有序kNN图的同步点定位和密度估计
Point Localization and Density Estimation from Ordinal kNN graphs using Synchronization
摘要：
研究了低维欧氏空间中无权有向k-近邻图的嵌入问题。每个顶点的k近邻提供了点之间距离的序数信息，但不是距离本身。我们使用这个序数信息以及低维来恢复点的坐标，直到任意相似性变换（刚性变换和缩放）。此外，我们还说明了通过总变差最大惩罚似然估计（TV-MPLE）方法鲁棒地恢复底层密度的可能性。我们使现有的方法可扩展的一个实例的本地到全球的算法的基础上组同步，最近在文献中提出的传感器网络定位和结构生物学的背景下，我们增加了缩放同步步骤。我们展示了我们的方法在大型图上的可扩展性，并展示了它如何与局部有序嵌入（LOE）算法进行比较，该算法是最近提出的，用于从稀疏距离集之间的成对顺序比较中恢复点云的配置。
英文摘要：
We consider the problem of embedding unweighted, directed k-nearest neighbor graphs in low-dimensional Euclidean space. The k-nearest neighbors of each vertex provides ordinal information on the distances between points, but not the distances themselves. We use this ordinal information along with the low-dimensionality to recover the coordinates of the points up to arbitrary similarity transformations (rigid transformations and scaling). Furthermore, we also illustrate the possibility of robustly recovering the underlying density via the Total Variation Maximum Penalized Likelihood Estimation (TV-MPLE) method. We make existing approaches scalable by using an instance of a local-to-global algorithm based on group synchronization, recently proposed in the literature in the context of sensor network localization and structural biology, which we augment with a scaling synchronization step. We demonstrate the scalability of our approach on large graphs, and show how it compares to the Local Ordinal Embedding (LOE) algorithm, which was recently proposed for recovering the configuration of a cloud of points from pairwise ordinal comparisons between a sparse set of distances.
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