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?

Question

BlackTom AI · Accepted Answer

We start by identifying the goal: represent the kernel k(x,y) as a inner product in some feature space, i.e., k(x,y) = φ(x) · φ(y).

Option a provides φ(x) = ( e^{x1 x2}, sqrt{2} * (x1/x2), 0.5 * x1^3 ). If we take the dot product φ(x) · φ(y) for this φ, we get:
- First component: e^{x1 x2} * e^{y1 y2} = e^{x1 x2} e^{y1 y2} = e^{(x1 x2) + (y1 y2)} since exponentials multiply when exponents add. This matches the first term e^{x1 x2 + y1 y2} in k(x,y).
- Second component: (sqrt{2} x1/x2) * (sqrt{2} y1/y2) = 2 * (x1/x2) * (y1/y2). This reproduces the second term 2 x1 y1 / (x2 y2).
- Third component: (0.5 x1^3) * (0.5 y1^3) = 0.25 * x1^3 * y1^3, which matches the third term 0.25 x1^3 y1^3.
Thus, the dot product φ(x) · φ(y) equals the given k(x,y).

Now evaluate the other options to see why they do not reproduce k(x,y) exactly:
- Option b suggests φ(x) = ( e^{x1}, e^{x2}, sqrt{2} * (x1/x2), 0.25 x1^3 ). The first term would be e^{x1} e^{x2} = e^{x1 + x2}, which does not match e^{x1 x2}. The second and fourth components also do not align with the coefficients and variables seen in k, so the overall dot product would not reproduce e^{x1 x2 + y1 y2} plus the given cross-terms.
- Option c proposes φ(x) = ( e^{x1 x2}, 2 x1 x2, 0.25 x1^3 ). The second component is 2 x1 x2, whereas the kernel requires a term proportional to x1/x2 multiplied by y1/y2, not x1 x2. The squared form in the dot product would yield 4 x1 x2 y1 y2, which is not present in k. Moreover, the third component has 0.25 x1^3, which would give 0.25 x1^3 y1^3 as intended, but the second term disrupts the matching with the original 2 x1 y1 /(x2 y2).
- Option d uses φ(x) = ( e^{x1 + x2}, sqrt{2} * (x1/x2), 0.5 x1^3 ). The first component would be e^{x1 + x2} rather than e^{x1 x2}, which breaks the first term alignment. The remaining terms also fail to correctly reproduce the required second term 2 x1 y1 /(x2 y2) via the dot product.

From these checks, option a is the one whose φ(x) components, when multiplied pairwise with φ(y), reproduce each term of k(x,y) exactly, with the correct coefficients and variable structure. The specific consistency across all three terms confirms the intended feature map.

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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类似问题

Let x=(x_1,x_2) and y=(y_1,y_2) and let kernel k be defined as follows: 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 \phi does this kernel correspond to?

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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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