## 4 Experimental Setup
### The Models--Verifiers  
Two popular binary classifiers, a hard-margin SVM and a Gentle Ada Boost Boosting Feature Selection algorithm (BFS), along with a Decision Stump Committee (DSC-BFS) [28; 37; 73; 74], were employed in order to independently build the WI signature verifier. Detailed operation of both the SVM and the DSC-BFS classifiers for both WD and WI signature verification can be found in the literature [28; 37; 75]. Both models operate under a similar training-validation protocol which will identify their optimal operating parameters. Algorithms 1 and 2 provide an overview of the basic functionality during the learning stage by means of two standard algorithms. In summary, sets of \(v_{\{\mathcal{S},\mathcal{S}\}}^{+}\) or \(dv_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\) vectors, which correspond to the training set \(\mathbb{T}_{\mathbb{R}}:\omega_{\mathbb{T}_{\mathbb{R}}}^{+}\) and \(\omega_{\mathbb{T}_{\mathbb{R}}}^{-100\forall RF}\) (or \(\omega_{\mathbb{T}_{\mathbb{R}}}^{-0\forall RF}\)), are fed to the classifiers, followed by the model training stage. Then, the validation set \(\mathbb{V}:\omega_{\mathbb{V}}^{+}\) and \(\omega_{\mathbb{V}}^{-100\forall RF}\) (or \(\omega_{\mathbb{V}}^{-0\forall RF}\)) in terms of the related \(v_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\) or \(dv_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\) is enabled for evaluation of the associated scores \(s\_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\) (or \(s\_{\mathcal{M}}_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\)) returned by the models. The optimal operating parameters are the ones that are associated with the maximum value of the Area under the Curve (AUC), a property of the Receiver Operating Characteristic (ROC) curve [76] of the \(s\_{\mathcal{V}}_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\) (or \(s\_{\mathcal{M}}_{\{\mathcal{S},\mathcal{S}\}}^{\pm}\)), evaluated only with the validation set \(\mathbb{V}\). The Learning Stage: the basic algorithms--SVM (Algorithm 1) and DSC-BFS (Algorithm 2). ```
0: The \(\omega^{+}=\{\omega^{+}_{\mathbb{T}_{R}},\omega^{+}_{V}\}\) and \(\omega^{-}=\{\omega^{-}_{\mathbb{T}_{R}},\omega^{-}_{V}\}\) by means of \(v^{\pm}\) or \(dv^{\pm}\)
0: VVM model \(\Delta_{SVM}(C_{opt},\,\gamma_{opt})\) with the optimal parameters for hard margin \(C_{opt}\) and kernel scale \(\gamma_{opt}\) BEGIN
1: FOR: A grid search on \(C\), \(\gamma\)
2: USE: \(\omega^{+}_{\mathbb{T}_{R}}\), \(\omega^{-}_{\mathbb{T}_{R}}\) by means of their vectored forms: \(v^{\pm}_{\mathbb{T}_{R}}\) or \(dv^{\pm}_{\mathbb{T}_{R}}\)
3: TRAIN: The current model \(\Delta_{SVM}(C,\gamma)\) with \(v^{\pm}_{\mathbb{T}_{R}}\) or \(dv^{\pm}_{\mathbb{T}_{R}}\)
4: USE: \(\omega^{+}_{V}\), \(\omega^{-}_{V}\) by means of their vectored forms: \(v^{\pm}_{V}\) or \(dv^{\pm}_{V}\)
5: EVAL: Scores s_\(v^{\pm}_{V}\) or \(s\_dv^{\pm}_{V}\) with current \(\Delta_{SVM}(C,\gamma,v^{\pm}_{V})\)
6: PERFORM: ROC analysis with \(s\_v^{\pm}_{V}\) or \(s\_dv^{\pm}_{V}\). 7: EVAL: \(AUC(C,\gamma)\) from ROC analysis
8: end_FOR
9: RETURN: \(\Delta_{SVM}(C_{opt},\,\gamma_{opt})\) which corresponds to the \(max(AUC)\). END
```  
**Algorithm 1**Learning a WI-SV verifier with the SVM. ```
0: The \(\omega^{+}=\{\omega^{+}_{\mathbb{T}_{R}},\omega^{+}_{V}\}\) and \(\omega^{-}=\{\omega^{-}_{\mathbb{T}_{R}},\omega^{-}_{V}\}\) by means of \(v^{\pm}\) (or \(dv^{\pm}\))
0: DSC-BFS model \(\Delta_{\text{DSC-BFS}}\big{(}T_{opt}\big{)}=sign\Big{[}\sum_{t=1}^{T_{opt}}f_ {t}(v^{\pm})\Big{]}\) with \(T_{opt}\) the optimal number of leafs. SET:\(T_{H}\): The early stopping criterion. \(T_{L}\): Maximum number of iterations.