PSD represents positive semi-definite.

1. Show that if k(u, v) and k2(u, v) are PSD kernels, then k(u, v) = 71ki(u, v) + 72k2(u, v) is a PSD kernel for 71,72 > 0. 2. Let f(x) : Rd + R be an arbitrary function on Rd. Show that if k(u, v) is a PSD kernel, then k'(u, v) = f(u)k(u, v)f(v) is also a PSD kernel. 3. Show that if k(u, v) and k2(u, v) are PSD kernels, then k'(u, v) = k1(u, v) k2(u, v) is a PSD kernel. (Hint: any N N symmetric positive semi-definite matrix can be written as VAVT = EN=1 Invnv, , and if you did part (b) correctly, you have already shown that if K is a symmetric PSD matrix, then K'(i,j) = v(i)K(i,j)v(j) is also symmetric PSD ...) 4. Suppose that ki(u, v), k2(u, v),... is an infinite sequence of PSD kernels that converge point- wise: lim ki(u, v) = k(u, v), for all u, v eRd. Argue that k(u, v) is a PSD kernel. 5. Argue that if k(u, v) is a PSD kernel, then k' (u, v) = exp(k(u, v)) is a PSD kernel. (Hint: Taylor, plus all the properties you proved above.) 6. Show that || U v|12 k(u, v) = exp 202 is a PSD kernel for all o > 0. (Hint: write the k above as k(u, v) = f(u)k'(u, v)f(v).) 1. Show that if k(u, v) and k2(u, v) are PSD kernels, then k(u, v) = 71ki(u, v) + 72k2(u, v) is a PSD kernel for 71,72 > 0. 2. Let f(x) : Rd + R be an arbitrary function on Rd. Show that if k(u, v) is a PSD kernel, then k'(u, v) = f(u)k(u, v)f(v) is also a PSD kernel. 3. Show that if k(u, v) and k2(u, v) are PSD kernels, then k'(u, v) = k1(u, v) k2(u, v) is a PSD kernel. (Hint: any N N symmetric positive semi-definite matrix can be written as VAVT = EN=1 Invnv, , and if you did part (b) correctly, you have already shown that if K is a symmetric PSD matrix, then K'(i,j) = v(i)K(i,j)v(j) is also symmetric PSD ...) 4. Suppose that ki(u, v), k2(u, v),... is an infinite sequence of PSD kernels that converge point- wise: lim ki(u, v) = k(u, v), for all u, v eRd. Argue that k(u, v) is a PSD kernel. 5. Argue that if k(u, v) is a PSD kernel, then k' (u, v) = exp(k(u, v)) is a PSD kernel. (Hint: Taylor, plus all the properties you proved above.) 6. Show that || U v|12 k(u, v) = exp 202 is a PSD kernel for all o > 0. (Hint: write the k above as k(u, v) = f(u)k'(u, v)f(v).)