## 3 Evaluation Method
### Global Metrics  
Global metrics aim to characterize the quality of a decision map with a single (scalar) value, much like the metrics used for direct projections such as trustworthiness, continuity, or normalized stress [36]. We propose five such metrics, as described below (see also Figure 2). **Classifier accuracy, \(ACC_{C}\)**: Accuracy is one of the most common metrics for evaluating the performance of a classifier and is defined as the ratio of correct predictions made by the model \(f\) to the total number of predictions, i.e.,  
\[ACC_{C}=\frac{|\{\mathbf{x}_{i}\in D\mid C(\mathbf{x}_{i})=f(\mathbf{x}_{i}) \}|}{|D|}, \tag{1}\]  
where \(|\cdot|\) indicates the set size, \(D\) is the sample set used for evaluation, and \(C(\mathbf{x}_{i})\) is \(\mathbf{x}_{i}\)'s ground-truth label \(y_{i}\) assigned by the trained model \(f\). We use these notations throughout this section when defining global metrics. In the following, we set \(D\) to either \(D_{t}\) (training set) or \(D_{T}\) (test set) and compute the corresponding classifier accuracies, which we denote as \(ACC_{C}^{\dagger}\) and \(ACC_{C}^{T}\), respectively. The range of \(ACC_{C}\) is \([0,1]\), where \(ACC_{C}=1\) indicates perfect classification. Although accuracy does not, as we already noted, gauge the quality of a decision map, it helps calibrate the understanding of subsequent metrics. For example, if we know a classifier is accurate, we expect its decision map to reflect this accordingly. **Map accuracy, \(ACC_{M}\)**: We define map accuracy as the fraction of data points (of a given dataset \(D\)) that are drawn in the correct decision zones. We define map accuracy as  
\[ACC_{M}=\frac{|\{\mathbf{x}_{i}\in D\mid C(\mathbf{x}_{i})=f(P^{-1}(P(\mathbf{ x}_{i})))\}|}{|D|}. \tag{2}\]  
Intuitively, this says that data points \(\mathbf{x}_{i}\), for which we have ground-truth labels \(y_{i}\), are indeed colored correctly (i.e., by \(y_{i}\)) in the computed decision map. As for class accuracy, we compute \(ACC_{M}^{\dagger}\) and \(ACC_{M}^{T}\) for the training and test sets, respectively. Note that map accuracy only evaluates the map pixels onto which the data in \(D\) projects since for all other pixels, we do not have ground-truth labels. The range of \(ACC_{M}\) is \([0,1]\), where \(ACC_{M}=1\) indicates that all data points are drawn on the pixels with the same color as their ground-truth labels. Another way to evaluate map accuracy is to employ an additional 2D classifier, as used in the DV evaluation (see \(Q_{RNN}\) in [20]). We do not use this option since we believe it introduces an additional degree of complexity in the selection and training of this additional classifier. **Data consistency, \(Cons_{d}\)**: In general, both the direct and inverse projections \(P\) and \(P^{-1}\) unavoidably introduce errors [32; 33; 37; 38]. That is, in general, \(P^{-1}\) is not an exact inverse of \(P\), i.e., \(P^{-1}(P(\mathbf{x}))\
eq\mathbf{x}\), for several data points \(\mathbf{x}\). Such errors can be evaluated using  
Figure 2: Illustration of the metrics used to evaluate decision maps (see Section 3). the mean square error \(MSE=\sum_{\mathbf{x}\in D}\|\mathbf{x}-P^{-1}(P(\mathbf{x}))\|^{2}/|D|\) computed over the set \(D\)[25]. However, for decision maps, the MSE is not the most relevant metric to consider: a pixel \(P(\mathbf{x})\) may be backprojected away from \(\mathbf{x}\), i.e., the MSE may be nonzero; still, if the backprojection has the same label as the original point \(\mathbf{x}\), then there is no visible error in the map. To account for this, we gauge whether the error introduced by the 'round-trip' direct and inverse projections creates inconsistencies in the decision maps.