ott.geometry.costs.RegTICost.h_transform#
- RegTICost.h_transform(f)[source]#
Compute the h-transform of a concave function.
Return a callable \(f_h\) defined as:
\[f_h(x) = \min_y h(x - y) - f(y)\]This is equivalent, up to a change of variables, \(z = x - y\), to define
\[\begin{split}\min_z h(z) - f(x - z). \\ \min_z h(z) + \tilde{f}(z, x).\end{split}\]where \(\tilde{f}(z, x) := -f(x - z)\).
By contrast to the
TICost.h_transform
, this is exploiting the structure ofh()
and requires having access to its proximal operator \(\prox_{\tau h}(x)\). Under the hood, this is solved usingproximal gradient descent
.