COMP 652 Study Guide - Quiz Guide: Linear Combination, Dot Product, Similarity Measure

• School: McGill University
• Department: Computer Science (Sci)
• Course: COMP 652
• Professor: Doina Precup

Document Summary

K(x, z) = ak1(x, z) + bk2(x, z), where a, b > 0 are real numbers. Since k1(x, z) and k2(x, z) are kernels over rn x rn, they must be symmetric and positive semi-definite. Let us consider a 2-dimensional example where: (cid:2869)=(cid:1876)(cid:883) (cid:1876)(cid:884) (cid:1876)(cid:884) (cid:1876)(cid:885) (cid:2870)=(cid:1877)(cid:883) (cid:1877)(cid:884) (cid:1877)(cid:884) (cid:1877)(cid:885) (cid:1853)(cid:2869)=(cid:1853)(cid:1876)(cid:883) (cid:1853)(cid:1876)(cid:884) (cid:1853)(cid:1876)(cid:884) (cid:1853)(cid:1876)(cid:885) (cid:1854)(cid:2870)=(cid:1854)(cid:1877)(cid:883) (cid:1854)(cid:1877)(cid:884) (cid:1854)(cid:1877)(cid:884) (cid:1854)(cid:1877)(cid:885) K = ak1 + bk2 = (cid:1853)(cid:1876)(cid:883)+(cid:1854)(cid:1877)(cid:883) (cid:1853)(cid:1876)(cid:884)+(cid:1854)(cid:1877)(cid:884) (cid:1853)(cid:1876)(cid:884)+(cid:1854)(cid:1877)(cid:884) (cid:1853)(cid:1876)(cid:885)+(cid:1854)(cid:1877)(cid:885) z = (cid:1878)(cid:883)(cid:1878)(cid:884) We see that k is symmetric for this 2-dimensional case. Let us consider whether it is positive semi- definite. = z12(ax1 + by1) + 2z1z2(ax2 + by2) + z22(ax3 + by3) Grouping the terms by a and be, we get: a(z12x1 + 2z1z2x2 + z22x3) + b(z12y1 + 2z1z2y2 + z22y3) Let a = a(z12x1 + 2z1z2x2 + z22x3), b = b(z12y1 + 2z1z2y2 + z22y3) This is because the original kernel matrices k1 and k2 > 0. To prove k is positive semi-definite for the general case: