Question

I would like to use the Jensen-Shannon divergence as a histogram distance function. I’m implementing a simple image similarity search, and the histograms are normalized RGB color distributions.

I have a question on the Kullback-Leibler divergence formula (on which JS is based on): what should I return when Pi or Qi are zero?

Here is the implementation in F#:

let dKL p q =
    Array.map2 (fun pi qi -> if pi = 0. then ?   // ?
                             elif qi = 0. then ? // ?
                             else pi * log (pi / qi)) p q
    |> Array.sum

and the Jensen-Shannon distance that uses it:

let dJS p q =
    let m = Array.map2 (fun pi qi -> (pi + qi) / 2.) p q
    (dKL p m) / 2. + (dKL q m) / 2.

Wikipedia says that it should returns 0 when pi=0 and qi>0, and is not defined when qi=0, but for histogram distance it does not make much sense.
What values would make sense in this case?

Edit

here’s the correct version as per Whatang’s answer, for future reference:

let dKL p q =
    Array.map2 (fun pi qi -> if pi = 0. && qi = 0. then 0.
                             else pi * log (pi / qi)) p q
    |> Array.sum

Answer 1

Since you’re using this to build the Jensen-Shannon divergence the only way that you can have qi equal to zero in the calculation of the Kullback-Leibler divergence is if the pi value is also zero. This is because really you’re calculating the average of dKL(p,m) and dKL(q,m), where m=(p+q)/2. So mi=0 implies both pi=0 and qi=0.

Expand the definition of dKL to be p log p - p log m, and use the convention/limit that 0 log 0 = 0 and you’ll see that there’s no problem: m can only be zero when p also is.

To make a long story short, when you call dKL from dJS the second clause elif qi = 0 will never be executed: put whatever you like in there (probably a good idea to make it zero unless you’re going to call dKL from somewhere else).