This paper studies the memorization capacity of deep neural networks with ReLU activation. Specifically, we investigate the minimal size of such networks to memorize any $N$ data points in the unit ball with pairwise separation distance $δ$ and discrete labels. Most prior studies characterize the memorization capacity by the number of parameters or neurons. We generalize these results by constructing neural networks, whose width $W$ and depth $L$ satisfy $W^2L^2= \mathcal{O}(N\log(δ^{-1}))$, that can memorize any $N$ data samples. We also prove that any such networks should also satisfy the lower bound $W^2L^2=Ω(N \log(δ^{-1}))$, which implies that our construction is optimal up to logarithmic factors when $δ^{-1}$ is polynomial in $N$. Hence, we explicitly characterize the trade-off between width and depth for the memorization capacity of deep neural networks in this regime.

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