Memorizing Gaussians with no over-parameterizaion via gradient decent on neural networks

Abstract: We prove that a single step of gradient decent over depth two network, with $q$ hidden neurons, starting from orthogonal initialization, can memorize $\Omega\left(\frac{dq}{\log<sup>4(d)}\right)$ independent and randomly labeled Gaussians in $\mathbb{R}<sup>d$. The result is valid for a large class of activation functions, which includes the absolute value.