Kernel embedding of measures and low-rank approximation of integral operators

Abstract We describe a natural coisometry from the Hilbert space of all Hilbert-Schmidt operators on a separable reproducing kernel Hilbert space $$\hbox { (RKHS)}\, \mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>(RKHS)</mml:mtext> <mml:mspace/> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> and onto the RKHS $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> associated with the squared-modulus of the reproducing kernel of $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> . Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> are isometrically represented as potentials in $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , and the quadrature approximation of these operators is equivalent to the approximation of integral functionals on $$\mathcal {G}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . We then discuss the extent to which the approximation of potentials in RKHSs with squared-modulus kernels can be regarded as a differentiable surrogate for the characterisation of low-rank approximation of integral operators.