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The nonnegative rank of a matrix A is the smallest integer r such that A canbe written as the sum of r rank-one nonnegative matrices. The nonnegative rankhas received a lot of attention recently due to its application inoptimization, probability and communication complexity. In this paper we studya class of atomic rank functions defined on a convex cone which generalizeseveral notions of "positive" ranks such as nonnegative rank or cp-rank (forcompletely positive matrices). The main contribution of the paper is a newmethod to obtain lower bounds for such ranks which improve on previously knownbounds. Additionally the bounds we propose can be computed by semidefiniteprogramming. The idea of the lower bound relies on an atomic norm approachwhere the atoms are self-scaled according to the vector (or matrix, in the caseof nonnegative rank) of interest. This results in a lower bound that isinvariant under scaling and that is at least as good as other existingnorm-based bounds.  We mainly focus our attention on the two important cases of nonnegative rankand cp-rank where our bounds satisfying interesting properties: For thenonnegative rank we show that our lower bound can be interpreted as anon-combinatorial version of the fractional rectangle cover number, while thesum-of-squares relaxation is closely related to the Lov\'asz theta number ofthe rectangle graph of the matrix. We also prove that the lower bound inheritsmany of the structural properties satisfied by the nonnegative rank such asinvariance under diagonal scaling, subadditivity, etc. We also apply our methodto obtain lower bounds on the cp-rank for completely positive matrices. In thiscase we prove that our lower bound is always greater than or equal the plainrank lower bound, and we show that it has interesting connections withcombinatorial lower bounds based on edge-clique cover number.

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