# Input Similarity from the Neural Network Perspective

arXiv: https://arxiv.org/abs/2102.05262 · HTML: https://mlsanity.com/paper/2102.05262/

## Key facts

- Cited by 79 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 26 references that are in the map.
- First posted on arXiv 10 Feb 2021.

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

- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [A Neural Algorithm of Artistic Style](https://mlsanity.com/paper/1508.06576/) (2015)
- [Representer Point Selection for Explaining Deep Neural Networks](https://mlsanity.com/paper/1811.09720/) (2018)
- [Learning from Noisy Labels with Distillation](https://mlsanity.com/paper/1703.02391/) (2017)
- [Riemannian metrics for neural networks I: feedforward networks](https://mlsanity.com/paper/1303.0818/) (2013)
- [Noisy Supervision for Correcting Misaligned Cadaster Maps Without Perfect Ground Truth Data](https://mlsanity.com/paper/1903.06529/) (2019)
- [Riemannian metrics for neural networks II: recurrent networks and learning symbolic data sequences](https://mlsanity.com/paper/1306.0514/) (2013)

## Built on by (most cited first, 30 of 79)

- [Datamodels: Predicting Predictions from Training Data](https://mlsanity.com/paper/2202.00622/) (2022)
- [Fishr: Invariant Gradient Variances for Out-of-Distribution Generalization](https://mlsanity.com/paper/2109.02934/) (2021)
- [If Influence Functions are the Answer, Then What is the Question?](https://mlsanity.com/paper/2209.05364/) (2022)
- [DataInf: Efficiently Estimating Data Influence in LoRA-tuned LLMs and Diffusion Models](https://mlsanity.com/paper/2310.00902/) (2023)
- [Intriguing Properties of Data Attribution on Diffusion Models](https://mlsanity.com/paper/2311.00500/) (2023)
- [Revisiting Methods for Finding Influential Examples](https://mlsanity.com/paper/2111.04683/) (2021)
- [An Empirical Comparison of Instance Attribution Methods for NLP](https://mlsanity.com/paper/2104.04128/) (2021)
- [Theoretical and Practical Perspectives on what Influence Functions Do](https://mlsanity.com/paper/2305.16971/) (2023)
- [Influence Tuning: Demoting Spurious Correlations via Instance Attribution and Instance-Driven Updates](https://mlsanity.com/paper/2110.03212/) (2021)
- [Understanding Influence Functions and Datamodels via Harmonic Analysis](https://mlsanity.com/paper/2210.01072/) (2022)
- [Diffusion Attribution Score: Evaluating Training Data Influence in Diffusion Models](https://mlsanity.com/paper/2410.18639/) (2024)
- [Do Influence Functions Work on Large Language Models?](https://mlsanity.com/paper/2409.19998/) (2024)
- [Supervision Complexity and its Role in Knowledge Distillation](https://mlsanity.com/paper/2301.12245/) (2023)
- [textttdattri: A Library for Efficient Data Attribution](https://mlsanity.com/paper/2410.04555/) (2024)
- [Adversarial Attacks on Knowledge Graph Embeddings via Instance Attribution Methods](https://mlsanity.com/paper/2111.03120/) (2021)
- [Bayesian Influence Functions for Hessian-Free Data Attribution](https://mlsanity.com/paper/2509.26544/) (2025)
- [Counterfactual Explanations for Misclassified Images: How Human and Machine Explanations Differ](https://mlsanity.com/paper/2212.08733/) (2022)
- [LayerIF: Estimating Layer Quality for Large Language Models using Influence Functions](https://mlsanity.com/paper/2505.23811/) (2025)
- [A Bayesian Approach To Analysing Training Data Attribution In Deep Learning](https://mlsanity.com/paper/2305.19765/) (2023)
- [How Many and Which Training Points Would Need to be Removed to Flip this Prediction?](https://mlsanity.com/paper/2302.02169/) (2023)
- [Outlier Gradient Analysis: Efficiently Identifying Detrimental Training Samples for Deep Learning Models](https://mlsanity.com/paper/2405.03869/) (2024)
- [On Training Data Influence of GPT Models](https://mlsanity.com/paper/2404.07840/) (2024)
- [Revealing the Parametric Knowledge of Language Models: A Unified Framework for Attribution Methods](https://mlsanity.com/paper/2404.18655/) (2024)
- [TarDiff: Target-Oriented Diffusion Guidance for Synthetic Electronic Health Record Time Series Generation](https://mlsanity.com/paper/2504.17613/) (2025)
- [GraSS: Scalable Data Attribution with Gradient Sparsification and Sparse Projection](https://mlsanity.com/paper/2505.18976/) (2025)
- [ALinFiK: Learning to Approximate Linearized Future Influence Kernel for Scalable Third-Party LLM Data Valuation](https://mlsanity.com/paper/2503.01052/) (2025)
- [Can neural networks extrapolate? Discussion of a theorem by Pedro Domingos](https://mlsanity.com/paper/2211.03566/) (2022)
- [Low-Dimensional Gradient Helps Out-of-Distribution Detection](https://mlsanity.com/paper/2310.17163/) (2023)
- [Sparse, Efficient and Explainable Data Attribution with DualXDA](https://mlsanity.com/paper/2402.12118/) (2024)
- [A Simple and Efficient Baseline for Data Attribution on Images](https://mlsanity.com/paper/2311.03386/) (2023)

## Abstract (arXiv)

We first exhibit a multimodal image registration task, for which a neural network trained on a dataset with noisy labels reaches almost perfect accuracy, far beyond noise variance. This surprising auto-denoising phenomenon can be explained as a noise averaging effect over the labels of similar input examples. This effect theoretically grows with the number of similar examples; the question is then to define and estimate the similarity of examples. We express a proper definition of similarity, from the neural network perspective, i.e. we quantify how undissociable two inputs A and B are, taking a machine learning viewpoint: how much a parameter variation designed to change the output for A would impact the output for B as well? We study the mathematical properties of this similarity measure, and show how to use it on a trained network to estimate sample density, in low complexity, enabling new types of statistical analysis for neural networks. We analyze data by retrieving samples perceived as similar by the network, and are able to quantify the denoising effect without requiring true labels. We also propose, during training, to enforce that examples known to be similar should also be seen as similar by the network, and notice speed-up training effects for certain datasets.