Minimax Lower Bounds for Nonnegative Matrix Factorization

The non-negative matrix factorization (NMF) problem consists in modeling data samples as non-negative linear combinations of non-negative dictionary vectors. While many algorithms for NMF have been proposed, fundamental performance limits of these algorithms are currently not available. This paper plugs this gap by providing lower bounds on the minimax risk (the minimum achievable worst case mean squared error) of estimating the non-negative dictionary matrix under a set of locality and statistical assumptions.