Witness Complex

Witness Complex

Jean-Daniel Boissonnat DataShape, INRIA

http://www-sop.inria.fr/datashape

Delaunay triangulations

Finite set of pointsP∈R^d

σ ∈DT(P) ⇔ ∃c_σ : kc_σ−pk ≤ kc_σ−qk ∀p∈σ and ∀q∈P

d

Motivations

Define Delaunay-like complexes in any metric space Walk around the curse of dimensionality

I The combinatorial complexity of DT depends exponentially on the ambient dimensiond

I The algebraic complexity depends ond

p0

p1 p2

p4

insphere(p0, . . . ,pd+1) =orient(ˆp0, . . . ,ˆpd+1)

=sign

1 . . . 1

p0 . . . pd+1

p²₀ . . . p²_d+1

Motivations

Define Delaunay-like complexes in any metric space Walk around the curse of dimensionality

I The combinatorial complexity of DT depends exponentially on the ambient dimensiond

I The algebraic complexity depends ond

p0

p4

insphere(p0, . . . ,pd+1) =orient(ˆp0, . . . ,ˆpd+1)

=sign

1 . . . 1

p . . . p

Witness Complex

La finite set of points (landmarks) vertices of the complex

W a dense sample (witnesses) pseudo circumcenters

The Witness Complex

Wit(L, W)

w

Definition

2Wit(L, W) () 8⌧ 9w2W with

d(w, p)d(w, q) 8p2 8q2L\

L:Landmarks (black dots); the vertices of the complex W :Witnesses (blue dots)

R. Dyer (INRIA) Del(L,M) =Wit(L, W) Assisi, EuroCG 2012 2 / 10

Letσbe a (abstract) simplex with vertices inL, and letw∈W. We say thatwis awitnessofσif

kw−pk ≤ kw−qk ∀p∈σ and ∀q∈L\σ

Thewitness complexWit(L,W)is the complex consisting of all simplexes σsuch that for any simplexτ⊆σ,τ has a witness inW

Witness Complex

La finite set of points (landmarks) vertices of the complex

W a dense sample (witnesses) pseudo circumcenters

The Witness Complex

Wit(L, W)

w

Definition

2Wit(L, W) () 8⌧ 9w2W with

d(w, p)d(w, q) 8p2 8q2L\

L:Landmarks (black dots); the vertices of the complex W :Witnesses (blue dots)

Letσbe a (abstract) simplex with vertices inL, and letw∈W. We say thatwis awitnessofσif

kw−pk ≤ kw−qk ∀p∈σ and ∀q∈L\σ

Witness Complex

La finite set of points (landmarks) vertices of the complex

W a dense sample (witnesses) pseudo circumcenters

The Witness Complex

Wit(L, W)

w

Definition

2Wit(L, W) () 8⌧ 9w2W with

d(w, p)d(w, q) 8p2 8q2L\

L:Landmarks (black dots); the vertices of the complex W :Witnesses (blue dots)

R. Dyer (INRIA) Del(L,M) =Wit(L, W) Assisi, EuroCG 2012 2 / 10

Letσbe a (abstract) simplex with vertices inL, and letw∈W. We say thatwis awitnessofσif

kw−pk ≤ kw−qk ∀p∈σ and ∀q∈L\σ

Thewitness complexWit(L,W)is the complex consisting of all simplexes σsuch that for any simplexτ⊆σ,τ has a witness inW

Easy consequences of the definition

The witness complex can be defined for any metric space and, in particular, for discrete metric spaces

IfW⁰⊆W, thenWit(L,W⁰)⊆Wit(L,W) Del(L)⊆Wit(L,R^d)

Easy consequences of the definition

The witness complex can be defined for any metric space and, in particular, for discrete metric spaces

IfW⁰⊆W, thenWit(L,W⁰)⊆Wit(L,W) Del(L)⊆Wit(L,R^d)

Easy consequences of the definition

The witness complex can be defined for any metric space and, in particular, for discrete metric spaces

IfW⁰⊆W, thenWit(L,W⁰)⊆Wit(L,W) Del(L)⊆Wit(L,R^d)

Construction of the k-skeleton of Wit(L, W )

Input: L,W, and∀w∈Wthe listN(w) = (p0(w), ...,p_k(w)) of theknearest landmarks ofw, sorted by distance fromw Wit(L,W) :=∅

W⁰ :=W

fori=0, ...,kdo foreachw∈W⁰ do

ifthei-simplexσ(w) = (p0(w),p₂(w), ...,p_i(w))6∈Wit(L,W) then

ifthe(i−1)-faces ofσ(w)are all inWit(L,W)then addσ(w)toWit(L,W)

else

W⁰ :=W⁰\ {w}

Output: the witness complexWit(L,W)

Construction of the k-skeleton of the witness complexes

Constructing the listsN(w):O(|W|log|W|+k|W|)[Callahan & Kosaraju 1995]

Construction ofWit(L,W):O (|Wit^k(L,W)|+|W|)k² log|L|

(using ST)

|Wit^k(L,W)≤k|W|

Algebraic complexity : comparisons of (squared) distances : degree 2 Implementation and experimental results : see the Gudhi library !

Delaunay and Witness complexes

Identity of Del

(L) and Wit(L, R

)

Theorem : Wit(L,W)⊆Wit(L,R^d) =Del(L) Remarks

I Faces of all dimensions have to be witnessed

I Wit(L,W)isembeddedinR^d ifLis in general position wrt spheres

Identity of Del

(L) and Wit(L, R

)

Theorem : Wit(L,W)⊆Wit(L,R^d) =Del(L) Remarks

I Faces of all dimensions have to be witnessed

I Wit(L,W)isembeddedinR^d ifLis in general position wrt spheres

Identity of Del

(L) and Wit(L, R

)

Theorem : Wit(L,W)⊆Wit(L,R^d) =Del(L) Remarks

I Faces of all dimensions have to be witnessed

I Wit(L,W)isembeddedinR^d ifLis in general position wrt spheres

Proof of de Silva’s theorem

τ = [p0, ...,pk]is ak-simplex ofWit(L)witnessed by a ballBτ (i.e.Bτ ∩L=τ) We prove thatτ∈Del(L)by induction onk

Clearly true fork=0

Bσ

c

w

Bτ

τ σ

Hyp.: true for k⁰≤k−1 B:=Bτ

σ:=∂B∩τ, l:=|σ|

//σ∈Del(L)by the hyp.

whilel+1=dimσ <kdo B←the ball centered on[cw]s.t.

- σ⊂∂B,

- Bwitnessesτ - |∂B∩τ|=l+1

(Bwitnessesτ)∧(τ⊂∂B) ⇒τ∈Del(L)

Proof of de Silva’s theorem

τ = [p0, ...,pk]is ak-simplex ofWit(L)witnessed by a ballBτ (i.e.Bτ ∩L=τ) We prove thatτ∈Del(L)by induction onk

Clearly true fork=0

Bσ

c

w

Bτ

τ σ

Hyp.: true for k⁰≤k−1 B:=Bτ

σ:=∂B∩τ, l:=|σ|

//σ∈Del(L)by the hyp.

whilel+1=dimσ <kdo B←the ball centered on[cw]s.t.

- σ⊂∂B,

- Bwitnessesτ

Case of sampled domains : Wit(L, W) 6 = Del(L)

W a finite set of points⊂Ω⊂R^d

Wit(L,W)6=Del(L), even ifWis a dense sample ofΩ

a b

Vor(a, b)

[ab]∈Wit(L,W) ⇔ ∃p∈W, Vor2(a,b)∩W 6=∅

Protection

δ

cσ

δ-protectionWe say that a Delaunay simplexσ⊂Lisδ-protected if

Protection implies Wit(L, W) = Del(L)

Lemma LetΩbe a convex subset ofR^dand letWandLbe two finite sets of points inΩ. IfW isε-dense inΩand if all simplices (of all dimensions) ofDel_|Ω(L)areδ-protected withδ≥2ε, then

Wit(L,W)) =Del_|Ω(L)

Proof It suffices to proveWit(L,W)⊇Del_|Ω(L)

∀σ ∈Del_|Ω(L),∃c∈Ωsuch that

∀p∈σ, ∀q∈L\σ, kc−pk ≤ kc−qk −δ

W ε-dense inΩ ⇒ ∃w∈Wsuch thatkw−ck ≤ε

∀p∈σ, ∀q∈L\σ kw−pk ≤ kw−ck+ (kc−qk −δ)

≤ kw−qk+2kw−ck −δ

≤ kw−qk+2ε−δ

Good links

A simplicial complexK is ak-pseudomanifold complexwithout boundary if

1 Kis a purek-complex

2 every(k−1)-simplex is the face of exactly twok-simplices

We say that a complexK⊂ ^d = ^d/ ^d with vertex setLhas

Good links

A simplicial complexK is ak-pseudomanifold complexwithout boundary if

1 Kis a purek-complex

2 every(k−1)-simplex is the face of exactly twok-simplices

We say that a complexK⊂T^d =R^d/Z^d with vertex setLhas good linksif

∀p∈L, link(p,K)is a(d−1)-pseudomanifold

Good links implies Wit(L, W ) = Del(L)

Lemma

IfKis a triangulation ofT^d andK⁰ ⊆K a simplicial complex with the same vertex set

then K⁰ =K ⇔ K⁰has good links Corollary

If all vertices ofWit(L,W)havegood links,Wit(L,W) =Del(L)

Good links implies Wit(L, W ) = Del(L)

Lemma

IfKis a triangulation ofT^d andK⁰ ⊆K a simplicial complex with the same vertex set

then K⁰ =K ⇔ K⁰has good links Corollary

If all vertices ofWit(L,W)havegood links,Wit(L,W) =Del(L)

Good links implies Wit(L, W ) = Del(L)

Lemma

IfKis a triangulation ofT^d andK⁰ ⊆K a simplicial complex with the same vertex set

then K⁰ =K ⇔ K⁰has good links Corollary

If all vertices ofWit(L,W)havegood links,Wit(L,W) =Del(L)

Turning witness complexes into Delaunay complexes

Input: L,W,ρ(perturbation radius) Init :L⁰ :=L; computeWit(L⁰,W)

while a vertexp⁰ ofWit(L⁰,W)has a bad linkdo perturbp⁰ and the points ofI(p⁰)

updateWit(L⁰,W)

Output: Wit(L⁰,W) =Del(L⁰)

Analysis : adapt Moser-Tardos algorithmic version of the Lov ´asz Local Lemma (last lecture)

Main result

Under the condition µ

4 ≥ρ≥ 24dε

¯

µJ where J⁻¹=

2

¯ µ

O(d²)

the algorithm terminates and returnsWit(L⁰,W) =Del(L⁰)

The expected complexity islinear in|L|

Main result

Under the condition µ

4 ≥ρ≥ 24dε

¯

µJ where J⁻¹=

2

¯ µ

O(d²)

the algorithm terminates and returnsWit(L⁰,W) =Del(L⁰)

The expected complexity islinear in|L|

Relaxed witness complex

Alpha-witnessLetσ be a simplex with vertices inL. We say that a pointw∈W is anα-witness ofσ if

kw−pk ≤ kw−qk+α ∀p∈σ and ∀q∈L\σ Alpha-relaxed witness complexTheα-relaxed witness complex Wit^α(L,W)is the maximal simplicial complex with vertex setLwhose simplices have anα-witness inW

Wit⁰(L,W) =Wit(L,W)

Filtration : α≤β ⇒ Wit^α(L,W)⊆Wit^β(L,W)

Construction

A_i ={z∈L: m_i ≤d(w,z)≤m_i+α}

m5

z0 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10

{z4, z5, z6, z7, z8} 3

!

w

m3 m3+ρ A3

Fori≤j+1,wα-witnesses all thej-simplices that contain

{z₀,· · ·,zi−1}and a(j+1−i)-subset ofAi(provided|Ai| ≥j+1−i) Allj-simplices that areα-witnessed byware obtained this way, and exactly once, wheniranges from0toj+1

Interleaving complexes

LemmaAssume thatWisε-dense inΩand letα≥2ε. Then Wit(L,W)⊆Del_|Ω(L)⊆Wit^α(L,W)

Proof

σ : ad-simplex ofDel(L), c_σ its circumcenter W ε-dense inΩ ∃w∈W s.t. kc_σ−wk ≤ε For anyp∈σandq∈L\σ, we then have

∀p∈σ and q∈L\σ kw−pk ≤ kc_σ−pk+kc_σ−wk

≤ kc_σ−qk+kc_σ−wk

≤ kw−qk+2kc_σ−wk