Persistent Homology
Cech complex
Let
be $n$-points in $ℝ^D$.
- The Cech complex of threshold $r ≥ 0$ on $P$:
-
is the simplicial complex $𝒞^r(P)$ satisfying:
σ = \lbrace p_{i_0}, …, p_{i_d} \rbrace ∈ 𝒞^r(P) \quad \text{ iff } \quad \bigcap\limits_{j=0}^d B^r(p_{i_j}) ≠ ∅
(cf. picture)
- Nerve lemma:
-
The union of balls $\bigcup\limits_{p ∈ P} B^r(p)$ and the domain of the Cech complex $\vert 𝒞^r(P) \vert$ are homotopy equivalent
- A filtration:
-
is a family of simplicial complexes
$\lbrace K_i \rbrace_{i ∈ [0, n]}$ such thatK_i ⊆ K_{i+1} \qquad \text{ for all } i=0, …, n-1
In particular, let $K$ be a finite complex and $f: K ⟶ ℝ$ that satisfies
then $f$ induces a filtration on $K$, where for all $α ∈ ℝ$:
Let $K = \lbrace σ_1, …, σ_m\rbrace$ (where subfaces appear before faces) st
Let $s_0, …, s_n ∈ ℝ$ satisfying
The filtration induced by $f$ on $K$ is:
For the $𝒞(P)$, there are finitely many radii $r$ where the topology of
changes.
Call these values $r_i, …, r_n$. Consider $s_0, …, s_n$ st
The Cech filtration is
Look at the homology of this complex ($\ast$ = any dimension):
NB:
- we use this multi-scale approach to get rid of the noise of small holes/artefacts.
- if we start from a finite set of points, then these homology vector spaces are finite dimensional.
Persistence Modules
Fix a field $𝔽$. In what follows, all v.s. are $𝔽$-vector spaces.
Homology has $𝔽$ coefficients. Here, let $𝔽 \; ≝ \; ℤ/2ℤ$
For a fixed set of indices $α_0 < α_1 < ⋯ < α_n$ (real numbers).
- A persistence module:
-
is a collection of finite dimensional vector spaces indexed over the $α_i$’s:
and linear maps
Actually, you look at it as a graded module \bigoplus_i V_{α_i}
and the ring acting on it is $𝔽[t]$:
Example: cf. picture
Morphisms on persistence modules
A morphism of persistence modules indexed on the same $α_i$’s
is a collection of linear maps
st these diagrams commute:
If all $ϕ_i$’s are isomorphisms, we say that $ϕ$ is an iso of persistence modules.
Persistence modules decomposition
A direct sum of persistence modules $𝕍 = (V_{α_i}, f_i)$ and $𝕎 = (W_{α_i}, g_i)$ is
- A persistence module $𝕍$ is indecomposable:
-
iff 𝕍 ≅ 𝕍_1 ⊕ 𝕍_2 \quad ⟹ \quad 𝕍_1 \text{ or } 𝕍_2 \text{ is trivial}
Th: Every persistence module $𝕍$ indexed over $α_0, …, α_n$ admits a unique decomposition into indecomposables, called interval modules:
𝕍 ≅ \bigoplus\limits_{j} 𝕀[b_j; d_j]where ($b$ stands for birth, $d$ stands for death):
𝕀[b; d] = 0 ⟶ ⋯ ⟶ 0 ⟶ \underbrace{𝔽}_{\text{at index } α_i = b} \overset{id}{⟶} ⋯ \overset{id}{⟶} \underbrace{𝔽}_{\text{at index } α_j = d} ⟶ 0 ⟶ ⋯ ⟶ 0
Example:
- Uniqueness property: Krull-Remak-Schmidt
- Decomposition: Gabriel
- cf picture
- cf second picture. Motto: « you kill the youngest. »
Pesistence Barcode
Let 𝕍 ≅ \bigoplus\limits_j 𝕀[b_j, d_j]
- Its persistence barcode:
-
is the collection of segments \lbrace [b_j, d_j] \rbrace_j
- Its persistence diagram:
-
is the collection of points \underbrace{\lbrace (b_j, d_j) \rbrace_j}_{⊆ ℝ^2} ∪ \underbrace{\lbrace (x, x) \rbrace_{x ∈ ℝ}}_{\text{diagonal } Δ \text{, added for convenience}}
Algorithm to compute persistence (i.e. persistent homology)
Wlog, all filtrations are of the form
for all $i= 0, …, n-1$:
NB: if you want to go back to the geometry, it’s enough to do, for any filtration function $f: K_n ⟶ ℝ$ (in cas of Cech: the radius of the smallest ciconscribing ball)
Proposition: Let $K \hookrightarrow K ∪ \lbrace σ\rbrace$, $\dim σ = d$, then
- either \dim H_{d-1}(K ∪ \lbrace σ \rbrace) = \dim H_{d-1}(K)-1
- or \dim H_d(K ∪ \lbrace σ \rbrace) = \dim H_d(K)+1
(cf. picture)
Proof: Note that for all $p$,
because the equations of the form $\partial c = 0$ and $∃d; c = \partial d$ remain true.
Let $d = \dim σ$
Also:
We know that
(kernel and image of the same map)
So
- either $\dim Z_d(K ∪ \lbrace σ \rbrace) = \dim Z_d(K) + 1$
- or $\dim B_{d-1}(K ∪ \lbrace σ \rbrace) = \dim B_{d-1}(K) + 1$
and we conclude with
Proposition:
K \hookrightarrow K ∪ \lbrace σ\rbrace \qquad \dim σ = d
$σ$ is a creator: \dim H_d(K ∪ \lbrace σ\rbrace) = \dim H_d(K) + 1 \quad ⟺ \quad [\partial σ] = 0 \text{ in } H_{d-1}(K)
$σ$ is a destructor:
\dim H_{d-1}(K ∪ \lbrace σ \rbrace) = \dim H_d(K) - 1 \quad ⟺ \quad [\partial σ] ≠ 0 \text{ in } H_{d-1}(K)
Proof:
Proof for the destructor case:
but $\partial σ ∈ B_{d-1}(K ∪ \lbrace σ\rbrace)$, which implies
Creator case:
If $[\partial σ] = 0$ in $K$, then $σ ∈ B_{d-1}(K)$, so there exists a chain $d$ in $K$ st
Consequently, $d - σ$ is a cycle in $K ∪ \lbrace σ\rbrace$, because
Additionally, if
then the family
is free, because none of the $c_i$’s contains $σ$ in their simplices.
Algorithm
Let
We construct a compatible basis
for $C(K_n)$, where
- $C(K_n) = \bigoplus_d C_d(K_n)$
- $Z(K) = \bigoplus_d Z_d(K)$
- $B(K) = \bigoplus_d B_d(K)$
and a partition
of the indices $\lbrace 1, …, n \rbrace$ satisfying
-
∀ i ∈ \lbrace 1, …, n\rbrace, \quad ⟨c_1, …, c_i⟩ = C(K_i) \\ ⟺ ∀ i, \quad c_i = ε_1 σ_1 + ⋯ + ε_{i-1} σ_{i-1} + \underbrace{σ_i}_{≠ 0} \qquad \text{ (where } ε_j ∈ 𝔽 \text{)}
-
∀ f ∈ F, \quad \partial c_f = 0
-
for all pairs $(g, h) ∈ 𝒫$, $g ∈ G$ and $h ∈ H$:
\partial c_h = c_g
Finally,
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