Full transport is cubic in the sample size and its gradient touches
every point, which is unusable inside a training loop. Averaging over
minibatches costs a constant per step, but it is a genuinely different
functional: the minibatch loss is not zero between a measure and
itself, and it does not vanish only at equality.
Usage
Otmm(X, Y, batch_size, n_batches, epsilon)
Arguments
- X, Y
Two point clouds.
- batch_size
Points per batch.
- n_batches
Number of batches.
- epsilon
Entropic strength, positive.
Value
List with loss, per_batch, batch_size,
n_batches, n, m, d.
Details
Formula: (1/M) sum_m OT_eps(X_m, Y_m) over M batches –
Fatras et al. (2020) eq. (3). The batches here are consecutive slices,
taken cyclically, so the estimate is deterministic.
References
Fatras, K., Zine, Y., Flamary, R., Gribonval, R. and
Courty, N. (2020). Proceedings of Machine Learning Research
108:2131-2141 (AISTATS).
Examples
set.seed(1)
r <- Otmm(X = rnorm(10), Y = rnorm(10), batch_size = 8L, n_batches = 8L, epsilon = 0.5); TRUE
#> [1] TRUE
