Confidence Sets for Persistent Diagrams Aleksandr Popov Eindhoven - - PowerPoint PPT Presentation

Confidence Sets for Persistent Diagrams

Aleksandr Popov

Eindhoven University of Technology

12th June 2018

1 / 31

What to expect

• 1. Introduction and motivation
• 2. Formal definition
• 3. Computation methods

2 / 31

Introduction and motivation

3 / 31

Back to the origins

Main idea of TDA: determine topology of underlying space, based on a point cloud. Point cloud can be viewed as a sample of the true space.

Random sample of 10 points.

4 / 31

Back to the origins

Main idea of TDA: determine topology of underlying space, based on a point cloud. Point cloud can be viewed as a sample of the true space.

Random sample of 10 points.

4 / 31

Čech complex

Čech complex on the 10 points.

5 / 31

Čech complex

Čech complex on the 10 points.

5 / 31

Čech complex

Čech complex on the 10 points.

5 / 31

Čech complex

Čech complex on the 10 points.

5 / 31

Čech complex

Čech complex on the 10 points.

5 / 31

Homology

Betui numbers: β0 = β1 = These will change if we take difgerent radius of balls!

6 / 31

Homology

Betui numbers: β0 = 1 β1 = 1 These will change if we take difgerent radius of balls!

6 / 31

Homology

Betui numbers: β0 = 1 β1 = 1 These will change if we take difgerent radius of balls!

6 / 31

A difgerent Čech complex

A difgerent Čech complex on the 10 points: β0 = 6, β1 = 0.

7 / 31

A difgerent Čech complex

A difgerent Čech complex on the 10 points: β0 = 6, β1 = 0.

7 / 31

A difgerent Čech complex

A difgerent Čech complex on the 10 points: β0 = 6, β1 = 0.

7 / 31

A difgerent Čech complex

A difgerent Čech complex on the 10 points: β0 = 6, β1 = 0.

7 / 31

A difgerent Čech complex

A difgerent Čech complex on the 10 points: β0 = 6, β1 = 0.

7 / 31

Persistent homology

1 2 3 4 1 2 3 4 Birth Death component loop

Persistence diagram from Vietoris–Rips complex for the 10 points.

8 / 31

So what’s wrong with this?

What here is ‘noise’ and what is ‘signal’? How well do we represent the homology of the space of interest?

9 / 31

What do we want?

1 2 3 4 1 2 3 4 Birth Death component loop

Persistence diagram from Vietoris–Rips complex for the 10 points with the confidence band.

10 / 31

Formal definition

11 / 31

Statistical model and basic definitions

Space of interest: manifold M. Sample set Sn X Xn following distribution P. P is concentrated on . Persistence diagram—multiset of points in

• plane. Denote by

.

Manifold M.

12 / 31

Statistical model and basic definitions

Space of interest: manifold M. Sample set Sn = {X1, . . . , Xn} following distribution P. P is concentrated on M. Persistence diagram—multiset of points in

• plane. Denote by

.

M with points sampled from P.

12 / 31

Statistical model and basic definitions

Space of interest: manifold M. Sample set Sn = {X1, . . . , Xn} following distribution P. P is concentrated on M. Persistence diagram—multiset of points in

• plane. Denote by

.

Sampled points Sn.

12 / 31

Statistical model and basic definitions

Space of interest: manifold M. Sample set Sn = {X1, . . . , Xn} following distribution P. P is concentrated on M. Persistence diagram—multiset of points in

• plane. Denote by P.

1 2 3 4 1 2 3 4 Birth Death component loop

Persistence diagram P for the example.

12 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A

D:

dA x inf

y A y

x i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: x dSn x for increasing from 0 to .

Čech filtration.

13 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A

D:

dA x inf

y A y

x i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: x dSn x for increasing from 0 to .

Čech filtration.

13 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A ⊂ RD: dA(x) = inf

y∈A∥y − x∥2 ,

i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: x dSn x for increasing from 0 to . A p

Distance from p to A.

13 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A ⊂ RD: dA(x) = inf

y∈A∥y − x∥2 ,

i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: {x : dSn(x) ≤ ε} , for ε increasing from 0 to ∞.

Regions for increasing ε.

13 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A ⊂ RD: dA(x) = inf

y∈A∥y − x∥2 ,

i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: {x : dSn(x) ≤ ε} , for ε increasing from 0 to ∞.

Regions for increasing ε.

13 / 31

Čech filtrations; distance

Čech filtration—sequence of Čech complexes with gradually increasing radius. Distance function for a set A ⊂ RD: dA(x) = inf

y∈A∥y − x∥2 ,

i.e. the distance to the closest point in A. Čech filtration corresponds to lower level set filtration of distance function: {x : dSn(x) ≤ ε} , for ε increasing from 0 to ∞.

Regions for increasing ε.

13 / 31

What are we trying to get?

We can construct the persistence diagram of Sn, ˆ P. We use it to estimate the persistence diagram of M, P. Can we guarantee with high probability that they are close for all the points? We need to express ‘closeness’ somehow.

14 / 31

What are we trying to get?

We can construct the persistence diagram of Sn, ˆ P. We use it to estimate the persistence diagram of M, P. Can we guarantee with high probability that they are close for all the points? We need to express ‘closeness’ somehow.

14 / 31

Botuleneck distance

Perfect matching M of sets A and B (bijection). Cost between points a ∈ A, b ∈ B: d∞(a, b) = max(|ax − bx|, |ay − by|) . Botuleneck distance: W A B min

M

max

a b M d

a b Botuleneck distance minimises the maximum pairwise cost. 2 3 a b

d∞(a, b) = 3.

15 / 31

Botuleneck distance

Perfect matching M of sets A and B (bijection). Cost between points a ∈ A, b ∈ B: d∞(a, b) = max(|ax − bx|, |ay − by|) . Botuleneck distance: W∞(A, B) = min

M

max

(a,b)∈M d∞(a, b) .

Botuleneck distance minimises the maximum pairwise cost. (0, 0) (0, 0.8) (4, 2) (4.8, 2.8) (2, 4) (2.8, 4)

W∞(A, B) =

15 / 31

Botuleneck distance

Perfect matching M of sets A and B (bijection). Cost between points a ∈ A, b ∈ B: d∞(a, b) = max(|ax − bx|, |ay − by|) . Botuleneck distance: W∞(A, B) = min

M

max

(a,b)∈M d∞(a, b) .

Botuleneck distance minimises the maximum pairwise cost. (0, 0) (0, 0.8) (4, 2) (4.8, 2.8) (2, 4) (2.8, 4)

W∞(A, B) =

15 / 31

Botuleneck distance

Perfect matching M of sets A and B (bijection). Cost between points a ∈ A, b ∈ B: d∞(a, b) = max(|ax − bx|, |ay − by|) . Botuleneck distance: W∞(A, B) = min

M

max

(a,b)∈M d∞(a, b) .

Botuleneck distance minimises the maximum pairwise cost. (0, 0) (0, 0.8) (4, 2) (4.8, 2.8) (2, 4) (2.8, 4)

W∞(A, B) = 0.8.

15 / 31

What are we trying to get: take 2

We can construct the persistence diagram of Sn, ˆ P. We use it to estimate the persistence diagram of M, P. Can we guarantee with high probability that they are close for all the points? Given confidence , find cn such that lim sup

n

W cn If the point in is further than cn from the diagonal, then with probability at least it is also not on the diagonal in , so it is significant.

16 / 31

What are we trying to get: take 2

We can construct the persistence diagram of Sn, ˆ P. We use it to estimate the persistence diagram of M, P. Can we guarantee with high probability that they are close for all the points? Given confidence α ∈ (0, 1), find cn such that lim sup

n→∞

P(W∞( ˆ P, P) > cn) ≤ α . If the point in is further than cn from the diagonal, then with probability at least it is also not on the diagonal in , so it is significant.

16 / 31

What are we trying to get: take 2

We can construct the persistence diagram of Sn, ˆ P. We use it to estimate the persistence diagram of M, P. Can we guarantee with high probability that they are close for all the points? Given confidence α ∈ (0, 1), find cn such that lim sup

n→∞

P(W∞( ˆ P, P) > cn) ≤ α . If the point in ˆ P is further than cn from the diagonal, then with probability at least 1 − α it is also not on the diagonal in P, so it is significant.

16 / 31

Botuleneck stability

L∞ distance between functions f, g : X → R: ∥f − g∥∞ = sup

x∈X

|f(x) − g(x)| . sin x sin(x + π)

17 / 31

Botuleneck stability

L∞ distance between functions f, g : X → R: ∥f − g∥∞ = sup

x∈X

|f(x) − g(x)| . sin x sin(x + π)

17 / 31

Botuleneck stability

L∞ distance between functions f, g : X → R: ∥f − g∥∞ = sup

x∈X

|f(x) − g(x)| . Botuleneck stability: W∞(P(f), P(g)) ≤ ∥f − g∥∞ , where P is a persistence diagram.

17 / 31

Hausdorfg distance

For two sets A, B ⊂ RD: H(A, B) = max { max

x∈A min y∈B∥x − y∥, max x∈B min y∈A∥x − y∥

} . This is maximum Euclidean distance from a point in one set to the closest point in the other set. (0, 0) (0, 0.8) (3, 2) (3, 3) (2, 4) (2.8, 4)

H(A, B) =

18 / 31

Hausdorfg distance

For two sets A, B ⊂ RD: H(A, B) = max { max

x∈A min y∈B∥x − y∥, max x∈B min y∈A∥x − y∥

} . This is maximum Euclidean distance from a point in one set to the closest point in the other set. (0, 0) (0, 0.8) (3, 2) (3, 3) (2, 4) (2.8, 4)

H(A, B) =

18 / 31

Hausdorfg distance

For two sets A, B ⊂ RD: H(A, B) = max { max

x∈A min y∈B∥x − y∥, max x∈B min y∈A∥x − y∥

} . This is maximum Euclidean distance from a point in one set to the closest point in the other set. (0, 0) (0, 0.8) (3, 2) (3, 3) (2, 4) (2.8, 4)

H(A, B) =

18 / 31

Hausdorfg distance

For two sets A, B ⊂ RD: H(A, B) = max { max

x∈A min y∈B∥x − y∥, max x∈B min y∈A∥x − y∥

} . This is maximum Euclidean distance from a point in one set to the closest point in the other set. (0, 0) (0, 0.8) (3, 2) (3, 3) (2, 4) (2.8, 4)

H(A, B) = 3 √ 2.

18 / 31

Bringing this all together

For two sets A, B ⊂ RD: H(A, B) = max { max

x∈A min y∈B∥x − y∥, max x∈B min y∈A∥x − y∥

} . L∞ distance between functions f, g : X → R: ∥f − g∥∞ = sup

x∈X

|f(x) − g(x)| . Therefore, ∥dSn − dM∥∞= H(Sn, M) .

19 / 31

Bringing this all together (pt. 2)

Using stability theorem, we get W∞(PSn, PM) ≤ ∥dSn − dM∥∞ = H(Sn, M) . Earlier we said we want to have lim sup

n

W cn Given confidence , we need to find cn such that lim sup

n

H Sn cn

20 / 31

Bringing this all together (pt. 2)

Using stability theorem, we get W∞(PSn, PM) ≤ ∥dSn − dM∥∞ = H(Sn, M) . Earlier we said we want to have lim sup

n→∞

P(W∞( ˆ P, P) > cn) ≤ α . Given confidence α, we need to find cn such that lim sup

n→∞

P(H(Sn, M) > cn) ≤ α .

20 / 31

Final definition

Given confidence α, we need to find cn such that lim sup

n→∞

P(H(Sn, M) > cn) ≤ α . Confidence set Cn is the set of persistence diagrams that are plausible as true persistence diagrams, given our estimate ˆ P: Cn = { ˜ P : W∞( ˆ P, ˜ P) ≤ cn} .

21 / 31

Illustration

1 2 3 4 1 2 3 4 Birth Death component loop

Persistence diagram from Vietoris–Rips complex for the 10 points with the confidence band.

22 / 31

Computation methods

23 / 31

Overview

Multiple methods. Simplest: subsampling. Betuer: concentration of measure. Difgerent, more robust: density estimation.

24 / 31

Subsampling

Let b be such that b = o (

n log n

) and bn → ∞ as n → ∞. We have n data points. Define N = (n

b

) .

• 1. Draw N subsamples from Sn, each of size b: Sb n

SN

b n.

• 2. Compute for each subsample Tj

H Sj

b n Sn .

• 3. Define Lb t

N N j

1 Tj t , so count the proportion of subsamples with Hausdorfg distance to the original above t.

• 4. Let cb

Lb .

25 / 31

Subsampling

Let b be such that b = o (

n log n

) and bn → ∞ as n → ∞. We have n data points. Define N = (n

b

) .

• 1. Draw N subsamples from Sn, each of size b: S1

b,n, . . . , SN b,n.

• 2. Compute for each subsample Tj = H(Sj

b,n, Sn).

• 3. Define Lb(t) = 1

N

∑N

j=1 1(Tj > t), so count the proportion of subsamples with

Hausdorfg distance to the original above t.

• 4. Let cb = 2L−1

b (α).

25 / 31

Why does this work?

As Sn is sampled from M, so our subsamples are sampled from Sn. We tune the cb so that most of our subsamples are close to Sn. This way Sn will be close to M. Formally: for all large n: W cb H Sn cb b n

26 / 31

Why does this work?

As Sn is sampled from M, so our subsamples are sampled from Sn. We tune the cb so that most of our subsamples are close to Sn. This way Sn will be close to M. Formally: for all large n: P(W∞( ˆ P, P) > cb) ≤ P(H(Sn, M) > cb) ≤ α + O (b n ) 1

4

.

26 / 31

Other methods

Subsampling method is conservative. Concentration of measure-based method is betuer, but complicated. Both use Čech filtration. Both are not robust to outliers. Density estimation-based methods are. Other methods require too much statistics to present here.

27 / 31

Some experimental results

28 / 31

Some experimental results (pt. 2)

29 / 31

Summary

We introduced the concept of confidence sets. Goal: distinguish ‘noise’ from ‘signal’ in persistence diagrams built on point sets. Implementation: confidence interval-like band around the diagonal. Points in the band are noise. Computation: various methods, subsampling the simplest.

30 / 31

References

David Cohen-Steiner, Herbert Edelsbrunner and John Harer. ‘Stability of Persistence Diagrams’. In: Discrete & Computational Geometry 37.1 (Jan. 2007),

• pp. 103–120. issn: 1432-0444. doi: 10.1007/s00454-006-1276-5.

Brituany Terese Fasy et al. ‘Confidence Sets for Persistence Diagrams’. Version 3. In: Annals of Statistics 42.6 (20th Nov. 2014), pp. 2301–2339. doi: 10.1214/14-AOS1252. arXiv: 1303.7117 [math.ST]. Brituany Terese Fasy et al. ‘Supplement to: Confidence Sets for Persistence Diagrams’. In: Annals of Statistics 42.6 (20th Nov. 2014). doi: 10.1214/14-AOS1252SUPP.

31 / 31