This paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call centered kernel ridge regression (CKRR), also known in the literature as kernel ridge regression with offset. This modified technique is obtained by accounting for the bias in the regression problem resulting in the old kernel ridge regression but with centered kernels. The analysis is carried out under the assumption that the data is drawn from a Gaussian distribution and heavily relies on tools from random matrix theory (RMT). Under the regime in which the data dimension and the training size grow infinitely large with fixed ratio and under some mild assumptions controlling the data statistics, we show that both the empirical and the prediction risks converge to a deterministic quantities that describe in closed form fashion the performance of CKRR in terms of the data statistics and dimensions. Inspired by this theoretical result, we subsequently build a consistent estimator of the prediction risk based on the training data which allows to optimally tune the design parameters. A key insight of the proposed analysis is the fact that asymptotically a large class of kernels achieve the same minimum prediction risk. This insight is validated with both synthetic and real data.

中文翻译：

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大维数据中心核岭回归的风险收敛性

本文对核岭回归的一种变体进行了大维度分析，我们称之为中心核岭回归 (CKRR)，在文献中也称为带偏移的核岭回归。这种修改后的技术是通过考虑回归问题中的偏差而获得的，该偏差导致旧的内核岭回归但具有居中的内核。分析是在数据来自高斯分布的假设下进行的，并且在很大程度上依赖于随机矩阵理论 (RMT) 的工具。在数据维度和训练大小以固定比率无限大的情况下，并在控制数据统计的一些温和假设下，我们表明，经验和预测风险都收敛到一个确定性数量，该数量以封闭形式描述 CKRR 在数据统计和维度方面的性能。受此理论结果的启发，我们随后基于训练数据构建了预测风险的一致估计器，从而可以优化调整设计参数。所提出的分析的一个关键见解是这样一个事实，即渐近地一大类内核实现了相同的最小预测风险。这种洞察力得到了合成数据和真实数据的验证。所提出的分析的一个关键见解是这样一个事实，即渐近地一大类内核实现了相同的最小预测风险。这种洞察力得到了合成数据和真实数据的验证。所提出的分析的一个关键见解是这样一个事实，即渐近地一大类内核实现了相同的最小预测风险。这种洞察力得到了合成数据和真实数据的验证。