Nice latent variable models have log-rank

Matrices of low rank are pervasive in big data, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure appears in the first place. We explain the abundance of low rank matrices in big data by proving that certain latent variable models associated to piecewise analytic functions are of log-rank. A large matrix from such a latent variable model can be approximated, up to a small error, by a low rank matrix.