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A real positive-semidefinite kernel is a symmetric function $k$ such that every finite Gram matrix $K_{ij}=k(x_i,x_j)$ is positive semidefinite. The representer theorem says that any minimizer in a Reproducing-kernel Hilbert space of an objective depending on $f$ only through $f(x_1),\dots ,f(x_n)$ and a strictly increasing function of $\Vert f{\Vert }_{\mathcal{H}}$ lies inIndeed, write $f=f_{\parallel }+f_{\perp }$ relative to this span. The reproducing property gives $f_{\perp }(x_i)=0$ for every $i$, while the Pythagorean theorem in an inner-product space gives $\Vert f{\Vert }^2=\Vert f_{\parallel }{\Vert }^2+\Vert f_{\perp }{\Vert }^2$. Removing a nonzero perpendicular component preserves all data values and strictly decreases the penalty, proving the theorem.

Apply this decomposition to both optimizers and write $f={\sum }_i{\alpha }_{i}k(X_i,\cdot )$ and $g={\sum }_i{\beta }_{i}l(Y_i,\cdot )$. If $K$ and $L$ are the two Gram matrices, thenWriting $u=K^{1/2}\alpha$ and $v=L^{1/2}\beta$, with pseudoinverses on the respective ranges, turns the supremum into

Solved by gpt-5.6-sol high.

A real positive-semidefinite kernel is a symmetric function $k:X\times X\to \mathbb{R}$ such that every finite Gram matrix $(k(x_i,x_j))$ is positive semidefinite. The Moore-Aronszajn theorem says that there are a Hilbert space $H$ and a feature map $\varphi :X\to H$ such thatEquivalently, $H$ can be chosen as the unique Reproducing-kernel Hilbert space with reproducing kernel $k$.

Solved by gpt-5.6-sol high.