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I was referring to this paper related to permutohedral lattice.

It states that the permutohedral lattice

$ A_{d^*}= \bigcup_{k=0}^{d}\{\vec{x}\in H_d\ |\ \vec{x}\text{ is a remainder-$k$ point}\} $

Where we call $\vec{x} \in H_d$ a remainder-$k$ point for some $k \in \{0, \dots , d\}$ iff all coordinates are congruent to $k$ modulo $d+1$.

I didn't get what this remainder-$k$ point means or being congruent to $k$ module $d+1$ means. I am just a beginner. I didn't get how the permutohedral lattice equation came to be like that in the end. Any suggestions?

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    Although the linked article refers to Gaussian filtering of data, both the article and the question appear to me to be more suitable for the mathematics site. I might be wrong but I see a lot of mathematics here but I don't see statistical, data mining or data visualization as a major part of it. My recommendation would be to migrate it to Math.2012-09-07

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Although its 5 years from when the question was asked, I hope the solution is useful.

As given in the paper, a point $x=\{x_0,x_1,...x_d\} \in R^{d+1}$ is said to be reminder-$k$ point in a given $(d+1)$ dimensional space, if and only if $ x_i\%(d+1) = k \quad \forall \quad i\in \{0,1,...,d\} $ This means that the remainder when you divide any coordinate $x_i$ of the point $x$ by the dimension $(d+1)$ of the space, is $k$.

For eg:-

Let $d=2$ (So, the space is $R^3$).

Let $x=\{1,1,-2\}$ ( i.e. $x_0=1,x_1=1,x_2=-2$).

Now, $x_i \% 3 = \{1,1,1\} \quad for \quad i\in \{0,1,2\}$.

Hence, it is a Remainder-1 point in $R^3$ space. Another example is $x=\{3,6,9\}$, which is a Remainder-0 point in $R^3$ space.