Infinite-width limit of deep linear neural networks

1 code implementation • 29 Nov 2022 • Lénaïc Chizat, Maria Colombo, Xavier Fernández-Real, Alessio Figalli

We finally study the continuous-time limit obtained for infinitely wide linear neural networks and show that the linear predictors of the neural network converge at an exponential rate to the minimal $\ell_2$-norm minimizer of the risk.

Free boundary regularity for almost every solution to the Signorini problem

no code implementations • 5 Dec 2019 • Xavier Fernández-Real, Xavier Ros-Oton

In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $(n-1-\alpha_\circ)$-dimensional for almost all times $t$, for some $\alpha_\circ > 0$.

Analysis of PDEs 35R35, 47G20