Interpolation with uncoupled separable matrix-valued kernels
Provides theoretical foundations and practical benefits for vector-valued function approximation, relevant to machine learning and approximation theory.
The paper derives error bounds for interpolation with matrix-valued reproducing kernels and introduces a subclass of kernels whose power-functions reduce to scalar-valued ones. Experiments on artificial data show that matrix-valued interpolation outperforms componentwise approaches.
In this paper we consider the problem of approximating vector-valued functions over a domain $Ω$. For this purpose, we use matrix-valued reproducing kernels, which can be related to Reproducing kernel Hilbert spaces of vectorial functions and which can be viewed as an extension to the scalar-valued case. These spaces seem promising, when modelling correlations between the target function components, as the components are not learned independently of one another. We focus on the interpolation with such matrix-valued kernels. We derive error bounds for the interpolation error in terms of a generalized power-function and we introduce a subclass of matrix-valued kernels whose power-functions can be traced back to the power-function of scalar-valued reproducing kernels. Finally, we apply these kind of kernels to some artificial data to illustrate the benefit of interpolation with matrix-valued kernels in comparison to a componentwise approach.