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.

中文翻译：

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用于解决线性代数、统计和图形问题的向量-矩阵-向量查询

我们考虑通过向量-矩阵-向量查询学习矩阵的一般问题。这些查询提供了 $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ 在固定字段 $\mathbb{F}$ 上的值，用于指定的向量对 $\粗体符号{u},\boldsymbol{v} \in \mathbb{F}^n$。为了激发这些查询，我们观察到它们概括了许多以前研究过的模型，例如独立集查询、切割查询和标准图查询。他们还专门研究了最近研究的矩阵向量查询模型。我们的工作具有探索性和广泛性，我们为各种问题提供了新的上下界，涵盖线性代数、统计和图形。我们的许多结果几乎都是紧凑的，我们使用了线性代数、随机算法和通信复杂性等多种技术。