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preconditioning the implicit function theorem linear system using a function

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@marcocuturi marcocuturi released this 29 Sep 16:39
· 122 commits to master since this release

In this release, we provide a new approach to stabilize the resolution of the linear system encountered when differentiating the optimal transport solution w.r.t. solutions.

To that end, we introduce a precondition_fun argument in the sinkhorn function to improve the resolution of the implicit function theorem differentiation of the outputs of the sinkhorn algorithm. Sinkhorn's first order condition (FOC) amounts to ∇ Energy = (marginal of transport) - (marginal fitting penalty) = 0.

This commit introduces the possibility of differentiating instead
precondition_fun(marginal of transport) - precondition_fun(marginal of transport) = 0.

For Sinkhorn, we draw inspiration from this paper to use by default the function 𝜀 log. This provides a numerically more stable approach.