# coVariance Neural Networks

arXiv: https://arxiv.org/abs/2205.15856 · HTML: https://mlsanity.com/paper/2205.15856/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 50 references that are in the map.
- First posted on arXiv 31 May 2022.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [Graph Neural Networks in Network Neuroscience](https://mlsanity.com/paper/2106.03535/) (2021)
- [Computing Graph Neural Networks: A Survey from Algorithms to Accelerators](https://mlsanity.com/paper/2010.00130/) (2020)
- [Graph Neural Networks: Architectures, Stability and Transferability](https://mlsanity.com/paper/2008.01767/) (2020)
- [Convergence and Stability of Graph Convolutional Networks on Large Random Graphs](https://mlsanity.com/paper/2006.01868/) (2020)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [PCANet: A Simple Deep Learning Baseline for Image Classification?](https://mlsanity.com/paper/1404.3606/) (2014)
- [Algorithms for Learning Graphs in Financial Markets](https://mlsanity.com/paper/2012.15410/) (2020)
- [A Low Effort Approach to Structured CNN Design Using PCA](https://mlsanity.com/paper/1812.06224/) (2018)

## Built on by (most cited first, 20 of 20)

- [Explainable Brain Age Prediction using coVariance Neural Networks](https://mlsanity.com/paper/2305.18370/) (2023)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Sparse Covariance Neural Networks](https://mlsanity.com/paper/2410.01669/) (2024)
- [Fair CoVariance Neural Networks](https://mlsanity.com/paper/2409.08558/) (2024)
- [Transferability of coVariance Neural Networks and Application to Interpretable Brain Age Prediction using Anatomical Features](https://mlsanity.com/paper/2305.01807/) (2023)
- [Predicting Brain Age using Transferable coVariance Neural Networks](https://mlsanity.com/paper/2210.16363/) (2022)
- [Towards a Foundation Model for Brain Age Prediction using coVariance Neural Networks](https://mlsanity.com/paper/2402.07684/) (2024)
- [Covariance Density Neural Networks](https://mlsanity.com/paper/2505.11139/) (2025)
- [Disentangling Neurodegeneration with Brain Age Gap Prediction Models: A Graph Signal Processing Perspective](https://mlsanity.com/paper/2510.12763/) (2025)
- [Bridging Input Feature Spaces Towards Graph Foundation Models](https://mlsanity.com/paper/2605.04834/) (2026)
- [Learning with Covariance Matrices: Principal Component Analysis Meets Learning with Graphs](https://mlsanity.com/paper/2609.10490/) (2026)
- [Explainable Brain Age Gap Prediction in Neurodegenerative Conditions using coVariance Neural Networks](https://mlsanity.com/paper/2501.01510/) (2025)
- [Precision Neural Networks: Joint Graph And Relational Learning](https://mlsanity.com/paper/2509.14821/) (2025)
- [Neural Tangent Kernels Motivate Graph Neural Networks with Cross-Covariance Graphs](https://mlsanity.com/paper/2310.10791/) (2023)
- [Sampling Transferable Graph Neural Networks with Limited Graph Information](https://mlsanity.com/paper/2410.16593/) (2024)
- [CoVariance Filters and Neural Networks over Hilbert Spaces](https://mlsanity.com/paper/2509.13178/) (2025)
- [Graph Variate Neural Networks](https://mlsanity.com/paper/2509.20311/) (2025)
- [Spatiotemporal Kronecker Covariance Neural Networks](https://mlsanity.com/paper/2609.25326/) (2026)
- [Joint Precision Neural Networks: Task-Aware Dependency and Predictive Learning](https://mlsanity.com/paper/2610.06023/) (2026)
- [Colored Graph Signal Diffusion Models](https://mlsanity.com/paper/2610.17282/) (2026)

## Abstract (arXiv)

Graph neural networks (GNN) are an effective framework that exploit inter-relationships within graph-structured data for learning. Principal component analysis (PCA) involves the projection of data on the eigenspace of the covariance matrix and draws similarities with the graph convolutional filters in GNNs. Motivated by this observation, we study a GNN architecture, called coVariance neural network (VNN), that operates on sample covariance matrices as graphs. We theoretically establish the stability of VNNs to perturbations in the covariance matrix, thus, implying an advantage over standard PCA-based data analysis approaches that are prone to instability due to principal components associated with close eigenvalues. Our experiments on real-world datasets validate our theoretical results and show that VNN performance is indeed more stable than PCA-based statistical approaches. Moreover, our experiments on multi-resolution datasets also demonstrate that VNNs are amenable to transferability of performance over covariance matrices of different dimensions; a feature that is infeasible for PCA-based approaches.