On the ability of neural nets to express distributions
This work addresses the theoretical gap in explaining how neural networks can model complex data distributions, which is incremental as it builds on existing theorems like Barron's Theorem.
The paper tackled the problem of understanding the expressivity of deep neural networks as generative models by providing a sufficient criterion for approximating functions with n hidden layers, translating this into a criterion for approximating probability distributions in Wasserstein distance. They demonstrated that compositions of Barron functions are more expressive than single Barron functions alone, building on recent lower bound work.
Deep neural nets have caused a revolution in many classification tasks. A related ongoing revolution -- also theoretically not understood -- concerns their ability to serve as generative models for complicated types of data such as images and texts. These models are trained using ideas like variational autoencoders and Generative Adversarial Networks. We take a first cut at explaining the expressivity of multilayer nets by giving a sufficient criterion for a function to be approximable by a neural network with $n$ hidden layers. A key ingredient is Barron's Theorem \cite{Barron1993}, which gives a Fourier criterion for approximability of a function by a neural network with 1 hidden layer. We show that a composition of $n$ functions which satisfy certain Fourier conditions ("Barron functions") can be approximated by a $n+1$-layer neural network. For probability distributions, this translates into a criterion for a probability distribution to be approximable in Wasserstein distance -- a natural metric on probability distributions -- by a neural network applied to a fixed base distribution (e.g., multivariate gaussian). Building up recent lower bound work, we also give an example function that shows that composition of Barron functions is more expressive than Barron functions alone.