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Affine-invariant Riemannian (AIRM) distance between two SPD matrices

Source: R/statistical_metrics_monte_carlo_overlap.R
spd_affine_invariant_distance.Rd

AIRM is the affine-invariant Riemannian metric on the cone of symmetric positive-definite matrices: \(d_R(A,B)=\lVert\log(A^{-1/2}BA^{-1/2})\rVert_F\). Do not use the Frobenius \(\lVert A-B\rVert_F\): that Euclidean chord leaves the cone, is not inversion-invariant, and suffers the swelling effect.

Usage

spd_affine_invariant_distance(a, b)

Arguments

a, b

Symmetric positive-definite matrices of equal size.

Value

Non-negative scalar.

See also

compute_average_riemannian()

Examples

# Two bivariate Gaussians: identity vs a correlated SPD covariance.
a <- diag(2)
b <- matrix(c(2, 0.5, 0.5, 1), nrow = 2)
spd_affine_invariant_distance(a, a)
#> [1] 0
d <- spd_affine_invariant_distance(a, b)
d > 0
#> [1] TRUE