# Overcoming the curse of dimensionality with Laplacian regularization in semi-supervised learning

arXiv: https://arxiv.org/abs/2009.04324 · HTML: https://mlsanity.com/paper/2009.04324/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2020.

## Built on by (most cited first, 14 of 14)

- [The SSL Interplay: Augmentations, Inductive Bias, and Generalization](https://mlsanity.com/paper/2302.02774/) (2023)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
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- [The Galerkin method beats Graph-Based Approaches for Spectral Algorithms](https://mlsanity.com/paper/2306.00742/) (2023)
- [Kernelized Diffusion maps](https://mlsanity.com/paper/2302.06757/) (2023)
- [Nonparametric Linear Feature Learning in Regression Through Regularisation](https://mlsanity.com/paper/2307.12754/) (2023)
- [Learning Globally Smooth Functions on Manifolds](https://mlsanity.com/paper/2210.00301/) (2022)
- [Active Labeling: Streaming Stochastic Gradients](https://mlsanity.com/paper/2205.13255/) (2022)
- [Joint Graph Learning and Model Fitting in Laplacian Regularized Stratified Models](https://mlsanity.com/paper/2305.02573/) (2023)
- [Curse of Dimensionality in Neural Network Optimization](https://mlsanity.com/paper/2502.05360/) (2025)
- [From Weakly Supervised Learning to Active Learning](https://mlsanity.com/paper/2209.11629/) (2022)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Semi-Supervised Learning with Noisy Proxy Covariates: Generalization Bounds and Distribution Regression](https://mlsanity.com/paper/2606.00512/) (2026)
- [Fast Learning Rates for Physics-Informed Kernel Methods](https://mlsanity.com/paper/2609.18901/) (2026)

## Abstract (arXiv)

As annotations of data can be scarce in large-scale practical problems, leveraging unlabelled examples is one of the most important aspects of machine learning. This is the aim of semi-supervised learning. To benefit from the access to unlabelled data, it is natural to diffuse smoothly knowledge of labelled data to unlabelled one. This induces to the use of Laplacian regularization. Yet, current implementations of Laplacian regularization suffer from several drawbacks, notably the well-known curse of dimensionality. In this paper, we provide a statistical analysis to overcome those issues, and unveil a large body of spectral filtering methods that exhibit desirable behaviors. They are implemented through (reproducing) kernel methods, for which we provide realistic computational guidelines in order to make our method usable with large amounts of data.