Let [math: x=(x1,x2)]x=(x_1,x_2) and [math: y=(y1,y2)]y=(y_1,y_2) and let kernel k be defined as follows: [math: k(x,y)=ex1x2+y1y2+2x1y1x2y2+0.25x13y13]k(x,y) = e^{x_1x_2+y_1y_2} +2 \frac{x_1 y_1}{x_2y_2} + 0.25 x_1^3 y_1^3 which transformation [math: ϕ]\phi does this kernel correspond to?

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We are given a kernel k(x,y) = e^{x1 x2 + y1 y2} + 2 (x1 y1)/(x2 y2) + 0.25 x1^3 y1^3 and asked which feature map ϕ corresponds to this kernel, i.e., k(x,y) = ϕ(x) · ϕ(y).

Option a proposes ϕ(x) = ( e^{x1 x2}, sqrt(2) x1/x2, 0.5 x1^3 ). Its inner product with y is:
- e^{x1 x2} e^{y1 y2} from the first components, which yields e^{x1 x2 + y1 y2} as in k.
- (sqrt(2) x1/x2) (sqrt(2) y1/y2) = 2 (x1 y1)/(x2 y2) from the second components, matching the second term.
- (0.5 x1^3) (0.5 y1^3) = 0.25 x1^3 y1^3 from the third components, matching the third term.
Therefore, the dot product of ϕ(x) and ϕ(y) reproduces k(x,y), so option a is a correct feature map.

Option b proposes ϕ(x) = ( e^{x1 + x2}, sqrt(2) x1/x2, 0.5 x1^3 ). The first component e^{x1 + x2} would yield e^{(x1 + x2) + (y1 + y2)} upon dotting with ϕ(y), which does not match e^{x1 x2 + y1 y2}; thus this option cannot reproduce k(x,y).

Option c proposes ϕ(x) = ( e^{x1 x2}, 2 x1 x2, 0.25 x1^3 ). The second component 2 x1 x2 would give a product term 4 x1 x2 y1 y2, which does not correspond to the required 2 (x1 y1)/(x2 y2) term, so this is incorrect. The first and third components match the first and third terms only if the cross-term were aligned, which it is not here.

Option d proposes ϕ(x) = ( e^{x1}, e^{x2}, sqrt(2) x1/x2, 0.25 x1^3 ). This vector has four components, and the first two components do not combine to give e^{x1 x2} when taking the dot product with a corresponding ϕ(y); furthermore, the presence of e^{x1} and e^{x2} separately leads to a mismatch with the required e^{x1 x2} term. Hence this option cannot reproduce k(x,y).

In summary, only option a yields an exact decomposition k(x,y) = ϕ(x) · ϕ(y) with the specified components, while options b, c, and d fail to reproduce at least one term of the kernel or introduce incompatible cross-terms.

COMP9417-Machine Learning & Data Mining - T3 2025

Let [math: x=(x1,x2)]x=(x_1,x_2) and [math: y=(y1,y2)]y=(y_1,y_2) and let kernel k be defined as follows: [math: k(x,y)=ex1x2+y1y2+2x1y1x2y2+0.25x13y13]k(x,y) = e^{x_1x_2+y_1y_2} +2 \frac{x_1 y_1}{x_2y_2} + 0.25 x_1^3 y_1^3 which transformation [math: ϕ]\phi does this kernel correspond to?

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