We study the problem of estimating the trace of a matrix $\mathbf{A}$ that can only be accessed through Kronecker-matrix-vector products. That is, for any Kronecker-structured vector $\boldsymbol{\mathrm{x}} = \otimes_{i=1}^k \boldsymbol{\mathrm{x}}_i$, we can compute $\mathbf{A}\boldsymbol{\mathrm{x}}$. We focus on the natural generalization of Hutchinson's Estimator to this setting, proving tight rates for the number of matrix-vector products this estimator needs to find a $(1\pm\varepsilon)$ approximation to the trace of $\mathbf{A}$. We find an exact equation for the variance of the estimator when using a Kronecker of Gaussian vectors, revealing an intimate relationship between Hutchinson's Estimator, the partial trace operator, and the partial transpose operator. Using this equation, we show that when using real vectors, in the worst case, this estimator needs $O(\frac{3^k}{\varepsilon^2})$ products to recover a $(1\pm\varepsilon)$ approximation of the trace of any PSD $\mathbf{A}$, and a matching lower bound for certain PSD $\mathbf{A}$. However, when using complex vectors, this can be exponentially improved to $\Theta(\frac{2^k}{\varepsilon^2})$. We show that Hutchinson's Estimator converges slowest when $\mathbf{A}$ itself also has Kronecker structure. We conclude with some theoretical evidence suggesting that, by combining Hutchinson's Estimator with other techniques, it may be possible to avoid the exponential dependence on $k$.

翻译：我们研究仅能通过克罗内克矩阵-向量乘积访问的矩阵$\mathbf{A}$的迹估计问题。具体而言，对于任意克罗内克结构向量$\boldsymbol{\mathrm{x}} = \otimes_{i=1}^k \boldsymbol{\mathrm{x}}_i$，我们可以计算$\mathbf{A}\boldsymbol{\mathrm{x}}$。我们聚焦于哈钦森估计器在该场景下的自然推广，并精确刻画该估计器为获取$\mathbf{A}$的$(1\pm\varepsilon)$近似迹所需矩阵-向量乘积次数的紧致界。通过推导使用高斯向量克罗内克积时估计器方差的精确方程，揭示了哈钦森估计器、部分迹算子和部分转置算子之间的深层关联。利用该方程表明：在使用实向量时，最坏情况下该估计器需要$O(\frac{3^k}{\varepsilon^2})$次乘积即可对任意半正定矩阵$\mathbf{A}$恢复$(1\pm\varepsilon)$近似迹，且对特定半正定矩阵存在匹配下界；而使用复向量时，该复杂度可指数级优化为$\Theta(\frac{2^k}{\varepsilon^2})$。我们进一步证明，当$\mathbf{A}$本身具有克罗内克结构时，哈钦森估计器的收敛速度最慢。最后，理论分析表明结合哈钦森估计器与其他技术，或可消除对$k$的指数依赖。