On approximating $\nabla f$ with neural networks

Consider a feedforward neural network $\psi: \mathbb{R}^d\rightarrow \mathbb{R}^d$ such that $\psi\approx \nabla f$, where $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth function, therefore $\psi$ must satisfy $\partial_j \psi_i = \partial_i \psi_j$ pointwise. We prove a theorem that a $\psi$ network with more than one hidden layer can only represent one feature in its first hidden layer; this is a dramatic departure from the well-known results for one hidden layer. The proof of the theorem is straightforward, where two backward paths and a weight-tying matrix play the key roles. We then present the alternative, the implicit parametrization, where the neural network is $\phi: \mathbb{R}^d \rightarrow \mathbb{R}$ and $\nabla \phi \approx \nabla f$; in addition, a "soft analysis" of $\nabla \phi$ gives a dual perspective on the theorem. Throughout, we come back to recent probabilistic models that are formulated as $\nabla \phi \approx \nabla f$, and conclude with a critique of denoising autoencoders.