# Semi-Structured Distributional Regression -- Extending Structured Additive Models by Arbitrary Deep Neural Networks and Data Modalities

arXiv: https://arxiv.org/abs/2002.05777 · HTML: https://mlsanity.com/paper/2002.05777/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2020.

## Built on by (most cited first, 24 of 24)

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- [Reflectance Estimation for Proximity Sensing by Vision-Language Models: Utilizing Distributional Semantics for Low-Level Cognition in Robotics](https://mlsanity.com/paper/2404.07717/) (2024)
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- [Semi-structured multi-state delinquency model for mortgage default](https://mlsanity.com/paper/2603.26309/) (2026)
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- [Anchor-MoE: A Mean-Anchored Mixture of Experts For Probabilistic Regression](https://mlsanity.com/paper/2508.16802/) (2025)
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## Abstract (arXiv)

Combining additive models and neural networks allows to broaden the scope of statistical regression and extend deep learning-based approaches by interpretable structured additive predictors at the same time. Existing attempts uniting the two modeling approaches are, however, limited to very specific combinations and, more importantly, involve an identifiability issue. As a consequence, interpretability and stable estimation are typically lost. We propose a general framework to combine structured regression models and deep neural networks into a unifying network architecture. To overcome the inherent identifiability issues between different model parts, we construct an orthogonalization cell that projects the deep neural network into the orthogonal complement of the statistical model predictor. This enables proper estimation of structured model parts and thereby interpretability. We demonstrate the framework's efficacy in numerical experiments and illustrate its special merits in benchmarks and real-world applications.