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Polynomial Kernel

Compute the polynomial kernel matrix between two sets of vectors.

The polynomial kernel is: $K(x, y) = (x \cdot y + c)^d$

Given two sets of vectors $A$ (m points) and $B$ (n points), compute the $m \times n$ kernel matrix where entry $(i,j) = K(A_i, B_j)$ .

Return the matrix with values rounded to 4 decimal places.

Input:

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]
c = 1, d = 2

Output:

[[324, 576], [1600, 2916]]

Reasoning:

First, we calculate the dot product of each vector in $A$ $A$ with each vector in $B$ $B$ :
- $A_1 \cdot B_1 = (1 \cdot 5) + (2 \cdot 6) = 17$ ,
- $A_1 \cdot B_2 = (1 \cdot 7) + (2 \cdot 8) = 23$ ,
- $A_2 \cdot B_1 = (3 \cdot 5) + (4 \cdot 6) = 39$ ,
- $A_2 \cdot B_2 = (3 \cdot 7) + (4 \cdot 8) = 53$
Then, we apply the polynomial kernel formula: $K(x, y) = (x \cdot y + c)^d$ $K (x, y) = (x \cdot y + c)^{d}$
- $K(A_1, B_1) = (17 + 1)^2 = 18^2 = 324$ ,
- $K(A_1, B_2) = (23 + 1)^2 = 24^2 = 576$ ,
- $K(A_2, B_1) = (39 + 1)^2 = 40^2 = 1600$ ,
- $K(A_2, B_2) = (53 + 1)^2 = 54^2 = 2916$
The final output is the $m \times n$ kernel matrix with the calculated values: $[[324, 576], [1600, 2916]]$

Editor

Python 3.13.1

0/0

Run code to see test results.

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