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

Abstract

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