# Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps

arXiv: https://arxiv.org/abs/2402.14985 · HTML: https://mlsanity.com/paper/2402.14985/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 63 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Builds on (most cited first, 19 of 19)

- [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)
- [Deep Neural Networks Learn Non-Smooth Functions Effectively](https://mlsanity.com/paper/1802.04474/) (2018)
- [Adaptive piecewise polynomial estimation via trend filtering](https://mlsanity.com/paper/1304.2986/) (2013)
- [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)
- [Total Variation Classes Beyond 1d: Minimax Rates, and the Limitations of Linear Smoothers](https://mlsanity.com/paper/1605.08400/) (2016)
- [Minimax Optimal Regression over Sobolev Spaces via Laplacian Regularization on Neighborhood Graphs](https://mlsanity.com/paper/2106.01529/) (2021)
- [Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation](https://mlsanity.com/paper/1903.01395/) (2019)
- [Spectral Analysis Of Weighted Laplacians Arising In Data Clustering](https://mlsanity.com/paper/1909.06389/) (2019)
- [Adaptive Estimation of Multivariate Piecewise Polynomials and Bounded Variation Functions by Optimal Decision Trees](https://mlsanity.com/paper/1911.11562/) (2019)
- [Bilevel Optimization, Deep Learning and Fractional Laplacian Regularization with Applications in Tomography](https://mlsanity.com/paper/1907.09605/) (2019)
- [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)
- [Efficiently Learning the Graph for Semi-supervised Learning](https://mlsanity.com/paper/2306.07098/) (2023)
- [Adaptive and non-adaptive minimax rates for weighted Laplacian-eigenmap based nonparametric regression](https://mlsanity.com/paper/2311.00140/) (2023)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [The Voronoigram: Minimax Estimation of Bounded Variation Functions From Scattered Data](https://mlsanity.com/paper/2212.14514/) (2022)

## Built on by (most cited first, 2 of 2)

- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [Graph Memory: Spectral Associative Memory via Dirichlet Energy](https://mlsanity.com/paper/2609.32365/) (2026)

## Abstract (arXiv)

We develop nonparametric regression methods for the case when the true regression function is not necessarily smooth. More specifically, our approach is using the fractional Laplacian and is designed to handle the case when the true regression function lies in an L2-fractional Sobolev space with order s∈ (0,1). This function class is a Hilbert space lying between the space of square-integrable functions and the first-order Sobolev space consisting of differentiable functions. It contains fractional power functions, piecewise constant or polynomial functions and bump function as canonical examples. For the proposed approach, we prove upper bounds on the in-sample mean-squared estimation error of order n^-frac2s2s+d, where d is the dimension, s is the aforementioned order parameter and n is the number of observations. We also provide preliminary empirical results validating the practical performance of the developed estimators.