We present an explicit deep neural network construction that transforms uniformly distributed one-dimensional noise into an arbitrarily close approximation of any two-dimensional Lipschitz-continuous target distribution. The key ingredient of our design is a generalization of the "space-filling" property of sawtooth functions discovered in (Bailey & Telgarsky, 2018). We elicit the importance of depth - in our neural network construction - in driving the Wasserstein distance between the target distribution and the approximation realized by the network to zero. An extension to output distributions of arbitrary dimension is outlined. Finally, we show that the proposed construction does not incur a cost - in terms of error measured in Wasserstein-distance - relative to generating $d$-dimensional target distributions from $d$ independent random variables.

We present Heapcraft: an open-source suite of tools for monitoring and improving collaboration in Minecraft. At the core of our system is a data collection and analysis framework for recording gameplay. We collected over 3451 player-hours of game behavior from 908 different players, and performed a general study of online collaboration. To make our game analytics easily accessible, we developed interactive information visualization tools and an analysis framework for players, administrators, and researchers to explore graphs, maps and timelines of live server activity. As part of our research, we introduce the collaboration index, a metric which allows server administrators and researchers to quantify, predict, and improve collaboration on Minecraft servers. Our analysis reveals several possible predictors of collaboration which can be used to improve collaboration on Minecraft servers. Heapcraft is designed to be general, and has the potential to be used for other shared online virtual worlds.