Editing Linear separability (section)

== Number of linear separations ==
Let <math>T(N, K)</math> be the number of ways to linearly separate ''N'' points (in general position) in ''K'' dimensions, then<ref name=":2">{{cite book |last=MacKay |first=David |url=https://books.google.com/books?id=AKuMj4PN_EMC&pg=PA483 |title=Information Theory, Inference and Learning Algorithms |date=2003-09-25 |publisher=[[Cambridge University Press]] |isbn=9780521642989 |page=483 |author-link=David J. C. MacKay}}</ref><math display="block">T(N, K)=\left\{\begin{array}{cc}
2^N & K \geq N \\
2 \sum_{k=0}^{K-1}\left(\begin{array}{c}
N-1 \\
k
\end{array}\right) & K<N
\end{array}\right.</math>When ''K'' is large, <math>T(N, K)/2^N</math> is very close to one when <math>N \leq 2K</math>, but very close to zero when <math>N> 2K</math>. In words, one [[perceptron]] unit can almost certainly memorize a random assignment of binary labels on N points when <math>N \leq 2K</math>, but almost certainly not when <math>N> 2K</math>.