Minimum-Distortion Embedding

We consider the vector embedding problem. are given a finite set of items, with goal assigning representative to each one, possibly under some constraints (such as collection vectors being standardized, i.e., have zero mean and unit covariance). data indicating that pairs items similar, optionally, other dissimilar. For similar we want corresponding be near other, for dissimilar pairs, not measured in Euclidean distance. formalize this by introducing distortion functions, defined items. Our is choose an minimizes total distortion, subject constraints. call minimum-distortion (MDE) The MDE framework simple but general. It includes wide variety methods, such spectral embedding, principal component analysis, multidimensional scaling, dimensionality reduction methods (like Isomap UMAP), force-directed layout, others. also new embeddings, provides principled ways validating historical embeddings alike. We develop projected quasi-Newton method approximately solves problems scales large sets. implement PyMDE, open-source Python package. In users can select from library functions or specify custom ones, making it easy rapidly experiment different embeddings. software sets millions tens functions. To demonstrate our method, compute several real-world sets, including images, academic co-author network, US county demographic data, single-cell mRNA transcriptomes.