标题： 带有变薄距离的树推断 显示英文标题

标题： Tree inference with varifold distances

摘要： 在本文中，我们考虑了一个树推断问题，该问题受到单细胞基因组学中关键问题的启发，即从测序数据中重建动态细胞过程。 特别是，给定从该过程中采样的细胞群体，我们感兴趣的是根据细胞在该过程中的进展对细胞进行排序的问题。 这被称为轨迹推断。 如果该过程是分化过程，这就相当于重建相应的分化树。 在实践中，一种实现方法是基于测序数据中观察到的细胞相似性来估计节点之间的最短路径距离。 最近的测序技术使得测量两种类型的数据成为可能：基因表达水平，以及RNA速度，这是一种预测基因表达变化的向量。 然后数据由一个离散的向量场组成，该向量场位于维度等于所考虑基因数量的欧几里得空间（或其子集）上。 通过整合这个速度场，我们追踪每个单细胞从某个初始阶段到当前阶段的基因表达水平的变化。 最终，我们假设在欧几里得空间中存在一个忠实的分化树嵌入，但我们只能通过从根到节点的路径曲线来观察它。 利用这些曲线之间的变面距离，我们定义了节点之间的相似性度量，并证明该度量近似于与目标树同构的树中的最短路径距离。 显示英文摘要

摘要： In this paper, we consider a tree inference problem motivated by the critical problem in single-cell genomics of reconstructing dynamic cellular processes from sequencing data. In particular, given a population of cells sampled from such a process, we are interested in the problem of ordering the cells according to their progression in the process. This is known as trajectory inference. If the process is differentiation, this amounts to reconstructing the corresponding differentiation tree. One way of doing this in practice is to estimate the shortest-path distance between nodes based on cell similarities observed in sequencing data. Recent sequencing techniques make it possible to measure two types of data: gene expression levels, and RNA velocity, a vector that predicts changes in gene expression. The data then consist of a discrete vector field on a (subset of a) Euclidean space of dimension equal to the number of genes under consideration. By integrating this velocity field, we trace the evolution of gene expression levels in each single cell from some initial stage to its current stage. Eventually, we assume that we have a faithful embedding of the differentiation tree in a Euclidean space, but which we only observe through the curves representing the paths from the root to the nodes. Using varifold distances between such curves, we define a similarity measure between nodes which we prove approximates the shortest-path distance in a tree that is isomorphic to the target tree.