Why are Big Data Matrices Approximately Low Rank?
This provides a theoretical foundation for the widespread use of low rank approximations in data science, addressing a gap in understanding why such structures appear in big data.
The paper explains why large datasets often exhibit low rank structure by proposing a generative model where matrix entries are piecewise analytic functions of latent variables, and proves that such matrices can be approximated with low rank matrices of order O(log(m+n)) up to a fixed error.
Matrices of (approximate) low rank are pervasive in data science, 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. Here, we explain the effectiveness of low rank models in data science by considering a simple generative model for these matrices: we suppose that each row or column is associated to a (possibly high dimensional) bounded latent variable, and entries of the matrix are generated by applying a piecewise analytic function to these latent variables. These matrices are in general full rank. However, we show that we can approximate every entry of an $m \times n$ matrix drawn from this model to within a fixed absolute error by a low rank matrix whose rank grows as $\mathcal O(\log(m + n))$. Hence any sufficiently large matrix from such a latent variable model can be approximated, up to a small entrywise error, by a low rank matrix.