Topological Data Analysis(TDA、位相的データ解析)のサーベイの目次

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](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf)**
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](https://users.cs.duke.edu/~edels/Papers/2008-B-02-PersistentHomology.pdf)**
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](http://www.ams.org/images/carlsson-notes.pdf)**
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](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.261.1298&rep=rep1&type=pdf)**
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](http://mathsoc.jp/publication/sugaku/)**
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
#### 7. 結び