How are linear representations learned? Exact solutions to the dynamics of abstraction

Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the Feature Learning Equation, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure - termed the Virtual Covariance - characterizes how representations evolve during training. Building on this perspective, we introduce Target Linearity, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to sequentially transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.

Neural networks acquire structured representations at specific moments during training, yet identifying these transitions typically relies on retrospective, label-dependent metrics. We introduce a bifurcation theory of representation dynamics to detect these moments in real time. Analyzing a passive GMM probe attached to the evolving encoder, we show the onset of structure corresponds to a supercritical pitchfork bifurcation driven by the loss Hessian. The system exhibits a theoretically predictable zero-crossing ($β_c$) that, compared to the network's current state ($β$), yields a dynamic ratio $β(t)/β_c(t)$: a universal, label-free phase coordinate for representation dynamics, computable entirely from hidden states. We empirically validate four distinct transition regimes predicted by this coordinate across diverse settings: SAEs on language models (Pythia), SSL (CIFAR), and grokking (modular arithmetic). Crucially, under finite dissipation, macroscopic symmetry-breaking can lag the initial zero-crossing by orders of magnitude, which providing a rigorous dynamical account of the delayed escape observed in grokking. Microscopically, the bifurcation creates a shared unstable subspace, forcing collective symmetry breaking. We term this the "feature lottery" in SAE training: a feature's terminal interpretability becomes predictable remarkably early. By only 5% of training, early atom purity robustly predicts final convergence purity, with top-decile early atoms achieving over 12x the baseline purity at convergence. Beyond explaining concept emergence, $β/β_c$ provides a practical early-warning indicator for training health, detecting the onset of usable structure, the crystallization of feature identity, and representational collapse epochs before downstream metrics react.

Grokking suggests that fitting the training data and learning a simple underlying rule may occur on different time scales. We formalize this phenomenon by separating the fast decay of the classification loss from the slower simplification of the learned representation, and we call the resulting pair of stopping times two training clocks. For deep linear networks, we show that a post-margin gap-growth or one-step tail-contraction condition reduces the cross-entropy loss to level epsilon on a logarithmic time scale. In contrast, when layerwise weight decay is present, the induced regularization on the end-to-end map can be expressed as a Schatten-type penalty; under a sharp late-time Kurdyka-Lojasiewicz tail, this structural energy closes on a polynomial time scale. The two clocks, therefore, separate fitting from representation simplification. We then explain how the same mechanism can appear in ReLU MLPs. In regions where the activation patterns on the training set remain fixed, the network reduces to a linear model in the active coordinates. In a two-layer ReLU embedding model, chain-rule estimates further show that the classifier head can receive larger effective gradients than the embedding block under controlled downstream norms. This supports a two-stage mechanism in which the classifier fits first, while the representation continues to simplify later. We use modular addition as the main experimental setting. The deep linear theory provides the rigorous core of the analysis. But the ReLU results are formulated as conditional reductions that account for empirical behavior without claiming a global proof for nonlinear training dynamics.