Abstract
We show that every d-dimensional probability distribution of bounded support can be generated through deep ReLU networks out of a 1-dimensional uniform input distribution. What is more, this is possible without incurring a cost—in terms of approximation error measured in Wasserstein-distance—relative to generating the d-dimensional target distribution from d independent random variables. This is enabled by a vast generalization of the space-filling approach discovered in Bailey and Telgarsky (in: Bengio (eds) Advances in neural information processing systems vol 31, pp 6489–6499. Curran Associates, Inc., Red Hook, 2018). The construction we propose elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero. Finally, we find that, for histogram target distributions, the number of bits needed to encode the corresponding generative network equals the fundamental limit for encoding probability distributions as dictated by quantization theory.
Highlights
Deep neural networks have been employed very successfully as generative models for complex natural data such as images [14,21] and natural language [4,26]
We show that every d-dimensional probability distribution of bounded support can be generated through deep ReLU networks out of a 1-dimensional uniform input distribution
The construction we propose elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero
Summary
Deep neural networks have been employed very successfully as generative models for complex natural data such as images [14,21] and natural language [4,26]. Notwithstanding the practical success of deep generative networks, a profound theoretical understanding of their representational capabilities is still lacking First results along these lines appear in [16], where it was shown that generative networks can approximate distributions arising from the composition of Barron functions [3]. We show that every target distribution supported on a bounded subset of Rd can be approximated arbitrarily well in terms of Wasserstein distance by pushing forward a 1-dimensional uniform source distribution through a ReLU network. We find the histogram distribution that best approximates it—for a given histogram resolution—in Wasserstein distance This histogram distribution is realized by a ReLU network driven by a uniform univariate input distribution. We find that, for histogram target distributions, the number of bits needed to encode the corresponding generative network equals the fundamental limit for encoding probability distributions as dictated by quantization theory [12]
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