# An Analytic Solution to Covariance Propagation in Neural Networks

arXiv: https://arxiv.org/abs/2403.16163 · HTML: https://mlsanity.com/paper/2403.16163/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 24 Mar 2024.

## Builds on (most cited first, 17 of 17)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Probabilistic Backpropagation for Scalable Learning of Bayesian Neural Networks](https://mlsanity.com/paper/1502.05336/) (2015)
- [A General Framework for Uncertainty Estimation in Deep Learning](https://mlsanity.com/paper/1907.06890/) (2019)
- [Lightweight Probabilistic Deep Networks](https://mlsanity.com/paper/1805.11327/) (2018)
- [Tractable Inference for Complex Stochastic Processes](https://mlsanity.com/paper/1301.7362/) (2013)
- [Semi-analytical approximations to statistical moments of sigmoid and softmax mappings of normal variables](https://mlsanity.com/paper/1703.00091/) (2017)
- [On the Sensitivity of Adversarial Robustness to Input Data Distributions](https://mlsanity.com/paper/1902.08336/) (2019)
- [How many perturbations break this model? Evaluating robustness beyond adversarial accuracy](https://mlsanity.com/paper/2207.04129/) (2022)
- [On The Distribution of Penultimate Activations of Classification Networks](https://mlsanity.com/paper/2107.01900/) (2021)

## Built on by (most cited first, 12 of 12)

- [Deterministic Uncertainty Propagation for Improved Model-Based Offline Reinforcement Learning](https://mlsanity.com/paper/2406.04088/) (2024)
- [Rethinking Approximate Gaussian Inference in Classification](https://mlsanity.com/paper/2502.03366/) (2025)
- [A Decision-driven Methodology for Designing Uncertainty-aware AI Self-Assessment](https://mlsanity.com/paper/2408.01301/) (2024)
- [A Guide to Failure in Machine Learning: Reliability and Robustness from Foundations to Practice](https://mlsanity.com/paper/2503.00563/) (2025)
- [ReLU Networks as Random Functions: Their Distribution in Probability Space](https://mlsanity.com/paper/2503.22082/) (2025)
- [Optimized Weight Initialization on the Stiefel Manifold for Deep ReLU Neural Networks](https://mlsanity.com/paper/2509.00362/) (2025)
- [Kalman Bayesian Transformer](https://mlsanity.com/paper/2509.10695/) (2025)
- [Uncertainty propagation through trained multi-layer perceptrons: Exact analytical results](https://mlsanity.com/paper/2601.16830/) (2026)
- [Noise Stability of Transformer Models](https://mlsanity.com/paper/2602.08287/) (2026)
- [Calibrated Sampling-Free Uncertainty Estimation in Bayesian Deep Learning](https://mlsanity.com/paper/2606.16214/) (2026)
- [Full-Covariance Smoothing of Bayesian Neural Networks for Online Adaptation](https://mlsanity.com/paper/2609.27244/) (2026)
- [Bounded Autonomy and Verifiable Safety for Agentic AI Enabled Automation](https://mlsanity.com/paper/2610.08815/) (2026)

## Abstract (arXiv)

Uncertainty quantification of neural networks is critical to measuring the reliability and robustness of deep learning systems. However, this often involves costly or inaccurate sampling methods and approximations. This paper presents a sample-free moment propagation technique that propagates mean vectors and covariance matrices across a network to accurately characterize the input-output distributions of neural networks. A key enabler of our technique is an analytic solution for the covariance of random variables passed through nonlinear activation functions, such as Heaviside, ReLU, and GELU. The wide applicability and merits of the proposed technique are shown in experiments analyzing the input-output distributions of trained neural networks and training Bayesian neural networks.