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Persistent homology is a leading tool in topological data analysis (TDA).Many problems in TDA can be solved via homological -- and indeed, linear --algebra. However, matrices in this domain are typically large, with rows andcolumns numbered in billions. Low-rank approximation of such arrays typicallydestroys essential information; thus, new mathematical and computationalparadigms are needed for very large, sparse matrices.  We present the U-match matrix factorization scheme to address this challenge.U-match has two desirable features. First, it admits a compressed storageformat that reduces the number of nonzero entries held in computer memory byone or more orders of magnitude over other common factorizations. Second, itpermits direct solution of diverse problems in linear and homological algebra,without decompressing matrices stored in memory. These problems include look-upand retrieval of rows and columns; evaluation of birth/death times, andextraction of generators in persistent (co)homology; and, calculation of basesfor boundary and cycle subspaces of filtered chain complexes. Such bases arekey to unlocking a range of other topological techniques for use in TDA, andU-match factorization is designed to make such calculations broadly accessibleto practitioners.  As an application, we show that individual cycle representatives inpersistent homology can be retrieved at time and memory costs orders ofmagnitude below current state of the art, via global duality. Moreover, thealgebraic machinery needed to achieve this computation already exists in manymodern solvers.

U-match factorization: sparse homological algebra, lazy cycle representatives, and dualities in persistent (co)homology