# NObSP: Functional Decomposition of Neural Networks via Oblique Subspace Projections

arXiv: https://arxiv.org/abs/2609.17825 · HTML: https://mlsanity.com/paper/2609.17825/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 40 references that are in the map.
- First posted on arXiv 15 Sep 2026.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [EfficientNet: Rethinking Model Scaling for Convolutional Neural Networks](https://mlsanity.com/paper/1905.11946/) (2019)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Learning Deep Features for Discriminative Localization](https://mlsanity.com/paper/1512.04150/) (2015)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Towards A Rigorous Science of Interpretable Machine Learning](https://mlsanity.com/paper/1702.08608/) (2017)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Designing Network Design Spaces](https://mlsanity.com/paper/2003.13678/) (2020)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [The Mythos of Model Interpretability](https://mlsanity.com/paper/1606.03490/) (2016)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Score-CAM: Score-Weighted Visual Explanations for Convolutional Neural Networks](https://mlsanity.com/paper/1910.01279/) (2019)
- [European Union regulations on algorithmic decision-making and a "right to explanation"](https://mlsanity.com/paper/1606.08813/) (2016)
- [Neural Additive Models: Interpretable Machine Learning with Neural Nets](https://mlsanity.com/paper/2004.13912/) (2020)
- [Visualizing the Effects of Predictor Variables in Black Box Supervised Learning Models](https://mlsanity.com/paper/1612.08468/) (2016)
- [Eigen-CAM: Class Activation Map using Principal Components](https://mlsanity.com/paper/2008.00299/) (2020)
- [Quantus: An Explainable AI Toolkit for Responsible Evaluation of Neural Network Explanations and Beyond](https://mlsanity.com/paper/2202.06861/) (2022)

## Abstract (arXiv)

Understanding how deep neural networks make decisions remains a fundamental challenge. We present NObSP (Nonlinear Oblique Subspace Projections), a framework that decomposes predictions into explicit per feature contribution functions and an interaction residual. NObSP exploits the linear final layer of a trained network and uses oblique projections in sample space to reduce double counting when learned feature subspaces overlap, thereby supporting both local explanations and global functional analysis. We establish connections to functional ANOVA and the Kolmogorov-Arnold representation theorem and derive an efficient partial regression algorithm for out of sample evaluation. For convolutional networks, NObSP-CAM produces class activation maps without backward passes after a one time calibration. Experiments on tabular and vision benchmarks show faithfulness comparable to established attribution methods. On a synthetic benchmark with known component functions, NObSP obtains a Function Reproduction Score of 0.989, compared with 0.966 for KernelSHAP and 0.922 for Integrated Gradients. On TinyImageNet, contribution vector embeddings improve mean nearest neighbor class purity from 0.654 for raw activations to 0.713 and reduce mean neighbor distance by more than half. These results indicate that NObSP complements scalar attribution methods by recovering functional contribution profiles with separable positive and negative evidence.