SummaryHigh-dimensional data is currently standard for biological inquiry. Biological systems are comprised of interrelated gene regulatory mechanisms, gene-gene interactions, and cellular interactions. These interactions induce low-dimensional structure within the high-dimensional data. Matrix factorization, also known as compressed sensing, learns low-dimensional mathematical representations from high-dimensional data. These factorization techniques can embed assumptions about pleiotropy, epistasis, inter-relationships between complex traits, and context-dependent interactions. They have been applied to uncover new biological knowledge in a breadth of topics ranging from pathway discovery to time course analysis. These techniques have been applied to data from diverse high-throughput omics technologies, including bulk and single-cell data. There are numerous computational techniques within the class of matrix factorization, each of which provides a unique interpretation of the processes in high-dimensional data. We review the visualization and applications of matrix factorization to systems-level analyses, which are diverse and require standardization to enable biological interpretation. Codifying the techniques to decipher biologically relevant features with matrix factorization enables their broad application to discovery beyond the limits of current biological knowledge-answering questions from high-dimensional data that we have not yet thought to ask.