发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定了低秩近似在矩阵-向量乘积查询模型下的最优查询复杂度，建立了匹配上下界，并揭示了范数参数与精度之间的关键转变。
AI 中文摘要
我们针对精确矩阵-向量乘积下的低秩近似建立了匹配的多项式查询界。给定一个未知矩阵 $A\in\mathbb{R}^{m\times n}$，在每一步中，随机化算法要么选择 $v\in\mathbb{R}^n$ 并接收 $Av$，要么选择 $u\in\mathbb{R}^m$ 并接收 $A^\top u$。该选择可以可测地依赖于所有先前的查询和回复以及算法的私有随机性；每个向量乘积花费一次查询。输出是一个秩为 $k$ 的右投影算子，其 Schatten-$p$ 残差至多为最优值的 $1+\varepsilon$ 倍。记 $N=\min\{m,n\}$，并令 $Q_p^*$ 表示在所有输入上成功概率为 $2/3$ 的最坏情况查询预算。对于每个 $1\le k<N$ 和足够小的 $\varepsilon$，我们的下界与现有的 Krylov 上界相结合，给出 $Q_p^*=\widetilde{\Theta}\left(\min\{N,k\min\{p^{1/6}\varepsilon^{-1/3},\varepsilon^{-1/2}\}\}\right)$（$2\le p<\infty$），$Q_\infty^*=\widetilde{\Theta}\left(\min\{N,k\varepsilon^{-1/2}\}\right)$。这些界具有普适常数，并允许 $p$ 随问题参数变化，识别出 $p\varepsilon\asymp1$ 处的转变。对于每个固定的 $1\le p<2$，一个补充结果给出 $\widetilde{\Theta}_p(\min\{N,k\varepsilon^{-1/3}\})$，其常数和精度阈值可能依赖于 $p$。综合这些结果，恢复了每个固定有限 $p$ 的固定范数速率，提供了先前下界中缺失的乘法秩依赖。波浪号隐藏了对数因子。证明将自适应 Wishart 延迟决策扩展到具有 $k$ 维零空间的矩形因子。后验重叠给出了一个简短的固定范数论证，而小压缩特征值的持续性控制了增长的 $p$ 和谱端点。精确范围恢复处理了阶为 $k$ 的目标成本；Wishart 族覆盖了其余机制。
英文摘要
We establish matching polynomial query bounds for low-rank approximation from exact matrix--vector products. Given an unknown matrix $A\in\mathbb{R}^{m\times n}$, at each step a randomized algorithm chooses either $v\in\mathbb{R}^n$ and receives $Av$, or $u\in\mathbb{R}^m$ and receives $A^\top u$. The choice may depend measurably on all previous queries and replies and on the algorithm's private randomness; each vector product costs one query. The output is a rank-$k$ right projector with Schatten-$p$ residual at most $1+\varepsilon$ times optimal. Write $N=\min\{m,n\}$ and let $Q_p^*$ denote the worst-case query budget for success probability $2/3$ on every input. For every $1\le k<N$ and sufficiently small $\varepsilon$, our lower bounds, combined with existing Krylov upper bounds, give $Q_p^*=\widetildeΘ\!\left(\min\{N,k\min\{p^{1/6}\varepsilon^{-1/3},\varepsilon^{-1/2}\}\}\right)$ $(2\le p<\infty)$, $Q_\infty^*=\widetildeΘ\!\left(\min\{N,k\varepsilon^{-1/2}\}\right)$. These bounds have universal constants and allow $p$ to vary with the problem parameters, identifying the transition at $p\varepsilon\asymp1$. A complementary result for each fixed $1\le p<2$ gives $\widetildeΘ_p(\min\{N,k\varepsilon^{-1/3}\})$, with constants and an accuracy threshold that may depend on $p$. Together, the results recover this fixed-norm rate for every fixed finite $p$, supplying the multiplicative rank dependence missing from previous lower bounds. Tildes suppress logarithmic factors. The proof extends adaptive Wishart deferred decisions to a rectangular factor with a $k$-dimensional nullspace. Posterior overlap gives a short fixed-norm argument, while persistence of small compression eigenvalues controls growing $p$ and the spectral endpoint. Exact range recovery handles target costs of order $k$; the Wishart family covers the remaining regimes.
