## 3 Results
### On Real-world Proteomic Data  
#### Experiment #1: Protein Function Classification  
Data Retrieval and Preprocessing  
The data retrieval process can be summarised as follows:  
1. the entire _Escherichia coli_ (str. K12) list of proteins has been retrieved from UniProt [84]
2. the list has been cross-checked with Protein Data Bank [85] in order to download PDB files for resolved proteins
3. proteins with multiple EC numbers have been discarded
4. in PDB files containing multiple structure models, only the first model is retained; similarly, for atoms having alternate coordinate locations, only the first location is retained. After this retrieval stage, a total number of 6685 proteins has been collected. From this initial set, all proteins without information regarding the measurement resolution have been discarded. Further, in order to consider only good quality structures (i.e., reliable atomic coordinates for building PCNs), all proteins whose measurement resolution is greater than 3A have been discarded as well. The 3A threshold has been selected by jointly considering the PCNs connectivity range and the  
Figure 1: Average accuracy on the test set amongst the dummy classifier, GRALG, WJK and the proposed embedding technique. Results are given in percentage. The colour scale has been normalised row-wise (i.e., for each dataset) from yellow (lower values) towards green (higher values, preferred). measurement resolution distribution within the dataset (Figure 2). This resolution-based filtering dropped the number of available proteins from 6685 to 5583. The classes distribution is summarised in Figure 3.  
Computational Results  
For a thorough investigation, this 7-classes problem has been cast into 7 one-against-all binary problems: the \(i\)th classifier sees the \(i\)th class as positive and all other classes as negatives. In order to take into account the intrinsic random behaviour for both classifiers' training phases, five stratified training-validation-test sets (Proportions: 50% for training set, 25% for validation set and 25% for test set. The stratified splitting thanks to \(\mathcal{L}\) is performed to preserve labels' distribution across splits. have been considered and the same splits are fed to both classifiers in order to ensure a fair comparison. Hereafter the average results across these five splits are shown. Again, the Clique complex has been considered in order to build simplicial complexes for PCNs since (by construction) the underlying graph is already available by scoring edges between \([4,8]\)A. The resulting alphabet size is reported in Table 1.  
Figure 3: Classes distribution within the final 5583 proteins set. Figure 2: Resolution distribution within the initial 6685 proteins set. Proteins with no resolution information are not considered. Tables 2 and 3 show the results on the test set for \(\ell_{1}\)-SVM and \(\
u\)-SVM (respectively) with \(\alpha=1\) and \(\alpha=0.5\) in the fitness function (13): the former case does not foster any feature selection during training (classifiers can choose as many features as they like), whereas the latter equally optimises performances and sparsity in selecting symbols from the alphabet. The rationale behind using \(\ell_{1}\)-SVMs alongside \(\
u\)-SVMs, despite their poor performances on benchmark data, stems from Section 3; by looking at Table 1 it is clear that this is a properly-said high-dimensional problem (converse to benchmark datasets whose maximum number of features reaches 1500), so it is also worth trying linear methods alongside non-linear ones. Performances on the test set are presented via the following parameters: accuracy (ACC), specificity (SPC), sensitivity (SNS), negative predictive value (NPV) and positive predictive value (PPV), along with the sparsity, defined as percentage of non-zero elements in \(\mathbf{w}\) (or \(\boldsymbol{\beta}\)); that is, the number of selected alphabet symbols over the entire alphabet size: the lower, the better.