发表机构
University of California at Santa Barbara; Université Laval（加州大学圣塔芭芭拉分校; 拉瓦尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在矩阵分析、图论与双曲几何交叉领域，提出双曲距离矩阵补全理论，利用锚定程序转化数据，以弦性为充要条件，并给出显式构造及应用。
AI 中文摘要
在矩阵分析、图论与双曲几何的交界处，发展了一套双曲距离数据的补全理论。Krein关于度量空间在洛巴切夫斯基空间中可嵌入性的刻画，引出了一种自然的锚定程序，将不定数据转化为正半定核。与正半定及欧几里得距离矩阵补全相类比，证明了规范图的弦性是局部Lorentz-Gram数据允许全局补全的充分必要条件。存在性通过显式构造加以补充。对于树，我们获得了测地线矫正补全和乘积距离补全；对于弦图，后者沿团树扩展为矩阵值转移。所得的规范补全以其逆的稀疏性和最大绝对行列式原理为特征。其度量失真表现出由团分隔符大小决定的尖锐二分性。开发了从稀疏双曲测量中精确恢复以及对层次和系统发育数据的应用。
英文摘要
A completion theory for hyperbolic distance data is developed at the interface of matrix analysis, graph theory, and hyperbolic geometry. Krein's characterization of the metric space embeddability in Lobachevsky space leads to a natural anchoring procedure that transforms the indefinite data into a positive semidefinite kernel. In analogy with positive semidefinite and Euclidean distance matrix completion, chordality of the specification graph is shown to be the necessary and sufficient condition for local Lorentz-Gram data to admit global completion. Existence is complemented by explicit constructions. For trees, we obtain geodesic-rectification and product-distance completions; for chordal graphs, the latter extends to matrix-valued transfers along clique-trees. The resulting canonical completion is characterized by sparsity of its inverse and by a maximum-absolute-determinant principle. Its metric distortion exhibits a sharp dichotomy governed by clique separator size. Applications to exact recovery from sparse hyperbolic measurements and to hierarchical and phylogenetic data are developed.
Comments62 pages, 9 figures
