Semiparametric regression models are used in several applications which require comprehensibility without sacrificing accuracy. Typical examples are spline interpolation in geophysics and nonlinear time series problems, where the system includes a linear and nonlinear component. We discuss here the use of a finite determinantal point process (DPP) for approximating semiparametric models. Recently, Barthelmé, Tremblay, Usevich, and Amblard introduced a novel representation of 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 semiparametric regression and interpolation. Also, a novel projected Nyström approximation is defined and used to derive a bound on the expected in-sample prediction error for the corresponding approximation of semiparametric regression. This work naturally extends similar results obtained for kernel ridge regression.