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Sinkhorn_debiaised_barycenter seems too diffuse #458
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very good point, I was a bit underwhelemd by the debiased barycenters that did not look that much bebiased in practice in practice, maybe it is an implementation problem? |
Yes it really looks like an implementation issue (or maybe something deeper, like the IBP not converging?). |
Of course we are all sawmped ;). thanks for looking into it ! |
Describe the bug
Theory (Theorem 3 in this paper ) tells us that the Sinkhorn barycenter between two Gaussian distribution with the same std$\sigma$ should be a Gaussian with std $\sigma$ .
However, when computing the barycenter between two Dirac-ish measures (Eulerian representation : measures are supported on a grid, with the mass concentrated on a single pixel), the barycentric interpolation using
convolutional_barycenter2d_debiased
returns a quite spread measure, which seems quite surprising.To Reproduce
Steps to reproduce the behavior:
convolutional_barycenter2d_debiased
with parameterScreenshots
The "interpolation" we get by computing the Sinkhorn debiaised barycenter as a minimizer of$\mu \mapsto (1-t) S_\epsilon(\mu, \mu_0) + t S_\epsilon(\mu, \mu_1)$ , with $\mu_0, \mu_1$ being two Dirac/very sharp Gaussian measures. The regularization parameter $\epsilon$ is set to
reg=0.1
.Note : a similar behavior occurs when considering 1D measures, suggesting that the issue is not intrinsic to the convolutional-2D approach (did not investigated this in detail). The following screenshot show the returned barycenter for two 1D-Gaussian distributions, with
reg=0.1
. The Gaussian distribution midway is too wide.Code sample
To reproduce the plot in 2D :
Expected behavior
The barycenter of two Dirac-ish measures should be (close to, up to numerical approximation) a Dirac-ish measure supported midway. More generally, the barycenter of two gaussian with std$\sigma$ should have std $\sigma$ .
Environment (please complete the following information):
Additional context
Has been discussed with @hichamjanati (who agrees on the issue). We plan to investigate this later. This issue is for documentation---discussion is welcome!
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