Body: <p>I trained a neural network which makes a regression to a <a href="https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model" rel="nofollow noreferrer">Poincarè Disk Model</a> with radius <span class="math-container">$r = 1$</span>.</p>
  
  <p>I want to optimize using the <em>hyperbolic distance</em> </p>
  
  <p><span class="math-container">$$
  \operatorname{arcosh} \left( 1 + \frac{2|pq|^2|r|^2}{(|r|^2 - |op|^2)(|r|^2 - |oq|^2)} \right)
  $$</span></p>
  
  <p>where <span class="math-container">$|op|$</span> and <span class="math-container">$|oq|$</span> are the distances of <span class="math-container">$p$</span> and respectively <span class="math-container">$q$</span> to the centre of the disk, <span class="math-container">$|pq|$</span> the distance between <span class="math-container">$p$</span> and <span class="math-container">$q$</span>, <span class="math-container">$|r|$</span> the radius of the boundary circle of the disk and  <span class="math-container">$\operatorname{arcosh}$</span> is the <a href="https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions" rel="nofollow noreferrer">inverse hyperbolic function</a> of hyperbolic cosine.</p>
  
  <p>But there is a problem</p>
  
  <ul>
  <li><p>In the Poincarè Disk Model with <span class="math-container">$r = 1$</span>, the distance is defined only for vectors which have norm less than <span class="math-container">$1$</span>.</p></li>
  <li><p>A neural network does not know this rule, so it can predict vectors with norm greater than <span class="math-container">$1$</span>.</p></li>
  </ul>
  
  <p>So, I tried to use the distance defined in a space with <span class="math-container">$r = 2$</span>, and it works very well for the learning task, but I'm doubtful because the distance doesn't scale in a linear way.</p>
  
  <p>Will there be unwanted effects, in your opinion?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-05-03T21:04:25.423Z
Comments.Text: Hi! Can you please explain a little bit more in detail the "regression to a Poincarè Disk Model" sentence?
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-05-04T01:55:01.560Z
Comments.Text: I don't understand all of the context of your question, but you could just normalize whatever output you get from your neural net.
Comments.UserDisplayName: Philip Raeisghasem
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-05-04T16:06:45.587Z
Comments.Text: 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
Comments.UserDisplayName: NooneBug
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-05-04T16:15:16.953Z
Comments.Text: 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
Comments.UserDisplayName: NooneBug
Tags: <machine-learning><math><objective-functions><geometric-deep-learning>
Title: Should I use the hyperbolic distance loss in the case of Poincarè Disk Model?
UserDisplayName: NooneBug
ThreadText: # Should I use the hyperbolic distance loss in the case of Poincarè Disk Model?
  
  (Asked by: username_0 on 2019-05-03)
  
  I trained a neural network which makes a regression to a [Poincarè Disk Model](https://en.wikipedia.org/wiki/Poincar%C3%A9_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](https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions) 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?
  
  **Tags:** machine-learning, math, objective-functions, geometric-deep-learning
  > **Comments:**
  > **username_1**: Hi! Can you please explain a little bit more in detail the "regression to a Poincarè Disk Model" sentence?  
  > **username_2**: I don't understand all of the context of your question, but you could just normalize whatever output you get from your neural net.  
  > **username_0**: 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  
  > **username_0**: 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  
  
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  ## 0 answers
  
  