Loss Landscapes of Regularized Linear Autoencoders
This provides theoretical insights for PCA algorithms and computational neuroscience, but it is incremental as it builds on known limitations of LAEs.
The paper tackles the problem that linear autoencoders (LAEs) cannot learn principal directions when trained for reconstruction, proving that L2-regularized LAEs become symmetric at critical points and learn these directions as decoder singular vectors, with empirical validation and connections to probabilistic PCA.
Autoencoders are a deep learning model for representation learning. When trained to minimize the distance between the data and its reconstruction, linear autoencoders (LAEs) learn the subspace spanned by the top principal directions but cannot learn the principal directions themselves. In this paper, we prove that $L_2$-regularized LAEs are symmetric at all critical points and learn the principal directions as the left singular vectors of the decoder. We smoothly parameterize the critical manifold and relate the minima to the MAP estimate of probabilistic PCA. We illustrate these results empirically and consider implications for PCA algorithms, computational neuroscience, and the algebraic topology of learning.