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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 featurein its first hidden layer; this is a dramatic departure from the well-knownresults 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 thenpresent the alternative, the implicit parametrization, where the neural networkis $\phi: \mathbb{R}^d \rightarrow \mathbb{R}$ and $\nabla \phi \approx \nablaf$; in addition, a "soft analysis" of $\nabla \phi$ gives a dual perspective onthe theorem. Throughout, we come back to recent probabilistic models that areformulated as $\nabla \phi \approx \nabla f$, and conclude with a critique ofdenoising autoencoders.

On approximating $\nabla f$ with neural networks