Answers.Body: <p>We are looking for coefficients <span class="math-container">$\alpha_1, \alpha_2$</span> and <span class="math-container">$\alpha_3$</span> such that
  <span class="math-container">$$
  \max(\max(a,b), c)=\alpha_1 \max(a, b) + \alpha_2 \max(a, c) +\alpha_3 \max(b, c) 
  $$</span>
  for all <span class="math-container">$a,b,c \in \mathbb R$</span>.
  Let's consider some simple cases.</p>
  <p>First, suppose <span class="math-container">$a=b=c=1$</span>. We then have
  <span class="math-container">$$
  1 = \max(\max(1,1), 1)=\alpha_1 \max(1, 1) + \alpha_2 \max(1, 1) +\alpha_3 \max(1, 1) = \alpha_1 +\alpha_2 +\alpha_3.
  $$</span>
  Now suppose <span class="math-container">$a=b=1$</span> and <span class="math-container">$c=2$</span>, we then have
  <span class="math-container">$$
  2=\max(\max(1,1), 2)=\alpha_1 \max(1, 1) + \alpha_2 \max(1, 2) +\alpha_3 \max(1, 2) = \alpha_1 +2\alpha_2 +2\alpha_3.
  $$</span>
  Hence, <span class="math-container">$\alpha_1 = 0$</span>. Note that we could've set <span class="math-container">$a=c=1$</span> and <span class="math-container">$b=2$</span> as well. So by the same argument, we must have <span class="math-container">$\alpha_2=0$</span>. And <span class="math-container">$\alpha_3 =0$</span>...</p>
  <p>So no, we cannot express the formula as linear combination of pairwise max functions.</p>
  
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Answers.CreationDate: 2024-03-27T00:37:32.543Z
Answers.LastActivityDate: 2024-03-27T00:37:32.543Z
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Body: <p>Like the title says, is there any way to simplify the piece wise linear function of max(max(a,b),c) into some linear combination of max(a,b), max(b,c), max(a,c)?</p>
  
Tags: <neural-networks><linear-algebra>
Title: Is it possible to simplify max(max(a,b),c)?
UserDisplayName: Xiaoyang Li
ThreadText: # Is it possible to simplify max(max(a,b),c)?
  
  (Asked by: username_0 on 2024-03-26)
  
  Like the title says, is there any way to simplify the piece wise linear function of max(max(a,b),c) into some linear combination of max(a,b), max(b,c), max(a,c)?
  
  **Tags:** neural-networks, linear-algebra
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2024-03-27. Score: 1
  
  We are looking for coefficients $\alpha\_1, \alpha\_2$ and $\alpha\_3$ such that
  $$
  \max(\max(a,b), c)=\alpha\_1 \max(a, b) + \alpha\_2 \max(a, c) +\alpha\_3 \max(b, c)
  $$
  for all $a,b,c \in \mathbb R$.
  Let's consider some simple cases.
  
  First, suppose $a=b=c=1$. We then have
  $$
  1 = \max(\max(1,1), 1)=\alpha\_1 \max(1, 1) + \alpha\_2 \max(1, 1) +\alpha\_3 \max(1, 1) = \alpha\_1 +\alpha\_2 +\alpha\_3.
  $$
  Now suppose $a=b=1$ and $c=2$, we then have
  $$
  2=\max(\max(1,1), 2)=\alpha\_1 \max(1, 1) + \alpha\_2 \max(1, 2) +\alpha\_3 \max(1, 2) = \alpha\_1 +2\alpha\_2 +2\alpha\_3.
  $$
  Hence, $\alpha\_1 = 0$. Note that we could've set $a=c=1$ and $b=2$ as well. So by the same argument, we must have $\alpha\_2=0$. And $\alpha\_3 =0$...
  
  So no, we cannot express the formula as linear combination of pairwise max functions.
  
  
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