We study algorithms for approximating the spectral density of a symmetric matrix $A$ that is accessed through matrix-vector product queries. By combining a previously studied Chebyshev polynomial moment matching method with a deflation step that approximately projects off the largest magnitude eigendirections of $A$ before estimating the spectral density, we give an $\epsilon\cdot\sigma_\ell(A)$ error approximation to the spectral density in the Wasserstein-$1$ metric using $O(\ell\log n+ 1/\epsilon)$ matrix-vector products, where $\sigma_\ell(A)$ is the $\ell^{th}$ largest singular value of $A$. In the common case when $A$ exhibits fast singular value decay, our bound can be much stronger than prior work, which gives an error bound of $\epsilon \cdot ||A||_2$ using $O(1/\epsilon)$ matrix-vector products. We also show that it is nearly tight: any algorithm giving error $\epsilon \cdot \sigma_\ell(A)$ must use $\Omega(\ell+1/\epsilon)$ matrix-vector products. We further show that the popular Stochastic Lanczos Quadrature (SLQ) method matches the above bound, even though SLQ itself is parameter-free and performs no explicit deflation. This bound explains the strong practical performance of SLQ, and motivates a simple variant of SLQ that achieves an even tighter error bound. Our error bound for SLQ leverages an analysis that views it as an implicit polynomial moment matching method, along with recent results on low-rank approximation with single-vector Krylov methods. We use these results to show that the method can perform implicit deflation as part of moment matching.

翻译：我们研究用于近似对称矩阵$A$谱密度的算法，该矩阵通过矩阵-向量乘积查询进行访问。通过将先前研究的切比雪夫多项式矩匹配方法与收缩步骤相结合——该步骤在估计谱密度前近似投影掉$A$的最大幅值特征方向——我们使用$O(\ell\log n+ 1/\epsilon)$次矩阵-向量乘积，在Wasserstein-$1$度量下给出谱密度的$\epsilon\cdot\sigma_\ell(A)$误差近似，其中$\sigma_\ell(A)$是$A$的第$\ell$大奇异值。在$A$呈现快速奇异值衰减的常见情况下，我们的界可能比现有工作强得多，后者使用$O(1/\epsilon)$次矩阵-向量乘积给出$\epsilon \cdot ||A||_2$的误差界。我们还证明该界近乎紧界：任何给出$\epsilon \cdot \sigma_\ell(A)$误差的算法必须使用$\Omega(\ell+1/\epsilon)$次矩阵-向量乘积。我们进一步证明流行的随机Lanczos求积法（SLQ）匹配上述界，尽管SLQ本身是无参数方法且不执行显式收缩。该界解释了SLQ强大的实际性能，并启发了一种实现更紧误差界的SLQ简单变体。我们对SLQ的误差界分析将其视为隐式多项式矩匹配方法，并结合了单向量Krylov方法低秩近似的最新结果。利用这些结果，我们证明该方法可以在矩匹配过程中执行隐式收缩。