A two-class dataset that is not linearly separable. The outer ring (cyan) is class ‘0’, while the inner ring (red) is class ‘1’
- Linearly inseparable training data. Unlike the linearly separable data, it is not possible to draw a straight (linear) line to separate the classes.
- Basic SVM would not be able to find a feasible solution here.
- But there is a way to extend the linear approach in this case!
无法用一条直线去区分两个分类区域，基础向量机无法提供可行解，但是存在一种可扩展的线性方法。
Kernel Trick
思路：通过将数据映射到高维空间来获得线性分离
The idea is to obtain linear separation by mapping the data to a higher dimensional space. Let’s see the example first:
> (Left) A dataset in 
, not linearly separable. (Right) The same dataset transformed by the transformation: 
Hyperplane (green plane) that linearly separates two classes in the higher dimensional space.
In the above example, we can train a linear SVM classifier that successfully finds a good decision boundary in 
.
However, we are given the dataset in 
. The challenge is to find a transformation 
, such that the transformed dataset is linearly separable in 
. 难点在于如何将低纬数据向高维转化。
, which after applied to every point in the original tuples yields the linearly separable dataset.
Assuming we have such a transformation 
, the new classification pipeline is as follows.
- First transform the training set
to 
with 
. 先把低维数据向高纬转化。 - Train a linear SVM on
to get a new SVM. 使用高维数据训练支持向量机。 - At test time, a new example
will first be transformed to 
during the testing time. 测试时，样例会被转化为高维。
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