AI 中文总结
该研究确定了估计矩阵核范数的随机无感知sketch所需线性测量数的近最优边界，解决了相关开放问题，给出了紧的多对数因子边界及对应的上下界构造方法。
AI 中文摘要
设$k_\epsilon(n)$为随机无感知sketch估计每个固定实$n\times n$矩阵的核范数且误差在$1\pm\epsilon$范围内、概率至少为$2/3$所需的最小实线性测量数。对于每个固定的$0<\epsilon<1$，已证明的结果为：对所有足够大的$n$，$\frac{n^2}{(\log n)^{A_\epsilon}} \leq k_\epsilon(n) \leq C_\epsilon\frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}$，其中$A_\epsilon$、$C_\epsilon$仅依赖于$\epsilon$。此前，一般线性sketch的最佳边界为$\Omega(n)$和平凡的$O(n^2)$上界（Li、Nguyen、Woodruff，2019）。因此该定理几乎解决了该工作留下的测量复杂度开放问题：上述下界和上界在多对数因子范围内是紧的，复杂度为$n^{2-o(1)}$，且对每个固定$c>0$，$O(n^{2-c})$测量是不可能的。上界通过固定高斯sketch获得，其解码器结合了隐式低秩恢复与高稳定秩残差的矩估计；下界通过构造矩匹配谱、随机化其奇异向量，并通过奇阶张量估计和费希尔信息路径论证比较每个低维观测值得到。
英文摘要
Let $k_{1,\varepsilon}(n)$ be the smallest number of real linear measurements needed by a randomized oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pm\varepsilon$, with probability at least $2/3$. For every fixed $0<\varepsilon<1$, we prove \[ \frac{n^2}{(\log n)^{A_\varepsilon}} \le k_{1,\varepsilon}(n) \le C_\varepsilon \frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}. \] Previously, the best unrestricted bounds for general linear sketches of the Schatten--1 norm were $Ω(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff'19), leaving a polynomial gap. Our bounds close that gap up to polylogarithmic factors and give a nontrivial logarithmic saving below the $n^2$-measurement storage bound. The result extends much further. Write $k_{p,\varepsilon}(n)$ for the analogous sketch dimension for the Schatten--$p$ norm. For every fixed finite $p>0$ that is not a positive even integer, there are positive constants $A_{p,\varepsilon},C_{p,\varepsilon},c_p$ such that \[ \frac{n^2}{(\log n)^{A_{p,\varepsilon}}} \le k_{p,\varepsilon}(n) \le C_{p,\varepsilon}\frac{n^2}{(\log n)^{c_p}}, \] so $k_{p,\varepsilon}(n)=n^{2-o(1)}$ throughout the non-even regime. Together with the known tight bounds $Θ_{p,\varepsilon}(n^{2-4/p})$ for positive even $p$ and $Θ_\varepsilon(n^2)$ for $p=\infty$ (Li, Woodruff'16), our results close the remaining polynomial gap across the Schatten family and complete, up to polylogarithmic factors, the polynomial-order classification of general linear sketches for all Schatten-$p$ norms.
Comments94 pages, added the turnstile streaming lower bound, upper/lower bound for all non-even p>0
