论文标题
使用椭圆形寻找歧管的同源性
Finding the Homology of Manifolds using Ellipsoids
论文作者
论文摘要
应用拓扑中的一个标准问题是如何从近似值的嘈杂点云中发现数据的拓扑不变性。我们考虑从适当嵌入的C1-Submanifold绘制样品的情况，在欧几里得空间中没有边界。我们表明，我们可以变形缩回椭圆形的结合，以样品点为中心并沿切线方向伸展到歧管。因此，歧管的同质类型，因此也是同源类型，与椭圆形覆盖神经复合物的同源类型相同。通过将样品点加厚到椭圆形而不是球，我们的结果比文献中的样品密度要小。他们还主张在持久同源性的条形码构建中使用细长的形状。
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.
