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I trained a neural network which makes a regression to a Poincarè Disk Model with radius $r = 1$.

I want to optimize using the hyperbolic distance

$$ \operatorname{arcosh} \left( 1 + \frac{2|pq|^2|r|^2}{(|r|^2 - |op|^2)(|r|^2 - |oq|^2)} \right) $$

where $|op|$ and $|oq|$ are the distances of $p$ and respectively $q$ to the centre of the disk, $|pq|$ the distance between $p$ and $q$, $|r|$ the radius of the boundary circle of the disk and $\operatorname{arcosh}$ is the inverse hyperbolic function of hyperbolic cosine.

But there is a problem

  • In the Poincarè Disk Model with $r = 1$, the distance is defined only for vectors which have norm less than $1$.

  • A neural network does not know this rule, so it can predict vectors with norm greater than $1$.

So, I tried to use the distance defined in a space with $r = 2$, and it works very well for the learning task, but I'm doubtful because the distance doesn't scale in a linear way.

Will there be unwanted effects, in your opinion?

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  • $\begingroup$ Hi! Can you please explain a little bit more in detail the "regression to a Poincarè Disk Model" sentence? $\endgroup$ – nbro♦ May 3 '19 at 21:04
  • $\begingroup$ I don't understand all of the context of your question, but you could just normalize whatever output you get from your neural net. $\endgroup$ – Philip Raeisghasem May 4 '19 at 1:55
  • $\begingroup$ Hi, I have points in R^100, points in H^10 and an unknown function f(R^100 -> H^10) to estimate. H^10 is a Poincarè Disk Model with dimension = 10 and radius = 1 but if I use the hyperbolic distance to evaluate the error on predicted points sometimes is not defined (because predicted points are outside the disk i.e. norm > 1) so I used the hyperbolic distance defined in a space with radius = 2 $\endgroup$ – NooneBug May 4 '19 at 16:06
  • $\begingroup$ To make an example, if we think in Euclidean space and instead of mse (mean squared error) I used mse/2, the collateral effect is that (probably) the learning will be slower because gradient will try to 'correct' the half value of error. But I'm a Computer Scientist and not a Matemathician so I don't know which collateral effects will happens if I don't use the rigth hyperbolic distance in a space, because hyperbolic does not scale in the same way as euclidean. Thanks a lot for the answers by the way $\endgroup$ – NooneBug May 4 '19 at 16:15

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