# Deep Networks Favor Simple Data

arXiv: https://arxiv.org/abs/2604.00394 · HTML: https://mlsanity.com/paper/2604.00394/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 22 references that are in the map.
- First posted on arXiv 1 Apr 2026.

## Builds on (most cited first, 18 of 18)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Likelihood Ratios for Out-of-Distribution Detection](https://mlsanity.com/paper/1906.02845/) (2019)
- [Do Deep Generative Models Know What They Don't Know?](https://mlsanity.com/paper/1810.09136/) (2018)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Why Normalizing Flows Fail to Detect Out-of-Distribution Data](https://mlsanity.com/paper/2006.08545/) (2020)
- [Input complexity and out-of-distribution detection with likelihood-based generative models](https://mlsanity.com/paper/1909.11480/) (2019)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Likelihood Ratios and Generative Classifiers for Unsupervised Out-of-Domain Detection In Task Oriented Dialog](https://mlsanity.com/paper/1912.12800/) (2019)
- [Neural Networks Learn Statistics of Increasing Complexity](https://mlsanity.com/paper/2402.04362/) (2024)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [Gaussian Embeddings: How JEPAs Secretly Learn Your Data Density](https://mlsanity.com/paper/2510.05949/) (2025)

## Abstract (arXiv)

Estimated density is often interpreted as indicating how typical a sample is under a model. Yet deep models trained on one dataset can assign higher density to simpler out-of-distribution (OOD) data than to in-distribution test data. We refer to this behavior as the OOD anomaly. Prior work typically studies this phenomenon within a single architecture, detector, or benchmark, implicitly assuming certain canonical densities. We instead separate the trained network from the density estimator built from its representations or outputs. We introduce two estimators: Jacobian-based estimators and autoregressive self-estimators, making density analysis applicable to a wide range of models. Applying this perspective to a range of models, including iGPT, PixelCNN++, Glow, score-based diffusion models, DINOv2, and I-JEPA, we find the same striking regularity that goes beyond the OOD anomaly: lower-complexity samples receive higher estimated density, while higher-complexity samples receive lower estimated density. This ordering appears within a test set and across OOD pairs such as CIFAR-10 and SVHN, and remains highly consistent across independently trained models. To quantify these orderings, we introduce Spearman rank correlation and find striking agreement both across models and with external complexity metrics. Even when trained only on the lowest-density (most complex) samples - or even a single such sample - the resulting models still rank simpler images as higher density. These observations lead us beyond the original OOD anomaly to a more general conclusion: deep networks consistently favor simple data. Our goal is not to close this question, but to define and visualize it more clearly. We broaden its empirical scope and show that it appears across architectures, objectives, and density estimators.