Question

Let us compute the t-SNE output for the isoceles triangle as in the previous problem.

Recall we are given 3 points in 2 dimensions, as shown in the figure below:


\mathbf{x}^{(1)} the center node , and \mathbf{x}^{(2)} and \mathbf{x}^{(3)} the boundary nodes. The 2 boundary nodes are at distance A to the center node, and are farther away from each other, at distance B (B>A) than they are from the center node. Here, we are given A=1, and B=\sqrt{2}.
The distribution \mathbf{P} in 2D remains the same as before, and for this small example, we will again try to minimize \text {KL}(\mathbf{P}||\mathbf{Q}) by solving for \text {KL}(\mathbf{P}||\mathbf{Q})=0.

What is different from before is that we will use the t-distribution to define \mathbf{Q} in the target space.

Assume again that the projection onto 1D is as follow with unknown distance a between the center and boundary nodes, but a distance 2a between the boundary nodes.


Write down q_{ij} in terms of the distance a and find a such that q_{12}=p_{12}, giving \text {KL}(\mathbf{P}||\mathbf{Q})=0.
(Enter an answer accurate to at least 3 decimal places.)

a=\quad