2023

Run video full screen to see the difference between Fisher-Rao geodesics (black, computationally time-consuming using geodesic shooting) and our proposed projected SPD geodesic curve which can be calculated fast (green). https://www.mdpi.com/1099-4300/25/4/654

Geodesic shooting from standard normal with initial value conditions

Let ℙ(d) denote the set of symmetric positive?definite (SPD) d×d matrices and (d) denote the set of multivariate normal distributions:

The Fisher-Rao distance between two normals N(μ_{1},Σ_{1}) and N(μ_{2},Σ_{2}) is the geodesic Riemannian distance on
the manifold (,g^{Fisher}) induced by the Fisher information metric:

where

and ds^{Fisher}(t) := _{c(t)} is the Fisher-Rao length element. The inner product ⟨v_{1},v_{2}⟩_{N} for
v_{1},v_{2} ∈ T_{N} at normal N is the called the Fisher-Rao norm (with tangent planes T_{N} is identified to
ℝ^{d} × Sym(d) where Sym(d) be the set of d × d symmetric matrices). The statistical model (d) is of dimension
m = dim(Λ(d)) = d + = and identifiable: there is a one-to-one correspondence λ ↔ p_{λ}(x) between
λ ∈ Λ(d) and N(μ,Σ) ∈(d).

- When d = 1, the Fisher-Rao distance is known in closed form:
where Δ(a,b;c,d) = is a Möbius distance and arctanh(u) := log for 0 ≤ u < 1. The Fisher-Rao geodesics are semi-ellipses with centers located on the x-axis:

- When the normal distributions belongs to the same submodel
_{μ}= {N(μ,Σ) : Σ ∈(d)}⊂ of normal distributions sharing the same mean μ, we have:_{i}(M) denotes the i-th generalized largest eigenvalue of matrix M, where the generalized eigenvalues are solutions of the equation |Σ_{1}- λΣ_{2}| = 0. The submanifold (_{μ},g^{Fisher}) is totally geodesic in (,g^{Fisher}). - When the normal distributions belongs to the same submodel
_{Σ}= {N(μ,Σ) : Σ ∈(d)}⊂ of normal distributions sharing the same covariance matrix Σ we havewhere Δ

_{Σ}is the Mahalanobis distance:

However, in the general case, the Fisher-Rao distance between normals is not known in closed form.

Calvo and Oller show how to embed N(μ,Σ) ∈(d) = into a SPD matrix of ℙ(d + 1):

so that the manifold ((d),g^{Fisher}) is isometrically embedded into the submanifold (,g^{trace}) of the cone
equipped with the trace metric

However, the submanifold ⊂ ℙ(d + 1) is not totally geodesic. Thus Calvo and Oller derived a lower bound on the Fisher-Rao distance:

which is also metric distance.

Our method consists in projecting the SPD geodesic γ_{ℙ(d+1)}(_{1}^{-1}_{2}) onto and then maps back the SPD
projected curve into by using f^{-1}:

Indeed, the geodesic γ_{ℙ(d+1)}(_{1}^{-1}_{2}) has closed-form equation

Now, we need to estimate the Fisher-Rao length of the curve c_{CO}(N_{1},N_{2};t) by discretizing the curve at T
positions:

and approximate for nearby normals their Fisher-Rao distances by the square root of their Jeffreys divergence:

where

The projection of a SPD matrix P ∈ ℙ(d + 1) onto is done as follows: Let β = P_{d+1,d+1} and write
P = . Then the orthogonal projection at P ∈ onto is:

and the SPD distance between P and _{⊥} is

Here are some examples of the curves c_{CO} (in green) compared to the Fisher-Rao geodesics (in
black):

More details and quantitative analysis: https://www.mdpi.com/1099-4300/25/4/654