# Understanding summation notation in the t-SNE paper

I am a graduate student reading classic papers on methodologies. The following is captured at the top of page 6 of the original paper for the t-SNE method (link).

My question is what does the label $l$ stands for? i.e. what are $x_l$ and $y_l$?

• This has nothing to do with t-SNE, it is a question about mathematical notation for sums. Commented Apr 27, 2017 at 13:38
• The $l$ also appears under the summation symbol, this is your clue. Commented Apr 27, 2017 at 14:21
• @MatthewDrury Thank you for your comment. I did notice it is also in summation index. Would you clarify the difference between using l and using i or j at the index? Are we looking at norms for all possible pairs except self-difference? Commented Apr 29, 2017 at 12:07

Think about what you want in the denominator or $p_{ij}$ and $q_{ij}$, you do not want to have the norm of a point with itself (e.g. $||x_2 - x_2||$.) So, we want to find the norm of $x_k$ with all the other possible $x$'s for each value of $k$. Have to call those $x$'s something so we choose $l$ to be that subscript and $l$ has the same range of $k$ everything is perfect with the norms $||x_k - x_l||$ except when $k=l$. So we take the sum over $k \neq l$
Edit: Simply, $i,j,k,l \in\left\{ {1,2...n}\right\}$ where $n = max(i) = max(j) = max(k) = max(l)$