发表机构
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对 Hutchinson 迹估计器随机比特开销大、Kronecker 变体方差指数增长的问题，提出基于递归 TensorSketch 的估计器，以较少随机比特实现无偏且方差多项式增长的迹估计。
AI 中文摘要
我们考虑估计隐式矩阵 $\mathbf{A} \in \mathbb{R}^{d^p\times d^p}$ 的迹的问题，该矩阵只能通过矩阵-向量乘积查询访问。Hutchinson 迹估计器是解决此问题的经典草图方法。其估计器 $H_{m}(\mathbf{A}) = \frac{1}{m} \sum_{i=1}^{m} {\mathbf{z}^{(i)}}^T \mathbf{A} \mathbf{z}^{(i)}$，其中 ${\mathbf{z}^{(i)}}\in \mathbb{R}^{d^p}$，且 $z^{(i)}_j \in N(0, 1), j\in [d^p]$，满足以下保证：(i) $\mathbb{E}[H_{m}(\mathbf{A})]=\operatorname{tr}(\mathbf{A})$，以及 (ii) $\mathrm{Var}[H_{m}(\mathbf{A})]=\frac{2}{m}||\mathbf{A}||_F^2$。生成一个查询向量 $\mathbf{z}^{(i)}$ 需要 $O(d^p)$ 个随机比特；因此，$m$ 次查询需要 $O(md^p)$ 个随机比特，这在大规模应用中可能令人望而却步。Meyer 等人最近的工作提出了一种 Hutchinson 迹估计器的变体，其中 $\mathbb{R}^{d^p}$ 中的每个查询向量被构造为 $\mathbb{R}^d$ 中 $p$ 个随机向量的 Kronecker 积，对于 $m$ 个查询向量需要 $O(mpd)$ 个随机比特。该估计器是无偏的；然而，其方差随 $p$ 呈指数增长。在这项工作中，我们通过提出一种基于草图的估计器来解决这一限制，该估计器需要 $O(p (d + m)\log m)$ 个随机比特，产生迹的无偏估计，同时实现随 $p$ 多项式增长的方差界。
英文摘要
We consider the problem of estimating the trace of an implicit matrix $\mathbf{A} \in \mathbb{R}^{d^p\times d^p}$ that can only be accessed through matrix-vector products queries. The \textit{Hutchinson trace estimator}% ~\cite{Girard1987algorithme, article-hutchinson} is a classical sketching method for this problem. Their estimator, $H_{m}(\mathbf{A}) = \frac{1}{m} \sum_{i=1}^{m} {\mathbf{z}^{(i)}}^T \mathbf{A} \mathbf{z}^{(i)}, \quad \text{where } \ {\mathbf{z}^{(i)}}\in \mathbb{R}^{d^p}$, and $z^{(i)}_j \in {N}(0, 1), j\in [d^p]$, satisfies the following guarantees: (i) $\mathbb{E}[H_{m}(\mathbf{A})]=\operatorname{tr}(\mathbf{A})$, and (ii) $\mathrm{Var}[H_{m}(\mathbf{A})]=\frac{2}{m}||\mathbf{A}||_F^2$. Generating one query vector $\mathbf{z}^{(i)}$ requires $O(d^p)$ random bits; thus, $m$ queries require $O(md^p)$ random bits, which can be prohibitive in large-scale applications. Recent work by Meyer et al.~\cite{meyer2025hutchinsonsestimatorbadkroneckertraceestimation} proposes a variant of the Hutchinson trace estimator in which each query vector in $\mathbb{R}^{d^p}$ is constructed as the Kronecker product of $p$ random vectors in $\mathbb{R}^d$, requiring $O(mpd)$ random bits for $m$ query vectors. The estimator of~\cite{meyer2025hutchinsonsestimatorbadkroneckertraceestimation} is unbiased; however, its variance grows exponentially with $p$. In this work, we address this limitation by proposing a sketching-based estimator that requires $O\!\big(p (d + m)\log m\big)$ random bits, yields an unbiased estimate of the trace, and simultaneously achieves a variance bound that grows polynomially with $p$.
