To understand WGAN better (hopefully maths in it) I followed this blog. While this is a great blog, I still couldn’t understand Fig 5 of it. Para below it states that: “....If we see the values $f(\xi)$ as connected with line segments, this means that the upward and the downward slope of these segments is limited. In our case, where we use Euclidian distances, these slope limits $1$ are $-1$..”
My doubts:
- Why are we seeing the values of $f$ as the slope of a line? I see that EMD is $\mathrm{EMD}(P_r, P_\theta) = \mathbf{f}^T P_r + \mathbf{g}^T P_\theta$. So this EMD (the joint distribution), is like an objective function. And f and g gives it a shape (shape of a surface) as $P_r$ and $P_\theta$ are fixed.
- Now as the difference between $f_i$ and $f_j$ is between $\pm D_{ij}$, and their sum needs to be $\leq D_{ij}$ so we need to limit them. But how is that coming out to be $\pm 1$ in our (the authors) case?
- Also I am not sure what has been explained in Fig 5.
- In Fig 6, we got the values using linprog, which makes sense, but I am not getting an intuition of the shape of $f$ and $g$ as we got them.
Sorry for such naive questions. Any help is highly appreciated. p.s; If this is not the right place to ask this, kindly let me know where can I ask it?