Conditional KRR: Injecting Unpenalized Features into Kernel Methods with Applications to Kernel Thresholding

Problem
This paper addresses a gap in the literature regarding the integration of unpenalized features into kernel methods, specifically through the lens of conditional kernel ridge regression (conditional KRR). The authors propose a novel approach to enhance the performance of kernel methods by leveraging conditionally positive definite (CPD) kernels, which have not been extensively explored in the context of KRR. This work is presented as a preprint and has not yet undergone peer review.

Method
The core technical contribution is the introduction of conditional KRR, which operates by first applying classical linear regression to a specified feature class $\mathcal{F}$, followed by KRR on the residuals of the target variable. The authors establish a theoretical framework that allows for the reduction of conditional KRR to KRR with a residual kernel, which is a fixed kernel derived from the original CPD kernel. The expected test risk of this method is shown to include an additional term bounded by $\mathcal{O}(1/\sqrt{N})$, where $N$ is the sample size. The analysis is conducted under two scenarios: (1) when $\mathcal{F}$ consists of the first $k$ principal eigenfunctions from the Mercer decomposition of the kernel $K$, and (2) when $\mathcal{F}$ comprises $k$ random features from a random feature representation of $K$. The authors demonstrate that these two settings are closely related and provide a theoretical basis for the performance improvements of conditional KRR over standard KRR.

Results
The experimental results indicate that conditional KRR consistently outperforms standard KRR when the $\mathcal{F}$-component of the regression function is more significant than the residual component. While specific numerical results are not detailed in the abstract, the authors imply that the effect sizes are substantial enough to warrant the proposed method’s adoption in practical applications. The performance gains are particularly pronounced in scenarios where the feature class $\mathcal{F}$ captures essential aspects of the underlying data distribution.

Limitations
The authors acknowledge that the additional term in the expected test risk, which scales with $1/\sqrt{N}$, may limit the method’s effectiveness in small-sample scenarios. They do not discuss potential computational complexities associated with the implementation of conditional KRR, particularly in high-dimensional settings or when the feature class $\mathcal{F}$ is large. Furthermore, the reliance on the structure of the kernel and the feature class may restrict the generalizability of the method to other types of kernels or feature representations.

Why it matters
This work has significant implications for the development of kernel methods in machine learning, particularly in scenarios where feature selection and dimensionality reduction are critical. By effectively integrating unpenalized features into the kernel framework, conditional KRR may enhance model interpretability and performance, especially in applications where the underlying relationships are complex. The theoretical insights provided could also pave the way for further research into hybrid models that combine the strengths of linear regression and kernel methods.

Authors: Rustem Takhanov, Zhenisbek Assylbekov
Source: arXiv:2605.26067
URL: https://arxiv.org/abs/2605.26067v1