*4.1. Isotropic Terminal Distribution*

Working with the full parameterization of a multivariate Gaussian distribution can be computationally expensive, especially when calculating their Fisher information matrix. For example, the number of parameters grows quadratically with dimension. However, isotropic Gaussian distributions, defined below in Equation (24) grow only linearly with the mean vector's size, orders of magnitude more favorable. In addition to being computationally efficient, isotropic Gaussian distributions provide a submanifold prescribable by a tractable transversality condition. Accordingly, we formulate the following variableendpoint problem: Given a general multivariate Gaussian distribution, what is the closest (in a geodesic sense) isotropic distribution?

Let Σ*<sup>i</sup>* be the covariance matrix of a multivariate Gaussian distribution with *n* mean components. This distribution is *isotropic* if

$$
\Sigma\_{\rm i} = \sigma^2 I\_{\rm n} \tag{24}
$$

where *In* is the *n*-dimensional identity matrix.

Formally, let *θ* capture all the parameters of a multivariate Gaussian distribution according to Equation (13). The functional to minimize is given be

$$\begin{aligned} \min \quad & \mathcal{F}[\theta] = \frac{1}{2} \int\_{\infty}^{\chi\_1} \dot{\theta}^T \mathcal{g}(\theta) \dot{\theta} d\mathbf{x} \\ \theta\_0 &= [\mu\_0, \Sigma\_0] \qquad \quad \quad \quad \theta\_1 = \phi(\mu\_1, \Sigma\_1) \end{aligned} \tag{25}$$

where *μ*<sup>0</sup> and Σ<sup>0</sup> are the known parameters of the starting distribution, but *μ*<sup>1</sup> and Σ<sup>1</sup> are identified by solving the Euler-Lagrange equations while satisfying the transversal condition in Equation (24).

For the bivariate Gaussian distribution, the terminal surface described in Equation (24) can be defined by

$$
\Phi(\sigma\_1^2, \sigma\_2^2) = \sigma\_1^2 - \sigma\_2^2 = 0,\tag{26}
$$

with *μ*<sup>1</sup> free and *σ*<sup>12</sup> = 0. Appling Equation (17) to this surface, we obtain the condition

$$(\sigma\_2^2)^2 \dot{\sigma}\_1^2 + (\sigma\_1^2)^2 \dot{\sigma}\_2^2 + \sigma\_{12}^2 \dot{\sigma}\_2^2 + \sigma\_{12}^2 \dot{\sigma}\_1^2 - 2\sigma\_{12}\sigma\_1^2 \dot{\sigma}\_{12} - 2\sigma\_{12}\sigma\_2^2 \dot{\sigma}\_{12} = 0. \tag{27}$$

Therefore, in addition to the Euler-Lagrange equations in Equation (19) through Equation (23), requiring the final distribution to be isotropic implies the geodesic must also satisfy Equation (27). Additionally, the terminal distribution must satisfy the conditions of constraint surface in Equation (26).

#### 4.1.1. Constant Mean Vector with Initial Isotropic Covariance

In this use case, we introduce a slight modification of the previous scenario, where now we set the mean vector of the initial and final distributions to be *μ*<sup>0</sup> = *μ*<sup>1</sup> = [0, 0]. We are still interested in finding the closed isotropic Gaussian, but now not allowing the curve evolution to move the distribution's mode.

For example, let us assume an initial distribution with

$$
\mu\_0 = [0,0], \quad \Sigma\_0 = \begin{bmatrix} \mathcal{T} & 0 \\ 0 & 2 \end{bmatrix} \tag{28}
$$

and the final distribution lie on the surface defined in Equation (26).

Applying the Euler-Lagrange equation with the transversality conditions to the problem above results in a final isotropic distribution with *σ*<sup>2</sup> = 3.74. Figure 1a shows the information path (dashed and curved) from the initial distribution to the chosen distribution on the isotropic constraint. A Euclidean path would end with a distribution with an isotropic variance equivalent to the average of the original variances. The geodesic path calculated under the Fisher information matrix is an indication of the curvature of the manifold in this region of the parameter space. It is possible in this special zero-mean case to analytically calculate the ending variance on the isotropic constraint surface. Given initial variances of *σ*<sup>2</sup> <sup>1</sup> , *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> , it can be shown that the final variance, *<sup>σ</sup>*<sup>2</sup> *<sup>f</sup>* is given by

$$
\sigma\_f^2 = \sqrt{\sigma\_1^2 \sigma\_2^2}.\tag{29}
$$

In Figure 1b, we can see the evolution of all the parameters, starting from the initial distribution to end. Noteworthy is that, even though *σ*<sup>12</sup> is not required to stay at 0, there is no benefit for it deviating from 0, as seen in Figure 1b. The Fisher information matrix is independent of the mean vector and, since the values of the mean vector are also not part of our isotropic constraint on the final distribution, the mean vector is not compelled to change from the original distribution, justifying the exclusion of the mean vector's path in Figure 1.

**Figure 1.** Shown above in (**a**) is the shortest path (dashed line) from a prescribed initial distribution with diagonal covariance matrix, *σ*<sup>2</sup> <sup>1</sup> = 7, *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> = 2, to the closest isotropic distribution. The final distribution has *σ*<sup>2</sup> <sup>1</sup> = *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> = 3.74. The red solid line above is the transversality constraint *<sup>σ</sup>*<sup>2</sup> <sup>1</sup> = *<sup>σ</sup>*<sup>2</sup> 2 , represents the isotropic submanifold. Additionally illustrated, in blue, is the Euclidean path, which is clearly the straight-line path resulting from an identity metric tensor. In (**b**) are the paths showing the value of each element of Σ as the distributions move towards the transversality constraint.
