4.1.2. Constant Mean Vector with Initial Full Covariance

In the example above, the distributions move along a 2-dimensional manifold parameterized just by the individual variances of the variables. Neither the mean vector, nor the off-diagonal values of the covariance matrices are considered by the Euler-Lagrange equation and therefore remain static. However, starting with a full covariance matrix changes the problem appreciably. For comparison, we will start with same diagonal elements of the covariance matrix but now incorporate the off-diagonal element.

The problem formulation is analogous to Equation (25) subject to the isotropic constraint in Equation (26). Furthermore, we define the initial distribution as

$$
\mu\_0 = [0, 0], \qquad \Sigma\_0 = \begin{bmatrix} \mathcal{T} & -3 \\ -3 & 2 \end{bmatrix} \tag{30}
$$

As seen in Figure 2, including *σ*<sup>12</sup> = −3 alters where the geodesic terminates on the transversality constraint, with the final covariance matrix having diagonal elements of *σ*2 <sup>1</sup> = *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> = 2.24. In Figure 2a, the effects of including the off-diagonal value *σ*<sup>12</sup> are clearly visible on the path of the geodesic causing it to deviate significantly. Figure 2b shows all values of the parameters along the geodesic. As mentioned before, the mean vector remains constant at all intermediate values of the distribution. Unlike before, the value for *σ*<sup>12</sup> must evolve to satisfy the terminal constraint surface, illustrated in Figure 2b and emphasized in Figure 2c.

(**c**) *σ*<sup>12</sup> path

**Figure 2.** In (**a**) is the shortest path (blue line) from a prescribed initial distribution with an offdiagonal covariance element of *σ*<sup>12</sup> = −3. Additionally, the path from (**a**) (dashed black) is shown to illustrate differences in the variances of the terminal distribution. The final distribution, when starting with a full covariance, has *σ*<sup>2</sup> <sup>1</sup> = *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> = 2.24. The red line above is the transversality constraint *σ*2 <sup>1</sup> = *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> , and represents the isotropic submanifold. Figure (**b**) captures the path of all parameters from the initial to the final distribution. Figure (**c**) highlights the values of *σ*<sup>12</sup> for each distribution in the geodesic on the manifold.
