LEARNING DISTRIBUTIONS GENERATED BY SINGLE-LAYER RELU NETWORKS IN THE PRESENCE OF ARBITRARY OUTLIERS

We consider a set of data samples such that a constant fraction of the samples are arbitrary outliers and the rest are the output samples of a single-layer neural network (NN) with rectified linear unit (ReLU) activation. The goal of this paper is to estimate the parameters (weight matrix and bias vector) of the NN assuming the bias vector to be non-negative. Our proposed method is a two-step algorithm. We first estimate the norms of the rows of the weight matrix and the bias vector using the gradient descent algorithm. Here, we also incorporate either the median or the trimmed mean based filters to mitigate the effect of the arbitrary outliers. Next, we estimate the angles between any two row vectors of the weight matrix. Combining the estimates of the norms and the angles, we obtain the final estimate of the weight matrix. Further, we prove that ${O}(\frac{1}{\epsilon p^4}\log\frac{d}{\delta})$ samples are sufficient for our algorithm to estimate the NN parameters within an error of $\epsilon$ with probability $1-\delta$ when the probability of a sample being uncorrupted is $p$ and the problem dimension is $d$. Our theoretical and simulation results provide insights on how the estimation of the NN parameters depends on the probability of a sample being uncorrupted, the number of samples, and the problem dimension.