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Optimal Transport for structured data with application on graphs
Titouan Vayer
Joint work with Laetitia Chapel, Remi Flamary, Romain Tavenard and Nicolas Courty
Contributions:
- Differentiable distance between labeled graphs.
Optimal Transport for structured data with application on graphs - - PowerPoint PPT Presentation
Optimal Transport for structured data with application on graphs - - PowerPoint PPT Presentation
Optimal Transport for structured data with application on graphs Titouan Vayer Joint work with Laetitia Chapel, Remi Flamary, Romain Tavenard and Nicolas Courty A novel distance between labeled graphs based on optimal transport Contributions:
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Contributions:
- Differentiable distance between labeled graphs.
Jointly considers the features and the structures Optimal transport: soft assignment between the nodes Distance = 1.41
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Contributions:
Computing average
- f labeled graphs
= +
1 2 (
)
- Differentiable distance between labeled graphs.
Jointly considers the features and the structures
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Structured data as probability distribution
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Structured data as probability distribution
Features
(ai)i
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Structured data as probability distribution
Features
(ai)i
nodes in the metric space of the graph
(xi)i
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Structured data as probability distribution
Features
(ai)i
nodes in the metric space of the graph
(xi)i
weighted by their masses (hi)i
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Optimal transport in a nutshell
Wasserstein distance
μX
νY
Gromov-Wasserstein distance
Compare two probability distributions by transporting one onto another
d(ai, bj)
μA νB
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Optimal transport in a nutshell
Wasserstein distance
μX
νY
Gromov-Wasserstein distance
Compare two probability distributions by transporting one onto another
d(ai, bj)
μA νB
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Optimal transport in a nutshell
Wasserstein distance
μX
νY
Gromov-Wasserstein distance
Compare two probability distributions by transporting one onto another
d(ai, bj)
μA νB
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Fused Gromov-Wasserstein distance
FGWq,α(μ, ν) = min
π∈Π(μ,ν) ∑ i,j,k,l
( (1 − α)d(ai, bj)q + α|C1(i, k) − C2(j, l)|q )πi,jπk,l
where is the soft assignment matrix
π
α is a trade-off features/structures
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Fused Gromov-Wasserstein distance
Properties
- Interpolate between Wasserstein distance on features and Gromov-Wasserstein distance on the structures
- Distance on labeled graph: vanishes iff graphs have same labels and weights at the same place up to a permutation
Optimization problem
- Non convex Quadratic Program: hard !
- Conditional Gradient Descent (aka Frank Wolfe)
- Suitable for entropic regularization + Sinkhorn iteraterations
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Applications
Noiseless graph Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6 Bary n= 15 Bary n= 7
Classification
Graph Barycenter + k-means clustering of graphs
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