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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.