AI 中文总结
该研究针对纯ε-差分隐私下的持续计数问题，证明了任意实矩阵分解代价的阶为Θ((log(n+1))^{3/2})，确定了优化误差界与p-核范数的阶，并给出匹配上界，解决了相关开放问题。
AI 中文摘要
设$T_n$为下三角前缀和矩阵，$c_{\text{F}}(T_n)$和$c_2(T_n)$为控制纯$\text{-差分隐私（$\text{Laplace矩阵机制下每坐标均方误差和最大均方误差的分解代价，其中$\text{ε>0}$。我们证明了$c_{\text{F}}(T_n),c_2(T_n)=\text{Θ}((\text{log}(n+1))^{3/2})$，该结果无符号、稀疏性或方阵性限制，且适用于任意有限内维数。\n 因此，在纯$\text{ε-DP}$矩阵机制类别中，优化后的最大均方误差和均方误差均为$\text{Θ}(\text{ε}^{-2}\text{log}^3(n+1))$。Arkhipov和Kalinin（arXiv:2607.08963v1）证明了元素为$\text{\u007b0,1\u007d$的因子的匹配低阶，并将任意因子扩展列为未解决问题，在他们的分解约定下，下述定理确立了任意实因子的阶。\n 下界通过$p$-核障碍得到：前缀链的聚合列宽估计$D_k(T_n)\text{asymp}n^{3/2}k^{-1/2}$在低秩范围$1\leq k\leq n/16$内成立，将其代入经典的Pietsch与Hinrichs--Pietsch逼近空间转换，在临界指数$p=2/3$处呈调和性，再通过Hölder不等式将其传递到两个分解代价上。\n 相同的计算确定了每个固定$0<p<1$时的$\text{mathfrak{n}}_p(T_n)$：$p$低于$2/3$时阶为$n$，等于$2/3$时为$n\text{log}n$，高于$2/3$时为$n^{3p/2}$。Fenwick区间分解提供了匹配的上界。上述结论仅限于纯$\text{ε-DP}$Laplace矩阵机制及所述的两个均方误差准则；不涵盖非矩阵持续机制、近似DP敏感度或跨坐标的期望最大值。
英文摘要
Let $T_n$ be the lower-triangular prefix-sum matrix and let $c_{\mathrm{F}}(T_n)$ and $c_2(T_n)$ be the factorization costs that govern the mean and maximum per-coordinate squared error of the Laplace matrix mechanism under pure $\varepsilon$-differential privacy, for $\varepsilon>0$. We prove $c_{\mathrm{F}}(T_n),c_2(T_n)=Θ((\log(n+1))^{3/2})$ with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure-$\varepsilon$-DP matrix-mechanism class, the optimized maximum and mean squared errors are both $Θ(\varepsilon^{-2}\log^3(n+1))$. Under the factorization contract of Arkhipov and Kalinin (arXiv:2607.08963v1), who prove the matching lower order for factors with entries in $\{0,1\}$ and state the arbitrary-factor extension as open, the theorem below establishes the order for arbitrary real factors. The lower bound runs through a $p$-nuclear obstruction: an aggregate column-width estimate $D_k(T_n)\asymp n^{3/2}k^{-1/2}$, valid in the low-rank range $1\leq k\leq n/16$, for the prefix chain, fed into the classical approximation-space conversion of Pietsch and Hinrichs--Pietsch, becomes harmonic at the critical exponent $p=2/3$, and Hölder's inequality transfers it to both factorization costs. The same computation determines $\mathfrak{n}_p(T_n)$ for each fixed $0<p<1$: order $n$ below $2/3$, $n\log n$ at $2/3$, and $n^{3p/2}$ above. A Fenwick interval factorization supplies matching upper bounds. The claims are confined to pure-$\varepsilon$-DP Laplace matrix mechanisms and the two stated squared-error criteria; they do not cover non-matrix continual mechanisms, approximate-DP sensitivity, or expected maxima across coordinates.
