PROJECT SUMMARY/ABSTRACT Immune profiling studies continue to increase in complexity, with multi-omic designs that encompass additional dimensions such as time, tissue, and spatial profiling becoming more commonplace. Unsupervised dimensionality reduction has been a widely used and valuable approach for extracting and understanding the major sources of variation in previous studies, but popular methods such as Principal Components Analysis (PCA) and Non-negative Matrix Factorization (NMF) cannot support these increases in data complexity, nor can existing multi-omic embedding methods, which are designed for static datasets. It is critical that algorithms be developed that incorporate the complex data structures inherent in state-of-the-art immune profiling multi-omics studies that include additional dimensions (e.g., time or space) in order to capture multi-resolution components of vaccination and infection. The goal of this project is to develop algorithms based on tensor frameworks - which are extensions of matrices beyond two dimensions. Tensors naturally represent complex data without flattening on any variable, and tensor decompositions can identify multi-index patterns of variation, analogous to PCA or NMF in higher dimensions. Tensor decomposition methodology is an active area of research in the applied mathematics community, but is under-developed for application to immune profiling data, and current methods face critical challenges that prevent them from being directly applied in immunology studies. This project brings together computational immunology and applied mathematics researchers to strengthen and develop novel approaches of tensor decomposition in order to make them beneficial to the immunology community. Aim 1 will reframe a tensor decomposition problem into a regularized NMF problem, thereby allowing tools developed for matrix analysis to be used on tensor data, and furthermore will extend the new algorithm to handle multi-omics data that has a temporal or spatial component. Aim 2 will directly improve tensor decomposition approaches by developing novel metrics for tensor decomposition quality, and by extending a multi-omic embedding method into the tensor space using a novel tensor-algebra. The resulting algorithm will be able to generate components associated with data that can include both multi-omic and multi-dimensional (e.g. time, space, tissue, etc.) designs. These components can be analyzed for association with clinical features and outcomes, allowing for discovery of novel biological mechanisms. The proposed project will result in a suite of complementary algorithms that will aid the immunology community in understanding complex pathogenic and treatment/vaccination processes using the increasingly complex study designs that are becoming common to immune profiling studies.