**2. Related Works**

Most efforts towards measuring distances on statistical manifolds build on the foundation started by Fisher in [17], in which he introduces the idea of the information matrix. In [18] Kullback and Leibler published a pioneering effort to describe this distance. Works such as [19,20], endowed statistical distributions with geometrical properties. However, it was Rao [21] that expanded on the ideas of Fisher that defined a metric for statistical models based on the information matrix. The information matrix satisfies the conditions of a metric on a Riemannian statistical manifold, and is widely used because of its invariance [22]. This connection between distance and distributions encouraged others to explore the distance between specific families of distributions [23]. Among these families include special cases of the multivariate normal model [24], the negative binomial distribution [25], the gamma distribution [26,27], Poisson distribution [28], among others.

In [29], the authors offer a detailed exploration of geodesics on a multivariate Gaussian manifold. They show that there exists a geodesic connecting any two distributions on a Gaussian manifold. However, a closed-form solution for the most general case remains an open problem.

In [30] and expanded on in [31], the authors offer a very detailed discussion, focusing primarily on the univariate normal distribution for which a closed-form solution for the Fisher–Rao distance is known. Here, the authors focus on a geometrical approach, abandoning the "proposition-proof" format offered in previous research. With this geometric approach, closed-form solutions to various special cases are derived: univariate Gaussian distributions, isotropic Gaussian distributions, and Gaussian distributions with diagonal covariance matrix.

Another novel application of geodesics on a Gaussian statistical manifold is explored in [32], where the authors use information geometry for shape analysis. Shapes were represented using a *K*-component Gaussian Mixture Model, with the number of components being the same for each shape. With this, each shape occupied a unique point on a common statistical manifold. Upon mapping two shapes to their points on this manifold, the authors use an iterative approach to calculate the geodesic between these two points, with the length of the geodesic offering a measure of similarity of the shapes. Furthermore, because of the iterative approach to solving for the geodesic, all intermediate points along path are revealed. These points can be mapped to their own unique shapes, essentially showing the evolution from one shape to another. This shape deformation exhibits the benefit of

analyzing more than just the distance between points on a manifold and that "walk" along the path has real substance.

In [33], the authors explore the complexity of Gaussian geodesic paths, with the ultimate goal of relating the complexity of a geodesic path on a manifold to the correlation of the variables labeling its macroscopic state. Specifically, the authors show that, if there is a dependence between the variables, the complexity of the geodesic path decreases. Complexity, for these purposes is defined as the volume of the manifold traversed by the geodesic connecting a known initial state to a future state, which is well defined. It is shown that this volume decays by a power law at a rate that is determined by the correlation between the variables on the Gaussian manifold.

In [34], the authors use the geometry of statistical manifolds to study how the quantum characteristics of a system are affected by its statistical properties. Similar to our work, the authors prescribe an initial distribution on the manifold of Gaussians and examine the geodesics emanating from it, without dictating a specific terminating distribution. The authors show that these paths tend to terminate at distributions that minimize Shannon entropy. However, unlike our work, these paths are free to roam on the manifold and are not required to terminate on a specific surface on the manifold. Furthermore, the most relevant part of the author's work considers only univariate Gaussians with a two-dimensional parameter manifold, without ever considering higher dimensions.

Though we have chosen to work with Riemannian geometry, it is worth mentioning that information geometry often employs dualistic geometries that can be established using divergence measures. In [35], the authors detail the use of divergence measures to obtain the dual coordinates for space of multivariate Gaussians. However, they point out that the choice of divergence measure is not unique and resulting geometries lack the same interpretative power of the natural parameterization.

Though these previous works operate in the space of multivariate Gaussians and deriving geodesics therein, they all require defining the initial and terminal distributions on the manifold. In this work, we address a novel problem of finding the geodesics when the terminal conditions are hypersurface constraints rather than a single point. Technically, these transversality conditions are variable boundary conditions placed on the initial and final distributions requiring them to reside on a parametric surface typically defined by constraining the coordinates. The usefulness of these variable boundary conditions has emerged in many areas including physics [36] in which the author studied wetting phenomenon on rough surfaces and in [37], where the authors studied the elasticity of materials. Additionally, in [8,38,39], transversality conditions were employed in economic optimal control problems with a free-time terminal condition. However, as practical as transversality conditions have been in the above fields, their application in information geometry literature is deficient.

### **3. Geodesics for Fixed-Endpoint Problems**

We begin by briefly developing standard calculus of variation results for discovering the shortest path between fixed points on a differentiable manifold. Then, the result is applied to the space of multivariate Gaussians, with detailed derivations for the case of bivariate Gaussians. We finally extend the formulation to show how to incorporate variable-endpoint boundary conditions.
