aa r X i v : . [ m a t h . A T ] M a y Divisive cover
Blaser, Nello ∗ and Brun, Morten † Department of Mathematics, University of BergenMay 29, 2018
Abstract
The aim of this paper is to present a method for computing persistent homologythat performs well at large filtration values. To this end we introduce the conceptof filtered covers. Given a parameter δ with 0 < δ ≤ δ -filtered cover and show that its filtered nerve is interleaved with the ˇCechcomplex. We introduce a particular δ -filtered cover, the divisive cover. The specialfeature of the divisive cover is that it is constructed top-down. If we disregardfine scale structure and X is a finite subspace of Euclidean space, then we obtaina filtered simplicial complex whose size makes computing persistent homologyfeasible. The concept of persistent homology was introduced in Edelsbrunner et al. [2000] andhas since been used in a wide range of applications. The persistent homology of a finitemetric space X can be approached by using several different constructions of filteredsimplicial complexes, such as the ˇCech complex, Vietoris-Rips complex or witnesscomplex. Several approximations of the Vietoris-Rips complex have recently beenproposed to speed up calculations Sheehy [2013], Oudot and Sheehy [2015], Dey et al.[2016].In this paper we construct a new approximation to the ˇCech complex computingpersistent homology down to a predefined threshold that can be chosen arbitrarily. Thecomplexity of our algorithm grows with the ratio between the radius of X and thethreshold. We also present a version with theoretical guarantees on size and time. If X is a subset of d -dimensional Euclidean space then the size of our approximationis bounded by an upper bound that is independent of the cardinality n of X and therequired computation time is linear in n . However the constants are so big that this isno improvement in practice. ∗ [email protected] † [email protected] δ -filtered covers and show that δ -filtered coversare interleaved with the ˇCech filtration. Section 3 introduces divisive covers, a particu-lar class of δ -filtered covers. Complexity estimates for divisive covers are presented inSection 4. In Section 5 we show how divisive covers can be applied to synthetic and toreal world data sets and in Section 6 we discuss our results. We introduce the notions of a filtered and δ -filtered cover of a bounded metric space.Throughout this section X = ( X , d ) will be a fixed but arbitrary bounded metric space.First recall the definition of a cover. Definition 2.1. A cover of X is a set U of subsets of X such that every point in X iscontained in a member of U .Recall that a simplicial complex K consists of a vertex set V and a set K of subsetsof V with the property that if σ is a member of K and if τ is a subset of σ , then τ is a member of K . Also recall that every simplicial complex K has an underlyingtopological space | K | . The book Lee [2011] may serve as gentle introduction andreference to abstract simplicial complexes. Definition 2.2.
Let U be a cover of X . The nerve N ( U ) of U is the simplicialcomplex with vertex set U defined as follows: A finite subset σ = { U , . . . , U n } of U is a member of N ( U ) if and only if the intersection of U ∩ · · · ∩ U n is non-empty.Note that the nerve construction U N ( U ) is functorial in the sense that if U ⊆ V is an inclusion of covers of X , then we have an induced inclusion N ( U ) ⊆ N ( V ) ofnevers.If B ⊆ A is an inclusion of partially ordered sets, we say that B is cofinal in A iffor every a ∈ A , there exists b ∈ B so that a ≤ b . Given a cover U , we consider it as apartially ordered set with partial order given by inclusion. We will need the followingresult several times: Lemma 2.3. If U ⊆ V are covers of X and if U is cofinal in V , then the geometricrealization of the inclusion N ( U ) ⊆ N ( V ) is a homotopy equivalence.Proof. Since U is cofinal in V there exists a map f : V → U such that V ⊆ f ( V ) for all V ∈ V . Note that the formula N f ( { V , . . . , V n } ) = { f ( V ) , . . . , f ( V n ) } definesa simplicial map N f : N ( V ) → N ( U ) . Similarly, the inclusion i : U → V induces asimplicial map Ni : N ( U ) → N ( V ) . 2ince V ⊆ f ( V ) for all V ∈ V , the composite N f ◦ Ni is contiguous with the identitymap on N ( U ) in the sense that for every face σ of N ( U ) , the set σ ∪ ( N f ◦ Ni ( σ )) is aface of N ( U ) . It follows that the geometric realization of N f ◦ Ni is homotopic to theidentity on the geometric realization of N ( U ) [Spanier, 1966, Lemma 2 p. 130]. Simi-larly the geometric realization of Ni ◦ N f is homotopic to the identity on the geometricrealization of N ( V ) .We are now ready to define and establish some basic properties of filtered basesand filtered nerves. Definition 2.4. A filtered basis of X is a basis U for the metric topology on X withthe property that X is a member of U . Given t > U t for the cover of X consisting of members of U contained in an open ball in X of radius t . Definition 2.5.
Let U be a filtered basis of X . The filtered nerve of U is the collection { N ( U t ) } t > together with the inclusions N ( U s ) ⊆ N ( U t ) induced by the inclusions U s ⊆ U t .Since X is bounded there exists T > N ( U t ) = N ( U ) for t ≥ T . Definition 2.6.
Let U be a filtered basis of X and let δ be a parameter satisfying0 < δ ≤
1. We say that U is a δ -filtered basis of X if for every x ∈ X and every r > A of U r containing B ( x , δ r ) . Example 2.7.
The ˇCech cover C = ˇ C ( X ) consisting of all balls in X is 1-filtered. Example 2.8.
Let 0 < δ < x ∈ X and R > X is contained inthe open ball B ( x , R ) . We claim that the subset U = ˇ C ( X , δ ) of the ˇCech cover ˇ C ( X ) consisting of balls of radius δ k R , where k is a nonnegative integer, is a δ -filtered basisof X . Indeed, let k be the nonnegative integer with δ k + R ≤ r ≤ δ k R . Since δ r ≤ δ k + R ,the set B ( p , δ r ) is contained in the member B ( p , δ k + R ) of U δ k + R for every p ∈ X . Wecan finish the argument by noting that since δ k + R ≤ r the cover U δ k + R is a subcoverof U r .We now introduce some notation regarding persistent homology. For the rest of thissection F will denote a fixed but arbitrary field.A persistence module V = ( V t ) t > consists of a F -vector space V t for each positivereal number t together with homomorphisms V s < t : V s → V t for s < t . These homomorphisms are subject to the condition that V s < t ◦ V r < s = V r < t whenever r < s < t . Given λ , λ ≥
1, two persistence modules V and W are multiplica-tively ( λ , λ ) -interleaved if there exist F -linear maps f t : V t → W λ t and g t : W t → V λ t for all real numbers t such that for all s < t the following relations hold f λ t ◦ g t = W t < λ λ t , g λ t ◦ f t = V t < λ λ t , g t ◦ W s < t = V λ s < λ t ◦ g s and f t ◦ V s < t = W λ s < λ t ◦ f s . K , we write H ∗ ( K ) for the homology of K with coeffi-cients in the field F .The following example justifies working with the intrisic ˇCech complex instead ofthe relative ˇCech complex. Example 2.9.
Let X be a subspace of a metric space M , let C ( X ) be the filtered basisfrom Example 2.7. Let C ( X , M ) be the relative ˇCech cover consisting of balls in M with center in X , that is, with C ( X , M ) t consisting of balls in M with center in X ofradius at most t .The homology of the intrinsic ˇCech chain complex C ∗ ( X ) t consisting of linearcombinations of subsets σ ⊆ X with the property that σ ⊆ B ( x , t ) for some x ∈ X isisomorphic to the homology of N ( C ( X ) t ) . Similarly, the homology of the ambientˇCech chain complex C ∗ ( X , M ) t consisting of linear combinations of subsets σ ⊆ X with the property that σ ⊆ B ( p , t ) for some p ∈ M is isomorphic to the homology of N ( C ( X , M ) t ) . By construction C ∗ ( X ) t ⊆ C ∗ ( X , M ) t , and by the triangle inequality C ∗ ( X , M ) t ⊆ C ∗ ( X ) t . Thus, the persistent homology of N ( C ( X )) is ( , ) -interleavedwith the persistent homology of N ( C ( X , M )) .By the Nerve Theorem [Hatcher, 2002, Corollary 4G.3], if all non-empty intersec-tions of balls in M are contractible, the geometric realization of the nerve N ( C ( X , M ) t ) of the cover C ( X , M ) t , consisting of balls in M with center in X of radius at most t , ishomotopy equivalent to the union of all balls in M of radius t with center in X . This isthe interior of the t -thickening of X in M . Theorem 2.10 (Relationship between δ -filtered basis and ˇCech complex) . Let C be theˇCech cover from Example 2.7, let U be a δ -filtered basis of X and N ( C ) and N ( U ) be their filtered nerves. Then the persistent homology of N ( U ) is multiplicatively ( , / δ ) -interleaved with the persistent homology of N ( C ) .Proof. By definition, the partially ordered set C r is cofinal in C r ∪ U r and U r is cofinalin C δ r ∪ U r . Thus, by Lemma 2.3, the homology H ∗ ( N ( C r )) is isomorphic to the ho-mology H ∗ ( N ( C r ∪ U r )) and the homology H ∗ ( N ( U r )) is isomorphic to the homology H ∗ ( N ( C δ r ∪ U r )) . Now the result follows from functoriality of the nerve constructionby considering the composites C δ r ∪ U r ⊆ C r ∪ U r ⊆ C r ∪ U r / δ and C r ∪ U r ⊆ C r ∪ U r / δ ⊆ C r / δ ∪ U r / δ . An easy diagram chase now gives:
Corollary 2.11.
If t > can be chosen so that in the situation of Theorem 2.10, the F -linear maps H ∗ ( N ( C δ t < t )) and H ∗ ( N ( C t < t / δ )) are both isomorphisms, then H ∗ ( N ( C t )) is isomorphic to the image of the homomorphism H ∗ ( N ( U t < t / δ )) . In [Chazal and Lieutier, 2005, Theorem 1] it is shown that if X is open in M = R d ,then the conditions of Corollary 2.11 are satisfied when 2 t / δ is smaller than the weak4eature size of X . In Chazal and Oudot [2008] these considerations have been extendedto similar results when X is a finite subset of a compact subset M of R d . Moreover,[Cohen-Steiner et al., 2005, Homological Inference Theorem] shows similar results forthe homological feature size of X .Next we introduce δ -filtered covers, which do not require the cover to be a basis. Definition 2.12.
Let U be a cover of X , and δ and r be parameters satisfying 0 < δ ≤ r ≥
0. We say that U is a δ -filtered cover of X of resolution r if there exists a filteredbasis V such that U s is cofinal in V s for all s ≥ r. Corollary 2.13.
Let X be a bounded metric space, r ≥ and U and V be as inDefinition 2.12. Then the persistent homology of N U t and the persistent homology ofN V t are isomorphic for t ≥ r.Proof. This is a direct consequence of Lemma 2.3.
In this section we discuss an algorithm to construct a δ -filtered cover of a boundedmetric space X . First it divides X into two smaller sets. It continues by dividing thebiggest of the resulting two sets into two, and then iteratively divides the biggest of theremaining sets in two.In order to describe the algorithm, we first define diameter and relative radius of asubset of a metric space. Definition 3.1.
Let X be a metric space and let Y be a subset of X .1. The diameter of Y is defined as d ( Y ) = sup { d ( y , y ) | y , y ∈ Y } .
2. The radius of Y relative to X is defined as r ( Y ) = inf { r > | Y ⊆ B ( x , r ) for some x ∈ X } Definition 3.2. A δ -division of a subset Y of radius r relative to a bounded metricspace X consists of a cover { Y , Y } of Y consisting of proper subsets of Y with theproperty that for every y ∈ Y the intersection Y ∩ B ( y , δ r ) is contained in at least one ofthe sets Y and Y . Definition 3.3.
Let X be a metric space. A δ -divisive cover of X of resolution r ≥ U of X containing X and a δ -division { Y , Y } of every Y ∈ U of radius r ( Y ) > r . Lemma 3.4.
Let U be a δ -divisive cover of resolution r ≥ of a bounded metric spaceX. If every non-empty subset of U has a minimal element with respect to inclusion,then U is a δ -filtered cover of resolution r. roof. Let x ∈ X and let s > r . Let Y ∈ U be minimal under the condition that B ( x , δ s ) ⊆ Y . Suppose that r ( Y ) > s and let { Y , Y } be a δ -division of Y containedin U . Since B ( x , δ s ) ⊆ B ( x , δ r ( Y )) we have that B ( x , δ s ) is contained in either Y or Y and Y and Y are proper subsets of Y . This contradicts the minimality of Y . Corollary 3.5. If U is a finite δ -divisive cover of X, then U is a δ -filtered cover. There exist many ways to construct δ -divisions. Here is an elementary one: Lemma 3.6.
Let Y be a subset of a bounded metric space X and suppose that y and y are points in Y of maximal distance. Given δ with < δ < / let f = ( − δ ) / ( + δ ) and let Y consist of the points y ∈ Y satisfying f d ( y , y ) ≤ d ( y , y ) . Similarly, let Y consist of the points y ∈ Y satisfying f d ( y , y ) ≤ d ( y , y ) . Then { Y , Y } is a δ -divisionof Y .Proof. Let x ∈ X and let r = r ( Y ) be the relative radius of Y . By symmetry we maywithout loss of generality assume that d ( x , y ) ≤ d ( x , y ) . We will show that if z ∈ B ( x , δ r ) ∩ Y , then z ∈ Y , that is, that f d ( z , y ) ≤ d ( z , y ) . Since the radius of Y issmaller than or equal to the diameter d ( y , y ) of Y it suffices to show that d ( x , z ) ≤ δ d ( y , y ) implies that f d ( z , y ) ≤ d ( z , y ) . However, since d ( y , y ) ≤ d ( y , x ) + d ( x , y ) ≤ d ( x , y ) , we have d ( z , y ) ≤ d ( z , x ) + d ( x , y ) ≤ δ d ( y , y ) + d ( x , y ) ≤ ( δ + ) d ( x , y ) and d ( x , y ) ≤ d ( x , z ) + d ( z , y ) ≤ δ d ( y , y ) + d ( z , y ) ≤ δ d ( x , y ) + d ( z , y ) Since f = ( − δ ) / ( + δ ) this gives f d ( z , y ) ≤ ( − δ ) d ( x , y ) ≤ d ( z , y ) . Given a bounded metric space X , a method for δ -division and r ≥
0, we construct inAlgorithm 1 a δ -divisive cover U r of X of resolution r . Thus the persistent homologyof ( U r ) s ≥ r is δ -interleaved with the persistent homology of ( C ) s ≥ r . For the study of complexity of Algorithm 1 we will restrict attention to the situationwhere X is a finite subset of R d with the L ∞ -metric d ∞ . For 1 ≤ i ≤ d , we let pr i : R d → R be the coordinate projection taking ( v , . . . , v d ) ∈ R d to v i .6 lgorithm 1: Divisive cover algorithm
Input :
A bounded metric space X , a method for δ -division and r ≥ Output: A δ -divisive cover U r of XX = X Create list L = { } i = while There exists a j ∈ L such that r ( X j ) > r do k = argmax j ∈ L { diameter of X j } Construct a δ -division ( X i + , X i + ) of X k remove k from L and add i + i + Li = i + end U r = { X , X , . . . , X i } Lemma 4.1 (Decision division) . Let X be a finite subset of R d equipped with the L ∞ -metric d ∞ and let x and x be points in X of maximal distance. Choose a coordinateprojection pr i so that d ∞ ( x , x ) = | pr i ( x − x ) | . Given δ with < δ < let X consistof the points x ∈ X satisfying | pr i ( x − x ) | ≤ + δ | pr i ( x − x ) | . Similarly, let X consistof the points in x ∈ X satisfying | pr i ( x − x ) | ≤ + δ | pr i ( x − x ) | . Then ( X , X ) is a δ -division of X.Proof. Let p ∈ X and let r be the relative radius of X . Note that d ( x , x ) = r inthe situation of the asserted statement. We have to show that the intersection of X with the ball centered in p of radius δ r is contained in one of X and X . Let us forconvenience write y = pr i ( x ) and y = pr i ( x ) and let us assume that y < y . Itsuffices by construction to show that the interval [ pr i ( p ) − r , pr i ( p ) + r ] is contained inone of the intervals [ y , y + ( + δ )( y − y ) / ] and [ y − ( + δ )( y − y ) / , y ] . Thisfollows from the fact that the intersection [ y − ( + δ )( y − y ) / , y + ( + δ )( y − y ) / ] of these intervals has length δ ( y − y ) = r δ . Theorem 4.2.
Let X be a finite subset of R d equipped with the L ∞ -metric d ∞ and lett > . If X has cardinality n, then the cover V of X obtained from Algorithm 1 isconstructed in O ( kd dn ) time, where k = ⌈ log + δ ( t / r ) ⌉ . The size of the cover V is atmost kd . The nerve of V can be constructed in O ( kd dn ) time. Note that for fixed d and δ , the term 2 kd is polynomial in the ratio r / t between theradius r of X and the threshold radius t .Let V be as in Theorem 4.2. Given s ≥ t we write V s for the cover of X given bymembers of V of radius less than s . By construction, for s ≥ t , the inclusion of V s in U s is cofinal. Thus by Lemma 2.3, for filtration values greater than t , the persistenthomology of the cover V coincides with the persistent homology of U . Proof of Theorem 4.2.
Note that in the L ∞ -metric, the radius of a subset of R d is givenby the maximum of the radii of its coordinate projections to R . A δ -decision division(4.1) reduces the radius of this coordinate projection by the factor + δ . Thus the radius7f any d -fold δ -divided part of X is at most r (cid:16) + δ (cid:17) , where r is the radius of X . Ifwe let k = ⌈ log + δ ( t / r ) ⌉ , then the radius of any kd -fold δ -divided part of X is at most r ( + δ ) k ≤ t . Since each δ -decision division consists of two parts, we conclude that V can be produced by making at most 2 kd δ -decision divisions. Since we work in the L ∞ metric, extremal points can be found by computing min- and max-values for thecoordinate projections of points in X . Similarly δ -decision division can be made bycomputing min- and max-values for the coordinate projections of points in X . Eachof these steps require O ( nd ) time, so the cover is of size at most 2 kd and it can beconstructed in O ( kd nd ) time.Finally, the nerve of the cover V is constructed by calculating intersections of mem-bers of V . Calculating the intersection of i ≤ d subsets of X can be done by, for eachelement x of X , deciding if x is a member of the intersection. The complexity of thisis O ( ni ) . Since the cardinality of V is at most 2 kd , independently of n , the time ofcalculating the nerve is O ( kd n ) .We shall use the following result to show that a δ -decision division of X ⊆ R d gives a d − / p δ -divisive cover of X in the L p -metric. This stems from the fact that all L p -metrics are equivalent. Proposition 4.3.
Let d and d be metrics on X and let α and β be positive numberssuch that for all x , y ∈ X the inequality α d ( x , y ) ≤ d ( x , y ) ≤ β d ( x , y ) holds. Then every δ -filtered cover of ( X , d ) is a δα / β -filtered cover of ( X , d ) .Proof. We emphasize the metrics d and d in the notation by writing U d t and U d t for the covers of X consisting of members of U contained in a closed ball of radius t in ( X , d ) and ( X , d ) respectively.By assumption, there are inclusions of balls B d ( x , t / β ) ⊆ B d ( x , t ) ⊆ B d ( x , t / α ) , so U d t / β ⊆ U d t ⊆ U d t / α . Given a point x ∈ X and a radius t >
0, we can find a set A ∈ U d t / β such that B d ( x , t δ / β ) ⊆ A since U is δ -filtered in ( X , d ) . Due to the above inclusions, A is also in U d t and B d ( x , t δα / β ) ⊆ B d ( x , t δ / β ) ⊆ A . Thus U is an δα / β -filteredcover of ( X , d ) .In the case where d is the L ∞ -metric and d is the L p -metric on R d the inequalitiesin Proposition 4.3 hold for α = β = d / p . Thus, if U is a δ -filtered cover of X with respect to the L ∞ -metric, then it is a d − / p δ -filtered cover of X with respect to the L p -metric. In particular it is δ / √ d -filtered with respect to the Euclidean metric.8 Examples
We used divisive cover with the δ -division of Lemma 3.6 to calculate the persistenthomology of a generated sphere. We generated 1000 data points with a radius normallydistributed with a mean of 1 and a standard deviation of 0.1 and uniform angle. Thetop panel of Figure 5.1.2 shows the resulting persistence barcodes. We calculated the persistent homology of a generated torus using divisive cover withthe δ -division of Lemma 3.6. We generated 400 data points on a torus. The toruswas generated as the product space of 20 points each on two circles of radius 1 withuniformly distributed angles. The second panel of Figure 5.1.2 shows the persistencebarcodes of the generated torus. The space of 3 by 3 high-contrast patches of natural images has been analysed usingwitness complexes before [Carlsson et al., 2008]. The authors analysed high-densitysubsets of 50,000 random 3 by 3 patches from a collection of 4 × patches presentedin Hateren and Schaaf [1998]. They denote the space X ( k , p ) of p percent highest den-sity patches using the k -nearest neighbours to estimate density and find that X ( , ) has the topology of a circle. We repeat this analysis using divisive cover with the δ -division of Lemma 3.6 and show that calculating persistent homology without land-marks is possible for real world data sets. The bottom panel of Figure 5.1.2 show thepersistence barcodes of X ( , ) . Filtered covers as the underlying structure for filtered complexes provides new insightsinto topological data analysis. It can be used as a basis for new constructions of sim-plicial complexes that are interleaved with the ˇCech nerve. We are not aware of anyprevious literature that made use of covers in such a way. Divisive covers are just onepossible way to create δ -filtered covers. Many other constructions are available, forexample optimized versions of the δ -filtered ˇCech cover we have presented.The idea of a divisive cover is conceptually simple and easy to implement. Com-pared to the ˇCech nerve, the nerve of a divisive cover can be substantially smaller.On the other hand, the witness complex is often considerably smaller than the divisivecover complex. Although we give theoretical guarantees that are linear in n , in prac-tice persistent homology calculations using the divisive cover algorithm proposed hereare not competitive with state of the art approximations to the Vietoris-Rips complexOudot and Sheehy [2015], Dey et al. [2016]. We see divisive covers as a new class of9 .00 0.25 0.50 0.75 1.00 Diameter Dimension
Diameter Dimension
Diameter Dimension
Figure 1: Persistence barcodes using divisive cover. All barcodes are shown for relativediameter between 0 . δ = .
05 and the second panel shows persistence barcodes ofa torus with δ = .
06. The third panel shows persistence barcodes of X ( , ) , with δ = .
025 10implicial complexes that can be studied in a fashion similar to Vietoris-Rips filtra-tions. It is possible to reduce the size of the divisive cover complex, for example byusing landmarks. We did not address such improvements in the present paper. It mightalso be possible to combine a version of the Vietoris-Rips complex for low filtrationvalues and a version of the divisive cover complex for high filtration values. The ver-sion of divisive cover we have presented is easy to implement and performs well atlarge filtration values.
Acknowledgements
This research was supported by the Research Council of Norway through grant 248840.
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