发表机构
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对核方法需CND距离的限制，提出稀疏地标嵌入核，通过紧支撑凸包嵌入任意距离为PSD核，理论保证并实验超越基线。
AI 中文摘要
核方法，特别是高斯过程（GPs），需要希尔伯特距离度量——其平方为条件负定（CND）——以保证核矩阵的正半定性（PSD）；这一条件在许多自然输入空间（包括光滑流形和概率分布空间）中无法满足。我们提出了稀疏地标嵌入（SLE）核，完全消除了这一要求。每个输入通过以所有|D|个训练点为中心的紧支撑凸包函数嵌入到稀疏特征向量中；在此嵌入空间中应用任何标准PSD核，都能为任意距离度量产生可证明的PSD核。紧支撑自动控制嵌入稀疏性，尽管环境维度很高，仍保持核矩阵良态且计算可行。我们提供了关于PSD、稀疏性、稳定性和通用逼近的理论保证，并使用测地距离和Wasserstein距离证明，SLE核在预测精度和不确定性量化方面均匹配或大幅超越领域特定基线。
英文摘要
Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure---one whose square is conditionally negative definite (CND)---to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.
