Minimax Optimal Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps

Zhaoyang Shi, Krishna Balasubramanian, Wolfgang Polonik
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:3826-3845, 2025.

Abstract

We develop minimax optimal estimators for nonparametric regression methods when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. 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 piecewise polynomial functions and bump function as canonical examples. We construct an estimator based on performing Principal Component Regression using Fractional Laplacian Eigenmaps and show that the in-sample mean-squared estimation error of this estimator is of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the order parameter and $n$ is the number of observations. We next prove a minimax lower bound of the same order, thereby establishing that no other estimator can improve upon the proposed estimator, up to context factors. We also provide preliminary empirical results validating the practical performance of the developed estimators.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-shi25a, title = {Minimax Optimal Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps}, author = {Shi, Zhaoyang and Balasubramanian, Krishna and Polonik, Wolfgang}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {3826--3845}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/shi25a/shi25a.pdf}, url = {https://proceedings.mlr.press/v286/shi25a.html}, abstract = {We develop minimax optimal estimators for nonparametric regression methods when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. 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 piecewise polynomial functions and bump function as canonical examples. We construct an estimator based on performing Principal Component Regression using Fractional Laplacian Eigenmaps and show that the in-sample mean-squared estimation error of this estimator is of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the order parameter and $n$ is the number of observations. We next prove a minimax lower bound of the same order, thereby establishing that no other estimator can improve upon the proposed estimator, up to context factors. We also provide preliminary empirical results validating the practical performance of the developed estimators.} }
Endnote
%0 Conference Paper %T Minimax Optimal Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps %A Zhaoyang Shi %A Krishna Balasubramanian %A Wolfgang Polonik %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-shi25a %I PMLR %P 3826--3845 %U https://proceedings.mlr.press/v286/shi25a.html %V 286 %X We develop minimax optimal estimators for nonparametric regression methods when the true regression function lies in an $L_2$-fractional Sobolev space with order $s\in (0,1)$. 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 piecewise polynomial functions and bump function as canonical examples. We construct an estimator based on performing Principal Component Regression using Fractional Laplacian Eigenmaps and show that the in-sample mean-squared estimation error of this estimator is of order $n^{-\frac{2s}{2s+d}}$, where $d$ is the dimension, $s$ is the order parameter and $n$ is the number of observations. We next prove a minimax lower bound of the same order, thereby establishing that no other estimator can improve upon the proposed estimator, up to context factors. We also provide preliminary empirical results validating the practical performance of the developed estimators.
APA
Shi, Z., Balasubramanian, K. & Polonik, W.. (2025). Minimax Optimal Nonsmooth Nonparametric Regression via Fractional Laplacian Eigenmaps. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:3826-3845 Available from https://proceedings.mlr.press/v286/shi25a.html.

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