Computation of spectral sequences of double complexes with applications to persistent homology

Transcription

1 Computation of spectral sequences of double complexes with applications to persistent homology Mikael Vejdemo-Johansson joint with David Lipsky, Dmitriy Morozov, Primoz Skraba School of Computer Science University of St Andrews Scotland Stanford Symposium Algebraic Topology: Applications and New Directions M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 1 / 24

2 Spectral sequence of a double complex Outline 1 Spectral sequence of a double complex 2 Higher differentials 3 Extension problem over k[t] 4 Applications to persistent homology M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 2 / 24

3 Spectral sequence of a double complex Double complex Fix some nice commutative ring R Everything in this talk will take place in R Mod; maybe R grmod De nition Let C, be a bigraded module over R C, is a double complex if there are maps d : C, C 1, and d : C, C, 1 such that d d = 0, d d = 0 and d d + d d = 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 3 / 24

4 Spectral sequence of a double complex Total complex De nition If C, is a double complex, Tot(C) is a chain complex with differential d = d + d Tot(C) n = p+q=n C p,q M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 4 / 24

5 Spectral sequence of a double complex Spectral sequence For every double complex C,, there are two spectral sequences with E 0 = C, Horizontal spectral sequence Vertical spectral sequence ḍ 0 = d d 1 induced from d ḍ 0 = d d 1 induced from d Homology of the total complex (Godement, 1958) For each of these spectral sequences, the E -page is the graded associated module of a ltration on H (Tot(C)) M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 5 / 24

6 Spectral sequence of a double complex How do we compute this? Bott and Tu Gives a few fundamental results; in a de Rham cohomology context Godement Gives some theorems, does not give higher differentials McCleary Gives some theorems, no explicit higher differentials Brown Ḥas double complex ss as an example Higher differentials are hard MacLane Gives some theorems on the double complex, no explicit higher differentials M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 6 / 24

7 Higher differentials Outline 1 Spectral sequence of a double complex 2 Higher differentials 3 Extension problem over k[t] 4 Applications to persistent homology M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 7 / 24

8 Higher differentials Independent de nition of cycles and boundaries Z r p,q Consists of α p such that there are α p 1,, α p r ful lling equations d 0 α p = 0 d 0 α p 1 = d 1 α p d 0 α p r = d 1 α p r+1 B r p,q Consists of α p such that there are β p, β p+1,, β p+r ful lling equations d 0 β p + d 1 β p+1 = α p d 0 β p+2 = d 1 β p+1 d 0 β p+r 1 = d 1 β p+r Partial collection of conditions on (α 0, α 1,, α p, 0,, 0) being a cycle in Tot(C); and on it being a boundary M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 8 / 24

9 Higher differentials Fundamental computational steps The higher differentials are decomposed into smaller computational steps: Across v Z r p,q maps by a map induced from d 1 to d 1 v Z r p 1,q Up d 0 has sections over B 1 p,q Hence, for r 1, w B r p,q maps by a map induced from d 0 to (d 0 ) 1 w Z r p,q+1 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 9 / 24

10 Higher differentials e E r differential α 3 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

11 Higher differentials e E r differential α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

12 Higher differentials e E r differential d 1 α 3 α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

13 Higher differentials e E r differential α 2 d 1 α 3 α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

14 Higher differentials e E r differential d 2 α 3 α 2 d 1 α 3 α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

15 Higher differentials e E r differential d 2 α 3 d 2 α 3 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

16 Higher differentials e E r differential α 0 d 3 α 3 α 1 d 2 α 3 α 2 d 1 α 3 α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

17 Higher differentials e E r differential α 0 α 1 α 2 β 3 α 3 β 4 0 β 5 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 10 / 24

18 Extension problem over k[t] Outline 1 Spectral sequence of a double complex 2 Higher differentials 3 Extension problem over k[t] 4 Applications to persistent homology M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 11 / 24

19 Extension problem over k[t] Primer on persistence language persistence module graded module over k[t] barcode decomposition of a persistence module into free and torsion summands Smith normal form is unreasonably effective for k[t] no GCD computations needed; lowest degree immediately implies divisibility M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 12 / 24

20 Extension problem over k[t] Representative chains in Tot(C) E has a basis on the shape ϕ(α p ) = (α 0,, α p 1, α p, 0,, 0) Write A p for such α in E p,q Problem t s α p B p,q does not necessarily mean ϕ(t s α p ) = 0 So behavior of barcodes for α p is different from behavior of barcodes for ϕ(α p ) M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 13 / 24

21 Extension problem over k[t] Finding coefficients α 0 α 1 α 2 α 3 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 14 / 24

22 Extension problem over k[t] Finding coefficients α 0 α 1 α 2 β 3 α 3 β 4 0 β 5 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 14 / 24

23 Extension problem over k[t] Finding coefficients α 0 α 1 α 2 β 3 α 3 β 4 0 β 5 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 14 / 24

24 Extension problem over k[t] Finding coefficients Consider ϕ(α p ) = (ϕ 0 (α p ),, ϕ p 1 (α p ), α p, 0,, 0) Suppose t s α p B There is some element β = (0,, β p,, β n ) Such that ϕ(t s α p ) Dβ = (t s ϕ 0 (α),, t s ϕ p 1 (α) d 1 β p, 0,, 0) M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 15 / 24

25 Extension problem over k[t] Finding coefficients Express this result with our preferred basis: t s ϕ(α) Dβ = (t s ϕ 0 (α),, t s ϕ p 1 (α) d 1 β p, 0,, 0) = c ϕ(α ) + c ϕ(α ) + + α A n 1 α A n 2 α (n) A 0 c (n) ϕ(α (n) ) Collect all these c, c,, c (n) into a matrix This matrix gives relations for H (Tot(C)) Smith normal form on this presentation matrix gives global barcodes M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 16 / 24

26 Applications to persistent homology Outline 1 Spectral sequence of a double complex 2 Higher differentials 3 Extension problem over k[t] 4 Applications to persistent homology M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 17 / 24

27 Applications to persistent homology Mayer-Vietoris spectral sequence M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 18 / 24

28 Applications to persistent homology Mayer-Vietoris spectral sequence M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 18 / 24

29 Applications to persistent homology Mayer-Vietoris spectral sequence M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 18 / 24

30 Applications to persistent homology Čech spectral sequence Double complex ith column Chain complex of all i-fold intersections vertical differential Boundary in chain complex horizontal differential Inclusion of deeper intersection into more shallow; coefficient from Čech complex boundary Generalizes the Mayer-Vietoris long exact sequence M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 19 / 24

31 Applications to persistent homology Čech spectral sequence eorem Ṭhis double complex computes H X using only a cover of X Proof sketch (following Brown; also Segal) C 2 0 C 2 1 C C 0 0 j=0 U σ j 0 σ j=0 U σ j C 2 0 σ M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 20 / 24

32 Applications to persistent homology Čech spectral sequence eorem Ṭhis double complex computes H X using only a cover of X Proof sketch (following Brown; also Segal) C 0 2 j=0 U C σ j 2 X 2 C 2 2 X C 0 0 j=0 U σ j 0 σ j=0 U C σ j C 2 0 X 0 σ Augment double complex M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 20 / 24

33 Applications to persistent homology Čech spectral sequence eorem Ṭhis double complex computes H X using only a cover of X Proof sketch (following Brown; also Segal) C 0 2 j=0 U C σ j 2 X 2 C 2 2 X C 0 0 j=0 U σ j 0 σ j=0 U C σ j C 2 0 X 0 σ Resolution (in chain complexes) because any simplex σ C X is σ k in the double complex Thus: associated graded H Tot = 0 M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 20 / 24

34 Applications to persistent homology Čech spectral sequence eorem Ṭhis double complex computes H X using only a cover of X Proof sketch (following Brown; also Segal) C 0 2 j=0 U C σ j 2 X 2 C 2 2 X C 0 0 j=0 U σ j 0 σ j=0 U C σ j C 2 0 X 0 σ Associated graded H Tot = 0 does not change with a change in ltration M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 20 / 24

35 Applications to persistent homology Čech spectral sequence eorem Ṭhis double complex computes H X using only a cover of X Proof sketch (following Brown; also Segal) C 0 2 j=0 U C σ j 2 X 2 C 2 2 X C 0 0 j=0 U σ j 0 σ j=0 U C σ j C 2 0 X 0 σ Because resolution: removing augmentation yields quasi-isomorphism M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 20 / 24

36 Applications to persistent homology Parallel/distributed computation of persistent homology Strategy Divide a ltered simplicial complex X into a cover of ltered simplicial subcomplexes U i Conquer by merging all H ( i U i) using the Mayer-Vietoris spectral sequence M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 21 / 24

37 Applications to persistent homology Parallel/distributed computation of persistent homology Problems Ẉe need to build tools for efficient computation Hope Algorithm asymptotically equally fast as persistent homology; but with memory consumption guarantees Results Ạchieved this for the two-column case deepest intersection pairwise Note: persistence can run in O(nm 2 ) time; for n simplices, m cycles We can now achieve O(n 2 m) for the two-column case Matrix ll-in and distributed module algebra our hardest problems M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 22 / 24

38 Applications to persistent homology More spectral sequences persistently Results summary Explicit higher differentials for double complex spectral sequences Explicit extensions for persistence modules Serre spectral sequences Dress (1967) constructs the Serre spectral sequence as a double complex spectral sequence All higher differentials follow Applicability in persistence follows Open question: how do we make this nite enough? M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 23 / 24

39 Applications to persistent homology ank you Happy birthdays! ṭo Gunnar Carlsson, Ralph Cohen, and Ib Madsen Acknowledgements Research joint with Dmitriy Morozov (Lawrence Berkeley Lab), Primoz Skraba (Jozef Stefan Institute), David Lipsky (University of Pennsylvania) My own research funded by EPSRC in the HPC-GAP project Input and discussions with Gunnar Carlsson, Robert Ghrist, Stephen Linton, Wojciech Chacholski Questions? M Vejdemo-Johansson (St Andrews) Computation of spectral sequences CCM-fest 24 / 24