Recent Algorithmic Advances in Topological Data Analysis · W I S S E N T E C H N I K L E I D E N S...

W I S S E N T E C H N I K L E I D E N S C H A F T

www.tugraz.at

Recent Algorithmic Advancesin Topological Data AnalysisMichael KerberGudhi workshop, Porquerolles, France, Oct 19, 2016

2 Computational Topology@TU Graz

Arnur Nigmetov Hannah Schreiber Aruni Choudhary(MPI Saarbrucken)

Gudhi workshop, Porquerolles, France, Oct 19, 2016

3 Our mission

“. . . to boldlycompute what notopologists hascomputed before.”

Algorithmic foundations, implementations, andsoftware in computational topology and geometry.

3 Our mission

“. . . to boldlycompute what notopologists hascomputed before.”

Algorithmic foundations, implementations, andsoftware in computational topology and geometry.

4 The algorithmic pipeline

1. Turn input into multi-scale representation

2. Compute topological invariants

3. Draw conclusions about the input

1. Turn input into multi-scale representation

2. Compute topological invariants

3. Draw conclusions about the input

5 Cech filtrations

Nested sequence of simplicial complexes

Important in topological data analysis

Vietoris-Rips complexes: Closely related

Problem: Size of k -skeleton is( n

)= O(nk+1)

6 Topological approximation

Persistence diagram of the Cech filtration:

Question: Can we find a small filtration whosepersistence diagram is provably close to the Cechdiagram?

6 Topological approximation

Persistence diagram of the Cech filtration:

Question: Can we find a small filtration whosepersistence diagram is provably close to the Cechdiagram?

7 Previous work

Sparse Rips complex [Sheehy 2012]

(1 + ε)-approximation of size

)O(∆k)

with ∆ the doubling dimension of the point set

Various related approaches [Dey, Fan, Wang 2012]

[K., Sharathkumar 2013] [Botnan, Spreemann 2015]

[Buchet et al. 2015] [Cavanna, Jahanseir, Sheehy 2015]

8 Our contributions [Choudhary,K.,Raghvendra, SoCG 2016]

6(d + 1)-approximation of size

n · 2O(d log k)

per scale for n points in Rd .

Combined with dimension reduction:O(log3/2 n)-approximation of size nO(1).

n · 2O(d log k)

9 Approximation by lattices I

Diameter of a cell: α ·√

Two non-adjacent cells areat least α apart

⇒√

d-approximation!

But highly degenerate: 2d

cells intersect in a point(leads to size n · 2O(dk))

⇒√

d-approximation!

⇒√

d-approximation!

⇒√

d-approximation!

⇒√

d-approximation!

10 Approximation by lattices II

Hexagonal grid

How to generalize in higherdimensions?

11 The permutahedron

Voronoi region of A∗d -lattice

Diameter: O(α ·√

Lemma: Non-intersectingcells are at leastα ·√

2/(d + 1) apart.

Size of the dual k -skeleton:n2O(d log k)

2/(d + 1) apart.

12 Interleaving

· · · //Rβ2(d+1)

g //Rβ8(d+1)3 // · · ·

· · · // Xβ

θ // Xβ4(d+1)2

// · · ·

13 The Johnson-Lindenstrauss Lemma

For a point set S ⊂ Rd of n points and 0 < ε < 1, thereis a map

f : Rd → Rm

with m = O( log nε2 ) such that for any two points s, t ∈ S:

(1− ε)‖s − t‖ ≤ ‖f (s)− f (t)‖ ≤ (1 + ε)‖s − t‖.

Moreover, a random (scaled) projection from Rd toRm has that property with a probability of at least 1

13 The Johnson-Lindenstrauss Lemma

For a point set S ⊂ Rd of n points and 0 < ε < 1, thereis a map

f : Rd → Rm

with m = O( log nε2 ) such that for any two points s, t ∈ S:

(1− ε)‖s − t‖ ≤ ‖f (s)− f (t)‖ ≤ (1 + ε)‖s − t‖.

Moreover, a random (scaled) projection from Rd toRm has that property with a probability of at least 1

14 Dimension reduction

Size per scale n · 2O(d log k)

[Johnson, Lindenstrauss 1984]: d ≈ log n (constant distortion)⇒ size nO(log k), total distortion O(log n)

[Matousek 1990]: d ≈ log nlog log n ((log n)-distortion)

⇒ size nO(1), total distortion O(log2 n)

[Bourgain 1985]: General metric space: Embed toO(log2 n) dimensions with distortion O(log n)

⇒size nO(1) and total distortion O(log3 n)

n · 2O(d log k)

Lower bound: Any (1 + δ)-approximation scheme hasto be of size nΩ(log log n) if δ < 1

96 log1.001 n.

1. Turn input into multi-scale representation2. Compute topological invariants3. Draw conclusions about the input

17 Distances between persistence diagrams

A1 A2 · · · An

A1 A2 . . . An−1 An

A1A2... d(Ai ,Aj)

An−1An

n diagrams⇒(n

)distances

Often the computationalbottleneck

18 Distance measures

One-to-one pairing of points

Every point must be paired

Pairing with diagonal allowed

Cost of a pair (p,q): ‖p − q‖∞Bottleneck distance: Minimize maximal cost1-Wasserstein distance: Minimize Σ of the costsStability [Cohen-Steiner et al. 2007]

Other distances

19 From diagram distance to graph matching

Weighted complete bipartite graph GWeight of an edge: L∞ distance of the pointsEXCEPT: weight is zero if both points are on diagonalUse graph matching algorithm (Hopcroft-Karp,Hungarian,. . .)

20 Does geometry help?

G is “almost” metric (modulo diagonal)Asymptotically faster algorithms are known for thiscase[Efrat, Itai, Katz: Geometry Helps in Bottleneck Matching... 2001]

[Vaidya: Geometry Helps in Matchings. 1989] (opt. assignment)Adaption to persistence diagrams straight-forward[Folklore? Mentioned in Edelsbrunner, Harer 2010]

AnswerYes, in theory

G is “almost” metric (modulo diagonal)Asymptotically faster algorithms are known for thiscase[Efrat, Itai, Katz: Geometry Helps in Bottleneck Matching... 2001]

[Vaidya: Geometry Helps in Matchings. 1989] (opt. assignment)Adaption to persistence diagrams straight-forward[Folklore? Mentioned in Edelsbrunner, Harer 2010]

AnswerYes, in theory

Our contributionYes, also in practice! [K., Morozov, Nigmetov, ALENEX 2016]

We compare geometric and non-geometricimplementations of bottleneck matchings and optimalassignment (for R2)We show experimentally that geometry improvesperformanceWe outperform Dionysus, the only publically availablesoftware for distances of persistence diagramsOur code is freely available:https://bitbucket.org/grey_narn/hera

22 The bottleneck case

Let G[α] be the graph G with all edges of weight > αdeleted

Observation: If G[α] has a perfect matching, thebottleneck distance is at most α.

23 The Wasserstein case

Assignment problem: Findperfect matching with minimalsum of costs.

Discrete optimal transport

Hungarian algorithm

23 The Wasserstein case

Assignment problem: Findperfect matching with minimalsum of costs.

Discrete optimal transport

Hungarian algorithm

24 The auction algorithm [Bertsekas 1988]

n bidders (left), n objects (right)

Maintain (partial) matchingObjects have (global) pricesBidders have (individual) appreciations

Repeat:

Pick an unassigned bidderValue of object: appreciation - pricePick best and 2nd-best valued objectsAssign bidder to best valued objectIncrease price by difference of values,plus ε > 0

n bidders (left), n objects (right)Maintain (partial) matching

Objects have (global) pricesBidders have (individual) appreciations

Repeat:

n bidders (left), n objects (right)Maintain (partial) matchingObjects have (global) prices

Bidders have (individual) appreciations

Repeat:

n bidders (left), n objects (right)Maintain (partial) matchingObjects have (global) pricesBidders have (individual) appreciations

Repeat:

Pick an unassigned bidder

Value of object: appreciation - pricePick best and 2nd-best valued objectsAssign bidder to best valued objectIncrease price by difference of values,plus ε > 0

Repeat:

Pick an unassigned bidderValue of object: appreciation - price

Pick best and 2nd-best valued objectsAssign bidder to best valued objectIncrease price by difference of values,plus ε > 0

Repeat:

Pick an unassigned bidderValue of object: appreciation - pricePick best and 2nd-best valued objects

Assign bidder to best valued objectIncrease price by difference of values,plus ε > 0

Repeat:

Pick an unassigned bidderValue of object: appreciation - pricePick best and 2nd-best valued objectsAssign bidder to best valued object

Increase price by difference of values,plus ε > 0

310+eps

Repeat:

25 Why auction?

Theorem [Bertsekas 1988]

Let opt denote the cost of the optimal assignment, and dthe cost returned by the auction. Then

opt− nε ≤ d ≤ opt

Large ε: Fast, rough approximationSmall ε: Slow, accurate approximationε-scaling [Bertsekas, Castanon 1991]

Remark: Getting exact result possible, but very slow

26 How geometry helps

−5−3

−5−4

Appreciation = -distance to object

Crucial query: Find the best andsecond best object for an unassignedbidder.

Our approach

k-d-tree with weight per node

Weight=minimal price

Prune search in subtrees if bettercandidates are known

26 How geometry helps

−5−3

−5−4

Appreciation = -distance to object

Crucial query: Find the best andsecond best object for an unassignedbidder.

Our approach

k-d-tree with weight per node

Weight=minimal price

Prune search in subtrees if bettercandidates are known

27 Experimental comparison

500 1,000 1,500 2,000

10−1

# points on a 4-sphere

Dionysusnon-geom.

Linear space (geometric) vs quadratic space(non-geometric)Exact distance (Dionysus) vs relative1%-approximation

28 The Hera library

URL: https://bitbucket.org/grey_narn/hera

Code for bottleneck (LGPL) andWasserstein (BSD)

Supports q-Wasserstein distanceand different choice of innermetric (instead of L∞)

Download it! Use it! Tell us yourexperience!

1. Turn input into multi-scale representation2. Compute topological invariants3. Draw conclusions about the input

Recent Algorithmic Advances in Topological Data Analysis · W I S S E N T E C H N I K L E I D E N S...

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