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We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k. Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.
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A Multicover Nerve for Geometric Inference
Don SheehyINRIA Saclay, France
Computational geometers use topology to certify geometric constructions.
Computational geometers use topology to certify geometric constructions.
Surface Reconstruction - homeomorphic
Computational geometers use topology to certify geometric constructions.
Surface Reconstruction - homeomorphic
Medial Axis Approximation - homotopy equivalence
Computational geometers use topology to certify geometric constructions.
Surface Reconstruction - homeomorphic
Medial Axis Approximation - homotopy equivalence
Topological Data Analysis - (persistent) homology
Computational geometers use topology to certify geometric constructions.
Surface Reconstruction - homeomorphic
Medial Axis Approximation - homotopy equivalence
Topological Data Analysis - (persistent) homology
Topological Inference
Topological Inference
Fixed Scale: Niyogi, Smale, Weinberger 2008
Topological Inference
Fixed Scale: Niyogi, Smale, Weinberger 2008Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009
Topological Inference
Fixed Scale: Niyogi, Smale, Weinberger 2008Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009All Scales (Persistence): Edelsbrunner, Letscher, Zomorodian 2002
Topological Inference
Fixed Scale: Niyogi, Smale, Weinberger 2008Variable Scale: Chazal, Cohen-Steiner, Lieutier 2009All Scales (Persistence): Edelsbrunner, Letscher, Zomorodian 2002Guarantees: Chazal and Oudot, 2008
The Nerve Theorem
The Nerve Theorem
Union of Shapes
The Nerve Theorem
Union of Shapes Simplicial Complex
The Nerve Theorem
Union of Shapes Simplicial Complex
The Nerve Theorem
Union of Shapes Simplicial Complex
The Nerve Theorem
Union of Shapes Simplicial Complex
Key Fact: Preserves Topology as long as intersections are empty or contractible.
The Nerve Theorem
Union of Shapes Simplicial Complex
Key Fact: Preserves Topology as long as intersections are empty or contractible.
Cech Complex
Geometric Persistent Homology
Geometric Persistent Homology
Input: P ! Rd
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Geometric Persistent Homology
P! =!
p!P
ball(p,!)
Input: P ! Rd
Offsets
Compute the Homology
Persistent
k-covered regions
k-covered regions
α
k-covered regions
α
Idea: Capture both mass and scale.
k-covered regions
α
Idea: Capture both mass and scale.
Goal: Build a simplicial complex homotopy equivalent to the k-covered regions.
The k-nerve.
k-nerve
The k-nerve.
k-nerve
The k-nerve.
The k-nerve already gives the right topology, but...
k-nerve
The k-nerve.
The k-nerve already gives the right topology, but...
k-nerve
...there is no easy relationship between the complexes for different values of k.
Barycentric Decomposition
Barycentric Decomposition
Barycentric Decomposition
0Barycentric Decomposition
1
0Barycentric Decomposition
1
0Barycentric Decomposition
2
1
0Barycentric Decomposition
2
2 1 0
1
0Barycentric Decomposition
2
2 1 03 2 1
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
Cech Complex Barycentric Decomposition Filtered
2,α-offsets 2-nerve Barycentric Decomposition
The kth barycentric Cech complex is homotopy equivalent to the k-nerve.
A persistent version.
A persistent version.
Input is a collection of filtrations, rather than a collection of sets.
A persistent version.
Input is a collection of filtrations, rather than a collection of sets.
The Result: Given a collection of convex filtrations, the persistent homology of the k-covered set is exactly that of the kth barycentric decomposition of the nerve of the filtrations.
What if we only have pairwise distances?
What if we only have pairwise distances?
The Rips complex at scale r is the clique complex of the r-neighborhood graph.(the edges are the same as those in the Cech complex)
What if we only have pairwise distances?
The Rips complex at scale r is the clique complex of the r-neighborhood graph.(the edges are the same as those in the Cech complex)
New Result: Applying the same barycentric trick to the Rips complexes gives a 2-approximation to the persistent homology of k-covered region of balls.
Conclusion
A filtered simplicial complex that captures the topology of the k-covered region of a collection of convex sets for all k.
Guaranteed correct persistent homology.
A guaranteed approximation via easier to compute Rips complexes.
Conclusion
A filtered simplicial complex that captures the topology of the k-covered region of a collection of convex sets for all k.
Guaranteed correct persistent homology.
A guaranteed approximation via easier to compute Rips complexes.
Thank you.
A 3-Step Process
StatisticsDe-noise and smooth the data.
GeometryBuild a complex.
Topology (Algebra)Compute the persistent homology.
1
2
3
A 3-Step Process
StatisticsDe-noise and smooth the data.
GeometryBuild a complex.
Topology (Algebra)Compute the persistent homology.
1
2
3
A 3-Step Process
StatisticsDe-noise and smooth the data.
GeometryBuild a complex.
Topology (Algebra)Compute the persistent homology.
1
2
3
Goal: No more tuning parameters
Goal: No more tuning parameters
i.e. Build a complex that works for every choice of de-noising parameters.
Capture both scale AND mass
Capture both scale AND mass
See Also: [Chazal, Cohen-Steiner, Merigot, 2009]
Capture both scale AND mass
See Also: [Chazal, Cohen-Steiner, Merigot, 2009][Guibas, Merigot, Morozov, Yesterday]
k-NN distance.
k-NN distance.
dP (x) = minp!P
|x! p| P! = d!1
P[0,!]
k-NN distance.
dP (x) = minp!P
|x! p|
dk(x) = minS!(Pk)
maxp!S
|x! p|
P! = d!1
P[0,!]
P!k = d!1
k[0,!]
α
k-NN distance.
dP (x) = minp!P
|x! p|
dk(x) = minS!(Pk)
maxp!S
|x! p|
P! = d!1
P[0,!]
P!k = d!1
k[0,!]
α
k-NN distance.
dP (x) = minp!P
|x! p|
dk(x) = minS!(Pk)
maxp!S
|x! p|
P! = d!1
P[0,!]
P!k = d!1
k[0,!]
a multifiltration with parameters α and k.
