Topological Data Analysis(TDA、位相的データ解析)に関する代表的なサーベイの目次を記載します。
複数のサーベイの目次を眺めることで、重要なキーワード等が見えるかと思います。
Barcodes: The Persistent Topology of Data
R. Ghrist, Bull. Amer. Math. Soc. (N.S.), vol 45 (1): 61 – 75 (2008).
1. The shape of data
1.1. Topological data analysis.
1.2. Clouds of data.
1.3. From clouds to complexes.
1.4. Which ε?
2. Algebraic topology for data
2.1. Persistence.
2.2. Persistent homology.
2.3. Barcodes.
2.4. Computation.
2.5. Other directions.
3. Example: natural images
3.1. “Round about the cauldron go”.
3.2. “Hover through the fog”.
3.3. “When shall we three meet again?”
3.4. “Come like shadows, so depart!”
Persistent Homology — a Survey
H. Edelsbrunner, J. Harer, In Surveys on discrete and computational geometry, volume 453 of Contemp. Math., 257 - 282. Amer. Math. Soc. (2008).
1. Introduction
Motivation.
History.
Outline.
2. Persistence
Single variable functions.
Homology.
Morse functions.
Simplicial complexes.
Tame functions.
Module structure.
3. Algorithm
Smith normal form.
Persistence pairing.
Generating cycles.
Sparse matrix implementation.
4. Variants
Relative homology.
Extended persistence.
Intersection homology.
Localized homology.
5. Spectral Sequences
Diagonal sweep.
Groups and maps.
Iteration.
6. Stability
Bottleneck distance.
Applications.
Time series.
Dynamic algorithm.
7. Discussion
Related developments.
Future directions.
Topology and Data
G. Carlsson, Bull. Amer. Math. Soc. (N.S.), vol. 46 (2):255 – 308 (2009).
1. Introduction
2. Persistence and homology
2.1. Introduction.
2.2. Building coverings and complexes.
2.3. Persistent homology.
2.4. Example: Natural image statistics.
2.5. Example: Electrode array data from primary visual cortex.
3. Imaging: mapper
3.1. Visualization.
3.2. A topological method.
3.3. Filters.
3.4. Scale space.
3.5. Examples.
4. Generalized forms of persistence
4.1. Multidimensional persistence.
4.2. Quivers and zigzags.
4.3. Tree based persistence.
5. Reasoning about clustering
6. What should the theorems be?
Topological Data Analysis
A. Zomorodian, Advances in applied and computational topology, Proc. Sympos. Appl. Math., vol. 70, 1 - 39, Amer. Math. Soc. (2012).
1. Introduction
1.1. Topology.
1.2. Data.
1.3. Analysis.
1.4. Pipeline.
2. Background
2.1. Topology.
2.2. Simplicial Complex.
2.3. Cover and Nerve.
3. Combinatorial Representations
3.1. Cech Complex.
3.2. Alpha Complex.
3.3. Vietoris-Rips Complex.
3.4. Witness Complex.
3.5. Cubical Complex.
3.6. Analysis.
4. Topological Invariants
4.1. Definition.
4.2. Euler Characteristic.
4.3. Simplicial Homology.
4.4. Single-Scale Analysis.
5. Multiscale Invariants
5.1. Multifiltration Model.
5.2. Persistent Homology.
5.3. Multidimensional Persistence.
5.4. Zigzag Persistence.
5.5. Multiscale Analysis.
6. Reduced Representations
6.1. Reductions.
6.2. Simplicial Sets.
6.3. Tidy Sets.
6.4. Tidy Analysis.
7. Conclusion
位相的データ解析とパーシステントホモロジー
平岡裕章, 数学, 第68巻 第4号, 361 – 380 (2016).
1. 序
2. 幾何モデル
2.1. チェック複体
2.2. ヴィートリス・リップス複体
2.3. その他の幾何モデル
3. パーシステントホモロジー
3.1. 次数付き加群
3.2. 箙の表現論とパースシステント加群の一般化
3.3. パーシステント加群のその他の拡張
4. 安定性定理
5. 計算ホモロジー
6. 応用
6.1. 工学
6.2. 材料科学
6.3. その他の応用