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Replace W_1 by a Hilbert norm on the difference of the measures

Source: R/otsobm.R
Otsobm.Rd

\(_1\) linearised around a common reference measure is exactly the H^{-1} norm of the difference, and unlike \(_1\) that norm is quadratic: it needs one linear solve, not a linear program, and it embeds into a Hilbert space so kernel methods apply directly. The price is that it is only a first-order approximation.

Usage

Otsobm(mu, nu, Laplace_inv)

Arguments

mu, nu

Two measures on the same support.

Laplace_inv

Inverse (or pseudo-inverse) of the Laplacian, symmetric positive semi-definite.

Value

List with W1_sob, quad_form, mass_gap, n.

Details

Formula: \(W_1(mu,nu) ~ ||mu - nu||_{H^{-1}} = sqrt((mu-nu)' (-Delta)^\{-1\} (mu-nu))\) – Peyre (2018), Section 2

References

Peyre, G. (2018). ESAIM: Control, Optimisation and Calculus of Variations 24(4):1489-1501. doi:10.1051/cocv/2017050 .

Examples

L <- matrix(c(1, -1, 0, -1, 2, -1, 0, -1, 1), 3)    # path-graph Laplacian
Otsobm(mu = c(0.5, 0.5, 0), nu = c(0, 0.5, 0.5), Laplace_inv = MASS::ginv(L))$W1_sob
#> [1] 0.7071068