Computation of Persistent Homology

Transcription

1 Computation of Persistent Homology Part 2: Efficient Computation of Vietoris Rips Persistence Ulrich Bauer TUM August 14, 2018 Tutorial on Multiparameter Persistence, Computation, and Applications Institute for Mathematics and its Applications 1 / 29

2 Vietoris Rips persistent homology

3 Vietoris Rips filtrations Consider a finite metric space (X, d). The Vietoris Rips complex is the simplicial complex Rips t (X) = {S X diam S t} 1-skeleton: all edges with pairwise distance t all possible higher simplices (flag complex) 2 / 29

4 Vietoris Rips filtrations Consider a finite metric space (X, d). The Vietoris Rips complex is the simplicial complex Goal: Rips t (X) = {S X diam S t} 1-skeleton: all edges with pairwise distance t all possible higher simplices (flag complex) compute persistence barcodes for H d (Rips t (X)) (in dimensions 0 d k) 2 / 29

5 Demo: Ripser Example data set: 192 points on S 2 persistent homology barcodes up to dimension 2 over 56 mio. simplices in 3-skeleton 3 / 29

6 Demo: Ripser Example data set: 192 points on S 2 persistent homology barcodes up to dimension 2 over 56 mio. simplices in 3-skeleton Comparison with other software: javaplex: 3200 seconds, 12 GB Dionysus: 533 seconds, 3.4 GB GUDHI: 75 seconds, 2.9 GB DIPHA: 50 seconds, 6 GB Eirene: 12 seconds, 1.5 GB 3 / 29

7 Demo: Ripser Example data set: 192 points on S 2 persistent homology barcodes up to dimension 2 over 56 mio. simplices in 3-skeleton Comparison with other software: javaplex: 3200 seconds, 12 GB Dionysus: 533 seconds, 3.4 GB GUDHI: 75 seconds, 2.9 GB DIPHA: 50 seconds, 6 GB Eirene: 12 seconds, 1.5 GB Ripser: 1.2 seconds, 152 MB 3 / 29

8 Ripser A software for computing Vietoris Rips persistence barcodes about 1000 lines of C++ code, no external dependencies support for coefficients in a prime field F p sparse distance matrices for distance threshold open source ( released in July 2016 online version ( launched in August 2016 most efficient software for Vietoris Rips persistence computes H 2 barcode for random points on a torus in 136 seconds / 9 GB (using distance threshold) 2016 ATMCS Best New Software Award (jointly with RIVET) 4 / 29

9 Design goals Goals for previous persistence software projects: PHAT [B, Kerber, Reininghaus, Wagner 2013]: fast persistence computation (matrix reduction only), comparing different algorithms and data sturctures DIPHA [B, Kerber, Reininghaus 2014]: distributed persistence computation, based on spectral sequence algorithm 5 / 29

10 Design goals Goals for previous persistence software projects: PHAT [B, Kerber, Reininghaus, Wagner 2013]: fast persistence computation (matrix reduction only), comparing different algorithms and data sturctures DIPHA [B, Kerber, Reininghaus 2014]: distributed persistence computation, based on spectral sequence algorithm Goals for Ripser: Use as little memory as possible Be reasonable about computation time 5 / 29

11 The four special ingredients Main improvements to the standard reduction algorithm: Clearing inessential columns [Chen, Kerber 2011] Computing cohomology [de Silva et al. 2011] Implicit matrix reduction Apparent and emergent pairs 6 / 29

12 The four special ingredients Main improvements to the standard reduction algorithm: Clearing inessential columns [Chen, Kerber 2011] Computing cohomology [de Silva et al. 2011] Implicit matrix reduction Apparent and emergent pairs Lessons from PHAT: Clearing and cohomology yield considerable speedup, but only when both are used in conjuction! 6 / 29

13 Filtrations and refinements

14 Simplexwise filtrations Call a filtration (K i ) i I of simplicial complexes (I totally ordered) essential if i j implies K i K j, simplexwise if for all i I with K i there is some index j < i I and some simplex σ K such that K i K j = {σ}. 7 / 29

15 Simplexwise filtrations Call a filtration (K i ) i I of simplicial complexes (I totally ordered) Note: essential if i j implies K i K j, simplexwise if for all i I with K i there is some index j < i I and some simplex σ K such that K i K j = {σ}. These properties are natural assumptions for computation. The Vietoris Rips filtration (indexed by R) is not simplexwise (and not essential). To compute Vietoris Rips persistence, we will reindex and refine. 7 / 29

16 Reindexing and refinement A filtration K I Simp is a reindexing of another filtration F R Simp if F = K r for some monotonic map r R I. If r is injective, we call K a refinement of F; if r is surjective, we call K a reduction of F. 8 / 29

17 Reindexing and refinement A filtration K I Simp is a reindexing of another filtration F R Simp if F = K r for some monotonic map r R I. If r is injective, we call K a refinement of F; if r is surjective, we call K a reduction of F. The Vietoris Rips filtration can be reindexed: reduced to an essential filtration (over the set of pairwise distances), and further refined to a simplexwise filtration 8 / 29

18 Reindexing and refinement A filtration K I Simp is a reindexing of another filtration F R Simp if F = K r for some monotonic map r R I. If r is injective, we call K a refinement of F; if r is surjective, we call K a reduction of F. The Vietoris Rips filtration can be reindexed: reduced to an essential filtration (over the set of pairwise distances), and further refined to a simplexwise filtration Ripser use the lexicographic refinement: simplices ordered by diameter, then dimension, then (decreasing) lexicographic order of vertices {v ik,..., v i0 } (induced by fixed total order v 0 < < v n 1 on vertices) 8 / 29

19 Enumerating (co)faces in lexicographic order There is an order-preserving bijection [v 2, v 1, v 0 ] [v 3, v 1, v 0 ] [v 3, v 1, v 0 ] [v 3, v 2, v 0 ] [v n 3, v n 2, v n 1 ] ( n k+1 1 from k-simplices (as ordered tuples, ordered lexicographically) to natural numbers (called combinatorial number system). One direction is given by [v ik,..., v i0 ] k j=0 ( i j j + 1 ) The other direction can also be computed quickly Using this bijection, faces and cofaces can be efficiently enumerated in (decreasing) lexicographic order 9 / 29

20 Persistence barcodes through reindexing Consider a filtration F R Simp with reindexing F = K r, K I Simp, r R [n]. The persistence barcode of K determines the persistence barcode of F : B(H (F )) = {r 1 (I) I B(H (K ))} / 29

21 Matrix reduction

22 Computing homology Computing homology H = Z /B : compute basis for boundaries B = im extend to basis for cycles Z = ker new (non-boundary) basis cycles generate quotient Z /B 11 / 29

23 Computing homology Computing homology H = Z /B : compute basis for boundaries B = im extend to basis for cycles Z = ker new (non-boundary) basis cycles generate quotient Z /B Computing persistent homology H = Z /B (for a simplexwise filtration K i K): compute filtered basis for boundaries B = im extend to basis for cycles Z = ker all basis cycles generate persistent homology 11 / 29

24 Persistence by matrix reduction Given: D: matrix of boundary d for a simplexwise filtration (K i ) i (for canonical basis in filtration order, indexed by I J) Wanted: persistence barcode of homology H d (K i ; F) for some (prime) field F, in dimensions d = 0,..., k Notation: m j : jth column of M, m ij : entry in ith row and jth column Pivot m j = min{i I m kj = 0 for all k > i} / 29

25 Persistence by matrix reduction Given: D: matrix of boundary d for a simplexwise filtration (K i ) i (for canonical basis in filtration order, indexed by I J) Wanted: persistence barcode of homology H d (K i ; F) for some (prime) field F, in dimensions d = 0,..., k Notation: m j : jth column of M, m ij : entry in ith row and jth column Pivot m j = min{i I m kj = 0 for all k > i}. Computation: barcode is obtained by matrix reduction of D R = D V reduced (non-zero columns have distinct pivots) V is regular upper triangular 12 / 29

26 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then 13 / 29

27 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, 13 / 29

28 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, Σ Z = Σ B {v i i I e } is a compatible basis of Z / 29

29 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, Σ Z = Σ B {v i i I e } is a compatible basis of Z. Persistent homology is generated by the basis cycles Σ Z / 29

30 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, Σ Z = Σ B {v i i I e } is a compatible basis of Z. Persistent homology is generated by the basis cycles Σ Z. Persistence intervals: {[i, j) i = Pivot r j } {[i, ) i I e } 13 / 29

31 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, Σ Z = Σ B {v i i I e } is a compatible basis of Z. Persistent homology is generated by the basis cycles Σ Z. Persistence intervals: {[i, j) i = Pivot r j } {[i, ) i I e } Matrix V not used for barcode 13 / 29

32 Compatible basis cycles For a reduced boundary matrix R = D V, call I b = {i r i = 0} birth indices, I d = {j r j 0} I e = I b Pivots R death indices, essential indices (note: i = Pivot r j implies r i = 0, thus Pivots R I b ). Then Σ B = {r j j I d } is a basis of B, Σ Z = Σ B {v i i I e } is a compatible basis of Z. Persistent homology is generated by the basis cycles Σ Z. Persistence intervals: {[i, j) i = Pivot r j } {[i, ) i I e } Matrix V not used for barcode Columns with indices Pivots R not used at all 13 / 29

33 Matrix reduction algorithm Require: D: I J matrix Ensure: R = D V: reduced, V: regular upper triangular, P: persistence pairs R = D V = Id(J J) for j J in increasing order do while k < j with i = Pivot r k = Pivot r j do λ = r ij r ik r j = r j λ r k eliminate pivot entry r ij v j = v j λ v k if i = Pivot r j 0 then append (i, j) to P return R, V 14 / 29

34 Clearing

35 Clearing non-essential positive columns Idea [Chen, Kerber 2011]: Don t reduce at non-essential birth indices Use the fact that i = Pivot r j implies r i = 0 Reduce boundary matrices of d C d C d 1 in decreasing dimension d = k + 1,..., 1 Whenever i = Pivot r j (in matrix for d ) Set r i to 0 (in matrix for d 1 ) 15 / 29

36 Clearing non-essential positive columns Idea [Chen, Kerber 2011]: Don t reduce at non-essential birth indices Use the fact that i = Pivot r j implies r i = 0 Reduce boundary matrices of d C d C d 1 in decreasing dimension d = k + 1,..., 1 Whenever i = Pivot r j (in matrix for d ) Set r i to 0 (in matrix for d 1 ) Set v i to r j (if V is needed) 15 / 29

37 Clearing non-essential positive columns Idea [Chen, Kerber 2011]: Don t reduce at non-essential birth indices Use the fact that i = Pivot r j implies r i = 0 Reduce boundary matrices of d C d C d 1 in decreasing dimension d = k + 1,..., 1 Whenever i = Pivot r j (in matrix for d ) Note: Set r i to 0 (in matrix for d 1 ) Set v i to r j (if V is needed) Still yields R = D V reduced, V regular upper triangular reducing birth columns typically harder than death columns: O(j 2 ) (birth) vs. O((j i) 2 ) (death) with clearing: need only reduce essential columns 15 / 29

38 Counting homology column reductions Consider k + 1-skeleton of n 1-simplex (Rips filtration) Number of columns in boundary matrix: k+1 ( n k+1 d=1 d + 1 ) = ( n 1 d=1 d ) total death k+1 + d=1 ( n 1 d + 1 ) birth 16 / 29

39 Counting homology column reductions Consider k + 1-skeleton of n 1-simplex (Rips filtration) Number of columns in boundary matrix: k+1 ( n k+1 d=1 d + 1 ) = ( n 1 k+1 d=1 d ) + d=1 total death k+1 = ( n 1 d=1 d ) + ( n 1 death ( n 1 d + 1 ) birth k + 2 ) essential k + d=1 ( n 1 d + 1 ) cleared 16 / 29

40 Counting homology column reductions Consider k + 1-skeleton of n 1-simplex (Rips filtration) Number of columns in boundary matrix: k+1 ( n k+1 d=1 d + 1 ) = ( n 1 k+1 d=1 d ) + d=1 total death k+1 = ( n 1 d=1 d ) + ( n 1 death ( n 1 d + 1 ) birth k + 2 ) essential k + d=1 ( n 1 d + 1 ) cleared k = 2, n = 192: = Clearing didn t help much! 16 / 29

41 Cohomology

42 Persistent (co)homology How is persistent cohomology related to persistent homology? Cohomology (over F) is vector space dual to homology: H p (K) H p (K) = Hom(H p (K), F) / 29

43 Persistent (co)homology How is persistent cohomology related to persistent homology? Cohomology (over F) is vector space dual to homology: H p (K) H p (K) = Hom(H p (K), F). Duality preserves ranks of linear maps (in finite dimensions): for f V W with dual f W V, rank f = rank f / 29

44 Persistent (co)homology How is persistent cohomology related to persistent homology? Cohomology (over F) is vector space dual to homology: H p (K) H p (K) = Hom(H p (K), F). Duality preserves ranks of linear maps (in finite dimensions): for f V W with dual f W V, rank f = rank f. The persistence barcode of a persistence module is uniquely determined by the ranks of the internal maps / 29

45 Persistent (co)homology How is persistent cohomology related to persistent homology? Cohomology (over F) is vector space dual to homology: H p (K) H p (K) = Hom(H p (K), F). Duality preserves ranks of linear maps (in finite dimensions): for f V W with dual f W V, rank f = rank f. The persistence barcode of a persistence module is uniquely determined by the ranks of the internal maps. Thus, persistent homology and persistent cohomology have the same barcode [de Silva et al 2011] / 29

46 Persistent (relative) homology How is persistent relative homology related to persistent homology? The short exact sequence of filtered chain complexes (with coefficients in F) 0 C p (K ) C p (K) C p (K, K ) 0 induces a long exact sequence in persistent homology H p+1 (K, K ) H p (K ) H p (K) H p (K, K ) and analogously for persistent cohomology / 29

47 Persistent (relative) homology barcodes Consider the long exact sequence H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) We have: B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) 19 / 29

48 Persistent (relative) homology barcodes Consider the long exact sequence H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) We have: B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) B(im(i)) = {[b, ) B(H p (K ))} 19 / 29

49 Persistent (relative) homology barcodes Consider the long exact sequence H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) We have: B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) B(im(i)) = {[b, ) B(H p (K ))} H p (K ) im( ) im(i) 19 / 29

50 Persistent (relative) homology barcodes Consider the long exact sequence H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) We have: B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) B(im(i)) = {[b, ) B(H p (K ))} H p (K ) im( ) im(i) B(coker(i)) = {(, b) [b, ) B(H p (K ))} = B(im(r)) 19 / 29

51 Persistent (relative) homology barcodes Consider the long exact sequence H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) We have: B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) B(im(i)) = {[b, ) B(H p (K ))} H p (K ) im( ) im(i) B(coker(i)) = {(, b) [b, ) B(H p (K ))} = B(im(r)) H p+1 (K, K ) im( ) im(r) 19 / 29

52 Persistent (relative) homology barcodes Consider the long exact sequence We have: H p+1 (K) r H p+1 (K, K ) H p (K ) i H p (K) B(ker(i)) = {[b, d) B(H p (K ))} = B(im( )) B(im(i)) = {[b, ) B(H p (K ))} H p (K ) im( ) im(i) B(coker(i)) = {(, b) [b, ) B(H p (K ))} = B(im(r)) H p+1 (K, K ) im( ) im(r) Thus, the absolute and relative persistence barcodes determine each other [de Silva et al 2011; Schmahl 2018] / 29

53 Cohomology and clearing Cohomology allows for clearing starting from dimension 0 (avoiding the initial overhead) 20 / 29

54 Cohomology and clearing Cohomology allows for clearing starting from dimension 0 (avoiding the initial overhead) Persistence in dimension 0 has special algorithms (Kruskal s algorithm for MST, union-find data structure) 20 / 29

55 Cohomology and clearing Cohomology allows for clearing starting from dimension 0 (avoiding the initial overhead) Persistence in dimension 0 has special algorithms (Kruskal s algorithm for MST, union-find data structure) Persistent cohomology arises from a reverse filtration C (K 0 ) C (K 1 ) C (K n ) 20 / 29

56 Cohomology and clearing Cohomology allows for clearing starting from dimension 0 (avoiding the initial overhead) Persistence in dimension 0 has special algorithms (Kruskal s algorithm for MST, union-find data structure) Persistent cohomology arises from a reverse filtration C (K 0 ) C (K 1 ) C (K n ) Persistent relative cohomology arises from a filtration C (K, K 0 ) C (K, K 1 ) C (K, K n ) 20 / 29

57 Cohomology and clearing Cohomology allows for clearing starting from dimension 0 (avoiding the initial overhead) Persistence in dimension 0 has special algorithms (Kruskal s algorithm for MST, union-find data structure) Persistent cohomology arises from a reverse filtration C (K 0 ) C (K 1 ) C (K n ) Persistent relative cohomology arises from a filtration C (K, K 0 ) C (K, K 1 ) C (K, K n ) Can be computed by reduction of coboundary matrix: transpose of boundary matrix, with reversed index orders 20 / 29

58 Counting cohomology column reductions Consider K: k + 1-skeleton of n 1-simplex, Rips filtration Number of columns in coboundary matrix: k ( n k d=0 d + 1 ) = ( n 1 k d=0 d + 1 ) + d=0 total death ( n 1 d ) birth 21 / 29

59 Counting cohomology column reductions Consider K: k + 1-skeleton of n 1-simplex, Rips filtration Number of columns in coboundary matrix: k ( n k d=0 d + 1 ) = ( n 1 k d=0 d + 1 ) + d=0 total death k = ( n 1 d=0 d + 1 ) + ( n 1 death ( n 1 d ) birth 0 ) essential k + d=1 ( n 1 d ) cleared 21 / 29

60 Counting cohomology column reductions Consider K: k + 1-skeleton of n 1-simplex, Rips filtration Number of columns in coboundary matrix: k ( n k d=0 d + 1 ) = ( n 1 k d=0 d + 1 ) + d=0 total death k = ( n 1 d=0 d + 1 ) + ( n 1 death ( n 1 d ) birth 0 ) essential k + k = 2, n = 192: = d=1 ( n 1 d ) cleared 21 / 29

61 Observations For a typical input: V has very few off-diagonal entries most death columns of D are already reduced 22 / 29

62 Observations For a typical input: V has very few off-diagonal entries most death columns of D are already reduced Previous example (k = 2, n = 192): Only = 122 out of = columns are actually modified in matrix reduction 22 / 29

63 Implicit matrix reduction

64 Implicit matrix reduction Standard approach: Boundary matrix D for filtration-ordered basis Explicitly generated and stored in memory 23 / 29

65 Implicit matrix reduction Standard approach: Boundary matrix D for filtration-ordered basis Explicitly generated and stored in memory Matrix reduction: store only reduced matrix R transform D into R by column operations 23 / 29

66 Implicit matrix reduction Standard approach: Boundary matrix D for filtration-ordered basis Explicitly generated and stored in memory Matrix reduction: store only reduced matrix R transform D into R by column operations Approach for Ripser: Boundary matrix D for lexicographically ordered basis Implicitly defined and recomputed when needed 23 / 29

67 Implicit matrix reduction Standard approach: Boundary matrix D for filtration-ordered basis Explicitly generated and stored in memory Matrix reduction: store only reduced matrix R transform D into R by column operations Approach for Ripser: Boundary matrix D for lexicographically ordered basis Implicitly defined and recomputed when needed Matrix reduction in Ripser: store only coefficient matrix V recompute previous columns of R = D V when needed Typically, V is much sparser and smaller than R Also used in Dionysus, Gudhi 23 / 29

68 Implicit boundary matrix reduction algorithm Require: D: I J matrix Ensure: R = D V: reduced, V: regular upper triangular, P: persistence pairs for j J in increasing order do v j = e j, r j = d j while k < j with i = Pivot r j and (i, k) P do λ = r ij r j = r j λ D v k eliminate pivot entry r ij v j = v j λ v k if i = Pivot r j 0 then v j = rij 1 v j make pivot entry r ij = 1 append (i, j) to P return V 24 / 29

69 Implicit boundary matrix reduction algorithm Require: D: I J matrix Ensure: R = D V: reduced, V: regular upper triangular, P: persistence pairs for j J in increasing order do v j = e j, r j = d j while k < j with i = Pivot r j and (i, k) P do λ = r ij r j = r j λ D v k eliminate pivot entry r ij v j = v j λ v k if i = Pivot r j 0 then v j = rij 1 v j make pivot entry r ij = 1 append (i, j) to P return V 24 / 29

70 Implicit boundary matrix reduction algorithm Require: D: I J matrix Ensure: R = D V: reduced, V: regular upper triangular, P: persistence pairs for j J in increasing order do v j = e j, r j = d j while k < j with i = Pivot r j and (i, k) P do λ = r ij r j = r j λ D v k eliminate pivot entry r ij v j = v j λ v k if i = Pivot r j 0 then v j = rij 1 v j make pivot entry r ij = 1 append (i, j) to P return V 24 / 29

71 Implicit boundary matrix reduction algorithm Require: D: I J matrix Ensure: R = D V: reduced, V: regular upper triangular, P: persistence pairs for j J in increasing order do v j = e j, r j = d j while k < j with i = Pivot r j and (i, k) P do λ = r ij r j = r j λ D v k eliminate pivot entry r ij v j = v j λ v k if i = Pivot r j 0 then v j = rij 1 v j make pivot entry r ij = 1 append (i, j) to P return V 24 / 29

72 Implementation details Current (working) columns v j, r j : priority queue (heap), comparison-based representing a linear combination of row basis elements elements: tuples consisting of coefficient row index diameter of corresponding simplex pivot entry lazily evaluated (when extracting top element) Previous (finalized) columns v k (k < j): sparse matrix data structure 25 / 29

73 Apparent and emergent pairs

74 Apparent pairs Definition In a simplexwise filtration, (σ, τ) is an apparent pair if σ is the youngest face of τ τ is the oldest coface of σ 26 / 29

75 Apparent pairs Definition In a simplexwise filtration, (σ, τ) is an apparent pair if σ is the youngest face of τ τ is the oldest coface of σ Lemma The apparent pairs are persistence pairs. Lemma The apparent pairs form a discrete gradient. Generalizes a construction proposed by [Kahle 2011] for the study of random Rips filtrations Apparent pairs also appear (independently) in Eirene [Henselmann 2016] and in [Mendoza-Smith, Tanner 2017] 26 / 29

76 Emergent persistent pairs A persistence pair (σ, τ) for a simplexwise filtration is an emergent face pair if σ is the youngest proper face of τ, an emergent coface pair if τ is the oldest proper coface of σ / 29

77 Emergent persistent pairs A persistence pair (σ, τ) for a simplexwise filtration is an emergent face pair if σ is the youngest proper face of τ, an emergent coface pair if τ is the oldest proper coface of σ. Lemma Consider the lexicographically refined Rips filtration. Assume that τ is the lexicographically minimal proper coface of σ with diam(τ) = diam(σ), and τ is not already in a persistence pair (ρ, τ) with ρ σ. Then (σ, τ) is a 0-persistence emergent coface pair / 29

78 Emergent persistent pairs A persistence pair (σ, τ) for a simplexwise filtration is an emergent face pair if σ is the youngest proper face of τ, an emergent coface pair if τ is the oldest proper coface of σ. Lemma Consider the lexicographically refined Rips filtration. Assume that τ is the lexicographically minimal proper coface of σ with diam(τ) = diam(σ), and τ is not already in a persistence pair (ρ, τ) with ρ σ. Then (σ, τ) is a 0-persistence emergent coface pair. Includes all apparent pairs with persistence 0 Can be identified without enumerating all cofaces of σ (shortcut for computation) 27 / 29

79 Speedup from emergent pairs shortcut Previous example (k = 2, n = 192): Only = out of = nonzero entries of the coboundary matrix are actually visited 28 / 29

80 Speedup from emergent pairs shortcut Previous example (k = 2, n = 192): Only = out of = nonzero entries of the coboundary matrix are actually visited Using implicit reduction (boundary matrix columns may be revisited multiple times): Only = out of = nonzero entries are actually visited Speedup: 1.2 s vs. 5.6 s (factor 4.7) 28 / 29

81 Ripser Live: users from 567 different cities 29 / 29