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Approximating Persistent Homology with DiscreteMorse Theory
Arkadi Schelling 12
University of Bremen
Thursday 25th July, 2019
1schelling@uni-bremen.de2joint with Alex Wagner
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 1 / 12
Persistent Homology
Persistent Homology
The main tool of topological data analysis.A multiscale description of data that shows geometric structure.
Figure: [KM16]
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 2 / 12
Persistent Homology
Persistent Homology
Finite filtered cell complexes (X , ∂,F ).
Z2-persistent homology
Computational complexity is O(nω).
The number of cells n can be very large.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 3 / 12
Discrete Morse Theory
Discrete Morse Theory: Basics
An acyclic matching (a.k.a discrete vector field) is a bijection ω : Q → Kbetween disjoint subsets of cells Q,K ⊂ X , such that it is
local: For all q ∈ Q we have q ∈ ∂ω(q)
acyclicity: The following relation generates a partial order on Q
q′ ≺ q ⇔ q 6= q′ and q′ ∈ ∂ω(q)
Theorem
An acyclic matching ω : Q → K on (X , ∂) induces a boundary operator ∂A
on A = X \ (Q ∪ K ), creating a cell complex (A, ∂A) which is homotopyequivalent to (X , ∂). [For98]
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 4 / 12
Discrete Morse Theory
Discrete Morse Theory: Filtrations
Discrete Morse theory can be adapted to filtrations by matching onlycells of the same filtration values. [MN13]
Complexity is O(npm), where n = #X , p much smaller than n and msmaller than #A2.
Can speed up the computation of persistent homology.
No speed up for cellwise filtrations, e.g. Cech complexes.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 5 / 12
Discrete Morse Theory
Unfiltered versus Filtered
How do general, i.e. unfiltered acyclic matchings change thepersistent homology of a filtered complex?
Reducing by an acyclic matching forgets both of the filtration values.
This can change the Wasserstein distance by arbitrarily high values.
Elder rule for merging cycles: Persistent homology keeps cycles thatexisted for a longer time.
Discrete Morse theory does not respect the elder rule!
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 6 / 12
Discrete Morse Theory
Unfiltered versus Filtered
How do general, i.e. unfiltered acyclic matchings change thepersistent homology of a filtered complex?
Reducing by an acyclic matching forgets both of the filtration values.
This can change the Wasserstein distance by arbitrarily high values.
Elder rule for merging cycles: Persistent homology keeps cycles thatexisted for a longer time.
Discrete Morse theory does not respect the elder rule!
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 6 / 12
Discrete Morse Theory
Unfiltered versus Filtered
How do general, i.e. unfiltered acyclic matchings change thepersistent homology of a filtered complex?
Reducing by an acyclic matching forgets both of the filtration values.
This can change the Wasserstein distance by arbitrarily high values.
Elder rule for merging cycles: Persistent homology keeps cycles thatexisted for a longer time.
Discrete Morse theory does not respect the elder rule!
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 6 / 12
Discrete Morse Theory
Unfiltered versus Filtered
How do general, i.e. unfiltered acyclic matchings change thepersistent homology of a filtered complex?
Reducing by an acyclic matching forgets both of the filtration values.
This can change the Wasserstein distance by arbitrarily high values.
Elder rule for merging cycles: Persistent homology keeps cycles thatexisted for a longer time.
Discrete Morse theory does not respect the elder rule!
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 6 / 12
Discrete Morse Theory
Unfiltered versus Filtered
How do general, i.e. unfiltered acyclic matchings change thepersistent homology of a filtered complex?
Reducing by an acyclic matching forgets both of the filtration values.
This can change the Wasserstein distance by arbitrarily high values.
Elder rule for merging cycles: Persistent homology keeps cycles thatexisted for a longer time.
Discrete Morse theory does not respect the elder rule!
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 6 / 12
Approximations
Approximations
A δ-approximation of a filtration changes the filtration values by nomore than δ > 0.
A global approximation is a monotonely increasing functiona : R→ R, that can be used to approximate a given filtration by a ◦F .
A binning is an approximation with a discrete image. Mapping to theclosest value in δZ is called the regular binning.
Lemma (Approximation lemma)
Let a be a global approximation, then
W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞
Proof.
Stability theorem of persistent homology.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 7 / 12
Approximations
Approximations
A δ-approximation of a filtration changes the filtration values by nomore than δ > 0.
A global approximation is a monotonely increasing functiona : R→ R, that can be used to approximate a given filtration by a ◦F .
A binning is an approximation with a discrete image. Mapping to theclosest value in δZ is called the regular binning.
Lemma (Approximation lemma)
Let a be a global approximation, then
W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞
Proof.
Stability theorem of persistent homology.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 7 / 12
Approximations
Approximations
A δ-approximation of a filtration changes the filtration values by nomore than δ > 0.
A global approximation is a monotonely increasing functiona : R→ R, that can be used to approximate a given filtration by a ◦F .
A binning is an approximation with a discrete image. Mapping to theclosest value in δZ is called the regular binning.
Lemma (Approximation lemma)
Let a be a global approximation, then
W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞
Proof.
Stability theorem of persistent homology.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 7 / 12
Approximations
Approximations
A δ-approximation of a filtration changes the filtration values by nomore than δ > 0.
A global approximation is a monotonely increasing functiona : R→ R, that can be used to approximate a given filtration by a ◦F .
A binning is an approximation with a discrete image. Mapping to theclosest value in δZ is called the regular binning.
Lemma (Approximation lemma)
Let a be a global approximation, then
W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞
Proof.
Stability theorem of persistent homology.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 7 / 12
Approximations
Matching Diagrams
A matching diagram P is a representation of an acyclic matchingω : Q → K on a filtered cell complex (X ,F ) given by a multiset ofintervals P = {[F (q),F (ω(q)) | q ∈ Q}.
Bonus: Matching diagrams can be graded, so that they coverpersistent homology pairs as well. This allows a graphical proof of a(filtered) Euler formula
Σk(−1)k#{x ∈ X k | x ≤ t} = Σk(−1)k#wk |t .
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 8 / 12
Approximations
Matching Diagrams
A matching diagram P is a representation of an acyclic matchingω : Q → K on a filtered cell complex (X ,F ) given by a multiset ofintervals P = {[F (q),F (ω(q)) | q ∈ Q}.Bonus: Matching diagrams can be graded, so that they coverpersistent homology pairs as well. This allows a graphical proof of a(filtered) Euler formula
Σk(−1)k#{x ∈ X k | x ≤ t} = Σk(−1)k#wk |t .
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 8 / 12
Approximations
Induced filtered acyclic matching
An acyclic matching induces a filtered acyclic matching by restrictionto matches of the same filtration value.
How to pick a δ-approximation, such that the number of matches inthe induced filtered acyclic matching is maximal?
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 9 / 12
Approximations
Diagram solutions are optimal
An optimal δ-close solution to a matching diagram P is a finite set of realnumbers B = {b0 < . . . < bk}, s. t. bi − bi−1 < δ which maximizes#P|B = #{[a, b) ∈ P | [a, b) does not intersect B}.
Theorem
Let X be a cell complex with an acyclic matching ω, a filtration F , andδ > 0. Let B = {b0, . . . , bk} be an optimal 2δ-close solution to theinduced matching diagram P. Then the map fB : R→ R withfB(x) = 0.5(di + di−1) if x ∈ [bi−1, bi ) is a global δ-approximation optimalw.r.t ω.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 10 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
Outlook
Outlook
How to design a good algorithm for solving a matching diagram?
Can we show similar results for general approximations?
Is it better to pick an optimized binning or go with the regularbinning of δZ? For very small and very large δ the number of reducedcells should be identical.
Small computational experiments on a VR-complex from a 2Dstandard normal distribution show the improved binningoutperforming the default binning up to δ = 0.5.
Can one show asymptotic optimality of possible reductions by theregular binning δZ for many samples given some reasonableassumptions on the sampling distribution?
Questions?
Thank you! Contact: schelling@uni-bremen.de
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 11 / 12
References
References I
Robin Forman.Morse theory for cell complexes.Advances of Mathematics, 134:90–145, 1998.
Firas Khasawneh and Elizabeth Munch.Chatter detection in turning using persistent homology.Mechanical Systems and Signal Processing, 70–71:527–541, 2016.
Konstantin Mischaikow and Vidit Nanda.Morse theory for filtrations and efficient computation of persistenthomology.Discrete Computational Geometry, 40:330–353, 2013.
Arkadi Schelling (University of Bremen) Approximating Persistent Homology with Discrete Morse TheoryThursday 25th July, 2019 12 / 12
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