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Topological Data Analysis

Leland McInnes
December 06, 2019

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.

Leland McInnes

December 06, 2019
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  1. 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]
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. We can build up a vast array of topological spaces

    in this purely combinatorial way
  11. · · · ! ! ! ! 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>
  12. 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>
  13. There are lots of related constructions for simplicial complexes: Delaunay

    complexes Alpha complexes Witness complexes …
  14. Topological data analysis for discovery in preclinical spinal cord injury

    and traumatic brain injury, Nielson et al, 2015
  15. Topology based data analysis identifies a subgroup of breast cancers

    with a unique mutational profile and excellent survival, Nicolau et al, 2011
  16. The resolution of the cover of the projection space can

    have a significant impact on results
  17. 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>
  18. for each column : while such that : add column

    to column j ∃j0 < j low(j0 ) = low(j) j0 j
  19. 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
  20. The matrix is sparse and significant practical speedups can be

    gained through careful use of sparse matrix data structures
  21. 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.
  22. 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
  23. 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
  24. Our Shared Digital Future Building an Inclusive, Trustworthy and Sustainable

    Digital Society, World Economic Forum Insight report, 2018
  25. 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)
  26. 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
  27. Matrix Factorization Principal Component Analysis Non-negative Matrix Factorization Latent Dirichlet

    Allocation Word2Vec GloVe Generalised Low Rank Models Linear Autoencoder
  28. 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>
  29. 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>
  30. 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
  31. Identifying galaxies, quasars and stars with machine learning: a new

    catalogue of classifications for 111 million SDSS sources without spectra, Clark et al, 2019
  32. Algorithm has two hard components: 1. Find near neighbours 2.

    Optimize according to the cross entropy
  33. 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