property SinkhornOutput.kl_reg_cost: float#

KL regularized OT transport cost.

This outputs

\[\langle P^{\star}, C \rangle + \varepsilon KL(P^{\star},ab^T) + \rho_a\text{KL}(P^{\star} 1|a) + \rho_b\text{KL}(1^T P^{\star}|b),\]

where \(P^{\star}, a, b\) are the coupling returned by the Sinkhorn algorithm and the two marginal weight vectors, respectively, and \(\rho_a=\varepsilon \tau_a / (1-\tau_a)\) and \(\rho_b=\varepsilon \tau_b / (1-\tau_b)\) are obtained when the problem is unbalanced from parameters tau_a and tau_b. Note that the last two terms vanish in the balanced case, when tau_a==tau_b==1. This quantity coincides with reg_ot_cost, which is computed using dual variables.