Machine learning Techniques-5-kernel Logistic Regression

5-kernel Logistic Regression

Last class, we learnt on soft margin and its application. Now, a new idea comes to us, could we

Apply the kernel trick to our old Frirend logistic regression?

Firstly, let's review those four concepts of margin handling:

As we can see, the differences between "hard" and "Soft" are showed from constant C, which are a bit similar to Regularizati Mnl

Since we define a new factor calledξ, we can use MAX function smoothly express the margin violation:

Thus the unconstrained form of Soft-margin SVM:

We can easily find the form of this subject are similar to L2 regularization.

However, there is no QP formation and the function of Max may leads to some place not differentiable.

Thus We get the idea, which apply SVM as a regularization model:

For the regularization FORM SVM, a larger C means the smaller influence from WTW, and which is the regularization factor.

Then, a comparition about error would be given,

The SVM error has different appearance with the middle point, which we called hinge error measure.

Now for this binary classification, could logreg and SVM is joint?

Because we know the advantage of SVM, which is able to simplify the computing by kernel, while the Logreg holds some other Benefits.

Here we apply the Platt ' s Scaling https://en.wikipedia.org/wiki/Platt_scaling

Which is found to being a nice method to better the binary problem.

We caculate the transforming of the SVM to get the W and B, and we all tool to find best A and B.

In conclusion, the structure of our demand are like that:

We want to use KERNEL, we need wt*z (to package into KERNEL), we need linear combination of Zn

Optimal W is represented by Zn:

Since the w| | Can is prpved to is the only subitem in W:

It can be proved, the l2-regularized linear model can be kernelized.

So, here we get a new represention called Kernel Logistic Regression (KLR),

There is something we should pay attention:

1. The dimention of this issue are subject to N of samples.

2. Theβcan is seen as a description toward the relationship between Xn and any other points in X space.

3.βn could not being zero, which means larger computing cost compared to the process of finding good w.

Machine learning Techniques-5-kernel Logistic Regression