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How do we derive/extrapolate the feature map for inhomogeneous polynomial kernels for degree $d$, define as $$K(x, x^\prime) = \bigg( (x \cdot x^\prime) + c\bigg)^d$$

I know we don't need to know a valid feature map to use the inhomogeneous polynomial kernels, I'm simply curious as to how one would derive/extrapolate it.

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Let $x, y \in \mathbb R^N, \mathbf \alpha \equiv (\alpha_0, \dots, \alpha_N) \equiv (\alpha_0, \mathbf \alpha'_0), \; \mathbf z^\alpha \equiv z_0^{\alpha_0} \dots z_N^{\alpha_N}$ and ${d \choose \alpha}$ be the multinomial coefficient. Then

$$\left(\left<\mathbf x, \mathbf y \right> + c\right)^d = \sum_{|\mathbf \alpha| = d} {d \choose \alpha} c^{\alpha_0} (\mathbf x \mathbf y)^{\mathbf \alpha'_0}$$

If you redistribute the coefficient ${d \choose \alpha} c^{\alpha_0}$ between $\Phi(\mathbf x)$ and $\Phi(\mathbf y)$, it follows that

$$\Phi(\mathbf z) = \left(\sqrt{ {d \choose \alpha} c^{\alpha_0}} \mathbf z^{\alpha'_0} \right)_{\forall |\mathbf \alpha| = d}$$

References

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