The effect of feature resolution on embedding dimension

High-dimensional data can be compressed into lower-dimensional embeddings while retaining a relatively large amount of relevant information, a phenomenon which, despite its widespread use, we struggle to fully explain. In this post, we use a common property of datasets - a limit on the number of features per data point - to show how a slight uniform dependence between features can be exploited to reduce the required dimensions by at least a third, while sacrificing no information about the features. To do so, we introduce the concepts of dataset resolution and feature composition of a dataset, and analyse how a set of orderings of the dataset affects the types of partitions we can create of the dataset.