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Fixed typo.
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Bill Bradley
Bill Bradley
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This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define $$ \mathbf{K}(\phi)_{i,j} = \sum_{i,j}\phi(x_i, x_j) $$

We will say that $\phi_1 \preceq \phi_2$ iff for any $N$, $x_1,...,x_N$, and $c_1,...,c_N\geq 0$, we have $$\sum_{i,j} c_i c_j (\phi_2(x_i,x_j)-\phi_1(x_i,x_j)) \geq 0$$ This gives a partial order on $\Phi$ that (by construction) respects the OP's property of interest.

Possibly the OP is less interested in the preceding construction and more interested in concrete parametric examples. If so, geodesics of the cone of PSD matrices provide some interesting examples. For example, take $K_1, K_2$ as two PSD kernels, and let $\Phi=[0,1]$. For a fixed $N$, let $A$ and $A$$B$ be the two corresponding PSD matrices. Then define $$K(x,y,\phi)=A^{1/2}\exp(\phi \log(A^{-1/2}BA^{-1/2}))A^{1/2}$$ More inspiration along that line might be found at Bonnabel and Sepulchre's "Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank"

This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define $$ \mathbf{K}(\phi)_{i,j} = \sum_{i,j}\phi(x_i, x_j) $$

We will say that $\phi_1 \preceq \phi_2$ iff for any $N$, $x_1,...,x_N$, and $c_1,...,c_N\geq 0$, we have $$\sum_{i,j} c_i c_j (\phi_2(x_i,x_j)-\phi_1(x_i,x_j)) \geq 0$$ This gives a partial order on $\Phi$ that (by construction) respects the OP's property of interest.

Possibly the OP is less interested in the preceding construction and more interested in concrete parametric examples. If so, geodesics of the cone of PSD matrices provide some interesting examples. For example, take $K_1, K_2$ as two PSD kernels, and let $\Phi=[0,1]$. For a fixed $N$, let $A$ and $A$ be the two corresponding PSD matrices. Then define $$K(x,y,\phi)=A^{1/2}\exp(\phi \log(A^{-1/2}BA^{-1/2}))A^{1/2}$$ More inspiration along that line might be found at Bonnabel and Sepulchre's "Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank"

This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define $$ \mathbf{K}(\phi)_{i,j} = \sum_{i,j}\phi(x_i, x_j) $$

We will say that $\phi_1 \preceq \phi_2$ iff for any $N$, $x_1,...,x_N$, and $c_1,...,c_N\geq 0$, we have $$\sum_{i,j} c_i c_j (\phi_2(x_i,x_j)-\phi_1(x_i,x_j)) \geq 0$$ This gives a partial order on $\Phi$ that (by construction) respects the OP's property of interest.

Possibly the OP is less interested in the preceding construction and more interested in concrete parametric examples. If so, geodesics of the cone of PSD matrices provide some interesting examples. For example, take $K_1, K_2$ as two PSD kernels, and let $\Phi=[0,1]$. For a fixed $N$, let $A$ and $B$ be the two corresponding PSD matrices. Then define $$K(x,y,\phi)=A^{1/2}\exp(\phi \log(A^{-1/2}BA^{-1/2}))A^{1/2}$$ More inspiration along that line might be found at Bonnabel and Sepulchre's "Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank"

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Bill Bradley
Bill Bradley
  • 4k
  • 20
  • 25

This example may be a little bit ridiculous, but suppose we take $\mathcal{X}=\mathbb{R}$ and let $\Phi$ be any parametric subset of the set of PSD kernels itself. We define $$ \mathbf{K}(\phi)_{i,j} = \sum_{i,j}\phi(x_i, x_j) $$

We will say that $\phi_1 \preceq \phi_2$ iff for any $N$, $x_1,...,x_N$, and $c_1,...,c_N\geq 0$, we have $$\sum_{i,j} c_i c_j (\phi_2(x_i,x_j)-\phi_1(x_i,x_j)) \geq 0$$ This gives a partial order on $\Phi$ that (by construction) respects the OP's property of interest.

Possibly the OP is less interested in the preceding construction and more interested in concrete parametric examples. If so, geodesics of the cone of PSD matrices provide some interesting examples. For example, take $K_1, K_2$ as two PSD kernels, and let $\Phi=[0,1]$. For a fixed $N$, let $A$ and $A$ be the two corresponding PSD matrices. Then define $$K(x,y,\phi)=A^{1/2}\exp(\phi \log(A^{-1/2}BA^{-1/2}))A^{1/2}$$ More inspiration along that line might be found at Bonnabel and Sepulchre's "Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank"