# Groups Don't Move as One: A Bregman Clustering View of Linearized Group Influence

arXiv: https://arxiv.org/abs/2610.14809 · HTML: https://mlsanity.com/paper/2610.14809/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 10 Oct 2026.

## Builds on (most cited first, 24 of 24)

- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [Studying Large Language Model Generalization with Influence Functions](https://mlsanity.com/paper/2308.03296/) (2023)
- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [Towards Efficient Data Valuation Based on the Shapley Value](https://mlsanity.com/paper/1902.10275/) (2019)
- [Influence Functions in Deep Learning Are Fragile](https://mlsanity.com/paper/2006.14651/) (2020)
- [On the Accuracy of Influence Functions for Measuring Group Effects](https://mlsanity.com/paper/1905.13289/) (2019)
- [If Influence Functions are the Answer, Then What is the Question?](https://mlsanity.com/paper/2209.05364/) (2022)
- [Practical Gauss-Newton Optimisation for Deep Learning](https://mlsanity.com/paper/1706.03662/) (2017)
- [Training Data Influence Analysis and Estimation: A Survey](https://mlsanity.com/paper/2212.04612/) (2022)
- [On Second-Order Group Influence Functions for Black-Box Predictions](https://mlsanity.com/paper/1911.00418/) (2019)
- [Input Similarity from the Neural Network Perspective](https://mlsanity.com/paper/2102.05262/) (2021)
- [A Swiss Army Infinitesimal Jackknife](https://mlsanity.com/paper/1806.00550/) (2018)
- [Theoretical and Practical Perspectives on what Influence Functions Do](https://mlsanity.com/paper/2305.16971/) (2023)
- [A scalable estimate of the extra-sample prediction error via approximate leave-one-out](https://mlsanity.com/paper/1801.10243/) (2018)
- [Revisiting inverse Hessian vector products for calculating influence functions](https://mlsanity.com/paper/2409.17357/) (2024)
- [Error Discovery by Clustering Influence Embeddings](https://mlsanity.com/paper/2312.04712/) (2023)
- [Efficient Parametric Approximations of Neural Network Function Space Distance](https://mlsanity.com/paper/2302.03519/) (2023)
- [Rescaled Influence Functions: Accurate Data Attribution in High Dimension](https://mlsanity.com/paper/2506.06656/) (2025)
- [Influence Functions for Edge Edits in Non-Convex Graph Neural Networks](https://mlsanity.com/paper/2506.04694/) (2025)
- [Interaction-Aware Influence Functions for Group Attribution](https://mlsanity.com/paper/2605.15675/) (2026)
- [On the Accuracy of Newton Step and Influence Function Data Attributions](https://mlsanity.com/paper/2512.12572/) (2025)
- [DeMix: Debugging Training Data with Mixed Data Error Types by Investigating Influence Vectors](https://mlsanity.com/paper/2606.11616/) (2026)

## Abstract (arXiv)

Influence functions provide an efficient alternative to retraining for assessing the effects of removing or reweighting training data. For groups, however, the standard first-order estimate reduces individual contributions to a sum that can conceal substantial differences among members, including opposing effects that cancel. We establish a Bregman centroid formulation of the linearized group influence objective, interpreting this aggregate as a single representative and using Bregman information (BI) to quantify the variation it omits. Extending this formulation to multiple clusters reveals distinct contributions whose sum recovers the original estimate. We further propose an estimator that jointly optimizes the weights of these cluster contributions, rather than simply summing them. Experiments uncover cluster-specific influence patterns and a positive association between target-specific BI and approximation error. The proposed estimator achieves higher R2 than standard group influence in predicting target changes under the nonlinear proximal Bregman response function. Simplified clustering and correction variants reduce computational cost. The centroid view thus explains what additive group influence leaves out and provides a basis for using that information to improve estimation.