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Dear authors,

I wanted to reach out and let you know that I found your works on negative sampling for recommender systems incredibly helpful for my own research on the same topic. However, I did encounter two issues while reading this paper, and I'd be grateful if you could provide some clarification on them.

Firstly, in the proof of Lemma 3.1, I had some trouble understanding the meaning of P'. It seems that P' is not a probability distribution over items, given that 1/p_i can be greater than 1. Yet, the set of items J is still sampled from P' as if it were a distribution. I'd be grateful if you could elaborate on the definition of P'.

Secondly, I noticed that the highlighted pop(i) should perhaps be replaced with pop(j)? Additionally, I was unable to deduce how pop(j) or pop(j) could be converted into |B|/|D|. I would appreciate any clarification you could provide on this point as well.

Thank you very much for your time, and I eagerly await your response.

Best regards,
Yatong