We investigate the computational complexity of several basic linear algebra primitives, including largest eigenvector computation and linear regression, in the computational model that allows access to the data via a matrix-vector product oracle. We show that for polynomial accuracy, $\Theta(d)$ calls to the oracle are necessary and sufficient even for a randomized algorithm. Our lower bound is based on a reduction to estimating the least eigenvalue of a random Wishart matrix. This simple distribution enables a concise proof, leveraging a few key properties of the random Wishart ensemble.

翻译：我们调查数种基本的线性代数原始元素的计算复杂性,包括最大的叶质代数计算和线性回归,这些原始元素的计算模型允许通过矩阵-矢量器产品或魔器访问数据。我们显示,对于多元精确度而言,$\Theta(d)美元呼叫神器是必要的,甚至对于随机的算法也足够。我们的下限基于对随机 Wishart 矩阵最小值的估计的缩减。这种简单分布可以提供简明的证明,利用随机Wishart 共性的一些关键属性。