# Test Sample Accuracy Scales with Training Sample Density in Neural Networks

arXiv: https://arxiv.org/abs/2106.08365 · HTML: https://mlsanity.com/paper/2106.08365/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 32 references that are in the map.
- First posted on arXiv 15 Jun 2021.

## Builds on (most cited first, 21 of 21)

- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [A Simple Unified Framework for Detecting Out-of-Distribution Samples and Adversarial Attacks](https://mlsanity.com/paper/1807.03888/) (2018)
- [Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks](https://mlsanity.com/paper/1706.02690/) (2017)
- [Can You Trust Your Model's Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift](https://mlsanity.com/paper/1906.02530/) (2019)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [Invariant Information Clustering for Unsupervised Image Classification and Segmentation](https://mlsanity.com/paper/1807.06653/) (2018)
- [CSI: Novelty Detection via Contrastive Learning on Distributionally Shifted Instances](https://mlsanity.com/paper/2007.08176/) (2020)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Inductive Biases for Deep Learning of Higher-Level Cognition](https://mlsanity.com/paper/2011.15091/) (2020)
- [SSD: A Unified Framework for Self-Supervised Outlier Detection](https://mlsanity.com/paper/2103.12051/) (2021)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization](https://mlsanity.com/paper/2106.06607/) (2021)
- [Classification-Based Anomaly Detection for General Data](https://mlsanity.com/paper/2005.02359/) (2020)
- [To Trust Or Not To Trust A Classifier](https://mlsanity.com/paper/1805.11783/) (2018)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Hybrid Models for Open Set Recognition](https://mlsanity.com/paper/2003.12506/) (2020)

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

- [Reliable Visual Question Answering: Abstain Rather Than Answer Incorrectly](https://mlsanity.com/paper/2204.13631/) (2022)
- [Real-Time Style Modelling of Human Locomotion via Feature-Wise Transformations and Local Motion Phases](https://mlsanity.com/paper/2201.04439/) (2022)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Sensitivity-Aware Amortized Bayesian Inference](https://mlsanity.com/paper/2310.11122/) (2023)
- [RSMT: Real-time Stylized Motion Transition for Characters](https://mlsanity.com/paper/2306.11970/) (2023)
- [A Novel Data Augmentation Technique for Out-of-Distribution Sample Detection using Compounded Corruptions](https://mlsanity.com/paper/2207.13916/) (2022)
- [Characterizing the Discrete Geometry of ReLU Networks](https://mlsanity.com/paper/2606.07728/) (2026)
- [Properties of Minimizing Entropy](https://mlsanity.com/paper/2112.03143/) (2021)
- [Complexity of Linear Regions in Self-supervised Deep ReLU Networks](https://mlsanity.com/paper/2604.24393/) (2026)

## Abstract (arXiv)

Intuitively, one would expect accuracy of a trained neural network's prediction on test samples to correlate with how densely the samples are surrounded by seen training samples in representation space. We find that a bound on empirical training error smoothed across linear activation regions scales inversely with training sample density in representation space. Empirically, we verify this bound is a strong predictor of the inaccuracy of the network's prediction on test samples. For unseen test sets, including those with out-of-distribution samples, ranking test samples by their local region's error bound and discarding samples with the highest bounds raises prediction accuracy by up to 20% in absolute terms for image classification datasets, on average over thresholds.