In the current era of vast data and transparent machine learning, it is essential for techniques to operate at a large scale while providing a clear mathematical comprehension of the internal workings of the method. Although there already exist interpretable semi-parametric regression methods for large-scale applications that take into account non-linearity in the data, the complexity of the models is still often limited. One of the main challenges is the absence of interactions in these models, which are left out for the sake of better interpretability but also due to impractical computational costs. To overcome this limitation, we propose a new approach using a factorization method to derive a highly scalable higher-order tensor product spline model. Our method allows for the incorporation of all (higher-order) interactions of non-linear feature effects while having computational costs proportional to a model without interactions. We further develop a meaningful penalization scheme and examine the induced optimization problem. We conclude by evaluating the predictive and estimation performance of our method.

In the age of big data and interpretable machine learning, approaches need to work at scale and at the same time allow for a clear mathematical understanding of the method's inner workings. While there exist inherently interpretable semi-parametric regression techniques for large-scale applications to account for non-linearity in the data, their model complexity is still often restricted. One of the main limitations are missing interactions in these models, which are not included for the sake of better interpretability, but also due to untenable computational costs. To address this shortcoming, we derive a scalable high-order tensor product spline model using a factorization approach. Our method allows to include all (higher-order) interactions of non-linear feature effects while having computational costs proportional to a model without interactions. We prove both theoretically and empirically that our methods scales notably better than existing approaches, derive meaningful penalization schemes and also discuss further theoretical aspects. We finally investigate predictive and estimation performance both with synthetic and real data.