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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BlackTom AI · Accepted Answer

First, restating the problem in my own words helps set the stage for analysis. 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 transformation φ corresponds to this kernel via k(x, y) = ⟨φ(x), φ(y)⟩ in the feature space.

Option a proposes φ(x) = (e^{x1 x2}, 2x1x2, 0.5 x1^3) with φ(x) = (e^{x1+x2}, sqrt{2} x1/x2, 0.5 x1^3). The first component would give e^{x1+x2} when paired with e^{y1+y2}, which does not match the kernel's first term e^{x1 x2 + y1 y2}. Moreover, the middle term would be (2 x1 x2)(2 y1 y2) = 4 x1 x2 y1 y2, which is not of the form 2 (x1 y1)/(x2 y2). Thus this option misaligns with all three terms.

Option c suggests φ(x) = (e^{x1 x2}, 2x1x2, 0.25 x1^3) with φ(x) = (e^{x1 x2}, sqrt{2} x1/x2, 0.25 x1^3). The first component would match the e^{x1 x2 + y1 y2} part only if paired with e^{y1 y2}, which is not what's provided; the second term would yield (sqrt{2} x1/x2)(sqrt{2} y1/y2) = 2 (x1 y1)/(x2 y2), which is correct for the second term, but the third term would be 0.25 x1^3 y1^3, not 0.25 x1^3 y1^3 if the third component is 0.25 x1^3; actually it would be 0.25 x1^3 * 0.25 y1^3 = 0.0625 x1^3 y1^3, which is incorrect. So this option is inconsistent for the first and third terms.

Option d presents φ(x) = (e^{x1}, e^{x2}, sqrt{2} x1/x2, 0.25 x1^3) and φ(x) = (e^{x1}, e^{x2}, sqrt{2} x1/x2, 0.25 x1^3). The first two components e^{x1} and e^{x2} would produce a cross-term e^{x1} e^{y1} or e^{x2} e^{y2} in the kernel if used in the inner product, which does not align with e^{x1 x2 + y1 y2}. This option introduces extraneous terms and mismatches with all three target terms.

Option b, on the other hand, sets φ(x) = (e^{x1 x2}, sqrt{2} x1/x2, 0.5 x1^3). Now, when we compute ⟨φ(x), φ(y)⟩, we get:
- First components: e^{x1 x2} e^{y1 y2} = e^{x1 x2 + y1 y2}, matching the first term of k.
- Second components: (sqrt{2} x1/x2)(sqrt{2} y1/y2) = 2 (x1 y1)/(x2 y2), matching the second term of k.
- Third components: (0.5 x1^3)(0.5 y1^3) = 0.25 x1^3 y1^3, matching the third term of k.
Thus, option b provides a φ whose inner product reproduces exactly the given kernel terms.

In summary:
- Option a and c fail due to mismatches in the first or third term (or both).
- Option d introduces extra components and misaligns with the required exponential and polynomial terms.
- Therefore, option b is the correct transformation φ that yields the kernel k via k(x, y) = ⟨φ(x), φ(y)⟩.

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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