# Tensor Product Neural Networks for Functional ANOVA Model

arXiv: https://arxiv.org/abs/2502.15215 · HTML: https://mlsanity.com/paper/2502.15215/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 55 references that are in the map.
- First posted on arXiv 21 Feb 2025.

## Builds on (most cited first, 19 of 19)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Fooling LIME and SHAP: Adversarial Attacks on Post hoc Explanation Methods](https://mlsanity.com/paper/1911.02508/) (2019)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [Neural Oblivious Decision Ensembles for Deep Learning on Tabular Data](https://mlsanity.com/paper/1909.06312/) (2019)
- [Introducing LETOR 4.0 Datasets](https://mlsanity.com/paper/1306.2597/) (2013)
- [NODE-GAM: Neural Generalized Additive Model for Interpretable Deep Learning](https://mlsanity.com/paper/2106.01613/) (2021)
- [Detecting Statistical Interactions from Neural Network Weights](https://mlsanity.com/paper/1705.04977/) (2017)
- [Neural Basis Models for Interpretability](https://mlsanity.com/paper/2205.14120/) (2022)
- [Purifying Interaction Effects with the Functional ANOVA: An Efficient Algorithm for Recovering Identifiable Additive Models](https://mlsanity.com/paper/1911.04974/) (2019)
- [Statistical Aspects of SHAP: Functional ANOVA for Model Interpretation](https://mlsanity.com/paper/2208.09970/) (2022)
- [Monotonic Neural Additive Models: Pursuing Regulated Machine Learning Models for Credit Scoring](https://mlsanity.com/paper/2209.10070/) (2022)
- [Neural Decomposition: Functional ANOVA with Variational Autoencoders](https://mlsanity.com/paper/2006.14293/) (2020)
- [Scalable Higher-Order Tensor Product Spline Models](https://mlsanity.com/paper/2402.01090/) (2024)

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

- [Tractable Shapley Values and Interactions via Tensor Networks](https://mlsanity.com/paper/2510.22138/) (2025)
- [Bayesian Neural Networks for Functional ANOVA model](https://mlsanity.com/paper/2510.00545/) (2025)
- [Twincher: Bijective Representation Learning for Robust Inversion of Continuous Systems](https://mlsanity.com/paper/2605.13470/) (2026)
- [Beyond Additive Decompositions: Interpretability Through Separability](https://mlsanity.com/paper/2605.31200/) (2026)

## Abstract (arXiv)

Interpretability for machine learning models is becoming more and more important as machine learning models become more complex. The functional ANOVA model, which decomposes a high-dimensional function into a sum of lower dimensional functions (commonly referred to as components), is one of the most popular tools for interpretable AI, and recently, various neural networks have been developed for estimating each component in the functional ANOVA model. However, such neural networks are highly unstable when estimating each component since the components themselves are not uniquely defined. That is, there are multiple functional ANOVA decompositions for a given function. In this paper, we propose a novel neural network which guarantees a unique functional ANOVA decomposition and thus is able to estimate each component stably and accurately. We call our proposed neural network ANOVA Tensor Product Neural Network (ANOVA-TPNN) since it is motivated by the tensor product basis expansion. Theoretically, we prove that ANOVA-TPNN can approximate any smooth function well. Empirically, we show that ANOVA-TPNN provide much more stable estimation of each component and thus much more stable interpretation when training data and initial values of the model parameters vary than existing neural networks do.