Multi-Field Persistent Homology

JGA 2013

Multi-Field Persistent Homology

Jean-Daniel Boissonnat & Cl´ ement Maria

1 Introduction

Motivation

Homology features:

components, holes, voids, etc.

but... that guy on the right is

“twisting”!

Conformation Space of the Cyclo-octane Molecule Image: [Martin, Thompson,

Coutsias, Watson]

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 3

Motivation

Homology features:

components, holes, voids, etc.

but... that guy on the right is

“twisting”!

Conformation Space of the Cyclo-octane Molecule Image: [Martin, Thompson,

Coutsias, Watson]

Homology with Z -coefficients

K ^p : set of p-simplices of K C p : Abelian group of formal sums of p-simplices with Z -coefficients

{ , , , , , , · · · }

{ e 1 , e 2 , e 3 , e 4 , e 5 , e 6 , · · · } e 1 + e 2 + e 3 + e 4 + e 5 + e 6

e

e

e

e

e

e

∂ p : C p → C p − 1 : Boundary operator ∂ ₂ = − + [a, b, c]

C ₁ = ( { P

i k _i e _i } , +)

[ab]

∂ 2 = − [bc] + [ca]

H p = ker ∂ p / im ∂ p+1

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 5

Homology and Torsion

Fundamental theorem of finitely generated abelian groups:

H p ( Z ) ∼ = Z ^β

^{( Z )} M

q prime

Z q

⊕ · · · ⊕ Z _q

vs H p ( F ) = F ^β

^{( F )}

I β _p ( Z ) and β _p ( F ) are Betti numbers: β _p ( Z ) 6 = β _p ( F ) in general

I k i > 0 and t(p , q) ≥ 0 over all prime numbers q

I if t(p, q) 6 = 0, q ^k

· · · q ^k

are torsion coefficients

Klein bottle:

Homology and Torsion

H p ( Z ) ∼ = Z ^β

^{( Z )} M

q prime

Z q

⊕ · · · ⊕ Z _q

vs H p ( F ) = F ^β

^{( F )}

H 0 ( Z ) = Z β 0 ( Z ) = 1 H 1 ( Z ) = Z ⊕ Z ² β 1 ( Z ) = 1 H ₂ ( Z ) = 0 β ₂ ( Z ) = 0 H 0 ( Z ² ) = Z ² β 0 ( Z ² ) = 1 H ₁ ( Z ² ) = ( Z ² ) ² β ₁ ( Z ² ) = 2 H 2 ( Z ² ) = Z ² β 2 ( Z ² ) = 1 H ₀ ( Z ³ ) = Z ³ β ₀ ( Z ³ ) = 1 H 1 ( Z ³ ) = Z ³ β 1 ( Z ³ ) = 1 H ₂ ( Z 3 ) = 0 β ₂ ( Z 3 ) = 0

Klein bottle:

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 7

Homology and Torsion

H p ( Z ) ∼ = Z ^β

^{( Z )} M

q prime

Z q

⊕ · · · ⊕ Z _q

vs H p ( F ) = F ^β

^{( F )}

H 0 ( Z ) = Z β 0 ( Z ) = 1 H 1 ( Z ) = Z ⊕ Z ² β 1 ( Z ) = 1 H ₂ ( Z ) = 0 β ₂ ( Z ) = 0 H 0 ( Z ² ) = Z ² β 0 ( Z ² ) = 1 H ₁ ( Z ² ) = ( Z ² ) ² β ₁ ( Z ² ) = 2 H 2 ( Z ² ) = Z ² β 2 ( Z ² ) = 1 H ₀ ( Z ³ ) = Z ³ β ₀ ( Z ³ ) = 1 H 1 ( Z ³ ) = Z ³ β 1 ( Z ³ ) = 1 H ₂ ( Z 3 ) = 0 β ₂ ( Z 3 ) = 0

Klein bottle???

o O ???

Homology and Torsion

H p ( Z ) ∼ = Z ^β

^{( Z )} M

q prime

Z q

⊕ · · · ⊕ Z _q

vs H p ( F ) = F ^β

^{( F )}

H 0 ( Z ) = Z β 0 ( Z ) = 1 H 1 ( Z ) = Z ⊕ Z ² β 1 ( Z ) = 1 H ₂ ( Z ) = 0 β ₂ ( Z ) = 0 H 0 ( Z ² ) = Z ² β 0 ( Z ² ) = 1 H ₁ ( Z ² ) = ( Z ² ) ² β ₁ ( Z ² ) = 2 H 2 ( Z ² ) = Z ² β 2 ( Z ² ) = 1 H ₀ ( Z ³ ) = Z ³ β ₀ ( Z ³ ) = 1 H 1 ( Z ³ ) = Z ³ β 1 ( Z ³ ) = 1 H ₂ ( Z 3 ) = 0 β ₂ ( Z 3 ) = 0

Klein bottle:

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 9

Remember Persistence?

A

A B

A

B A

B A

Death

Birth

Very useful!

General

Stable w.r.t noise Efficient Algorithm BUT

Strictly restricted to FIELD coefficients!

(algebraic reason)

Remember Persistence?

A

A B

A

B A

B A

Death

Birth

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 11

Remember Persistence?

A

A B

A

B A

B A

Death

Birth

B B B

x

y

Remember Persistence?

A

A B

A

B A

B A

Death

Birth

B B B

1 cycle alive ”during”

[x : y]

y

x

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 13

Remember Persistence?

A

A B

A

B A

B A

Death

Birth

B B B

x y

2 cycles alive ”during”

[x : y]

Cl´ement Maria - Multi-Field Persistent Homology

2 Multi-Field Persistence Diagram

Multi-Field Persistence Diagram

Z 2

Z 3

only in Z ² only in Z 3

in Z 2 and Z 3

Multi-Field Persistence Diagram

Z 2

Z 3

only in Z ² only in Z 3

in Z 2 and Z 3

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 17

Multi-Field Persistence Diagram

Z ⊕ Z ²

Z ³ ⊕ Z 2

Z ²

Z 2

Z 3

only in Z ² only in Z 3

in Z 2 and Z 3

Homology Group Reconstruction

We compute the Multi-Field Persistence Diagram for the fields:

{Z ^q

, · · · , Z ^q

} for the family of first r distinct primes q 1 , · · · , q r

Suppose we know β p ( Z ^q

), ∀ 1 ≤ s ≤ r (with H p ( Z ^q

) ∼ = Z ^{β q}

^{( Z}

⁾ ).

H _p ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z q

⊕ · · · ⊕ Z _q

H p−1 ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z _q

⊕ · · · ⊕ Z

q

0 t(p−1,q)

The Universal Coefficient Theorem of Homology allows to compute H p (K, F ) from H p (K, Z ) and H p−1 (K, Z ).

With q _r “big enough”, we partially reverse the computation:

t(p, q s ) = β p ( Z ^q

) − β p ( Z ^q

) − t(p − 1, q s )

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 19

Homology Group Reconstruction

We compute the Multi-Field Persistence Diagram for the fields:

{Z ^q

, · · · , Z ^q

} for the family of first r distinct primes q 1 , · · · , q r

Suppose we know β p ( Z ^q

), ∀ 1 ≤ s ≤ r (with H p ( Z ^q

) ∼ = Z ^{β q}

^{( Z}

⁾ ).

H _p ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z q

⊕ · · · ⊕ Z _q

H p−1 ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z _q

⊕ · · · ⊕ Z

q

0 t(p−1,q)

The Universal Coefficient Theorem of Homology allows to compute H p (K, F ) from H p (K, Z ) and H p−1 (K, Z ).

With q _r “big enough”, we partially reverse the computation:

t(p, q s ) = β p ( Z ^q

) − β p ( Z ^q

) − t(p − 1, q s )

Homology Group Reconstruction

We compute the Multi-Field Persistence Diagram for the fields:

{Z ^q

, · · · , Z ^q

} for the family of first r distinct primes q 1 , · · · , q r

Suppose we know β p ( Z ^q

), ∀ 1 ≤ s ≤ r (with H p ( Z ^q

) ∼ = Z ^{β q}

^{( Z}

⁾ ).

H _p ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z q

⊕ · · · ⊕ Z _q

H p−1 ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z _q

⊕ · · · ⊕ Z

q

0 t(p−1,q)

The Universal Coefficient Theorem of Homology allows to compute H p (K, F ) from H p (K, Z ) and H p−1 (K, Z ).

With q _r “big enough”, we partially reverse the computation:

t(p, q s ) = β p ( Z ^q

) − β p ( Z ^q

) − t(p − 1, q s )

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 21

Homology Group Reconstruction

We compute the Multi-Field Persistence Diagram for the fields:

{Z ^q

, · · · , Z ^q

} for the family of first r distinct primes q 1 , · · · , q r

Suppose we know β p ( Z ^q

), ∀ 1 ≤ s ≤ r (with H p ( Z ^q

) ∼ = Z ^{β q}

^{( Z}

⁾ ).

H _p ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z q

⊕ · · · ⊕ Z _q

H p−1 ( Z ) ∼ = Z ^β

^{( Z )} L

q prime

Z _q

⊕ · · · ⊕ Z

q

0 t(p−1,q)

The Universal Coefficient Theorem of Homology allows to compute H p (K, F ) from H p (K, Z ) and H p−1 (K, Z ).

With q _r “big enough”, we partially reverse the computation:

t(p, q s ) = β p ( Z ^q

) − β p ( Z ^q

) − t(p − 1, q s )

Homology Group Reconstruction

H 0 ( Z ) = Z H 1 ( Z ) = Z ⊕ Z ² H ₂ ( Z ) = 0

Z ⊕ Z 2 ^∗ Z ³ ⊕ Z 2 ^∗

Z ²

only in Z

only in Z

in Z

and Z

Birth

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 23

Multi-Field Persistence Diagram

We compute the Multi-Field Persistence Diagram for the fields:

{Z ^q

, · · · Z ^q

} for a family of distinct primes q ₁ , · · · , q _r , with q _r strict upper bound on the primary divisors of torsion coefficients.

H _p ( Z ) ∼ = Z ^β

^{( Z )} M

q prime

Z q

⊕ · · · ⊕ Z _q

Let:

I K be a filtered simplicial complex with m simplices

I | D(K, F ) | = Θ(m) be the number of points in the persistence diagram for any F (field) coefficients

I | D(K, Z q

· · · Z q

) | be the number of distinct points in the

superimposition of diagrams D(K, Z ^q

), · · · D(K, Z ^q

)

Multi-Field Persistence Diagram

Usually:

I q ₁ , · · · , q _r are the first r prime numbers, and r ≤ 100

I m is huge!

I m ' | D(K, F ) | '

| D(K, Z ^q

· · · Z ^q

) | r × m

Z ⊕ Z 2

Z ³ ⊕ Z 2

Z ²

only inZ2

only inZ3

inZ2andZ3 Death

Birth

We design algorithms such that:

O(f ( | D(K, F ) | )) becomes O(f ( | D(K, Z ^q

· · · Z ^q

) | ) × A) for some small A.

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 25

3 Modular Reconstruction

for Gaussian Elimination

Remember the Persistent Cohomology Algorithm?

t ₀ t ₁ t ₂ t ₃ t ₄ t ₅ t ₆ t ₇

K ⁰ ⊆ K ¹ ⊆ K ² ⊆ K ³ ⊆ K ⁴ ⊆ K ⁵ ⊆ K ⁶ ⊆ K ⁷

I Insert the simplices in the order of the filtration

I Update the cohomology groups accordingly

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 27

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∈ Z q

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

Inverse in Z ^q

a ∂σ _i+1

0 6 =

0 0

Algo overview in Z ^q

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∈ Z ^q

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

Inverse in Z q

a ∂σ _i+1

0 6 =

0 0

Algo overview in Z q

Gaussian Elimination in a field

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 29

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

∈ Z q ₁ × · · · × Z q r

(u ₁ , · · · , u _r ) × (v ₁ , · · · , v _r ) + (w ₁ , · · · , w _r ) (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

=

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

∈ Z q ₁ × · · · × Z q r

(u ₁ , · · · , u _r ) × (v ₁ , · · · , v _r ) + (w ₁ , · · · , w _r ) (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

=

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 31

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

∈ Z q ₁ × · · · × Z q r

(u ₁ , · · · , u _r ) × (v ₁ , · · · , v _r ) + (w ₁ , · · · , w _r ) (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

=

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

(u ₁ , · · · , u _r ) × (v ₁ , · · · , v _r ) + (w ₁ , · · · , w _r ) (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

=

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 33

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

c _k = (u ₁ , · · · , u _r ) with u _s lowest 6 = 0

in Z q s for s ∈ S ⊆ [r] of a _∂σ (u ₁ ^S , · · · , u _r ^S ) = c ⁻ _k ¹

u _s ⁻¹ if s ∈ S

0 o.w.

partial inverse

w.r.t. S

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

Multi-Field? ← Algo in Z ^q ₁ × · · · × Z ^q r

c _k = (u ₁ , · · · , u _r ) with u _s lowest 6 = 0

in Z q s for s ∈ S ⊆ [r] of a _∂σ

(u ₁ ^S , · · · , u _r ^S ) = c ⁻ _k ¹ u _s ^S =

u ⁻¹ _s if s ∈ S

0 o.w.

partial inverse w.r.t. S

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 35

Z ^q

× · · · × Z ^q

∼ = Z ^q

×···× q

Theorem (Chinese Remainder Theorem)

For a family q 1 , · · · , q r of r distinct prime numbers, there is a ring isomorphism

ψ : Z ^q

× · · · × Z ^q

→ Z ^q

×···×q

s.t. the restriction ψ ^× : Z ^× q

× · · · × Z ^× q

→ Z ^× q

×···×q

is a group isomorphism.

for

I Z q

×···×q

with pointwise addition/multiplication

I R ^× the multiplicative group of invertible elements of ring R

Moreover, ψ is easily constructible.

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

=

Multi-Field? ← Algo in Z ^q _{1 ×···×} ^q r

ψ(u ₁ , · · · , u _r ) × ψ(v ₁ , · · · , v _r ) + ψ(w ₁ , · · · , w _r ) ψ (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

ψ ψ

∈ Z q ₁ ×···× q r

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 37

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

=

Multi-Field? ← Algo in Z ^q _{1 ×···×} ^q r

ψ(u ₁ , · · · , u _r ) × ψ(v ₁ , · · · , v _r ) + ψ(w ₁ , · · · , w _r ) ψ (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

ψ ψ

∈ Z q ₁ ×···× q r

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

=

Multi-Field? ← Algo in Z ^q _{1 ×···×} ^q r

ψ(u ₁ , · · · , u _r ) × ψ(v ₁ , · · · , v _r ) + ψ(w ₁ , · · · , w _r ) ψ (u ₁ × v ₁ + w ₁ , · · · , u _r × v _r + w _r )

ψ ψ

∈ Z q ₁ ×···× q r

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 39

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

with u _s lowest 6 = 0

in Z q s for s ∈ S ⊆ [r] of a _∂σ

(u ₁ ^S , · · · , u _r ^S ) = c ⁻ _k ¹ u _s ^S =

u ⁻¹ _s if s ∈ S

0 o.w.

partial inverse w.r.t. S

Multi-Field? → Algo in Z ^q _{1 ×···×} ^q r

c _k = ψ(u ₁ , · · · , u _r )

ψ

ψ

ψ

Multi-Field Persistent Cohomology

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

with u _s lowest 6 = 0

in Z q s for s ∈ S ⊆ [r] of a _∂σ

(u ₁ ^S , · · · , u _r ^S ) = c ⁻ _k ¹ u _s ^S =

u ⁻¹ _s if s ∈ S

0 o.w.

partial inverse w.r.t. S

Multi-Field? → Algo in Z ^q _{1 ×···×} ^q r

c _k = ψ(u ₁ , · · · , u _r )

ψ ψ ψ

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 41

Partial Inverse Construction

Define Q S = P

s∈S q s for S ⊆ [r ]. We prove that:

Data: x, Q _S

Q _R ← gcd(x, Q _S ); via the euclidean algorithm: O(A

(Q));

Q

← Q

/Q

;

v ← Extended-Euclidean-Algorithm (x, Q

); such that;

vx + wQ

= 1;

v ← v mod Q

;

L

← D(Q

); some preprocessed constant;

x

← (v × L

) mod Q;

return x

;

computes the partial inverse of x w.r.t. S. Cost: O(A) in Z ^q

···q

Partial Inverse Construction

Define Q S = P

s∈S q s for S ⊆ [r ]. We prove that:

Data: x, Q _S

Q _R ← gcd(x, Q _S ); via the euclidean algorithm: O(A

(Q));

Q

← Q

/Q

;

v ← Extended-Euclidean-Algorithm (x, Q

); such that;

vx + wQ

= 1;

v ← v mod Q

;

L

← D(Q

); some preprocessed constant;

x

← (v × L

) mod Q;

return x

;

computes the partial inverse of x w.r.t. S. Cost: O(A) in Z ^q

···q

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 43

3 Complexity Analysis

and Experiments

Arithmetic Complexity

I For an integer z, let λ(z) = b log ₂ z /w c + 1 the number of w-bits memory words to store z.

I Let Q = q ₁ × · · · × q _r be the product of the first r primes

I For B = λ(z), we have:

• Addition A ₊ (z ) = O(B)

• Multiplication A × (z) = O(M(B))

• Division A ÷ (z ) = O(M (B) log B) with M (B) = O(B log B 2 ^O(log

^B) ) Moreover:

λ(Q) <

1.46613r ln(r ln r) w

+ 1

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 45

Complexity Analysis

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∈ Z q

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

Inverse in Z ^q

a ∂σ _i+1

0 6 =

0 0

Algo overview in Z ^q

C a

C G

Complexity: O( | D(K, Z ^q ) | × [ C a _∂ + C G ])

Complexity Analysis

φ

φ

τ

τ

∂σ

a

=

τ∈∂σi+1

1 ( ) a

φ

φ

τ

τ

φ

0 0 0 0 0 0 0 τ

6

= 0

= 0

ψ

ψ

σ

σ

0

0

0 0 1

σ

∀ j

φ j ← φ j − a ∂σ

[j]c

_k ¹

× φ k

c k

a ∂σ _i+1

0 0

C a

C G

Algo overview in Z ^q _{1 ×···×} ^q r

O( | D(K, Z ^q ₁ · · · Z ^q r ) | × [ C a _∂ + C G ] × A)

∈ Z q 1 ×···×q r

Partial inverse

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 47

Experiments

0 2 4 6 8 10 12 14 16 18 20

0 1,000 2,000 3,000 4,000 5,000

number of primes r

time (s.)

modular reconstruction naive

0 6 12 18 24 30

ratio

ratio x = y

Figure: Timings for the modular reconstruction algo vs naive.

O( | D(K, Z ^q

· · · Z ^q

) |× [ C a

+ C G ] × A) vs O(r ×| D(K, Z ^q ) |× [ C a

+ C G ])

Cl´ement Maria - Multi-Field Persistent Homology

Conclusion

Why Multi-Field Persistence?

Not for today but: we also have

I Multi-Field Bottleneck Distance between diagrams

I Generalized algorithm for maximal point set matching

I Efficient algorithm in O(m ^03/2 log m ⁰ √

tA) (instead of O(m ^3/2 log m))

1 4

{1, 2, 3}

{1,2}

{ 1, 3, 4 } { 4 } {2}

{ 1,3 } {4}

1 3 2

{1}

G G

Why Multi-Field Persistence?

The Multi-Field Persistence Diagram

I is more accurate:

characterizes torsion

I admits a fast construction algorithm

I admits a generalized bottleneck distance with a fast algorithm

Z ⊕ Z 2

Z ³ ⊕ Z 2

Z ²

only inZ2

only inZ3

inZ2andZ3 Death

Birth

Code available soon.

Cl´ement Maria - Multi-Field Persistent Homology December 17, 2013- 51

Thank you!

Question?