The regularised cost OT_eps(mu, mu) is not zero, so
OT_eps alone is not a divergence and a generative model trained
on it drifts towards a blurred target. Subtracting half of each
self-cost fixes that: the result is zero exactly when the two measures
coincide.
Usage
Otdiv(a, b, Cab, Caa, Cbb, epsilon, max_iter = 200)
Arguments
- a, b
The two histograms.
- Cab
Cross cost, n by m.
- Caa
Cost of a against itself, n by n.
- Cbb
Cost of b against itself, m by m.
- epsilon
Regularisation strength, positive.
- max_iter
Sinkhorn sweeps.
Value
List with S_eps, OT_ab, OT_aa,
OT_bb, n, m.
Details
Formula: \(S_eps(mu,nu) = OT_eps(mu,nu) - 0.5 (OT_eps(mu,mu) +
OT_eps(nu,nu))\) with \(OT_eps(mu,nu) = <T*,C> + eps KL(T* | mu x
nu)\) – Genevay, Peyre and Cuturi (2018) eq. (3)-(4).
References
Genevay, A., Peyre, G. and Cuturi, M. (2018). Proceedings
of Machine Learning Research 84:1608-1617 (AISTATS).
Examples
x <- 1:4
C <- outer(x, x, function(a, b) (a - b)^2)
a <- c(0.4, 0.3, 0.2, 0.1)
Otdiv(a, a, C, C, C, epsilon = 0.5)$S_eps # 0 for identical measures
#> [1] 0
Otdiv(a, rev(a), C, C, C, epsilon = 0.5)$S_eps
#> [1] 1.178869
