Topological Persistence

Transcription

1 Topological Persistence

2 Topological Persistence (in a nutshell) X topological space f : X f persistence X Dg f signature: persistence diagram encodes the topological structure of the pair (X, f) 1

3 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f t f 1 ((, t]) 2

4 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

5 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

6 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

7 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

8 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

9 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

10 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family f 2

11 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family Finite set of intervals (barcode) encodes births/deaths of topological features f 2

12 Topological Persistence (in a nutshell) Inside the black box: Nested family (filtration) of sublevel-sets f 1 ((, t]) for t ranging over Track the evolution of the topology (homology) throughout the family Finite set of intervals (barcode) encodes births/deaths of topological features Alternate representation as a (multi-) set of points in the plane (diagram). f β α β α 2

13 Example: Distance Function f P : 2 x min p P x p

14 Example: Distance Function f P : 2 x min p P x p

15 Example: Distance Function f P : 2 x min p P x p

16 Example: Distance Function f P : 2 x min p P x p

17 Example: Distance Function f P : 2 x min p P x p

18 Example: Distance Function f P : 2 x min p P x p

19 Example: Distance Function f P : 2 x min p P x p

20 Example: Distance Function f P : 2 x min p P x p

21 1 (u, v) 7 (cos(2πu), sin(2πu), cos(2πv), sin(2πv)) Example: sampled periodic curve in the 2d flat torus, 2 S3 4 ( mod Z)2 Example: Distance Function source: 4

22 Example: Distance Function 1 (u, v) 7 (cos(2πu), sin(2πu), cos(2πv), sin(2πv)) Example: sampled periodic curve in the 2d flat torus, 2 S3 4 ( mod Z)2 (spiral winding around the flat torus) source: 4

23 Example: Distance Function (spiral winding around the flat torus) 4

24 Mathematical viewpoint: homology + quivers Filtration: F 1 F 2 F 3 F 4 F 5 Note: for simplicity I am indexing the family over the integers, but in fact it can be indexed ov Example 1: offsets filtration (nested family of unions of balls, cf. previous slide) 5

25 Mathematical viewpoint: homology + quivers Filtration: F 1 F 2 F 3 F 4 F 5 Note: for simplicity I am indexing the family over the integers, but in fact it can be indexed ov Example 1: offsets filtration (nested family of unions of balls, cf. previous slide) Example 2: simplicial filtration (nested family of simplicial complexes) a b a b a b a b a b a b d c d c d c d c d c F 1 F 2 F 3 F 4 F 5 F 6 5

26 Mathematical viewpoint: homology + quivers Filtration: F 1 F 2 F 3 F 4 F 5 Note: for simplicity I am indexing the family over the integers, but in fact it can be indexed ov Example 1: offsets filtration (nested family of unions of balls, cf. previous slide) Example 2: simplicial filtration (nested family of simplicial complexes) Example 3: sublevel-sets filtration (family of sublevel sets of a function f : X ) α F α := f 1 ((, α]) X 5

27 Mathematical viewpoint: homology + quivers Filtration: F 1 F 2 F 3 F 4 F 5 Note: for simplicity I am indexing the family over the integers, but in fact it can be indexed ov topological level (homology functor) the functor is parametrized k by a field of coefficients, omitted in the notations algebraic level Persistence module: H (F 1 ) H (F 2 ) H (F 3 ) H (F 4 ) H (F 5 ) 5

28 Mathematical viewpoint: homology + quivers Example: k (degree-1 homology) k ( 1 0 ) k 2 ( 0 1 ) k ( 0 1 ) k 2 ( ) k 2 5

29 Mathematical viewpoint: homology + quivers ThereTheorem. is a natural way Letto Mextend be a persistence path quivers module with consistent over an index orientations set T. to Then, arbitrary ind M decomposes as a direct sum of interval modules k b,d : id id k k 0 }{{}}{{}}{{ 0} Note: the stars t< b,d indicate limits from below or b, above, d depending on whether t> b,d the interval b in the following cases: T is finite [Gabriel 1972] [Auslander 1974], M is pointwise finite-dimensional (every space M t has finite dimension) forward means that all arrows i j satisfy i j, with the relation i j kĩ k [Webb 1985] [Crawley-Boevey 2012]. Moreover, when it exists, the decomposition is unique up to isomorphism and permutation of the terms [Azumaya 1950]. k ( 1 0 ) k 2 ( 0 1 ) k ( 0 1 ) k ( ) k 2 (Note: this is independent of the choice of field k.) M j J k bj,d j (the barcode is a complete descriptor of the algebraic structure of M) 5

30 Mathematical viewpoint: homology + quivers ThereTheorem. is a natural way Letto Mextend be a persistence path quivers module with consistent over an index orientations set T. to Then, arbitrary ind M decomposes as a direct sum of interval modules k b,d : id id k k 0 }{{}}{{}}{{ 0} Note: the stars t< b,d indicate limits from below or b, above, d depending on whether t> b,d the interval b in the following cases: T is finite [Gabriel 1972] [Auslander 1974], M is pointwise finite-dimensional (every space M t has finite dimension) forward means that all arrows i j satisfy i j, with the relation i j kĩ k [Webb 1985] [Crawley-Boevey 2012]. Moreover, when it exists, the decomposition is unique up to isomorphism and permutation of the terms [Azumaya 1950]. (Note: this is independent of the choice of field k.) 5

31 Mathematical viewpoint: homology + quivers Example: k (degree-1 homology) k ( 1 0 ) k 2 ( 0 1 ) k ( 0 1 ) k 2 ( ) k 2 5

32 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration

33 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration 3 Output: boundary matrix

34 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration 3 Output: boundary matrix reduced to column-echelon form

35 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration 3 Output: boundary matrix reduced to column-echelon form simplex pairs give finite intervals: [2, 4), [3, 5), [6, 7) unpaired simplices give infinite intervals: [1, + )

36 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration Output: boundary matrix reduced to column-echelon form PLU factorization: cubic worst-case time, near linear in practice because the boundary matrices tend to remain Gaussian elimination adapted from [Bunch, Hopcroft 1974] subcubic worst-case time, very complicated + very fast matrix multiplication (divide-and-conquer) [Bunch, Hopcroft 1974] random projections? 6

37 Computation of barcodes: matrix reduction [Edelsbrunner, Letscher, Zomorodian 2002] [Carlsson, Zomorodian 2005]... Input: simplicial filtration Output: boundary matrix reduced to column-echelon form PLU factorization: cubic worst-case time, near linear in practice because the boundary matrices tend to remain Gaussian elimination - PLEX / JavaPLEX ( - Dionysus ( - Perseus ( - Gudhi ( - PHAT ( code/phat) - DIPHA ( - CTL ( 6

38 Stability of persistence barcodes Theorem: For any pfd functions f, g : X, d (Dg f, Dg g) f g f g X 7

39 Space of persistence diagrams Persistence diagram finite multiset in the open half-plane >0 (2) (1) (1) 8

40 Space of persistence diagrams Persistence diagram finite multiset in the open half-plane >0 Given a partial matching M : X Y : cost of a matched pair (x, y) M: c p (x, y) := x y p cost of an unmatched point z X Y : c p (z) := z z p cost of M: c p (M) := (x, y) matched c p (x, y) + z unmatched c p (z) 1/p (2) z x y z 8

41 Space of persistence diagrams Persistence diagram finite multiset in the open half-plane >0 Given a partial matching M : X Y : cost of a matched pair (x, y) M: c p (x, y) := x y p cost of an unmatched point z X Y : c p (z) := z z p cost of M: c p (M) := (x, y) matched c p (x, y) + z unmatched c p (z) 1/p (2) Def: p-th diagram distance (extended metric): z x y d p (X, Y ) := inf c p(m) M:X Y z Def: bottleneck distance: d (X, Y ) := lim p d p(x, Y ) 8

42 X topological space f : X f persistence Lipschitz X Dg f signature: persistence diagram encodes the topological structure of the pair (X, f) 8