The paper discusses the development of low-rank smoothers for $d \geq 1$-dimensional data, which can be fitted using regression or penalized regression methods. These smoothers are constructed by transforming and truncating the basis from the solution of the thin plate spline smoothing problem, aiming to minimize the perturbation of the thin plate spline smoothing problem given the dimension of the basis used. The approach is computationally efficient, allowing for the use of appropriate regression models to model interaction terms in generalized additive models (GAMs). The method provides a way to incorporate smooth functions of multiple variables into non-linear models and improves the computational efficiency of penalized likelihood models incorporating thin plate splines. The paper also addresses the issue of knot placement, which is a common problem in regression spline modeling, by providing a method that allows for hypothesis testing-based model selection. The results show that the thin plate regression splines perform well in terms of computational efficiency and model performance, especially for large datasets. The method is demonstrated through simulations and an example using fisheries data, showing its practical advantages over traditional knot-based regression splines and full thin plate splines.The paper discusses the development of low-rank smoothers for $d \geq 1$-dimensional data, which can be fitted using regression or penalized regression methods. These smoothers are constructed by transforming and truncating the basis from the solution of the thin plate spline smoothing problem, aiming to minimize the perturbation of the thin plate spline smoothing problem given the dimension of the basis used. The approach is computationally efficient, allowing for the use of appropriate regression models to model interaction terms in generalized additive models (GAMs). The method provides a way to incorporate smooth functions of multiple variables into non-linear models and improves the computational efficiency of penalized likelihood models incorporating thin plate splines. The paper also addresses the issue of knot placement, which is a common problem in regression spline modeling, by providing a method that allows for hypothesis testing-based model selection. The results show that the thin plate regression splines perform well in terms of computational efficiency and model performance, especially for large datasets. The method is demonstrated through simulations and an example using fisheries data, showing its practical advantages over traditional knot-based regression splines and full thin plate splines.