## 1 Introduction
Recently, a number of methodologies have been proposed for WI-SV that adhere to the assumption that the signature descriptors comply with geometrically curved spaces [52; 53; 54; 55; 56]. Specifically, the signature descriptors, in the form of signature covariance matrices, are considered to be points that belong to the Symmetric Positive Definite (SPD) \(\mathbf{P}_{d}\) manifold, a subset of the native vector space \(\mathbb{R}^{d\times d}\). Therefore, inspired by (a) the use of the DT on signature verification and (b) the success of applying geometrically inspiring algorithms which exploit curved spaces, this work provides a comprehensive mathematical framework regarding the use of an equivalent expression of the Dichotomy Transform in curved and non-Euclidean spaces. The objective of this work is to clearly show that simple Riemannian approaches, compared to computationally intense verifiers, can provide low verification errors as well. Bearing in mind that we consider the term dissimilarity to relate to the Dichotomy Transform, by means of the subtraction operator and not to the distance between two entities, the main contributions of this paper are below:  
1. We provide a mathematical framework for modeling the leap from the Euclidean-oriented DT to its Riemannian equivalent, Riemannian dissimilarity vectors (RDVs) \(\Psi\), which are employed for the first time in the literature for addressing WI-SV. The Euclidean-oriented DT is typically performed with the use of the subtraction operator \((-)\) between vector descriptors. In the context of the Riemannian framework, the concept of a bipoint, i.e., oriented pairs of points, which is an antecedent of a vector, offers a novel perspective on the interpretation of subtractions [57]. As a result,the proposed RDVs \(\Psi\) are formed as the result of the Riemannian equivalent for vector space subtraction between two SPD manifold entities \(\Psi\leftarrow(\mathbf{X}-\mathbf{Y})_{\mathbf{\mathcal{M}}}\). The RDVs \(\Psi\) are expressed by a manifold dissimilarity function, \(\Psi\leftarrow\Psi_{\mathcal{M}}(\cdot,\cdot):\,P_{d}\times P_{d}\rightarrow \mathcal{T}_{P_{d}}\), which is a map to the tangent bundle of the SPD manifold. For classification tasks, the resulting RDVs (in the form of symmetric matrices) are converted to a vectored from \((v_{\mathbf{X},\mathbf{Y}}\,\,\text{or}\,\,dv_{\mathbf{X},\mathbf{Y}})\) with the use of a vector operator \(vec_{(I)}(\Psi)\). 2. We present and compare two alternative methodologies for constructing the RDVs between two signature covariance matrices, \(\mathbf{X},\mathbf{Y}\), namely the local and the global common pole approach. In the case of the local approach, the RDV is formed by the local tangent vectors \(\Psi_{\mathbf{X},\mathbf{Y}}\,\,\epsilon T_{\mathbf{X}}\). Intuitively, the local RDV approach encodes the notion of the dissimilarity between \(\mathbf{X},\mathbf{Y}\) by means of an appropriate subtraction operation \((\mathbf{X}-\mathbf{Y})_{\mathbf{\mathcal{M}}}\rightarrow\Psi_{\mathbf{X},\mathbf{Y}}\,\,\epsilon T _{\mathbf{X}}\). In the case of the global common pole approach, the \(\mathbf{I}_{d}\) common pole is used to evaluate the RDV dissimilarity \((\mathbf{I}-\mathbf{X})_{\mathbf{\mathcal{M}}}\rightarrow\Psi_{\mathbf{L}\mathbf{X}}\,\,\epsilon T _{\mathbf{I}}\), \((\mathbf{I}-\mathbf{Y})_{\mathbf{\mathcal{M}}}\rightarrow\Psi_{\mathbf{L}\mathbf{Y}}\,\,\epsilon T _{\mathbf{I}}\) for each one of the \(\mathbf{X},\mathbf{Y}\) with respect to the identity matrix \(\mathbf{I}_{d}\). Then, the Euclidean-based DT, applied directly to the \(\Psi_{\mathbf{L}\mathbf{X}}\), \(\Psi_{\mathbf{I}\mathbf{Y}}\), evaluates the global common pole RDV \(\Psi_{\mathbf{X},\mathbf{Y}}^{\mathbf{V}}=|\Psi_{\mathbf{I},\mathbf{X}}-\Psi_{\mathbf{I},\mathbf{Y}}|\). To both local and global common pole RDVs, we then apply the \(vec_{(I)}(\Psi)\) operator in order to create any of the two \(v\) or \(dv\) vectored forms for classification purposes. 3.