Recovering the homology of immersed manifoldsRaphael Tinarrage

arxiv.org/abs/1912.03033

Recovering the homology of immersed manifoldsRaphael Tinarrage

embedded manifold

(may self-intersect)

arxiv.org/abs/1912.03033

Statement of the problem

We are observing an immersed manifold M⊂ Rn.

Abstract manifold Immersed manifold

M0 M = u(M0)

Immersion

u⊂ Rn

Statement of the problem

We are observing an immersed manifold M⊂ Rn.

Abstract manifold Immersed manifold

M0 M = u(M0)

Immersion

u

Klein bottle

Klein bottle ∪ sphere

[Martin, Coutsias, Thompson, Topology of Cyclooctane Energy Landscape]

⊂ Rn

Statement of the problem

We are observing an immersed manifold M⊂ Rn.

Abstract manifold Immersed manifold

M0 M = u(M0)

Immersion

u

Question 1: Given M, compute the (singular)homology groups of M0.

Question 2: Given X ⊂ Rn close to M, compute thehomology groups of M0.

H0 = Z/2ZH1 = Z/2Z

M X

⊂ Rn

Statement of the problem

We are observing an immersed manifold M⊂ Rn.

Abstract manifold Immersed manifold

M0 M = u(M0)

Immersion

u

A bit of context

[Arias-Castro, Ery, Gilad Lerman and Teng Zhang. Spectral clustering based on localPCA.]

[Dıaz Martınez, Memoli and Mio. The shape of data and probability measures.]

[Memoli, Smith and Wan. The Wasserstein transform.]

⊂ Rn

Our method

We will use persistent homology.

Unfortunately, the persistent homology of the Cech filtration of M doesnot reveal the homology of M0.

H0

H1

Barcodes

H0

Our method

We will use persistent homology.

Unfortunately, the persistent homology of the Cech filtration of M doesnot reveal the homology of M0.

We will try to lift M in a higher dimensional space, where the Cechfiltration reveals a circle.

H0

H1

Barcodes

H0

Our method

How to lift M?

M0 M⊂ Rn

x0 u(x0)

(u(x0),

Choose f such that u is an embedding.

u

uM0 M ⊂ Rn

x0 f (x0))

×Rm

Our method

How to lift M?

M0 M⊂ Rn

x0 u(x0)

(u(x0),

Choose f such that u is an embedding.

u

u

Our choice is

u is an embedding under a reasonable assumption

[T., Computing Stiefel-Whitney classes of line bundles]

we are actually estimating the tangent bundle of M0

M0 M ⊂ Rn

x0

f : x0 7−→ Tx0M0 (tangent space ofM0 at x0)

f (x0))

×Rm

lifted manifoldlift space

Our method

Notations:u : M0 →M⊂ Rn is an immersion

For x0 ∈M0, x = u(x0)

For x0 ∈M0, TxM denotes the tangent space of M0 seen in Rn

M(Rn) denotes the space of n× n matrices

pTxM ∈M(Rn) denotes the orthogonal projection matrix on TxM

Lift space: Rn ×M(Rn)

Lifted manifold : M = {(x, pTxM) , x0 ∈M0} ⊂ Rn ×M(Rn)

Lifting map: u : M0 → M

M0

(x, pTxM)

M ⊂ Rn ×M(Rn)

M⊂ Rnu

u

u : x0(PCA dimension reduction)

Recipe in practice

We observe a point cloud X ⊂ Rn close toM.

Let r > 0 be a parameter.For every x ∈ X, compute a local covariance matrix

ΣX(x, r) =1

|A|∑y∈A

(x− y)⊗2

where A = {y ∈ X, ‖x− y‖ ≤ r}.

Consider the set

X = {(x,ΣX(x, r)) , x ∈ X} ⊂ Rn ×M(Rn).

∈M(Rn)

x

r

Recipe in practice

X

X

Consider the set

X = {(x,ΣX(x, r)) , x ∈ X} ⊂ Rn ×M(Rn).

x

r

x

r

X

X

Consider the set

X = {(x,ΣX(x, r)) , x ∈ X} ⊂ Rn ×M(Rn).

Recipe in practice

bad estimation oftangent space

M and X are not close inHausdorff distance :(

Recipe (that does not work) in practice

X

M = {(x, pTxM), x0 ∈M0}

X = {(x,ΣX(x, r)), x ∈M}

A measure-theoretic setting

M0 M⊂ Rnµ0 measure on M0

u

µ push-forward

measure on M

M ⊂ Rn ×M(Rm)

u

µ0 push-forward

measure on Mwhere u : x0 7−→(x, 1

d+2pTxM)

µ0 can be defined as follows: for every test function φ : Rn ×M(Rn)→ R,∫φ(x,A) · dµ0(x,A) =

∫φ

(x,

1

d + 2pTxM

)· dµ0(x0).

A measure-theoretic setting

M0 M⊂ Rnµ0 measure on M0

u

µ push-forward

measure on M

M ⊂ Rn ×M(Rm)

u

µ0 push-forward

measure on Mwhere u : x0 7−→(x, 1

d+2pTxM)

µ0 can be defined as follows: for every test function φ : Rn ×M(Rn)→ R,∫φ(x,A) · dµ0(x,A) =

∫φ

(x,

1

d + 2pTxM

)· dµ0(x0).

Now, we are observing a measure ν close to µ

Define ν as follows: for every test function φ : Rn ×M(Rn)→ R,∫φ(x,A) · dν(x,A) =

∫φ

(x,

1

r2Σν(x, r)

)· dν(x),

where Σν(x, r) is the local covariance matrix.

A measure-theoretic setting

µ0 can be defined as follows: for every test function φ : Rn ×M(Rn)→ R,∫φ(x,A) · dµ0(x,A) =

∫φ

(x,

1

d + 2pTxM

)· dµ0(x0).

Now, we are observing a measure ν close to µ

Define ν as follows: for every test function φ : Rn ×M(Rn)→ R,∫φ(x,A) · dν(x,A) =

∫φ

(x,

1

r2Σν(x, r)

)· dν(x),

where Σν(x, r) is the local covariance matrix.

Theorem:Let r > 0. Suppose that W1(µ, ν) is small enough. Under severalassumptions on M0 and µ0, we have

Wp(ν, µ0) ≤ constant · r1p

where Wp denote the p-Wasserstein distance.

Persistent homology for measures

[Anai, Chazal, Glisse, Ike, Inakoshi, T., Umeda. DTM-based filtrations]

Usual Cech filtration: with X ⊂ Rk,

Xt =⋃x∈XB(x, t)

DTM-filtration: with µ measure on Rk,

V t =⋃

x∈supp(µ)

B(x, t− dµ(x))

where dµ is the distance-to-measure (DTM) associated to µ

Persistent homology for measures

observation X ⊂ R2

lift in R2 ×M(R2)

H0

H1

DTM-filtration

A last illustration

observation X ⊂ R2

lift in R6

H0

H1

measure-basedfiltrations