Generation Matrix：一种层次树的可嵌入矩阵表示
Generation Matrix: An Embeddable Matrix Representation for Hierarchical Trees
摘要：
从局部结构出发研究层次树是一种常用的研究方法。然而，繁琐的分析和描述使得朴素方法难以适应日益复杂的层次树问题。为了提高层次树研究的效率，我们提出了一种层次树的可嵌入矩阵表示，称为生成矩阵。它可以将抽象的层次树转化为具体的矩阵表示，然后将层次树作为一个整体进行研究，大大降低了研究的复杂性。数学分析表明，生成矩阵可以模拟各种递归算法，而无需访问局部结构，并提供了多种可解释的矩阵运算，支持层次树的研究。将生成矩阵应用于差分隐私层次树发布，提出了一种基于生成矩阵的最优一致发布算法（GMC）。它提供了一个非常简洁的过程描述，使我们可以描述其核心步骤作为一个简单的矩阵表达式，而不是像现有的算法多个复杂的递归过程。我们的实验表明，GMC只需要几秒钟就可以完成超过1000万个节点的大规模数据集的发布。计算效率比现有方案提高了100倍。
英文摘要：
Starting from the local structures to study hierarchical trees is a common research method. However, the cumbersome analysis and description make the naive method challenging to adapt to the increasingly complex hierarchical tree problems. To improve the efficiency of hierarchical tree research, we propose an embeddable matrix representation for hierarchical trees, called Generation Matrix. It can transform the abstract hierarchical tree into a concrete matrix representation and then take the hierarchical tree as a whole to study, which dramatically reduces the complexity of research. Mathematical analysis shows that Generation Matrix can simulate various recursive algorithms without accessing local structures and provides a variety of interpretable matrix operations to support the research of hierarchical trees. Applying Generation Matrix to differential privacy hierarchical tree release, we propose a Generation Matrix-based optimally consistent release algorithm (GMC). It provides an exceptionally concise process description so that we can describe its core steps as a simple matrix expression rather than multiple complicated recursive processes like existing algorithms. Our experiments show that GMC takes only a few seconds to complete a release for large-scale datasets with more than 10 million nodes. The calculation efficiency is increased by up to 100 times compared with the state-of-the-art schemes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
