6.2 Show that the following kernels K are PDS: (a) K(x; y) = cos(x ???? y) over...
Question:
6.2 Show that the following kernels K are PDS:
(a) K(x; y) = cos(x ???? y) over R R.
(b) K(x; y) = cos(x2 ???? y2) over R R.
(c) For all integers n > 0;K(x; y) =
PN i=1 cosn(x2i
???? y2 i ) over RN RN.
(d) K(x; y) = (x + y)????1 over (0;+1) (0;+1).
(e) K(x; x0) = cos\(x; x0) over Rn Rn, where \(x; x0) is the angle between x and x0.
(f) 8 > 0; K(x; x0) = exp
????
???? [sin(x0 ???? x)]2
over R R.
(Hint: rewrite [sin(x0 ???? x)]2 as the square of the norm of the dierence of two vectors.)
(g) 8 > 0;K(x; y) = e????kx????yk over RN RN.
(Hint: you could show that K is the normalized kernel of a kernel K0 and show that K0 is PDS using the following equality: kx ???? yk =
1 2????( 1 2 )
R +1 0
1????e????tkx????yk2 t
32 dt valid for all x; y.)
(h) K(x; y) = min(x; y) ???? xy over [0; 1] [0; 1].
(Hint: you could consider the two integrals R 1 R 0 1t2[0;x]1t2[0;y]dt and 1 0 1t2[x;1]1t2[y;1]dt.)
(i) K(x; x0) = 1 p1????(xx0)
over x; x0 2 X = fx 2 RN : kxk2 < 1g.
(Hint: one approach is to nd an explicit expression of a feature mapping
by considering the Taylor expansion of the kernel function.)
(j) 8 > 0;K(x; y) = 1 1+kx????yk2
2 over RN RN.
(Hint: the function x 7!
R +1 0 e????sxe????sds dened for all x 0 could be useful for the proof.)
(k) 8 > 0;K(x; y) = exp
PN i=1 min(jxij;jyij)
2
over RN RN.
(Hint: the function (x0; y0) 7!
R +1 0 1t2[0;jx0j]1t2[0;jy0j]dt dened over R R could be useful for the proof.)
Step by Step Answer:
Foundations Of Machine Learning
ISBN: 9780262351362
2nd Edition
Authors: Mehryar Mohri, Afshin Rostamizadeh