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The polynomial kernel is a powerful similarity measure used extensively in kernel-based machine learning algorithms, particularly Support Vector Machines (SVMs) and kernel methods. Unlike simple linear similarity measures, the polynomial kernel enables learning of complex, non-linear decision boundaries by implicitly projecting input features into higher-dimensional spaces—without the computational expense of explicitly computing the transformation.
This technique leverages the kernel trick: instead of computing the feature mapping φ(x) and then taking the inner product <φ(x), φ(y)>, we directly compute K(x, y), which yields the same result while avoiding the potentially infinite-dimensional feature space computation.
Mathematical Definition:
Given two input vectors x and y, the polynomial kernel is defined as:
$$K(x, y) = (\gamma \cdot \langle x, y \rangle + c_0)^d$$
Where:
Your Task:
Implement a function that computes the polynomial kernel similarity between two input vectors. The function should:
x = [1.0, 2.0, 3.0]
y = [4.0, 5.0, 6.0]
degree = 3
gamma = 1.0
coef0 = 1.035937.0Step 1: Compute the dot product ⟨x, y⟩ = (1.0 × 4.0) + (2.0 × 5.0) + (3.0 × 6.0) ⟨x, y⟩ = 4.0 + 10.0 + 18.0 = 32.0
Step 2: Apply the polynomial kernel formula K(x, y) = (γ × ⟨x, y⟩ + c₀)^d K(x, y) = (1.0 × 32.0 + 1.0)³ K(x, y) = 33³ = 35937.0
The polynomial kernel projects these vectors into a space where their similarity is 35937.0.
x = [1.0, 0.0]
y = [0.0, 1.0]
degree = 2
gamma = 1.0
coef0 = 1.01.0Step 1: Compute the dot product ⟨x, y⟩ = (1.0 × 0.0) + (0.0 × 1.0) = 0.0
The vectors are orthogonal (perpendicular), so their dot product is zero.
Step 2: Apply the polynomial kernel formula K(x, y) = (1.0 × 0.0 + 1.0)² = 1² = 1.0
Note: Even though the vectors are orthogonal in the original space, the kernel value is 1.0 (not 0) due to the additive constant coef0. This demonstrates how the polynomial kernel creates non-trivial similarity even for orthogonal vectors.
x = [2.0, 2.0]
y = [2.0, 2.0]
degree = 2
gamma = 0.5
coef0 = 0.016.0Step 1: Compute the dot product ⟨x, y⟩ = (2.0 × 2.0) + (2.0 × 2.0) = 4.0 + 4.0 = 8.0
The vectors are identical, resulting in the maximum possible similarity for these vectors.
Step 2: Apply the polynomial kernel formula K(x, y) = (0.5 × 8.0 + 0.0)² = 4² = 16.0
With coef0 = 0 (homogeneous polynomial kernel), orthogonal vectors would yield exactly 0, making this a stricter similarity measure. The gamma = 0.5 scaling reduces the dot product's influence before exponentiation.
Constraints