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The big data era of science and technology motivates statistical modeling ofmatrix-valued data using a low-rank representation that simultaneouslysummarizes key characteristics of the data and enables dimension reduction fordata compression and storage. Low-rank representations such as singular valuedecomposition factor the original data into the product of orthonormal basisfunctions and weights, where each basis function represents an independentfeature of the data. However, the basis functions in these factorizations aretypically computed using algorithmic methods that cannot quantify uncertaintyor account for explicit structure beyond what is implicitly specified via datacorrelation. We propose a flexible prior distribution for orthonormal matricesthat can explicitly model structure in the basis functions. The prior is usedwithin a general probabilistic model for singular value decomposition toconduct posterior inference on the basis functions while accounting formeasurement error and fixed effects. To contextualize the proposed prior andmodel, we discuss how the prior specification can be used for various scenariosand relate the model to its deterministic counterpart. We demonstrate favorablemodel properties through synthetic data examples and apply our method to seasurface temperature data from the northern Pacific, enhancing our understandingof the ocean's internal variability.

A flexible class of priors for conducting posterior inference on structured orthonormal matrices