Topological Data Analysis Leland McInnes CMS Winter Meeting 2019

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 [email protected]

I want to empower users to explore their data

Data Analysis

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

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

What questions do you want answered?

What questions should you be asking?

Look at your data!

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

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

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

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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

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There are many different lenses through which to view data

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

Each approach reveals some things and hides other things

Look at your data!

Exploring your data through many different lenses can help you find the right questions to ask

Beyond plotting the most powerful tools come from machine learning

Machine Learning

Classical Machine Learning

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Are we really just fitting multivariate polynomials?

Image by Randall Munroe: https://xkcd.com/1838/

Unsupervised Learning

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Two clusters?

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Look at your data!

What is going to determine which way the data might cluster?

How do you compare two data samples?

For unsupervised learning dissimilarity is what matters

What is the geometry of the data under a chosen dissimilarity?

Some Topology

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

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Continuity, infinity, and the continuum can be hard to work with.

Can we make topology finite and combinatorial?

Simplicial Complexes

Build simple pieces of varying dimension in a combinatorial way

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We can build up a vast array of topological spaces in this purely combinatorial way

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Face maps provide the combinatorics to glue simplicial complexes together

· · · ! ! ! ! X2 ! ! ! X1 ! !X0 AAAC3niclVJNSxtBGJ5dP5rGr2iPvQwGwYthZ6Oot0AvHi00Gthd4uxkkgzOziwz71bCkoMXDxXx6u/qrT+k987GFFoxiC8MPDzP+/1OmkthIQh+ef7S8srqh9rH+tr6xuZWY3vnwurCMN5lWmrTS6nlUijeBQGS93LDaZZKfplef6n0y+/cWKHVN5jkPMnoSImhYBQc1W/8jtlAg8WxLVILlF2XsdRqZMRoDNQYfRPH0QHhWbKAxi94vEiY4l4/fLvMu/KR9+erwoJ6v9EMWiQ8PSQhduAkaJ9WoE3ax+ERJq1gZk00t/N+42c80KzIuAImqbURCXJISmpAMMmn9biwPHdt0BGPHFQ04zYpZ+eZ4j3HDPBQG/cU4Bn7b0RJM2snWeo8Mwpj+1KryNe0qIDhSVIKlRfAFXsuNCwkBo2rW+OBMJyBnDhAmRGuV8zG1FAG7kdUS/g7KV4MLsIWcSv6etjsnM3XUUOf0S7aRwQdow46Q+eoi5gXebfeD+/ev/Lv/Af/8dnV9+Yxn9B/5j/9ARVD6z8= 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

This provides us with simple combinatorial and algebraic tools for working with topological spaces.

Homology

Boundary map ∂i : ℤ[Xi ] → ℤ[Xi−1 ]

∂i (∑ k ak σk) = ∑ k,ℓ ak (−1)ℓdℓ (σk )

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The sign can provide a direction for a simplex

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Boundaries form cycles What if we apply the boundary map to a cycle?

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The boundary map will map cycles to zero

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Cycles form a group

∂1 : ℤ[X1 ] → ℤ[X0 ]

ker(∂1 ) ◃ ℤ[X1 ]

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There is a group of cycles modulo filled triangles

H1(X) = ker(@1) im(@2) AAACTHicbVBNT9tAEF2HFkJoS4Ajl1UjpORi2XGA5ICEaA9cqoJEIFIcWevNGFZZ26vdNSKy/Dv4NVzh1nv/BzeExDpEtHw8aaSnNzP7dl4oOFPacf5alYVPnxeXqsu1lS9fv63W19ZPVZpJCn2a8lQOQqKAswT6mmkOAyGBxCGHs3Dyo+yfXYJULE1O9FTAKCbnCYsYJdpIQd09DNzmoIX3sB9JQnN/ArLpCyI1IzxwW0Xus/if0G4VQb3h2O225+1uY8f23E635xnS2fG6joNd25mhgeY4CtasBX+c0iyGRFNOlBq6jtCjvHyScihqfqZAEDoh5zA0NCExqFE+u63AW0YZ4yiVphKNZ+r/GzmJlZrGoZmMib5Qb3ul+FFvmOmoO8pZIjINCX02ijKOdYrLoPCYSaCaTw0hVDLzV0wviMlImzhfuVxNBaPmjJ9gzpPwy1j9FiCJTmWZXpGbqs1i65XYeQnpPTlt265nd447jf2DeYBVtIm+oyZy0S7aR4foCPURRdfoBt2iO+uPdW89WI/PoxVrvrOBXqGy+AQsXLLV

Hi(X) = ker(@i) im(@i+1) 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

What does ∂0 do? What should ∂0 do?

Topology from Data

Cech Complexes

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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. 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

Extracting the Cech complex preserves most topological information

Vietoris Rips Complexes

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Vietoris-Rips complexes are very similar to Cech complexes, but are entirely determined by the 1-skeleton

There are lots of related constructions for simplicial complexes: Delaunay complexes Alpha complexes Witness complexes …

In all cases: Dissimilarity matters

Mapper

Motivation from Morse Theory

Try to understand a manifold by looking at functions defined on that manifold

Look at the function in terms of level-sets

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Mapper

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Applications

Mapper tends to get used for clustering and exploratory analysis

Identification of type 2 diabetes subgroups through topological analysis of patient similarity, Li et al, 2015

Topological data analysis for discovery in preclinical spinal cord injury and traumatic brain injury, Nielson et al, 2015

Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, Nicolau et al, 2011

From 5 to 13: Redefining the Positions in Basketball, Alagappan, 2012

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Summary

Provides a novel topological view of data

The choice of function matters a lot

The resolution of the cover of the projection space can have a significant impact on results

https://kepler-mapper.scikit-tda.org/

Persistent Homology

Consider a Vietoris- Rips complex…

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For any given radius there is homology we can compute

By fixing a radius we induce topology and forget metric information

How can we get metric information back into our computations?

Vr (X) ⊆ Vr+ϵ (X) ⊆ Vr+2ϵ (X)

Vr(X) // Vr+✏(X) // Vr+2✏(X) Hi(Vr(X)) // Hi(Vr+✏(X)) // Hi(Vr+2✏(X)) 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

Compute homology for varying radii!

How do we describe the result?

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Computing Persistent Homology

First, to make things easier, let’s work over a field

∂i : [Xi ] → [Xi−1 ] Boundary map

Boundary maps are between vector spaces Linear algebra

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Reduce to Smith normal form …

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So simple linear algebra is enough to compute everything

What about persistence?

There is a partial order on simplices based on inclusion and arrival time

Construct a boundary matrix over all simplices in order

For simplicity we’ll work in 2

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Reduce via column additions from left to right

for each column : while such that : add column to column j ∃j0 < j low(j0 ) = low(j) j0 j

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We can now read off the barcode or persistence diagram

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This can be extended to work for other fields, but this is the core idea

Applications

Persistent homology has seen use in a diverse range of applications

Using Persistent Homology to Quantify a Diurnal Cycle in Hurricane Felix, Tymochko et al, 2019

Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis, Perea and Harer, 2014

Topological Eulerian Synthesis of Slow Motion Periodic Videos, Tralie and Berger, 2018

Coverage in sensor networks via persistent homology, De Silva and Ghrist, 2007

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

Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing, Zhu, 2013

Implementation

Naively this algorithm is worst case O(m3)

The matrix is sparse and significant practical speedups can be gained through careful use of sparse matrix data structures

Specializing to specific cases (e.g. only Vietoris-Rips complexes) can provide significant gains

https://github.com/scikit-tda/ripser.py

Clustering

Find groups of points that are similar

What counts as a group?

What makes things similar enough to group together?

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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.

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How can we do such a thing in practice?

HDBSCAN

We need an effective density function over sample points

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Connected components is something persistent homology could tell us

Can we weave density and connectedness together?

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Mutual Reachability Distance

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Applications

Untangling the Galaxy. I. Local Structure and Star Formation History of the Milky Way, Kounkel and Covey, 2019

Unsupervised star, galaxy, qso classification Application of HDBSCAN, Logan and Fotopoulou, 2019

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

Uncovering Large-Scale Conformational Change in Molecular Dynamics without Prior Knowledge, Melvin et al, 2016

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

Parallels in the sequential organization of birdsong and human speech, Sainburg et al, 2019

https://www.esri.com/arcgis-blog/products/geoanalytics-server/analytics/geoanalytics-detect-delays-public-transit/

Our Shared Digital Future Building an Inclusive, Trustworthy and Sustainable Digital Society, World Economic Forum Insight report, 2018

Implementation

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)

Specializations described before can get it down to O(N4)

Can we do any better?

Connected components of a weighted graph Minimum spanning trees

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Spatial Indexing Ram, Lee, Ouyang, Gray 2009

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

With a little finessing this can provide the core for HDBSCAN in time O(N log(N)α(N))

https://github.com/scikit-learn-contrib/hdbscan

Dimension Reduction

Find the “latent” features in your data

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Matrix Factorization Neighbour Graphs

Matrix Factorization Principal Component Analysis Non-negative Matrix Factorization Latent Dirichlet Allocation Word2Vec GloVe Generalised Low Rank Models Linear Autoencoder

Neighbour Graphs Locally Linear Embedding Laplacian Eigenmaps Hessian Eigenmaps Local Tangent Space Alignment t-SNE UMAP Isomap JSE

PCA is the prototypical matrix factorization

PCA on MNIST digits

PCA on Fashion MNIST

t-SNE is the current state-of-the art for neighbour graphs

t-SNE on MNIST digits

t-SNE on Fashion MNIST

UMAP

UMAP builds mathematical theory to justify the graph based approach

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. 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If the data is uniformly distributed on the manifold then the cover will be “good”

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Vary the notion of distance according to the density

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We can do this all formally using some category theory sleight-of-hand

Suppose we were given a low dimensional representation

We can apply the same process to get a probabilistic graph!

Now measure the distance between the graphs using cross- entropy and optimize

X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1 µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ 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

We are just embedding the graph

X a2A µ(a) log ✓ µ(a) ⌫(a) ◆ + (1 µ(a)) log ✓ 1 µ(a) 1 ⌫(a) ◆ 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 AAADFXicfVHLjtMwFHXCY4byamHJxqJC6ghRJSloZnaDQIgNYoB2ZqS6qlz3JrHGcSLbYaisbPgJvoYdYsuaJX+Ck2YEHR5Xsnx0jo/u9T2LQnBtguC751+6fOXq1va1zvUbN2/d7vbuHOm8VAwmLBe5OllQDYJLmBhuBJwUCmi2EHC8OH1W68fvQWmey7FZFTDLaCJ5zBk1jpp3LZkSXWZzSwmX+GmFSVYO6A4ReUIExGZAYkWZXbOVJbK5ieJJanbwQzwI8aPW84fpl1St8YaZzObdfjAMo/3HYYQd2AtG+zUYhaPd6AkOh0FTfdTW4bznfSTLnJUZSMME1XoaBoWZWaoMZwKqDik1FJSd0gSmDkqagZ7ZZksVfuCYJY5z5Y40uGF/d1iaab3KFu5lRk2qL2o1+TdtWpp4b2a5LEoDkq0bxaXAJsf1yvGSK2BGrBygTHE3K2YpdRsyLpiLXUyauX9IODMp5Aoy296VHbegQzS4nGViUksMfDBnfOkms2HEavE5uNUoeOXGfF2AoiZXlrzg8i1Q4QJsxo/tOfEfwzsukw1DQ1QutfNo8L/BUTQMXaZvov7BuM1vG91D99EAhWgXHaCX6BBNEEM/vC2v6/X8T/5n/4v/df3U91rPXbRR/refSHP9Ew== 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 Get the clumps right Get the gaps right

UMAP on MNIST digits

UMAP on Fashion MNIST

Applications

Exploring Neural Networks with Activation Atlases, Carter et al, 2019

Visualizing and Measuring the Geometry of BERT, Coenen et al, 2019

The single-cell transcriptional landscape of mammalian organogenesis, Cao et al, 2019

A lineage-resolved molecular atlas of C. elegans embryogenesis at single cell resolution, Packer et al, 2019

Dimensionality reduction for visualizing single-cell data using UMAP, Becht et al, 2019

UMAP reveals cryptic population structure and phenotype heterogeneity in large genomic cohorts, Diaz-Papkovich et al, 2019

OpenSyllabus Galaxy, McClure, 2019

Modeling the Structure of Recent Philosophy, Noichl, 2019

TimeCluster: dimension reduction applied to temporal data for visual analytics, Ali et al, 2019

Identifying galaxies, quasars and stars with machine learning: a new catalogue of classifications for 111 million SDSS sources without spectra, Clark et al, 2019

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Implementation

Algorithm has two hard components: 1. Find near neighbours 2. Optimize according to the cross entropy

Near neighbours via NN-Descent

Optimization via Stochastic Gradient Descent with negative sampling

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

https://github.com/lmcinnes/umap

Conclusions

Understanding your data is critical for any analysis

Understanding data is an unsupervised learning problem

Unsupervised Inter-relationships Topology

Topological techniques provide powerful methods for exploratory data analysis

TDA Session: Saturday 8:30-10:30 and 16:00-18:00