<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>

CC BY-SA 4.0
2021-09-21T03:59:31.330Z
2021-09-21T03:59:31.330Z
John Rawls
<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>

<machine-learning><proofs><perceptron>
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?
John Rawls
# 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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