arXiv Machine Learning
Sep 25

Pointwise Generalization in Deep Neural Networks

The paper introduces a pointwise generalization theory for fully connected deep neural networks, using a pointwise Riemannian Dimension derived from eigenvalues of learned feature representations across layers. This framework provides hypothesis-dependent, representation-aware generalization bounds that are significantly tighter than traditional size- or norm-based approaches, both theoretically and experimentally. The authors analytically identify structural properties that explain deep networks’ tractability and empirically show that the pointwise Riemannian Dimension captures feature compression, over‑parameterization effects, and optimizer bias.

By Shaojie Li, Yunbei Xu
arXiv Machine Learning
Aug 31

Towards a mathematical theory of superposition

The paper develops a mathematical theory of superposition in neural networks using frame theory and compressed sensing. It shows that a sparse binary vector of active features can be encoded by an overcomplete dictionary and recovered via a ReLU operation with a suitable bias. The authors prove recovery theorems for both random-support and worst-case support settings, providing high-probability guarantees for low-coherence dictionaries and a sharp criterion for sparsity levels, with explicit results for Gaussian random matrices and equiangular tight frames.

By Michael I. Ivanitskiy, John Jasper, Emily J. King, Dustin G. Mixon
arXiv Machine Learning
Oct 9

Stop Probing, Start Coding: Why Linear Probes and Sparse Autoencoders Fail at Compositional Generalisation

The paper argues that the linear representation hypothesis—where neural activations encode concepts as linear mixtures—breaks down under superposition, causing linear decision boundaries to lose linearity after projection. Classical sparse coding with per‑sample inference can recover latent factors, but sparse autoencoders (SAEs) amortise this inference into a fixed encoder, creating a systematic gap that persists across training sizes, latent dimensions, and sparsity levels. Experiments show that the failure of SAEs under out‑of‑distribution compositional shifts stems from poor dictionary learning rather than inference, and that a good dictionary can solve the problem at all tested scales, highlighting scalable dictionary learning as the key open challenge.

By Vit\'oria Barin-Pacela, Shruti Joshi, Isabela Camacho, Simon Lacoste-Julien, David Klindt