On the Intrinsic Dimensions of Data in Kernel Learning

TL;DR

This paper investigates two notions of intrinsic dimension in kernel learning, analyzing their relationship, impact on generalization bounds, and proposing algorithms to estimate these dimensions from data.

Contribution

It introduces a novel analysis of intrinsic dimensions in kernel learning, relating Minkowski and effective dimensions to eigenvalue decay and generalization performance.

Findings

Eigenvalues decay characterized by Kolmogorov n-widths

Effective dimension d_K can be smaller than Minkowski dimension d_ρ

Proposed algorithms estimate upper bounds on n-widths from samples

Abstract

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 $d_\rho$, 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 $d_K$, is the effective dimension, derived from the decay rate of Kolmogorov $n$-widths associated with $K$ on $\Omega$. Given a probability measure $\mu$ on $\Omega$, we analyze the relationship between these $n$-widths and eigenvalues of the integral operator $\phi \to \int_\Omega K(\cdot,x)\phi(x)d\mu(x)$. We show that, for a fixed domain $\Omega$, the Kolmogorov $n$-widths characterize the worst-case eigenvalue decay across all probability measures $\mu$ 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 $O(n^{-\frac{2+d_K}{2+2d_K} + \epsilon})$ for any $\epsilon > 0$, 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 $\mu$. For distributions close to uniform, we prove that $\epsilon$-accurate upper bounds on all $n$-widths can be computed with high probability using at most $O\left(\epsilon^{-d_\rho}\log\frac{1}{\epsilon}\right)$ samples, with fewer required for small $n$. Finally, we compute the effective dimension $d_K$ for various fractal sets and present additional numerical experiments. Our results show that, for kernels such as the Laplace kernel, the effective dimension $d_K$ can be significantly smaller than the Minkowski dimension $d_\rho$, even though $d_K = d_\rho$ provably holds on regular domains.