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Inception score is used to evaluate the generative models. It is a score given based on quality and diversity of images generated.

I have doubt about the range of inception score because of the reason that an article mentions about the possibility of range $[0, \infty]$ and still talks about upper bound in practical setting

The lowest score possible is zero. Mathematically the highest possible score is infinity, although in practice there will probably emerge a non-infinite ceiling. For a ceiling to the IS, imagine that our generators produce perfectly uniform marginal label distributions and a single label delta distribution for each image — then the score would be bounded by the number of labels.

Suppose I have 1000 classes/labels in my task, then is it possible to get an inception score of 2000? Or is it mandatory that the inception score must lie in $[1, 1000]$?

To be concise: Is bounding inception score to a particular range $[1, \text{number of classes}]$ optional or mandatory?

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Yes. You are right. The IS is bound by the number of classes.

This paper titled "A Note on the Inception Score" clearly shows a formal proof of the same. Please head to section 3.3 of the main text for a description and the appendix for the proof.

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