On approximating nabla f with neural networks

Consider a feedforward neural network psi: ℝdrightarrow ℝd such that psi≈ nabla f, where f:ℝd rightarrow ℝ is a smooth function, therefore psi must satisfy partialj psii = partiali psij 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 φ: ℝd rightarrow ℝ and nabla φ ≈ nabla f; in addition, a "soft analysis" of nabla φ gives a dual perspective on the theorem. Throughout, we come back to recent probabilistic models that are formulated as nabla φ ≈ nabla f, and conclude with a critique of denoising autoencoders.