Recently, I am reading the paper "Neighbourhood Components Analysis". At the last paragraph on the second page of the paper,

"Notice that by learning the overall scale of A as well as the relative directions of its rows we are also effectively learning a real-valued estimate of the optimal number of neighbours (K). This estimate appears as the

effective perplexityof the distributions $p_{ij}$"

1.Is effective perplexity the perplexity?

2.I really can't see the reason. How may I get the formulation of K in your paper and connect it to effective perplexity please?

3.The equation of KL-divergence on the Euq.(6) seems different as in other books. How should I understand it?

I am so sorry if this is not the right place to ask questions about a paper. I asked this question on StackOverflow as well. I contacted authors of the paper but get no answers so far as usual.

Thank you in advance for your help.