Simon Martin, Francis Bach, Giulio Biroli · 2026 · HAL (Le Centre pour la Communication Scientifique Directe)
DOI: 10.48550/arxiv.2610.29864 · cited by 0 works (OpenAlex) · open-access PDF
We study gradient flow in a two-layer linear network in the high-dimensional regime where the input, inner and output dimensions diverge proportionally. Under whitened inputs and isotropic imbalance, we combine a closed-form solution of the dynamics with prior random matrix theory results to derive a high-dimensional limit for the dynamics. Our method yields exact asymptotic spectral distributions for the weight matrices at all times and characterizes the evolution of the learned subspace. In the small-initialization limit, we identify a silent alignment phenomenon: the singular directions of the network align on an order-one timescale, while the singular values and empirical loss remain at their initial values. Visible learning occurs later, on a logarithmic timescale, as singular values progressively escape zero and the loss decreases. This phase is characterized by a saddle-to-saddle dynamics, which becomes a continuous process of spectral learning in the high-dimensional limit.
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Topics: Mathematics, Limit (mathematics), Singular value, Perspective (graphical), Logarithm, Matrix (chemical analysis)
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