流形上高斯核的低维嵌入
Low-Dimensional Embeddings for Gaussian Kernels on Manifolds
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中文总结 AI 辅助
针对流形上的高斯核距离近似问题，提出基于随机傅里叶特征的均匀相对误差嵌入定理，证明特征数量仅对数依赖于环境维数和流形参数，并保持持久同调。
中文摘要 AI 辅助
高斯核是一种广泛使用的相似性度量，是核主成分分析和谱聚类等核方法的基础，但计算许多点对之间的高斯核距离可能代价高昂。Chen 和 Phillips [ALT 2017] 使用随机傅里叶特征（RFF）证明，对于 ${\mathbb R}^N$ 中 $d$ 维欧几里得球内的点，$t=\Omega((d/\varepsilon^2)\log(dR/\varepsilon))$ 个特征足以高概率地保持所有成对高斯核距离在 $(1\pm\varepsilon)$ 因子内。我们针对更一般的任意正到达子流形 $\mathcal M\subset{\mathbb R}^N$（内蕴维数为 $d$）建立了均匀相对误差嵌入定理。我们证明，$t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}\delta)))$，或近似 $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilon\delta))))$ 个 RFF 就足以以概率 $1-\delta$ 将每对流形点之间的高斯核距离保持到相对误差 $\varepsilon$ 内。因此，该界限仅对数依赖于环境维数以及体积和到达等流形参数，同时保留了 $1/\varepsilon^2$ 的欧几里得速率。我们还证明了一个拓扑推论：在相同的 RFF 嵌入下，持久同调得以保持，即由高斯核幂距离构建的加权 Cech 和 Rips 过滤是 $(1\pm\varepsilon_\star)$ 交错的，其中 $\varepsilon_\star$ 同时考虑了距离畸变和核权重近似。
英文摘要
The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.
发表机构
- Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
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