# Learning manifold diffusion semigroups from graph transition matrices

arXiv: https://arxiv.org/abs/2605.25383 · HTML: https://mlsanity.com/paper/2605.25383/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 35 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator](https://mlsanity.com/paper/1801.10108/) (2018)
- [Lipschitz regularity of graph Laplacians on random data clouds](https://mlsanity.com/paper/2007.06679/) (2020)
- [Eigen-convergence of Gaussian kernelized graph Laplacian by manifold heat interpolation](https://mlsanity.com/paper/2101.09875/) (2021)
- [Manifold learning with bi-stochastic kernels](https://mlsanity.com/paper/1711.06711/) (2017)
- [Spectral Convergence of the connection Laplacian from random samples](https://mlsanity.com/paper/1306.1587/) (2013)
- [Convergence of Graph Laplacian with kNN Self-tuned Kernels](https://mlsanity.com/paper/2011.01479/) (2020)
- [Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension](https://mlsanity.com/paper/1706.09396/) (2017)
- [Bi-stochastically normalized graph Laplacian: convergence to manifold Laplacian and robustness to outlier noise](https://mlsanity.com/paper/2206.11386/) (2022)
- [Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds](https://mlsanity.com/paper/2506.00171/) (2025)
- [Improved convergence rate of kNN graph Laplacians: differentiable self-tuned affinity](https://mlsanity.com/paper/2410.23212/) (2024)

## Abstract (arXiv)

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup Qt = etDelta can be approximated directly by iterating the graph transition matrix P, under only low regularity assumptions on the test function f, including the case f ∈ L^∞. We bound | Pn f - Qt f | in ∞-norm, with the operator application to f properly defined, and we recover the classical graph-Laplacian pointwise rate O(N-2/(d+6)) up to logarithmic factors, for diffusion times t  up to O(1) and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of Qt f at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density p is C3 and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.