Beyond Linear and Overcomplete Regimes: A Mean-Field Analysis of Bottleneck Autoencoders

Theoretical studies of machine learning models commonly consider different limiting regimes in which the learning dynamics of gradient descent becomes theoretically tractable. It is, however, desirable to have a systematically obtained picture of all qualitatively different extreme learning regimes for a particular type of models. In this paper we propose such a picture for large weight-tied linear autoencoders characterized by input and latent dimensions, initialization magnitude, and training set size. This model is nonlinear in the weights and its gradient flow does not have a general theoretical solution. We show that at the level of the formal loss-expansion hierarchy, its extreme regimes are naturally associated with faces of a triangular prism. In particular, there are five basic extreme regimes associated with the 2-faces of the prism: (1) large-data, (2) small-data, (3) mean-field, (4) narrow-latent, and (5) free. For regimes (1,2,3,4), we derive explicit expressions for both train and population limiting loss evolutions under gradient flow, obtaining very good agreement with experimental results.

Autoencoders are widely used for nonlinear dimensionality reduction and manifold learning. While most common implementations rely on both nonlinear encoders and decoders, we investigate the specific role of the encoder and the extent to which it can be constrained to be linear without reducing accuracy. We conduct a comparative study on four autoencoder architectures: standard fully nonlinear autoencoders (AE), linear-encoder autoencoders (Lenc-AE), linear-decoder autoencoders (Ldec-AE), and fully linear autoencoders (LAE), evaluated on synthetic manifolds, computational mechanics data sets, and real-world image data sets including MNIST. We demonstrate that imposing a linear encoder preserves most of the representational capacity of the autoencoder, provided the decoder remains nonlinear. In particular, Lenc-AE consistently outperforms both Ldec-AE and LAE, and achieves reconstruction quality comparable to fully nonlinear AE, while offering advantages in terms of parsimony and interpretability of the latent representation. These results suggest that the nonlinear decoder is the critical component for manifold learning, rather than the encoder. A geometric interpretation of this finding is developed, which identifies the precise conditions under which a linear encoder is sufficient, and the specific manifold configurations that expose its limitations.

Infinite-width limits are a standard way to reason about neural networks, but it is not automatic that the limiting learner has the same complexity-theoretic inductive bias as large finite networks. We study this question for Bayesian neural networks at the mean-field, or critical feature-learning, scaling. The central quantity is the \emph{reduced entropy} $$s_\infty(y,\varepsilon)=\limsup_N -\frac{1}{N}\log π_N^0(L\le \varepsilon),$$ the intensive prior cost of representing a target function $y$ to population mean-squared error $\varepsilon$. Our main result is a width-robust learnability theorem. At fixed depth, a family of Boolean-cube targets is learnable from polynomially many samples at infinite width if and only if it is learnable at polynomial width, if and only if its reduced entropy is polynomially bounded. Equivalently, up to polynomial slack in accuracy, the Bayesian mean-field learner generalizes exactly on the targets that can be represented by polynomial-size networks. The forward direction is proved by a form of subsampling: from the infinitely many hidden neurons in the mean-field solution, one can select polynomially many representatives and still preserve the learned function on every input simultaneously. At the critical scaling this subsampling has both an active'' component, which keeps the data-dependent low-dimensional statistics, and a lazy'' component, which resamples the entropy-dominated directions from the prior. Thus the infinite-width mean-field limit gives a clean analytic description of learning without introducing spurious width-dependent generalization power.