<p>I have a collection <span class="math-container">$X$</span> of 10 million <span class="math-container">$(x,y,z)$</span> 3-tuples, where <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, and <span class="math-container">$z$</span> are all numbers between 0 and 1.  The distribution of <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, and <span class="math-container">$z$</span> values are complex, and the distributions of <span class="math-container">$x$</span> vs. <span class="math-container">$y$</span> vs. <span class="math-container">$z$</span> are not independent.  </p>

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

<ol>
<li><p><span class="math-container">$|f(x_i) - f(x_j)| + |f(y_i) - f(y_j)| + |f(z_i) - f(z_j)|$</span> is a valid distance metric according to the usual rules on any pair of 3-tuples <span class="math-container">$(x_i, y_i, z_i)$</span> and <span class="math-container">$(x_j, y_j, z_j)$</span></p></li>
<li><p>When I take the distribution of the 10 million pairs <span class="math-container">$\{f(x)\}_{x\in X}$</span>, it is as close as possible to Uniform(0,1) x Uniform(0,1) x Uniform(0,1).</p></li>
</ol>

<p>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.</p>

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

CC BY-SA 4.0
2018-10-06T03:43:14.603Z
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.
D.W.
CC BY-SA 4.0
2018-10-06T03:45:15.030Z
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).
D.W.
CC BY-SA 4.0
2018-10-06T03:45:37.487Z
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?
D.W.
CC BY-SA 4.0
2018-10-06T03:48:15.207Z
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.
D.W.
<machine-learning><st.statistics>
Function that maps non-linear distribution to normal distribution while maintaining distance
lightning
# Function that maps non-linear distribution to normal distribution while maintaining distance

(Asked by: username_0 on 2018-10-04)

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!)

**Tags:** machine-learning, st.statistics
> **Comments:**
> **username_1**: 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.  
> **username_1**: 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).  
> **username_1**: 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?  
> **username_1**: 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.  

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## 0 answers

