# Algorithms for Lipschitz Learning on Graphs

arXiv: https://arxiv.org/abs/1505.00290 · HTML: https://mlsanity.com/paper/1505.00290/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2015.

## Built on by (most cited first, 16 of 16)

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- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Laplacian-based Semi-Supervised Learning in Multilayer Hypergraphs by Coordinate Descent](https://mlsanity.com/paper/2301.12184/) (2023)
- [Interface Laplace Learning: Learnable Interface Term Helps Semi-Supervised Learning](https://mlsanity.com/paper/2408.05419/) (2024)
- [Enhancement of Quantum Semi-Supervised Learning via Improved Laplacian and Poisson Methods](https://mlsanity.com/paper/2508.02054/) (2025)
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## Abstract (arXiv)

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large p of p-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time widetildeO (m n). The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform l0-regularization on the given values in polynomial time and l1-regularization on the initial function values and on graph edge weights in time widetildeO (m3/2).