# EigenGame: PCA as a Nash Equilibrium

arXiv: https://arxiv.org/abs/2010.00554 · HTML: https://mlsanity.com/paper/2010.00554/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2020.

## Built on by (most cited first, 30 of 39)

- [Trust Region Policy Optimisation in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2109.11251/) (2021)
- [Remember the Past: Distilling Datasets into Addressable Memories for Neural Networks](https://mlsanity.com/paper/2206.02916/) (2022)
- [Heterogeneous-Agent Reinforcement Learning](https://mlsanity.com/paper/2304.09870/) (2023)
- [Enhancing multimodal cooperation via sample-level modality valuation](https://mlsanity.com/paper/2309.06255/) (2023)
- [NeuralEF: Deconstructing Kernels by Deep Neural Networks](https://mlsanity.com/paper/2205.00165/) (2022)
- [On the Limitations of Elo: Real-World Games, are Transitive, not Additive](https://mlsanity.com/paper/2206.12301/) (2022)
- [Neural Eigenfunctions Are Structured Representation Learners](https://mlsanity.com/paper/2210.12637/) (2022)
- [EigenGame Unloaded: When playing games is better than optimizing](https://mlsanity.com/paper/2102.04152/) (2021)
- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [Spectral Representation Learning for Conditional Moment Models](https://mlsanity.com/paper/2210.16525/) (2022)
- [A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms](https://mlsanity.com/paper/2211.12386/) (2022)
- [A Novel Stochastic Gradient Descent Algorithm for Learning Principal Subspaces](https://mlsanity.com/paper/2212.04025/) (2022)
- [Pursuit of a Discriminative Representation for Multiple Subspaces via Sequential Games](https://mlsanity.com/paper/2206.09120/) (2022)
- [Game Theoretic Rating in N-player general-sum games with Equilibria](https://mlsanity.com/paper/2210.02205/) (2022)
- [One Rank at a Time: Cascading Error Dynamics in Sequential Learning](https://mlsanity.com/paper/2505.22602/) (2025)
- [Developing, Evaluating and Scaling Learning Agents in Multi-Agent Environments](https://mlsanity.com/paper/2209.10958/) (2022)
- [Representations and Exploration for Deep Reinforcement Learning using Singular Value Decomposition](https://mlsanity.com/paper/2305.00654/) (2023)
- [Extending Kernel PCA through Dualization: Sparsity, Robustness and Fast Algorithms](https://mlsanity.com/paper/2306.05815/) (2023)
- [PixelGame: Infrared small target segmentation as a Nash equilibrium](https://mlsanity.com/paper/2205.13124/) (2022)
- [Bridging Neural and Symbolic Representations with Transitional Dictionary Learning](https://mlsanity.com/paper/2308.02000/) (2023)
- [On the Error-Propagation of Inexact Hotelling's Deflation for Principal Component Analysis](https://mlsanity.com/paper/2310.04283/) (2023)
- [Rethinking Teacher-Student Curriculum Learning through the Cooperative Mechanics of Experience](https://mlsanity.com/paper/2404.03084/) (2024)
- [Generalizable Spectral Embedding with an Application to UMAP](https://mlsanity.com/paper/2501.11305/) (2025)
- [Provable Model-Parallel Distributed Principal Component Analysis with Parallel Deflation](https://mlsanity.com/paper/2502.17615/) (2025)
- [A Non-Negative Matrix Factorization Game](https://mlsanity.com/paper/2104.05069/) (2021)
- [Speeding up PCA with priming](https://mlsanity.com/paper/2109.03709/) (2021)
- [Self-consistent Gradient-like Eigen Decomposition in Solving Schr"odinger Equations](https://mlsanity.com/paper/2202.01388/) (2022)
- [Calibration of Derivative Pricing Models: a Multi-Agent Reinforcement Learning Perspective](https://mlsanity.com/paper/2203.06865/) (2022)
- [The Symmetric Generalized Eigenvalue Problem as a Nash Equilibrium](https://mlsanity.com/paper/2206.04993/) (2022)
- [Stochastic Parallelizable Eigengap Dilation for Large Graph Clustering](https://mlsanity.com/paper/2207.14589/) (2022)

## Abstract (arXiv)

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm -- which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization -- is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.