The paper presents theoretical results for the computation of the matching distance between 2-parameter persistence modules. The key insights are:
The matching distance can be computed from a finite set of lines, rather than considering all lines with positive slope in the parameter space. This is achieved by partitioning the lines into equivalence classes based on their reciprocal position with respect to a finite set of critical values and switch points.
The critical values capture all the changes in homology occurring throughout the 2-parameter filtration. They can be used to determine a finite set of lines that are sufficient for computing the matching distance.
The switch points are additional points in the parameter space that are necessary to refine the equivalence relation on lines. This ensures that within each equivalence class, there is at least one matching that achieves the bottleneck distance for all lines in that class.
The matching distance is shown to be attained either on a line through two distinct points, one from the set of critical values and switch points, and the other from the union of critical values and switch points, or on a diagonal line through exactly one point in this union.
The paper provides a geometric interpretation of the different types of lines, including horizontal, vertical, and diagonal lines, and their contribution to the matching distance computation. This leads to an implementable algorithm for the exact computation of the matching distance between 2-parameter persistence modules.
他の言語に翻訳
原文コンテンツから
arxiv.org
深掘り質問