E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The...

Ripser

E?cient Computation of Vietoris–Rips Persistence Barcodes

Ulrich Bauer

March 23, 2017

Computational and Statistical Aspects of Topological Data Analysis

Alan Turing Institute

http://ulrich-bauer.org/ripser-talk.pdf 1 / 25

Persistent homology

0.1 0.2 0.4 0.8δ

Vietoris–Rips persistence

Vietoris–Rips Bltrations

Consider a Bnite metric space (X, d).

The Vietoris–Rips complex is the simplicial complex

Ripst(X) = {S ⊆ X ∣ diam S ≤ t}

• 1-skeleton: all edges with pairwise distance ≤ t• all possible higher simplices (Dag complex)

• compute persistence barcodes for Hd(Ripst(X))(in dimensions 0 ≤ d ≤ k)

Vietoris–Rips Bltrations

Consider a Bnite metric space (X, d).

The Vietoris–Rips complex is the simplicial complex

Ripst(X) = {S ⊆ X ∣ diam S ≤ t}

• 1-skeleton: all edges with pairwise distance ≤ t• all possible higher simplices (Dag complex)

• compute persistence barcodes for Hd(Ripst(X))(in dimensions 0 ≤ d ≤ k)

Demo: Ripser

Example data set:

• 192 points on S2

• persistent homology barcodes up to dimension 2• over 56 mio. simplices in 3-skeleton

Comparison with other software:

• javaplex: 3200 seconds, 12 GB

• Dionysus: 533 seconds, 3.4 GB

• GUDHI: 75 seconds, 2.9 GB

• DIPHA: 50 seconds, 6 GB

• Eirene: 12 seconds, 1.5 GB

Ripser: 1.2 seconds, 152MB

Demo: Ripser

Example data set:

Demo: Ripser

Example data set:

Ripser

A software for computing Vietoris–Rips persistence barcodes

• about 1000 lines of C++ code, no external dependencies

• support for

• coe?cients in a prime Beld Fp• sparse distance matrices for distance threshold

• open source (http://ripser.org)

• released in July 2016

• online version (http://live.ripser.org)

• launched in August 2016

• most e?cient software for Vietoris–Rips persistence

• computes H2barcode for 50 000 random points on a torus in 136 seconds /

9 GB (using distance threshold)

• 2016 ATMCS Best New Software Award (jointly with RIVET)

Ripser

• support for

• coe?cients in a prime Beld Fp

• sparse distance matrices for distance threshold

Ripser

• support for

Ripser

• support for

Ripser

• support for

Ripser

• support for

Ripser

• support for

Design goals

Goals for previous projects:

• PHAT [B, Kerber, Reininghaus,Wagner 2013]:

fast persistence computation (matrix reduction only)

• DIPHA [B, Kerber, Reininghaus 2014]:

distributed persistence computation

Goals for Ripser:

• Use as little memory as possible

• Be reasonable about computation time

Design goals

Goals for previous projects:

• PHAT [B, Kerber, Reininghaus,Wagner 2013]:

fast persistence computation (matrix reduction only)

• DIPHA [B, Kerber, Reininghaus 2014]:

distributed persistence computation

Goals for Ripser:

• Use as little memory as possible

• Be reasonable about computation time

The four special ingredients

The improved performance is based on 4 insights:

• Clearing inessential columns [Chen, Kerber 2011]

• Computing cohomology [de Silva et al. 2011]

• Implicit matrix reduction

• Apparent and emergent pairs

Lessons from PHAT:

• Clearing and cohomology yield considerable speedup,

• but only when both are used in conjuction!

The four special ingredients

The improved performance is based on 4 insights:

• Clearing inessential columns [Chen, Kerber 2011]

• Computing cohomology [de Silva et al. 2011]

• Implicit matrix reduction

• Apparent and emergent pairs

Lessons from PHAT:

• Clearing and cohomology yield considerable speedup,

• but only when both are used in conjuction!

Matrix reduction

Matrix reduction algorithm

Setting:

• Bnite metric space X, n points

• persistent homologyHd(Ripst(X);F2) in dimensions d ≤ kNotation:

• D: boundary matrix of Bltration

• Ri: ith column of R

Algorithm:

• R = D,V = I• while ∃i < jwith pivotRi = pivotRj

• add Ri to Rj, add Vi to Vj

Result:

• R = D ⋅V is reduced (unique pivots)

• V is full rank upper triangular

Setting:

• Ri: ith column of RAlgorithm:

Result:

Setting:

• Ri: ith column of RAlgorithm:

Result:

Compatible basis cycles

For a reduced boundary matrix R = D ⋅V , call

P = {i ∶ Ri = 0} positive indices,

N = {j ∶ Rj ≠ 0} negative indices,

E = P ∖ pivotsR essential indices.Then

Σ̃Z = {Vi ∣ i ∈ P} is a basis of Z∗,ΣB = {Rj ∣ j ∈ N} is a basis of B∗,

ΣZ = ΣB ∪ {Vi ∣ i ∈ E} is another basis of Z∗.

Persistent homology is generated by the basis cycles ΣZ.

• Persistence intervals: {[i, j) ∣ i = pivotRj} ∪ {[i,∞) ∣ i ∈ E}• Columns with non-essential positive indices never used!

Σ̃Z = {Vi ∣ i ∈ P} is a basis of Z∗,

ΣB = {Rj ∣ j ∈ N} is a basis of B∗,

Σ̃Z = {Vi ∣ i ∈ P} is a basis of Z∗,ΣB = {Rj ∣ j ∈ N} is a basis of B∗,

Σ̃Z = {Vi ∣ i ∈ P} is a basis of Z∗,ΣB = {Rj ∣ j ∈ N} is a basis of B∗,ΣZ = ΣB ∪ {Vi ∣ i ∈ E} is another basis of Z∗.

Clearing

Clearing non-essential positive columns

Idea [Chen, Kerber 2011]:

• Don’t reduce at non-essential positive indices

• Reduce boundary matrices of ∂d ∶ Cd → Cd−1in decreasing dimension d = k + 1, . . . , 1

• Whenever i = pivotRj (in matrix for ∂d)

• Set Ri to 0 (in matrix for ∂d−1)

• Set Vi to Rj

• Still yields R = D ⋅V reduced, V full rank upper triangular

• reducing positive columns typically harder than negative

• with clearing: need only reduce essential positive columns

• Set Ri to 0 (in matrix for ∂d−1)• Set Vi to Rj

Cohomology

Persistent cohomology

We have seen: many columns of R = D ⋅V are not needed

• Skip those inessential columns in matrix reduction

For persistence barcodes in low dimensions d ≤ k:

• Number of skipped indices for reducing DT (cohomology) is much larger

than for D (homology)

• reducing boundary matrix produces basis for Hk+1(Kk+1), which is not needed

• The resulting persistence barcode is the same [de Silva et al. 2011]

Persistent cohomology

We have seen: many columns of R = D ⋅V are not needed

• Skip those inessential columns in matrix reduction

For persistence barcodes in low dimensions d ≤ k:

• Number of skipped indices for reducing DT (cohomology) is much larger

than for D (homology)

• reducing boundary matrix produces basis for Hk+1(Kk+1), which is not needed

• The resulting persistence barcode is the same [de Silva et al. 2011]

Counting homology column reductions

• standard matrix reduction:

k+1∑d=1

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+k+1∑d=1

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k = 2, n = 192: 56050096 = 1 161 471 + 54 888625• using clearing:

k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+ (n − 1k + 2

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶

dimHk+1(K)

=k+2∑d=1

(n − 1d

k = 2, n = 192: 54 888 816 = 1 161 471 + 53 727 345

k+1∑d=1

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+k+1∑d=1

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k = 2, n = 192: 56050096 = 1 161 471 + 54 888625

• using clearing:k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+ (n − 1k + 2

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶

dimHk+1(K)

=k+2∑d=1

(n − 1d

k = 2, n = 192: 54 888 816 = 1 161 471 + 53 727 345

k+1∑d=1

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+k+1∑d=1

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+ (n − 1k + 2

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶

dimHk+1(K)

=k+2∑d=1

(n − 1d

k = 2, n = 192: 54 888 816 = 1 161 471 + 53 727 345

k+1∑d=1

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+k+1∑d=1

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k+1∑d=1

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd−1(K)

+ (n − 1k + 2

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶

dimHk+1(K)

=k+2∑d=1

(n − 1d

k = 2, n = 192: 54 888 816 = 1 161 471 + 53 727 345

Counting cohomology column reductions

k∑d=0

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+k∑d=0

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k = 2, n = 192: 1 179 808 = 18 337 + 1 161 471• using clearing:

k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+ (n − 10

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimH0(K)

+ =k+1∑d=0

(n − 1d

k = 2, n = 192: 1 161 472 = 1 + 1 161 471

k∑d=0

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+k∑d=0

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k = 2, n = 192: 1 179 808 = 18 337 + 1 161 471

• using clearing:k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+ (n − 10

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimH0(K)

+ =k+1∑d=0

(n − 1d

k = 2, n = 192: 1 161 472 = 1 + 1 161 471

k∑d=0

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+k∑d=0

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+ (n − 10

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimH0(K)

+ =k+1∑d=0

(n − 1d

k = 2, n = 192: 1 161 472 = 1 + 1 161 471

k∑d=0

( nd + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimCd(K)

=k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+k∑d=0

(n − 1d

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimZd(K)

k∑d=0

(n − 1d + 1

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimBd+1(K)

+ (n − 10

)´¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¶dimH0(K)

+ =k+1∑d=0

(n − 1d

k = 2, n = 192: 1 161 472 = 1 + 1 161 471

Observations

For a typical input:

• V has very few o>-diagonal entries

• most negative columns of D are already reduced

Previous example (k = 2, n = 192):

• Only 845 out of 1 161 471 columns have to be reduced

Observations

For a typical input:

• V has very few o>-diagonal entries

• most negative columns of D are already reduced

Previous example (k = 2, n = 192):

• Only 845 out of 1 161 471 columns have to be reduced

Implicit matrix reduction

Standard approach:

• Boundary matrixD for Bltration-ordered basis

• Explicitly generated and stored in memory

• Matrix reduction: store only reduced matrix R• transform D into R by column operations

Approach for Ripser:

• Boundary matrixD for lexicographically ordered basis

• Implicitly deBned and recomputed when needed

• Matrix reduction in Ripser: store only coe?cient matrix V• recompute previous columns of R = D ⋅V when needed

• Typically, V is much sparser and smaller than R

Standard approach:

• Typically, V is much sparser and smaller than Rhttp://ulrich-bauer.org/ripser-talk.pdf 16 / 25

Oblivious matrix reduction

Algorithm variant:

• R = D• for j = 1, . . . , n

• while ∃i < jwith pivotRi = pivotRj• add Di to Rj

Requires only

• current column Rj

• pivots of previous columns Ri

to obtain the persistence intervals: {[i, j) ∣ i = pivotRj} ∪ {[i,∞) ∣ i ∈ E}.

Corollary

The rank of anm × nmatrix can be computed inO(n) memory.

Algorithm variant:

• R = D• for j = 1, . . . , n

Requires only

Corollary

Algorithm variant:

• R = D• for j = 1, . . . , n

Requires only

Corollary

Apparent and emergent pairs

Natural Bltration settings

Typical assumptions on the Bltration:

• general Bltration persistence (in theory)

• Bltration by singletons or pairs discreteMorse theory

• simplexwise Bltration persistence (computation)

Conclusion:

• DiscreteMorse theory sits in the middle

between persistence and persistence (!)

Natural Bltration settings

Typical assumptions on the Bltration:

• general Bltration persistence (in theory)

• Bltration by singletons or pairs discreteMorse theory

• simplexwise Bltration persistence (computation)

Conclusion:

• DiscreteMorse theory sits in the middle

between persistence and persistence (!)

DiscreteMorse theory

DeBnition (Forman 1998)

A discrete vector Beld on a cell complex is a partition of the set of simplices into

• singleton sets {ϕ} (critical cells), and

• pairs {σ , τ}, where σ is a facet of τ.

A function f ∶ K → R on a cell complex is a discreteMorse function if6

• sublevel sets are subcomplexes, and

• level sets form a discrete vector Beld.

A function f ∶ K → R on a cell complex is a discreteMorse function if

Fundamental theorem of discreteMorse theory

Let f be a discreteMorse function on a cell complex K.

Theorem (Forman 1998)

If (s, t] contains no critical value of f , then the sublevel setKt collapses toKs .

Corollary

K ≃M for some cell complexM built from the critical cells of f .

This homotopy equivalence is compatible with the Bltration.

Corollary

K and M have isomorphic persistent homology (with regard to the sublevel sets of f ).

Corollary

Morse pairs and persistence pairs

Consider aMorse Bltration (one or two simplices at a time).

Morse pair (σ , τ):

• inserting σ and τ simultaneously does not change the homotopy type

Consider a simplexwise Bltration (one simplex at a time).

Persistence pair (σ , τ):

• inserting simplex σ creates a new homological feature

• inserting τ destroys that feature again

Morse pairs and persistence pairs

Consider aMorse Bltration (one or two simplices at a time).

Morse pair (σ , τ):

• inserting σ and τ simultaneously does not change the homotopy type

Consider a simplexwise Bltration (one simplex at a time).

Persistence pair (σ , τ):

• inserting simplex σ creates a new homological feature

• inserting τ destroys that feature again

Apparent pairs

DeBnition

In a simplexwise Bltration, (σ , τ) is an apparent pair if

• σ is the youngest face of τ• τ is the oldest coface of σ

Any apparent pairs is a persistence pair.

The apparent pairs form a discrete gradient.

• Generalizes a construction proposed by [Kahle 2011] for the study of random

Rips Bltrations

FromMorse theory to persistence and back

Proposition (fromMorse to persistence)

The pairs of aMorse Bltration are apparent 0-persistence pairs for the canonical

simplexwise reBnement of the Bltration.

Proposition (from persistence toMorse)

Consider an arbitrary Bltrationwith a simplexwise reBnement. The apparent

0-persistence pairs yield aMorse Bltration

• reBning the original one, and

• reBned by the simplexwise one.

FromMorse theory to persistence and back

Proposition (fromMorse to persistence)

The pairs of aMorse Bltration are apparent 0-persistence pairs for the canonical

simplexwise reBnement of the Bltration.

Proposition (from persistence toMorse)

Consider an arbitrary Bltrationwith a simplexwise reBnement. The apparent

0-persistence pairs yield aMorse Bltration

• reBning the original one, and

• reBned by the simplexwise one.

Emergent persistent pairs

Consider the lexicographically reBned Rips Bltration:

• increasing diameter, reBned by

• lexicographic order

This is the simplexwise Bltration for computations in Ripser.

Assume that

• τ is the lexicographicallyminimal proper coface of σ with diam(τ) = diam(σ),

• and there is no persistence pair (ρ, τ)with σ ≺ ρ.

Then (σ , τ) is an emergent persistence pair.

• Includes all apparent persistence 0 pairs

• Can be identiBed without enumerating all cofaces of σ• Provides a shortcut for computation

Emergent persistent pairs

Consider the lexicographically reBned Rips Bltration:

• increasing diameter, reBned by

• lexicographic order

This is the simplexwise Bltration for computations in Ripser.

Assume that

• τ is the lexicographicallyminimal proper coface of σ with diam(τ) = diam(σ),

• and there is no persistence pair (ρ, τ)with σ ≺ ρ.

Then (σ , τ) is an emergent persistence pair.

• Includes all apparent persistence 0 pairs

• Can be identiBed without enumerating all cofaces of σ• Provides a shortcut for computation

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E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The...

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