in the pairing [8], Step 4, the cokernel. Consider ﬁnally the maintenance that is, both transposing simplices are responsible for deaths of Rcok = Dcok Vcok . For reasons that will become ap- of homology classes. If the two transposing simplices are parent shortly, we switch to maintaining the decomposition both negative then there are no columns exchanges in , −1 Dker Dcok = Rcok Ucok , where Ucok = Vcok . We might as , and . However, the rows of and ex- Rker Vker Dker Rker well since the matrix Vcok plays no role in the construc- change if both σi and σi+1 are in L or in K − L. The former tion of the persistence diagrams. It is a curious property of case is easy because the transposing simplices do not con- the algorithm in [8] maintaining D = RU (or equivalently tain the lowest one in their rows. However, in the latter case, R = DV ) that the only time the columns of V correspond- similarly to Case 1.1 of [8], the matrix Rker may become ing to positive simplices can change is in the preprocessing non-reduced. Denote by and (both in ) the simplices σl σk L step of Case 1, namely when σi and σi+1 are both positive in in Rker that have in their columns the lowest ones in the po- R and V [i,i +1]=1. Applied to the decomposition of Df , sitions of simplices σi and σi+1, respectively. We can reduce we add column Vg[i] to Vg[i + 1] whenever this happens. To by adding the preceding column to the succeeding col- Rker offset the resulting change in Dcok , we add the row of σi+1 umn. We thus obtain tothe rowof σi in Ucok , notingthat the two rowsare not necessarily adjacent. We can update Ucok in linear time while P ′R Sk = (P ′D )(V Sk) or ker l ker ker l the equivalent ﬁx to Vcok would require a quadratic num- ′ l ′ l P Rker Sk = (P Dker )(Vker Sk). ber of operations. The only remaining changes to Dcok are

13 transpositions of rows and columns which we handle directly using the algorithm in [8]. It follows that the maintenance of the decomposition Dcok = Rcok Ucok takes linear time per operation.