# Scalable Higher-Order Tensor Product Spline Models

arXiv: https://arxiv.org/abs/2402.01090 · HTML: https://mlsanity.com/paper/2402.01090/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 46 references that are in the map.
- First posted on arXiv 2 Feb 2024.

## Builds on (most cited first, 10 of 10)

- [InterpretML: A Unified Framework for Machine Learning Interpretability](https://mlsanity.com/paper/1909.09223/) (2019)
- [Neural Collaborative Filtering vs. Matrix Factorization Revisited](https://mlsanity.com/paper/2005.09683/) (2020)
- [NODE-GAM: Neural Generalized Additive Model for Interpretable Deep Learning](https://mlsanity.com/paper/2106.01613/) (2021)
- [Higher-Order Factorization Machines](https://mlsanity.com/paper/1607.07195/) (2016)
- [Towards a Better Understanding of Linear Models for Recommendation](https://mlsanity.com/paper/2105.12937/) (2021)
- [deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression](https://mlsanity.com/paper/2104.02705/) (2021)
- [A New PHO-rmula for Improved Performance of Semi-Structured Networks](https://mlsanity.com/paper/2306.00522/) (2023)
- [Interaction-aware Factorization Machines for Recommender Systems](https://mlsanity.com/paper/1902.09757/) (2019)
- [Factorized Structured Regression for Large-Scale Varying Coefficient Models](https://mlsanity.com/paper/2205.13080/) (2022)
- [Accelerated Componentwise Gradient Boosting using Efficient Data Representation and Momentum-based Optimization](https://mlsanity.com/paper/2110.03513/) (2021)

## Built on by (most cited first, 4 of 4)

- [Tensor Product Neural Networks for Functional ANOVA Model](https://mlsanity.com/paper/2502.15215/) (2025)
- [LIT-LVM: Structured Regularization for Interaction Terms in Linear Predictors using Latent Variable Models](https://mlsanity.com/paper/2506.15492/) (2025)
- [tensorFM: Low-Rank Approximations of Cross-Order Feature Interactions](https://mlsanity.com/paper/2602.15229/) (2026)
- [Structured Nonparametric Variational Inference for Dependent Latent Modeling](https://mlsanity.com/paper/2606.15458/) (2026)

## Abstract (arXiv)

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.