How is the smooth dice loss differentiable?

Question

I am training a U-Net in keras by minimizing the dice_loss function that is popularly used for this problem: adapted from here and here

def dsc(         


        

Answer 1

Adding smooth to the loss does not make it differentiable. What makes it differentiable is
1. Relaxing the threshold on the prediction: You do not cast y_pred to np.bool, but leave it as a continuous value between 0 and 1
2. You do not use set operations as np.logical_and, but rather use element-wise product to approximate the non-differenetiable intersection operation.

You only add smooth to avoid devision by zero when both y_pred and y_true do not contain any foreground pixels.