# AGOP as Explanation: From Feature Learning to Per-Sample Attribution in Image Classifiers

arXiv: https://arxiv.org/abs/2605.12816 · HTML: https://mlsanity.com/paper/2605.12816/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 13 of 13)

- [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)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [SmoothGrad: removing noise by adding noise](https://mlsanity.com/paper/1706.03825/) (2017)
- [Grad-CAM++: Improved Visual Explanations for Deep Convolutional Networks](https://mlsanity.com/paper/1710.11063/) (2017)
- [Evaluating the visualization of what a Deep Neural Network has learned](https://mlsanity.com/paper/1509.06321/) (2015)
- [Ground Truth Evaluation of Neural Network Explanations with CLEVR-XAI](https://mlsanity.com/paper/2003.07258/) (2020)
- [Mechanism of feature learning in convolutional neural networks](https://mlsanity.com/paper/2309.00570/) (2023)
- [xRFM: Accurate, scalable, and interpretable feature learning models for tabular data](https://mlsanity.com/paper/2508.10053/) (2025)
- [XAI-TRIS: Non-linear image benchmarks to quantify false positive post-hoc attribution of feature importance](https://mlsanity.com/paper/2306.12816/) (2023)

## Built on by (most cited first, 1 of 1)

- [AGOP-IxG: A Gradient Covariance Filter for Local Feature Attribution on Tabular Data, with a Controlled Benchmark](https://mlsanity.com/paper/2605.15700/) (2026)

## Abstract (arXiv)

The Average Gradient Outer Product (AGOP) governs feature learning in neural networks: the Neural Feature Ansatz states that weight Gram matrices at each layer align with the corresponding AGOP matrices computed over the training distribution. We ask a complementary question: can this same quantity serve as a post-hoc attribution method for explaining individual predictions? We introduce AGOP-Weighted: a novel attribution method that multiplies the per-sample gradient by sqrt(diag(M) / max diag(M)), a training-distribution prior that suppresses gradient noise and amplifies consistently important pixels -- a combination not present in any prior attribution method. We formalise two companion variants -- AGOP-Local (per-sample gradient, equivalent to VanillaGrad) and AGOP-Global (diag(M) directly as a zero-cost saliency map) -- and implement an efficient training-time accumulation hook; AGOP-Global then requires zero inference cost (disk lookup) while AGOP-Weighted requires only a single gradient pass. We conduct the first rigorous comparison of AGOP attribution against Integrated Gradients (IG), SmoothGrad, GradCAM, and VanillaGrad across two benchmarks with pixel-level ground truth: (i) the synthetic XAI-TRIS benchmark (four classification scenarios, 8x8 images, CNN8by8) and (ii) the photorealistic CLEVR-XAI benchmark (ResNet-18 fine-tuned from ImageNet). AGOP-Weighted achieves 44% higher mIoU than IG on linear tasks; AGOP-Global achieves 7x higher mIoU than IG on multiplicative tasks (where IG falls below random) at zero inference cost. Both findings generalise to ResNet-18 on CLEVR-XAI (+18% and +37% respectively). We further show that GradCAM fails on small-resolution images due to spatial resolution collapse, and that diag(M) quality improves monotonically throughout training even after classification accuracy has plateaued.