We present an explicit deep network construction that transforms uniformly distributed one-dimensional noise into an arbitrarily close approximation of any two-dimensional target distribution of finite differential entropy and Lipschitz-continuous pdf. The key ingredient of our design is a generalization of the "space-filling” property of sawtooth functions introduced in (Bailey Telgarsky, 2018). We elicit the importance of depth in our construction in driving the Wasserstein distance between th…