<p>I don't think there's any rationale behind the usage of the letter <span class="math-container">$z$</span> to denote the noise (which sometimes is also denoted by <span class="math-container">$\epsilon$</span> in other contexts), apart from the fact that <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are already being used and that the letters <span class="math-container">$x$</span>, <span class="math-container">$y$</span>, <span class="math-container">$z$</span> and <span class="math-container">$w$</span> are often used to denote variables in mathematics. In particular, in machine learning, <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are often used to denote the inputs and outputs (or labels) respectively, while <span class="math-container">$w$</span> often denotes the parameters (although <span class="math-container">$\theta$</span> is also used for that).</p>
<p>In other words, it's just a convention (e.g. <span class="math-container">$z$</span> is also used to denote the hidden variable in <a href="https://arxiv.org/pdf/1312.6114.pdf" rel="nofollow noreferrer">the VAE paper</a>). Even if it wasn't and someone used <span class="math-container">$z$</span> as a mnemonic letter or for some particular reason, I wouldn't lose too much time on this issue, as another author may very well use any other letter to refer to the same concept. It's important to be a bit flexible when it comes to notation in mathematics, otherwise, you may easily get lost.</p>

CC BY-SA 4.0
2021-08-02T10:06:07.757Z
2021-08-02T10:11:13.663Z
2021-08-02T10:11:13.663Z
nbro
<p>Most of the notations in Artificial Intelligence are borrowed from the mathematics.</p>
<p><span class="math-container">$x$</span> stands for input (vector), <span class="math-container">$y$</span> stands for output (vector) etc., and the list is long.</p>
<p>But, I am not sure whether <span class="math-container">$z$</span> has any (widely used) role in mathematics.</p>
<p>Is there any reason behind the usage of letter <span class="math-container">$z$</span> to represent a noise vector?  Or is it just selected randomly without any reason?</p>

CC BY-SA 4.0
2021-08-01T06:56:11.363Z
Letter in mathematical discplines are often chosen at random, maybe zzzzz - resembled the noise to the inventor)
spiridon_the_sun_rotator
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2021-08-01T11:59:37.990Z
Please try to comment whether down-votes reflect the low quality of the question or off-topic-ness of the question or some other. It will be helpful for me and the community.
hanugm
<history><notation><etymology><noise>
Why is noise vector represented by letter $z$?
hanugm
# Why is noise vector represented by letter $z$?

(Asked by: username_0 on 2021-08-01)

Most of the notations in Artificial Intelligence are borrowed from the mathematics.

$x$ stands for input (vector), $y$ stands for output (vector) etc., and the list is long.

But, I am not sure whether $z$ has any (widely used) role in mathematics.

Is there any reason behind the usage of letter $z$ to represent a noise vector? Or is it just selected randomly without any reason?

**Tags:** history, notation, etymology, noise

> **Comments:**
> **username_1**: Letter in mathematical discplines are often chosen at random, maybe zzzzz - resembled the noise to the inventor)  
> **username_0**: Please try to comment whether down-votes reflect the low quality of the question or off-topic-ness of the question or some other. It will be helpful for me and the community.  

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## 1 answers

### Answer by username_2 on 2021-08-02. Score: 3

I don't think there's any rationale behind the usage of the letter $z$ to denote the noise (which sometimes is also denoted by $\epsilon$ in other contexts), apart from the fact that $x$ and $y$ are already being used and that the letters $x$, $y$, $z$ and $w$ are often used to denote variables in mathematics. In particular, in machine learning, $x$ and $y$ are often used to denote the inputs and outputs (or labels) respectively, while $w$ often denotes the parameters (although $\theta$ is also used for that).

In other words, it's just a convention (e.g. $z$ is also used to denote the hidden variable in [the VAE paper](https://arxiv.org/pdf/1312.6114.pdf)). Even if it wasn't and someone used $z$ as a mnemonic letter or for some particular reason, I wouldn't lose too much time on this issue, as another author may very well use any other letter to refer to the same concept. It's important to be a bit flexible when it comes to notation in mathematics, otherwise, you may easily get lost.


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