EMB: Non-Euclidean embeddings for microbiome classification

The human body is home to vast communities of microbes, collectively called the microbiome, that shape digestion, immunity, and susceptibility to disease. Scientists can now take a census of these communities, recording which kinds of microbe are present and in what proportions. Turning such a census into understanding requires arranging the data so that similar communities sit close together and different ones far apart. The arrangements in common use assume that the underlying space is flat, like a sheet of paper, yet the biology points elsewhere. Microbes are joined by an ever-branching tree of descent, and a tree that keeps dividing quickly runs out of room on a flat sheet, so a curved space with room to spare can hold it far more faithfully. Communities also rise and fall with natural rhythms, such as the menstrual cycle or the daily rhythm of activity and rest, and these loops are captured better by a rounded, closed space than by a straight line. This project builds arrangements whose shape is chosen to match these biological patterns, promising sharper insight into how microbial communities relate to health and how they shift over time, with potential benefit for diagnosis and treatment. The work also broadens participation in research that unites mathematics and biology, engaging middle school and high school students in New York City, undergraduates at Barnard College, and summer research participants, and letting learners explore microbial communities through an immersive virtual reality environment. Placing microbial abundance measurements in flat, zero-curvature spaces discards biologically meaningful structure, since the evolutionary relationships among microbial types are tree-shaped and many community dynamics are periodic. This project develops representation-learning methods that fit the embedding space's curvature to the data rather than fixing it beforehand. The work proceeds through three complementary lines of inquiry. First, abundance vectors are placed in negatively curved spaces whose tree-like geometry mirrors shared ancestry, with curvature learned jointly with the coordinates and resilience to the sampling noise, missing values, and mislabeled strains of real measurements. Second, the embeddings are widened to products of spaces combining negative, zero, and positive curvature, so that one representation separates hierarchical structure among microbial types from cyclic structure across samples; the number, dimension, and curvature of the components are inferred through variational methods and optimization on curved manifolds, with an added objective aligning each recovered dimension to an interpretable biological process. Third, a contrastive-learning strategy treats evolution as a source of function-preserving variation, forming training pairs from evolutionarily related sequence fragments to learn a representation that factors what a microbe does apart from its lineage, supporting tasks such as sorting fragments by lineage and predicting host health from microbial genomic variation, including for lineages absent from the training data. Performance is judged against established flat-space methods on simulated data with known ground truth and on curated public microbial collections spanning time and individuals, and the resulting software extends an openly available toolkit for learning non-Euclidean representations. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.