The equation of the hyperplane Why do we use the hyperplane equation w'x instead of y=ax+b? For two reasons 1. it is easier to work in more than two dimensions with this notation 2. the vector w will always be normal to the hyperplane 1/27/2021 PATTERN RECOGNITION
The equation of the hyperplane Why do we use the hyperplane equation wTx instead of y=ax+b? 1/27/2021 PATTERN RECOGNITION 11 For two reasons: 1. it is easier to work in more than two dimensions with this notation, 2. the vector w will always be normal to the hyperplane
What is a separating hyperplane? We could trace a line and then all the data points representing men will be above the line, and all the data points representing women will be below the line Such a line is called a separating hyperplane, or a decision boundary o Women R 8 165 Size(cm) 1/27/2021 PATTERN RECOGNITION
What is a separating hyperplane? We could trace a line and then all the data points representing men will be above the line, and all the data points representing women will be below the line. Such a line is called a separating hyperplane, or a decision boundary 1/27/2021 PATTERN RECOGNITION 12
What is a separating hyperplane? An hyperplane is a generalization of a plane in one dimension an hyperplane is called a point o in two dimensions it is a line in three dimensions, it is a plane in more dimensions you can call it an hyperplane ::E-F L G---H--I--J 1/27/2021 PATTERN RECOGNITION
What is a separating hyperplane? An hyperplane is a generalization of a plane. ◦ in one dimension, an hyperplane is called a point ◦ in two dimensions, it is a line ◦ in three dimensions, it is a plane ◦ in more dimensions you can call it an hyperplane 1/27/2021 PATTERN RECOGNITION 13
Compute signed distance from a point to the hyperplane We have an hyperplane, which separates two group of data The equation of the hyperplane is wx=0 W W is the unit normal vector We would like to compute the distance between the point a and the hyperplane C x+b=0 1/27/2021 PATTERN RECOGNITION
Compute signed distance from a point to the hyperplane We have an hyperplane, which separates two group of data The equation of the hyperplane is 𝒘𝑇𝒙 = 0 𝒘∗ = 𝒘 𝒘 is the unit normal vector We would like to compute the distance between the point a and the hyperplane 1/27/2021 PATTERN RECOGNITION 14
Compute signed distance from a point to the hyperplane We want to find distance between a and line in direction of w If we define point ao on the line, then this distance corresponds to length of a-ao in direction of w", which equals w(a 0 Since wa0=-b, the distance equals 1(wa+b) C x+b=0 1/27/2021 PATTERN RECOGNITION
Compute signed distance from a point to the hyperplane We want to find distance between a and line in direction of 𝒘∗ If we define point a0 on the line, then this distance corresponds to length of 𝒂 − 𝒂0 in direction of 𝒘∗ , which equals 𝒘∗𝑇 (𝒂 − 𝒂0 ) Since 𝒘𝑇𝒂0 = −𝑏, the distance equals 1 𝒘 (𝒘𝑇𝒂 + 𝑏) 1/27/2021 PATTERN RECOGNITION 15