# Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions

arXiv: https://arxiv.org/abs/2601.14515 · HTML: https://mlsanity.com/paper/2601.14515/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 37 references that are in the map.
- First posted on arXiv 20 Jan 2026.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Continuum limit of total variation on point clouds](https://mlsanity.com/paper/1403.6355/) (2014)
- [Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms](https://mlsanity.com/paper/1805.09450/) (2018)
- [Asymptotic behavior of ℓp-based Laplacian regularization in semi-supervised learning](https://mlsanity.com/paper/1603.00564/) (2016)
- [Autoencoders in Function Space](https://mlsanity.com/paper/2408.01362/) (2024)

## Abstract (arXiv)

Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy.