So sánh phương pháp

	Lý thuyết đồng luân bền vững	Locally Linear Embedding (LLE)	Thuật toán Mapper
Lĩnh vực	Tô pô học	Học máy	Tô pô học
Họ	Machine learning	Machine learning	Machine learning
Năm ra đời	2002	2000	2007
Người khởi xướng	Edelsbrunner, Letscher & Zomorodian	Sam Roweis & Lawrence Saul	Singh, Mémoli & Carlsson
Loại	Topological feature extraction algorithm	Nonlinear manifold dimensionality reduction	Graph-based topological summarization
Công trình gốc	Edelsbrunner, H., Letscher, D., & Zomorodian, A. (2002). Topological persistence and simplification. Discrete & Computational Geometry, 28(4), 511–533.	Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326.	Singh, G., Mémoli, F., & Carlsson, G. (2007). Topological methods for the analysis of high dimensional data sets and 3D object recognition. Eurographics Symposium on Point-Based Graphics, 91–100.
Tên gọi khác	Topological Persistence, Persistence Barcodes, Persistent Betti Numbers, Kalıcı Homoloji	LLE, manifold learning, nonlinear dimensionality reduction, yerel doğrusal gömme	Topological Mapper, TDA Mapper, Reeb Graph Approximation, Eşleyici Algoritma
Liên quan	2	3	2
Tóm tắt	Persistent homology is a method in topological data analysis that quantifies the multi-scale topological structure of data by tracking connected components, loops, and voids as a scale parameter varies. Introduced by Edelsbrunner, Letscher, and Zomorodian in 2002, it encodes topological features through their birth and death scales, producing persistence diagrams or barcodes that serve as compact, coordinate-free descriptors of shape. The approach is robust to noise and provides a mathematically rigorous bridge between discrete data and algebraic topology.	Locally linear embedding, introduced by Sam Roweis and Lawrence Saul in 2000, is a manifold-learning method for nonlinear dimensionality reduction. It assumes that although data may curve through a high-dimensional space, each point and its neighbours lie approximately on a flat patch. LLE captures each point as a weighted combination of its neighbours and then finds a low-dimensional layout that preserves those same local relationships, unrolling curved structure into a faithful low-dimensional map.	The Mapper algorithm is a method in topological data analysis (TDA) that produces a graph-based summary of the shape of high-dimensional point cloud data. Introduced by Singh, Mémoli, and Carlsson in 2007 at the Eurographics Symposium on Point-Based Graphics, Mapper constructs a simplicial complex — typically a graph — that captures the global topological and geometric structure of a dataset without requiring a fixed embedding or metric assumption.