扩散映射核岭回归
Diffusion Maps Kernel Ridge Regression
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中文总结 AI 辅助
本文提出采用扩散映射（DM）核的核岭回归，证明DM核收敛到热核，其RKHS与高斯核等距同构，且数值验证了DM核在带边界流形监督学习中的优势。
中文摘要 AI 辅助
本文研究采用数据驱动的扩散映射（DM）核的核岭回归，该核通过扩散映射算法的代数运算构造而成。在数据位于流形上且均匀采样的假设下，我们证明当数据集规模增大且核带宽被适当缩放时，对于足够大的时间，经适当缩放的DM核会一致收敛到流形上的热核。因此，由DM核诱导的极限再生核希尔伯特空间（RKHS）与流形上热核对应的RKHS一致。我们进一步证明，DM核诱导的RKHS与高斯核的RKHS等距同构，该高斯核的RKHS连续嵌入到Matérn核的RKHS中，而Matérn核的范数等价于适当的索伯列夫范数。这一结果意味着适用于Matérn核的核岭回归的标准风险界也适用于DM核。最后，我们提供数值结果：（1）验证热核近似的收敛性；（2）表明在带边界流形上的监督学习中，与高斯核相比，DM核能表达更大类的函数，表现力更强；（3）表明在学习具有不同频率和余维数的函数时，DM核比高斯核更具优势。
英文摘要
In this paper, we study kernel ridge regression using the data-driven diffusion maps (DM) kernel, which is constructed through algebraic manipulations of the diffusion maps algorithm. Under the assumptions that the data lie on a manifold and are sampled uniformly, we prove that the appropriately scaled DM kernel converges uniformly to the heat kernel on the manifold for sufficiently large times as the dataset size increases and the kernel bandwidth is scaled appropriately. Consequently, the limiting Reproducing Kernel Hilbert Space (RKHS) induced by the DM kernel coincides with the RKHS associated with the heat kernel on the manifold. We further show that the RKHS induced by the DM kernel is isometrically isomorphic to the RKHS of the Gaussian kernel, which is continuously embedded in the RKHS of a Matérn kernel whose norm is equivalent to an appropriate Sobolev norm. This result implies that standard risk bounds for kernel ridge regression applicable to Matérn kernels also apply to the DM kernel. Finally, we provide numerical results that (1) validate the convergence of the heat kernel approximation, (2) demonstrate the greater expressiveness of the DM kernel compared to the Gaussian kernel for supervised learning over a larger class of functions on manifolds with boundary, and (3) demonstrate the advantage of the DM kernel over the Gaussian kernel in learning functions with varying frequencies and co-dimensions.
