[IOP / Proba-Stats] Attention-based PCA

We study attention mechanisms through the lens of a canonical unsupervised problem: principal component analysis (PCA). We show that, when trained on Gaussian data, both softmax and linear attention layers learn parameters that align with the principal eigenvectors of the covariance matrix, thereby establishing a direct and explicit connection with PCA. Our analysis covers both finite and infinite prompt regimes. In the infinite-prompt limit, we prove convergence to globally optimal solutions aligned with the leading spectral direction, while in the finite prompt setting we show that the same behavior emerges up to sampling effects. These results demonstrate that attention inherently performs PCA-like computations under unsupervised objectives, providing a theoretical foundation for its representation-learning capabilities.