Persistent Homology in ℓ ∞ Metric
PPersistent Homology in (cid:96) Metric
Gabriele Beltramo ˚ , Primoz Skraba School of Mathematical SciencesQueen Mary University of London, London, E1 4NS
Abstract
Proximity complexes and filtrations are a central construction in topological data analysis. Built usingdistance functions or more generally metrics, they are often used to infer connectivity information from pointclouds. We investigate proximity complexes and filtrations built over the Chebyshev metric, also known asthe maximum metric or (cid:96) metric, rather than the classical Euclidean metric. Somewhat surprisingly, the (cid:96) case has not been investigated thoroughly. In this paper, we examine a number of classical complexesunder this metric, including the ˇCech, Vietoris-Rips, and Alpha complexes. We define two new families offlag complexes, which we call the Alpha flag and Minibox complexes, and prove their equivalence with ˇCechcomplexes in homological dimensions zero and one. Moreover, we provide algorithms for finding Miniboxedges for two, three and higher dimensional points. Finally we run computational experiments on randompoints, which show that Minibox filtrations can often be used to reduce the number of simplices included inˇCech filtrations, and so speed up persistent homology computations. Keywords:
Topological data analysis, Persistent homology, Chebyshev distance, Delaunay triangulation
1. Introduction
Topological data analysis (TDA) has been the subject of intense research over the last decade [1, 2, 3].Persistent (co)homology is by far the most popular and studied algebraic invariant considered in TDA.Informally, this is an invariant assigned to a filtration – an increasing sequence of spaces. In our setting, anatural filtration arises from the sub-level sets of the distance to a finite sample of a space under consideration.Most commonly, the finite sample is on or near a manifold embedded in Euclidean space, R d . In thestandard Euclidean setting, the ˇCech and the Alpha filtrations [4, 5, 6] directly capture the topology of thecorresponding sub-level sets. Relatedly, the Vietoris-Rips filtration [7] provides an approximation to thistopology. In particular, the corresponding filtrations in Euclidean space may be related via a sandwichingargument [8].In this paper we study the ˇCech persistent homology of a finite set of points S in (cid:96) metric space.Given n points in d -dimensional space, the ˇCech filtration has Θ p n d ` q simplices. In the Euclidean setting,the number of simplices to be considered can be reduced by using Alpha filtrations, restricting simplicesto those of the Delaunay triangulation of S . Furthermore, the Alpha filtration is known to carry the sametopological information as the ˇCech filtration (via homotopy equivalence ). On the other hand, we will seethat the structure of (cid:96) -Voronoi regions makes Alpha filtrations an unsuitable candidate for the computationof general ˇCech persistence diagrams. Moreover, (cid:96) -Voronoi regions and their dual Delaunay triangulationshave been studied primarily from a geometric standpoint and/or in low dimension [9, 10]. To overcome someof these limitations, we define two novel families of complexes: Alpha flag complexes and Minibox complexes. ‹ This work was partially funded by the School of Mathematical Sciences at Queen Mary University of London and by theSSHRC-NFRF and DSTL/Turing Institute grant DS-015. ˚ Corresponding author
Email addresses: [email protected] (Gabriele Beltramo), [email protected] (Primoz Skraba)
Preprint submitted to COMP GEOM-THEOR APPL October 9, 2020 a r X i v : . [ c s . C G ] O c t hese are both flag complexes defined on a subset of the edges of ˇCech complexes. Our contributions can besummarised as follows: • Under genericity assumptions, i.e. general position, we prove that Alpha complexes are equivalentto ˇCech complexes for two-dimensional points, i.e. filtrations built with these complexes produce thesame persistence diagrams. Moreover, we give a counterexample to this equivalence for points inhigher-dimensions. • For arbitrary dimension, we prove the equivalence of Alpha flag and Minibox complexes with ˇCechcomplexes in homological dimensions zero and one. • We give algorithms for finding edges contained in Minibox complexes for two, three, and higher-dimensional points. In two dimensions, using a sweeping algorithm, we show a running time bound of O p n q (which is optimal). In three dimensions, we achieve a bound of O p n log p n qq by extending thetwo-dimensional algorithm. In higher dimensions, using orthogonal range queries, we achieve a runningtime of O p n log d ´ p n qq . • We show that for randomly sampled points in R d the expected number of Minibox edges is proportionalto Θ ´ d ´ p d ´ q ! n log d ´ p n q ¯ . This is an improvement over the quadratic number of edges contained in ˇCechcomplexes, and results in smaller filtrations. Interestingly, this implies that Minibox complexes are onlya polylogarthmic factor larger than (cid:96) -Delanauy complexes for random points. • We provide experimental evidence for speed ups in computation of persistence diagrams by means ofMinibox filtrations.While there is not as large a body of work on complexes in (cid:96) metric, as there is for (cid:96) metric, there areseveral relevant related works. In particular, approximations of (cid:96) -Vietoris-Rips complexes are studied in[11]. Moreover, the equivalence of the different complexes in zero and one homology is related to the resultsof [12]. In this work offset filtrations of convex objects in two and three-dimensional space are considered.As in our case, an equivalence of filtrations is proven in homological dimensions zero and one by restrictingoffsets with Voronoi regions. While this result holds for general convex objects, Minibox filtrations can beused to reduce the size of (cid:96) - ˇCech filtration in dimensions higher than three. We also note that our approachis similar in spirit to the preprocessing step via collapses of [13], but works directly on the geometry of thegiven finite point set S . Finally, we remark that all of our results hold for finite embedded points in (cid:96) metricas well. As mentioned for the two-dimensional case in [9], a finite set of points can be preprocessed so as totransform (cid:96) -balls into (cid:96) -balls. Outline.
After introducing background information in Section 2, we study Alpha complexes and their prop-erties in the (cid:96) setting in Section 3. In Section 4 and 5, we introduce our two new families of flag complexes,and present the proofs of their equivalence with ˇCech complexes in homological dimensions zero and one.Algorithms for finding Minibox edges, and results on worst-case and expected number of Minibox edges aregiven in Section 6. Finally, in Section 7 we expose the results of computational experiments using Alpha flagand Minibox complexes.
2. Preliminaries
We introduce the relevant definitions we will use in later sections. While we aim for completeness, wepoint the reader to [14] for a more detailed introduction to homology and persistent homology [15, 16].
Simplicial Complexes.
In this work we limit ourselves to simplicial complexes built on a finite set of pointsin R d . We denote the simplicial complex by K . We say that K is a flag complex if it is the clique complexof its 1-skeleton, i.e. it contains a simplex σ if and only if it contains all the one-dimensional faces of σ . Wenow introduce several constructions we will use. Let τ denote a simplex in K .2 a) (b) (c) Figure 1: (a)
Four points in R whose Delaunay complex is three-dimensional. (b) Degenerate intersection of (cid:96) -Voronoiregions. (c) (cid:96) -Voronoi regions are not convex. • The nerve of a finite collection of closed sets t A i u i P I in R d is the abstract simplicial complex Nrv pt A i u i P I q “ (cid:32) σ Ď I | Ş i P σ A i ‰ H ( . • The star of τ in K is the subset of simplices of K defined by St p τ q “ t σ P K | τ ď σ u . • The closed star Cl p St p τ qq of τ in K is the smallest subcomplex of K containing St p τ q . • The link of τ in K is Lk p τ q “ t σ P Cl p St p τ qq | τ X σ “ Hu . Balls and Boxes in (cid:96) Metric.
Given p, q P R d , the (cid:96) distance, also known as maximum distance orChebyshev distance, is defined by d p p, q q “ max ď i ď d t| p i ´ q i |u . Given a point p P p R d , d q and r ě
0, the open ball of radius r and center p is B r p p q “ t x P R d | d p x, p q ă r u .We denote the closed ball of radius r and center p by B r p p q , its boundary by B B r p p q . We have ε p B r p p qq “ B r ` ε p p q , where ε p A q “ t x P R d | d p x, A q ď ε u is the ε -thickening of A Ď R d . Moreover, an open ball B r p p q consists of the points x satisfying the constraints p i ´ r ă x i and x i ă p i ` r for 1 ď i ď d . Thus B r p p q “ ś di “ I pi , where I pi “ p p i ´ r, p i ` r q , which is the interior of an axis-parallel hypercube centered at p , with sides of length 2 r . In general we call any such Cartesian product of intervals a d -dimensional box .Here we recall two properties of boxes, which we often refer to in the rest of the paper. Their proofs followdirectly from the facts that Cartesian products and intersections commute, and that the intersection of afinite number of intervals is either empty or an interval. Proposition 2.1.
Let B be a finite collection of either open or closed boxes in R d . (i) The intersection of the boxes in B is equal to the Cartesian product of the intersections of intervalsdefining these boxes. So this intersection is either empty or a box. (ii) The intersection of any subset of boxes in B is non-empty if and only if all the pairwise intersectionsof these boxes are non-empty.Voronoi diagrams and Delaunay triangulations. These constructions have been extensively studied in com-putational geometry [17], primarily for Euclidean space. We refer the reader to [18] for a reference for generalVoronoi diagrams and Delaunay triangulations.
Definition 2.2.
Let S be a finite set of points in p R d , d q . The (cid:96) - Voronoi region of a point p P S is V p “ ! x P R d | d p p, x q ď d p q, x q , @ q P S ) . The set of (cid:96) -Voronoi regions t V p u p P S is the (cid:96) - Voronoi diagram of S .3 igure 2: Alpha filtration of points in R . Definition 2.3.
The
Delaunay complex of a finite set of points S is the simplicial complex K D “ ! σ Ď S | č p P σ V p ‰ H ) . The structure of intersections of Voronoi regions defined using general polyhedral norms may be degener-ate. For instance, as in the Euclidean case, d ` R d can have (cid:96) -Voronoi regions with a non-emptyintersection. This is illustrated by the four points in R of Figure 1a. Moreover, without assuming any hy-pothesis on S the intersection of two (cid:96) -Voronoi regions may be a d -dimensional subset of R d . For example,given any two collinear points p and q in R , V p X V q is the union of a line segment and two cones, the shadedareas in Figure 1b. To avoid such cases, we can impose conditions on S . In [19] the structure of Voronoiregions of polyhedral distances is studied in light of different types (weak and strong) of general positionassumptions. We give different definitions of this concept in different dimensions. Definition 2.4.
Let S be a finite set of points in p R d , d q . We say that S is in general position if thedistances between pairs of points in S are all distinct. Moreover, for d “
2, we require no four points in S tolie on the boundary of a square, no three points to be collinear, and no two points to have the same x or y coordinate.We impose more conditions on points in dimension two, so that in this case S is in weak general position.This guarantees that the intersection of three Voronoi regions is either empty or a point, by [19, Corollary3.18]. In Section 3, we make use of this result in our discussion of Alpha complexes in (cid:96) metric space.Furthermore we give the following definition. Definition 2.5.
The
Delaunay triangulation of a finite set of points S in general position in p R , d q is thegeometric realisation of the Delaunay complex K D of S , which is the set of convex hulls of simplices of K D .Finally, we conclude by noting that (cid:96) -Voronoi regions are not generally convex. To see this, we consider p “ p , q and q “ ` , ˘ and the intersection of their (cid:96) -Voronoi regions, as in Figure 1c. These are such that z “ ` , ˘ , z “ ` , ˘ P V p , V q , but the middle point on the line segment from z to z is z ` z “ ` , ˘ which belongs to V p only. Thus V q is not convex, so that the standard way of proving the equivalence ofAlpha and ˇCech filtrations does not work in (cid:96) metric. We study this problem in Section 3. Persistent Homology. A filtration of a simplicial complex K parameterized by R is a nested sequence ofsubcomplexes K R “ t K r Ď K r Ď . . . Ď K r m u , where R “ t r i u mi “ a finite set of monotonically increasingreal values. We list three types of complexes used to define filtrations on a finite set of points S . • The
Vietoris-Rips complex with radius r of S is K V Rr “ (cid:32) σ Ď S | max p,q P S d p p, q q ď r ( . • The ˇCech complex with radius r of S is K ˇ Cr “ (cid:32) σ Ď S | Ş p P σ B r p p q ‰ H ( . • The
Alpha complex with radius r of S is K Ar “ (cid:32) σ Ď S | Ş p P σ ` B r p p q X V p ˘ ‰ H ( .4e note that for each of the complexes above we have K ‚ r Ď K ‚ r if r ă r . So given a monotonicallyincreasing set of real values R , we have the filtration K ‚ R “ t K ‚ r Ď K ‚ r Ď . . . Ď K ‚ r m u . An exampleillustrating a sequence of three nested Alpha complexes of points in R is given in Figure 2. Besides wenote that from the definition of ˇCech and Vietoris-Rips complexes and Proposition 2.1 (ii) we have that K ˇ Cr “ K V Rr for any r P R . Thus in (cid:96) metric ˇCech complexes are flag complexes and the filtration parameterof any ˇCech simplex σ is ¯ r “ max p,q P σ d p p,q q .Given a filtration K R , we obtain the k -th persistence module M k p K R q “ t H k p K r ; F q Ñ H k p K r ; F q Ѩ ¨ ¨ Ñ H k p K r m ; F qu by applying the k -th homology functor H k p´ ; F q , with coefficients in a field F , to itselements. This admits a unique decomposition, as shown in [20], which is in bijiection with a set of intervalsof the form r r i , r j q and r r i , `8q . Mapping these intervals into the points p r i , r j q and p r i , `8q , we obtain the k -th persistence diagram Dgm k p K R q of the filtration K R . This is a multi-set of points in the extended plane R , where R “ R Yt`8u . The bottleneck distance between two persistence diagrams Dgm k p K R q , Dgm k p K R q is d B p Dgm k p K R q , Dgm k p K R qq “ inf η : X Ñ Y sup x P X d p x, η p x qq , where the infimum is taken over the set of all possible bijections η : X Ñ Y , X “ Dgm k p K R q Y D , Y “ Dgm k p K R q Y D , and D is the set of diagonal points counted with infinite multiplicity. Importantly, theStability Theorem of [21] implies that the persistence diagrams of the ˇCech filtrations obtained from S and S are close in bottleneck distance if the finite point sets are close in Hausdorff distance.In practice, the persistent homology algorithm, first described in [22], takes a filtration, and outputsits persistence diagrams up to a fixed homological dimension. A substantial amount of work has beendone on the computational complexity of computing persistent homology, with a large number of results[23, 24, 25, 26, 27, 28, 29] and heuristics [30, 31, 32, 33] which have greatly sped up computations in practice[34]. The standard algorithm has a complexity of O p m q , which can be reduced to O p m ω q where m is thenumber of simplices in the input filtrations and ω is the matrix multiplication exponent. However, it has beenobserved that the majority of computation time is spent constructing the filtration. Thus smaller complexesgenerally result in faster computation. For instance, in the case of ˇCech filtrations, we have to consider Θ p n k ` q simplices in order to compute their k -th persistence diagram. In Euclidean metric ˇCech persistenthomology can be computed using Alpha filtrations, which greatly reduces the number of simplices to beconsidered. In the next section, we study how this translates in the (cid:96) metric space case.
3. Alpha Complexes
Given a finite set of points S in Euclidean space, it is know that the filtration K A R of produces the samepersistence diagrams of K ˇ C R . This is proven in [15, Section 3.4] by means of the Nerve Theorem. This appliesbecause K Ar is the nerve of the collection (cid:32) B r p p q X V p ( p P S , the elements of which are all convex and closed.Together with the Persistence Equivalence Theorem of [15, Section 7.2], this proves the equivalence of K A R and K ˇ C R . Thus a way of speeding up the computation of ˇCech persistent homology of S Ď p R d , d q is to useits Alpha filtration. This restricts the simplices in the filtration to those in the Delaunay complex K D , andrequires computing K D of S Ď R d , which contains O p n r d s q d -simplices, and can be done efficiently only inlow-dimensions [35]. In this section we investigate Alpha filtrations in (cid:96) metric, and their possible use forthe computation of ˇCech persistent homology. Alpha Filtrations in R . Given a two-dimensional set of points S in general position, we prove the equivalenceof its Alpha and ˇCech filtrations for the computation of persistence diagrams. For this we make use of thefollowing version of the Nerve Theorem. Theorem 3.1 (Theorem 10.7 [36]) . Let X be a triangulable space and t A i u i P I a locally finite family ofopen subsets (or a finite family of closed subsets) such that X “ Ť i P I A i . If every non-empty intersection A i X A i X . . . X A i t is contractible, then X and the nerve Nrv pt A i u i P I q are homotopy equivalent. a) (b) Figure 3: Voronoi diagrams and Delaunay triangulations of four points in R , with Euclidean and (cid:96) metric in (a) and (b) respectively. We also need the following characterization (cid:96) -Delaunay edges, making use of the concept of witnesspoints. Its proof is given in Appendix A. Definition 3.2. A witness point of σ P K Ar is a point z such that z P Ş p P σ V p ‰ H and d p z, p q “ max p,q P σ d p p,q q for each p P σ . Proposition 3.3.
Let S be a finite set of points in p R d , d q . Given a subset e “ t p, q u Ď S , we define A ¯ re “ B B ¯ r p p q X B B ¯ r p q q , where ¯ r “ d p p,q q . We have that A ¯ rσ “ B ¯ r p p q X B ¯ r p q q is a non-empty box. Moreover,the set of witness points of e is Z e “ A ¯ re z ` Ť y P S z e B ¯ r p y q ˘ , and e belongs to the (cid:96) -Delaunay complex of S ifand only if Z e is non-empty. Theorem 3.4.
Let S be a finite set of points in p R , d q in general position. The Alpha and ˇCech filtrationsof S are equivalent, i.e. produce the same persistence diagrams.Proof. Alpha complexes K Ar are nerves of collections of closed sets t B r p p q X V p u p P S for r P R . We show thatany intersection of k elements in any such collection is either empty or contractible. • k “
2. Let p, q be two points of S , and ¯ r “ d p p,q q . We show that L “ B r p p q X V p X B r p q q X V q is eitherempty or contractible. In R we have that A ¯ re “ B ¯ r p p q X B ¯ r p q q is a line segment of length strictly lessthan 2¯ r , by our general position assumption. If this line segment is covered by Ť y P S zt p,q u B ¯ r p y q , then byProposition 3.3 we have that V p X V q is empty, so that L is empty. Moreover L is empty if r ă ¯ r , because B r p p qX B r p q q is. On the other hand, if r ě ¯ r and A “ A ¯ re z Ť y P S zt p,q u B ¯ r p y q is a non-empty line segment,and we can show that L is contractible. First we define a deformation retraction φ of V p X V q onto A .This is obtained by taking the Euclidean projection of p V p X V q q z A onto p V p X V q q X A . This can bedone because p V p X V q q z A contains a maximum of two line segments, defined by the union of pointsin B B ¯ r ` ε p p q X B B ¯ r ` ε p q q not contained in Ş y P S zt p,q u B ¯ r ` ε p y q for any ε ą
0. For instance, considered thebisector V p X V q in Figure 1c, φ retracts the two line segments oriented at a forty-five degree angle ontothe horizontal line segment. Moreover φ restricts to L , by the convexity of B r p p q X B r p q q , and the factthat this contains A for r ě ¯ r . Hence L has the same homotopy type of A , which is a line segmentand so contractible. • k “
3. These intersections can either be empty or contain a single point by the general position of S . • k ą
3. Any such intersection is empty, again by the general position of S .6 a) Projection along x and y axes. (b) Projection along y and z axes. Figure 4: Five points in R realising a counterexample to Delaunay complexes being flag complexes in dimensions higher thantwo. Thus we can apply the Nerve Theorem 3.1 obtaining that X “ Ť p P S ` B r p p q X V p ˘ and K Ar are homotopyequivalent for any r P R . Besides X “ Ť p P S B r p p q , and by applying the Nerve Theorem to the collection t B r p p qu p P S , we have that X is homotopy equivalent to K ˇ Cr as well. So K Ar » K ˇ Cr for any r P R , and thedesired equivalence of Alpha and ˇCech filtrations follows by applying the Persistence Equivalence Theoremof [15, Section 7.2].This is similar to the results of [12], which proves that the nerve of offsets of convex shapes is equivalent tothe union of the shapes for zero and one dimensional homology in two and three dimensions. Our argumentusing general position implies that no higher-dimensional homology can appear in the nerve. In particular,the theorem implies that Alpha filtrations of two-dimensional points produce equivalent persistence diagramsto ˇCech filtrations. Hence, the above result ensures that the two-dimensional homology of Alpha complexesof S Ď R is trivial, because it equals the one of the two-dimensional sets Ť p P S B r p p q . At the end of thissection, we show that in general this is not the case for three-dimensional points, and so for any set of pointsin dimension d ě Alpha Filtrations in R . In order to construct the Alpha filtration of S Ď p R , d q in general position, weneed the (cid:96) -Delaunay triangulation of S . These can be found with the O p n log p n qq plane-sweep algorithmof [9]. Moreover, it is necessary to find the radius parameter r i of each simplex σ P K Ar , i.e. the minimum r i ą Ş p P σ ` B r i p p q X V p ˘ ‰ H . For this we have the following proposition. Its proof is providedin Appendix A, and Figure 3 gives an example with four points in R showing that the same does not holdin Euclidean metric. Proposition 3.5.
Let S be a finite set of points in general position in p R , d q and r ě . Both the Delaunaycomplex K D and the Alpha complex K Ar of S are flag complexes and e “ t p, q u P K Ar if and only if d p p,q q ď r . Thus the radius parameter of σ P K Ar is max p,q P σ d p p,q q , i.e. half the edge length of the longest edge in σ . Counterexample: The Alpha Complex is not Flag in Higher Dimensions.
We show that Proposition 3.5 doesnot hold for points in dimension three or higher. Given S “ t x i u i “ Ď p R , d q , where x “ r , , s , x “ r , , s , x “ r . , . , ´ . s , x “ r . , ´ . , ´ . s , and x “ r . , . , . s , we can prove thatthe Alpha complex K A of S is not a flag complex. In particular, we use witness points to show that t x , x u , t x , x u , t x , x u P K A , and t x , x , x u R K A . One can check that: • p , , q is a witness of t x , x u at distance 1 from x and x . • p . , . , . q is a witness of t x , x u at distance 0 . x and x .7 a) Projection along x and y axes. (b) Projection along y and z axes. Figure 5: Counterexample to the equivalence of Alpha and ˇCech persistent homology in (cid:96) metric. The two circumcenters ofthe tetrahedron t x , x , x , x u are the red square markers. The boundaries of cubes centered in the vertices of t x , x , x , x u are shown as dashed lines.Table 1: Coordinates of points S Ď p R , d q giving a counterexample to the equivalence of Alpha and ˇCech filtrations. x y z x x x x x x x x • p . , . , . q is a witness of t x , x u at distance 0 . x and x .Thus the pairs t x , x u , t x , x u , t x , x u are edges of the Delaunay complex K D , and edges of K A byProposition 3.5. On the other hand τ “ t x , x , x u is not a triangle in K D , and so does not belong to anyAlpha complex. This follows from the fact that A τ “ B B p x q X B B p x q X B B p x q is formed by the twoline segments, plotted as thickened lines in Figure 4, with endpoints p , . , q , p , . , . q and p , . , . q , p , , . q , which are covered by B p x q Y B p x q . The ε -thickenings of these line segments contain A ` ετ forany ε ě
0, by the properties of ε -thickenings used in the proof of Proposition 3.3 in Appendix A. In turn,the ε -thickenings of the two line segments are contained in ε p B p x q Y B p x qq “ B ` ε p x q Y B ` ε p x q . Thisimplies that it does not exist a point z P V x X V x X V x , as this would require A ` ετ z ` B ` ε p x q Y B ` ε p x q ˘ to be non-empty for some ε ě Counterexample: Non-Equivalence in Higher Dimensions.
We conclude this section by providing a coun-terexample to the equivalence of Alpha and ˇCech filtrations in homological dimension two. We show aconfiguration of eight points S “ t x i u i “ Ď R such that their Delaunay complex contains the four faces ofthe tetrahedron t x , x , x , x u , but not the tetrahedron itself. This way the Alpha complexes of S nevercontain t x , x , x , x u as a simplex, but for a big enough radius parameter they contain its the four faces.Moreover, the Delaunay complex of S also does not contain other tetrahedra that could possibly fill in thetwo-dimensional void created by the faces of t x , x , x , x u . We list the coordinates of the points giving acounterexample in Table 1, and plot them in Figure 5 by projecting along two of the three coordinate axes.These were found by randomly sampling many sets of eight points in R , and testing whether their Alphaand ˇCech persistence diagrams were equal. The existence of such a counterexample can be thought of as a8onsequence of the non-convexity of general (cid:96) -Voronoi regions, even if one may hope the nerve of generalVoronoi regions to be well behaved enough to prevent this from happening.One can check that there are six tetrahedra belonging to the Delaunay complex K D of S : t x , x , x , x u , t x , x , x , x u , t x , x , x , x u , t x , x , x , x u , t x , x , x , x u , and t x , x , x , x u . This can be done byfinding the circumcenters of any four given points, and checking that the circumspheres of these (which inthis case are cubes) do not contain any of the other points. It is important to note that in (cid:96) metric four three-dimensional points might have two distinct circumcenters. For instance this is the case for t x , x , x , x u ,the circumcenters of which are represented as red square markers in Figures 5a and 5b, having coordinates w “ p . , . , . q and w “ p . , . , . q . On the other hand, in Euclidean metric four affinelyindependent three-dimensional points have exactly one circumcenter. Moreover, w and w are not witnessesof t x , x , x , x u , because they are closer to x and x than to the vertices of this tetrahedron. Thus t x , x , x , x u R K D . Regarding the faces of t x , x , x , x u , we have that: • p . , . , . q is a witness of t x , x , x u at distance 3 . x , x , and x . • p . , . , . q is a witness of t x , x , x u at distance 3 .
55 from x , x , and x . • p . , . , . q is a witness of t x , x , x u at distance 3 .
55 from x , x , and x . • p . , . , . q is a witness point of t x , x , x u at distance 3 . x , x , and x .The tetrahedra belonging to the Delaunay complex of S (listed in the above discussion) do not create aboundary to the two-dimensional homology class created by adding t x , x , x u , t x , x , x u , t x , x , x u ,and t x , x , x u into K Ar , for r ą S has a point at infinity, i.e. an homology class that never dies. On the other hand, thetwo-dimensional persistence diagrams of the ˇCech filtration of S cannot have such a point, because ˇCechcomplexes have trivial homology for a big enough radius.
4. Alpha Flag Complexes
In the previous section we have seen that Alpha filtrations can be used to compute ˇCech persistencediagrams of points in R . On the other hand, already in three dimensions there exists a set of points S having different Alpha and ˇCech persistence diagrams in homological dimensions two. Moreover, in Section2 we show that in (cid:96) metric ˇCech filtrations are sequences of flag complexes. In particular a simplex σ belongs to K ˇ Cr if and only if max p,q P σ d p p, q q ď r . The new family of complexes we define here has thesame properties. Definition 4.1.
The
Alpha flag complex of S with radius r is K AFr “ (cid:32) σ Ď S | max p,q P σ d p p, q q ď r and t p, q u P K D for each p, q P σ ( In this section we prove that Alpha flag and ˇCech persistence diagrams coincide in homological dimensionszero and one. In particular, we think of ˇCech filtrations as a sequence of complexes where a single edge isadded when going from K ˇ Cr i to K ˇ Cr i ` . We prove that at each such step the zero and one-dimensional homologygroups of Alpha flag and ˇCech complexes remain isomorphic. To deal with the problem of multiple edgeshaving equal length, we assume that the (cid:96) distances between pairs of points of S are all distinct, i.e. S is in general position. In case this property does not hold, the finite set of points S can be infinitesimallyperturbed to obtain it. Importantly, the Stability Theorem of persistent homology, mentioned in Section 2,guarantees that the persistence diagrams of the original and perturbed points are close in bottleneck distance.Note that to obtain the next two theorems we make use of a number of technical results, which arepresented in Appendix B. Besides, we omit the field F , used in Section 2, when referring to the homologyof complexes, and say that a pair of points t p, q u Ď S is a non-Delaunay edge if it does not belong to theDelaunay complex of S . 9 a) (b) (c) Figure 6: (a)
Balls centered in the points of ¯ Y “ t y , y , y , y u covering A ¯ re . (b) K “ Nrv ` t B ¯ r p y qu y P ¯ Y q . (c) K , the unionof the cones from K to p and q . Theorem 4.2.
Let S be a finite set of points in p R d , d q in general position, and K ˇ Cr the ˇCech complex of S with radius r ą . If e “ t p, q u Ď S is a non-Delaunay edge contained in K ˇ Cr , then H k p K ˇ Cr z St p e qq and H k p K ˇ Cr q are isomorphic in homological dimensions zero and one.Proof. We can apply the reduced Mayer-Vietoris sequence, as given in [37, Section 4.6], with A “ Cl p St p e qq Ď K ˇ Cr and B “ K ˇ Cr z St p e q , because A X B “ Cl p St p e qqz St p e q ‰ H . We obtain ¨ ¨ ¨ Ñ ˜ H k p A X B q Ñ ˜ H k p A q ‘ ˜ H k p B q Ñ ˜ H k p A Y B q Ñ ˜ H k ´ p A X B q Ñ ¨ ¨ ¨ó¨ ¨ ¨ Ñ ˜ H k p Cl p St p e qqz St p e qq Ñ ˜ H k p K ˇ Cr z St p e qq Ñ ˜ H k p K ˇ Cr q Ñ ˜ H k ´ p Cl p St p e qqz St p e qq Ñ ¨ ¨ ¨ where ˜ H k p A q cancels out, because it is trivial by definition of A . Thus showing that ˜ H k p Cl p St p e qqz St p e qq istrivial in homological dimensions k and k ´
1, implies that ˜ H k p K ˇ Cr z St p e qq Ñ ˜ H k p K ˇ Cr q is an isomorphism,from the exactness of the Mayer-Vietoris sequence above.By definition of nerve, Proposition 2.1 (ii) , and the fact that K ˇ Cr is a flag complexes, it follows that A “ Cl p St p e qq “ Nrv ` t B r p y qu y P Y ˘ , where Y “ t y P S | d p y, p q ď r and d p y, q q ď r u . Defined A ¯ re “ B B ¯ r p p q X B B ¯ r p q q , where ¯ r “ d p p,q q ď r , we have that A ¯ re is covered by Ť y P S z e B ¯ r p y q byProposition 3.3. We can restrict this union of open balls to those centered in the points of¯ Y “ t y P S | d p y, p q ă r and d p y, q q ă r u Ď Y , because B ¯ r p y q X A ¯ re “ H if y R ¯ Y . So A ¯ re must be covered by Ť y P ¯ Y B ¯ r p y q andNrv ` t B ¯ r p y qu y P ¯ Y ˘ Ď Nrv ` t B r p y qu y P Y ˘ z St p e q “ Cl p St p e qqz St p e q Ď K ˇ Cr . In Appendix B, we prove that Nrv ` t B ¯ r p y qu y P ¯ Y q has the homotopy type of A ¯ re , and so trivial homology.The idea is that Nrv ` t B ¯ r p y qu y P ¯ Y q has the same homotopy type of Ť y P ¯ Y B ¯ r p y q by the Nerve Theorem, and thatthis union of balls retracts onto A ¯ re . Then, given the simplices in Cl p St p e qqz St p e q and not in Nrv ` t B ¯ r p y qu y P ¯ Y q ,we prove that adding them into Nrv ` t B ¯ r p y qu y P ¯ Y q does not alter its zero and one-dimensional homology.Regarding zero-dimensional homology we know that Nrv ` t B ¯ r p y qu y P ¯ Y ˘ consists of one connected compo-nent. Also, the vertices in Cl p St p e qqz St p e q not in Nrv ` t B ¯ r p y qu y P ¯ Y ˘ , that could potentially create a homology10lass in ˜ H p Cl p St p e qqz St p e qq , are the points Y z ¯ Y . We have that p, q P Y z ¯ Y , and these are fully connectedto the points in ¯ Y , so do not create any connected component. Moreover, all other points in Y z ¯ Y are con-nected to both p and q by definition of Y . So Cl p St p e qqz St p e q cannot contain a connected component not inNrv ` t B ¯ r p y qu y P ¯ Y ˘ , and ˜ H p Cl p St p e qqz St p e qq must be trivial.For one-dimensional homology, we define K “ Nrv ` t B ¯ r p y qu y P ¯ Y ˘ and K n “ Nrv ` t B r p y qu y P Y ˘ z St p e q , andshow the existence of a filtration K Ď K Ď . . . Ď K n , such that at each step K i Ď K i ` no one-dimensionalhomology class is created. We start by defining K as the union of the cones from K to p and q . Figures6b and 6c illustrate this step for the points in Figure 6a. So going from K to K one-dimensional homologyremains trivial, because adding these cones cannot create any new 1-cycle. Then we add the points of y P Y zp ¯ Y Y t p, q uq into K one by one, obtaining a new complex K i ` of the filtration above each time.Furthermore, at each such step K i Ď K i ` , we also add two triangles t p, y , ¯ y u and t q, y , ¯ y u , where ¯ y P ¯ Y .This can be done because A ¯ re is covered by Ť y P ¯ Y B ¯ r p y q , so that there exist ¯ y P ¯ Y such that B r p y q X B r p ¯ y q Ě B r p y qX B ¯ r p ¯ y q ‰ H , because B r p y qX A ¯ re ‰ H . Hence both B r p p qX B r p y qX B r p ¯ y q and B r p q qX B r p y qX B r p ¯ y q must be non-empty, by Proposition 2.1 (ii) , so that t p, y , ¯ y u , t q, y , ¯ y u P K ˇ Cr . Thus by Proposition B.3 no one-dimensional homology class is created going from K i to K i ` “ K i Yt y uYt p, y uYt q, y uYt p, y , ¯ y uYt q, y , ¯ y u .We denote the K i ` having Y as its set of vertices by K n ´ . Finally, we add all the simplices in K n z K n ´ in the last filtration step. Again we can apply Proposition B.3, because for each edge t y , y u , with y , y P Y added into K n , there must be a triangle t p, y , y u P K n by definition of Y . Hence we can conclude that K n has trivial reduced one-dimensional homology, i.e. ˜ H p Cl p St p e qqz St p e qq is trivial.The proof follows from the exactness of the reduced Mayer-Vietoris sequence as mentioned above, andthe fact that isomorphisms in reduced homology translate into isomorphisms in non-reduced homology. Theorem 4.3.
Let S be a finite set of points in p R d , d q in general position. Given r ą and ε ą such that K ˇ Cr ` ε contains exactly one edge not in K ˇ Cr , if i kr : H k p K AFr q Ñ H k p K ˇ Cr q is an isomorphism, then i kr ` ε : H k p K AFr ` ε q Ñ H k p K ˇ Cr ` ε q is also an isomorphism for k “ , .Proof. Let e “ t p, q u Ď S be the only edge added to K ˇ Cr by increasing the radius parameter of ε ą
0. Theneither e is a Delaunay edge, so that e P K AFr and e P K ˇ Cr , or e is not a Delaunay edge, so that e R K AFr and e P K ˇ Cr . We split the proof in two parts, dealing with these two cases separately.We use the notation of Proposition 3.3, meaning that r ă ¯ r “ d p p,q q ď r ` ε and A ¯ re “ B ¯ r p p q X B ¯ r p q q .Also, as in the proof of Theorem 4.2, we define ¯ Y “ t y P S | d p y, p q ă r and d p y, q q ă r u , so that if e isa non-Delaunay edge, then A ¯ re must be covered by Ť y P ¯ Y B ¯ r p y q . CASE 1: e is DelaunayFor r ą K AFr and K ˇ Cr contain the same vertices by definition. Also, because the homomor-phism induced by the inclusion of complexes H p K AFr q Ñ H p K ˇ Cr q is an isomorphism, K AFr and K ˇ Cr havethe same connected components. Thus after e is added in both K AFr and K ˇ Cr either connected componentsdo not change or the same connected component is merged in both. In the first case zero-dimensional ho-mology remains unchanged, while in the second case the same zero-dimensional homology class is deletedin H p K AFr q and H p K ˇ Cr q . In both cases i r : H p K AFr ` ε q Ñ H p K ˇ Cr ` ε q is an isomorphism induced by theinclusion K AFr ` ε Ď K ˇ Cr ` ε .We now look at one-dimensional homology. Adding a single edge e and the cliques it forms into K AFr and K ˇ Cr can result in the creation or deletion of one-dimensional homology classes. We have two subcases.1. The edge e adds nothing but itself to the Alpha flag complex K AFr ;2. The edge e adds itself and one or more triangles to the Alpha flag complex K AFr . Subcase 1.1
We start by proving that the edge e is the only simplex added into K ˇ Cr as well. To show this, let us suppose bycontradiction that increasing the radius parameter from r to r ` ε results into adding e and a triangle t p, q, y u into K ˇ Cr . This means t p, y u , t q, y u P K ˇ Cr , so that they are strictly shorter than t p, q u from our hypothesis ondistances between pairs of points in S , i.e. general position. Given 0 ă δ ă r ´ max t d p p, y q , d p q, y qu , wehave d p p, y q ă r ´ δ and d p q, y q ă r ´ δ . Moreover d p p, q q “ r , so the three axis-parallel hypercubes11 ¯ r p p q , B ¯ r p q q , and B ¯ r ´ δ p y q have non-empty pairwise intersections. Their triple intersection is also non-empty,by Proposition 2.1 (ii) , and it follows that A ¯ re X B ¯ r p y q “ B ¯ r p p q X B ¯ r p q q X B ¯ r p y q ‰ H . Hence the set of points Y contains at least one point, and because e is a Delaunay edge, we have that A ¯ re z ` Ť y P ¯ Y B ¯ r p y q ˘ is non-empty.Thus the closed set ` Ť y P ¯ Y B ¯ r p y q ˘ c needs to intersect A ¯ re , which is a closed box. So there exist a point z of A ¯ re belonging to the boundary of the closure of Ť y P ¯ Y B ¯ r p y q , otherwise A ¯ re would need to be disconnected, i.e. A ¯ re XB ` Ť y P Y B ¯ r p y q ˘ ‰ H . Furthermore, z P A ¯ re X ` Ť y P Y B B ¯ r p y q ˘ , because B ` Ť y P Y B ¯ r p y q ˘ Ď ` Ť y P Y B B ¯ r p y q ˘ .In conclusion z P B B ¯ r p y q for some y P Y , and t p, q, y u is a Delaunay triangle with z as a witness point,which belongs to K AFr ` ε . This contradicts the hypothesis of Subcase 1.1, because the Alpha flag complex K AFr ` ε cannot contain any triangles of which t p, q u is an edge. Thus, when increasing the radius parameter from r to r ` ε , the edge e “ t p, q u is the only simplex added in both K AFr and K ˇ Cr .In general adding a single edge to an abstract simplicial complex can result in either the deletion ofa connected component or the creation of a one-dimensional homology class. The former of these twocases is dealt within the discussion of zero-dimensional homology above, and does not affect one-dimensionalhomology. On the other hand, if e does not merge connected components in K AFr , then it also does not mergeconnected components in K ˇ Cr , because as already discussed zero-dimensional homology remains isomorphic.Thus both H p K AFr ` ε q and H p K ˇ Cr ` ε q contain a new homology class. In this case i r ` ε : H p K AFr ` ε q Ñ H p K ˇ Cr ` ε q is the isomorphism induced by the inclusion, which extends i r : H p K AFr q Ñ H p K ˇ Cr q by mapping the one-dimensional homology class created by e in K AFr ` ε into the one created by e in K ˇ Cr ` ε . Subcase 1.2
Adding e “ t p, q u to both K AFr and K ˇ Cr results in one or more triangles t τ ¯ rj u j P J added to the Alpha flagcomplex K AFr ` ε . Moreover, by the definition of flag complex, the same triangles are added to K ˇ Cr ` ε . Also,there might be triangles t ˇ τ ¯ rj u j P ˇ J added to K ˇ Cr ` ε , which are not added to K AFr ` ε . These t ˇ τ ¯ rj u j P ˇ J contain t p, q u as an edge, and at least one non-Delaunay edge among their other edges.To begin with, we note that e does not create any one-dimensional homology class in K AFr ` ε and K ˇ Cr ` ε byProposition B.3. It remains to prove that a one-dimensional homology class r γ s P H p K AFr q is deleted atradius r ` ε if and only if i r pr γ sq “ r ˇ γ s P H p K ˇ Cr q is also deleted.The first direction holds because if a homology class is deleted in the Alpha flag complex, then the sameformal sum of triangles is a boundary for the same homology class of the ˇCech complex.For the opposite direction, let us suppose that r ˇ γ s P H p K ˇ Cr q is deleted at radius r ` ε , and that r γ s remains open in the Alpha flag complex with radius r ` ε . We can think of adding the triangles t τ ¯ rj u j P J and t ˇ τ ¯ rj u j P ˇ J one by one in K ˇ Cr in any order, obtaining a new ˇ K i Ď K ˇ Cr ` ε at each step. At some point one of thesemust be creating a boundary deleting r ˇ γ s in ˇ K i . If this is a triangle τ ¯ rj (containing Delaunay edges only), thenits edges form a formal sum which is homologous to both r γ s and r ˇ γ s . Moreover, τ ¯ rj bounds this formal sumsin both complexes, so that r γ s R H p K AFr ` ε q , which is a contradiction. On the other hand, if a non-Delaunaytriangle ˇ τ ¯ rj is creating a boundary deleting r ˇ γ s , we can apply Theorem 4.2 to one of the non-Delaunay edgesˇ e of ˇ τ ¯ rj . We have a contradiction with the assumption of ˇ τ ¯ rj deleting r ˇ γ s , because K ˇ Cr ` ε z St p ˇ e q and K ˇ Cr ` ε needto have the same one-dimensional homology and ˇ τ ¯ rj P St p ˇ e q .In conclusion the same one-dimensional homology classes are deleted in both complexes by the sametriangles, and so i r ` ε : H p K AFr ` ε q Ñ H p K ˇ Cr ` ε q is an isomorphism induced by the inclusion K AFr ` ε Ď K ˇ Cr ` ε . CASE 2: e is non-DelaunayBy applying Theorem 4.2, we have that H k p K ˇ Cr ` ε z St p e qq Ñ H k p K ˇ Cr ` ε q is an isomorphism for k “ , H k p K AFr q H k p K ˇ Cr ` ε z St p e qq H k p K AFr ` ε q H k p K ˇ Cr ` ε q – – – obtained by applying the homology functor to the inclusion maps between complexes commutes, because K ˇ Cr “ K ˇ Cr ` ε z St p e q and K AFr “ K AFr ` ε , proving that H k p K AFr ` ε q Ñ H k p K ˇ Cr ` ε q is an isomorphism for k “ , orollary 4.4. Let S be a finite set of points in p R d , d q in general position. Given a finite set of mono-tonically increasing real-values R “ t r i u mi “ , the Alpha flag K AF R and ˇCech filtrations K ˇ C R of S have the samepersistence diagrams in homological dimensions zero and one.Proof. Given the two parameterized filtrations K ACr Ď K ACr Ď . . . Ď K ACr m and K ˇ Cr Ď K ˇ Cr Ď . . . Ď K ˇ Cr m . Wehave that H k p K AFr i q Ñ H k p K ˇ Cr i q is an isomorphism for each 1 ď i ď m and k “ , • For r i ď K AFr i and K ˇ Cr i are empty. • For r i ą
0, we can think of K AFr i and K ˇ Cr i as the result of adding one edge at a time, and the cliques theseform into K AC and K ˇ C . Theorem 4.3 ensures that each new edge added preserves the isomorphismbetween the zero and one-dimensional homology groups of the Alpha flag and ˇCech complexes.The proof follows by applying the Persistence Equivalence Theorem, as stated in [15, Section 7.2]The above result extends to generic dimension d the equivalence of zero and one-dimensional persistencediagrams proven in [12] for two and three dimensional points.
5. Minibox Complexes
In this section we introduce yet another family of complexes (again flag complexes), which we proveto have the same property of Alpha flag complexes, i.e. they can be used to compute the ˇCech persistencediagrams of S in homological dimensions zero and one. We conclude with a discussion regarding the expectednumber of edges these complexes can contain. In the next section, we describe algorithms for finding theseedges. Definition 5.1.
Let p, q be two points in p R d , d q . The minibox of p and q isMini pq “ d ź i “ ` min t p i , q i u , max t p i , q i u ˘ , that is to say the interior of the minimal bouding box of p and q . Proposition 5.2.
Let S be a finite set of points in p R d , d q , e “ t p, q u a pair of points of S , and Mini pq the minibox of p and q . If it exists y P S z e such that y P Mini pq , then e is not an edge of the (cid:96) -Delaunaycomplex of S .Proof. Given ¯ r “ d p p,q q , we have A ¯ re “ B ¯ r p p qX B ¯ r p q q by Proposition 3.3. Equivalently A ¯ re “ ś di “ r b i ´ ¯ r, a i ` ¯ r s , where a i “ min t p i , q i u and b i “ max t p i , q i u for each 1 ď i ď d . Then, given y P Mini pq , it follows that a i ă y i ă b i for each 1 ď i ď d , implying y i ´ ¯ r ă b i ´ ¯ r and a i ` ¯ r ă y i ` ¯ r . Thus r b i ´ ¯ r, a i ` ¯ r s Ă p y i ´ ¯ r, y i ` ¯ r q for each 1 ď i ď d , and A ¯ re Ă B ¯ r p y q . The result follows applying Proposition 3.3. Definition 5.3.
The
Minibox complex of S with radius r is K Mr “ (cid:32) σ Ď S | max p,q P σ d p p, q q ď r and Mini pq X p S zt p, q uq “ H for each p, q P σ ( Theorem 5.4.
Let S be a finite set of points in p R d , d q in general position. Given the Alpha flag K AFr andMinibox K Mr complexes with radius r , then H k p K AFr q and H k p K Mr q are isomorphic in homological dimensionszero and one. roof. We have K AFr Ď K Mr Ď K ˇ Cr , and we know that H k p K AFr q Ñ H k p K ˇ Cr q is an isomorphism for k “ , H k p K AFr q Ñ H k p K Mr q is injective for k “ , r P R . K AFr K ˇ Cr K Mr ùñ H k p K AFr q H k p K ˇ Cr q H k p K Mr q – To conclude our proof we need to show the surjectivity of this homomorphism for k “ , k “
0, because K AFr and K Mr have the same set of vertices, and K Mr might contain more edges, itfollows that K AFr has the same or more connected components than K Mr . So in homological dimension zerothe homomorphism induced by the inclusion K AFr Ď K Mr , must be surjective.To prove the surjetivity of i r : H p K AFr q Ñ H p K Mr q , we show that for any r γ s P H p K Mr q a 1-cycle γ representing it has to be homologous to a 1-cycle γ containing Delaunay edges of length less than or equalto 2 r , so that i r pr γ sq “ r γ s .Let γ be a 1-cycle in K Mr representing r γ s P H p K Mr q , and e “ t p, q u the non-Delaunay edge in γ ofmaximum length. We have A ¯ re “ B ¯ r p p q X B ¯ r p q q , where ¯ r “ d p p,q q by Proposition 3.3. Defined ¯ Y “ t y P S | d p y, p q ă r and d p y, q q ă r u , we equivalently have ¯ Y “ S X B r p p q X B r p q q and ¯ Y “ S X ¯ r p A ¯ re q ,because ε p A ¯ re q equals B ¯ r ` ε p p q X B ¯ r ` ε p q q by the properties of boxes described in Appendix A. For points in R , these sets are illustrated in Figure 7, where A ¯ re is represented by a thickened vertical line between p and q . Moreover, given c “ p ` q , we have Mini pq Ď ¯ r p c q Ď ¯ r p A ¯ re q , because c Ď A ¯ re , taking ε -thickenings preservesinclusions, and Mini pq has sizes of length less than or equal to 2¯ r and center c . Then, because e is not aDelaunay edge, A ¯ re must be covered by the union of balls centered in the points of S zt p, q u by Proposition3.3. Thus at least one y P S zt p, q u is such that B ¯ r p y q intersects A ¯ re , i.e. ¯ Y ‰ H . Defined ¯ y P ¯ Y to be thepoint realizing min y P ¯ Y d p y, Mini pq q , we have that Mini p ¯ y and Mini q ¯ y do not contain points in S zt p, q, ¯ y u , as we can show a contradiction otherwise.In particular suppose there exist either y P S z ¯ Y or y P ¯ Y belonging to one of these two miniboxes. Withoutloss of generality, we assume either y Ď Mini p ¯ y or y Ď Mini p ¯ y . In the former case we have Mini p ¯ y Ď ¯ r p A ¯ re q ,because p is on the boundary of ¯ r p A ¯ re q and ¯ y in its interior. So y P ¯ r p A ¯ re q , implying that y P ¯ Y , which is acontradiction. In the latter case, it must be that d p y , Mini pq q ă d p ¯ y, Mini pq q by definition of Mini p ¯ y and d , which is in contradiction with ¯ y being the closest point of ¯ Y to Mini pq .So there exists a vertex ¯ y of the Minibox complex connected to p and q by the edges t p, ¯ y u and t q, ¯ y u .These are shorter than 2¯ r so that t p, ¯ y u , t q, ¯ y u Ď K Mr . Swapping t p, ¯ y u and t q, ¯ y u for e in γ , we obtain a1-cycle homologous to γ with the property of having a shorter longest non-Delaunay edge. This procedurecan be repeated only a finite number of times, as we have a finite number of non-Delaunay edges, and ateach iteration the maximum non-Delaunay edge length in the current 1-cycle decreases. When the procedurecannot be repeated, we have a 1-cycle γ in K Mr homologous to γ , containing only Delaunay edges. Hence γ represents a one-dimensional homology class in the Alpha flag complex which is mapped into r γ s by i r : H p K AFr q Ñ H p K Mr q . Corollary 5.5.
Let S be a finite set of points in p R d , d q in general position. Given a finite set of mono-tonically increasing real-values R “ t r i u mi “ , the Alpha flag K AF R and Minibox filtrations K M R of S have thesame persistence diagrams in homological dimensions zero and one.Proof. Follows from the Persistence Equivalence Theorem of [15, Section 7.2] as for Corollary 4.4.
Number of Minibox Edges.
We conclude this section by studying the number of edges that a Minibox complex K Mr can contain. We are able to show that for randomly sampled points the expected number of emptyminiboxes on the points of S is proportional to n ¨ polylog p n q , where n in the number of points in S .14 a) (b) Figure 7: (a)
The pair p p, q q is not a Delaunay edge, but is a Minibox edge. Mini pq is the gray region having p and q as twovertices. The set ¯ Y consists of four y i points contained in the rectangle ¯ r p A ¯ re q , whose boundary is represented by a dash-dotline. (b) Expected number of Minibox edges of randomly sampled points compared to all possible edges (dashed line).
We start by noting that in the worst case a Minibox complex can contain O p n q edges. For example theunion of S “ (cid:32) p i “ ` ´ in , ´ n ˘( ni “ and S “ (cid:32) q j “ ` ´ jn , ´ jn ˘( nj “ is a set of 2 n points in R , onparallel line segments, such that all the miniboxes Mini p i q j for 1 ď i ď j ď n do not contain any point in S Y S . Thus the Minibox complex of S Y S will contain more than n p n ´ q points for a large enough radiusparameter.Next, given S to be a random points in R d , we can derive the expected number of edges contained in anymaximal Minibox complex. Definition 5.6.
Let p and q be points in R d . We say that p dominates q if each of the coordinates of p isgreater than the corresponding coordinate of q . Given a finite set of points S Ď R d , we say that p directlydominates q if p dominates q and there is no other point y P S such that p dominates y and y dominates q . Proposition 5.7.
Let S be a finite set of uniformly distributed random points in p R d , d q . The expectednumber of edges contained in the maximal Minibox complex of S is Θ ´ d ´ p d ´ q ! n log d ´ p n q ¯ , where n “ | S | .Proof. We have that if p directly dominates q , then Mini pq X S “ H . On the other hand, Mini pq X S “ H does not imply that either p directly dominates q or q directly dominates p . However, for each pair t p, q u there exists a sequence of a maximum of d reflections about the coordinate hyperplanes that transforms S into a set of points such that q dominates p . There are 2 d possible such sequences of reflections, one for eachorthant, and each produces a set of points S k with a set of directly dominated pairs disjoint from those ofthe other S k s. Moreover, if t p, q u is not a directly dominated pair in any S k for 1 ď k ď d , then Mini pq X S must be non-empty. So if the expected number of directly dominated pairs in S k is m , then the expectednumber of empty miniboxes on S is 2 d ´ ¨ m , because each edge t p, q u is counted twice in the 2 d transformedpoint sets S k . In [38] it is shown that for n random points in R d the expected number of directly dominatedpairs is Θ ´ p d ´ q ! n log d ´ p n q ¯ . Thus, in dimension d there are Θ ´ d ´ p d ´ q ! n log d ´ p n q ¯ pair of points t p, q u such that Mini pq X S “ H . The proof follows from the definition of Minibox complex.Figure 8b plots the expected number of minibox edges for random points in dimension 2 ď k ď n in the range r , s . This is an empirical estimate obtained by randomly sampling points in the unithypercube and counting the number of edges found with the algorithms of the next section.15 a) Alpha flag edges. (b) Minibox edges. (c) ˇCech edges. Figure 8: Comparison of Alpha flag (i.e. Delaunay), Minibox, and ˇCech edges of thirty random points in R .
6. Algorithms
We present algorithms for finding all pairs of points t p, q u Ď S such that Mini pq X S is empty. By definitionthese are all the edges a Minibox complex can contain. We study the two-dimensional, three-dimensional,and higher dimensional cases separately. For d “ d “
3, we present a plane-sweep and a space-sweepalgorithm respectively. These maintain front data structures that can be used to efficiently determine whetherMini pq X S is empty or not. For general dimension d , we see the problem of finding all empty miniboxes on S as of an offline orthogonal range emptiness problem with n p n ´ q range queries, and reference know resultson range queries.We also provide an implementation of our algorithms in the form of the persty Python package, the sourcecode of which is available at https://github.com/gbeltramo/persty. Points in two-dimensions.
We start by taking S to be a finite set of points in p R , d q . For this case wedescribe a O p n q algorithm, whose pseudocode is given in Algorithm 1. This is worst case optimal by thediscussion on the number of Minibox edges at the end of Section 5. An example of the edges contained inthe maximal Minibox complex on thirty random points in R is given in Figure 8b. This can be compared tothe edges in Figures 8a and 8c showing the edges contained in the maximal Alpha flag and ˇCech complexeson the same points.The algorithm works by sweeping the plane form left to right for each point p “ p p x , p y q , starting from p x . Then for each point q in the half plane p p x , `8q ˆ p´8 , `8q it checks whether p p, q q is a Minibox edge.This is done on line 8 of Algorithm 1. For this it uses a front, which consists of two points f ront Ò and f ront Ó . These are updated on line 10, and are such that f ront Ò has a y -coordinate greater than p y , while f ront Ó has a y -coordinate less than or equal to p y . Moreover defined X to be the set of points in S with x -coordinate in the range p p x , q x q , f ront Ò has a y -coordinate smaller than any other point x P X such that x y ą p y , and f ront Ó has a y -coordinate larger than any other point x P X such that x y ď p y . Given q y ą p y , by definition Mini pq is non-empty if and only if there exist of point x P S such that x y ą p y and q dominates x y . But such a point exists only if q dominates f ront Ò . Thus if q y ą p y , either f ront Ò P Mini pq or Mini pq X X “ Mini pq X S “ H . The same applies to q with q y ď p ´ y using f ront Ó . This is illustrated inFigure 9, where X “ t x , x , x , x u and f ront Ò “ x .The algorithm complexity is O p n q because we loop on all possible n p n ´ q edges, and at each iterationwe need O p q operations to check whether Mini pq is empty and update f ront Ò , f ront Ó . Points in three-dimensions.
For a finite set of points S in three dimensions, we present Algorithm 2, whichuses a space-sweep strategy.Given p, q P S , we define the sweep-plane to be the yz -plane with origin p p y , p z q . We have that a point y P Mini pq must be such that its projection onto the sweep-plane belongs to the same quadrant as the16 lgorithm 1 Minibox edges of a finite set of points S in two-dimensions. input: array points , the finite set of points S in p R , d q edges Ð empty list of two-tuples of integers Sort points on their x -coordinate f ront Ò , f ront Ó Ð p p r s , `8q , p p r s , ´8q , where p “ points r s for i “ | S | ´ do for j “ i ` | S | ´ do p, q Ð points r i s , points r j s if Mini pq does not contain f ront Ò or f ront Ó then Add p i, j q to edges Set f ront Ò “ q if p r s ă q r s , or f ront Ó “ q if p r s ą“ q r s end if end for end for return edges (a) (b) Figure 9: Illustration of the plane-sweep algorithm for Minibox edges in R , with X “ t x , x , x , x u and front Ò “ x . In (a) t p, q u is not a Minibox edge, because front Ò P Mini pq . In (b) Mini pq is empty, so in this case t p, q u is a Minibox edge. projection of q . Hence, without loss of generality, we always assume the projection of q to belong to the firstquadrant of the sweep-plane. In Algorithm 2 this reflects into the definition of p and q on line 8. The ideais to maintain a front data structure for each quadrant of the sweep-plane, and use it to test whether t p, q u is a Minibox edge or not.At each step of the inner loop on lines 6 ´
21, we have that t p, q u is a Minibox edge if and only if Mini p q does not contain a point y in the sweep-plane. Because we restrict ourselves to the first quadrant, we onlyneed to check whether or not q dominates any y projected from a y P S with y x in the range p p x , q x q . Tospeed this up we can store the points y as we sweep on p p x , q x q in a black-red tree front, sorting them ontheir first coordinate, and then check if Mini p q is empty by searching among the points in this front. Inparticular, we only store the points q which are adding a Minibox edge, i.e. those that do not dominatepoints in the front. This happens on lines 13 , ,
20 of Algorithm 2. The other points q , dominating anotherpoint y already in the front, are not needed. This is due to the fact that if a future q dominates q , thenit must also dominate y . Furthermore, it may happen for q to be dominated by points previously stored inthe front. In this case these are no longer needed, as for q above, and we can replace them with q , whichhappens on lines 13 and 16. A consequence of the way points are stored in and deleted from the red-blacktree front is that these are the vertices of a staircase in the first quadrant of the sweep-plane, i.e. sorting thepoints along their first coordinate their second coordinates are monotonically decreasing. This disposition of17 lgorithm 2 Minibox edges of a finite set of points S in three-dimensions. input: array points , the finite set of points S in p R , d q edges Ð empty list of two-tuples of integers Sort points on their x -coordinate for i “ | S | ´ do f ronts Ð list of four empty red-black trees, one per quadrant for j “ i ` | S | ´ do p, q Ð points r i s , points r j s p , q Ð p , q , p| q r s ´ p r s| , | q r s ´ p r s|q k Ð index such that p q r s , q r sq is in the k -th quadrant of the sweep-plane with origin p p r s , p r sq if f ronts r k s is non-empty then y Ð first element to the left of q in f ronts r k s bisecting on q r s if y does not exist then Delete the points in f ronts r k s that dominate q , add q in f ronts r k s , and add p i, j q in edges else if y R Mini p q then Delete the points in f ronts r k s that dominate q , add q in f ronts r k s , and add p i, j q in edges end if end if else Add q in f ronts r k s , and add p i, j q in edges end if end for end for return edges points is similar to those in the examples given for the two-dimensional case in Figure 9. The difference isthat q can be dominated by one of the points already in the front. Thus, to find y dominated by q in thefront we can bisect on the first coordinate values of its points. This follows because if q dominates any pointin the front, then it also has to dominate the point in the front directly to its left, by the fact that the frontis a staircase.The inner loop may require to delete and add O p n q points into a red-black tree, and to bisect on the sametree O p n q times. Since either deleting, adding, or bisecting on a red-black tree requires O p log p n qq operations,we conclude that the inner loop takes a total of O p n log p n qq operations. Hence Algorithm 2 has O p n log p n qq complexity. Higher dimensions.
For points in general dimension d ě
4, we propose different strategies, using decreasingamount of additional storage, to test whether Mini pq X S is empty for each pair of points in S .For instance, high-dimensional range trees with fractional cascading [17, Section 5.6] can be used to answerorthogonal range emptiness queries in O p log d ´ p n qq time, at the additional cost of O p n log d ´ p n qq storage.By testing all pairs of points in S , we have a O p n log d ´ p n qq algorithm. Similarly, kd -trees [17, Section 5.2]can be used to answer the same query in O p n ´ d q time, only taking O p n q additional storage, resulting in a O p n ´ d q algorithm for finding all the edges contained in any Minibox complex. Furthermore, we note thatby the curse of dimensionality, if d becomes too big it might be faster to test each of the n p n ´ q pairs of pointsin S via a brute force strategy, searching all points in S sequentially, which takes O p n q time. This results ina O p dn q total time algorithm, but does not require storing any additional data structure. The choice amongthese options depends on the amount of memory that can be spared for storing additional data structures.Moreover, we note that each of the above strategies could take advantage of parallel implementations usingthe independence of tests on each pair of points in S .Finally, we also mention that in the Word RAM model of computation the offline orthogonal range18ounting algorithm of [39] can be used to find all empty miniboxes on S in constant dimension d ě O p n log d ´ ` d p n qq . Anyway, as remarked in [39], for this algorithm to be applicable to floating-pointnumbers one needs to assume that the word size is at least as large as both log p n q and the maximum size ofan input number.
7. Computational Experiments
In this final section, we present computational experiments giving empirical evidence of the speed upobtained by using Minibox filtrations in the calculation of zero and one-dimensional ˇCech persistence diagramsof S in (cid:96) metric. Moreover, we compute the persistence diagrams of Alpha flag, Minibox, and ˇCech filtrationsobtained using randomly sampled points in r , s Ď p R d , d q . These allow us to illustrate the similaritiesand dissimilarities between two-dimensional diagrams of these filtrations.We use the implementation of the persistent homology algorithm provided by the Ripser.py [40] Pythonpackage, in combination with the algorithms of the persty Python package. All computations were run ona laptop with Intel Core i7-9750H CPU with six physical cores clocked at 2.60GHz with 16GB of RAM.
Size of Minibox filtrations.
First we study the expected size of Minibox filtrations versus the size of ˇCechfiltrations. Our filtrations contain vertices, edges, and triangles, because we only need to compute zero andone dimensional persistence diagrams. So we have that the ˇCech filtration contains Θ p n q simplices. Giventhe edges in the maximal Minibox complex of S , the clique triangles on these can be found in O p nk q time,where k is the maximum degree of any point in S , i.e. the maximum number of Minibox edges a point iscontained in. Moreover O p nk q is also an upper bound on the number of possible Minibox triangles, and byProposition 5.7 it follows that the expected value of k for a uniformly distributed finite set of random points isΘ ´ d ´ p d ´ q ! log d ´ p n q ¯ . Hence, we expect the Minibox filtration of S to contain less simplices compared to theˇCech filtation. We give empirical evidence of this by calculating the expected number of Minibox simplicesfor 500, 1000, 1500, and 2000 uniformly distributed random points, averaging over five runs. Table 2 presentsour results for Minibox filtrations in two, three and four dimensions. The number of simplices contained inthe ˇCech filtrations are listed for comparison. Table 2: Average number of simplices contained in the Minibox and ˇCech filtrations for different input sizes. n = 500 n = 1000 n = 1500 n = 2000Minibox 2D 0 . ˆ . ˆ . ˆ . ˆ Minibox 3D 0 . ˆ . ˆ . ˆ . ˆ Minibox 4D 1 . ˆ . ˆ . ˆ . ˆ ˇCech 20 . ˆ . ˆ . ˆ . ˆ Running Time and Memory Usage.
Next we explore the use of Minibox filtrations for the computation ofˇCech persistence diagrams of S Ď p R d , d q in homological dimensions zero and one. As already mentioned, wemake use of the Ripser.py package, which provides a Python interface to Ripser [41] C++ code. In particular,we think of Minibox filtrations as of sparse filtrations, and feed into the persistent homology algorithm aprecomputed sparse matrix in coordinate format. We give timing and memory usage results for points in therange r , s for Minibox filtrations, averaging over five runs. In the case of ˇCech filtrations we limitour experiments to a maximum of 8000 points, because of memory constraints. Moreover we consider onlypoints in R , as results are similar in higher dimensions.We list our results in Tables 3, 4, 5, and 6, where columns correspond to different sizes of the input pointsset S , and times are given in seconds. In particular, we use Algorithm 1 for edges in Table 3, Algorithm 219or edges in Table 4, and a brute force algorithm for edges in Table 5. We also report the average total peakmemory use in megabytes. In all the experiments, the reduced number of simplices of Minibox filtrations results in a substantialimprovement in memory usage over ˇCech filtrations, and in a speed up in the computation of Dgm andDgm . This allows to increase the maximum size of inputs of the persistence algorithm, given a fixed amountof available memory. The price is having to precompute Minibox edges. We note that this computation couldalso take advantage of implementations parallelizing the inner loops of Algorithms 1 and 2, or the individualchecks on edges of any brute force algorithm, as already mentioned in Section 6. Table 3: Timing (seconds) and memory usage (MB) with Minibox filtrations of points in R . n = 500 n = 1000 n = 2000 n = 4000 n = 8000 n = 16000 n = 32000Edges time 0.008 0.016 0.047 0.117 0.289 0.891 2.852Sparse matrix time 0.023 0.070 0.141 0.312 0.734 1.562 3.406Dgm , time 0.008 0.016 0.031 0.078 0.172 0.477 1.148Total time 0.039 0.102 0.219 0.507 1.195 2.929 7.406Peak memory usage 2.92 5.52 11.51 25.15 53.50 112.07 246.28 Table 4: Timing (seconds) and memory usage (MB) with Minibox filtrations of points in R . n = 500 n = 1000 n = 2000 n = 4000 n = 8000 n = 16000 n = 32000Edges time 0.062 0.188 0.586 2.047 7.500 27.898 110.641Sparse matrix time 0.117 0.281 0.742 21.871 4.609 11.289 26.555Dgm , time 0.016 0.055 0.211 0.547 1.664 4.516 12.336Total time 0.195 0.523 1.539 4.429 13.773 43.703 149.531Peak memory usage 9.22 21.87 54.91 137.25 329.25 770.80 1848.01 Table 5: Timing (seconds) and memory usage (MB) with Minibox filtrations of points in R . n = 500 n = 1000 n = 2000 n = 4000 n = 8000 n = 16000 n = 32000Edges time 0.273 1.648 9.430 54.164 307.078 1657.852 8866.555Sparse matrix time 0.258 0.727 2.055 6.250 15.680 43.516 107.773Dgm , time 0.070 0.227 0.797 2.539 9.320 27.016 107.273Total time 0.601 2.601 12.281 62.953 332.078 1728.383 9081.601Peak memory usage 19.194 51.18 155.44 410.41 1122.84 2841.05 7960.18 Table 6: Timing (seconds) and memory usage (MB) with ˇCech filtrations of points in R . n = 500 n = 1000 n = 2000 n = 4000 n = 8000Sparse matrix time 0.656 2.758 11.047 44.789 178.727Dgm , time 0.133 0.602 2.958 13.312 66.219Total time 0.789 3.359 14.005 58.101 244.945Peak memory usage 42.05 151.14 614.13 2532.38 10340.73 In Windows this was measured using the Win32 function
GetProcessMemoryInfo() to obtain the
PeakWorkingSetSize memory attribute of the Python process building sparse matrices and computing persistence diagrams. a) Alpha flag diagrams of S . (b) Minibox diagrams of S . (c) ˇCech diagrams of S .(d) Alpha flag diagrams of S . (e) Minibox diagrams of S . (f) ˇCech diagrams of S . Figure 10: Persistence diagrams of finite sets of three-dimensional points in (cid:96) metric space. Each row contains the diagramsof a different finite point set. These empirically show the equality of diagrams in dimensions zero and one, and illustrate thepossible differences between diagrams of Alpha flag, Minibox, and ˇCech filtrations in homological dimension two. Differences in higher-dimensional diagrams.
We present two examples of Alpha flag, Minibox, and ˇCechpersistence diagrams, obtained from distinct S , S Ď p R d , d q . These finite point sets were obtained byrandomly sampling fifty points in r , s . The persistence diagrams were calculated with Ripser.py passingin the appropriate space matrix. For the Alpha flag case the edges belonging to the Delaunay complex of S and S were computed with a brute force strategy using the result of Proposition 3.3, i.e. checking if A ¯ re iscovered by Ť y P S z e B ¯ r p y q for each pair p, q P S .The first row in Figure 10 contains the diagrams of S . In this case Dgm p K M R q contains a point atinfinity, while Dgm p K AF R q does not. Furthermore, both contain additional off-diagonal points, which do notcoincide. In the second row of Figure 10 we have the diagrams of S , and in this case it is Dgm p K AF R q that contains a point at infinity, while Dgm p K M R q only has an additional off-diagonal point. This showsthat it is possible to obtain Alpha flag and Minibox diagrams with off-diagonal points not contained in thecorresponding ˇCech diagrams in homological dimensions higher than one. Furthermore, Dgm p K AF R q andDgm p K M R q are generally different, and are not one a subset of the other.
8. Discussion
In this paper, we prove that Alpha and ˇCech filtrations are equivalent for points in p R , d q , and showa counterexample to this equivalence for higher-dimensional points. Then, we introduce two new types ofproximity filtrations: the Alpha flag and Minibox filtrations. We are able to prove that both of these producethe same persistence diagrams of ˇCech filtrations in homological dimensions zero and one. We also describe21lgorithms for finding the edges of Minibox complexes. In particular, we present an O p n q algorithm fortwo-dimensional points, a O p n log p n qq algorithm for three-dimensional points, and reference [17] to obtaina O p n log d ´ p n qq algorithm for d -dimensional points. Moreover, in the case of randomly sampled points, weprove that the expected number of these edges is proportional to n ¨ polylog p n q . Thus in many cases Miniboxfiltrations can be seen as a tool to drastically reduce the number of simplices to be considered in order tocompute (cid:96) - ˇCech persistence diagrams in homological dimensions zero and one. We also provide a numberof computational experiments involving Minibox and ˇCech filtrations of randomly sampled points in two,three, and four-dimensional space. These show that the reduced number of simplices contained in Miniboxfiltrations results in a speed up of persistent homology computations, as well as in less memory being usedfor the same number of points. 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Bauer, Ripser: efficient computation of vietoris-rips persistence barcodes, arXiv e-prints (2019). arXiv:1908.02518 . 24 . (cid:96) -Delaunay Edges In this section we provide a proof of the characterization of (cid:96) -Delaunay edges in terms of witness points.Using this, we also show that Delaunay and Alpha complexes of two-dimensional points in (cid:96) metric are flagcomplexes.We recall that we define a box to be an axis-parallel hyperrectangle, i.e. the Cartesian product of d intervals in R d . As discussed in Section 2, we have that (cid:96) -balls are boxes with sizes of length 2 r and thata finite set of (cid:96) -balls has a non-empty intersection if and only if all pairwise intersections of (cid:96) -balls arenon-empty. Moreover, we use the notion of general position defined in the main paper. So the intersections of (cid:96) -Voronoi regions of points in R are never degenerate. We start by showing some properties of ε -thickenings,which we use throughout this section. Proposition A.1. (i)
Let I , I Ď R be two non-empty closed intervals. If I X I ‰ H , then ε p I X I q “ ε p I q X ε p I q . (ii) Let B , B Ď R be two non-empty boxes. If B X B ‰ H , then ε p B X B q “ ε p B q X ε p B q . (iii) Taking ε -thickenings preserves inclusions. (iv) Let A “ t A u i P I be a finite collection of sets. The ε -thickening of the union of sets in A is equal to theunion of the ε -thickenings of sets in A .Proof. (i) We have I “ r a , b s and I “ r a , b s , with I X I ‰ H . So either one of the two intervals iscontained in the other or they share a common subinterval. In the first case, we can suppose without loss ofgenerality I Ď I . Then ε p I X I q “ ε p I q “ r a ´ ε, b ` ε s “ r a ´ ε, b ` ε s X r a ´ ε, b ` ε s “ ε p I q X ε p I q .In the latter case, we can assume without loss of generality that I X I “ r a , b s , and we have ε p I X I q “r a ´ ε, b ` ε s “ r a ´ ε, b ` ε s X r a ´ ε, b ` ε s “ ε p I q X ε p I q . (ii) Follows from property (i) and the definition of box. (iii)
We consider
A, B Ď R d such that A Ď B . Given any x P ε p A qz A , by the definition of ε -thickeningthere exists a P A such that d p x, a q ď ε . We have that x P ε p B q , because a P B and d p x, a q ď ε . So ε p A qz A Ď ε p B q , and because A Ď ε p B q it follows that ε p A q Ď ε p B q . (iv) For any set A Ď R d , we have ε p A q “ Ť x P A B ε p x q . Thus ε ˜ď i P I A i ¸ “ ď x P Ť i P I A i B ε p x q “ ď i P I ď x P A i B ε p x q “ ď i P I ε p A i q We now give the proof of the main result of this section, and state the definition of witness point asprovided in the main paper for completeness.
Definition A.2. A witness point of σ P K Ar is a point z such that z P Ş p P σ V p ‰ H and d p z, p q “ max p,q P σ d p p,q q for each p P σ . Proposition A.3.
Let S be a finite set of points in p R d , d q . Given a subset e “ t p, q u Ď S , we define A ¯ re “ B B ¯ r p p q X B B ¯ r p q q , where ¯ r “ d p p,q q . We have that A ¯ rσ “ B ¯ r p p q X B ¯ r p q q is a non-empty box. Moreover,the set of witness points of e is Z e “ A ¯ re z ` Ť y P S z e B ¯ r p y q ˘ , and e belongs to the (cid:96) -Delaunay complex of S ifand only if Z e is non-empty.Proof. A ¯ re is the intersection of the boundaries of the closed balls B ¯ r p p q and B ¯ r p q q , which are axis-parallelhypercubes. So we have A ¯ re Ď B ¯ r p p q X B ¯ r p q q , because B B ¯ r p p q Ď B ¯ r p p q and B B ¯ r p q q Ď B ¯ r p q q . Moreover B ¯ r p p q X B ¯ r p q q Ď B B ¯ r p p q X B B ¯ r p q q “ A ¯ re , because we can show a contradiction if ` B ¯ r p p q X B ¯ r p q q ˘ z A ¯ re isnon-empty. In particular, given y P ` B ¯ r p p q X B ¯ r p q q ˘ z A ¯ re we have d p y, p q ď ¯ r and d p y, q q ď ¯ r , and atleast one of these two distances must be strictly less than r , i.e. d p y, p q ă ¯ r or d p y, q q ă ¯ r . Applying thetriangular inequality to these distances, we obtain ¯ r ` ¯ r ą d p p, y q ` d p q, y q ě d p p, q q “ r , which is the25 igure A.11: The ε -thickening of the non-empty intersection of two squares equals the intersection of the ε -thickenings of thesquares. desired contradiction. Thus A ¯ re “ B ¯ r p p q X B ¯ r p q q , which is the Cartesian product of the intervals defining B ¯ r p p q and B ¯ r p q q , because Cartesian product and intersections commute, and is non-empty by definition of¯ r . Furthermore A ¯ r ` εe “ B B ¯ r ` ε p p q X B B ¯ r ` ε p q qĎ B ¯ r ` ε p p q X B ¯ r ` ε p q q“ ε p B ¯ r p p qq X ε p B ¯ r p q qq “ ε p A ¯ re q , because we can apply Proposition A.1 (ii) with ε p A ¯ re q “ ε p B ¯ r p p q X B ¯ r p q qq . Hence A ¯ r ` εe Ď ε p A ¯ re q for any ε ě
0. With this result we can show the equivalence of e being a (cid:96) -Delaunay edge and Z e being non-empty.( ñ ) The pair e “ t p, q u is a Delaunay edge, so V p X V q ‰ H . Equivalently there exist ε ě z P R d such that z P A ¯ r ` εe z ` Ť y P S z e B ¯ r ` ε p y q ˘ , where ¯ r “ d p p,q q .We prove this direction of the result by contradiction. Let us suppose A ¯ re is covered by Ť y P S z e B ¯ r p y q , i.e. Z e is empty. We know that A ¯ r ` εe Ď ε p A ¯ re q from the discussion above, and applying Proposition A.1 (iii) and (iv) we derive the following inclusions A ¯ r ` εe Ď ε ˆ A ¯ re ˙ Ď ε ˆ ď y P S z e B ¯ r p y q ˙ “ ď y P S z e B ¯ r ` ε p y q , for any ε ě
0. Thus A ¯ r ` εe Ď Ť y P S z e B ¯ r ` ε p y q , which contradicts the existence of z P A ¯ r ` εe z ` Ť y P S z e B ¯ r ` ε p y q ˘ for any ε ě ð ) Any point in Z e ‰ H belongs to V p X V q , so that e P K D .The above result is illustrated in Figure A.12, which shows how this characterization of Delaunay edgesdoes not hold in Euclidean metric. Proposition A.4.
Let S be a finite set of points in general position in p R , d q and r ě . Both theDelaunay complex K D and the Alpha complex K Ar of S are flag complexes and e “ t p, q u P K Ar if and onlyif d p p,q q ď r .Proof. Let us consider three points x , x , x Ď S , such that t x , x u , t x , x u and t x , x u are Delaunayedges. Without loss of generality, we can assume t x , x u to be the longest edge. Defined ¯ r “ d p x ,x q ,and A ¯ rx x “ B B ¯ r p x q X B B ¯ r p x q , by Proposition A.3 we have that A ¯ rx x “ B ¯ r p x q X B ¯ r p x q , which is anon-empty axis-parallel line segment of length less than 2¯ r . Moreover, by definition of ¯ r , the intersections B ¯ r p x q X B ¯ r p x q , B ¯ r p x q X B ¯ r p x q , and B ¯ r p x q X B ¯ r p x q are non-empty, so that by the properties of (cid:96) -balls A ¯ rx x X B ¯ r p x q is. If A ¯ rx x z B ¯ r p x q “ H , then A ¯ rx x is covered by B ¯ r p x q , which is in contradiction26 a) (b)(c) (d) Figure A.12: In (a)
Euclidean balls centered in p , q intersect in a point which is covered by the ball centered in y . As the radiusgrows in (b) this intersection is not covered by the ball centered in y , so that z is a witness point of e “ t p, q u . In (c) (cid:96) -ballscentered in p , q intersect in A ¯ re which is covered by the (cid:96) -ball centered in y . Again the radius grows in (d) , but in this casethe (cid:96) -ball centered in y covers A ¯ r ` εe . with t x , x u being a Delaunay edge from Proposition A.3. On the other hand, if A ¯ rx x z B ¯ r p x q ‰ H ,then line segment A ¯ rx x must intersect the boundary of the square B ¯ r p x q . Defined τ “ t x , x , x u and A ¯ rτ “ A ¯ rx x X B B ¯ r p x q , we have that the set of witness points of τ is Z τ “ A ¯ rτ z ` Ť y P S z τ B ¯ r p y q ˘ .Hence if Z τ is non-empty, we can conclude that the Delaunay complex of S is a clique complex from thedefinition of witness point. We suppose by contradiction that Z τ “ H , and show that in every possible caseone between t x , x u , t x , x u , and t x , x u cannot be a Delaunay edge.We know that the axis-parallel square B ¯ r p x q intersects A ¯ rx x without covering it, so that A ¯ rτ is a pointby our general position assumption. To simplify the exposition, we assume without loss of generality A ¯ rx x to be a vertical line segment in R , and B ¯ r p x q to be intersecting A ¯ rx x from below. More precisely, given x “ p x , x q , x “ p x , x q , and x “ p x , x q , we assume d p x , x q “ | x ´ x | “ r ě | x ´ x | , and that x ď min t x , x u . This implies A ¯ rτ Ď A ¯ rx x X B ¯ r p x q “ r a , a s ˆ r a , ˆ b s where a “ max t x , x u ´ ¯ r , a “ max t x , x u ´ ¯ r , b “ min t x , x u ` ¯ r , and ˆ b “ x ` ¯ r . So A ¯ rτ “ p a , ˆ b q ,and because we are assuming by contradiction that A ¯ rτ is covered by balls of radius ¯ r centered in the pointsof S z τ , there exists y P S z τ such that p a , ˆ b q P B ¯ r p y q . Finally, either B ¯ r p y q intersects A ¯ rx x from aboveor from below. These two cases are illustrated in Figures A.13a and A.13b, where the boundary of B ¯ r p y q isrepresented as a dashed line, and A ¯ rτ “ p a , ˆ b q as a red square marker. In the former case B ¯ r p x q Y B ¯ r p y q covers A ¯ rx x , which is in contradiction with t x , x u being a Delaunay edge, by Proposition A.3. In the latter27 a) (b) Figure A.13: Illustration of the last two cases of the proof of Proposition ?? . In (a) the red square marker represents point p a , ˆ b q on A ¯ rx x , which is covered by B ¯ r p y q from above. In (b) the same point is covered by B ¯ r p y q from below. In both (a) and (b) the boundary of B ¯ r p y q is drawn as a dashed line. case, given y “ p y , y q , we have min t x , x u ă y ă max t x , x u , and x ă y ă min t x , x u , because B ¯ r p y q intersects A ¯ rx x without covering it, and contains p a , ˆ b q . As a result y is either in Mini x x orMini x x , depending on y being on the right or left of x , and either t x , x u or t x , x u is a non-Delaunayedge by Proposition 5.2, which again is a contradiction. Thus K D is a flag complex.We conclude by showing that K Ar is also a flag complex. By Proposition A.3 any edge e “ t p, q u is addedinto K Ar at r “ d p p,q q . Moreover, when the longest edge of any Delaunay triangle τ is added at radius ¯ r , also τ is added in K A ¯ r , because from the discussion above there exist a point A ¯ rτ at distance ¯ r from the verticesof τ , which is a witness of this triangle. B. Supporting Results for Proof of Alpha Flag and ˇCech Equivalence
In this section we present results used in the proof of Theorem 4.3.
Proposition B.1.
Let B and B be two boxes in R d . If B X B is non-empty, then the Euclidean projection π B : B Ñ B , defined by mapping each x P B to its closest points in Euclidean distance on B , is suchthat π B p B q Ď B X B .Proof. Let B “ ś di “ r a B i , b B i s and B “ ś di “ r a B i , b B i s such that B X B ‰ H . Because Cartesianproducts and intersections of intervals commute, we have that r a B i , b B i s X r a B i , b B i s “ r ¯ a i , ¯ b i s ‰ H for each1 ď i ď d , and B X B “ ś di “ r ¯ a i , ¯ b i s .Given x P B , we suppose by contradiction that y “ π B p x q P B is such that y R B X B . Thus y R ś di “ r ¯ a i , ¯ b i s , and there exists 1 ď ˆ i ď d such that y ˆ i R r ¯ a ˆ i , ¯ b ˆ i s . The intervals r a B ˆ i , b B ˆ i s and r a B ˆ i , b B ˆ i s canintersect in four possible ways: (i) r a B ˆ i , b B ˆ i s intersects r a B ˆ i , b B ˆ i s on the left, i.e. a B ˆ i ď a B ˆ i ď b B ˆ i ď b B ˆ i . Thus a B ˆ i ď x ˆ i ď b B ˆ i ă y ˆ i , andwe define y “ r y , . . . , b B ˆ i , . . . , y d s ; (ii) r a B ˆ i , b B ˆ i s intersects r a B ˆ i , b B ˆ i s on the right, i.e. a B ˆ i ď a B ˆ i ď b B ˆ i ď b B ˆ i . Thus y ˆ i ă a B ˆ i ď x ˆ i ď b B ˆ i ,and we define y “ r y , . . . , a B ˆ i , . . . , y d s ; (iii) r a B ˆ i , b B ˆ i s is contained in r a B ˆ i , b B ˆ i s , i.e. a B ˆ i ď a B ˆ i ď b B ˆ i ď b B ˆ i . Thus a B ˆ i ď x ˆ i ď b B ˆ i ă y ˆ i or y ˆ i ă a B ˆ i ď x ˆ i ď b B ˆ i , and in the first case we define y “ r y , . . . , b B ˆ i , . . . , y d s and in the second y “ r y , . . . , a B ˆ i , . . . , y d s ; 28 iv) r a B ˆ i , b B ˆ i s contains r a B ˆ i , b B ˆ i s , i.e. a B ˆ i ď a B ˆ i ď b B ˆ i ď b B ˆ i .In case (iv) we have a contradiction as y ˆ i P r a B ˆ i , b B ˆ i s “ r ¯ a ˆ i , ¯ b ˆ i s S y ˆ i . In the other three cases, taken either y or y we have d p x, y q “ gffe p x ˆ i ´ b B ˆ i q ` d ÿ i “ ,i ‰ ˆ i p x i ´ y i q ă gffe d ÿ i “ p x i ´ y i q “ d p x, y q , (B.1) d p x, y q “ gffe p x ˆ i ´ a B ˆ i q ` d ÿ i “ ,i ‰ ˆ i p x i ´ y i q ă gffe d ÿ i “ p x i ´ y i q “ d p x, y q . (B.2)because p x ˆ i ´ b B ˆ i q ă p x ˆ i ´ y ˆ i q in Equation (B.1), and p x ˆ i ´ a B ˆ i q ă p x ˆ i ´ y ˆ i q in Equation (B.2). The prooffollows because this contradicts y being the closest point in Euclidean distance to x in B . Proposition B.2.
Let S be a finite set of points in p R d , d q . Given e “ t p, q u Ď S , we have thatNrv pt B ¯ r p y qu y P ¯ Y q has the homotopy type of A ¯ re , where ¯ r “ d p p,q q and ¯ Y “ t y P S | d p y, p q ă r and d p y, q q ă r u .Proof. From the Nerve Theorem 3.1 it follows that Nrv pt B ¯ r p y quq y P ¯ Y q and Ť y P ¯ Y B ¯ r p y q are homotopy equiv-alent, because convex sets are contractible. We show how to define a deformation retraction φ : ˆ ď y P ¯ Y B ¯ r p y q ˙ ˆ r , s Ñ A ¯ re . Given φ , we have that the sets Ť y P ¯ Y B ¯ r p y q has the homotopy type of A ¯ re , which is contractible by itsconvexity. To obtain φ , we first define φ y : B ¯ r p y q ˆ r , s Ñ A ¯ re for each y P ¯ Y . Given the Euclideanprojection π B ¯ r p y q : B ¯ r p y q Ñ A ¯ re , we set φ y p x, t q “ p ´ t q ¨ x ` t ¨ π B ¯ r p y q p x q , for every x P B ¯ r p y q and t P r , s . From Proposition B.1 we have π B ¯ r p y qq p x q P B ¯ r p y q X A ¯ re , so that thestraight line segment from x to π B ¯ r p y q p x q is fully contained in B ¯ r p y q , by the convexity of this set. Thus φ y iswell-defined and continuous by the continuity of π B ¯ r p y q . Then we set φ p x, t q “ φ ˆ y p x, t q , for every x P Ť y P ¯ Y B ¯ r p y q and t P r , s , with ˆ y P ¯ Y such that x P B ¯ r p ˆ y q . This might not be well-defined,because for a given x all the φ ˆ y corresponding to a point in ¯ Y x “ t ˆ y P ¯ Y | x P B ¯ r p ˆ y qu can be used to define φ p x, t q for any t P r , s . Luckily, given R “ Ş ˆ y P ¯ Y x B ¯ r p ˆ y q , which is a box containing x , Proposition B.1guarantees that π R : R Ñ A ¯ re is such that π R p R q Ď R X A ¯ re . Thus φ is well-defined because the straight linesegment defined by p ´ t q ¨ x ` t ¨ π R p x q for t P r , s is contained within R , again by convexity. Furthermore, φ is continuous by the continuity of the Euclidean projections π B ¯ r p y q , and is a deformation retraction onto A ¯ re because A ¯ re Ď Ť y P ¯ Y B ¯ r p y q . Proposition B.3.
Let K and K be two abstract simplicial complexes such that K Ď K . If there is onlyone edge e contained in K and not in K , and it exists a triangle τ P K of which e is a face, then H p K q cannot contain an homology class r γ s not in H p K q .Proof. Any 1-cycle representing an homology class r γ s such that r γ s P H p K q and r γ s R H p K q must contain e . But given e “ t p, q u and τ “ t p, q, y u , any such 1-cycle would be homologous to a formal sum containing t p, y u and t y, q u in place of e . Thus it would exist a 1-cycle representing r γ s containing edges in K only,which is in contradiction with r γ s R H p K q . 29 . NotationPreliminaries • St p τ q star of τ Ď K , Lk p τ q link of τ Ď K . • B r p p q open ball, B r p p q closed ball, B B r p p q boundary of closed ball. • V p Voronoi region, K D Delaunay complex.
Persistent Homology • K R filtration of K parameterized by R “ t r i u mi “ . • M k p K R q is the k -th persistence module of K R . • d B p Dgm k p K q , Dgm k p K qq is the bottleneck distance. • K V R R VR filtration, K ˇ C R ˇCech filtration, K A R Alpha filtration.