AI 中文总结
本研究在拓扑数据分析领域，针对1维同调类，引入闭链总绝对曲率作为代价函数，构建角度最优同源闭链问题并转化为二次二元优化问题，在人工玩具数据上开展实验，为同调代表元优化提供新方法。
AI 中文摘要
在拓扑数据分析，尤其是持续同调分析中，提取同调类的“最优”代表元对识别感兴趣的几何区域至关重要。现有研究中，最优性通常以最小化长度或体积来定义。本研究聚焦于单个1维同调类，引入闭链的总绝对曲率作为代价函数。我们证明，该基于闭链边之间夹角的代价函数，会惩罚闭链代表元偏离平面性、凸性和简单性的情况。我们构建了“角度最优同源闭链问题”，将其重写为二次二元优化问题，并展示了在人工玩具数据上的实验结果。
英文摘要
In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.
Comments26 pages
