# Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition

arXiv: https://arxiv.org/abs/2204.07856 · HTML: https://mlsanity.com/paper/2204.07856/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 69 references that are in the map.
- First posted on arXiv 16 Apr 2022.

## Builds on (most cited first, 9 of 9)

- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Optimal Rates for Spectral Algorithms with Least-Squares Regression over Hilbert Spaces](https://mlsanity.com/paper/1801.06720/) (2018)
- [Optimal rates for the regularized learning algorithms under general source condition](https://mlsanity.com/paper/1611.01900/) (2016)
- [Relating Leverage Scores and Density using Regularized Christoffel Functions](https://mlsanity.com/paper/1805.07943/) (2018)
- [Nystr"om landmark sampling and regularized Christoffel functions](https://mlsanity.com/paper/1905.12346/) (2019)
- [Sobolev Norm Learning Rates for Conditional Mean Embeddings](https://mlsanity.com/paper/2105.07446/) (2021)
- [Kernel regression, minimax rates and effective dimensionality: beyond the regular case](https://mlsanity.com/paper/1611.03979/) (2016)
- [Kernel-based Outlier Detection using the Inverse Christoffel Function](https://mlsanity.com/paper/1806.06775/) (2018)

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

- [On the Optimality of Misspecified Spectral Algorithms](https://mlsanity.com/paper/2303.14942/) (2023)
- [On the Optimality of Misspecified Kernel Ridge Regression](https://mlsanity.com/paper/2305.07241/) (2023)

## Abstract (arXiv)

We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis class, and most notably does not employ the conventional textita priori assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of ``tight'' L∞(X) embeddings of suitable Hilbert scales. Our analysis utilizes a new Fourier isocapacitary condition, which captures the interplay of the kernel Dirichlet capacities and small ball probabilities via the optimal Hilbert scale function.