  Semi-parametric regression models are used in several applications which
require comprehensibility without sacrificing accuracy. Typical examples are
spline interpolation in geophysics, or non-linear time series problems, where
the system includes a linear and non-linear component. We discuss here the use
of a finite Determinantal Point Process (DPP) for approximating semi-parametric
models. Recently, Barthelm\'e, Tremblay, Usevich, and Amblard introduced a
novel representation of some finite DPPs. These authors formulated extended
L-ensembles that can conveniently represent partial-projection DPPs and suggest
their use for optimal interpolation. With the help of this formalism, we derive
a key identity illustrating the implicit regularization effect of determinantal
sampling for semi-parametric regression and interpolation. Also, a novel
projected Nystr\"om approximation is defined and used to derive a bound on the
expected risk for the corresponding approximation of semi-parametric
regression. This work naturally extends similar results obtained for kernel
ridge regression.
