$hp$-自适应树用于图信号逼近
$hp$-adaptive trees for graph signal approximation
- IDSIA USI-SUPSI, Università della Svizzera italiana（IDSIA USI-SUPSI，瑞士卢加诺大学）
- Department of Mathematics “Tullio Levi-Civita” and Padova Neuroscience Center (PNC), Università di Padova（帕多瓦大学“图利奥·莱维-奇维塔”数学系和帕多瓦神经科学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出基于$hp$-细化的图信号逼近策略，通过将细化视为背包问题提取高效子树，并优化多项式次数剪枝，结合局部嵌入，在保持误差的同时降低成本。
AI中文摘要：
树编码的图划分是图信号分解和逼近的基本工具。为了高效逼近此类图信号，我们通过结合区域分解与使用更高次多项式的改进局部逼近，开发了基于$hp$-细化（$hp$-refinement）的策略。这样，从给定的图划分树中提取出一棵更高效的子树，在保持相同总误差的同时，显著降低了信号逼近的成本。为此，我们将细化过程解释为二元背包问题，以确定一棵增强的划分树。我们进一步研究了一种后验（a-posteriori）策略，该策略通过优化子域上的多项式次数来剪枝划分树。为了使多项式基系统适用于一般图或高维数据，我们提出了将图局部嵌入到低维欧几里得空间的方法。我们通过广泛的数值测试验证了算法的效率，这些测试仔细评估了所应用的细化和优化策略的影响。
英文摘要：
Tree-encoded partitionings of graphs are fundamental tools for the decomposition and approximation of graph signals. For the efficient approximation of such graph signals, we develop strategies based on $hp$-refinement by combining domain decomposition with an improved local approximation using polynomials of higher degree. In this way, from a given graph partitioning tree, a more efficient subtree is extracted in which the cost of the signal approximation is considerably reduced by still maintaining the same total error. To this end, we interpret the refinement process as a binary knapsack problem to determine an enhanced partitioning tree. We further study an a-posteriori strategy which prunes the partitioning tree by optimizing the polynomial degrees over the subdomains. To make polynomial basis systems accessible for general graphs or high-dimensional data, we propose local embeddings of graphs into low dimensional Euclidean spaces. We underpin the efficiency of our algorithms with extensive numerical tests which carefully assess the impact of the applied refinements and optimization strategies.
