## 4 Lumawig in Digit Classification  
With new access to a fast algorithm for computing dimension zero bottleneck distance, we leverage persistence and other clustering-based diagrams to craft features for digit classification. This application illustrates how the significant computational speed-up for the dimension 0 bottleneck distance affords a way to examine intrinsic differences in the multi-scale clustering dynamics of point clouds from the perspective of persistent homology and in conjuction with hierarchical clustering algorithms. In addition, we will show that information captured by dimension 0 bottleneck distance can be a source of a good feature base for point cloud classification.  
We classify 10,000 28\(\times\)28-pixel digit images in the MNIST data set via a random forest classifier. Similar to [PERSON] and [PERSON] [16], we train the classifier using features based on topological summaries. However, we depart from [PERSON] and [PERSON]'s approach in that we only extract features from dimension zero persistence diagrams and other related clustering-based diagrams. In particular, we craft statistical summaries from distributions of bottleneck distances computed from diagrams resulting from dimension zero persistent homology and clustering of multiple sub-collections of points. It is in this light that the eventual classifier performance must be viewed -- the classification of digits, viewed as point clouds with higher dimensional characteristics, is done using lower dimensional clustering information. We summarize this procedure next. For a detailed account of this procedure, we point the interested reader to [21] where it is used to recover higher dimensional shape information of digits from intrinsic clustering behavior.  
The first step in the procedure is to generate multiple collections of points from the digits via samples extracted based on point distributions referenced from nine pre-selected landmark points. We use the same landmark points introduced in [16]. Sampling is also done across multiple resolutions by varying the number of points selected in every bin of every distribution histogram. Then, for each sampled sub-collection of points in each sampling resolution, persistent homology and clustering algorithms are respectively used to generate persistence and clustering diagrams. We gather diagrams by their sampling setting and compute pairwise bottleneck distances using Lumawig. Finally, we compute statistical summaries from the distributions of computed bottleneck distances, and use these to train a random forest classifier with 1000 trees.  
We perform a 10-fold cross validation on our training set of 10,000 digit images from MNIST, and report the summary of obtained \(F_{1}\) scores in Table 2. The average class predictions of the random forest are summarized in the confusion matrix in Figure 9.  
The results above show that the random forest classifier is able to use our crafted bottleneck-based features to classify, at a respectable level of accuracy, the 10 digits in the MNIST data set despite all digits possessing the same dimension zero topological signature of having only one connected component. In particular, we infer from the exceptionally high score on the classification of the simplest digit 1, that differences captured by the bottleneck distance in the clustering behavior across multiple point samples of this digit is outstandingly subtle, and hence different, from the rest.