UML is known to be challenging: Even learning linear mixtures requires highly nontrivial analytical tools, e.g., independent component analysis or nonnegative matrix factorization.

Graphs are central to the chemical sciences, providing a natural language to describe molecules, proteins, reactions, and industrial processes. They capture interactions and structures that underpin materials, biology, and medicine. This primer, Graph Data Modeling: Molecules, Proteins, & Chemical Processes, introduces graphs as mathematical objects in chemistry and shows how learning algorithms (particularly graph neural networks) can operate on them. We outline the foundations of graph design, key prediction tasks, representative examples across chemical sciences, and the role of machine learning in graph-based modeling. Together, these concepts prepare readers to apply graph methods to the next generation of chemical discovery.

This paper proposes two algorithms for estimating square Wasserstein distance matrices from a small number of entries. These matrices are used to compute manifold learning embeddings like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are extremely costly to compute. We analyze matrix completion from upper triangular samples and Nyström completion in which $\mathcal{O}(d\log(d))$ columns of the distance matrices are computed where $d$ is the desired embedding dimension, prove stability of MDS under Nyström completion, and show that it can outperform matrix completion for a fixed budget of sample distances. Finally, we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is stable on data embedded from the Nyström estimation of the distance matrix even when only 10\% of the columns are computed.