# Multiplicative-Additive Constrained Models:Toward Joint Visualization of Interactive and Independent Effects

arXiv: https://arxiv.org/abs/2509.21923 · HTML: https://mlsanity.com/paper/2509.21923/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 25 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [DeepSeek-Coder: When the Large Language Model Meets Programming -- The Rise of Code Intelligence](https://mlsanity.com/paper/2401.14196/) (2024)
- [This Looks Like That: Deep Learning for Interpretable Image Recognition](https://mlsanity.com/paper/1806.10574/) (2018)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Generative Image Dynamics](https://mlsanity.com/paper/2309.07906/) (2023)
- [GAMI-Net: An Explainable Neural Network based on Generalized Additive Models with Structured Interactions](https://mlsanity.com/paper/2003.07132/) (2020)
- [Supersparse Linear Integer Models for Optimized Medical Scoring Systems](https://mlsanity.com/paper/1502.04269/) (2015)
- [Neural Basis Models for Interpretability](https://mlsanity.com/paper/2205.14120/) (2022)
- [Sparse Interaction Additive Networks via Feature Interaction Detection and Sparse Selection](https://mlsanity.com/paper/2209.09326/) (2022)
- [ProtoNAM: Prototypical Neural Additive Models for Interpretable Deep Tabular Learning](https://mlsanity.com/paper/2410.04723/) (2024)

## Abstract (arXiv)

Interpretability is one of the considerations when applying machine learning to high-stakes fields such as healthcare that involve matters of life safety. Generalized Additive Models (GAMs) enhance interpretability by visualizing shape functions. Nevertheless, to preserve interpretability, GAMs omit higher-order interaction effects (beyond pairwise interactions), which imposes significant constraints on their predictive performance. We observe that Curve Ergodic Set Regression (CESR), a multiplicative model, naturally enables the visualization of its shape functions and simultaneously incorporates both interactions among all features and individual feature effects. Nevertheless, CESR fails to demonstrate superior performance compared to GAMs. We introduce Multiplicative-Additive Constrained Models (MACMs), which augment CESR with an additive part to disentangle the intertwined coefficients of its interactive and independent terms, thus effectively broadening the hypothesis space. The model is composed of a multiplicative part and an additive part, whose shape functions can both be naturally visualized, thereby assisting users in interpreting how features participate in the decision-making process. Consequently, MACMs constitute an improvement over both CESR and GAMs. The experimental results indicate that neural network-based MACMs significantly outperform both CESR and the current state-of-the-art GAMs in terms of predictive performance.