Abstract
We consider the vector embedding problem. We are given a finite set of items, with the goal of assigning a representative vector to each one, possibly under some constraints (such as the collection of vectors being standardized, i.e., having zero mean and unit covariance). We are given data indicating that some pairs of items are similar, and optionally, some other pairs are dissimilar. For pairs of similar items, we want the corresponding vectors to be near each other, and for dissimilar pairs, we want the vectors to not be near each other, measured in Euclidean distance. We formalize this by introducing distortion functions, defined for some pairs of items. Our goal is to choose an embedding that minimizes the total distortion, subject to the constraints. We call this the minimum-distortion embedding (MDE) problem.
The MDE framework is simple but general. It includes a wide variety of specific embedding methods, such as spectral embedding, principal component analysis, multidimensional scaling, Euclidean distance problems, dimensionality reduction methods (like Isomap and UMAP), semi-supervised learning, sphere packing, force-directed layout, and others. It also includes new embeddings, and provides principled ways of validating or sanity-checking historical and new embeddings alike.
In a few special cases, MDE problems can be solved exactly. For others, we develop a projected quasi-Newton method that approximately minimizes the distortion and scales to very large data sets, while placing few assumptions on the distortion functions and constraints. This monograph is accompanied by an open-source Python package, PyMDE, for approximately solving MDE problems. Users can select from a library of distortion functions and constraints or specify custom ones, making it easy to rapidly experiment with new embeddings. Because our algorithm is scalable, and because PyMDE can exploit GPUs, our software scales to problems with millions of items and tens of millions of distortion functions. Additionally, PyMDE is competitive in runtime with specialized implementations of specific embedding methods. To demonstrate our method, we compute embeddings for several real-world data sets, including images, an academic co-author network, US county demographic data, and singlecell mRNA transcriptomes.