## 3 Results
### On Benchmark Data
Alongside GRALG and WJK, the accuracy of the dummy classifier is also included [82]: the latter serves as a baseline solution and quantifies the performance obtained by a purely random decision rule. Indeed, the dummy classifier outputs a given label, say \(l_{i}\) with a probability related to the relative frequency of \(l_{i}\) amongst the training patterns and, by definition, does not consider the information carried out by the pattern descriptions (input domain) in training data. In Figure 1, the accuracy on the test set is shown for the five competitors: the dummy classifier, WJK, GRALG and the proposed embedding procedure using both non-linear \(\
u\)-SVM and \(\ell_{1}\)-SVM. In order to take into account intrinsic randomness in stratified training/test splitting and in genetic optimisation, the results presented here have been averaged across five different runs. Clearly, for the tested datasets, a linear classifier performs poorly: it is indeed well-known that especially for high-dimensional datasets non-linear and linear methods have comparable performances [31, 83]. As a matter of fact, for these datasets, Peking_1 lead to the largest embedding space (approx. 1500 symbols), followed by MSRC_9 (approx. 220 symbols). From Figure 1, it emerges that WJK is generally the best performing method, followed by the proposed embedding procedure with \(\
u\)-SVM which is, in turn, followed by GRALG. Indeed, WJK exploits the entire simplicial complexes to the fullest, by considering only simplices belonging to the two simplicial complexes to be matched and without 'discarding' any simplices due to the explicit (and optimised) embedding procedure, as proposed in this work. Amongst the three methods, WJK is also the fastest to train: the kernel matrix can be pre-evaluated using very fast vectorised statements and the only hyperparameter that needs to be tuned is the \(\
u\)-SVM regularisation term, which can done by performing a plain random search in \((0,1]\). Amongst the two information granulation-based techniques, the proposed system outperforms GRALG in the vast majority of the cases. This not only has to be imputed to the modelling capabilities offered by hypergraphs but also has a merely computational facet: the number of simple paths is much greater than the number of simplices (A graph with \(n\) vertices has \(\mathcal{O}(n! )\) paths, whereas the number of cliques goes like \(\mathcal{O}(3^{n/3})\)), hence GRALG needs a 'compression stage' (i.e., a clustering procedure) to return a feasible number of alphabet symbols. This compression stage not only may impact the quality of the embedding procedure, but also leads to training times that are incredibly high with respect to the proposed technique in which simplices can be interpreted as granules themselves. Another interesting aspect that should be considered for comparison relies on the model interpretability. Despite WJK seems the most appealing technique due to high training efficiency and remarkable generalisation capabilities, it basically relies on pairwise evaluations of a positive-definite kernel function between pair of simplicial complexes which can then be fed into a kernelised classifier. This _modus operandi_ does not make the model interpretable and no knowledge discovery phase can be pursued afterwards. The same is not true for Granular Computing-based pattern recognition systems such as GRALG or the one proposed in this paper, as will be confirmed in Section 4.