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

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

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