# The Loss Kernel: A Geometric Probe for Deep Learning Interpretability

arXiv: https://arxiv.org/abs/2509.26537 · HTML: https://mlsanity.com/paper/2509.26537/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 46 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [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)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [BERT Busters: Outlier Dimensions that Disrupt Transformers](https://mlsanity.com/paper/2105.06990/) (2021)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition](https://mlsanity.com/paper/2310.06301/) (2023)
- [You Are What You Eat -- AI Alignment Requires Understanding How Data Shapes Structure and Generalisation](https://mlsanity.com/paper/2502.05475/) (2025)
- [Dynamics of Transient Structure in In-Context Linear Regression Transformers](https://mlsanity.com/paper/2501.17745/) (2025)
- [Training Data Attribution (TDA): Examining Its Adoption & Use Cases](https://mlsanity.com/paper/2501.12642/) (2025)
- [Structure Development in List-Sorting Transformers](https://mlsanity.com/paper/2501.18666/) (2025)
- [Programs as Singularities](https://mlsanity.com/paper/2504.08075/) (2025)
- [W-Kernel and Its Principal Space for Frequentist Evaluation of Bayesian Estimators](https://mlsanity.com/paper/2311.13017/) (2023)

## Built on by (most cited first, 9 of 9)

- [Influence Dynamics and Stagewise Data Attribution](https://mlsanity.com/paper/2510.12071/) (2025)
- [Feature Identification via the Empirical NTK](https://mlsanity.com/paper/2510.00468/) (2025)
- [From Density Matrices to Phase Transitions in Deep Learning: Spectral Early Warnings and Interpretability](https://mlsanity.com/paper/2603.29805/) (2026)
- [Towards Worst-Case Guarantees with Scale-Aware Interpretability](https://mlsanity.com/paper/2602.05184/) (2026)
- [Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning](https://mlsanity.com/paper/2605.07980/) (2026)
- [Mechanistic Anomaly Detection via Functional Attribution](https://mlsanity.com/paper/2604.18970/) (2026)
- [Dead Directions: Geometric Singular Learning](https://mlsanity.com/paper/2606.05957/) (2026)
- [Differentiable Kernel Ridge Regression for Deep Learning Pipelines](https://mlsanity.com/paper/2605.02313/) (2026)
- [Patterning in Practice: Debiasing Reward Models with Susceptibilities](https://mlsanity.com/paper/2609.00699/) (2026)

## Abstract (arXiv)

We introduce the loss kernel, an interpretability method for measuring similarity between data points according to a trained neural network. The kernel is the covariance matrix of per-sample losses computed under a distribution of low-loss-preserving parameter perturbations. We first validate our method on a synthetic multitask problem, showing it separates inputs by task as predicted by theory. We then apply this kernel to Inception-v1 to visualize the structure of ImageNet, and we show that the kernel's structure aligns with the WordNet semantic hierarchy. This establishes the loss kernel as a practical tool for interpretability and data attribution.