How to "use" Yolo Loss Function

Question

I am dealing with Yolo Loss Function (the following). $$\begin{align} &\lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(x_i-\hat{x}_i)^2 + (y_i-\hat{y}_i)^2 ] \\&+ \lambda_{coord} \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}[(\sqrt{w_i}-\sqrt{\hat{w}_i})^2 +(\sqrt{h_i}-\sqrt{\hat{h}_i})^2 ]\\ &+ \sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{obj}(C_i - \hat{C}_i)^2 + \lambda_{noobj}\sum_{i=0}^{S^2}\sum_{j=0}^B \mathbb{1}_{ij}^{noobj}(C_i - \hat{C}_i)^2 \\ &+ \sum_{i=0}^{S^2} \mathbb{1}_{i}^{obj}\sum_{c \in classes}(p_i(c) - \hat{p}_i(c))^2 \\ \end{align}$$

I cannot understand how to use it. That is, the "hat" variables are the right ones (related to the training set), while the standard variables are the predicted ones. However, it has some sense if there is a one-to-one corrispondence between the two kind of variables.

But what should happens if my model predicts an airplane while there are not?

I know the predicted center ($x_i,y_i$) and the predicted whidth and height ($w_i,h_i$) but since there are no airplanes what sould be the related hat variables?

Also, what is the meaning of $\hat{p}_i(c)$ since I have the certainess that that object belong to that class? It should be 1?

Thanks in advance :)

Answer 1

YOLOv1 from Scratch explains the loss function design and implementation. e.g. Hat ^ means a predicted value, so $y - \hat{y}$ means subtracting predicted value.

If there is no object in the cell, then the identify function $\mathbb{1}_{ij}^{obj}$ returns 0 and the bounding boxes has no contribution to the loss.

It explains other considerations e.g. questioned in YOLO loss function width and height component explanation.