高维空间中,点的超平面的距离
首先来看看三维的:
如下图所示,在三维空间中,假设平面\(U=\{x \in R^3|0=W^T·x+b, b \in R, W \in R^2\}\),\(W\)为\(U\)的法向量;平面外一点\(X\),\(U\)内一点\(X'\),那么\(X\)到\(U\)的距离为:

然后我们来看看高维的:
在\(n\)维空间中,假设平面\(U=\{x \in R^n|0=W^T·x+b, b \in R, W \in R^n \}\),\(W\)为\(U\)的法向量;平面外一点\(X \in R^n\),\(U\)内一点\(X'\),那么\(X\)到\(U\)的距离为:
\[\begin{aligned}
L &= \frac{|W·(X-X')|}{|W|} \\
\because 0 &= W^T·x+b, x \in U,X' \in U \\
\therefore L &= \frac{|W·X-W·X'|}{|W|}\\
&= \frac{|W·X+b|}{|W|}
\end{aligned}
\]