<h1>ReLU is non-linear by definition</h1>

<blockquote>
  <p>In calculus and related areas, a <a href="https://en.wikipedia.org/wiki/Linear_function" rel="nofollow noreferrer">linear function</a> is a function whose graph is a straight line, that is a polynomial function of degree one or zero.</p>
</blockquote>

<p>Since the graph of the ReLU function <span class="math-container">$f(x) = \max(0,x)$</span> is not a straight line (equivalently, it cannot be expressed in the form <span class="math-container">$f(x) = mx + c$</span>), by definition it is not linear.</p>

<h1>ReLU <em>is</em> piecewise linear</h1>

<p>ReLU <em>is</em> <a href="https://en.wikipedia.org/wiki/Piecewise_linear_function" rel="nofollow noreferrer">piecewise linear</a> on the bounds <span class="math-container">$(-\inf,0]$</span> and <span class="math-container">$[0,\inf)$</span>:</p>

<p><span class="math-container">$$
f(x) = \max(0,x) = \begin{cases}
               0               &amp; x \le 0\\
               x               &amp; x \gt 0\\
       \end{cases}
$$</span></p>

<p>But this is still non-linear on the entire domain:</p>

<p><a href="https://i.sstatic.net/Gg4bV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Gg4bV.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2020-05-26T16:15:50.760Z
2020-05-26T16:34:38.610Z
2020-05-26T16:34:38.610Z
brazofuerte
<p><strong>Short Answer:</strong> Yes</p>

<p><strong>Visually:</strong></p>

<p><a href="https://i.sstatic.net/Yqgclm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Yqgclm.png" alt="enter image description here"></a></p>

<p>if you see the image from <a href="https://en.wikipedia.org/wiki/Linear_function" rel="nofollow noreferrer">wikipedia</a>, it shown that ReLU (the blue line) is non-Linear (the line is not straight, it turns in 0). You can also check "visual" definition of linear function in <a href="https://en.wikipedia.org/wiki/Linear_function" rel="nofollow noreferrer">wikipedia</a>:</p>

<blockquote>
  <p>"In calculus and related areas, a linear function is a function whose graph is a straight line"</p>
</blockquote>

<p><strong>Mathematically:</strong></p>

<p>Linear function of one variable can be defined as: </p>

<p><span class="math-container">$ f(x) = ax + b $</span> </p>

<p>If you plot that function in 2D, it will give you a straight line. Then, the form of linear function with multi variables:</p>

<p><span class="math-container">$ f(x_1, x_2, ..., x_n) = a_1x_1 + a_2x_2 + ... + a_nx_n + b $</span></p>

<p>If you again plot that function in the correct dimension it also give you a straight line. And if you that function carefully, it similar with calculation that happen in a neuron. That's why neuron addition and multiplication is a linear function:</p>

<p><span class="math-container">$ f(x_1, x_2, ..., x_n) = w_1x_1 + w_2x_2 + ... + w_nx_n + b $</span></p>

<p>Adding more layer of linear functions doesn't make the function become "complex" for example, if you have <span class="math-container">$f(x)$</span> like below and then you put another layer of linear function <span class="math-container">$g(x)$</span> on top of it:</p>

<p><span class="math-container">$f(x) = ax + b$</span></p>

<p><span class="math-container">$g(x) = cf(x) + d = cax + cb + d$</span></p>

<p>as the neural network is trained to find the value of <span class="math-container">$a,b,c,d$</span>, we can group the constant from the formula above, and then rewrite to:</p>

<p><span class="math-container">$h(x) = mx + n$</span></p>

<p>with <span class="math-container">$m=ca$</span> and <span class="math-container">$n=cb+d$</span>. So without non-linear function the layer of neural network is useless, it only give you another "simple" linear function</p>

<p>ReLU formula is a <span class="math-container">$f(x)=max(0,x)$</span>, it produces non-linearity as you can't write to linear function format. Using this function will give you "complexity" when you add more layer on top of it.</p>

CC BY-SA 4.0
2020-03-29T02:27:13.590Z
2020-03-29T02:55:43.340Z
2020-03-29T02:55:43.340Z
malioboro
<p>According to <a href="https://www.deeplearning-academy.com/p/ai-wiki-activation-functions" rel="nofollow noreferrer">this blog post</a></p>

<blockquote>
  <p>The purpose of an activation function is to add some kind of non-linear property to the function</p>
</blockquote>

<p>The sigmoid is typically used as an activation function of a unit of a neural network in order to introduce non-linearity.</p>

<p>Is ReLU a non-linear activation function? And why? If not, then why is it used as an activation function of neural networks?</p>

<neural-networks><deep-learning><math><activation-functions><relu>
Is ReLU a non-linear activation function?
slowmonk
# Is ReLU a non-linear activation function?

(Asked by: username_0 on 2020-03-29)

According to [this blog post](https://www.deeplearning-academy.com/p/ai-wiki-activation-functions)

> The purpose of an activation function is to add some kind of non-linear property to the function

The sigmoid is typically used as an activation function of a unit of a neural network in order to introduce non-linearity.

Is ReLU a non-linear activation function? And why? If not, then why is it used as an activation function of neural networks?

**Tags:** neural-networks, deep-learning, math, activation-functions, relu


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## 2 answers

### Answer by username_1 on 2020-05-26. Score: 3

# ReLU is non-linear by definition

> In calculus and related areas, a [linear function](https://en.wikipedia.org/wiki/Linear_function) is a function whose graph is a straight line, that is a polynomial function of degree one or zero.

Since the graph of the ReLU function $f(x) = \max(0,x)$ is not a straight line (equivalently, it cannot be expressed in the form $f(x) = mx + c$), by definition it is not linear.

# ReLU *is* piecewise linear

ReLU *is* [piecewise linear](https://en.wikipedia.org/wiki/Piecewise_linear_function) on the bounds $(-\inf,0]$ and $[0,\inf)$:

$$
f(x) = \max(0,x) = \begin{cases}
0 & x \le 0\\
x & x \gt 0\\
\end{cases}
$$

But this is still non-linear on the entire domain:

[![enter image description here](https://i.sstatic.net/Gg4bV.png)](https://i.sstatic.net/Gg4bV.png)


---

### Answer by username_2 on 2020-03-29. Score: 2

**Short Answer:** Yes

**Visually:**

[![enter image description here](https://i.sstatic.net/Yqgclm.png)](https://i.sstatic.net/Yqgclm.png)

if you see the image from [wikipedia](https://en.wikipedia.org/wiki/Linear_function), it shown that ReLU (the blue line) is non-Linear (the line is not straight, it turns in 0). You can also check "visual" definition of linear function in [wikipedia](https://en.wikipedia.org/wiki/Linear_function):

> "In calculus and related areas, a linear function is a function whose graph is a straight line"

**Mathematically:**

Linear function of one variable can be defined as:

$ f(x) = ax + b $

If you plot that function in 2D, it will give you a straight line. Then, the form of linear function with multi variables:

$ f(x\_1, x\_2, ..., x\_n) = a\_1x\_1 + a\_2x\_2 + ... + a\_nx\_n + b $

If you again plot that function in the correct dimension it also give you a straight line. And if you that function carefully, it similar with calculation that happen in a neuron. That's why neuron addition and multiplication is a linear function:

$ f(x\_1, x\_2, ..., x\_n) = w\_1x\_1 + w\_2x\_2 + ... + w\_nx\_n + b $

Adding more layer of linear functions doesn't make the function become "complex" for example, if you have $f(x)$ like below and then you put another layer of linear function $g(x)$ on top of it:

$f(x) = ax + b$

$g(x) = cf(x) + d = cax + cb + d$

as the neural network is trained to find the value of $a,b,c,d$, we can group the constant from the formula above, and then rewrite to:

$h(x) = mx + n$

with $m=ca$ and $n=cb+d$. So without non-linear function the layer of neural network is useless, it only give you another "simple" linear function

ReLU formula is a $f(x)=max(0,x)$, it produces non-linearity as you can't write to linear function format. Using this function will give you "complexity" when you add more layer on top of it.


---

