# Understanding Likelihood of Normalizing Flow and Image Complexity through the Lens of Out-of-Distribution Detection

arXiv: https://arxiv.org/abs/2402.10477 · HTML: https://mlsanity.com/paper/2402.10477/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 52 references that are in the map.
- First posted on arXiv 16 Feb 2024.

## Builds on (most cited first, 30 of 31)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [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)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [VOS: Learning What You Don't Know by Virtual Outlier Synthesis](https://mlsanity.com/paper/2202.01197/) (2022)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [OpenOOD: Benchmarking Generalized Out-of-Distribution Detection](https://mlsanity.com/paper/2210.07242/) (2022)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [MOS: Towards Scaling Out-of-distribution Detection for Large Semantic Space](https://mlsanity.com/paper/2105.01879/) (2021)
- [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)
- [Understanding Failures in Out-of-Distribution Detection with Deep Generative Models](https://mlsanity.com/paper/2107.06908/) (2021)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Provable Guarantees for Understanding Out-of-distribution Detection](https://mlsanity.com/paper/2112.00787/) (2021)
- [Understanding and Mitigating Exploding Inverses in Invertible Neural Networks](https://mlsanity.com/paper/2006.09347/) (2020)
- [Understanding Anomaly Detection with Deep Invertible Networks through Hierarchies of Distributions and Features](https://mlsanity.com/paper/2006.10848/) (2020)
- [Compression with Flows via Local Bits-Back Coding](https://mlsanity.com/paper/1905.08500/) (2019)
- [Further Analysis of Outlier Detection with Deep Generative Models](https://mlsanity.com/paper/2010.13064/) (2020)
- [Out-of-distribution Detection via Frequency-regularized Generative Models](https://mlsanity.com/paper/2208.09083/) (2022)

## Built on by (most cited first, 5 of 5)

- [Zero-Shot Image Anomaly Detection Using Generative Foundation Models](https://mlsanity.com/paper/2507.22692/) (2025)
- [Understanding Sensitivity of Differential Attention through the Lens of Adversarial Robustness](https://mlsanity.com/paper/2510.00517/) (2025)
- [Mitigating the Likelihood Paradox in Flow-based OOD Detection via Entropy Manipulation](https://mlsanity.com/paper/2602.09581/) (2026)
- [Why the Counterintuitive Phenomenon of Likelihood Rarely Appears in Tabular Anomaly Detection with Deep Generative Models?](https://mlsanity.com/paper/2602.09593/) (2026)
- [Hybrid Latent-Structural Fusion (HLSF) for Cyber Anomaly Detection](https://mlsanity.com/paper/2607.18479/) (2026)

## Abstract (arXiv)

Out-of-distribution (OOD) detection is crucial to safety-critical machine learning applications and has been extensively studied. While recent studies have predominantly focused on classifier-based methods, research on deep generative model (DGM)-based methods have lagged relatively. This disparity may be attributed to a perplexing phenomenon: DGMs often assign higher likelihoods to unknown OOD inputs than to their known training data. This paper focuses on explaining the underlying mechanism of this phenomenon. We propose a hypothesis that less complex images concentrate in high-density regions in the latent space, resulting in a higher likelihood assignment in the Normalizing Flow (NF). We experimentally demonstrate its validity for five NF architectures, concluding that their likelihood is untrustworthy. Additionally, we show that this problem can be alleviated by treating image complexity as an independent variable. Finally, we provide evidence of the potential applicability of our hypothesis in another DGM, PixelCNN++.