CS4601: Intro. to AI
Homework 1 for Pattern Recognition Due date: Dec 2, 1998

1.

(20%) The L_p norm (or the Minkowski distance metrics) of a vector ${\bf x}$ is defined by

$$L_p({\bf x}) = \left[ \sum_{i=1}^n \vert x_i\vert^p \right]^{1/p},$$

where ${\bf x}= [x_1, x_2, \ldots, x_n]$. Prove that

$$\lim_{p \rightarrow \infty} L_p({\bf x}) = \max_i \vert x_i\vert.$$

2.

(60%) A three-class classifier is defined by a set of discriminant functions:

$$\left\{ \begin{array} {rcl} D_1 & = & x - y \\ D_2 & = & 2x +y - 6 \\ D_3 & = & y \\ \end{array} \right.$$

(a)

(10%) Plot the three lines corresponding to D₁=0, D₂=0, and D₃=0, respectively.

(b)

(30%) Plot the decision boundaries of the classifier. Label the decision region for each class.

(c)

(20%) Does the decision boundaries always converge to one point for this type of three-class two-feature classifier? Why or why not?