I was referring to this paper here related to permutohedral lattice. I didn't get what this remainder $k$ point means when defining the permutohedral lattice.
I am beginner. Can anyone provide me any pointers?
I am confused at this line
$ A_{d^*}= \bigcup_{k=0}^{d}\{\vec{x}\in H_d\ |\ \vec{x}\text{ is a remainder-$k$ point}\} $
we call $\vec{x}\in H_d$ a remainder-$k$ point for some $k \in \{0,\ldots,d\}$ iff all coordinates are congruent to $k$ modulo $d +1$.
I am having difficulty in visualizing it.
Well I got this example of the lattice for $d=2$. However, I didn't get how they plotted the points. The points are in the $Z^3$ space. I cannot actually get how they plotted the points $(2,-1,-1)$ and $(1,1,-2)$ in this space. I suppose they are wrongly placed or something. I am not sure though. I mean suppose the normal 3D space. point $(2,-1,-1)$ should be somewhere else isn't it?
Here is something that I drew for 2D space, where $H_d$ is a line, and I have drawn the remainder-$k$ points with their remainder.