Persistent Homology: Persistence Modules Andrey Blinov 6 October 2017 Andrey Blinov Persistent Homology: Persistence Modules
Big picture Andrey Blinov Persistent Homology: Persistence Modules
Homology Homology — handy invariant of topological spaces. Informally speaking — holes in the space. Andrey Blinov Persistent Homology: Persistence Modules
Homology Homology — handy invariant of topological spaces. Informally speaking — holes in the space. Singular homology: good for theorems; difficult to compute. Andrey Blinov Persistent Homology: Persistence Modules
Homology Homology — handy invariant of topological spaces. Informally speaking — holes in the space. Singular homology: good for theorems; difficult to compute. Siimplicial homology: easier to compute; if defined, then coincides with the singular. Andrey Blinov Persistent Homology: Persistence Modules
Homology Homology — handy invariant of topological spaces. Informally speaking — holes in the space. Singular homology: good for theorems; difficult to compute. Siimplicial homology: easier to compute; if defined, then coincides with the singular. Data point sets: Not clear what is the space; Abstract simplicial complexes are handy. Andrey Blinov Persistent Homology: Persistence Modules
Complexes Andrey Blinov Persistent Homology: Persistence Modules
Complexes: comparison ˇ Cech complexes: nerve theorem; bad in computations. Andrey Blinov Persistent Homology: Persistence Modules
Complexes: comparison ˇ Cech complexes: nerve theorem; bad in computations. Nerve theorem — for suitable ε , the abstract ˇ Cech complex have the same homology as the corresponding union of balls. Andrey Blinov Persistent Homology: Persistence Modules
Complexes: comparison ˇ Cech complexes: nerve theorem; bad in computations. Nerve theorem — for suitable ε , the abstract ˇ Cech complex have the same homology as the corresponding union of balls. Vietoris-Rips complexes: topologically worse; better in memory usage. Andrey Blinov Persistent Homology: Persistence Modules
Persistence There are inclusions: C ǫ ( S ) → ˇ ˇ C ǫ ′ ( S ) for ǫ ≤ ǫ ′ . VR ǫ ( S ) → VR ǫ ′ ( S ) for ǫ ≤ ǫ ′ . VR ǫ ( S ) → ˇ C √ 2 ǫ ( S ) → VR √ 2 ǫ ( S ). Andrey Blinov Persistent Homology: Persistence Modules
Persistence There are inclusions: C ǫ ( S ) → ˇ ˇ C ǫ ′ ( S ) for ǫ ≤ ǫ ′ . VR ǫ ( S ) → VR ǫ ′ ( S ) for ǫ ≤ ǫ ′ . VR ǫ ( S ) → ˇ C √ 2 ǫ ( S ) → VR √ 2 ǫ ( S ). Idea: perceive the family of complexes (with morphisms between them) as one object. Andrey Blinov Persistent Homology: Persistence Modules
Barcodes and persistence Andrey Blinov Persistent Homology: Persistence Modules
One more example of persistence There is another way to get a persistence module. consider a (good) function f : X → R ; take level sets X t = f − 1 ( −∞ , t ). Andrey Blinov Persistent Homology: Persistence Modules
One more example of persistence There is another way to get a persistence module. consider a (good) function f : X → R ; take level sets X t = f − 1 ( −∞ , t ). (One can also take X t = f − 1 ( −∞ , t ].) Andrey Blinov Persistent Homology: Persistence Modules
One more example of persistence There is another way to get a persistence module. consider a (good) function f : X → R ; take level sets X t = f − 1 ( −∞ , t ). (One can also take X t = f − 1 ( −∞ , t ].) There are inclusions X s → X t for s < t . Andrey Blinov Persistent Homology: Persistence Modules
One more example of persistence There is another way to get a persistence module. consider a (good) function f : X → R ; take level sets X t = f − 1 ( −∞ , t ). (One can also take X t = f − 1 ( −∞ , t ].) There are inclusions X s → X t for s < t . Andrey Blinov Persistent Homology: Persistence Modules
Persistence module There is no one best ǫ . The inclusion ˇ C ǫ ( S ) → ˇ C ǫ ′ ( S ) for ǫ ≤ ǫ ′ . Also VR ǫ ( S ) → VR ǫ ′ ( S ). All that produces the following definition: Andrey Blinov Persistent Homology: Persistence Modules
Persistence module There is no one best ǫ . The inclusion ˇ C ǫ ( S ) → ˇ C ǫ ′ ( S ) for ǫ ≤ ǫ ′ . Also VR ǫ ( S ) → VR ǫ ′ ( S ). All that produces the following definition: Definition A persistence module is a family of modules { V t , t ∈ R } over a ring A with morphisms v s t : V s → V t whenever s ≤ t such that v t t = id V t for every t ; v r t = v s t ◦ v r s for every triple r ≤ s ≤ t . Andrey Blinov Persistent Homology: Persistence Modules
Persistence module We can define persistence modules for any partially ordered set T . Often T = { 1 , 2 , 3 , . . . , n } , or T = N , or T = Z . Andrey Blinov Persistent Homology: Persistence Modules
Category of persistence modules We can define morphisms P between persistence modules: v s t V s V t P s P t w s t W s W t Andrey Blinov Persistent Homology: Persistence Modules
Category of persistence modules We can define morphisms P between persistence modules: v s t V s V t P s P t w s t W s W t There is a notion of isomorphic modules. Andrey Blinov Persistent Homology: Persistence Modules
Category of persistence modules We can define morphisms P between persistence modules: v s t V s V t P s P t w s t W s W t There is a notion of isomorphic modules. There are also kernels, images, and direct sums. Andrey Blinov Persistent Homology: Persistence Modules
Persistence module Very Important Remark: From now, we consider A = k to be a field. Andrey Blinov Persistent Homology: Persistence Modules
Persistence module Very Important Remark: From now, we consider A = k to be a field. Definition We call the persistence module V = { V t , t ∈ R } pointwise finite-dimensional ( pfd ) if all V t are finite-dimensional. Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? V 1 . Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? V 1 . V 1 → V 2 . Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? V 1 . V 1 → V 2 . We actually know the answer for these cases! • : vector spaces. • → • : operators between vector spaces. Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? V 1 . V 1 → V 2 . We actually know the answer for these cases! • : vector spaces. • → • : operators between vector spaces. There is a connection between those two problems: 1. Isomorphism classes of finite persistence modules; 2. Decomposition into direct sums. Andrey Blinov Persistent Homology: Persistence Modules
Isomorphism classes Isomorphism classes of pfd persistence modules for simple T ? V 1 . V 1 → V 2 . We actually know the answer for these cases! • : vector spaces. • → • : operators between vector spaces. There is a connection between those two problems: 1. Isomorphism classes of finite persistence modules; 2. Decomposition into direct sums. Definition We call a persistent module V ∗ indecomposable if every decomposition of V ∗ into a direct sum is trivial, i.e. if V ∗ = U ∗ ⊕ W ∗ then either U ∗ = 0 or W ∗ = 0. Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Definition We call I ⊆ T an interval if for all x , y ∈ I and each x < z < y holds z ∈ T . Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Definition We call I ⊆ T an interval if for all x , y ∈ I and each x < z < y holds z ∈ T . For T = { 1 , 2 , . . . n } all intervals are closed. Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Definition We call I ⊆ T an interval if for all x , y ∈ I and each x < z < y holds z ∈ T . For T = { 1 , 2 , . . . n } all intervals are closed. For T = R all the sets [ p , q ) , [ p , q ] , ( p , q ) , ( p , q ] are intervals. Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Definition We call I ⊆ T an interval if for all x , y ∈ I and each x < z < y holds z ∈ T . For T = { 1 , 2 , . . . n } all intervals are closed. For T = R all the sets [ p , q ) , [ p , q ] , ( p , q ) , ( p , q ] are intervals. Definition The interval module k [ I ] is the following persistent module: � k , t ∈ I ; k [ I ] t = 0 otherwise. with k [ I ] t s = id if s , t ∈ I and 0 otherwise. Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Proposition End ( k [ I ]) = k . Andrey Blinov Persistent Homology: Persistence Modules
Interval modules Proposition End ( k [ I ]) = k . Proof. For t ∈ I , every endomorphism of k [ I ] t = k is a multiplication by a scalar λ . The corresponding diagram is commutative, so λ is the same for all t ∈ I . Andrey Blinov Persistent Homology: Persistence Modules
Recommend
More recommend
