# Nonparametric regression on random geometric graphs sampled from submanifolds

arXiv: https://arxiv.org/abs/2405.20909 · HTML: https://mlsanity.com/paper/2405.20909/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 58 references that are in the map.
- First posted on arXiv 31 May 2024.

## Builds on (most cited first, 21 of 21)

- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [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)
- [Analysis of p-Laplacian Regularization in Semi-Supervised Learning](https://mlsanity.com/paper/1707.06213/) (2017)
- [Spectral convergence of diffusion maps: improved error bounds and an alternative normalisation](https://mlsanity.com/paper/2006.02037/) (2020)
- [Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs](https://mlsanity.com/paper/1910.13476/) (2019)
- [Rates of Convergence for Laplacian Semi-Supervised Learning with Low Labeling Rates](https://mlsanity.com/paper/2006.02765/) (2020)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [The density of expected persistence diagrams and its kernel based estimation](https://mlsanity.com/paper/1802.10457/) (2018)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [Minimax Rates for Estimating the Dimension of a Manifold](https://mlsanity.com/paper/1605.01011/) (2016)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Estimating densities with nonlinear support using Fisher-Gaussian kernels](https://mlsanity.com/paper/1907.05918/) (2019)
- [Unlabeled Data Help in Graph-Based Semi-Supervised Learning: A Bayesian Nonparametrics Perspective](https://mlsanity.com/paper/2008.11809/) (2020)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Some aspects of symmetric Gamma process mixtures](https://mlsanity.com/paper/1504.00476/) (2015)
- [Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps](https://mlsanity.com/paper/2402.14985/) (2024)

## Built on by (most cited first, 1 of 1)

- [Node Regression on Latent Position Random Graphs via Local Averaging](https://mlsanity.com/paper/2410.21987/) (2024)

## Abstract (arXiv)

We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.