ott.problems.linear.potentials.EntropicPotentials.transport
ott.problems.linear.potentials.EntropicPotentials.transport#
- EntropicPotentials.transport(vec, forward=True)#
Transport
vecaccording to Brenier formula [Brenier, 1991].Uses Theorem 1.17 from [Santambrogio, 2015] to compute an OT map when given the Legendre transform of the dual potentials.
That OT map can be recovered as \(x- (\nabla h)^{-1}\circ \nabla f(x)\) For the case \(h(\cdot) = \|\cdot\|^2, \nabla h(\cdot) = 2 \cdot\,\), and as a consequence \(h^*(\cdot) = \|.\|^2 / 4\), while one has that \(\nabla h^*(\cdot) = (\nabla h)^{-1}(\cdot) = 0.5 \cdot\,\).
When the dual potentials are solved in correlation form (only in the Sq. Euclidean distance case), the maps are \(\nabla g\) for forward, \(\nabla f\) for backward.
- Parameters
vec (
Array) – Points to transport, array of shape[n, d].forward (
bool) – Whether to transport the points from source to the target distribution or vice-versa.
- Return type
Array- Returns
The transported points.