# On the Intrinsic Dimensions of Data in Kernel Learning

arXiv: https://arxiv.org/abs/2601.16139 · HTML: https://mlsanity.com/paper/2601.16139/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 42 references that are in the map.
- First posted on arXiv 22 Jan 2026.

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

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [On the Inductive Bias of Neural Tangent Kernels](https://mlsanity.com/paper/1905.12173/) (2019)
- [Neural Tangents: Fast and Easy Infinite Neural Networks in Python](https://mlsanity.com/paper/1912.02803/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS](https://mlsanity.com/paper/2009.10683/) (2020)
- [Kernel Alignment Risk Estimator: Risk Prediction from Training Data](https://mlsanity.com/paper/2006.09796/) (2020)
- [Stability and Deviation Optimal Risk Bounds with Convergence Rate O(1/n)](https://mlsanity.com/paper/2103.12024/) (2021)
- [Fast Neural Kernel Embeddings for General Activations](https://mlsanity.com/paper/2209.04121/) (2022)
- [Sample complexity and effective dimension for regression on manifolds](https://mlsanity.com/paper/2006.07642/) (2020)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [A spectral-based analysis of the separation between two-layer neural networks and linear methods](https://mlsanity.com/paper/2108.04964/) (2021)
- [On the speed of uniform convergence in Mercer's theorem](https://mlsanity.com/paper/2205.00487/) (2022)
- [Multi-layer random features and the approximation power of neural networks](https://mlsanity.com/paper/2404.17461/) (2024)

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

- [Conditional KRR: Injecting Unpenalized Features into Kernel Methods with Applications to Kernel Thresholding](https://mlsanity.com/paper/2605.26067/) (2026)

## Abstract (arXiv)

The manifold hypothesis suggests that the generalization performance of machine learning methods improves significantly when the intrinsic dimension of the input distribution's support is low. In the context of KRR, we investigate two alternative notions of intrinsic dimension. The first, denoted dρ, is the upper Minkowski dimension defined with respect to the canonical metric induced by a kernel function K on a domain Omega. The second, denoted dK, is the effective dimension, derived from the decay rate of Kolmogorov n-widths associated with K on Omega. Given a probability measure μ on Omega, we analyze the relationship between these n-widths and eigenvalues of the integral operator φ → int_Omega K(·,x)φ(x)dμ(x). We show that, for a fixed domain Omega, the Kolmogorov n-widths characterize the worst-case eigenvalue decay across all probability measures μ supported on Omega. These eigenvalues are central to understanding the generalization behavior of constrained KRR, enabling us to derive an excess error bound of order O(n^-frac2+dK2+2dK + ε) for any ε > 0, when the training set size n is large. We also propose an algorithm that estimates upper bounds on the n-widths using only a finite sample from μ. For distributions close to uniform, we prove that ε-accurate upper bounds on all n-widths can be computed with high probability using at most Oleft(ε-dρlogfrac1εright) samples, with fewer required for small n. Finally, we compute the effective dimension dK for various fractal sets and present additional numerical experiments. Our results show that, for kernels such as the Laplace kernel, the effective dimension dK can be significantly smaller than the Minkowski dimension dρ, even though dK = dρ provably holds on regular domains.