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 for any such $\psi$ networks, and for any depth $L>2$, all the input weights must be parallel to each other. In other words, $\psi$ can only represent $one$ feature in its first hidden layer. The proof of the theorem is straightforward, where two backward paths (from $i$ to $j$ and $j$ to $i$) and a weight-tying matrix (connecting the last and first hidden layers) play the key roles. We thus make a strong theoretical case in favor of the $implicit$ parametrization, where the neural network is $\phi: \mathbb{R}^d \rightarrow \mathbb{R}$ and $\nabla \phi \approx \nabla f$. Throughout, we revisit two recent unnormalized probabilistic models that are formulated as $\psi \approx \nabla f$ and also discuss the denoising autoencoders in the end.