Body: <p>I am new to machine learning. I am reading <a href="http://me-ramesh.blogspot.com/p/machine-learning.html" rel="nofollow noreferrer">this blog post</a> on the VC dimension.</p>
  
  <p><span class="math-container">$\mathcal H$</span> consists of all hypotheses in two dimensions <span class="math-container">$h: R^2 → \{−1, +1 \}$</span>, positive inside some square boxes and negative elsewhere.</p>
  
  <p>An example.</p>
  
  <p><a href="https://i.sstatic.net/cKgKt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cKgKt.png" alt="enter image description here"></a></p>
  
  <p>My questions:</p>
  
  <ol>
  <li><p>What is the maximum number of dichotomies for the 4 data points? i.e calculate mH(4)</p></li>
  <li><p>It seems that the square can shatter 3 points but not 4 points. The <span class="math-container">$\mathcal V \mathcal C$</span> VC dimension of a square is 3. What is the proof behind this?</p></li>
  </ol>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-07T17:23:17.490Z
Comments.Text: "What is the maximum number of dichotomies for the 4 data points?", but in the picture you're showing, there are 5 data points (or numbers).
Comments.UserDisplayName: nbro
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2019-11-08T02:17:25.473Z
Comments.Text: The picture is just an example showing one of the possibilities of data points distributions. For point INSIDE the square, it is +1, For point OUTSIDE the square, it is -1.  My question is asking 4 points, it is correct and unrelated to 5 points in my drawing.
Comments.UserDisplayName: Rain
Tags: <machine-learning><computational-learning-theory><vc-dimension>
Title: What is the maximum number of dichotomies in a square?
UserDisplayName: Rain
ThreadText: # What is the maximum number of dichotomies in a square?
  
  (Asked by: username_0 on 2019-11-04)
  
  I am new to machine learning. I am reading [this blog post](http://me-ramesh.blogspot.com/p/machine-learning.html) on the VC dimension.
  
  $\mathcal H$ consists of all hypotheses in two dimensions $h: R^2 → \{−1, +1 \}$, positive inside some square boxes and negative elsewhere.
  
  An example.
  
  [![enter image description here](https://i.sstatic.net/cKgKt.png)](https://i.sstatic.net/cKgKt.png)
  
  My questions:
  
  1. What is the maximum number of dichotomies for the 4 data points? i.e calculate mH(4)
  2. It seems that the square can shatter 3 points but not 4 points. The $\mathcal V \mathcal C$ VC dimension of a square is 3. What is the proof behind this?
  
  **Tags:** machine-learning, computational-learning-theory, vc-dimension
  
  > **Comments:**
  > **username_1**: "What is the maximum number of dichotomies for the 4 data points?", but in the picture you're showing, there are 5 data points (or numbers).  
  > **username_0**: The picture is just an example showing one of the possibilities of data points distributions. For point INSIDE the square, it is +1, For point OUTSIDE the square, it is -1.  My question is asking 4 points, it is correct and unrelated to 5 points in my drawing.  
  
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  ## 0 answers
  
  