AAPPROXIMATE TRIANGULATIONS OF GRASSMANNMANIFOLDS
KEVIN P. KNUDSON
Abstract.
We define the notion of an approximate triangulation for amanifold M embedded in euclidean space. The basic idea is to build anested family of simplicial complexes whose vertices lie in M and usepersistent homology to find a complex in the family whose homologyagrees with that of M . Our key examples are various Grassmann man-ifolds G k ( R n ). Introduction
Smooth manifolds admit piecewise-linear triangulations [14]. However,there are many subsequent questions one might ask: How many simplicesare required? What is the minimal number of vertices? Is there an algorithmto construct a triangulation?A great deal of work in algebraic topology has been devoted to thesequestions. The question of the number of simplices required to triangulatea given manifold is often attacked by sophisticated cohomological methodsinvolving characteristic classes (such arguments also often yield estimateson the minimal embedding dimension for the manifold). Surprisingly, muchof this work is very recent [6], [7]. A main result in [7] is the following.
Theorem 1.1 ([7], Theorem 3.10) . Every triangulation of the Grassmannmanifold G k ( R n + k ) must have at least [( n + k )( n + k + 1) − kn ] · (2 kn +1 − simplices. For example, any triangulation of the manifold G ( R ) must have at least372 simplices. The Grassmann manifolds will be defined in Section 2.1 be-low. These are important spaces to study because of their utility in algebraictopology, especially with respect to the study of characteristic classes [11].Unfortunately, most results along these lines are not constructive; that is,the proofs do not yield an explicit triangulation of the manifold. In fact, ifone seeks a triangulation of a Grassmannian G k ( R n ) the end result is usuallydisappointment. For the smallest nontrivial space, G ( R ) = R P , there are Date : June 26, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Grassmannian; persistent homology; Vietoris-Rips complex;witness complex; triangulation. a r X i v : . [ m a t h . A T ] J un KEVIN P. KNUDSON many well-known small triangulations, and even an algorithm to generate atriangulation from any collection of points in general position [1]. Beyondthat, however, results are sparse.In this paper, we develop a procedure to find what we call an approximatetriangulation of the manifold G k ( R n ) (Definition 2.8). The basic idea is tofirst generate a sample of points on G k ( R n ). This already leads to technicaldifficulties involving embeddings of these spaces into a euclidean space R N ,but we are able to solve this. We then build a nested family of simplicialcomplexes on the point cloud, parametrized by the positive real numbers.The persistent homology of this family is then computed and we identifyan interval of parameters for which the mod 2 homology of the complexesin that range agrees with that of G k ( R n ). Such a complex is then a viablemodel for the manifold: its vertices lie in G k ( R n ) ⊂ R N and it has the cor-rect homology. We then implement this procedure for the following spaces: R P ⊂ R , R P ⊂ R , R P ⊂ R , and G ( R ) ⊂ R . Computational limi-tations have so far prohibited further calculations; we discuss this in Section4. Acknowledgments.
This problem was suggested to me by Vidit Nanda;I thank him for the inspiration and helpful conversations. Henry Adamsprovided useful tips for Javaplex. I am also grateful to Mikael Vejdemo-Johansson for the use of his rather powerful computer.2.
Materials and Methods
Further details and proofs of the results in Subsections 2.1 and 2.2 maybe found in [11].2.1.
Grassmann manifolds.
Denote by R n the euclidean space of dimen-sion n . By a k -frame in R n we mean a k -tuple of linearly independentvectors; denote by V k ( R n ) the collection of k -frames in R n . This is an opensubset of the k -fold cartesian product R n × · · · × R n . Definition 2.1.
The
Grassmann manifold G k ( R n ) is the set of all k -dimensionalplanes through the origin in R n . It is topologized via the quotient map V k ( R n ) → G k ( R n ) which takes a k -frame to the k -plane it spans.When k = 1, we see that G ( R n ) is the real projective space R P n − , amanifold of dimension n −
1. In general we have the following result.
Lemma 2.2.
The Grassmannian G k ( R n ) is a compact manifold of dimen-sion k ( n − k ) . The map X → X ⊥ , which takes a k -plane to its orthogonalcomplement is a diffeomorphism between G k ( R n ) and G n − k ( R n ) . Schubert cells.
Grassmannians have a well-known cell decompostioninto Schubert cells. Consider the sequence of subspaces of R n : R ⊂ R ⊂ R ⊂ · · · ⊂ R n , where R i consists of the vectors of the form ( a , . . . , a i , , . . . , k -plane X gives rise to a sequence of integers0 ≤ dim( X ∩ R ) ≤ dim( X ∩ R ) ≤ · · · ≤ dim( X ∩ R n ) = k. PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 3
Consecutive integers differ by at most 1.
Definition 2.3. A Schubert symbol σ = ( σ , . . . , σ k ) is a sequence of k integers satisfying 1 ≤ σ < σ < · · · < σ k ≤ n. Given a Schubert symbol σ , let e ( σ ) ⊂ G k ( R n ) denote the set of k -planes X such that dim( X ∩ R σ i ) = i, dim( X ∩ R σ i − ) = i − . Each X ∈ G k ( R n ) belongs to precisely one of the sets e ( σ ). Lemma 2.4. e ( σ ) is an open cell of dimension d ( σ ) = ( σ −
1) + ( σ −
2) + · · · + ( σ k − k ) . In terms of matrices, X ∈ e ( σ ) if and only if it can be described as therow space of a k × n matrix of the form ∗ · · · ∗ · · · · · · · · · ∗ · · · ∗ ∗ ∗ · · · ∗ · · · · · · ∗ · · · ∗ ∗ ∗ · · · ∗ ∗ ∗ · · · ∗ · · · where the i -th row has σ i -th entry positive (say equal to 1) and all subsequententries zero. Equivalently, we could (and do in the sequel) consider thecolumn space of the transpose of this matrix. Theorem 2.5.
The (cid:0) nk (cid:1) sets e ( σ ) form the cells of a CW-decomposition of G k ( R n ) . Proposition 2.6.
The number of r -cells in G k ( R n ) is equal to the numberof partitions of r into at most k integers each of which is ≤ n − k . For example, the possible Schubert symbols and cells for G ( R ) are asfollows. Such a symbol has the form σ = ( σ , σ ) where 1 ≤ σ < σ ≤ σ d ( σ )(1 ,
2) 0(1 ,
3) 1(1 ,
4) 2(2 ,
3) 2(2 ,
4) 3(3 ,
4) 4The mod 2 homology of G k ( R n ) is easily computed from the Schubertcell decomposition: since the induced boundary maps are all either 0 ormultiplication by 2, the mod 2 homology has basis corresponding to thecells. KEVIN P. KNUDSON
Continuing the example of G ( R ), we have H i ( G ( R ) , Z /
2) = Z / i = 0 Z / i = 1 Z / ⊕ Z / i = 2 Z / i = 3 Z / i = 42.3. Persistent Homology.
Suppose we are given a finite nested sequenceof finite simplicial complexes K R ⊂ K R ⊂ · · · ⊂ K R p , where the R i are real numbers R < R < · · · < R p . For each homologicaldegree (cid:96) ≥
0, we then obtain a sequence of homology groups and inducedlinear transformations (homology with Z / H (cid:96) ( K R ) → H (cid:96) ( K R ) → · · · → H (cid:96) ( K R p ) . Since the complexes are finite, each H (cid:96) ( K R i ) is a finite-dimensional vectorspace. Thus, there are only finitely many distinct homology classes. Aparticular class z may come into existence in H (cid:96) ( K R s ), and then one oftwo things happens. Either z maps to 0 (i.e., the cycle representing z getsfilled in) in some H (cid:96) ( K R t ), R s < R t , or z maps to a nontrivial element in H (cid:96) ( K R p ). This yields a barcode , a collection of interval graphs lying abovean axis parametrized by R . An interval of the form [ R s , R t ] corresponds to aclass that appears at R s and dies at R t . Classes that live to K R p are usuallyrepresented by the infinite interval [ R s , ∞ ) to indicate that such classes arereal features of the full complex K R p .As an example, consider the tetrahedron T with filtration T ⊂ T ⊂ T ⊂ T ⊂ T ⊂ T = T defined by T = { v , v , v , v } , T = T ∪ { all edges } , T = T ∪ [ v v v ], T = T ∪ [ v v v ], T = T ∪ [ v v v ], and T = T . The barcodes for thisfiltration are shown in Figure 1. Note that initially, there are 4 components( β = 4), which get connected in T , when 3 independent 1-cycles are born( β = 3). These three 1-cycles die successively as triangles get added in T , T , and T . The addition of the final triangle in T creates a 2-cycle( β = 1).For analyzing point cloud data, one needs a simplicial complex modelingthe underlying space. Since it is impossible to know a priori if a complex is“correct”, one builds a nested family of complexes approximating the datacloud, computes the persistent homology of the resulting filtration, and looksfor homology classes that exist in long sections of the filtration. We discusstwo popular methods for doing this in the next subsection. PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 5
Figure 1.
The barcodes for a filtration of the tetrahedron2.4.
Vietoris-Rips and witness complexes.
Now suppose we are givena discrete set X of points in some metric space (typically a Euclidean space R m ). The standard example of such an object is a sample of points fromsome geometric object M . We would like to recover information about M from the sample X , and the first step is to obtain an approximation of M using only the point cloud X . There are many such techniques; perhaps themost classical is the Delaunay triangulation of X . This is defined as follows.Say X = { x , x , . . . , x r } ⊂ R m . The Voronoi decomposition of R m relativeto X is the partition of R m into cells V ( x i ), i = 1 , . . . , r , defined by V ( x i ) = { x ∈ R m : || x − x i || ≤ || x − x j || , j (cid:54) = i } . The corresponding Delaunay triangulation, Del( X ), is the nerve of theVoronoi decomposition; that is, a collection V ( x i ) , . . . , V ( x i (cid:96) ) forms an (cid:96) -simplex in Del( X ) if ∩ (cid:96)j =0 V ( x i j ) (cid:54) = ∅ . One obtains a geometric realizationof Del( X ) via the map V ( x i ) (cid:55)→ x i . See Figure 2 for an example.While the Delaunay triangulation provides a good approximation to theunderlying space M , it has several disadvantages. If the point cloud X islarge, there will be a very large number of simplices in Del( X ). Also, Del( X )suffers from the “curse of dimensionality;” that is, if the ambient dimension( m ) is large, calculating the Voronoi decomposition is computationally ex-pensive.There are many popular alternatives to the Delaunay triangulation. Theone used most often is the Vietoris-Rips complex , which is built as follows.Consider the point cloud X and let r >
0. The Vietoris-Rips complex withparameter r is the simplicial complex V R ( X, r ) whose k -simplices are { ( x , . . . , x k ) : d ( x i , x j ) < r, i (cid:54) = j } . That is, if one imagines a ball of radius r/ x ∈ X , thenwe join the points x i and x j with an edge if the balls intersect. Observe thatif r < r (cid:48) then there is an inclusion of complexes V R ( X, r ) ⊂ V R ( X, r (cid:48) ). We
KEVIN P. KNUDSON
Figure 2. ( a ) A Delaunay triangulation of a collection ofpoints in the plane with the corresponding Voronoi diagram,and ( b) two associated witness complexestherefore have a nested sequence of complexes { V R ( X, r ) } r ≥ and we maystudy the persistent homology of this filtration. The corresponding barcodesyield information about the topology of the underlying space M .Many software packages support the calculation of Vietoris-Rips persis-tence on point clouds. In this paper, we use the Eirene package developedby Gregory Henselman [9]. Other popular programs include Ulrich Bauer’sRipser [2] and Vidit Nanda’s Perseus [12].In Section 3.5, we shall use the witness complexes of de Silva and Carlsson[5]. The idea is to model the Delaunay triangulation on a smaller set of points L ⊂ X , called landmarks , in such a way that the topology of the underlyingobject is well-approximated. Moreover, the definition makes sense in anymetric space, so assume that X is a metric space with distance function d (e.g., X could be a finite point cloud in R m with the usual Euclideandistance). Choose a subset L = { (cid:96) , (cid:96) , . . . , (cid:96) n } of X = { x , x , . . . , x N } andlet R ≥ witness complex W ( X, L, R ) is defined as follows: • The vertex set of W ( X, L, R ) is L ; PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 7 • (cid:96), (cid:96) (cid:48) ∈ L span an edge if there exists an x ∈ X , called a witness , suchthat d ( x, (cid:96) ) , d ( x, (cid:96) (cid:48) ) ≤ R + min { d ( x, (cid:96) (cid:48)(cid:48) ) : (cid:96) (cid:48)(cid:48) ∈ L − { (cid:96), (cid:96) (cid:48) }} ; • A collection (cid:96) , . . . , (cid:96) p ∈ L spans a p -simplex if { (cid:96) i , (cid:96) j } span an edgefor all i (cid:54) = j .Examples of witness complexes are shown in Figure 2( b ) alongside theassociated Delaunay triangulation. Four landmark points were chosen usingthe maxmin procedure described below. The complex on the left has R = . R = . R yields a complex with more simplices. Also, note that the witnesscomplex is a coarse approximation of the Delaunay triangulation.We make some observations about this definition. Let D be the n × N matrix of distances from points in L to points in X . • If R = 0, then (cid:96), (cid:96) (cid:48) ∈ L form an edge if there is an x i ∈ X such that d ( x i , (cid:96) ) and d ( x i , (cid:96) (cid:48) ) are the two smallest entries in the i -th columnof D . This is analogous to the existence of an edge in the Delaunaytriangulation Del( L ). • For
R >
0, one may think of relaxing the boundaries of the Voronoidiagram of L and taking the nerve of the resulting covering of X . • If 0 ≤ R < R (cid:48) , then there is an inclusion of simplicial complexes W ( X, L, R ) ⊆ W ( X, L, R (cid:48) ).By a theorem of de Silva and Carlsson [5], this complex is a naturalanalogue of the Delaunay triangulation for a space represented by pointcloud data.Suppose that X is a sample of points from some object M ⊂ R m . There isno guarantee that W ( X, L, R ) recovers the topology of M , but experimentson familiar geometric objects [5] (spheres, for example) suggest that for asuitable range of values of R and good choices of landmarks L , the topologyof W ( X, L, R ) is the same as that of M . This begs the questions:(1) How should the landmark set L be chosen?(2) What is the correct value of R ?The second question is best handled via the use of persistent homology,which we discussed in Section 2.3 above. As for the choice of landmarks,there are three standard options:(1) Select landmarks at random.(2) Use the maxmin procedure: Choose a seed (cid:96) at random. Then if (cid:96) , . . . , (cid:96) n have been chosen, let (cid:96) n +1 ∈ X − { (cid:96) , . . . , (cid:96) n } be the pointwhich maximizes the function z (cid:55)→ min { d ( z, (cid:96) ) , d ( z, (cid:96) ) , . . . , d ( z, (cid:96) n ) } . (3) Use a density-based strategy.The maxmin procedure yields more evenly-spaced landmarks, but tendsto emphasize extremal points. It is generally more reliable than a random KEVIN P. KNUDSON selection [5]. Another useful resource is [3]. In our experiments in Section3.5 below we use the maxmin process to generate landmarks.2.5.
Sampling procedures.
To build a Vietoris-Rips or witness complexon points in G k ( R n ), we need to develop a sampling procedure. The firstquestion to be asked is in which euclidean space do we embed G k ( R n )?This is highly nontrivial. Even in the case of projective spaces ( k = 1)it is not so obvious how to proceed. A whole industry has been devotedto the question of the minimal embedding dimension of R P n [4], but theproof of the minimality of any particular embedding rarely comes with anexplicit formula for the map. An exception is if one insists on an isometric embedding [15], but the minimal dimension of such an embedding for R P n is n ( n + 3) /
2, which grows rather quickly.For arbitrary Grassmannians, one could try to use the Pl¨ucker embedding G k ( R n ) → P ( (cid:86) k ( R n )) = R P ( nk ) − defined by( x , . . . , x k ) (cid:55)→ [ x ∧ · · · ∧ x k ](where [ v ] denotes the line spanned by the vector v ) and then embed thetarget projective space into euclidean space. Of course this explodes thedimension further, making this an impractical solution. Aside from somelow dimensional projective spaces, we will instead approach this problemvia the following result. Proposition 2.7.
The manifold G k ( R n ) is diffeomorphic to the smoothmanifold consisting of all n × n symmetric, idempotent matrices of trace k . The map ϕ realizing this takes a k -plane X to the operator defined byorthogonal projection onto X .Proof. If X is a k -plane with orthonormal basis x , . . . , x k , denote by A the n × k matrix having the x i as columns. Define a map ϕ : G k ( R n ) → M n ( R ) by X (cid:55)→ AA T . This map is clearly smooth since it consists ofpolynomials in the entries of the various x i . Note that choosing a differentbasis for X amounts to conjugating AA T by the corresponding change ofbasis matrix. The matrix AA T is symmetric: ( AA T ) T = ( A T ) T A T = AA T .It is idempotent: ( AA T ) = AA T AA T = AI k A T = AA T (note that A T A = I k , the k × k identity matrix, since the columns of A are orthonormal).Finally, the trace of AA T is k since its rank is k and its only eigenvaluesare 0 and 1. Thus the image of ϕ lies in the set of symmetric, idempotentmatrices of trace k . To see that ϕ surjects onto this set, note that such amatrix B is projection onto a k -dimensional subspace X and there exists abasis x , . . . , x k with ϕ ( X ) = B . Injectivity of ϕ follows since the subspacedetermined by a projection is unique. (cid:3) Now, to generate a sample of points on which to build a Vietoris-Rips orwitness complex, we will use the embedding ϕ . A crude sampling is thenobtained by the following procedure. • Select k random vectors in R n . PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 9 • Perform the Gram-Schmidt orthogonalization algorithm to yield anorthornomal set x , . . . , x k . Let A be the matrix with x i as columns. • Compute AA T .One immediate problem with this process is that the k -plane it constructslives in the top-dimensional Schubert cell with probability 1. However, sincewe know the space we are interested in, and we know its homology, we canbias our sample to ensure we include points from each Schubert cell. Thefollowing procedure implements this idea. • Determine the percentage of sample points desired from each Schu-bert cell. For example, one might choose 5% from a 1-cell, 10% froma 2-cell, and so on. • Elements of a given Schubert cell correspond to the column space ofa particular matrix form. Generate such a matrix B using randomvectors of the required form. • Generate a random n × n orthogonal matrix X . • Add the matrix A = X ( BB T ) X T to the point cloud.Note the final step above. If we merely took the matrix B , we wouldnot end up with a well-distributed sample. For example, in the case of G ( R ), such a matrix lying in the 1-cell of the Schubert decomposition hasthe following form B = ∗ The corresponding point in R would have most coordinates equal to 0,which is clearly not what we want. Conjugating the various BB T by arandom orthogonal matrix X (a different X for each B ) yields a widerdistribution of points in G k ( R n ).The MATLAB files we used to generate samples in various projectivespaces and Grassmannians are available at https://github.com/niveknosdunk/grassmann .2.6. Approximate triangulations.
We are now ready to search for sim-plicial complexes modeling the spaces G k ( R n ). The procedure we employ isas follows. • Construct a sample of points on G k ( R n ). • Construct a collection of Vietoris-Rips or witness complexes on thepoint cloud. • Compute the persistent homology of this filtration. • Determine a range of parameters where the homology of the com-plexes agrees with that of G k ( R n ). Definition 2.8.
Let K r denote either V R ( X, r ) or W ( X, L, r ). If thereexists a parameter r > K r agrees with that of G k ( R n ), then we call K r an approximate triangulation of G k ( R n ). Figure 3.
Vietoris-Rips persistence diagrams for 100points on R P ( a ) H persistence and ( b ) H persistence Figure 4.
Vietoris-Rips persistence diagrams for 200points on R P ( a ) H persistence and ( b ) H persistenceNote that K r is a subcomplex of the euclidean space in which we haveembedded G k ( R n ). However, it does not necessarily lie inside the embedded G k ( R n ). Still, its vertices do lie on G k ( R n ) and so we can think of this asbeing close to a triangulation of this manifold.3. Results R P , Part I. Let us begin by embedding R P into R using the map ψ : S → R defined by ψ : ( x, y, z ) (cid:55)→ ( xy, xz, y − z , yz ) . Note that ψ ( − x, − y, − z ) = ψ ( x, y, z ) and so it descends to a map R P → R . Generate a sample of 100 points on S and then use this map to getthe points in R . The persistence diagrams are shown in Figure 3. There isa tiny window, around r = 0 .
87 where we get the correct homology.Now generate a sample of 200 points. As expected the Vietoris-Rips com-plex has the correct homology for a longer range of parameters, as indicatedin Figure 4. Here we see a long interval 0 . < r < .
87 where we get thecorrect homology. So the Vietoris-Rips complex built on these 200 points in R is a good approximation to R P . PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 11
Figure 5.
Vietoris-Rips persistence diagrams for 100points on R P , using the isometric embedding into R ( a ) H persistence and ( b ) H persistence Figure 6.
Vietoris-Rips persistence diagrams for 200points on R P , using the isometric embedding into R ( a ) H persistence and ( b ) H persistence3.2. R P , Part II. The embedding of R P into R is not an isometricembedding, though. For that we need R :( x, y, z ) (cid:55)→ (cid:18) yz, xz, xy,
12 ( x − y ) , √ x + y − z ) (cid:19) If we then generate 100 random points on this surface, we obtain theVietoris-Rips barcodes in Figure 5. This works better than the embeddinginto R ; we get the correct answer for 0 . < r < . R P . We use the fact that R P is diffeomorphic to SO (3), the spaceof 3 × R , we find that there is only a tiny window where β = 1, so 100 points probably is not enough to yield a good approximatetriangulation. The H barcode is shown in Figure 8.If we now sample 200 points at random on R P (computation time 6:54)we obtain the barcodes in Figures 9 and 10. Note that we get the correcthomology for 2 . < r < . Figure 7.
Vietoris-Rips persistence diagrams for 100points on R P , realizing it as the Lie group SO (3) ⊂ R ( a ) H persistence and ( b ) H persistence Figure 8.
The H barcode for 100 points on R P Figure 9.
Vietoris-Rips persistence diagrams for 200points on R P ( a ) H persistence and ( b ) H persistence3.4. G ( R ) , Part I. We now consider the first Grassmannian that is not aprojective space. Embed the 4-manifold G ( R ) as the space of symmetricidempotent 4 × PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 13
Figure 10.
The H barcode for 200 points on R P Table 1.
Computation times for Vietoris-Rips persistenceup to dimension 4 for G ( R ). An X indicates that the soft-ware could not complete the calculation. Figure 11.
Vietoris-Rips persistence diagrams for 150points on G ( R ) ( a ) H persistence and ( b ) H persistenceEirene could compute homology for 200 points up to dimension 3 in about3 minutes, producing a parameter value of r = 0 .
95 where the homology iscorrect in these dimensions. It seems that H is the sticking point. Thebarcodes for 150 points are shown in Figures 11 and 12. At r = 0 .
96, thehomology is correct up to dimension 3, but H = 0 there.In a quest for more memory, we received an offer from Mikael Vejdemo-Johannson to use his machine. It has 256GB RAM. We began the 200 pointVietoris-Rips calculation in Eirene in the background and logged out. After10 hours it was still processing and was using 97% of the system memory.The next morning the process was complete; the output file (in JLD2 format)was 74 GB (!). Since Eirene uses PlotlyJS to render barcodes, they cannotbe viewed remotely. Even if the file could be retrieved, it is unclear that our Figure 12.
Vietoris-Rips persistence diagrams for 150points on G ( R ) ( a ) H persistence and ( b ) H persistencelaptop could even open it, nor is there any guarantee that the barcodes arecorrect.3.5. G ( R ) , Part II. We then took a different approach. The Vietoris-Rips complex is nice because it is easy to compute, but it suffers fromcombinatorial explosion. We turned to witness complexes and made theassociated computations using the Javaplex package [10] in MATLAB.The initial attempt simply generated elements of G ( R ) by taking a pairof orthonormal vectors in R and using them to build a certain 4 × M of points on G ( R ), we took 5% from the 1-cell, 15% from eachof the 2-cells, 25% from the 3-cell, and 40% from the 4-cell. One couldchoose different proportions, of course.This worked remarkably well. We generated 5000 points on G ( R ) andconstructed the witness complex on 100 landmarks chosen using the max-min process. The barcodes for one such trial are shown in Figure 13. Notethat we get the correct homology for r > . G ( R ). The point cloud and witness points are available as text files at https://github.com/niveknosdunk/grassmann .4. Conclusions
In this paper we demonstrated the utility of using Vietoris-Rips and wit-ness complexes to obtain approximate triangulations of the Grassmann man-ifolds G k ( R n ). We were able to construct such spaces with relatively fewvertices, but some questions remain for further study.(1) How small of a sample can we use to generate an approximate trian-gulation? For example, a result in [7] asserts that any triangulationof G ( R ) must have at least 14 vertices. We built an approximatetriangulation using a witness complex on 100 landmarks. Surely our PPROXIMATE TRIANGULATIONS OF GRASSMANN MANIFOLDS 15
Figure 13.
Barcodes for a witness complex on 100 pointsin a 5000-point sample on G ( R )algorithm will not work with only 14 points, but we plan to inves-tigate how few we can get away with. A theorem of Niyogi-Smale-Weinberger [13] provides lower bounds on the number of points re-quired to compute homology correctly with high probability, butthese are certainly too high and can be improved in practice.(2) Can we push the computations further? The next Grassmannian tostudy is G ( R ). This is a 6-manifold, and using our procedure wewould embed it in R . The machine used to compute the persistenthomology of the witness complexes on G ( R ) in MATLAB ran outof memory on 100 landmarks in G ( R ). We therefore need eithera bigger machine running MATLAB, or software that can handlewitness complexes. The GUDHI package [8] is one option, but wehave not attempted it yet.(3) The author expects to gain access to a new GPU based supercom-puter at his institution in the next year. This may allow for similarcomputations on higher-dimensional G k ( R n ). References [1] Aanjaneya, M. and Teillaud, M. Triangulating the real projective plane,
MACISproceedings (2007).[2] Bauer, U. Ripser, a software package for computing persistent homology, https://github.com/Ripser/ripser , accessed 06/22/2020.[3] Chazal, F. and Oudot, S. Towards persistence-based reconstruction in Euclideanspaces,
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