Persistent Homology: A Unified Framework for Characterizing Local and Global Structures in Disordered Systems
This paper introduces a novel framework based on persistent homology (PH) for characterizing both local and global structures in disordered systems, using a unified mathematical approach to bridge the gap between microscopic and macroscopic properties.
Stable Similarity Comparison of Persistent Homology Groups Using a Novel Pseudometric
This research paper introduces a novel pseudometric, d(p) S, for comparing persistent homology groups, demonstrating its stability under conformal linear transformations and its ability to classify similar objects effectively by focusing on essential features rather than congruence.
Approximating Topological Prevalence in High Dimensional Point Clouds Using Cross-Matched Interval Prevalence
This paper introduces a novel algorithm for approximating topological prevalence in large, high-dimensional point clouds, addressing the limitations of traditional persistent homology methods in terms of noise sensitivity and computational complexity.
Multiple Cylinder of Relations for Finite Spaces and a Stronger Nerve Theorem for Strong-Good Covers
This paper introduces the "multiple cylinder of relations" for comparing topological spaces and a stronger version of the Nerve Theorem using "strong-good covers" where intersections are collapsible, not just contractible. This provides new tools for analyzing homotopy types in finite spaces and simplicial complexes.
矩形持久性模組交織距離的封閉公式
本文提出了一個封閉公式，僅使用底層矩形的幾何形狀來計算矩形持久性模組之間的交織距離，並將其擴展到計算矩形可分解持久性模組的瓶頸距離。
직사각형 지속 모듈의 인터리빙 거리에 대한 폐쇄 공식
이 논문에서는 기본 직사각형의 기하학적 정보만을 사용하여 두 개의 직사각형 지속 모듈 간의 인터리빙 거리를 계산하는 공식을 제시합니다.
Computing the Interleaving Distance Between Rectangle Persistence Modules Using a Closed Formula
This research paper presents a novel closed formula for calculating the interleaving distance between rectangle persistence modules, a fundamental concept in topological data analysis, based solely on the geometric properties of the underlying rectangles.
On $\ell_p$-Vietoris-Rips Complexes (Topological Data Analysis using $\ell_p$ Norms)
This paper introduces and analyzes the $\ell_p$-Vietoris-Rips complex, a novel tool for topological data analysis that generalizes the classical Vietoris-Rips complex using $\ell_p$ norms, offering potential advantages in stability and sensitivity to data variations.
Persistent Homology with Spectral Distances for High-Dimensional Data Analysis
Traditional persistent homology struggles with high-dimensional data due to noise sensitivity, but spectral distances like effective resistance and diffusion distance on kNN graphs offer a robust solution for accurate topology detection.
정보 전파에서의 에이전트 상태에 대한 위상적 식별: 금융 시장 적용 - 기회주의적 거래자 식별에 초점
머신러닝 기법 중 하나인 TDA의 Mapper 알고리즘을 활용하여 금융 시장에서 내부 정보를 이용하는 기회주의적 투자자를 식별할 수 있다.
