<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>

CC BY-SA 4.0
2019-11-07T17:23:17.490Z
"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).
nbro
CC BY-SA 4.0
2019-11-08T02:17:25.473Z
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.
Rain
<machine-learning><computational-learning-theory><vc-dimension>
What is the maximum number of dichotomies in a square?
Rain
# 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

