# Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression

arXiv: https://arxiv.org/abs/2311.00140 · HTML: https://mlsanity.com/paper/2311.00140/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 69 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 14 of 14)

- [The Laplacian in RL: Learning Representations with Efficient Approximations](https://mlsanity.com/paper/1810.04586/) (2018)
- [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)
- [Trend Filtering on Graphs](https://mlsanity.com/paper/1410.7690/) (2014)
- [A variational approach to the consistency of spectral clustering](https://mlsanity.com/paper/1508.01928/) (2015)
- [Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms](https://mlsanity.com/paper/1805.09450/) (2018)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [Spectral Analysis Of Weighted Laplacians Arising In Data Clustering](https://mlsanity.com/paper/1909.06389/) (2019)
- [A maximum principle argument for the uniform convergence of graph Laplacian regressors](https://mlsanity.com/paper/1901.10089/) (2019)
- [A Spectral Series Approach to High-Dimensional Nonparametric Regression](https://mlsanity.com/paper/1602.00355/) (2016)
- [Topologically penalized regression on manifolds](https://mlsanity.com/paper/2110.13749/) (2021)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Eigenmaps on Neighborhood Graphs](https://mlsanity.com/paper/2111.07394/) (2021)
- [Data driven estimation of Laplace-Beltrami operator](https://mlsanity.com/paper/1612.09434/) (2016)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)

## Built on by (most cited first, 5 of 5)

- [On the Rate of Convergence of Kolmogorov-Arnold Network Regression Estimators](https://mlsanity.com/paper/2509.19830/) (2025)
- [Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps](https://mlsanity.com/paper/2402.14985/) (2024)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Node Regression on Latent Position Random Graphs via Local Averaging](https://mlsanity.com/paper/2410.21987/) (2024)
- [Graph Memory: Spectral Associative Memory via Dirichlet Energy](https://mlsanity.com/paper/2609.32365/) (2026)

## Abstract (arXiv)

We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods, when the true regression function belongs to a Sobolev space and the sampling density is bounded from above and below. The adaptation methodology is based on extensions of Lepski's method and is over both the smoothness parameter (s∈mathbbN+) and the norm parameter (M>0) determining the constraints on the Sobolev space. Our results extend the non-adaptive result in citegreen2021minimax, established for a specific normalized graph Laplacian, to a wide class of weighted Laplacian matrices used in practice, including the unnormalized Laplacian and random walk Laplacian.