Machine Learning: Difference between revisions

Machine Learning (view source)

47 bytes removed , 19 May 2020

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 ; Polynomial Kernel
 * See [[Wikipedia:Polynomial kernel]]
-* <math>K(x,z) = (c+x^Tz)^d</math>
+* \(K(x,z) = (c+x^Tz)^d\)
-* For <math>d=2</math>
+* For \(d=2\), we have
-** we have <math>(1+x^Tz)^2 = 1 + 2(x^Tz) + (x^Tz)^2</math>
+\[
-** <math>= 1 + 2 \sum x_i z_i + (\sum x_i z_i)(\sum x_j z_j)</math>
+\begin{align}
-** <math>= 1 + 2 \sum x_i z_i + 2\sum_{i < j} (x_i x_j) (z_i z_j) + \sum (x_i^2)(z_i)^2</math>
+(1+x^Tz)^2 &= 1 + 2(x^Tz) + (x^Tz)^2\\
-** <math>= 1 + \sum (\sqrt{2}x_i) (\sqrt{2}z_i) + \sum_{i < j} (\sqrt{2} x_i x_j)(\sqrt{2} z_i z_j) + \sum x_i^2 z_i^2</math>
+&= 1 + 2 \sum x_i z_i + (\sum x_i z_i)(\sum x_j z_j)\\
+&= 1 + 2 \sum x_i z_i + 2\sum_{i < j} (x_i x_j) (z_i z_j) + \sum (x_i^2)(z_i)^2\\
+&= 1 + \sum (\sqrt{2}x_i) (\sqrt{2}z_i) + \sum_{i < j} (\sqrt{2} x_i x_j)(\sqrt{2} z_i z_j) + \sum x_i^2 z_i^2
+\end{align}
+\]
 * The dimension of the feature map associated with this kernel is exponential in d
-** There are <math>1+n+\binom{n}{2} + ... + \binom{n}{d} = O(n^d)</math> terms
+** There are \(1+n+\binom{n}{2} + ... + \binom{n}{d} = O(n^d)\) terms
 ==Learning Theory==