I have a collection $X$ of 10 million $(x,y,z)$ 3-tuples, where $x$, $y$, and $z$ are all numbers between 0 and 1. The distribution of $x$, $y$, and $z$ values are complex, and the distributions of $x$ vs. $y$ vs. $z$ are not independent.

I want to find some function $f$ that maps each $(x,y,z)$ pair to another $n$-tuple (I don't care about the final arity, let's assume mapping 3-tuples to 3-tuples for now) such that:

  1. $|f(x_i) - f(x_j)| + |f(y_i) - f(y_j)| + |f(z_i) - f(z_j)|$ is a valid distance metric according to the usual rules on any pair of 3-tuples $(x_i, y_i, z_i)$ and $(x_j, y_j, z_j)$

  2. When I take the distribution of the 10 million pairs $\{f(x)\}_{x\in X}$, it is as close as possible to Uniform(0,1) x Uniform(0,1) x Uniform(0,1).

How would I go about this? The closest thing I can think of is a variational autoencoder, but I don't know if that maintains distance properties, and results in a normal and not uniform distribution.

(I don't even know what tags this would fit under - suggestions in comments are welcome!)

  • 1
    $\begingroup$ Your question doesn't typecheck. What is the type signature of $f$? In one sentence you say it maps a 3-tuple to another 3-tuple, i.e., $f:\mathbb{R}^3 \to \mathbb{R}^3$. However when you write $|f(x_i)-f(x_j)|$, that seems to imply that it maps a real number to a real number, i.e., $f:\mathbb{R} \to \mathbb{R}$. So, I am confused about what you are looking for. $\endgroup$ – D.W. Oct 6 '18 at 3:43
  • $\begingroup$ As for your first requirement, if you mean $f:\mathbb{R} \to \mathbb{R}$, then every function $f$ meets the first requirement (i.e., that expression will always form a valid distance metric). $\endgroup$ – D.W. Oct 6 '18 at 3:45
  • $\begingroup$ You mention pairs in multiple places. Do you mean 3-tuples? I don't see any pairs anywhere. Also, what does $f(x)$ mean for $x \in X$? Do you mean $f(x,y,z)$ for $(x,y,z) \in X$? But then is that a 3-tuple or a real number? I find it confusing to use $x$ in one place to refer to a real number and in another place to refer to a 3-tuple. Can you edit your question to correct these points? $\endgroup$ – D.W. Oct 6 '18 at 3:45
  • $\begingroup$ For your second requirement, how do you propose to measure "close"? By what measure? Note that it might not be possible to get close: for example, if all of the $x$-values are 0 and all of the $y$-values are 1, you can't get close to uniform. $\endgroup$ – D.W. Oct 6 '18 at 3:48

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.