Topological Data Analysis

An introduction to topological data analysis presented as a mini-course at the CMS winter meeting 2019. Some of the animated content did not survive the translation to PDF.

Transcript

1. Topological Data Analysis Leland McInnes CMS Winter Meeting 2019
2. Leland McInnes Researcher at the Tutte Institute for Mathematics and Computing Maintainer for many machine learning packages umap-learn, hdbscan, pynndescent, enstop Scikit-learn and Scikit-TDA contributor @leland_mcinnes leland.mcinnes@gmail.com
3. I want to empower users to explore their data
4. Data Analysis
5. Name Birthdate Height Weight Alice 1985-10-12 178 55 Bob 1991-01-02 189 85 Carmen 1978-05-18 170 54 David 1996-11-30 175 72 Eva 1975-09-21 159 45 Frank 1999-06-28 192 80 Gertrude 1943-10-19 181 63 Harold 1982-11-08 176 65
6. Name Birthdate Height Weight Mohsen March 21st 1985 5’8” 55 kg Type Name Here 12/31/1991 almost 6 feet 178 Rahul Pushpakumara 30-05-2001 170 cm ??? Luuk Sander van der Berg 02/07/05 1.75 metres N/A Yumiko❤✨ Feb 31 1978 short Akwesi Olatunji 1970-01-01 5 foot 11 80 王秀英 5’9” 130 Robert Garcia-Smith Jr. January I think? 176 65
7. What questions do you want answered?
8. What questions should you be asking?
9. Look at your data!
10. Name Birthdate Height Weight Mohsen March 21st 1985 5’8” 55 kg Type Name Here 12/31/1991 almost 6 feet 178 Rahul Pushpakumara 30-05-2001 170 cm ??? Luuk Sander van der Berg 02/07/05 1.75 metres N/A Yumiko❤✨ Feb 31 1978 short Akwesi Olatunji 1970-01-01 5 foot 11 80 王秀英 5’9” 130 Robert Garcia-Smith Jr. January I think? 176 65
11. Petal width Petal length Sepal width Sepal length Species 5.1 3.5 1.4 0.2 Setosa 4.9 3 1.4 0.2 Setosa 4.7 3.2 1.3 0.2 Setosa 4.6 3.1 1.5 0.2 Setosa 5 3.6 1.4 0.2 Setosa 5.4 3.9 1.7 0.4 Setosa 4.6 3.4 1.4 0.3 Setosa 5 3.4 1.5 0.2 Setosa 4.4 2.9 1.4 0.2 Setosa 4.9 3.1 1.5 0.1 Setosa 5.4 3.7 1.5 0.2 Setosa 4.8 3.4 1.6 0.2 Setosa 4.8 3 1.4 0.1 Setosa
12. Petal width Petal length Sepal width Sepal length Species 5.1 3.5 1.4 0.2 Setosa 4.9 3 1.4 0.2 Setosa 4.7 3.2 1.3 0.2 Setosa 4.6 3.1 1.5 0.2 Setosa 5 3.6 1.4 0.2 Setosa 5.4 3.9 1.7 0.4 Setosa 4.6 3.4 1.4 0.3 Setosa 5 3.4 1.5 0.2 Setosa 4.4 2.9 1.4 0.2 Setosa 4.9 3.1 1.5 0.1 Setosa 5.4 3.7 1.5 0.2 Setosa 4.8 3.4 1.6 0.2 Setosa 4.8 3 1.4 0.1 Setosa 4.3 3 1.1 0.1 Setosa 5.8 4 1.2 0.2 Setosa 5.7 4.4 1.5 0.4 Setosa 5.4 3.9 1.3 0.4 Setosa 5.1 3.5 1.4 0.3 Setosa 5.7 3.8 1.7 0.3 Setosa 5.1 3.8 1.5 0.3 Setosa 5.4 3.4 1.7 0.2 Setosa 5.1 3.7 1.5 0.4 Setosa 4.6 3.6 1 0.2 Setosa 5.1 3.3 1.7 0.5 Setosa 4.8 3.4 1.9 0.2 Setosa 5 3 1.6 0.2 Setosa 5 3.4 1.6 0.4 Setosa 5.2 3.5 1.5 0.2 Setosa 5.2 3.4 1.4 0.2 Setosa 4.7 3.2 1.6 0.2 Setosa 4.8 3.1 1.6 0.2 Setosa 5.4 3.4 1.5 0.4 Setosa 5.2 4.1 1.5 0.1 Setosa 5.5 4.2 1.4 0.2 Setosa 4.9 3.1 1.5 0.2 Setosa 5 3.2 1.2 0.2 Setosa 5.5 3.5 1.3 0.2 Setosa Petal width Petal length Sepal width Sepal length Species 4.9 3.6 1.4 0.1 Setosa 4.4 3 1.3 0.2 Setosa 5.1 3.4 1.5 0.2 Setosa 5 3.5 1.3 0.3 Setosa 4.5 2.3 1.3 0.3 Setosa 4.4 3.2 1.3 0.2 Setosa 5 3.5 1.6 0.6 Setosa 5.1 3.8 1.9 0.4 Setosa 4.8 3 1.4 0.3 Setosa 5.1 3.8 1.6 0.2 Setosa 4.6 3.2 1.4 0.2 Setosa 5.3 3.7 1.5 0.2 Setosa 5 3.3 1.4 0.2 Setosa 7 3.2 4.7 1.4 Versicolor 6.4 3.2 4.5 1.5 Versicolor 6.9 3.1 4.9 1.5 Versicolor 5.5 2.3 4 1.3 Versicolor 6.5 2.8 4.6 1.5 Versicolor 5.7 2.8 4.5 1.3 Versicolor 6.3 3.3 4.7 1.6 Versicolor 4.9 2.4 3.3 1 Versicolor 6.6 2.9 4.6 1.3 Versicolor 5.2 2.7 3.9 1.4 Versicolor 5 2 3.5 1 Versicolor 5.9 3 4.2 1.5 Versicolor 6 2.2 4 1 Versicolor 6.1 2.9 4.7 1.4 Versicolor 5.6 2.9 3.6 1.3 Versicolor 6.7 3.1 4.4 1.4 Versicolor 5.6 3 4.5 1.5 Versicolor 5.8 2.7 4.1 1 Versicolor 6.2 2.2 4.5 1.5 Versicolor 5.6 2.5 3.9 1.1 Versicolor 5.9 3.2 4.8 1.8 Versicolor 6.1 2.8 4 1.3 Versicolor 6.3 2.5 4.9 1.5 Versicolor 6.1 2.8 4.7 1.2 Versicolor Petal width Petal length Sepal width Sepal length Species 6.4 2.9 4.3 1.3 Versicolor 6.6 3 4.4 1.4 Versicolor 6.8 2.8 4.8 1.4 Versicolor 6.7 3 5 1.7 Versicolor 6 2.9 4.5 1.5 Versicolor 5.7 2.6 3.5 1 Versicolor 5.5 2.4 3.8 1.1 Versicolor 5.5 2.4 3.7 1 Versicolor 5.8 2.7 3.9 1.2 Versicolor 6 2.7 5.1 1.6 Versicolor 5.4 3 4.5 1.5 Versicolor 6 3.4 4.5 1.6 Versicolor 6.7 3.1 4.7 1.5 Versicolor 6.3 2.3 4.4 1.3 Versicolor 5.6 3 4.1 1.3 Versicolor 5.5 2.5 4 1.3 Versicolor 5.5 2.6 4.4 1.2 Versicolor 6.1 3 4.6 1.4 Versicolor 5.8 2.6 4 1.2 Versicolor 5 2.3 3.3 1 Versicolor 5.6 2.7 4.2 1.3 Versicolor 5.7 3 4.2 1.2 Versicolor 5.7 2.9 4.2 1.3 Versicolor 6.2 2.9 4.3 1.3 Versicolor 5.1 2.5 3 1.1 Versicolor 5.7 2.8 4.1 1.3 Versicolor 6.3 3.3 6 2.5 Virginica 5.8 2.7 5.1 1.9 Virginica 7.1 3 5.9 2.1 Virginica 6.3 2.9 5.6 1.8 Virginica 6.5 3 5.8 2.2 Virginica 7.6 3 6.6 2.1 Virginica 4.9 2.5 4.5 1.7 Virginica 7.3 2.9 6.3 1.8 Virginica 6.7 2.5 5.8 1.8 Virginica 7.2 3.6 6.1 2.5 Virginica 6.5 3.2 5.1 2 Virginica Petal width Petal length Sepal width Sepal length Species 6.4 2.7 5.3 1.9 Virginica 6.8 3 5.5 2.1 Virginica 5.7 2.5 5 2 Virginica 5.8 2.8 5.1 2.4 Virginica 6.4 3.2 5.3 2.3 Virginica 6.5 3 5.5 1.8 Virginica 7.7 3.8 6.7 2.2 Virginica 7.7 2.6 6.9 2.3 Virginica 6 2.2 5 1.5 Virginica 6.9 3.2 5.7 2.3 Virginica 5.6 2.8 4.9 2 Virginica 7.7 2.8 6.7 2 Virginica 6.3 2.7 4.9 1.8 Virginica 6.7 3.3 5.7 2.1 Virginica 7.2 3.2 6 1.8 Virginica 6.2 2.8 4.8 1.8 Virginica 6.1 3 4.9 1.8 Virginica 6.4 2.8 5.6 2.1 Virginica 7.2 3 5.8 1.6 Virginica 7.4 2.8 6.1 1.9 Virginica 7.9 3.8 6.4 2 Virginica 6.4 2.8 5.6 2.2 Virginica 6.3 2.8 5.1 1.5 Virginica 6.1 2.6 5.6 1.4 Virginica 7.7 3 6.1 2.3 Virginica 6.3 3.4 5.6 2.4 Virginica 6.4 3.1 5.5 1.8 Virginica 6 3 4.8 1.8 Virginica 6.9 3.1 5.4 2.1 Virginica 6.7 3.1 5.6 2.4 Virginica 6.9 3.1 5.1 2.3 Virginica 5.8 2.7 5.1 1.9 Virginica 6.8 3.2 5.9 2.3 Virginica 6.7 3.3 5.7 2.5 Virginica 6.7 3 5.2 2.3 Virginica 6.3 2.5 5 1.9 Virginica 6.5 3 5.2 2 Virginica
14. 0.59 0.59 0.61 0.59 0.6 0.57 0.54 0.54 0.6 0.69 0.67 0.65 0.61 0.6 0.57 0.53 0.47 0.46 0.49 0.51 0.54 0.57 0.39 0.51 0.42 0.39 0.38 0.42 0.52 0.59 0.59 0.6 0.6 0.63 0.65 0.65 0.64 0.62 0.61 0.48 0.26 0.23 0.22 0.22 0.22 0.67 0.63 0.61 0.56 0.54 0.53 0.53 0.58 0.64 0.68 0.65 0.61 0.58 0.57 0.56 0.53 0.52 0.5 0.51 0.53 0.56 0.57 0.39 0.48 0.49 0.4 0.42 0.37 0.38 0.52 0.54 0.58 0.6 0.63 0.65 0.65 0.64 0.64 0.58 0.4 0.33 0.32 0.32 0.31 0.31 0.73 0.69 0.64 0.54 0.47 0.46 0.52 0.61 0.64 0.59 0.6 0.58 0.57 0.53 0.51 0.51 0.53 0.52 0.51 0.51 0.51 0.55 0.49 0.52 0.53 0.45 0.42 0.42 0.44 0.48 0.48 0.55 0.63 0.66 0.64 0.63 0.64 0.61 0.47 0.4 0.39 0.39 0.38 0.38 0.37 0.71 0.7 0.65 0.52 0.47 0.52 0.57 0.62 0.59 0.57 0.61 0.56 0.51 0.54 0.56 0.55 0.49 0.47 0.49 0.55 0.54 0.55 0.49 0.48 0.56 0.53 0.42 0.46 0.52 0.47 0.48 0.51 0.6 0.66 0.65 0.62 0.63 0.57 0.46 0.44 0.44 0.43 0.43 0.42 0.42 0.71 0.69 0.67 0.51 0.48 0.57 0.63 0.61 0.57 0.59 0.58 0.54 0.55 0.59 0.54 0.51 0.51 0.49 0.54 0.58 0.58 0.52 0.55 0.56 0.48 0.53 0.44 0.5 0.51 0.42 0.5 0.54 0.6 0.65 0.63 0.62 0.63 0.56 0.48 0.47 0.47 0.46 0.46 0.46 0.46 0.77 0.75 0.71 0.55 0.52 0.57 0.64 0.61 0.58 0.57 0.55 0.57 0.59 0.59 0.53 0.51 0.52 0.49 0.52 0.58 0.59 0.49 0.54 0.64 0.48 0.51 0.47 0.49 0.47 0.51 0.54 0.59 0.62 0.63 0.63 0.61 0.64 0.54 0.49 0.49 0.49 0.49 0.49 0.48 0.48 0.8 0.79 0.73 0.58 0.59 0.61 0.64 0.62 0.6 0.57 0.55 0.59 0.63 0.61 0.6 0.56 0.57 0.51 0.53 0.6 0.62 0.51 0.58 0.69 0.51 0.52 0.5 0.54 0.46 0.6 0.56 0.58 0.63 0.6 0.65 0.62 0.57 0.48 0.47 0.49 0.5 0.5 0.5 0.5 0.5 0.8 0.79 0.72 0.61 0.62 0.58 0.57 0.59 0.6 0.59 0.59 0.62 0.65 0.66 0.68 0.63 0.61 0.59 0.58 0.63 0.62 0.51 0.64 0.73 0.58 0.55 0.5 0.59 0.55 0.64 0.6 0.57 0.65 0.62 0.64 0.61 0.5 0.48 0.47 0.49 0.5 0.51 0.51 0.51 0.51 0.79 0.77 0.69 0.58 0.58 0.56 0.54 0.47 0.41 0.48 0.54 0.61 0.66 0.64 0.53 0.48 0.45 0.46 0.53 0.63 0.63 0.55 0.69 0.72 0.65 0.61 0.59 0.68 0.66 0.67 0.65 0.57 0.66 0.65 0.64 0.62 0.58 0.57 0.56 0.53 0.5 0.5 0.5 0.52 0.53 0.77 0.76 0.68 0.55 0.55 0.54 0.49 0.42 0.46 0.52 0.59 0.66 0.66 0.44 0.38 0.46 0.36 0.19 0.22 0.42 0.62 0.62 0.68 0.67 0.66 0.66 0.67 0.73 0.72 0.7 0.69 0.59 0.67 0.66 0.65 0.63 0.64 0.65 0.63 0.55 0.46 0.44 0.47 0.51 0.53 0.75 0.72 0.6 0.53 0.54 0.46 0.39 0.49 0.59 0.58 0.61 0.7 0.72 0.45 0.52 0.53 0.32 0.16 0.21 0.32 0.42 0.64 0.68 0.65 0.62 0.64 0.69 0.73 0.71 0.65 0.67 0.61 0.63 0.62 0.67 0.67 0.65 0.67 0.65 0.54 0.4 0.37 0.43 0.49 0.51 0.66 0.57 0.51 0.48 0.45 0.41 0.49 0.61 0.64 0.63 0.59 0.65 0.71 0.5 0.51 0.49 0.19 0.1 0.27 0.4 0.27 0.59 0.69 0.66 0.6 0.6 0.68 0.74 0.68 0.64 0.6 0.39 0.37 0.37 0.51 0.66 0.63 0.62 0.6 0.49 0.36 0.38 0.43 0.47 0.48 0.45 0.45 0.52 0.5 0.49 0.59 0.65 0.68 0.67 0.63 0.58 0.58 0.63 0.61 0.46 0.5 0.27 0.17 0.38 0.43 0.25 0.53 0.7 0.68 0.62 0.59 0.67 0.73 0.67 0.61 0.34 0.19 0.33 0.45 0.49 0.6 0.59 0.62 0.6 0.52 0.43 0.43 0.45 0.46 0.45 0.34 0.47 0.56 0.58 0.6 0.65 0.68 0.72 0.69 0.61 0.56 0.53 0.53 0.61 0.54 0.5 0.49 0.44 0.46 0.32 0.22 0.47 0.71 0.71 0.65 0.61 0.67 0.72 0.66 0.54 0.23 0.12 0.35 0.51 0.59 0.7 0.69 0.72 0.72 0.69 0.65 0.61 0.57 0.51 0.47 0.38 0.5 0.58 0.59 0.56 0.64 0.7 0.73 0.7 0.64 0.59 0.56 0.55 0.53 0.6 0.58 0.48 0.38 0.31 0.3 0.25 0.43 0.7 0.73 0.67 0.64 0.66 0.68 0.64 0.53 0.32 0.22 0.45 0.48 0.67 0.72 0.69 0.73 0.76 0.76 0.76 0.76 0.73 0.69 0.64 0.38 0.49 0.56 0.58 0.51 0.54 0.68 0.71 0.7 0.65 0.57 0.57 0.57 0.55 0.57 0.59 0.65 0.66 0.65 0.57 0.39 0.41 0.67 0.71 0.67 0.66 0.68 0.67 0.64 0.52 0.43 0.43 0.42 0.52 0.71 0.63 0.58 0.6 0.6 0.61 0.64 0.68 0.71 0.7 0.69 0.41 0.5 0.55 0.57 0.55 0.49 0.55 0.61 0.62 0.6 0.53 0.52 0.55 0.56 0.57 0.56 0.59 0.63 0.65 0.59 0.49 0.42 0.57 0.65 0.67 0.7 0.72 0.71 0.62 0.49 0.48 0.47 0.57 0.66 0.6 0.53 0.51 0.52 0.49 0.46 0.47 0.5 0.5 0.5 0.52 0.53 0.53 0.56 0.57 0.58 0.52 0.55 0.54 0.51 0.49 0.49 0.52 0.54 0.55 0.55 0.56 0.57 0.6 0.6 0.6 0.53 0.44 0.52 0.62 0.67 0.71 0.75 0.77 0.62 0.49 0.65 0.65 0.64 0.57 0.54 0.51 0.49 0.48 0.48 0.46 0.43 0.43 0.45 0.46 0.47 0.54 0.54 0.59 0.59 0.6 0.55 0.54 0.61 0.6 0.58 0.56 0.56 0.55 0.54 0.51 0.55 0.55 0.59 0.63 0.62 0.55 0.5 0.5 0.57 0.65 0.7 0.75 0.78 0.67 0.51 0.58 0.57 0.57 0.55 0.53 0.52 0.49 0.47 0.47 0.47 0.46 0.49 0.58 0.62 0.62 0.54 0.53 0.61 0.62 0.62 0.59 0.53 0.55 0.6 0.63 0.62 0.63 0.64 0.62 0.59 0.59 0.61 0.6 0.62 0.6 0.56 0.55 0.48 0.48 0.59 0.67 0.72 0.77 0.66 0.51 0.57 0.56 0.54 0.54 0.56 0.55 0.51 0.5 0.52 0.5 0.5 0.6 0.7 0.73 0.73 0.52 0.52 0.59 0.65 0.64 0.62 0.57 0.53 0.53 0.59 0.62 0.65 0.67 0.67 0.66 0.68 0.66 0.63 0.63 0.63 0.63 0.62 0.54 0.47 0.54 0.63 0.7 0.75 0.63 0.49 0.54 0.54 0.52 0.56 0.63 0.65 0.63 0.64 0.65 0.59 0.56 0.65 0.72 0.74 0.75 0.49 0.51 0.56 0.64 0.67 0.65 0.62 0.59 0.55 0.56 0.62 0.65 0.67 0.69 0.69 0.69 0.67 0.66 0.64 0.58 0.65 0.66 0.58 0.46 0.48 0.58 0.64 0.69 0.62 0.52 0.54 0.56 0.59 0.6 0.63 0.69 0.73 0.74 0.73 0.68 0.61 0.66 0.7 0.72 0.74 0.49 0.5 0.52 0.62 0.67 0.68 0.64 0.6 0.58 0.54 0.54 0.61 0.66 0.68 0.71 0.72 0.68 0.63 0.58 0.58 0.65 0.69 0.59 0.38 0.37 0.45 0.53 0.58 0.53 0.53 0.56 0.58 0.62 0.58 0.56 0.57 0.67 0.73 0.73 0.68 0.59 0.65 0.69 0.72 0.74 0.48 0.46 0.49 0.61 0.67 0.7 0.68 0.66 0.64 0.62 0.56 0.56 0.62 0.66 0.71 0.72 0.71 0.7 0.67 0.61 0.64 0.72 0.69 0.44 0.34 0.42 0.49 0.48 0.43 0.55 0.55 0.58 0.61 0.6 0.56 0.52 0.6 0.66 0.69 0.63 0.56 0.64 0.7 0.73 0.74 0.48 0.46 0.46 0.55 0.66 0.7 0.7 0.69 0.68 0.66 0.62 0.56 0.57 0.64 0.7 0.7 0.68 0.65 0.62 0.62 0.66 0.71 0.74 0.66 0.4 0.3 0.45 0.44 0.55 0.62 0.58 0.63 0.62 0.62 0.6 0.6 0.61 0.57 0.6 0.57 0.55 0.64 0.7 0.74 0.74 0.51 0.47 0.47 0.52 0.64 0.7 0.72 0.71 0.7 0.7 0.68 0.64 0.59 0.59 0.64 0.67 0.66 0.64 0.62 0.64 0.67 0.69 0.7 0.69 0.56 0.34 0.37 0.5 0.62 0.59 0.58 0.62 0.61 0.58 0.57 0.59 0.59 0.55 0.51 0.51 0.58 0.67 0.72 0.74 0.74 0.51 0.49 0.47 0.53 0.65 0.71 0.73 0.73 0.72 0.71 0.69 0.67 0.65 0.62 0.61 0.64 0.63 0.62 0.59 0.59 0.6 0.61 0.6 0.6 0.54 0.29 0.41 0.57 0.58 0.57 0.58 0.61 0.6 0.55 0.54 0.59 0.65 0.65 0.54 0.52 0.64 0.71 0.73 0.74 0.75 0.5 0.52 0.52 0.58 0.67 0.72 0.73 0.73 0.73 0.72 0.71 0.69 0.66 0.65 0.64 0.63 0.62 0.61 0.58 0.55 0.53 0.53 0.52 0.51 0.47 0.41 0.5 0.53 0.53 0.55 0.59 0.58 0.56 0.57 0.63 0.7 0.73 0.7 0.59 0.58 0.68 0.71 0.7 0.7 0.71 0.47 0.53 0.56 0.6 0.67 0.71 0.73 0.73 0.73 0.72 0.71 0.69 0.68 0.66 0.65 0.64 0.63 0.62 0.6 0.57 0.55 0.53 0.52 0.5 0.5 0.51 0.55 0.57 0.57 0.58 0.59 0.58 0.6 0.68 0.73 0.75 0.74 0.68 0.56 0.5 0.57 0.62 0.62 0.63 0.66 0.42 0.52 0.57 0.61 0.66 0.7 0.72 0.73 0.73 0.72 0.71 0.69 0.68 0.67 0.65 0.64 0.63 0.62 0.61 0.6 0.57 0.57 0.55 0.55 0.56 0.58 0.6 0.6 0.58 0.56 0.57 0.63 0.71 0.75 0.76 0.75 0.73 0.66 0.49 0.38 0.48 0.65 0.7 0.7 0.69
16. There are many different lenses through which to view data
17. Pairs plot Heatmap Flat clustering Topic modelling Dimension reduction Correlation plot Density plot Vector quantization Persistence barcode Cluster tree Data table Scatter plot Outlier analysis Histograms Tree map Fourier analysis Time series matrix profiling Clustergram Swarm plot
18. Each approach reveals some things and hides other things
19. Look at your data!
20. Exploring your data through many different lenses can help you find the right questions to ask
21. Beyond plotting the most powerful tools come from machine learning
22. Machine Learning
23. Classical Machine Learning
26. Are we really just fitting multivariate polynomials?
27. Image by Randall Munroe: https://xkcd.com/1838/
28. Unsupervised Learning
32. Two clusters?
35. Look at your data!
36. What is going to determine which way the data might cluster?
37. How do you compare two data samples?
38. For unsupervised learning dissimilarity is what matters
39. What is the geometry of the data under a chosen dissimilarity?
40. Some Topology
41. Topology (noun): 1. The study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures. 2. The way in which constituent parts are interrelated or arranged. — Oxford English Dictionary
43. Continuity, infinity, and the continuum can be hard to work with.
44. Can we make topology finite and combinatorial?
45. Simplicial Complexes
46. Build simple pieces of varying dimension in a combinatorial way
52. We can build up a vast array of topological spaces in this purely combinatorial way
55. Face maps provide the combinatorics to glue simplicial complexes together
56. · · · ! ! ! ! X2 ! ! ! X1 ! !X0 <latexit sha1_base64="oMjPuGNawGKAJKwWOlxY6qAd9Fw=">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</latexit> <latexit sha1_base64="oMjPuGNawGKAJKwWOlxY6qAd9Fw=">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</latexit> <latexit sha1_base64="oMjPuGNawGKAJKwWOlxY6qAd9Fw=">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</latexit> <latexit sha1_base64="oMjPuGNawGKAJKwWOlxY6qAd9Fw=">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</latexit>
57. This provides us with simple combinatorial and algebraic tools for working with topological spaces.
58. Homology
59. Boundary map ∂i : ℤ[Xi ] → ℤ[Xi−1 ]
60. ∂i (∑ k ak σk) = ∑ k,ℓ ak (−1)ℓdℓ (σk )
63. The sign can provide a direction for a simplex
67. Boundaries form cycles What if we apply the boundary map to a cycle?
73. The boundary map will map cycles to zero
77. Cycles form a group
78. ∂1 : ℤ[X1 ] → ℤ[X0 ]
79. ker(∂1 ) ◃ ℤ[X1 ]
86. There is a group of cycles modulo filled triangles
87. H1(X) = ker(@1) im(@2) <latexit sha1_base64="cyiBqCgeSl4MAbgU+hMacCKsRn0=">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</latexit>
88. Hi(X) = ker(@i) im(@i+1) <latexit sha1_base64="pX/E5Oij4V5HHNP5cRAmZ1Lax6Q=">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</latexit>
89. What does ∂0 do? What should ∂0 do?
90. Topology from Data
91. Cech Complexes
96. Theorem 1 (Nerve theorem). Let U = {Ui }i2I be a cover of a topological space X. If, for all ⇢ I T i2 Ui is either contractible or empty, then N(U) is homtopically equivalent to X. <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit>
97. Extracting the Cech complex preserves most topological information
98. Vietoris Rips Complexes
103. Vietoris-Rips complexes are very similar to Cech complexes, but are entirely determined by the 1-skeleton
104. There are lots of related constructions for simplicial complexes: Delaunay complexes Alpha complexes Witness complexes …
105. In all cases: Dissimilarity matters
106. Mapper
107. Motivation from Morse Theory
108. Try to understand a manifold by looking at functions defined on that manifold
109. Look at the function in terms of level-sets
114. Mapper
126. Applications
127. Mapper tends to get used for clustering and exploratory analysis
128. Identification of type 2 diabetes subgroups through topological analysis of patient similarity, Li et al, 2015
129. Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury, Nielson et al, 2015
130. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Nicolau et al, 2011
131. From 5 to 13: Redefining the Positions in Basketball, Alagappan, 2012
133. Summary
134. Provides a novel topological view of data
135. The choice of function matters a lot
136. The resolution of the cover of the projection space can have a significant impact on results
137. https://kepler-mapper.scikit-tda.org/
138. Persistent Homology
139. Consider a Vietoris- Rips complex…
141. For any given radius there is homology we can compute
142. By fixing a radius we induce topology and forget metric information
143. How can we get metric information back into our computations?
144. Vr (X) ⊆ Vr+ϵ (X) ⊆ Vr+2ϵ (X)
145. Vr(X) // Vr+✏(X) // Vr+2✏(X) Hi(Vr(X)) // Hi(Vr+✏(X)) // Hi(Vr+2✏(X)) <latexit sha1_base64="Pc65o8G5/iJWo6a0zgn0koUYzak=">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</latexit>
146. Compute homology for varying radii!
147. How do we describe the result?
149. Computing Persistent Homology
150. First, to make things easier, let’s work over a field
151. ∂i : [Xi ] → [Xi−1 ] Boundary map
152. Boundary maps are between vector spaces Linear algebra
154. Reduce to Smith normal form …
158. So simple linear algebra is enough to compute everything
159. What about persistence?
160. There is a partial order on simplices based on inclusion and arrival time
161. Construct a boundary matrix over all simplices in order
162. For simplicity we’ll work in 2
164. Reduce via column additions from left to right
165. for each column : while such that : add column to column j ∃j0 < j low(j0 ) = low(j) j0 j
167. We can now read off the barcode or persistence diagram
170. This can be extended to work for other fields, but this is the core idea
171. Applications
172. Persistent homology has seen use in a diverse range of applications
173. Using Persistent Homology to Quantify a Diurnal Cycle in Hurricane Felix, Tymochko et al, 2019
174. Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis, Perea and Harer, 2014
175. Topological Eulerian Synthesis of Slow Motion Periodic Videos, Tralie and Berger, 2018
176. Coverage in sensor networks via persistent homology, De Silva and Ghrist, 2007
177. Topological Feature Vectors for Chatter Detection in Turning Processes, Yesilli et al, 2019 Chatter Diagnosis in Milling Using Supervised Learning and Topological Features Vector, Yesilli et al, 2019
178. Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing, Zhu, 2013
179. Implementation
180. Naively this algorithm is worst case O(m3)
181. The matrix is sparse and significant practical speedups can be gained through careful use of sparse matrix data structures
182. Specializing to specific cases (e.g. only Vietoris-Rips complexes) can provide significant gains
183. https://github.com/scikit-tda/ripser.py
184. Clustering
185. Find groups of points that are similar
186. What counts as a group?
187. What makes things similar enough to group together?
189. A connected component of a super-level-set of the probability density function of the underlying (and unknown) distribution from which our data samples are drawn.
195. How can we do such a thing in practice?
196. HDBSCAN
197. We need an effective density function over sample points
201. Connected components is something persistent homology could tell us
202. Can we weave density and connectedness together?
204. Mutual Reachability Distance
209. Applications
210. Untangling the Galaxy. I. Local Structure and Star Formation History of the Milky Way, Kounkel and Covey, 2019
211. Unsupervised star, galaxy, qso classification Application of HDBSCAN, Logan and Fotopoulou, 2019
212. Manifold learning of four-dimensional scanning transmission electron microscopy, Li et al, 2019 Machine learning for the structure–energy–property landscapes of molecular crystals, Musil et al, 2017
213. Uncovering Large-Scale Conformational Change in Molecular Dynamics without Prior Knowledge, Melvin et al, 2016
214. Unsupervised clustering of temporal patterns in high-dimensional neuronal ensembles using a novel dissimilarity measure, Grossberger et al, 2018 Computational analysis of laminar structure of the human cortex based on local neuron features, Štajduhar et al, 2019
215. Parallels in the sequential organization of birdsong and human speech, Sainburg et al, 2019
216. https://www.esri.com/arcgis-blog/products/geoanalytics-server/analytics/geoanalytics-detect-delays-public-transit/
217. Our Shared Digital Future Building an Inclusive, Trustworthy and Sustainable Digital Society, World Economic Forum Insight report, 2018
218. Implementation
219. If we have points then there are 1-simplices and persistent homology computations will be worst case N O(N2) O((N2)3) = O(N6)
220. Specializations described before can get it down to O(N4)
221. Can we do any better?
222. Connected components of a weighted graph Minimum spanning trees
225. Spatial Indexing Ram, Lee, Ouyang, Gray 2009
226. March, Ram, Gray 2010 Curtin, March, Ram, Anderson, Gray, Isbell 2013 Dual Tree Boruvka for Euclidean Minimum Spanning Trees Where and are a data dependent constants and is the inverse Ackermann function O(max{c6, c2 p , c2 l }N log(N)α(N)) c, cp cl α
227. With a little finessing this can provide the core for HDBSCAN in time O(N log(N)α(N))
228. https://github.com/scikit-learn-contrib/hdbscan
229. Dimension Reduction
230. Find the “latent” features in your data
233. Matrix Factorization Neighbour Graphs
234. Matrix Factorization Principal Component Analysis Non-negative Matrix Factorization Latent Dirichlet Allocation Word2Vec GloVe Generalised Low Rank Models Linear Autoencoder
235. Neighbour Graphs Locally Linear Embedding Laplacian Eigenmaps Hessian Eigenmaps Local Tangent Space Alignment t-SNE UMAP Isomap JSE
236. PCA is the prototypical matrix factorization
237. PCA on MNIST digits
238. PCA on Fashion MNIST
239. t-SNE is the current state-of-the art for neighbour graphs
240. t-SNE on MNIST digits
241. t-SNE on Fashion MNIST
242. UMAP
243. UMAP builds mathematical theory to justify the graph based approach
244. Theorem 1 (Nerve theorem). Let U = {Ui }i2I be a cover of a topological space X. If, for all ⇢ I T i2 Ui is either contractible or empty, then N(U) is homtopically equivalent to X. <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit> <latexit sha1_base64="8ITSuq3xcb28tfscSBtUyYXmYf8=">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</latexit>
248. If the data is uniformly distributed on the manifold then the cover will be “good”
250. Vary the notion of distance according to the density
252. We can do this all formally using some category theory sleight-of-hand
253. Suppose we were given a low dimensional representation
254. We can apply the same process to get a probabilistic graph!
255. Now measure the distance between the graphs using cross- entropy and optimize
256. X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1 µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit>
257. We are just embedding the graph
258. X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1 µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> <latexit sha1_base64="u7fUXwg3iBtccLqdNJCqNBZ5RiA=">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</latexit> Get the clumps right Get the gaps right
259. UMAP on MNIST digits
260. UMAP on Fashion MNIST
261. Applications
262. Exploring Neural Networks with Activation Atlases, Carter et al, 2019
263. Visualizing and Measuring the Geometry of BERT, Coenen et al, 2019
264. The single-cell transcriptional landscape of mammalian organogenesis, Cao et al, 2019
265. A lineage-resolved molecular atlas of C. elegans embryogenesis at single cell resolution, Packer et al, 2019
266. Dimensionality reduction for visualizing single-cell data using UMAP, Becht et al, 2019
267. UMAP reveals cryptic population structure and phenotype heterogeneity in large genomic cohorts, Diaz-Papkovich et al, 2019
268. OpenSyllabus Galaxy, McClure, 2019
269. Modeling the Structure of Recent Philosophy, Noichl, 2019
270. TimeCluster: dimension reduction applied to temporal data for visual analytics, Ali et al, 2019
271. Identifying galaxies, quasars and stars with machine learning: a new catalogue of classifications for 111 million SDSS sources without spectra, Clark et al, 2019
273. Implementation
274. Algorithm has two hard components: 1. Find near neighbours 2. Optimize according to the cross entropy
275. Near neighbours via NN-Descent
276. Optimization via Stochastic Gradient Descent with negative sampling
277. Performance Comparison t-SNE UMAP COIL20 20 seconds 7 seconds MNIST 22 minutes 98 seconds Fashion MNIST 15 minutes 78 seconds GoogleNews 4.5 hours 14 minutes UMAP speed up over t-SNE COIL20 3x MNIST 13x Fashion MNIST 11x GoogleNews 19x
278. https://github.com/lmcinnes/umap
279. Conclusions
280. Understanding your data is critical for any analysis
281. Understanding data is an unsupervised learning problem
282. Unsupervised Inter-relationships Topology
283. Topological techniques provide powerful methods for exploratory data analysis
284. TDA Session: Saturday 8:30-10:30 and 16:00-18:00