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

NeurIPS 2026 · Architectures and efficiency › RNNs and normalization · approach: Mechanistic circuit interpretability · 0 citations (OpenAlex)

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Develops an exact dynamical theory of "abstraction" (cosine alignment of concept difference vectors) during gradient-flow training in linear and infinite-width nonlinear two-layer/deep networks, showing terminal, maximum, depth and nonlinearity (attenuation) laws, with checks in ResNets, DINOv3, Gemma 4 and macaque data.

Problem

Linear representation hypothesis and abstract representations are usually analyzed only at end of training; how concept-direction alignment (abstraction) evolves during training, and how depth, initialization, data geometry, and nonlinearity affect it, is poorly understood.

Approach

Reduce the two-layer linear network via variable projection on the readout so the feature kernel obeys a Riccati-type ODE; under a two-factor symmetry assumption the kernel diagonalizes into five modes and abstraction reduces to a scalar ODE in the signal-to-noise ratio, giving exact implicit trajectories. Extends to infinite-width erf and leaky-ReLU networks via NNGP kernel maps, proving an atten

Key results

• Terminal abstraction in linear networks follows alpha_inf = (1 - sqrt(nu_X nu_Y))/(1 + sqrt(nu_X nu_Y)), depending only on input and target inverse-SNRs.
• Maximum abstraction is strictly decreasing in initialization scale kappa, approaching 1 as kappa to 0 and alpha_inf as kappa to infinity.
• Attenuation law: for erf and leaky ReLU, feature abstraction is alpha_K = A_phi(Q) alpha_Q with 0 <= A_phi <= 1, so nonlinearities weaken abstraction relative to preactivations.

Surprising findings

• Pure ReLU networks can reach complete anti-abstraction (alpha -> -1) when input inverse-SNR is large, even with noiseless targets.
• Under ReLU, abstraction depends more on input geometry than on target geometry, contrary to the linear theory where noiseless targets force perfect abstraction.

Limitations

• Exact results rely on strong assumptions: variable-projected readout and two-factor symmetry on kernels.
• Extension to deep nonlinear networks is not analytically solved; layerwise balancing is assumed.
• Empirical validation in realistic models departs from assumptions and is qualitative.

Experimental setup

Datasets: 3dshapes Majaj-Hong macaque V4/IT recordings

Benchmarks: 3dshapes-derived shape/color task Majaj-Hong macaque V4/IT

Metrics: abstraction score (cosine similarity) linear probe generalization (CCGP)

Baselines: frozen-readout linear network theory tanh vs ReLU empirical comparison

Builds on: linear network learning dynamics (Saxe et al.) linear representation hypothesis NNGP kernels rich and lazy learning

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