Body: <p>I am trying to understand the solution to part 4 of problem 3 from the midterm exam <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-867-machine-learning-fall-2006/exams/midterm_f03soln.pdf" rel="nofollow noreferrer">6.867 Machine learning: Mid-term exam (October 15, 2003)</a>.</p>
  <p>For reproducibility, here is problem 3.</p>
  <blockquote>
  <p>We consider here linear and non-linear support vector machines (SVM) of the form:
  <span class="math-container">$$
  \begin{equation}
  \begin{aligned}
  \min w_{1}^{2} / 2 &amp; \text { subject to } y_{i}\left(w_{1} x_{i}+w_{0}\right)-1 \geq 0, \quad i=1, \ldots, n, \text { or } \\
  \min \mathbf{w}^{T} \mathbf{w} / 2 &amp; \text { subject to } y_{i}\left(\mathbf{w}^{T} \Phi_{i}+w_{0}\right)-1 \geq 0, \quad i=1, \ldots, n
  \end{aligned}
  \end{equation}
  $$</span>
  where <span class="math-container">$\Phi_{i}$</span> is a feature vector constructed from the corresponding real-valued input <span class="math-container">$x_{i}$</span>. We wish to compare the simple linear SVM classifier <span class="math-container">$\left(w_{1} x+w_{0}\right)$</span> and the non-linear classifier
  <span class="math-container">$\left(\mathbf{w}^{T} \Phi+w_{0}\right)$</span>, where <span class="math-container">$\Phi=\left[x, x^{2}\right]^{T}$</span>.</p>
  </blockquote>
  <p>Here is part 4 of the same problem.</p>
  <blockquote>
  <p>In general, is the margin we would attain using scaled feature vectors
  <span class="math-container">$\Phi=\left[2 x, 2 x^{2}\right]^{T}$</span></p>
  <ul>
  <li>greater</li>
  <li>equal</li>
  <li>smaller</li>
  <li>any of the above</li>
  </ul>
  </blockquote>
  <p>The correct answer is the first (<strong>greater</strong>). Why is that the case?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-01-09T07:35:16.547Z
Comments.Text: @DuttaA would you clarify a bit more? I didn't get the point.
Comments.UserDisplayName: user871621
Tags: <machine-learning><classification><support-vector-machine><homework><kernel-trick>
Title: Why is the margin attained with $\Phi=\left[2 x, 2 x^{2}\right]^{T}$ greater than the margin attained with $\Phi=\left[x, x^{2}\right]^{T}$?
UserDisplayName: user871621
ThreadText: # Why is the margin attained with $\Phi=\left[2 x, 2 x^{2}\right]^{T}$ greater than the margin attained with $\Phi=\left[x, x^{2}\right]^{T}$?
  
  (Asked by: username_0 on 2021-01-08)
  
  I am trying to understand the solution to part 4 of problem 3 from the midterm exam [6.867 Machine learning: Mid-term exam (October 15, 2003)](https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-867-machine-learning-fall-2006/exams/midterm_f03soln.pdf).
  
  For reproducibility, here is problem 3.
  
  > We consider here linear and non-linear support vector machines (SVM) of the form:
  > $$
  > \begin{equation}
  > \begin{aligned}
  > \min w\_{1}^{2} / 2 & \text { subject to } y\_{i}\left(w\_{1} x\_{i}+w\_{0}\right)-1 \geq 0, \quad i=1, \ldots, n, \text { or } \\
  > \min \mathbf{w}^{T} \mathbf{w} / 2 & \text { subject to } y\_{i}\left(\mathbf{w}^{T} \Phi\_{i}+w\_{0}\right)-1 \geq 0, \quad i=1, \ldots, n
  > \end{aligned}
  > \end{equation}
  > $$
  > where $\Phi\_{i}$ is a feature vector constructed from the corresponding real-valued input $x\_{i}$. We wish to compare the simple linear SVM classifier $\left(w\_{1} x+w\_{0}\right)$ and the non-linear classifier
  > $\left(\mathbf{w}^{T} \Phi+w\_{0}\right)$, where $\Phi=\left[x, x^{2}\right]^{T}$.
  
  Here is part 4 of the same problem.
  
  > In general, is the margin we would attain using scaled feature vectors
  > $\Phi=\left[2 x, 2 x^{2}\right]^{T}$
  >
  > * greater
  > * equal
  > * smaller
  > * any of the above
  
  The correct answer is the first (**greater**). Why is that the case?
  
  **Tags:** machine-learning, classification, support-vector-machine, homework, kernel-trick
  > **Comments:**
  > **username_0**: @DuttaA would you clarify a bit more? I didn't get the point.  
  
  ---
  
  ## 0 answers
  
  