# Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms

arXiv: https://arxiv.org/abs/1805.09450 · HTML: https://mlsanity.com/paper/1805.09450/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

## Built on by (most cited first, 22 of 22)

- [Graph-based Semi-supervised Learning: A Comprehensive Review](https://mlsanity.com/paper/2102.13303/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [On Data-Augmentation and Consistency-Based Semi-Supervised Learning](https://mlsanity.com/paper/2101.06967/) (2021)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [Poisson Reweighted Laplacian Uncertainty Sampling for Graph-based Active Learning](https://mlsanity.com/paper/2210.15786/) (2022)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs](https://mlsanity.com/paper/2111.07394/) (2021)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Bayesian Learning via Q-Exponential Process](https://mlsanity.com/paper/2210.07987/) (2022)
- [Consistency of semi-supervised learning, stochastic tug-of-war games, and the p-Laplacian](https://mlsanity.com/paper/2401.07463/) (2024)
- [Manifold Filter-Combine Networks](https://mlsanity.com/paper/2307.04056/) (2023)
- [Gaussian Process Regression under Computational and Epistemic Misspecification](https://mlsanity.com/paper/2312.09225/) (2023)
- [Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps](https://mlsanity.com/paper/2402.14985/) (2024)
- [Models for information propagation on graphs](https://mlsanity.com/paper/2201.07577/) (2022)
- [Graph Laplacian-based Bayesian Multi-fidelity Modeling](https://mlsanity.com/paper/2409.08211/) (2024)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Manifold limit for the training of shallow graph convolutional neural networks](https://mlsanity.com/paper/2601.06025/) (2026)
- [A-Optimal Active Learning](https://mlsanity.com/paper/2110.09585/) (2021)
- [A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity data](https://mlsanity.com/paper/2304.04862/) (2023)
- [Barycentric model aggregation in the Wasserstein space of distributions and a variational approach to consistency](https://mlsanity.com/paper/2507.11719/) (2025)
- [Topology-Aware Active Learning on Graphs](https://mlsanity.com/paper/2510.25892/) (2025)
- [Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions](https://mlsanity.com/paper/2601.14515/) (2026)

## Abstract (arXiv)

Scalings in which the graph Laplacian approaches a differential operator in the large graph limit are used to develop understanding of a number of algorithms for semi-supervised learning; in particular the extension, to this graph setting, of the probit algorithm, level set and kriging methods, are studied. Both optimization and Bayesian approaches are considered, based around a regularizing quadratic form found from an affine transformation of the Laplacian, raised to a, possibly fractional, exponent. Conditions on the parameters defining this quadratic form are identified under which well-defined limiting continuum analogues of the optimization and Bayesian semi-supervised learning problems may be found, thereby shedding light on the design of algorithms in the large graph setting. The large graph limits of the optimization formulations are tackled through Gamma-convergence, using the recently introduced TLp metric. The small labelling noise limits of the Bayesian formulations are also identified, and contrasted with pre-existing harmonic function approaches to the problem.