## 3 Evaluation Method
### Global Metrics
For this, we define the projection consistency metric  
\[Cons_{d}=\frac{|\{\mathbf{x}_{i}\in D\mid f(P^{-1}(P(\mathbf{x}_{i})))=f( \mathbf{x}_{i})\}|}{|D|}. \tag{3}\]  
Simply put, \(Cons_{d}\) measures the fraction of so-called consistent projections, i.e., samples, \(\mathbf{x}_{i}\), from a given set \(D\) that keep the same classification label after one round trip given by the projection \(P\) and inverse projection \(P^{-1}\). This metric is also used in DV's evaluation, denoted as \(Q_{data}\)[20]. As for class and map accuracy, we compute \(Cons_{d}^{t}\) and \(Cons_{d}^{T}\) for the training and test sets, respectively. Note that in contrast to \(ACC_{C}\) and \(ACC_{M}\), \(Cons_{p}\) does not use ground-truth labels \(y_{i}\). That is, consistency only assesses how much the decision map can represent a given classifier and not whether the DBM is correct with respect to the ground-truth labels. The range of \(Cons_{d}\) is \([0,1]\), where \(Cons_{d}=1\) indicates perfect consistency. **Map consistency**, \(Cons_{p}\): All the above metrics only evaluate a decision map at locations where an actual data point in \(D\) would project. However, as already noted, most pixels in such a map are not covered by data points. We extend \(Cons_{d}\) to cover all pixels in a map by  
\[Cons_{p}=\frac{|\{\mathbf{p}\in I\mid f(P^{-1}(P(P^{-1}(\mathbf{p}))))=f(P^{-1 }(\mathbf{p}))\}|}{|I|}, \tag{4}\]  
where \(\mathbf{p}\) is a pixel of the decision-map image \(I\). \(Cons_{p}\) calculates the fraction of consistent pixels in the decision map, that is, pixels whose corresponding data points (obtained by inverse projection \(P^{-1}\)) have the same classification label after a round trip of projection \(P\) and inverse projection \(P^{-1}\). Note that \(Cons_{p}\) extends the idea of using the round trip to visually evaluate an inverse projection [25] with a quantitative metric used to evaluate a decision map. \(Cons_{p}\) ranges in \([0,1]\), where \(Cons_{p}=1\) implies perfect pixel-level consistency in a decision map. **Class stability**, \(\bar{S}\): One application of decision maps concerns improving a classifier, for instance, by adding extra labeled training samples by backprojecting selected decision-map pixels. In this case, _multiple_ round trips between the 2D map space and data space occur. Since \(P^{-1}\) is not an exact inverse of \(P\), several such round trips can increasingly accumulate errors. We measure this as follows. Given a pixel in the decision map, we apply \(P^{-1}\) to obtain a data point and then apply the classifier to it. Next, we project this point to the 2D space and repeat the backprojection and labeling process until the class label changes or a maximum number of iterations (set to \(k_{max}=10\) in our experiments) is reached. Let \(S\) be an image recording this maximum iteration count (normalized by \(k_{max}\)), which keeps the class label constant at every map pixel. We then define the class stability \(\bar{S}\) as the average of \(S\) over all pixels, with values in \([0,1]\), where a value of \(\bar{S}\) close to \(1\) indicates a stable decision map with consistent class assignments through multiple \(P\) and \(P^{-1}\) round trips. Conversely, a value of \(\bar{S}\) close to \(0\) suggests an unstable decision map, sensitive to distortions introduced by (inverse) projection, potentially misrepresenting the classifier's decisions. As explained in Section 3.2, we also directly visualize \(S\) to obtain local insights into the class stability over the map.