Simplicial Complexes

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Simplicial Complexes

Jean-Daniel Boissonnat Geometrica, INRIA

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

Winter School, University of Nice Sophia Antipolis January 26-30, 2015

Winter School 1 Simplicial Complexes Sophia Antipolis 1 / 39

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Examples of simplicial complexes

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Geometric simplices

Ak-simplexσ is the convex hull ofk+1points ofR^d that are affinely independent

σ =conv(p0, ...,pk) ={x∈R^d, x= Xk

i=0

λi pi, λi∈[0,1], Xk

i=0

λi=1}

k=dim(aff(σ))is called the dimension ofσ

1-simplex = line segment 2-simplex = triangle 3-simplex = tetrahedron

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Faces of a simplex

V(σ) =set of vertices of ak-simplexσ

∀V⁰⊆V(σ),conv(V⁰)is afaceofσ

ak-simplex has

k+1 i+1

faces of dimensioni

total nb of faces=Pd i=0

k+1 i+1

=2^k+1−1

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Geometric simplicial complexes

A finite collection of simplicesKcalled thefacesofKsuch that

∀σ∈K,σ is a simplex σ ∈K,τ ⊂σ ⇒τ ∈K

∀σ, τ ∈K, eitherσ∩τ =∅orσ∩τ is a common face of both

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Geometric simplicial complexes

Thedimensionof a simplicial complexKis the max dimension of its simplices

A subset ofKwhich is a complex is called asubcomplexofK

Theunderlying space|K| ⊂R^d ofKis the union of the simplices ofK

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Example 1 : Triangulation of a finite point set of R

^d

A simpliciald-complexKispureif every simplex inK is the face of ad-simplex.

Atriangulationof a finite point setP∈R^dis a pure geometric simplicial complexKs.t. vert(K) =P and |K|=conv(P).

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Example 1 : Triangulation of a finite point set of R

^d

A simpliciald-complexKispureif every simplex inK is the face of ad-simplex.

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Example 2 : triangulation of a polygonal domain of R

²

Atriangulationof a polygonal domainΩ⊂R²is a pure geometric simplicial complexK s.t. vert(K) =vert(Ω) and |K|= Ω.

Exercises

I Show that such a triangulation exists for anyΩ

I Propose an algorithm of complexityO(nlogn)to compute it where n=]vert(Ω)

I Show that some polyhedral domains ofR³do not admit a triangulation

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Example 2 : triangulation of a polygonal domain of R

²

Atriangulationof a polygonal domainΩ⊂R²is a pure geometric simplicial complexK s.t. vert(K) =vert(Ω) and |K|= Ω.

Exercises

I Show that such a triangulation exists for anyΩ

I Propose an algorithm of complexityO(nlogn)to compute it where n=]vert(Ω)

I Show that some polyhedral domains ofR³do not admit a triangulation

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Example 3 : the boundary complex of the convex hull of a finite set of points in general position

Polytope

conv(P) ={x∈R^d, x=Pk

i=0 λip_i, λi ∈[0,1], Pk

i=0λi =1} Supporting hyperplaneH :

H∩P6=∅, Pon one side ofH Faces:conv(P)∩H,Hsupp. hyp.

Pis ingeneral positioniff no subset ofk+2points lie in ak-flat IfPis in general position, all faces ofconv(P)are simplices

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Example 3 : the boundary complex of the convex hull of a finite set of points in general position

Polytope

i=0 λip_i, λi ∈[0,1], Pk

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Example 3 : the boundary complex of the convex hull of a finite set of points in general position

Polytope

i=0 λip_i, λi ∈[0,1], Pk

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Abstract simplicial complexes

Given a finite set of elementsP, an abstract simplicial complexKwith vertex setPis a set of subsets ofPs.t.

1 ∀p∈P, p∈K

2 ifσ∈Kandτ ⊆σ, thenτ ∈K

The elements ofK are called the (abstract) simplices or faces ofK The dimension of a simplexσisdim(σ) =]vert(σ)−1

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Nerve of a finite cover U = { U

1

, ..., U

_n

} of X

An example of an abstract simplicial complex

Computational Topology (Jeff Erickson) Examples of Cell Complexes

Corollary 15.1.For any points setPand radius�, the Aleksandrov-ˇCech complexAˇC_�(P)is homotopy- equivalent to the union of balls of radius�centered at points inP.

Aleksandrov-ˇCech complexes and unions of balls for two different radii. 2-simplices are yellow; 3-simplices are green.

15.1.2 Vietoris-Rips Complexes: Flags and Shadows

Theproximity graph N_�(P)is the geometric graph whose vertices are the pointsPand whose edges join all pairs of points at distance at most 2�; in other words,N_�(P)is the 1-skeleton of the Aleksandrov-ˇCech complex. TheVietoris-Rips complex VR_�(P)is theflag complexorclique complexof the proximity graphN_�(P). A set ofk+1 points inPdefines ak-simplex inV R_�(P)if and only if every pair defines an edge inN_�(P), or equivalently, if the set has diameter at most 2�. Again, the Vietoris-Rips complex is an abstractsimplicial complex.

The Vietoris-Rips complex was used by Leopold Vietoris[57]in the early days of homology theory as a means of creating finite simplicial models of metric spaces.²The complex was rediscovered by Eliayu Rips in the 1980s and popularized by Mikhail Gromov[35]as a means of building simplicial models for group actions. ‘Rips complexes’ are now a standard tool in geometric and combinatorial group theory.

The triangle inequality immediately implies the nesting relationshipACˇ_�(P)⊆V R_�(P)⊆ACˇ_2�(P) for any�, where⊆indicates containmentas abstract simplicial complexes. The upper radius 2�can be reduced to�3�/2 if the underlying metric space is Euclidean[21], but for arbitrary metric spaces, these bounds cannot be improved.

One big advantage of Vietoris-Rips complexes is that they determined entirely by their underlying proximity graphs; thus, they can be applied in contexts like sensor-network modeling where the underlying metric is unknown. In contrast, the Aleksandrov-ˇCech complex also depends on the metric of the ambient space that containsP; even if we assume that the underlying space is Euclidean, we need the lengths of the edges of the proximity complex to reconstruct the Aleksandrov-ˇCech complex.

On the other hand, there is no result like the Nerve Lemma for flag complexes. Indeed, it is easy to construct Vietoris-Rips complexes for pointsin the Euclidean planethat contain topological features of arbitrarily high dimension.

2Vietoris actually defined a slightly different complex. LetU={U1,U₂, . . .}be a set of open sets that cover some topological spaceX. TheVietoris complexofUis the abstract simplicial complex whose vertices are points inX, and whose simplices are finite subsets ofXthat lie in some common setU_i. Thus, the Vietoris complex of an open cover is the dual of its Aleskandrov-ˇCech nerve. Dowker[25]proved that these two simplicial complexes have isomorphic homology groups.

2

ThenerveofU is the simplicial complexK(U)defined by σ = [Ui₀, ...,Ui_k]∈K(U) ⇔ ∩^ki=1Ui_j 6=∅

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Realization of an abstract simplicial complex

Arealizationof an abstract simplicial complexKis a geometric simplicial complexKgwhose corresponding abstract simplicial complex isisomorphictoK, i.e.

∃bijectivef :vert(K)→vert(Kg) s.t. σ∈K ⇒ f(σ)∈Kg

Any abstract simplicial complexKcan be realized inRⁿ

Hint : v_i →p_i = (0, ...,0,1,0, ...0)∈Rⁿ (n=]vert(K)) σ=conv(p₁, ...,p_n) (canonical simplex) Kg⊆σ

Realizations are not unique but are alltopologically equivalent (homeomorphic)

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Topological equivalence

Two subsetsXandY ofR^dare said to betopologically equivalentor homeomorphicif there exists a continuous, bijective mapf :X→Y with continuous inversef⁻¹

Topological disks Nota topological disk

No need for the conditionf⁻¹to be continuous ifX is compact andY is Hausdorff

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Topological equivalence

Two subsetsXandY ofR^dare said to betopologically equivalentor homeomorphicif there exists a continuous, bijective mapf :X→Y with continuous inversef⁻¹

Topological disks Nota topological disk

No need for the conditionf⁻¹to be continuous ifX is compact andY is Hausdorff

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Are these objects homeomorphic ?

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Are these objects homeomorphic ?

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Are these objects homeomorphic ?

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Triangulated balls and spheres

A triangulatedd-ball ((d−1)-sphere) is a simplicial complex whose realization is homeomorphic to the unitd-ball ((d−1)-sphere) ofR^d

Examples

I a triangulated simple polygon

I the boundary complex of a simpliciald-polytope is a triangulated (d−1)-sphere

I a triangulated polyhedron without hole

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Triangulated balls and spheres

A triangulatedd-ball ((d−1)-sphere) is a simplicial complex whose realization is homeomorphic to the unitd-ball ((d−1)-sphere) ofR^d

Examples

I a triangulated simple polygon

I the boundary complex of a simpliciald-polytope is a triangulated (d−1)-sphere

I a triangulated polyhedron without hole

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A weaker notion of topological equivalence

LetXandY be two subsets ofR^d. Two mapsf₀,f₁:X→Y are said to behomotopicif there exists a continuous mapH: [0,1]×X →Y s.t.

∀x∈X, H(0,x) =f₀(x) ∧ H(1,x) =f₁(x)

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Homotopy equivalence

XandY are said to behomotopy equivalentif there exist two continuous mapsf :X→Y andg:Y →X such thatf◦g(g◦f) is homotopic to the identity map inY (X)

Deformation retract: r:X→Y ⊆X is a d.r. if it is homotopic toId XandY then have the same homotopy type

Xis said to becontractibleif it has the same homotopy type as a point

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Homotopy equivalence

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Homotopy equivalence

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Nerve of a finite cover U = { U

1

, ..., U

_n

} of X

Corollary 15.1.For any points setPand radius�, the Aleksandrov-ˇCech complexACˇ_�(P)is homotopy- equivalent to the union of balls of radius�centered at points inP.

2Vietoris actually defined a slightly different complex. LetU={U₁,U₂, . . .}be a set of open sets that cover some topological spaceX. TheVietoris complexofUis the abstract simplicial complex whose vertices are points inX, and whose simplices are finite subsets ofXthat lie in some common setUi. Thus, the Vietoris complex of an open cover is the dual of its Aleskandrov-ˇCech nerve. Dowker[25]proved that these two simplicial complexes have isomorphic homology groups.

ThenerveofU is the simplicial complexK(U)defined by σ = [Ui0, ...,U_i_k]∈K(U) ⇔ ∩^ki=1U_i_j 6=∅

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Nerve of a cover

Corollary 15.1.For any points setPand radius�, the Aleksandrov-ˇCech complexAˇC_�(P)is homotopy- equivalent to the union of balls of radius�centered at points inP.

2Vietoris actually defined a slightly different complex. LetU={U1,U₂, . . .}be a set of open sets that cover some topological spaceX. TheVietoris complexofUis the abstract simplicial complex whose vertices are points inX, and whose simplices are finite subsets ofXthat lie in some common setU_i. Thus, the Vietoris complex of an open cover is the dual of its Aleskandrov-ˇCech nerve. Dowker[25]proved that these two simplicial complexes have isomorphic homology groups.

2

Nerve Theorem (Leray)

If any intersection of theU_i is either empty or contractible, thenX and K(U)have the same homotopy type

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Example 1: Cech complex of a point set P ⊂ R

^d

σ⊆P∈C(P, α) ⇔ ∩^p∈σB(p, α)6=∅

SIMPLICIAL HOMOLOGY OF RANDOM CONFIGURATIONS 7

Sensor network coverage ˘Cech complex representation

S1 v1

S1 S2 v1 v2

S1 S2

S3

v1 v2

v3

S1 S2

S2

v1 v2

v3

S1 S2 S3 S4

!

!!

v1

v2

v3

v4

Table 1.Topological representation of the coverage of a sensor network. Each nodevrepresents a sensor. From top to bottom, the highest order simplex is a vertex, an edge, a triangle, three edges, a tetrahedron.

intensityλin a Polish spaceY. The space of configurations onY, is the set of locally finite simple point measures (cf [18]):

Ω^Y=

! ω=

"n k=0

δ(xk) : (xk)^k=n_k=0⊂Y, n∈N∪ {∞}

# ,

whereδ(x)denotes the Dirac measure atx∈ Y. Simple measure means that ω({x})≤1for anyx∈Y. Locally finite means thatω(K)<∞for any compactK ofY. It is often convenient to identify an elementωofΩ^Ywith the set corresponding to its support, i.e.,$n

k=0δ(xk)is identified with the unordered set{x1,· · ·, xn}.

ForA∈ B(Y), we haveδ(xk)(A) =1[xk∈A], so

ω(A) ="

xk∈ω

1[xk∈A]=

%

A

dω(x),

counts the number of atoms inA. The configuration spaceΩ^Yis endowed with the vague topology and its associatedσ-algebra denoted byF^Y. Sinceωis a Poisson

hal-00578955, version 2 - 12 Jul 2011

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Exercises

Show thatσ ∈C(P, α) ⇔ R(minball(P))≤α Propose an algorithm to compute minball(P) (O(#P)time complexity for fixed dimensiond) Involves computing radii of circumscribing spheres

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Example 2 : Rips complex of P

σ⊆P∈R(P, α) ⇔ ∀p,q∈σ kp−qk ≤α ⇔ B(p,α

2)∩B(q,α 2)6=∅

64 ROBERT GHRIST

Figure 2. A fixed set of points [upper left] can be completed to a Cech complexˇ C![lower left] or to a Rips complexR![lower right]

based on a proximity parameter![upper right]. This ˇCech complex has the homotopy type of the!/2 cover (S¹∨S¹∨S¹), while the Rips complex has a wholly different homotopy type (S¹∨S²).

stored as a graph and reconstituted instead of storing the entire boundary operator needed for a ˇCech complex. This virtue — that coarse proximity data on pairs of nodes determines the Rips complex — is not without cost. The penalty for this simplicity is that it is not immediately clear what is encoded in the homotopy type ofR. In general, it is neither a subcomplex ofEⁿ nor does it necessarily behave like ann-dimensional space at all (Figure 2).

1.4. Which!? Converting a point cloud data set into a global complex (whether Rips, ˇCech, or other) requires a choice of parameter !. For! sufficiently small, the complex is a discrete set; for!sufficiently large, the complex is a single high- dimensional simplex. Is there an optimal choice for ! which best captures the topology of the data set? Consider the point cloud data set and a sequence of Rips complexes as illustrated in Figure 3. This point cloud is a sampling of points on a planar annulus. Can this be deduced? From the figure, it certainly appears as though an ideal choice of !, if it exists, is rare: by the time!is increased so as to remove small holes from within the annulus, the large hole distinguishing the annulus from the disk is filled in.

2. Algebraic topology for data

Algebraic topology offers a mature set of tools for counting and collating holes and other topological features in spaces and maps between them. In the context of high-dimensional data, algebraic topology works like a telescope, revealing objects and features not visible to the naked eye. In what follows, we concentrate on ho-

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Exercises

Show thatR(P, α)⊆C(P, α)⊆R(P,2α)

ComputingR(P, α)reduces to computing the graphG (vertices+edges) ofR(P, α)and computing the cliques ofG

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Nerves of Euclidean Voronoi diagrams

Voronoi cell V(pi) ={x:kx−pik ≤ kx−pjk, ∀j}

Voronoi diagram(P) ={collection of all cellsV(pi),p_i ∈ P }

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Nerves of Euclidean Voronoi diagrams

The nerve ofVor(P)is called the Delaunay complexDel(P) Del(P)cannot always be realized inR^d

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Triangulation of a finite point set of R

^d

A simplicialk-complexK ispureif every simplex inKis the face of ak-simplex.

Problem : show that the Delaunay triangulation of a finite point set of R^d is a triangulation under some mild genericity assumption

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Triangulation of a finite point set of R

^d

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Triangulation of a finite point set of R

^d

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Stars and links

LetK be a simplicial complex with vertex setP. Thestarofp∈P is the set of simplices ofK that havepas a vertex

Thelinkofpis the set of simplicesτ ⊂σsuch thatσ ∈star(p,K) butτ 6∈star(p,K)

IfKis a triangulation of a point set

the link of any vertex ofK\∂K is a triangulated(k−1)-sphere the link of any vertex of∂Kis a triangulated(k−1)-ball

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Data structures to represent simplicial complexes

Atomic operations

Look-up/Insertion/Deletionof a simplex Thefacetsandsubfacesof a simplex Thecofacesof a simplex

Edge contractions Elementary collapses

Explicit representation of all simplices ? of all incidence relations ?

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The incidence graph

G(V,E) σ∈V ⇔ σ∈K (σ, τ)∈E ⇔ σ⊂τ

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5

2 3

4 5 5 5

3

5 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

1 1 2 2 2 3 3 4 6 6 7 7

2

1 23

4 3 2

4

3 67

3

2 24

∅

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The Hasse diagram

G(V,E) σ∈V ⇔ σ∈K

(σ, τ)∈E ⇔ σ⊂τ ∧ dim(σ) =dim(τ)−1

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5

2 3

4 5 5 5

3

5 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

1 1 2 2 2 3 3 4 6 6 7 7

2

1 23

4 3 2

4

3 67

3

2 24

∅

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The simplex tree

^{[C. Maria]}

1 Select a specific spanning tree of the Hasse diagram s.t. the chosen incidences respect the lexicographic order

2 Keep only the biggest vertex in each simplex. The vertices of a simplex are encountered in the path from the root to its node

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5

2 3

4 5 5 5

3

5 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

1 1 2 2 2 3 3 4 6 6 7 7

2

1 23

4 3 2

4

3 67

3

2 24

∅

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The simplex tree

^{[C. Maria]}

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5

2 3

4 5 5 5

3

5 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

1 1 2 2 2 3 3 4 6 6 7 7

2

1 23

4 3 2

4

3 67

3

2 24

∅

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The simplex tree

^{[C. Maria]}

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5

2 3

4 5 5 5

3

5 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

1 1 2 2 2 3 3 4 6 6 7 7

2

1 23

4 3 2

4

3 67

3

2 24

∅

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The simplex tree

^{[C. Maria]}

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5 2 3

4 5 5 5

3

5 3

3

3 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

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The simplex tree is a trie

1 index the vertices ofK

2 associate to each simplexσ ∈K, the sorted list of its vertices

3 store the simplices in atrie.

1

2 3

4

5

3 4 5

2 1

3 4 5 4 5 5 2 3

4 5 5 5

3

5 3

3

3 3

6 7 8 9

0

9 8 9 7 9

8 6

9 7

0

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Performance of the simplex tree

Explicit representation of all simplices

#nodes = #K

Memory complexity:O(1)per simplex.

depth = dim(K) +1

#children(σ) ≤ #cofaces(σ) ≤ deg(last(σ))

4.1 Memory Performance of the Simplex Tree 15

Data |P| D d r k Tg |E| TRips |K| Ttot Ttot/|K|

Bud 49,990 3 2 0.11 3 1.5 1,275,930 104.5 354,695,000 104.6 3.0·10 ⁷ Bro 15,000 25 ? 0.019 25 0.6 3083 36.5 116,743,000 37.1 3.2·10 ⁷ Cy8 6,040 24 2 0.4 24 0.11 76,657 4.5 13,379,500 4.61 3.4·10 ⁷ Kl 90,000 5 2 0.075 5 0.46 1,120,000 68.1 233,557,000 68.5 2.9·10 ⁷ S4 50,000 5 4 0.28 5 2.2 1,422,490 95.1 275,126,000 97.3 3.6·10 ⁷ Data |L| |W| D d ⇢ k Tnn TWit^⇢ |K| Ttot Ttot/|K|

Bud 10,000 49,990 3 2 0.12 3 1. 729.6 125,669,000 730.6 12·10 ³ Bro 3,000 15,000 25 ? 0.01 25 9.9 107.6 2,589,860 117.5 6.5·10 ³ Cy8 800 6,040 24 2 0.23 24 0.38 161 997,344 161.2 23·10 ³ Kl 10,000 90,000 5 2 0.11 5 2.2 572 109,094,000 574.2 5.7·10 ³ S4 50,000 200,000 5 4 0.06 5 25.1 296.7 163,455,000 321.8 1.2·10 ³

Figure 8: Data, timings (in s.) and statistics for the construction of Rips complexes (TOP) and relaxed witness complexes (BOTTOM).

We use a variety of both real and synthetic datasets. Budis a set of points sampled from the surface of theStanford BuddhainR³. Brois a set of5⇥5high-contrast patchesderived from natural images, interpreted as vectors inR²⁵, from the Brown database (with parameterk= 300 and cut30%) [12, 6]. Cy8is a set of points inR²⁴, sampled from the space of conformations of the cyclo-octane molecule [14], which is the union of two intersecting surfaces.Klis a set of points sampled from the surface of the figure eight Klein Bottle embedded inR⁵. FinallyS4is a set of points uniformly distributed on the unit4-sphere inR⁵. Datasets are listed in Figure 8 with details on the sets of pointsPor landmarksLand witnessesW, their size|P|or|L|and

|W|, the ambient dimensionD, the intrinsic dimensiondof the object the sample points belong to (if known), the parameterror⇢, the dimensionkup to which we construct the complexes, the timeTg to construct the Rips graph or the timeTnn to compute the lists of nearest neighbors of the witnesses, the number of edges|E|, the time for the construction of the Rips complexTRips or for the construction of the witness complexTWit^⇢, the size of the complex|K|, and the total construction timeTtot and average construction time per faceTtot/|K|.

We test the performance of our algorithms on these datasets, and compare them to theJPlex library[16] which is a Java software package which can be used withMatlab.JPlexis widely used to construct simplicial complexes and to compute their homology. We also provide an exper- imental analysis of the memory performance of our data structure compared to other representa- tions. Unless mentioned otherwise, all simplicial complexes are computed up to the embedding dimension, because the homology is trivial in dimenson higher than the ambient dimension. All timings are averaged over10independent runs. Due to the lack of space, we cannot report on the performance of each algorithm on each dataset but the results presented are a faithful sample of what we have observed on other datasets.

As illustrated in Figure 8, we are able to construct and represent both Rips and relaxed witness complexes of up to several hundred million faces in high dimensions, on all datasets.

4.1 Memory Performance of the Simplex Tree

In order to represent the combinatorial structure of an arbitrary simplicial complex, one needs to mark all maximal faces. Indeed, from the definition of a simplicial complex, we cannot infer

hal-00707901, version 1 - 14 Jun 2012

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Exercises

Show how to implement the atomic operations on a ST Show how to represent a Rips complex

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Computing the min. enclosing ball mb(P) of P ⊂ R

^d

Properties

mb(P)is unique

mb(P)is determined by at mostd+1points IfB=mb(P\ {p}) andp6∈B, thenp∈∂mb(P) same results for mb(P,Q), the min ballBsuch that

P⊂intB and Q∈∂B (if it exists)

IfB=mb(P\ {p},Q)) andp6∈B, then

I p∈∂mb(P,Q)(if it exists)

I ⇔mb(P,Q) =mb(P\ {p},Q∪ {p})

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Computing the min. enclosing ball mb(P) of P ⊂ R

^d

Algorithm inputP

Q:=∅ // points on∂mb(P) mb(P) := miniball(P,Q)

stop

Algorithm miniball(P,Q)

1 ifP=∅then compute directlyB:=mb(Q)

2 else

1 choose a randomp∈P

2 B:=miniball(P\ {p},Q)

3 ifp6∈BthenB:=miniball(P\ {p},Q∪ {p}) //p∈∂B

3 returnB

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Complexity analysis

LetT(n,j) =expected number of testsp6∈B with #P=n and j=d+1−#Q T(0,j) =0andT(n,0) =O(1)

sincepis any point amongPand#(B∩∂B) =j, proba (p6∈B)≤ n^j

T(n,j)≤T(n−1,j) +O(1) +_n^j T(n−1,j−1)

⇒ T(n,j)≤(j+1)!n

Complexity of mb(P)=O(d)T(n,d+1) =O(n)for fixedd