A fixed grid forces the barycentre onto points that may all be far from
the data, which in more than two dimensions is fatal. Letting the atoms
move turns the problem into an alternating scheme – solve the
transports, then move each atom to the weighted average of the mass it
received – which is Lloyd's algorithm with a transport plan in place
of the nearest-point assignment.
Usage
Otbarfree(X_list, weights, n_supp, max_iter = 20)
Arguments
- X_list
List of input point clouds, each n_k by d.
- weights
Barycentric weights; rescaled to sum to one.
- n_supp
Number of support atoms.
- max_iter
Alternations.
Value
List with Y, weights_y, cost,
n_supp, d, K, iters.
Details
Formula: alternate T_k = argmin <T, C(Y, X_k)> and
Y <- n sum_k w_k T_k X_k – Cuturi and Doucet (2014) Section 4.
Uniform weights are kept on the support throughout.
References
Cuturi, M. and Doucet, A. (2014). Proceedings of Machine
Learning Research 32:685-693 (ICML).
Examples
set.seed(9)
X1 <- matrix(rnorm(20, -2), 10, 2); X2 <- matrix(rnorm(20, 2), 10, 2)
b <- Otbarfree(list(X1, X2), weights = c(0.5, 0.5), n_supp = 5)
round(colMeans(b$Y), 2) # near the midpoint (0, 0)
#> [1] 0.13 -0.09
