This paper discusses the construction of low rank smoothers for d-dimensional data, which can be fitted using regression or penalized regression methods. These smoothers are derived from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to minimize the perturbation of the thin plate spline smoothing problem given the basis dimension. The basis change and truncation are computationally efficient using Lanczos iteration. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems associated with 'knot placement', provide a sensible way of modeling interaction terms in generalized additive models (GAMs), provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. The approach produces spline-like models with a sparse basis, allowing the incorporation of unpenalized spline-like terms in linear and generalized linear models. Keywords: Generalized additive model; Regression spline; Thin plate spline. The paper introduces a method for constructing low rank thin plate spline-like smoothers that are optimal in terms of minimizing the perturbation of the thin plate spline smoothing problem. The method involves truncating the eigenbasis of the matrix E, which results in a low rank approximation that minimizes the worst-case changes in both the goodness-of-fit term and the penalty term. This approach allows for efficient computation using Lanczos iteration and is suitable for large data sets. The method is demonstrated through simulation studies and an example using generalized additive models (GAMs) to model fisheries data. The results show that the thin plate regression spline approach provides better performance in terms of computational efficiency and model selection compared to traditional knot-based regression splines and full thin plate splines. The method is also shown to be effective in modeling complex functions with multiple variables and provides a natural way to incorporate smooth functions into non-linear models. The paper concludes that the thin plate regression spline approach offers a computationally efficient and statistically sound method for modeling complex data with smooth functions.This paper discusses the construction of low rank smoothers for d-dimensional data, which can be fitted using regression or penalized regression methods. These smoothers are derived from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to minimize the perturbation of the thin plate spline smoothing problem given the basis dimension. The basis change and truncation are computationally efficient using Lanczos iteration. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems associated with 'knot placement', provide a sensible way of modeling interaction terms in generalized additive models (GAMs), provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. The approach produces spline-like models with a sparse basis, allowing the incorporation of unpenalized spline-like terms in linear and generalized linear models. Keywords: Generalized additive model; Regression spline; Thin plate spline. The paper introduces a method for constructing low rank thin plate spline-like smoothers that are optimal in terms of minimizing the perturbation of the thin plate spline smoothing problem. The method involves truncating the eigenbasis of the matrix E, which results in a low rank approximation that minimizes the worst-case changes in both the goodness-of-fit term and the penalty term. This approach allows for efficient computation using Lanczos iteration and is suitable for large data sets. The method is demonstrated through simulation studies and an example using generalized additive models (GAMs) to model fisheries data. The results show that the thin plate regression spline approach provides better performance in terms of computational efficiency and model selection compared to traditional knot-based regression splines and full thin plate splines. The method is also shown to be effective in modeling complex functions with multiple variables and provides a natural way to incorporate smooth functions into non-linear models. The paper concludes that the thin plate regression spline approach offers a computationally efficient and statistically sound method for modeling complex data with smooth functions.