In the teacher-student setting, variable-projection training of two-layer networks is shown to match a weighted ultra-fast diffusion in the zero-regularization limit, giving linear convergence of the learned feature distribution.
Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case
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abstract
In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strict minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations.
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Ultra-fast feature learning for the training of two-layer neural networks in the two-timescale regime
In the teacher-student setting, variable-projection training of two-layer networks is shown to match a weighted ultra-fast diffusion in the zero-regularization limit, giving linear convergence of the learned feature distribution.