## 3 Materials and Methods
### Theoretical Elements and Mathematical Tools of the SPD Manifold  
The symmetric positive definite manifold \(\mathbf{P}_{d}\) is the set of all real matrices \(\{\mathbf{X}\in\mathbb{R}^{d\times d}\}\) such that the matrix is symmetric \(\mathbf{X}-\mathbf{X}^{\prime}=0\) and strictly positive, \(\mathbf{\upsilon}^{\prime}\mathbf{X}\mathbf{\upsilon}>0,\forall\upsilon\in\mathbb{R}^{d} \smallsetminus\{0^{d}\}\). Intuitively, points that belong to the SPD manifold are contained within the interior of a convex cone in a \(n(n+1)/2\) dimensional Euclidean space. The tangent space \(\mathcal{T}_{\mathbf{X}}\) at any manifold point \(\mathbf{X}\in\mathbf{P}_{d}\) is the set of symmetric matrices \(\Psi\in\{S_{d}\}\), and it comprises all the possible derivatives (i.e., the tangent vectors) on the manifold at \(\mathbf{X}\); therefore, \(\mathcal{T}_{\mathbf{X}}\) is a vector space. The SPD is a Riemannian manifold; thus, it is a differentiable manifold equipped with a smoothly varying inner product \(\left\langle\cdot\right\rangle_{\mathbf{X}}\). The geometric perspective of the SPD manifold is endowed with the use of a Riemannian metric (or norm) of a tangent vector \(\Psi\in\mathcal{T}_{\mathbf{X}}\), defined by \(\left\|\Psi\right\|_{\mathbf{X}}^{2}=\left\langle\Psi,\Psi\right\rangle_{\mathbf{X}}\). On the SPD manifold and its tangent plane on point \(\mathbf{X}\), two mappings are defined: the first is the exponential map \(\exp_{\mathbf{X}}(\Psi):\mathcal{T}_{\mathbf{X}}\to P_{d}\), in which a tangent vector, \(T\), with origins in a manifold point (or pole), \(\mathbf{X}\), is projected back on the manifold; the second is the logarithmic map, \(log_{\mathbf{X}}(\mathbf{T})=\exp_{\mathbf{X}}^{-1}(\mathbf{T}):P_{d}\to\mathcal{T}_{\mathbf{X}}\), which projects a manifold point, \(\mathbf{T}\), to the local tangent plane of the pole, \(\mathbf{X}\), by means of its local tangent vector, \(T\). The logarithmic map is considered to be the equivalent \((-)_{\mathcal{M}}\) of the Euclidean subtraction operator \((-)\)[57]. A Riemannian metric is defined as a smoothly varying inner product on the tangent space at each point of the manifold. The Riemannian metric is of particular interest since it is employed to measure arc lengths upon the manifold. A curve of zero acceleration that connects two points, \(\mathbf{X}\) and \(\mathbf{Y}\), in the SPD manifold is called a geodesic, and it is analogous to straight lines in \(\mathbb{R}^{d}\), while its length is called the geodesic distance. The geometric perspective of the SPD manifold is often endowed with the use of the related Affine Invariant Riemannian Metric (AIRM) [57], defined for \(\mathbf{X}\in\mathbf{P}_{d}\) and \(Y,W\in\mathcal{T}_{\mathbf{X}}\) as follows:  
\[\left\langle Y,W\right\rangle_{\mathbf{X}}\triangleq\left\langle\mathbf{X}^{-1/2}Y\bm {X}^{-1/2},\mathbf{X}^{-1/2}W\mathbf{X}^{-1/2}\right\rangle=Tr(\mathbf{X}^{-1}Y\mathbf{X}^{- 1}W) \tag{1}\]  
which induces the notion of a distance, formally termed geodesic distance \(\delta_{g}(\cdot,\cdot)\): \(P_{d}\times\mathbf{P}_{d}\to\mathbb{R}^{+}\), between the manifold points \(\mathbf{X},\mathbf{Z}\in\mathbf{P}_{d}\) as  
\[\delta_{g}(\mathbf{X},\mathbf{Z})=\left\|\mathbf{logm}(\mathbf{X}^{-1/2}\mathbf{X}^{-1/2})\right\| _{F} \tag{2}\]  
where \(logm\) is the matrix logarithm function expressed by  
\[logm(\mathbf{X})=\mathbf{U}diag(\log\left(\lambda_{\mu}\right))\mathbf{U}^{\prime} \tag{3}\]  
and \(\lambda_{\mu}\) is the \(\mu\)-th eigenvalue, \(\mu=1:d\), derived from the eigenvalue analysis of \(\mathbf{X}=\mathbf{U}diag\left(\lambda_{\mu}\right)\mathbf{U}^{\prime}\). The geodesic distance \(\delta_{g}\) is considered to be the most popular distance measure in the SPD manifold. In addition, a number of metrics or divergences can also be defined over the SPD manifold. In this work, we also exploit two additional symmetric Bregman divergences to act as distance measures between two points on the SPD manifold.