In this paper we consider the problem of estimating a dense depth map from a set of sparse LiDAR points. We use techniques from compressed sensing and the recently developed Alternating Direction Neural Networks (ADNNs) to create a deep recurrent auto-encoder for this task. Our architecture internally performs an algorithm for extracting multi-level convolutional sparse codes from the input which are then used to make a prediction. Our results demonstrate that with only two layers and 1800 parameters we are able to out perform all previously published results, including deep networks with orders of magnitude more parameters.