# Neural Decomposition: Functional ANOVA with Variational Autoencoders

arXiv: https://arxiv.org/abs/2006.14293 · HTML: https://mlsanity.com/paper/2006.14293/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Interpretability with full complexity by constraining feature information](https://mlsanity.com/paper/2211.17264/) (2022)
- [Ultra-marginal Feature Importance: Learning from Data with Causal Guarantees](https://mlsanity.com/paper/2204.09938/) (2022)
- [Tensor Product Neural Networks for Functional ANOVA Model](https://mlsanity.com/paper/2502.15215/) (2025)
- [Contrastive Mixture of Posteriors for Counterfactual Inference, Data Integration and Fairness](https://mlsanity.com/paper/2106.08161/) (2021)
- [ANOVA-boosting for Random Fourier Features](https://mlsanity.com/paper/2404.03050/) (2024)
- [Neural dSCA: demixing multimodal interaction among brain areas during naturalistic experiments](https://mlsanity.com/paper/2106.02948/) (2021)
- [CUBE: Contrastive Understanding by Balanced Experiments](https://mlsanity.com/paper/2509.10825/) (2025)
- [Bayesian Neural Networks for Functional ANOVA model](https://mlsanity.com/paper/2510.00545/) (2025)
- [TimeSAE: Causal Sparse Decoding for Faithful Explanations of Black-Box Time Series Models](https://mlsanity.com/paper/2601.09776/) (2026)

## Abstract (arXiv)

Variational Autoencoders (VAEs) have become a popular approach for dimensionality reduction. However, despite their ability to identify latent low-dimensional structures embedded within high-dimensional data, these latent representations are typically hard to interpret on their own. Due to the black-box nature of VAEs, their utility for healthcare and genomics applications has been limited. In this paper, we focus on characterising the sources of variation in Conditional VAEs. Our goal is to provide a feature-level variance decomposition, i.e. to decompose variation in the data by separating out the marginal additive effects of latent variables z and fixed inputs c from their non-linear interactions. We propose to achieve this through what we call Neural Decomposition - an adaptation of the well-known concept of functional ANOVA variance decomposition from classical statistics to deep learning models. We show how identifiability can be achieved by training models subject to constraints on the marginal properties of the decoder networks. We demonstrate the utility of our Neural Decomposition on a series of synthetic examples as well as high-dimensional genomics data.