Learning Network Parameters in the ReLU Model

Rectified linear units, or ReLUs, have become a preferred activation function for artificial neural networks. In this paper we consider the problem of learning a generative model in the presence of nonlinearity (modeled by the ReLU functions). Given a set of signal vectors $\mathbf{y}^i \in \mathbb{R}^d, i =1, 2, \dots , n$, we aim to learn the network parameters, i.e., the $d\times k$ matrix $A$, under the model $\mathbf{y}^i = \mathrm{ReLU}(A\mathbf{c}^i +\mathbf{b})$, where $\mathbf{b}\in \mathbb{R}^d$ is a random bias vector, and {$\mathbf{c}^i \in \mathbb{R}^k$ are arbitrary unknown latent vectors}. We show that it is possible to recover the column space of $A$ within an error of $O(d)$ (in Frobenius norm) under certain conditions on the distribution of $\mathbf{b}$.

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