## 3 Comparison to Previous Work and Additional Theoretical Outcomes
### FS-Ordinal versus SVM-Ordinal Structures  
As already mentioned above, our first approach to provide a working definition for ordinal class structures was based on pairwise resubstitution accuracies of linear SVM models. In [21], we justified the choice for such a working definition and discussed its potential, in combination with a numerical evaluation. Moreover, we also discussed the limitations of an SVM-based definition of ordinal structures. More precisely, we showed that in the case of a 3-class classification task, in which all of the possible class pairs are linearly separable, the provided detection algorithm (see Figure 3) always fails to find SVM-ordinal class structures. However, what we did not consider in [21] is that the detection algorithm already fails if two out of the three possible class pairs of a 3-class classification task are linearly separable, as we will show shortly. This is due to the fact that the (resubstitution) accuracy is bounded to 1 (100%). As an illustrative example, let us consider a two-dimensional 3-class toy data set that is depicted in Figure 4.  
Figure 3: Detectionof FS-ordinal structures. If the given task \(X_{\Omega}\) constitutes an FS-ordinal classification task, then the output includes exactly two permutations, which represent the ordinal structure of the current task (This figure is adapted from our previous work [21]). From Figure 4, we can observe that the data points from class \(\omega_{1}\) are linearly separable from the data points from both remaining classes \(\omega_{2}\) and \(\omega_{3}\), respectively. Moreover, the data points from class \(\omega_{2}\) are not linearly separable from the data points from class \(\omega_{3}\), in the provided two-dimensional feature space. However, the toy data set clearly constitutes an ordinal class structure with the order \(\omega_{1}\prec\omega_{2}\prec\omega_{3}\), and its reversed order \(\omega_{3}\prec\omega_{2}\prec\omega_{1}\). Note that for the definition of SVM-ordinal class structures, the PSM originates from the resubstitution accuracies between the corresponding class pairs. As the class pairs \((\omega_{1},\omega_{2})\) and \((\omega_{1},\omega_{3})\) are linearly separable, it follows that \(\mu_{1,2}=\mu_{1,3}=1\), and due to the symmetry (see Equation (4) for the definition of \(\mu_{i,j}\)), \(\mu_{2,1}=\mu_{3,1}=1\). Moreover, for the class pair \((\omega_{2},\omega_{3})\), it clearly holds that \(\mu_{2,3}=\mu_{3,2}=\alpha\), with \(\alpha\in(0,1)\). Now, let us define \(\tau\in\mathcal{T}^{3}\) as \(\tau:(1,2,3)\mapsto(1,3,2)\), and thus \(-\tau:(1,2,3)\mapsto(2,3,1)\). Then, for the PSMs, with \(\mu\in\mathcal{M}^{2}\) defined as the resubstitution accuracy based on linear SVMs, we obtain the following matrices:  
\[M^{(id)}= \begin{pmatrix}\omega_{1}&\omega_{2}&\omega_{3}&\omega_{3}&\omega_{ 2}&\omega_{1}\\ 0&1&1&1\\ 1&0&\alpha\\ 1&\alpha&0\end{pmatrix} ,\quad M^{(-id)}= \begin{pmatrix}0&1&1\\ 1&0&\alpha\\ 1&\alpha&0\end{pmatrix} ,\quad\text{and}\] \[M^{(\tau)}= \begin{pmatrix}\omega_{1}&\omega_{3}&\omega_{2}&\omega_{2}&\omega _{3}&\omega_{1}\\ 0&1&1&1\\ 1&0&\alpha\\ 1&\alpha&0\end{pmatrix} ,\quad M^{(-\tau)}= \begin{pmatrix}0&1&1\\ 1&0&\alpha\\ 1&\alpha&0\end{pmatrix} .\]  
Obviously, both matrices \(M^{(id)}\) and \(M^{(\tau)}\), and therefore their corresponding reversed counterparts, \(M^{(-id)}\) and \(M^{(-\tau)}\), fulfil the properties of Equation (6), and thus constitute ordinal arrangements by Definition 2. Thus, the uniqueness property of Definition 3 is violated and no (SVM-)ordinal class structure is found. The main issue for not detecting the obvious ordinal structure of the 3-class toy data set depicted in Figure 4 is the choice of the LSM mapping \(\mu\). More precisely, as we briefly discussed above, the resubstitution accuracy is bounded to the value 1. Therefore, for 3-class classification tasks, the SVM-based working definition fails in cases where at least two of the three possible class pairs are linearly separable.