A covariance is not just an ellipse
Same endpoints. Different geometry.
Compare square-root Frobenius and Bures interpolation. Each outline is a unit-Mahalanobis covariance ellipse: its principal radii are square roots of eigenvalues, oriented along the eigenvectors.
0 · covariance Acovariance B · 1
01
Square-root Frobenius
Interpolate the symmetric square roots
Σroot(t)
02
Bures–Wasserstein
Align the factors, then interpolate
ΣB(t)
A B current t t = 0.1, …, 0.9 Equal axes · shared, time-independent scale
Rotate red B; blue A keeps its axes fixed.
Change both ellipses’ axis ratio; 1 gives circles.
droot(A, B)
dB(A, B)
droot / dB
‖AB − BA‖F / (‖A‖F‖B‖F)