Approximating Persistent Homology with Discrete Morse Theory · 2019-08-07 · Approximating Persistent Homology with Discrete Morse Theory Arkadi Schelling 12 University of Bremen

Approximating Persistent Homology with DiscreteMorse Theory

Arkadi Schelling 12

University of Bremen

Thursday 25th July, 2019

[email protected] 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]


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.


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]



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.



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!


































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.


Approximations

Approximations






W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞

Proof.



Approximations

Approximations






W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞

Proof.



Approximations

Approximations






W∞((X ,F ), (X , a ◦ F )) ≤ ||a− id ||∞

Proof.



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 .


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 .


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?


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.


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: [email protected]


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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.


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Approximating Persistent Homology with Discrete Morse Theory · 2019-08-07 · Approximating Persistent Homology with Discrete Morse Theory Arkadi Schelling 12 University of Bremen