Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank

The nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of positive ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks. Additionally the bounds we propose can be computed by semidefinite programming using sum-of-squares relaxations. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that enjoys other interesting structural properties. For the case of the nonnegative rank we show that our bound has an appealing connection with existing combinatorial bounds and other norm-based bounds. For example we show that our lower bound is a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lovasz number of the rectangle graph of the matrix. We also prove that the lower bound is always greater than or equal to the hyperplane separation bound (and other similar norm-based bounds). We also discuss the case of the tensor nonnegative rank as well as the cp-rank, and compare our bound with existing results.