= Common representer coefficients for a sum of kernels
For a real-valued <loss function> depending only on evaluations at $x_1,\ldots,x_n$, and a positive squared-norm penalty, any minimizing tuple in a <sum of reproducing-kernel Hilbert spaces> has the form
$$
\widehat f_j=\sum_{i=1}^n\widehat\alpha_i k_j(\cdot,x_i)
$$
with the same coefficient <vector> for every component. The tuple must be the unique minimum-norm decomposition of its sum. Apply the <representer theorem> to that sum and note that the displayed tuple has total squared <norm> $\sum_j\widehat\alpha^TK_j\widehat\alpha=\widehat\alpha^T(\sum_jK_j)\widehat\alpha$, exactly the <norm> of the sum. This argument is conditional on existence of a <minimizer>; arbitrary <loss functions> need not attain their infimum.
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