论文标题
Morse理论信号压缩和链复合物的重建
Morse Theoretic Signal Compression and Reconstruction on Chain Complexes
论文作者
论文摘要
在拓扑数据分析（TDA）和机器学习的交集中，近年来细胞信号处理的领域迅速发展。在这种情况下，使用组合laplacian和所得的Hodge分解来处理复合物细胞上的每个信号。同时，离散的摩尔斯理论已被广泛用于通过减少复合物的大小，同时保留其全球拓扑特性来加快计算的速度。在本文中，我们为链复合物提供了一种信号压缩和重建方法，该方法利用代数离散摩尔斯理论的工具。主要目标是通过变形缩回并在其细胞上的一组信号中减少和重建一个基于链的复合物，并尽可能保留复合物和信号的全局拓扑结构。我们首先证明，基于有限程度的链链复合物的任何变形缩回都等同于摩尔斯匹配。然后，我们将研究信号在特定类型的摩尔斯匹配下如何变化，显示其重建误差在霍奇分解的特定组件上是微不足道的。此外，我们提供了一种算法来计算最小重建误差的Morse匹配。
At the intersection of Topological Data Analysis (TDA) and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian, and the resultant Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this paper, we provide an approach to signal compression and reconstruction on chain complexes that leverages the tools of algebraic discrete Morse theory. The main goal is to reduce and reconstruct a based chain complex together with a set of signals on its cells via deformation retracts, preserving as much as possible the global topological structure of both the complex and the signals. We first prove that any deformation retract of real degree-wise finite-dimensional based chain complexes is equivalent to a Morse matching. We will then study how the signal changes under particular types of Morse matching, showing its reconstruction error is trivial on specific components of the Hodge decomposition. Furthermore, we provide an algorithm to compute Morse matchings with minimal reconstruction error.
