On approximating $\nabla f$ with neural networks

Abstract

Consider a feedforward neural network $\psi :{\mathbb{R}}^d\to {\mathbb{R}}^d$ such that $\psi \approx \nabla f$, where $f:{\mathbb{R}}^d\to \mathbb{R}$ 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 $\varphi :{\mathbb{R}}^d\to \mathbb{R}$ and $\nabla \varphi \approx \nabla f$; in addition, a "soft analysis" of $\nabla \varphi$ gives a dual perspective on the theorem. Throughout, we come back to recent probabilistic models that are formulated as $\nabla \varphi \approx \nabla f$, and conclude with a critique of denoising autoencoders.

arXiv comment10 pages

References in corpus (4)

• Deep Learning in Neural Networks: An Overview
• The Expressive Power of Neural Networks: A View from the Width
• Deep Energy Estimator Networks
• Neural Empirical Bayes

Cited by in corpus (8)

• How to Learn a Useful Critic? Model-based Action-Gradient-Estimator Policy Optimization
• Neural Empirical Bayes
• Provable Robust Classification via Learned Smoothed Densities
• Gradient Origin Networks
• Recomposing the Reinforcement Learning Building Blocks with Hypernetworks
• Learning and Inference in Imaginary Noise Models
• Input Convex Gradient Networks
• Unnormalized Variational Bayes