Reducing complexes in Multidimensional Persistence Claudia Landi - - PowerPoint PPT Presentation

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Reducing complexes in Multidimensional Persistence

Claudia Landi

University of Modena and Reggio Emilia joint work with M. Allili and T. Kaczynski

GETCO 2015

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.
• An alternative strategy:
• reduce the initial complex using geometric and combinatorial methods

before computing persistence

• use reductions that preserve persistent homology groups

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.
• An alternative strategy:
• reduce the initial complex using geometric and combinatorial methods

before computing persistence

• use reductions that preserve persistent homology groups
• Case of 1D persistence in homology degree 0: [Frosini-Pittore 1999]

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.
• An alternative strategy:
• reduce the initial complex using geometric and combinatorial methods

before computing persistence

• use reductions that preserve persistent homology groups
• Case of 1D persistence in homology degree 0: [Frosini-Pittore 1999]
• Case of multiD persistence in homology degree 0: [Cerri et al 2006]:

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.
• An alternative strategy:
• reduce the initial complex using geometric and combinatorial methods

before computing persistence

• use reductions that preserve persistent homology groups
• Case of 1D persistence in homology degree 0: [Frosini-Pittore 1999]
• Case of multiD persistence in homology degree 0: [Cerri et al 2006]:
• Case of 1D persistence in any degree: [Nanda-Mischaikow 2013]

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Motivation

• The most common algorithm used for computing 1-D persistent

homology has complexity O(n3) [Zomorodian-Carlsson 2005]

• Computation of multi-D persistent homology has exponential

complexity and polynomial complexity for one-critical filtrations [Carlsson et al 2010]

• We know of no way to improve the worst case complexity of the
• problem. For massive datasets, this can be a severe limitation.
• An alternative strategy:
• reduce the initial complex using geometric and combinatorial methods

before computing persistence

• use reductions that preserve persistent homology groups
• Case of 1D persistence in homology degree 0: [Frosini-Pittore 1999]
• Case of multiD persistence in homology degree 0: [Cerri et al 2006]:
• Case of 1D persistence in any degree: [Nanda-Mischaikow 2013]
• This talk: apply this strategy for multiD persistence in any degree

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Outline

Introduction S-Complexes Multidimensional Persistent Homology

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Outline

Introduction S-Complexes Multidimensional Persistent Homology Partial Matching & Reductions in the framework of persistence

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Outline

Introduction S-Complexes Multidimensional Persistent Homology Partial Matching & Reductions in the framework of persistence A matching algorithm for multidimensional persistence

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Outline

Introduction S-Complexes Multidimensional Persistent Homology Partial Matching & Reductions in the framework of persistence A matching algorithm for multidimensional persistence Conclusion

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S-Complexes

• 1. Let S be a finite set with a gradation Sq such that Sq = ∅ for q < 0.

For every element σ ∈ S there exists a unique number q such that σ ∈ Sq. This number is called the dimension of σ and denoted dim σ.

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S-Complexes

• 1. Let S be a finite set with a gradation Sq such that Sq = ∅ for q < 0.

For every element σ ∈ S there exists a unique number q such that σ ∈ Sq. This number is called the dimension of σ and denoted dim σ.

• 2. Let κ : S × S → R, R a PID, be a function such that, if κ(σ, τ) = 0,

then dim σ = dim τ + 1. κ is called the coincidence index.

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S-Complexes

• 1. Let S be a finite set with a gradation Sq such that Sq = ∅ for q < 0.

For every element σ ∈ S there exists a unique number q such that σ ∈ Sq. This number is called the dimension of σ and denoted dim σ.

• 2. Let κ : S × S → R, R a PID, be a function such that, if κ(σ, τ) = 0,

then dim σ = dim τ + 1. κ is called the coincidence index.

• 3. Cq(S) := R(Sq), the free module over R generated by Sq.

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S-Complexes

• 1. Let S be a finite set with a gradation Sq such that Sq = ∅ for q < 0.

For every element σ ∈ S there exists a unique number q such that σ ∈ Sq. This number is called the dimension of σ and denoted dim σ.

• 2. Let κ : S × S → R, R a PID, be a function such that, if κ(σ, τ) = 0,

then dim σ = dim τ + 1. κ is called the coincidence index.

• 3. Cq(S) := R(Sq), the free module over R generated by Sq.
• 4. We say that (S, κ) is an S-complex if (C∗(S), ∂κ

∗ ) with

∂κ

q : Cq(S) → Cq−1(S) defined on generators σ ∈ S by

∂κ(σ) :=

• τ∈S

κ(σ, τ)τ is a free chain complex with base S.

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S-Complexes

• 1. Let S be a finite set with a gradation Sq such that Sq = ∅ for q < 0.

For every element σ ∈ S there exists a unique number q such that σ ∈ Sq. This number is called the dimension of σ and denoted dim σ.

• 2. Let κ : S × S → R, R a PID, be a function such that, if κ(σ, τ) = 0,

then dim σ = dim τ + 1. κ is called the coincidence index.

• 3. Cq(S) := R(Sq), the free module over R generated by Sq.
• 4. We say that (S, κ) is an S-complex if (C∗(S), ∂κ

∗ ) with

∂κ

q : Cq(S) → Cq−1(S) defined on generators σ ∈ S by

∂κ(σ) :=

• τ∈S

κ(σ, τ)τ is a free chain complex with base S.

• 5. By the homology of an S-complex (S, κ) we mean the homology of

the chain complex (C∗(S), ∂κ

∗ ), and we denote it by H∗(S, κ) or simply

by H∗(S).

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:
• 1. assume an ordering of S0 is given and every simplex σ in S is coded as

[v0, v1, . . . vq], where the vertices v0, v1, . . . vq are listed according to the prescribed ordering of S0.

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:
• 1. assume an ordering of S0 is given and every simplex σ in S is coded as

[v0, v1, . . . vq], where the vertices v0, v1, . . . vq are listed according to the prescribed ordering of S0.

• 2. define

κ(σ, τ) :=    (−1)i if σ = [v0, v1, . . . , vq] and τ = [v0, v1, . . . , vi−1, vi+1, . . . , vq]

• therwise.

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:
• 1. assume an ordering of S0 is given and every simplex σ in S is coded as

[v0, v1, . . . vq], where the vertices v0, v1, . . . vq are listed according to the prescribed ordering of S0.

• 2. define

κ(σ, τ) :=    (−1)i if σ = [v0, v1, . . . , vq] and τ = [v0, v1, . . . , vi−1, vi+1, . . . , vq]

• therwise.
• 3. obtain an S-complex whose chain complex is the classical simplicial chain

complex used in simplicial homology

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:
• 1. assume an ordering of S0 is given and every simplex σ in S is coded as

[v0, v1, . . . vq], where the vertices v0, v1, . . . vq are listed according to the prescribed ordering of S0.

• 2. define

κ(σ, τ) :=    (−1)i if σ = [v0, v1, . . . , vq] and τ = [v0, v1, . . . , vi−1, vi+1, . . . , vq]

• therwise.
• 3. obtain an S-complex whose chain complex is the classical simplicial chain

complex used in simplicial homology

• cubical complexes

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Examples of S-Complexes

Main examples of S-complexes:

• simplicial complexes:
• 1. assume an ordering of S0 is given and every simplex σ in S is coded as

[v0, v1, . . . vq], where the vertices v0, v1, . . . vq are listed according to the prescribed ordering of S0.

• 2. define

κ(σ, τ) :=    (−1)i if σ = [v0, v1, . . . , vq] and τ = [v0, v1, . . . , vi−1, vi+1, . . . , vq]

• therwise.
• 3. obtain an S-complex whose chain complex is the classical simplicial chain

complex used in simplicial homology

• cubical complexes
• cell complexes

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Multi-filtration of an S-complex

• In Rk consider the partial order a = (ai) b = (bi) if and only if

ai ≤ bi for all i = 1, 2, . . . , k;

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Multi-filtration of an S-complex

• In Rk consider the partial order a = (ai) b = (bi) if and only if

ai ≤ bi for all i = 1, 2, . . . , k;

• A k-filtration of S is a family F = {Sa}a∈Rk of subsets of S with the

following properties:

• F is nested with respect to inclusions:

Sa ⊆ Sb, for every a b

• F is non-increasing on faces:

if σ ∈ Sa and τ is a face of σ then τ ∈ Sa

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Multi-filtration of an S-complex

• In Rk consider the partial order a = (ai) b = (bi) if and only if

ai ≤ bi for all i = 1, 2, . . . , k;

• A k-filtration of S is a family F = {Sa}a∈Rk of subsets of S with the

following properties:

• F is nested with respect to inclusions:

Sa ⊆ Sb, for every a b

• F is non-increasing on faces:

if σ ∈ Sa and τ is a face of σ then τ ∈ Sa

• Given a function f : S0 → Rk, the sublevel set filtration is defined by

Sa = {σ = [v0, v1, . . . , vq] ∈ S | f(vi) a, i = 0, . . . , q}.

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Multidimensional Persistent Homology

Persistence analyzes the homological changes of the filtration as a varies:

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Multidimensional Persistent Homology

Persistence analyzes the homological changes of the filtration as a varies:

• for a b, consider the homomorphism

H∗(j(a,b)) : H∗(Sa) → H∗(Sb). induced by the inclusion map j(a,b) : Sa ֒ → Sb.

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Multidimensional Persistent Homology

Persistence analyzes the homological changes of the filtration as a varies:

• for a b, consider the homomorphism

H∗(j(a,b)) : H∗(Sa) → H∗(Sb). induced by the inclusion map j(a,b) : Sa ֒ → Sb.

• The i-th persistent homology group of the filtration at (a, b) is image
• f the map Hi(j(a,b)):

Ha,b

i

(S) := im Hi(j(a,b))

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Partial matchings

Given an S-complex (S, κ), a partial matching on (S, κ) is a quadruplet (A, B, C, m ) where

• A, B, C is a partition of S, and
• m : A → B is a map such that, for each τ ∈ A, κ(m (τ), τ) is invertible.

In particular, m (τ) is a primary coface of τ.

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Partial matchings

Given an S-complex (S, κ), a partial matching on (S, κ) is a quadruplet (A, B, C, m ) where

• A, B, C is a partition of S, and
• m : A → B is a map such that, for each τ ∈ A, κ(m (τ), τ) is invertible.

In particular, m (τ) is a primary coface of τ. (A, B, C, m ) is conveniently represented by arrows:

• For all pairs (m (τ), τ), draw an arrow from τ to m (τ)

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Partial matchings

Given an S-complex (S, κ), a partial matching on (S, κ) is a quadruplet (A, B, C, m ) where

• A, B, C is a partition of S, and
• m : A → B is a map such that, for each τ ∈ A, κ(m (τ), τ) is invertible.

In particular, m (τ) is a primary coface of τ. (A, B, C, m ) is conveniently represented by arrows:

• For all pairs (m (τ), τ), draw an arrow from τ to m (τ)
• A partial matching is called acyclic if there is no non-trivial sequence
• f simplices

σ0

m

− → τ0

>

− → σ1

m

− → τ1

>

− → . . .

>

− → σn

m

− → τn

>

− → σ0

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Reductions

Let (A, B, C, m ) be a partial matching (not necessarily acyclic) on (S, κ). A single pair (m (σ), σ) can be removed so to obtain again an S-complex:

• For σ ∈ A, define (S, κ) where S = S \ {m (σ), σ}, and κ : S × S → R,

κ(η, ξ) = κ(η, ξ) − κ(η, σ)κ(m (σ), ξ) κ(m (σ), σ) .

• (S, κ) is an S-complex.

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Reductions

Let (A, B, C, m ) be a partial matching (not necessarily acyclic) on (S, κ). A single pair (m (σ), σ) can be removed so to obtain again an S-complex:

• For σ ∈ A, define (S, κ) where S = S \ {m (σ), σ}, and κ : S × S → R,

κ(η, ξ) = κ(η, ξ) − κ(η, σ)κ(m (σ), ξ) κ(m (σ), σ) .

• (S, κ) is an S-complex.

Proposition

If (A, B, C, m ) is acyclic then, for any τ ∈ A \ {σ}, κ(m (τ), τ) is invertible. Furthermore, κ(m (τ), τ) = κ(m (τ), τ).

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Reductions

Let (A, B, C, m ) be a partial matching (not necessarily acyclic) on (S, κ). A single pair (m (σ), σ) can be removed so to obtain again an S-complex:

• For σ ∈ A, define (S, κ) where S = S \ {m (σ), σ}, and κ : S × S → R,

κ(η, ξ) = κ(η, ξ) − κ(η, σ)κ(m (σ), ξ) κ(m (σ), σ) .

• (S, κ) is an S-complex.

Proposition

If (A, B, C, m ) is acyclic then, for any τ ∈ A \ {σ}, κ(m (τ), τ) is invertible. Furthermore, κ(m (τ), τ) = κ(m (τ), τ).

Theorem

Fixed σ ∈ A, define A = A \ {σ}, B = B \ {m (σ)}, m = m |A, and C = C. If (A, B, C, m ) is an acyclic partial matching, then (A, B, C, m ) is an acyclic partial matching too.

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Reductions preserve homology [KMS 1998]

• Define π : C∗(S) → C∗(S) and ι : C∗(S) → C∗(S) on generators by

setting π(τ) =    if τ = m (σ) −

ξ∈S κ(m (σ),ξ) κ(m (σ),σ)ξ

if τ = σ τ

• therwise

and ι(τ) = τ − κ(τ, σ) κ(m (σ), σ)m (σ).

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Reductions preserve homology [KMS 1998]

• Define π : C∗(S) → C∗(S) and ι : C∗(S) → C∗(S) on generators by

setting π(τ) =    if τ = m (σ) −

ξ∈S κ(m (σ),ξ) κ(m (σ),σ)ξ

if τ = σ τ

• therwise

and ι(τ) = τ − κ(τ, σ) κ(m (σ), σ)m (σ).

• π and ι are chain equivalences with the chain homotopy

D∗ : C∗(S) → C∗+1(S) given on generators τ ∈ Sq, q ∈ Z, by Dq(τ) =

• 1

κ(m (σ),σ)m (σ)

if τ = σ

• therwise

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Filtration-preserving matching

A partial matching (A, B, C, m ) on a filtered S-complex is said to preserve the filtration when, for every a ∈ Rk, If σ ∈ Sa then m (σ) ∈ Sa.

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Filtration-preserving matching

A partial matching (A, B, C, m ) on a filtered S-complex is said to preserve the filtration when, for every a ∈ Rk, If σ ∈ Sa then m (σ) ∈ Sa. A filtration-preserving partial matching on S naturally induces a filtration

• n S: for each τ ∈ S,

τ ∈ Sa ⇐ ⇒ τ ∈ Sa.

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Filtration-preserving matching

A partial matching (A, B, C, m ) on a filtered S-complex is said to preserve the filtration when, for every a ∈ Rk, If σ ∈ Sa then m (σ) ∈ Sa. A filtration-preserving partial matching on S naturally induces a filtration

• n S: for each τ ∈ S,

τ ∈ Sa ⇐ ⇒ τ ∈ Sa.

Proposition

Let (S, κ) be obtained from (S, κ) by reduction of the pair (m (σ), σ). Then, for each q ∈ Z, π(Cq(Sa)) ⊆ Cq(Sa), ι(Cq(Sa)) ⊆ Cq(Sa), Dq(Cq(Sa)) ⊆ Cq+1(Sa)

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Reductions preserve persistent homology

Proposition

The maps π|C∗(Sa) : C∗(Sa) → C∗(Sa) and ι|C∗(Sa) : C∗(Sa) → C∗(Sa) defined by restriction are chain homotopy equivalences. Moreover, the diagram H∗(Sa)

H∗(j(a,b))

− → H∗(Sb)   ∼

=

  ∼

=

H∗(Sa)

H∗(j(a,b))

− → H∗(Sb) commutes.

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Reductions preserve persistent homology

Proposition

The maps π|C∗(Sa) : C∗(Sa) → C∗(Sa) and ι|C∗(Sa) : C∗(Sa) → C∗(Sa) defined by restriction are chain homotopy equivalences. Moreover, the diagram H∗(Sa)

H∗(j(a,b))

− → H∗(Sb)   ∼

=

  ∼

=

H∗(Sa)

H∗(j(a,b))

− → H∗(Sb) commutes.

Theorem

For every a b ∈ Rk, Ha,b

∗ (S) is isomorphic to Ha,b ∗ (S).

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The matching algorithm: introduction

• We start from the matching algorithm of [King-Knudson-Mramor

2005] for homology

• input: a simplicial complex and an R-valued function on its vertices
• output: an acyclic partial matching
• recursive
• vertex-based and lower-link-based
• cancels pairs of critical cells to achieve optimality

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The matching algorithm: introduction

• We start from the matching algorithm of [King-Knudson-Mramor

2005] for homology

• input: a simplicial complex and an R-valued function on its vertices
• output: an acyclic partial matching
• recursive
• vertex-based and lower-link-based
• cancels pairs of critical cells to achieve optimality
• and extend it for multidimensional persistent homology
• input: a simplicial complex, an Rk-valued function and an ordering on its

vertices

• output: a filtration-preserving acyclic partial matching
• recursive
• vertex-based and lower-link-based
• without cancellation

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w).

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w). Construction of I:

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w). Construction of I:

• f : S0 → Rk induces a partial order on the vertices

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w). Construction of I:

• f : S0 → Rk induces a partial order on the vertices
• represent it by a Directed Acyclic Graph

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w). Construction of I:

• f : S0 → Rk induces a partial order on the vertices
• represent it by a Directed Acyclic Graph
• apply the topological sorting algorithm

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Indexing map for vertices

Lemma

There exists an injective function I : S0 → N such that, for each v, w ∈ S0 with v = w, f(v) f(w) implies I(v) < I(w). Construction of I:

• f : S0 → Rk induces a partial order on the vertices
• represent it by a Directed Acyclic Graph
• apply the topological sorting algorithm
• this algorithm has linear complexity

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B.

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}.

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The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A.

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A. let f ′ : S′

0(v) → Rk and I′ : S′ 0(v) → N be the restrictions of f and I.

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A. let f ′ : S′

0(v) → Rk and I′ : S′ 0(v) → N be the restrictions of f and I.

Call Partition (recursively): (A′, B′, C′, m ′) = Partition(S′(v), f ′, I′).

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A. let f ′ : S′

0(v) → Rk and I′ : S′ 0(v) → N be the restrictions of f and I.

Call Partition (recursively): (A′, B′, C′, m ′) = Partition(S′(v), f ′, I′). Set D′ = {w ∈ C′

0 | f(w) is minimal in C′ 0 w.r.t. }.

Set w0 as the vertex with smallest index I′ in D′. Add [v, w0] to B and define m (v) = [v, w0]. For each σ ∈ C′ \ {w0}, add v ∗ σ to C. For each σ ∈ A′, add v ∗ σ to A, v ∗ m ′(σ) to B, and set m (v ∗ σ) = v ∗ m ′(σ).

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A. let f ′ : S′

0(v) → Rk and I′ : S′ 0(v) → N be the restrictions of f and I.

Call Partition (recursively): (A′, B′, C′, m ′) = Partition(S′(v), f ′, I′). Set D′ = {w ∈ C′

0 | f(w) is minimal in C′ 0 w.r.t. }.

Set w0 as the vertex with smallest index I′ in D′. Add [v, w0] to B and define m (v) = [v, w0]. For each σ ∈ C′ \ {w0}, add v ∗ σ to C. For each σ ∈ A′, add v ∗ σ to A, v ∗ m ′(σ) to B, and set m (v ∗ σ) = v ∗ m ′(σ).

• 3. For each σ ∈ S \ (A ∪ B ∪ C), add σ to C.

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Figs/BU-logo-purple-lowres

The matching algorithm for multiD persistence

Input: A finite simplicial complex S with a function f : S0 → Rk and an indexing I : S0 → N on its vertices. Output: Three lists A, B, C of simplices of S, and a function m : A → B. function Partition (complex S, function f, indexing map I)

• 1. Initially, set A, B, C = ∅.
• 2. For each v ∈ S0,

Let S′(v) = {τ ∈ S | v ∗ τ ∈ S ∧ ∀ vertex w ∈ τ, f(w) f(v)}. If S′(v) is empty, then add v to C. Else

add v to A. let f ′ : S′

0(v) → Rk and I′ : S′ 0(v) → N be the restrictions of f and I.

Call Partition (recursively): (A′, B′, C′, m ′) = Partition(S′(v), f ′, I′). Set D′ = {w ∈ C′

0 | f(w) is minimal in C′ 0 w.r.t. }.

Set w0 as the vertex with smallest index I′ in D′. Add [v, w0] to B and define m (v) = [v, w0]. For each σ ∈ C′ \ {w0}, add v ∗ σ to C. For each σ ∈ A′, add v ∗ σ to A, v ∗ m ′(σ) to B, and set m (v ∗ σ) = v ∗ m ′(σ).

• 3. For each σ ∈ S \ (A ∪ B ∪ C), add σ to C.
• 4. return A, B, C, m .

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Correctness, reduction, and complexity

Theorem

The matching algorithm produces a partial matching (A, B, C, m ) that is

• acyclic. Moreover, if σ ∈ Sa then m (σ) ∈ Sa.

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Correctness, reduction, and complexity

Theorem

The matching algorithm produces a partial matching (A, B, C, m ) that is

• acyclic. Moreover, if σ ∈ Sa then m (σ) ∈ Sa.

Corollary

For every a b ∈ Rk, Ha,b

∗ (S) is isomorphic to Ha,b ∗ (C).

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Correctness, reduction, and complexity

Theorem

The matching algorithm produces a partial matching (A, B, C, m ) that is

• acyclic. Moreover, if σ ∈ Sa then m (σ) ∈ Sa.

Corollary

For every a b ∈ Rk, Ha,b

∗ (S) is isomorphic to Ha,b ∗ (C).

Theorem

The algorithm produces (A, B, C, m ) in less than 2γd(d + 1)!N steps, where d : dimension of S N : cardinality of S0 γ : upper bound on the number of cofaces of any simplex in S

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Numerical tests

• Considered four simplicial complexes.
• Each complex was filtered by the R2-valued function defined on

vertices by f(v) = (|x|, |y|).

• Present the results in a table where
• row 1 shows the number of vertices, edges, faces, and the total number of

cells of each considered mesh S,

• while row 2 shows the same quantities referred to the cell complex C
• btained by using our matching algorithm to reduce S.
• Finally, row 3 shows the ratio between the second and the first rows,

expressing them in percentage points. In other words, the lower are those ratios, the higher is the reduction rate.

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Numerical tests

tie space_shuttle x_wing space_station #S0 #C0 % 2014 228 11.3 2376 121 5.1 3099 175 5.6 5749 1879 32.7 #S1 #C1 % 5944 3343 56.2 6330 3699 58.4 9190 3605 39.2 15949 11158 70.0 #S2 #C2 % 3827 3012 78.7 3952 3576 90.5 6076 3415 56.2 10237 9316 91.0 #S #C % 11785 6583 55.9 12658 7396 58.4 18365 7195 39.2 31935 22353 70.0

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Conclusions

• The reduction algorithm has linear complexity vs polynomial

complexity of persistence computation

• May be a convenient pre-processing technique
• Full paper available at arXiv:1310.8089v2
• On-going research focuses on improving the results obtained so far:
• using a simplex-based and lower-star-based algorithm inspired from

[Robins et al. 2010]

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Figs/BU-logo-purple-lowres

Conclusions

• The reduction algorithm has linear complexity vs polynomial

complexity of persistence computation

• May be a convenient pre-processing technique
• Full paper available at arXiv:1310.8089v2
• On-going research focuses on improving the results obtained so far:
• using a simplex-based and lower-star-based algorithm inspired from

[Robins et al. 2010]

Thank you for your attention!

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