## 2 Formalisation and Generalised Working Definition for Ordinal Class Structures
### Feature Space-Based Working Definition for Ordinal Classification Tasks  
As a prior step to our proposed working definition for ordinal classification tasks, we first introduce the term _ordinal arrangement_, which we present in Definition 2. Definition 2 (Ordinal Arrangements): Let \(S\in\mathbb{R}^{c\times c}\) be a symmetric matrix with elements \((s_{i,j})_{i,j=1}^{c}\), with \(c>2\). Matrix \(S\) represents an ordinal arrangement, if and only if, \(\forall i,j,k\in\{1,\ldots,c\}\), it holds that \begin{table}
\begin{tabular}{l l} \hline \hline
**Variable** & **Description** \\ \hline \(X\subset\mathbb{R}^{d}\) & \(d\)-dimensional data set, \(d\in\mathbb{N}\) \\ \(\Omega=\{\omega_{1},\ldots,\omega_{c}\}\) & set of class labels, with \(c>2\), \(c\in\mathbb{N}\) \\ \(I=\{1,\ldots,c\}\) & index set \\ \(\mathcal{T}^{c}\) & set of all permutations \(\tau\) of the set \(I\) \\ \(\mu\in\mathcal{M}^{d}\) & mapping for measuring the level of separability \\ \(\mu_{i,j}\in\mathbb{R}_{\geq 0}\) & level of separability between classes \(\omega_{i}\) and \(\omega_{j}\) \\ \(M^{(\tau)}=(\mu_{\tau(i),\tau(j)})_{i,j=1}^{c}\) & symmetric pairwise separability matrix (PSM) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of applied notations. \[s_{i,j}\geq s_{i,k} \forall j<k\leq i,\] \[\wedge s_{i,j}\leq s_{i,k} \forall i\leq j<k. \tag{6}\]  
Note that the properties of Equation (6) can be summarised as follows. Let \(S\in\mathbb{R}^{c\times c}\), \(c>2\), constitute an ordinal arrangement by Definition 2. Then, the relations between the elements of \(S\) can be symbolically depicted as  
\[S=S^{\mathbb{T}}\widehat{\Xi} \begin{pmatrix}\mathbf{0}&\leq&*&\leq&\cdots& \leq&*\\ *&\geq&\mathbf{0}&\leq&\cdots&\leq&*\\ \vdots&&\ddots&\ddots&\ddots&&\vdots\\ *&\geq&\cdots&\geq&\mathbf{0}&\leq&*\\ *&\geq&\cdots&\geq&*&\geq&\mathbf{0}\end{pmatrix} , \tag{7}\]  
whereby \(S^{T}\) denotes the transpose of \(S\). Note that for each symmetric matrix \(S\in\mathbb{R}^{c\times c}\), \(c>2\), property \(s_{i,j}\geq s_{i,k}\,\forall j<k\leq i\) is equivalent to \(s_{j,i}\geq s_{k,i}\,\forall j<k\leq i\), and statement \(s_{i,j}\leq s_{i,k}\,\forall i\leq j<k\) is equivalent to \(s_{j,i}\leq s_{k,i}\,\forall i\leq j<k\) (which is implied in Equation (7) by including \(S^{\mathbb{T}}\)). Based on the concept of ordinal arrangements introduced above, we provide the working definition for ordinal classification tasks as follows. **Definition 3** (Working Definition for Ordinal Classification Tasks). : _Let \(X_{\Omega}\), \(X\subset\mathbb{R}^{d}\), \(\Omega=\{\omega_{1},\ldots,\omega_{c}\}\), constitute a \(c\)-class classification task, with \(c>2\). Let \(\mu\in\mathcal{M}^{d}\) be an LSM mapping. Furthermore, let \(\mathcal{T}^{c}\) be the set of all permutations of the set \(\{1,\ldots,c\}\) and \(\
u\in\mathcal{T}^{c}\). We denote \(X_{\Omega}\) as feature space-based ordinal (FS-ordinal), with respect to the order \(\omega_{\pm(1)}\prec\ldots\prec\omega_{\pm v(c)}\) and mapping \(\mu\), _if and only if_, \(\forall\tau\in\mathcal{T}^{c}\), _for the corresponding PSMs_, \(M^{(\tau)}\), _it holds that_  
\[M^{(\tau)}\]  
(8) and \[M^{(\tau)}\]  
violates the properties of Equation (6), \[\text{if }\tau\
eq\pm v,\] (uniqueness). \[\bigg{\}}\]  
Note that, as defined in Section 2.1, \(-\
u\in\mathcal{T}^{c}\) denotes the reversed permutation of \(\
u\in\mathcal{T}^{c}\), e.g., \(id:(1,\ldots,c)\mapsto(1,\ldots,c)\) and \(-id:(1,\ldots,c)\mapsto(c,c-1,\ldots,1)\). If the task \(X_{\Omega}\) constitutes an FS-ordinal classification task, with respect to the order \(\omega_{\pm v(1)}\prec\ldots\prec\omega_{\pm v(c)}\) and mapping \(\mu\in\mathcal{M}^{d}\), we simply say that the task \(X_{\Omega}\) is FS-ordinal specific to \((\mu,\pm v)\). Figure 2 illustrates the properties of an ordinal classification task, based on a two-dimensional 5-class toy data set.