Gromov–Wasserstein Distances and the Metric Approach to Object Matching
Citations
1,355 citations
Cites methods from "Gromov–Wasserstein Distances and th..."
...23) and as explained in [Mémoli, 2011], it is possible to rewrite equivalently the GH distance using couplings as follows:...
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Cites background or methods from "Gromov–Wasserstein Distances and th..."
...This definition (4) of GW extends slightly the one considered by (Mémoli, 2011), since we consider an arbitrary loss L (rather than just the L2 squared loss)....
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...For L = L2, if the (Cs)s are positive semidefinite (PSD) matrices, the iteratesC produced by the algorithm are also PSD. Proof....
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...To relax the matching problem to a continuous—but still non-convex—optimization problem, the seminal work of Mémoli considers matching between metric-measure spaces, incorporating a probability distribution on the ground space that can account for some form of uncertainty (Mémoli, 2011)....
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...For the square loss L = L2, the update (13) becomes C ← 1 pp> ∑ s λsT > s CsTs. (14) This formula highlights an important property of the method: Proposition 4....
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...In the case L = L2, (Mémoli, 2011) proves that GW1/2 defines a distance on the space of metric measure spaces quotiented by measure-preserving isometries....
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188 citations
Cites background or methods from "Gromov–Wasserstein Distances and th..."
...GW is a distance between metric measure spaces, i.e., metric spaces equipped with a probability distribution (see [Mémoli 2011; Mémoli 2014; Sturm 2012] for more details)....
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...We study an entropically-regularized version of the Gromov-Wasserstein (GW) mapping objective function from [Mémoli 2011] measuring the distortion of geodesic distances....
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...[Mémoli 2011] and subsequent works mention two optimization algorithms (without regularization): gradient descent and alternation....
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...Although [Mémoli 2011] and subsequent work identified the possibility of using GW distances for geometric correspondence, computational challenges hampered their practical application....
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...Following [Mémoli 2011], define the space M(µ0, µ) of measure couplings as the set of measures γ ∈ Prob(Σ0 × Σ) satisfying γ(S0 × Σ) = µ0(S0) and γ(Σ0 × S) = µ(S) for all measurable S0 ⊆ Σ0 and S ⊆ Σ....
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References
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"Gromov–Wasserstein Distances and th..." refers background or methods in this paper
...Indeed, transportation problems, also known as Hitchcock transportation problems are quite common in the optimization literature and standard references such as [88] describe specialized algorithms for solving these problems which run in polynomial time....
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...There exist standard specialized algorithms for numerically computing the minimal value above [88], see Sect....
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6,693 citations
"Gromov–Wasserstein Distances and th..." refers background in this paper
...In [4, 74, 91], similar invariants were proposed where to each point a ∈ A one attaches the histogram or distribution of distances ‖a − a′‖ for all a′ ∈ A....
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...The so called shape context [4, 23, 97, 102] invariant is closely related to hX ....
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...[54]; the spin images of Johnsson [60]; the shape distributions of [86]; the canonical forms of [37]; the Hamza–Krim approach [53]; the spectral approaches of [94, 100]; the integral invariants of [30, 74, 91]; the shape contexts of [4]....
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6,334 citations
4,908 citations