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 \(\hbox { (RKHS)}\, \mathcal {H}\) and onto the RKHS \(\mathcal {G}\) associated with the squared-modulus of the reproducing kernel of \(\mathcal {H}\). Through this coisometry, trace-class integral operators defined by general measures and the reproducing kernel of \(\mathcal {H}\) are isometrically represented as potentials in \(\mathcal {G}\), and the quadrature approximation of these operators is equivalent to the approximation of integral functionals on \(\mathcal {G}\). 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.