理解线性代数的克罗内克矩阵-向量复杂度
Understanding the Kronecker Matrix-Vector Complexity of Linear Algebra
AI总结：
本文研究仅通过克罗内克结构向量的矩阵-向量乘积访问矩阵的计算模型，证明估计迹和最大特征值等性质的指数下界，揭示小字母表查询与高斯查询的复杂度差异，其根源是克罗内克结构随机向量内积的指数级减小。
摘要：
我们研究了一种计算模型，其中只能通过计算矩阵-向量乘积$\boldsymbol{A}\boldsymbol{x}$来访问矩阵$\boldsymbol{A}$，且向量$\boldsymbol{x}$的形式为$\boldsymbol{x} = \boldsymbol{x}_1 \bigotimes \cdots \bigotimes \boldsymbol{x}_q$。我们证明了估计$\boldsymbol{A}$的迹和最大特征值等多种性质所需查询次数的指数下界。在算法查询满足温和条件假设的前提下，我们的证明适用于所有自适应算法。我们进一步证明，若查询向量来自小字母表（例如$\boldsymbol{x}_i \in \{\pm1\}^n$），则无法以多项式复杂度测试$\boldsymbol{A}$是否恒为零，尽管使用高斯向量的单次查询能以概率1解决该问题。与非克罗内克情形形成鲜明对比的是，这表明使用相同次高斯范数的不同分布对$\boldsymbol{A}$进行草图绘制时，查询复杂度可能存在指数级差异。我们的证明源于以下观察：具有克罗内克结构的随机向量的内积比其非克罗内克对应向量的内积小指数倍。
英文摘要：
We study the computational model where we can access a matrix $\mathbf{A}$ only by computing matrix-vector products $\mathbf{A}\mathrm{x}$ for vectors of the form $\mathrm{x} = \mathrm{x}_1 \otimes \cdots \otimes \mathrm{x}_q$. We prove exponential lower bounds on the number of queries needed to estimate various properties, including the trace and the top eigenvalue of $\mathbf{A}$. Our proofs hold for all adaptive algorithms, modulo a mild conditioning assumption on the algorithm's queries. We further prove that algorithms whose queries come from a small alphabet (e.g., $\mathrm{x}_i \in \{\pm1\}^n$) cannot test if $\mathbf{A}$ is identically zero with polynomial complexity, despite the fact that a single query using Gaussian vectors solves the problem with probability 1. In steep contrast to the non-Kronecker case, this shows that sketching $\mathbf{A}$ with different distributions of the same subguassian norm can yield exponentially different query complexities. Our proofs follow from the observation that random vectors with Kronecker structure have exponentially smaller inner products than their non-Kronecker counterparts.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
