Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations) are commonly used for the analysis. A large class of feature maps, which we call linear, depends on some weight functions, the choice of which is a critical issue. An important criterion to choose a weight function is to ensure stability of the feature maps with respect to Wasserstein distances on diagrams. We improve known results on the stability of such maps, and extend it to general weight functions. We also address the choice of the weight function by considering an asymptotic setting; assume that $\mathbb{X}_n$ is an i.i.d. sample from a density on $[0,1]^d$. For the \v{C}ech and Rips filtrations, we characterize the weight functions for which the corresponding feature maps converge as $n$ approaches infinity, and by doing so, we prove laws of large numbers for the total persistences of such diagrams. Those two approaches (stability and convergence) lead to the same simple heuristic for tuning weight functions: if the data lies near a $d$-dimensional manifold, then a sensible choice of weight function is the persistence to the power $\alpha$ with $\alpha \geq d$.
翻译:可见度图是点云表面学的高效描述符。 标准统计方法并非自然属于Hilbert 空间, 因此无法直接应用标准统计方法。 相反, 通常使用地貌地图( 或表达方式) 来进行分析 。 我们称之为线性的一大批地貌地图取决于某种重量函数, 其选择是一个关键问题 。 选择权重函数的一个重要标准是确保地貌地图相对于图上瓦西斯坦距离的稳定性。 我们改进了这些地图稳定性的已知结果, 并将之扩展至一般重量函数 。 我们还通过考虑一个无症状设置来解决权重函数的选择问题; 假设 $\ mathb{X ⁇ n$ 是 $1, 1, 1\ dd. d. d. 的密度样本。 对于 & v{C} 精度和分数过滤来说, 我们将相应的地貌地图的重量功能定性为 $( $) 接近度, 并且通过这样做, 我们通过考虑一个无压的设置, $ 总持久性的值法度函数的法度。 这些方法( ) 和 $ max strual strual res strualtial stru strital) code code the the the sqmal stritaltial max the sqmal stritaltial strital strital strital maxild) a max max max maxil maxy max max max le le) max max maxital max 。 max max max max max max maxil 。