前缀矩阵分解的近最优下界
A Near-Optimal Lower Bound for Prefix-Matrix Factorizations
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中文总结 AI 辅助
该研究针对下三角全1矩阵Q，利用Gemini智能体系统证明了前缀矩阵分解的近最优下界，其结果可应用于转置流估计与差分隐私计数的相关界。
中文摘要 AI 辅助
对于n×n的下三角全1矩阵Q，我们证明了近最优下界：γ₂,₁(Q) := inf_{Q=AB} ||A||_{2→∞}||B||_{1→1} = Ω(log^(3/2)n / (log log n)^(3/2))，其中下确界遍历任意有限内维的实分解。该代价是转置流中基于分解的秩与分位数估计的空间界，以及纯差分隐私下持续计数的矩阵机制误差界的核心参数。证明结合了右侧Haar投影与B行的尺度相关数值稀疏分解：在每个尺度，秩-Frobenius论证表明数值稀疏行无法覆盖所有所需的Schatten 2/3质量，而Haar投影估计界定了剩余行的贡献；对二进尺度求和这些界得到结果。该证明使用Google内部开发的完全自动化Gemint智能体系统完成，作者验证了证明并做了微小修订。
英文摘要
For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ γ_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = Ω\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.
