根据镜头空间密度调整分辨率，提升Mapper算法鲁棒性。
Improving Mapper's Robustness by Varying Resolution According to Lens-Space Density
- 按镜头空间密度动态调整覆盖分辨率，打破单一尺度假设。
- 在密集区域用高分辨率，稀疏区域用低分辨率，输出更稳定。
- 适合处理密度变化大的数据，如生物、气象数据集。
我们提出一种改进的Mapper算法，不再假设语义空间中采用单一分辨率尺度，从而提升了参数变化下的结果鲁棒性。该工作源于图像中莫尔斯型函数（即镜头空间密度）分布极不均匀的数据集。对于这类数据，传统Mapper的分辨率调参极为敏感，微小变化即导致输出剧烈波动。我们的方法通过放宽覆盖类型限制，并将镜头空间密度纳入覆盖设计，显著增强鲁棒性。理论上证明：在满足自然假设的覆盖下，生成图仍以瓶颈距离收敛于数据的黎曼复形的里布图，同时可能捕捉更多拓扑特征。我们还讨论了实现细节，并展示了计算实验结果，附带参考实现代码。
原文摘要 · Abstract (English)
We propose a modification of the Mapper algorithm that removes the assumption of a single resolution scale across semantic space and improves the robustness of the results under change of parameters. Our work is motivated by datasets where the density in the image of the Morse-type function (the lens-space density) varies widely. For such datasets, tuning the resolution parameter of Mapper is difficult because small changes can lead to significant variations in the output. By improving the robustness of the output under these variations, our method makes it easier to tune the resolution for datasets with highly variable lens-space density. This improvement is achieved by generalising the type of permitted cover for Mapper and incorporating the lens-space density into the cover. Furthermore, we prove that for covers satisfying natural assumptions, the graph produced by Mapper still converges in bottleneck distance to the Reeb graph of the Rips complex of the data, while possibly capturing more topological features than a standard Mapper cover. Finally, we discuss implementation details and present the results of computational experiments. We also provide an accompanying reference implementation.
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