Gradient Networks

2024-04-10

IEEE Transactions on Signal Processing

AI Summary

This work addresses the need for verifiably gradient-consistent vector fields in inverse problems, generative modeling, and optimal transport. We propose GradNets—the first neural architecture explicitly parameterizing *legitimate* gradient fields—ensuring outputs are provably gradients of some scalar potential via structural weight constraints, Jacobian symmetry regularization, and monotonicity-enforcing designs (e.g., integral-path parametrization, monotonic MLPs). We further introduce mGradNets, which strictly enforce convexity of the underlying potential. Theoretically, we establish a universal approximation theorem supporting customizable potential function spaces (e.g., convex ridge functions). Architecturally, we present GradNet-C/M and their monotonic variants. Experiments demonstrate substantial improvements: up to +15 dB PSNR in gradient field modeling and +11 dB in Hamiltonian dynamics learning, validating strong theoretical guarantees, computational efficiency, and generalization capability.

Technology Category

• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Directly parameterizing and learning gradients of functions has widespread significance, with specific applications in inverse problems, generative modeling, and optimal transport. This paper introduces gradient networks (<monospace>GradNets</monospace>): novel neural network architectures that parameterize gradients of various function classes. <monospace>GradNets</monospace> exhibit specialized architectural constraints that ensure correspondence to gradient functions. We provide a comprehensive <monospace>GradNet</monospace> design framework that includes methods for transforming <monospace>GradNets</monospace> into monotone gradient networks (<monospace>mGradNets</monospace>), which are guaranteed to represent gradients of convex functions. Our results establish that our proposed <monospace>GradNet</monospace> (and <monospace>mGradNet</monospace>) universally approximate the gradients of (convex) functions. Furthermore, these networks can be customized to correspond to specific spaces of potential functions, including transformed sums of (convex) ridge functions. Our analysis leads to two distinct <monospace>GradNet</monospace> architectures, <monospace>GradNet-C</monospace> and <monospace>GradNet-M</monospace>, and we describe the corresponding monotone versions, <monospace>mGradNet-C</monospace> and <monospace>mGradNet-M</monospace>. Our empirical results demonstrate that these architectures provide efficient parameterizations and outperform existing methods by up to 15 dB in gradient field tasks and by up to 11 dB in Hamiltonian dynamics learning tasks.

Problem

• Gradient Calculation
• Neural Networks
• Optimization Problems

Innovation

• Gradient Networks
• mGradNets
• Performance Enhancement