recovering the homology of immersed manifolds
TRANSCRIPT
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