Hi, I'm Jagong from SungKyunKwan University. From this time we learn about support-vector machine through three videos. This is the first video. Today, I want to explain about the linear support vector machine with what each linear support vector machine and learning method with linear support vector machine. Let's think about this. We have a two-dimensional space like this and we also have two classes like blue and red. The question is we want to make a binary classification that means we want to make a decision boundary on here and this would be the candidate of our answer. It's a linear boundary and how about the other one like this. It's also satisfying the linear boundary. All of this candidate would be the linear decision boundary for these corresponding two classes. What would be the best linear decision boundary on here? If we select this one, we had input on the data and on here. In our intuition did unknown data would be qualified to read but based on that decision boundary, it is classified two-dot blue. Which decision boundary would be better for us? I can say obviously the right side. Our question on here is the decision boundary would be better if it had a more in-betweenness class. How to evaluate the in-betweenness of each class. Like this. We can see very easily, the in-betweenness of the left side is poor than the right side. If we can define the margin, the decision boundary would be better if the margin would be maximized on that condition. Let's find the boundary. We maximizing the margins. How can we maximum margin to find it? It's good for human intuition of making classification boundary. One way to find the maximization of the margin is to find the support vectors like this and these vectors can identify the maximum margin of decision boundary. We can know about when we have the decision boundary is on here. The upper of the decision boundary would be class one and the below of the decision boundary would be class zero empirically walking where for the corresponding two classes. Find the maximum margin boundary, is our goal. We can assume this optimal boundary is wx plus b equals zero and we can also get the upper boundary and lower boundary with plus n minus one boundary. With these conditions, what would be the w and the bias b? If we can find the proper w and b, we can find the optimal boundary with this binary classification. What's the margin? We can have a concept. If we had a linear boundary like this, we also can define the plus zone with a plus 1 and minus zone with a minus 1, and the margin would be the distance with the plus zone and minus zone with the perpendicular lines. I want to explain about the sequence to find the margins. First, the vector w is perpendicular to the lines. Let's say a and b are on the line, and we can get this formula from these assumptions. Also, we can say x plus is on the plus line and x minus is the corresponding point on the minus line. It's perpendicular to the lines of this? Yes, we can say. We can do its M, it means margin. We also know about these things with the Lambda values. We can expand these things, and finally, we can make this formula. With these three formulas, we can make the sixth formula like this. With this one, we can almost define the margin with the Lambda values. I can explain the margin is equal of 2 over the distance of vector w and as we defined with the margin is this, and we can also maximize the linear boundary with this assumptions. Based on this assumption the upper on the plus boundary, it's a plus zone and below of this minus zone, and we labeled it as our plus 1 and minus 1. If we can have the plus label, the boundary of the plus boundary would be like this, and if the data sample x minus labeled it also have this formula. We also expand this one with the linear boundary formulas, and then we merge it to the based on the data samples, so the older data sample from 1-n would be merged with that formula. Now, I can formalization with the linear SVM. When the dataset is given like this, we have our input data vector x and the label or with y and number of n data samples down here, we can define the margin like this and we want to maximize it with the 2w and b based on this or we can express this equation by embossing like this. Now I want to summarize of this video. Today, we've already learned about the linear support vector machine. It's from to aim about clarifying digit two classification problem. By learning this linear support vector machine, we define the margin and maximize it, and with this, we know the decision boundary would be fellow on our human intuitions. Thank you.
