mlsanityData to 14 Oct 2026

Input Convex Gradient Networks

arXiv 2111.12187 ↗First posted 23 Nov 2021cs.LG stat.MLcited by 17

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Abstract (arXiv)

The gradients of convex functions are expressive models of non-trivial vector fields. For example, Brenier's theorem yields that the optimal transport map between any two measures on Euclidean space under the squared distance is realized as a convex gradient, which is a key insight used in recent generative flow models. In this paper, we study how to model convex gradients by integrating a Jacobian-vector product parameterized by a neural network, which we call the Input Convex Gradient Network (ICGN). We theoretically study ICGNs and compare them to taking the gradient of an Input-Convex Neural Network (ICNN), empirically demonstrating that a single layer ICGN can fit a toy example better than a single layer ICNN. Lastly, we explore extensions to deeper networks and connections to constructions from Riemannian geometry.

Key facts. Cited by 17 papers in the map (data through 14 Oct 2026). Builds on 19 of its 32 references that are in the map. First posted on arXiv 23 Nov 2021.

Builds on

19 of 32 references.

References · 19

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2017Improved Training of Wasserstein GANscited by 2,235
  3. 2019Deep Equilibrium Modelscited by 615
  4. 2016Input Convex Neural Networkscited by 414
  5. 2017Learning Generative Models with Sinkhorn Divergencescited by 314
  6. 2018Sorting out Lipschitz function approximationcited by 213
  7. 2019Optimal transport mapping via input convex neural networkscited by 206
  8. 2017Sobolev Training for Neural Networkscited by 143
  9. 2018Improving GANs Using Optimal Transportcited by 123
  10. 2019Wasserstein-2 Generative Networkscited by 97
  11. 2020Neural Spatio-Temporal Point Processescited by 93
  12. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 90
7 more
  1. 2021Gradients are Not All You Needcited by 86
  2. 2021Large-Scale Wasserstein Gradient Flowscited by 77
  3. 2020Learned convex regularizers for inverse problemscited by 55
  4. 2021Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networkscited by 53
  5. 2020Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networkscited by 50
  6. 2020An Inductive Bias for Distances: Neural Nets that Respect the Triangle Inequalitycited by 24
  7. 2019On approximating nabla f with neural networkscited by 18

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 2; 2023: 5; 2024: 5; 2025: 4; 2026: 1.

17 papers.

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  1. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Stepcited by 78
  2. 2022Supervised Training of Conditional Monge Mapscited by 61
  3. 2023The Monge Gap: A Regularizer to Learn All Transport Mapscited by 32
  4. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  5. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 25
  6. 2022Neural Unbalanced Optimal Transport via Cycle-Consistent Semi-Couplingscited by 17
  7. 2024Differentiation Through Black-Box Quadratic Programming Solverscited by 10
  8. 2023Learning Gradients of Convex Functions with Monotone Gradient Networkscited by 7
  9. 2024JacNet: Learning Functions with Structured Jacobianscited by 7
  10. 2025Learning Brenier Potentials with Convex Generative Adversarial Neural Networkscited by 3
  11. 2023Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applicationscited by 1
  12. 2024A solution for the mean parametrization of the von Mises-Fisher distributioncited by 1
5 more
  1. 2026Amortized Maximum Inner Product Search with Learned Support Functionscited by 1
  2. 2024Gradient Networkscited by 0
  3. 2025Convex Physics Informed Neural Networks for the Monge-Amp`ere Optimal Transport Problemcited by 0
  4. 2025GradNetOT: Learning Optimal Transport Maps with GradNetscited by 0
  5. 2025Universal Representation of Generalized Convex Functions and their Gradientscited by 0