On the Intrinsic Dimensions of Data in Kernel Learning
This provides theoretical insights into kernel learning for researchers, but it is incremental as it builds on existing manifold hypothesis and dimension concepts.
The paper tackles the problem of understanding generalization in kernel ridge regression by analyzing two notions of intrinsic dimension, showing that the effective dimension can be smaller than the Minkowski dimension for certain kernels, and deriving an excess error bound of order O(n^{-(2+d_K)/(2+2d_K) + ε}) for large training sets.
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 $d_ρ$, is the upper Minkowski dimension defined with respect to the canonical metric induced by a kernel function $K$ on a domain $Ω$. The second, denoted $d_K$, is the effective dimension, derived from the decay rate of Kolmogorov $n$-widths associated with $K$ on $Ω$. Given a probability measure $μ$ on $Ω$, we analyze the relationship between these $n$-widths and eigenvalues of the integral operator $φ\to \int_ΩK(\cdot,x)φ(x)dμ(x)$. We show that, for a fixed domain $Ω$, the Kolmogorov $n$-widths characterize the worst-case eigenvalue decay across all probability measures $μ$ supported on $Ω$. These eigenvalues are central to understanding the generalization behavior of constrained KRR, enabling us to derive an excess error bound of order $O(n^{-\frac{2+d_K}{2+2d_K} + ε})$ 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 $O\left(ε^{-d_ρ}\log\frac{1}ε\right)$ 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 $d_ρ$, even though $d_K = d_ρ$ provably holds on regular domains.