Eigen-CAM

Mathematical model for Eigen-CAM

Eigen decomposition-based CAM (Eigen-CAM) computes and visualizes the principle components of the final layer convolutional features for generating class activation maps. In other words, Eigen-CAM uses the elements of the highest principle component vector as weights for the linear combination in the CAM equation.

Let’s assume that $\mathcal{F}_l \in \R^{h \times w}$ is the $l^{th}$ feature map obtained from the penultimate convolutional layer ($h$ and $w$ are the height and width of the feature map) and $\mathcal{F}_l(i,j)$ represents the activation of the $(i,j)^{th}$ neuron in the $l^{th}$ feature map (total $L$ feature maps in $\mathcal{F}$).

The algorithm first transforms each feature map $\mathcal{F}_l \in \R^{h \times w}$ into a flattened feature map $\mathcal{F}'_l \in \R^{hw}$. It then combines these feature maps to create a feature matrix $\mathcal{F}' \in \R^{L \times hw}$ as follows: