The MNIST dataset is a collection of images of handwritten digits. Before the birth of Convolutional Neural Networks most machine learning approaches arranged each 28x28 pixel image into a 758-dimensional vector. Does this make sense? Does natural data have these coordinates? What if I told you this is seen as standard procedure in “machine learning”. Are we cramming vectors, coordinates and metrics where they have no business in?

Complex data requires complex models, right? But does it really? Classifying two concentric circles is challenging not because they are circles but because they are concentric. The complexity of the decision boundary measures how the classes are entangled and is the closest approximation to the intrinsic difficulty of a classification problem. But it all starts by sampling the decision boundary. (Or is it a decision boundary?)

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Imagine that you wanted to compare the persistent homology of two different metric spaces. The filtration parameter of standard Vietoris-Rips filtration is the metric, but each space has its own metric. So, how do we compare their persistent homology?

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