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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: ˇ Cǫ(S) → ˇ Cǫ′(S) for ǫ ≤ ǫ′. VRǫ(S) → VRǫ′(S) for ǫ ≤ ǫ′. VRǫ(S) → ˇ C√

2ǫ(S) → VR√ 2ǫ(S).

Andrey Blinov Persistent Homology: Persistence Modules

Persistence

There are inclusions: ˇ Cǫ(S) → ˇ Cǫ′(S) for ǫ ≤ ǫ′. VRǫ(S) → VRǫ′(S) for ǫ ≤ ǫ′. VRǫ(S) → ˇ C√

2ǫ(S) → VR√ 2ǫ(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 Xt = 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 Xt = f −1(−∞, t). (One can also take Xt = 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 Xt = f −1(−∞, t). (One can also take Xt = f −1(−∞, t].) There are inclusions Xs → Xt 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 Xt = f −1(−∞, t). (One can also take Xt = f −1(−∞, t].) There are inclusions Xs → Xt for s < t.

Andrey Blinov Persistent Homology: Persistence Modules

Persistence module

There is no one best ǫ. The inclusion ˇ Cǫ(S) → ˇ Cǫ′(S) for ǫ ≤ ǫ′. Also VRǫ(S) → VRǫ′(S). All that produces the following definition:

Andrey Blinov Persistent Homology: Persistence Modules

Persistence module

There is no one best ǫ. The inclusion ˇ Cǫ(S) → ˇ Cǫ′(S) for ǫ ≤ ǫ′. Also VRǫ(S) → VRǫ′(S). All that produces the following definition: Definition A persistence module is a family of modules {Vt, t ∈ R} over a ring A with morphisms vs

t : Vs → Vt whenever s ≤ t such that

vt

t = idVt for every t;

vr

t = vs t ◦ vr 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: Vs Vt Ws Wt

vs

t

Ps Pt ws

t Andrey Blinov Persistent Homology: Persistence Modules

Category of persistence modules

We can define morphisms P between persistence modules: Vs Vt Ws Wt

vs

t

Ps Pt ws

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: Vs Vt Ws Wt

vs

t

Ps Pt ws

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 = {Vt, t ∈ R} pointwise finite-dimensional (pfd) if all Vt 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? V1.

Andrey Blinov Persistent Homology: Persistence Modules

Isomorphism classes

Isomorphism classes of pfd persistence modules for simple T? V1. V1 → V2.

Andrey Blinov Persistent Homology: Persistence Modules

Isomorphism classes

Isomorphism classes of pfd persistence modules for simple T? V1. V1 → V2. 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? V1. V1 → V2. 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? V1. V1 → V2. 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[I]t =

• k, t ∈ I;

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

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. Proposition Interval modules are indecomposable.

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. Proposition Interval modules are indecomposable. Proof. If k[I] = A ⊕ B, then the projection p : A ⊕ B → B is an

• idempotent. Since k = End(k[I]), we get p = 0 or p = 1.

Andrey Blinov Persistent Homology: Persistence Modules

Azumaya theorem

If there is a decomposition into interval modules, it must be unique: Theorem (Krull-Remak-Schmidt-Azumaya) Suppose a persistence module over T ⊆ R can be expressed as a direct sum of interval modules in two different ways: V =

• ℓ∈L

k[Jℓ] =

• m∈M

k[Km] Where L and M are families of sets. Then there is a bijection σ : L → M such that Jℓ = Kσ(ℓ).

Andrey Blinov Persistent Homology: Persistence Modules

Quivers

Definition A quiver is an oriented graph (V ; E). Its representation is the set (Av; fe) of vector spaces Av over k indexed by v ∈ V , and linear maps fe : Av → Aw, where e ∈ E is the edge going from v to w.

Andrey Blinov Persistent Homology: Persistence Modules

Quivers

Definition A quiver is an oriented graph (V ; E). Its representation is the set (Av; fe) of vector spaces Av over k indexed by v ∈ V , and linear maps fe : Av → Aw, where e ∈ E is the edge going from v to w. Note: that we do not require fe to commute along different paths.

Andrey Blinov Persistent Homology: Persistence Modules

Quivers

Definition A quiver is an oriented graph (V ; E). Its representation is the set (Av; fe) of vector spaces Av over k indexed by v ∈ V , and linear maps fe : Av → Aw, where e ∈ E is the edge going from v to w. Note: that we do not require fe to commute along different paths. Note: there can be loops and multiple edges between vertices.

Andrey Blinov Persistent Homology: Persistence Modules

Quivers

Definition A quiver is an oriented graph (V ; E). Its representation is the set (Av; fe) of vector spaces Av over k indexed by v ∈ V , and linear maps fe : Av → Aw, where e ∈ E is the edge going from v to w. Note: that we do not require fe to commute along different paths. Note: there can be loops and multiple edges between vertices. The persistence modules for finite T ⊂ R are representations of the corresponding quivers.

Andrey Blinov Persistent Homology: Persistence Modules

Gabriel’s theorem

Theorem (Gabriel, 1972) A quiver has only finitely many isomorphism classes of indecomposable representations if and only if it comes from one of the following graphs called ADE Dynkin diagrams:

Andrey Blinov Persistent Homology: Persistence Modules

Generalizations of Gabriel’s theorem

Theorem (Gabriel, Auslander, Ringel-Tachikawa, Webb, Crawley-Boevey) Let V be a persistence module over T ⊆ R. Then V can be decomposed as a direct sum of interval modules in either of the following situations: T is finite; each Vt is finite-dimensional. Additionally, there is an example of infinite T ⊆ R with infinite-dimensional V which does not admit the interval decomposition.

Andrey Blinov Persistent Homology: Persistence Modules

Webb’s (counter)example

Consider the following persistent module with T = Z≤0: W0 = {sequences (x0, x1, . . . ) of scalars}; W−n = {such sequences with x1 = x2 = · · · = xn = 0}.

Andrey Blinov Persistent Homology: Persistence Modules

Webb’s (counter)example

Consider the following persistent module with T = Z≤0: W0 = {sequences (x0, x1, . . . ) of scalars}; W−n = {such sequences with x1 = x2 = · · · = xn = 0}. This module is not a direct sum of interval modules. The dimensions of W−n are uncountable, though.

Andrey Blinov Persistent Homology: Persistence Modules

Lesnick’s (counter)example

Even better counterexample with T = Z≤0: L0 = k; L−1 = {eventually-zero sequences (x1, x2, x3, . . . ) of scalars}; L−n = {such sequences with x1 = x2 = · · · = xn−1 = 0} (n < 1).

Andrey Blinov Persistent Homology: Persistence Modules

Lesnick’s (counter)example

Even better counterexample with T = Z≤0: L0 = k; L−1 = {eventually-zero sequences (x1, x2, x3, . . . ) of scalars}; L−n = {such sequences with x1 = x2 = · · · = xn−1 = 0} (n < 1). Again, not a direct sum of interval modules. All dimensions are countable.

Andrey Blinov Persistent Homology: Persistence Modules

Interval modules

Interval modules — building blocks of persistent modules.

Andrey Blinov Persistent Homology: Persistence Modules

Interval modules

Interval modules — building blocks of persistent modules. Extended reals — R ∪ {−∞, +∞}.

Andrey Blinov Persistent Homology: Persistence Modules

Interval modules

Interval modules — building blocks of persistent modules. Extended reals — R ∪ {−∞, +∞}. A pair of extended reals (p, q) produces up to four intervals: [p, q), [p, q], (p, q), (p, q].

Andrey Blinov Persistent Homology: Persistence Modules

Interval modules

Interval modules — building blocks of persistent modules. Extended reals — R ∪ {−∞, +∞}. A pair of extended reals (p, q) produces up to four intervals: [p, q), [p, q], (p, q), (p, q]. There are two ways to resolve the discrepancy: consider only one type (i.e. p and q produce [p, q)); decorate reals.

Andrey Blinov Persistent Homology: Persistence Modules

Decorated extended reals

Decorate reals with ± upperscripts: (p−, q−) denotes [p, q); (p−, q+) denotes [p, q]; (p+, q−) denotes (p, q); (p+, q+) denotes (p, q].

Andrey Blinov Persistent Homology: Persistence Modules

Decorated extended reals

Decorate reals with ± upperscripts: (p−, q−) denotes [p, q); (p−, q+) denotes [p, q]; (p+, q−) denotes (p, q); (p+, q+) denotes (p, q]. Also, −∞ = −∞+ = −∞− and ∞ = ∞+ = ∞−.

Andrey Blinov Persistent Homology: Persistence Modules

Decorated extended reals

Decorate reals with ± upperscripts: (p−, q−) denotes [p, q); (p−, q+) denotes [p, q]; (p+, q−) denotes (p, q); (p+, q+) denotes (p, q]. Also, −∞ = −∞+ = −∞− and ∞ = ∞+ = ∞−. Additionally, consider p− < p < p+. That way we get: Definition (p∗, q∗) := {t ∈ R, p∗ < t < q∗}

Andrey Blinov Persistent Homology: Persistence Modules

Decorated persistence diagram

Andrey Blinov Persistent Homology: Persistence Modules

Decorated persistence diagram

Andrey Blinov Persistent Homology: Persistence Modules

Application: clustering

Andrey Blinov Persistent Homology: Persistence Modules