E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The...
Transcript of E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The...
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Ripser
E?cient Computation of Vietoris–Rips Persistence Barcodes
Ulrich Bauer
TUM
March 23, 2017
Computational and Statistical Aspects of Topological Data Analysis
Alan Turing Institute
http://ulrich-bauer.org/ripser-talk.pdf 1 / 25
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Persistent homology
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0.1 0.2 0.4 0.8δ
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0.1 0.2 0.4 0.8δ
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0.1 0.2 0.4 0.8δ
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0.1 0.2 0.4 0.8δ
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0.1 0.2 0.4 0.8δ
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0.1 0.2 0.4 0.8δ
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Vietoris–Rips persistence
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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)
Goal:
• compute persistence barcodes for Hd(Ripst(X))(in dimensions 0 ≤ d ≤ k)
http://ulrich-bauer.org/ripser-talk.pdf 4 / 25
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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)
Goal:
• compute persistence barcodes for Hd(Ripst(X))(in dimensions 0 ≤ d ≤ k)
http://ulrich-bauer.org/ripser-talk.pdf 4 / 25
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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
http://ulrich-bauer.org/ripser-talk.pdf 5 / 25
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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
http://ulrich-bauer.org/ripser-talk.pdf 5 / 25
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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
http://ulrich-bauer.org/ripser-talk.pdf 5 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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)
http://ulrich-bauer.org/ripser-talk.pdf 6 / 25
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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
http://ulrich-bauer.org/ripser-talk.pdf 7 / 25
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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
http://ulrich-bauer.org/ripser-talk.pdf 7 / 25
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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!
http://ulrich-bauer.org/ripser-talk.pdf 8 / 25
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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!
http://ulrich-bauer.org/ripser-talk.pdf 8 / 25
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Matrix reduction
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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
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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 RAlgorithm:
• 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
http://ulrich-bauer.org/ripser-talk.pdf 9 / 25
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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 RAlgorithm:
• 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
http://ulrich-bauer.org/ripser-talk.pdf 9 / 25
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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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
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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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
![Page 43: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/43.jpg)
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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
![Page 44: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/44.jpg)
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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
![Page 45: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/45.jpg)
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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
![Page 46: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/46.jpg)
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!
http://ulrich-bauer.org/ripser-talk.pdf 10 / 25
![Page 47: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/47.jpg)
Clearing
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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
Note:
• reducing positive columns typically harder than negative
• with clearing: need only reduce essential positive columns
http://ulrich-bauer.org/ripser-talk.pdf 11 / 25
![Page 49: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/49.jpg)
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
Note:
• reducing positive columns typically harder than negative
• with clearing: need only reduce essential positive columns
http://ulrich-bauer.org/ripser-talk.pdf 11 / 25
![Page 50: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/50.jpg)
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
Note:
• reducing positive columns typically harder than negative
• with clearing: need only reduce essential positive columns
http://ulrich-bauer.org/ripser-talk.pdf 11 / 25
![Page 51: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/51.jpg)
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
Note:
• reducing positive columns typically harder than negative
• with clearing: need only reduce essential positive columns
http://ulrich-bauer.org/ripser-talk.pdf 11 / 25
![Page 52: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/52.jpg)
Cohomology
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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]
http://ulrich-bauer.org/ripser-talk.pdf 12 / 25
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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]
http://ulrich-bauer.org/ripser-talk.pdf 12 / 25
![Page 55: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/55.jpg)
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
http://ulrich-bauer.org/ripser-talk.pdf 13 / 25
![Page 56: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/56.jpg)
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
http://ulrich-bauer.org/ripser-talk.pdf 13 / 25
![Page 57: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/57.jpg)
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
http://ulrich-bauer.org/ripser-talk.pdf 13 / 25
![Page 58: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/58.jpg)
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
http://ulrich-bauer.org/ripser-talk.pdf 13 / 25
![Page 59: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/59.jpg)
Counting cohomology column reductions
• standard matrix reduction:
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
http://ulrich-bauer.org/ripser-talk.pdf 14 / 25
![Page 60: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/60.jpg)
Counting cohomology column reductions
• standard matrix reduction:
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
http://ulrich-bauer.org/ripser-talk.pdf 14 / 25
![Page 61: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/61.jpg)
Counting cohomology column reductions
• standard matrix reduction:
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
http://ulrich-bauer.org/ripser-talk.pdf 14 / 25
![Page 62: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/62.jpg)
Counting cohomology column reductions
• standard matrix reduction:
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
http://ulrich-bauer.org/ripser-talk.pdf 14 / 25
![Page 63: E C V R P Ba U Ba - Ulrich BauerVietoris–Rips Bltrations Consider aBnite metricspace (X,d). The Vietoris–Rips complexis the simplicial complex Ripst (X)={S ⊆X SdiamS ≤t} •](https://reader035.fdocuments.us/reader035/viewer/2022070722/5f01c22d7e708231d400e56c/html5/thumbnails/63.jpg)
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
http://ulrich-bauer.org/ripser-talk.pdf 15 / 25
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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
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Implicit matrix reduction
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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
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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
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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
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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 Rhttp://ulrich-bauer.org/ripser-talk.pdf 16 / 25
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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.
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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.
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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.
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Apparent and emergent pairs
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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 (!)
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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 (!)
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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
45
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 if
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 if
2 1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 if
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 if
4
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 if
45
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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
45
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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
45
2
3
1
• sublevel sets are subcomplexes, and
• level sets form a discrete vector Beld.
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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 .
645
2
3
1
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 ).
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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 .
645
2
3
1
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 ).
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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 .
45
2
3
1
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 ).
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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 .
45
2
3
1
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 ).
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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 .
45
2
3
1
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 ).
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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 .
45
2
3
1
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 ).
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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
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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
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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
Lemma
Any apparent pairs is a persistence pair.
Lemma
The apparent pairs form a discrete gradient.
• Generalizes a construction proposed by [Kahle 2011] for the study of random
Rips Bltrations
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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.
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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.
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Emergent persistent pairs
Consider the lexicographically reBned Rips Bltration:
• increasing diameter, reBned by
• lexicographic order
This is the simplexwise Bltration for computations in Ripser.
Lemma
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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Emergent persistent pairs
Consider the lexicographically reBned Rips Bltration:
• increasing diameter, reBned by
• lexicographic order
This is the simplexwise Bltration for computations in Ripser.
Lemma
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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