Compatible basis, Stability of persistent homology, and Homology inference

\[K_0 = ∅ \overset{σ_1}{\hookrightarrow} K_1 \overset{σ_2}{\hookrightarrow} K_2 \overset{σ_3}{\hookrightarrow} ⋯ \overset{σ_n}{\hookrightarrow} K_n = \lbrace σ_1, …, σ_n\rbrace\\ K_{i-1} \overset{σ_i}{\hookrightarrow} K_i = K_{i-1} ∪ \lbrace σ_i\rbrace\]

Compatible basis

Compatible basis at $K_n$:
  • cycles $c_1, …, c_n ∈ 𝒞(K_n)$

  • a partition $\lbrace 1, …, n\rbrace = F \sqcup G \sqcup H$

  • bijection $G ≅ H$, denoted by $𝒫 = \lbrace (g, h)\rbrace$

such that

  1. \[∀ i, ⟨c_1, …, c_i⟩ = 𝒞(K_i) \\ c_i = \underbrace{ε_1}_{∈ ℤ/2} σ_1 + ⋯ + ε_{i-1} σ_{i-1} + \underbrace{σ_i}_{≠ 0} \qquad \text{ in } ℤ/2\]
  2. \[∀ f ∈ F, \partial c_f = 0\]
  3. \[∀ (g, h) ∈ 𝒫, \partial c_h = c_g \\ ⟹ \partial c_g = 0\]

Properties:

1.

\[∀ i ∈ \lbrace 1, …, n\rbrace, \lbrace c_f \rbrace_{f ≤ i,\; f ∈ F} ∪ \lbrace c_g \rbrace_{g ≤ i,\; g ∈ G} \text{ generate } Z(K_i)\]

Consider the restriction map

\[R: \begin{cases} 𝒞(K_n) &⟶ 𝒞(K_i) \\ σ &⟼ σ && \text{ if } σ ∈ K_i\\ τ &⟼ 0 && \text{ if } τ ∈ K_n \backslash K_i\\ c_j \quad (j ≤ i) &⟼ c_j \end{cases}\]

cf. picture

2.

\[∀ i, \lbrace c_g\rbrace_{(g,h) ∈ 𝒫, \; g < h ≤ i} \quad \text{ generate } B(K_i)\]

3.

\[\lbrace [c_f] \rbrace_{f ∈ F, \; f ≤ i} ∪ \lbrace [c_g] \rbrace_{(g, h) ∈ 𝒫, \; g ≤ i < h} \quad \text{ generate } H(K_i)\]

Example: cf. picture

You can basically read the decomposition of the persistence module from a compatible basis.

Th:

Let $K_0 \hookrightarrow ⋯ \hookrightarrow K_n$ and a compatible basis $\lbrace c_1, …, c_n\rbrace$, $F \sqcup G \sqcup H = \lbrace1, …, n\rbrace$, $𝒫 ⊆ G × H$

Then the persitence module is:

\[ℍ(K_n) ≅ \bigoplus_{f ∈ F} 𝕀[f; n] \bigoplus_{(g, h) ∈ 𝒫} 𝕀[g; h-1]\]

cf. picture

Sketch of proof:

Consider one $c_i$:

\[𝒞(K_0) ⟶ 𝒞(K_1) ⟶ ⋯ ⟶ 𝒞(K_n)\]

(cf picture)

Iterative algorithm

Suppose we have a compatible basis for the filtration

\[K_0 = ∅ \overset{σ_1}{⟶} K_1 \overset{σ_2}{⟶} K_2 \overset{σ_3}{⟶} ⋯ \overset{σ_i}{⟶} K_i\\\]

$\lbrace c_1, …, c_n\rbrace$, $F \sqcup G \sqcup H = \lbrace1, …, n\rbrace$, $𝒫 ⊆ G × H$

Consider the filtration

\[∅ \overset{σ_1}{⟶} K_1 ⋯ ⟶ K_i \overset{σ_{i+1}}{⟶} K_{i+1}\]

update the compatible basis to $K_{i+1}$.

Similar to what we did during the previous class: creator or destructor?

  1. Creator: if $[\partial σ_{i+1}] = 0$ in $H(K_i)$, then $\partial σ_{i+1} ∈ B(K_i)$

    Consequently:

    \[\partial σ_{i+1} = \sum\limits_{ g ∈ G } ε_g c_g = \sum\limits_{ (g, h) ∈ 𝒫 } ε_g \partial c_h\\ ⟹ \partial \Big(σ_{i+1} - \sum\limits_{ (g, h) ∈ 𝒫 } ε_g c_h\Big) = 0\]

    So $c_{i+1}$ is a cycle:

    \[c_{i+1} = σ_{i+1} - \sum\limits_{ (g, h) ∈ 𝒫 } ε_g c_h ∈ Z(K_{i+1})\]
    • $c_{i+1}$ is independent of the other $c_j$’s in the compatible basis:
    \[⟨c_1, …, c_{i+1}⟩ = 𝒞(K_{i+1})\]
    • and $c_{i+1}$ cannot be the boundary of another chain because $σ_{i+1}$ has no coface.

    Set

    \[F ← F ∪ \lbrace i + 1\rbrace\]

    compatible basis is $c_1, …, c_i, c_{i+1}$

  2. Destructor: if $[\partial σ_{i+1}] ≠ 0$ in $H(K_i)$, then

    \[\partial σ_{i+1} = \sum\limits_{ g ∈ G } ε_g c_g + \underbrace{\sum\limits_{ f ∈ F } ε_f c_f}_{≠ 0}\]

    since

    \[\lbrace c_f \rbrace_{f ≤ i,\; f ∈ F} ∪ \lbrace c_g \rbrace_{g ≤ i,\; g ∈ G} \text{ generate } Z(K_i)\]

    Pick

    \[f_0 \; ≝ \; \max \lbrace f ∈ F \; \mid \; ε_f ≠ 0\rbrace\]

    As

    \[\partial \Big(σ_{i+1} - \sum\limits_{ (g, h) ∈ 𝒫 } ε_g c_h\Big) = \sum\limits_{ f ∈ F } ε_f c_f\]

    we can set

    \[c_{f_0} ← \sum\limits_{ f ∈ F } ε_f c_f \qquad \text{ leading term remains } σ_{f_0}\\ c_{i+1} ← σ_{i+1} - \sum\limits_{ (g, h) ∈ 𝒫 } ε_g c_h \qquad \text{ leading term remains } σ_{i+1}\]

    (Condition 1 of the compatible bases remains true)

    Now, pair $f_0$ and $i+1$:

    \[H ← H ∪ \lbrace i + 1\rbrace \\ F ← F \backslash \lbrace f_0 \rbrace\\ G ← G ∪ \lbrace f_0 \rbrace\\ 𝒫 ← 𝒫 ∪ \lbrace (f_0, i+1) \rbrace\]

    (like last time, we “kill the youngest” ($f_0$))

Example: cf. picture

NB: picking $f_0$ to be the max amounts to killing off (i.e. putting it in $G$ ⇒ trivial homology) the “youngest” (most recent) $c_i$ in the persistence module (cf. last week)

Matrix Implementation

We don’t really need to keep track of $H$ actually.

Let $M$ be the matrix of

\[\partial: \underbrace{𝒞(K_n)}_{= \bigoplus_d C_d(K_n)} ⟶ 𝒞(K_n)\]

in the canonical basis $\lbrace σ_1, …, σ_n\rbrace$ (boundary map extended to all dimensions, convenient way to only deal with one matrix).

Define $low(i)$ to be the index of the lowest non-zero coefficient in the $i$-th column $col_i$ of $M$.

If $col_i = 0$, then $low(i)$ is undefined.

Algorithm

for i = 1 … n:
    while ∃ j < i s.t. low(j) = low(i):
        col_i ← col_i + col_j (in ℤ/2) # amounts to writing ∂σ_{i+1}  = ∑_{g∈G} ε_g c_g + ∑_{f∈F} ε_f c_f
    if col_i ≠ 0: # you have c_f's
        add (low(i), i-1) to the persistence diagram # what we called (f_0 = g, h-1) before, we pair f_0 = low(i) with index i

# Tackle the (f, n)
Add all (i, n) s.t. col_i = 0 and there is no column col_j with low(j) = i to the persistence diagram

The algorithm maintains the following property: at the end of the $i$-th iteration of the for loop, the matrix of $\partial$ looks like this: cf. picture

Stability of persistent homology

Let

\[D = \lbrace (b_i, d_i)\rbrace_{i ∈ I} ∪ \lbrace (x, x) \; \mid \; x ∈ ℝ\rbrace\\ D' = \lbrace (b_j', d_j')\rbrace_{j ∈ J} ∪ \lbrace (x, x) \; \mid \; x ∈ ℝ\rbrace\\\]

be two persistence diagrams.

To compare them, we will introduce a notion of “distance” between them.

The bottleneck distance between $D$ and $D’$:

is defined as

\[d_B(D, D') \; ≝ \; \inf\limits_{ϕ: D \, ≅ \, D'} \sup_{p ∈ D} \Vert p - ϕ(p) \Vert_∞\]

Note that bijections $ϕ$ exist thanks to the addition of the diagonal (that’s the only reason we added the diagonal to persistence diagrams in the first place).

cf. picture

How are the original of geometry and the one of the persistence diagrams related?

Stability theorem

Th (Stability): Let $f, g: K ⟶ ℝ$, where $K$ is a finite simplicial complex, such that $f$ and $g$ induce a level-set filtration on $K$.

\[K_α = f^{-1} (-∞, α] \quad \text{ is a simplicial complex}\]

Let $Dgm(f)$ be the persistence diagram of the filtration

\[∅ ⟶ K_{α_1} ⟶ K_{α_2} ⟶ ⋯ ⟶ K_{α_n}\]

where the $α_i$’s are in the image of $f$.

Define similarly $Dgm(g)$. Then

\[d_B(Dgm(f), Dgm(g)) ≤ \Vert f - g \Vert_∞ = \max_{σ ∈ K} \vert f(σ) - g(σ) \vert\]

Continuous version:

Th (Stability 2): Let $f, g: ℳ ⟶ ℝ$ where $ℳ$ is a metric space, such that $f$ and $g$ are tame.

$f$ and $g$ define level-set filtrations

\[\lbrace f^{-1}(-∞, r]\rbrace_{r ∈ ℝ} \qquad \text{ and } \qquad \lbrace g^{-1}(-∞, r]\rbrace_{r ∈ ℝ}\]

then

\[d_B(Dgm(f), Dgm(g)) ≤ \Vert f - g \Vert_∞\]

Philosophy of the proof:

\[\Vert f - g \Vert_∞ ≤ ε\]

so

\[∀ r, \quad f^{-1}(-∞, r] ⊆ g^{-1}(-∞, r+ε] ⊆ f^{-1}(-∞, r + 2ε] ⊆ ⋯\]

cf. picture

Application: Homology inference

Back to our very first motivation: we’re given a set of points, how do we unravel their underlying topology?

Reminder:

Haussdorff distance between compact sets $X, Y ⊆ ℝ^N$:
\[d_H(X, Y) = \max \lbrace \sup_{x ∈ X} d(x, Y), \sup_{y ∈ Y} d(y, X)\rbrace\]

where \(d(x, Y) = \inf_{y ∈ Y} d(x, Y)\)

(cf. picture)

A set of points $P$ is an $ε$-sample of a compact set $K ⊆ ℝ^N$:

if \(d_H(P, K) ≤ ε\)

Note that for every compact set $K ⊆ ℝ^N$ induces a “distance-to-this-compact” function

\[d_K: \begin{cases} ℝ^N &⟶ ℝ \\ x &⟼ d(x, K) \end{cases}\]

Level-set

\[d^{-1}(-∞, r] = \lbrace \text{ all points of } ℝ^N \text{ that are at distance at most } r \text{ from } K\rbrace\]

Measure of “complexity” of the geometry of a compact

Let $K$ be a compact.

A critical point for $d_K$:

is a point $x ∈ ℝ^N$ with at least two nearest neighbors on $K$. I.e.

\[∃ y_1, y_2 ∈ K, \quad y_1 ≠ y_2 \text{ s.t. } d(x, y_1) = d(x, y_2) = d_K(x)\]
The reach of $K$:

is \(reach(K) \; ≝ \; \inf \; \lbrace d_K (x) \; \mid \; x \text{ is critical}\rbrace\)

Inference theorem

Let $K ⊆ ℝ^N$ be a “nice” compact set of reach $ε > 0$.

Let $P$ be a $ν$-sample of $K$, with $ν « ε$.

Then there is a one-to-one correspondence between the points of the persistence diagram of the Cech filtration on $P$ that are $ν$-away from the diagonal and the homology features of $K$.

cf. picture

Sketch of proof:

Claims:

  1. $d_K^{-1}(-∞, r]$ has the same homotopy type as $K$ for all $r < reach(K)$

    (cf. picture)

  2. if $P$ is a $ν$-sample, i.e. $d_H(P, K) ≤ ν$, then

    \[\Vert d_P - d_K \Vert_∞ ≤ ν\]

  3. Stability theorem: \(d_B(Dgm(d_K), Dgm(d_P)) ≤ ν = \Vert d_K - d_P\Vert_∞\)

  4. \[d_P^{-1}(-∞, r] = \bigcup\limits_{p∈P} \overline{B}(p, r) \overset{\text{Nerve lemma}}{≅} 𝒞^r_{Cech}(P)\]
  5. Persistent homolgy of $d_K$ will be expressed as something like $\bigoplus 𝕀[0, ε]$

cf. picture

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