# Separable kernel vs depthwise separable convolution

I would like to compare separable kernel with depthwise separable convolution. In the first case (eg for 2D) we have:

$$y[m, n] = h[m, n] * x[m, n] = (h_1[m] \cdot h_2[n]) * x[m, n] = \\ h_1[m] * (h_2[n] * x[m ,n])$$

So the factorization of the kernel introduces the possibility of breaking one convolution into two separate ones. In this article, we have the opportunity to learn about the depthwise separable convolution. In equations (2) and (3) we can see something very similar to the above-stated factorization but containing the mysterious compound operation instead of the multiplication. What is the operation and is it possible to break equations (2) and (3) into independent convolutions as in the case above ?