# Seg-Eigen-CAM: Eigen-Value-Based Visual Explanations for Semantic Segmentation Models
## 2 Related Work
### Overview of Traditional XAI Methods in Image Classification
#### 2.1.2 Eigen-CAM  
Eigen-CAM is another CAM-based method that does not rely on the backpropagation of gradients. Its core idea is to perform singular value decomposition on an activation map and use the first principal component as the final linear representation. Specifically, Eigen-CAM generates the localization map \(L_{\text{Eigen-CAM}}\), which is computed as follows:  
\[O_{L=k}=W_{L=k}^{T}I \tag{3}\]  
\[O_{L=k}=U\Sigma V^{T} \tag{4}\]  
\[L_{\text{Eigen-CAM}}=O_{L=k}V_{1} \tag{5}\]  
First, the activation map \(I\) is projected onto the \(k\)-th convolutional layer, and \(W_{L=k}\) denotes the combined weight matrix of the first k layers, with dimensions \((m,n)\). Next, SVD is performed on \(O_{L=k}\) to decompose the feature maps into orthogonal components, where the first principal component captures the direction of maximum variance in the activation patterns. Where \(U\) is an orthogonal matrix of size \(m\times m\), and the column of  
Figure 1: Weaknesses of traditional CAM methods. (**a**) Object localization capability in the image, where the generated saliency map, while highlighting the target category (e.g., person), also includes non-target categories, thus reducing the accuracy of object localization. (**b**) Complex backgrounds or high-density target objects in the image, and not all target categories (e.g., persons) are visible. The generated saliency map fails to cover all the target objects.  
are the left singular vectors, \(\Sigma\) is a diagonal matrix of size \(m\times n\) with singular values along the diagonal, and \(V\) is an orthogonal matrix of size \(n\times n\), and the columns of \(V\) are the right singular vectors. Finally, the localization map \(L_{\text{Eigen-CAM}}\) is generated by projecting \(O_{L=k}\) onto the first principal component, where \(V_{1}\) represents the first eigenvector of matrix \(V\).  
As an improved variant of Grad-CAM, Eigen-CAM distinguishes itself by generating visual explanations without relying on backpropagation of gradients, class relevance scores, maximum activation locations, or any other form of feature weighting. Instead, it derives CAMs solely from convolutional layer outputs, enabling stable and reliable visualizations even in scenarios where model accuracy is suboptimal or adversarial noise is present. However, when applied to image segmentation models, Eigen-CAM still faces several challenges, as shown in Figure 1. These challenges stem from the fundamental differences between image classification and segmentation tasks. Image classification models output a single-category prediction, allowing the first principal component analysis to assign higher weights to the most relevant regions. In contrast, image segmentation models must handle spatial judgments between multiple classes simultaneously. This distinction transforms Eigen-CAM's independence from class relevance scores from an advantage into a limitation, reducing its ability to precisely localize target objects of specific categories. Furthermore, SVD calculation introduces sign ambiguity [21; 22; 23; 24]: for each singular vector \(U_{i}\) and \(V_{i}\), their directions can be arbitrarily assigned as positive or negative. Although this does not affect the mathematical correctness of the decomposition, it can hinder the interpretability and comparability of the results. To address this, [PERSON] et al. [24] proposed determining the sign of the singular vector based on the alignment between the singular vector and the data vector, ensuring consistency with the direction of the primary data and thereby improving the interpretability and comparability of the explanations. However, when handling multi-objective tasks, a dedicated sign correction strategy is required to ensure that high-relevance regions in the saliency map accurately correspond to the intended class. This prevents erroneous emphasis on background regions while maintaining a positive correlation between the explanation results and the decision-making process of the model.