A roadmap for the computation of persistent homology

Transcription

1 A roadmap for the computation of persistent homology Nina Otter Mathematical Institute, University of Oxford (joint with Mason A. Porter, Ulrike Tillmann, Peter Grindrod, Heather A. Harrington) 2nd School and Conference on Topological Data Analysis Querétaro, 11 December 2015 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

2 What is out there? Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

3 Perseus javaplex Dionysus phom TDA GAP persistence PHAT DIPHA GUDHI Kenzo module The persistence landscape toolbox SimpPers jholes HoPeS RIVET Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

4 A brief (biased) history of PH software PH algorithm H. Edelsbrunner, D. Letscher, G. Carlsson and A. Zomorodian 2005 Plex V. de Silva, P. Perry, L. Kettner, A. Zomorodian 2011 javaplex A. Tausz, M. Vejdemo-Johansson, H. Adams 2012 Perseus 2013 PHAT V. Nanda M. Kerber, J. Reininghaus, U. Bauer, H. Wagner 2014 DIPHA 2014 GUDHI 2016 RIVET Nina Otter (Oxford) M. Kerber, J. Reininghaus, U. Bauer C. Maria, J.D. Boissonnat, M. Glisse, M. Yvinec M. Lesnick, M. Wright Roadmap for PH computation Quere taro, 11 December / 45

5 Pipeline for PH computation Data (1) Filtered complex (2) (3) Barcodes Interpretation Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

6 Step 1: from data to a filtered complex Point cloud data Point cloud S Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

7 Step 1: from data to a filtered complex Point cloud data Point cloud S Filtered simplicial complex Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

8 Čech complex C ɛ (S) = { σ S x σ B x(ɛ) } Nerve theorem: C ɛ (S) x S B x(ɛ) Complexity: 2 O(n) Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

9 Vietoris-Rips complex VR ɛ (S) = {σ S x, y σ : d(x, y) 2ɛ} Nerve theorem does not hold Complexity: 2 O(n) For all ɛ 0 we have C ɛ (S) VR ɛ (S) C 2ɛ (S). Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

10 α complex α ɛ (S) = { σ S x σ B x(ɛ) V x } Nerve theorem: α ɛ (S) x S B x(ɛ) Complexity: n O( d/2 ) (n points in R d ) For all ɛ 0 we have α ɛ (S) C ɛ (S) VR ɛ (S). Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

11 Optimisations and speed-ups (A) Sparse filtrations: construct sparse filtered complexes which approximate homology of a given complex. Witness complex [de Silva, Carlsson 2004] Graph induced complex [Dey, Fang, Wang 2014] Linear size approximation of VR [Sheehy 2013] Linear size approximation of Čech [Kerber, Sharathkumar 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

12 Optimisations and speed-ups (A) Sparse filtrations: construct sparse filtered complexes which approximate homology of a given complex. Witness complex [de Silva, Carlsson 2004] Graph induced complex [Dey, Fang, Wang 2014] Linear size approximation of VR [Sheehy 2013] Linear size approximation of Čech [Kerber, Sharathkumar 2013] Graph induced complex on point data, Dey, Fan, Wang 2014 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

13 Optimisations and speed-ups (A) Sparse filtrations: construct sparse filtered complexes which approximate homology of a given complex. Witness complex [de Silva, Carlsson 2004] Graph induced complex [Dey, Fang, Wang 2014] linear size approximation of VR [Sheehy 2013] linear size approximation of Čech [Kerber, Sharathkumar 2013] (B) Heuristics: given a filtered simplicial complex, reduce number of simplices without changing homology. Tidy set [Zomorodian 2010] Morse reductions [Mischaikow, Nanda 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

14 Optimisations summary Complex K Size of K (A) Approximation (B) Reduction Čech O(2 n ) linear size approx. tidy set Morse red. Vietoris Rips (VR) O(2 n ) linear size approx. tidy set (lazy) witness Morse red. GIC α Data structures: n O( d/2 ) (n points in R d ) Simplex tree [Boissonnat, Maria 2012] - tidy set Morse red. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

15 Implementations Complex K (A) Approximation (B) Reduction Čech Dionysus linear size approx.? tidy set not open source Morse red. Perseus Vietoris Rips (VR) Dionysus,Perseus, PHAT DIPHA, javaplex, GUDHI α Dionysus, HoPeS (1-D) linear size approx.? (lazy) witness javaplex GIC? - tidy set Morse red. tidy set Morse red. Data structure: Simplex tree GUDHI Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

16 Other data types 1 Undirected (weighted) networks Weight rank clique filtration [Petri et al 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

17 Other data types 1 Undirected (weighted) networks Weight rank clique filtration [Petri et al 2013] 2 Image data: cubical complexes Lower star filtration [Wagner et al 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

18 Other data types: implementations 1 Undirected (weighted) networks Weight rank clique filtration jholes 2 Image data: cubical complexes DIPHA, Perseus Lower star filtration DIPHA Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

19 Step 2: from a filtered complex to barcodes Standard algorithm Step 1: Order the simplices so that the total order is compatible with the filtration. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

20 Step 2: from a filtered complex to barcodes Standard algorithm Step 1: Order the simplices so that the total order is compatible with the filtration. Step 2: Construct the boundary matrix. Let n be the number of simplices. The boundary matrix B is an n n-matrix defined by: { 1 σ i σ j and dim(σ i ) = dim(σ j ) 1 B(i, j) = 0 otherwise Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

21 Step 2: from a filtered complex to barcodes Standard algorithm Step 1: Order the simplices so that the total order is compatible with the filtration. Step 2: Construct the boundary matrix. Let n be the number of simplices. The boundary matrix B is an n n-matrix defined by: { 1 σ i σ j and dim(σ i ) = dim(σ j ) 1 B(i, j) = 0 otherwise Step 3: Reduce the matrix using column additions from left to right. Step 4: Read the endpoints of the persistence intervals from the matrix to get the barcode. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

22 Standard algorithm Step 1. Given a filtered simplicial complex = K 0 K 1 K m = K put an order on its simplices such that: A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

23 Standard algorithm Step 1. Given a filtered simplicial complex = K 0 K 1 K m = K put an order on its simplices such that: A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. Example: K 1 K 2 K 3 K 4 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

24 Standard algorithm Put an order on the simplices such that: σ 1 A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. σ 2 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

25 Standard algorithm Put an order on the simplices such that: A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. σ 2 σ 1 σ 4 σ 1 σ 2 σ 3 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

26 Standard algorithm Put an order on the simplices such that: A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. σ 2 σ 1 σ 2 σ 2 σ 4 σ 1 σ 3 σ 4 σ5 σ 1 σ 6 σ 3 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

27 Standard algorithm Put an order on the simplices such that: A face of a simplex precedes the simplex. A simplex in K i precedes simplices in K \ K i. σ 2 σ 1 σ 4 σ 6 σ 2 σ 2 σ 7 σ 1 σ 3 σ 4 σ5 σ 1 σ 3 σ 1 σ 2 σ 4 σ5 σ 6 σ 3 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

28 Standard algorithm Step 2. We obtain the boundary matrix: For j = 1,..., 7 define low(j) = i if i is the position of the lowest 1 in column j. If the column is zero leave low(j) undefined. e.g. low(4) = 2 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

29 Standard algorithm Step 3. Reduce the boundary matrix. Let n be the number of simplices. Algorithm: for j = 1 to n do while there exists i < j with low(i) = low(j) do add column i to column j end while end for Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

30 Standard algorithm Step 3. Reduce the boundary matrix. Algorithm: for j = 1 to n do while there exists i < j with low(i) = low(j) do add column i to column j end while end for Add column 5 to column 6: Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

31 Standard algorithm Step 3. Reduce the boundary matrix. Algorithm: for j = 1 to n do while there exists i < j with low(i) = low(j) do add column i to column j end while end for Add column 4 to column 6: Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

32 Standard algorithm Step 4. Read the persistence pairs. If low(j) = i then σ j is negative and paired with the positive σ i. If low(j) is undefined then σ j is positive. If there exists k such that low(k) = j then σ j is paired with the negative simplex σ k. If no such k exists σ j is unpaired σ 1 positive, unpaired σ 2 positive, paired with σ 4 σ 3 positive, paired with σ 5 σ 4 negative, paired with σ 2 σ 5 negative, paired with σ 3 σ 6 positive, paired with σ 7 σ 7 negative, paired with σ 6 Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

33 Standard algorithm σ 2 σ 1 σ 4 σ 6 σ 2 σ 2 σ 2 σ 7 σ 1 σ 3 σ 4 σ5 σ 1 σ 3 σ 4 σ5 σ 1 σ 6 σ 3 σ 1 positive, unpaired interval [1, ) in H 0. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

34 Standard algorithm σ 2 σ 1 σ 4 σ 6 σ 2 σ 2 σ 2 σ 7 σ 1 σ 3 σ 4 σ5 σ 1 σ 3 σ 4 σ5 σ 1 σ 6 σ 3 σ 1 positive, unpaired interval [1, ) in H 0. σ 2 positive, paired with σ 4 no interval, since σ 2 and σ 4 enter at the same time in the filtration Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

35 Standard algorithm σ 2 σ 1 σ 4 σ 6 σ 2 σ 2 σ 2 σ 7 σ 1 σ 3 σ 4 σ5 σ 1 σ 3 σ 4 σ5 σ 1 σ 6 σ 3 σ 1 positive, unpaired interval [1, ) in H 0. σ 2 positive, paired with σ 4 no interval, since σ 2 and σ 4 enter at the same time in the filtration σ 3 positive, paired with σ 5 interval [2, 3) in H 0. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

36 Standard algorithm σ 2 σ 1 σ 4 σ 6 σ 2 σ 2 σ 2 σ 7 σ 1 σ 3 σ 4 σ5 σ 1 σ 3 σ 4 σ5 σ 1 σ 6 σ 3 σ 1 positive, unpaired interval [1, ) in H 0. σ 2 positive, paired with σ 4 no interval, since σ 2 and σ 4 enter at the same time in the filtration σ 3 positive, paired with σ 5 interval [2, 3) in H 0. σ 6 positive, paired with σ 7 interval [3, 4) in H 1. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

37 Twist algorithm Note: If column j is reduced and low(j) = i > 0 then simplex σ i is positive and reducing column i will result in setting column i to zero. σ i is a codimension 1 face of σ j Algorithm: Reduce columns from right to left If column j is reduced and low(j) = i > 0 then set column i to zero. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

38 Twist algorithm Previous example: Column 7 is reduced and low(7) = 6, so set column 6 to zero. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

39 Twist algorithm Previous example: Column 7 is reduced and low(7) = 6, so set column 6 to zero Now we are done. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

40 Spectral sequence algorithm Let k j denote the number of simplices in subcomplex K j. Let B j denote the columns numbered k j to k j. Let B i denote the rows numbered k i to k i. Idea: Reduce the matrix in phases: in each phase r, reduce columns in B j by adding columns in the blocks from B j r+1 to B j. Optimisation: The reduction in each block and each phase is independent, and can be executed in parallel. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

41 Spectral sequence algorithm Phase r = 1: Phase r = 2: B B j B k B B j B k B B B j B j B k B k Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

42 Spectral sequence algorithm Phase r = 3: Phase r = k: B B j B k B B j B k B B B j B j B k B k Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

43 Heuristic speed-ups Sequential: Dual algorithm [de Silva, Morozov, Vejdemo-Johansson 2011] Twist algorithm [Chen, Kerber 2011] Parallel: Spectral sequence algorithm [Edelsbrunner, Harer 2008] Chunk algorithm [Bauer, Kerber, Reininghaus 2013] Distributed algorithm [Bauer, Kerber, Reininghaus 2014] Blow-up complex [Morozov, Lewis 2015] Data structure: Bit tree pivot column [Bauer, Kerber, Reininghaus, Wagner 2013] Compressed annotation matrix [Boissonnat, Dey, Maria 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

44 Speed-ups: implementations Sequential: Dual algorithm Dionysus, javaplex, GUDHI, PHAT, DIPHA Twist algorithm PHAT,DIPHA Parallel: Spectral sequence algorithm PHAT Chunk algorithm PHAT Distributed algorithm DIPHA Blow-up complex open source? Data structure: Bit tree pivot column PHAT, DIPHA Compressed annotation matrix GUDHI Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

45 Step 3: interpretation (A) Comparing diagrams: p-th Wasserstein distance W p [σ](x, Y ) := inf φ: X Y [ ] 1/p σ(x, φ(x)) p x X for 1 p < and σ a metric on the plane. W [σ](x, Y ) := Example: W [L ] is the bottleneck distance. inf sup σ(x, φ(x) φ: X Y x X Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

46 (B) Statistics: The space (D, W p [σ]) (D, W p [L ]) is a Polish space. Fréchet expectations exist but are not unique nor continuous. [Mileyko et al 2011] Same is true for (D, W 2 [L 2 ]). Algorithm for estimation of Fréchet mean [Turner et al 2014] Fréchet mean for continuously varying diagrams as a probability distribution for (D, W 2 [L 2 ]). [Munch et al 2015] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

47 (C) Alternative: map persistence diagrams to simpler space Persistence landscapes [Bubenik 2014] Space of algebraic functions [Adcock et al 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

48 Implementations Dionysus and javaplex: bottleneck distance TDA package: bottleneck and Wasserstein distances, persistence landscapes, Kernel distances,... Persistence Landscape Toolbox: persistence landscapes (means, norms,... ), T -student test,... Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

49 Summary 1 of 2 Software Precomp. filt. Compl. Parallel PH algorithms Interpr. javaplex VR, LW, W, CW Perseus VR Dionysus α, VR, Čech PHAT shared DIPHA VR, lower star distr. GUDHI VR standard, dual zig-zag Morse reductions + standard standard, dual zig-zag standard, dual, twist chunk, spectral seq. dual, twist distributed multifield dual bottleneck bottleneck Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

50 Summary 2 of 2 Software Precomp. filt. Compl. Parallel PH algorithms Interpr. jholes WRCF shared standard (javaplex) HoPeS α (only 1-D) TDA VR dual (GUDHI, Dionysus, PHAT) Wasserstein, landscapes, bootstraps, etc. phom VR,LW standard, dual Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

51 Benchmarking We study the libraries javaplex, Perseus, Dionysus, DIPHA and GUDHI Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

52 Benchmarking We study the libraries javaplex, Perseus, Dionysus, DIPHA and GUDHI... by using complexes and algorithms implemented by the largest number of libraries: Vietoris-Rips complex and standard and dual algorithms Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

53 Methods We study the software from three different points of view: 1 Performance measured in CPU and real time 2 Memory usage 3 Maximum size of simplicial complex allowed by the software Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

54 Data Synthetic data Points sampled from Klein bottle Random Vietoris Rips complexes Real world data Genomic sequence of HIV virus [Chan et al, 2013] 3D scans of Stanford dragon C. Elegans neuronal network [Petri et al, 2013] Human genome network [Petri et al, 2013] Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

55 Machines Cluster 1 : 1728 (180*16) cores of 2.0GHz RAM : 64 GiB x80 nodes, 128 GiB x 4 nodes 1 Advanced Research Computing (ARC), Oxford Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

56 Results: performance Data set C. elegans Klein HIV Dragon 1 Dragon 2 Size of complex javaplex st Dionysus st DIPHA st Perseus Dionysus d DIPHA d GUDHI Table: Performance measured in wall-time and CPU seconds for the computation of PH with the Vietoris Rips complex. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

57 Results: memory usage Data set C. elegans Klein HIV Dragon 1 Dragon 2 Size of complex javaplex st < 5 < 15 > 120 > 120 > 120 Dionysus st DIPHA st Perseus Dionysus d DIPHA d GUDHI Table: Memory usage in GB. For javaplex, we indicate the value of the maximum heap size that was sufficient to perform the computation. For DIPHA, we indicate the maximum memory used by a single core. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

58 Results: maximal size of simplicial complex Software javaplex Dionysus DIPHA Perseus GUDHI st st d st d st d max. size Table: Maximal size of simplicial complex supported by the software. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45

59 Conclusion Future directions: Creation of a computational topology library. User-friendly interfaces. Uniformization of input type across different implementations. Focus on step (1) and (3) of pipeline: implementation and construction of simpler simplicial complexes and interpretation of results. Multi-parameter PH. Nina Otter (Oxford) Roadmap for PH computation Querétaro, 11 December / 45