Deriving Decoder-Free Sparse Autoencoders from First Principles

• URL: http://arxiv.org/abs/2601.06478v1
• Date: Sat, 10 Jan 2026 08:13:58 GMT
• Title: Deriving Decoder-Free Sparse Autoencoders from First Principles
• Abstract summary: Theory predicts collapse without volume control analogous to the log-determinant in Gaussian mixture models.<n>We instantiate the theory in a single-layer encoder with an LSE objective and InfoMax regularization for volume control.<n>Experiments confirm the theory's predictions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Gradient descent on log-sum-exp (LSE) objectives performs implicit expectation--maximization (EM): the gradient with respect to each component output equals its responsibility. The same theory predicts collapse without volume control analogous to the log-determinant in Gaussian mixture models. We instantiate the theory in a single-layer encoder with an LSE objective and InfoMax regularization for volume control. Experiments confirm the theory's predictions. The gradient--responsibility identity holds exactly; LSE alone collapses; variance prevents dead components; decorrelation prevents redundancy. The model exhibits EM-like optimization dynamics in which lower loss does not correspond to better features and adaptive optimizers offer no advantage. The resulting decoder-free model learns interpretable mixture components, confirming that implicit EM theory can prescribe architectures.

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