核心概念
Study the interleaving distance on Rd-mapper graphs using loss functions to compute distances efficiently.
要約
The content discusses the computation of the interleaving distance for mapper graphs using a loss function. It covers the encoding of data structures, construction of graphs, and mapping between functors. The analysis focuses on basis elements and provides a method to compute distances in polynomial time.
Data Structures:
- Construct graphs for F and G based on functor representations.
- Encode natural transformations φ and ψ as set maps between vertices and edges.
Basis Elements:
- Define basis unnatural transformations for functors H and H'.
- Introduce basis n-assignment to focus computations on basic open sets.
Computation:
- Develop algorithms to compute loss functions LB(φ, ψ) efficiently.
- Demonstrate encoding methods for d = 1 case before extending to higher dimensions.
統計
NP-hardであることが多い間隔距離の計算を効率的に行う方法を提供します。
計算量が多項式であることを示すアルゴリズムを開発します。
基本要素に焦点を当てた基礎的なオープンセット上での計算方法を紹介します。