RNA profiling is an excellent phenotype of cellular responses and tissue states, but can be costly to generate at the massive scale required for studies of regulatory circuits, genetic states or perturbation screens. Here, we draw on a series of advances over the last decade in the field of mathematics to establish a rigorous link between biological structure, data compressibility, and efficient data acquisition. We propose that very few random composite measurements - in which gene abundances are combined in a random linear combination - are needed to approximate the high-dimensional similarity between any pair of gene abundance profiles. We then show how finding latent, sparse representations of gene expression data would enable us to 'decompress' a small number of random composite measurements and recover high-dimensional gene expression levels that were not measured (unobserved). We present a new algorithm for finding sparse, modular structure, which improves the ability to interpret samples in terms of small numbers of active modules, and show that the modular structure we find is sufficient to recover gene expression profiles from composite measurements (with ~100-fold fewer composite measurements than genes). Moreover, the knowledge that sparse, modular structures exist allows us to recover expression profiles from composite measurements, even without access to any training data. Finally, we present a proof-of-concept experiment for making composite measurements in the laboratory, involving the measurement of linear combinations of RNA abundances. Altogether, our results suggest new compressive modalities in experimental biology that can form a foundation for massive scaling in high-throughput measurements, while also offering new insights into the interpretation of high-dimensional data. A recorded seminar presentation of this work is available at: https://www.youtube.com/watch?v=2dBZEOXqKHs
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