# Learning and Recovery in the ReLU Model

Proceedings of 57th Annual Allerton Conference on Communication, Control, and Computing, 2019 (to appear)

## Abstract

Rectified linear units, or ReLUs, have become a preferred activation function for artificial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a simple network with ReLU activations. The first problem we study corresponds to 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. 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 $\mathbf{b}$.

The second problem we consider is that of robust recovery of the signal in the presence of outliers. In this setting, we are interested in recovering the latent vector $\mathbf{c}$ from its noisy nonlinear images of the form $\mathbf{v} = \mathrm{ReLU}(A\mathbf{c}) + \mathbf{e}+\mathbf{w}$, where $\mathbf{e} \in \mathbb{R}^d$ denotes the outliers with sparsity $s$ and $\mathbf{w} \in \mathbb{R}^d$ denote the dense but small noise. We show that the LASSO algorithm recovers $\mathbf{c} \in \mathbb{R}^k$ within an $\ell_2$-error of $O\big(\sqrt{{((k+s)\log d})/{d}}\big)$ when $A$ is a random Gaussian matrix.