标题： RTD-Lite：用于学习任务中比较加权图的可扩展拓扑分析 显示英文标题

标题： RTD-Lite: Scalable Topological Analysis for Comparing Weighted Graphs in Learning Tasks

摘要： 用于比较加权图的拓扑方法在各种学习任务中很有价值，但在大规模数据集上常常存在计算效率低的问题。 我们引入了RTD-Lite，这是一种可扩展的算法，能够高效地比较两个加权图的拓扑特征，特别是顶点一一对应情况下的连通性或聚类结构，在任意尺度上。 使用辅助图中的最小生成树，RTD-Lite以$O(n^2)$的时间和内存复杂度捕捉拓扑差异。 这种效率使其适用于降维和神经网络训练等任务。 在合成和现实数据集上的实验表明，RTD-Lite能够有效识别拓扑差异，同时相比现有方法显著减少了计算时间。 此外，将RTD-Lite集成到神经网络训练中作为损失函数的一部分，可以增强学习表示中拓扑结构的保留。 我们的代码可在 https://github.com/ArGintum/RTD-Lite 公开获取。 显示英文摘要

摘要： Topological methods for comparing weighted graphs are valuable in various learning tasks but often suffer from computational inefficiency on large datasets. We introduce RTD-Lite, a scalable algorithm that efficiently compares topological features, specifically connectivity or cluster structures at arbitrary scales, of two weighted graphs with one-to-one correspondence between vertices. Using minimal spanning trees in auxiliary graphs, RTD-Lite captures topological discrepancies with $O(n^2)$ time and memory complexity. This efficiency enables its application in tasks like dimensionality reduction and neural network training. Experiments on synthetic and real-world datasets demonstrate that RTD-Lite effectively identifies topological differences while significantly reducing computation time compared to existing methods. Moreover, integrating RTD-Lite into neural network training as a loss function component enhances the preservation of topological structures in learned representations. Our code is publicly available at https://github.com/ArGintum/RTD-Lite