2.1.Definition of Perceptron Definition(Perceptron):Assume that the input space (feature space)is C R",and the output space is={+1,-1}.Input x E denotes the feature vector of a sample,corresponding to a point in the input space;output ye) denotes the class of the sample.Perceptron is a mapping function from input space to output space: y=f(x)=sign(w·x+b) g8-{杜安8 where: ew·x is inner product. o w,b are perceptron parameters o sign()is sign function 10/66
2.1. Definition of Perceptron ▶ Definition (Perceptron): Assume that the input space (feature space) is X ⊆ R n , and the output space is Y = {+1, −1}. Input x ∈ X denotes the feature vector of a sample, corresponding to a point in the input space; output y ∈ Y denotes the class of the sample. Perceptron is a mapping function from input space to output space: y = f (x) = sign(w · x + b) sign(z) = +1, z ≥ 0 −1, z < 0 where: w · x is inner product. w, b are perceptron parameters sign(·) is sign function 10 / 66
Perceptron is a linear classification model and belongs to the discriminant model. Hypothesis space of perceptron model is the set of functions defined by all linear classification models or linear classifiers in the feature space: {flf(x)=w·x+b} 11/66
▶ Perceptron is a linear classification model and belongs to the discriminant model. ▶ Hypothesis space of perceptron model is the set of functions defined by all linear classification models or linear classifiers in the feature space: {f | f (x) = w · x + b} 11 / 66
Outline (Level 1-2) Perceptron concept o Definition of Perceptron Geometric interpretation of perceptron 12/66
Outline (Level 1-2) 2 Perceptron concept Definition of Perceptron Geometric interpretation of perceptron 12 / 66
2.2.Geometric interpretation of perceptron w w.x+b=0 The linear equation w.x+b=0 corresponds to a hyperplane S in the feature space,where w is the normal vector of S and b is the intercept of S Hyperplane S(separating hyperplane)divides the feature space into two parts,namely positive (w's direction)and negative classes. 13/66
2.2. Geometric interpretation of perceptron The linear equation w · x + b = 0 corresponds to a hyperplane S in the feature space, where w is the normal vector of S and b is the intercept of S Hyperplane S (separating hyperplane) divides the feature space into two parts, namely positive (w’s direction) and negative classes. 13 / 66
Outline (Level 2-3) o Geometric interpretation of perceptron Distance of point to hyperplane o Geometric margin o Functional margin 14/66
Outline (Level 2-3) Geometric interpretation of perceptron Distance of point to hyperplane Geometric margin Functional margin 14 / 66