I am working my way through this paper and while I have now managed to obtain all the results contained in it, I am stuck with the rather fundamental problem that I do not understand where equation (1) comes from. Specifically, the author states:
The similarity of datapoint $x_j$ to datapoint $x_i$ is the conditional probability, $p_{j|i}$ , that $x_i$ would pick $x_j$ as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at $x_i$. ... Mathematically, the conditional probability $p_{j|i}$ is given by $$p_{j\mid i} = \frac{\exp{\left(-\left\|x_i-x_j\right\|^2/2\sigma_j^2\right)}} {\sum_{k\neq i}\exp\left(-\left\|x_j-x_k\right\|^2/2\sigma_i^2\right)}$$
which makes perfect sense to me at an intuitive level, however my problem is that since I don't come from a statistics background, I am not sure how one actually would develop this result. In particular, the data I am working with is not distributed as a Gaussian, nor can I use a 2-norm to measure distance. Thus, I need to re-formulate $p_{j|i}$ in a manner that is applicable to me, but I don't know how to go about actually doing this.
Any pointers on either how one may go about developing the above equation, or steering me to an appropriate read on the subject would be greatly appreciated.
Thanks!