ott.problems.linear.potentials.EntropicPotentials.transport#
- EntropicPotentials.transport(vec, forward=True)#
Transport
vec
according to Gangbo-McCann Brenier [Brenier, 1991].Uses Proposition 1.15 from [Santambrogio, 2015] to compute an OT map when applying the inverse gradient of cost.
When the cost is a general cost, the operator uses the
twist_operator()
associated of the correspondingCostFn
.When the cost is a translation invariant
TICost
cost, \(c(x,y)=h(x-y)\), and the twist operator translates to the application of the convex conjugate of \(h\) to the gradient of the dual potentials, namely \(x- (\nabla h^*)\circ \nabla f(x)\) for the forward map, where \(h^*\) is the Legendre transform of \(h\). For instance, in the case \(h(\cdot) = \|\cdot\|^2, \nabla h(\cdot) = 2 \cdot\,\), one has \(h^*(\cdot) = \|.\|^2 / 4\), and therefore \(\nabla h^*(\cdot) = 0.5 \cdot\,\).Note
When the dual potentials are solved in correlation form, and marked accordingly by setting
corr
toTrue
, the maps are \(\nabla g\) for forward, \(\nabla f\) for backward map. This can only make sense when using the squared-EuclideanSqEuclidean
cost.