持续同调：持久性与抗性的结合——强化硬度研究
Persistence Meets Resistance: Doubling Down on Hardness
AI总结：
本文在持续同调领域，针对特定度量空间滤过的稳定不变量近似计算，提出线性时间算法与常数时间概率近似方法，还证明矩阵秩计算可归约为相关条码近似问题。
AI中文摘要：
本文针对有限度量空间滤过的稳定不变量近似计算问题，在持续同调领域取得了相关成果。我们在n点度量空间中建立了新颖的近似算法，该空间的加倍维度增长为o(log n)且直径有界。在单参数情形下，通过以新方式重新审视已知的贪心排列技术，我们推导了首个线性时间算法，用于计算任意稳定条码的加性ε近似。通过推导均匀样本的收敛速率与近似质量的界，我们将该方法扩展至选定的多参数滤过。我们证明，对于归一化测度双滤过（包括多覆盖和细分-Rips双滤过），任何稳定不变量都可在n的常数时间内进行概率近似，算法运行时间中的常数取决于加倍维度、直径和成功概率。我们还从细粒度复杂性视角研究该问题，证明矩阵秩的计算可归约为近似Vietoris–Rips或Čech滤过的条码问题，我们提出了两种归约变体，一种适用于足够好的加性近似，另一种适用于任意常数因子的乘性近似。
英文摘要：
We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of $n$-point metric spaces where the growth of the doubling dimension is in $o(\log n)$ and the diameter is bounded. In the $1$-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive $\varepsilon$-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in $n$. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris--Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.
