AI 中文总结
本文针对Nyström方法处理对称不定矩阵时的失效问题，提出基于随机草图矩阵的双边草图最小二乘算法，在高斯或杠杆得分采样下获得理论保证，通过合成示例与核方法应用验证了结果。
AI 中文摘要
Nyström方法将矩阵A近似为A≈A(:,I)A(I,I)†A(:,I)ᵀ=CA(I,I)†Cᵀ，其中C:=A(:,I)∈ℝⁿˣʳ是A的列子集矩阵。当将该方法应用于对称不定矩阵时，它可能会失效，因为核心矩阵A(I,I)可能会严重低估A的特征值，并且可能变为（近乎）奇异矩阵。我们通过开发并分析一种算法来解决此问题，该算法通过求解双边草图最小二乘问题minₘ‖X(A−CMCᵀ)Xᵀ‖_F，仔细选择M̂∈ℝʳˣʳ以替代A(I,I)†，其中X∈ℝᵗˣⁿ是随机草图矩阵。我们详细研究了X为高斯矩阵或杠杆得分采样（LSS）矩阵的情况，结果表明，当过采样t>r时，残差‖A−CM̂Cᵀ‖_*与minₘ‖A−CMCᵀ‖_*相当。对于高斯草图，我们需要t=𝒪(r)个样本；对于LSS，我们证明理论保证需要t=𝒪(r log r)个样本，LSS方法的优势在于，一旦确定了t个行索引，在给定C的情况下，求M̂仅需要t²个矩阵元素计算。我们通过合成示例和在核方法中的应用来说明我们的结果。
英文摘要
The Nyström method approximates $A\approx A(\,:\,,I)A(I,I)^{\dagger} A(\,:\,,I)^{\top}=CA(I,I)^{\dagger} C^{\top}$, where $C:=A(:,I)\in\mathbb{R}^{n\times r}$ is a column subset matrix of $A$. When applied to symmetric but indefinite matrices, the Nyström method can fail because the core matrix $A(I,I)$ may severely underestimate the eigenvalues of $A$ and may become (nearly) singular. We address this issue by developing and analyzing an algorithm that carefully chooses $\widehat{M}\in\mathbb{R}^{r\times r}$ in place of $A(I,I)^\dagger$ by solving the two-sided sketched least-squares problem $\min_{M}\|X(A-CMC^{\top})X^{\top}\|_F$, where $X\in\mathbb{R}^{t\times n}$ is a random sketch matrix. We study in detail the cases where $X$ is a Gaussian or a leverage score sampling (LSS) matrix, and show that with oversampling $t>r$ the residual $\|A-C\widehat{M}C^{\top}\|_*$ is comparable to $\min_{M}\|A-CMC^{\top}\|_*$. For the Gaussian sketch, we require $t=\mathcal{O}(r)$ samples; for LSS, we show that $t=\mathcal{O}(r \log r)$ samples suffice for the theoretical guarantee, with the LSS approach carrying the advantage that once a set of $t$ row indices is identified, the approximation requires only $t^{2}$ matrix-entry evaluations to find $\widehat{M}$, given $C$. We illustrate our results with synthetic examples and applications to kernel methods.
Comments16 pages, 4 figures
