Abstract:
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 the target distribution and its approximation realized by the proposed neural network to zero. Finally, we outline how our construction can be extended to output distributions of arbitrary dimension.
