# Learning Globally Smooth Functions on Manifolds

arXiv: https://arxiv.org/abs/2210.00301 · HTML: https://mlsanity.com/paper/2210.00301/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 80 references that are in the map.
- First posted on arXiv 1 Oct 2022.

## Builds on (most cited first, 30 of 42)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Robustness via curvature regularization, and vice versa](https://mlsanity.com/paper/1811.09716/) (2018)
- [Disentangling Adversarial Robustness and Generalization](https://mlsanity.com/paper/1812.00740/) (2018)
- [Constrained Reinforcement Learning Has Zero Duality Gap](https://mlsanity.com/paper/1910.13393/) (2019)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [An Overview of Deep Semi-Supervised Learning](https://mlsanity.com/paper/2006.05278/) (2020)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Neural Lander: Stable Drone Landing Control using Learned Dynamics](https://mlsanity.com/paper/1811.08027/) (2018)
- [Safe Policies for Reinforcement Learning via Primal-Dual Methods](https://mlsanity.com/paper/1911.09101/) (2019)
- [Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application](https://mlsanity.com/paper/1309.1541/) (2013)
- [A law of robustness for two-layers neural networks](https://mlsanity.com/paper/2009.14444/) (2020)
- [Logically-Constrained Reinforcement Learning](https://mlsanity.com/paper/1801.08099/) (2018)
- [Achieving Zero Constraint Violation for Constrained Reinforcement Learning via Primal-Dual Approach](https://mlsanity.com/paper/2109.06332/) (2021)
- [Probably Approximately Correct Constrained Learning](https://mlsanity.com/paper/2006.05487/) (2020)
- [Neural Manifold Clustering and Embedding](https://mlsanity.com/paper/2201.10000/) (2022)
- [On the Geometry of Adversarial Examples](https://mlsanity.com/paper/1811.00525/) (2018)
- [A case for new neural network smoothness constraints](https://mlsanity.com/paper/2012.07969/) (2020)
- [Constrained Learning with Non-Convex Losses](https://mlsanity.com/paper/2103.05134/) (2021)
- [Learning Optimal Resource Allocations in Wireless Systems](https://mlsanity.com/paper/1807.08088/) (2018)
- [Adversarial Robustness with Semi-Infinite Constrained Learning](https://mlsanity.com/paper/2110.15767/) (2021)
- [CLIP: Cheap Lipschitz Training of Neural Networks](https://mlsanity.com/paper/2103.12531/) (2021)
- [KNODE-MPC: A Knowledge-based Data-driven Predictive Control Framework for Aerial Robots](https://mlsanity.com/paper/2109.04821/) (2021)
- [Learning Hybrid Control Barrier Functions from Data](https://mlsanity.com/paper/2011.04112/) (2020)

## Built on by (most cited first, 4 of 4)

- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions](https://mlsanity.com/paper/2409.05191/) (2024)
- [Convolutional Filtering on Sampled Manifolds](https://mlsanity.com/paper/2211.11058/) (2022)
- [Graph Semi-Supervised Learning for Point Classification on Data Manifolds](https://mlsanity.com/paper/2506.12197/) (2025)

## Abstract (arXiv)

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.