We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection $L$ which preserves the persistent diagram of a point cloud $\mathbb{X}$ via simulated annealing. The projection $L$ induces a set of canonical simplicial maps from the Rips (or \v{C}ech) filtration of $\mathbb{X}$ to that of $L\mathbb{X}$. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism $\mu_{\operatorname{quasi-iso}}$ or strong homotopy equivalence $\mu_{\operatorname{equiv}}$. These $\mu_{\operatorname{quasi-iso}}$ and $\mu_{\operatorname{equiv}}$ measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.
翻译:我们引入了一种线性维度降低技术, 通过持久性同质学来保存表层特征。 方法的设计是寻找线性投影 $L$, 通过模拟整流来保存点云的持久性图 $\ mathbb{X} $。 投影 $L$ 能够直接比较利普( 或\\\ v{C} {C} ech) 的精度简单化图和 $L\ mathbb{X} 的过滤。 除了持续图之间的距离外, 投影还引出一个在过滤器之间绘制的地图, 叫做 过滤法共和态 。 使用过滤式同质性共性图, 人们可以测量两种过滤法的形状之间的差异, 直接比较与准软化的精度复杂度 $\\\ { mathb{ {quab{X} 美元 或强烈的同质等同性等同性等同性等同性等同性框架 $\ =quasiqual desimasi- proqual deal deal dequimatimatial deal deal deal deal deal depressal matize roqual roqual roqual roqual matipeal roqual siabal ma rotique ma ma ma ro ro ro ro rotipe rotipe ro ma ma ma ma rotime rotical ma ma ro ma ro ro ro ro ro ro ro ro ro ma ro ro ro ro ro ro ro ro ro ro ro ro ro ro si ro ro ro ro si ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro si ro