Determinantal Point Processes Implicitly Regularize Semi-parametric Regression Problems

Semi-parametric regression models are used in several applications which require comprehensibility without sacrificing accuracy. Examples are spline interpolation in geophysics, or non-linear time series problems, where the system includes for instance a linear and non-linear component. We discuss here the use of a finite Determinantal Point Process (DPP) sampling for approximating semi-parametric models in two cases. On the one hand, in the case of large training data sets, DPP sampling is used to reduce the number of model parameters. On the other hand, DPPs can determine experimental designs in the case of the optimal interpolation models. Recently, Barthelm\'e, Tremblay, Usevich, and Amblard introduced a novel representation of finite DPP's. They formulated extended $L$-ensembles that can conveniently represent for instance 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.