Implement kernel SVM

Introduction

日本語 ver

Today, I implement the kernel SVM.
Oputimization is interror point method.
This post is written about Implementation.
I will write Theorem of kernel SVM in another post.
I will put when writing its post finished.

# I finished writing theorem of SVM.
Theorem of SVM part1

My computer is windows.
Also, OS is windows.
I implement by Python3.

Overview

• introduce kernel
• introduce dataset
• result of implementation

kernel

The kernel is the method of solving the nonlinear problem.
kernel is map converting data so that linear can separate class of data.

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Converting data is expressed \(\phi(x)\).
\[\phi:x -> \phi(x)\]
but, kernel function is used \[K(x,y)=\phi(x)^T \phi(y)\] in SVM,
Because, SVM can cumpute only \(\phi(x)^T \phi\).
The famous kernel is RBF and polynomial.

• RBF
  \[K(x,y) = \exp(-\gamma ||x-y||^2)\]
  
• polynomial
  \[K(x,y) = (x^Ty+c)^p\]
  

If \(K(x,y) = x^Ty\) , This is linear SVM.

Dataset

Today, I use following two data generated from normal distribution by np.random.randn.

and

The code making this dataset is upload in github.
code of making dataset

Implementation

I used RBF kernel.
gamma is 0.5
Regularization coefficient is 50.

next, I use RBF kernel.
gamma is 0.5.
Regularization coefficient is 100.

My code is published github.
kernel SVM code

Reference

https://qiita.com/ta-ka/items/e6fd0b6fc46dbab4a651
http://aidiary.hatenablog.com/entry/20100501/1272712699
https://www.amazon.co.jp/%E3%82%B5%E3%83%9D%E3%83%BC%E3%83%88%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E3%83%9E%E3%82%B7%E3%83%B3-%E6%A9%9F%E6%A2%B0%E5%AD%A6%E7%BF%92%E3%83%97%E3%83%AD%E3%83%95%E3%82%A7%E3%83%83%E3%82%B7%E3%83%A7%E3%83%8A%E3%83%AB%E3%82%B7%E3%83%AA%E3%83%BC%E3%82%BA-%E7%AB%B9%E5%86%85-%E4%B8%80%E9%83%8E/dp/4061529064