Generative networks have experienced great empirical successes in distribution learning.

These authors contributed equally to this work.

We prove an exponential separation between depth 2 and depth 3 neural networks, when approximating an $\mathcal{O}(1)$-Lipschitz target function to constant accuracy, with respect to a distribution with support in $[0,1]^{d}$, assuming exponentially bounded weights. This addresses an open problem posed in \citet{safran2019depth}, and proves that the curse of dimensionality manifests in depth 2 approximation, even in cases where the target function can be represented efficiently using depth 3. Previously, lower bounds that were used to separate depth 2 from depth 3 required that at least one of the Lipschitz parameter, target accuracy or (some measure of) the size of the domain of approximation scale polynomially with the input dimension, whereas we fix the former two and restrict our domain to the unit hypercube. Our lower bound holds for a wide variety of activation functions, and is based on a novel application of an average- to worst-case random self-reducibility argument, to reduce the problem to threshold circuits lower bounds.

Generative networks have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that generative networks can generate high-dimensional complex data from a low-dimensional easy-to-sample distribution. However, this phenomenon can not be justified by existing theories. The widely held manifold hypothesis speculates that real-world data sets, such as natural images and signals, exhibit low-dimensional geometric structures. In this paper, we take such low-dimensional data structures into consideration by assuming that data distributions are supported on a low-dimensional manifold. We prove statistical guarantees of generative networks under the Wasserstein-1 loss. We show that the Wasserstein-1 loss converges to zero at a fast rate depending on the intrinsic dimension instead of the ambient data dimension. Our theory leverages the low-dimensional geometric structures in data sets and justifies the practical power of generative networks. We require no smoothness assumptions on the data distribution which is desirable in practice.

We solve an open question from Lu et al. (2017), by showing that any target network with inputs in $\mathbb{R}^d$ can be approximated by a width $O(d)$ network (independent of the target network's architecture), whose number of parameters is essentially larger only by a linear factor. In light of previous depth separation theorems, which imply that a similar result cannot hold when the roles of width and depth are interchanged, it follows that depth plays a more significant role than width in the expressive power of neural networks. We extend our results to constructing networks with bounded weights, and to constructing networks with width at most $d+2$, which is close to the minimal possible width due to previous lower bounds. Both of these constructions cause an extra polynomial factor in the number of parameters over the target network. We also show an exact representation of wide and shallow networks using deep and narrow networks which, in certain cases, does not increase the number of parameters over the target network.

Existing depth separation results for constant-depth networks essentially show that certain radial functions in $\mathbb{R}^d$, which can be easily approximated with depth $3$ networks, cannot be approximated by depth $2$ networks, even up to constant accuracy, unless their size is exponential in $d$. However, the functions used to demonstrate this are rapidly oscillating, with a Lipschitz parameter scaling polynomially with the dimension $d$ (or equivalently, by scaling the function, the hardness result applies to $\mathcal{O}(1)$-Lipschitz functions only when the target accuracy $\epsilon$ is at most $\text{poly}(1/d)$). In this paper, we study whether such depth separations might still hold in the natural setting of $\mathcal{O}(1)$-Lipschitz radial functions, when $\epsilon$ does not scale with $d$. Perhaps surprisingly, we show that the answer is negative: In contrast to the intuition suggested by previous work, it \emph{is} possible to approximate $\mathcal{O}(1)$-Lipschitz radial functions with depth $2$, size $\text{poly}(d)$ networks, for every constant $\epsilon$. We complement it by showing that approximating such functions is also possible with depth $2$, size $\text{poly}(1/\epsilon)$ networks, for every constant $d$. Finally, we show that it is not possible to have polynomial dependence in both $d,1/\epsilon$ simultaneously. Overall, our results indicate that in order to show depth separations for expressing $\mathcal{O}(1)$-Lipschitz functions with constant accuracy -- if at all possible -- one would need fundamentally different techniques than existing ones in the literature.

We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (B\"olcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with their recently established universal approximation property of affine function systems (B\"olcskei et al., 2018), this shows that deep neural networks approximate vastly different signal structures generated by the affine group, the Weyl-Heisenberg group, or through warping, and even certain fractals, all with approximation error decaying exponentially in the number of neurons. We also prove that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.