Intersecting manifold segmentation has been a focus of research, where individual manifolds, that intersect with other manifolds, are separated to discover their distinct properties. The proposed method is based on the intuition that when a manifold in $D$ dimensional space with an intrinsic dimension of $d$ intersects with another manifold, the data variance grows in more than $d$ directions. The proposed method measures local data variances and determines their vector directions. It counts the number of vectors with non-zero variance, which determines the manifold's intrinsic dimension. For detection of the intersection region, the method adapts to the changes in the angular gaps between the corresponding direction vectors of the child and parent using exponential moving averages using a tree structure construction. Accordingly, it includes those data points in the same manifold whose neighborhood is within the adaptive angular difference and eventually identifies the data points in the intersection area of manifolds. Data points whose inclusion in the neighborhood-identified data points increases their intrinsic dimensionality are removed based on data variance and distance. The proposed method performs better than 18 SOTA manifold segmentation methods in ARI and NMI scores over 14 real-world datasets with lesser time complexity and better stability.

翻译：相交流形分割一直是研究热点，其目标是通过分离与其他流形相交的各个流形，以揭示其独特性质。本文所提方法基于以下直观认识：当$D$维空间中本征维度为$d$的流形与另一流形相交时，数据方差会在超过$d$个方向上增长。该方法通过测量局部数据方差并确定其向量方向，统计具有非零方差的向量数量，从而确定流形的本征维度。为检测相交区域，该方法利用树结构构造中的指数移动平均，自适应地处理子节点与父节点对应方向向量间角度间隙的变化。因此，它将邻域位于自适应角度差范围内的数据点归入同一流形，并最终识别出流形相交区域的数据点。基于数据方差和距离度量，该方法会剔除那些加入邻域识别数据点会导致本征维度增加的数据点。在14个真实数据集上，所提方法在ARI和NMI指标上优于18种当前最优的流形分割方法，同时具有更低的时间复杂度和更好的稳定性。