Answers.Body: <p>We have a vector <span class="math-container">$w*$</span> such that it has separated all of the data points. This implies if the correct classification of a point is -1, then <span class="math-container">$w^{*T}x_n$</span> is also negative. If <span class="math-container">$y_n$</span> is positive, then <span class="math-container">$w^{*T}x_n$</span> is positive. Thus is because we've stated every point is correctly classed. Thus <span class="math-container">$\rho&gt;0$</span></p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-09-21T03:59:31.330Z
Answers.LastActivityDate: 2021-09-21T03:59:31.330Z
Answers.UserDisplayName: John Rawls
Body: <p>In the book <a href="https://rads.stackoverflow.com/amzn/click/com/1600490069" rel="nofollow noreferrer" rel="nofollow noreferrer">Learning from Data written</a> (by Abu Mostafa), we have the following exercise:</p>
  <blockquote>
  <p>Let <span class="math-container">$\rho$</span> be minimum attainable from <span class="math-container">$y_n(W^{*T}X_n)$</span> where <span class="math-container">$W^*$</span> is the vector that separates the data. Show <span class="math-container">$\rho &gt; 0$</span>. Also assume the Perceptron Learning Algorithm is initialized with the 0 vector.</p>
  </blockquote>
  <p>How to prove the above statement?</p>
  <p>I thought that it could be negative since a Perceptron function returns either +/-1?</p>
  <p>Even I wonder if I comprehend this proof question correctly.</p>
  
Tags: <machine-learning><proofs><perceptron>
Title: How to show $\rho > 0$ when $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$, where $W^*$ the vector that separates the data?
UserDisplayName: John Rawls
ThreadText: # How to show $\rho > 0$ when $\rho$ be minimum attainable from $y_n(W^{*T}X_n)$, where $W^*$ the vector that separates the data?
  
  (Asked by: username_0 on 2021-09-20)
  
  In the book [Learning from Data written](https://rads.stackoverflow.com/amzn/click/com/1600490069) (by Abu Mostafa), we have the following exercise:
  
  > Let $\rho$ be minimum attainable from $y\_n(W^{\*T}X\_n)$ where $W^\*$ is the vector that separates the data. Show $\rho > 0$. Also assume the Perceptron Learning Algorithm is initialized with the 0 vector.
  
  How to prove the above statement?
  
  I thought that it could be negative since a Perceptron function returns either +/-1?
  
  Even I wonder if I comprehend this proof question correctly.
  
  **Tags:** machine-learning, proofs, perceptron
  
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  ## 1 answers
  
  ### Answer by username_0 on 2021-09-21. Score: 0 | Accepted answer
  
  We have a vector $w\*$ such that it has separated all of the data points. This implies if the correct classification of a point is -1, then $w^{\*T}x\_n$ is also negative. If $y\_n$ is positive, then $w^{\*T}x\_n$ is positive. Thus is because we've stated every point is correctly classed. Thus $\rho>0$
  
  
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