解线性代数、统计和图形问题的向量-矩阵-向量矩阵法
Vector-Matrix-Vector Queries for Solving Linear Algebra, Statistics, and Graph Problems
摘要：
我们考虑通过向量-矩阵-向量查询学习矩阵的一般问题。这些查询在用于指定向量对$\boldsymbol{u},\boldsymbol{v} \in \mathbb{F}^n$的固定字段$\mathbb{F}$上提供$\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$的值。为了激发这些查询，我们观察到，他们概括了许多以前研究的模型，如独立集查询，割查询，标准图查询。他们还专门研究了最近研究的矩阵向量查询模型。我们的工作是探索性的和广泛的，我们提供了各种各样的问题，跨越线性代数，统计和图形新的上限和下限。我们的许多结果几乎是紧密的，我们使用不同的技术，从线性代数，随机算法，和通信复杂性。
英文摘要：
We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ over a fixed field $\mathbb{F}$ for a specified pair of vectors $\boldsymbol{u},\boldsymbol{v} \in \mathbb{F}^n$. To motivate these queries, we observe that they generalize many previously studied models, such as independent set queries, cut queries, and standard graph queries. They also specialize the recently studied matrix-vector query model. Our work is exploratory and broad, and we provide new upper and lower bounds for a wide variety of problems, spanning linear algebra, statistics, and graphs. Many of our results are nearly tight, and we use diverse techniques from linear algebra, randomized algorithms, and communication complexity.
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。
