3. (10 pts) (Constructing Kernels) One convenient property of kernels is that they can be combined together to build new kernels. In this questions, we consider building a kernel K(u, v), u, v E Rd, using existing kernels G(u, v) and H(u, v). Remember that a function K(u, v) is a (Mercer) kernel if and only if it satisfies the following two conditions: Symmetry K(u, v) = K(v, u) Positive semi-definiteness: Di=;D;=1 ;a; K (2;, ;) > 0, for any data points (1) ... , In and any values a1, ..., An E R. And, under such conditions, there exists a mapping o such that K(u, v) = (u)'(x). = n = 2 = Conversely, if one can show such a mapping o exists such that K(u, y) = $(u)To(u), then this also shows it is a valid kernel. Show that the following are valid kernels: (a) (2 pts) (Scaling) K(u, v) = c.G(u, v), with c> 0 (b) (3 pts) (Sum) K(u, v) = G(u, v) + H(u, v) (c) (3 pts) (Product) K(u, v) = G(u, v) H(u, v) (d) (2 pts) (Polynomial) K(u, v) = 2:40 a;G(u, v)', with ai > 0 = 3. (10 pts) (Constructing Kernels) One convenient property of kernels is that they can be combined together to build new kernels. In this questions, we consider building a kernel K(u, v), u, v E Rd, using existing kernels G(u, v) and H(u, v). Remember that a function K(u, v) is a (Mercer) kernel if and only if it satisfies the following two conditions: Symmetry K(u, v) = K(v, u) Positive semi-definiteness: Di=;D;=1 ;a; K (2;, ;) > 0, for any data points (1) ... , In and any values a1, ..., An E R. And, under such conditions, there exists a mapping o such that K(u, v) = (u)'(x). = n = 2 = Conversely, if one can show such a mapping o exists such that K(u, y) = $(u)To(u), then this also shows it is a valid kernel. Show that the following are valid kernels: (a) (2 pts) (Scaling) K(u, v) = c.G(u, v), with c> 0 (b) (3 pts) (Sum) K(u, v) = G(u, v) + H(u, v) (c) (3 pts) (Product) K(u, v) = G(u, v) H(u, v) (d) (2 pts) (Polynomial) K(u, v) = 2:40 a;G(u, v)', with ai > 0 =